Renormalization of initial conditions and the trans-Planckian problem of inflation

TL;DR

The paper addresses how initial-state information can propagate and be renormalized during inflation. It develops an EFT for arbitrary initial states in flat space using the Schwinger-Keldysh formalism, demonstrating that short-distance features of the initial state generate divergences confined to the initial surface and can be removed by boundary counterterms, with their coefficients running under the renormalization scale via the Callan-Symanzik equation. It classifies initial states as renormalizable (boundary relevant/marginal) or nonrenormalizable (boundary irrelevant) and shows how these distinctions map onto a boundary EFT that controls potential trans-Planckian imprints on the CMB. The framework provides a principled, scalable way to parametrize Planck-scale physics in inflation and outlines extensions to expanding backgrounds and backreaction, highlighting observational relevance through scale hierarchies $H$ and $M$.

Abstract

Understanding how a field theory propagates the information contained in a given initial state is essential for quantifying the sensitivity of the cosmic microwave background to physics above the Hubble scale during inflation. Here we examine the renormalization of a scalar theory with nontrivial initial conditions in the simpler setting of flat space. The renormalization of the bulk theory proceeds exactly as for the standard vacuum state. However, the short distance features of the initial conditions can introduce new divergences which are confined to the surface on which the initial conditions are imposed. We show how the addition of boundary counterterms removes these divergences and induces a renormalization group flow in the space of initial conditions.

Figures (4)

• Figure 1: The leading contributions to the running of the running of the mass $m$ and the coupling $\lambda$ in a $\varphi^4$ theory. The solid lines represent propagating $\psi$ fields while the dashed lines correspond to the zero mode $\phi$.
• Figure 2: Further graphs obtained by expanding the exponential in Eq. (\ref{['nolinfull']}) to second order. The last of these graphs contains the leading nontrivial correction to the wavefunction renormalization.
• Figure 3: The shaded blob corresponds to the following two graphs. The time derivatives act on the classical $\phi(t)$ field.
• Figure 4: The time contour---or equivalently the field content---can be formally doubled in order to write the time evolution of the initial and the final states, both defined at $t=t_0$, in terms of a single time evolution operator.