**3.3. Nonrelativistic evolution in an external gravitational field**

In this section we consider hot dark matter made of nonrelativistic
massive neutrinos with
_{}
<< so that
their self-gravity is unimportant. The gravitational potential
(** x**,
) (using
comoving coordinates) is assumed to be given from other sources
such as cold dark matter in a mixed hot and cold dark matter model.

We can solve the Vlasov equation (3.9) approximately by replacing
*f* /
** p** with
the unperturbed term

A quadrature solution of the Vlasov equation can be obtained provided
that we change the time variable from
to
*s* =
*d* / *a* =
*dt* /
*a*^{2} and then Fourier transform the spatial variable:

(3.16) |

The gravitational potential
is transformed
similarly. Integrating eq. (3.9) over *s*, we obtain the solution

(3.17) |

where
** u** =

To understand the behavior of the free-streaming solution, let us examine the integral term of eq. (3.17), which is proportional to

(3.18) |

where
*k*^{.} ** p** /

If, however,
*y*
>> 1, corresponding to neutrinos traveling
across many wavelengths of a perturbation, the rapid oscillations of the
exponential lead to cancellation in the integrand of eq. (3.18)
and suppression of the neutrino phase space density perturbation.
This effect, known as free-streaming damping, occurs because neutrinos
that are initially at the crests or troughs of density waves move so far
that they distribute themselves almost uniformly. The small gravitational
acceleration induced by the external potential is inadequate to collect
the fast-moving neutrinos in dense regions.

Thus, perturbations can grow only for the neutrinos that move less than about one wavelength per Hubble time. Our analysis confirms the rough picture we sketched in the beginning of this lecture.

We can obtain the net density perturbation (in Fourier space) by integrating eq. (3.17) over momenta:

(3.19) |

where
*n*_{0} =
*d*^{3}*p* *f*_{0}(*p*) is the mean
comoving number density
and *F* is the Fourier transform - with respect to the momentum!
- of the unperturbed distribution function:

(3.20) |

For the relativistic Fermi-Dirac distribution appropriate to hot dark
matter, *F* has the series representation
(Bertschinger & Watts
1988)

(3.21) |

where
(3) =
1.202... is the Riemann zeta function and *F*(0) = 1.

Equation (3.19) does not give much insight into free-streaming
damping. To get a better feel for the physics, as well as a simpler
approximation for treating hot dark matter, we now show how to
convert eq. (3.19) into a differential equation for the
evolution of the hot dark matter density perturbation similar to
eq. (2.12) for a perfect collisional fluid. This may seem
impossible a priori - how can the dispersive behavior of a collisionless
gas be represented by fluid-like differential equations? - but we
shall see that it is possible if we approximate
*f*_{0}(*p*) by a form differing
slightly from the Fermi-Dirac distribution. The results, although not
exact, will give us additional insight into the behavior of collisionless
damping.

The first step is to rewrite eq. (3.19) for the Fourier
transform of the density fluctuation
_{}:

(3.22) |

Next, we differentiate twice with respect to the time coordinate *s*:

(3.23) |

(3.24) |

Note the appearance of a non-integrated source term in the second
derivative, arising because *d* (*qF*) / *dq* does not
vanish at *s* = *s'* (*q* = 0) while *qF* does.

Next, we note that if
*d*^{2}(*qF*) / *dq*^{2} were to equal a
linear combination
of *d* (*qF*) / *dq* and (*qf* ), then we could
write the integral in equation (3.24) as a linear combination of
_{} /
*s*
and _{}. Unfortunately, this is not
the case for *F*(*q*) given
by eq. (3.21). However, it is true for the family of
distribution functions whose Fourier transforms are

(3.25) |

for any dimensionless constant . This defines the family of phase space density distributions

(3.26) |

For this form of unperturbed distribution we have

(3.27) |

Combining eqs. (3.22)-(3.24) and (3.27), we get

(3.28) |

To put this result into a form similar to the acoustic wave equation we derived for a collisional fluid, we define the characteristic proper thermal speed

(3.29) |

Next, we change the time variable from *s* back to
with
*d* / *ds* =
*a*. Finally, we assume that the source term gravitational potential
is
given by the Poisson equation for a perturbation
_{c} in a
component with mean mass density
_{c}
(e.g., cold dark matter - recall that we are neglecting the self-gravity
of the neutrinos). Dropping the hat on
_{}, the result is

(3.30) |

This equation was first derived by
Setayeshgar (1990).
It is approximate
(not exact) for the linear evolution of massive neutrinos because we
replaced the Fermi-Dirac distribution by eq. (3.26). It is not
difficult to show that eq. (3.26) is the only form of the
distribution function for which eq. (3.17) can be reduced
to a differential equation for
_{}(** k**,
). (Even the
Maxwell-Boltzmann distribution fails - a collisionless gas with this
distribution initially does not evolve the same way as a collisional
gas with the Maxwell-Boltzmann distribution function for all times.)
One should also bear in mind that

Even if eq. (3.30) is not exact for massive neutrinos and
does not fully specify the perturbations, it provides an extremely
helpful pedagogic guide to the physics of collisionless damping. We
see at once that a gravitational source can induce density perturbations
in a collisionless component, but the source competes agains acoustic
(*k*^{2}
*c*_{}^{2})
and damping ( / *a* +
2*kc*_{})
terms. Roughly
speaking, hot dark matter behaves like a collisional gas with an extra
free-streaming damping term.

Does the *k*^{2}
*c*_{}^{2}
term imply that a collisionless gas can support
acoustic oscillations? To check this we consider the limit
*kc*_{}
>> 1 so that the
Hubble damping and gravitational source terms are negligible. We then have

(3.31) |

Because _{} changes very slowly with
time compared with the oscillation timescale
^{-1},
eq. (3.31) is a
linear differential equation with constant coefficients and is easily
solved to give the two modes

(3.32) |

Neither solution oscillates! The first one begins to grow but is rapidly
damped on a timescale
_{}, after the typical neutrino
has had time to cross one wavelength.

Because the damping time (*kc*_{})^{-1} is proportional to the wavelength,
short-wavelength perturbations are damped most strongly. At any given
time , perturbations of
comoving wavelength less than about
*c*_{} are attenuated. This is
precisely the free-streaming distance we introduced in the beginning of
this lecture, equation (3.2).

Our results enable us to understand why the hot dark matter transfer
function is similar to that of cold dark matter for long wavelengths but
cuts off sharply for short wavelengths
(Bond & Szalay 1983).
During the radiation-dominated era,
*a*()
. While the massive
neutrinos were relativistic,
*c*_{}
*c* was
constant. The comoving free-streaming distance increased,
*c*_{}
*a*,
with hot dark matter perturbations being erased on scales up to the
Hubble distance. After the neutrinos became nonrelativistic, however,
*c*_{} is given by
eq. (3.29),
*c*_{}
*a*^{-1}.
Thus, the free-streaming distance saturates at the Hubble distance
when the neutrinos become nonrelativistic. During the matter-dominated
era, *a*()
_{2} (while
1) so that the
free-streaming distance decreases:
*c*_{}
*a*^{-1/2}.
However, free-streaming has already erased the hot dark matter
perturbations on scales up to the maximum free-streaming distance,
eq. (3.3). Only if the perturbations are re-seeded,
e.g. by cold dark matter or topological defects, will small-scale power
be restored to the hot dark matter.