5.4. Observational Constraints

There are several important observational constraints, on large and small scales, which are now available. The plethora of structure formation scenarios just considered can be weighed against the data to make some appear implausible. No structure formation theory to date, however, can satisfy all the constraints. Still, significant progress is made when theories can be falsified by good data. We being with the largest scale constraints.

5.4.1. Large Scale Constraints

COBE: In the early universe matter and radiation were coupled. In this circumstance, any density fluctuation in the matter would represent gravitational potential wells in which the radiation (photons) would have to climb out of in order to escape. This effect, called the Sachs-Wolfe effect (discussed in Chapter 3), causes the photons that are in these potential wells to lose a small bit of energy and become redshifted with respect to photons that are not in the vicinity of a potential well. At the surface of last scattering, these small energy differences will be manifest in the CMB as small temperature anisotropies. The structure and amplitude of the temperature anisotropy map of the CMB directly reflects the spectrum of initial density fluctuations that produced the structure observed today. As there is a spectrum of density perturbations, then each successive perturbation the photon encounters may be either of smaller or larger amplitude than the perturbation previously encountered. Thus, photons can either gain (blueshift) or lose (redshift) energy through these repeated encounters. Under inflationary cosmology, density perturbations are generated through initial quantum fluctuations in the inflation field. These are predicted to be highly Gaussian in nature. In this case, the corresponding temperature fluctuations in the CMB, over sufficient angular scale, will also be Gaussian. However, even fluctuations that are non-Gaussian will again, when averaged over many horizons, produce mostly Gaussian temperature fluctuations. Hence, the detection of any higher order departure from a purely Gaussian temperature fluctuation spectrum in the COBE data would be highly significant. To date (see Hinshaw et al. 1995) none have been convincingly detected and this would seem to rule out many exotic models which appeal to non-Gaussian initial density fluctuations.

For Gaussian fluctuations there is a simple way to estimate the approximate expected anisotropy. The relation between scale factor and redshift in the matter dominated Universe is:

Hence, a(t) has grown by a factor of 30,000 from the redshift epoch z = 1000 (where decoupling occurs) until z = 0. Since the observed Universe exhibits factors of two over density (e.g., / = 1) on large scales, then its clear that a perturbation, of amplitude 30,000^-1 could have been present at the surface of last scattering. These leads to the simple expectation that the fluctuation level should be a few x 10^-5. Based on 4 years worth of data analyzed, the observed fluctuation level is 1.5 x 10^-5 (see Bennett et al. 1996). In addition, the measured spectral index is n 1.2 ± 0.3, perfectly consistent with the n = 1 prediction from inflation and scale invariant fluctuations. Overall the COBE data provides an accurate normalization for of structure on large scales. Any structure formation scenario, regardless of its nature, must be firmly anchored to this normalization. Furthermore, the fact that anisotropy was detected, at about the expected level, really provides excellent confirmation that the gravitational instability paradigm must be basically correct.

At recombination, the angular size of the horizon is approximately 2 degrees but the COBE beam has a resolution of 7 degrees and hence the measured anisotropy is beam-averaged over a few horizon scales. Within one horizon, there can be considerably larger amplitude fluctuations which correspond to some of the large scale structure features previously discussed. Fluctuations on the degree scale are particularly important to detect, as these may well correspond with the largest scale features in the observed power spectrum of galaxies. Detection of anisotropy on smaller angular scales (e.g., 0.5 - 2 degrees) by using terrestrial techniques (e.g., balloon borne detectors or direct observation with radio telescopes) provides yet more evidence in favor of the gravitational instability paradigm. The recent observations with the Cambridge Anisotropy Telescope (Netterfield et al. 1997), have strongly confirmed the COBE results on these smaller scales. However, as pointed out by Kogut and Hinshaw (1996), there is considerable disagreement between various experiments on this angular size scale. An overall mean of the anisotropy measurements on the degree scale is approximately 3.5 ± 1.0 x 10^-5.

In sum, the COBE results rule out any model which predicts anisotropies of order 10^-4 or greater. Because of that, these models are not discussed here. The COBE normalization of the power spectrum would also seem to strongly rule out the standard CDM model. The highly Gaussian nature of the temperature fluctuations is inconsistent with topological defects and similar models. Hu and Sugiyama (1995) also show that the COBE anisotropy is fairly inconsistent with PBI. The surviving models are the variations of CDM all of which predict a nearly scale-invariant spectrum as predicted by inflation and observed by COBE.

The value of H₀

While many CDM based models do survive the COBE test, most of them are unlikely to survive the H₀ constraint if H₀ is above 70. Low H₀ and MDM models will be ruled out. PBI is not ruled out. Furthermore, if H₀ is above 80, then a positive cosmological constant is required if we believe the ages of globular clusters. If this is the case, then obviously non-zero models are favored.

