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In this brief concluding chapter, we summarize what it is that has been learned from redshift and peculiar velocity surveys, and put the results into the context of the larger field of observational cosmology.

9.1. The Initial Power Spectrum

We have put constraints on the power spectrum directly from measurements of the distribution of galaxies (Section 5.3). The fact that power spectra derived from redshift surveys in different areas of the sky, using different samples, agree, implies that it is meaningful to define a power spectrum in the first place. That is, we do not live in a simple fractal universe, in which a mean density depends on the scale on which it is measured. Moreover, our samples are starting to become big enough that our statistical measures are not completely dominated by sampling fluctuations, at least for measures probing relatively small scales (50 h-1 Mpc and smaller). This is not to say that improvements in the statistical errors in the power spectrum on these scales are not needed!

The redshift survey data strongly rule out the standard CDM model, and are fit much better by a Gamma = 0.20 - 0.30 model (Fig. 9). This is in accord with analyses of the small-scale velocity dispersion of galaxies, their large-scale angular clustering, observations of bulk flows on large scales, as well as constraints from the CMB fluctuations (Efstathiou et al. 1992; Kamionkowski & Spergel 1994 ; Kamionkowski, Spergel, & Sugiyama 1994 ), the distribution and mass spectrum of clusters (Bahcall & Cen 1992 ; 1993), and a host of other constraints. Unfortunately, we cannot conclude from this that the dark matter problem is solved. A number of the different suggested power spectra are degenerate over the scales probed by redshift surveys, with very different implications for the nature of the dark matter (compare the range of Fig. 9 to that of Fig. 2). In particular, the data we have presented are also consistent with the Mixed Dark Matter model and the Tilted Cold Dark Matter model, which are two of the more popular models being discussed.

The Mixed Dark Matter model has a long history, starting with the idea that adiabatic damping of the power spectrum on small scales by baryons will cause a turn-down in the power spectrum (Silk 1968 ; Dekel 1981); its recent incarnation is in terms of a mix of hot and cold dark matter (Schaefer, Shafi, & Stecker 1989 ; Schaefer & Shafi 1992 ; Taylor & Rowan-Robinson 1992 ; Davis, Summers, & Schlegel 1992 ; Klypin et al. 1993). In particular, the hot dark matter suppresses the power spectrum on small scales, decreasing the small-scale velocity dispersion relative to standard CDM, and increasing the amount of large-scale power for a given normalization on small scales. However, the model perhaps suppresses small-scale power overly much: galaxies cannot form on these small scales until very late, which is difficult to reconcile with observations of galaxies and quasars at very high redshifts (Cen & Ostriker 1994 ; cf., Efstathiou & Rees 1988 for a similar critique of standard CDM).

The tilted CDM model was suggested simultaneously by a number of workers (Cen et al. 1992 ; Lidsey & Coles 1992 ; Lucchin, Matarrese & Mollerach 1992; Liddle, Lyth & Sutherland 1992; Adams et al. 1993 ; Cen & Ostriker 1993). It also increases the amount of power on large scales relative to small, in this case by changing the slope of the primordial power spectrum. However, Muciaccia et al. (1993) showed that standard CDM is preferred over tilted models when velocity field data and CMB fluctuations are taken into account as well.

There are a number of further constraints on the power spectrum, some of which we've only touched upon on this review. The most important of these is the fluctuations in the CMB. On the largest scales, the fluctuations as detected by COBE appear to be consistent with a primordial power spectrum of index n = 1 (the inflationary prediction) (Górski et al. 1994), with an amplitude that matches well that predicted from the Gamma = 0.2 CDM model (Fig. 9). Observations of the CMB fluctuations on sub-COBE scales (a few degrees) are just beginning to yield reproducible results; because more than the Sachs-Wolfe effect is operating on these smaller scales, these data can potentially yield information on Omega0, the density parameter in baryons, as well as the power spectrum (Hu & Sugiyama 1995). In addition, these probe scales now being reached by the largest peculiar velocity and redshift surveys, spanning much of the gap seen in Fig. 2. In this regard, the bulk flow results of Lauer & Postman (1994) (Section 7.1.4) remain unexplained in the context of models for large-scale structure. It is vitally important that this result be checked, as a number of workers are now doing (Section 9.7). The comparison between large-scale flows and CMB fluctuations has the potential to check gravitational instability theory directly, independent of the power spectrum (Juszkiewicz, Górski, & Silk 1987 ; Tegmark, Bunn & Hu 1994).

