Particle physics and the universe: Proceedings of Nobel Symposium 109 : Haga Slott, Enkoping, Sweden, August 20-25, 1998

Mp in bang loses its dominant position, being removed to the the theory (m2/2)cj)2). This result is based on the investi- indefinite past. gation of the probability of quantum jumps with amplitude From this new perspective many old problems of 5(j) » H/2n. cosmology, including the problem of initial conditions, look Until now we have considered the simplest inflationary much less profound than they seemed before. In many model with only one scalar field, which had only one mini- versions of inflationary theory it can be shown that the mum of its potential energy. Meanwhile, realistic models fraction of the volume of the universe with given properties of elementary particles propound many kinds of scalar (with given values of fields, with a given density of matter, fields. For example, in the unified theories of weak, strong etc.) does not depend on time, both at the stage of inflation and electromagnetic interactions, at least two other scalar and even after it. Thus each part of the universe evolves © Physica Scripta 2000

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in time, but the universe as a whole may be stationary, and the properties of its parts do not depend on the initial conditions [12]. Of course, this happens only for the (rather broad) set of initial conditions which lead to self-reproduction of the universe. However, only finite number of observers live in the universes created in a state with initial conditions which do not allow self-reproduction, whereas infinitely many observers can live in the universes with the conditions which allow self-reproduction. Thus it seems plausible that we (if we are typical, and live in the place where most observers do) should live in the universe created in a state with initial conditions which allow self-reproduction. On the other hand, stationarity of the self-reproducing universe implies that an exact knowledge of these initial conditions in a self-reproducing universe is irrelevant for the investigation of its future evolution [12].

requires the universe to be exponentially large, flat and homogeneous prior to the stage of the pre-big-bang inflation [15,16]. Thus it would be very difficult to replace the usual post-big-bang inflation by the pre-big-bang one. And if the post-big-bang inflation is indeed necessary, then the pre-big-bang stage will have no observational manifestations. Therefore it is very desirable to find those versions of string theory and supergravity where a consistent post-big-bang inflationary theory can be developed. This is a difficult problem, but not hopeless. Perhaps the easiest way to find a consistent inflationary model in supersymmetric theories is based on the hybrid inflation scenario [17]. The simplest version of this scenario is based on chaotic inflation in the theory of two scalar fields with the effective potential

V{a, 0) = 1 (M2 - key2)2 + y 4>2 + y * V •

(8)

The effective mass squared of the field a is equal to —M2 +g2(j>2. Therefore for > c = M/g the only miniEven though the main principles of inflationary cosmology mum of the effective potential V(a, (f>) is at a = 0. The curby now are well understood, it is not so simple to construct vature of the effective potential in the <7-direction is much inflationary models in a context of realistic theories of greater than in the -direction. Thus at the first stages of elementary particles, such as supergravity and string theory. expansion of the universe the field a rolled down to Another problem may arise if observational data will indi- a = 0, whereas the field <j> could remain large for a much cate that the universe is not flat but open or closed. In these longer time. section we will briefly describe some recent ideas which At the moment when the inflaton field 0 becomes smaller may help us to cope with these problems. than 4>c = M/g, the phase transition with the symmetry breaking occurs. If m2<j>2 = m2M2/g2 «C M4/k, the Hubble 5.1. Towards stringy inflation constant at the time of the phase transition is given by 2 4 2 2 (km2/g) The main problem here is that the effective potential for the H = (2nM /3kM ). If one assumes that M » 2 2 inflaton field in supergravity typically is too curved, growing and that m <§; H , then the universe at <j> > 4>c undergoes as exp(C2/Mp) at large values of 4>. The typical value of the a stage of inflation, which abruptly ends at = (f>c. One of the advantages of this scenario is the possibility to parameter C is 0(1), which makes inflation impossible because the inflaton mass in this theory becomes of the order obtain small density perturbations even if coupling conof the Hubble constant. The situation is not much better in stants are large, k, g = 0{\). This scenario works if the effecstring theory. Typical shape of the scalar field potential tive potential has a relatively flat (^-direction. But flat in string theory is exp(C/Mp) with C = 0(1), which also directions often appear in supersymmetric theories. This makes hybrid inflation an attractive playground for those prevents inflation for > Mp. As a result, after many years of development of who wants to achieve inflation in supergravity. Another advantage of this scenario is a possibility to have supersymmetric theories we still do not have a consistent cosmological theory based on supergravity and superstrings. inflation at (/> «; Mv. This helps to avoid problems which Of course, interpretation of string theory (or M-theory?) may appear if the effective potential in supergravity and changes every year, so one may simply postpone investi- string theory blows up at 4> > Mp. Several different models gation of this problem and study only globally super- of hybrid inflation in supergravity have been proposed during the last few years (F-term inflation [18], D-term symmetric models. Hopefully, this problem is only temporary; it may be inflation [19], etc.) A detailed discussion of various versions related to our limited understanding of string theory (or of hybrid inflation in supersymmetric theories can be found M-theory), which rapidly changes every year. It might be in [20]. possible also that inflation can be implemented in the simplest versions of string theory, but in a rather nontrivial 5.2. Open inflation way. One of the often discussed possibilities is the pre-big-bang Until very recently, we did not have any consistent scenario developed in [13]. This scenario assumes that cosmological models describing a homogeneous open inflation occurred in a stringy phase prior to the big bang. universe. Even though the Friedmann open universe model Unfortunately, it is extremely difficult to match the stage did exist, it did not appear to make any sense to assume that of the pre-big-bang inflation and the subsequent post- all parts of an infinite universe can be created simultaneously big-bang evolution [14]. It is difficult to obtain density and have the same value of energy density everywhere. A physically consistent model of open universe was properturbations with a nearly-flat spectrum in this scenario, and it is not quite clear how one can solve the primordial posed only after the invention of inflationary cosmology. monopole problem and avoid excessive production of (This is somewhat paradoxical, because most of inflationary gravitinos. Recently it was found also that this scenario models predict that the universe must be flat.) The main idea 5. Recent versions of inflationary theory

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nor of the type of [26]. A consistent model of one-field open inflation was proposed only very recently [32]. Effective potential in this model has a sharp peak at large 4>. During the tunneling and soon after it the condition \V"\ n without any need for the Coleman-De Luccia bubble observational consequences of such models can be found formation [26]. Unfortunately, all existing versions of this in [36]. scenario lead to a structureless universe with Q = 10~2 All these models are rather complicated, and it is certainly [26,27]. The only exception is the model proposed by true that the models which lead to Q = 1 are much more Barvinsky, which is based on investigation of the one-loop abundant and natural. Therefore flatness of the universe effective action in a theory of a scalar field with an extremely remains a robust prediction of most of the inflationary large (and fine-tuned) nonminimal coupling to gravity [28]. models. However, it is encouraging that inflationary theory However, this model, just as the original model of Ref. [26], is versatile enough to include models with all possible values is based on the assumption that the quantum creation of of Q. the universe is described by the Hartle-Hawking wave function [29]. Meanwhile, according to [6,12,27,30], the HartleHawking wave function describes the ground state of the 6. Reheating after inflation universe rather than the probability of the quantum creation The theory of reheating of the universe after inflation is the of the universe. Instead of describing the creation of the most important application of the quantum theory of paruniverse and its subsequent relaxation to the minimum of ticle creation, since almost all matter constituting the the effective potential, which is the essence of inflationary universe was created during this process. theory, it asserts that a typical universe from the very beginAt the stage of inflation all energy is concentrated in a ning is in the ground state corresponding to the minimum of classical slowly moving inflaton field <j>. Soon after the V((p). This is the main reason why the Hartle-Hawking wave end of inflation this field begins to oscillate near the minifunction fails to predict a long stage of inflation and reason- mum of its effective potential. Eventually it produces many ably large Q in most of inflationary models. Another prob- elementary particles, they interact with each other and come lem of this scenario is related to the singular nature of to a state of thermal equilibrium with some temperature T . r the Hawking-Turok instanton [27,31]. Elementary theory of this process was developed many Thus, for a long time we did not have any consistent and years ago [37]. It was based on the assumption that the unambiguous realization of the one-field open universe oscillating inflaton field can be considered as a collection model predicting 0.2 < Q < 1, neither of the type of [23], of noninteracting scalar particles, each of which decays sepwas to use the well known fact that the bubbles created in the process of quantum tunneling tend to have spherically symmetric shape, and homogeneous interior, if the tunneling probability is strongly suppressed. Bubble formation in the false vacuum is described by the Coleman-De Luccia (CDL) instantons [21]. Any bubble formed by this mechanism looks from inside like an infinite open universe [21,22]. If this universe continues inflating inside the bubble, then we obtain an open inflationary universe. Then by a certain fine-tuning of parameters one can get any value of Q in the range 0 < Q < 1 [23]. Even though the basic idea of this scenario was pretty simple, it was very difficult to find a realistic open inflation model. The general scenario proposed in [23] was based on investigation of chaotic inflation and tunneling in the theories of one scalar field 4>- There were many papers containing a detailed investigation of density perturbations, anisotropy of the microwave background radiation, and gravitational wave production in this scenario. However, no models where this scenario could be successfully realized have been proposed until very recently. As it was shown in [24], in the simplest models with polynomial potentials of the type of (m2/2)<£2 - (<5/3)03 + (A/4)4 the tunneling occurs not by bubble formation, but by jumping onto the top of the potential barrier described by the Hawking-Moss instanton [25]. This process leads to formation of inhomogeneous domains of a new phase, and the whole scenario fails. The main reason for this failure was in fact rather generic: CDL instantons exist only if \V"\ > H2 during the tunneling. Meanwhile, inflation, which, according to [23], begins immediately after the tunneling, typically requires \V"\
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arately in accordance with perturbation theory of particle particles produced each time will be proportional to the decay. However, recently it was understood that in many number of particles produced before. This leads to explosive inflationary models the first stages of reheating occur in a process of particle production [38]. regime of a broad parametric resonance. To distinguish this Bosons produced at that stage are far away from thermal stage from the subsequent stages of slow reheating and equilibrium and have enormously large occupation thermalization, it was called preheating [38]. The energy numbers. Explosive reheating leads to many interesting transfer from the inflaton field to other bose fields and effects. For example, specific nonthermal phase transitions particles during preheating is extremely efficient. may occur soon after preheating, which are capable of To explain the main idea of the new scenario we will con- restoring symmetry even in the theories with symmetry 16 sider first the simplest model of chaotic inflation with the breaking on the scale ~ 10 GeV [39]. These phase 2 2 effective potential V(<j>) = (m /2)(j) , and with the inter- transitions are capable of producing topological defects such action Lagrangian —{\/2)g2(j)2x1 — h}jj\jJx. We will take as strings, domain walls and monopoles [40]. Strong m = 10~ 6 M p , as required by microwave background ani- deviation from thermal equilibrium and the possibility of sotropy [6], and in the beginning we will assume for sim- production of superheavy particles by oscillations of a relaplicity that x particles do not have a bare mass, i.e. tively light inflaton field may resurrect the theory of GUT baryogenesis [41] and may considerably change the mx(4>) = g\ 0.3MP [6]. Suppose way baryons are produced in the Affleck-Dine scenario [42], for definiteness that initially 4> is large and negative, and and in the electro weak theory [43]. inflation ends at <j> ~ — 0.3M p . After that the field (/> rollsUsually only a small fraction of the energy of the inflaton to 4> = 0, then it grows up to 10 _1 M P ~ 1018 GeV, and finally field ~ lO - 2 ^ 2 is transferred to the particles x when the field rolls back and oscillates about = 0 with a gradually (j) approaches the point <j> = 0 for the first time [44]. The role decreasing amplitude. of the parametric resonance is to increase this energy We will assume that g > 10~5 [38], which implies exponentially within several oscillations of the inflaton field. gMp > 102m for the realistic value of the mass m ~ But suppose that the particles x interact with fermions \j/ with 10~6Mp. Thus, immediately after the end of inflation, when the coupling h^/^x- If this coupling is strong enough, then x cj) ~ Mp/3, the effective mass g\4>\ of the field / is much particles may decay to fermions before the oscillating field greater than m. It decreases when the field moves down, returns back to the minimum of the effective potential. If this happens, parametric resonance does not occur. but initially this process remains adiabatic, \mx\ <3C m2. Particle production occurs at the time when the However, as we will show, something equally interesting adiabaticity condition becomes violated, i.e. when may occur instead of it: The energy density of the x particles I"1*! ~ S\4>\ becomes greater than m2 = g2^2. This happens at the moment of their decay may become much greater than their energy density at the moment of their creation. This only when the field (f> rolls close to <j> = 0. The velocity of 7 may be sufficient for a complete reheating. the field at that time was |0 O | s a w M p / 1 0 « 10~ M P . The 2 2 process becomes nonadiabatic for g ^ < g\(j)0\, i.e. for Indeed, prior to their decay the number density of x —$„ < < „, where $„ particles produced at the point <j> = 0 remains practically ~ >/Soi/f> P8]- N ° t e tnat f ° r *» constant [38], whereas the effective mass of each x particle 10~5 the interval — „ < (j> < <£„ is very narrow: * <£grows as mx = gcj) when the field <j> rolls up from the miniM p /10. As a result, the process of particle production occurs mum of the effective potential. Therefore their total energy density grows. One may say that x particles are "fattened," nearly instantaneously, within the time being fed by the energy of the rolling field (f>. The fattened A '»~7XT~c^oir 1/2 <9) X particles tend to decay to fermions at the moment when l = 0. P universe, so all effects related to the expansion of the At that moment x particles can decay to two fermions with universe can be neglected during the process of particle mass up to m^ ~ (g/2)10 _1 M p , which can be as large as production. The uncertainty principle implies in this case 5 x 1017 GeV for g ~ 1. This is 5 orders of magnitude greater that the created particles will have typical momenta than the masses of the particles which can be produced by k ~ (A;*) -1 ~ df|<^ol)1/2- The occupation number nk of x the usual decay of 0 particles. As a result, the chain reaction particles with momentum k is equal to zero all the time when 4> -*• x -*• $ considerably enhances the efficiency of transfer it moves toward = 0. When it reaches 0 = 0 (or, more of energy of the inflaton field to matter. exactly, after it moves through the small region More importantly, superheavy particles ij/ (or the prod~4>t < (j) < 0 J the occupation number suddenly (within ucts of their decay) may eventually dominate the total energy the time At*) acquires the value [38] density of matter even if in the beginning their energy density was relatively small. For example, the energy density of the „ t = exp(--=K-Y (10) oscillating inflaton field in the theory with the effective potential (l/4)4 decreases as a~A in an expanding universe and this value does not change until the field (f> rolls to the with a scale factor a(t). Meanwhile the energy density stored in the nonrelativistic particles \jj (prior to their decay) point (j> = 0 again. 3 The main idea of the scenario of broad parametric decreases only as a~ . Therefore their energy density rapidly resonance is that each time when the field 0 approaches becomes dominant even if originally it was small. A subthe point (j) = 0, new / particles are being produced. Bose sequent decay of such particles leads to a complete reheating statistics implies, roughly speaking, that the number of of the universe. Physica Scripta T85

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Inflationary Cosmology This is exactly what happens in the theories where the post-inflationary motion of the inflaton field occurs along a flat direction of the effective potential. In such theories the standard scenario of reheating does not work because the field does not oscillate. Until the invention of the instant preheating scenario the only mechanism of reheating discussed in the context of such models was based on the gravitational production of particles [45]. The mechanism of instant preheating is much more efficient. After the moment when % particles are produced their energy density grows due to the growth of the field . Meanwhile the energy density of the field cp moving along a flat direction of F($) decreases extremely rapidly, as a~6(t). Therefore very soon all energy becomes concentrated in the particles produced at the end of inflation, and reheating completes. 7. Conclusions During the last 20 years inflationary theory gradually became the standard paradigm of modern cosmology. But this does not mean that all difficulties are over and we can relax. First of all, inflation is still a scenario which changes with every new idea in particle theory. Do we really know that inflation began at Planck density 1094 g/cm 3 ? What if our space has large internal dimensions, and energy density could never rise above 1025 g/cm 3 [46]? Was there any stage before inflation? Is it possible to implement inflation in string theory/M-theory? Inflationary theory evolved quite substantially, from old inflation to the theory of chaotic eternal self-reproducing universe. We learned that in most versions of inflationary theory the universe must be flat, and the spectrum of density perturbations should be also flat ("flat paradigm"). But we also learned that there are models where the spectrum of density perturbations is not flat. If necessary, it is possible to obtain not only adiabatic perturbations, but also isocurvature nongaussian perturbations with a very complicated spectrum. It is even possible to have inflation with This situation makes some observers unhappy. Few years ago it seemed that one can easily kill inflationary theory if one finds, for example, that the density of the universe is not equal to the critical density. It would be an important scientific result. Now the situation changed. If we find that Q = 1, it will be a confirmation of inflationary theory because 99% of inflationary models predict that Q=\, and no other theories make this prediction. On the other hand, if observations will show that Q ^ 1, it will not disprove inflation. Indeed, the only consistent theory of a large homogeneous universe with Q ^ 1 that is available now is based on inflationary cosmology. Still inflationary models are falsifiable. Each particular inflationary model can be tested, and many of them have been already ruled out by comparison of their predictions with observational data. A new generation of precision experiments in cosmology are going to make our life even more complicated and interesting. However, it is difficult to disprove the basic idea of inflation. To make my position more clear, I would like to use an analogy from the history of the standard model of electroweak interactions [47]. Even though this model was developed by Glashow, Weinberg and Salam in the 60's, © Physica Scripta 2000

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it became popular only in 1972, when it was realized that gauge theories with spontaneous symmetry breaking are renormalizable [48]. However, it was immediately pointed out that this model is far from being perfect. In particular, it was not based on the simple group of symmetries, and it had anomalies. Anomalies could destroy the renormalizability, and therefore it was necessary to invoke a mechanism of their cancellation by enlarging the fermion sector of the theory. This did not look very natural, and therefore Georgi and Glashow in 1972 suggested another model [49], which at the first glance looked much better. It was based on the simple group of symmetry 0(3), and it did not have any anomalies. However, after the discovery of neutral currents, which could not be described by the 0(3) model, everybody forgot about the issues of naturalness and simplicity and returned back to the more complicated Glashow-Weinberg-Salam model. This model has about twenty free parameters which so far did not find an adequate theoretical explanation. Some of these parameters may appear rather unnatural. The best example is the coupling constant of the electron to the Higgs field, which is 2 x 10~6. It is a pretty unnatural number which is fine-tuned in such a way as to make the electron 2000 lighter than the proton. It is important, however, that all existing versions of the electroweak theory are based on two fundamental principles: gauge invariance and spontaneous symmetry breaking. As far as these principles hold, we can adjust our parameters and wait until they get their interpretation in a context of a more general theory. We do not call it "fine tuning," we call it "fitting." This is the standard way of development of the elementary particle physics. For a long time cosmology developed in a somewhat different way, because of the scarcity of reliable observational data. Ten years ago many different cosmological models (HDM, CDM, Q=l, £ 2 « 1 , etc.) could describe all observational data reasonably well. The main criterion for a good theory was its beauty and naturalness. Now the situation is changing. Cosmology gradually becomes a normal experimental science, where the results of observations play a more important role than the considerations of naturalness. But in our search for a correct theory we cannot give up the requirement of its internal consistency. In particle physics the main principle which made this theory internally consistent was gauge invariance. It seems that in cosmology something like inflation is needed to make the universe large and homogeneous and resolve a pletora of other cosmological problems. It is encouraging that even if we find that the universe is open, we will be able to fit it into inflationary cosmology. But it seems equally important that so far we did not find any other theoretical framework apart from inflation which would explain the extraordinary homogeneity of an open universe. Thus, even if we find that Q ^ 1, we will rule out a very large class of inflationary models but we will be unable to rule out the idea of inflation. What's about microwave background anisotropy and the theory of large scale structure of the universe? Again, observational data may confirm inflationary theory because most of inflationary models predict perturbations with specific properties. However, if it so happens that inflationary perturbations are in a conflict with observational data, then one can easily propose inflationary Physica Scripta T85

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models where inflation solves homogeneity, isotropy and other problems, but produces extremely small density perturbations. Then one will be free to use his best noninflationary mechanism for production of density perturbations (strings, textures, etc.). Perhaps there is a specific way to disprove inflationary cosmology by comparing its predictions with observational data: If we find that the universe rotates as a whole, or has any other anisotropy which is described by vector perturbations of metric, then it will be very difficult to make it compatible with inflation [50]. Indeed we know that scalar and tensor perturbations can be produced during inflation, but vector perturbations, just like other vector fields, can hardly be produced. But what if we find out a nontrivial way of producing vector perturbations, just like we found a way to produce an inflationary universe with Q ^ 1? It seems that the only sure way to kill inflationary cosmology (if one really wants to do it) is to suggest a better cosmological theory.

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18. Copeland, E. J., Liddle, A. R., Lyth, D. H., Stewart, E. D. and Wands, D., Phys. Rev. D, 49, 6410 (1994); Dvali, G., Shafi, Q. and Schaefer, R. K., Phys. Rev. Lett. 73, 1886 (1994); Linde, A. D. and Riotto, A., Phys. Rev. D 56, 1841 (1997). 19. Binetruy, P. and Dvali, G., Phys. Lett. B 388, 241 (1996); Halyo, E., Phys. Lett. B 387, 43 (1996). 20. Lyth, D. H. and Riotto, A., hep-ph/9807278. 21. Coleman, S. and De Luccia, F., Phys. Rev. D 21, 3305 (1980). 22. Gott, J. R. Nature 295, 304 (1982); Gott, J. R. and Statler, T. S., Phys. Lett. 136B, 157 (1984). 23. Bucher, M., Goldhaber, A. S. and Turok, N., Phys. Rev. D 52, 3314 (1995); Yamamoto, K., Sasaki, M. and Tanaka, T., Astrophys. J. 455, 412 (1995). 24. Linde, A. D., Phys. Lett. B 351, 99 (1995); Linde, A. D. and Mezhlumian, A., Phys. Rev. D 52, 6789 (1995). 25. Hawking, S. W. and Moss, I. G., Phys. Lett. HOB, 35 (1982). 26. Hawking, S. W. and Turok, N., Phys. Lett. B 425, 25 (1998); Hawking, S. W. and Turok, N., Phys. Lett. B432 271 (1998), hep-th/9803156. 27. Linde, A. D., Phys. Rev. D58, 083514 (1998), gr-qc/9802038. 28. Barvinsky, A. O., hep-th/9806093, gr-qc/9812058. 29. Hartle, J. B. and Hawking, S. W., Phys. Rev. D 28, 2960 (1983). 30. Linde, A. D., Zh. Eksp. Teor. Fiz. 87, 369 (1984) [Sov. Phys. J. Exp. Theor. Phys. 60, 211 (1984)]; Lett. Nuovo Cim. 39, 401 (1984); Zeldovich, Ya.B. and Starobinsky, A. A., Sov. Astron. Lett. 10, 135 (1984); Rubakov, V. A., Phys. Lett. 148B, 280 (1984); Vilenkin, A., Phys. Rev. D 30, 549 (1984). 31. Vilenkin, A., hep-th/9803084; Unruh, W., gr-qc/9803050; Bousso, R. and Linde, A. D., gr-qc/9803068; Garriga, J., hep-th/9803210, hep-th/9804106; Bousso, R. and Chamblin, A., hep-th/9805167. 32. Linde, A. D., Phys. Rev. D 59, 023503 (1999). 33. Linde, A. D., Sasaki, M. and Tanaka, T., astro-ph/9901135. 34. Garcia-Bellido, J., hep-ph/9803270. 35. Garcia-Bellido, J., Garriga, J. and Montes, X., Phys. Rev. D 57, 4669 (1998). 36. Garcia-Bellido, J., Garriga, J. and Montes, X., hep-ph/9812533. 37. Dolgov, A. D. and Linde, A. D., Phys. Lett. 116B, 329 (1982); Abbott, L. F., Fahri, E. and Wise, M., Phys. Lett. 117B, 29 (1982). 38. Kofman, L. A., Linde, A. D. and Starobinsky, A. A., Phys. Rev. Lett. 73, 3195 (1994); Kofman, L., Linde, A. A. and Starobinsky, A. A., Phys. Rev. D 56, 3258-3295 (1997). 39. Kofman, L. A., Linde, A. D. and Starobinsky, A. A., Phys. Rev. Lett. 76, 1011 (1996); Tkachev, I., Phys. Lett. B 376, 35 (1996). 40. Tkachev, I., Kofman, L. A., Linde, A. D., Khlebnikov, S. and Starobinsky, A. A., in preparation. 41. Kolb, E. W., Linde, A. and Riotto, A., Phys. Rev. Lett. 77,4960 (1996). 42. Anderson, G. W., Linde, A. D. and Riotto, A., Phys. Rev. Lett. 77, 3716 (1996). 43. Garcia-Bellido, J., Grigorev, D., Kusenko, A. and Shaposhnikov, M., hep-ph/9902449; Garcia-Bellido, J. and Linde, A., Phys. Rev. D 57, 6075 (1998). 44. Felder, G., Kofman, L. A. and Linde, A. D., hep-ph/9812289. 45. Ford, L. H., Phys. Rev. D 35, 2955 (1987); Spokoiny, B., Phys. Lett. B 315, 40 (1993); Joyce, M., Phys. Rev. D 55, 1875 (1997); Joyce, M. and Prokopec, T., Phys. Rev. D 57, 6022 (1998); Peebles, P. J. E. and Vilenkin, A., astro-ph/9810509. 46. Arkani-Hamed, N., Dimopoulos, S., Kaloper, N. and March-Russell, J., hep-ph/9903224. 47. Glashow, S. L., Nucl. Phys. 22, 579 (1961); Weinberg, S., Phys. Rev. Lett. 19, 1264 (1967); Salam, A., in: "Elementary Particle Theory," (edited by N. Svartholm), (Almquist and Wiksell, Stockholm, 1968), p. 367. 48. 't Hooft, G., Nucl. Phys. B 35, 167 (1971); Lee, B. W., Phys. Rev. D 5, 823 (1972); Lee, B. W. and Zinn-Justin, J., Phys. Rev. D 5,3121 (1972); 't Hooft, G. and Veltman, M., Nucl. Phys. B 50, 318 (1972); Tyutin, I. V. and Fradkin, E. S., Sov. J. Nucl. Phys. 16, 464 (1972); Kallosh, R. E. and Tyutin, I. V., Sov. J. Nucl. Phys. 17, 98 (1973) [Yad. Fiz. 17, 190 (1973)]. 49. Georgi, H. and Glashow, S. L., Phys. Rev. Lett. 28, 1494 (1982). 50. Ellis, J. and Olive, K. A., Nature 303, 379 (1983).

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Physica Scripta.Vol. T85, 177-182, 2000

Quintessence and the Missing Energy Problem Paul J. Steinhardt Department of Physics, Princeton University, Princeton, NJ 08650, USA Received November 29, 1998; accepted August 2, 1999 PACS

Ref: 98.80.-k, 98.70Vc, 98.65.Dx, 98.80.Cq

Abstract Recent evidence suggests that the total matter density of the universe is significantly less than the critical density. The shortfall may be explained by curvature (an open universe), vacuum energy density (a cosmological constant), or quintessence (a time-evolving, spatially inhomogeneous component with negative pressure). In all three cases, a key problem is to explain the initial conditions required to have the energy density nearly coincident with the matter density today. A possible solution is "tracker fields," a form of quintessence with an attractor-like solution which leads to the same conditions today for a very wide range of initial conditions. Tracker field make quintessence a more viable candidate for the missing energy component and produce more distinctive predictions.

region lies far from Qm = 1. Explaining the shortfall constitutes the missing energy problem. As convincing as the case for Qm < 1 may appear to be, the conclusion should be accepted reluctantly. Every resolution of the missing energy problem entails introducing a profound and bizarre addition to our cosmological model compared to the Qm = 1 case. Especially for those whose research is infieldsoutside cosmology, a reasonable reaction is to ignore the problem until yet further evidence is gathered. At the same time, the strength of the current evidence summarized in Figs. 1 and 2 may explain why some cosmologists, including myself, have felt compelled to look ahead and explore solutions to the problem [4].

