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

Abstract

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 $γ$-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 a 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 $γ$-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 energy density converted into turbulence, $\varepsilon_{\rm turb}=0.1$ and $1$, we provide the range of first-order phase transition parameters, together with the corresponding range of magnetic field strength $B$ and correlation length $λ$, 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, but an IGMF is still generated. We find that for values as small as $\varepsilon_{\rm turb} \sim 10^{-13}$ or $10^{-9}$, respectively helical or non-helical magnetic fields can provide IGMF compatible with MAGIC's lower bound.

Figures (4)

• Figure 1: Spectra of the different components of the SGWB for $T_*=100\hbox{GeV}$, $\alpha=0.5$, $\beta=10H_*$, $v_w = 0.95$, and $\varepsilon_{\rm turb}=1$, compared to LISA's power law sensitivity from Caprini:2019egz with a signal-to-noise ratio threshold of 10. The SGWB spectra from sound waves are based on the SSM of Hindmarsh:2019phv (dashed red) and on the fit from HL simulations of Jinno:2022mie and Caprini:2024gyk (solid red). The spectrum of the turbulence is based on the model developed and numerically validated in RoperPol:2022iel. The vertical lines indicate the relevant frequencies of the turbulence template ($\delta t_{\rm fin}^{-1}$ and $f_{\rm turb}$) in blue, of the sound-wave template ($\lambda_*^{-1}$ and $\delta \lambda_*^{-1}$) in red, and the conformal Hubble frequency ${\cal H}_*$ at the time of GW generation in black.
• Figure 2: Contours of the parameters $\alpha$ and $\beta/H_*$ that lead to a signal above LISA's power law sensitivity with a signal-to-noise ratio threshold of 10 (see Fig. \ref{['fig:example']}). The top panels take a fixed value of the wall velocity $v_w = \{0.4, 0.6, 0.8, 1\}$ (indicated in different colors) and scan over temperatures to construct the contours, while the lower panels fix the temperature in GeV $T_\ast = \{10, 10^2, 10^3, 10^4, 10^5, 10^6\}$ and scan over wall velocities to construct the contours. Left and right panels correspond to $\varepsilon_{\rm turb} = 0.1$ and 1, respectively. The results shown correspond to the sound-wave template based on the fit from the Higgsless simulations (cf. Sec. \ref{['gw_sw']}), as this choice does not affect the magnetic field parameters in Fig. \ref{['fig:GW_B_signatures']}. Colored contours correspond to regions of the parameter space where $H_* \tau_{\rm nl} \lesssim 1$, while their continuation by dashed contours corresponds to regions where $H_* \tau_{\rm nl} \gtrsim 1$.
• Figure 3: Range of the comoving magnetic field strength $\tilde{B}$ and correlation length $\tilde{\lambda}_B$ corresponding to the contours in the lower panels of Fig. \ref{['fig:alpha_beta_GW_signatures']}, for $\varepsilon_{\rm turb} = 0.1$ (top panel) and $1$ (bottom panel). The colored numbers indicate the value of the temperature $T_\ast$ in GeV. For values of $\varepsilon_{\rm turb}$ smaller than 0.1, the range of observable phase transition parameters is unaffected, and the contour of magnetic field parameters just shifts to smaller amplitudes proportional to $\sqrt{\varepsilon_{\rm turb}}.$
• Figure 4: Expected initial and final comoving strength and correlation length of a cosmological magnetic field generated at a first-order phase transition with parameters leading to an SGWB detectable by LISA. The black and grey contours in the upper left corner of the figure show the parameter space of initial conditions if one assumes that, respectively, the totality ($\varepsilon_{\rm turb}=1$) and 10% ($\varepsilon_{\rm turb}=0.1$) of the sound-wave kinetic energy is eventually transferred to MHD turbulence (i.e., magnetic and vortical kinetic energy) at the time of nonlinearities development. The orange-shaded region is the one found in RoperPol:2022iel (RPCNS 22) assuming that the SGWB is sourced exclusively by MHD turbulence. The blue star corresponds to the phase transition parameters for which the SGWB spectrum is shown in Fig. \ref{['fig:example']}, considered for illustrative purposes. The inclined arrows show the envelopes of the evolutionary tracks. The dash-dotted purple arrows apply to helical fields, $B \sim \lambda^{-0.5}$, as given in Banerjee:2004df; the solid green arrows apply to non-helical fields, $B \sim \lambda^{-1.25}$, as given in Hosking:2022umv; the dashed green arrows correspond to the hypothesis of "selective decay" of short-range modes for non-helical fields, $B \sim \lambda^{-2.5}$Banerjee:2004df. The black and red dash-dotted lines show the possible endpoints of the magnetic field evolution, corresponding to the IGMF present in the voids today, assuming respectively that the reconnection time scale dominates the magnetic field dynamics Hosking:2022umv (HS 22) and that the Alfvénic time scale dominates the magnetic field dynamics Banerjee:2004df (BJ 04). The grey-shaded region at the bottom left of the plot is excluded by the lower bound on the IGMF established by the MAGIC $\gamma$-ray observatory MAGIC:2022piy. The dark blue and black thin lines show the upper limit on the IGMF from, respectively, ultra-high-energy cosmic rays Neronov:2021xua and Faraday rotation measures Pshirkov:2015tua. The blue-shaded area shows the range of IGMF parameters that will be probed by the $\gamma$-ray observatory CTA Korochkin:2020pvg. The red and black ticks over the BJ 04 and HS 22 recombination lines correspond to the range of magnetic field strengths obtained in Galli:2021mxk, which would induce enough baryon clumping to help alleviate the Hubble tension, as proposed in Jedamzik:2020krr. The green and purple areas denote, respectively, the range of non-helical and helical magnetic field parameters that would arise from a first-order phase transition occurring at $T_* \sim 100$ GeV and sourcing a SGWB detectable by LISA, fixing the smallest possible value $\varepsilon_{\rm turb}$ while still satisfying MAGIC's lower bound. These values of $\varepsilon_{\rm turb}$ are ${\cal O} (10^{-9})$ (non-helical) and ${\cal O} (10^{-13})$ (helical).