Constraints can be put on the power spectrum from observations of galaxies at high redshift. The amount of small-scale power determines at what epoch galaxies will form; measures of the age of galaxies and their evolution thus tell us something about the power spectrum. Balancing the need for enough small-scale power to allow galaxies to form early, as seems to be required by observations of high-redshift galaxies and quasars, against the requirement of not too much small-scale power, in order to restrict the small-scale velocity dispersion (Section 5.2.1) has not yet been self-consistently done for any model. Similarly, observations of clusters of galaxies and their evolution also have the power to constrain cosmological models (eg., Peebles, Daly, & Juszkiewicz 1989). Finally, detection of the evolution of clustering in the universe would be a tremendously important observation. In an open universe, clustering ceases to grow when Omega deviates significantly from unity (Section 2.2); unambiguous detection of this effect would be a sensitive measure of Omega0.

Perhaps the most dramatic constraint one could imagine on the power spectrum would be the laboratory detection of dark matter (Primack et al. 1988). It would be a tremendous triumph of our theoretical framework if a dark matter particle were discovered with properties consistent with the best-fit power spectrum from astronomical data. The HDM model has been out of favor for some time, given its unphysically late formation epoch for galaxies (e.g., White, Frenk, & Davis 1983), but if a definite non-zero mass for the muon neutrino were measured appropriate to close the universe, HDM models would certainly enjoy a resurgence of popularity!

9.2. The Distribution Function of the Initial Fluctuations

All the tests we have described for the random-phase hypothesis have yielded positive results; there is no direct evidence for non-Gaussian fluctuations in the initial density field. Perhaps the strongest such claim comes from the direct measure of the distribution function of initial fluctuations as found by the time machine of Nusser & Dekel (1993) . Similar conclusions are found in analyses of the COBE CMB fluctuations (e.g., Hinshaw et al. 1994). However, as we emphasized in Section 5.4, this by no means allows us to conclude that all non-Gaussian models are dead. Each of the tests described in Section 5.4 refer to a specific smoothing scale, and a model that is non-Gaussian on a given scale need not be so on another. What is needed is a systematic test of each non-Gaussian model proposed against the various observational constraints. This has been done to a certain extent for models of cosmic strings (Bennett, Stebbins, & Bouchet 1992) and texture models (Pen, Spergel, & Turok 1994), mostly in the context of non-Gaussian signatures in CMB fluctuations.

9.3. The Gravitational Instability Paradigm

The results we have presented here are all consistent with the gravitational instability picture. In particular, we have seen that there exist physically plausible power spectra which can simultaneously match the observed large-scale distribution of galaxies, large-scale flows, and the CMB fluctuations (36) . When redshift surveys began to reveal extensive structures such as giant voids and the Great Wall, many questioned whether this could be explained in the context of gravitational instability theory. However, simulations by Weinberg & Gunn (1990) , and Park (1990) , as well as arguments based on the Zel'dovich approximation (Shandarin & Zel'dovich 1989), showed that these structures were not unexpected given gravitational instability and plausible models for the power spectrum.

The most direct test of gravitational instability comes from the comparison of peculiar velocity and redshift surveys. Dekel et al. (1993) in particular claim that the Mark II peculiar velocity data are consistent with the velocity field predicted from the distribution of IRAS galaxies, and gravitational instability theory. However, this is not a proof; Babul et al. (1994) demonstrate models with velocities due to non-gravitational forces (in particular, large-scale explosion models) that the Dekel et al. tests would not rule out.

