Self-interacting dark matter with observable $\Delta N_{\rm eff}$

We propose a GeV-scale self-interacting dark matter (SIDM) candidate within a dark $U(1)_D$ gauged extension of the Standard Model (SM), addressing small-scale structure issues in $\Lambda$CDM while predicting an observable contribution to $\Delta N_{\rm eff}$ in the form of dark radiation. The model introduces a fermionic DM candidate $\chi$ and a scalar $\phi$, both charged under an unbroken $U(1)_D$ gauge symmetry. The self-interactions of $\chi$ are mediated by a light vector boson $X^\mu$, whose mass is generated via the Stueckelberg mechanism. The relic abundance of $\chi$ is determined by thermal freeze-out through annihilations into $X^\mu$, supplemented by a non-thermal component from the late decay of $\phi$. Crucially, $\phi$ decays after the Big Bang Nucleosynthesis (BBN) but before the Cosmic Microwave Background (CMB) epoch, producing additional $\chi$ and a dark radiation species ($\nu_S$). This late-time production compensates for thermal underabundance due to efficient annihilation into light mediators, while remaining consistent with structure formation constraints. The accompanying dark radiation yields a detectable $\Delta N_{\rm eff}$, compatible with Planck 2018 bounds and within reach of next-generation experiments such as SPT-3G, CMB-S4, and CMB-HD.