**3.4. Nonrelativistic evolution including self-gravity**

Now that we have developed the basic techniques for solving the
linearized nonrelativistic Vlasov equation, adding self-gravity of the
collisionless particles is easy. We simply add a contribution to
arising from _{}. In eq. (3.17), if we have a
mixture of hot and cold dark matter,
(_{c} +
_{}); additional contributions
may be added as appropriate. Equation (3.22) becomes

(3.33) |

This equation was first derived (in a slightly different form) by
Gilbert (1966)
and is known as the Gilbert equation. Note that in
the self-gravitating case
_{} appears both inside and
outside an integral. Equation (3.33) is a Volterra integral equation of
the second kind.
Bertschinger & Watts
(1988)
present a numerical quadrature solution method.

Using the same trick as in the previous subsection, we can convert the
Gilbert equation to a differential equation for
_{}, if the
unperturbed phase space density distribution is approximated by the
form *f*_{}(*p*) of eq. (3.26). The result is

(3.34) |

With a suitable choice for the parameter , the solution of eq. (3.34) provides a good match (to within a few percent, in general) to the solution of the Gilbert equation using the correct Fermi-Dirac distribution for massive neutrinos (Setayeshgar 1990). Therefore, it may be used for obtaining quick estimates of the density perturbations of nonrelativistic hot dark matter.