The HDM spectrum.

In the hot dark matter (HDM) model the mass contained inside the horizon when constituent particles become non relativistic is much larger than the typical mass of a galaxy ( $\sim 10^{11}M_\odot$), so that particles with a low mass are required. A natural candidate for HDM constituent is the massive neutrino. If a neutrino species has non vanishing mass, then its contribution to the mean matter density is

$$\Omega_\nu \simeq  \left({m_\nu\over 100eV}\right) h^{-2} ,$$ (33)

where $m_\nu$ is expressed in $eV$. Although in the classical version of the Standard Model for the electroweak interaction the neutrino is considered massless, nevertheless there exists no fundamental reason which fixes $m_\nu =0$. Viceversa, several theoretical models have been proposed to generate a non-vanishing neutrino mass [37,14]. Although recent experimental evidences are actually indicating that at least one neutrino species should be massive [32], its mass could be too small to be cosmologically relevant.

For a neutrino of mass $m_\nu$, the redshift at which it becomes non-relativistic can be estimated to be $z_\nu \simeq 6\times 10^4 (m_\nu/30 eV)$, which corresponds to

$$M_{\nu,\mbox{\tiny H}} \simeq  2\times 10^{15} \left({m_\nu\over 30 eV}\right) M_\odot$$ (34)

for the mass contained inside the horizon at that epoch. Numerical calculations carried out by Bond & Szalay [6] give the transmission factor

$$T(k) = 10^{-(k/k_\nu)^{1.5}}         ;        k_\nu\simeq 0.4 \Omega_o h^2 {\rm Mpc}^{-2} ,$$ (35)

which suppresses all the fluctuation modes at wavelengths $\lambda < \lambda_\nu = 2\pi/k_{\nu} \simeq 40 (m_\nu/30 eV)^{-1}$ Mpc. In the HDM scenario, the smallest fluctuations surviving recombination are roughly on the same scale as large galaxy clusters. Accordingly, structure formations proceeds in a ``top-down" way; first large pancakes of mass $\sim 10^{15} M_\odot$ form, while galaxies originate later via fragmentation of structures at larger scales. Numerical simulations of structure formation in HDM dominated Universe have been done [36] and show the development of cellular structures, which are promisingly similar to those displayed by the redshift galaxy surveys. Big voids forms on scales comparable to the characteristic scale $\lambda_\nu$, which are surrounded by galaxies and clusters forming at the intersection between three of such cells. Unfortunately, the agreement with the observed galaxy distribution is only apparent. In fact, since the characteristic size of the earliest forming structures is $\lambda_\nu$, the variance of the matter distribution at this scale should be around one. This is quite difficult to reconcile with the much smaller correlation length displayed by the galaxy distribution, $r_o\simeq 5 h^{-1}{\rm Mpc}$. In order to alleviate this problem, we can either assume that galaxy formation occurred very recently (at $z\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$<$} }1$) so as to give no time for structures to become overclustered, or invoke some mechanism to reduce the galaxy clustering with respect to that of the underlying matter. It is however clear that, whatever way out we choose, additional problems arise. A too recent galaxy formation seems to be quite difficult to reconcile with the detection of high-redshift ( $z\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$>$} }3$) quasars. Also requiring that galaxies are less clustered than matter is at variance with the expectation that dissipative galaxy formation should preferentially occur in the deep wells of the gravitational potential field, which would give an increase of their correlation with respect to the DM distribution. It is however clear that, if a neutrino species were discovered to have a non-vanishing mass $m_\nu \simeq 30 eV$, we are obliged to consider HDM as responsible for structure formation and try to overcome in some way all the above difficulties.