Cosmology

3 downloads 0 Views 2MB Size Report
matibre-rayonnement) dans l'Univers primordial rEsulte en une prediction unique .... discovery of the cosmic microwave background radiation in 1964. Gamow ...
C. R. Acad. Sci. Paris, t. 327, S~rie II b, p. 829-840, 1999

Cosmologie/Cosmology

Version fram;aise abr~g$e - Pour le ~ Big Bang Le module cosmologique du ~>repose sur des fondements solides. Le plus impressionnant est peut-&re la mesure remarquablement pr6cise de la temp6rature du rayonnement cosmique de fond de ciel, dont le spectre ne diff'ere pas appr6ciablement d'un spectre de corps noir, avec une pr6cision meilleure qu'un centi~me de pourcentage. Une origine de type du rayonnement (avant le dEcouplage matibre-rayonnement) dans l'Univers primordial rEsulte en une prediction unique concernant la dependance angulaire de l'amplitude des fluctuations de temperature. ,h, l'Echelle du ~ superhorizon ~, les fluctuations sont invariantes, puisque l'inflation s'est produite. Mais, une fois ~ l'intErieur de l'horizon en expansion, la gravit6 joue par le dEveloppement de l'amplitude des fluctuations, en dEclenchant des mouvements particuliers, et en causant des ondes acoustiques (de compression) dans le fluide de matiSre-rayonnement. La ~ derni6re diffusion ~r se produit en un moment bien dEfini, celui auquel l'Univers a vu la recombinaison de l'hydrogSne se faire, g une Epoque d'environ 300 000 ans aprbs le ~ Big Bang ~. Alors, l'horizon sous-tendait un angle d'environ 1°. Cet angle dEfinit l'Echelle angulaire au-dessus de laquelle l'Echelle des fluctuations est si grande, qu'elle ne peut 8tre due qu'h la prediction acausale de l'Univers primordial postErieur h l'inflation. L'inflation est en fait comprise comme 6tant la source de ces fluctuations, en raison du renforcement des fluctuations quantiques. Alors que l'inflation se produit, les fluctuations quantiques se dEveloppent de faqon exponentielle dans l'Echelle physique vers des 6chelles de plus en plus grandes, allant nettement au-del~ de l'horizon. L'inflation se termine ; et l'expansion usuelle de Friedmann recommence alors, et c'est seulement bien plus tard que ces fluctuations pEnbtreront l'horizon. Des theories inflationnaires simples prEdisent une distribution des fluctuations invariante d'Echelle. L'invariant de courbure, qui est de fait l'amplitude de la fluctuation de densitE au passage de l'horizon, est inddpendant de l'Echelle de l'horizon. A l'intErieur d'une barre d'erreur Evidemment assez grande, due surtout ~t la variance cosmique, cette prediction est 830

consistante avec les donnres relatives aux fluctuations ~t grande 6chelle du rayonnement cosmologique de fond de ciel, telles qu'elles ont 6t6 drtectres par le satellite.COBE en 1992. Cependant, h des 6chelles angulaires petites, la physique intervient pour drvelopper encore ces fluctuations en quelque sorte fossiles. Des ondes sonores ont le temps de se drvelopper dans le plasma de rayonnement et de baryons, antrrieurement au drcouplage de la matibre et du rayonnement. Les fluctuations de densit6 peuvent alors &re reprrsentres comme des ondes acoustiques sur le demier horizon de diffusion du modble cosmologique standard. L'amplification en rdsultant pour les modes Doppler et acoustiques a pour effet net de produire un accroissement de l'anisotropie, aux 6chelles de l'ordre du degrr. Cela en fait a maintenant 6t6 mesurr, en conformit6 avec les prrdictions du plus simple des modules de ~. Les donnres sur les pics acoustiques, drduits de la distribution des fluctuations observres, confirment jusqu'h prrsent le premier pic. Cependant, plusieurs pics sont drtectables avant que le lissage radiatif n'efface la structure angulalre, au-dessous d'une 6chelle angulaire de quelques minutes d'arc. La drtection de ces pics par de futures exprriences constituera une vrrification de la throrie inflationnaire, en ce qui concerne l'origine des fluctuations de densitr. N'importe quelle origine causale rrsulterait en un mrlange des modes et les phases ne seraient plus cohrrentes, si bien qu'aucune structure angulaire ne serait drtectable. Cependant, l'inflation prrdit une origine acausale pour les fluctuations, qui interagissent seulement apr~s avoir travers6 l'horizon, et grn~rent de ce fait une srrie d'ondes dont les phases sont cohrrentes au moment de la traversre de l'horizon, c'est-h-dire au moment o?a les demibres interactions par diffusion du rayonnement sur la mati~re interviennent. La cohrrence rrsulte en une srrie de pics dans le spectre de puissance du rayonnement. I1 est prrmatur6 actuellement de dire que l'inflation a 6t6 confirmre, mais les exprriences futures MAP et PLANCK, darts le domaine des micro-ondes, foumiront la base de donnres qui permettra d'examiner h fond les prrdictions de la cosmologie inflationnaire.

