Gravitational waves from dark domain walls

Written 27th of December 2023 Last updated 8th of January 2023


Structure formation

Locally in smaller regions, there are large differences between largest densities in the cosmic filaments and the smaller densities in the voids that the filaments encloses. The voids can be seen to act repulsively, seemingly pushing the residual matter into the filaments.

Gravitational waves

To create the sound, at first a white noise signal is generated, and then a band-pass filter is applied to match it to the power \(\dot{h}_{ij}\dot{h}^{ij}\) spectrum amplitudes. The filter is continuously adjusted as the video progresses, following the changes in the power spectrum. The frequency was shifted by 18 orders of magnitude, such that \(10^{-15}\) Hz becomes 1 kHz. This is a handy way to visualise the simulation spectra, but the correspondence to the videos should not be exact, because 1) videos are of slices of the simulation volume, whereas the spectra are of the entire box, and 2) the power spectra remove all non-Gaussian information originally contained within the fields' statistics.


For the standard cosmological model, \(\Lambda\)CDM, we are seeing a mostly adiabatic evolution, but also a sourcing from the clustering cold dark matter. The amplitude for small scale modes is however very subleading with respect to the case of the other models.

One can also see faint ripples from fixed locations, throughout the simulation. These have been identified to be owing to the discretised mass distribution, and are caused by sudden displacements of particles when projected onto the lattice. The ripples' intensity was reduced significantly by going from a Nearest Grid Point (NGP) scheme to a higher order Cloud-In-Cell (CIC) scheme, and has been found to be unimportant with respect to the wave mode scales that we are interested in.

Model Z

\(\ (L_C, z_*, \bar \beta, \Delta \beta/\bar\beta)=(1\) Mpc/h\(\ , 2, 1, 0) \)

For a first demonstration of the production of gravitational waves from the scalar field, we show the case of a relatively early phase transition, at \(z_{SSB}=z_*=2\). In this event, we see a much more violent phase transition that is less confined by the environment and cosmic filaments. Instead, the resultant domain walls wander accross the simulation volume and eventually collapse, producing large ripples in both the scalar field and in the gravitational waves. The scale of these large domain collapses are comparatively larger, so that we would need a much larger box to resolve their statistics, but are instead resolving a single event here. Additionally, there is GW production from the vibrations of the walls that we are resolving the statistics of, and that we study in the other simulations below.

Model Ⅰ

\(\ (L_C, z_*, \bar \beta, \Delta \beta/\bar\beta)=(1\) Mpc/h\(\ , 0.1, 8, 0) \)

Model Ⅱ

\(\ (L_C, z_*, \bar \beta, \Delta \beta/\bar\beta)=(0.75\) Mpc/h\(\ , 0.1, 8, 0) \)

In this case, we look for the effect of reducing the Compton wavelength \(L_C\) corresponding the to the Lagrangian mass parameter \(\mu\) and relating to the scale of correlation for the field when in vacuum. In this case, we expect a shift of the spectrum towards smaller scales. We are also seeing a different collapse of the scalar field due to different phases at the time of the symmetry breaking.

Model Ⅲ

\(\ (L_C, z_*, \bar \beta, \Delta \beta/\bar\beta)=(1\) Mpc/h\(\ , 0.1, 8, 10\%) \)

For the final animation that we display here, we show the effect of including an asymmetric potential \(\Delta\beta/\bar\beta=10\)%. The main effect seems to be more unstable domain walls, and therefore more collapses in the late-time universe, when the domains intersect more strongly.

Finally, for this same parameter choice, we demonstrate the behaviour of the tensor perturbation, contracted with the direction vector of the line of sight for an observer at equidistant shells moving from a distance \(D=370\) Mpc and source redshift \(z=0.08\) to an inner distance \(D=0\) Mpc and observer redshift \(z=0\). The contraction of the time-differentiated perturbation with the line of sight vector, is what should be integrated over the light path from the pulsars to the observer, to give the observable \(\Delta P/P\) that NANOGrav is measuring and doing statistics on (where \(P\) is the pulsar period).