Density and Gravity in expanding Universe Be $a(t)$ the expansion factor and $\rho$ a density, say of an ideal gas or something. In various papers the relation $\overline{\rho}=a^3\rho$ is stated. But why should this be the case? When space expands, shouldn't density decline? The mass doesn't change but the volume grows, doesn't it?
Then, a somewhat related question: what happens to gravitational acceleration ${g}$ (a vector), when space expands with $a(t)$?
The second question is related to the first one, because a steady state solution to a relativistic fluid should be $\nabla P = \rho g$, with $P$ the pressure. How would this look like in a non-steady state?
Thanks a lot in advance. :)
 A: This answer of mine is relevant to your question .
The expansion of the universe as it has been measured with the Hubble constant, does not affect "say of an ideal gas or something". All matter is held together by forces much stronger than the effective"force" of expansion. Thus the ideal gas law will hold because the atom are strongly held together by the four forces , gravity, electromagnetic, weak and strong. As I say in the link, which provides links for these statements, :

Evidently, by construction, if everything was expanding, then one would never know it, because the units would be expanding accordingly. Meters and seconds are a matter of definition after all. If the defining measure is expanding there would be no way to know it.

Galaxies themselves are tightly bound by gravity. Their density is something measurable and

the three possible types of expanding universes are called open, flat, and closed universes. If the universe were open, it would expand forever. If the universe were flat, it would also expand forever, but the expansion rate would slow to zero after an infinite amount of time. If the universe were closed, it would eventually stop expanding and recollapse on itself, possibly leading to another big bang. In all three cases, the expansion slows, and the force that causes the slowing is gravity.
For the last eighty years, astronomers have been making increasingly accurate measurements of two important cosmological parameters: Ho - the rate at which the universe expands - and w - the average density of matter in the universe. Knowledge of both of these parameters will tell which of the three models describes the universe we live in, and thus the ultimate fate of our universe. The Sloan Digital Sky Survey, with its large systematic measurement of the galaxy density in the Universe, should enable astronomers to precisely measure the density parameter w.

....

Then, a somewhat related question: what happens to gravitational acceleration g (a vector), when space expands with a(t)?

Nothing in the vicinity of the earth, as matter is tightly bound and the expansion cannot affect it. For cosmological distances one does not use g, but general relativity equations .
( at the moment the hyperphsyics links do not work, but I hope  the site will recover. Also the negative vote comes with the comment which I have answered, a misunderstanding imo)
