Gravitational instability

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 (39

) to (41

) 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 (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 $δ$ 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

Then the equation for $δk$ reduces to

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:

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.