### 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.