Why does power spectrum is calculated at the radiation/matter epochs?

[chronnox]

In the standard inflationary scenario, the power spectrum is evaluated at the cosmological time when one assumes an equation of state $ P= \omega \rho$ , that is, one is assuming a particular radiation or matter dominated universe. Why does it has to be in these cosmological epochs? does it involve something about the physical scale of the perturbations reentering the Hubble horizon and then one treats the perturbations as classical fields?.

 

[Chalnoth]

In the standard inflationary scenario, the power spectrum is evaluated at the cosmological time when one assumes an equation of state $ P= \omega \rho$ , that is, one is assuming a particular radiation or matter dominated universe. Why does it has to be in these cosmological epochs? does it involve something about the physical scale of the perturbations reentering the Hubble horizon and then one treats the perturbations as classical fields?.

The primary constraint on the contents of the very early universe stems from Big Bang Nucleosynthesis, where if the universe had different contents, the relative ratios of the primordial light elements would have been different. Because of this, we can be quite confident that at and before the emission of the cosmic microwave background, our universe was almost entirely dominated by normal matter (including dark matter) and radiation: dark energy played little to no role during that epoch.

The reason why it works out this way is that the energy density of the different components of the universe scales differently with the expansion. Normal matter drops off with the cube of the scale factor: you're just diluting out the matter. Radiation drops off with the fourth power of the scale factor: not only are you diluting the number density of photons, but you're also redshifting them. Dark energy, by contrast, doesn't dilute much (if at all) as the universe expands, and it's this feature which makes it cause the expansion to accelerate.

What this means is that as time goes on, dark energy's contribution to the total energy density grows compared to matter and radiation: in the distant past, both matter and radiation were far more dense, but dark energy wasn't. And so the early expansion of the universe can be reproduced extremely accurately by just considering normal matter and radiation.