Cosmology and structure formation

ELSEVIER

Nuclear PhysicsB (Proc. Suppl.) 81 (2000) 3-10

PROCEEDINGS SUPPLE~NTS www.elsevier.nl/locate/npe

Cosmology and Structure Formation Joseph Silka aAstrophysics, University of Oxford, Keble Road, Oxford OXl 3RH There are ten critical cosmological parameters that monitor cosmological models and structure formation. In this review, I summarize the current status of these parameters and discuss the issues that are presently unresolved.

1. I N T R O D U C T I O N The great achievement of Friedmann and Lema~tre was to realise that Einstein's equations simplify under the assumptions of homogeneity and isotropy to an equation for the scale factor a(t) that is the ratio of physical to coordinate (or comoving) distance. One can characterise the solutions for a(t) by four parameters H0, fl0, Tr,0 and flA. There are other parameters that are independently derived but depend on the primary parameters, thereby enabling one to develop a self-consistent model. These are the age to, deceleration q0 and curvature k. In addition there is the contribution to f 0 from baryons, fibf o refers to matter that is cold (that is, gravitationally clustered). I add two additional parameters that characterise the fluctuations: n and as, to give 10 parameters. The cosmological model is overspecified because of the two FriedmannLema~tre equations for/i and d which specify q0 and k as functions of H0, flo and flA. Nevertheless because of the systematic uncertainties in several of the observational derivations of these parameters, we cannot yet specify the cosmological model with any certainty. I shall argue that we cannot yet distinguish between flat or negatively curved space-like hyposuffaces, and that the underlying density fluctuations may be inadequately described by a primordial power-law power spectrum.

which is measured to a precision that is unequalled in nature for a blackbody source, with no distortions detected to an uncertainty of less than 50#K [1]. Useful limits on structure formation come from upper limits on spectral distortions. The Compton y parameter is measured on 7 degree angular scales to have an upper limit (95% confidence) Y

./ne(kT/mec2)cdt

< 3 x 10-6.

(1)

This limit strongly constrains late (matter era) energy input, and early (radiation era) energy input is constrained by the limit on the chemical potential < 6 x 10-5

(2)

3. A G E One of the historically controversial parameters has been the age of the universe. This is best constrained, independently of cosmological model, from the age of the oldest stars. Modern stellar evolution tracks and dates on globular cluster stars yields a stellar age of 12 + 2 Gyr [2]. Adding on 2 Gyr for the period prior to formation of the Milky Way, the age of the universe is estimated to be to = 14 4- 2 Gyr

(3)

4. H U B B L E ' S C O N S T A N T 2. T E M P E R A T U R E

The microwave background is the most precisely measured of any cosmological component. It is a blackbody at a temperature Tr,0 = 2.728 K

Even more controversial historically has been the determination of Hubble's constant. Long the focus of efforts by distinct groups that offered determination differing by five or more standard deviations of estimated error, the Hubble constant

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J. Silk~Nuclear Physics B (Proc. Suppl.) 81 (2000) 3-10

controversy has practically disappeared. In large part, this is due to improved calibration, with HST of Cepheid variables and supernova light curves in nearby galaxies. A value for H0 of 65+10 k m s - l M p c -1 generously encompasses recent determinations centering on 60km s-1Mpc-1 [3] and 72 ± 1 0 k m s - l M p c -1 [4]. Combination of H0 and to means that even an Einstein-de Sitter universe, with to = (2/3)Ho 1, is close to being consistent with the observed age of the universe. With the preferred ~ ,-~ 0.3 universe that is currently measured, an age as long as 17 Gyr is allowed. 5. B A R Y O N D E N S I T Y P A R A M E T E R It is customary to define the mean present density of the universe relative to the Einstein-de Sitter density, 3 H 2 / 8 ~ G . The baryon density ~'~b is determined in two ways. The most precise determination comes from comparison of primordial nucleosynthesis predictions with the observations of as primitive He 4, 2H and Li6 as one can find. High redshift determinations of 2H, while limited to only two lines of sight, have converged, as has the extrapolation of He 4 in low metallicity galaxies to zero metallicity [5]. The net result is ~tbh 2 = 0.015(±0.005), so that ~b ----0.04(±0.01). Modelling of the Lyman alpha forest at z ~ 3 has established that ~b(Z ~ 3) ~ 0.03. Most of the gas is photoionized by the radiation field from quasars. The robustness of the modelling, via simulation and analytic techniques, depends on the fact that ~'~b depends only on the square root of the ionizing photon flux, the neutral component being directly measured [6]. At low redshift, the luminous regions of galaxies account for ~b ~ 0.003. The correction for gas at z ~ 0 is uncertain. In halos and in clusters of galaxies, the contribution of atomic and ionized diffuse gas to ~-~b is negligible. Attempts to account for halo dark matter as cold molecular gas remain unresolved. Halos, with M / L ~ 50 relative to the universal value M / L = 1000 ~ for the measured mean luminosity density, account for ~halo ~ 0.05 in dark matter. Halos could mostly consist of dark baryons, although there are probably not enough baryons to account for all of the

