Making a scale factor invariant *density* in FRW spacetime

Question

For a timelike observer in an FRW spacetime with a perfect fluid, the timelike energy density is given by $T_{\mu\nu}U^\mu U^\nu = \rho(t)$ for a comoving observer.

I want to be able to track changes to $\rho(t)$ but not include changes due to the scale factor. Let's assume only dust for this simple example, and occasionally some of the dust is stolen into a different universe or something.
Integrating over a flat or open universe will yield an infinite value, and a closed universe would be some arbitrary but finite value. So it seems like integrating won't yield anything useful.

Could I use something like $$\rho(t)\sqrt{-g}$$ to represent a scale factor independent quantity? Because $\sqrt{-g} = a^3$ for a simple cartesian example, it would make physical sense that the $a^{-3}$ is cancelled by the $a^3$.

My concern is that $\rho(t)\sqrt{-g}$ is a scalar density, and so is not generally covariant like good GR objects should be. To fix this, you would multiply by the opposite weight, which is -1, so to turn it into a scalar it would simply be $$\rho(t)\sqrt{-g}\frac{1}{\sqrt{-g}}$$ And we're back to where we started with $\rho(t)$

Now that i've written this out, I guess what I'm asking is if it is possible to define a scale factor independent quantity and not have it be a scalar or tensor density.

Answer 1

You are looking for a scalar representation of what is normally called the "comoving density" (or density per comoving volume), $\rho a^3$, where $\rho$ is the rest-frame energy density of the fluid and $a$ is the global FLRW scale factor. The problem is that if you could find such a scalar, then you could take the ratio between that and the density to get (some power of) the global scale factor $a$ as a scalar. That is impossible because $a$ does not correspond to a generally covariant quantity (see https://physics.stackexchange.com/a/771021/180843 and links therein). On a conceptual level, the local physics do not "know" about the global expansion of the universe.

However, if you don't care about the global scale factor and simply want to factor out the local expansion or contraction of your fluid, then that can be done. One possibility is to divide the rest-frame energy density by the rest-frame particle number density. For dust, this just gives the average rest mass of the constituent particles, but for relativistic fluids you can get something more interesting.

That might not be exactly what you want, since it would not inform you as to whether

occasionally some of the dust is stolen into a different universe or something.

Perhaps you want to test instead whether the fluid's density changes commensurately with its velocity divergence, since the latter describes how it is expanding. That amounts to testing if the continuity equation holds. However, the continuity equation corresponds to the $\mu=0$ component of the expression for conservation of the stress-energy tensor, ${T^{\mu\nu}}_{;\nu}=0$, which is automatically true in general relativity (it's enforced by the Einstein field equations). So these "disappearing particles" are not possible in general relativity, but if you wanted to test for them anyway, the appropriate scalar quantity might be $$U_\mu {T^{\mu\nu}}_{;\nu}.$$