1. The problem The missing energy problem is emerging as one of the most profound puzzles of cosmology. The problem is best illustrated by displaying the current observational constraints in a plot of Qm versus Ho, where Qm is the ratio of the baryonic plus dark matter density to the critical density and Ho is the the Hubble constant HQ = 100/z km/s/Mpc [1,2]. See Fig. 1. The observational constraints cut a series of swaths through this plane. In Fig. 2, the swaths have been removed and all that remains is the region which is in concordance with all current observations [1,3]. Given more than a dozen independent constraints, the existence of a region of concordance is remarkable. Equally significant is the fact that two or more independent constraints limit each boundary of the concordance region. That is, the concordance region can move by a significant amount in any direction only if two or more independent constraints are incorrect. The most notable feature is that the concordance

2. The Solutions One possible explanation for the missing energy problem is that the universe is open - space is curved and there is no missing energy needed to explain the difference between Qm and unity. A theoretical difficulty with this option is that it contradicts inflationary cosmology.[5-8] To be sure, cosmological models can be rigged so that there is just the right amount of inflation to make an Qm = 0.3 open universe today [9]. However, inflation is a rapid, runaway process which removes the curvature exponentially fast. As a result, open inflation requires extraordinary tuning

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cluster abundance & evolution: baryon fraction 0.5

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Hubble Constant (x 100 km/sec/Mpc) Fig. 2. A plot showing the range of Qm and the Hubble constant which are in concordance with all current observations (the shaded region between the vertical dashed lines). Along the boundary are named the constraints that fix the given boundary section. Shown separately with dashed lines are current Fig. 1. Some of the current observational constraints in the Qm-h plane, illus- constraints on the Hubble constant (65 ± 10 km s_1 Mpc-1). Note that each boundary is fixed by two or more constraints. trating how the constraints cut swaths through parameter space. Hubble Expansion Rate (x 100 km/sec/Mpc)

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of parameters to control the amount of inflation precisely; with this tuning, inflation loses its simplicity and power as a scientific theory. If inflation is to remain a compelling, explanatory theory, then it is essential that the flatness be verified observationally. In fact, the flat universe is somewhat preferred by current observations. Evidence from Type IA supernovae [25] and from the cosmic microwave background anisotropy [26] disfavor the open option. Both sets of results are inconclusive, but near-future developments should be definitive. A second possibility is that the universe is flat, but there is a non-zero vacuum density or cosmology constant (A). A cosmological constant is, by definition, static and spatially uniform. At present, cosmological models with a mixture of A and cold dark matter (ACDM) fit all current observations well. A third possibility, the focus of this paper, is that the universe is flat and the missing energy consists of quintessence, [10] a dynamical, evolving, spatially inhomogeneous component. The term "quintessence" derives from the ancient term for the fifth element after earth, air, fire and water, an all-pervasive, weakly interacting component. In the current context, quintessence refers to a weakly interacting component in addition to the four already known to influence the evolution of the universe: baryons, leptons, photons, and dark matter. More specifically, quintessence refers to a component with negative pressure p and an equation-of-state (w = p/p) with 0 > w > — 1. (A cosmological constant has w precisely equal to —1.) Unlike a cosmological constant, the quintessential pressure and energy density evolve in time, and w may also. Furthermore, because the quintessence component evolves in time, quintessence is necessarily spatially inhomogeneous. The quintessential concept is related to the notion of a time-dependent cosmological constant, which dates back to the early days of relativistic cosmology, when it was ascribed political as well as scientific significance [27]. In these and some later treatments, an energy component was assumed whose density decreases more slowly than the matter density but which is smoothly distributed. However, it is important to note that a time-dependent cosmological constant, in the sense of a spatially uniform but temporally varying energy, is inconsistent with the equivalence principle [10]. Quintessence, which has non-negligible spatial fluctuations on large scales (> 100 Mpc), is the closest approximation that is physically consistent.

describe the inflaton field in inflationary cosmology [6,7]. The major difference is that the energy scale associated with quintessence is much lower so that it remains a subdominant contribution to the Friedmann equations that describe the expansion of the universe until recently. In this framework, quintessence can be viewed as the onset of a low-energy, late inflationary epoch. The pressure of the scalar field, p — 1 Q2 ~ ^ ( 2 ) is negative if the field rolls slowly enough that the kinetic energy density is less than the potential energy density. The ratio of kinetic-to-potential energy is determined by equation-of-motion for the scalar field: Q + 3HQ+V(Q)

=0

(1)

where

*-$-¥«•+>*

<2)

a(t) is the Robertson-Walker scale factor, and G is Newtons' constant. Depending on the detailed form of V(Q), the equation-of-state w can vary between 0 and —1. For most potentials, w evolves slowly over time. The field is assumed to couple only gravitationally to matter. In all cases, the Q-energy density decreases with time as l/a 3 ' 1+H, e) 5 so negative pressure corresponds to a density which decreases more slowly than 1/a3. Spatial fluctuations in Q evolve over time due to the gravitational interaction between Q and clustering matter. The perturbations are important because they can leave a distinguishable imprint on the CMB and large-scale structure. To determine how the perturbations evolve, specifying w is insufficient. One must know the response of the component to perturbations. This can be defined by specifying the sound speed cs as a function of wavenumber k or, alternatively, by specifying the equations-of motion (from which the perturbative equations can be derived). Note that it is possible, in principle, to have two fluids with the same w but different cs, which would lead to distinct observational predictions. For a scalar field, the equation-of-motion for the perturbations SQ in synchronous gauge is: SQ + 3HSQ + (k2 + a2 V"(Q))5Q = -\hQ,

(3)

where the dot represents the derivative with respect to conformal time, the prime represents the derivative with respect to Q, and hk is the kth fourier mode of the perturbed metric. The source term in Eq. (3) has several important properties. First, any realistic cosmological model has clustering 3. Quintessential basics matter, so hk must be non-zero. Also, except for the limit The general description of quintessence given in the previous of a constant V(Q) (which corresponds to a true section incorporates many proposals for the missing energy cosmological constant), Q is non-zero. Hence, the source that have been proposed over the past decade or so [10-19]. term must be non-zero overall for any p > — p (w > — 1). For example, under some conditions, light, nonabelian This is significant because it ensures that Q cannot be cosmic defects, strings or walls, evolve into a frozen network smoothly spread. Even if SQ is non-zero initially, the source whose equation-of-state corresponds to w = —1/3 and term causes perturbations to grow. (If the universe consisted w = —2/3, respectively [12]. The example considered here initially of a uniform distribution of quintessence, hk could and in most papers is the energy density associated with remain zero and Q could remain smoothly spread. However, a scalar field Q rolling down the potential V(Q), which once there is a mixture of components and at least one can have an equation-of-state anywhere between zero and clusters, smoothness is no longer possible.) A further conse— 1. The equations are nearly identical to the equations that quence of the source term is that the perturbations in Q Physica Scripta T85

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Quintessence and the Missing Energy Problem 1.0 r-

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equation-of-state w Fig. 3. For QCDM models, the range of Qm and w in concordance with all current measurements. The full region represents the combination of COBE constraints from measurements of the microwave background anisotropy on large angular scales and other astrophysical observations. The right end represents the parameter range cut off by current measurements of the microwave background anisotropy at small angular scales.

observed today are extremely insensitive to the initial conditions for 8Q. Assuming that SpQ/pQ is comparable to the perturbations in other energy components, the transient solution to the perturbation equation is negligible today compared to the particular solution set by the source term.

4. Quintessential motivation Why consider quintessence if its effect on the expansion of the universe is similar to a cosmological constant? The reasons are that: (a) quintessence has different implications for fundamental physics; (b) quintessence may fit the observational data better than cosmological constant; and (c) quintessence may solve cosmoological problems which a cosmological constant cannot. Distinguishing quintessence from a cosmological constant is important both for cosmology and for fundamental physics. A vacuum density or cosmological constant (A) is static and spatially uniform. Its value is set once and for all in the very early universe. Hence, A is tied directly to quantum gravity physics near the Planck scale. Quintessence is new dynamics at ultra-low energies (energy scale ~ 1 meV today), perhaps a harbinger of a whole spectrum of new low-energy phenomena. At present, both models with cosmological constant and cold dark matter (ACDM) and models with a mixture of quintessence and cold dark matter (QCDM) agree with current data [3]. However, QCDM fits marginally better. Figure 3 illustrates the range of Qm and WQ which fit all current observations. Note that the allowed region only includes negative pressure whether or not recent supernovae observations are included. The allowed region encompasses cosmological constant w = — 1 and a substantial region with w > — 1. The best-fit model [3] corresponds to a quintessence model with w & —2/3, although the difference in fit between this model and the best-fit A model is small. © Physica Scripta 2000

179

For quintessence composed of scalar fields, there is the added observational constraint that the coupling to ordinary matter be sufficiently suppressed to evade fifth force and other constraints on light fields [20]. This constraint does not project into the Qm-w plane. Whatever form the missing energy takes, two new cosmological problems arise. First, the component must be comparable in density today to the critical density, 10~47 GeV 4 . This small value is forced by the observation that QQ and Qm are comparable today. We will refer to the puzzle of explaining this tiny energy as the "fine-tuning problem." A second problem arises when the cosmological model is extrapolated back in time to the very early universe, at the end of inflation, say. The curvature, the cosmological constant, and, in general, quintessence decrease at different rates than the matter density. At the end of inflation, it appears that the ratio of missing energy to matter and radiation energy must be specially fixed to an extraordinarily small value (of order 10~100 for the case of vacuum density) in order to have the ratio evolve to be of order unity today. Accounting for the special ratio in the early universe will be referred to as the "coincidence problem [23]." The coincidence problem is a generalization of the flatness problem pointed out by Dicke and Peebles [24]. The fine-tuning and coincidence problems are often treated as one and the same because they occur together for models with vacuum density or curvature. Vacuum density and curvature are fixed by a single, unvarying constant set at the beginning of the universe. The constant must be tiny compared to particle physics scales (the fine-tuning problem) and must be set to an infinitesimal fraction of the initial total energy density to have the correct density today (the coincidence problem). For quintessence, though, the two problems separate [21,22]. Because quintessence evolves temporally and can interact with matter gravitationally or directly, the quintessential field might be able to begin with natural initial conditions and, through its dynamics, evolve to the correct value today. Indeed, recent progress in resolving the coincidence problem along these lines has been made, as described in the next section. The fine-tuning problem requires a separate solution, as described in Section 5.2.

5. A quintessential solution to the cosmic coincidence problem Recently, Zlatev et al. have introduced the notion of tracking fields and tracking potentials as a quintessential solution to the coincidence problem. Tracker fields have an equation-of-motion with attractor-like solutions in which a very wide range of initial conditions rapidly converge to a common, cosmic evolutionary track. The initial value of PQ can vary by nearly 100 orders of magnitude without altering the cosmic history. The acceptable initial conditions include the natural possibility of equipartition after inflation - nearly equal energy density in Q as in the other 100-1000 degrees of freedom (i.e., QQI % 10~3). Furthermore, the resulting cosmology has desirable properties. The equation-of-state WQ varies according to the background equation-of-state WB- When the universe is radiation-dominated (WB = 1/3), then WQ is less than or Physica Scripta T85

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equal to 1 /3 and pQ decreases less rapidly than the radiation density. When the universe is matter-dominated (WB = 0), then WQ is less than zero and pQ decreases less rapidly than the matter density. Eventually, pQ surpasses the matter density and becomes the dominant component. At this point, Q slows to a crawl and WQ -> — 1 as QQ -*• 1 and the universe is driven into an accelerating phase. These properties seem to match current observations well [3]. 5.1. Trackers and initial conditions The key property of tracker fields is that they rapidly converge to the tracker solution for an extraordinarily wide range of initial conditions. The tracker solution differs from a classical dynamics attractor because it is time-dependent: during tracking, QQ increases steadily. This contrasts with the "self-adjusting" solutions based on exponential potentials recently discussed by Ferreira and Joyce [14,15,18] in which QQ is constant during the matter dominated-epoch. For the self-adjusting solution, QQ is constrained to be small (QQ < 0.15) at the beginning of matter domination in order that large-scale structure be formed, but then it cannot change thereafter. Hence, the quintessence remains a small, subdominant component. The tracker solution is more desirable, though, because it enables the g-energy to eventually overtake the matter density and induce a period of accelerated expansion, which produces a cosmology more consistent with measurements of the matter density, large scale structure, and supernovae observations [3,25].

Z+1 Fig. 4. Energy density versus red shift for the evolution of a tracker field. For computational convenience, z = 1012 (rather than inflation) has been arbitrarily chosen as the initial time. The white bar on left represents the range of initial PQ that leads to undershoot and the grey bar represents overshoot, combining for a span ofmore than 100 orders of magnitude if we extrapolate back to inflation. The solid black circle represents the unique initial condition if the missing energy is vacuum energy density. The solid thick curve represents an "overshoot" in which p e begins from a value greater than the tracker solution value, decreases rapidly and freezes, and eventually joins the tracker solution.

oscillating around and then rapidly converging to the tracker solution. Next, let's consider the "overshoot" case in which pQ is To study how the tracker solution is approached, it is initially much greater than the tracker value pQ. This case useful to rewrite the equation-of-motion Eq. (1) in the form: includes equipartition where, for 100 — 1000 degrees of freedom, say, pQ « 10~(2~3)pB. Now, the initial Q is at a high I d lnx V (4) point in the potential where V/ V is much greater than the V 1 + WQ 6d In a tracker solution. At this steep point in the potential, the field where x = (1 4- WQ)/(1 - WQ) = \ Q^/V is the ratio of the is rapidly accelerated to the point where its energy is kinetic to potential energy density for Q and prime means dominantly kinetic and the potential is irrelevant. The field derivative with respect to Q. The ± sign depends on whether overshoots the tracker by a calculable amount [21,22] that V > 0 or V < 0, respectively. The tracking solution (to does not depend on the potential. Eventually, it freezes when which general solutions converge) has the property that its kinetic energy is red shifted away. From that point on, it WQ is nearly constant and lies between WB and — 1. For behaves just as in the undershoot case above. An example 1 + WQ = 0(1), Q2 « QQH2 and the equation-of-motion, of an overshoot solution is illustrated in Fig. 4. Eq. (4), dictates that Only if Q dominates over the radiation density initially, PQ » pB, does tracking fail. In this case case, Q rolls rapidly H 1 (5) down hill and overshoots the attractor solution to an ~v" /Or. ~Q extraordinary degree and is "deep-frozen." The field freezes for a tracking solution; we shall refer to this as the "tracker at a value so far downhill that V'/V « : H/Q today and condition." In order to maintain this condition, it is necess- the attractor solution has not been reached yet. However, ary that V'/V be decreasing as Q rolls downhill and QQ exempting the initial condition pQ » pB still leaves a range of over 100 orders of magnitude of viable initial conditions increases. Solutions to the equation-of-motion are drawn towards a including the most likely case, equipartition. common solution, Q(t) whose energy density is Pg(t). First, let's consider the "undershoot" case where pQ is initially much smaller than the tracker value pQ. See Fig. 4. In this 5.2. Testing for trackers case, the initial Q is at a low point on the potential where Hence, testing for the existence of tracking solutions reduces V'/V is much smaller than the tracker value. This value to a simple condition on V(Q) without having to solve the is so small that the Hubble damping term dominates the equation-of-motion directly. It suffices to check whether >\ and is nearly constant equation-of-motion and the field is nearly frozen in place. T =V"V/(V')2 The frozen field cannot evolve until the universe ages (and (|d(r - \)/Hdt\ «; \r - 11) over the range of plausible initial H decreases) to the point where V'/V = H/Q; that is, Q Q. Here we are assuming WQ < wB; that is, the energy density is now at the ideal point on the potential required by the in Q decreases more slowly than the background density. tracker solution for the given H. The field begins to move, The range of initial Q extends from 2max> where K(Q max ) Physica Scripta T85

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Quintessence and the Missing Energy Problem is equal to the initial background energy density p B , to Qmm where V{Qm\n) is equal to the background density at matter-radiation equality. The energy density spans more than 100 orders of magnitude over this range. The condition that r« constant can be evaluated by testing whether

r

r(v/v)

« i

(6)

over the range between Q max and Qm\n as V decreases. These conditions encompass an extremely broad range of potentials, including inverse power-law potentials (V(Q) = M 4+a IQ* for a > 0) and combinations of inverse power-law terms {e.g., V(Q) = M 4 e x p ( M / 0 ) . Some potentials of this form are suggested by particle physics models with dynamical symmetry breaking or nonperturbative effects [29-34].

181

V(Q). Now consider the entire family of tracker solutions for each given V(Q) and consider whether QQ is more likely to dominate late in the universe for one V(Q) or another. In general, QQ is proportional to a3(-WB~w^ oc t2(WB-WQ)K\+WB)^ w h e r e i t c a n b e s h o w n t hat [22] WB -

WQ

=

2(r - \)(wB +1) i + 2(r-i)

(7)

Hence, we find QQ OC tp where P =

4(r-i) l + 2(r - i) "

(8)

A satisfying explanation as to why QQ dominates at late times, rather than early, would be if QQ automatically changes behavior at late times. This is not the case for two special cases (which have the been the focus of earlier papers): K ~ l / 2 a and V ~ exp(/?£?). As the universe evolves, T — 1 and, hence, P are constant; consequently, 5.3. Tracking and the fine-tuning problem Some tracker solutions do not require small mass par- QQ grows as the same function of time throughout the ameters in the scalar potential to obtain a small energy den- radiation- and matter-dominated epochs. However, these sity needed today to explain Qm, which is relevant to the are the exception, rather than the rule. For more general potentials, P increases as the universe fine-tuning problem. The one free dimensional parameter in V(Q), M, is determined by the observational constraint, ages. Consider first a potential which is the sum of two QQ « 0 . 7 today. To have Qn«0.7 today requires inverse power-law terms with exponents ai < o<2. The term 47 4 V(Q « Mp) « pm, where pm « 10" GeV is the current with the larger power is dominant at early times when Q matter density. The condition imposes the constraint is small, but the term with the smaller power dominates M«(pmAf«)1/("+4). For low values of a or for the at late times as Q rolls downhill and obtains a larger value. exponential potential, this forces M to be a tiny mass, as Hence, the effective value of a decreases and T — 1 oc 1/a low as 1 meV for the exponential case. However, we note increases; the result is that P increases at late times. For that M > 1 GeV — comparable to particle physics scales more general potentials, such as V ~ exp(l/g), the effective — is possible for a > 2. This case exemplifies how a small value of a decreases continuously and P increases with time. density may be achieved by functional forms combining a Fig. 5 illustrates the comparison in the growth of P. An increasing P means that QQ grows more rapidly as the typical particle physics mass scale with a large expectation universe ages. Figure 6 compares a tracker solution for a value for Q. Hence, while this is not the real aim of tracking, pure inverse power-law potential (V ~ l/Q6) model with it is interesting to note that the tracker solution does not require the introduction of a new mass hierarchy in funda- a tracker solution for V ~ e x p ( l / 0 , where the two solutions have been chosen to begin at the same value of QQ. (The start mental parameters. time has been chosen arbitrarily at z = 1017 for the purposes of this illustration.) Following each curve to the right, there 5.4. The QQ-WQ relation is a dramatic (10 orders of magnitude) difference between An important consequence of the tracker solutions is the the time when the first solution (solid line) meets the backprediction of a relation between WQ and QQ today [22]. ground density versus the second solution (dot-dashed line). Because tracker solutions are insensitive to initial con- That is, beginning from the same QQ, the first tracker solditions, both WQ and QQ only depend on V(Q). Hence, ution dominates well before matter-radiation equality and for any given V(Q), once QQ is measured, WQ is determined. the second (generic) example dominates well after In general, the closer that QQ is to unity, the closer WQ is matter-domination. The difference is less dramatic as a to —1. However, the fact that Qm > 0.2 today means that increases for the pure inverse power-law model; also, as a. WQ cannot be so close to —1. We find that WQ > - 0 . 8 for increases, the value of WQ today (given Q > 0.2) approaches m practical models.[21,22] This QQ-WQ condition on the zero and the universe does not enter a period of acceleration equation-of-state forces WQ to be sufficiently different from by the present epoch. Hence, a significant conclusion is that — 1 that the tracker field proposal can be distinguished from the pure exponential (exp(/J0) and inverse power-law the cosmological constant, e.g., by measurements of the (M 4+(X /2 o: ) models are atypical; the generic potential expansion history of the universe using Type IA (V(M/Q)) has properties that make it more plausible that supernovae. QQ dominates late in the history of the universe and induces a recent period of accelerated expansion. 5.5. Why now? Why should QQ only begin to dominate and initiate a period of accelerated expansion late in the history of the universe? The tracker field notion offers some explanation. Thus far, 6. Conclusions we have imagined fixing M to guarantee that Any resolution of the missing energy problem requires a QQ = 1 — Qm has the measured value today. The choice dramatic departure from the simplicity of the standard cold amounts to considering one tracker solution for each dark matter picture. Within a few years, improved obser© Physica Scripta 2000

Physica Scripta T85

182

Paul J. Steinhardt not require a small mass parameter. This progress makes the notion of quintessence as the missing matter a much more interesting and compelling concept. In addition, tracking, like inflation, is a general idea which may find other applications in cosmology not yet anticipated.

10

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£6

Acknowledgements

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V - exp(1/Q) - 1

The work described in this paper is the result of collaborations with Limin Wang, Ivaylo Zlatev and Rob Caldwell. This research was supported by the US Department of Energy grant DE-FG02-95ER40893 (Penn).

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z+1 Fig. 5. A plot of P/P0 versus /, where QQ a tp and P0 is the initial value of P. The plot compares pure inverse power-law (V ~ l/Q") potentials for which P is constant with a generic potential (e.g. K~exp ( 1 / 0 ) for which P increases with time.

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z+1 Fig. 6. A plot comparing two tracker solutions for the case of a V ~ \IQ potential (solid line) and a V ~ e x p ( 1 / 0 potential (dot-dash). The dashed line is the background density The two tracker solutions were chosen to have the same energy density initially. The tracker solution for the generic example ( K ~ e x p ( l / 0 ) reaches the background density much later than for the pure inverse-power law potential. Hence, QQ is more likely to dominate late in the history of the universe in the generic case.

vations will determine whether cosmologists are forced to this consideration. In the meantime, the imaginative exploration of possible solutions to the missing energy problem may lead to new concepts in cosmology. Quintessence serves as an instructive example. As a general notion, quintessence has many degrees of freedom and no clear advantage compared to the cosmological constant solution. Now, however, the tracker field form of quintessence introduces a new approach to two seemingly intractable issues: the problem of initial conditions and the issue of why pQ is dominating today rather than at some early epoch. It leads to a new, unanticipated prediction a relation between Qm = 1 - QQ and WQ today that makes tracker fields distinguishable from a cosmological constant. And, it may Physica Scripta T85

1. See, for example, Ostriker, J. P. and Steinhardt, P. J. Nature 377, 600 (1995), and references therein. 2. Turner, M. S., Steigman, G. and Krauss, L. Phys. Rev. Lett. 52, 2090 (1984). 3. Wang, L., Caldwell, R. R., Ostriker, J. P. and Steinhardt, P. J., in preparation. 4. See paper by Turner, M. in this volume for a fuller discussion of the current observational evidence. 5. Guth, A. H., Phys. Rev. D23, 347, (1981). 6. Linde, A. D., Phys. Lett. 108B, 389 (1982). 7. Albrecht, A. and Steinhardt, P. J., Phys. Rev. Lett. 48, 1220 (1982). 8. Guth, A. H. and Steinhardt, P. J., "The Inflationary Universe" in "The New Physics," (ed. by P. Davies), (Cambridge U. Press, Cambridge, 1989) pp. 34-60. 9. See. for example, Hawking, S. W. and Turok, N., Phys. Lett. B425, 25 (1998); and Bucher, M., Goldhaber, A. S. and Turok, N., Phys. Rev. Lett. Phys. Rev. D52, 3314 (1995). 10. Caldwell, R. R., Dave, R. and Steinhardt, P. J., Phys. Rev. Lett. 80, 1582 (1998). 11. Turner, M. S. and White, M., Phys. Rev. D 56, R4439 (1997). 12. Spergel, D. and Pen, U.-L., Astrophys. J. 491, L67, (1997). 13. Weiss, N., Phys. Lett. B 197, 42 (1987). 14. Ratra, B. and Peebles, P. J. E., Phys. Rev. D 37, 3406 (1988); Peebles, P. J. E. and Ratra, B., Astrophys. J. 325, L17 (1988). 15. Wetterich, C , Astron. Astrophys. 301, 32 (1995). 16. Frieman, J. A., et al., Phys. Rev. Lett. 75, 2077 (1995). 17. Coble, K. Dodelson, S. and Frieman, J., Phys. Rev. D 55, 1851 (1995). 18. Ferreira, P. G. and Joyce, M. Phys. Rev. Lett. 79, 4740 (1997); Phys. Rev. D 58, 023503 (1998). 19. Copeland, E. J., Liddle, A. R. and Wands, D. Phys. Rev. D57, 4686 (1998). 20. Carroll, S., Phys. Rev. Lett. 81, 3067 (1998). 21. Zlatev, I. Wang, L. and Steinhardt, P. J., Phys. Rev. Lett. 82, 896 (1999). 22. Steinhardt, P., Wang, L. and Zlatev, I., submitted to PRD (1998). 23. Steinhardt, P., in "Critical Problems in Physics," (ed. by V.L. Fitch and D.R. Marlow), (Princeton U. Press, 1997). 24. Dicke, R. H. and Peebles, P. J. E. in "General Relativity: An Einstein Centenary Survey," (ed. by S.W. Hawking & W. Israel), (Cambridge Univ. Press 1979). 25. Perlmutter, S., etal., Nature 391, 51 (1998); A. G. Riess, et al, Astron. J. 116, 1009 (1998). 26. Tegmark, M., Ap. J. 514, L69 (1999) and Ap. J. 519, 513 (1999). 27. See Ref. 33 in Overduin, J. M. and Cooperstock, F. I., Phys. Rev. D58 043506 (1998); Bronstein, M., Phys. Z. Sowjet Union 3, 73 (1933). 28. Binetruy, P., Phys. Rev. D 60, 063522 (1999). 29. Anderson, G. W. and Carroll, S. M., astro-ph/9711288. 30. Barreiro, T., de Carlos, B. and Copeland, E. J., Phys. Rev. D57, 7354 (1998). 31. Binetruy, P., Gaillard, M. K. and Wu, Y.-Y., Phys. Lett. B412, 288 (1997); Nucl. Phys. B493, 27 (1997). 32. Barrow, J. D. Phys. Lett. B235, 40 (1990). 33. Hill, C. and Ross, G. G., Nucl. Phys. B 311, 253 (1988); Phys. Lett. B 203, 125 (1988). 34. Affleck, I. et al., Nuc. Phys. B 256, 557 (1985).

© Physica Scripta 2000

Physica Scripta.Vol. T85, 183-190, 2000

Ultra High Energy Cosmic Rays - an Enigma A. A. Watson* Department of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK Received March 3, 1999; accepted August 2, 1999

PACS Ref: 96.40.De, 96.40.Pq, 98.70.Sa

Abstract The methods of detection of high energy cosmic rays are briefly described and it is shown that results on the energy spectrum and arrival distribution above 4 x 1 0 " eV confound theoretical expectation. Some of the ideas which have been put forward to explain the data are outlined. There is an urgent need for better statistics as only about 12 events above 1020 eV have so far been detected. Furthermore the limit to the energy which cosmic rays can have is not known. In addition to the Pierre Auger Observatory, which is briefly described, the Hi-Res fluorescence detector is about to take data and an ambitious satellite observatory (Airwatch/OWL) is under discussion.

1. Introduction The cosmic ray energy spectrum extends from 10 9 eV to at least 3 x 1020eV with no evidence that the highest energy recorded thus far is Nature's upper limit. Although cosmic rays have been known since 1912 it remains uncertain as to where they come from. We believe that the majority of the particles are the nuclei of the common elements and at low energies, were direct observations can be made, the distribution of the elements is strikingly similar to that found in ordinary material in the solar system with the exception of elements such as Li, Be and B. These are the products of fragmentation of heavier nuclei and are very over-abundant by comparison with their local galactic abundance. The energy spectrum is nearly featureless lacking the lines or dips which would characterise an electromagnetic spectrum covering so many decades. It is because the particles are charged and travel through the tangled magnetic fields in our galaxy and beyond that it so difficult to trace them back to their origins. Only at the very highest energies does it seem reasonable to contemplate cosmic ray astronomy but the studies that have been possible so far have shown only a very unexpected isotropy. This result, coupled with the surprisingly great energies of the rarest particles, presents an enigma. In this short review the methods of detection of the highest energy cosmic rays will be described and results on the energy spectrum, mass composition and arrival direction distribution outlined. The enigma presented by the data will be highlighted and some of the ideas which have been suggested to resolve the puzzle will be reviewed. New methods of detection are needed and some of those planned will be outlined. There have been a number of recent reviews of the subject [1-5]. 2. Detection methods for ultra high energy cosmic rays Ultra High Energy Cosmic Rays (UHECR) are detected through the cascades of particles which are produced when * e-mail: [email protected] © Physica Scripta 2000

a high energy photon, proton or heavier nucleus interacts in the earth's atmosphere. The cascades are know as extensive air showers, the phenomenon having been discovered by Pierre Augur in 1938. His work established that the showers he was able to detect were initiated by entities (he thought that they were photons) of energy 10 15 eV and this changed the upper energy limit of energetic particles by nearly 106. The showers provide a detection route because the atmosphere acts both as a particle amplifier and a particle spreader. A primary of 1019eV will have about 10 10 particles in the resulting cascade when it reaches ground level while Coulomb scattering of the shower particles (mainly electrons) and the transverse momentum associated with the hadronic interactions spread the particles over a wide area. A shower from a 1019eV primary has a footprint on the ground of over 10 km 2 . The generic method of detection of air showers is a development of the technique used by Auger and his colleagues in their discovery work. The procedure is to spread an number of particle detectors (scintillators and water Cerenkov detectors are the current devices of choice) in a regular array on the ground. To study showers produced by primaries of 1019eV and above the spacing between the detectors can be about 1 km. The direction of the events can be measured to better than 3° by determining the relative time of arrival of the shower disk at the detectors. The particle density pattern can be used to infer the number of particles which reach the ground. To determine the primary energy it is necessary to use the predictions of a calculation which makes assumptions about the development of the shower in the atmosphere using extrapolations of the properties of interactions studied at accelerators. It is, of course, uncertain as to what the characteristics of hadronic interactions are at energies well above accelerator energies. The need to rely on extrapolations of known particle physics to estimate the primary energy is a major drawback of the generic method and has raised doubts about the accuracy of the energy estimates which derive from such studies. However the shower particles also excite the 2 + band of nitrogen and the resulting fluorescence is in the 350 - 450 nm band. Although only about 4.5 photons per metre of electron track are radiated, the light is detectable from showers above about 3 x 10 7 eV with arrays of photomultipliers on clear, moonless nights. This allows a calorimetric measurement of the shower energy in a manner similar to a technique familiar from particle physics. The resulting cascade shape gives the particle number as a function of depth and the energy of the primary can be obtained rather directly from the area under the curve. The angular accuracy of the fluorescence technique is very similar to that achieved with arrays of particle detectors. The fluorescence technique also allows the Physica Scripta T85

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A. A. Watson

depth of maximum of the shower, Xmax to be determined to within ±50 gcm~ 2 . The particle array technique has been realised on scales > 10 km 2 at 5 sites world wide, Volcano Ranch (USA), Haverah Park (UK), Narribri (Australia), Yakutsk (Russia) and AG AS A (Japan). For a variety of reasons there are problems with data from the Narribri and Yakutsk experiments at the very highest energies (see [1,6] for a discussion) and in what follows data from the other three arrays will be emphasised. The fluorescence technique has so far been implemented only in the Dugway desert by the Fly's Eye group from the University of Utah: this has provided a key set of data. 3. The energy spectrum of high energy cosmic rays The arrays at Volcano Ranch and Haverah Park were set up with the primary motivation of measuring the arrival direction distribution of cosmic rays above 10 18 eV. In the late 1950s it was expected that the distribution would be anisotropic, possibly showing features associated with the magnetic field structure of our galaxy. No such anisotropics have so far been detected but in 1966 a prediction, following upon the discovery of the 2.7 K background radiation, led to a change of focus and to intense interest in measuring the spectral shape above 10 19 eV. It was recognised by Greisen [7] and independently by Zatsepin and Kuzmin [8] that cosmic rays of high enough energy would suffer serious energy losses as they passed through the microwave background radiation field. The interaction process is the excitation of the A + resonance if the highest energy cosmic rays are protons: the energies of interest are above 4 x l 0 1 9 e V . The protons lose energy rapidly and at 4 x 1019 eV 50% of the particles which reach the earth will have travelled less that 150 Mpc: at 10 20 eV the 50% distance is only 19 Mpc [5]. This effect has become known as the GZK effect. Heavier nuclei will also lose energy rapidly by photo-disintegration (the diffuse infra-red background is also important) and photons interacting with radio photons lose energy rapidly through pair production. The results from Haverah Park, Yakutsk, AGASA and Fly's Eye show that the cosmic ray energy spectrum steepens at around 2 x 1018 eV and then flattens above about 1019 eV. These spectral details were first claimed by the Haverah Park group [9] and were confirmed by later experiments with superior statistics. A very widely held interpretation of this result (e.g. [60]) is that galactic cosmic ray accelerators are no longer efficient above about 2 x 1018eV and that an extragalactic component begins to dominate at the highest energies. The origin of the extragalactic component remains unknown. Evidence for trans-GZK particles has been accumulating for many years. The first detection of a particle with energy above 10 20 eV was at the pioneering Volcano Ranch array [10] before the GZK prediction was made. Subsequent re-assessment of the energy has shown the original estimate to have been remarkably accurate. At Haverah Park 4 events above 1020 eV were reported from the exposure between 1967 and 1987 [11]. The deduction of the primary energy from the density pattern observed with ground arrays relies on the Monte Carlo calculations describing the development of the showers. While ground parameters were Physica Scripta T85

measured that have low susceptibility to uncertainties about the underlying particle physics, the astrophysical significance of the claimed energies was such that there was significant debate about their accuracy. In particular the first Fly's Eye spectrum [12], measured with a single eye, suggested evidence of a cut-off. However in 1993 the Fly's Eye group, using binocular data, reported a spectrum that agreed well with that measured at Haverah Park and, in addition, the group described the detection of an event with energy 3 x 1020 eV [13]. The calorimetric nature of the fluorescence method and the agreement with earlier measurements between 3 x 1017 and 3 x 1019 eV, where statistics are more numerous, removed many of the doubts about earlier claims for particles beyond the cut-off. In 1994 the AGASA group in Japan reported an event of 2 x 10 20 eV [14] and this group has recently published the results from a 7 year exposure with their 100 km 2 array. Six events with energies above 10 20 eV, well above the trans-GZK cut-off, are described [15]. Their data are reproduced in Fig. 1 where they are also compared with the final Haverah Park and Fly's Eye measurements. The agreement is extremely impressive bearing in mind the different techniques used and demonstrates that the systematic errors are understood at the 15% level. In Table I the exposures at 10 20 eV for the various instruments and the number of events recorded above that energy are shown. It is clear that within the very low statistics

