Geodesic Completion of Big Bangs from Emergent Geometry

A theorem by Borde, Guth, and Vilenkin requires any cosmology that is always expanding to be past-incomplete, having, e.g., a big bang singularity. A non-singular cosmology is thought to require either an asymptotically stationary cosmology (emergent universe scenario), or an initially contracting cosmology (bouncing or cyclic universe scenarios). Here we demonstrate that there is a third possibility: Einstein-frame cosmic time can be a dynamical quantity that is able to spontaneously reverse direction. This occurs in the presence of phantom Chaplygin gas, a perfect fluid defined by an equation of state where pressure is inversely and negatively related to energy density. The acoustic cone of Chaplygin gas defines a more fundamental "causal-frame" geometry, in which the causal-frame light cone contains the Einstein-frame light cone. The resulting spacetime is geodesically complete, both in causal-frame and, surprisingly, in Einstein frame, despite the fact that the Einstein-frame metric has a local minimum of cosmic time, when the Einstein-frame lapse function smoothly changes sign. A Big Brake singularity that would occur as the energy density tends towards zero is avoided by choosing finite causal frame lapse function, rather than choosing the Einstein-frame lapse function. We dub this soft "singularity" a Big Benjamin Button, since although gravity (and any matter minimally coupled to the Einstein-fram metric) experiences time reversal, the Chaplygin gas continues to evolve forward in time, realizing a transient mismatch in the thermodynamic arrows of time between the Chaplygin gas and gravitational waves. The causal-frame geometry created by the Chaplygin gas provides a sensible notion of causality for all degrees of freedom, and predicts a non-singular bounce that is robust against arbitrary additions of any types of matter.