ISW during matter dominant era

[Mordred]

I've been reading reading an older article on ISW (integrated Sache-wolfe effect). One line struck me as odd in the paper.
http://arxiv.org/abs/0801.4380v2

on page two the line " In particular, we know that during the matter dominated era the gravitational potential stays constant "

I can understand that during the matter dominant era, the cosmological constant effect is greatly reduced. However wouldn't there be an ISW potential due to dynamics of the matter dominant era?

 

[bapowell]

The gravitational potential referred to is the analog Newtonian potential associated with the matter perturbations (recall that the Newtonian potential, $\Phi$, vanishes in any homogeneous cosmology). We start with Poisson's Eq.,
$$\nabla ^2 \Phi = 4\pi G \delta \rho_m$$
where $\delta \rho_m$ is the matter perturbation. Usually a different quantity is actually solved for -- the density contrast: $$\delta = \delta \rho/\bar{\rho}$$ where the bar denotes an average, so we instead have
$$\nabla^2 \Phi = 4\pi G \bar{\rho}\delta_m$$ In order to solve Poisson's equation, we work in Fourier space so that the $\nabla^2$ brings down two factors of the physical wavenumber, $(k/a)^2$. Putting it all together we have
$$\Phi_k = 4 \pi G\left(\frac{a^2}{k^2}\right)\bar{\rho}\delta_{m,k}$$
Now, during matter domination we know that $\bar{\rho} \sim a^{-3}$, and there is a growing mode $\delta_k \sim a$ (this is found by solving the perturbation equation in Newtonian gauge). We find then
$$\Phi_k = 4\pi G \left(\frac{a^2}{k^2}\right) \frac{a}{a^3} = const$$.

EDIT: The important conceptual point here is that $\Phi$ is not associated with the background density, $\rho$ (or really $\bar{\rho}$), but with the perturbations. In a homogeneous cosmology, there is no gravitational potential/field.

 

[Mordred]

Thanks a ton that excellent explanation, answered the question beautifully.