Minimal modifications of the primordial power spectrum from an adiabatic short distance cutoff

TL;DR

The paper tackles the trans-Planckian problem in inflation by proposing a minimal short-distance model in which each mode is initialized in its instantaneous adiabatic vacuum at a fixed cutoff scale $M$ and then evolves freely. By formulating the problem in a power-law inflation background, it derives the Bogoliubov coefficients relating the instantaneous vacuum to the asymptotic positive-frequency solution and computes the resulting modification to the curvature perturbation power spectrum. The leading correction is found to be of order $(H/M)^3$, with an oscillatory modulation in wavenumber $k$ of the form $\delta_k \propto \sigma_0^3 \cos(2x_k+\chi_k)$ where $x_k=-q/\sigma_k$ and $\sigma_k=\sigma_0 (k/M)^{-\epsilon}$; in the exact de Sitter limit ($\epsilon=0$) one has $|\beta_k^{\eta_k}| \simeq \tfrac{1}{2}\sigma_0^3$ and $\delta_k$ acquires a simple amplitude. This framework ties the oscillation frequency to the slow-roll parameter $\epsilon$ and the amplitude to the high-energy ratio $\sigma_0=H/M$, offering a concrete, testable signature of Planck-scale physics in the CMB and a lower bound on possible trans-Planckian modifications.

Abstract

As a simple model for unknown Planck scale physics, we assume that the quantum modes responsible for producing primordial curvature perturbations during inflation are placed in their instantaneous adiabatic vacuum when their proper momentum reaches a fixed high energy scale M. The resulting power spectrum is derived and presented in a form that exhibits the amplitude and frequency of the superimposed oscillations in terms of H/M and the slow roll parameter epsilon. The amplitude of the oscillations is proportional to the third power of H/M. We argue that these small oscillations give the lower bound of the modifications of the power spectrum if the notion of free mode propagation ceases to exist above the critical energy scale M.