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log(ENERGY in eV) Fig. 1. A comparison of the spectrum reported1 by the AGASA group [15] with final spectra measured by the Haverah Park [11] and Fly's Eye [13]. I am grateful to Professor M. Nagano for providing me with this figure. © Physica Scripta 2000

Ultra High Energy Cosmic Rays - an Enigma Table I. Exposures of UHECR km 2 sr year > 1020 eV Volcano Ranch (1962) Haverah Park (1987) Fly's Eye (stereo) Fly's Eye (mono) (1993) AGASA (1998)

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the rates from the different experiments are quite consistent. In addition the Yakutsk group have reported an event of 1.2 x 1020 eV [16]. The exposure for this array is not known with certainty as there are doubts about its efficiency for detecting high energy events [6,17]. We may conclude that the shape of the spectrum is now reasonably well known to about 5 x 1019 eV (Fig. 1) and that about 12 events have been detected above 1020 eV (Table I). Even if the energies of these events have been over-estimated by 20% it appears rather certain that the GZK cut-off is not observed. The rate of events above 10 20 eV is about 1 per km 2 per sr per century but this figure is uncertain to ±30%. 4. The arrival direction distribution The existence of events beyond the GZK cut-off implies that the sources of the events must be nearby. If a source could accelerate a proton to 1021 eV then, because of the loss processes associated with photo-pion production, the energy will fall to 3 x 1020 eV at 30 Mpc from it. For an iron nucleus the energy after travelling the same path would be even lower and for a photon lower still. Little is know about the magnetic field over such distances but, following Kronberg [18], it is often supposed that it has a mean strength of 10~9 G and is structured in cells of ~ 1 Mpc within which the field orientation changes. Such a configuration would deflect a proton of 1020 eV about 3° from a straight line over 30 Mpc. The celestial co-ordinates of the UHECR can be determined to better than 3° and one might therefore expect correlations between the arrival direction of the most energetic events with possible sources such as active radio galaxies. However one should look only at those objects which appear exceptional as there are many catalogues and there is always the possibility of finding some correlations from excessive searching when none really exists. One of the most careful pieces of work in this regard is that of Elbert and Sommers [19] who made a systematic but unsuccessful search for possible objects which might be the source of the 3 x 1020 eV event recorded by Fly's Eye. They argue that the objects of interest should be strong radio galaxies in which there would be extensive magnetic fields along with intense fluxes of high energy particles. They used the catalogue of strong radio sources (radio power 1041 ergs s ') compiled by Burbidge and Crowne [20] Within z < 0.125 (100 Mpc for H0 = 75 km s"1 Mpc" 1 ) © Physica Scripta 2000

185

there are 24 candidate objects, including well-known powerful systems such as For A, M82, M84, M87 and Cen A, but none lies close to the direction of the highest energy event. A more general search, using only data from the ground arrays, has been reported recently by Hillas [5]. His result is shown in Fig. 2. The restriction to ground array material, for which the exposure in right ascension is essentially uniform, allows a radial scale to be chosen so that an area in any part of the plot has the same exposure as an equal area in any other part of the plot. Events with energies above 1020 eV are shown with larger symbols. No correlations are found with galactic or extra-galactic features: the distribution of events is isotropic. The comprehensive plot shown in Fig. 2 has been preceded by other studies with smaller data sets. Using a data set dominated by Haverah Park events, Stanev et al. [21] claimed that cosmic rays above 4 x 1019 eV showed a correlation with the direction of the Super Galactic Plane: the level of significance was 2.5 — 2.8 a. Later studies with AGASA data [22] and with Fly's Eye data [23] did not support this claim. Very recently the AGASA group [24] have released details of 581 events above 10 19 eV recorded by them. Of these 47 are above 4 x 1019 eV and 7 are above 1020 eV. Within this self-consistent data set they find some evidence of clustering on an angular scale of 2.5° there are three doublets and one triplet, the chance occurrence of which is calculated as less than 1%. If such clusters are established in very much larger data sets they will have profound implications for our ideas on cosmic ray origin. It is interesting to note that one of the doublets becomes a triplet when a Haverah Park event of 10 20 eV is added: both triplets lie close to the Super Galactic Plane.

Fig. 2. The arrival directions of events with energy > 4 x 1019 eV The events are from the ground arrays for which the exposure in right ascension is uniform. The radial scale has been chosen so that equal areas in different parts of the plot correspond to equal exposures. The larger circles mark the directions of the events with energy above 1020eV. The super galactic plane is represented by the heavy dashed line running nearly vertically down the diagram. The solid line is the galactic plane: 'A' marks the galactic anticentre direction. The heavily stippled areas indicate regions where there are concentrations of radio galaxies and quasars [5]. Physica Scripta T85

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5. Mass composition of UHECR Interpretation of the data on UHECR is hampered by lack of knowledge of the mass of the incoming particles. There are data from several experiments (AGASA, Fly's Eye and Haverah Park) which might be interpreted as showing a change from a dominantly iron beam near 3 x 10 I7 eV to a dominantly proton beam at 10 19 eV but there is still debate (see [25] for a recent discussion). The data are too limited to allow any reliable statement at energies beyond this. It is unlikely that the majority of the 12 events so far detected above 10 20 eV have photon parents as the showers seem to have normal numbers of muons (the tracers of hadronic primaries) [14] and the cascade profile of the most energetic fluorescence event is inconsistent with a photon primary [26]. It is also unlikely that the majority of events are created by neutrinos as the distribution of zenith angles would be different from what is observed. Indeed events of 1020eV look like events of 10 19 eV but ten times larger. However whether the particles are protons, iron nuclei or photons does not alter the problem posed by the existence of events above 10 20 eV with arrival directions that are distributed isotropically: this is the enigma of ultra high energy cosmic rays.

6. Theoretical interpretations The UHECR enigma is attracting significant theoretical attention and papers describing possible ways round the difficulty have appeared at the rate of 1 per week over the last 15 months. Many of the methods propose some form of electromagnetic process while others invoke ideas which demand new physics. 6.1. Electromagnetic

processes

It is important to recognise that the energy in a source capable of accelerating particles to 1020 eV and beyond must be extremely large as very general arguments demonstrate. A simple analysis [27,28], in which the size of the acceleration region is assumed to be comparable to the Larmor radius of the particle and the magnetic field is sufficiently weak to constrain synchrotron losses, shows that the energy in the magnetic field of the source grows as y5 where y is the Lorentz factor of the particle. For 10 20 eV the energy in the magnetic field must be » 10 57 ergs with B <0.1 Gauss. Such sources are likely to be strong radio emitters with radio power S> ,1041 ergs s"1, unless hadrons are being accelerated and electrons are not. It is worth emphasising that the present upper limit of 3 x 1020 eV is very probably set by the exposure rather than by any natural limit: the magnetic energy in the source of such a particle must be » 2 x 1059 ergs. The above analysis does not specify what acceleration process might be involved. Currently it is widely believed that cosmic rays with energies up to about 1015 eV are energised by diffusive shock acceleration with supernovae explosions identified as the most likely sites. At higher energies it is argued by some (e.g. [29]) that the same process continues but with the particles being accelerated by interaction with multiple supernova remnants as they move through the interstellar medium. This extended acceleration is supposed Physica Scripta T85

to take particles to energies close to 10 18 eV with the higher charge nuclei reaching the higher energies and becoming more dominant in the primary beam. This idea is consistent with the limited evidence on mass composition and with the steepening of the energy spectrum near 1018 eV. However there is, as yet, no direct evidence of acceleration of protons by supernova remnants at any energy. The diffusive shock acceleration process has been extensively studied since its conception in the late 1970s. In an accessible review Drury [30] has shown that the maximum energy attainable is given by E — kZeBRfic, where B is the magnetic field in the region of the shock, R is the size of the shock region and A: is a constant less than 1. The same result has been obtained by several authors including Hillas [31] who has used a simple but elegant plot of B vs. R to show that very few objects satisfy the conditions needed to achieve the maximum energy. The limit has been reconfirmed by Cesar sky [32] and by Blandford [33]. However some authors (e.g. [34]) claim that the diffusive shock acceleration process can be modified to give much higher energies than suggested by Drury's result and that radio galaxy lobes in particular are the acceleration sites. It is difficult to see how an energy of 3 x 1020 eV can be accounted for if the size of the shock region is 10 kpc and the magnetic field is 10 uG as even the optimum estimate of the energy is lower by a factor of 3 than this observational upper limit. Additionally Biermann and colleagues have speculated [35] that an energy spectrum and arrival direction distribution consistent with the data can be obtained if the magnetic field in the local Supercluster is 10~7 G. Thus two pieces of astrophysical understanding would need to be revised to explain the events: the parameters in the lobes of radio galaxies and the strength of the magnetic field in the local Supercluster. It should be noted that Elbert and Sommers [19] also proposed an unexpectedly strong extra-galactic magnetic field so that objects such as Cen A might be considered as candidate sources for the most energetic Fly's Eye event. There have been proposals on both theoretical and observational grounds that the highest energy cosmic rays are created when pairs of galaxies interact. Cesarsky [32] and Cesarsky and Ptushkin [36] were the first to examine this idea. They propose that the converging flows contain shock fronts which can accelerate the particles and claim that energies of ~ 10 20 eV can be reached if the primary nuclei are iron. They identify the Antennae system (NGC 4038/39) as a possible UHECR source. Observationally Al-Dargazelli et al. [37] have noted the association of a group of cosmic rays with energies above 10 19 eV with the galaxy pair VV338 while Takeda et al. [24] have tentatively identified an interacting galaxy Mrk 40 at z = 0.02 with a triplet of high energy events, the lowest energy of which is 5.35 x 1019 eV. It somewhat discouraging to find two colliding systems so identified: it lacks conviction. Furthermore Jones [38] has discussed the possibility of such colliding systems being sources and concluded that 10 17 eV is the maximum energy which is reasonably attainable. Amongst the most violent shocks in the Universe are the accretion shocks formed during the gravitational collapses which may lead to the formation of clusters of galaxies. These have been studied as a UHECR source by Kang, © Physica Scripta 2000

Ultra High Energy Cosmic Rays - an Enigma Ryu and Jones [39] but the limiting energy for protons is claimed to be only 6 x 1019 eV even if the field is as strong as 1 uG. They also claim that cosmic rays of around 1019eV could be produced in the Virgo cluster in such shocks. A further source of UHECR which has been considered is GRBs [40]. Here shocks with high Lorentz factors may give rise to conditions conducive to acceleration to the energies observed and it is argued that the energetics are such that attenuation by the microwave background radiation is not a problem. However the observational constraints on what is possible in GRBs may tighten in a few years and limit the energies reached in a way that we cannot envisage. In summary it seems difficult to find places where electromagnetic acceleration processes can raise particles to the energies observed. It should be stressed, however, that if the primaries were heavy nuclei there is some alleviation of the problem but much greater attention then has to be given to the photon environment in the acceleration region and to the possibility of fragmentation while they are being transported to earth: heavy nuclei of this energy are very fragile against photodisintegration. 6.2. Non-electromagnetic processes A number of proposals have been made which dispense with the need for electromagnetic acceleration. In general attention has been focused on the very highest energy events (> 10 20 eV) but in my view any mechanism that explains these events must also account for the events above about 1018 eV, where the galactic component probably disappears: the spectrum above this energy is too smooth to admit of two or more radically different components. Also the proposals must produce particles at the top of the atmosphere that can generate showers of the type we see: the characteristics of events about 10 20 eV are strikingly similar to those produced by primaries one or two decades lower in energy. One approach has been to invent mechanisms to evade the energy losses in the 2.7 K radiation field. For example Kolb, Farrar and collaborators [41] have speculated that a stable, massive, supersymmetric hadron, the S° (a combination of uds quarks and a gluino), may be responsible for producing the largest air showers. Such a particle, and similar massive particles dubbed 'uhecrons', might have a lower cross-section for the production of resonance particles at the energies observed. One positive feature of this proposal is that the particles would produce air showers rather similar in type to what is observed. As the S° is neutral the events would be expected to come from identifiable objects and Farrar and Biermann [42] have claimed an association with radio-loud QSOs for 5 of the most energetic events. Their statistical analysis has recently been challenged by Hoffman [43] and the idea is now capable of an independent test with the precise directions of the new AGASA events mentioned above [24]. Such an identification would stretch even further our understanding of particle acceleration processes as uhecrons of 3 x 1020eV would presumably be produced in conventional hadronic interactions that would require parent particles of even greater energy. Accelerator limits to the production cross-section of the S° imply an extremely severe energetics problem at the source, bearing in mind the y5 factor in the estimate of the magnetic field energy. © Physica Scripta 2000

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Thus I find this hypothesis rather improbable on the grounds of energetics alone. Similarly vortons [44,45] represent a solution which is limited to the very highest energies. A similar idea, which I believe has the same generic difficulty with energetics, is the suggestion that ultra high energy neutrinos (10 22 eV) created in very distant sources produce cascades of photons, protons and electrons when they interact with background neutrinos of mass a few eV [46]. In addition to the production problem the predicted proton spectrum does not fit the experimental data down to 10 18 eV so that a variety of sources would be required to explain the observations above this energy. Kephart and Weiler [47] have revived an old idea [48] and proposed that monopoles of mass < 10 10 GeV might be the source of the high energy events. The monopoles would be accelerated in galactic magnetic fields and are not so numerous as to violate the Parker bound. However it is hard to see why an anisotropy associated with the galactic plane would not be expected and such monopoles have too low a Lorentz factor to produce the kind of showers observed. The later difficulty has recently been addressed by Huguet and Peter [49] who suggest that the monopoles have an internal structure which allows them to generate air showers in the conventional manner. Such entities would also be produced in sources and would not be in conflict with the observation of clusters of events from the same point in the sky. There is also speculation that UHECR arise from the decay of superheavy relic particles. In this picture cold dark matter is supposed to contain a small admixture of long lived superheavy particles with a mass > 1012 GeV and a lifetime greater than the age of the Universe [50]. It is argued that such particles can be produced during reheating following inflation or through the decay of hybrid topological defects such as monopoles connected by strings. It is hard to judge how speculative these ideas are but the decay cascade from a particular candidate [51] has been studied in some detail [52]. It is possible to produce the observed spectrum of cosmic rays, at least in the region above about 4 x 1019 eV, from the decay cascade and in addition the isotropy of the most energetic particles could be understood - within the limited statistics currently available - in terms of a galactic halo distribution of superheavy relic particles ([53], see also [5]). A further feature of the decay cascade is that an accompanying flux of photons and neutrinos is predicted which may be detectable with a large enough installation. The anisotropy question has been examined in some detail by Benson et al. [54] Berezinsky and Mikhailov [55] and Medina-Tanco and Watson [56]. The latter authors have made specific predictions for the anisotropy which would be seen by a Southern Hemisphere air shower array. The observation of the predicted anisotropy, plus the identification of appropriate numbers of neutrinos and photons, would be suggestive of a super heavy relic origin. However the existence of multiplets would not be expected in such a picture. Other ideas have been proposed including the suggestion [57] that there may be a departure from strict Lorentz invariance at the energies in question. Any of the classes of explanation outlined above would require a major reappraisal of our understanding of astrophysics, particle physics or cosmology. However without additional data we cannot hope to solve the puzzle Physica Scripta T85

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set by the existence of UHECR and it is to provide these additional data that new projects have been planned.

7. New projects 7.1. The Pierre Auger Observatory The Pierre Auger Observatory has been conceived to measure the properties of the highest energy cosmic rays with unprecedented statistical precision. It will eventually consist of two instruments, constructed in the Northern and Southern Hemispheres, each covering an area of 3000 km 2 . The project will commence in late 1999 with the construction of an engineering array at Pampa Amarilla in Mendoza Province, Argentina. Subsequently work will start on the northern instrument in Millard County, Utah, USA. The Observatory has been designed by a consortium drawn from 18 nations and makes extensive use of the experience with previous ground arrays and with the Fly's Eye fluorescence detector. As well as providing an unsurpassed number of events, the Auger Observatory is designed to measure the energy and arrival direction of each event very accurately and to separate neutrinos and photons from hadronic primaries. Separation of hadronic primaries will be harder but there is a reasonable expectation of obtaining the fraction of protons and the fraction of iron nuclei as a function of energy. The design for the Observatory, which is the product of a six month planning operation supported principally by the Fermi National Accelerator Laboratory and UNESCO, calls for a hybrid detector system with 1600 particle detector elements and three or four fluorescence detectors at each of the sites. The principles of operation of a surface array and of a fluorescence detector separately were outlined in Section 2. The particle detectors will be 10 m 2 x 1.2 m deep water-Cerenkov tanks arranged on a 1.5 km hexagonal grid. These detectors are similar to those used at the Haverah Park array and have been selected because the water acts as a very effective absorber of the multitude of low energy electrons and photons found at distances of about 1 km from the shower axis. Both Monte Carlo calculations and experiments have shown that the mean energy of photons and electrons is about lOMeV. By contrast the muons, which are much less numerous, carry, on average, about 1 GeV of energy. At the Southern site fluorescence detectors will be set up at four locations, one near the centre of the particle array with the others on small promontories at the array edge: the site is close to the town of Malargue. During clear moonless nights signals will be recorded in both the fluorescence detectors and the particle detectors while for roughly 90% of the time only particle detector data will be available. It is much too costly to connect detectors by cables or optical fibres over an area of 3000 km 2 . Instead each detector will operate in a "stand-alone" mode with trigger data being sent to the central station using a Wireless LAN radio link operating at 915 MHz. The power for the surface detectors will be obtained from solar cells while data at all detectors will be time-stamped using GPS receivers. The GPS timestamp is sufficiently precise to allow accurate reconstruction of the direction of each incoming event. A major advantage of the use of detectors of two types - the hybrid array is that independent measurements of some parameters will Physica Scripta T85

be made for about 10% of the showers. For example the energy calibration of a surface array depends on the details of the interaction model used to convert the ground level measurements to the energy input at the top of the atmosphere. By contrast the fluorescence detector has direct energy calibration although correction must be made for the component of the detected light which is scattered Cerenkov light and for the transparency of the atmosphere, which needs to be monitored on a very regular basis. This later feature means that the aperture of the fluorescence detector grows as the energy increases in contrast to that of the surface detector which (see Fig. 3) is constant above an energy dictated only by the adopted spacing of the detectors. Thus for 1.5 km spacing the surface detector aperture remains constant at energies above 10 19 eV. The fluorescence detector can be used to measure the depth of shower maximum directly from the shape of the cascade curve. This is an important parameter but its interpretation does depend on the shower model used to describe the data. The ground array can measure the risetime of the signal at each detector, the lateral distribution of the signal amplitudes and the curvature of the shower front, each of which is related to the depth of shower maximum. The combination of the ground array and the fluorescence detector will also allow the core position of the shower to be found to within about 50 m, more precisely than with either detector alone. It seems probable that this combination of shower parameters, very accurately measured in 10% of events, will help decide which model is appropriate to describe the data and hence allow the primary mass to be extracted. In addition the water-tanks are equipped with Flash ADCs which enable the temporal form of the signal to be measured. It has been shown, both with Monte Carlo calculations and with a prototype 10 m 2 tank operated at the AGASA array, that the time-spread is sufficient to allow the large signals from through-going muons to be detected against the more numerous but smaller signals produced

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Ultra High Energy Cosmic Rays - an Enigma by the lower energy electromagnetic components. This provides a powerful method of identifying the fraction of muons in the shower, an important number which will be used to aid the derivation of the mass composition of the primary beam. This augurs particularly well for the identification of photon- initiated primaries. Neutrino events are expected to be detected most readily at angles greater than 75° from the vertical. The showers will be similar to those produced by hadrons in the near vertical direction and will contrast strongly with the muon dominated events which will characterise the remnants of showers produced by hadrons at such large angles. Detailed simulations of the performance of the ground array for energy and direction measurement have been carried out. The results are shown in Fig. 3. At 4 x 1019eV the energy resolution, with the ground array of particle detectors alone, will be ~ 10% and the angular resolution will be ~ 1.5° : on average about 11 detectors will be struck. The energy resolution and angular accuracy improves as the energy increases. The first stage of the Auger project will produce unique data as the Southern sky has not been studied with cosmic rays of these energies. There are some reasons to suppose that the spectrum and arrival direction distributions may not be the same as measured from the Northern hemisphere. These include the different distribution of luminous matter, the diffuse infrared photon flux (which has an influence on propagation of heavy nuclei) and the distribution of matter in the galactic halo. The Southern Hemisphere Auger detector thus offers the opportunity of unique science which, as explained above, could have a major impact on cosmology and/or particle physics. The engineering array is expected to be completed in late 2000. Although construction of the 3000 km 2 detector will take a further 4 years, exciting new data will be available as soon as 100 km 2 is operational with one fluorescence detector. 7.2. Other detectors: AGASA, Hi-Res and Airwatch/OWL At present the only experiment taking data above 10 19 eV is the AGASA instrument in Japan which has an aperture of 170 km 2 sr. From mid-1999 a successor to the Fly's Eye instrument, known as Hi-Res [58] started to take data in the Northern Hemisphere with a time averaged aperture of 340 km 2 sr at 1019 eV and 1000 km 2 sr at 10 20 eV. This is a stereo system which is expected to be able to measure the depth of shower maximum to within 30gem - 2 on an event-by-event basis. This precision is smaller than the expected fluctuations in depth of maximum for proton or Fe initiated showers. By contrast a single Auger observatory will have an aperture of 7000 km 2 sr above 1019 eV and will thus be 20 times as powerful as Hi-Res at the lower energy and 7 times more effective at the higher energy. The rate of events with fluorescence and ground array information from Auger will be comparable to the rate from Hi-Res. Achieving an exposure greater than that promised by the Auger Observatory is a formidable challenge. A promising line is the development of an idea of Linsley [59]. The concept is to observe fluorescence light produced by showers from space with satellite borne equipment. It is proposed to monitor ~ 10 5 km 2 sr (after allowing for the estimated 8% on-time). Preliminary design studies have been carried out in Italy [61] and the USA [62] and the groups interested © Physica Scripta 2000

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have combined to plan an project known as Airwatch/OWL. Two satellites might be used to observe the fluorescence emission. The project requires considerable technological development but may be the only way to push to energies beyond whatever limit is found with the Auger instruments.

Acknowledgements I am gratefUl to the organising committee of the Nobel Symposium for inviting me to the meeting and for their generous hospitality and support.

References 1. Watson, A. A., Nucl. Phys. B (Proc. Suppl.) 22B, 116 (1990). 2. Watson, A. A., Nucl. Phys. B (Proc. Suppl.) 28B, 3 (1992). 3. Sokolsky, P., Sommers, P. and Dawson, B. R., Phys. Rep. 217, 225 (1995). 4. Yoshida, S. and Dai, H„ J. Phys. G 24, 905 (1998). 5. Hillas, A. M., Nature 395, 15 (1998). 6. Watson, A. A., Proc. Int. Symp. on Extremely High Energy Cosmic Rays: Astrophysics and Future Observatories (editor: M. Nagano: ICRR, Tokyo), 1996, p. 362. 7. Griesen, K., Phys. Rev. Lett. 16, 748 (1966). 8. Zatsepin, G. T. and Kuzmin, V. A., Sov. Phys. JETP Lett. 4, 78 (1966). 9. Cunningham, G. et al, Astrophys. J. 236, L71 (1980). 10. Linsley, J., Phys. Rev. Lett. 10, 146 (1963). 11. Lawrence, M. A., Reid, R. J. O. and Watson, A. A., J. Phys. G 17, 733 (1991). 12. Baltrusaitis, R. M. et al, Phys. Rev. Lett. 54, 1875 (1985). 13. Bird, D. J. et al, Astrophys. J. 424 491 (1994). 14. Hayashida, N. et al, Phys. Rev. Lett. 73, 3491 (1994). 15. Takeda, M. et al, Phys. Rev. Lett. 81, 1163 (1998). 16. Efimov, N. N. et al, Proc. ICCR International Symposium (Kofu): Astrophysical Aspects of the Most Energetic Cosmic rays, (World Scientific, 1990) (editors: M. Nagano and F. Takahara), p. 20. 17. Bower, A. J. et al, J. Phys. G 9, L53 (1982). 18. Kronberg, P., Rep. Prog. Phys. 57, 325 (1994). 19. Elbert, J. W. and Sommers, P., Astrophys. J. 441, 151 (1995). 20. Burbidge, G. and Crowne, A. H., Astrophys. J. Suppl. 40, 583 (1979). 21. Stanev, T. et al, Phys. Rev. Lett. 75, 3056 (1995). 22. Hayashida, N. et al, Phys. Rev. Lett. 77, 1000 (1996). 23. Bird, D. J. et al, Astrophys. J. 511, 739 (1999). 24. Takeda, M. et al, Astrophys. J. 522, 225 (1999). 25. Dawson, B. R., Meyhandan, R. and Simpson, K. M., Astroparticle Phys. 9, 331 (1998). 26. Halzen, F. et al, Astroparticle Phys. 3, 151 (1995). 27. Griesen, K., Proc. 19th ICRC (London) 2, 609 (1965). 28. Cavallo, G., Astron. Astrophys. 65, 415 (1978). 29. Ip, W. H. and Axford, W. I., AIP Conference Proc. 264 (Particle Acceleration in Cosmic Plasmas, 1991), p. 400. 30. Drury, L. O ' C , Contemporary Phys. 35, 232 (1994). 31. Hillas, A. M., Ann. Rev. Astron. Astrophys. 22, 425 (1984). 32. Cesarsky, C. J., Nucl. Phys. 288, 51 (1992). 33. Blandford, R. D., These proceedings (1999). 34. Biermann, P., J. Phys. G 23, 1 (1997). 35. Biermann, P. et al, astro-ph/9806283 (1998). 36. Cesarsky, C. J. and Ptushkin, V., Proc. 23rd ICRC (Calgary) 2,341 (1993). 37. Al-Dargazelli, S. S. et al, Proc. 25th ICRC (Durban) 4, 465 (1997). 38. Jones, F. C , AIP Conference Proc. 433 (Workshop on Observing Giant Air Showers from > 1020 eV particles from Space), 1998, p. 37. 39. Kang, H., Ryu, D. and Jones, T. W., Astrophys. J. 456, 422 (1996). 40. Waxman, E„ Phys. Rev. Lett. 75, 386 (1995). 41. Kolb, E. W., Farrar, G. H. et al, 1997, 1998, astro-ph/9707036 and hep-ph/9805288. 42. Farrar, G. H. and Biermann, P., Phys. Rev. Lett. 81, 3579 (1998). 43. Hoffman, C , astro-ph/9901026 (1999). 44. Bonazzola, S. and Peters, P., Astroparticle Phys. 7, 161 (1997). 45. Masperi, L. and Silva, G., Astroparticle Phys. 8, 173 (1998). Physica Scripta T85

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Yoshida, S., Sigl, G. and Lee, S., hep-ph/9808324 (1998). Kephart, T. W. and Weiler, T. J., Astroparticle Phys. 4, 271 (1996). Porter, N . A., Nuovo Cimento 16, 958 (1960). Huguet, E. and Peter, P., submitted to Astroparticle Phys. (1999). Berezinsky, V., Kachelriess, M. and Vilenkin, A., Phys. Rev. Lett. 79, 4302 (1997). Benkali, K., Ellis, J. and Nanopoulos, D. V., hep-ph/9803333, Phys. Rev. D (in press) (1998). Birkel, M. and Sarkar, S„ Astroparticle Phys. 9, 297 (1998). Dubovsky, S. L. and Tinyakov, P. G., J.E.T.P. Lett. 68, 107 (1998). Benson, A., Smialkowski, A. and Wolfendale, A. W., Astroparticle Phys. 10, 313 (1998). Berezinsky, V. and Mikhailov, A. A., astro-ph/9810277, Phys. Lett. (submitted) (1998).

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56 Medina-Tanco, G. A. and Watson, A. A., Astroparticle Phys. 12, 25 (1999). 57. Coleman, S. and Glashow, S. L., he-ph/9808446 (1998). 58. Sokolsky, P., AIP Conference Proceedings 433 (Workshop on Observing Giant Air Showers from > 1020 eV particles from Space), 1998, p. 65. 59. Linsley, J., 1979, USA Astronomy Survey Committee (Field Committee) Documents and Proc. 19th Int. Cos. Ray. Conf. (La Jolla) 3, 438 (1985). 60. Axford, W. I., Astrophys. J. Suppl. 90, 937 (1994). 61. Demarzo, C. N., AIP Conference Proc. 433 (Workshop on Observing Giant Air Showers from > 1020 eV particles from Space), 1998, p. 87. 62. Streitmatter, R. E., AIP Conference Proc. 433 (Workshop on Observing Giant Air Showers from > 1020 eV particles from Space), 1998; p. 95.

