The Corley-Jacobson dispersion relation and trans-Planckian inflation

TL;DR

This work investigates whether trans-Planckian physics can alter the inflationary perturbation spectrum by analyzing Corley–Jacobson dispersion with $n_{ m eff}^2 = n^2 + n^2 b_1 \left(\frac{\ell_C}{\lambda}\right)^2$. It introduces a new exact analytic solution based on parabolic cylinder functions $E(a,x)$, where $a = -\frac{\pi}{\epsilon \sqrt{b_1}}$ and $\epsilon = \frac{\ell_C}{\ell_0}$, providing a smooth connection between sub-Planckian and low-energy regimes and cross-validating with the WKB approach in adiabatic cases. The key findings are that for $b_1>0$ the Harrison-Zeldovich spectrum is preserved when the matching is performed at the correct frequency, while for $b_1<0$ the spectrum can be modified with oscillations or exponential terms depending on initial conditions. Overall, the results illustrate the high sensitivity of inflationary predictions to sub-Planckian physics and underscore the need for a complete UV theory to make robust, testable predictions.

Abstract

In this Letter we study the dependence of the spectrum of fluctuations in inflationary cosmology on possible effects of trans-Planckian physics, using the Corley/Jacobson dispersion relations as an example. We compare the methods used in previous work [1] with the WKB approximation, give a new exact analytical result, and study the dependence of the spectrum obtained using the approximate method of Ref. [1] on the choice of the matching time between different time intervals. We also comment on recent work subsequent to Ref. [1] on the trans-Planckian problem for inflationary cosmology.