The Observed Power Spectrum

Redshift surveys of galaxies and the generation of the power spectrum is now something of a cottage industry. Large surveys in the Northern and Southern Hemispheres are well underway and, as of early 1995, an all sky, "magnitude-limited" (but see Chapter 6 for the myth of magnitude limited galaxy samples) sample of 15,000 galaxies was available. Various data sets can be constructed over various magnitude limited ranges and angular regions of the sky. To date, all derived power spectra agree within the errors out to a scale of 100 h^-1 Mpc and the power spectra keep rising. The disagreement between various surveys or samples occurs on scales of 200-400 h^-1 Mpc range and in the particular scale the power spectrum turns over. The existence of power on scales of at least 100 h^-1 Mpc is no longer in dispute (see Landy et al. 1996).

However, its clear that not enough data exists on large scale to have a definitive representation of the power spectrum on these length scales. One of the primary goals of the Sloan Digital Sky Survey, which expects to obtain 10⁶ redshifts over 5 years, is to obtain this large scale data. Hence, any attempt at constraining structure formation scenarios by bridging the gap between the well-sampled region out to 100 h^-1 Mpc and the COBE Scale ( 1000 h^-1 Mpc) is premature. Rather than reproducing tons of power spectra, it suffices to say that most strongly resemble that shown in Figure 5-1. Following the most recent treatment by Lin et al. (1996) the classes of models that remain consistent with the power spectrum are:

Flat CDM models with ₀ 0.4-0.5 , 0.5-0.6, and H₀ 50. These models have very little bias (e.g., b = 0.9).

Open CDM models with ₀ 0.5 and H₀ as large as 80 and have b = 0.9.

Flat ₀ = 1 models which have mixed dark matter such that the contribution of neutrinos to is 0.2 These models require H₀ 50 and are mildly anti-biased (b = 0.8).

Flat ₀ = 1 models with a tilted spectrum (n 0.7). These also require H₀ 50 and are mildly biased (b = 1.1-1.3).

Even though these models do survive the constraint imposed by the observed power spectrum of the galaxy distribution, most of them could also be eliminated if H₀ is greater than 70.

The clear result is that CDM models, if they are to be salvaged, require exotic modifications such as non-zero , an open Universe, a mixture of HDM or a tilted spectrum. The standard CDM model is no longer viable. These conclusions are consistent with another way to characterize large scale structure, namely by the observed topology. Topology is a measure of the connectedness of high and low density regions in the Universe. In the most recent treatment which incorporates all the available data, Gott et al. (1996) conclude that the standard CDM model is also ruled out while the non-zero and/or CDM open Universe models are consistent with the data.

Clusters of Galaxies

The number density of rich clusters, their baryonic mass fractions, the amount of substructure that they contain, the cluster-cluster correlation function and epoch of virialization of clusters are all probes of and structure formation scenarios. In general, most of the structure formation models under consideration do not over-produce rich clusters and are consistent with the cluster-cluster correlation function. Cluster baryonic mass fractions, however, have become a recent concern. Since most of the baryons in a cluster are not in the member galaxies, but rather in the hot intracluster medium (ICM), accurate cluster masses as inferred from X-ray observations are required.

Simon White and collaborators (White et al. 1993) have shown that the ratio _b / ₀ measured for a cluster should not be significantly different than the Universal value. The baryonic mass in clusters consists of two forms, a visible component (e.g. luminous galaxies and cluster X-ray emission) denoted by f_b and a dark component (e.g., stellar remnants, low mass stars). The total baryonic density _b is inferred from primordial nucleosynthesis as previously discussed. Hence, if f_b can be determined for clusters then ₀ can be inferred from the relation ₀ = _b / f_b. Current observations indicate that f_b 0.04h^-3/2. When combined with the nucleosynthesis limits (_b ~ 0.015h^-2 - Walker et al. 1991), this leads to ₀ 0.3h^-1/2. To reconcile this with = 1 models requires either H₀ 30 or the possibility that total cluster masses have been systematically underestimated. The latter possibility does not appear to be the case (Evrard et al. 1996) and hence the measured values of f_b in clusters appears quite inconsistent with = 1.

But a low value of appears to be inconsistent with the substructure arguments that suggest the formation of clusters is still on going (or at least terminated rather recently). Late cluster formation requires high . Identifying the formation epoch of clusters also is a strong constraint as in any CDM model, clusters form after galaxies have formed and hence we would not expect much clustering at high redshift. Recent data obtained with the Hubble Space Telescope is beginning to suggest that galaxies are somewhat clustered at redshifts z = 2-3 which is consistent with deep ground based images of fields around high redshift QSOs that show they are often surrounded by other galaxies. While this is not strongly inconsistent with the gravitational instability paradigm for the formation of clusters of galaxies, it does serve as a reminder that other important non-linear effects may come in to play that effectively speed up the formation process. One of those processes might be the statistical biasing discussed earlier.

