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On loops in inflation. (English) Zbl 1294.83099
Summary: We study loop corrections to correlation functions of inflationary perturbations. Previous calculations have found that in certain theories the two-point function can have a logarithmic running of the form \(\log(k/\mu)\), where \(k\) is the wavenumber of the perturbation, and \(\mu\) is the renormalization scale. We highlight that this result would have profound consequences for both eternal inflation and the predictivity of standard inflation. We find a different result in the same cases. We consider two sets of theories: one where the inflaton has a large cubic self-interaction and one where the inflaton interacts gravitationally with \(N\) massless spectator scalar fields. We find that there is a logarithmic running but of the form log(\(H/\mu\)), where \(H\) is the Hubble constant during inflation. We find this result in three independent ways: by performing the calculation with a sharp cutoff in frequency-momentum space, in dimensional regularization and by the simple procedure of making the loop integral dimensionless. For the simplest of our theories we explicitly renormalize the correlation function proving that the divergencies can be reabsorbed and that the correlation function for super-horizon modes does not depend on time (once the tadpole terms have been properly taken into account). We prove the time-independence of the super-horizon correlation function in several additional ways: by doing the calculation of the correlation function at finite time using both the regularizations and by developing a formalism which expresses loop corrections directly in terms of renormalized quantities at each time. We find this last formalism particularly helpful to develop intuition which we then use to generalize our results to higher loops and different interactions. In particular we argue correlation functions have no long-term time dependence even if the spectator fields have a potential.

MSC:
83F05 Cosmology
81V22 Unified quantum theories
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T20 Quantum field theory on curved space or space-time backgrounds
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