Abstract
Abstract We show that if the universe began as a vacuum fluctuation instead of a singular Big Bang, it must undergo at least one period of cosmic acceleration. This is required by a “cosmological sum rule” derived using the Schwarzian form of the Friedmann equations. We discuss possible connections to conformal and Möbius transformations, and also compute that the best fit present cosmic data is consistent with the necessary crossing of the Schwarzian through zero having already occurred (while any late time crossing would not yet have happened in a $$\Lambda $$ Λ CDM cosmology).