Making predictions in the multiverse

I describe reasons to think we are living in an eternally inflating multiverse where the observable "constants" of nature vary from place to place. The major obstacle to making predictions in this context is that we must regulate the infinities of eternal inflation. I review a number of proposed regulators, or measures. Recent work has ruled out a number of measures by showing that they conflict with observation, and focused attention on a few proposals. Further, several different measures have been shown to be equivalent. I describe some of the many nontrivial tests these measures will face as we learn more from theory, experiment, and observation.Comment: 20 pages, 3 figures; invited review for Classical and Quantum Gravity; v2: references improve

Towards a gauge invariant volume-weighted probability measure for eternal inflation

An improved volume-weighted probability measure for eternal inflation is proposed. For the models studied in this paper it leads to simple and intuitively expected gauge-invariant results.Comment: 16 pages, 3 figs, few misprints corrected, comments adde

Anthropic prediction in a large toy landscape

The successful anthropic prediction of the cosmological constant depends crucially on the assumption of a flat prior distribution. However, previous calculations in simplified landscape models showed that the prior distribution is staggered, suggesting a conflict with anthropic predictions. Here we analytically calculate the full distribution, including the prior and anthropic selection effects, in a toy landscape model with a realistic number of vacua, $N \sim 10^{500}$. We show that it is possible for the fractal prior distribution we find to behave as an effectively flat distribution in a wide class of landscapes, depending on the regime of parameter space. Whether or not this possibility is realized depends on presently unknown details of the landscape.Comment: 13 page

Measures for a Transdimensional Multiverse

The multiverse/landscape paradigm that has emerged from eternal inflation and string theory, describes a large-scale multiverse populated by "pocket universes" which come in a huge variety of different types, including different dimensionalities. In order to make predictions in the multiverse, we need a probability measure. In $(3+1)d$ landscapes, the scale factor cutoff measure has been previously shown to have a number of attractive properties. Here we consider possible generalizations of this measure to a transdimensional multiverse. We find that a straightforward extension of scale factor cutoff to the transdimensional case gives a measure that strongly disfavors large amounts of slow-roll inflation and predicts low values for the density parameter $\Omega$, in conflict with observations. A suitable generalization, which retains all the good properties of the original measure, is the "volume factor" cutoff, which regularizes the infinite spacetime volume using cutoff surfaces of constant volume expansion factor.Comment: 30 pages, 1 figure Minor revisions, reference adde

Cosmological Measures without Volume Weighting

Many cosmologists (myself included) have advocated volume weighting for the cosmological measure problem, weighting spatial hypersurfaces by their volume. However, this often leads to the Boltzmann brain problem, that almost all observations would be by momentary Boltzmann brains that arise very briefly as quantum fluctuations in the late universe when it has expanded to a huge size, so that our observations (too ordered for Boltzmann brains) would be highly atypical and unlikely. Here it is suggested that volume weighting may be a mistake. Volume averaging is advocated as an alternative. One consequence may be a loss of the argument that eternal inflation gives a nonzero probability that our universe now has infinite volume.Comment: 15 pages, LaTeX, added references for constant-H hypersurfaces and also an idea for minimal-flux hypersurface

A status report on the observability of cosmic bubble collisions

In the picture of eternal inflation as driven by a scalar potential with multiple minima, our observable universe resides inside one of many bubbles formed from transitions out of a false vacuum. These bubbles necessarily collide, upsetting the homogeneity and isotropy of our bubble interior, and possibly leading to detectable signatures in the observable portion of our bubble, potentially in the Cosmic Microwave Background or other precision cosmological probes. This constitutes a direct experimental test of eternal inflation and the landscape of string theory vacua. Assessing this possibility roughly splits into answering three questions: What happens in a generic bubble collision? What observational effects might be expected? How likely are we to observe a collision? In this review we report the current progress on each of these questions, improve upon a few of the existing results, and attempt to lay out directions for future work.Comment: Review article; comments very welcome. 24 pages + 4 appendices; 19 color figures. (Revised version adds two figures, minor edits.

Cosmological equations and Thermodynamics on Apparent Horizon in Thick Braneworld

We derive the generalized Friedmann equation governing the cosmological evolution inside the thick brane model in the presence of two curvature correction terms: a four-dimensional scalar curvature from induced gravity on the brane, and a five-dimensional Gauss-Bonnet curvature term. We find two effective four-dimensional reductions of the Friedmann equation in some limits and demonstrate that they can be rewritten as the first law of thermodynamics on the apparent horizon of thick braneworld.Comment: 25 pages, no figure, a definition corrected, several references added, more motivation and discussio

Anthropic prediction for a large multi-jump landscape

The assumption of a flat prior distribution plays a critical role in the anthropic prediction of the cosmological constant. In a previous paper we analytically calculated the distribution for the cosmological constant, including the prior and anthropic selection effects, in a large toy ``single-jump'' landscape model. We showed that it is possible for the fractal prior distribution we found to behave as an effectively flat distribution in a wide class of landscapes, but only if the single jump size is large enough. We extend this work here by investigating a large ($N \sim 10^{500}$) toy ``multi-jump'' landscape model. The jump sizes range over three orders of magnitude and an overall free parameter $c$ determines the absolute size of the jumps. We will show that for ``large'' $c$ the distribution of probabilities of vacua in the anthropic range is effectively flat, and thus the successful anthropic prediction is validated. However, we argue that for small $c$, the distribution may not be smooth.Comment: 33 pages, 7 figures Minor revisions made and references added. arXiv admin note: substantial text overlap with arXiv:0705.256

Loop Quantum Cosmology: A Status Report

The goal of this article is to provide an overview of the current state of the art in loop quantum cosmology for three sets of audiences: young researchers interested in entering this area; the quantum gravity community in general; and, cosmologists who wish to apply loop quantum cosmology to probe modifications in the standard paradigm of the early universe. An effort has been made to streamline the material so that, as described at the end of section I, each of these communities can read only the sections they are most interested in, without a loss of continuity.Comment: 138 pages, 15 figures. Invited Topical Review, To appear in Classical and Quantum Gravity. Typos corrected, clarifications and references adde