We have not discussed features of the velocity field on small scales, smaller than are resolved by the POTENT-IRAS comparison, with its 1200 km s-1 Gaussian smoothing. Burstein (1990) presents evidence that the very local velocity field, as measured with the Aaronson et al. (1982a) TF data, differ qualitatively from the IRAS predictions, a conclusion that continues to hold with the Tormen & Burstein (1995) reanalysis of the Aaronson et al. data. Similarly, the complete lack of an infall signature in the spiral galaxies around the Coma cluster found by Bernstein et al. (1994) is worrisome, and remains unexplained. Understanding these results remains a task for the future.

9.4. The Value of Omega0

Gravitational instability theory has given us a tool to measure the cosmological density parameter, by comparing peculiar velocities with the density distribution (Eq. 30), although in most applications, galaxy biasing means that we constrain only beta ident Omega00.6 / b. In Table 3, we summarize the various constraints on Omega0 that we have discussed in this review.

Table 3

Is there some consensus in the literature as to the value of beta? One way to assess this, at least qualitatively, is to plot each of the determinations in Table 3 as a series of Gaussians of unit integral, with means and standard deviations given by the numbers in the table. For simplicity, asymmetric error bars have been symmetrized, and those determinations without quoted error bars are not included. Determinations based on IRAS and optical samples are plotted with different symbol types. We now simply add the Gaussians together, to yield the two heavy curves in the plots. Note that this procedure tends to give lower weight to those determinations with more realistic (i.e., larger) error bars. No attempt has been made to assess the relative quality of these different determinations. Note also that because many of these determinations are from common datasets, they are not independent. Thus this form of qualitative summary gives an unprejudiced view of literature of determinations of beta from redshift and peculiar velocity surveys. The heavy curves have a mean of 0.78 and standard deviation of 0.33 (IRAS) and a mean of 0.71 and standard deviation of 0.25 (optical). These values are actually in quite close agreement, although that seems more coincidental than anything else, given the large spread of individual determinations.

The community is clearly not quite ready to settle on a single value for beta for the IRAS galaxies. The determinations range from 0.45 (Fisher et al. 1994b) to 1.28 (Dekel et al. 1993 , although the latter is likely to come down slightly with the Mark III data; Dekel, private communication). This is reflected in the large standard deviation, larger than any individual determination, and the flat top to the heavy curve in Fig. 20 . The optical beta shows a smaller spread, perhaps simply because there are fewer individual determinations of it. The odd man out is the determination of Omega0 by Shaya et al. (1994), although their determination is heavily affected by their modeling of the background density within 3000 km s-1 and the density field beyond there. Moreover, their work remains in flux (compare with Shaya et al. 1992) and it is not clear where their final results will lie.

Figure 20

Figure 20. The distribution of determinations of beta discussed in this review. Each determination is shown as a Gaussian of unit integral with mean and standard deviation given by the values listed in Table 3. IRAS and optical determinations are plotted with different line types. The (renormalized) sums of all curves (optical and IRAS separately) are shown as the heavy curves.

Most of the references in Table 3 are very recent, and we have not done a thorough job of reviewing the earlier literature, especially on Virgocentric infall. However, the common impression that estimates of Omega0 have taken a dramatic upturn in recent years is wrong. Davis et al. (1980) used observations of Virgocentric flow to find beta = 0.6 ± 0.1, in good agreement with the values for optical galaxies here. The value from the Cosmic Virial Theorem from Davis & Peebles (1983b) is difficult to interpret in terms of a biasing model, but corresponds to beta = 0.4 for an unbiased model.

9.5. The Relative Distribution of Galaxies and Mass

We have very few handles on the biasing parameter independent of beta. One approach has been to assume a model for the power spectrum, normalize it to the COBE fluctuations, and then compare the results predicted for the galaxy fluctuations at 8 h-1 Mpc with observations. This approach is by definition model-dependent; for standard CDM, one finds that optical galaxies are unbiased and that IRAS galaxies are anti-biased, while a model like Gamma = 0.2 CDM gives a normalization that leaves the IRAS galaxies unbiased.