1. Introduction The Big Bang is the modem version of the creation myth. It is a theory and a model that can be firmly anchored in the context of modem observations. Above all, it is a simplification of the astronomical complexities of nature that reveals an elegant underlying symmetry and beauty. In the best traditions of science, it is a highly predictive theory, and one that has been systematically refined as the observational data base grows. Consider first the simplification aspects of the theory. We assume that the laws and constants of physics are unchanged throughout cosmic time. Einstein' s theory of gravitation and the Planck-inspired quantum theory tell us all that we need to know to describe space and time. The Universe is observed to be highly inhomogeneous. Yet if one filters the observed structure, homogeneity appears once the filter bandpass exceeds a few tens of megaparsecs. The Universe is approximately homogeneous. It is also isotropic, there being no apparent preferred direction. Of course, these observations are made from our vantage point. The cosmological principle generalizes the appearance of homogeneity and isotropy to a set of observers distributed through the Universe. One motivation behind the cosmological principle is the need to dethrone us as being privileged observers from the vantage point of the earth. The Universe is assumed to be statistically isotropic at all times for sets of fundamental observers. One consequence is that the Universe must be statistically homogeneous.

831

Observations of the cosmic microwave background have vindicated the cosmological principle, originally applied by Einstein in his first derivation of a static Universe. The cosmic microwave background is isotropic to approximately 1 part in 105. It originates from the early Universe, and demonstrates that the matter distribution satisfied a similar level of homogeneity during the first million years of cosmic history. There is one aspect of cosmological precepts that bears on the role of the observer, indeed the existence of observers. The necessary congeniality of the Universe for the origin and evolution of life imposes severe restrictions within the set of all possible Universes. One could pursue the Goldilocks analogy: the Universe can be neither too hot nor too cold, too young or too small, too smooth or too chaotic. We seem to have acquired a well-designed Universe. Is there any physics to be determined from our heritage? Some cosmologists apply anthropic arguments to learn about the initial moments of the Universe. Perhaps if one can constrain the range of possibilities one can make more precise predictions about the outcome as well as try to obtain a better understanding of the initial regime at the edge of known physics. The more scientific response is that the ultimate Theory of Everything will provide the answer. A theory of the beginning will account for the initial conditions of the observed Universe.

2. Evidence for the Big Bang The Big Bang rests on four rather impressive pillars. Two of these burst into the world with such dramatic effect that they have been called the "golden moments" of cosmology. These were the discoveries of the expansion of the Universe and of the cosmic microwave background radiation. Both were predicted. The expanding Universe is described by the Friedmann-Lema~tre cosmological model. This elegant and simple theory, derived independently by Friedmann in 1924 and Lemattre in 1927, predicted the expansion of the Universe that was observed by Hubble in 1929. According to this model, usually referred to as the Big Bang theory, the rate of deceleration of a sufficiently large shell of matter in the Universe is determined by the total mass contained within the shell. Isotropy leads to a linear expansion law, v o¢ a ( t ) . Only such a linear function of a ( t ) preserves shapes, otherwise the expansion is anisotropic. One deduces that Vexpansion -----n0 d where H0 is Hubble's constant, named for the proportionality constant in the linear law that Hubble discovered in 1929. The break-through in cosmology that is the hallmark of the modern era commenced with the discovery of the cosmic microwave background radiation in 1964. Gamow and two of his collaborators predicted that the Universe should now be at a temperature of a few degrees above absolute zero, as a relic of its hot past. The fossil radiation from the early Universe is expected to be that of a blackbody at this low temperature, whose radiation is detectable as a diffuse glow in the sky at microwave frequencies. Arno Penzias and Robert Wilson discovered the radiation serendipitously in 1964 (Penzias and Wilson, 1965). The spectrum was measured in 1990 to be that of a perfect blackbody, with spectral deviations of at most one part in 104. Only the very early Universe could have provided the near-perfect furnace in the sky that could have generated such a spectral energy distribution, and this must have occurred within the first month of the Big Bang. We have a direct glimpse of radiation that probes the Universe after its birth. Another prediction was that the microwave background radiation should not be completely uniform but contain fluctuations that are the fossilized seeds of galaxies and galaxy clusters. The fluctuations were first detected in 1992. 2.1. H u b b l e ' s Constant H o Determination of the Hubble constant relies on using Type Ia supernovae as standard candles that can be seen to great distances. These are used in conjunction with Cepheid variable stars as the more

