266 Gravitational instability in Newtonian theory
smooth unclustered energy component, like radiation or vacuum energy, influences
the growth of inhomogeneities in the cold matter component. Finally, we derive a
few exact solutions which describe the behavior of perturbations with certain spatial
geometrical symmetries into the nonlinear regime. Based on these solutions, we are
able to explain the general features of the matter distribution on nonlinear scales.
6.1 Basic equations
On large scales matter can be described in a perfect fluid approximation. This means
that at any given moment of time it can be completely characterized by the energy
density distributionε(x,t), the entropy per unit massS(x,t), and the vector field
of 3-velocitiesV(x,t). These quantities satisfy the hydrodynamical equations and
we begin with a brief reminder of their derivation.
Continuity equationIf we consider afixed volume elementVin Euler (nonco-
moving) coordinatesx,then the rate of change of its mass can be written as
dM(t)
dt=
∫
V∂ε(x,t)
∂tdV. (6.1)On the other hand, this rate is entirely determined by the flux of matter through the
surface surrounding the volume:
dM(t)
dt=−
∮
εV·dσ=−∫
V∇(εV)dV. (6.2)These two expressions are consistent only if
∂ε
∂t+∇(εV)= 0. (6.3)Euler equationsThe accelerationgof a smallmatter elementof massMis
determined by the gravitational force
Fgr=−M·∇φ, (6.4)whereφis the gravitational potential, and by the pressurep:
Fpr=−∮
p·dσ=−∫
V∇pdV−∇p·V. (6.5)With
g≡dV(x(t),t)
dt=
(
∂V
∂t)
x+
dxi(t)
dt(
∂V
∂xi)
=
∂V
∂t