⚛️ high-energy theory

QCD Crossover Transfer Functions for Scalar-Induced Gravitational Waves in the PTA Band

This paper demonstrates that the softening of the equation of state during the QCD crossover significantly alters the transfer functions for scalar-induced gravitational waves, modifying their spectral amplitude by up to 55% within the PTA band and necessitating its inclusion to avoid biased inference of primordial curvature perturbations from current and future pulsar timing array data.

Original authors: Gabriele Franciolini, Xavier Pritchard, Yuichiro Tada

Published 2026-09-25

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CC BY 4.0

Original authors: Gabriele Franciolini, Xavier Pritchard, Yuichiro Tada

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep in the quiet rhythm of the universe, there exists a faint, persistent hum known as the gravitational wave background. Unlike the sharp, violent chirps of colliding black holes that have captured headlines, this background is a continuous ripple, a collective whisper from countless cosmic events occurring throughout history. For the first time, massive observatories called pulsar timing arrays have detected evidence of this hum, specifically in a very low frequency range measured in billionths of a hertz. While the source of this signal remains a mystery, one leading theory suggests it was created by the violent birth of the early universe itself. If this theory is correct, the signal carries a hidden message about the state of the cosmos when it was just a fraction of a second old, a time when the fundamental forces of nature were still settling into their current forms. Understanding this signal requires knowing exactly how the universe expanded and cooled during those first moments, because the way space stretched back then determines how these ancient ripples sound to us today.

A team of researchers has now provided a crucial piece of this puzzle by calculating how the universe's expansion changed during a specific, chaotic era known as the quantum chromodynamics, or QCD, crossover. This was a moment roughly a ten-thousandth of a second after the Big Bang, when the hot soup of fundamental particles cooled enough for quarks to bind together and form protons and neutrons. During this transition, the universe did not behave like a simple, smooth fluid; instead, its internal pressure softened significantly, much like a balloon that suddenly loses some of its stiffness. The researchers, Gabriele Franciolini, Xavier Pritchard, and Yuichiro Tada, set out to solve the complex equations that describe how gravitational waves are generated when the universe undergoes such a change. They focused on a specific type of wave called a scalar-induced gravitational wave, which is created when tiny fluctuations in the density of the early universe collide and interact to produce ripples in space-time.

The team's work reveals that ignoring this brief period of softening leads to a significant misunderstanding of the signal. They found that the height of the gravitational wave signal can be altered by as much as 55 percent depending on whether the waves were generated just before or just after this transition. If the waves were created before the universe softened, the signal appears stronger; if they were created after, it appears weaker. This is not a minor adjustment but a major shift that changes the entire shape of the expected signal across the frequency range that current detectors can hear. The researchers solved the equations governing these waves across the entire thermal history of the early universe, creating a detailed set of lookup tables that other scientists can use to interpret their data accurately. These tables act as a new standard, replacing older, simpler models that assumed the universe expanded in a perfectly uniform way.

To test the importance of their findings, the team applied these new calculations to real data collected by the NANOGrav collaboration, which has been monitoring the timing of distant pulsars for fifteen years. When they fitted their new, more accurate model to the data, the results shifted slightly but meaningfully. The peak of the signal, which indicates the most intense part of the gravitational wave background, was found to be at a slightly lower frequency and with a slightly lower amplitude than previously thought when using the old models. While the shift is currently small compared to the margin of error in the data, the researchers warn that as detectors become more sensitive and the data becomes clearer, this correction will become essential. Without it, scientists might draw incorrect conclusions about the nature of the early universe, such as overestimating the number of primordial black holes that could have formed from these fluctuations.

The study also clarifies how the signal behaves at different frequencies, particularly at the lower end of the spectrum. Previous theories suggested that the signal would follow a predictable pattern based on the speed of sound in the early universe, but the researchers found that the complex thermal history of the QCD era breaks this pattern. The signal does not simply fade away in a standard way; instead, it carries a unique imprint of the universe's changing temperature and pressure. This means that the gravitational wave background is not just a generic noise but a detailed record of the universe's infancy. As future observatories come online with greater sensitivity, they will be able to hear these subtle variations more clearly, potentially revealing new physics beyond our current understanding. For now, the work of this team ensures that when we finally decode the message from the early universe, we are listening with the right ears, accounting for the moment the cosmos took a deep breath and changed its shape.

Technical Summary: QCD Crossover Transfer Functions for Scalar-Induced Gravitational Waves in the PTA Band

Problem Statement
Pulsar Timing Array (PTA) collaborations have reported evidence for a stochastic gravitational wave (GW) background in the nanohertz (nHz) band. A leading cosmological explanation for this signal involves Scalar-Induced Gravitational Waves (SIGWs), generated at second order by enhanced primordial curvature perturbations. However, reconstructing the underlying curvature power spectrum from PTA data requires a transfer function that accurately accounts for the Standard Model (SM) thermal history.

