2.6 Gravitational
instability
We have presented the zero-th order description of the Universe
neglecting the inhomogeneity or spatial variation of matter inside.
Now we are in a position to consider the evolution of matter in the
Universe. For simplicity we focus on the non-relativistic regime
where the Newtonian approximation is valid. Then the basic
equations for the self-gravitating fluid are given by the
continuity equation, Euler’s equation, and the Poisson equation:
We would like to rewrite those equations in the comoving frame. For
this purpose, we introduce the position
in the comoving
coordinate, the peculiar velocity
, density fluctuations
, and the gravitational potential
which are defined as
respectively. Then Equations (39) to (41) reduce to
where the dot and
in the above equations are the time
derivative for a given
and the spatial derivative
with respect to
, i.e., defined in the comoving
coordinate (while those in Equations (39, 40, 41) are defined in the
proper coordinate).
A standard picture of the cosmic structure
formation assumes that the initially tiny amplitude of density
fluctuation grow according to Equations (46, 47, 48). Also the Universe
smoothed over large scales approaches a homogeneous model. Thus at
early epochs and/or on large scales, the nonlinear effect is small
and one can linearize those equations with respect to
and
:
where
is the sound velocity squared.
As usual, we transform the above equations in
space using
Then the equation for
reduces to
If the signature of the third term is positive,
has an unstable, or, monotonically increasing
solution. This condition is equivalent to the Jeans criterion:
namely, the wavelength of the fluctuation is larger than the Jeans
length
which characterizes the scale that the sound wave
can propagate within the dynamical time of the fluctuation
. Below the scale, the pressure wave can suppress the
gravitational instability, and the fluctuation amplitude
oscillates.