A self-interacting dark-matter (SIDM) density spike around merging supermassive black holes may be able to supply the dynamical friction needed to shrink binaries from $\sim 1$ 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 emission and SIDM drag, compute the resulting nanohertz gravitational-wave 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.

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• Classical black holes
• Gravitational wave sources
• Particle dark matter