Inflation and topology from the no-boundary state

TL;DR

Changing the spatial topology to a 3-torus resolves the no-boundary wavefunction's preference for small universes, favoring a large inflating universe consistent with observations and affecting the CMB spectrum.

Contribution

Demonstrates that using a 3-torus topology and automorphic forms yields a wavefunction favoring large inflationary universes, addressing previous small universe bias.

Findings

Wavefunction favors large inflating universe with over 250 e-folds.

Sum over SL(3,Z) geometries computed using automorphic forms.

Torus moduli fluctuations induce corrections to the CMB power spectrum.

Abstract

The no-boundary wavefunction for slow-roll inflation on the 3-sphere exponentially favors a small universe, in sharp disagreement with observations. We show that this problem is resolved by changing the spatial topology to the 3-torus. The sum over the ${\rm SL}(3,\mathbb{Z})$ family of geometries, computed using the theory of automorphic forms for ${\rm GL}(3)$, produces a wavefunction favoring a large inflating universe with ${\cal N} \gtrsim 250$ $e$-folds. We also compute corrections to the CMB power spectrum induced by torus moduli fluctuations.