halo dark matter. Most baryons are evidently dark, and have become dark between z ~ 3 and z ~ 0. This may be partly an effect of observational selection: it is exceedingly difficult to map the local counterparts of the high redshift Lyman alpha forest. There could be a substantial mass of (,-~ 106 K) gas, outside clusters and groups, in the filaments and sheets that characterize large-scale structure. Such gas has not yet been detected but its presence is predicted by high resolution simulations of large-scale structure [7] as well as suggested by OVI absorption towards high redshift quasars [8] and indications of a gas component with b-values as high as ~ 80 kms -1 [9]. Another likely sink for at least some dark baryons is in the form of MACHOs. These compact baryonic objects are likely to be either white dwarfs or brown dwarfs. Primordial black holes are possible MACHOs that can account for gravitational microlensing signals, but would not contribute to the baryon budget at primordial nucleosynthesis. However the MACHO interpretation of microlensing events has also been attributed to self lensing by either SMC or LMC stars, most plausibly for the 2 events detected towards the SMC [10]. There is possible evidence for old white dwarfs as MACHO candidates, based on proper motions of faint HDF stars [11]. Such objects could be sufficiently common to account for up to half of the halo dark matter density, and have masses consistent with the measured timescales of the LMC microlensing events if these are interpreted as being due to halo objects. 6. M A T T E R D E N S I T Y P A R A M E T E R Consider next the total density of nonrelativistic matter in the universe, denoted at the present epoch by ~0. There are several observational indications which support a value ~t0 = 0.3(+0.1). In order of robustness, these are the relative change in frequency of rich clusters between z = 0 and z -- 1, large-scale bulk flows, the gas fraction in clusters, and the present number density of rich clusters. The growth of density fluctuations is arrested at z < ~ o 1, and indeed systemically becomes suppressed by a fac-

J Silk~Nuclear Physics B (Proc. Suppl.) 81 (2000) 3-10

tot fl0°'6 relative to the t 2/3 growth in a k = 0 universe. Galaxy clusters are rare objects, forming in the exponential tail of a gaussian distribution of density fluctuations, and objects moreover whose formation is well understood in terms of gravitional instability and collapse. It is therefore a straightforward prediction of a high 120 universe that there is a strong decrease in the number of clusters above a specified mass with increasing redshiff. A low f~0 universe, in contrast, has little evolution in cluster frequency at z <~ 1. The discovery of at least one rich cluster at z ,,~ 0.8, for which x-ray temperatures are measured, and the probable existence of massive clusters at z > 1, provide strong arguments for a model with f}0 ~ 0.3 [12], although not all authors agree with this conclusion [13]. Large-scale bulk flows are predicted, for cold dark matter models, to amount to ,-~ 1000~r8~ 0"6 kms -1,

(4)

where the inverse bias parameter as is the rms density fluctuation amplitude at 8 h - l M p c , the scale at which galaxy number count fluctuations have unit amplitude. Measured bulk flows are ,-~ 300 kms -1 on scales of I ~ 40 Mpc [14]. This favours ~0 ~ 0.3, but the uncertainties are large. Detailed comparisons of the peculiar velocity and mass density fields are less conclusive. There is a trade-off between raising bias or lowering fi, and the empirical evidence cannot do much better than argue for 0.5 ~< ~'~0"60"8 ~ 1. Weak lensing measurements suggest that as ~ 1, as does the reconstruction of the density fluctuation power spectrum in the region of overlap between CMB and galaxy redshift survey probes [15]. The gas fraction in rich clusters is about 0.15, and adding the stellar component gives a cluster baryon fraction of 0.18. For this to be reconciled with the nucleosynthesis baryon abundance of 0.03 requires D0 ~ 0.15, provided that we assume that galaxy clusters fairly sample the universal baryon fraction [16]. The rich cluster number density is higher than observed by a factor 10-100 if ~2 -- 1 at the present epoch. However comparison with observations of optical clusters requires one to assume a universal galaxy formation efficiency in clusters. Moreover

