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The Trans-Planckian Problem of Inflationary Cosmology

Jerome Martin, Robert H. Brandenberger

TL;DR

The paper investigates whether the standard inflationary prediction of a scale-invariant spectrum of fluctuations is robust when trans-Planckian physics is taken into account. It models possible Planck-scale effects by introducing time-dependent dispersion relations for a free scalar perturbation and analyzes two representative classes (Unruh-type and Corley–Jacobson-type) in an expanding de Sitter background. Through quantization with energy-density-minimizing initial states and region-wise analytical solutions, it shows that Unruh-type dispersion largely preserves scale invariance for the minimum-energy state, while Corley–Jacobson dispersion can induce tilt, oscillations, or exponential growth depending on the dispersion's real/complex character and the chosen initial state. The findings imply that inflationary predictions may depend sensitively on high-energy physics and motivate further work on connecting Planck-scale theories to cosmological observables. This has implications for unifying fundamental physics with early-Universe cosmology and for interpreting CMB/LSS data in light of possible new physics at the Planck scale.

Abstract

In most current models of inflation based on a weakly self-coupled scalar matter field minimally coupled to gravity, the period of inflation lasts so long that, at the beginning of the inflationary period, the physical wavelengths of comoving scales which correspond to the present large-scale structure of the Universe were smaller than the Planck length. Thus, the usual computations of the spectrum of fluctuations in these models involve extrapolating low energy physics (both in the matter and gravitational sector) into regions where this physics is not applicable. In this paper we demonstrate that the usual predictions of inflation for the spectrum of cosmological fluctuations do indeed depend on the hidden assumptions about super-Planck scale physics. We introduce a class of modified dispersion relations to mimic possible effects of super-Planck scale physics, and show that in some cases important deviations from the usual predictions of inflation are obtained. Some implications of this result for the unification of fundamental physics and early Universe cosmology are discussed.

The Trans-Planckian Problem of Inflationary Cosmology

TL;DR

The paper investigates whether the standard inflationary prediction of a scale-invariant spectrum of fluctuations is robust when trans-Planckian physics is taken into account. It models possible Planck-scale effects by introducing time-dependent dispersion relations for a free scalar perturbation and analyzes two representative classes (Unruh-type and Corley–Jacobson-type) in an expanding de Sitter background. Through quantization with energy-density-minimizing initial states and region-wise analytical solutions, it shows that Unruh-type dispersion largely preserves scale invariance for the minimum-energy state, while Corley–Jacobson dispersion can induce tilt, oscillations, or exponential growth depending on the dispersion's real/complex character and the chosen initial state. The findings imply that inflationary predictions may depend sensitively on high-energy physics and motivate further work on connecting Planck-scale theories to cosmological observables. This has implications for unifying fundamental physics with early-Universe cosmology and for interpreting CMB/LSS data in light of possible new physics at the Planck scale.

Abstract

In most current models of inflation based on a weakly self-coupled scalar matter field minimally coupled to gravity, the period of inflation lasts so long that, at the beginning of the inflationary period, the physical wavelengths of comoving scales which correspond to the present large-scale structure of the Universe were smaller than the Planck length. Thus, the usual computations of the spectrum of fluctuations in these models involve extrapolating low energy physics (both in the matter and gravitational sector) into regions where this physics is not applicable. In this paper we demonstrate that the usual predictions of inflation for the spectrum of cosmological fluctuations do indeed depend on the hidden assumptions about super-Planck scale physics. We introduce a class of modified dispersion relations to mimic possible effects of super-Planck scale physics, and show that in some cases important deviations from the usual predictions of inflation are obtained. Some implications of this result for the unification of fundamental physics and early Universe cosmology are discussed.

Paper Structure

This paper contains 14 sections, 110 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Equivalence between cosmological perturbations and a fictitious scalar field
  3. Time dependent dispersion relations
  4. Unruh's dispersion relation
  5. Generalized Corley/Jacobson dispersion relation
  6. Quantization of a massive scalar field
  7. Lagrangian and Hamiltonian formalisms
  8. Fixing the initial conditions
  9. Analytical solutions
  10. Unruh's case
  11. The Corley/Jacobson case
  12. The case $s=-1$, $b_m<0$
  13. The case $s=1$, $b_m>0$
  14. Discussion and Conclusions

Figures (3)

  • Figure 1: Sketch of the different dispersion relations.
  • Figure 2: Sketch of the evolution of a physical mode throughout the three regions defined in the text. The dashed region is the region where the dispersion relation is modified.
  • Figure 3: Sketch of the time evolution of the adiabaticity coefficient $\alpha$ for the dispersion relations.