Cosmology Lecture 13

Jeans (Gravitational) Instability (in a Static Universe)

In a non-relativistic, non-dissipative, static fluid described by $\rho ,P$ , the fluid velocity ${\vec {v}}$ , and gravitational potential $\phi$ , we can write three equations to describe motion:

The Continuity equation (describing mass conservation):

{\partial \rho  \over \partial t}+{\vec {\nabla }}(\rho {\vec {v}})=0\,\!

(Continuity Equation)

Euler’s Equation:

\underbrace {{\partial {\vec {v}} \over \partial t}+({\vec {v}}\cdot {\vec {\nabla }}){\vec {v}}} _{{d{\vec {v}} \over dt}=({\partial  \over \partial t}+{\dot {x}}_{i}{d \over dx_{i}}){\vec {v}}}=\underbrace {-{1 \over \rho }{\vec {\nabla }}P-{\vec {\nabla }}\phi } _{force \over mass}+other\ terms\,\!

(Euler’s Fluid Equation) where “ $other/terms$ ” 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:

\nabla ^{2}\phi =4\pi G\rho \,\!

(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 $P$ to $\rho$ . 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 $\rho _{0}=const$ to describe the “zeroth order” of mass density. We’ll also say $P_{0}=const$ , and ${\vec {v}}_{0}=0$ . We have to be careful about $\phi _{0}$ , though: Euler’s equation (2) tells us that $\div \phi _{0}=0$ , but if $\phi _{0}=const=0$ , then Poisson’s Equation (3) says:

\div \phi _{0}={4\pi  \over 3}G\rho _{0}{\vec {x}}\neq 0\,\!

(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 $(\rho _{1},P_{1},{\vec {v}}_{1},\phi _{1})$ such that:

{\begin{matrix}\rho =\rho _{0}+\rho _{1}&P=P_{0}+P_{1}\\{\vec {v}}={\vec {v}}_{0}+{\vec {v}}_{1}={\vec {v}}_{1}&\phi =\phi _{0}+\phi _{1}\\\end{matrix}}\,\!

We will define the Adiabatic Sound Speed as $v_{s}^{2}\equiv {\partial P \over \partial \rho }{\big |}_{s}={P_{1} \over \rho _{1}}{\big |}_{s}$ , so our equations become:

{\partial P \over \partial t}+\rho _{0}(\div \cdot {\vec {v}})=0\,\!

{\partial {\vec {v}}_{1} \over \partial t}={-v_{s}^{2} \over \rho _{0}}\div \rho _{1}-\div \phi _{1}\,\!

\nabla ^{2}\phi _{1}=4\pi G\rho _{1}\,\!

We can combine (1) and (2) to get:

{\partial P_{1} \over \partial t^{2}}+\rho _{0}\left(-{v_{s}^{2} \over \rho _{0}}\nabla ^{2}\rho _{1}-\nabla ^{2}\phi _{1}\right)=0\,\!

Then using (3) we get:

{{\partial ^{2}P_{1} \over \partial t^{2}}-v_{s}^{2}\nabla ^{2}\rho _{1}-4\pi G\rho _{0}\rho _{1}=0}\,\!

Given that this equation looks like a wave equation, we can guess there is a solution where $\rho _{1}({\vec {r}},t)=\int {d^{3}ka_{k}e^{i({\vec {k}}{\vec {r}}-\omega t)}}$ . This gives us a dispersion relation:

{\omega ^{2}=v_{s}^{2}k^{2}-4\pi G\rho _{0}}\,\!

This looks similar to the dispersion relation of plasma physics. However, in a plasma, $\omega ^{2}=v_{s}^{2}k^{2}+{4\pi n_{e}e^{2} \over m_{e}}$ . Therefore, in plasma, $\omega ^{2}>0$ , so it is always oscillating. In our case, we can rewrite $\omega$ as:

\omega ^{2}=v_{s}^{2}(k^{2}-k_{J}^{2})\,\!

where $k_{J}$ is:

{k_{J}\equiv {\sqrt {4\pi G\rho _{0} \over v_{s}^{2}}}}\,\!

(Jeans Wavenumber) There is also a Jeans Wavelength: $\lambda _{J}={2\pi \over k_{J}}$ , and a Jeans Mass: $M_{J}={4\pi \over 3}({\pi \over k_{J}})^{3}\rho _{0}$ .