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Physica Scripta.Vol. T85, 191-194, 2000

Acceleration of Ultra High Energy Cosmic Rays R. D. Blandford 130-33 Caltech Pasadena CA 91125 USA Received March 3, 1999; accepted August 2, 1999

extending from the proton rest mass, ~ 1 GeV to the "knee" at ~ 100 TeV -10 PeV. (10 GeV cosmic rays are about 10 m Some general features of cosmic ray acceleration are summarized along with apart and have an energy density comparable with that some inferences that can be drawn concerning the origin of the UHE of the microwave background. The spectrum steepens above component. The UHE luminosity density is found to be similar to that derived for GeV cosmic rays and its slope suggests a distinct origin. Reports of the knee: C / ( £ ) ~ 4 x 10" 18 (£/10 P e V ) " u J m" 3 . It then clustering on small angular scale, if confirmed, would rule out most proposed dips and flattens around the "ankle" (~ 1 — 10 EeV). source models. More generally, it is argued that the highest energy particles UHE cosmic rays - the toenail clippings of the universe 20 can only be accelerated in sites that can sustain an EMF £ :> 3 x 10 V and an associated power Lmiii :> £2/Z ~ 1039 W, where Z is the characteristic, - are observed up to 300 EeV and, with a little imagination, U(E) ~ 1.5 x 10"21CE/10 EeV)" 0 5 J m" 3 , comparable with electrical impedance, typically <100fi Shock acceleration, unipolar induction and magnetic flares are the three most potent, observed, accelerthe estimated, integrated background from y-ray bursts. ation mechanisms and radio jet termination shocks, •y-ray blast waves, dor(Despite the large uncertainty, and the fact that the number mant black holes in galactic nuclei and magnetars are the least density has fallen by ~ 10 orders of magnitude, we do implausible, "conventional" manifestations of these mechanisms that have measure the EeV spectrum better than the MeV spectrum, been invoked to explain the UHE cosmic rays. Each of these models presents of which we are, quite decently, ignorant.) problems and deciding between these and "exotic" origins for UHE cosmic rays, including those involving new particles or defects will require improved The ~ 1 GeV-100 TeV cosmic rays are of Galactic origin. statistical information on the energies, arrival times and directions, as should The ratio of Li, Be, B secondaries to C, N, O primaries be provided by the AUGER project. measures their range to be 1(E) ~ 100(£/10 GeV) - 0 6 kg m~2 (eg [6]). The cosmic ray luminosity of the Galaxy is then estimated as ~ MdU(E)c/l(E) ~ 2 x 10 33 (£yi GeV)" 0 1 W 1. Introduction where Mi is the gas mass of the disk. Scaling from the local I have been asked to summarize "conventional" schemes for galaxy luminosity density (per In is and assuming the acceleration of UHE cosmic rays, though any physical h = 0.6), we derive an average, cosmological, luminosity process capable of endowing a subatomic particle with density (per l n £ ) , < C > (E) ~ 4 x 10" 37 (^/10 GeV) -0 ' 1 W the kinetic energy of a well-hit baseball/cricketball can m~ 3 , for 10 GeV < E < 100 TeV. (For comparison, the hardly be considered conventional. This means that I shall stellar luminosity density is ~ 10~33 W m~ 3 .) leave others to review mechanisms that attribute the origin The UHE particles are almost surely extragalactic. As of these particles to topological defects, strings, monopole with y-ray bursts, there is no good evidence for disk, halo, decay, supersymmetric hadrons, cosmic necklaces, cryptons cluster or supercluster anisotropy (despite some tantalising and so on. Indeed, I suspect that the "hidden agenda" is hints in the past) [7]. Furthermore, magnetic confinement for me to fail at my appointed task, and, like my colleagues by the Galaxy is impossible - the Larmor radius r^(E) of on the MACHO experiment, to make the world safe for a 300 EeV cosmic ray in a uG field is ~ 300 kpc. If we assume elementary particle theorists. I shall not disappoint. that UHE cosmic rays are protons, (and assuming that they Many of the issues that I will cover have been recognised are not, only makes matters worse), then they have a short for some time and have been well-discussed in several excel- lifetime to photo-pion production on the microwave backlent reviews including Hillas [1] and Cronin [2] and the many ground [8,9]. The characteristic lifetime of a ~ 60— 300 EeV relevant contributions to the recent conference on this sub- cosmic ray is, very roughly, T(E) ~ 0.1 (is/300 EeV) -2 Gyr. ject [3], including, especially, the lively summary by the late This implies that the luminosity density increases with David Schramm. The conference proceedings edited by energy < £ > ( £ ) - U(E)/T(E)~ 10" 37 (£/300 EeV) 15 Chupp & Benz [4] is also relevant. Wm~ 3 . At the highest measured energy, the estimated cosmological luminosity density is not significantly different from that of 10 GeV cosmic rays. The change in slope in 2. The cosmic ray spectrum the source spectrum, above ~ 60 EeV, is a strong indication In order to give this topic some context, consider the com- that these UHE cosmic rays comprise a quite distinct plete cosmic ray spectrum (eg [5]). This extends over nearly component from their lower energy counterparts. In order to investigate this further, it is necessary to take twelve decades of energy from the proton rest mass, ~ 1 GeV, where their energy density is that of the microwave account of the fluctuations in energy loss. Taking the 17 background, to at least 300 EeV (= 50 J = 3 x 10" 8 m P i). We events reported by the AGASA collaboration above can consider the cosmic ray spectral energy density inferred 60 EeV, it is possible to derive a maximum likelihood estiat the solar system, U(E) = (4n/c)(dI/d\nE) ~ 5xl0~ 1 4 mate of the unnormalized energy density, uncorrected for -07 3 (is/10 G e V ) J m " (correcting for solar modulation) biases in detection efficiency. I find that if U(E) oc E"; Abstract

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E > .Emin = 60 EeV, then a = —1.2 ± 0.5.1 then calculate the probability that a particle of energy EQ has energy > E after time t, P(E, t\ Eo), following Aharonian and Cronin [10] and Bahcall and Waxman [11]. If we assume a power law for the luminosity density C(E) oc E&; E > Em{n = 60 EeV, then the logarithm of the likelihood for obtaining the observed events is

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dP 3\nE

(1)

Maximizing this function with respect to variation of /?, gives the estimate /? = 0.3 ± 0.2, and may even be consistent with a single, "top down" source with energy well above 300 EeV. A more sophisticated computation that takes into account the detection probabilities of the different events should be performed. There have been reports that UHE cosmic rays are significantly clustered on the sky. Specifically, in a sample of 47 events observed with AGASA, there are three pairs and one triple above 40 EeV with separations <2.5°, comparable with the positional errors [7]. (There are two more coincidences with events drawn from other samples.) There is no clear pattern for the associated particles to be ordered in energy and, in particular, one double has a 106 EeV particle arriving over 3 yr. after a 44 EeV particle. If these associations are real and typical, then there are three important implications. Firstly, as particles are likely to be deflected by intergalactic magnetic field through an angle S6 ~ (D£B)y2/rL(E) then they will be delayed by ~ D2lB/rL(E)2c oc E~2, where iB is the field correlation length and D ~ c7\150EeV) ~ 30 Mpc is the supposed source distance (cf [12]). Even Aesop would be challenged to explain how ~ 40 EeV cosmic rays precede ~ 100 EeV cosmic rays if they started at the same time and we would conclude that the source persists for several years, at least. This would rule out all particle/defect and y-ray burst models. Secondly, the small deflection angles at low energy limit the intergalactic field strength to B<20 (lB/\Mpc)~l/2 fT, far smaller than generally supposed, though probably not excludable by direct observation. Thirdly, the presence of three ~ 40 EeV cosmic rays associated with high energy cosmic rays of much shorter range, implies that the background of low energy cosmic rays not associated with high energy events must be larger than the incidence of clustered events by roughly the ratio of their typical lifetimes ~ 30, which more than accounts for the remainder of the low energy sample. This, in turn, implies that the high energy cosmic rays must come from a very few sources which are, consequently, quite energetic: E ~ lO^r/Syr) J, where T is their lifetime. (If the low energy cosmic rays are scattered through ~ 2.5°, then x > 105 yr and E ~ 3 x 1048 J.) However, this clustering hypothesis, which is necessarily a posteriori when expressed in detail, is only supported with modest confidence. (A simple, Monte Carlo simulation distributing 47 points at random on half the sky and looking for similar patterns is quite instructive.) Particle/defect/ burst explanations of UHE cosmic rays need not yet be rejected on these grounds. Physica Scripta T85

3. Cosmic ray acceleration The standard model of bulk cosmic ray production is first order Fermi acceleration at strong, super-Alfvenic, shocks associated with supernova remnants and, possibly, winds from hot stars {eg [13]). A typical relativistic proton will cross a shock, travelling with speed u, 0(c/u) times, gaining energy AE/E = 0(u/c) each traversal through scattering by hydromagnetic waves moving slowly with respect to the converging fluid flows on either side of the front. The net mean relative energy gain is 0(1), but the process is statistical and a kinetic calculation shows that the transmitted spectrum will be a power law in momentum, f(p) oc p-3r/(r-i)^ w h e r e rt ( = 4 for a strong shock), is the compression ratio. This mechanism can account, broadly, for the power (eg [14]), the slope (eg [6]) and the composition (eg [15]) of GeV cosmic rays. Shock acceleration is also, arguably, observed directly in SN1006 [16], as well as in the solar system (eg [17]). The maximum energy to which a particle can be accelerated at a shock front is dictated by the scattering mean free path, 1(E) ~ (B/SB)2LrL(E), where SB is the amplitude of resonant hydromagnetic waves with wavelength matched to the particle Larmor radius. The diffusion scale-length of cosmic rays ahead of the shock is ~ Ic/u and, assuming that this is limited by the size of the shock ~ R we arrive at the unsurprising result that the maximum energy achievable in shock acceleration, assuming SB < B and the presence of a large scale magnetic field, is Emax = e£ ~ euBR ~ ed<$>/&t, the product of the charge and the motional potential difference across the whole shock. Equivalently, we conclude that in order to accelerate a proton by this mechanism to an energy £e, the rate of dissipation of energy exceeds £„,;„ ~ £2/Z, where Z = \iQu — \XQE/B is the effective impedance of the accelerator in SI units. Imposing this condition for a supernova remnant in the interstellar medium leads to an estimate .Emaxsnr ~ 30 TeV (eg [6]), close to the knee. An additional source is needed between the knee and the ankle, where the source is generally supposed to be metagalactic. Larger shocks, especially those at Galactic wind termination shocks (eg [18]) and associated with gas flows around groups and clusters of galaxies have been invoked (eg [19]). These shocks are likely to be relatively weak and therefore to transmit steeper spectra, as observed. The major uncertainty is the strength of the magnetic field. If B ~ 30 pT at a galactic shock and ~ 10 pT at a cluster shock, then £ m a x ~ 10, 100 PeV respectively. Neither site is likely to accelerate the highest energy particles. An alternative accelerator is the unipolar inductor (eg [20]). The archetypical example is a pulsar - a spinning, magnetised, neutron star. The surface field will be quite complex but a certain quantity of magnetic flux $ can be regarded as "open" and tracable to large distances from the star, (well beyond the light cylinder). As the star is an excellent conductor, an EMF will be electromagnetically induced across these open field lines £ ~ £2$, where $ is the total, open magnetic flux. This EMF will cause currents to flow along the field and as the inertia of the plasma is likely to be insignificant the only appreciable impedance in the circuit is related to the electromagnetic impedance of free space Z ~ 0.3|ioc ~ 100 Q. The maximum energy © Physica Scripta 2000

Acceleration of Ultra High Energy Cosmic Rays to which a particle can be accelerated is Emax ~ e£ and the total rate at which energy is extracted from the spin of the pulsar is Lmin ~ £2/Z. Taking the Crab pulsar as an example, Emax ~ 30 PeV for protons and Z^n ~ 1031 W. As the stellar surface may well comprise iron, even the Crab pulsar has the capacity to accelerate ~EeV cosmic rays. However, it is not obvious that all of this potential difference will actually be made available for particle acceleration. In particular, this is unlikely to happen in the pulsar magnetosphere as a large electric field parallel to the magnetic field will be shorted out by electron-positron pairs, which are very easy to produce, and radiative drag is likely to be severe. A more reasonable site is the electromagnetic pulsar wind and the surrounding nebula where particles can gain energy as they undergo gradient drift between the pole and the equator [21]. Pulsars may well contribute to the spectrum of intermediate energy cosmic rays. A third, protoypical accelerator is a flare, for example one occuring on the solar surface or the Earth's magnetotail. Here magnetic instability leads to a catastrophic rearrangement, which must be accompanied by a large inductive EMF. Unless the instabilities are explosive, the effective impedance is again ~ \i0u, where u is a characteristic speed. Non-relativistic flares generally convert most of the dissipated magnetic energy into heat and are notoriously inefficient in accelerating high energy particles. Other acceleration mechanisms have been proposed and may contribute to the acceleration of the bulk of Galactic cosmic rays and relativistic electrons in non-thermal sources. These include a variety of second order processes and steady, magnetic reconnection. Many of them can be observed to operate within the solar system. However, they are thought to be too slow to be relevant to the acceleration of the highest energy cosmic rays.

4. Zevatrons Having argued that the three most potent, observed acclerators are shocks, unipolar inductors and flares, let us see how they can be modified to account for ~ ZeV cosmic rays. Firstly, note that, as u ~ c, mildly relativistic shocks minimise the power that has to be invoked to attain high energy. Specifically, we need a power > 1039 W to account for 300 EeV cosmic rays and this exceeds the bolometric luminosity of a powerful quasar. One of the few sites where such a large potential difference can be achieved is the termination shock of a powerful radio jet like that associated with Cygnus A (eg [22]). Stretching the numbers a little, we combine a field strength ~ 10 nT, with a speed ~ c and a transverse scale ~ 3 kpc which gives Em^ ~ 300 EeV. The problem with this model is that observed UHE cosmic rays are not positionally identified with the few known radio sources within D{E) ~ 30 Mpc that might be powerful enough to account for them (c/[23]). A more elaborate shock accelerator is the y-ray burst blast wave (eg [24]). Here, the shocks (assumed to be spherical) are ultrarelativistic with Lorentz factor f. The maximum energy, measured in the frame of the explosion, to which a proton can be accelerated in a dynamical timescale from an ultrarelativistic shock of radius R is Emm ~ eB'Rc, where Bl is the comoving field strength. The explosion power, © Physica Scripta 2000

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adopting the most elementary of assumptions, is then Lmin ~ 4nr2(Emax/e)2/\iQC. Observed bursts have typical explosion powers estimated to be L exp ~ 1045 W, which can be consistent with 300 EeV proton acceleration as long as r < 300, which is just compatible with existing models. A serious physical constraint is the avoidance of radiative loss in this environment. An observational concern with this model is the improbability of having enough active bursts close to supply the highest energy particles roughly isotropically (cf [25]). The most relevant variant on unipolar induction is magnetic energy extraction from spinning, black holes, where the magnetic field is supported by external current, and the horizon is an imperfect conductor with resistance ~ 100Q (eg [26,27]). This impedance is matched to the electromagnetic load so that roughly half of the available spin energy ends up in the irreducible mass of the hole, the remainder being made available for particle acceleration. The total electromagnetic power needed to account for ~ 300 EeV acceleration is, once more, ~ 1039 W. A rapidly spinning, ~ 109 M© hole endowed with a field strength > 3 T or a 105 M G hole threaded by a > 3 x 104 T field suffices to accelerate 300 EeV particles. The major concern with this model is that the radiation background must be extremely low in order that catastrophic loss due to pion and pair production be avoided. Specifically, it is necessary that the microwave luminosity in an acceleration zone, of size R, be < 1034(i?/1014m) W, far smaller than the electromagnetic power. The best generalization of flare acceleration involves "magnetars" which are young, spinning neutron stars endowed with a ~ 10 — 100GT surface magnetic field as first postulated by Thompson and Duncan [28]. Now, the observation of 5 — 7 s period pulsations from three "soft gamma repeaters" effectively confirms their identification as old magnetars that have been decelerated by electromagnetic torque and which are now powered by magnetic energy which is released in a series of giant flares [29]. The inductive EMFs associated with an electromagnetic flare from a magnetar can be as high as AV ~ 3 x 1019 V, making them candidate UHE accelerators because the surface composition is likely to be Fe. However, the available reservoir of magnetic energy is only ~ 1040 J and the magnetar birthrate is no more than ~ 10 - 3 yr _ 1 in the Galaxy. This rules them out as an extragalactic source. Only if UHE cosmic rays have a Galactic origin, (and the large scale isotropy observations suggest quite strongly that do not), can there be enough power in magnetars to account for the UHE energy density.

5. Discussion I have argued, tentatively, that UHE cosmic rays are created in a new population of extragalactic sources with an average luminosity density that approaches that of Galactic cosmic rays. I have also described problems with each of the candidate "conventional" mechanisms for accelerating protons to these high energies. Quite different, general inferences have been drawn here, from the same data, by Waxman, and elsewhere by others. All of this underscores the need for better statistics which should be met by the AUGER proPhysica Scripta T85

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ject. Perhaps the most pressing need is to understand if particles of very different energy have a common origin. If true, this must rule out essentially all primordial particle/topological defect, neutron star, y-ray burst explanations, leaving only massive black holes and radio source models among the possibilities discussed above. In this case, it will be possible to seek identifications, especially at the highest energies, where the positions will be most accurate and the delays due to magnetic scattering the smallest. If, alternatively, clustering and its implications are not substantiated, then the next best clues will probably come from composition studies and detailing the large scale distribution on the sky. The most exciting outcome of all of this is that we are dealing with a new particle or defect with energy well out of the range of terrestrial accelerators. (For example, if there is a particle of energy Ex which decays with half life xx into N protons, then the cosmological energy density of these particles must be Qx ~ 3 x lO^AT'CExVlAeV)-1 (HQTX)~1.) Whatever happens, in a subject where the dullest and most conventional theories involve massive, spinning, black holes, ultrarelativistic blast waves and 100 GT fields threading nuclear matter, the future is guaranteed to be interesting.

Acknowledgements I am indebted to John Bahcall, Jim Cronin, Michael Hillas, Martin Rees, Alan Watson and Eli Waxman for stimulating discussions and the editors for their forbearance. I also gratefuly acknowledge the hospitality of the Insitute for Advanced Study (through the Sloan Foundation) and the Institute of Astronomy (through the Beverly and Raymond Sackler Foundation) as well as NASA grant 5-2837.

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References 1. Hillas, M., Ann. Rev. Astron. Astrophys. 22, 245 (1984). 2. Cronin, J. "Unsolved Problems in Astrophysics" (Eds. J. Bahcall and J. Ostriker) (Princeton: Princeton University Press) p. 325. 3. Krizmanic, J. F., Ormes, J. F. and Streitmatter, R. E. (Eds.) (New York: AIP, 1998). 4. Chupp, E. L. and Benz, A. O. (Eds.) Astrophys. J. Suppl. 90, 511 (1994). 5. Berezinski, V. S. et al., "Astrophysics of Cosmic Rays" (Amsterdam North Holland, 1990). 6. Axford, W. I., Astrophys. J. Suppl. 90, 937 (1994). 7. Takeda, M. et al., astro-ph/9902239 (1999). 8. Greisen, K., Phys. Rev. Lett. 16, 748 (1996). 9. Zatsepin, G. T. and Kuz'min, V. A., J. Exp. Theor. Phys. 4, L78 (1966). 10. Aharonian, F. and Cronin, J., Phys. Rev. D 50, 1892 (1994). 11. Bahcall, J. N. and Waxman, E., 1999 (preprint). 12. Miralda-Escude, J. and Waxman, E„ Astrophys. J. 462, L59 (1996). 13. Blandford, R. D. and Eichler, D., Phys. Rep. 154, 1 (1987). 14. Malkov, M. A., Astrophys. J. 511, L53 (1999). 15. Ellison, D., Drury, L. O'C. and Meyer, J.-P., Astrophys. J. 487, 197 (1997). 16. Koyama, K. et al., Nature 378, 255 (1995). 17. Erdbs, G. and Balogh, A., Astrophys. J. Suppl. 90, 553 (1994). 18. Jokipii, J. R. and Morrill, G., Astrophys. J. 312, 170 (1987). 19. Norman, C. A., Melrose, D. B. and Achterberg, A., Astrophys. J. 454, 60 (1995). 20. Goldreich, P. and Julian, W. H., Astrophys. J. 157, 869 (1967). 21. Bell, A. R., Mon. Notes R. Astron. Soc. 257, 493 (1992). 22. Cavallo, G., Astron. Astrophys. 65, 415 (1978). 23. Farrar, G. R. and Biermann, P. L., Phys. Rev. Lett. 81, 3579 (1998). 24. Waxman, E., Phys. Rev. Lett. 75, 386 (1995). 25. Waxman, E. and Miralda-Escude, J., Astrophys. J. 472, 489 (1996). 26. Thorne, K. S., Price, R. M. and MacDonals, D., 'Black Holes: The Membrane Paradigm" (New Haven: Yale University Press, 1986). 27. Boldt, E. and Ghosh, P., astro-ph/9902342 (1999). 28. Thompson, C. A. and Duncan, R. C , Mon. Notes R. Astron. Soc. 275, 255 (1995). 29. Kouveliotou, C. et al, Nature 393, 235 (1998).

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Physica Scripta.Vol. T85, 195-209, 2000

Particle Astrophysics with High Energy Photons T. C. Weekes* Whipple Observatory, Harvard-Smithsonian CfA, P.O. Box 97, Amado, AZ 85645-0097, U.S.A. Received January 19, 1999; accepted August 2, 1999

PACS Ref: 95.55Ka; 98.70Rz; 98.58MJ; 98.54Cm

Abstract The development of the atmospheric Cherenkov imaging technique has led to significant advances in y-ray detection sensitivity in the energy range from 200 GeV to 50 TeV The Whipple Observatory 10m reflector has detected the first galactic and extragalactic sources in the Northern Hemisphere; the Crab Nebula has been established as the standard candle for ground-based y-ray astronomy. The highly variable Active Galactic Nuclei, Markarian 421 and Markarian 501, have proved to be particularly interesting. A new generation of telescopes with improved sensitivity has the promise of interesting measurements of fundamental phenomena in physics and astrophysics. VERITAS (the Very Energetic Radiation Imaging Telescope Array System) is one such next generation system; it is an array of seven large atmospheric Cherenkov telescopes planned for a site in southern Arizona.

1. The relativistic universe It has to be with a certain sense of apology that one introduces the subject of photon astronomy to a symposium devoted to particle physics. Photons are, by any definition, rather dull specimens of the cosmic particle zoo. However one can argue that their very dullness, their lack of charge and mass, their infinite lifetime, their appearance as a decay product in many processes, their predictability, all combine to make them a valuable probe of the behavior of more exotic particles in distant, and therefore difficult to study, regions of the universe. Certainly no one can argue that photon astronomy at low energies (optical, radio and X-ray) has not had a major influence in our perception of the physical universe! Our universe is dominated by objects emitting radiation via thermal processes. The blackbody spectrum dominates, be it from the microwave background, the sun or the accretion disks around neutron stars. This is the ordinary universe, in the sense that anything on an astronomical scale can be considered ordinary. It is tempting to think of the thermal universe as THE UNIVERSE and certainly it accounts for much of what we see. However to ignore the largely unseen, non-thermal, relativistic, universe is to miss a major component and one that is of particular interest to the physicist, particularly the particle physicist. The relativistic universe is pervasive but largely unnoticed and involves physical processes that are difficult to emulate in terrestrial laboratories. The most obvious local manifestation of this relativistic universe is the cosmic radiation, whose origin, 86 years after its discovery, is still largely a mystery (although it is generally accepted, but not proven, that much of it arises in shock waves from galactic supernova explosions). The existence of a steady rain of particles, whose power law spectrum attests to their non-thermal origin and whose highest ener* e-mail: [email protected] © Physica Scripta 2000

gies extend far beyond that achievable in man-made particle accelerators, attests to the strength and reach of the forces that power this strange relativistic radiation. If thermal processes dominate the "ordinary" universe, then truly relativistic processes illuminate the "extraordinary" universe and must be studied, not just for their contribution to the universe as a whole but as the denizens of unique cosmic laboratories where physics is demonstrated under conditions to which we can only extrapolate. The observation of the extraordinary universe is difficult, not least because it is masked by the dominant thermal foreground. In places, we can see it directly such as in the relativistic jets emerging from AGNs but, even there, we must subtract the foreground of thermal radiation from the host elliptical galaxy. Polarization leads us to identify the processes that emit the radio, optical and X-ray radiation as synchrotron emission from relativistic particles, probably electrons, but polarization is not unique to synchrotron radiation and the interpretation is not always unambiguous. The hard, power-law, spectrum of many of the non-thermal emission processes immediately suggests the use of the highest radiation detectors to probe such processes. Hence hard X-ray and y-ray astronomical techniques must be the observational disciplines of choice for the exploration of the relativistic universe. Because the earth's atmosphere has the equivalent thickness of a meter of lead for this radiation, its exploitation had to await the development of space platforms for X-ray and y-ray telescopes. Although the primary purpose of the astronomy of hard photons is the search for new sources, be they point-like, extended or diffuse, it opens the door to the investigation of more obscure phenomenon in high energy astrophysics and even in cosmology and particle physics. Astronomy at energies up to 10 GeV has made dramatic progress since the launch of the Compton Gamma Ray Observatory in 1991 and that work has been summarized [1]. Beyond 10 GeV it is difficult to efficiently study y-rays from space vehicles, both because of the sparse fluxes which necessitate large collection areas and the high energies which make containment a serious problem. The development of techniques whereby y-rays of energy 100 GeV and above can be studied from the ground, using indirect, but sensitive, techniques is relatively new and has opened up a new area of high energy photon astronomy with some exciting possibilities and some preliminary results. The latter include the detection of TeV photons from supernova remnants and from the relativistic jets in AGNs. Such observations seriously constrain the models for such sources and in many cases lead to the development of a new paradigm. There remains the possibility that the annihilation lines from neutralinos might be discovered in the GeV-TeV region, that the evaporation of Physica Scripta T85

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primordial black holes might be manifest by the emission of bursts of TeV photons, that the infrared density of intergalactic space might be probed by its absorbing effect on TeV photons from distant sources, and even (in some models) that the fundamental quantum gravity energy scale might be constrained by the observation of short-term TeV flares in extragalactic sources. 2. Detection technique The techniques of ground-based Very High Energy (VHE) y-ray astronomy are not new but only achieved credibility in the late eighties with the detection of the Crab Nebula. The most sensitive technique, the atmospheric Cherenkov imaging technique, is the one that has been most successful and is now in use at some eight observatories. Its history and present status has been reviewed elsewhere [2]. It is an optical "telescope" technique and thus suffers the usual limitations associated with optical astronomy: limited duty cycle, weather dependence, limited field of view. But it also has the advantage that it is relatively inexpensive because it uses the same detector technology (photomultipliers) as optical astronomy, the same optical reflectors that borrow from solar energy investigations, and the same pulse processing techniques that are routinely used in high energy particle physics. In addition the Cherenkov technique operates in an energy regime where the physics of particle

interactions is relatively well understood and where there exist advanced Monte Carlo programs for the simulation of particle cascades. In recent years, VHE y-ray astronomy has been dominated by two advances in technique: the development of the atmospheric Cherenkov imaging technique, which led to the efficient rejection of the hadronic background, and the use of arrays of atmospheric Cherenkov telescopes to measure the energy spectra of y-ray sources. The former is exemplified by the Whipple Observatory 10m telescope (Fig. 1) with more modern versions CAT, the French telescope in the Pyrenees, and the Japanese-Australian CANGAROO telescope in Woomera, Australia. The most significant examples of the latter are thefivetelescope array of imaging telescopes on La Palma in the Canary Islands which is run by the Armenian-German-Spanish collaboration, HEGRA, and the four, soon to be seven, Telescope Array in Utah which is operated by a group of Japanese institutions. These techniques are relatively mature and the results from observations with overlapping telescopes are in good agreement. Vigorous observing programs are now in progress at all of these facilities; the vital observing threshold has been achieved whereby both galactic and extragalactic sources have been reliably detected. Many exciting results are anticipated as more of the sky is observed with this generation of telescopes. 3. Galactic sources

It is a measure of the maturity of this new discipline that the existence and study of galactic sources of TeV radiation is now considered ordinary and relatively uncontroversial. This is a dramatic change from only a decade ago when the existence of any galactic sources at all was hotly contested. These sources were always variable and difficult to confirm or refute [4]; it was not until the observation of steady sources, in particular, the observation of the Crab Nebula (which has become the standard candle), that the relative sensitivity of the different techniques could be assessed and some standards of credibility set. The Crab Nebula has been observed by some eight independent groups and no evidence for variability has been detected. It has been seen at energies from 200 GeV to more than 50 TeV and accurate energy spectra have been determined [3]. Originally predicted by Gould [5] as a TeV energy source based on a Compton-synchrotron model, the complete y-ray spectrum can now befittedby an updated version of the same model [3]. The variable parameter in this model is the magnetic field which is set by the TeV observations at 16±1 nanotesla, somewhat smaller than the value estimated from the equipartition of energy (Fig. 2). In practice, recent optical observations reveal a complex structure at the center of the nebula (where the TeV photons are believed to originate) and more sophisticated models are certainly called for. VHE y-rays have also been detected from other galactic sources. All of these detections are of sources with negative declinations, best seen in the Southern Hemisphere where there are fewer VHE observatories and hence the detections have largely been by one group. The exception is the y-ray Fig. 1. The Whipple Observatory 10 m reflector which will be the prototype pulsar PSR1706-44 which was discovered by the CANGAROO group [6] and confirmed by the Durham for the telescopes in VERITAS. Physica Scripta T85

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certainly have sufficient energy and their occurrence rate is about right; also there is a known mechanism associated with shock fronts to explain acceleration. Hence when EGRET detected a small number of y-ray sources at 10 GeV energies which appeared to coincide with known SNRs [11], it was widely believed that the cosmic ray origin problem had been solved. However Drury et al. [12] had shown that the y-ray spectrum of such sources should be rather flat power-laws that would extend to TeV energies. Extensive observations by the Whipple collaboration have failed to find any evidence for TeV emission [13]. The upper limits 10 are shown in Fig. 3 along with the EGRET points. More elaborate models have been constructed that can be made to fit the observations [14]. It is also possible that the o Method 1 1988-9 EGRET source/SNR identifications are in error since the o Method 1 1995-6 sources are not strong and the galactic y-ray plane is a con* Method 2 1988-9 fused region at MeV/GeV energies. Either way, it would • EGRET A CANGAR00 be reassuring for theories of cosmic ray origins to see definite 10 detections from some shell-type SNRs where the emission is 10 10 10 10 10 10 E (TeV) consistent with n production in the shell. The next generation of VHE detectors should provide these definitive Fig. 2. TeV spectra of the Crab and the predicted inverse-Compton spectrum observations. for three magnetic field strengths (from [3]). Shown are data from EGRET, CANGAROO, and three different epochs/data analysis methods for Whipple.

