[Paper Review] Footprints of the QCD Crossover on Cosmological Gravitational Waves at Pulsar Timing Arrays

Gabriele Franciolini, Davide Racco+1 more|arXiv (Cornell University)|Jun 29, 2023

Cosmology and Gravitation Theories4 citations

TL;DR

This paper demonstrates that the QCD crossover in the early universe imprints a distinctive spectral feature on the low-frequency tail of cosmological gravitational wave backgrounds detectable by pulsar timing arrays (PTAs). By modeling the causality-limited gravitational wave spectrum with QCD-induced changes in the equation of state, the authors show a significant Bayesian improvement over standard power-law fits, suggesting that Standard Model physics can decisively influence interpretation of nHz gravitational wave signals.

ABSTRACT

Pulsar Timing Arrays (PTAs) have reported evidence for a stochastic gravitational wave (GW) background at nHz 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 $T\gtrsim 1$ GeV is distinctively affected by confinement of strong interactions (QCD), due to the corresponding sharp decrease in the number of relativistic species. 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. This suggests that the inclusion of Standard Model effects on GWs can have a potentially decisive impact on model selection.

Motivation & Objective

• To investigate whether Standard Model physics, particularly the QCD crossover, leaves a detectable imprint on the low-frequency tail of primordial gravitational wave backgrounds.
• To improve the modeling of causality-limited gravitational wave signals in PTAs by incorporating QCD-induced changes in the equation of state and relativistic degrees of freedom.
• To assess whether the inclusion of QCD effects leads to a statistically significant improvement in fitting current PTA data compared to standard power-law models.
• To provide a model-independent framework for interpreting nHz gravitational wave signals in terms of early-universe dynamics and SM physics.

Proposed method

• The authors derive the spectral shape of the causality tail (CT) of gravitational waves produced during early-universe sources with time scales comparable to the Hubble time, accounting for the QCD crossover's impact on the equation of state.
• They model the evolution of tensor modes $ h_k(t) $ in the low-frequency regime, where GWs are insensitive to source details and depend only on the expansion history and GW propagation.
• Using lattice QCD results for the number of relativistic degrees of freedom, they compute the redshifted energy density $ \Omega_{\textsc{gw}}(f) $, showing that the $ f^3 $ power-law tilt is modified by the QCD crossover.
• They perform Bayesian model comparison using the IPTA DR2 and NANOGrav 15-year datasets, implementing the new CT model in the enterprise and enterprise_extensions pipelines with PTMCMC sampling.
• The analysis includes noise parameters (white, red, dispersion measure) and GWB parameters with priors informed by cosmological bounds and data-driven constraints.
• They compare the new QCD-modified CT model against standard power-law and broken power-law models, computing Bayes factors to assess model preference.

Experimental results

Research questions

• RQ1 Does the QCD crossover in the early universe leave a detectable imprint on the low-frequency tail of cosmological gravitational wave backgrounds?
• RQ2 Can the inclusion of QCD-induced changes in the equation of state improve the fit of gravitational wave data from pulsar timing arrays compared to standard power-law models?
• RQ3 What is the quantitative impact of SM physics on model selection for nHz stochastic gravitational wave backgrounds?
• RQ4 How do the spectral features of causality-limited gravitational wave signals change due to the rapid drop in relativistic degrees of freedom during the QCD crossover?

Key findings

• The inclusion of QCD crossover effects in the causality tail model leads to a significant Bayesian improvement over standard cubic power-law fits in both IPTA DR2 and NANOGrav 15-year datasets.
• For IPTA DR2, the posterior median amplitude of the QCD-modified CT signal is $ \log_{10}A_{\mathrm{CT}} = -13.67^{+0.08}_{-0.07} $, with a spectral index consistent with the expected $ f^3 $ behavior.
• For NANOGrav 15-year data, the posterior median amplitude is $ \log_{10}A_{\mathrm{CT}} = -13.85^{+0.06}_{-0.05} $, indicating a strong signal preference over noise.
• The Bayesian evidence favors the QCD-modified CT model over the standard power-law model, suggesting that SM effects can play a decisive role in interpreting current PTA data.
• The analysis shows that the transition from matter to radiation domination, influenced by the QCD crossover, modifies the high- and low-frequency spectral indices, with $ \gamma_{\text{MD}} = 4 $ and $ \delta_{\text{MD}} = 2 $, respectively.
• The results imply that future model comparisons must include QCD effects to avoid biased inference, especially in distinguishing cosmological from astrophysical origins of nHz GW signals.