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

We study the evolution of maximally symmetric $p$-branes with a $S_{p-i}\otimes \mathbbm{R}^i$ 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 $H \neq {\rm constant}$ in models where the evolution of the universe is driven by a perfect fluid with constant equation of state parameter, $w={\cal P}_p/\rho_p$, 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.