More on IR divergences and decoherence in inflationary universe

I happened recently to dig through a couple of interesting papers by Yuko Urakawa and Takahiro Tanaka about IR divergences that cosmological perturbation theory in quasi-dS space features, namely “No influence on observation from IR divergence during inflation — Single field inflation –” and “Influence on observation from IR divergence during inflation — Multi field inflation –“.

Since the subject is close to my heart, I’ll make here a couple of comments for own future reference.

1. In the first paper, the authors suggest an interesting idea to cure the problem of IR divergences plaguing inflationary perturbation theory. The idea is the following. When we construct perturbation theory in quasi-de Sitter space, we pick a gauge and choose an appropriate time slicing of the background spacetime. For example, we can choose slicing. This is not an appropriate thing to do, they say, since it contradicts causality. A given observer does not have access to the whole hypersurface but only to her own light cone part of it. Instead of choosing constant t slicing, one has to choose a gauge and slicing that does not contradict causality in this sense.

What does it mean exactly? Well, in cosmological perturbation theory we have to take care of not only dynamical equations of motion for degrees of freedom, but for constraints as well. The latter typically have the form like (symbolically)

,

i.e., elliptical equations. They have to be supplied with appropriate boundary conditions, and the authors suggest to choose them for the observer’s light cone (in other words, the gauge is only chosen inside of an observer’s causal patch, while no restrictions are introduced beyond it).
If they choose the gauge in this way, they claim that IR divergences in the perturbation theory are automatically absent.

I am not sure whether I am comfortable with this point of view. Consider for example a gauge field theory in Minkowski spacetime. The counterpart of the authors’ proposal to this situation would be the idea of choosing the gauge only inside the light cone with no restrictions applied to the theory outside it. I find this a bit strange and counter-intuitive

Second, even if IR divergences are absent in the gauge chosen by the authors, they will be present in some other gauge as the authors note themselves. One can say that IR divergences are gauge artifact, but what is physical is not IR divergent behavior of correlation functions of the inflaton and curvature perturbation, but the running present in these correlation functions. I wonder how running looks like in the gauge chosen by the authors.

2. In the second paper the authors expand their analysis to the multifield case. One of the most interesting claims is that decoherence should lead to spatial variation of background value from path to patch. Although the authors do not introduce any particular mechanism of decoherence, I think I agree with the claim: the time scale for decoherence to happen for the given mode in inflationary universe is several efoldings after the mode leaves horizon. The scale for description of the inflationary universe to be described in terms of Starobinsky-Fokker-Planck equation is more or less the same, and the very existence of description in terms of the distribution means spatial variation of .
In this respect, I would like to mention the paper by Prokopec and Rigopoulos, where a particular mechanism of the decoherence is introduced in multifield models – one traces out unobservable isocurvature mode.