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We derive an anthropic bound on the extra neutrino species, $\Delta N_{\rmeff}$, based on the observation that a positive $\Delta N_{\rm eff}$ suppressesthe growth of matter fluctuations due to the prolonged radiation dominated era,which reduces the fraction of matter that collapses into galaxies, hence, thenumber of observers. We vary $\Delta N_{\rm eff}$ and the positive cosmologicalconstant while fixing the other cosmological parameters. We then show that theprobability of finding ourselves in a universe satisfying the current bound isof 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 otherhand, if the upper bound on $\Delta N_{\rm eff}$ is significantly improved byfuture observations, such simple anthropic consideration does not explain thesmall value of $\Delta N_{\rm eff}$. We also study simple models where darkradiation consists of relativistic particles produced by heavy scalar decays,and show that the prior probability distribution sensitively depends on thenumber of the particle species.

Anthropic Bound on Dark Radiation and its Implications for Reheating