Definition of Energy in Friedmann equations?

[jcap]

The first Friedmann equation for a flat Universe is given by:

$$\bigg(\frac{\dot{a}(t)}{a(t)}\bigg)^2 = \frac{8 \pi G}{3} \rho(t)$$

The energy density $\rho(t)$ is given by:

$$\rho(t) \propto \frac{E(t)}{a(t)^3}$$

where $E(t)$ is the energy of the cosmological fluid in a co-moving volume.

Is the energy $E$ the energy measured by a local observer at time $t$ or is it the energy measured with respect to a (global) reference observer at the present time $t_0$ where $a(t_0)=1$?

 

[PeterDonis]

The energy density $\rho(t)$ is given by:
$$
\rho(t) \propto \frac{E(t)}{a(t)^3}
$$
where $E(t)$ is the energy of the cosmological fluid in a co-moving volume.

No, you have it backwards. The energy density $\rho(t)$ is a direct observable (it is the energy density measured by a comoving observer); it isn't derived from anything. If you want to try to define a "total energy" $E(t)$, then you could do it along the lines of $E(t) \propto \rho(t) a(t)^3$. But the physical meaning, if any, of such an $E(t)$ would be an interesting question.