I The Expanding Universe and a stationary frame of reference

[darkdave3000]

What is the rate of expansion of the universe and what is it's acceleration rate of expansion? Exactly?

I recall it's about 67km/s at 1 mega parsecs? But then what's the acceleration rate? This exact information I cannot find.

If the universe is expanding, and this expanding is accelerating, wouldn't that mean that any object moving slower than the rate of expansion (67km/s) relative to objects 1 megaparsecs away is becoming more and more stationary?

Sort of like not having escape velocity to escape a planet. If you have less than the required speed you will fall back down or in this case approach zero velocity gradually relative to other objects 1 megaparsecs away(initially).

Even if it take an eternity, all objects moving at less than the speed of the expansion 67km/s relative to each other (minimum of 1 megaparsec distance) will approach zero velocity over an eternity.Here's a thought experiment, if I want to reach a planet "A" 1 megaparsecs away I must approach it at beyond 67km/s or ill never reach there, any less and ill never reach there and in fact the object will receded from me if less than 67km/s. Now imagine planet "B" was behind me at the same distance in the beginning of the experiment (also 1 megaparsec away), if I move less than 67km/s toward planet A and at the same speed away from "B" after an eternity, I will still eventually be more and more of an equal distance between A and B and my relative velocity between them will approach zero km/s. Wouldn't this mean I am approaching a stationary frame of reference(as eternity drags on)?

 

[Bandersnatch]

What is the rate of expansion of the universe

That's easy, it's just the Hubble constant. So, depending on which measurements you prefer (i.e. long-distance or short-distance) it's either close to 68km/s/Mpc, or 73km/s/Mpc. The first value is generally though to be the better bet as to the actual global rate of expansion. It corresponds to 1/144 % growth per million years.
Of course, it doesn't actually stay constant, but goes down with time (only slowly, in this epoch).

But then what's the acceleration rate? This exact information I cannot find.

Right, so I'm not sure how to represent this in a straightforward manner, like it can be done easily with the expansion rate. The main issue here is that acceleration and expansion rate are not really directly related, like e.g. the familiar velocity and acceleration are related in kinematics.
For one, accelerated expansion does not mean the expansion rate is growing. And in fact, it is currently going down, and (in the current best model) will remain doing so forever, approaching a constant rate in the far future - which corresponds to exponentially accelerating expansion. Acceleration requires only that the rate doesn't go down too fast.
There were two recent threads where the acceleration rate was being discussed that might, maybe, shed some more light on this:
https://www.physicsforums.com/threads/what-is-the-acceleration-of-the-universe.957237/
https://www.physicsforums.com/threads/how-much-does-the-universe-accelerate.959704/
So, after reading those threads, you can perhaps see that current acceleration is expressed by the deceleration parameter $q$, whose current value of -0.53 can be in some sense interpreted as half way through between no acceleration (steady expansion) and exponential expansion.

But anyway.
Something like what you describe does happen, although it doesn't require accelerated expansion. The effect the expansion has on light is redshifting its frequency. Massive objects, on the other hand, have their momentum w/r to the Hubble flow reduced. I.e., over time, objects join the flow of the expansion, becoming stationary w/r to this particular frame, called comoving frame. (more on this effect can be found by looking up 'tethered galaxy problem', including the paper by Lineweaver and Davis that discusses this mathematically)
I don't think it's a good idea to call it universal rest frame, though, as it immediately conjures up the specter of ether. Which would be a red herring, since while it's a convenient frame, there's nothing physically special about it, apart from the fact that the universe looks a certain (convenient) way when viewed from it.
Again, this doesn't require accelerated expansion, just expansion. The acceleration/deceleration determines how exactly the Hubble flow is joined.

 

[kimbyd]

What is the rate of expansion of the universe and what is it's acceleration rate of expansion? Exactly?

I recall it's about 67km/s at 1 mega parsecs? But then what's the acceleration rate? This exact information I cannot find.

It's not terribly easy to model. You can do it numerically, but it's best just to think about it in approximate terms.

The current rate of expansion can be estimated as being roughly 70km/s/Mpc. Over time, as matter continues to dilute, that rate is going to decrease, leaving nothing but dark energy. Dark energy is currently about 68.5% of the energy density of the universe, and the rate of expansion scales with the square root of the density. So eventually the density will be 68.5% of its current value, and the rate of expansion will be about 83% of its current value, or roughly 58 km/s/Mpc. Assuming the dark energy is a cosmological constant, it the rate will stay there forever.

If the universe is expanding, and this expanding is accelerating, wouldn't that mean that any object moving slower than the rate of expansion (67km/s) relative to objects 1 megaparsecs away is becoming more and more stationary?

I think the way you've stated it is a little inaccurate, but in essence this is true.

As objects move through the universe, they will "catch up" to objects which are expanding away from them. Thus their speed relative to the local expansion drops over time. This effect occurs no matter what the rate of expansion is, or how the rate of expansion changes over time. All that you need is uniform expansion, and that expansion will act as a sort of friction on the motions of everything in the universe. This effect can even be used to describe how light redshifts (light doesn't lose speed, but it does lose momentum). The speed at which objects are moving is also irrelevant. Faster-moving objects will just take longer to slow down.

Eventually, many the objects will slow down enough compared to the local material to fall into some gravitational well or other, and then their motions will be determined by the dynamics of that gravity well rather than the overall expansion.