Geodesic completion of big bangs from emergent geometry

Abstract

Past-geodesically-complete cosmologies are thought to require either contraction, or an asymptotically static past. We introduce a third possibility: Einstein-frame time can dynamically attain a local minimum. This time-reversal is caused by phantom Chaplygin gas, whose acoustic cone defines a `causal-frame' geometry that is geodesically-complete. While gravity experiences time-reversal, the Chaplygin gas always evolves forward in time, realizing a transient mismatch in thermodynamic arrows of time. Time-reversal affects the scale-factor, enforcing a non-singular bounce in causal frame that is robust against any additional matter.

Figures (2)

• Figure 1: The Chaplygin gas energy density vs. scale factor. The branch point where $\rho_\lambda$ changes sign represents a Big Brake singularity in Einstein frame, but is smoothly navigated in causal frame, as shown by the $t$-parameterized gray arrows. For $H(t)>0$, causal frame cosmic time $t$ flows in the direction of decreasing $\rho_\lambda$, in agreement with eq. (\ref{['eq:rhosol']}).
• Figure 2: Illustration of the time-reversal mechanism during radiation domination, with $H_\ast \equiv H(t_\ast) \gg \sqrt{G_N\lambda}$. Einstein-frame cosmic time $T = \int N{\rm d} t$ attains a local minimum at causal-frame cosmic time $t = t_\ast$ when $N(t)$ changes sign. For the same reason, the scale factor $a(t)$ exhibits a non-singular bounce without $H(t)$ changing sign.