Bayesian Evidence for Angular Symmetry and Spectral Curvature in the Nanohertz Gravitational-Wave Background

Pulsar timing arrays have detected a nanohertz signal exhibiting the Hellings–Downs angular correlation, the expected angular symmetry of an isotropic stochastic gravitational-wave background. This angular symmetry fixes the tensor correlation class of the signal, so its physical origin must be inferred from the frequency spectrum. Using the public NANOGrav 15-year free-spectrum products, we compare five spectral hypotheses through Bayesian evidence: the canonical scale-free power law from purely gravitational-wave-driven supermassive black hole binaries (SMBHBs), a free-slope power law, an environmental SMBHB turnover model, a first-order phase transition template, and an effective cosmic string spectrum. The evidence favors spectra with physical curvature or a characteristic scale, while the strict scale-free SMBHB law is strongly disfavored. Within the tested physical templates, the phase transition model gives the largest compressed spectral evidence; within astrophysical source models, environmental SMBHB hardening is the leading interpretation and links the spectral bend to parsec-scale nuclear stellar densities. The effective cosmic string spectrum shows little evidence gain under the baseline prior. An orbit- and foreground-aware LISA continuation forecast gives this ranking a multi-band check: the PTA-selected QCD-scale phase transition posterior has no appreciable millihertz continuation, whereas an unchanged broad cosmic string extrapolation is LISA-bright and needs additional spectral structure.