5

x-ray selected cluster samples are gas-biased. One has to assume that the intergalactic gas has not undergone excessive preheating, in which case gas infall to rich clusters, and especially into cluster cores, would be suppressed. Neither of these assumptions can be rigorously justified, and only a gravitational shear-selected cluster sample will be able to definitively address this test. 7. C O S M O L O G I C A L

CONSTANT

First introduced and then rejected by Einstein, the cosmological constant has subsequently been in and out of fashion. More recently, it has been reinterpreted as a vacumn density, for which very early universe spontaneous symmetry breaking phase transitions have provided evidence of existence, although at an energy density that is of order (100 GeV) 4 as compared to the observed or constrained current epoch value of around (0.001 eV) 4. The corresponding equation of state is p = - p , and from the Friedmann-Lema~tre equation for the scale factor /~ 47rG a 3 (p + 3p) (5) one infers that acceleration of the universe is inevitable even for less extreme values of negative pressure, provided that p = w p with w < -½. There is no means at present of distinguishing between alternative formulations of the generalized equation of state for vacuum energy, including the quintessence models in which w is a function of time. However observations of supernovae of type Ia have provided strong support for a cosmological constant, provided that one accepts the case that SNIa are good standard candles. Acceleration is inferred and provides a measure of ftm - f~h, which is found to be negative ,,~ - 0 . 2 [17],[18]. Here ~h is the normalised vacuum density, and the classical deceleration parameter q0 = .~a_~2h. The dimming of high redshift relative to low redshift supernovae could be an artefact of intervening dust. However no colour differences are found, and one would have to appeal to ~ 0.2 mag of grey extinction, with no apparent change in the dispersion of supernova peak magnitudes, between z = 0 and z = 1. A more serious concern is

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J Silk~NuclearPhysics B (Proc. Suppl.) 81 (2000) 3-10

the possibility of evolution of the supernova itself. This could easily happen because the precursor history .of the merging or accreting white dwarf is unknown for SNIa, and could plausibly vary as the age decreases and main sequence turn-off mass increases of the parent population. It is difficult to say if this would affect the supernova light curve: however there are as yet no 3-dimensional calculations of white dwarf mergers, so that the theory is incomplete. Moreover, only about half of the He/C/O core implodes and is a source of energy for the light curve, so that evolution could conceivably result in the 20 percent modulation needed to seriously change the interpretation of the observed dimming. The most recent controversy centres on the fact that a difference in precursor rise times has been reported between high and low redshift supernovae [19]. This is the first evidence for possible evolutionary differences as a function of cosmic epoch, and, if this result holds up, would suggest that the systematic uncertainties in the relative calibration of high and redshift supernovae may be unknown.

8. S P A T I A L C U R V A T U R E The Friedmann-Lema~treenergy equation, iz2 8~G R a-~ = ~ +A

k aS,

(6)

yields an equation for the curvature k = (hA +

- 1)H0

(7)

Curvature is directly measured by integrating along light rays and measuring proper density or the associated angular separation of objects, which depend in turn on the proper distance ra(t) = a(t) [ c dt

J

(8)

Specific curvature measurements include determination of the angular sizes of objects of known proper size, gravitational lensing of distant galaxies or quasars by foreground galaxy clusters or galaxies, and measurement of the comoving density of well-specified classes of objects. However the most promising and most straightforward curvature determination utilizes the first acoustic