The frequencies probed by current PTAs ( $f \sim 1\text{--}100$ nHz) correspond to modes that re-entered the Hubble horizon when the plasma temperature was near the Quantum Chromodynamics (QCD) crossover ( $T_{\text{QCD}} \simeq 0.15$ GeV). At this epoch, the equation of state (EoS) of the radiation plasma softens significantly ( $w$ drops from $1/3$ to $\approx 0.23$ ), deviating from the standard radiation-domination (RD) assumption. The paper addresses the lack of precise transfer functions that incorporate this SM effect, noting that neglecting it could bias the inference of primordial parameters and the associated constraints on Primordial Black Hole (PBH) overproduction.

Methodology
The authors solve the coupled tensor-scalar system of equations across the exact SM thermal history, specifically focusing on the QCD crossover era.

Thermodynamic Background: The evolution of the universe is modeled using lattice QCD results interpolated via a cubic spline to describe the temperature dependence of the energy density $\rho(T)$ , entropy density $s(T)$ , EoS parameter $w(T)$ , and sound speed squared $c_s^2(T)$ . The background expansion is obtained by integrating the Friedmann and continuity equations.
Equations of Motion:
- Scalar Sector: The Bardeen equation is solved numerically for the curvature perturbation transfer function $\Phi_k(\eta)$ , accounting for the time-dependent $w(T)$ and $c_s^2(T)$ .
- Tensor Sector: The second-order tensor perturbation equation is solved using a Green's function method. The source term depends on the scalar perturbations and the time-dependent EoS.
Numerical Techniques:
- WKB Continuation: To handle the computational cost of sub-horizon evolution (particularly in the squeezed limit where internal momenta $q \gg k$ ), the authors employ a WKB approximation for modes deep inside the horizon. Matching to the numerical solution occurs at a time $\eta_{\text{WKB}}$ defined by an adiabaticity criterion.
- Kernel Integration: The induced tensor power spectrum is calculated by integrating the product of scalar transfer functions and tensor mode functions over loop variables. The time integration is performed on a logarithmic grid dense enough to resolve the fastest internal frequencies.
Tabulation: The oscillation-averaged kernel $I_2(k, t, s)$ is tabulated on a fixed grid of external wavenumbers $k$ (spanning $f \in [10^{-3}, 10^4]$ nHz) and internal loop variables $(t, s)$ . This allows for rapid evaluation of the induced spectrum for arbitrary primordial power spectra without re-solving the differential equations.

Key Contributions

Tabulated Transfer Functions: The primary deliverable is a set of ready-to-use, tabulated kernels that incorporate the SM QCD crossover effect. These are made publicly available to replace standard RD kernels in existing PTA analysis pipelines.
Quantification of SM Effects: The paper provides the first detailed quantification of how the QCD crossover modifies the SIGW spectrum relative to the pure RD expectation.
Data Analysis Application: The authors apply these transfer functions to fit the NANOGrav 15-year data using a broken-power-law curvature spectrum, comparing the results against a standard RD analysis.

Results

Spectral Modification: The QCD crossover induces a characteristic "two-sided" pattern in the SIGW spectrum. Modes crossing the horizon during the onset of the crossover (around $f \approx 50$ nHz) are suppressed by up to $\approx 35\%$ , while modes crossing just before the transition (around $f \approx 2$ nHz) are enhanced by a similar amount. The total modification across the PTA band can reach $\approx 55\%$ .
Physical Origin: The effect arises primarily from the scalar transfer function. The softening of the EoS creates a non-zero potential term in the scalar evolution equation that changes sign, leading to a temporary boost or dip in the scalar amplitude depending on the phase of the mode relative to the transition. The tensor modes themselves only exhibit a boost due to the adiabatic response to the changing $w$ .
Infrared Tail: The infrared tail of the induced spectrum ( $f \ll f_*$ ) does not follow the standard causality scaling ( $\Omega_{\text{GW}} \propto k^3$ ) or the RD logarithmic template. Instead, the tail's normalization is set by the dilution factor evaluated at the peak frequency $f_*$ , as the emission window overlaps the crossover.
Impact on Parameter Inference: Fitting the NANOGrav 15-year data with the QCD kernel shifts the inferred peak amplitude ( $A$ ) and peak scale ( $k_*$ ) relative to the RD fit: $\Delta \log_{10} A = -0.09$ and $\Delta \log_{10} k_* = -0.19$ . While these shifts are currently within the statistical uncertainties of the 15-year dataset (approx. $0.26\sigma$ and $0.53\sigma$ ), they represent a correlated shift along the degeneracy direction.

Significance and Claims
The paper claims that the QCD crossover is a Standard Model effect that cannot be neglected in the interpretation of a potential scalar-induced origin for the PTA signal. While the current statistical uncertainty of the NANOGrav 15-year data is large enough that the crossover does not alter qualitative conclusions, the authors assert that this systematic bias will become increasingly relevant as future, more sensitive PTA datasets reduce statistical errors.

Specifically, the paper notes that neglecting this effect could significantly bias the inference of the underlying curvature power spectrum and, consequently, the derived bounds on the abundance of Primordial Black Holes. The authors emphasize that as statistical uncertainties shrink, the inclusion of these SM transfer functions will be essential for accurate parameter estimation. The work also suggests that similar techniques could be applied to probe Beyond Standard Model (BSM) physics that predicts softenings of the equation of state at higher energy scales (and thus higher frequencies) accessible to future observatories like LISA and the Einstein Telescope.

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