p-brane dynamics in N+1-dimensional FRW universes

Abstract

We study the evolution of maximally symmetric $p$-branes with a $S_{p-i}\otimes \text{mathbbm}{R}^i$ topology in flat expanding or collapsing homogeneous and isotropic universes with $N+1$ dimensions (with $N\ge 3$, $p<N$, $0\le i<p$). 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 $H\ne \mathrm{constant}$ in models where the evolution of the universe is driven by a perfect fluid with constant equation of state parameter, $w={\mathcal{P}}_p/{\rho }_p$, and show that a critical radius, $r_c$, can still be defined for $-1\le w<w_c$ with $w_c=(2-N)/N$. We further show that for $w\sim w_c$ the critical radius is given approximately by $r_{c}H\propto (w_c-w{)}^{{\gamma }_c}$ with ${\gamma }_c=-1/2$ ($r_{c}H\to \infty$ when $w\to w_c$). Finally, we discuss the impact that the large scale dynamics of the universe can have on the macroscopic evolution of very small loops.

Comments: 6 pages, 3 figures