The fate of bound systems through Sudden Future Singularities

Abstract

Sudden singularities occur in FRW spacetimes when the scale factor remains finite and different from zero while some of its derivatives diverge. After proper rescaling, the scale factor close to such a singularity at $t=0$ takes the form $a(t)=1+c|t{|}^{\hat{I}}\cdot$ (where $c$ and $\hat{I}\cdot$ are parameters and $\hat{I}\cdot \ge 0$). We investigate analytically and numerically the geodesics of free and gravitationally bound particles through such sudden singularities. We find that even though free particle geodesics go through sudden singularities for all $\hat{I}\cdot \ge 0$, bound systems get dissociated for a wide range of the parameter $c$. For $\hat{I}\cdot <1$ bound particles receive a diverging impulse at the singularity and get dissociated for all positive values of the parameter $c$. For $\hat{I}\cdot >1$ (Sudden Future Singularities (SFS)) bound systems get a finite impulse that depends on the value of $c$ and get dissociated for values of $c$ larger than a critical value $c_{cr}(\hat{I}\cdot ,\ddot{I}{‰}_0)>0$ that increases with the value of $\hat{I}\cdot$ and the rescaled angular velocity $\ddot{I}{‰}_0$ of the bound system. We obtain an approximate equation for the analytical estimate of $c_{cr}(\hat{I}\cdot ,\ddot{I}{‰}_0)$. We also obtain its accurate form by numerical derivation of the bound system orbits through the singularities. Bound system orbits through Big Brake singularities ($c<0$, $1<\hat{I}\cdot <2$) are also derived numerically and are found to get disrupted at the singularity. However, they remain bound for all values of the parameter $c$ considered.

10 pages, 8 figures, comments added, typo in eq. (1.1) corrected, references added. Accepted in Phys. Rev. D (to appear)