832

local distance measure. Only with supernovae can one then go to more distant galaxies beyond the Cepheid range and use galaxies at great enough distances that there is minimal uncertainty from the effect of local streaming motions on H0. The calibration of the supernova luminosity at maximum light has been established for supernovae in six nearby galaxies for which Cepheid variables have been monitored using the Hubble Space Telescope. A major source of distance uncertainty is removed once Cepheid variables are studied in host galaxies where supernovae have been found. The Cepheid luminosity varies with period, and the known period-luminosity relation enables one to infer a distance. Two rival groups of astronomers assign different weights to the calibration procedure, which is compounded by subtle intrinsic variations between the supernovae themselves. The influx of new Cepheid data has brought the two determinations of H0 into reasonable agreement. Sandage and Tammann (Tammann, 1998) find that H 0 = 55 _+ 10 klTl'S-I'Mpc -1, whereas Freedman and collaborators argue for H 0 = 73 + 14 krn.s-l-Mpc-1, overlapping to within one sigma or so (Freedman et al., 1998). 2.2. Age of the Universe From the oldest stars, one can infer an age of 12 (+ 2) Gyr (Chaboyer, 1998), where I have averaged over several recent determinations. With a gestation period for galaxies of 1 Gyr, the true age of 15 (+ 2) Gyr can be compared with the Hubble time Ho 1, equal to 15 Gyr for H0 = 65 km.s-l.Mpc -1. The age of the Universe is approximately equal to H o 1 if deceleration is unimportant and the Universe is open, a model which is thereby favored. In a fiat Universe, deceleration is significant, and the age is 2/3 H 0, barely reconcilable with stellar ages even when one adopts the lowest feasible stellar age scale. 2.3. Primordial synthesis of the light elements While Lema~tre was convinced that the Big Bang had a hot beginning, it was Gamow who quantified the temperature history of the Universe as a function of the density and realized that during the first minutes the density and temperature could be high enough to include thermonuclear cookery of elements. Nuclear equilibrium in the first second guaranteed a mixture of neutrons and protons, with about one neutron for every five protons. Protons collide at high enough speed to overcome the electromagnetic repulsion between a proton pair and can fuse together to produce, along with neutrons, deuterium, or heavy hydrogen and helium, along with trace abundances of other light elements. The Universe has a high enough density during the first few minutes that essentially all the neutrons are converted into helium: the result is roughly one helium nucleus (containing 2 neutrons and two protons) for every 8 protons. Stars synthesize helium but can only have generated a small fraction of what is seen in the Sun. Twenty-five percent of the mass of stars is helium, and the Big Bang accounts for the precise amount. The agreement with the observed helium and deuterium abundances (Schramm and Turner, 1998) provides an important verification of a prediction of the Big Bang theory. 2.4. Density of the Universe The mean density of the Universe 1"2is measured relative to the density of an Einstein de Sitter, or spatially fiat, Universe. The luminous matter in the Universe amounts to a density in the form of stars 12.---0.005. The contribution in the form of intergalactic gas is determined by measuring absorbing clouds toward distant quasars, and provides a contribution that amounts to 12gas= 0.04. An indirect method appeals to primordial nucleosysthesis of light elements such as helium, deuterium and lithium in the first minutes of the Big Bang. Fewer baryons would result in underproduction of helium and overproduction of deuterium; a higher baryon fraction would result

833

in overproduction of helium and underproduction of deuterium. The helium abundance as well as the abundances of deuterium and lithium, are fossils from the Big Bang that measure the baryon density. Comparison with observed abundances yields the total baryon density, which must include both stars and intergalactic gas, as g2B = 0.04, implying that one has accounted for most of the baryons in the early Universe. Where the baryons are today is more of a mystery. The gas density at high redshift measured in diffuse absorbing clouds of hydrogen seen towards distant quasars (Lyman alpha clouds) amounts to (28 ---0.04, in agreement with the indirect inference from nucleosynthesis, whereas the density of luminous matter, mostly in stars, is much less. The local baryons may be mostly in the form of diffuse, hot intergalactic gas, hitherto undetected. The rotation curves of galaxies suggest that the dark halos contain about ten times more matter than is seen in stars. It is usually conjectured that this dark matter is nonbaryonic. However another possibility is that most baryons are in the form of dark compact baryonic objects in our galactic halo, for which there is some evidence from gravitational microlensing studies. The total matter density is ~¢'2m = 0 . 3 , and is mostly outside galaxies. Dynamical measurements provide a robust approach to measuring dark matter on large scales. These methods arise from "weighing" large regions of the Universe, notably galaxy clusters and superclusters. Determination of galaxy peculiar motions enables one to estimate the local mass that is responsible for the overdensity in the Universe that drives these motions. Without such an overdensity, one would have only a uniform Hubble flow. Studies of galaxy clusters measure the mass on scales of about 1 Mpc, but one can use galaxy pecular motions to measure the mass density on scales of up to 30 Mpc. One infers that f2m is at least 0.2 and probably less than 0.5. Most of the matter in the Universe is therefore not baryonic. The favored candidate for the bulk of the dark matter is that of a weakly interacting elementary particle. Such particles, generally massive, act as cold dark matter: they aggregate freely under gravity on all scales as density fluctuations amplify by gravitational instability. Since only the component of mass density is measured that is non-uniform over the sampled region, one could have an even higher density in the form of a uniform dark matter component: this would be in the form of hot dark matter. However hot dark matter can only amount to at most twenty percent of the critical density, or else structure formation would be adversely affected.

