Inflation with a stringy minimal length, reworked

Abstract

In this paper we revisit the formulation of scalar field theories on de Sitter backgrounds subject to the generalized uncertainty principle (GUP). The GUP arises in several contexts in string theory, but is most readily thought of as resulting from using strings as effective probes of geometry, which suggests an uncertainty relation incorporating the string scale l s . After reviewing the string theoretic case for the GUP, which implies a minimum length scale l s , we follow in the footsteps of Kempf and concern ourselves with how one might write down field theories which respect the GUP. We uncover a new representation of the GUP, which unlike previous studies, readily permits exact analytical solutions for the mode functions of a scalar field on de Sitter backgrounds. We find that scalar fields cannot be quantized on inflationary backgrounds with a Hubble radius H −1 smaller than the string scale, implying a sensibly stringy (as opposed to Planckian) cutoff on the scale of inflation resulting from the GUP. We also compute (Hl s ) 2 corrections to the two point correlation function analytically and comment on the future prospects of observing such corrections in the fortunate circumstance our universe is described by a very weakly coupled string theory.

Features and new physical scales in primordial observables: Theory and observation

“…A general context in which modified dispersion relations finds theoretical grounding is in the of high energy Lorentz invariance violation, either considered phenomenologically [267][268][269] or in the context of a specific construction (such as Hořava gravity [270][271][272][273]). Related contexts where short distance modifications to mode propagation lead to Lorentz invariance violation as a corollary includes non-commutative geometry [274][275][276][277][278][279][280][281][282][283][284][285][286][287][288] and inflation in the presence of a minimal length scale [289][290][291][292][293][294][295][296][297][298][299][300][301][302][303].…”

Primordial power spectra of cosmological fluctuations with generalized uncertainty principle and maximum length quantum mechanics

“…An alternative approach in deriving the effects of a GUP on the primordial perturbation spectrum involves the generalization of the position and momentum operators as described in the Introduction, but with an ultraviolet rather than infrared cutoff, while keeping the field theoretical commutation relations unchanged [88,89]. According to [88,89], this approach would also lead to a modification of the evolution of the field perturbation modes eq. (3.24) even though this equation is derived before quantization at the classical level.…”

Generalized non-commutative inflation

“…elements: π(Ψ), U N P π(Φ) → Ψ, U P Φ ) to the associated low energy Poincaré representation if all intermediate states of direct integral decomposition converge as (9), which corresponds to states of sufficiently low energy and momentum. Many phenomenological models of Trans-Planckian physics are based on deformations of the energymomentum relations (see for example: [25], [26], [27], [28], [29] and [30]). As we do not have a consensus on the correct non-commutative version of space-time, or the high energy deformation of Poincaré symmetry, we could consider the cosmological consequences associated with the correspondent deformed energy-momentum relation and thereby, in principle, put cosmological constraints on physical principles beyond the standard model.…”

“…Many phenomenological models of trans-Planckian physics are based on deformations of the energy-momentum relations (see for example: [25][26][27][28][29] and [30]). As we do not have a consensus on the correct non-commutative version of spacetime, or the high-energy deformation of the Poincaré symmetry, we could consider the cosmological consequences associated with the correspondent deformed energy-momentum relation and thereby, in principle, put cosmological constraints on physical principles beyond the standard model.…”