Anthropic prediction in a large toy landscape

TL;DR

This paper analytically investigates the distribution of vacua in a toy landscape model to assess the validity of anthropic predictions for the cosmological constant, highlighting the conditions under which the prior appears flat.

Contribution

It provides an analytical calculation of the full distribution in a large toy landscape, exploring conditions for an effectively flat prior distribution.

Findings

Fractal prior distribution can behave as flat in certain landscape regimes

The behavior depends on unknown landscape details

Analytical approach extends previous simplified models

Abstract

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.