The CDM spectrum.

The dark matter content of the Universe is said to be cold if particles becomes non-relativistic at sufficiently early times, so that the mass contained within the horizon at that time is much smaller than the typical galaxy mass. Thus, in the cold dark matter (CDM) scenario the free-streaming cut-off scale is to small to be of any cosmological relevance. The low velocity required for CDM particles can arise for two different reasons. Firstly, the particle mass is so large that they become non relativistic at a high temperature. This cannot be the case for massive neutrinos, for which a mass $\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$>$} }100 eV$ give an exceedingly high contribution to the density parameter (see eq.[33]). However, supersymmetric theories provide a large variety of exotic CDM candidates, such as photinos or gravitinos, with masses above $1 GeV$. Secondly, there can be particles, like axions, that never were in thermal equilibrium, so they have a very low thermal velocity, despite their small mass ( $\sim 10^{- 5}eV$). See, e.g., ref.[12] for a review of CDM candidates.

Although for CDM fluctuations neither free-streaming nor Silk damping introduce characteristic scales, a distortion of the spectrum is however generated by the Meszaros effect [23], which suppresses the growth of fluctuations of non-relativistic matter which cross the horizon before non-relativistic matter start dominating. In order to see how this happens, let us consider the equation for the evolution of non-relativistic matter fluctuations in a relativistic background:

$${d^2\delta \over dt^2}+2H {d\delta \over dt}-4\pi G\bar \rho_m\delta=0 .$$ (36)

Here the relativistic background enters only in determining the cosmic expansion rate $a(t)$. By introducing the new time variable $\tau = \bar \rho_m /\bar \rho_r$ ( $\tau \propto a$ since $\bar \rho_m \propto a^{-3}$ and $\bar \rho_r \propto a^{-4}$), eq.(36) can be rewritten as

$${d^2\delta \over d\tau^2}+{2+3\tau \over 2\tau(1+\tau)} {d\delta \over d\tau}-{3\over 2} {\delta\over \tau(1+\tau)}=0 .$$ (37)

The growing-mode solution of the above equation turns out to be $\delta \propto 1+ {3\over 2} \tau$. Thus, during radiation domination ($\tau \ll 1$), the fluctuation amplitude is frozen. Only when non-relativistic matter dominates ($\tau \gg 1$) the matter fluctuations start growing as $\delta \propto a$, as expected on the grounds of linear evolution. In order to evaluate the resulting distortion of the spectrum, let $\delta_k(t_i)$ be the amplitude of the fluctuation mode with wavenumber $k$ at some initial time $t_i$ before matter-radiation equality, and suppose that it crosses the horizon at $t_H$, also before equality. During this period the amplitude grows by a factor $t_H/t_i$. Since $t_H\propto k^{-2}$ during radiation domination, the amplitude of the perturbation after horizon crossing is frozen at the value $\delta_k( t_i)t_i^{-1}k^{-2}$ until matter starts dominating. Viceversa, fluctuations outside the horizon continue to grow according to linear theory so that no distortion of the spectrum occurs at such scales. The characteristic scale at which we expect a feature in the spectrum is that corresponding to the horizon size at $t_{eq}$,

$$\lambda_{eq}  \simeq  10 (\Omega_oh^2)^{-1} {\rm Mpc} .$$ (38)

If $P(k)\propto k^n$ was the primordial spectrum, its shape after $t_{eq}$ will be preserved at scales larger than $\lambda_{eq}$, while for $\lambda \ll \lambda_{eq}$ the freezing of the fluctuation amplitude tilts the spectral index to the value $n-4$. Precise computations of the processed spectrum in a CDM dominated Universe have been done by several groups (see, e.g., refs.[11] and references therein). A useful fit to the CDM transfer function can be given in the form

$$T(k) = [1+(ak+(bk)^{1.5}+(ck)^2)^\nu]^{-1/\nu}$$ (39)

[ $a=6.4 (\Omega_oh^2)^{-1}$ Mpc, $b=3.0 (\Omega_oh^2)^{-1}$ Mpc, $c=1.7 (\Omega_oh^2)^{-1}$ Mpc, $\nu=1.13$], obtained by Bond & Efstathiou [5], assuming the presence of three species of massless neutrinos and negligible contribution from the baryonic component ( $\Omega_b\ll \Omega_{CDM}$). According to eq.(39), as $k\to 0$, we have $T(k)\simeq 1$ and the primordial spectrum $P(k)\propto k$ is left unchanged. At small scales, $T(k) \propto k^{-2}$, so that $P(k)\propto k^{-3}$.

As opposed to the HDM scenario, a considerable amount of small-scale power is now present, so that the first fluctuations reaching non-linearity are at small scales. The resulting clustering proceeds in a ``bottom-up" way, with structures of increasing size forming from the tidal interaction and the merging of smaller structures. It is clear that the possibility for small-scale structures not to be disrupted as the hierarchical clustering goes on depends on the ability of the baryonic component to cool down and fully virialize before being incorporated within larger DM fluctuations. Taking into account dissipative effects allows one to identify the CDM fluctuations where galaxy formation takes place. Detailed investigations of galaxy formation in the CDM scenario (see, e.g., ref.[30]) have shown that the observed variety of galaxy morphology and the relative morphological segregation of the clustering can be nicely reproduced.

A series of detailed numerical simulations of structure formation in a CDM Universe have been realized (see, e.g., ref.[19]). The N-body experiments show that the primordial CDM spectrum is able to account for many aspects of the observed galaxy clustering at small and intermediate scales. Once a suitable prescription is assigned to identify galaxies in a purely dissipationless simulation, a CDM model with $\Omega_0=0.3$ and flatness restored by a cosmological constant term provides the correct correlation amplitudes, small-scale velocity dispersions and the mean number density of both galaxies and clusters.