LISA and

Aims. We study two possible cosmological consequences of a first-order phase transition in the temperature range of 1 GeV to 103 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 γ-ray telescopes on intergalactic magnetic fields (IGMF) at present time. Methods. The SGWB spectrum is evaluated adopting semi-analytical models, accounting for both the contributions from sound waves and magnetohydrodynamic (MHD) turbulence in the aftermath of the first-order phase transition. Turbulence is assumed to arise only after an initial period of sound waves, and the magnetic field is assumed to be in equipartition with the turbulent kinetic energy. Several paths are considered for the magnetic field evolution throughout the radiation-dominated era, in order to predict the amplitude and correlation length scale of the resulting IGMF today. Comparing the SGWB level with the sensitivity of LISA and the IGMF parameters with the sensitivity reach of the CTA γ-ray telescope, we identify a range of first-order phase transition parameters providing observable signatures at both detectors. Results. If even a small fraction of the kinetic energy in sound waves is converted into MHD turbulence, a first-order phase transition occurring at a temperature between 1 and 106 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 γ-ray telescope MAGIC, for all proposed evolutionary paths of the magnetic fields throughout the radiation-dominated era (i.e., for both helical and non-helical magnetic fields). For the following fractions of the energy density converted into turbulence, εturb = 0.1 and 1, we provide the range of first-order phase transition parameters (strength α, duration β−1, bubble wall speed vw, and temperature T*), together with the corresponding range of magnetic field strength B and correlation length λ, which would lead to the SGWB and IGMF being 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 εturb ≪ 1, the SGWB is only sourced by sound waves, but an IGMF is still generated with a magnetic field amplitude proportional to $ \sqrt{\varepsilon_{\mathrm{turb}}} $. We find that for values as small as $ \varepsilon_{\mathrm{turb}} \sim {\cal O}(10^{-13}) $ or 𝒪 (10−9), respectively, helical or non-helical primordial magnetic fields provide IGMF compatible with MAGIC’s lower bound.