Exiting Inflation with a Smooth Scale Factor

The expectation that the physical expansion of space occurs smoothly may be expressed mathematically as a requirement for continuity in the time derivative of the metric scale factor of the Friedmann–Robertson–Walker cosmology. We explore the consequences of imposing such a smoothness requirement, examining the forms of possible interpolating functions between the end of inflation and subsequent radiation- or matter-dominated eras, using a straightforward geometric model of the interpolating behavior. We quantify the magnitude of the cusp found in a direct transition from the end of slow-roll inflation to the subsequent era, analyze the validity of several smooth interpolator candidates, and investigate equation-of-state and thermodynamic constraints. We find an order-of-magnitude increase in the size of the universe at the end of the transition to a single-component radiation or matter era. We also evaluate the interpolating functions in terms of the standard theory of preheating and determine the effect on the number of bosons produced.