How Slowly can the Early Universe Expand?

TL;DR

This paper investigates the theoretical possibility and stability of ultra-slow expansion phases in the early universe, analyzing various models and highlighting their limitations and conditions for such expansion to occur.

Contribution

The paper systematically examines different models for ultra-slow expansion, identifying their stability issues and the need for non-standard physics or exotic components.

Findings

Scalar and barotropic models are unstable to evolution toward faster expansion.

Braneworld models can produce ultra-slow expansion but require modifications to standard equations.

Loitering models can achieve quasi-static expansion but need exotic negative-density components.

Abstract

When the expansion of the universe is dominated by a perfect fluid with equation of state parameter $w$ and a sound speed $c_s$ satisfying $w = c_s^2 \le 1$, the Hubble parameter $H$ and time $t$ satisfy the bound $Ht \ge 1/3$. There has been recent interest in "ultra-slow" expansion laws with $Ht < 1/3$ (sometimes described as "fast expanding" models). We examine various models that can produce ultra-slow expansion: scalar fields with negative potentials, barotropic fluids, braneworld models, or a loitering phase in the early universe. Scalar field models and barotropic models for ultra-slow expansion are unstable to evolution toward $w = 1$ or $w \rightarrow \infty$ in the former case and $w \rightarrow \infty$ in the latter case. Braneworld models can yield ultra-slow expansion but require an expansion law beyond the standard Friedman equation. Loitering early universe models can produce a quasi-static expansion phase in the early universe but require an exotic negative-density component. These results suggest that appeals to an ultra-slow expansion phase in the early universe should be approached with some caution, although the loitering early universe may be worthy of further investigation. These results do not apply to ultra-slow contracting models.