What Comes First - Galaxy-size or Cluster-size Potentials?
While the real Universe is likely some complex hybrid of HDM and CDM, we can consider three limiting cases of structure formation that arise under the gravitational instability paradigm. Each of these three cases assumes that a single kind of particle dominates the mass in the Universe.
The Top-down Scenario:
Under this scenario, all fluctuations which are smaller than a horizon size are erased by the free streaming motion of relativistic particles. This scenario best applies in a HDM dominated universe in which the HDM particle is a neutrino. A convenient way to express the critical density in units of energy density is
The freestreaming of neutrinos stops when they become non-relativistic. This occurs through expansion and cooling of the Universe. When the universe has cooled to the point where kT is equal to the rest mass energy of the neutrino they become non-relativistic.
At this point the universe has some horizon size, Rhor , and variations in neutrino density can only occur on scales larger than Rhor.
The total mass within the horizon is approximately
where mnu is the neutrino mass.
For mnu = 30 eV, the horizon mass is 1016 solar massses, which is similar to the mass of the putative Great Attractor. For a neutrino dominated Universe, potentials of this mass would be the first to form. The formation of smaller scale structure would occur from fragmentation of gas within these potentials. This scenario then predicts that all smaller scale structures should be embedded in larger scale structures, which is in good qualitative agreement with the observations. The greatest strength of this scenario lies in its natural ability to produce power on large scales. Its greatest weakness lies in the supreme difficulty of producing small scale structure early on in the Universe.
The Bottom-up Scenario:
This is the scenario that Newton would have preferred as structure in the Universe is built by the hierarchical gravitational clustering of subunits. The minimum mass of these subunits is set by the Jeans instability criterion previously described. In a CDM dominated Universe, the hypothetical CDM particles are not subject to radiation drag in the early Universe and thus can begin to clump via gravitational instability at very early times (perhaps as early as the end of the inflationary epoch).
Amplification of these seeds will then produce density fluctuations which can acrete baryonic material after recombination. As this material continues to flow into the density fluctuation, it continues to grow in size thus sweeping up more material in the vicinity. Eventually galaxy size objects are made via this gravitational coalescence of subunits and then clusters of galaxies are made later in the Universe via the continuation of this gravitational clustering hierarchy.
The greatest strength of the CDM dominated scenario is the natural production of small scale structure that should be embedded in a large scale distribution of dark matter. Furthermore, galaxy formation is something that occurs early on. The greatest weakness of the CDM model lies in its inability to produce the truly large scale structure that is observed.
An interesting consequence of the bottom-up scenario is the suggestion that their may be totally dark galaxies, that is, CDM dominated potentials which were unable to trap and confine baryonic gas that subsequently fragmented into stars to produce a luminous galaxy to mark the location of that potential. In addition, the bottom-up scenario also predicts that cluster formation via gravitational merging of subunits is continuing at the present epoch. This would provide a natural source to generate the observed peculiar velocities.
The detection of high redshift clusters would not be expected under this scenario.
The Baryon Dominated Scenario:
This is a variant of the bottom-up scenario that would occur in a low OMEGA Universe that is dominated by baryons. Of course, baryonic fluctuations can not easily grow during the radiation dominated error so the relevant issue is the amplitude of the Jeans length at the time of recombination. While it is quite unlikely that the Universe is baryon dominated, this scenario naturally produces old globular clusters whose formation is difficult to understand in the other two scenarios (because globular clusters appear to contain very little Dark Matter).
It also seems likely that the formation of galaxies via the gravitational clustering of Jeans mass density enhancements is not very efficient, and we would not expect to find many baryons in galaxies. Hence, a test of this scenario would involve determining the ratio of baryons that are in galaxies compared to those that are distributed in an intergalactic population.
While there is some evidence for an intergalactic population of baryons (see chapter 6) it is fairly clear that there is not an order of magnitude more baryons outside of galactic potentials than inside of them.
The concept of linear biasing between the distribution of light and mass was introduced in Chapter 3. Since the structure formation models that we are considering all predict the distribution of mass, the role of biasing is pivotal if mapping the observed light distribution back into the model.
One of the first indicators that biasing does exist came from comparing the galaxy-galaxy correlation function to the cluster-cluster correlation function. The observations indicate that while both correlation functions have the same power-law slope, the cluster-cluster correlation function has an amplitude about 20 times larger.
If galaxies and clusters have both arisen due to gravitational instability and amplification of fluctuations in the primordial density field, then both scales should trace large scale structure equally well. The large difference in correlation amplitude, however, is quite inconsistent with this expectation.
To reconcile this discrepancy, Nick Kaiser (1984) shaped the following physical argument. Very rich clusters of galaxies are obviously the most massive objects that have collapsed by the present epoch into some kind of equilibrium state. These clusters (e.g. the Coma cluster) are also extremely rare as the number density of rich clusters is orders of magnitude smaller than the number density of galaxies.
Since we are assuming Gaussian fluctuations, then clusters of galaxies have to form at rare, high-sigma peaks in the distribution. A general statistical property of Gaussian random fields is that the rare high-sigma peaks tend to occur near other high peaks. Quantitatively, the relatively weak correlation on large scales is amplified with respect to the background matter density by a factor of v2 where v is the number of sigma away from the mean. Since the difference in correlation amplitude is 20 thyen v ~ 4 .
In this way the observed number density of rich clusters is an observational constraint on the models.
Biasing however, has had a not very robust history. When the concept was first introduced in the mid-80's, a bias parameter of b = 2.4 was required to reproduce the strong small scale clustering of galaxies. When more data was acquired, it became clear that b = 2.4 could not hold on larger scales. Today most studies indicate that b ~ 1 (little or no bias).
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 \etal 1995) none have been convincingly detected and this would seem to rule out many exotic models which appeal to non-Gaussian initial density 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. 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 Ho:
While many CDM based models do survive the COBE test, most of them are unlikely to survive the Ho if Ho > 70. Low Ho and Mixed Dark Matter models will be ruled out.
The Power Spectrum of Galaxies
This was discussed last time. The most recent treatment of the problem by Lin \etal 1996 suggests the following classes of models that remain consistent with the power spectrum: