How slowly can the early Universe expand?

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 $w={c}_{s}^{2}\ensuremath{\le}1$, the Hubble parameter $H$ and time $t$ satisfy the bound $Ht\ensuremath{\ge}1/3$. There has been recent interest in ``ultraslow'' expansion laws with $Ht<1/3$ (sometimes described as ``fast expanding'' models). We examine various models that can produce ultraslow 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 ultraslow expansion are unstable to evolution toward $w=1$ or $w\ensuremath{\rightarrow}\ensuremath{\infty}$ in the former case and $w\ensuremath{\rightarrow}\ensuremath{\infty}$ in the latter case. Braneworld models can yield ultraslow expansion but require an expansion law beyond the standard Friedman equation. Loitering early universe models can produce a quasistatic expansion phase in the early Universe but require an exotic negative-density component. These results suggest that appeals to an ultraslow 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 ultraslow contracting models.