Ricci flows and expansion in axion-dilaton cosmology

TL;DR

This paper explores how Ricci flows relate to cosmological models in axion-dilaton theories, revealing solvable mathematical structures and describing universe evolution with big-bang expansion and isotropization.

Contribution

It establishes a connection between Ricci flows and cosmological dynamics in axion-dilaton models, including solvable systems and time-dependent solutions with cosmological implications.

Findings

Ricci flows correspond to renormalization-group equations in sigma-models.

Exact solutions involve modular forms and describe universe expansion.

Anisotropies diminish over time, leading to isotropic cosmologies.

Abstract

We study renormalization-group flows by deforming a class of conformal sigma-models. We consider overall scale factor perturbation of Einstein spaces as well as more general anisotropic deformations of three-spheres. At leading order in alpha, renormalization-group equations turn out to be Ricci flows. In the three-sphere background, the latter is the Halphen system, which is exactly solvable in terms of modular forms. We also analyze time-dependent deformations of these systems supplemented with an extra time coordinate and time-dependent dilaton. In some regimes time evolution is identified with renormalization-group flow and time coordinate can appear as Liouville field. The resulting space-time interpretation is that of a homogeneous isotropic Friedmann-Robertson-Walker universe in axion-dilaton cosmology. We find as general behaviour the superposition of a big-bang (polynomial) expansion with a finite number of oscillations at early times. Any initial anisotropy disappears during the evolution.