Footprints of the QCD Crossover on Cosmological Gravitational Waves at Pulsar Timing Arrays

Pulsar timing arrays (PTAs) have reported evidence for a stochastic gravitational wave (GW) background at nanohertz frequencies, possibly originating in the early Universe. We show that the spectral shape of the low-frequency (causality) tail of GW signals sourced at temperatures around $T\gtrsim 1\mathrm{GeV}$ is distinctively affected by confinement of strong interactions (QCD), due to the corresponding sharp decrease in the number of relativistic species, and significantly deviates from $\sim f^3$ commonly adopted in the literature. Bayesian analyses in the NANOGrav 15 years and the previous international PTA datasets reveal a significant improvement in the fit with respect to cubic power-law spectra, previously employed for the causality tail. While no conclusion on the nature of the signal can be drawn at the moment, our results show that the inclusion of standard model effects on cosmological GWs can have a decisive impact on model selection.

Figure 1

Top: EOS during the QCD crossover. The inset shows $w$ in an expanded range of temperatures. Bottom: impact of the variation of $w(T)$ and $g_{\star }(T)$ on the CT of a primordial GWB (plotted with an arbitrary amplitude). The dashed line shows the $f^3$ scaling obtained in a pure radiation-dominated universe.

Figure 2

The GW spectrum for the CT (solid line), $n_T=3$ (dashed line), and free power-law (dotted line) models, selecting the maximum posterior values obtained with IPTA-DR2 [11] (blue) and NG15 [2] (green) datasets. The shaded region shows the $\mathrm{CMB}+\mathrm{BAO}$ bound on $\mathrm{\Delta }{N}_{\mathrm{eff}}$, and the dotted line the reach of CMB-S4 experiments. The lower limits of the posteriors are determined by the priors of [2, 11]. See Fig. 3 for parameter posteriors.

Figure 3

Upper: 2D posterior for a power-law model for IPTA-DR2 [11] and NG15, the latter enforcing Hellings-Downs (HD) correlations in the analysis [2, 4]. The vertical lines indicate approximations of the CT. The yellow star shows the best-fitting CT amplitude. We show current (dashed line) and future (dot-dashed line) $\mathrm{\Delta }{N}_{\mathrm{eff}}$ bounds affecting any power-law signal extending up to $f_{\star }=1,3f_{\mathrm{yr}}$. The excluded region lies above the corresponding line. Lower: 1D posteriors on $A$ for the power-law model (marginalizing over the second parameter $n_T$) and for the CT and $f^3$ models. The smaller amplitude observed for the free $n_T$ case follows from the $A-n_T$ degeneracy observed in the upper panel, and the arbitrary choice of defining the spectral amplitude at $f_{\mathrm{yr}}$; see Eq. (5).