According to the book Cosmology by Daniel Baumann, the Poisson equation for a static universe (a universe without expansion) where gravitation is neglected is the following, denoting by $\phi$ the gravitational potential and by $\rho$ the density of the fluid that fills the universe:
$$\nabla^2\phi=4\pi G\rho$$
The book makes a brief commentary about how this causes an inconsistency with the equations that determine the evolution of the fluid, since there are no static self-gravitating fluids. So, the next step is to include the expansion of the universe, which implies using phsyical coordinates $\vec{r}=a\vec{x}$ instead of comoving coordinates $\vec{x}$, where $a$ is the scale factor that measures the expansion of the universe. According to the book, this yields:
$$\nabla^2\phi=4\pi G a^2\rho$$
This is what I am trying to prove. My reasoning is the following:
How the nabla operator changes
$$\vec{r}=a\vec{x}\ \ \Rightarrow\ \ r_i=ax_i\Rightarrow\dfrac{\partial}{\partial r_i}=\dfrac{\partial}{\partial x_i}\dfrac{dx_i}{dr_i}\ \ \Rightarrow\ \ \dfrac{\partial}{\partial r_i}=\dfrac{1}{a}\dfrac{\partial}{\partial x_i}\ \ \Rightarrow\ \ \vec{\nabla}_\vec{r}=\dfrac{1}{a}\vec{\nabla}_\vec{x}$$
where $\vec{\nabla}_\vec{r}$ and $\vec{\nabla}_\vec{x}$ are calculated at constant time $t$.
How the gravitational potential changes
This is probably wrong since this expression is for masses that are considered a point:
$$\phi_x=-\dfrac{GM}{x}=-\dfrac{GM}{r/a}=-a\dfrac{GM}{r}=a\phi_r$$
How the density changes
$$\rho_x=\dfrac{m}{x^3}=\dfrac{m}{(r/a)^3}=a^3\dfrac{m}{r^3}=\rho_r$$
So the final result is…
$$\nabla^2_x\phi_x=4\pi G\rho_x\ \ \Rightarrow\ \ (a^2\nabla^2_r)(a\phi_r)=4\pi G(a^3\rho_r)\ \ \Rightarrow\ \ \nabla^2_r\phi_r=4\pi G\rho_r$$
This is obviously not the correct result, as the form of the equation does not change at all and the scale factor is not present. Can anyone help me identify the mistakes I'm making? Any help would be greatly appreciated!
Best Answer
The transformations of the gravitational potential and the energy density is not quite correct. For example, given that $\phi = \phi(t, {\bf r})$, in terms of the comoving coordinates ${\bf x} = {\bf r}/a(t)$ one has $\phi = \phi(t, {\bf r} = a(t){\bf x})$. This notation is a bit unfortunate, but basically puts in an equation the statement that the function values themselves don't change under a coordinate transformation, only the coordinate points where these values are attained. As a result, neither $\phi$ nor $\rho$ contribute to the transformation of the Poisson equation. Transforming the gradient then yields a factor of $a^2$, consistent with the claim in Baumann's Cosmology.