Validity of Newtonian gravity inside the Hubble radius

Question

When dealing with structure formation in Cosmology, it's often said that well inside the Hubble radius we can use Newtonian gravity and therefore the starting point to obtain equations for the perturbations in matter density is the well known Poisson equation.

However, I don't see the relation between both things: Hubble radius and whether I can use Newton's laws or not. Actually, inside the Hubble radius galaxies are interacting among themselves so in principle that should be ruled by General Gravity unless the fields are really weak. Is that the reason why? Are galaxies so far away from each other that the gravitational fields are truly weak, hence we can make use of Newtonian gravity?

But in that case, it would be the distance among galaxies and not the Hubble radius the reason for flat spacetime.

Answer 1

As far as I know the Newtonian approximation is valid only for $\delta=\frac{\delta \rho}{\rho_0} \ll 1$. This corresponds to early universe and in the early universe the background was nearly homogeneous and isotropic. In this case I think its valid to use the Poission equation.

Currently the density constrast is $\delta \approx 10^{6}$. So you cannot talk about the newtonian approximation.

Answer 2

You're taking the argument overly literally. The statement is only "the effects of the cosmology are weak for distance scales much smaller than the hubble radius".

Why this is the case is made most obvious if you approximate the cosmology as de Sitter, and write it in the "schwarzschild-like" coordinates of the form:

$$ds^{2} = -A(r)^{2} dt^{2} + B^{2}(r)dr^{2} + r^{2}\left(d\theta^{2} + sin^{2}\theta d\phi^{2}\right)$$

when you do this, you will see that $A$ and $B$ differ from $1$ only by terms proportional to $\Lambda r^{2} = \left(\frac{r}{R_{H}}\right)^{2}$, so for $r \ll R_{H}$, this is Minkowski spacetime.