Even if most of the dark matter is of the cold variety, a little hot
dark matter can have a dramatic effect on the predicted distribution of
galaxies. In the early universe, the free streaming of the fast-moving
neutrinos washes out any inhomogeneities in their spatial distribution
on the scales that will later become galaxies. If these neutrinos are
a significant fraction of the total mass of the universe, then although
the density inhomogeneities will be preserved in the cold dark matter,
their growth rates will be slowed. As a result, the amplitude of the
galaxy-scale inhomogeneities today is less with a little hot dark
matter than if the dark matter is only cold. (With the tilt *n* of
the primordial spectrum *P*_{p}(*k*) = *A
k*^{n} fixed - which as we
discuss below is not necessarily reasonable - the fractional
reduction in the power on small scales is
*P / P*
8
_{} /
_{m}
[58]. See
Fig. 1 for examples of how
the power spectrum *P(k)* is affected by the addition of hot dark
matter in
_{m} = 0.4 flat
cosmologies.) Since the main problem with
_{m} = 1 cosmologies
containing only cold dark matter is that the
amplitude of the galaxy-scale inhomogeneities is too large compared to
those on larger scales, the presence of a little hot dark matter
appeared to be possibly just what was needed. And, as was mentioned at
the outset, a CHDM model with
_{m} = 1,
_{} = 0.2, and
Hubble parameter *h* = 0.5 is perhaps the best fit to the galaxy
distribution in the nearby universe of any cosmological model. The
effects of the relatively small amount of hot dark matter in a CHDM
model on the distribution of matter compared to a purely CDM model are
shown graphically in
[59];
cf. also [60]. As
expected, within galaxy halos the distribution of cold and hot
particles is similar. But the hot particles are more widely
distributed on larger scales, and the hot/cold ratio is significantly
enhanced in low-density regions.

The first step in working out the theory of structure formation is to
use linear perturbation theory, which is valid since cosmic microwave
background measurements show that density fluctuations are small at the
redshift of recombination, *z*_{r} ~ 10^{3}. The
most extensive early
calculations of this sort were carried out by Holtzman
[61,
62],
who concluded that the most promising
cosmological models were CHDM and
CDM
[63].
The most efficient method of
computing the linear evolution of fluctuations now is that used in the
CMBFAST code
[64].
An alternative Monte Carlo treatment of
the evolution of neutrino density fluctuations was given by
[65],
but the differences from the usual treatment appear to
be small. Detailed analytic results have been given by
[66,
67]
and reviewed in [60].
But the key
point can be understood simply: there is less structure in CHDM models
on small scales because the growth rate of cold dark matter
fluctuations is reduced on the scales where free streaming has wiped
out neutrino fluctuations. Let us define the fluctuation growth rate
*f* by

(9) |

where (*k*) is the
amplitude of the fluctuations of wave number
*k* = 2 /
in cold dark matter, and as
usual *a* = 1 / (1 + *z*) is the scale factor. For
_{m} = 1 CDM
fluctuations, the growth rate
*f* = 1. This is also true for fluctuations in CHDM, for *k*
sufficiently small that free-streaming has not significantly
decreased the amplitude of neutrino fluctuations. However, in the
opposite limit *k* ->
[43,
60],

(10) |

assuming that _{c} +
_{} = 1. For example, for
_{} = 0.2,
*f*_{} =
0.87. Even though the growth rate
is only a little lower for these large-*k* (i.e., short-wavelength)
modes, the result is that their amplitude is decreased substantially
compared to longer-wavelength modes.

The next step in determining the implications for structure formation
is to work out the effects on nonlinear scales using N-body
simulations. This is harder for Cold+Hot models than for CDM because
the higher velocities of the neutrinos require more particles to
adequately sample the neutrino phase space. The simulations must
reflect the fact that the neutrinos initially have a redshifted
Fermi-Dirac phase space distribution
[68]. These CHDM
simulations were compared with observational data using various
statistics. CHDM with
_{} = 0.3, the value indicated by
approximate analyses
[63,
69], was shown to
lead to groups of galaxies having substantially lower velocity
dispersions than CDM, and in better agreement with observations
[70].
But it also leads to a Void Probability Function (VPF)
with more intermediate-sized voids than are observed
[71].
This theory had so little small-scale power that a quasi-linear
analysis using the Press-Schechter approximation
showed that there would not be enough of the high-column-density
hydrogen clouds at high redshift *z* ~ 3 known as damped
Lyman- systems
[72,
73,
74]. But CHDM with
_{} = 0.2 suppresses small-scale fluctuations less and therefore
has a better chance of avoiding this problem
[75]. Simulations
[76]
showed that this version of CHDM also has a VPF in good
agreement with observations
[77]. The group
velocity
dispersions also remained sufficiently small to plausibly agree with
observations, but it had become clear that the N-body simulations used
lacked sufficient resolution to identify galaxies so that this
statistic could be measured reliably
[78].

A resolution problem also arose regarding the high-redshift damped
Lyman- systems. Earlier
research had been based on the idea
that these systems are rather large disk galaxies in massive halos
[79],
but then high-resolution hydrodynamical simulations
[80]
showed that relatively small
gaseous protogalaxies moving in smaller halos provide a good match to
the new, detailed kinematic data
[81]. It thus
appeared possible that CHDM models with
_{}
0.2 might
produce enough damped Lyman-
systems. With the low Hubble parameter
*h* ~ 0.5 required for such
_{m} = 1 models, the
total neutrino
mass would then be 5 eV.

While neutrino oscillation experiments can determine the differences of squared neutrino masses, as we will briefly review next, cosmology is sensitive to the actual values of the neutrino masses - for any that are larger than about 1 eV. In that case, cosmology can help to fill in the neutrino mass matrix.

One example of this is the fact that if the hot DM mass is roughly
evenly shared between two or three neutrino species, the neutrinos
will be lighter than if the same mass were all in one species, so that
the free streaming length will be longer. A consequence is that, for
the same total neutrino mass and corresponding
_{}, the power
spectrum will be approximately 20% lower on the scale of galaxy
clusters if the mass is shared between two neutrino species
[1].
Since the amplitude and ``tilt'' *n* of the power
spectrum in CDM-type models is usually fixed by comparison with COBE
and cluster abundance, this has the further consequence that higher
*n* (i.e., less tilt) is required when the neutrino mass is divided
between comparable-mass neutrino species. Less tilt means that there
will be more power on small scales, which appeared to be favorable for
the CHDM model, for example because it eased the problems with damped
Lyman- systems
[1,
82].