A generic estimate of trans-Planckian modifications to the primordial power spectrum in inflation

TL;DR

This work provides a general, boundary-condition-based framework for trans-Planckian modifications to the primordial power spectrum during inflation. It shows that long-wavelength spectra are modulated by a factor determined by the vacuum choice at a fixed short-distance cutoff, with the background expansion shaping the k-dependence through $y_c(k)=p_c/H$ and $y_c(k)\propto k^ ext{ε}$. By applying Danielsson's adiabatic vacuum to arbitrary backgrounds, the authors derive explicit expressions for the mode-coefficient pair $(C_+,C_-)$ and demonstrate that the leading modulation scales as $H/M$, offering a simple phenomenological path to constrain short-distance physics with cosmological data. The results emphasize that the amplitude, not just the presence, of such trans-Planckian signatures hinges on the choice of boundary condition, and that a full theory of short-distance physics is needed to fix these conditions from first principles.

Abstract

We derive a general expression for the power spectra of scalar and tensor fluctuations generated during inflation given an arbitrary choice of boundary condition for the mode function at a short distance. We assume that the boundary condition is specified at a short-distance cutoff at a scale $M$ which is independent of time. Using a particular prescription for the boundary condition at momentum $p = M$, we find that the modulation to the power spectra of density and gravitational wave fluctuations is of order $(H/M)$, where $H$ is the Hubble parameter during inflation, and we argue that this behavior is generic, although by no means inevitable. With fixed boundary condition, we find that the shape of the modulation to the power spectra is determined entirely by the deviation of the background spacetime from the de Sitter limit.

Figures (2)

• Figure 1: Power spectrum $P^{1/2}(k)$ as a function of $k$, for $\epsilon = 0.01$ and $y_c(k_0) = 100$.
• Figure 2: The modulation $\Delta P^{1/2} / P^{1/2} = 1 - \left\vert C_{+} + C_{-} \right\vert$ as a function of wavenumber $k$, for $\epsilon = 0.01$ and $y_c(k_0) = 100$.