Validity of Newtonian gravity inside the Hubble radius

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.

• I don't know of any reliable sources that use the word "radious". Where are you getting your information from? Feb 17 '20 at 10:06
• For the "radious" you would deserve a fat downvote, but nevertheless: in cosmology we assume that everything is homogenous, isotropic and the galaxies to be flowing with the hubble flow, so locally at rest. If you want to calculate the trajectories for objects with high peculiar velocities (relative to the CMB) then you have to use relativity, but otherwise you can derive the evolution of the universe with Newton as well, see youtube.com/… Feb 17 '20 at 10:26
• English is not my first language so it's normal to commit mistakes, take it easy Feb 17 '20 at 19:06
• No problem, and by the way, I was not the one who downvoted, that was meant to be a joke. Another lecture where the derivation via Newton's law is explained can be found here: youtube.com/watch?v=vKLqWj0FRyc&t=058s Feb 17 '20 at 21:36

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.

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.

• Yes, this isn't a complete answer to this, but it's a heuristic to show why the conclusion is reasonable. Feb 18 '20 at 16:18