group [7]; both of these groups operate from Australia. The source is detected by EGRET at MeV-GeV energies as 100% pulsed. There is no evidence in the TeV signal for pulsations but there is weak evidence that the pulsar is in a plerion which may be the source of the TeV y-rays. The CANGAROO group also report the detection of an unpulsed TeV signal from a location close to the Vela pulsar [8]; the position coincides with the birthplace of the pulsar and hence the signal may originate in a weak plerion left after the ejection of the pulsar. Another interesting result is the detection of Cen X-3 by the Durham group [9]. Perhaps the most surprising (and controversial) result is the detection of a TeV source that is coincident with one part of the shell of the supernova remnant, SN1006 [10]. X-ray observations had shown that there is non-thermal emission from two parts of the shell that is consistent with synchrotron emission from electrons with energy up to 100 TeV; hence the TeV y-ray detection is not a surprise. The TeV emission is consistent with inverse Compton emission from electrons which have been shock accelerated in the shell. However it is not clear why it should be seen from only one region. Because this represents the first direct detection of SNR shell emission this result, when confirmed, has great significance. Not only can the magnetic field be estimated but also the acceleration time; these two parameters are very important for shock acceleration theory. More sensitive observations may reveal the detailed energy spectrum, whether or not the source is extended, and the relative strength of the TeV emission from each shell. Ideally, of course, one would like to see direct evidence from VHE y-ray astronomy of emission from hadron collisions in SNR shells. These SNRs are widely believed to be the source of the hadronic cosmic rays seen in the solar system (at least up to proton energies of 100 TeV) which fill the galaxy. However this canonical model mostly rests on circumstantial evidence and it is highly desirable to find the smoking gun that would clinch the issue. Supernovae © Physica Scripta 2000

4. Extragalactic sources 4.1. Relativistic Jets One of the most surprising results to come from VHE y-ray astronomy has been the discovery of TeV-emitting blazars. Unlike the observation of galactic supernovae such as the Crab Nebula, which are essentially standard candles, the light-curves of blazars are highly variable. In Fig. 4 the nightly averages of the TeV flux from Markarian 421 (Mkn 421) in 1995 are shown as observed at the Whipple Observatory [15]. Although AGN variability was a feature of the AGNs observed by EGRET on the Compton Gamma Ray Observatory at energies from 30 MeV to 10 GeV, the weaker signals (because of the finite collection area) do not allow such detailed monitoring, particularly on short time-scales. Active galactic nuclei (AGN) are the most energetic on-going phenomena that we see in extragalactic astronomy. The canonical model of these objects is that they contain massive black holes (often at the center of elliptical galaxies) surrounded by accretion disks and that relativistic jets emerge perpendicular to the disks; these jets are often the most prominent observational feature. Blazars are an important sub-class of AGNs because they seem to represent those AGNs which have one of their jets aligned in our direction. Observations of such objects are therefore unique. The VHE y-ray astronomer is thus in the position of the particle physicist who is offered the opportunity to observe the accelerator beam, either head-on or from the side. For the obvious reason that there is more energy transferred in the forward direction the particle physicist usually chooses to put his most important detectors directly in the direction of the beam (or close to it) and its high energy products. While such observations give the best insight into the energetic processes in the jet, they do not give the best pictorial representation. Hence just as it is difficult to visualize the working of a cannon by looking down its barrel, it is difficult to get a picture of the jet by looking at it Physica Scripta T85

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head-on. Observations at right angles to the jet give us our best low energy view of the jet phenomenon and indeed provide us with the spectacular optical pictures of jets from nearby AGNs (such as M87). 4.2. Sources

Mkn 421 is the closest example of an AGN which is pointing in our direction. It is a BL Lac object, a sub-class of blazars, so-called because they resemble the AGN, BL Lacertae which is notorious because of the lack of emission lines in its optical spectrum. Because such objects are difficult, and somewhat uninteresting, for the optical astronomer they Physica Scripta T85

were largely ignored until they were found to be also strong and variable sources of X-rays and y-rays. Mkn 421 achieved some notoriety largely because it was the first extragalactic source to be identified as a TeV y-ray emitter [16]. At discovery, its average VHE flux was «s 30% of the VHE flux from the Crab Nebula. Markarian 501 (Mkn 501), which is similar to Mkn 421 in many ways, was detected as a VHE source by the Whipple group in May 1995 [17]. It was only 8% of the level of the Crab Nebula and was near the limit of detectibility of the technique at that time. The discovery was made as part of an organized campaign to observe objects that were similar to Mkn 421 and were at small redshifts. This same campaign © Physica Scripta 2000

Particle Astrophysics with High Energy Photons

hours. The very large collection areas (> 10,000 m2) associated with atmospheric Cherenkov Telescopes is ideally suited for the investigation of short term variability. The VHE emission from the two best observed sources, Mkn 421 and Mkn 501 (Fig. 5), varies by a factor of a hundred. Although many hundreds of hours have now been devoted to their study, the variations are so complex that it is still difficult to characterize their emissions. It has been suggested [15] that for Mkn 421 the emission is consistent with a series of short flares above a baseline that falls below the threshold of the Whipple telescope (Fig. 4); the average flare duration is one day or shorter. The most important observations of Mkn 421 were in May, 1996 when it was found to be unusually active. On May 7, a flare was observed with the largest flux ever

later yielded the detection of the BL Lac object, 1ES 2344+514 ([18] which is also close (z = 0.044). Recently the Durham group has announced the detection of the BL Lac object, PKS2155-304 [19] which is also at a small redshift (z = 0.116). A more controversial, but potentially more important detection, is that of 3C 66A reported by the Crimean group [20]. These sources are summarized in Table I. Whereas the first two sources have been seen by a number of groups, the last three are reported by only one group and require confirmation. 4.3. Variability Perhaps the most exciting aspect of these detections is the observation of variability on time-scales from minutes to

Table I. Properties of the

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recorded from a VHE source. The observations began when the flux was already several times that of the Crab Nebula and it continued to rise over the next two hours before levelling off (Fig. 6). Observations were terminated as the moon rose but the following night it was observed at its quiescent level. One week later (May 15) a smaller, but shorter, flare was detected; in this case the complete flare was observed and the doubling time in the rise and fall was ^ 1 5 minutes. This is the shortest time variation seen in any extragalactic y-ray source at energies >10MeV (apart from in a y-ray burst). Mkn 501 is also variable, but as at other wavelengths, the characteristic time seems longer. Its baseline emission has varied by a factor of 15 over four years [26] (Fig. 5). Hour-scale variability has also been detected but its most important time variation characteristic appears to be the slow variations seen over the five months in 1997. Physica Scripta T85

4.4. Spectrum The atmospheric Cherenkov signal is essentially calorimetric and hence it should be possible to derive the y-ray energy spectrum from the observed light pulse spectrum. In practice it is more difficult because, unless an array of detectors is used, the distance to the shower core (impact parameter) is unknown. Although the extraction of a spectrum from even a steady and relatively steady source as the Crab Nebula required considerable effort and the development of new techniques, it was relatively easy to measure the spectra of Mkn 421 and Mkn 501 in their high state because the signal was so strong. The general features of the spectra derived from the Whipple observations are in agreement with those derived at the HEGRA telescopes [28]. The May 7, 1996 flare of Mkn 421 provided an excellent data base for the extraction of a spectrum; the data can be fit by a simple power-law (dN/dE oc E~26). There is © Physica Scripta 2000

Particle Astrophysics with High Energy Photons

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the highest energy end of the spectrum. Large zenith angle observations at Whipple [27] and observation by HEGRA [28] confirm the absence of a cutoff out to 10 TeV. The generally high state of Mkn 501 throughout 1997 give data from the Whipple telescope that can be best fit by a curved spectrum of the form: dN/dE and £-2.20-o.45/og10£ [29] (Fig. 7). Here the spectrum extends to at least 10 TeV. The curvature in the spectrum could be caused by the intrinsic emission mechanism or by absorption in the source. Since Mkn 421 and Mkn 501 are virtually at the same redshift it is unlikely that it could be due to intergalactic absorption since Mkn 421 does not show any curvature [30]. 4.5. Multiwavelength

observations

The astrophysics of the y-ray emission from the jets of AGNs are best explored using multiwavelength observations. These are difficult to organize and execute because of the different observing constraints on radio, 10 Photon Energy (TeV) optical, X-ray, space-based y-ray and ground-based y-ray observatories. Of necessity observations are often Fig. 7. VHE spectra of Mkn 421 (filled circles) and Mkn 501 (open stars) as incomplete and, when complete coverage is arranged, the measured with the Whipple Observatory telescope [30]. source does not always cooperate by behaving in an interesting way! The first multiwavelength campaign on Mkn 421 no evidence of a cutoff up to energies of 5 TeV [24] (Fig. 7). Because of the possibility of a high energy cutoff due to coincided with a TeV flare on May 14-15, 1994 and showed intergalactic absorption there is considerable interest in some evidence for correlation with the X-ray band; however © Physica Scripta 2000

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no enhanced activity was seen in EGRET [31]. A year later, in a longer campaign, there was again correlation between the TeV flare and the soft X-ray and UV data but with an apparent time lag of the latter by one day [15] (Fig. 8). The variability amplitude is comparable in the X-ray and TeV emission («s 400%) but is smaller in the EUV («200%) and optical (%20%) bands. In April, 1998 there was again a correlation seen between an X-ray flare observed by SAX and Whipple; in this case the TeV flare was much shorter (a few hours) compared to the X-ray (a day) [32]. The first multiwavelength campaign on Mkn501 was undertaken when the TeV signal was seen to be at a high level. The surprising result was that the source was detected by the OSSE experiment on CGRO in the 50-150 kev band (Fig. 8). This was the highest flux ever recorded by OSSE from any blazar (it has not detected Mkn 421) but the amplitude of the X-ray variations (^200%) was less than those of the TeV y-rays («400%) [33].

of Compton-synchrotron models, e.g., the Crab Nebula. Whereas the synchrotron peak in Mkn 421 occurs near 1 keV, that of Mkn 501 occurs beyond 100 keV which is the highest seen from any AGN. In 1998 the synchrotron spectrum peak in Mkn 501 shifted back to 5 keV and the TeV flux fell below the X-ray flux. 4.7. Implications The sample of VHE emitting AGNs is still very small but it is possible to draw some conclusions from their properties (summarized in Table I). •

4.6. Multiwavelength power spectra Because of the strong variability in the TeV blazars it is dif- • ficult to represent their multiwavelength spectra. In Fig. 9 we show the fluxes plotted as power (v x Fv) from Mkn 421 and Mkn 501 during flaring as well as the average fluxes. Both sources display the two peak distribution characteristic Physica Scripta T85

The first three objects, all detected by the Whipple group, are the three closest BL Lacs in the northern sky. Some 20 other BL Lacs have been observed with z < 0.10 without detectable emission. This could be fortuitous, because they are standard candles and these are closest (but the distance differences are small), or because they suffer the least absorption (but there is no cutoff apparent in their spectra). All of the objects are BL Lacs; because such objects do not show emission lines and therefore probably do not have strong optical/infrared absorption close to the source, it is suggested that BL Lacs are preferentially VHE emitters. © Physica Scripta 2000

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Four of the five sources are classified as XBLs which indicates that they are strong in the X-ray region and that the synchrotron spectrum most likely peaks in that range (and that the Compton spectrum peaks in the VHE y-ray range). The fifth, 3C 66A, is an RBL, like many of the blazars detected by EGRET; it is believed that these blazars have synchrotron spectra that peak at lower energies and Compton spectra that peak in the HE y-ray region. Only three (Mkn 421, PKS 2155-304 and 3C 66A) are listed in the Third EGRET Catalog; there is a weak detection reported by EGRET for Mkn 501. If 3C 66A is confirmed (and to a lesser extent PKS 2155-305), then the intergalactic absorption is significantly less than had been suggested from galactic evolution models. There is evidence for variability in all of the sources. The rapid variability seen in Mkn 421 indicates that the emitting region is very small which might suggest it is close to the black hole. In that case the local absorption must be very low (low photon densities). It seems more likely that the region is well outside the dense core.

There are three basic classes of model considered to explain the high energy properties of BL Lac jets: Synchrotron Self Compton (SSC), Synchrotron External Compton (SEC) and Proton Cascade (PC) Models. In the first two the progenitor particles are electrons, in the third they are protons. VHE y-ray observations have constrained the types of models that are likely to produce the y-ray emission but still do not allow any of them to be eliminated. For instance, the correlation of the X-ray and the VHE flares is consistent with the first two models where the same population of electrons radiate the X-rays and y-rays. There is little evidence for the IR component in BL Lac objects which would be necessary in the SEC models as the targets for Compton-scattering, so this particular type of model may not be likely for these objects. The PC models which produce © Physica Scripta 2000

the y-ray emission through e + e~ cascades also have great difficulty explaining the rapid cooling observed in the TeV emission from Mkn 421. Also the high densities of unbeamed photons near the nucleus, such as the accretion disk or the broad line region, are required to initiate the cascades and these cause high pair opacities to TeV y-rays [34]. Significant information comes from the multiwavelength campaigns (although thus far these have been confined to Mkn 421 and Mkn 501). Simultaneous measurements constrain the magnetic field strength (B) and Doppler factor (<5) of the jet when the electron cooling is assumed to be via synchrotron losses. The correlation between the VHE y-rays and optical/UV photons observed in 1995 from Mkn 421 indicates both sets of photons are produced in the same region of the jet; S > 5 is required for the VHE photons to escape significant pair-production losses [15]. If the VHE y-rays are produced in the synchrotronself-Compton process, 5 = 15 - 40 and B = 0.03 - 0.9G for Mrk 421 [35], [36] and d < 15 and B = 0.08 - 0.2G for Mkn 501 [29], [36]. On the other hand by assuming protons produce the y-rays in Mkn 421, Mannheim [37] derives 5 = 16 and B = 90G. The Mkn 421 values of 5 and B are extreme for blazars, but they are still within allowable ranges and are consistent with the extreme variability of Mkn 421. 5. Intergalactic absorption Thus far it has not been possible to make a direct measurement of the infrared background radiation at wavelengths more than 3.5 microns and less than 140 microns. This is unfortunate since the background potentially contains valuable information for cosmology, galaxy formation and particle physics. The problem for direct measurement is the presence of foreground local and galactic sources. However the infrared background can make its presence felt by the absorption it produces on the spectra of VHE Physica Scripta T85

204

T. C. Weekes

y-ray sources when they are at great distances. The absorption is via the yy -» e + e~ process, the physics of which is well understood. The maximum absorption occurs when the product of the energy of the two photons (y-ray and infrared) is approximately equal to the product of the rest masses of the electron-pair. Hence a 1 TeV y-ray is most heavily absorbed by O.leV (1.2 micron) infrared photon in head-on collisions. The importance of this effect for VHE and UHE y-ray astronomy was first pointed out by Nikishov [38]; its potential for making an indirect measurement of the infrared background was pointed out by Gould and Schreder [39] and, more recently, in the aftermath of the EGRET detections of AGNs, by Stecker and de Jager [40]. At the redshift of the AGNs detected at VHE energies to date (0.03 to 0.5) if the infrared density has the value assumed in some models [40], the effect is appreciable and should be apparent in carefully measured energy spectra in the range 1 to 50 TeV. Ideally for such a measurement the intrinsic emission spectrum of the y-rays from the distant source should be known. In practice this is not the case although thus far all the AGNs detected in the GeV-TeV range appear to have very smooth power-law spectra. Biller et al. [41] have made a conservative derivation of upper limits on the infrared spectrum based on the measured y-ray spectrum from 0.5 to 10 TeV from Mkn 421 and Mkn 501 by the Whipple and HEGRA groups (Table II). These upper limits apply to infrared energies from 0.025 to 1.0 eV; they are the best upper limits over this range. At some wavelengths, these limits are as much as an order of magnitude below the upper limits set by the DIRBE/COBE satellite (see Fig. 10). For the derivation the following assumptions are made:

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•

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Then infrared densities are calculated such that they do not cause the shape of the observed VHE spectrum to deviate from the bounds set from the VHE measurements. This approach has the effect of anchoring the lower energy TeV data to the appropriate infrared upper limits and then extending these bounds so that they are consistent with those based on the shape and extent of the AGN spectra at the higher energies. Thus the maximum energy density in each interval of infrared energy is determined; these limits are given in Table II where the maximum energy in the measured y-ray spectrum is taken to be 10 TeV. They are plotted in Fig. 10; also shown are the upper limits from other methods. These upper limits do not conflict with the predictions of the infrared background based on detailed models of galactic evolution [42]. They do however allow some more cosmological possibilities to be eliminated. In particular in one scenario, density fluctuations in the early universe (z ^ 1000) could have produced very massive objects which would collapse to black holes at later times and could explain the dark matter. However although undetectable now, they would have produced an amount of infrared radiation that would have exceeded the above limits [41]. These limits also place some constraints on radiative neutrino decay.

6. Gamma ray bursts The contribution of TeV observations to the physics of y-ray bursts is at once the most speculative and most important (potentially) of all the scientific topics considered here. As yet, there is no positive detection of TeV photons during or immediately after a classical y-ray burst (GRB) (although there is one tantalizing but unconfirmed observation [43]). However since there is no turn over seen in the spectra of GRBs detected by EGRET at energies > 30 GeV, there is the potential for interesting observations at VHE energies. The observed EGRET spectra are power

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© Physica Scripta 2000

Particle Astrophysics with High Energy Photons

A feature of the EGRET GRB observations was that there was evidence for delayed emission (up to 1.5 hours) from the burst site [46]. This may indicate a different component at these energies. Some models [47] predict that this delayed emission could persist for days and could hence be easily observed with narrow field of view instruments. The detection of a TeV y-ray component in a GRB would be a serious parameter for the emission models, in particular the Lorentz bulk motion in the source would be constrained. It would also be an independent distance indicator since the source would have to show absorption if the redshift was > 0.1.

y-ray source at the Galactic Center [49] at energies above 300 MeV (Fig. 12). There is no significant evidence for variability and the spectrum is unlike other galactic sources; this suggests that the source is unusual and unlike other galactic sources. Preliminary observations (4.5 hours) by the Whipple group [50] did not give a significant signal at 1.5 TeV energies. In a recent paper, Bergstrom, Ullio and Buckley [51] have estimated the flux from the annihilation radiation of neutrinos in the Galactic Center using the most recent models of the galactic mass distribution. The predicted line has a relative width of 10~3. Neither space nor ground-based detectors have energy resolution of this quality (even in the next generation of detectors) but the intensity of the line is such that it might be detectable even with relatively crude energy resolution. A broad spectrum of secondary y-rays would also be detected from the other annihilation modes giving a spectrum of the shape shown in Fig. 11 The wide range of possible parameters in supersymmetric space give a range of predicted y-ray fluxes. These are shown in Fig. 13 where the sensitivity of various ground-based y-ray experiments are also shown. The line labelled Whipple is the current sensitivity of the Whipple experiment and is seen to be unable to constrain the predictions. GRANITE III represents the expected sensitivity of a current upgrade to the Whipple telescope which should be completed in 1999. VERITAS is a next generation array of telescopes (see below) planned for southern Arizona. The sensitivity for a similar array in the Southern Hemisphere is also shown; the increased sensitivity of the telescope in the Northern Hemisphere comes from the enhanced performance of atmospheric Cherenkov telescopes when operated at large zenith angles.

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The next generation y-ray satellite experiment, GLAST will give good coverage of parameter space at energies less than 300 GeV. Hence together GLAST and VERITAS will allow a sensitive search over the full range of allowed neutralino mass. 8. Quantum gravity Some quantum gravity models predict the refractive index of light in vacuum to be dependent on the energy of the photon. This effect, originating from the polarization of space-time, causes an energy-dependance to the velocity of light. Effectively, the quantum fluctuations are on distance scales near the Planck length, (Z,p ~ 10 - 3 3 cm), (corresponding to time-scales of l / £ p , the Planck mass (~ 1019GeV)). Different models of quantum gravity give widely differing predictions for the amount of time dispersion. In one model Physica Scripta T85

where A? is the time delay relative to propagation at the velocity of light, c, I; is a model-dependent factor of order 1, E is the energy of the observed photons, EQQ is the quantum energy scale, and L is the distance from the source. In most models £QG % EP but, in recent work in the context of string theory, it can be as low as 1016GeV [52]. Recently it has been suggested that astrophysical observations of transient high energy emission from distant sources might be used to measure (or limit) the quantum gravity energy scale. Amelino-Camelia et al. [53] suggested that BATSE observations of GRBs would provide a powerful method of probing this fundamental constant if variations on time-scales of milliseconds could be measured in the MeV signal in a GRB which was measured to be at a cosmological distance. Such time-scales and distances have been measured in GRBs but so far not in the same GRB. The absence of time dispersion in flares of TeV y-rays from AGNs at known distances provides an even more sensitive measure. Biller et al. [54] have used the sub-structure observed in the 15 minute flare in Mkn 421 observed by the Whipple group on April 15, 1996 [25] to derive a lower limit on EQQ. On a time-scale of 280 seconds there is weak (2 2 TeV. For a Hubble Constant of 85 km/s/Mpc, the distance L is 1.1 x 1016 light-seconds. This gives a lower limit to -£QG of > 4 X 1016GeV assuming £, is «a 1. This is the most convincing lower limit on £ Q G to date. Because VHE y-ray astronomy is still in its infancy and the exposure time on AGNs still limited, it is likely that much more sensitive measurements will lead to better limits on EQG as a new generation of detectors comes on-line and permits the detection of shorter time-variations and/or more distant sources. © Physica Scripta 2000

Particle Astrophysics with High Energy Photons 9. Future prospects It is clear that to fully exploit the potential of ground-based y-ray astronomy the detection techniques must be improved. This will happen by extending the energy coverage of the technique and by increasing its flux sensitivity. Ideally one would like to do both but in practice there must be trade-offs. Reduced energy threshold can be achieved by the use of larger but cruder mirrors and this approach is currently being exploited using existing arrays of solar heliostats (STACEE [61] and CELESTE [59]). A German-Spanish project (MAGIC) to build a 17 m aperture telescope using state-of-the-art technology has also been proposed. These projects may achieve thresholds as low as 20-30 GeV where they will effectively close the current gap in the y-ray spectrum from 20-200 GeV. Ultimately this gap will be covered by GLAST, the next generation y-ray space telescope (which will use solid-state detectors) which is scheduled for launch in 2005 by an international collaboration. Extension to even higher energies can be achieved by the atmospheric Cherenkov telescopes working at large zenith angles and by particle arrays at very high mountain altitudes. An interesting telescope that will soon come on line and will complement these techniques is the MILAGRO water Cherenkov detector in New Mexico which will operate 24 hours a day with wide field of view and will have good sensitivity to y-ray bursts and transients. VERITAS, with seven 10 m telescopes arranged in a hexagonal pattern with 80 m spacing (Fig. 14), will aim for the middle ground, with its primary objective being high sensitivity in the 100 GeV to 10 TeV range. It will be located in southern Arizona and will be the logical development of the Whipple telescope. It is hoped to begin construction in 1999 and to complete the array by 2004. The German-French HESS (initially four, and eventually perhaps sixteen, 10 m class telescopes) will be built in

Namibia and the Japanese NEW CANGAROO array (with three to four telescopes in Australia) will have similar objectives. In each case the arrays will exploit the high sensitivity of the imaging ACT and the high selectivity of the array approach. The relative flux sensitivities of the present and next generation of VHE telescopes as a function of energy are shown in Fig. 15, where the sensitivities of the wide field detectors are for one year and for the ACT for 50 hours; in all cases a 5
S 499 pixel cameras

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Et (GeV) Fig. 15. Comparison of the point source sensitivity of VERITAS to Whipple Fig. 14. Layout of the seven 10 m telescopes in VERITAS. Each telescope will [56], MAGIC [57], CELESTE/STACEE [59]; [60]; [61]; HEGRA [58], GLAST [62], EGRET [63], and MILAGRO [64]. have a camera with 499 pixels. © Physica Scripta 2000

Physica Scripta T85

208

T. C. Weekes

(electrons should show cut-offs which correlate with lower energy spectra, protons would not show a simple correlation). In addition, the recent efforts [65] to unify the different classes of blazar into different manifestations of the same object type can be tested. In addition the infrared background will be probed by the detection of sources over a range of redshifts. SNRs The existing data clearly indicate that in order to resolve the contributions of the various y-ray emission mechanisms, one needs more accurate measurements over a more complete range of energies. The NGGRTs and GLAST will be a powerful combination to address these issues. The excellent angular resolution of the NGGRTs will allow detailed mapping of the emission in SNRs. The sensitivity and energy resolution, combined with observations at lower y-ray and X-ray energies help to elucidate the y-ray emission mechanism. This may lead to direct the confirmation or elimination of SNRs as the source of cosmic rays. Gamma-ray pulsars The detection of VHE y-rays would be decisive in favoring the outer gap model over the polar cap model. Six pulsars are detected at EGRET energies and their high energy emission is already seriously constrained by the VHE upper limits. The detection of a pulsed y-ray signal above 50 GeV would be a major breakthrough.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

Unidentified galactic EGRET sources The legacy of EGRET may be more than 70 unidentified sources, many of which are in the Galactic plane. The positional uncertainty of these sources make identifications with sources at longer wavelengths unlikely. In the galactic plane, probable sources are SNRs and pulsars, particularly in regions of high IR density (e.g., OB associations), but some may be new types of objects. The N G G R T should have the sensitivity and low energy threshold necessary to detect many of these objects. Detailed studies of these objects with the excellent source location capability of the NGGRTs could lead to many identifications with objects at longer wavelengths. The energy resolution and energy coverage could also go a long way to explaining the y-ray emission mechanisms in any new types of source identified. In addition, any variability in these objects would be easily identified and measured with the NGGRT.

35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

Acknowledgements Research in VHE y-ray astronomy at the Whipple Observatory is supported by a grant from the U.S.D.O.E. Helpful comments from Mike Catanese, Stephen Fegan and Vladimir Vassiliev are also acknowledged.

References 1. Proceedings of the Fourth Compton Symposium, Williamsburg, Virginia, USA, April 1997, (Eds. C. D. Dermer, M. S. Strickman and J. D. Kurfess) AIP 410. 2. Ong, R. A., Physics Reports 305, 93 (1998). 3. Hillas, A. M. et al, Astrophys. J. 503, 744 (1998).

Physica Scripta T85

51. 52. 53. 54. 55. 56. 57. 58. 59.

Weekes, T. C , Space Sci. Rev. 59, 315 (1991). Gould, R. J., Phys. Rev. Lett. 15, 577 (1965). Kifune, T. et al, Astrophys. J. 438, L91 (1995). Chadwick, P. M. et al, Proc. 25th Int. Cos. Ray Conf., Durban 3, 189 (1997). Yoshikoshi, T. et al., Astrophys. J. 487, L65 (1997). Chadwick, P. M. et al., Astrophys. J. 503, 391 (1998). Tanimori, T. et al., Astrophys. J. 497, L25 (1998). Esposito, J. A. et al, Astrophys. J. 461, 820 (1996). Drury, L. O ' C , Aharonian, F. A. and Volk, H. J. Astron. Astrophys. 287, 959 (1994). Buckley, J. H. et al, Astron. Astrophys. 329, 639 (1998). Gaisser, T. K., Protheroe, R. J. and Stanev, T., Astrophys. J. 492, 219 (1998). Buckley, J. H. et al, Astrophys. J. 472, L9 (1996). Punch, M. et al, 358, 477 (1992). Quinn, J. et al, Astrophys. J. Lett. 456, L83 (1996). Catanese, M. et al, Astrophys. J. 501, 616 (1998). Chadwick, P. M. et al, Astropart. Phys. 9, 131 (1998). Neshpor, Yu. I. et al, Astron. Lett. 24, 134 (1998). Perlman, E. S. et al, Astrophys. J. Suppl. 104, 251 (1996). Mukherjee, R. et al, 490, 116 (1997). Kataoka, J. et al, Astrophys. J. 514, 138 (1999). Zweerink, J. et al, Astrophys. J. Lett. 490, L141 (1997). Gaidos, J. A. et al, Nature 383, 319 (1996). Quinn, J. et al, Astrophys. J. 518, 693 (1999). Krennrich, F. et al, Astrophys. J. 481, 758 (1997). Lorenz, E., Proc. of Workshop on TeV Astrophysics of Extragalactic Sources, Cambridge, MA, 131 (1998). Samuelson, F. W. et al, Astrophys. J. Lett. 501, L17 (1998). Krennrich, F. et al, Astrophys. J. 511, 149 (1999). Macomb, D. J. et al, Astrophys. J. 449, L99 (1995). Maraschi, L. et al, Proc. of Workshop on TeV Astrophysics of Extragalactic Sources, Cambridge, MA, 189 (1998). Catanese, M. et al, Astrophys. J. 487, L143 (1997). Coppi, P. S., Kartje, J. F. and Konigl, A. 1993, in Proc. Compton Symposium, (Eds. M. Friedlander, N. Gehrels and D. J. Macomb) (New York: AIP), 559. Catanese, M., Proc. of Symposium on BL Lac Phenomenon, Turku, Finland, 243 (1998). Tavecchio, F., Maraschi, L. and Ghiselline, G. Astrophys. J. 509, 608 (1998). Mannheim, K., Astron. Astrophys. 269, 67 (1993). Nikishov, A. J., Soviet Physics J.E.T.P. 14, 393 (1962). Gould, R. P. and Schreder, G. P. Phys. Rev. 155, 1408 (1967). Stecker, F. W. and De Jager, O. C. Astrophys. J. 415, L71 (1993). Biller, S. D. et al, Phys. Rev. Lett. 80, 2992 (1998a). MacMinn, D. and Primack, J. R., Space Sc. Rev. 75, 413 (1996). Padilla, L. et al, Astron. Astrophys. 337, 43 (1998). Dingus, B. et al, Proc. 4th Huntsville GRB Symposium, (AIP 428), (Eds. C. A. Meegan, R. D. Preece, T. M. Koshut) 349 (1998). Connaughton, V. et al, Astrophys. J. 479, 859 (1997). Hurley, K. et al, Nature 372, 652 (1994). Totani, T., Astrophys. J. 502, L13 (1998). Dermer, C. D., Chiang, J., Bottcher, M., Astrophys. J. (in press) (1998). Mayer-Hasselwander, H. A. et al, Astron. Astrophys. 335, 161 (1998). Buckley, J. H. et al, Proc. 25th Int. Cos. Ray Conf., Durban 3, 237 (1997). Bergstrbm, L., Ullio, P. and Buckley, J. H., Astropart. Phys. 9, 137 (1998). Witten, E., Nucl. Phys. B 471, 135 (1996). Amelino-Camelia, G. et al, Nature 383, 319 (1998). Biller, S. D. et al, Phys. Rev. Lett. 83, 2108 (1999). Vassiliev, V. V., Proc. of Workshop on TeV Astrophysics of Extragalactic Sources, Cambridge, MA, (in press). Weekes, T. C. et al, Astrophys. J. 342, 370 (1989). Barrio, J. A. et al, "The Magic Telescope", design study (MPI-PhE/98-5) (1998). Daum, A. et al, Astropart. Phys. 8, 1 (1997). Quebert, J. et al, in "Towards a Major Atmospheric Cherenkov Detector IV", (Padova, Italy), (Ed. M. Cresti), 248 (1995).

© Physica Scripta 2000

Particle Astrophysics with High Energy Photons 60. Bhat, C. L., Proc. 25th Int. Cos. Ray Conf., Durban 8, 211 (1997). 61. Ong, R. A. and Covault, C. E., in "Towards a Major Atmospheric Cherenkov Detector IV", (Padova, Italy), (Ed. M. Cresti), 247 (1998). 62. Leonard, P. J. T., Nature, 383, 394 (1996).

