LISA and $\gamma$-ray telescopes as multi-messenger probes of a first-order cosmological phase transition

We study two possible cosmological consequences of a first-order phase transition in the temperature range of $1$ GeV to $10^3$ TeV: the generation of a stochastic gravitational wave background (SGWB) within the sensitivity of the Laser Interferometer Space Antenna (LISA) and, simultaneously, primordial magnetic fields that would evolve through the Universe's history and could be compatible with the lower bound from $\gamma$-ray telescopes on intergalactic magnetic fields (IGMF) at present time. We find that, if even a small fraction of the kinetic energy in sound waves is converted into MHD turbulence, a first order phase transition occurring at temperature between $1$ and $10^6$ GeV can give rise to an observable SGWB signal in LISA and, at the same time, an IGMF compatible with the lower bound from the $\gamma$-ray telescope MAGIC, for all proposed evolutionary paths of the magnetic fields throughout the radiation dominated era. For two values of the fraction of energy density converted into turbulence, $\varepsilon_{\rm turb}=0.1$ and $1$, we provide the range of first-order phase transition parameters (strength $\alpha$, duration $\beta^{-1}$, bubbles wall speed $v_w$, and temperature $T_*$), together with the corresponding range of magnetic field strength $B$ and correlation length $\lambda$, that would lead to the SGWB and IGMF observable with LISA and MAGIC. The resulting magnetic field strength at recombination can also correspond to the one that has been proposed to induce baryon clumping, previously suggested as a possible way to ease the Hubble tension. In the limiting case $\varepsilon_{\rm turb} \ll 1$, the SGWB is only sourced by sound waves, however, an IGMF is still generated. We find that values as small as $\varepsilon_{\rm turb} \sim O(10^{-13})$ (helical) and $O (10^{-9})$ (non-helical) can provide IGMF compatible with MAGIC's lower bound.