Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Trans-Planckian Effects in Inflationary Cosmology and the Modified Uncertainty Principle

S. F. Hassan, Martin S. Sloth

TL;DR

This work develops a principled framework for incorporating Planck-scale physics into inflationary perturbations via a minimum length uncertainty principle implemented through modified commutators. The resulting theory yields a non-linear, time-dependent dispersion relation $\omega_{phys}=\rho=\frac{n/a}{f(n/a)}$ and a Jacobian that reduces high-energy degrees of freedom, producing trans-Planckian damping. In de Sitter inflation, these effects fix the initial state and yield a scale-invariant (flat) power spectrum, while suppressing back reaction from ultra-high-energy modes. The approach provides a Lorentz-violating effective description that links fundamental Planck-scale physics to observable cosmological perturbations and offers a concrete mechanism to suppress trans-Planckian concerns in the early universe.

Abstract

There are good indications that fundamental physics gives rise to a modified space-momentum uncertainty relation that implies the existence of a minimum length scale. We implement this idea in the scalar field theory that describes density perturbations in flat Robertson-Walker space-time. This leads to a non-linear time-dependent dispersion relation that encodes the effects of Planck scale physics in the inflationary epoch. Unruh type dispersion relations naturally emerge in this approach, while unbounded ones are excluded by the minimum length principle. We also find red-shift induced modifications of the field theory, due to the reduction of degrees of freedom at high energies, that tend to dampen the fluctuations at trans-Planckian momenta. In the specific example considered, this feature helps determine the initial state of the fluctuations, leading to a flat power spectrum.

Trans-Planckian Effects in Inflationary Cosmology and the Modified Uncertainty Principle

TL;DR

This work develops a principled framework for incorporating Planck-scale physics into inflationary perturbations via a minimum length uncertainty principle implemented through modified commutators. The resulting theory yields a non-linear, time-dependent dispersion relation and a Jacobian that reduces high-energy degrees of freedom, producing trans-Planckian damping. In de Sitter inflation, these effects fix the initial state and yield a scale-invariant (flat) power spectrum, while suppressing back reaction from ultra-high-energy modes. The approach provides a Lorentz-violating effective description that links fundamental Planck-scale physics to observable cosmological perturbations and offers a concrete mechanism to suppress trans-Planckian concerns in the early universe.

Abstract

There are good indications that fundamental physics gives rise to a modified space-momentum uncertainty relation that implies the existence of a minimum length scale. We implement this idea in the scalar field theory that describes density perturbations in flat Robertson-Walker space-time. This leads to a non-linear time-dependent dispersion relation that encodes the effects of Planck scale physics in the inflationary epoch. Unruh type dispersion relations naturally emerge in this approach, while unbounded ones are excluded by the minimum length principle. We also find red-shift induced modifications of the field theory, due to the reduction of degrees of freedom at high energies, that tend to dampen the fluctuations at trans-Planckian momenta. In the specific example considered, this feature helps determine the initial state of the fluctuations, leading to a flat power spectrum.

Paper Structure

This paper contains 16 sections, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The minimum length uncertainty principle
  3. The dynamical nature of the modification
  4. Modified commutation relations in flat space
  5. Scalar fields in flat space with modified commutators
  6. The modified commutator in flat RW space-time
  7. Trans-Planckian effects in the inflationary era
  8. Fields in flat RW space with modified commutators
  9. Time-dependent non-linear dispersion relations
  10. Modified red-shift and evolution of wavelengths
  11. Trans-Planckian damping and the initial state
  12. Review of the $J=1$ case
  13. $J\neq 1$ and trans-Planckian damping
  14. The initial state
  15. Solution in regions I and 0
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Dispersion relations
  • Figure 2: Time evolution of wavelength $\lambda$ associated with momentum mode $n$
  • Figure 3: $\rho^2$ and $a^{-2}\partial_\eta^2 a_J/a_J$ ($a_J\equiv a{\sqrt J}$) for the Unruh dispersion relation for $\gamma =3$, as a function of momentum $v$