Regularizing the induced GW spectrum with dissipative effects

Finite mean free paths of light particles, like photons and neutrinos, lead to dissipative effects and damping of small-scale density fluctuations in the early universe. We study the impact of damping on the spectral density of gravitational waves induced by primordial fluctuations in the radiation-dominated universe. We show that the most important effects of damping are $(i)$ regularization of the resonant frequency and $(ii)$ a far low-frequency tail with no logarithmic running. The exact location of the break frequency below which the logarithmic running is lost depends on the damping rate. Both effects stem from the effective finite lifetime of the gravitational wave source caused by damping. Interestingly, we find that, for the standard model of particles, the effects of damping are most relevant at around or below the nHz frequencies. Our results showcase the importance of including the damping of primordial fluctuations in future analysis of induced gravitational waves. We provide detailed analytical formulas and approximations for the kernel of induced gravitational waves. Lastly, we discuss possible implications of damping in alleviating the gauge issue of induced gravitational waves and in suppressing the so-called poltergeist mechanism.