# de Sitter, $\alpha'$-Corrections & Duality Invariant Cosmology

## TL;DR

This paper explores de Sitter solutions in string cosmology with $oldsymbol{ ext{O}(d,d)}$-duality invariance, deriving conditions involving an unknown function that determine the existence of such solutions, and analyzing their properties in string and Einstein frames.

## Contribution

It derives the Einstein frame equations from duality-invariant string cosmology and identifies a second order nonlinear ODE as a key condition for de Sitter solutions.

## Key findings

- De Sitter solutions require the unknown function to satisfy a specific nonlinear ODE.
- Solutions do not admit simple power series expansions compatible with supergravity.
- All potential solutions involve a running dilaton, with constant dilaton solutions being power-law cosmologies.

## Abstract

Demanding $O(d,d)$-duality covariance, Hohm and Zwiebach have written down the action for the most general cosmology involving the metric, $b$-field and dilaton, to all orders in $\alpha'$ in the string frame. Remarkably, for an FRW metric-dilaton ansatz the equations of motion turn out to be quite simple, except for the presence of an unknown function of a single variable. If this unknown function satisfies some simple properties, it allows de Sitter solutions in the string frame. In this note, we write down the Einstein frame analogues of these equations, and make some observations that make the system tractable. Perhaps surprisingly, we find that a necessary condition for de Sitter solutions to exist is that the unknown function must satisfy a certain second order non-linear ODE. The solutions of the ODE do not have a simple power series expansion compatible with the leading supergravity expectation. We discuss possible interpretations of this fact. After emphasizing that all (potential) string and Einstein frame de Sitter solutions have a running dilaton, we write down the most general cosmologies with a constant dilaton in string/Einstein frame: these have power law scale factors.