- Open Access
- Access by Xinjiang University
Gamma rays and gravitational waves from inelastic Higgs portal dark matter
Phys. Rev. D 113, 115009 – Published 3 June, 2026
DOI: https://doi.org/10.1103/4vqv-vmxz
Abstract
We explore a simple and predictive dark matter scenario involving a complex scalar field, , coupled to the Higgs portal with no additional field content. In the UV, the field possesses a global symmetry which is broken by mass terms and Higgs portal interactions. In the mass basis, the complex field splits into a pair of real scalars with a small mass splitting (in analogy to pseudo-Dirac fermions), such that the Higgs portal acquires both diagonal and off-diagonal terms with respect to these eigenstates. In the parameter space where the off-diagonal interaction predominates, this scenario is safe from direct detection constraints. Moreover, this model provides a viable explanation for the longstanding Galactic Center -ray excess. Additionally, this model influences the Higgs potential in a way that could facilitate a strong first-order electroweak phase transition in the early Universe, potentially leading to a stochastic gravitational wave background that could fall within the reach of upcoming space-based detectors.
Physics Subject Headings (PhySH)
Article Text
References (145)
- V. Silveira and A. Zee, Scalar phantoms, Phys. Lett. 161B, 136 (1985).
- J. McDonald, Gauge singlet scalars as cold dark matter, Phys. Rev. D 50, 3637 (1994).
- C. P. Burgess, M. Pospelov, and T. ter Veldhuis, The minimal model of nonbaryonic dark matter: A singlet scalar, Nucl. Phys. B619, 709 (2001).
- D. O’Connell, M. J. Ramsey-Musolf, and M. B. Wise, Minimal extension of the standard model scalar sector, Phys. Rev. D 75, 037701 (2007).
- J. M. Cline, K. Kainulainen, P. Scott, and C. Weniger, Update on scalar singlet dark matter, Phys. Rev. D 88, 055025 (2013); 92, 039906(E) (2015).
- M. Duerr, P. Fileviez Perez, and J. Smirnov, Scalar singlet dark matter and gamma lines, Phys. Lett. B 751, 119 (2015).
- M. Duerr, P. Fileviez Pérez, and J. Smirnov, Scalar dark matter: Direct vs Indirect detection, J. High Energy Phys. 06 (2016) 152.
- M. Duerr, P. Fileviez Pérez, and J. Smirnov, Gamma-ray excess and the minimal dark matter model, J. High Energy Phys. 06 (2016) 008.
- D. Hooper, G. Krnjaic, A. J. Long, and S. D. Mcdermott, Can the inflaton also be a weakly interacting massive particle?, Phys. Rev. Lett. 122, 091802 (2019).
- K. Fraser, A. Parikh, and W. L. Xu, A closer look at -violating Higgs portal dark matter as a candidate for the GCE, J. High Energy Phys. 03 (2021) 123.
- N. Aghanim et al. (Planck Collaboration), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).
- J. M. Cline, P. Scott, K. Kainulainen, and C. Weniger, Update on scalar singlet dark matter, Phys. Rev. D 88, 055025 (2013).
- J. A. Casas, D. G. Cerdeño, J. M. Moreno, and J. Quilis, Reopening the Higgs portal for single scalar dark matter, J. High Energy Phys. 05 (2017) 036.
- B. Díaz Sáez, J. Lahiri, and K. Möhling, Coscattering in the extended singlet-scalar Higgs portal, J. Cosmol. Astropart. Phys. 10 (2024) 001.
- P. De La Torre Luque, J. Smirnov, and T. Linden, Gamma-ray lines in 15 years of Fermi-LAT data: New constraints on Higgs portal dark matter, Phys. Rev. D 109, L041301 (2024).
- A. Djouadi, O. Lebedev, Y. Mambrini, and J. Quevillon, Implications of LHC searches for Higgs–portal dark matter, Phys. Lett. B 709, 65 (2012).
- G. Arcadi, A. Djouadi, and M. Raidal, Dark matter through the Higgs portal, Phys. Rep. 842, 1 (2020).
- G. Krnjaic, Probing light thermal dark-matter with a Higgs portal mediator, Phys. Rev. D 94, 073009 (2016).
- K. Ghorbani and H. Ghorbani, Scalar split WIMPs in future direct detection experiments, Phys. Rev. D 93, 055012 (2016).
- J. Guo, Y. He, J. Liu, and X.-P. Wang, Heavy long-lived coannihilation partner from inelastic dark matter model and its signatures at the LHC, J. High Energy Phys. 04 (2022) 024.
- J. A. Casas, G. A. Gómez Vargas, J. M. Moreno, J. Quilis, and R. Ruiz de Austri, Extended Higgs-portal dark matter and the Fermi-LAT galactic center excess, J. Cosmol. Astropart. Phys. 06 (2018) 031.
- M. Ackermann et al. (Fermi-LAT Collaboration), The Fermi galactic center GeV excess and implications for dark matter, Astrophys. J. 840, 43 (2017).
