Sailing through the big crunch-big bang transition

In a recent series of papers, we have shown that theories with scalar fields coupled to gravity (e.g., the standard model) can be lifted to a Weyl-invariant equivalent theory in which it is possible to unambiguously trace the classical cosmological evolution through the transition from big crunch to big bang. The key was identifying a sufficient number of finite, Weyl-invariant conserved quantities to uniquely match the fundamental cosmological degrees of freedom across the transition. In doing so we had to account for the well-known fact that many Weyl-invariant quantities diverge at the crunch and bang. Recently, some authors rediscovered a few of these divergences and concluded based on their existence alone that the theories cannot be geodesically complete. In this paper, we show that this conclusion is invalid. Using conserved quantities we explicitly construct the complete set of geodesics and show that they pass continuously through the big crunch-big bang transition.

Figure 1

A typical massive particle geodesic, showing the continuous passage through the big crunch and big bang singularities. In this example, the coordinate velocity $v^3={\stackrel{.}{x}}^3$ goes to zero at the crunch and infinity at the bang. The proper speed of the particle with respect to particles comoving with the $x^i$ coordinates, $v_{\text{proper}}=\sqrt{g_{3ij}{\stackrel{.}{x}}^i{\stackrel{.}{x}}^j}/\sqrt{g_{3ij}{\stackrel{.}{x}}^i{\stackrel{.}{x}}^j+m^2{a}^2}$, never exceeds unity and touches zero at the crunch and unity at the bang. Likewise, the coordinate $x^3$ is finite and continuous throughout. The numerical values of the parameters used to generate this plot are $p_1=-1/4$, $p_2=0$, $p_3=1$, $T_1=1$, $T_2=1$, $T_3=1$, $k_1=0$, $k_2=0$, $k_3=1$, $g_p=1$, ${\rho }_r=0.4$.