Improved cosmological limits on $Z^\prime$ models with light right-handed neutrinos

We improve limits on $Z^\prime$ extensions of the Standard Model (SM) with light right-handed neutrinos. The presence of shared gauge interactions between the light right-handed neutrinos and other SM fermions allows for production of $\nu_R$ in the early Universe and we use the excess in the effective number of neutrino species $\Delta N_\text{eff}$ to place limits. Our benchmark model is a minimal gauged $U(1)_{B-L}$ that often arises as a building block in other models, and we discuss applicability to more general $U(1)$ extensions. We devise an improved Monte Carlo integration scheme convenient for implementation of generic integrated Boltzmann equations with minimal simplifying assumptions. We sketch our numerical implementation in detail for future reference. Using the new ACT DR6 limit $\Delta N_\text{eff}<0.17$, we improve constraints on the gauge coupling for $1\mathrm{\,GeV} < m_{Z^\prime} < 100\mathrm{\,TeV}$ by orders of magnitude and find the strongest limits thus far, surpassing even current and future colliders, and explore the potential of future CMB experiments to test $U(1)$ extensions up to the GUT scale. We perform a detailed analysis of the robustness of cosmological limits within standard and non-standard thermal histories and find that a strong first order phase transition, early dark energy or early matter domination could dilute $\nu_R$ abundances beyond detection. We investigate the effect of reheating on $\nu_R$-genesis and provide results and prescriptions to apply our bounds to non-standard thermal histories. Limits are generically weakened for reheating $T_\text{reh}\ll m_{Z^\prime}$. Our results suggest that projected limits on $Z^\prime$ with Dirac neutrinos can only be accommodated for in non-standard thermal histories, thus limiting the options to include dark matter candidates or Dirac leptogenesis.