peak of the temperature fluctuations in the cosmic microwave background [20]. The maximum sound horizon is essentially cosmological modelindependent. Inflationary fluctuations originate as superhorizon metric or curvature fluctuations, and consequently there is a peak wavelength that reaches its first peak on the horizon scale at last scattering. All scales that are of interest for structure formation are acoustic waves prior to matterradiation decoupling, and eventually dissipate by radiative diffusion on subhorizon scales. However a series of coherent peaks develops at last scattering, corresponding to the compressions and rarefractions of sound waves between the horizon and damping scales at wavenumber 7rnvatLS for n = 1, 2 .... The most pronounced peak (kLs~rvsts8) is visible, when projected into spherical harmonics on the celestial sphere, at l ~ 200 or about 1°. Negative curvature shifts the first acoustic peak by a factor of order f~0~ towards smaller angular scales. Current experiments constrain the spatial curvature to be near zero [21], with a liberal interpretation of the data allowing 2 ~> ~A + ~0 ~> 0.3. However it is anticipated that future experiments will greatly increase the sensitivity of this curvature constraint. Of course this conclusion does assume an inflationary origin for the fluctuations. If the fluctuations are, for example, of the isocurvature mode as in string models, the most prominent peak can occur not at last scattering on the maximum sound horizon VstLS but, in at least some models, can be delayed because the induced velocity perturbations are 7r/2 out of phase with the primary adiabatic peturbations at the horizon scale. Hence the canonical peak location in a zero curvature background can, in principle, be duplicated by a low density isocurvature model with ~0 ~ 4 / r 2. Gravitational lensing has provided an alternative approach to measuring curvature, and more specifically f/A to the extent that this determines f/0. Quasar and compact extragalactic radio source splittings argue [22] for a modest flA ~ 0.6, in weak contradiction with the SNIa result of f~h ~ 0.7. The frequency of giant arcs actually favours low f~0 and negligible ~A [23]. In effect,

J. Silk~Nuclear Physics B (Proc. Suppl.) 81 (2000) 3-10

gravitational lensing measures comoving volume, which at large redshift is dominated by the matter density: the combination ~h + (1 + z)3~0 is being constrained. Ultimately, we might hope to measure the frequency of galaxies at very high redshift. This would provide a strong constraint on the comoving volume available above a specified redshift. 9. D E N S I T Y F L U C T U A T I O N S Primordial density fluctuations are an essential ingredient in the standard cosmological model. Quantum fluctuations imprinted by inflationary cosmology on to macroscopic scales provided a major advance in the predictability of cosmological models. While scale-invariant density fluctuations had previously been touted, they were one of various alternatives on the cosmological market. The generic prediction of many (but not all) inflationary models was that nearly scalevariant curvature fluctuations are the source of large-scale power in the universe. The natural consequence of scale-invariant curvature fluctuations is a bottom-up or hierarchical theory of structure formation in which progressively large scales become unstable and collapse with increasing epoch. The hierarchical prediction arises because the metric perturbations corresponding to mass fluctuations ~M on scale L can be expressed as

Lc-----~ -

P

"~

'

(9)

and correspond to the amplitude of the density fluctuation at horizon crossing. Subhorizon growth beyond the initial value at horizon crossing only occurs during the matter-dominated phase of the expansion, thereby generating a change in slope in the predicted density fluctuation spectrum from 6PIP o¢ L -2 on large scales to 6pip c¢ constant on scales much less than the horizon size at matter-radiation equality, about 12(f/0h2) -1 Mpc. This is true for cold dark matter: introduction of hot dark matter suppresses fluctuations via the associated free-streaming on scales 100(1 eV/m~)Mpc. For an admixture of hot dark

7

matter with cold dark matter predominating, the only possibility for hot dark matter to play any role in structure formation, partial suppression occurs on small scales, and neutrino masses of 1 - 2eV are allowed by most of the observational constraints apart possibly from that of early structure formation. If f/0 = 1, the standard cold dark matter fails to account for the simplest parametrization of the power spectrum. This utilizes the slope defined by ~k 0( k n and normalization at 8h-lMpc, where the galaxy count fluctuations have unit amplitude. It is conventional to define

\ng ] I

8h-'Mpc

The power spectrum is defined by I((~) 2) =

f k3p(k)-~l and ~ = f 6keikrd3k. The slope is well determined by CMB observations, most notably COBE, to be n = 1 ~= 0.2. However as is model-dependent. It is most directly characterized from large-scale flows and cluster abundances which require asf~°'6 ~ 0.7(±0.2). An alternative scheme follows from a combination of redshift survey data, which measure the fluctuations in luminous matter, with CMB anisotropy analysis, which can be reconstructed to yield the fluctuations in dark matter for a specified model. The CMB-normalized power spectrum requires as ~ 1 for IRAS and blue-selected galaxy samplea. These are essentially unbiased. Red-selected samples, in which galaxy clusters are more prominent, display a small bias, as "~ 0.8. Mass therefore seems to trace fight on ,~ 10 Mpc scales for field galaxies, while more highly clustered objects such as galaxy clusters represent rarer, and hence more biased, peaks, in the initially gaussian distribution of density fluctuations. The f~ = 1 standard CDM model predicts excessive power at 1 0 - 100 Mpc scales once CMB-normalized. This manifests itself as antibias, which would be unphysical, and generates both extensive bulk flow velocities on 10 - 30 Mpc scales, and excessive cluster abundances, effectively on 10 Mpc scales. There is also a shape problem at 1 0 - 30 h-aMpc, with the

