Annu. Rev. Astron. Astrophys. 1994. 32: 319-70 Copyright © 1994 by Annual Reviews. All rights reserved

2.2. Structure Formation Theories

The standard model of structure formation, which we shall use to explore the CMB fluctuations on degree scales, has been the Cold Dark Matter (CDM) model (e.g. Peebles 1982a, Blumenthal et al 1984, Davis et al 1985, Frenk 1991, Ostriker 1993). In this model, ₀ = 1, with a variable fraction _B residing in baryons and the rest in massive (nonrelativistic) dark matter. The initial fluctuations are assumed to be Gaussian distributed, adiabatic, scalar density fluctuations with a Harrison-Zel'dovich spectrum on large scales, i.e. P_mat(k) kⁿ with n = 1.

Under these assumptions the matter power spectrum can be calculated. It depends only on _B and the Hubble constant. The main feature is a turnover from the P_mat(k) k form at large scales (small k) to a k^-3 falloff at small scales (large k), which occurs at k 0.03h Mpc^-1, the horizon size at matter-radiation equality. The reason for this turnover is that perturbations that enter the horizon before the universe becomes matter-dominated do not grow. This retards the growth of fluctuations on small scales, which spend longer periods inside the horizon during the radiation-dominated era. It is assumed the baryons fall into dark matter potentials that formed due to gravitational instability once the photon drag becomes small enough. The standard (₀ = 1, n = 1, h = 0.5) CDM matter spectrum (Holtzman 1989) has been plotted as the solid line in Figure 1 along with the power spectrum for the other models discussed in this section.

Figure 1. The master power spectrum for a range of models of structure formation. The models are: CDM ( solid), tilted CDM (dotted), HDM ( short dashed), CDM ( long dashed), MDM ( dot-short-dashed), and BDM ( dot-long-dashed). All models have been arbitrarily normalized to 1 at k = 0.2h Mpc-1, which corresponds roughly to 8 normalization.

An alternative to CDM, now out of favor, is the Hot Dark Matter (HDM) model (e.g. Zel'dovich 1970, Bond et al 1980, Bond & Szalay 1983, Centrella et al 1988, Anninos et al 1991, Cen & Ostriker 1992). In this model, the difference between _B and ₀ = 1 is made up of relativistic particles. These relativistic particles have a minimum scale on which gravitational instability can cause overdensities due to their free streaming. This leads to a small-scale (large k) cutoff in the power spectrum as shown by the short-dashed line in Figure 1 (Holtzman 1989) for = 0.94 and h = 1/2(m = 22 eV). In this model, galaxies form by the fragmentation of larger structures unless there is some extra small-scale power (e.g. Dekel 1983, Brandenberger et al 1990, Gratsias et al 1993).

A currently popular idea (e.g. Schaefer & Shafi 1992, M. Davis et al 1992, van Dalen & Schaefer 1992, Taylor & Rowan-Robinson 1992, Holtzman & Primack 1993, Klypin et al 1993) is the Mixed Dark Matter (MDM) model (Shafi & Stecker 1984, Achilli et al 1985, Umemura & Ikeuchi 1985, Valdarnini & Bonometto 1985, Bardeen et al 1987, Schaefer et al 1989, Holtzman 1989) which combines the above two scenarios, and may account for the extra power in large-scale structure measurements over the predictions of CDM (e.g. Maddox et al 1990, Kaiser et al 1991). The current favorite has an HDM component (of relic, massive neutrinos say, with m 7 eV) of 30% of closure density and a CDM component (plus baryons) making up the rest. It is unclear whether this gives the required amplitude of fluctuations at smaller scales (e.g. Bartlett & Silk 1993, Kauffmann & Charlot 1994, Pogosyan & Starobinskii 1993, Mo & Miralda-Escude 1994). The radiation power spectrum for this model is very similar to that of CDM except on the very smallest scales where it is already strongly damped by the thickness of last scattering (see Section 3.2). An approximation to the matter power spectrum is shown by the dot-short-dashed line of Figure 1 (Holtzman 1989).