Alternatively, one can constrain biasing by looking for non-linear effects to break the degeneracy between Omega0 and b. The skewness is one such effect. Fry & Gaztañaga (1993; 1994) and Frieman & Gaztañaga (1994) claim that the beautiful agreement between the measured higher-order moments of the APM counts-in-cells with that predicted given the power spectrum, implies that biasing of optical galaxies is very weak, and to the extent that there is biasing, that it is local. Dekel et al. (1993) attempted to look for non-linear effects in the IRAS-POTENT comparison; they could only show that the data are inconsistent with very strong non-linearities, thereby ruling out very small values of b.

Finally, one can look for relative biasing of different types of galaxies, as we described in Section 5.10. The effects are subtle: outside of clusters, there are no two populations of galaxies known that have qualitatively different large-scale distributions. The lack of such effects have motivated several workers (Valls-Gabaud et al. 1989; Peebles 1993) to argue that biasing cannot be acting at all. But differential effects are seen between galaxies of different luminosities and morphological types. It is time for a detailed comparison of these observed effects with hydrodynamic simulations, in order to see what constraints these put on general biasing schemes.

In any case, the consensus of the community is that biasing is relatively weak; few authors are arguing for b > 1.5 these days. This is quite a contrast to a decade ago, when the idea of biasing was first introduced; values of b = 2.5 or higher were popular (e.g., Davis et al. 1985). Thus we conclude that beta5/3 < Omega0 < 2beta5/3; the results of Table 3 are still consistent with values in the range Omega0 = 0.3 to Omega0 = 1. It has been quite popular in recent years to argue for the lower value, given the coincidence with the value of Omega0 needed to match the Gamma = 0.25 value preferred by the power spectrum (Coles & Ellis 1994).

9.6. Is the Big Bang Model Right?

One tests the Big Bang model with redshift and peculiar velocity data only to the extent that they give results which can be fit into our grander picture of the evolution of the universe, with input from all the subjects we did not discuss: observations of distant galaxies and quasars, measurements of individual galaxy properties, abundances of the light elements, and so on. We should point out one serious problem which we see on the horizon. The data we have discussed point towards a value of Omega0 close to unity, implying an age of the universe given roughly by t0 = 2/3 H0-1 (Eq. 17). With recent determinations of the Hubble Constant of the order of 80 km s-1 Mpc-1 (Jacoby et al. 1992 ; Pierce et al. 1994 ; Freedman et al. 1994), this gives an age of 8 billion years, less than half the currently accepted ages of the oldest globular clusters (e.g., Chaboyer, Sarajedini, & Demarque 1992). Note that this would be a problem even if Omega0 -> 0, for which t0 = 1 / H0 = 12 billion years. We may find ourselves invoking theoretically awkward models in which Omega0 approx OmegaLambda ltapprox 1. In any case, the next few years should be very exciting, as we come to grips with this rather basic problem.

9.7. The Future

We conclude this review with a quick discussion of the various on-going and planned redshift surveys and peculiar velocity surveys of which we are aware. As these become available, we can look forward to applying the statistics developed so far to vastly superior datasets; moreover, these will allow us to do analyses of much more subtle statistics.

There are a number of large-scale peculiar velocity surveys in progress. Giovanelli, Haynes, and collaborators are doing a Tully-Fisher survey of Sc I galaxies from the Northern sky drawn from the UGC catalog, together with calibrating galaxies drawn from a number of clusters. They have data for roughly 800 galaxies. In the meantime, Mathewson & Ford (1994) have extended their Tully-Fisher survey in the Southern Hemisphere to smaller diameters, as reported in Section 7.1.3; their sample now includes a total of 2473 galaxies.

At higher redshift, a team of eight astronomers started by three of the original 7 Samurai (Burstein, Davies, and Wegner), has extended the 7 Samurai Dn-sigma survey of elliptical galaxies to a further ~ 500 galaxies in clusters at redshifts ~ 10, 000 km s-1 (Colless et al. 1993).