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J. Silk~Nuclear Physics B (Proc. Suppl.) 81 (2000) 3-10

predicted power being flatter than the observed slope. There are three standard solutions that correct the power spectrum normalization and shape fit problems at 10 Mpc scales. These are an admixture of 20 percent hot dark matter ( ~ m~ = 6eV), a tilt (n ~ 0.9) or lowering of fl0 to f~0 ~ 0.3. The latter forces a power spectrum renormalization (by a factor ,~ f~-2 due to the suppressed growth) combined with a shift in the power spectrum peak wavelength (c< 12-2) to larger scales. The former two solutions, have too little power at subgalactic scales to allow sufficient early formation of Lyman alpha clouds and star-forming galaxies, required at z ~> 4. Hence low fl0 models are required, with or without A which has little effect on P(k) as fit to the matter fluctuations derived from redshift surveys. The reconstruction of P(k) from CMB fluctuations is sensitive to A, which however at specified curvature is degenerate with f~b. A combination of low redshift weak lensing surveys with CMB fluctuations will be required to self-consistently derive 12h as well as fl0 and 12b. The general agreement of CMB and large-scale structure observations, especially in the region where similar ~ 100 Mpc scales are probed, with the predicted P(k) for ~ -- 0.3 is remarkable. It argues that we cannot be far from the correct model, given the concordance between independent data sets that probe the universe at z ,,~ 1000 and z ~ 0. However the fit is not perfect, and at least one of the redshift surveys (the only survey in physical as opposed to redshift space) has a shape near the 100 Mpc peak that is significantly more peaked than the simplest inflationary theories. Either the current data suffers from hitherto unidentified systematic errors or we may be required to develop a more precise parametrization of P(k) that takes account of detailed shape variations. One could imagine a primordial feature in the power spectrum near the peak. This has been suggested to arise, for example from the effects of multiple scalar fields driving multiple stages of inflation [24], [25] or with an incomplete first order phase transition responsible for production of large-scale bubbles, whose size and frequency must be tuned to fit the data [26]. One conse-

quence of such tinkering with canonical inflation is that non-gaussianity may be introduced. This would further complicate the determination of cosmological parameters. With or without nonganssianity, excess power in P(k) near 100 Mpc could play a role in accounting for results from narrow angle surveys that appear to show evidence for anomalous clustering power near this scale. The current situation with regard to data will soon improve, with the advent of ,~ 250,000 2DF galaxy redshifts and ~ 106 SDSS redshifts. It is presently premature to devote too much attention to possible anomalous features in the galaxy survey data. 10. G A L A X Y F O R M A T I O N The most serious unsolved problem in the standard model of cosmology is that of galaxy formation. The theory is phenomenological, and is largely based on local measurements of star formation rates and the initial mass function of stars. There is no fundamental theory of local star formation or of the origin of the mass function of stars. Whether these local properties that are essential to galaxy formation operate unchanged in extreme environments is unknown. There are some discouraging aspects. For example, magnetic fields play a central role in local star formation by supporting clouds against premature collapse. The cosmic history of magnetic fields is completely uncertain, since there is no consensus on the sites of the dynamos that generated pregalactic, protogalactic and galactic fields. Disk formation is understood in terms of slow gas accretion and cooling, and spheroids form by mergers. Merger theory, in terms of dark halos, is well understood via numerical simulations. However the accompanying role of star formation is less secure. Whether the star formation precedes most of the merging and occurs in small subunits or mostly occurs monolithically in the last major merger is uncertain. Simulations are best adapted to handle the dark halo evolution, and have been supplemented by semi-analytical techniques to follow the actual process of luminous galaxy formation. There are a considerable number of parameters that enter

