Cosmology Lecture 13

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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):

(Continuity Equation)

  • Euler’s Equation:

(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:

(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:

where is:

(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.