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Recently I've been reading up a bit on inflation and the subsequent reheating of the early universe. What has left me a bit confused, however, is that in all of the notes/papers that I've read so far the author simply states that at the end of inflation the inflaton field $\phi$ has formed a condensate of coherently oscillating zero-momentum inflatons, but gives no justification for why this is the case?!

From my naive undestanding, during inflation the dynamics of the inflaton and the cosmological background are governed by the following set of coupled differential equations: $$\ddot{\phi}+3H(t)\dot{\phi} +m^{2}\phi=0\\ H^{2}=\frac{1}{3M_{P}^{2}}\left(\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}m^{2}\phi^{2}\right)$$ where $H(t)=\frac{\dot{a}}{a}$ is the Hubble parameter, and I have assumed for simplicity that $V(\phi)=\frac{1}{2}m^{2}\phi^{2}$ (also assuming natural units).

I can't see from this how one can conclude that at the end of inflation the inflaton field $\phi$ has formed a condensate of zero-momentum inflatons, other than the fact that at this point it has started to oscillate about the minimum of its potential, centred around $\phi =0$. Is the point that this is so far a classical description, but quantum mechanically this can be interpreted as a collection of coherently oscillating inflaton quanta?

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The initial stage of reheating, also known as preheating, can involve highly non-perturbative processes, during which the universe gets populated via parametric resonances. They cannot be described with the usual perturbative expansions in coupling constants, even in cases with weak couplings. Such resonances arise as the inflaton condensate (or generally any light scalar, that has attained a non-zero vacuum expectation value during inflation) begins to oscillate about the minimum of its potential, soon after inflation. The oscillations induce an effective time dependence in the couplings of the inflaton to the other species of matter.

Credit :- "Lectures on reheating after inflation by Kaloian Lozanov"

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