Identifying cluster formation, however, is a very ambiguous problem. Substructure in nearby clusters, for instance, indicates that they are still acreting material. This kind of cluster augmentation provides clear evidence of merger processes involving smaller structures which have shorter collapse times. If there is sufficient power on large scales so that these smaller units are available to infall at later times (as appears to be the case), then cluster formation is a process that occurs on a much longer timescale than galaxy formation. Still, there are examples of distant clusters (z 0.9) which look very much like the core of the Coma cluster (see figure 3-3) looks now (see Postman et al. 1996). This brings up a possible very powerful test of cluster formation and virialization that was first proposed by Perrenrod and Henry in 1980.

Most nearby clusters with strongly virialized cores are also sources of X-ray emission. The x-ray emission is a consequence of intracluster gas being heated by the cluster potential to its virial temperature. For typical clusters, the virial temperature is a few million degrees leading to strong x-ray emission by the gas in the energy range 0.5-5 kev. Equilibrium should occur approximately on a dynamical timescale. The origin of the intracluster gas is unclear although likely sources are 1) tidally liberated gas caused by interactions between proto-galaxies as they form in the overall cluster potential and 2) gas which has been driven out of galaxy potential wells into the cluster potential by energetic internal processes such as star formation and supernova heating and 3) left over gas that didn't get incorporated into any galaxy size potentials. For most clusters, the x-ray gas contains strong lines of ionized iron indicative of metal abundances which are near solar. This indicates processing of the gas in galaxies and subsequent expulsion (via supernova heating).

Observations of the evolution of the X-ray luminosity function of clusters as a function of redshift may reveal the epoch of cluster core virialization. To date, the sensitivity of various X-ray satellites (e.g., EINSTEIN, ROSAT, ASCA) has allowed the detection of X-ray emission in clusters of galaxies out to z 1 (Hattori et al. 1997). To date, the most distant cluster detected in X-rays has z = 1.0 and was detected on the basis of an emission line at 3.35 keV which is thought to be the redshifted 6.7 keV iron line (Hattori et al. 1997). There is not yet enough data to see if their is a characteristic redshift at which most clusters "turn-on". However, with future increases in X-ray satellite sensitivity (e.g., AXAF) it may be possible to either detect or strongly constrain this "turn-on" epoch to redshifts less than some value. In fact, while the existence of some clustering at high redshift may not be too surprising, the detection of substantial x-ray emission originating from a virialized cluster core at redshifts z 2 would seem to either strongly rule out the formation of these cores via gravitational instability or indicate a new population of significantly denser clusters than currently are known.

Other aspects of clusters of galaxies can also act as a constraint on structure formation models. Zabludoff and Geller (1994) use kinematic observations of the densest clusters of galaxies to show that models which match the power on large scales do not match the observed distribution of velocity dispersions. Moreover, biased models predict too few high velocity dispersion clusters compared to the number of low velocity dispersion clusters. In fact, they conclude that no model matches both the statistics of the galaxy distribution on large scales and the small scale velocity dispersion characteristics of clusters of galaxies. Crone and Geller (1995) consider the effects of merging on the evolution of cluster velocity dispersions. Their models show that the abundance of clusters with _v 1200 km s^-1 increases with time, while the number of groups decrease with time. The particular evolutionary rates depend upon choice of cosmogenic scenario. The models which match the data best are either ₀ = 0.2 or biased = 1. All models, however, predict fewer low velocity dispersion systems than is actually observed (see Zabludoff et al. 1993). Finally, Dell'Antonio et al. (1995) show that the baryonic fraction in low velocity dispersion clusters is approximately one-half that observed for higher velocity dispersions clusters (see also Evrard et al. 1996). This is a curious result as it suggests baryonic and dark matter may be segregated in different ways depending upon the overall depth of the potential well. This conclusion is strongly at odds with the good match between the X-ray and optical distributions for both high and low _v clusters.

QSO absorption lines

A possibly strong constraint on structure formation timescale comes from from observations of QSO absorption lines. As baryonic gas inside a dark matter potential collapses and forms stars, any massive stars rapidly evolve and feedback heavy elements into the gas. The production of the first heavy elements can be equated to the "epoch of galaxy formation". To detect these metals, the line of sight to a QSO must pass through one of these "forming galaxies". The probability of this occurring is directly proportional to the size of the protogalaxy and the number density of QSOs at high redshift. To date, metal lines, specifically those of Carbon (e.g., Carbon IV) can be identified in QSO spectra back to a redshift of 4 (Steidel 1992). The distribution of metal line strengths with redshifts (reproduced in Figure 5-2) indicates the following:

Prior to z = 2.5-3, metal line strength is fairly low indicating little processing.

A major episode of metal-production, which can plausibly be identified with the formation of the disk components of galaxies seems to be occurring between z = 1.5 - 3. During this time the mean metallicity appears to increase from 0.01 solar to 0.1 solar.

Figure 5-2: Distribution of the number density of various QSO absorption line systems. The solid crosses are for pure hydrogen asbsorption systems whose number density, as expected, increases with increasing redshift. Of interest here, is the behavior of the Carbon IV systems, dashed crosses, whose number density evolution shows a strong monotonic decrease with increasing redshift, approaching zero by redshift 3.5. Figure courtesy of Chuck Steidel (see Steidel 1992).