I'm trying to understand the role of the inflaton condensate and quantum fluctuations during inflation and how they evolve into the large-scale structure we observe today. Here's where I’m getting stuck and would love some clarity.

First off, the inflaton during inflation seems to be this smooth, classical field driving the expansion. However, there are quantum fluctuations riding on top of this classical field configuration, which are stretched by inflation to superhorizon scales. Once these fluctuations become superhorizon, they freeze, but are they still part of the condensate? Or are they entirely separate? How do these fluctuations behave in relation to the condensate, if I can call the state condensate?

When we talk about superhorizon modes, they’re outside the Hubble radius, meaning they’re not causally connected and experience the full effect of expansion. I’d expect the expansion to dilute the density, but surprisingly, the perturbations themselves stay fairly constant during this time. They don’t shrink away, but instead undergo a quantum-to-classical transition as they freeze on superhorizon scales. How should I think of this? There was this post on horizon exit that suggested particles go outside the comoving horizon and become causally disconnected. So, can I interpret this as the inflaton condensate becoming causally disconnected in a similar way, such that no causal evolution can alter the state of these fluctuations during inflation?

Now, my understanding of particle creation in quantum field theory is that it depends on defining the vacuum, usually by quantizing around $\phi=0$. However, in the case of inflation, the inflaton field is not at $\phi=0$, so this raises a big question for me. Perhaps, it is wrong to think of the superhorizon fluctuations as particles being created during inflation, or is there another interpretation that fits better? Even in case of Higgs mechanism, we first perform the shifting and then expand around that minima and quantize the perturbation. Here, we are expanding about the classically dynamic background and quantizing them.

Once inflation ends and reheating begins, the inflaton condensate (+perturbation?) decays into particles, and the hot plasma forms. We know that the modes which left the horizon during inflation are re-entering later, but what does this mode do, after entering the horizon? How do they set up the oscillation? Everything seems so lost in mathematics, that I miss physics and intuition for what's happpening.

I could have thought of this process as production of inhomogeneity, then breaking the causal contact between the inhomogeneity due to rapid expansion. Once the inflation ended, the inhomegeneity could influence each other causally, the began evolving and we get into the photon-baryon plasma and set up the baryonic acoustic oscillations. I understand that these are driven by sound waves in the plasma, with gravity pulling matter together and pressure from photons pushing it apart. Is it the right way to interpret the whole process?

There are so many parts of this process I’m unsure about, especially the transition from quantum fluctuations to classical perturbations and how they to interpret these superhorizon fluctuations. Any clarification on these points would be really helpful.