Hubble Constant: Force Acting on Hydrogen Atom Due to Expansion of Universe

[Dogya]

TL;DR Summary: Express through the Hubble constant the force that acts on the hydrogen atom due to the expansion of the universe

We have a hydrogen atom, in a gravitationally bound system nothing interesting happens to it. Let's put it in an empty world with only an electron and a proton. There are several forces acting on it: gravitational forces, electromagnetic forces, and since we only have two bodies, the expansion of the universe should work. The question is how to express through the Hubble constant and calculate the force of this expansion.
As pointed out to me, to solve this problem, we need to consider the case where we have two particles without electromagnetic interaction, such as 2 neutrons.
As I understood the solution has to be sought from Friedman equations.
This is difficult for me as I am only in my 2nd year of university.

 

[Ibix]

In an empty universe there is no expansion, and even in spacetimes where there is expansion it isn't a force anyway.

Is there some context to this question? It doesn't seem like a coherent scenario to me.

 

[Dogya]

In an empty universe there is no expansion, and even in spacetimes where there is expansion it isn't a force anyway.

Is there some context to this question? It doesn't seem like a coherent scenario to me.

I meant that apart from the proton and the electron there is nothing else around that affects them in any meaningful way. So there will be expansion. And force, in its usual sense, really does not exist, but as we know the expansion of the universe happens with acceleration and so we can translate it into "newtons", getting the force

 

[Ibix]

So you've got an FLRW universe everywhere filled with uniform density neutral matter and a non-zero cosmological constant, plus one electron and one proton?

The answer is still zero. The only sensible way to measure the acceleration of either particle is to measure its proper acceleration, and this will depend on the EM field at its location and nothing else (ignoring drag from the medium). The value of the EM field will depend on the cosmological constant, although the effect will be utterly negligible. I don't think there's a meaningful way to break the EM field down into "what it would be without the cosmological constant" and "the extra bit", though.

the expansion of the universe happens with acceleration

Not in the sense you seem to mean. The acceleration isn't the second derivative of position as needed for Newton's laws.