Jeans (Gravitational) Instability (in a Static Universe)
In a non-relativistic, non-dissipative, static fluid described by , the fluid velocity , and gravitational potential , we can write three equations to describe motion:
- The Continuity equation (describing mass conservation):
(Euler’s Fluid Equation) where “” could come from other forces like magnetic fields or viscous force. The full version of the above equation is called the Wavier-Stokes equation. This equation describes how the fluid velocity changes with force.
(Poisson’s Equation) This is just a Gauss’ Law for a gravitational field.
At this point, we would like to write an equation of state which relates to . To add a little bit of generality to this derivation so that we can perturb the equation for a lumpy universe, we’ll start using to describe the “zeroth order” of mass density. We’ll also say , and . We have to be careful about , though: Euler’s equation (2) tells us that , but if , then Poisson’s Equation (3) says:
(Jeans Swindle) There is really no way to fix this. On PS#6 we will show that although this fails for the completely static case, when we start doing first-order perturbations to this equation, things work out better.
For small perturbations to the static uniform solution such that:
We will define the Adiabatic Sound Speed as , so our equations become:
We can combine (1) and (2) to get:
Then using (3) we get:
Given that this equation looks like a wave equation, we can guess there is a solution where . This gives us a dispersion relation:
This looks similar to the dispersion relation of plasma physics. However, in a plasma, . Therefore, in plasma, , so it is always oscillating. In our case, we can rewrite as:
(Jeans Wavenumber) There is also a Jeans Wavelength: , and a Jeans Mass: .
We have two regimes for the Jeans Wavenumber:
- (or ):
So is imaginary, and thus:
(Jeans Instability) So the density of waves can grow or decay exponentially.
- (or ):
is real, so oscillates stably.
Thus, in a static universe, small scale density waves are stable, but for larger scale waves, we have runaway growth.