Several groups are attempting to check the large-scale bulk flow measured by Lauer & Postman (1994) . The same authors, in collaboration with Strauss, are in the process of extending the survey to include the BCG's of all Abell clusters to z = 0.08, a total of over 600 clusters. They expect to complete the gathering of the data by mid-1996. Fruchter & Moore are measuring distances to the same clusters as the original Lauer & Postman (1994) dataset by fitting Schechter functions to the luminosity distributions in the clusters. In a complementary effort, Willick is measuring accurate distances to 15 clusters around the sky at redshifts of approx 10, 000 km s-1, using TF and Dn-sigma distances to spirals and ellipticals in each cluster. Finally, Hudson, Davies, Lucey, and Baggley are measuring Dn-sigma parameters of 6-10 ellipticals in each Lauer-Postman cluster with redshift less than 12,000 km s-1. This will result in distance errors of approx 8% per cluster.

There are two major new redshift survey projects in preparation. A British collaboration led by Ellis plans to measure redshifts for 250,000 galaxies to bJ = 19.7 selected from the APM galaxy catalog in a series of fields in the Southern Sky, using the 2dF 400-fiber spectrograph on the Anglo-Australian telescope (Gray et al. 1992). The survey geometry consists of two long strips in the Fall and Spring skies, plus 100 randomly placed fields of 2° diameter, totaling 0.53 ster. The principal motivation is to measure the large-scale power spectrum of the galaxy distribution, redshift space distortions to constrain Omega0, and evolutionary effects.

The Sloan Digital Sky Survey (SDSS) will use a dedicated 2.5m telescope to survey 3 ster around the Northern Galactic Cap with CCD's in five photometric colors. A multi-object spectrograph with 640 fibers will be used to carry out a flux-limited redshift survey of galaxies to roughly R = 18.0. Over five years, this survey will measure redshifts for ~ 106 galaxies, with a median redshift of approx 31, 000 km s-1. The survey will see first light in the second half of 1995. Details may be found in Gunn & Knapp (1993) , and Gunn & Weinberg (1995) . The SDSS is one of the few large-scale surveys in in which the photometric data from which the redshift galaxy sample will be selected is obtained as part of the survey itself. The use of CCD data and careful calibration guarantees that it will be the best calibrated of these surveys. It does not go as deep as the 2dF survey mentioned above, but covers much more area.

Thus we look forward to tremendous growth in the quantity and quality of both peculiar velocity and redshift data. We set forth a series of questions in the beginning of this review (Section 2.5) which we hoped to address with the data available. We have reviewed the analyses that have been done with redshift and peculiar velocity surveys to answer these questions. However, as we have summarized in this concluding chapter, there are few of these questions for which we now have definitive answers. Indeed, most of the quantities we hope to measure are known to within a factor of two at best, and more often only within an order of magnitude. We expect that the next decade will be a period of intense activity in this branch of observational cosmology, during which superior data and a more complete understanding of the theoretical issues will allow us to make observational cosmology a precision science; there is no doubt qualitatively new science to be discovered when we measure the power spectrum on large scales, the value of Omega0, the bias parameter of different galaxy types, and many other quantities, to 10% accuracy.


We thank Alan Dressler and Sandra Faber for comments and suggestions on parts of the text. Avishai Dekel supplied two of the figures. Karl Fisher, Mike Hudson, and David Weinberg read through the entire paper and made many valuable comments; in addition, we received useful comments and suggestions from Yehuda Hoffman, David Burstein, Roman Juszkiewicz, and an anonymous referee. Maggie Best helped tremendously in the compilation of the references. JAW thanks his collaborators on the Mark III project for permission to discuss aspects of this work prior to publication. MAS acknowledges the support of the WM Keck Foundation during the writing of this review.

36 An obvious exception to this statement is the Lauer-Postman (1994) bulk flow; if it is confirmed by further observations, we may find ourselves questioning the gravitational instability paradigm. Back.

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