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63. Kurfess, J. D. et al., Proceedings of the Fourth Compton Symposium, Williamsburg, Virginia, USA, April 1997, (Eds. C. D. Dermer, M. S. Strickman and J. D. Kurfess), (AIP 410), 509. 64. Sinnis, G. et al, Nucl. Phys. B (Proc. Suppl.) 43, 141 (1995). 65. Ghisellini, G. et al., Mon. Notes R. Astron. 301, 451 (1998).

Physica Scripta T85

Physica Scripta.Vol. T85, 210-220, 2000

Dark Matter and Dark Energy in the Universe Michael S. Turner Departments of Astronomy & Astrophysics and of Physics, Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637-1433, USA NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, IL 60510-0500, USA Received January 10, 1999; accepted August 2, 1999

PACS Ref: 95.35.+d

Abstract

when the expansion rate,

For the first time, we have a plausible and complete accounting of matter and energy in the Universe. Expressed a fraction of the critical density it goes like this: neutrinos, between 0.3% and 15%; stars, between 0.3% and 0.6%; baryons (total), 5% ± 0 . 5 % ; matter (total), 40% ± 10%; smooth, dark energy, 80% ± 2 0 % ; totaling to the critical density (within the errors). This accounting is consistent with the inflationary prediction of a flat Universe and measurements of the anisotropy of the CBR. It also defines three "dark problems": Where are the dark baryons? What is the nonbaryonic dark matter? What is the nature of the dark energy? The leading candidate for the (optically) dark baryons is diffuse hot gas; the leading candidates for the nonbaryonic dark matter are slowly moving elementary particles left over from the earliest moments (cold dark matter), such as axions or neutralinos; the leading candidates for the dark energy involve fundamental physics and include a cosmological constant (vacuum energy), a rolling scalar field (quintessence), and a network of light, frustrated topological defects.

/r2 = ^ J >

1. Introduction The quantity and composition of matter and energy in the Universe is of fundamental importance in cosmology. The fraction of the critical energy density contributed by all forms of matter and energy, Qo =

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(1)

determines the geometry of the Universe: R2 curv

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Here, subscript '0' denotes the value at the present epoch, p crit = 3H$/SnG ~ 1.88/r2 x 10" 29 g cm" 3 , O, is the fraction of critical density contributed by component i (e.g., baryons, photons, stars, etc) and Ho = 100/ikms" 1 M p c - 1 . The sign of i? 2 urv specifies the spatial geometry: positive for a 3-sphere, negative for a 3-saddle and 0 for flat space. The present rate of deceleration of the expansion depends upon Oo as well as the composition of matter and energy,

1

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vanishes and R

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is less than zero. If there were matter alone, a positively curved universe (Q0 > 1) eventually recollapses and a negatively curved universe (Qo < 1) expands forever. However, exotic components complicate matters: a positively curved universe with positive vacuum energy can expand forever, and a negatively curved universe with negative vacuum energy can recollapse. The quantity and composition of matter and energy in the Universe is also crucial for understanding the past. It determines the relationship between age of the Universe and redshift, when the Universe ended its early radiation dominated era, and the growth of small inhomogeneities in the matter. Ultimately, the formation and evolution of large-scale structure and even individual galaxies depends upon the composition of the dark matter and energy. Measuring the quantity and composition of matter and energy in the Universe is a challenging task. Not just because the scale of inhomogeneity is so large, around 10 Mpc; but also, because there may be components that remain exactly or relatively smooth (e.g., vacuum energy or relativistic particles) and only reveal their presence by their influence on the evolution of the Universe itself.

2. A complete inventory of matter and energy

2.1. Preliminaries 2.1.1. Radiation Because the cosmic background radiation (CBR) is known to be black-body radiation to very high precision (better than 0.005%) and its temperature is known to four signifi1 _ 0 WR\ = ^ o + - £ > , • Wi. (3) cant figures, T0 = 2.7277 ± 0.002 K, its contribution is very qo=2" Hi precisely known, Qyh2 = 2.480 x 10~5. If neutrinos are _4 The pressure of component /, pt = w,-p,-; e.g., for baryons massless or very light, mv <£ 10 eV, their energy density wt = 0, for radiation H>, = 1/3, and for vacuum energy is equally well known because it is directly related to that of the photons, Qv = | ( 4 / l l ) 4 / 3 O y (per species) (there is a Wi = — 1 . The fate of the Universe - expansion forever or recollapse small 1% positive correction to this number; see [1]). It is possible that additional relativistic species contribute - is not directly determined by Qo and/or qo. It depends upon precise knowledge of the composition of all components of significantly to the energy density, though big-bang matter and energy, for all times in the future. Recollapse nucleosynthesis (BBN) severely constrains the amount (the occurs only if there is a future turning point, that is an epoch equivalent of less than 0.2 of a neutrino species; see e.g., [2]) Physica Scripta T85

i Physica Scripta 2000

Dark Matter and Energy in the Universe

211

MATTER / ENERGY in the UNIVERSE

unless they were produced by the decay of a massive particle after the epoch of BBN. In any case, we can be confident that the radiation component of the energy density today is small. The matter contribution (denoted by £2M), consisting of particles that have negligible pressure, is the easiest to determine because matter clumps and its gravitational effects are thereby enhanced (e.g., in rich clusters the matter density typically exceeds the mean density by a factor of 1000 or more). With this in mind, I will decompose the present matter/energy density into two components, matter and vacuum energy,

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ignoring the contribution of the CBR and ultrarelativistic neutrinos. I will use vacuum energy as a stand in for a smooth component of energy (more later). Vacuum energy and a cosmological constant are indistinguishable: a cosmological constant corresponds to a uniform energy density of magnitude p vac = A/inG.

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2.2. Q0 = 1 ± 0 . 2 There is a growing consensus that the anisotropy of the CBR offers the best means of determining the curvature of the Universe and thereby Qo, cf., Eq. (2). This is because the method is intrinsically geometric - a standard ruler on the last-scattering surface - and involves straightforward physics at a simpler time (see e.g., [3]). At last scattering baryonic matter (ions and electrons) was still tightly coupled to photons; as the baryons fell into the dark-matter potential wells the pressure of photons acted as a restoring force, and gravity-driven acoustic oscillations resulted. These oscillations can be decomposed into their Fourier modes; Fourier modes with k ~ lH$/2 determine the multipole amplitudes a/m of CBR anisotropy. Last scattering occurs over a short time, making the CBR is a snapshot of the Universe at t\s ~ 300 000 yrs. Each mode is "seen" in a well defined phase of its oscillation. (For the density perturbations predicted by inflation, all modes the have same initial phase because all are growing-mode perturbations.) Modes caught at maximum compression or rarefaction lead to the largest temperature anisotropy; this results in a series of acoustic peaks beginning at / ~ 200 (see Fig. 2). The wavelength of the lowest frequency acoustic mode that has reached maximum compression, ^max ~ vst\s, is the standard ruler on the last-scattering surface. Both l m a x and the distance to the last-scattering surface depend upon Q0, and the position of the first peak / ~ 200/V^o- This relationship is insensitive to the composition of matter and energy in the Universe (see [4]). CBR anisotropy measurements, shown in Figs. 2 and 3, now cover three orders of magnitude in multipole number and involve more than twenty experiments. COBE is the most precise and covers multipoles / = 2 — 20; the other measurements come from balloon-borne, Antarctica-based and ground-based experiments using both low-frequency (/~<100GHz) HEMT receivers and high-frequency (f > 100 GHz) bolometers. Taken together, all the measurements are beginning to define the position of the first acoustic peak, at a value that is consistent with a flat Universe. Various analyses of the extant data have been carried out, indicating QQ ~ 1 ± 0.2 (see e.g., [5]). It is certainly too early to draw definite conclusions or put too much weigh © Physica Scripta 2000

STARS T 00O5-/-O.0O2

0.001 —

—

Q Fig. 1. Summary of matter/energy in the Universe. The right side refers to an overall accounting of matter and energy; the left refers to the composition of the matter component. The contribution of relativistic particles, CBR photons and neutrinos, QTe\h2 = 4.170 x 10~5, is not shown. The upper limit to mass density contributed by neutrinos is based upon the failure of the hot dark matter model of structure formation [26,64] and the lower limit follows from the evidence for neutrino oscillations [69]. Here Ho is taken to be 6 5 k m s - 1 Mpc - 1 .

in the error estimate. However, a strong case is developing for a flat Universe and more data is on the way (Python V, Viper, TOCO, Maxima, Boomerang, DASI, and others). Ultimately, the issue will be settled by NASA's MAP (launch late 2000) and ESA's Planck (launch 2007) satellites which will map the entire CBR sky with 30 times the resolution of COBE (around 0.1°) (see [6]). 2.3. Matter 2.3.1. Baryons. For more than twenty years big-bang nucleosynthesis has provided a key test of the hot big-bang cosmology as well as the most precise determination of the baryon density. In 1995 comparison of the primeval abundances of D, 3 He, 4 He and 7 Li with their big-bang predictions defined a concordance interval, Q^h2 = 0.007 - 0.024 (see e.g., [7]; for another view, see [8]). Of the four light elements produced in the big bang, deuterium is the most powerful "baryometer" - its primeval abundance depends strongly on the baryon density (oc 1/pg7) - and the evolution of its abundance since the big bang is simple - astrophysical processes only destroy deuterium. Until recently deuterium could not be exploited Physica Scripta T85

212

Michael S. Turner 120

100 « i

-

o

+

1

10

M Tegmark, April 1998

100

1000

1

as a baryometer because its abundance was only known locally, where roughly half of the material has been through stars with a similar amount of the primordial deuterium destroyed. In 1998, the situation changed dramatically, launching BBN into the precision era of cosmology. Over the past four years there have been claims of upper limits, lower limits, and determinations of the primeval deuterium abundance, ranging from ( D / H ) = 1 0 -55 to ( D / H ) = 3 x 10 4 . In short, the situation was confusing. In 1998 Buries and Tytler clarified matters and established a strong case for (D/H)P = (3.4 ± 0.3) x 10~5 [9]. That case is based upon the deuterium abundance measured in four high-redshift hydrogen clouds seen in absorption against distant QSOs, and the remeasurement and reanalysis of other putative deuterium systems. In this enterprise, the Keck I 10-meter telescope and its HiRes Echelle Spectrograph have played the crucial role. The Buries - Tytler measurement turns the previous factor of three concordance range for the baryon density into a 10% determination of the baryon density, p B = (3.8 ±0.4)x 10"31 g e m - 3 or QBh2 = 0.02 ± 0.002 (see Fig. 4), with about half the error in p B coming from the theoretical error in predicting the BBN yield of deuterium. [A very recent analy-

Fig. 2. Summary of all CBR anisotropy measurements, where the temperature variations across the sky have been expanded in spherical harmonics, 5T(6, (j>) = ]T\ atm YimiO, <j>) and Q = (|a/m|2). In plain language, this plot shows the size of the temperature variation between two points on the sky separated by angle 8 vs. multipole number / = 200°/6 (1 = 2 corresponds to 100°, / = 200 corresponds to 0 = 1°, and so on). The curves illustrate the predictions of CDM models with fio = 1 (curve with lower peak) and J2o = 0.3 (darker curve). Note: the preference of the data for aflatUniverse and the evidence for the first of a series of "acoustic peaks." The presence of these acoustic peaks is a key signature of the density perturbations of quantum origin predicted by inflation (Figure courtesy of M. Tegmark).

B o n d . Jaffe a n d Knox

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Fig. 4. Predicted abundances of He (mass fraction), D, 3He, and 7Li (number relative to hydrogen) as a function of the baryon density; widths of the curves indicate "2a" theoretical uncertainty. The dark band highlights Fig. 3. The same data as in Fig. 2, but averaged and binned to reduce error the determination of the baryon density based upon the recent measurement 2 bars and visual confusion. The theoretical curve is for the ACDM model of the primordial abundance of deuterium [2,9] QBh = 0.019 ± 0.0024 (95% _1 -1 with Ho = 65kms Mpc and QM = 0.4; note the goodness of fit (Figure cl); the baryon 22density is related to the baryon-to-photon ratio by pB = 6.88f7 x 10" (from [2]). courtesy of L. Knox). Physica Scripta T85

© Physica Scripta 2000

Dark Matter and Energy in the Universe sis has reduced the theoretical error significantly, and improved the accuracy of the determination of the baryon density, QBh2 = 0.019 ± 0.0012 [2].] This precise determination of the baryon density, made at a simpler time and based upon the early Universe physics of BBN, is consistent with two other measures of the baryon density, based upon entirely different physics. By comparing measurements of the opacity of the Lyman-a forest toward high-redshift quasars with high-resolution, hydrodynamical simulations of structure formation, several groups [10-12] have inferred a lower limit to the baryon density, Q^h2 > 0.015. The second test involves the height of the first acoustic peak: it rises with the baryon density (the higher the baryon density, the stronger the gravitational force driving the acoustic oscillations). Current CBR measurements are consistent with the Buries - Tytler baryon density; the MAP and Planck satellites should ultimately provide a 5% or better determination of the baryon density, based upon the physics of gravity-driven acoustic oscillations when the Universe was 300,000 yrs old (see e.g., [4]). This will be an important cross check of the BBN determination. 2.3.2. Weighing the dark matter: Qu = 0.4 ± 0.1. Since the pioneering work of Fritz Zwicky and Vera Rubin, it has been known that there is far too little material in the form of stars (and related material) to hold galaxies and clusters together, and thus, that most of the matter in the Universe is dark (see e.g. [13]). Weighing the dark matter has been the challenge. At present, I believe that clusters provide the most reliable means of estimating the total matter density. Rich clusters are relatively rare objects - only about 1 in 10 galaxies is found in a rich cluster - which formed from density perturbations of (comoving) size around lOMpc. However, because they gather together material from such a large region of space, they can provide a "fair sample" of matter in the Universe. Using clusters as such, the precise BBN baryon density can be used to infer the total matter density [14]. (Baryons and dark matter need not be well mixed for this method to work provided that the baryonic and total mass are determined over a large enough portion of the cluster.) Most of the baryons in clusters reside in the hot, X-ray emitting intracluster gas and not in the galaxies themselves, and so the problem essentially reduces to determining the gas-to-total mass ratio. The gas mass can be determined by two methods: (1) measuring the x-ray flux from the intracluster gas and (2) mapping the Sunyaev - Zel'dovich CBR distortion caused by CBR photons scattering off hot electrons in the intracluster gas. The total cluster mass can be determined three independent ways: (1) using the motions of clusters galaxies and the virial theorem; (2) assuming that the gas is in hydrostatic equilibrium and using it to infer the underlying mass distribution; and (3) mapping the cluster mass directly by gravitational lensing [15]. Within their uncertainties, and where comparisons can be made, the three methods for determining the total mass © Physica Scripta 2000

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agree (see e.g., [15]); likewise, the two methods for determining the gas mass are consistent. Mohr et al. [16] have compiled the gas to total mass ratios determined from X-ray measurements for a sample of 45 clusters; they find / g a s = (0.075 ± 0.002)/r 3/2 . Carlstrom [17], using his S-Z gas measurements and X-ray measurements for the total mass for 27 clusters, finds / g a s = (0.06 ± 0.006) h _ 1 . (The agreement of these two numbers means that clumping of the gas, which could lead to an overestimate of the gas fraction based upon the X-ray flux, is not a problem.) Invoking the "fair-sample assumption," the mean matter density in the Universe can be inferred: GM = i2B//gas = (0.3 ± 0.05) h" 1 ' 2 (Xray) = (0.25 ± 0.04) h" 1 (S - Z)

(7)

= 0.4 ± 0.1 (my summary). I believe this to be the most reliable and precise determination of the matter density. It involves few assumptions, and most of them have now been tested (clumping, hydrostatic equilibrium, variation of gas fraction with cluster mass). There is one implicit assumption: that all the cluster baryons are in stars or hot gas. For this reason some authors quote the value from this method as an upper limit to QM. 2.3.3. Supporting evidence for QM = 0.4 ± 0.1. This result is consistent with a variety of other methods that involve different physics. For example, based upon the evolution of the abundance of rich clusters with redshift, Henry [18] finds Qu = 0.45 ± 0.1 (also see, [19,20]). Dekel and Rees [21] place a low limit Qu > 0.3 (95% cl) derived from the outflow of material from voids (a void effectively acts as a negative mass proportional to the mean matter density). The analysis of the peculiar velocities of galaxies provides an important probe of the mass density averaged over very large scales (of order several hundred Mpc). By comparing measured peculiar velocities with those predicted from the distribution of matter revealed by redshift surveys such as the IRAS survey of infrared galaxies, one can infer the quantity /? = QM6/°i where b\ is the linear bias factor that relates the inhomogeneity in the distribution of IRAS galaxies to that in the distribution of matter (in general, the bias factor is expected to be in the range 0.7 to 1.5; IRAS galaxies are expected to be less biased, b\ « 1.). Recent work by Willick and Strauss [22] finds )S = 0.5 ± 0.05, while Sigad et al. [23] find /? = 0 . 9 ± 0 . 1 . The apparent inconsistency of these two results and the ambiguity introduced by bias preclude a definitive determination of Qu by this method. However, Dekel [24] quotes a 95% confidence lower bound, Qu > 0.3, and the work of Willick & Strauss seems to strongly indicate that Qu is much less than 1; both are consistent with Qu ~ 0.4. Finally, there is strong, but circumstantial, evidence from structure formation that Qu is around 0.4 and significantly greater than QB. It is based upon two different lines of reasoning. First, there is no viable model for structure formation without a significant amount of nonbaryonic dark matter. The underlying reason is simple: in a baryons-only model, density perturbations grow only from the time of decoupling, z ~ 1000, until the Universe becomes curvature Physica Scripta T85

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dominated, z ~ Q^1 ~ 20; this is not enough growth to produce all the structure seen today with the size of density perturbations inferred from CBR anisotropy. With nonbaryonic dark matter and QM 3> ®B, dark-matter perturbations begin growing at matter - radiation equality and continue to grow until the present epoch, or nearly so, leading to significantly more growth and making the observed large-scale structure consistent with the size of the density perturbations inferred from CBR anisotropy. Second, the transition from radiation domination at early times to matter domination at around 10,000 yrs leaves its mark on the shape of the present power spectrum of density perturbations, and the redshift of matter - radiation equality depends upon Qu- Measurements of the shape of the present power spectrum based upon redshift surveys indicate that the shape parameter r = Quh ~ 0.25 ± 0.05 (see e.g., [25]). For h ~ 2/3, this implies QM ~ 0.4. (If there are relativistic particles beyond the CBR photons and relic neutrinos, the formula for the shape parameter changes and Qu ^> 0.4 can be accommodated; see [26]).

end of star formation in the Universe: 80% of star formation took place at a redshift greater than unity; see Fig. 6.) While the value for Qu derived from the cluster baryon fraction also relies upon clusters, the underlying assumption is far weaker and much more justified, namely that clusters provide a fair sample of matter in the Universe. Even if mass-to-light ratios were measured in the red (they typically are not), where the light is dominated by low-mass

2.4. Mass-to-light ratios and Qu-amazingly, the glass is half full! The most mature approach to estimating the matter density involves the use of mass-to-light ratios, the measured luminosity density, and the deceptively simple equation, (pM) = (M/L)C,

(8) 8

where L = 2 Ah x 10 LBQ Mpc~ is the measured (B-band) luminosity density of the Universe. Once the average mass-to-light ratio for the Universe is determined, Qu follows by dividing it by the critical mass-to-light ratio, (M/L) crit = 1200 h (in solar units). Though it is tantalizingly simple this method does not provide an easy and reliable method of determining QM- While computing a massto-light ratio is easy, actually determining the average mass-to-light ratio is probably as hard as determining the average mass density. Based upon the mass-to-light ratios of the bright, inner regions of galaxies, (M/L)t ~ few, the fraction of critical density in stars (and closely related material) has been determined, fl* ~ (0.003 ± 0 . 0 0 1 ) / r ' (see e.g., [27]). Persic and Salucci [28] derive a similar value based upon the observed stellar-mass function. Luminous matter accounts for only a tiny fraction of the total mass density and only about a tenth of the baryons. CNOC [29,30] have done a very careful job of determining a mean cluster mass-to-light ratio, (M/L) cluster = 240 ± 50, which translates to an estimate of the mean matter density, ^cluster = 0.20 ± 0.04. Because clusters contain thousands of galaxies and cluster galaxies do not seem radically different from field galaxies, one is tempted to take this estimate of the mean matter density seriously. However, it is significantly smaller than the value I advocated earlier, Qu = 0.4 ± 0 . 1 . Which estimate is right? I believe the higher number, based upon the cluster baryon fraction, is more reliable and that we should be surprised that the CNOC number is so close, closer than we had any right to expect! After all, only a small fraction of galaxies are found in clusters and the luminosity density £ itself evolves strongly with redshift and corrections for this effect are large and uncertain. (We are on the tail Physica Scripta T85

4

3

6

8

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Tx [keV] Fig. 5. Cluster gas fraction as a function of cluster gas temperature for a sample of 45 galaxy clusters [16]. While there is some indication that the gas fraction decreases with temperature for T < 5 keV, perhaps because these lower-mass clusters lose some of their hot gas, the data indicate that the gas fraction reaches a plateau at high temperatures, _/^as =0.212 ±0.006 for h = 0.5 (Figure courtesy of Joe Mohr).

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Fig. 6. Star formation history as a function of cosmic time (adapted from [58]). The two curves bracket the estimated uncertainty and the ordinate is given by the star formation rate times cosmic time. Note that today we are on the tail end of star formation. © Physica Scripta 2000

Dark Matter and Energy in the Universe 215 stars and reflects the integrated history of star formation rather than the current star-formation rate as it does in the blue, one would still require the fraction of baryons converted into stars in clusters to be identical to that in the field to have agreement between the CNOC estimate and that based upon the cluster baryon fraction. Apparently, the fraction of baryons converted into stars in the field and in clusters is similar, but not identical. To put this in perspective and to emphasize the shortcomings of the mass-to-light technique, had one used the cluster mass-to-x-ray ratio and the x-ray luminosity density, one would have inferred fiM ~ 0-05. A factor of two discrepancy based upon optical mass-to-light ratios does not seem so bad. Enough said.

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The results Oo = 1 ± 0.2 and QM = 0.4 ± 0.1 are in apparent contradiction, suggesting that one or both are wrong. However, prompted by a strong belief in a flat Universe, theorists have explored the remaining logical possibility: a dark, exotic form of energy that is smoothly distributed and contributes 60% of the critical density [31,32]. Being smoothly distributed its presence would not have been detected in measurements of the matter density. The properties of this missing energy are severely constrained by other cosmological facts, including structure formation, the age of the Universe, and CBR anisotropy. So much so, that a smoking-gun signature for the missing energy was predicted: accelerated expansion [32]. To begin, let me parameterize the bulk equation of state of this unknown component: w = px/px. This implies that its energy density evolves as px oc R~" where n = 3(1 + w). The development of the structure observed today from density perturbations of the size inferred from measurements of the anisotropy of the CBR requires that the Universe be matter dominated from the epoch of matter - radiation equality until very recently. Thus, to avoid interfering with structure formation, the dark-energy component must be less important in the past than it is today. This implies that n must be less than 3 or w < 0; the more negative w is, the faster this component gets out of the way (see Fig. 7). More careful consideration of the growth of structure implies that w must be less than about —5 [33]. Next, consider the constraint provided by the age of the Universe and the Hubble constant. Their product, Hoto, depends the equation of state of the Universe; in particular, Hoto increases with decreasing w (see Fig. 8). To be definite, I will take ?0 = 14±1.5Gyr and i/ 0 = 65 ± 5 km s~' M p c - 1 (see e.g., [34,35]); this implies that H0t0 = 0.93± 0.13. Fig. 8 shows that w < — \ is preferred by age/Hubble constant considerations. To summarize, consistency between QM ~ 0.4 and O0 ~ 1 along with other cosmological considerations implies the existence of a dark-energy component with bulk pressure more negative than about —px/2. The simplest example of such is vacuum energy (Einstein's cosmological constant), for which w = — 1. The smoking-gun signature of a smooth, dark-energy component is accelerated expansion since qo = 0.5 + \.5wQx - 0.5 + 0.9w < 0 for w < 5

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2.6. Missing energy found! In 1998 evidence for the smoking gun signature of dark was found. It came in the form of the magnitude - redshift (Hubble) diagram for fifty-some type l a supernovae (SNe la) with redshifts out to nearly 1. Two groups, the Supernova Cosmology Project [36,37] and the High-z Supernova Search Team [38,39], working independently and using different Physica Scripta T85

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methods of analysis, each found evidence for accelerated expansion. Perlmutter et al. [36] summarize their results as a constraint to a cosmological constant (see Fig. 9), 4„

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For £ 2 M ~ 0 . 4 ± 0 . 1 , this implies QA = 0.85 ± 0.2, or just what is needed to account for the missing energy! (A simple explanation of the SN la results may be useful. If galactic distances and velocities were measured today they would obey a perfect Hubble law, o0 = Hod, because the expansion of the Universe is simply a rescaling of all distances. Because we see distant galaxies at an earlier time, their velocities should be larger than predicted by the Hubble law, provided the expansion is slowing due to the attractive force of gravity. Using SNe la as standard candles to determine the distances to faraway galaxies, the two groups in effect found the opposite: distant galaxies are moving slower than predicted by the Hubble law, implying the expansion is speeding up!) Recently, two other studies, one based upon the x-ray properties of rich clusters of galaxies [40] and the other based upon the properties of double-lobe radio galaxies [41], have reported evidence for a cosmological constant (or similar dark-energy component) that is consistent with the SN la results (i.e., QA ~ 0.7). There is another powerful test of an accelerating Universe whose results are more ambiguous. It is based upon the fact that the frequency of multiply lensed QSOs is expected to be significantly higher in an accelerating universe [42]. Kochanek [43] has used gravitational lensing of QSOs to place a 95% cl upper limit, QA < 0.66; and Waga and Miceli [44] have generalized it to a dark-energy component with negative pressure: Qx < 1.3 + 0.55w (95% cl), both results for a flat Universe. On the other hand, Chiba and Yoshii

[45] claim evidence for a cosmological constant, QA = 0.7+^, based upon the same data. From this I conclude: (1) lensing excludes QA larger than 0.8, and (2) when larger objective surveys of gravitational-lensed QSOs are carried out (e.g., the Sloan Digital Sky Survey), there is the possibility of uncovering another smoking-gun for accelerated expansion. By far, the strongest evidence for dark energy is the SN l a

data. The statistical errors reported by the two groups are smaller than possible systematic errors. Thus, the behevability of the results turns on the reliability of SNe la as one-parameter standard candles. SNe la are thought to be associated with the nuclear detonation of Chandrasekhar-mass white dwarfs. The one parameter is the rate of decline of the light curve: The brighter ones decline more slowly (the so-called Phillips relation; see [46]). The lack of a good theoretical understanding of this (e.g., what is the physical parameter?) is offset by strong empirical evidence for the relationship between peak brightness and rate of decline, based upon a sample of nearby SNe la. It is reassuring that in all respects studied, the distant sample of SNe la appear to be similar to the nearby sample. For example, distribution of decline rates and dispersion about the Phillips relationship. The local sample spans a range of metallicity, likely spanning that of the distant sample, and further, suggesting that metallicity is not an important second parameter. At this point, it is fair to say that if there is a problem with SNe la as standard candles, it must be subtle. Cosmologists are even more inclined to believe the SN la results because of the preexisting evidence for a "missing-energy component" that led to the prediction of accelerated expansion.

2.7. Cosmic concordance

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With the SN la results we have for the first time a complete and self-consistent accounting of mass and energy in the Universe (see Fig. 1). The consistency of the matter/ energy accounting is illustrated in Fig. 9. Let me explain this exciting figure. The SN la results are sensitive to the acceleration (or deceleration) of the expansion and constrain the combination | O M - QA- (Note, qo = \QM - &A, 5&M - QA corresponds to the deceleration parameter at redshift z ~ 0.4, the median redshift of these samples). The (approximately) orthogonal combination, QQ — Qu+ &A is constrained by CBR anisotropy. Together, they define a concordance region around Qo ~ 1, QM ~ 1/3, and QA ~ 2/3. The constraint to the matter density alone, Qu — 0.4 ± 0 . 1 , provides a cross check, and it is consistent with these numbers. Cosmic concordance!

But there is more. We also have a consistent and well motivated picture for the formation of structure in the Universe, ACDM. The ACDM model, which is the cold dark n 7\ 11 N 1 1 1 1 1 1 INI 1 1 1 II 1 II 1 f matter model with QB ~ 0.05, QCDM ~ 0-35 and QA ~ 0.6, is 2.5 a very good fit to all cosmological constraints: large-scale 0 .5 1.5 structure, CBR anisotropy, age of the Universe, Hubble conM stant and the constraints to the matter density and Fig. 9. Two-
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can be turned around to infer £2gas at the time clusters formed, redshifts z ~ 0 — 1, Qgash2 =fgasQMh2

0.4

0.5 0.6 0.7 H0 / 100 k m s~! Mpc-1

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Fig. 10. Constraints used to determine the best-fit CDM model: PS = large-scale structure + CBR anisotropy; AGE = age of the Universe; CBF = cluster-baryon fraction; and HQ = Hubble constant measurements. The best-fit model, indicated by the darkest region, has Ho ~ 60— 65kms _ 1 Mpc _ 1 and QA — 0.55 — 0.65. Evidence for its smoking-gun signature - accelerated expansion - was found in 1998 (adapted from [47]).

characteristic of the Gaussian, curvature perturbations predicted by inflation. Until 1998, /lCDM's only major flaw was the absence of evidence for accelerated expansion. Not now. 3. Three dark matter problems While stars are very interesting and pretty to look at - and without them, astronomy wouldn't be astronomy and we wouldn't exist - they represent a tiny fraction of the cosmic mass budget, only about 0.5% of the critical density. As we have known for several decades, the bulk of the matter and energy in the Universe is dark. The present accounting defines three dark matter/energy problems; none is yet fully addressed. 3.1. Dark baryons By a ten to one margin, the bulk of the baryons are dark and not in the form of stars. With the exception of clusters, where the "dark" baryons exist as hot, X-ray emitting intracluster gas, the nature of the dark baryons is not known. Clusters only account for around 10% or so of the baryons in the Universe [49], and the (optically) dark baryons elsewhere, which account for 90% or more of all the baryons, could take on a different form. The two most promising possibilities for the dark baryons are diffuse hot gas and "dark stars" (white dwarfs, neutron stars, black holes or objects of mass around or below the hydrogen-burning limit). I favor the former possibility for a number of reasons. First, that's where the dark baryons in clusters are. Second, the cluster baryon fraction argument © Physica Scripta 2000

= 0.023 (£2M/0.4)(/!/0.65)1/2 .

