**3.3. Free Streaming**

The most salient feature of hot DM is the erasure of small
fluctuations by free streaming. Thus even collisionless particles
effectively exhibit a Jeans mass. It is easy to see that the
minimum mass of a surviving fluctuation is of order
*M*^{3}_{Pl} /
*m*^{2}_{}
[42,
43].
Let us suppose that some process in the very early
universe - for example, thermal fluctuations subsequently
vastly inflated in the inflationary scenario -
gave rise to adiabatic fluctuations on all scales. In adiabatic
fluctuations, all the components - radiation and matter -
fluctuate together. Neutrinos of nonzero mass
*m*_{} stream relativistically
from decoupling until the temperature drops to
*T* ~ *m*_{},
during which time they traverse a distance
*d*_{} =
*R*_{H}(*T = m*_{}) ~ *M*_{Pl},
*m*_{}^{-2}. In
order to
survive this free streaming, a neutrino fluctuation must be
larger in linear dimension than
*d*_{}. Correspondingly,
the minimum mass in neutrinos of a surviving fluctuation is
*M*_{J,}
~ *d*^{3}_{}
*m*_{}
*n*_{}(*T* =
*m*_{}) ~
*d*^{3}_{}
*m*^{4}_{} ~
*M*^{3}_{Pl}
*m*_{}^{-2}. By
analogy with Jeans's
calculation of the minimum mass of an ordinary fluid
perturbation for which gravity can overcome pressure, this
is referred to as the (free-streaming) Jeans mass.

A more careful calculation [43, [44] gives

(7) |

that is, *d*_{} = 41
(*m*_{} / 30
eV)^{-1} Mpc in comoving
coordinates, and correspondingly

(8) |

which is the mass scale of superclusters. Objects of this size are the first to form in a -dominated universe, and smaller scale structures such as galaxies can form only after the initial collapse of supercluster-size fluctuations.

When a fluctuation of total mass ~ 10^{15}
*M*_{} enters the
horizon at *z* ~
10^{4}, the density contrast
_{RB} of the
radiation plus baryons ceases growing and instead starts
oscillating as an acoustic wave, while that of
the massive neutrinos
_{} continues to grow
linearly with the scale factor *R* = (1 + *z*)^{-1} since the
Compton drag that prevents growth of
_{RB} does
not affect the neutrinos. By recombination, at *z*_{r}
~ 10^{3}, _{RB} /
_{}
10^{-1}, with
possible additional suppression of
_{RB} by Silk
damping. Thus the hot DM scheme with adiabatic primordial fluctuations
predicts small-angle fluctuations in the microwave
background radiation that are lower than in the
adiabatic baryonic cosmology, which was one of the reasons HDM
appealed to Zel'dovich and other theorists. Similar considerations
apply in the warm and cold DM schemes. However, as we will discuss in
a moment, the HDM top-down
sequence of cosmogony is wrong, and with the COBE
normalization hardly any structure would form by the present.

In numerical simulations of dissipationless gravitational
clustering starting with a fluctuation spectrum
appropriately peaked at ~
*d*_{} (reflecting
damping by free streaming below that size and less time for
growth of the fluctuation amplitude above it),
the regions of high density form a network of
filaments, with the highest densities occurring at the
intersections and with voids in between
[45,
46,
47,
48].
The similarity of these
features to those seen in observations was cited as evidence
in favor of HDM [49].