# Cosmology Lecture 13

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### Jeans (Gravitational) Instability (in a Static Universe)

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

• The Continuity equation (describing mass conservation):
${\displaystyle {\partial \rho \over \partial t}+{\vec {\nabla }}(\rho {\vec {v}})=0\,\!}$

(Continuity Equation)

• Euler’s Equation:
${\displaystyle \underbrace {{\partial {\vec {v}} \over \partial t}+({\vec {v}}\cdot {\vec {\nabla }}){\vec {v}}} _{{d{\vec {v}} \over dt}=({\partial \over \partial t}+{\dot {x}}_{i}{d \over dx_{i}}){\vec {v}}}=\underbrace {-{1 \over \rho }{\vec {\nabla }}P-{\vec {\nabla }}\phi } _{force \over mass}+other\ terms\,\!}$

(Euler’s Fluid Equation) where “${\displaystyle other/terms}$” 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:
${\displaystyle \nabla ^{2}\phi =4\pi G\rho \,\!}$

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

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

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

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

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

${\displaystyle {\partial P \over \partial t}+\rho _{0}(\div \cdot {\vec {v}})=0\,\!}$
${\displaystyle {\partial {\vec {v}}_{1} \over \partial t}={-v_{s}^{2} \over \rho _{0}}\div \rho _{1}-\div \phi _{1}\,\!}$
${\displaystyle \nabla ^{2}\phi _{1}=4\pi G\rho _{1}\,\!}$

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

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

Then using (3) we get:

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

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

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

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

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

where ${\displaystyle k_{J}}$ is:

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

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

We have two regimes for the Jeans Wavenumber:

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

So ${\displaystyle \omega }$ is imaginary, and thus:

${\displaystyle \rho _{1}\propto e^{\pm |\omega |t}\,\!}$

(Jeans Instability) So the density of waves can grow or decay exponentially.

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

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

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