Reheating of the early universe and the inflaton field

Question

I've recently been reading this set of introductory notes on the physics of reheating of the early universe. In it the author briefly mentions that, at the end of inflation, the inflaton field can be treated as a condensate of inflaton particles with zero momentum and oscillating at frequency $\omega =m$.

My question is, what is the reasoning why one can treat the inflaton field as a condensate of inflaton quanta at zero momentum? Is it simply because the inflaton field is oscillating around the minimum of its potential and so the momentum of each field mode is subdominant, such that each mode is oscillating at approximately the same frequency at (approximately) zero momentum and in this sense can be considered as a condensate of inflaton quanta in the zero momentum state?

Answer 1

In Equation 2.3 the author proves that a scalar field in a quadratic potential behaves like dust. Dust doesn't exert any pressure. The pressure, and thus the momenta, is negligible compared to its density. That's how he gets to 2.4, in page 4

Pressureless dust is the way a matter dominated universe is modeled in GR also. Just a lot of particles but no pressure resisting compression. A perfect fluid with rho and p is called dust when p is zero. Radiation is totally different, significant pressure, and momentum.

Answer 2

As we can see from eq.(3.1) (i.e., the Fourier expansion of the field) of the article mentioned, the classical inflaton field (zeroth mode of the inflaton field $\phi_0(t)$) can be similarly written as a series of zero momentum modes.