# 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: $M_{J}={4\pi \over 3}({\pi \over k_{J}})^{3}\rho _{0}$ .

We have two regimes for the Jeans Wavenumber:

• $k (or $\lambda >\lambda _{J}$ ):
$\omega ^{2}=v_{s}^{2}(k^{2}-k_{J}^{2})<0\,\!$ So $\omega$ is imaginary, and thus:

$\rho _{1}\propto e^{\pm |\omega |t}\,\!$ (Jeans Instability) So the density of waves can grow or decay exponentially.

• $k>k_{J}$ (or $\lambda <\lambda _{J}$ ):

$\omega$ is real, so $\rho _{1}$ oscillates stably.

Thus, in a static universe, small scale density waves are stable, but for larger scale waves, we have runaway growth.