How the early Universe’s changing particle soup could reshape gravitational waves

The question is not simply whether the early Universe produced gravitational waves. It is whether their pattern across frequencies could reveal how its hot particle mixture changed as it cooled.

In a preprint, Peizhi Du, Xiang-Xi Zeng and Jiahang Zhong calculate how those changes affect scalar-induced gravitational waves: ripples in spacetime generated through the nonlinear evolution of primordial variations in density and curvature. Their results predict distinctive changes in wave strength across frequency, but their analysis of existing observations finds only a very weak preference for including one important transition.

A changing medium changes the prediction

A common simplified calculation treats the early Universe as an ideal radiation-dominated fluid with fixed properties. Its pressure is one-third of its energy density, and disturbances travel through it at a constant sound speed.

The real thermal history is less uniform. As the Universe cools, particles become nonrelativistic, annihilate or become bound into different forms. These processes change both the pressure relative to energy density—the equation of state—and the speed at which pressure disturbances propagate.

One important example is the quantum chromodynamics, or QCD, crossover, when quarks and gluons became confined into hadrons. A crossover is a smooth change rather than an abrupt transition. The authors also include the electroweak crossover and electron–positron annihilation.

These changes matter because the primordial variations that generate gravitational waves evolve within this changing medium. The study concerns that indirect imprint, not a separate calculation of waves produced directly by colliding transition bubbles.

Making the thermal history consistent

The researchers numerically follow the evolving primordial perturbations and the gravitational waves they generate. They first use a simplified particle mixture to isolate the effects of a single thermal change, then calculate predictions using the Standard Model’s thermal history.

Two methodological corrections are central.

First, the calculation assumes that entropy within an expanding parcel of the cosmic fluid is conserved. The authors reconstruct the QCD thermodynamic inputs so that they obey this requirement while remaining within the quoted uncertainties of lattice-QCD calculations—numerical studies of the strong interaction.

Second, they account for the earlier expansion history when setting the perturbations’ initial values. Starting every mode with the usual ideal-radiation values can introduce an artificial enhancement if the calculation begins during a transition. Their history-dependent initialization removes that spurious feature.

Why a step appears

For an assumed primordial spectrum with equal curvature power across scales, the strongest predicted feature is a step between two levels of gravitational-wave strength.

The explanation is relative dilution. When massive particles transfer entropy to lighter particles, the radiation bath is heated relative to freely propagating gravitational waves. Waves generated before that transfer therefore contribute a smaller fraction of the later energy budget than otherwise comparable waves generated afterward.

Changing pressure and sound speed add nearby peaks and dips by altering how the source perturbations evolve. These features depend on the assumed primordial spectrum; they are not a universal template independent of the waves’ origin.

The QCD imprint appears around 10 nanohertz, within the range accessible to pulsar timing arrays. These arrays search for gravitational waves through correlated changes in the arrival times of pulses from rapidly rotating stars.

Existing data do not establish the imprint

The authors fit the NANOGrav 15-year data using two possible primordial spectra: an idealized narrow peak and a spectrum that rises before becoming flat. They compare predictions including the QCD crossover with those using the simpler radiation background.

The resulting Bayes factors are 1.06 and 1.36. These measure relative statistical support between the compared models; values this close to one provide little separation. This is not a detection of the QCD imprint, nor evidence that the observed background must be primordial. Supermassive black-hole binaries remain the paper’s conservative interpretation of the pulsar-timing signal.

What remains conditional

The predicted signal requires sufficiently strong primordial fluctuations on small scales, whose strength and shape are uncertain. The authors also acknowledge that primordial explanations fitting pulsar-timing data typically risk producing too many primordial black holes. They do not resolve that problem here.

Their higher-energy example, involving composite-Higgs physics, places features near millihertz frequencies covered by planned missions such as LISA, Taiji and TianQin. But it uses a simplified thermal model and assumes negligible entropy injection during the transition.

The study supplies more consistent predictions to test, not an established reconstruction of cosmic history. Extracting that history will require distinguishing the source spectrum from the thermal imprint and from competing gravitational-wave sources.