Understanding comoving frame in an expanding universe? A particle is moving in a universe that is expanding with a constant acceleration (i.e. expansion of universe is accelerating). I am considering 2 cases where the particle is (1) constantly accelerating and (2) moving with constant velocity. What would we observe the motion of the particle to be like in a comoving frame in both of the above cases?
Attempt: I think (1) if acceleration is same as acceleration of expansion of universe, it will be observed to have constant velocity (2) observed to decelerate. I am not sure and my understanding of comoving vs proper coordinates is fuzzy. If someone would clarify the concepts and provide possible solutions with explanation/equations if applicable that will be great.
 A: Comoving Distance & Proper Distance
You can think of the expanding space as a grid expanding at a certain rate. To keep it simple, lets consider a two dimensional, square grid. Each side of a square is a constant $d$. If we wish to measure a distance between two points in space we might say, the distance between $A$ and $B$ is $3d$ if the sides of three squares separate between them. This distance remains the same as the grid stretches, since no new squares are added to it and the separation between $A$ and $B$ will remain $3d$. This is the comoving distance which remains constant.
But we also say that the grid is expanding, therefore there is a measure that grows larger as the stretching of the grid progresses, it is known as the proper distance.  In order to get the proper distance between the two points we have to multiply the comoving coordinate, $d$, by what is known as the scale factor $a(t)$ which does increase with time. If you are interested, the scale factor comes from the FLRW metric and can be calculated using Hubble Law. The proper distance is given by $$d_p(t) = a(t)d$$
Expansion of the Universe & Hubble Law
The expansion of the universe, is usually likened to that of the surface of an inflating balloon. Every point is expanding away from every other point with a rate that is dependent on the separation distance. This is given by Hubble Law
$$\dot a(t) = H a(t)$$
You can multiply both sides of the formula by the comoving $d$ to get the proper quantities. In general, the Hubble constant $H$ is not a constant but depends on time with a changing manner throughout the history of the universe. But the Hubble constant today $H_0$ varies very slowly, that for a lot of intents and purposes it is effectively a constant
Peculiar Velocities
An object can have a motion relative to its comoving frame. We usually refer to it as a peculiar velocity and it fits into the Hubble equation as such
$$\dot{r}(t) = Hr(t) + v_p(t)$$ where the quantities here are of proper distance. You can use it to answer questions about bodies with motion relative to their comoving frame as observed from a certain point, and so on.
These are all very simplified explanations, but I hope they manage clarify some points.
