Measures for a multidimensional multiverse

Abstract

We explore the phenomenological implications of generalizing the causal patch and fat geodesic measures to a multidimensional multiverse, where the vacua can have differing numbers of large dimensions. 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 time scales for the existence of observers, for vacuum domination, and for curvature domination (${t}_{\text{obs}},{t}_{\mathrm{\ensuremath{\Lambda}}}$, and ${t}_{c}$, respectively.) In this work we restrict ourselves to bubbles with positive cosmological constant, $\mathrm{\ensuremath{\Lambda}}$. We find that in the case of the causal patch cutoff, when the bubble universes have $p+1$ large spatial dimensions with $p\ensuremath{\ge}2$, the shape of the probability distribution is such that we obtain the coincidence of time scales ${t}_{\text{obs}}\ensuremath{\sim}{t}_{\mathrm{\ensuremath{\Lambda}}}\ensuremath{\sim}{t}_{c}$. 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 $p\ensuremath{\ge}3$. 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 $p\ensuremath{\ge}2$, and we once again obtain the coincidence ${t}_{\text{obs}}\ensuremath{\sim}{t}_{\mathrm{\ensuremath{\Lambda}}}\ensuremath{\sim}{t}_{c}$. 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 time scales 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.