Scalar induced gravitational waves as probes of dark QCD

This paper investigates scalar induced gravitational waves (SIGWs) as probes of a dark QCD crossover. Motivated by twin Higgs and asymmetric twin baryon dark matter scenarios, the authors consider a dark QCD sector with a confinement scale approximately 5.5 times the Standard Model (SM) QCD scale, i.e., ΛdQCD ≃ 5.5 ΛQCD. They construct the effective energy and entropy degrees of freedom for the SM supplemented by dark QCD sectors containing either three light dark quark flavors (SM + dQCD3) or all six dark quark flavors (SM + dQCD6). The resulting equation of state parameter w(T) and sound speed c2s(T) are then used to solve the first-order scalar perturbations and the second-order SIGWs through the SM and dark QCD crossover epochs.

For a monochromatic primordial curvature power spectrum, Pζ(k) = Aζ k∗ δ(k − k∗), the authors first demonstrate that the realistic SM thermal history modifies the SIGW spectrum relative to the idealized radiation-dominated case. Specifically, compared with the radiation-dominated background, the exact zero of the radiation-dominated result is lifted to a finite local minimum, while the logarithmic resonant singularity is smoothed into a finite cusp. The size of this modification depends on when the scalar mode enters the horizon relative to the QCD-induced dip in w and c2s.

The authors then show that a dark QCD crossover generates an additional frequency-shifted distortion when the enhanced scalar mode reenters the horizon near the dark confinement scale. For the benchmark cases d and e, which probe the dark QCD crossover region, the finite dip values of omegaGW(k, ηc)/A2ζ in the SM + dQCD3 case are significantly larger than those in the corresponding SM thermal history. This behavior is consistent with the trend observed when c2s(ηcancel) lies near the bottom of a sound speed dip, lifting the zero of the radiation-dominated spectrum to a larger finite value. By contrast, when the mode enters the horizon below the dark QCD crossover region (Case f, at about 0.3 GeV), the SM and SM + dQCD3 spectra remain close to each other, since the relevant values of w and c2s are similar at the cancellation time.

The characteristic location of the dark QCD imprint is set by the horizon-entry scale associated with the dark QCD crossover. For the benchmark considered, the corresponding feature is shifted to a larger wavenumber, or equivalently to a higher present-day frequency, relative to the SM QCD feature. Using a common reference scale kQCD or fQCD, the pure SM QCD feature appears near k/kQCD ∼ 1, while the dark QCD feature appears near k/kQCD ∼ 5.5, up to corrections from the changing effective degrees of freedom and entropy evolution.

The authors also checked that the six-flavor dark QCD benchmark SM + dQCD6 gives a SIGW spectrum qualitatively similar to the SM + dQCD3 benchmark in the temperature range most relevant for the dark QCD crossover, since the corresponding w(T) and c2s(T) profiles are very close below 1.2 GeV.

The results demonstrate that SIGWs provide a complementary cosmological probe of hidden confining sectors, with characteristic spectral features shifted to higher frequencies relative to the SM QCD imprint. Since horizon reentry around the QCD scale corresponds to frequencies relevant for pulsar timing arrays (PTA), future PTA observations may be sensitive not only to the SM QCD imprint but also to frequency-shifted features associated with a dark QCD crossover. A statistically robust search would require fitting SIGW templates incorporating the SM and dark QCD thermal histories to PTA spectral data and comparing them with smooth astrophysical backgrounds and templates based on the SM thermal history alone. The analysis developed in this work can also be extended to other well-motivated theories containing different dark confining sectors.

Improvements to AI systems based on this paper:

1. Physics-informed template generation for gravitational wave data analysis

• The AI can automatically generate accurate SIGW spectral templates that incorporate realistic SM + dark QCD thermal histories (including w(T) and c2 s(T) profiles) for arbitrary confinement scales and flavor counts.

• It can interpolate between benchmark cases (e.g., SM+dQCD3, SM+dQCD6) to produce continuous frequency-dependent predictions for hidden confining sectors, enabling direct fitting to PTA data.

2. Bayesian model selection with multi-sector thermal histories

• The AI can perform nested sampling or MCMC over a parameter space that includes the dark QCD confinement scale (Λ dQCD), the number of light dark flavors, and the primordial curvature power spectrum amplitude A ζ and shape (e.g., monochromatic or log-normal).

• It can compute Bayes factors comparing (i) pure SM thermal history, (ii) SM+dQCD3, (iii) SM+dQCD6, and (iv) smooth astrophysical background models, thereby quantifying evidence for a dark QCD crossover from PTA data.

3. Horizon-entry mapping and frequency-shift prediction

• The AI can automatically map any dark confinement scale Λ d to the corresponding present-day frequency shift relative to the SM QCD feature (e.g., f dQCD ≈ 5.5 × f QCD for Λ d = 5.5 Λ QCD), including corrections from entropy and effective degrees-of-freedom evolution.

• It can predict the exact location of the finite local minimum and cusp features in omega GW(k) as a function of the sound speed dip timing, enabling targeted searches in specific frequency bands.

4. Uncertainty quantification for spectral distortions

• The AI can propagate uncertainties in the QCD equation of state (e.g., lattice QCD errors) and in the dark sector parameters into the predicted SIGW spectrum, producing credible intervals for the dip amplitude and cusp height.

• It can identify which parameter regions yield the largest deviations from the radiation-dominated result, guiding observational strategies.

5. Automated detection of hidden confining sectors in PTA datasets

• The AI can scan PTA frequency spectra for the characteristic double-feature pattern (SM QCD + dark QCD shifted) using matched filtering with the generated templates.

• It can distinguish between a single QCD feature and two overlapping features by exploiting the frequency ratio and amplitude ratio predicted by the model, even when the dark feature is partially degenerate with noise.

6. Transfer learning to other dark confining theories

• The AI can generalize the framework to arbitrary hidden gauge groups (e.g., SU(N) with different N, or composite Higgs models) by re-scaling the confinement scale and flavor content, without re-deriving the perturbation equations from scratch.

• It can provide a modular pipeline that takes as input any w(T) and c2 s(T) table (from lattice or effective models) and outputs the SIGW spectrum, enabling rapid exploration of many beyond-SM scenarios.

7. Real-time spectral prediction for multi-messenger observations

• The AI can combine SIGW predictions with other cosmological probes (e.g., BBN, CMB, or dark matter abundance constraints from asymmetric twin baryons) to produce joint likelihoods, improving the overall sensitivity to dark QCD sectors.

• It can generate mock PTA datasets with injected dark QCD signals to train detection algorithms, validating their statistical robustness before application to real data.