bloby said:
To use Einstein fields equations you have to choose a coordonate systeme(c.s.). The FRW metric is the metric in a c.s. in which the matter appears isotropic and homogeneouse. This c.s. is the comoving c.s. in which galaxies keep their own time independent coordinate. The R-H-S of Einstein equations must be expressed in the same c.s. Now the energy-momentum tensor takes the form of a perfect fluid at rest. How are defined pressure and density? In a "comoving volume" the quantity of matter will remains the same...
Well, as you said, the pressure and density depend upon the coordinate system. Just as a particularly dumb example, I can express water as being 1g/cm^3, or 1000kg/m^3. In the end, though, the relationship between pressure and energy density is determined by the type of matter you have. For example, normal matter and dark matter effectively experience zero pressure on cosmological scales (e.g. there is no pressure between galaxies).
By contrast, light has a pressure equal to one third its energy density.
The relationship between pressure and energy density is actually what determines how the matter evolves within a comoving volume. For instance, with normal matter, which has no pressure on cosmological scales, an expanding volume always has the same average amount of matter, and so the total amount doesn't change, and the density just decreases along with the increase in volume.
Photons, however, are different. Because they experience pressure, expansion actually removes energy from the system, so while an expanding gas of photons may keep the same number of photons per unit volume, the individual photons themselves reduce in energy. This is the cosmological redshift.
Now, as you correctly point out, this does depend upon the coordinate system you use. Everything I have said in this post assumes FRW coordinates, the ones you talked about. I could, in principle, use any other coordinate system I chose, and things would appear somewhat different in those coordinate systems, but none of the actual physics would change. One can thus think of this description of one possible way of talking about it, but not the only correct way: just as before, I can correctly talk about water having 1g/cm^3 density, or 1000kg/m^3 density. Neither is more valid than the other, though one may be more useful in certain contexts than the other.
