Noncommutative Spacetime, Stringy Spacetime Uncertainty Principle, and Density Fluctuations

TL;DR

The paper develops a noncommutative spacetime framework that realizes the stringy spacetime uncertainty relation (SSUR) without breaking homogeneous isotropic FRW symmetries and studies its impact on cosmological perturbations. By deforming the field theory with a star-product and introducing a $k$-dependent smearing factor $z_k(\tilde{η})$ and a new time coordinate $\tilde{η}$, the authors derive a nonlocal coupling between fluctuation modes and the background and a mode-specific formation time $\tilde{η}^0_k$ for each wavenumber $k$. They find that UV modes (inside the Hubble radius) reproduce the standard commutative spectrum, while IR modes (generated outside the Hubble radius) acquire a blue tilt $P_k \propto k^{3/(n+1)}$ for power-law expansion, with exponential inflation yielding a scale-invariant spectrum whose amplitude depends on the ratio $H/M_s$. This framework links string-scale physics to observable density fluctuations and offers insights into trans-Planckian concerns in inflationary cosmology, highlighting how SSUR-induced nonlocality can modify the amplitude and tilt of cosmological perturbations across different expansion histories.

Abstract

We propose a variation of spacetime noncommutative field theory to realize the stringy spacetime uncertainty relation without breaking any of the global symmetries of the homogeneous isotropic universe. We study the spectrum of metric perturbations in this model for a wide class of accelerating background cosmologies. Spacetime noncommutativity leads to a coupling between the fluctuation modes and the background cosmology which is nonlocal in time. For each mode, there is a critical time at which the spacetime uncertainty relation is saturated. This is the time when the mode is generated. These effects lead to a spectrum of fluctuations whose spectral index is different from what is obtained for commutative spacetime in the infrared region, but is unchanged in the ultraviolet region. In the special case of an exponentially expanding background, we find a scale-invariant spectrum. but with a different magnitude than in the context of commutative spacetime if the Hubble constant is above the string scale.