Abstract
The NANOGrav 15-year data provides compelling evidence for a stochastic
gravitational-wave (GW) background at nanohertz frequencies. The simplest
model-independent approach to characterizing the frequency spectrum of this
signal consists in a simple power-law fit involving two parameters: an
amplitude A and a spectral index \gamma. In this paper, we consider the next
logical step beyond this minimal spectral model, allowing for a running (i.e.,
logarithmic frequency dependence) of the spectral index, \gamma_run(f) = \gamma
+ \beta \ln(f/f_ref). We fit this running-power-law (RPL) model to the NANOGrav
15-year data and perform a Bayesian model comparison with the minimal
constant-power-law (CPL) model, which results in a 95% credible interval for
the parameter \beta consistent with no running, \beta \in [-0.80,2.96], and an
inconclusive Bayes factor, B(RPL vs. CPL) = 0.69 +- 0.01. We thus conclude
that, at present, the minimal CPL model still suffices to adequately describe
the NANOGrav signal; however, future data sets may well lead to a measurement
of nonzero \beta. Finally, we interpret the RPL model as a description of
primordial GWs generated during cosmic inflation, which allows us to combine
our results with upper limits from big-bang nucleosynthesis, the cosmic
microwave background, and LIGO-Virgo-KAGRA.