Holographic discreteness of inflationary perturbations

Abstract

The cosmological holographic principle, which states that the total observable entropy of de Sitter space (including gravitational quanta) is bounded by S < π/H 2 , where H is the expansion rate, is used to estimate the magnitude of quantum-gravitational effects on inflationary perturbations. The constraint is shown to imply that the initial states of the vacua for perturbation modes are not independent, and that field theory is not a reliable tool to study transplanckian effects on mode amplitudes and phases. It is argued that holographic discreteness alters the continuous, random-phase gaussian distribution predicted by standard field theory for fluctuations in the inflaton field that lead to cosmic background anisotropy. A toy model, applied in the context of scalar inflaton perturbations produced during standard slow-roll inflation, and assuming that horizonscale perturbations "freeze out" in discrete steps separated by one bit of observable information, predicts discrete steps in anisotropy separated by ∆T ≈ 10 −10 K. It is conjectured that the Hilbert space of a typical observable perturbation is equivalent to that of no more than about 10 5 binary spins (approximately the inverse of the final scalar metric perturbation amplitude, independent of H and other parameters), and that some manifestations of this discreteness may be observable.

Initial Conditions for Inflation

“…The rate will be approximately e 2α ∼ 10 −10 , which is independent of 8 Of course, it has been proposed by many authors that a solution to the cosmological constant problem requires mixing between the scale of the present horizon and the fundamental UV scales of the theory. However, the best guess for the effects of this mixing during inflation would be that they make the number of degrees of freedom in an inflating patch finite [45,46,47]. The effects on observations will be much smaller than the effects discussed here [48].…”

Discrete spectrum of inflationary fluctuations

“…The combination of inflatonpotential parameters H ÿ3 V 0 , in slow-roll inflation, is approximately given by the inverse of the final (observed) amplitude of gravitational metric scalar perturbations. Up to factors of the order of unity [13], n V H ÿ3 V 0 2 10 10 . This gives an estimate N H for modes of the inflaton on the horizon scale at any given time -approximately, the number of possible classical values the frozen-in inflaton field amplitude can adopt during an e-folding of inflation.…”

Role of the cosmological constant in the holographic description of the early universe

“…quasi-de Sitter spacetime [3] and during the slow-roll period, the equation of state can be approximated by the vacuum state as p ≈ −ρ like p Λ = ωρ Λ , ω = −1 for the cosmological constant Λ [4]. The other is that an accelerating universe (with positive cosmological constant) has recently proposed to interpret the astronomical data of supernova.…”