Probing the Scotogenic Dirac Model with FIMP Dark Matter and $\Delta N_{\rm eff}$

We study a feebly interacting massive particle realization of the Scotogenic Dirac Model in which the lightest neutral fermion $N_1$ serves as a dark matter candidate, produced via the freeze-in or super-WIMP mechanism. The model generates Dirac neutrino masses at one loop, resulting in a rank-2 mass matrix that predicts one nearly massless neutrino. We analyze the DM relic density for various next-to-lightest odd particles (NLOPs), finding that coannihilation effects and enhanced annihilation channels are crucial for achieving the correct thermal freeze-out abundance of the NLOP. We provide a detailed analysis of the model's implications for the effective number of relativistic species, $\Delta N_{\mathrm{eff}}$, which receives contributions from both a thermal bath of right-handed neutrinos and non-thermal energy injection due to late NLOP decays. Through an extensive parameter scan, we identify viable parameter space for all NLOP candidates that satisfies constraints from DM relic density, lepton flavor violation, Big Bang Nucleosynthesis, Cosmic Microwave Background, and $\Delta N_{\mathrm{eff}}$.