3. Foundations of the Big Bang theory The basic idea underlying a mathematical model for the Universe is that space should be described by a metric. The cosmological principle forces uniformity and isotropy, and allows one to define a universal time t. The metric has: d s 2 = C2 d t 2 _ d l 2

where dl is the spatial line element. The homogeneity and isotropy of space-like surfaces allowed Robertson and Walker to derive a line element which scales conformally with a scale factor a ( t ) and has three possible forms corresponding to the spatial geometry. Three geometries are allowable, corresponding to spherical space (k = 1), Euclidean or flat space (k = 0), and hyperbolic space (k = - 1). The scale factor can be computed from Einstein's field equations. One defines the proper or physical distance as d = ra( t ), where r is the coordinate or comoving distance. The frequency of light from a distant galaxy is reduced due to the Doppler shift by the expansion of space. The wavelength stretches out as a, and the frequency redshifts as vo~ 1/a( t ). The redshift is defined to be the factor by which the Universe has expanded between light emission and observation at to, so that

1 +z=a(to)/a(t).

834

The constant k can be thought of in Newtonian terms as the total energy of a shell (kinetic energy d 2/2 + gravitational potential energy 2 G M (a)). Einstein interprets k as curvature. An infinite k = - 1 Universe expands forever, whereas a finite k = 1 Universe must collapse. The result of appplying the cosmological principle to Einstein's equations leads to a simple equation, derived not by Einstein but by Friedmann and Lemaltre, for the scale factor: a~ _ a2 -

8~Gp

3

_

a2

One can interpret a/a as the Hubble parameter, equal to Hubble's constant at the present epoch. Once the matter density is prescribed one can solve the Friedmann-Lema~tre equation for the evolution of the scale factor of the Universe. Evaluation of the present values of the Hubble and density terms allows one to estimate the curvature of the Universe. A third parameter must also be specified, since there may be contributions to the density that are not in the form of ordinary matter. In particular, the vacuum may have an intrinsic energy density that is important over cosmological scales. Einstein first introduced such a concept in the form of the cosmological constant A as a repulsion force to explain why a static Universe would avoid collapse. He regretted having introduced the cosmological constant after the expansion of the Universe was discovered, to little avail. The cosmological constant remains as a parameter of cosmology that is equivalent to a contribution f2n ( - A / 3 H i ) to the density parameter that tends to accelerate the Universe. In the absence of a cosmological constant the deceleration of the Universe is specified by ~2,,/2. More generally it is the combination 12m/2 - 12A that must be positive if the Universe is decelerating. The redshift of a galaxy moving away from us with the expansion of space is the fractional amount by which the light is shifted to the red because of the Doppler shift. Supernovae of type Ia (the most luminous variety) have been detected out to redshift 0.8. Utilization of these supernovae as standard candles provides a measure of 12m and f2A in combination. Approximately 50 Type-Ia supernovae have been studied out to this distance. The high redshift supernovae appear to be dimmer than expected in the open Friedman-Lema~tre model Universe. If the Universe is assumed to be flat ((2~ = 0), the currently available data (Perlmutter et al., 1998; Schmidt et al., 1998) yield ~r~ m ~ 0.7. Because the luminosity distance to a distant supernova varies with a changing dependence on ~'-~m and £2~ as a function of redshift, one can in principle measure both 12m and 12~ independently. There is considerable uncertainty because of possible systematic errors. Grey extinction by interstellar or intergalactic dust would dim distant supernovae. Evolution could also result in a dimming if for example the nature of the energy release systematically changed because of a systematic change in the white dwarf mass. The only argument against the latter possibility is that it seems contrived. In any case, this effect will be detectable because the effect of A diminishes at z > 1. It seems that 12A is very likely to be positive, thereby implying that the Universe is accelerating. From independent tests one can measure the curvature of the Universe, £2~. The most promising of these is provided by the angular correlations of the cosmic microwave background. These directly map the angular scale on the sky of the maximum sound horizon of the Universe, which behaves as a geometry-invariant measuring rod. Its angular projection, on degree scales, probes geometry, as will be described below. 4. Structure formation In 1981, a major advance occurred in theoretical cosmology. A new theory appeared for the very early Universe that removed much of the mystery about the initial conditions of the Big Bang. Inflation of the Universe occurred, an unexpected growth in the scale factor at an epoch of t = 10-35 s. The

