Atomic clock in expanding space

Question

I have read these questions:

Is there any paper analyzing the validity of Relativity in expanding space?

The Great Atomic Clock of Kansas

photons in expanding space: how is energy conserved?

Is an atomic clock itself affected by gravity?

https://en.wikipedia.org/wiki/Speed_of_light

https://en.wikipedia.org/wiki/Second

And it made me curious.

The definition of c depends on the second.

The definition of second depends on the atomic clock.

So the local definition of speed of light depends on the atomic clock.

Now we know that the gravitational field has an effect on the atomic clock's speed, relatively when viewed from a far away observer, it will tick differently (Shapiro delay).

But none of these answers were talking about the effects of expanding space on an atomic clock.

The vacuum in the voids of the inter galaxy cluster space, where gravity has no effect, and dark energy is dominant, and space is expanding, in those regions, we do not know how atomic clocks tick.

I do not know whether atomic clocks would tick differently in those regions of space.

Question:

1. Do we know of atomic clock tick differently in expanding space (relatively), has there been any experiment on this?

2. will this affect the speed of light there (relatively)?

Answer 1

Do we know of atomic clock tick differently in expanding space (relatively), has there been any experiment on this?

Yes. The light we receive from far away galaxies was emitted by atoms. Whereby the measured redshift is due to the relative increase of the scale factor between emission and absorption. So from our perspective those atoms tick more slowly accordingly. The same is true for a supernova far away, the full process seems slower compared to a supernova in our vicinity. So anything which happens far away including peculiar velocities e.g. the speed of light seems slower in our accelerated expanding universe from our view.