Acceleration from a phase of entropic balance

Abstract We discuss the notion of generating a cosmic inflation without any big bang singularity. It has recently been proved by Good and Linder ( arXiv:2503.02380 [gr-qc]) that such an expansion of the universe can be driven by quantum fluctuations embedded in vacuum. The rate of expansion is guided by a cosmological sum rule defined through the Schwarzian derivative. We explore the thermodynamic roots of Schwarzian and connect it with the surface gravity associated with an apparent horizon. In General Relativity the cosmological sum rule can be enforced only if the early universe is a Milne vacuum. We show that this restriction can be removed by considering an entropic source term in the Einstein–Hilbert action.