835

temperature, 1015 GeV, corresponds to the grand unification scale. As the Universe cools, the symmetry breaks, and there is a first order phase transition. The release of latent heat provides a source of vacuum energy. While the vacuum energy dominates, the Friedmann-Lema~tre equation reduces to a2/a2= constant, or a ~ e Ht where H= ( 8 7 ~ G p v a c ] 3 ) 1/2. Physical scales which are proportional to a ( t ) also increase exponentially. In this way one accounts for the size and the flatness of the Universe. Any curvature scale is inflated to a scale much larger than the present horizon. Quantum fluctuations are present on the horizon scale prior to inflation, and are imprinted after inflation onto macroscopic scales containing galaxy masses and even the horizon mass. These provide the resolution of the origin of the observed fluctuations. Indeed structure formation requires primordial density fluctuations. In a cold static medium, density fluctuations grow at an exponential rate, fip/p ~ exp( t ~ ). This means that one can have sizable growth from infinitesimal fluctuations. However once the medium is expanding, the growth rate becomes a power law in time, c~p/p o~ ill3 ~ a for the expanding Universe during the matterdominated regime. Primordial fluctuations are essential. Effective growth only occurs during the matter-dominated regime, during which the Universe expands by a factor of about 104. Hence primordial fluctuations Op/p ,,~ 10-4 are required. Inflation provides density fluctuations, and predicts the distribution of strengths with mass scale. The prediction is that the fluctuations have equal strength on all scales. However, inflation does not predict the fluctuation amplitude. The breakthrough came with discovery of fluctuations in the temperature of the cosmic microwave background radiation.

5. Temperature fluctuations One sees the microwave background radiation in the standard cosmological model back to a redshift of about 1 000, at which the last scattering occurred of photons by ionized gas. Subsequently, the diffuse matter in the Universe is too sparse to induce any further scatterings. By studying the variations between different regions on the sky of the cosmic background temperature, one is in effect mapping the primordial density fluctuations in the early Universe. Density fluctuations are required to be present, since all large-scale structure in the Universe has formed by gravitational instability of primordial density fluctuations. Indeed, this leads to the prediction that associated temperature fluctuations A T / T , , ~ 10-5 must be induced at the last scattering of the radiation. Dark matter plays an important role in this prediction: its dominance enhances fluctuation growth and so assures that the temperature fluctuations are approximately a factor of 10 smaller than they would be in the simplest models of a purely baryonic Universe. Temperature fluctuations were predicted in 1967 but were not discovered until 1992. They are inevitable in the Big Bang theory in order for a proper accounting to be given of the large-scale structures in the Universe. There are three components that contribute to the fluctuations (Hu et al., 1997). Gravitational potential fluctuations redshift or blueshift photons emerging from the last scattering with electrons. These last scatterings occurred approximately 300 000 years after the Big Bang: only then was the electron density sufficiently high. Intrinsic fluctuations arise from the adiabatic compression of matter and radiation. Doppler shifts arise because the scattering matter acquires rotation due to the gravitational pull of fluctuations. All of the fluctuations add up to give a rich prediction of temperature variations over a broad range of angular scales, from 90 ° to a few minutes of arc. These primary fluctuations are imprinted on the last scattering surface together with an additional contribution that arises because in a low density Universe the gravitational potential varies slowly with time. Photons traversing a time-varying gravity field experience either a differential blueshift or a redshift as they traverse density fluctuations.

