Jeans Instability in an Expanding Universe

[Arman777]

I am trying to derive the equation for a case, where we have a dust(zero-pressure) in an expanding universe.

There are 4 equations but I think exercising on one of them would be helpful for me.

I am trying to derive the equation for a case, where we have a dust(zero-pressure) in an expanding universe.

There are 4 equations but I think exercising on one of them would be helpful for me.

So we have a continuity equation

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v})=0$$

In an expanding universe

$\rho = \rho_0a^{-3}$

$\vec{v} = \frac{\dot{a}}{a}\vec{r}$

$\vec{r} = \vec{r_0}a$

where $a=a(t)$

So the problem is I am not sure how to define the perturbation

I thought I can write

$$\rho = \rho_0a^{-3} + \delta \rho$$
$$\vec{v} = \frac{\dot{a}}{a}\vec{r} + \delta \vec{v}$$

If I put it into the equation I get

$$\frac{\partial }{\partial t}(\rho_0a^{-3} + \delta \rho) + \nabla \cdot ((\rho_0a^{-3} + \delta \rho)(\frac{\dot{a}}{a}\vec{r} + \delta \vec{v}))=0$$

$$\frac{\partial }{\partial t}(\rho_0a^{-3}) + \frac{\partial }{\partial t}(\delta \rho) + \nabla \cdot (\rho_0a^{-3}\frac{\dot{a}}{a}\vec{r} + \rho_0a^{-3}\delta \vec{v}+ \delta \rho\frac{\dot{a}}{a}\vec{r})=0$$

since $\rho = \rho_0a^{-3}$

$$\frac{\partial }{\partial t}(\rho_0a^{-3})+\frac{\partial }{\partial t}(\delta \rho) + \nabla \cdot (\rho\frac{\dot{a}}{a}\vec{r} + \rho\delta \vec{v}+ \delta \rho\frac{\dot{a}}{a}\vec{r})=0$$

$$\frac{\partial }{\partial t}(\rho_0a^{-3})+\frac{\partial }{\partial t}(\delta \rho) + \rho\frac{\dot{a}}{a} (\nabla \cdot \vec{r}) + \rho(\nabla \cdot \delta \vec{v})+ \delta \rho\frac{\dot{a}}{a} (\nabla \cdot \vec{r})=0$$The answer is

$$\frac{\partial }{\partial t}(\delta \rho) + 3\delta \rho\frac{\dot{a}}{a}+ \rho(\nabla \cdot \delta \vec{v})+ \delta \rho\frac{\dot{a}}{a} (\vec{r} \cdot \nabla )=0$$

Also $(\nabla \cdot \vec{r}) = 3 ?$

For reference : https://www.amazon.com/dp/0471489093/?tag=pfamazon01-20 Page 216

 

[AndyW]

Thank you for reaching out with your question. It is great to see that you are trying to understand the equations for a dust-filled expanding universe. I will try to guide you through the derivation process and provide some clarification on the steps you have taken so far.

Firstly, let's define our variables. We have the density of the dust, $\rho$, which is a function of time, $t$, and the position vector, $\vec{r}$. We also have the velocity of the dust, $\vec{v}$, which is also a function of time and position. Finally, we have the scale factor of the universe, $a$, which is only a function of time.

Now, the continuity equation you have written is correct, and it describes the conservation of mass for a fluid. However, in this case, we are dealing with a dust-filled universe, which has zero pressure. This means that the dust particles do not interact with each other, and their motion is solely determined by the expansion of the universe. This leads to a simplified continuity equation:

$$\frac{\partial \rho}{\partial t} + 3\frac{\dot{a}}{a}\rho=0$$

where $\dot{a}$ is the time derivative of the scale factor. This equation is derived by substituting your expressions for $\rho$ and $\vec{v}$ into the original continuity equation and simplifying.

Now, let's move on to the perturbation part. You are correct in assuming that we can write the density and velocity as the sum of their background values and a perturbation term. However, there is a small mistake in your expressions. The perturbation terms should be written as:

$$\delta\rho=\rho-\rho_0a^{-3}$$

$$\delta\vec{v}=\vec{v}-\frac{\dot{a}}{a}\vec{r}$$

This ensures that the perturbation terms are zero when the background values are used.

Substituting these expressions into the continuity equation, we get:

$$\frac{\partial }{\partial t}(\delta \rho) + \nabla \cdot (\delta \rho \vec{v} + \rho_0a^{-3}\delta \vec{v}) + \frac{\partial }{\partial t}(\rho_0