- A. McDaniel, M. Ajello, C. M. Karwin, M. Di Mauro, A. Drlica-Wagner, and M. A. Sánchez-Conde, Legacy analysis of dark matter annihilation from the Milky Way dwarf spheroidal galaxies with 14 years of Fermi-LAT data, Phys. Rev. D 109, 063024 (2024).
- M. Di Mauro, M. Stref, and F. Calore, Investigating the effect of Milky Way dwarf spheroidal galaxies extension on dark matter searches with Fermi-LAT data, Phys. Rev. D 106, 123032 (2022).
- A. Cuoco, M. Krämer, and M. Korsmeier, Novel dark matter constraints from antiprotons in light of AMS-02, Phys. Rev. Lett. 118, 191102 (2017).
- G. Giesen, M. Boudaud, Y. Génolini, V. Poulin, M. Cirelli, P. Salati, and P. D. Serpico, AMS-02 antiprotons, at last! secondary astrophysical component and immediate implications for dark matter, J. Cosmol. Astropart. Phys. 09 (2015) 023.
- I. Cholis, T. Linden, and D. Hooper, A robust excess in the cosmic-ray antiproton spectrum: Implications for annihilating dark matter, Phys. Rev. D 99, 103026 (2019).
- M. Trodden, Electroweak baryogenesis, Rev. Mod. Phys. 71, 1463 (1999).
- G. W. Anderson and L. J. Hall, The electroweak phase transition and baryogenesis, Phys. Rev. D 45, 2685 (1992).
- P. Huet and A. E. Nelson, Electroweak baryogenesis in supersymmetric models, Phys. Rev. D 53, 4578 (1996).
- D. E. Morrissey and M. J. Ramsey-Musolf, Electroweak baryogenesis, New J. Phys. 14, 125003 (2012).
- P. Athron, C. Balázs, A. Fowlie, L. Morris, and L. Wu, Cosmological phase transitions: From perturbative particle physics to gravitational waves, Prog. Part. Nucl. Phys. 135, 104094 (2024).
- J. Aalbers et al. (LZ Collaboration), Dark matter search results from 4.2 tonne-years of exposure of the LUX-ZEPLIN (LZ) experiment, Phys. Rev. Lett. 135, 011802 (2025).
- D. Tucker-Smith and N. Weiner, Inelastic dark matter, Phys. Rev. D 64, 043502 (2001).
- R. J. Hill and M. P. Solon, Standard model anatomy of WIMP dark matter direct detection II: QCD analysis and hadronic matrix elements, Phys. Rev. D 91, 043505 (2015).
- A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, feynrules 2.0—A complete toolbox for tree-level phenomenology, Comput. Phys. Commun. 185, 2250 (2014).
- G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, micromegas 2.0: A program to calculate the relic density of dark matter in a generic model, Comput. Phys. Commun. 176, 367 (2007).
- J. Billard, L. Strigari, and E. Figueroa-Feliciano, Implication of neutrino backgrounds on the reach of next generation dark matter direct detection experiments, Phys. Rev. D 89, 023524 (2014).
- P. Gondolo and G. Gelmini, Cosmic abundances of stable particles: Improved analysis, Nucl. Phys. B360, 145 (1991).
- G. Krnjaic, Freezing in, heating up, and freezing out: Predictive nonthermal dark matter and low-mass direct detection, J. High Energy Phys. 10 (2018) 136.
- A. Ghosh, A. Ibarra, T. Mondal, and B. Mukhopadhyaya, Gamma-ray signals from multicomponent scalar dark matter decays, J. Cosmol. Astropart. Phys. 01 (2020) 011.
- T. R. Slatyer, Energy injection and absorption in the cosmic dark ages, Phys. Rev. D 87, 123513 (2013).
- T. R. Slatyer and C.-L. Wu, General constraints on dark matter decay from the cosmic microwave background, Phys. Rev. D 95, 023010 (2017).
- L. Forestell, D. E. Morrissey, and G. White, Limits from BBN on light electromagnetic decays, J. High Energy Phys. 01 (2019) 074.
- P. F. Depta, M. Hufnagel, and K. Schmidt-Hoberg, Updated BBN constraints on electromagnetic decays of MeV-scale particles, J. Cosmol. Astropart. Phys. 04 (2021) 011.
- M. Baryakhtar, A. Berlin, H. Liu, and N. Weiner, Electromagnetic signals of inelastic dark matter scattering, J. High Energy Phys. 06 (2022) 047.
- M. Carrillo González and N. Toro, Cosmology and signals of light pseudo-Dirac dark matter, J. High Energy Phys. 04 (2022) 060.
- A. Berlin, G. Krnjaic, and E. Pinetti, Reviving MeV-GeV indirect detection with inelastic dark matter, Phys. Rev. D 110, 035015 (2024).
- L. Goodenough and D. Hooper, Possible evidence for dark matter annihilation in the inner milky way from the Fermi Gamma Ray Space Telescope, arXiv:0910.2998.
- D. Hooper and L. Goodenough, Dark matter annihilation in the galactic center as seen by the Fermi Gamma Ray Space Telescope, Phys. Lett. B 697, 412 (2011).
- D. Hooper and T. Linden, On the origin of the gamma rays from the galactic center, Phys. Rev. D 84, 123005 (2011).
