In the first moments of its life, the universe resembled a boiling cauldron. Matter wasn't spread evenly — clumps and voids were brewing within it, and their violent motion gave rise to a trembling of space itself: gravitational waves. Today, we try to catch their echoes (for instance, with the NANOGrav network of radio telescopes) and compare them with predictions. To do this, physicists run numerical simulations on supercomputers, recreating that ancient storm.
But there's a catch. When you compute the spectrum directly from the simulation, the distribution of wave energy across frequencies, a fine jitter appears at low frequencies. It looks like ripples that were never expected. The cause isn't physics but mathematics: when averaging squared amplitudes, doubled frequencies sneak into the signal. It's as if an echo bounced off a wall and superimposed on itself. The authors call these 'parasitic wiggles' and show that they have nothing to do with real waves. Such a spectrum cannot be shown to observers: it's misleading.
The solution came from a simple idea: let the simulation 'cool down.' When the source of waves — for example, a network of domain walls (hypothetical folds in spacetime)—disappears, the waves continue to roam the universe for a while. If you extend the calculation for several Hubble times (periods of doubling of the universe's size) after the source is turned off, the parasitic oscillations smooth out by themselves. The physicists tested this with the CosmoLattice code, simulating exactly a network of domain walls. The extended simulation allowed them to recover the true, smooth infrared slope of the spectrum — the very one that carries the memory of the most violent stage of wave birth.
The authors warn: artificially stretching the equations using the well-known Press–Riddick–Spergel (PRS) trick won't work here. It gives incorrect spectra, even if you carefully renormalize the sources. Only an honest physical continuation of the calculation, which took a few extra Hubble times, brings the true signal to light. The method is quite universal and applicable to any numerical model of hypothetical gravitational wave sources in the early universe.
as written by the authors arXiv:2607.25073
Numerical simulations of gravitational waves (GW) production during violent evolution of matter inhomogeneities in the early Universe yield highly wiggle infrared parts of the spectra. Obtained directly from the two-point correlation function, these wiggles are nonphysical, corresponding to the parasitic, double frequency terms in naively averaged squared oscillation amplitudes of GW, and hence must be washed out. The deep infrared behavior can be predicted on general grounds, e.g. fixed by causality considerations. However, what matters for real observations, e.g. like that of NANOGrav, and what can be only inferred from numerical simulations, is the infrared slope near the maximum of the spectrum. It reflects the dynamics responsible for the GW production in its heyday, and hence must be accurately predicted. We illustrate the problem with numerical simulations of the Domain Wall network performed with the help of code CosmoLattice. We suggest a numerical procedure to smooth out these parasitic wiggles, which allows us to recover the true spectrum. Being quite generic, it may be applied to numerical simulations of other hypothetical sources of GW possibly operating in the early Universe. The procedure requires to extend the simulation by a few Hubble times after termination of the GW production. We checked with numerical simulations, that the technically natural long-time extension of simulations, which becomes available via artificial scaling of different parts in the scalar sector equations provided by PRS prescription, gives wrong GW spectra even if the source terms are properly rescaled.
A network of domain walls is a set of hypothetical 'wrinkles' in spacetime that could have formed as the universe cooled, passing through phase transitions, much like water turning into ice.
In Carl Sagan's novel 'Contact,' an alien signal contains the blueprint for a machine that uses gravitational waves to travel through spacetime.