Discriminating Between Models of the Nanohertz Gravitational-Wave Background with Pulsar Timing Arrays

TL;DR

The paper develops a Bayesian framework to discriminate among three plausible origins of the nanohertz gravitational-wave background detected by pulsar timing arrays: supermassive black hole binary mergers, first-order early-Universe phase transitions, and cosmic strings. It jointly models PTA timing data, intrinsic pulsar noise, dispersion-measure variations, and the Hellings–Downs spatial correlations, and computes posteriors and evidences for each model. The analysis yields a GWB amplitude of $A_{ m GWB} \approx 2.4\times10^{-15}$ with a spectral slope near $\gamma_{\rm GWB} \approx 13/3$, consistent with the SMBHB expectation, while cosmological templates can mimic the signal with Bayes factors that are not decisively favorable. The results underscore a robust common-spectrum, HD-correlated signal that is most naturally explained by SMBHBs but remain compatible with cosmological origins under current uncertainties; future PTA data, broader pulsar samples, and multi-band observations will be essential to break degeneracies and pinpoint the true origin.

Abstract

Recent pulsar timing array results, including the NANOGrav 15-year data set, show evidence for a stochastic gravitational-wave background (GWB) in the nanohertz band. We present a Bayesian framework to compare three possible origins: (i) a background from supermassive black hole binary mergers, (ii) a first-order phase transition in the early Universe, and (iii) a network of cosmic strings. We derive the PTA likelihood with the Hellings-Downs angular correlation and model intrinsic pulsar red noise and dispersion-measure variations. Using Bayesian model selection, we infer posteriors for the GWB amplitude and spectral slope and compute marginal likelihoods for each scenario. We confirm a common-spectrum process with Hellings-Downs spatial correlations and recover a characteristic strain amplitude at f_yr = 1/year of A_GWB approx 2.4e-15, with a slope consistent with gamma approx 13/3 as expected for supermassive black hole binaries. While fully consistent with an astrophysical origin, cosmological sources are not excluded: cosmic strings with Gmu ~ 1e-11 to 1e-10 and phase transitions peaking near 1e-8 to 1e-7 Hz can reproduce the observed amplitude within allowed parameter ranges. Current Bayes factors do not show a decisive preference among these scenarios. We discuss noise-mitigation implications and prospects for discrimination with future PTA observations.

Figures (7)

• Figure 1: Hellings--Downs correlation in the NANOGrav 15-year data. Measured correlation coefficients between timing residuals of pulsar pairs are plotted versus their angular separation on the sky (blue points with 1$\sigma$ error bars). The dashed line shows the theoretical Hellings--Downs curve $\Gamma(\gamma)$ from general relativity Hellings1983, which is normalized to $0.5$ at $\gamma=0^\circ$ and falls to $-0.25$ at $\gamma=180^\circ$. The observed correlations follow the expected quadrupolar trend. For comparison, a common but uncorrelated noise process would correspond to zero correlation at all angles (horizontal line at $\Gamma=0$, not shown). The detection of this pattern is evidence for a gravitational-wave background as opposed to other sources of common noise.
• Figure 2: Hellings--Downs cross-check using PINT and the wideband public data set. Each point corresponds to a pulsar pair; color indicates the number of overlapping 30-day bins contributing to the correlation estimate. Black circles show angle-binned averages with standard errors, while the crimson line is the best-fitting Hellings--Downs template scaled by the naive amplitude $A_{\mathrm{HD}}^{\mathrm{naive}}=0.46$. Although simplified, this independent analysis still recovers a positive quadrupolar trend at ${>}4\sigma$.
• Figure 3: Characteristic strain spectra $h_c(f)$ for three template models in the PTA band: SMBHB power-law with $\alpha=-2/3$, a representative cosmic string slope $\beta=-1/2$, and a broken-power-law phase-transition template with a peak near $f\sim 5\times10^{-8}$ Hz, all normalized around $A\sim 2\times10^{-15}$ at $f_{\rm yr}$.
• Figure 4: Model validation diagnostics: left, residual power spectra compared with model and whitened residual agreement with unity; right, posterior predictive distribution for the Hellings--Downs correlation vs. angle.
• Figure 5: Posterior distribution for the common-spectrum GWB parameters assuming a power-law spectrum (SMBHB model). The two-dimensional joint posterior for the strain amplitude $A_{\mathrm{GWB}}$ (at $f_{\mathrm{yr}}=1/\text{yr}$) and the spectral index $\gamma_{\mathrm{GWB}}$ (where $P(f)\propto f^{-\gamma_{\mathrm{GWB}}}$) is shown as contour levels enclosing 68% and 95% credible regions. One-dimensional marginalized posteriors (probability density functions) for each parameter are displayed along the top and right. The amplitude posterior (top) peaks around $2\times10^{-15}$ and is well-separated from zero. The spectral index posterior (right) peaks near $\gamma_{\mathrm{GWB}}\approx4.3$ (vertical dashed line indicates $13/3\approx4.33$ for a nominal SMBHB inspiral background). The contours indicate a mild anti-correlation between $A_{\mathrm{GWB}}$ and $\gamma_{\mathrm{GWB}}$. Overall, the data favor a strain spectrum consistent with the $-2/3$ power-law slope expected from SMBHBs, with an amplitude $A_{\mathrm{GWB}}\sim2$--$3\times10^{-15}$.