Alternatively, the difference between the CDM component and ₀ 1 could be made up of a cosmological constant (e.g. Peebles 1984, Turner et al 1984, Efstathiou et al 1990, Kofman et al 1993). Including a nonzero has little effect on large-scale structure, other than the enhancement of large-scale power due to lowering , an effect that is used to account for the galaxy-galaxy correlations found in deep surveys (Maddox et al 1990, Loveday et al 1992) as well as to allow an older universe (e.g. Gunn & Tinsley 1975). The matter power spectrum for = 0.8 and h = 1 (Efstathiou et al 1992) is shown in Figure 1.

On all but the smallest scales, the CDM, HDM, CDM, and MDM models give very similar radiation power spectra, showing that large- and intermediate-scale CMB fluctuations are not very sensitive to the form of the dark matter that makes ₀ = 1 in these models. Since the shape of the matter spectra are changed, however, the relation between the large-scale power and the bias is changed, affecting comparison with large-scale structure studies.

Instead of modifying the matter content of the model, one variant sticks with ₀ = 1 in CDM, but modifies the initial power spectrum away from the Harrison-Zel'dovich form (e.g. see Vittorio et al 1988, Salopek et al 1989, Cen et al 1992, Lucchin et al 1992, Fry & Wang 1992, Liddle et al 1992, Adams et al 1992, Cen & Ostriker 1993, Gottlober & Mucket 1993, Muciaccia et al 1993, Sasaki 1993, and Suto et al 1990 for CMB constraints). Such "tilted" spectra can be (but are not always) associated with a stochastic background of gravitational waves. The most commonly discussed model has a spectral slope of n 0.7 (to be compared with the standard n = 1 slope) and is shown by the dotted line in Figure 1.

The Baryonic Dark Matter model (BDM, also known as PBI or PIB, see Section 7.2) differs from its cold and hot dark matter rivals in that it does not assume ₀ = 1 and postulates isocurvature rather than adiabatic fluctuations (Peebles 1987a, b; Cen et al 1993). The fluctuation spectrum is not taken to be Harrison-Zel'dovich, but the slope of the spectrum of entropy fluctuations (P_ent(k) k^m) m is a free parameter that is adjusted to fit observational data. The transfer function T(k) k² at small k and T(k) 1 at large k. The principal feature of the (matter) power spectrum is a large "bump" at the matter-radiation Jeans length (Doroshkevich et al 1978, Hogan & Kaiser 1983, Jørgensen et al 1993), the height of which will depend on the ionization history and the values of _B and h. The "bump" at large scales may help explain some of the large-scale velocity measurements (Groth et al 1989). Figure 1 shows a particular model (Cen et al 1993) which has m = - 0.5, ₀ = _B = 0.1, h = 0.8, and x_e = 0.1 (dot-dash line).

An orthogonal approach to structure formation is that of the defect models, e.g. global monopoles and textures or cosmic string models. In these models, field configurations known as "defects" which arise due to a phase transition in the early universe form the seeds for matter and radiation fluctuations. The prototypical defect model is the cosmic string model where the "defect" is one dimensional. In this model all the string properties are described by the mass per unit length of the string µ. Cosmologically interesting strings have Gµ ~ 10^-6. In this model most of the fluctuations are imprinted at high redshift. The fluctuations are non-Gaussian, having strong phase correlations which lead to sharp line discontinuities on the sky (Kaiser & Stebbins 1984, Brandenberger & Turok 1986, Stebbins 1988, Bouchet et al 1988). Most defect models being discussed now (e.g. D. Bennett et al 1992, Bouchet et al 1992; Veeraraghavin & Stebbins 1992, Hara et al 1993, Perivolaropoulos 1993a, Bennett & Rhie 1993, Pen et al 1994, Hindmarsh 1993, Coulson et al 1994, Vollick 1993, Durrer et al 1994, Stebbins & Veeraraghavan 1993, Brandenberger 1993) will produce a roughly scale-invariant CMB fluctuation spectrum as required. The major difference lies in their non-Gaussian nature. Since on large scales (e.g. COBE), many defects contribute to the observed fluctuations, the central limit theorem suggests that the fluctuations will look Gaussian. One has to go to smaller scales (< 1°) to observe significantly non-Gaussian structure. It is not clear whether there will be an easily detectable difference between the Gaussian and non-Gaussian models currently being considered.