836

The temperature fluctuations were measured by the DMR experiment on the COBE satellite, to be within a factor of 2 of the predicted level. The COBE satellite measured fluctuations over angular scales of 10°-90 ° , and found approximately equal power spread over the different scales. This is the prediction of inflationary cosmology, the standard model of the very early Universe. On a scale of about 1°, however, the inflationary model predicts that there should be an enhancement in the form of a distinct peak in the temperature fluctuations. One can depict the growth of density fluctuations as being controlled by the interplay between gravity, which provides the driving force, and radiation pressure, which in the early Universe when scattering occurred, tends to disperse the fluctuations. After the last scattering epoch, density fluctuations grow freely by gravitational instability and eventually form galaxies and large-scale structures. A natural scale is imprinted on the distribution of primordial temperature fluctuations that corresponds to the horizon scale of the Universe at last scattering. The horizon scale is the distance light has travelled since the Big Bang. Only on this scale, and on smaller scales, at last scattering do density fluctuations have time to respond at the sound (or acoustic) speed to the release of pressure as photons first travel freely. There is a resulting enhancement in temperature fluctuations, by about a factor of 3, that occurs on the angular scale of ~ 1 ° subtended by the horizon of last scattering. The finite duration of the period over which scattering ceases means that on smaller angular scales, the fluctuations in temperature are erased. There have now been more than 20 experiments that have mapped the cosmic microwave background at angular resolutions as high as a few arcminutes. The "acoustic peak" predicted in the temperature fluctuations has probably been seen at an angular scale that is near 1°. This angular scale directly measures the curvature of the Universe. The angular projection of any physical scale imprinted in the early Universe depends only on geometry and the angular size of the associated feature scales as f2 x/2 Last scattering occurred at an epoch of about 300 000 yr, and the horizon size was 300 000 light years. The angular scale of the acoustic peak is simply this scale at a redshift of 1 000 seen projected on the Universe as viewed by us. One can therefore read off the curvature of the Universe from a map of the cosmic microwave background radiation. Current observations above an angular scale of 1° restrict (2~ < 1, and tentative evidence of a decline in AT/T at 0.25 ° compels I2~ to exceed 0.3 (Hancock et al., 1998). Greatly refined data relative to presently available maps is needed for a definitive measurement, and this should be forthcoming in future experiments that are planned for launch by NASA in 2001 and ESA in 2007. K

"

6. D a r k m a t t e r

Dark matter accounts for at least 90 %, and possibly for 99 % of the matter in our Universe. Elucidating its nature is one of the most urgent problems in astrophysics that has attracted the attention of astronomers and particle theorists. Its existence is predicated on our theory of gravitation. Various attempts have been made to tinker with Einstein's theory, modifying the law of gravitation to be able to account for, say, fiat rotation curves without introducing dark matter. However, such modified laws of gravity are generally ad hoc, ugly, and not self-consistent when applied to strong gravity fields or the very early Universe. The dark matter problem was posed by Fritz Zwicky in 1933. He noted that observation of galaxy velocities in the Coma cluster of galaxies implied a mass-to-light ratio of about 400 M®/L o, as compared to the more modest value of a few that seemed appropriate for galaxies composed of Sun-like stars. Remarkably, Zwicky's estimate has not changed significantly in sixty years, despite an increase in the distance scale by an order of magnitude. Dark matter is constrained by astronomical observations over scales from our local solar neighborhood up to practically the scale of the visible Universe.

837

Dark matter candidate masses span a range of some 80 orders of magnitude. One can divide this into two domains, those of elementary particle physics and astrophysics. The particle candidates range from axions, near 10-4 eV, to neutralinos, stable supersymmetric relics of mass near the SUSY symmetrybreaking scale of ,-~ 100 GeV. The generic name for such particles is WIMPs, at least in the more massive end of the range, for weakly interacting massive particles. Current accelerator bounds plus the cosmological density bound requires the lower bound on the neutralino mass to be about 40 GeV in minimal SUSY (Ellis et al., 1997). An estimate of the upper limit in minimal SUSY in order to avoid excessively large g2 is likely to be about (Ellis et al., 1998) 600 GeV. The astrophysical candidates are generically referred to as MACHOs (massive astrophysical compact halo objects), in part because diffuse gas is not a significant contributor to the dark matter content of the Universe, and in part because the logical and likely environment for dark baryons is in galaxy halos, surrounding the luminous baryons. At least two distinct arguments point to the existence of a substantial amount of baryonic dark matter. Primordial nucleosynthesis of such light elements as 2H, 4He, 3He, and 7Li requires an amount of baryons that significantly exceeds the mass in luminous galaxies, in the form of stars and interstellar matter. Studies of quasar absorption lines measure intergalactic gas clouds, both of high hydrogen column density (damped Lyman alpha clouds) and low column density (Lyman alpha forest). Modelling (Rauch et al., 1999), especially of the latter, yields a substantial baryon density of g2 -- 0.3 at redshift ~ 3--4 that exceeds by far the low redshift mass fraction in luminous matter, g2 -- 0.05. MACHOs have been detected in the galactic halo. The experimental technique is that of gravitational microlensing. From the galactic rotation curve, one knows the local dark matter density in the halo. MACHOs occasionally cross the line-of-sight to a distant star, the light from which is amplified as the gravity field of the MACHO acts like a gravitational lens. The best target is the Large Magellanic Cloud, where there are many source stars for which one can estimate the lensing probability. If the halo consists entirely of MACHOs, the optical depth to microlensing is (Vrot/C)2 10-6, where Vrot is the galactic rotation velocity. Some 20 events have now been detected towards the LMC. If these events are due to halo MACHOs, the event durations imply that the MACHO mass is in the range 0.1 - 1 M o, and from the event rate, one concludes that between 20 and 80 % of our halo, between us and the LMC (at 50 kpc distance), consists of MACHOs. Because of incompleteness, the derived mass fraction is likely to represent a lower bound on the actual halo mass fraction in MACHOs. Possible candidates include primordial black holes, brown dwarfs and white dwarfs, in decreasing order of plausibility. However the discovery of one and possibly two star-star binary lensing events in the SMC has cast some doubt over the MACHO interpretation. The extended nature of the SMC along the line of sight means that star-star lensing events in this direction are about as frequent as MACHO events, in a MACHO-dominated halo. One does not know how extended the LMC is along the line of sight, thereby complicating any direct analogy with the SMC result. It is premature to draw any firm conclusions about the existence of MACHOs. 7. G a l a x y f o r m a t i o n

