Density Perturbations and the Cosmological Constant from Inflationary Landscapes

TL;DR

The paper investigates how scanning of inflationary parameters in a cosmological landscape affects the anthropic explanation for the cosmological constant and the amplitude of density perturbations. It shows that volume weighting from slow-roll inflation generically makes the observed perturbation amplitude $\boldsymbol{\sigma}$ exponentially unlikely to match the COBE value, creating a robust \"σ problem\" unless the initial volume distribution or the inflationary scanning is restricted. Through chaotic and hybrid inflation ensembles, the authors demonstrate the exponential sensitivity of the volume factor to $\boldsymbol{\sigma}$, and thus the tension with Weinberg-style CC predictions. They propose two main routes to avoid the σ problem: (i) a sharply peaked initial distribution ${\cal I}$ that effectively selects a single anthropically allowed vacuum, and (ii) a class of landscapes where ${\cal IV}$ (the a priori volume) yields a mild or flat dependence on $\boldsymbol{\sigma}$, possibly by decoupling $N_e$ and $\boldsymbol{\sigma}$ or by invoking a perturbation mechanism not tied to inflaton fluctuations. In the best-case scenarios, these approaches can restore a non-negligible likelihood for a small CC (a few percent), while preserving a natural explanation for the flat inflaton potential and for the observed spectrum, with significant dependence on the assumed anthropic priors and landscape structure.

Abstract

An anthropic understanding of the cosmological constant requires that the vacuum energy at late time scans from one patch of the universe to another. If the vacuum energy during inflation also scans, the various patches of the universe acquire exponentially differing volumes. In a generic landscape with slow-roll inflation, we find that this gives a steeply varying probability distribution for the normalization of the primordial density perturbations, resulting in an exponentially small fraction of observers measuring the COBE value of 10^-5. Inflationary landscapes should avoid this "σproblem", and we explore features that can allow them to do that. One possibility is that, prior to slow-roll inflation, the probability distribution for vacua is extremely sharply peaked, selecting essentially a single anthropically allowed vacuum. Such a selection could occur in theories of eternal inflation. A second possibility is that the inflationary landscape has a special property: although scanning leads to patches with volumes that differ exponentially, the value of the density perturbation does not vary under this scanning. This second case is preferred over the first, partly because a flat inflaton potential can result from anthropic selection, and partly because the anthropic selection of a small cosmological constant is more successful.

Figures (2)

• Figure 1: Parameter space of the chaotic inflation landscape. The density perturbation $\sigma$ depends on only one parameter of this model, $m$, while the volume increase due to slow-roll inflation is determined by $\phi_i$. $N_e$ and $\sigma$ are related by an $m$-dependent upper bound on $\phi_i$.
• Figure 2: Schematic parameter space of hybrid inflation models for fixed $\phi_i$ and $\phi_f$. Sufficient e-folding is not obtained in the lower-right region, while the density perturbation is too large in the upper-left region. Directions normal to the contours of $\sigma$ and $N_e$ are indicated by two arrows in the figure, and are slightly different. Thus, on a contour of $\sigma$, the e-folding number $N_e$ increases in the direction shown by the broken arrow. For a given $\sigma$, $(M(\sigma),m(\sigma))$ on the line $M=M_{min}$ provides the largest e-folding number $N_e(M(\sigma),m(\sigma))$. With $\phi_i \sim M_{\rm pl}$ and $\phi_f \sim M_{\rm pl}/e$ the volume increase factor is roughly given by $e^{3N_e(M(\sigma),m(\sigma))}$.