Measures for a Multidimensional Multiverse

TL;DR

This paper investigates how measures in a multidimensional multiverse influence the probability distribution of observations, revealing that certain timescale coincidences are robust and independent of the universe's dimensionality or measure choice.

Contribution

It generalizes measure calculations to a multidimensional multiverse and shows the robustness of timescale coincidences across different dimensions and measures.

Findings

Timescale coincidences are robust in multidimensional multiverses.

The shape of probability distributions varies with the number of large spatial dimensions.

The observed timescale coincidence cannot distinguish between different measures or dimensions.

Abstract

We explore the phenomenological implications of generalizing measures to a multidimensional multiverse. We consider a simple model in which the vacua are nucleated from a $D$-dimensional parent spacetime through dynamical compactification of the extra dimensions, and compute the geometric contribution to the probability distribution of observations within the multiverse for each measure. We then study how the shape of this probability distribution depends on the timescales for the existence of observers, for vacuum domination, and for curvature domination ($t_{obs}, t_{\Lambda},$ and $t_c$, respectively.) In this work we restrict ourselves to bubbles with positive cosmological constant, $\Lambda$. In the case of the causal patch cutoff, when the bubble universes have $p+1$ large spatial dimensions with $p \geq 2$, the shape of the probability distribution is such that we obtain the coincidence of timescales $t_{obs} \sim t_{\Lambda} \sim t_c$. Moreover, the size of the cosmological constant is related to the size of the landscape. However, the exact shape of the probability distribution is different in the case $p = 2$, compared to $p \geq 3$. In the case of the fat geodesic measure, the result is even more robust: the shape of the probability distribution is the same for all $p \geq 2$, and we once again obtain the coincidence $t_{obs} \sim t_{\Lambda} \sim t_c$. These results require only very mild conditions on the prior probability of the distribution of vacua in the landscape. Our work shows that the observed double coincidence of timescales is a robust prediction even when the multiverse is generalized to be multidimensional; that this coincidence is not a consequence of our particular universe being (3+1)-dimensional; and that this observable cannot be used to preferentially select one measure over another in a multidimensional multiverse.