Dark matter controls the potential wells from which galaxies form, and also determines the fate of the Universe. The measurement of fluctuations in temperature by the COBE satellite can be used to infer the strength of the primordial density fluctuations in the dark matter. The density fluctuations measured on the horizon scale are found to be of constant amplitude 6p/p. Now the strength of a fluctuation is measured by the gravitational potential energy, and there is a trade-off between size and ~p/p: the smaller the size the larger is c~p/p. Since 6p/p grows with time, it is clear that smaller

838

scales, with larger amplitudes contract and collapse to form self-gravitating clouds before larger scales collapse. We have a bottom-up theory of structure formation, in which galaxies form before clusters. This sequence is consistent with observations. The outcome is contraction and merging of increasingly more massive clouds of dark matter as hierarchical clustering proceeds. Tidal torquing between adjacent, quasilinear fluctuations results in acquisition of a dimensionless angular momentum per galaxy halo that is comparable to what is inferred from observations of disks with simple assumptions about baryonic collapse and angular momentum conservation within the dark halo. Excellent agreement with the general form of rotation curves for massive galaxies is found (Navarro et al., 1997). On larger scales, structure formation ab initio accounts for galaxy clustering as described by the correlations in the galaxy distribution (Jenkins et al., 1998), and for the abundance and structure of galaxy clusters (Thomas et al., 1998). A notable accomplishment has been the explanation of the distribution of intergalactic gas as measured in Lyman alpha clouds in the redshift range z = 2 - 4 (Katz et al., 1996). However there are some less well understood issues. The superficial success in deriving a characteristic cold gas mass, and inferred luminosity, that corresponds to L. in the Schechter function for the distribution of galaxy luminosities has not held up. The deeper the cold dark matter potential well, the greater is the mass of cooled gas. Cooling must be terminated. The predicted slope of ot = 2 is steeper than the observed slope of ot---1 for red-selected galaxies. Interestingly, blue-selected galaxy samples (e.g. Zucca et al., 1997) have an upturn ( a = - 1.5 ) at M8 > -16. Infall of all or most of the diffuse matter surrounding forming galaxies results in a ratio of mass-to-luminosity that is excessive, relative to the mass-luminosity ratios inferred for spirals or ellipticals from the normalization of the Tully-Fisher and fundamental plane correlations, respectively. Appeal must be sought to feedback from star formation and death that effectively makes star formation progressively less efficient in lower mass galaxies. Various feedback self-regulation models have been explored with differing degrees of success. Star formation prescriptions have been applied that account for the observed luminosity function, but at the price of anomalously high mass-toluminosity ratios for bright galaxies. Conversely, models that satisfy the observed mass-toluminosity ratios cannot easily be reconciled with the bright galaxy normalisation (and even low mass tail) of the galaxy luminosity function (c.f. Steinmetz and Navarro, 1998). Another, potentially more serious, difficulty with galaxy formation models has arisen with the derivation of the size of galaxy disks in hydrodynamical simulations (Steinmetz and Muller, 1995). Clumpy collapse and baryonic cooling result in effective dynamical friction on the dense baryonic clumps and efficient angular momentum transfer. The resulting disk size is found from simulations to be about 20 % of that predicted for a homogenous collapse of baryons in an initially slowly rotating dark halo, with initial rotation characterized by the dimensionless angular momentum parameter 2 predicted from tidal torque theory to be about 0.07. A typical disk half-mass radius is found to be ~0.2 2 R i, where R i is the dark halo radius of ~ 50-100 kpc, and is far smaller than the ~ 5-10 kpc measured for typical disks. Presumably feedback associated with star formation will help resolve the disk crisis, but a plausible and generic prescription for this remains to be developed. Clearly, the cold dark matter theory of galaxy formation has provided some remarkable successes in accounting for the structure of galaxy halos, the clustering of galaxies, the distribution of intergalactic gas and of galaxies in different environments, and the clustering of galaxies. Precisely how the detailed morphologies of galaxies arise depends on the theory of star formation, a topic which is still poorly understood even in the nearby Universe. Nevertheless, a general view is that once self-gravitating clouds form, there is sufficient angular momentum acquired by mutual tidal torques between neighboring clouds to allow collapse to a disk. The disk fragments into stars: a spiral galaxy is born. In dense regions, disks merge: this heralds the formation of elliptical galaxies.

