Cosmological perturbations and short distance physics from Noncommutative Geometry

TL;DR

This paper explores how noncommutative geometry at short distances could influence inflationary perturbations by introducing a covariant curved $*$-product with a fixed noncommutativity scale $\Lambda$. By expanding to $\mathcal{O}(\Theta^2)$, it derives a modified dispersion relation for inflaton fluctuations that depends on the perpendicular component $k_\perp$ and a preferred spatial direction, controlled by $\epsilon^2 = H^4/(8\Lambda^4)$. The resulting primordial spectrum gains a directional correction while maintaining near-Gaussianity, and propagates to the CMB as a quadrupole-type anisotropy with off-diagonal correlations between multipoles separated by two, scaling as $\epsilon^2$ (i.e., $(H^4/\Lambda^4)$). The work suggests that future CMB measurements could test quantum spacetime scales if $\Lambda$ is not far below the inflationary horizon.

Abstract

We investigate the possible effects on the evolution of perturbations in the inflationary epoch due to short distance physics. We introduce a suitable non local action for the inflaton field, suggested by Noncommutative Geometry, and obtained by adopting a generalized star product on a Friedmann-Robertson-Walker background. In particular, we study how the presence of a length scale where spacetime becomes noncommutative affects the gaussianity and isotropy properties of fluctuations, and the corresponding effects on the Cosmic Microwave Background spectrum.