(10)

That is, at the time clusters formed, the mean gas density was essentially equal to the baryon density (unless Quh1^2 is very small), thereby accounting for the bulk of baryons in gaseous form. Third, numerical simulations suggest that most of the baryons should still be in gaseous form today [11,50]. There are two arguments for dark stars as the baryonic dark matter. First, the gaseous baryons not associated with clusters have not been detected. Second, the results of the microlensing surveys toward the LMC and SMC [51] are consistent with about one-third of our halo being in the form of half-solar mass white dwarfs. I find neither argument compelling; gas outside clusters will be cooler ( r ~ 1 0 5 - 1 0 6 K ) and difficult to detect, either in absorption or emission. There are equally attractive explanations for the Magellanic Cloud microlensing events (e.g., self lensing by the Magellanic Clouds, lensing by stars in the spheroid, or lensing due to disk material that, due to flaring and warping of the disk, falls along the line of sight to the LMC; see [52-56]). The white-dwarf interpretation for the halo has a host of troubles: Why haven't the white dwarfs been seen [57]? The star formation rate required to produce these white dwarfs - close to 100yr _ 1 Mpc~ 3 - far exceeds that measured at any time in the past or present (see [58]). Where are the lower-main-sequence stars associated with this stellar population and the gas, expected to be 6 to 10 times that of the white dwarfs, that didn't form into stars [59]? Finally, there is evidence that the lenses for both SMC events are stars within the SMC [60-62] and at least one of the LMC events is explained by an LMC lens. The SMC/LMC microlensing puzzle can be stated another way. The lenses have all the characteristics of ordinary, low-mass stars (e.g., mass and binary frequency). If this is so, they cannot be in the halo (they would have been seen); the puzzle is to figure out where they are located. 3.2. Cold dark matter The second dark-matter problem follows from the inequality Qu — 0.4 » QB — 0.05: There is much more matter than there are baryons, and thus, nonbaryonic dark matter is the dominant form of matter. The evidence for this very profound conclusion has been mounting for almost two decades. This past year, the Burles-Tytler deuterium measurement anchored the baryon density and allowed the cleanest determination of the matter density, through the cluster baryon fraction, making the case for nonbaryonic matter almost airtight. Particle physics provides an attractive solution to the nonbaryonic dark matter problem: relic elementary particles left over from the big bang (see [63]). Long-lived or stable particles with very weak interactions can remain from the earliest moments of particle democracy in sufficient numbers to account for a significant fraction of critical density (very weak interactions are needed so that their annihilations cease before their numbers are too small). Physica Scripta T85

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The three most promising candidates are a neutrino of mass 30 eV or so (or J2i mv,~ 30 eV), an axion of mass 10~ 5±1 eV, and a neutralino of mass between 50GeV and 500 GeV. All three are motivated by particle-physics theories that attempt to unify the forces and particles of Nature. The fact that such particles can also account for the nonbaryonic dark matter is either a big coincidence or a big hint. Further, the fact that these particles interact with each other and ordinary matter very weakly, provides a simple and natural explanation for dark matter being more diffusely distributed. At the moment, there is significant circumstantial evidence against neutrinos as the bulk of the dark matter. Because they behave as hot dark matter, structure forms from the top down, with superclusters fragmenting into clusters and galaxies [64], in stark contrast to the observational evidence that indicates structure formed from the bottom up. (Hot + cold dark matter is still an outside possibility, with Qv ~ 0.15 or less; see [26,65]) Second, the evidence for neutrino mass based upon the atmospheric [66] and solar-neutrino [67,68] data suggests a neutrino mass pattern with the tau neutrino at 0.1 eV, the muon neutrino at 0.001 eV to 0.01 eV and the electron neutrino with an even smaller mass. In particular, the factor-of-two deficit of atmospheric muons neutrinos with its dependence upon zenith angle is very strong evidence for a neutrino mass difference squared between two of the neutrinos of around 10~2eV2 [69]. This sets a lower bound to neutrino mass of about 0.1 eV, implying neutrinos contribute at least as much mass as bright stars. WOW! The question is no longer whether there is nonbaryonic dark matter, but how much nonbaryonic dark matter and in what forms. Both the axion and neutralino behave as cold dark matter; the success of the cold dark matter model of structure formation makes them the leading particle dark-matter candidates. Because they behave as cold dark matter, they are expected to be the dark matter in our own halo; in fact, there is nothing that can keep them out [70]. As discussed above, 2/3 of the dark matter in our halo - and probably all the halo dark matter - cannot be explained by baryons in any form. The local density of halo material is estimated to be 10~ 24 gcm~ 3 , with an uncertainty of slightly less than a factor of 2 [71]. This makes the halo of our galaxy an ideal place to look for cold dark matter particles! An experiment at Livermore National Laboratory with sufficient sensitivity to detect halo axions is currently taking data [72,73] and experiments at several laboratories around the world are beginning to search for halo neutralinos with sufficient sensitivity to detect them [74]. The particle dark-matter hypothesis is compelling, and more importantly, it is now being tested. Finally, while the axion and the neutralino are the most promising particle dark-matter candidates, neither one is a "sure thing." Moreover, any sufficiently heavy particle relic (mass greater than a GeV or so) will behave like cold dark matter. A host of more exotic possibilities have been suggested, from solar-mass primordial black holes produced at the quark/hadron transition (see e.g., [75,76]) that masquerade as MACHOs in our halo to supermassive (mass greater than 10 10 GeV) particles produced by nonthermal processes at the end of inflation (see e.g., [77]). Lest we Physica Scripta T85

become overconfident, we should remember that Nature has many options for the particle dark matter.

3.3. Dark energy I have often used the term exotic to refer to particle dark matter. That term will now have to be reserved for the dark energy that is causing the accelerated expansion of the Universe - by any standard, it is more exotic and more poorly understood. Here is what we do know: it contributes about 60% of the critical density; it has pressure more negative than about — p/2; and it does not clump (otherwise it would have contributed to estimates of the mass density). The simplest possibility is the energy associated with the virtual particles that populate the quantum vacuum; in this case p = —p and the dark energy is absolutely spatially and temporally uniform. This "simple" interpretation has its difficulties. Einstein "invented" the cosmological constant to make a static model of the Universe and then he discarded it; we now know that the concept is not optional. The cosmological constant corresponds to the energy associated with the quantum vacuum. However, there is no sensible calculation of that energy (see e.g., [78-80]), with estimates ranging from 10122 to 1055 times the critical density. Some particle physicists believe that when the problem is understood, the answer will be zero. Spurred in part by the possibility that cosmologists may have actually weighed the vacuum (!), particle theorists are taking a fresh look at the problem (see e.g., [81,82]). Sundrum's proposal, that the energy of the vacuum is close to the present critical density because the graviton is a composite particle with size of order 1 cm, is indicative of the profound consequences that a cosmological constant has for fundamental physics. Because of the theoretical problems mentioned above, as well as the checkered history of the cosmological constant, theorists have explored other possibilities for a smooth, component to the dark energy (see e.g., [33]). Wilczek and I pointed out that even if the energy of the true vacuum is zero, as the Universe as cooled and went through a series of phase transitions, it could have become hung up in a metastable vacuum with nonzero vacuum energy [83]. In the context of string theory, where there are a very large number of energy-equivalent vacua, this becomes a more interesting possibility: perhaps the degeneracy of vacuum states is broken by very small effects, so small that we were not steered into the lowest energy vacuum during the earliest moments. Vilenkin [84] has suggested a tangled network of very light cosmic strings (also see, [85]) produced at the electroweak phase transition; networks of other frustrated defects (e.g., walls) are also possible. In general, the bulk equationof-state of frustrated defects is characterized by w = —N/3 where N is the dimension of the defect (N = \ for strings, = 2 for walls, etc.). The SN la data almost exclude strings, but still allow walls. An alternative that has received a lot of attention is the idea of a "decaying cosmological constant", a termed coined by the Soviet cosmologist Matvei Petrovich Bronstein in 1933 [86]. (Bronstein was executed on Stalin's orders in 1938, presumably for reasons not directly related to the cosmological constant; see [87].) The term is, of course, © Physica Scripta 2000

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While we have many urgent questions, we can see a flood of precision cosmological and laboratory data coming that will help to answer these questions: High-resolution maps of CBR anisotropy (MAP and Planck); large redshift surveys (SDSS and 2dF); more SN la data; experiments to directly detect halo axions and neutralinos; more microlensing data (MACHO, EROSII, OGLE, AGAPE, and super MACHO); accelerator experiments at Fermilab and CERN, searching for the neutralino and its supersymmetric friends and further evidence for neutrino and its equation of motion by (see e.g., [88]) mass, and at the KEK and SLAC B-factories, revealing more about the nature of CP violation; and nonaccelerator '<j> + 3HJ>+V(k) = 0. (12) experiments that will shed further light on neutrino mass, The basic idea is that energy of the true vacuum is zero, but particle dark matter, new forces, and the nature of gravity. These are exciting times in cosmology! not all fields have evolved to their state of minimum energy. This is qualitatively different from that of a metastable vacuum, which is a local minimum of the potential and is classically stable. Here, the field is classically unstable Acknowledgments This work was supported by the DoE (at Chicago and Fermilab) and by the and is rolling toward its lowest energy state. Two features of the "rolling-scalar-field scenario" are NASA (at Fermilab by grant NAG 5-7092). I wish to thank the organizers, L. Bergstrom, P. Carlson, and C. Fransson, both for their graciousness as worth noting. First, the effective equation of state, hosts and for putting on such a fine meeting. 2 2 w — (5 (j) — V)/(\ (j) + V), can take on any value from 1 to — 1. Second, w can vary with time. These are key features that may allow it to be distinguished from the other References possibilities. In fact, there is some hope that more SNe 1. Dodelson, S. and Turner, M. S., Phys. Rev. D 42, 3372 (1992). la will be able to do this and perhaps even permit the rec2. Buries, S., Nollett, K., Truan, J. and Turner, M. S., Phys. Rev. Lett. 82, onstruction of the scalar-field potential [89]. 4176 (1999). 3. Kamionkowski, M., Spergel, D. N. and Sugiyama, N., Astrophys. J. The rolling scalar field scenario (aka mini-inflation or Lett., 426, L57 (1994). quintessence) has received a lot of attention over the past 4. Spergel, D. N., this volume (2000). decade [90-96]. It is an interesting idea, but not without 5. Lineweaver, C , Astrophys. J. Lett. 505, 69 (1998). its own difficulties. First, one must assume that the energy 6. Wilkinson, D. T., this volume (2000). of the true vacuum state (4> a t the minimum of its potential) 7. Copi, C. J., Schramm, D. N. and Turner, M. S., Science 267,192 (1995). 8. Hata, N. et al, Phys. Rev. Lett. 75, 3977 (1995). is zero; i.e., it does not address the cosmological constant 9. Tytler, D., this volume (2000). problem. Second, as Carroll [97] has emphasized, the scalar field is very light and can mediate long-range forces. This 10. Meiksin, A. and Madau, P., Astrophys. J. 412, 34 (1993). 11. Rauch, M. et al, Astrophys. J. 489, 7 (1997). places severe constraints on it. Finally, with the possible 12. Weinberg, D. et al, Astrophys. J. 490, 546 (1997). exception of one model [93], none of the scalar-field models 13. Trimble, V., Ann. Rev. Astron. Astrophys. 25, 425 (1987). address how fitsinto the grander scheme of things and 14. White, S. D. M. et al, Nature 366, 429 (1993). 15. Tyson, J. A., this volume (2000). why it is so light (m ~ 10 - 3 3 eV). an oxymoron; what people have in mind is making vacuum energy dynamical. The simplest realization is a dynamical, evolving scalar field. If it is spatially homogeneous, then its energy density and pressure are given by

4. Concluding remarks 1998 was a very good year for cosmology. We now have a plausible and complete accounting of matter and energy in the Universe; in ACDM, a model for structure formation that is consistent with all the data at hand; and the first evidence for the key tenets of inflation (flat Universe and adiabatic density perturbations). One normally conservative cosmologist has gone out on a limb by stating that 1998 may be a turning point in cosmology as important as 1964, when the CBR was discovered [98]. We still have important questions to address: Where are the dark baryons? What is the dark matter? What is the nature of the dark energy? What is the explanation for the complicated pattern recipe for our universe: neutrinos (0.3%), baryons (5%), cold dark matter particles (35%) and dark energy (60%)? Especially puzzling is the ratio of dark energy to dark matter: because they evolve differently with time, the ratio of dark matter to dark energy was higher in the past and will be smaller in the future; only today are they comparable. WHY NOW? © Physica Scripta 2000

16. Mohr, J., Mathiesen, B. and Evrard, A. E. Astrophys. J. 517, 627 (1998). 17. Carlstrom, J., this volume (2000). 18. Henry, P., BAAS 31, 1301H (1999). 19. Bahcall, N. and Fan, X., Astrophys. J. 504, 1 (1998). 20. Bahcall, N., this volume (2000). 21. Dekel, A. and Rees, M., Astrophys. J. Lett. 422, LI (1994). 22. Willick, J. and Strauss, M., Astrophys. J. 507, 64 (1998). 23. Sigad, et al, Astrophys. J. 495, 516 (1998). 24. Dekel, A., Ann. Rev. Astron. Astrophys., 32, 371 (1994). 25. Peacock, J. and Dodds, S., Mon. Not. R. Astron. Soc. 267, 1020 (1994). 26. Dodelson, S., Gates, E. I. and Turner, M. S., Science 274, 69 (1996). 27. Faber, S. and Gallagher, J., Ann. Rev. Astron. Astrophys. 17, 135 (1979). 28. Persic, M. and Salucci, P., Mon. Not. R. Astron. Soc. 258, 14p (1992). 29. Carlberg, R. et al, Astrophys. J. 436, 32 (1996). 30. Carlberg, R. et al, Astrophys. J. 478, 462 (1997). 31. Turner, M. S., Steigman, G. and Krauss, L., Phys. Rev. Lett. 52, 2090 (1984); Peebles, P. J. E. Astrophys. J. 284, 439 (1984). 32. Turner, M. S., Physica Scripta T36, 167 (1991). 33. Turner, M. S. and White, M., Phys. Rev. D 56, R4439 (1997). 34. Chaboyer, B. et al, Astrophys. J. 494, 96 (1998). 35. Freedman, W., this volume (2000). 36. Perlmutter, S. et al, Astrophys. J. 517, 565 (1999). 37. Goobar, A., this volume (2000). 38. Riess, A. et al, Astron. J. 116, 1009 (1998). 39. Schmidt, B. et al, Astrophys. J. 507, 46 (1998). Physica Scripta T85

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40. Mohr, J. et al., in preparation (2000). 41. Guerra, E. J., Daly, R. A. and Wan, L., Astrophys. J. submitted (astro-ph/9807249) (1998). 42. Turner, E. L„ Astrophys. J. Lett. 365, L43 (1990). 43. Kochanek, C , Astrophys. J. 466, 638 (1996). 44. Waga, I. and Miceli, A. P. M. R., Phys. Rev. D 59, 103507 (1999). 45. Chiba, M. and Yoshii, Y. Astrophys. J. 510, 42 (1999). 46. Phillips, M. M , Astrophys. J. Lett. 413, L105 (1993). 47. Krauss, L. and Turner, M. S., Gen. Rel. Grav. 27, 1137 (1995). 48. Ostriker, J. P. and Steinhardt, P. J., Nature 377, 600 (1995). 49. Turner, M. S., in Critical Dialogues in Cosmology, ed. N. Turok (World Scientific, Singapore 1997), p. 555. 50. Ostriker, J. P., this volume (2000). 51. Spiro, M., this volume (2000). 52. Sahu, K. C , Nature 370, 265 (1994). 53. Evans, W., Gyuk, G., Turner, M. S. and Binney, J., Astrophys. J. Lett. 501, L45 (1998). 54. Gates, E. I., Gyuk, G., Holder, G. and Turner, M. S., Astrophys. J. Lett. 500, 145 (1998). 55. Zaritsky, D. and Lin, D. N. C , Astron. J. 114, 254 (1997). 56. Zhao, H.-S., Mon. Not. R. Astron. Soc. 294, 139 (1998). 57. Graff, D. S., Laughlin, G. and Freese, K., Astrophys. J. 499, 7 (1998). 58. Madau, P., this volume (2000). 59. Fields, B., Mathews, G. J. and Schramm, D. N., Astrophys. J. 483, 625 (1997). 60. Alcock, C. et al, Astrophys. J. 518, 44 (1999). 61. EROS Collaboration, Astron. Astrophys. 332, 1 (1998). 62. EROS Collaboration, Astron. Astrophys. 337, L17 (1998). 63. Ellis, J.E., this volume (2000). 64. White, S. D. M., Frenk, C. and Davis, M., Astrophys. J. Lett. 274, LI (1983). 65. Gawiser, E. and Silk, J., Science 280, 1405 (1998). 66. Totsuka, Y., this volume (2000). 67. Kirsten, T., this volume (2000). 68. Bahcall, J., this volume (2000).

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69. Fukuda, Y. et al. (SuperKamiokande Collaboration), Phys. Rev. Lett. 81, 1562 (1998). 70. Gates, E. I. and Turner, M. S., Phys. Rev. Lett. 72, 2520 (1994). 71. Gates, E. I., Gyuk, G. and Turner, M. S., Astrophys. J. Lett. 449, 123 (1995). 72. van Bibber, K. et al, Phys. Rev. Lett. 80, 2043 (1998). 73. Rosenberg, L. and van Bibber, K., Phys. Rep., in press (2000). 74. Sadoulet, B., this volume (2000). 75. Jedamzik, K., Phys. Rept. 307, 155 (1998). 76. Jedamzik, K. and Niemeyer, J., Phys. Rev. Lett. 80, 5481 (1998). 77. Kolb, E.W., this volume (2000). 78. Zel'dovich, Ya.B., JETP 6, 316 (1967). 79. Bludman, S. and Ruderman, M. Phys. Rev. Lett. 38, 255 (1977). 80. Weinberg, S., Rev. Mod. Phys. 61, 1 (1989). 81. Harvey, J., Phys. Rev. D 59, 026002 (1999). 82. Sundrum, R., JHEP 9907, 001 (1999). 83. Turner, M. S. and Wilczek, F., Nature 298, 633 (1982). 84. Vilenkin, A., Phys. Rev. Lett. 53, 1016 (1984). 85. Spergel, D. N. and Pen, U.-L., Astrophys. J. Lett. 491, L67 (1997). 86. Bronstein, M.P., Phys. Zeit. der Sowjetunion, 3, 73 (1933). 87. Kragh, H., "Cosmology and Controversy", (Princeton Univ. Press, Princeton, NJ, 1996). 88. Turner, M. S., Phys. Rev. D 28, 1243 (1983). 89. Huterer, D. and Turner, M. S., Phys. Rev. D 60, 081301 (1999). 90. Freese, K. et al, Nucl. Phys. B, 287, 797 (1987). 91. Ozer, M. and Taha, M. O., Nucl. Phys. B 287, 776 (1987). 92. Ratra, B. and Peebles, P. J. E., Phys. Rev. D 37, 3406 (1988). 93. Frieman, J., Hill, C , Stebbins, A. and Waga, I., Phys. Rev. Lett. 75, 2077 (1995). 94. Coble, K., Dodelson, S. and Frieman, J. A., Phys. Rev. D 55, 1851 (1996). 95. Caldwell, R., Dave, R. and Steinhardt, P.J., Phys. Rev. Lett. 80, 1582 (1998) 96. Steinhardt, P. J., this volume (2000). 97. Carroll, S., Phys. Rev. Lett. 81, 3067 (1998). 98. Turner, M. S., Pub. Astron. Soc. P a c , 111, 264 (1999).

© Physica Scripta 2000

Physica Scripta.Vol. T85, 221-230, 2000

Particle Candidates for Dark Matter John Ellis Theoretical Physics Division, CERN CH-1211 Geneva 23, Switzerland Received December 11, 1998; accepted August 2, 1999

PACS Ref: 12.10.Dm, 12.60.Jv, 14.60.Pq, 14.80.Ly, 95.85.Ry, 96.40.Pq, 98.80.Cq, 98.80.Es

Abstract Some particle candidates for dark matter are reviewed in the light of recent experimental and theoretical developments. Models for massive neutrinos are discussed in the light of the recent atmospheric-neutrino data, and used to motivate comments on the plausibility of different solutions to the solar neutrino problem. Arguments are given that the lightest supersymmetric particle should be a neutralino % and accelerator and astrophysical constraints used to suggest that 50GeV <mx< 600 GeV. Minimizing the fine tuning of the gauge hierarchy favours Qxh2 ~ 0.1. The possibility of superheavy relic particles is mentioned, and candidates from string and M theory are reviewed. Finally, the possibility of non-zero vacuum energy is discussed: its calculation is a great opportunity for a quantum theory of gravity, and the possibility that it is time dependent should not be forgotten.

It is of interest for our subsequent discussion to review in more detail [2] the upper limit (3) on the mass of a cold dark matter particle. One may write Qx =

Px 2x

mxnx Wh2T2'

(4)

where 7b ~ 2.73K is the present effective temperature of the cosmic microwave background radiation. To a good approximation, the comoving number density has remained essentially constant since the freeze-out temperature 7} at which annihilation terminated: T2 mp

«£ nx(T{) nx(T{)((Jznn(XX)vx)=-~-±.. 3 T ~ a 1. Introduction There is a wide range of possible masses for a candidate dark matter particle [1]. If it was once in thermal equilibrium, its number density n is almost independent of its mass, as long as the latter is <£ 1 MeV. Thus, for neutrinos with masses in the range discussed in the next section, nv ~ constant and hence pv = mvnv oc mv, leading to

°.»-&y-nm-

a)

where h is the present Hubble expansion rate in units of 100 kms _ 1 Mpc _ 1 and pc is the critical density. There is a region of masses for neutrinos, or similar particles, between 0(100)eV and 0(3)GeV where Qv > 1. Above this range, the cosmological density px of a particle X may be sufficiently suppressed by mutual annihilation: m

Px = xnx

• nx oc

a{xx^...y

(2)

that Qx % 1. In the case of neutrinos or similar particles, annihilation is particularly efficient when mv ~ ntz/2, leading to a local minimum in the relic density with Qv «; 1. Above this mass, the relic density in general rises again, depending on the behaviour of the annihilation cross-section. Thus one has three regions where the relic density of a neutrino or similar particle may be of cosmological interest: Qv ~ 1: 30eV, ~ fewGeV, or ~ lOOGeV.

(3)

The first of these possibilities corresponds to hot dark matter, and is of active interest for neutrinos as discussed in Section 2, whilst the other two correspond to cold dark matter, and are of interest for supersymmetric particles, as discussed in Section 3. The middle region has been essentially excluded by LEP, as we discuss in more detail later, whilst the third may be in the realm of the LHC. © Physica Scripta 2000

(5)

where a is the cosmological scale factor and mp ~ 1.2 x 1019 GeV is the Planck mass. For relic particles of interest, one typically finds that mx/T( ~ 20 to 30 and hence Qxh1-

10" (GzUXX^x)

10" \T0mP)

<
n

(XX)vx)TeVl

(6)

The TeV scale emerges naturally as the geometric mean of T0~2J3K and m P ~ 1 . 2 x l 0 1 9 GeV! Using the dimensional estimate (Ozrm(XX)vx) ~ C • a2/mx, where a is a generic coupling strength and C is a model-dependent numerical coefficient, we find mx

(16aVC)A/^TeV,

(7)

which yields the expectation that mx ;S 1 TeV. We will discuss in Section 3 the extent to which this argument implies that the LHC is "guaranteed" to discover a cold dark matter particle. For the moment, we just point out that the above discussion assumed that the particle was at one time in thermal equilibrium, which is not necessarily the case. Counter-examples include the axion discussed here by Turner [3], and the superheavy relic particles discussed here by Kolb [4], and in Section 4 of this talk. 2. Neutrinos Particle physics experiments [5] tell us that neutrino masses must be much smaller than those of the corresponding charged leptons and quarks: mVe <3.5eV vs. me ~ 0.511 MeV mVfi < 160keV vs. wM ~ 105 MeV,

(8)

wVt < 18MeV vs. mT ~ 1.78 GeV. Physica Scripta T85

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John Ellis

There is another difference between neutrinos and other particles, namely that only left-handed neutrinos are known to exist, produced by the familiar V — A charged current:

following approximate eigenstates and eigenvalues: v

whereas both quarks and charged leptons have both left- and right-handed states ^L,R, ^L,R- Thus "Dirac" masses mP coupling them are possible: gh{THAI=y2,AL=0fRfh =^mf = gni[(0\HAI=y2,AL=0\0),

(10)

where the quantum numbers of the Standard Model Higgs have been indicated explicitly. The following puzzles then arise. If right-handed neutrinos VR exist, why are the mv (10) so small? If vR do not exist, can neutrinos acquire masses: wv^0? Most particle theorists believe that particles should be massless only if there is an exact gauge symmetry to guarantee this, as is the case for the photon and gluon. However, there is no such candidate symmetry to guarantee mv = 0, so most of us expect mv ^ 0. The fact that the v have no exact gauge quantum numbers enables them to have Majorana masses mM, since / R in (12) is replaced byf[ =f^C, where C is an antisymmetric matrix:

0 ( ^ ) v

R

:

W L

= ^ M - K )

t

= ^ ) , (15)

(9)

Jn = eyM(i - y 5 K + mv(l - 75K + ^ O - TsK,

L + V

R + ° ^ 7 F ) V L : W R = M.

M

Thus there are naturally very light neutrinos if M 3> mw, as would be expected in a GUT with M = 0(MQUJ). If the Dirac masses of the different neutrino generations scale like the corresponding quark or charged-lepton masses, one would expect ml or m2, m,

(16)

M,

and hence mVe «.mv

(17)

«mVt,

if the heavy Majorana mass matrix is diagonal, and if its eigenvalues M, are approximately the same. As an example, putting mj5 ~ 100 GeV for the third generation, one finds mV3 - 0 . 1 eV if M 3 ~ 1014 GeV. Before the advent of the atmospheric neutrino data [7-9], one might also have expected small neutrino mixing angles, analogous to those for quarks, which originate from Dirac mass matrices. M, This and the prejudice (17) are not necessarily supported m™v< vL = m„M.,T > i C v L = mfv • v . (11) h L by the atmospheric neutrino data [7-9], which suggest a large This is not possible for quarks and charged leptons, because mixing angle: sin2 26^ > 0.8 and a mass-squared difference both <7L • #L and 1-L • i-h have Qem ^ 0, whilst q^ • q^ also Amltmo ~ (1°~ 2 t o 10" 3 )eV 2 . We also recall that the solar has non-zero colour. Such a Majorana neutrino mass (11) neutrino data [10,11] favour one of three possible solutions: would require lepton-number violation: AL = 2 and weak the large-angle MSW solution with isospin AI = 1, which does contradict any sacred theoretical principles, but is not provided by the Higgs fields in the Stan- Am, lO- 5 eV 2 ,sin 2 20>O.8, (18a) solar dard Model or in the minimal SU(5) GUT. A Majorana mass term (13) could in principle be provided by a suitable the small-angle MSW solution with "exotic" Higgs field: 5 2 2 2 (18b) Am,solar 10~ eV ,sin 29~ 10" ,

gHw#d/=i,/iL=2VL • VL =*• mv = gH„WHAI=iiAL=2\0},

(12) or vacuum oscillations with

as appears in some non-minimal GUT models such as SO(10) with a 126 Higgs representation. However, there are difficulties with sich a scenario, since one would have expected a AI = 1 Majoron particle which should have been detected via invisible Z° decays. Alternatively, one can obtain m™ from a non-renormalizable coupling to the Standard Model Higgs:

M

(HAI=l/2Vh) • (HM^I/2VL)

•

rrC =gs-

M

— (13)

The question then arises: what could be the origin of the large mass parameter M? The most natural possibility is the exchange of a massive singlet neutrino field, traditionally called a right-handed neutrino vR, though this nomenclature is somewhat anachronistic. Given a VR, a Dirac mass m® =gHvv(0|^/=i/2|0) is also possible, and one arrives at the famous see-saw mass matrix [6]: 0 (o
I - M (VL,VR)

m

D

(14)

\mmrf v M Mj\vRJOne would expect the M^ entries in (16) to be of order mq or mt < {Mw), so the matrix diagonalization yields the 1

Physica Scripta T85

Am:solar

lO-' u eV 2 ,sin z 20£O.6.

(18c)

Which of these solutions might be favoured by postsuper-Kamiokande models of neutrino masses, and how plausible is neutrino hot dark matter of cosmological and astrophysical significance, which would require w v > l eV for at least one neutrino species? A first comment is that large neutrino mixing is (in retrospect) not at all implausible [12]. For one thing, it is not at all necessary that m ° oc mq or mi. For example, in a specific flipped SU (5) model used earlier to discuss quark and lepton masses, we found [13]

m:

(0(n)

0(1)

o '

V o

o

oil),

cm o(i) o

(19)

where r\ is a small parameter, which would yield at least one large neutrino mixing angle. Moreover, there is no good reason why the heavy singlet Majorana mass matrix should be (approximately) diagonal in the same basis. For example, in models with a U(l) flavour symmetry one expects matrix elements My a £"<+"; M G U T

(20) © Physica Scripta 2000

Particle Candidates for Dark Matter

Fig. 1. Light-neutrino mixing in a simple 2 x 2 model (22), as a function of the ratios of mass-matrix elements [13].

-10

223

Fig. 3. Possible renormalization-group running of the light-neutrino mixing angle [13].

sin 023 without fine-tuning of the ratios b/d, c/d, for example if b/d ~ 0.5 and c/d ~ 1.5. Moreover, the light-neutrino mixing angle may be enhanced by renormalization-group effects between the GUT and electro weak scales [15]:

0.25

b/d=0.5\

U -20

d ,2.mi+m22 16nl j (sinz 023) = -2(sin^ 023)(cos'! 023) x (A| - XA2) m 33 - m22 (23)

Pi

-30

as seen in Fig. 3. The renormalization-group enhancement is particularly important if A2 — X\ is large, as may happen at large tan /? in supersymmetric models, and/or if the diagonal 0.5 entries w 33 and W22 are almost equal. We infer that the hier1.5 archy (21) is plausible if Am2oXM ~ 10~5 eV2 as in the largeFig. 2. Light-neutrino mass hierarchy in a simple 2 x 2 model (22), as a funcand small-angle MSW solutions (18a,b). However, it seems tion of the ratios of mass-matrix elements [13]. difficult to stretch the hierarchy (24) as far as would be required by the vacuum solution (18c) to the solar neutrino problem. Our present inclination is therefore to favour where e is an expansion parameter. If n, = — rij, one gets a large off-diagonal entry My = 0{\) x MGUT, which looks (24) (lo -1 to I0"15)ev, m2 ~ l r r V v at least as plausible as having n, = 0, which would be mv required to give a large diagonal entry M„ = C ( M G U T ) - Such and the remaining question is whether the large- or a large off-diagonal entry would be another source of large small-angle MSW solution (18a,b) is to be favoured. In neutrino mixing. our specific flipped SU(5) model [13], we found not only large With so many different features contributing to the light off-diagonal entries in m® (22), but also large off-diagonal neutrino mass matrix, there is no obvious symmetry or other entries (23) in the singlet-neutrino mass matrix: reason why near-degeneracy w, ~ m; » |w, — mj\ should occur. Therefore, we expect that there may be a hierarchy of masses: (25) -40

m3 ~ y/Amltmo

>m2~

y^m 2 0 iar > mx.