839

8. Conclusions The Big Bang theory is in remarkable accord with observations. There are no contradictions between theory and data. This situation is in sharp contrast to what prevailed in the past. During the 1930's, for example, the Hubble time scale was consistently found to be less than the age of the earth inferred by radioactive data. The finding of a subcritical density guarantees expansion. The counterparts of the fossil density fluctuations, as imprints on the microwave sky, that generate structure has been identified. Nevertheless one can be cautious about how well the Big Bang can be justified. Direct observation probes the last few billion years. The birth epoch of galaxies has yet to be identified. Observations of the microwave background fluctuations probe the Universe back to an age of 300 000 yr. Less directly, the abundances of the light elements constrain the Universe back to an age of a second to have been close to the Friedmann-Lemaltre model. But this is about as far back in the Big Bang can be taken. Ordinary matter condensed at 10-4 s, and the electromagnetic interactions were distinct from nuclear interactions at 10-~° s. This is about as far back as the particle physics can be reliably examined, by experiments in particle accelerators, and understood in the context of the standard model of elementary particles. At higher energies or earlier epochs, we venture into unknown territory. We speculate that baryon synthesis occurred prior to 10-~° s, and that inflation occurred at 10-35 s. The ultimate mystery is the Big Bang singularity at 10-43 s, for which theory is completely lacking. Nevertheless there is sufficient late time concordance between theory and observational data that we can be confident that the Big Bang is a close approximation to reality. The ultimate theory may emerge in the future, when quantum gravity is understood, but the Big Bang is likely to constitute at least the low energy approximation to a future Theory of Everything.

References Chaboyer B., 1998. The age of the Universe, Phys. Rep.-Rev. Section Phys. Lett., 307 (1~-), 23-30. Ellis J., Falk T., Olive K.A., Schmitt M., 1996. Supersymmetric dark matter, in the light of Lep 1.5, Phys. Lett., B 888, 97-105. Ellis J., Falk T., Olive K., 1998. Phys. Lett. B 444, 367. Freedman W., Mould J.R., Kennicutt R., Madore B., 1998. in: Sato K. (Ed.), IAU Symposium 183, Cosmological Parameters and the Evolution of the Universe, Kluwer, Dordrecht. Friedmann A., 1924. Ober die Mrglichkeit einer Welt mit Konstanter negativer Krtimmung des Raumes, Z. Phys., 21,326-332. Hancock S., Rocha J., Lesinby A., Gutierrez C., 1998. Constraints on cosmological parameters from recent measurements of cosmic microwave background anisotropy, MNRAS, 294/1, L1-6. Hu W., Sugiyama N., Silk J., 1997. The Physics of microwave background anisotropies, Nature, 386 (6620) 37-43. Hubble E.P., 1929. Proc. Nat. Acad. Sci. (USA), 15, 168. Jenkins A., et al. 1998. Evolution of structure in cold matter universes, ApJ., 499, 201. Katz N., Weinberg D., Hernquist L., Miralde-Escude J., 1996. ApJ., 457, L57. Lema3tre G., 1927. Ann. Soc. Brux., 478, 49. Navarro J.F., Frenk C.S., White S.D.M., 1997. An universal density profile from hierarchical clustering, ApJ., 490, 493-508. Penzias A., Wilson R., 1965. A measurement of excess antenna temperature at 4080 Mc/S, ApJ., 142, L1,419-421. Perlmutter S., et al., 1999. ApJ., 517, 565. Rauch M., et al. 1997. The opacity of the Ly-Alpha forest and implications for Omega (B) and the ionizing background, ApJ., 489, 7-20. Schmidt B., et al. AJ., 1998. ApJ., 507, 465. Schramm D., Turner M., 1998. Big Bang nucleosynthesis enters the precision era, IRMP, 70, 303-318. Steinmetz M., Navarro J., 1998. ApJ., 513, 555. Steinmetz M., Muller E., 1995. The formation of disc galaxies in a cosmological context: structure and kinematics, MNRAS, 276, 549-562. Tammann J., 1998. in: General Relativity, 8th Marcel Grossmann Symposium, Piran T. (Ed.), World Scientific Singapore, in press. Thomas P.A., et al., 1998. The structure of galaxy clusters in various cosmologies, MNRAS, 296, 1061-1071. Zucca E., et al., 1997. A.&A., 326, 477. Zwicky F., 1933. Helv. Phys. Acta, 6, 110.

840