Self-interacting dark-matter spikes and the final-parsec problem: Bayesian constraints from the NANOGrav 15-year gravitational-wave background

A self-interacting dark-matter (SIDM) density spike around merging supermassive black holes (SMBHs) may be able to supply the dynamical friction needed to shrink binaries from $\sim 1\, \mathrm{pc}$ to $\sim 10^{-2} \,\mathrm{pc}$, thereby resolving the long-standing "final-parsec problem". Embedding the binary-halo system in a cosmological population model, we evolve the inspiral under the combined influence of gravitational-wave (GW) emission and SIDM drag, compute the resulting nanohertz GW background, and confront it with the NANOGrav 15-year pulsar-timing data. A six-parameter Bayesian analysis, performed with a Gaussian-process-accelerated Markov chain Monte Carlo, yields posterior constraints on the cross-section per unit mass and maximum circular velocity values that were consistent with independent galaxy-rotation and cluster-lensing limits. Within this parameter space, the SIDM spike remains intact, supplies sufficient friction to overcome the stellar depletion barrier, and produces a characteristic-strain spectrum that matches the NANOGrav signal as well as phenomenological astrophysical models.