J Silk~NuclearPhysics B (Proc. Suppl.) 81 (2000) 3-10 galaxy formation theory, and the net result is that any specific observational issue can be addressed with apparent success. W h a t is more difficult is to simultaneously account for all observational constraints on galaxy formation in the context of currently available models. For example, the theory of dark halo merging generically predicts a tail proportional to M -2 in the halo mass spectrum, yet the observed galaxy luminosity fluctuation is significantly flatter, varying approximately as L -13 to at least MB = --14. High resolution simulations confirm the persistence of halo substructure [27]. The dense clumps, also present in the dark matter, also present a potentially serious problem for overheating the disk. The standard resolution is to introduce feedback from supernova explosions, which drives a wind that preferentially disrupts the gas reservoir and thereby inhibits star formation in low mass galaxies with low escape velocities. However once a feedback prescription is adopted that preferentially affects low mass galaxies, one can no longer modify the luminosities of massive galaxies. The result is that the normalisation of the Thlly-Fisher relation, between galaxy luminosity and maximum rotation velocity, fails: massive galaxies are found to be underluminous for their mass [28]. This is likely to be a generic problem, as might be inferred from any cold dark matter model in which on scales of say 10 Mpc, mass traces light. On these scales, the effective massto-light ratio for the typical object is 1000 ~mThis is what is found for clusters, for which the model is fully consistent, but one would expect the ratio of mass-to-light for massive galaxies to be a factor of (Mg/Mcl) 1/3 or ,-, 10 less than this, namely 30 with ~'~m = 0.3. This should be compared to the observed value of --, 10. High resolution simulations confirm this qualitative argument, finding that M / L for L. galaxies is too high by ~ 3 [29]. The problem may be attributed to the universal dark matter profile p(r) o~ r - l ( 1 + ar2) -1 found for dark halos [30]. A similar excess of dark mass, also by a factor ~ 3, is found for the Milky Way within the solar circle. The univer-

9

sal dark matter profile has also encountered difficulties in accounting for the rotation curves of dwarf spiral galaxies. These objects, unlike L. galaxies, are everywhere dark matter-dominated, and hence should provide good laboratories for studying dark matter. Soft cores are found, in contradiction to the predicted r -1 cores. However numerical simulations of the dwarf galaxy halos have not necessarily reached the resolution required to be confident of the predicted cuspy profiles [33],[31]. The numerical simulations have unveiled an especially serious contradiction with observations of disk galaxies. This is the prediction of disk sizes. Earher work, both analytic and numerical, found that the initial dimensionless angular momentum acquired by a protogalaxy, A ~ 0.06, is conserved at any given radius as the baryons cooled and contracted in the dark halo. The final disk size was predicted to be ~ Arh, where rh is the halo radius, and agrees well with the observed disk sizes of ~ 5 kpc. The high resolution simulations, however, find that the infall is clumpy and angular momentum is transferred outwards as dense clumps of baryon sink by dynamical friction into the halo core. The consequence is that the predicted disk sizes are too small, by almost an order of magnitude [32]. It is likely that all of these problems have straightforward resolutions requiring more detailed physical input. Baryonic disk formation, if sufficiently non-axisymmetric may grossly perturb the inner dark halo profile. Feedback from early star formation may suffice to prevent most of the protodisk gas from losing excessive angular momentum before most disk stars have formed. Nevertheless, in the absence of quantitative modelling of inner halo structure and disk sizes, it is premature to attach much significance to the ab initio predictions of young galaxy properties in the early universe. 11. S U M M A R Y

VV'eare close to converging on the definitive cosmological model. Most parameters of the background model are under control. The age is to = 14 ± 2Gyr, the Hubble constant H0 =

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J. Silk~Nuclear Physics B (Proc. Suppl.) 81 (2000) 3-10

65 ± 1 0 k m s - l M p c , the baryon density Db = 0.04 ± 0.02, and the density parameter is D = 0.3 ± 0.1. The urgent model issue is 12a which is found to be 0.7 but with uncertain systematic errors because of possible evolutionary dimming. Current data allows the density fluctuations to be represented by n = 1 ± 0.2 and as = 0.8 ± 0.2, with an upper limit on a possible tensor or gravitational wave component of T / S < 0.2. There are indications of possible additional power in the data relative to the standard scale-invariant prescriptions, as evidenced by the slope of P(k) near 100 Mpc, excess power at .~ 100 Mpc in deep pencil beam surveys, and the height of the acoustic peak in the CMB anisotropy spectrum. However the rapidly improving data sets for both CMB and galaxy redshift surveys to z ~ 0.2 suggest that it may be unwise to invest much effort in exploring current data sets which are of inadequate size and are almost certainly contaminated by systematic errors. More fundamental issues that merit more theoretical effort include galaxy formation. This currently amounts to phenomenology driven by observations. One may hope that higher resolution simulations with improved star formation physics will eventually improve the present situation. Another important model issue is the topology of the universe. There are no predictions for this quantity but if the topological scale is of the order of the curvature scale, as simple arguments suggest, observable global anisotropies are generated in the CMB. Even in the area of quantum gravity, observation may be ahead of theory.

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