Anthropic Bound on Dark Radiation and its Implications for Reheating

We derive an anthropic bound on the extra neutrino species, $\Delta N_{\rm eff}$, based on the observation that a positive $\Delta N_{\rm eff}$ suppresses the growth of matter fluctuations due to the prolonged radiation dominated era, which reduces the fraction of matter that collapses into galaxies, hence, the number of observers. We vary $\Delta N_{\rm eff}$ and the positive cosmological constant while fixing the other cosmological parameters. We then show that the probability of finding ourselves in a universe satisfying the current bound is of order a few percents for a flat prior distribution. If $\Delta N_{\rm eff}$ is found to be close to the current upper bound or the value suggested by the $H_0$ tension, the anthropic explanation is not very unlikely. On the other hand, if the upper bound on $\Delta N_{\rm eff}$ is significantly improved by future observations, such simple anthropic consideration does not explain the small value of $\Delta N_{\rm eff}$. We also study simple models where dark radiation consists of relativistic particles produced by heavy scalar decays, and show that the prior probability distribution sensitively depends on the number of the particle species.