Back-Reaction and the Trans-Planckian Problem of Inflation Revisited

TL;DR

This work revisits the trans-Planckian problem in inflation, arguing that back-reaction from ultraviolet fluctuations, when modeled with a modified dispersion relation, does not obstruct inflation. A concrete toy model with a non-linear dispersion relation is used to compute the UV energy density and pressure, showing that the UV sector behaves like a cosmological-constant–type source up to small corrections of order $H_{ m inf}/m_C$. The authors demonstrate that, in a de Sitter background, the UV energy density is time-translation invariant and the total ultra-violet equation of state remains close to $-1$, implying back-reaction renormalizes the inflationary cosmological constant rather than terminating inflation. They also acknowledge limitations of their approach and call for a covariant regularization and inclusion of metric fluctuations to solidify the conclusions, while noting possible observable signatures in the cosmic microwave background for small occupation numbers.

Abstract

It has recently been suggested that Planck scale physics may effect the evolution of cosmological fluctuations in the early stages of cosmological inflation in a non-trivial way, leading to an excited state for modes whose wavelength is super-Planck but sub-Hubble. In this case, the issue of how this excited state back-reacts on the background space-time arises. In fact, it has been suggested that such back-reaction effects may lead to tight constraints on the magnitude of possible deviations from the usual predictions of inflation. In this note we discuss some subtle aspects of this back-reaction issue and point out that rather than preventing inflation, the back-reaction of ultraviolet fluctuations may simply lead to a renormalization of the cosmological constant driving inflation.

Figures (2)

• Figure 1: Sketch of the dispersion used for the study of the toy model. The five scales $k_0$, $k_1$, $k_2$, $\Lambda _1$ and $\Lambda _2$ are defined in the text. $H_{\rm inf}$ is the Hubble parameter during inflation while $H_0$ is the Hubble parameter today. Clearly, this is not a scaled figure since $H_0$ should be much smaller than is represented on the plot.
• Figure 2: Left panel: Equation of state versus $H_{\rm inf}/m_{_{\mathrm C}}$ for $\alpha =0.01$ and $\Upsilon=0.7003$ (dotted line) computed according to Eq. (\ref{['statenearuv']}). The dashed line is an approximation of the equation of state, valid for small values of $H_{\rm inf}/m_{_{\mathrm C}}$, and derived in Eq. (\ref{['statenearuvapprox']}). Right panel: same as left panel but with $\alpha =0.0001$ and $\Upsilon=0.700003$.