P-brane dynamics inN+1-dimensional FRW universes

Abstract

We study the evolution of maximally symmetric $p$-branes with a ${S}_{p\ensuremath{-}i}\ensuremath{\bigotimes}{\mathbb{R}}^{i}$ topology in flat expanding or collapsing homogeneous and isotropic universes with $N+1$ dimensions (with $N\ensuremath{\ge}3$, $plN$, $0\ensuremath{\le}ilp$). We find the corresponding equations of motion and compute new analytical solutions for the trajectories in phase space. For a constant Hubble parameter $H$ and $i=0$ we show that all initially static solutions with a physical radius below a certain critical value ${r}_{c}^{0}$ are periodic while those with a larger initial radius become frozen in comoving coordinates at late times. We find a stationary solution with constant velocity and physical radius ${r}_{c}$ and compute the root mean square velocity of the periodic $p$-brane solutions and the corresponding (average) equation of state of the $p$-brane gas. We also investigate the $p$-brane dynamics for $H\ensuremath{\ne}\mathrm{\text{constant}}$ in models where the evolution of the universe is driven by a perfect fluid with constant equation of state parameter $w={\mathcal{P}}_{p}/{\ensuremath{\rho}}_{p}$ and show that a critical radius ${r}_{c}$ can still be defined for $\ensuremath{-}1\ensuremath{\le}wl{w}_{c}$ with ${w}_{c}=(2\ensuremath{-}N)/N$. We further show that for $w\ensuremath{\sim}{w}_{c}$ the critical radius is given approximately by ${r}_{c}H\ensuremath{\propto}({w}_{c}\ensuremath{-}w{)}^{{\ensuremath{\gamma}}_{c}}$ with ${\ensuremath{\gamma}}_{c}=\ensuremath{-}1/2$ (${r}_{c}H\ensuremath{\rightarrow}\ensuremath{\infty}$ when $w\ensuremath{\rightarrow}{w}_{c}$). Finally, we discuss the impact that the large-scale dynamics of the universe can have on the macroscopic evolution of very small loops.