Magnetogenesis from a sawtooth coupling: Gravitational wave probe of reheating

The detection of gravitational waves (GWs) by LIGO-Virgo and pulsar timing arrays has opened a new window into early Universe cosmology. Yet, the origin of large-scale magnetic fields and the dynamics of the reheating epoch remain poorly understood. In this work, we study the generation of secondary GWs (SGWs) sourced by primordial magnetic fields produced via a Sawtooth-type coupling during reheating with a general background evolution. We show that the reheating equation of state significantly influences the spectral shape and amplitude of the magnetic fields. While a scale-invariant spectrum is typically needed to match observational bounds, this coupling naturally produces a strongly blue-tilted spectrum that remains consistent with current constraints. Crucially, the magnetic field continues to grow during reheating, leading to a GW signal with a broken power-law spectrum and a distinctive blue tilt on superhorizon scales. This SGW signal can fall within the sensitivity of upcoming detectors such as LISA, DECIGO, and BBO. The unique spectral features make this scenario distinguishable from other sources, offering a viable mechanism for cosmic magnetogenesis and a novel probe of the reheating era through GW observations.