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*)
*k*^{n}
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* 0.03*h*
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.

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*)
*k*^{2} 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
*G**µ* ~ 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.