The book Cosmology by Daniel Baumann states that the Poisson equation for a universe where we consider the effects of both gravity and expansion, expressed in physical coordinates $\vec{r}=a\vec{x}$, is the following:
$$\nabla^2\phi=4\pi Ga^2\rho$$
where $\phi$ is the gravitational potential, $a$ is the scale factor and $\rho$ is the density field of the fluid that fills the universe. If we now consider perturbation theory and write:
$$\phi=\bar{\phi}+\delta\phi\ \ \ \ \text{and}\ \ \ \ \rho=\bar{\rho}(1+\delta)$$
where $\delta\phi$ is the perturbation in the gravitational potential and $\delta$ is the density contrast, then it is stated multiple times in the book that the Poisson equation becomes:
$$\nabla^2\phi=4\pi Ga^2\bar{\rho}\delta$$
However, this is not what I get when I do the calculations. My reasoning is the following:
$$\nabla^2\phi=4\pi Ga^2\rho\ \ \Rightarrow\ \ \nabla^2(\bar{\phi}+\delta\phi)=4\pi Ga^2\bar{\rho}(1+\delta)\ \ \Rightarrow\ \ \nabla^2\bar{\phi}+\nabla^2\delta\phi=4\pi Ga^2\bar{\rho}+4\pi Ga^2\bar{\rho}\delta$$
The only way to get what the book states would be to set $4\pi Ga^2\bar{\rho}=0$, which is impossible, as it would mean that $\bar{\rho}=0$, and the background density is not zero. And if we consider $\nabla^2\bar{\phi}=4\pi Ga^2\bar{\rho}$ (I am not sure whether this is true or not), then what we get is $\nabla^2\delta\bar{\phi}=4\pi Ga^2\bar{\rho}\delta$, which is not quite the desired expression either.
Any help would be greatly appreciated!
Best Answer
The definition of $\phi$ has been changed, so that it sources peculiar gravitational forces rather than the total gravitational force. That is:
This potential sources peculiar velocities $\vec u$, such that $\dot{\vec{u}} = -\nabla\phi$.
This potential sources physical velocities $\vec v=H\vec r+\vec u$, so that $\dot{\vec{v}} = -\nabla\phi$. If you split this into 0th and 1st order sets of equations, you should find that the 0th order equations determine the cosmic expansion (at least in a matter-dominated universe) while the 1st order equations govern the perturbations. Cosmological perturbation theory is only concerned with the 1st order equations.