We show that the recent detection of a gravitational wave (GW) background reported by various pulsar timing array (PTA) collaborations including NANOGrav-15yr, PPTA, EPTA, and CPTA can be explained in terms of first order phase transitions (FOPTs) from dark sector models (DSM). Specifically, we explore a model for first order phase transitions that involves the majoron, a Nambu-Goldstone boson that is emerging from the spontaneous symmetry breaking of a U (1)L or U (1)B−L symmetry. We show how the predicted GW power spectrum, with a realistic choice of the FOPT parameters, is consistent with 1-σ deviations from the estimated parameters of the background detected by the PTA collaborations.must be included in the Lagrangian.

We derive a universal expression for the gravitational radiation energy spectrum dEGW/dω at sub-leading order emitted from a generic gravitational hard scattering of multi-particles or multi-bodies. Our result includes all $$ \mathcal{O} $$ O (ω) corrections to the gravitational radiation flux from a generic 2 → N collision, in both the cases of massless and massive particles/bodies. We also show the dependence of the radiation energy flux by the quantum spin in case of particle collisions. Then, we consider the specific case of a gravitational elastic scattering of two massive bodies, i.e. m + M → m + M with m, M the masses of the two bodies respectively. We demonstrate that in this case all $$ \mathcal{O} $$ O (ω) contributions to the energy flux exactly cancel each others. Nevertheless, we also show that, for a 2 → 2 inelastic scattering, the inclusion of sub-leading soft gravitons leads to a not zero radiation flux, having a simple expression in certain asymptotic regimes. Our results can be applied to the case of Black Hole collisions with possible testable implications in gravitational waves physics.

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