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:
∂ρ ---+ ∇ ⋅ (ρu ) = 0, (39 ) ∂t ∂u-+ (u ⋅ ∇ )u = − 1-∇p − ∇ Φ, (40 ) ∂t ρ ∇2 Φ = 4πG ρ. (41 )
We would like to rewrite those equations in the comoving frame. For this purpose, we introduce the position x in the comoving coordinate, the peculiar velocity v, density fluctuations δ (t,x ), and the gravitational potential ϕ(t,x) which are defined as
x = -r--, (42 ) a(t) ˙ v = a(t)x, (43 ) ρ(t,x)- δ(t,x ) = ρ¯(t) − 1, (44 ) 1 ϕ (t,x ) = Φ + --a¨a|x|2, (45 ) 2
respectively. Then Equations (39View Equation) to (41View Equation) reduce to
1 ˙δ + --∇ ⋅ [(1 + δ)v] = 0, (46 ) a ˙v + 1-(v ⋅ ∇ )v + ˙av = − 1-∇p − 1∇ ϕ, (47 ) a a ρa a ∇2ϕ = 4πG ¯ρa2δ, (48 )
where the dot and ∇ in the above equations are the time derivative for a given x and the spatial derivative with respect to x, i.e., defined in the comoving coordinate (while those in Equations (39View Equation, 40View Equation, 41View Equation) 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 (46View Equation, 47View Equation, 48View Equation). 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 δ and v:

˙ 1- δ + a∇ ⋅ v = 0, (49 ) ˙a c2 1 v˙+ -v = − -s∇ δ − -∇ ϕ, (50 ) a a a ∇2 ϕ = 4πG ¯ρa2δ, (51 )
where 2 cs ≡ (∂p∕∂ρ ) is the sound velocity squared.

As usual, we transform the above equations in k space using

∫ δ (t) ≡ -1 δ(t,x )exp(ik ⋅ x )dx. (52 ) k V
Then the equation for δk reduces to
a˙ ( c2k2 ) ¨δk + 2--˙δk + s-2-− 4πG ¯ρ δk = 0. (53 ) a a
If the signature of the third term is positive, δk has an unstable, or, monotonically increasing solution. This condition is equivalent to the Jeans criterion:
2π ∘ -π-- λ ≡ ---> λJ ≡ cs ---, (54 ) k G ¯ρ
namely, the wavelength of the fluctuation is larger than the Jeans length λJ which characterizes the scale that the sound wave can propagate within the dynamical time of the fluctuation ∘ π∕G-¯ρ-. Below the scale, the pressure wave can suppress the gravitational instability, and the fluctuation amplitude oscillates.
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