Trans-Planckian Physics and Inflationary Cosmology

TL;DR

The paper addresses the trans-Planckian problem in inflationary cosmology by examining two principled approaches: modified dispersion relations and space-time noncommutativity (via a string-inspired space-time uncertainty). It shows that Planck-scale physics can modify the generation and evolution of cosmological perturbations, potentially leaving observable imprints in the curvature power spectrum and offering a window into quantum gravity effects. Additionally, it explores a novel inflation mechanism driven by trans-Planckian physics within a non-commutative framework, where inflation arises from a modified radiation-like medium and can produce a near-scale-invariant spectrum under suitable conditions. The work emphasizes the importance of back-reaction, boundary conditions, and compatibility with high-energy constraints in assessing the viability and observability of trans-Planckian effects.

Abstract

Due to the quasi-exponential red-shifting which occurs during an inflationary period in the very early Universe, wavelengths which at the present time correspond to cosmological lengths are in general sub-Planckian during the early stages of inflation. This talk discusses two approaches to addressing this issue which both indicate that the standard predictions of inflationary cosmology - made using classical general relativity and weakly coupled scalar matter field theory - are not robust against changes in the physics on trans-Planckian scales. One approach makes use of modified dispersion relations for a usual free field scalar matter theory, the other uses some properties of space-time non-commutativity - a feature expected in string theory. Thus, it is possible that cosmological observations may be used as a window to explore trans-Planckian physics. We also speculate on a novel way of obtaining inflation based on modified dispersion relations for ordinary radiation.

Figures (1)

• Figure 1: Space-time diagram (physical distance vs. time) showing how inflationary cosmology provides a causal mechanism for producing cosmological fluctuations. The line labeled a) is the physical wavelength associated with a fixed comoving scale $k$. The line b) is the Hubble radius or horizon in SBB cosmology. Note that at $t_{\rm rec}$ the fluctuation is outside the Hubble radius. Curve c) shows the Hubble radius during inflation. As depicted, at early times during inflation the curve a) is inside the Hubble radius, thus allowing for a causal generation mechanism for fluctuations on the corresponding scale. The horizon in inflationary cosmology is shown in curve d). The graph also demonstrates the trans-Planckian problem of inflationary cosmology: at very early times, the wavelength is smaller than the Planck scale $\ell _{\rm Pl}$ (Phase I), at intermediate times it is larger than $\ell _{\rm Pl}$ but smaller than the Hubble radius $H^{-1}$ (Phase II), and at late times during inflation it is larger than the Hubble radius (Phase III).