(21)

If this is indeed the case, even the heaviest light neutrino would weigh < 0.1 eV, and would not be of great astrophysical and cosmological interest [14]. A corollary question is the extent to which large mixing is compatible with a large neutrino mass hierarchy. To address this, we may consider a simple two-state model [13] m

*=(d

c)

(22)

and diagonalize it to obtain eigenvalues m2 and m3 and a mixing angle 023- We see in Figs. 1 and 2 that a hierarchy mj, = O(10)w2 is quite compatible with large mixing © Physica Scripta 2000

where all of the indicated non-zero entries might a priori be comparable in magnitude. We therefore find the large-angle MSW solution (18a) to be at least as plausible as the small-angle MSW solution (18b), perhaps even more so. A final comment concerns the magnitudes of the entries in (28): we estimate their natural order of magnitude to be O(10 1 3 ± 2 ) GeV, corresponding to wV3 = 10 0±2 eV, overlapping comfortably with the desired range (21). In my view, although the super-Kamiokande data [7] make a very strong case for atmospheric neutrino oscillations, particle physicists will not be completely convinced until they have verified the effect using a beam with controllable energy, spectrum and flavour content, as provided by an accelerator neutrino beam. One such project, Physica Scripta T85

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John Ellis

the K2K experiment [16], is to start taking data in 1999, and another, the NUMI/MINOS project [17], has also been approved. A third project, neutrino Gran Sasso (NGS), has been studied by a joint CERN-INFN working group [18], and is ready for construction if funding is approved. The K2K project is at relatively low energy, insufficient to produce the x lepton. The NUMI/MINOS and NGS projects are at higher energies, and should be able to reach down to Am2 ~ 10~3 eV2, though without much margin. Nevertheless, in a few years we may expect to know whether accelerator experiments confirm the super-Kamiokande results. However, as mentioned earlier, it currently seems unlikely that neutrinos will turn out to be an important component of the dark matter [14].

3. The lightest supersymmetric particle The motivation for supersymmetry at an accessible energy is provided by the gauge hierarchy problem [19], namely that of understanding why m w <§: mp, the only candidate for a fundamental mass scale in physics. Alternatively and equivalently, one may ask why (?F ~ g V ^ w ^ ^ N = 1/wp, where Mp is the Planck mass, expected to be the fundamental gravitational mass scale. Or one may ask why the Coulomb potential inside an atom is so much larger than the Newton potential, which is equivalent to why e2 = 0(1) » WpWe/wp, where mp,e are the proton and electron masses. One might think it would be sufficient to choose the bare mass parameters: mw <£ mp. However, one must then contend with quantum corrections, which are quadratically divergent: <5<w = OQA2

(26)

which is much larger than m w , if the cutoff A representing the appearance of new physics is taken to be 0(mp). This means that one must fine-tune the bare mass parameter so that it is almost exactly cancelled by the quantum correction (26) in order to obtain a small physical value of mw. This seems unnatural, and the alternative is to introduce new physics at the TeV scale, so that the correction (26) is naturally small. At one stage, it was proposed that this new physics might correspond to the Higgs boson being composite [20]. However, calculable scenarios of this type are inconsistent with the precision electroweak data from LEP and elsewhere. The alternative is to postulate approximate supersymmetry [21], whose pairs of bosons and fermions produce naturally cancelling quantum corrections: 5m\l = oQ\ml-ml\

(27)

that are naturally small: Sm2^ < m\, if |m|-m||
(28)

There are many other possible motivations for supersymmetry, but this is the only one that gives reason to expect that it might be accessible to the current generation of accelerators and in the range (7) expected for a cold dark matter particle. Physica Scripta T85

The minimal supersymmetric extension of the Standard Model (MSSM) has the same gauge interactions as the Standard Model, and the Yukawa interactions are very similar: Xd QDCH + X\LECH + XUQ UCH + /iHH

(29)

where the capital letters denote supermultiplets with the same quantum numbers as the left-handed fermions of the Standard Model. The couplings ld,i,u give masses to down quarks, leptons and up quarks respectively, via distinct Higgs fields H and H, which are required in order to cancel triangle anomalies. The new parameter in (22) is the bilinear coupling /i between these Higgs fields, that plays a significant role in the description of the lightest supersymmetric particle, as we see below. The gauge quantum numbers do not forbid the appearance of additional couplings XLLEC + X'LQDC + X UCDCDC

(30)

but these violate lepton or baryon number, and we assume they are absent. One significant aspect of the MSSM is that the quartic scalar interactions are determined, leading to important constraints on the Higgs mass, as we also see below. Supersymmetry must be broken, since supersymmetric partner particles do not have identical masses, and this is usually parametrized by scalar mass parameters m2, |0,-|2, gaugino masses \ Ma Va • Va and trilinear scalar couplings •AijkXyk4>ijk- These are commonly supposed to be inpu from some high-energy physics such as supergravity or string theory. It is often hypothesized that these inputs are universal: mo, = mo, Ma = M\/2, Ayk = A, but these assumptions are not strongly motivated by any fundamental theory. The physical sparticle mass parameters are then renormalized in a calculable way: m

l = ml + Cm2l/2, Ma = (-^— )mi/2

(31)

\aGUT/

where the C, are calculable coefficients [22] and MSSM phenomenology is then parametrized by fi, mo, m\/z, A and tan/? (the ratio of Higgs v.e.v.'s). Precision electroweak data from LEP and elsewhere provide two qualitative indications in favour of supersymmetry. One is that the inferred magnitude of quantum corrections favour a relatively light Higgs boson [23] m/, = 66+™±10GeV

(32)

which is highly consistent with the value predicted in the MSSM: m/, < 150 GeV [24] as a result of the constrained quartic couplings. (On the other hand, composite Higgs models predicted an effective Higgs mass > 1 TeV and other unseen quantum corrections.) The other indication in favour of low-energy supersymmetry is provided by measurements of the gauge couplings at LEP, that correspond to sin2 0w — 0.231 in agreement with the predictions of supersymmetric GUTs with sparticles weighing about 1 TeV, but in disagreement with non-supersymmetric GUTs that predict sin2 0 W ~ 0.21 to 0.22 [25]. Neither of these arguments provides an accurate estimate of the sparticle mass scales, however, since they are both only logarithmically sensitive to mo and/or m\/2© Physica Scripta 2000

Particle Candidates for Dark Matter The lightest supersymmetric particle (LSP) is expected to be stable in the MSSM, and hence should be present in the Universe today as a cosmological relic from the Big Bang [26]. This is a consequence of a multiplicativelyconserved quantum number called R parity, which is related to baryon number, lepton number and spin: 3B+L+2S

It is easy to check that R = +1 for all Standard Model particles and R = — 1 for all their supersymmetric partners. The interactions (33) would violate R, but not a Majorana neutrino mass term or the other interactions in SU(5) or SO(10) GUTs. There are three important consequences of R conservation: (i) sparticles are always produced in pairs, e.g., pp -> qgX, e + e~ ->• Ji+Jr, (ii) heavier sparticles decay into lighter sparticles, e.g., q -*• qg, ji-*- /iy, and (iii) the LSP is stable because it has no legal decay mode. If such a supersymmetric relic particle had either electric charge or strong interactions, it would have condensed along with ordinary baryonic matter during the formation of astrophysical structures, and should be present in the Universe today in anomalous heavy isotopes. These have not been seen in studies of H, He, Be, Li, O, C, Na, B and F isotopes at levels ranging from 10~" to 10~29 [27], which are far below the calculated relic abundances from the Big Bang: > 1(T6 to 10

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for relics with electromagnetic or strong interactions. Except possibly for very heavy relics, one would expect these primordial relic particles to condense into galaxies, stars and planets, along with ordinary bayonic material, and hence show up as an anaomalous heavy isotope of one or more of the elements studied. There would also be a "cosmic rain" of such relics [28], but this would presumably not be the dominant source of such particles on earth. The conflict with (37) is sufficiently acute that the lightest supersymmetric relic must presumably be electromagnetically neutral and weakly interacting [26]. In particular, I believe that the possibility of a stable gluino can be excluded. This leaves as scandidates for cold dark matter a sneutrino v with spin 0, some neutralino mixture of y/H°/Z with spin 1/2, and the gravitino G with spin 3/2. LEP searches for invisible Z° decays require mj ^ 4 3 GeV [29], and searches for the interactions of relic particles with nuclei then enforce wj > few TeV [30], so we exclude this possibility for the LSP. The possibility of a gravitino G LSP has attracted renewed interest recently with the revival of gauge-mediated models of supersymmetry breaking [31], and could constitute warm dark matter if MA ~ 1 keV. In this talk, however, I concentrate on the y/H /Z° neutralino combination x, which is the best supersymmetric candidate for cold dark matter. The neutralinos and charginos may be characterized at the tree level by three parameters: m\/2, n and tan/?. The lightest neutralino % simplifies in the limit m\/2 -*• 0 where it becomes essentially a pure photino y, or n ->• 0 where it becomes essentially a pure higgsino H. These possibilities are excluded, however, by LEP and the FNAL Tevatron collider [29]. From the point of view of astrophysics and © Physica Scripta 2000

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(b) Fig. 4. Regions of the (ji, Mi) plane in which the supersymmetric relic density may lie within the interesting range 0.1 < Qxh2 < 0.3 [32].

cosmology, it is encouraging that there are generic domains of the remaining parameter space where Qxh2 ~ 0.1 to 1, in particular in regions where % is approximately a U(l) gaugino B, as seen in Fig. 4 [32]. Purely experimental searches at LEP enforce w x J>30 GeV, as seen in Fig. 5 [33]. This bound can be strengthened by making various theoretical assumptions, such as the Physica Scripta T85

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and LEP should eventually be able to establish or exclude mx up to about 50 GeV. As seen in Fig. 7, LEP has already explored almost all the parameter space available for a Higgsino-like LSP, and this possibility will also be thoroughly explored by LEP [33]. Should one be concerned that no sparticles have yet been seen by either LEP or the FNAL Tevatron collider? One way to quantify this is via the amount of fine-tuning of © Physica Scripta 2000

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is possible. In the past, it was thought that all the Physica Scripta T85

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E<5 x 1019 GeV, because of the reaction P + 7CMBR -*• ^ + [42]. However, no such GreisenZatsepin-Kuzmin cut-off is seen in the data! [43] The sasooj ultra-high-energy cosmic rays must originate nearby, and !'L«It= lOOfb"' — ^ should point back to any point-like sources such as AGNs. A,, = 0 , » B p - 2 , | i < 0 -A \ However, no such sources have been seen. v " Could the ultra-high-energy cosmic rays be due to the ^\ % decays of superheavy relic particles? These should be \ clustered in galactic haloes (including our own), and hence "- . ^x \ gaooo) 21 OS i x^ give an anisotropic flux [44], but there would be no obvious \ point sources. There have been some reports of anisotropies -. 2!SS "^~rrr——_ "^ in high-energy cosmic rays, but it is not clear whether they could originate in superheavy relic decays. We have analyzed [45] recently possible superheavy relic ' . - -__.. ,—-:: ;•-•••; ." 8(1500) candidates in string [46] and/or M theory. One expects Kaluza-Klein states when six excess dimensions are compactified: 10 —>• 4 or 11 —> 5, which we call hexons. However, , / - ~ — ^ these are expected to weigh > 1016 GeV, which may be too heavy, and there is no particular reason to expect hexons WOOD) \ to be metastable. In M theory, one expects massive states associated with a further compactification: 5 -* 4 dimensions, which we call pentons. Their mass could be ~ 1013 GeV, which would be suitable, but there is again 8(5001 \.l no good reason to expect them to be metastable. We are left with bound states from the hidden sector of string IM theory, which we call cryptons [46]. These could also have masses ~ 1013 GeV, and might be metastable for much the same reason as the proton in a GUT, decaying via higher-dimensional multiparticle operators. For example, m0 (GeV) in a flipped SU(5) model we have a hidden-sector Fig. 11. The region of the (mo, mi/2) plane accessible to sparticle searches at SU(4) x SO(10) gauge group, and the former factor confines four-constituent states which we call tetrons. Initial the LHC [40]. studies [46,45] indicate that the lightest of these might well have a lifetime >10 17 y, which would be suitable for the cosmologically-preferred region of MSSM parameter space' decays of superheavy dark matter particles. Detailed could be explored by the LHC [40], as seen in Fig. 11, but it simulations have been made of the spectra of particles of their decay now seems possible that there may be a delicate region close produced by the fragmentation products [47,48], and the ultra-high-energy cosmic-ray data to the upper bound (38). This point requires further study. are consistent with the decays of superheavy relics weighing ~ 1012 GeV, as seen in Fig. 12 [48]. Issues to be resolved here include the roles of supersymmetric particles in the fragmentation cascades, and the relative fluxes of 7, v and 4. Superheavy Relic Particles p among the ultra-high-energy cosmic rays. The expectation (7), exemplified by the MSSM range (41), was based on the assumption that the cold dark matter particles were at one time in thermal equilibrium. As dis- 5. Vacuum energy cussed here by Kolb [4], much heavier relic particles are possible if one invokes non-thermal production mechanisms. Data on large-scale structure [49] and high-redshift Non-thermal decays of inflatons in conventional models of supernovae [50] have recently converged on the suggestion cosmological inflation could yield Qx ~ 1 for mx ~ 1013 that the energy of the vacuum may be non-zero. In my view, GeV. Preheating via the parametric resonance decay of this represents a wonderful opportunity for theoretical the inflaton could even yield Qx ~ 1 for mx ~ 1015 GeV. physics: a number to be calculated in the Theory of EveryOther possibilities include a first-order phase transition at thing including quantum gravity. The possibility that the the end of inflation, and gravitational relic production vacuum energy may be non-zero may even appear more induced by the rapid change in the scale factor in the early natural than a zero value, since there is no obvious symmetry Universe [41]. It is therefore of interest to look for possible or other reason known why it should vanish. In the above paragraph, I have used the term vacuum experimental signatures of superheavy dark matter. energy rather than cosmological constant, because it may One such possibility is offered by ultra-high-energy cosmic not actually be constant. This option has been termed rays. Those coming from distant parts of the Universe quintessence here by Steinhardt [51], who has discussed a (D>100Mpc) are expected to be cut off at an energy classical scalar-field model that is not strongly motivated 1 There has recently been progress in implementing the constraints from the ab- by the Standard Model, supersymmetry or GUTs, though something similar might emerge from string theory. I prefer sence of charge and colour-breaking minima [39]. m.„ (GeV)

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to think that a varying vacuum energy might emerge from a quantum theory of gravity, as the vacuum relaxes towards an asymptotical value (zero?) in an infinitely large and old Universe. We have recently given [52] an example of one such possible effect which yields a contribution to the vacuum energy that decreases as l/t2. This is compatible with the high-redshift supernova data, and one may hope that these could eventually discriminate between such a possibility and a true cosmological constant.

References 1. Griest, K. Jungman, G. and Kamionkowski, M., Phys. Rept. 267, 195 (1996). 2. Dimopoulos, S. Phys. Lett. B246, 347 (1990). 3. Turner, M. S. talk at this meeting; see also astro-ph/9811364, astro-ph/9811366, astro-ph/9811447, astro-ph/9811454. 4. Kolb, E. W. talk at this meeting; see also hep-ph/9810362. 5. Particle Data Group, Particle Data Group, C. Caso et al, Eur. Phys. J. C3, 1 (1998). 6. Yanagida, T., Proc. Workshop on the Unified Theory and the Baryon Number in the Universe (KEK, Japan, 1979); Slansky, R., Talk at the Sanibel Symposium, Caltech preprint CALT-68-709 (1979). 7. Super-Kamiokande collaboration, Fukuda, Y., et al., Phys. Rev. Lett. 81, 1562 (1998). 8. Kafka, T., for the Soudan II collaboration, Nucl. Phys. Proc. Suppl. 70, 340 (1999). 9. MACRO collaboration, Ambrosio, M., et al, Phys. Lett. B434, 451 (1998). 10. Bahcall, J. N., talk at this meeting; see also astro-ph/9808162; Bahcall, J. N., Krastev, P. I. and Yu. Smirnov, A., Phys.Rev. D58, 096016 (1998). 11. E. Lisi, talk at this meeting; see also Fogli, G. L., Lisi, E., Marrone, A. and Scioscia, G., Phys. Rev. D59, 033001 (1999). 12. For some recent work, see: King, S. F., Phys. Lett. B439, 350 (1998); Elwood, J. K., Irges, N. and Ramond, P., Phys. Rev. Lett. B81, 5064 (1998); Barger, V., et al, Phys. Rev. D58, 093016 (1998); Osland, P. and Vigdel, G. Phys. Lett. B438, 129 (1998); Joshipura, A. and Smirnov, A., Phys. Lett. B439, 103 (1998); Tanimoto, M., Phys. Rev. D59, 017304 (1999); Lazarides, G. and Vlachos, N., Phys. Lett. B441, 46 (1998); Barger, V., Weiler, T. and Whisnant, K., Phys. Lett. B440, 1 (1998); Pati, J., Nucl. Phys. Rev. Supp. 77, 299 (1999); Gonzalez-Carcia, M., et al., Nucl. Phys. B543, 3 (1999); Altarelli, G. and Feruglio, F., Phys. Lett. B439, 112 (1998). 13. Ellis, J., Leontaris, G., Lola, S. andNanopoulos, D. V. Eur. Phys. J. C9, 389 (1999). © Physica Scripta 2000

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14. However, combinations of structure-formation and cosmic microwave-background measurements may be sensitive to smaller neutrino masses: Tegmark, M., talk at this meeting; see also Hu, W., Eisenstein, D. J., Tegmark, M. and White, M., Phys. Rev. D59, 023512 (1999). 15. Tanimoto, M., Phys. Lett. B360, 41 (1995). 16. Oyama, Y., for the K2K collaboration, hep-ex/9803014. 17. MINOS collaboration, Abies, E., et al., Fermilab Proposal P-875 (1995). 18. Acquistapace, G., et al., CERN report 98-02. 19. Maiani, L., Proc. Summer School on Particle Physics, Gif-sur-Yvette, 1979 (IN2P3, Paris, 1980) p. 3; G 't Hooft, in: (G 't Hooft et al, eds.), "Recent Developments in Field Theories" (Plenum Press, New York, 1980); Witten, E., Nucl. Phys. B188, 513 (1981); Kaul, R. K., Phys. Lett. 109B, 19 (1982). 20. For a review, see: Farhi, E. and Susskind, L. Phys. Rep. 74C, 277 (1981); Dimopoulos, S. and Susskind, L., Nucl. Phys. B155 237 (1979); Eichten, E. and Lane, K. Phys. Lett. B90, 125 (1990); Peskin, M. E. and Takeuchi, T., Phys. Rev. D46, 381 (1992); Altarelli, G., Barbieri, R. and Jadach, S., Nucl. Phys. B369, 3 (1992). 21. Fayet, P. and Ferrara, S., Phys. Rep. 32, 251 (1997); Haber, H. E. and Kane, G. I., Phys. Rep. 117, 75 (1985). 22. See, e.g., Kounnas, C , Lahanas, A. B., Nanopoulos, D. V. and Quiros, M., Phys. Lett. 132B, 95 (1983). 23. Gruunewald, M. and Karlen, D., talks at International Conference on High-Energy Physics, Vancouver 1998, http://www.cern.ch/ LEPEWWG/misc. 24. Okada, Y., Yamaguchi, M. and Yanagida, T., Progr. Theor. Phys. 85,1 (1991); Ellis, J., Ridolfl, G. andZwirner, F., Phys. Lett. B257, 83, (1991) Phys. Lett. B262, (1991) 477; Haber, H. E. and Hempfling, R., Phys. Rev. Lett. 66, 1815 (1991); Barbieri, R., Frigeni, M. and Caravaglios, F. Phys. Lett. B258, 167 (1991); Okada, Y., Yamaguchi, M. and Yanagida, T., Phys. Lett. B262, 54 (1991). 25. Ellis, J., Kelley, S. and Nanopoulos, D. V., Phys. Lett. B249,441 (1990) and Phys. Lett. B260, 131 (1991); Amaldi, U., de Boer, W. and Furstenau, H., Phys. Lett. B260, 447 (1991); Langacker, P. and Luo, M., Phys. Rev. D44, 817 (1991). 26. Ellis, J., Hagelin, J. S., Nanopoulos, D. V., Olive, K. A. and Srednicki, M., Nucl. Phys. B238, 453 (1994). 27. Rich, J., Spiro, M. and Lloyd-Owen, J., Phys. Rep. 151, 239 (1987); Smith, P. F., Contemp. Phys. 29, 159 (1998); Hemmick, T. K., et al, Phys. Rev. D41, 2074 (1990). 28. Mohapatra, R. N. and Nussinov, S., Phys. Rev. D57, 1940 (1998). 29. Ellis, J., Falk, T., Olive, K. and Schmitt, M., Phys. Lett. B388,97 (1996) and Phys. Lett. B413, 355 (1997). 30. Klapdor-Kleingrothaus, H. V. and Ramachers, Y., Eur. Phys. J. A3, 85 (1998). 31. Giudice, G. and Rattazzi, R., hep-ph/9801271. 32. Ellis, J., Falk, T., Olive, K. and Schmitt, M. Phys. Lett. B413, 355 (1997). 33. LEP Experiments Committee meeting, Nov. 12th, 1998, http:www.cern.ch/Committees/LEPC/minutes/LEPC50.html. This source provides some preliminary updates of this and other experimental plots from papers contributed to the International Conference on High-Energy Physics, Vancouver 1998: http://ichep. triumf.ca / main. asp. 34. J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Mod. Phys. Lett. Al, (1986) 57; Giudice, G. F. and Barbieri, R. Nucl. Phys. B306, (1988) 63. 35. Chankowski, P. H., Ellis, J. and Pokorski, S., Phys. Lett. B423, (1998) 327; Chankowski, P. H., Ellis, J., Olechowski, M. and Pokorski, S., Nucl. Phys. B544, 39 (1999). 36. Chankowski, P. H„ Ellis, J., Olive, K. A. and Pokorski, S., Phys. Lett. B452, 28 (1999). 37. Olive, K. A. and Srednicki, M. Phys. Lett. B230, (1989) 78 and Nucl. Phys. B355, 208 (1991); Griest, K., Kamionkowski, M., and Turner, M. S., Phys. Rev. D41, 3565 (1990). 38. Ellis, J., Falk, T. and Olive, K. A., Phys. Lett. B444, 367 (1998). 39. Abel, S. and Falk, T., Phys. Lett. B444, 427 (1998). 40. Abdullin, S. and Charles, F., Nucl. Phys. B547, 60 (1999). 41. Kolb, E. W., Chung, D. J. H. and Riotto, A., hep-ph/9810361 and references therein. 42. Greisen, K., Phys. Rev. Lett. 16 (1966) 748; Zatsepin, G. T. and Kuzmin, V. A. Pis'ma Zh. Eksp. Teor. Fiz. 4 114 (1966). Physica Scripta T85

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43. Watson, A. A., talk at this meeting; Takeda, M., et al, Phys. Rev. Lett. 81, 1163 (1998) and references therein. 44. Berezinsky, V. and Mikhailov, A., Phys. Lett. B449,237 (1999); Medina Tanco, G. A. and Watson, A. A., "Dark Matter Halos and the Anisotropy of Ultra-High Energy Cosmic Rays," (November 1998). 45. Benakli, K., Ellis, J. and Nanopoulos, D. V., Phys. Rev. D59, 047301 (1999). 46. Ellis, J., Lopez, J. and Nanopoulos, D. V., Phys. Lett. B247,257 (1990); Ellis, J., Gelmini, G., Lopez, J., Nanopoulos, D. V. and Sarkar, S., Nucl. Phys. B373, 399 (1992).

Physica Scripta T85

47. Berezinsky, V., Kachelriess, M. and Vilenkin, A., Phys. Rev. Lett. 79, 4302 (1997); Berezinsky, V. and Kachelriess, M. Phys. Lett. B434, 61 (1998). 48. Birkel, M. and Sarkar, S. Astropart. Phys. 9, 297 (1998). 49. Bahcall, N., talk at this meeting; see also Bahcall, N. A. and Fan, X. H., astro-ph/9804082. 50. Riess, A. G. et al, Astron. J. 116, 1009 (1998); Perlmutter, S., et al, astro-ph/9812133. 51. Steinhardt, P. J., talk at this meeting; see also Zlatev, I., Wang, L. W. and Steinhardt, P. J., Phys. Rev. Lett. 82, 896 (1999). 52. Ellis, J., Mavromatos, N. E. and Nanopoulos, D. V., gr-qc/9810086.

© Physica Scripta 2000

Physica Scripta. Vol. T85, 231-245, 2000

Early-Universe Issues: Seeds of Perturbations and Birth of Dark Matter Edward W. Kolb NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, Illinois 60510-0500, USA and Department of Astronomy and Astrophysics, Enrico Fermi Institute, The University of Chicago, Chicago, Illinois 60637-1433, USA Received January 26, 1999; accepted August 2, 1999

PACS Ref: 98.80 Cg

Abstract The signatures of events that occurred in the early universe can be seen today on the sky in the form of large-scale structures and fluctuations in the temperature of the cosmic background radiation. If we can read the signatures, we can learn what occurred in the early universe and something about the physics involved.

1. Introduction The theme of this meeting, "Particles and the Universe" represents perhaps the most striking illustration of the true unity of science. At first glance it is not at all clear that "particles" have anything to do with "the Universe." The realm of "particle" physics is nature on the smallest scales, while cosmology is concerned with the universe on the largest scales. Although the two fields are separated by the sizes of the objects they study, they are joined by the fact that it is impossible to understand the origin and evolution of large-scale structures in the universe without understanding the "initial conditions" that led to their development. The initial data was set in the very early universe when the fundamental particles and forces acted to produce the perturbations in the cosmic density field. A complete understanding of the present structure of the universe will also be impossible without accounting for the dark component in the density field. The most likely possibility is that this ubiquitous dark component is an elementary particle relic from the early universe. The study of the structure of the present universe may reveal insights into events that occurred in the early universe, and hence, into the nature of the fundamental forces and particles at an energy scale far beyond the reach of terrestrial accelerators. Perhaps the early universe, the ultimate particle accelerator, will provide the first glimpse of physics at the scale of Grand Unified Theories (GUTs), or even the Planck scale. The fact that "particles" and "the universe" is joined, may well be best expressed by a quote from the great American naturalist and conservationist John Muir (1838-1914): When one tugs at a single thing in nature, he finds it attached to the rest of the universe. In this paper I will concentrate on two events which occurred in the early universe. The first is the generation of perturbations in the density field during an early period of rapid expansion known as cosmic inflation. The second is the genesis of dark matter. The record of these events is written in the arrangement of galaxies, galaxy clusters, © Physica Scripta 2000

and the imperfections in the isotropy of the cosmic microwave background radiation. If we really understood particle physics we could predict the nature of those patterns. If we really knew how to read the story in the structures, we would learn something about particle physics. The story is there on the sky, patiently waiting for our wits to become sharp enough to read it. 2. Inflation One of the striking features of the CBR temperature fluctuations is that they appear to be noncausal. The CBR fluctuations were largely imprinted at the time of last-scattering, about 300 000 years after the bang. However, there are fluctuations on length scales much larger than 300 000 light years! How could a causal process imprint fluctuations on scales larger than the light-travel distance since the time of the bang? The answer is inflation. Sometime during the early evolution of the universe the expansion must have been such that a > 0. From Einstein's equations, the "acceleration" is related to the energy density and the pressure as a oc —{p + 3p). Therefore, an acceleration (positive a) requires an unusual equation of state with p + 3/7 < 0. This is referred to as "accelerated expansion" or "inflation." Including the inflationary phase, our best guess for the different epochs in the history of the universe is given in Table I. There is basically nothing known about the stringy phase, if indeed there was one. The earliest phase we have information about is the inflationary phase. As we shall see, the information we have is from the quantum fluctuations during inflation, which were imprinted upon the metric, and can be observed as CBR fluctuations and the departures from homogeneity and isotropy in the matter distribution, e.g., the power spectrum. Inflation was wonderful, but all good things must end. A lot of effort has gone into studying the end of inflation (For Table I. Different epochs in the history of the universe and the associated tempos of the expansion rate. tempo

passage

age

P

P

P + 3p

prestissimo presto

string dominated vacuum dominated (inflation) matter dominated radiation dominated matter dominated

< 10"43s ~ 10"38s

?

9

9

Pv

~P\

—

~ 10" s < 104yr

HjA

0

> 10V

Pmatter

allegro andante largo

36

r

+ 7*73 + 0 +

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Edward W. Kolb

a review, see Kofman et al. [1]) It was likely that there was a inflaton potential energy density, fluctuations in the inflaton brief period of matter domination before the universe field lead to fluctuations in the energy density. Because of became radiation dominated. Very little is known about this the rapid expansion of the universe during inflation, these period after inflation. Noteworthy events that might have fluctuations in the energy density are frozen into superoccurred during this phase include baryogenesis, phase Hubble-radius-size perturbations. Later, in the radiation transitions, and generation of dark matter. We do know that or matter-dominated era they will come within the Hubble the universe was radiation dominated for almost all of the radius as if they were noncausal perturbations. first 10 000 years. The best evidence of the radiationThe spectrum and amplitude of perturbations depend dominated era is primordial nucleosynthesis, which is a relic upon the nature of the inflaton potential. Mukhanov [2] of the radiation-dominated universe in the period 1 second has developed a very nice formalism for the calculation to 3 minutes. The earliest picture of the matter-dominated of density perturbations. One starts with the action for gravera is the CBR. ity (the Einstein-Hilbert action) plus a minimally-coupled One important issue is how does one imagine a universe scalar inflatonfield4>\ dominated by vacuum energy making a transition to a matter-dominated or radiation-dominated universe. A -n\j^R-[g"\'t>U+Vi4') (1) simple way to picture this is by the action of a scalar field • / * (j) with potential V(). Let's imagine the scalar field is displaced from the minimum of its potential as illustrated in Here R is the Ricci curvature scalar. Quantum fluctuations Fig. 1. If the energy density of the universe is dominated result in perturbations in the metric tensor and the inflaton by the potential energy of the scalar field (j), known as field the inflaton, then p + 3p will be negative. The vacuum FRW <j> -*• 0 + 5 (2) •faP> o/iv energy disappears when the scalar field evolves to its gliv minimum. The amount of time required for the scalar field FRW T where g£ is the Friedmann-Robertson-Walker metric, v to evolve to its minimum and inflation to end (or even more and 0O(O is the classical solution for the homogeneous, useful, the number of e-folds of growth of the scale factor) isotropic evolution of the inflaton. The action describing can be found by solving the classical field equation for the dynamics of the small perturbations can be written as the evolution of the inflaton field