- K. N. Abazajian and M. Kaplinghat, Detection of a gamma-ray source in the galactic center consistent with extended emission from dark matter annihilation and concentrated astrophysical emission, Phys. Rev. D 86, 083511 (2012); 87, 129902(E) (2013).
- D. Hooper and T. R. Slatyer, Two emission mechanisms in the Fermi bubbles: A possible signal of annihilating dark matter, Phys. Dark Universe 2, 118 (2013).
- C. Gordon and O. Macias, Dark matter and pulsar model constraints from galactic center Fermi-LAT gamma ray observations, Phys. Rev. D 88, 083521 (2013); 89, 049901(E) (2014).
- T. Daylan, D. P. Finkbeiner, D. Hooper, T. Linden, S. K. N. Portillo, N. L. Rodd, and T. R. Slatyer, The characterization of the gamma-ray signal from the central milky way: A case for annihilating dark matter, Phys. Dark Universe 12, 1 (2016).
- B. Zhou, Y.-F. Liang, X. Huang, X. Li, Y.-Z. Fan, L. Feng, and J. Chang, GeV excess in the Milky Way: The role of diffuse galactic gamma-ray emission templates, Phys. Rev. D 91, 123010 (2015).
- F. Calore, I. Cholis, and C. Weniger, Background model systematics for the Fermi GeV excess, J. Cosmol. Astropart. Phys. 03 (2015) 038.
- M. Ajello et al. (Fermi-LAT Collaboration), Fermi-LAT observations of high-energy -ray emission toward the galactic center, Astrophys. J. 819, 44 (2016).
- I. Cholis, Y.-M. Zhong, S. D. McDermott, and J. P. Surdutovich, Return of the templates: Revisiting the galactic center excess with multimessenger observations, Phys. Rev. D 105, 103023 (2022).
- M. Di Mauro, Characteristics of the galactic center excess measured with 11 years of -LAT data, Phys. Rev. D 103, 063029 (2021).
- D. Hooper, R. K. Leane, Y.-D. Tsai, S. Wegsman, and S. J. Witte, A systematic study of hidden sector dark matter: Application to the gamma-ray and antiproton excesses, J. High Energy Phys. 07 (2020) 163.
- M.-Y. Cui, Q. Yuan, Y.-L. S. Tsai, and Y.-Z. Fan, Possible dark matter annihilation signal in the AMS-02 antiproton data, Phys. Rev. Lett. 118, 191101 (2017).
- A. Cuoco, J. Heisig, L. Klamt, M. Korsmeier, and M. Krämer, Scrutinizing the evidence for dark matter in cosmic-ray antiprotons, Phys. Rev. D 99, 103014 (2019).
- J. M. Cline and K. Kainulainen, Electroweak baryogenesis and dark matter from a singlet Higgs, J. Cosmol. Astropart. Phys. 01 (2013) 012.
- V. Vaskonen, Electroweak baryogenesis and gravitational waves from a real scalar singlet, Phys. Rev. D 95, 123515 (2017).
- B. Grzadkowski and D. Huang, Spontaneous -violating electroweak baryogenesis and dark matter from a complex singlet scalar, J. High Energy Phys. 08 (2018) 135.
- J. Ellis, M. Lewicki, M. Merchand, J. M. No, and M. Zych, The scalar singlet extension of the standard model: gravitational waves versus baryogenesis, J. High Energy Phys. 01 (2023) 093.
- S. Roy, Dark matter and electroweak baryogenesis with spontaneous violation in the early universe, Phys. Rev. D 113, 055040 (2026).
- A. D. Linde, On the vacuum instability and the Higgs meson mass, Phys. Lett. 70B, 306 (1977).
- A. D. Linde, Phase transitions in gauge theories and cosmology, Rep. Prog. Phys. 42, 389 (1979).
- A. D. Linde, Decay of the false vacuum at finite temperature, Nucl. Phys. B216, 421 (1983); B223, 544(E) (1983).
- J. S. Langer, Statistical theory of the decay of metastable states, Ann. Phys. (N.Y.) 54, 258 (1969).
- S. R. Coleman, The fate of the false vacuum. 1. Semiclassical theory, Phys. Rev. D 15, 2929 (1977); 16, 1248(E) (1977).
- I. Affleck, Quantum statistical metastability, Phys. Rev. Lett. 46, 388 (1981).
- A. Mazumdar and G. White, Review of cosmic phase transitions: Their significance and experimental signatures, Rep. Prog. Phys. 82, 076901 (2019).
- M. Quiros, Finite temperature field theory and phase transitions, in ICTP Summer School in High-Energy Physics and Cosmology (1999), Vol. 1, pp. 187–259; arXiv:hep-ph/9901312.
- C. L. Wainwright, cosmotransitions: Computing cosmological phase transition temperatures and bubble profiles with multiple fields, Comput. Phys. Commun. 183, 2006 (2012).
- J. R. Espinosa, B. Gripaios, T. Konstandin, and F. Riva, Electroweak baryogenesis in non-minimal composite Higgs models, J. Cosmol. Astropart. Phys. 01 (2012) 012.
- S. Blasi and A. Mariotti, Domain walls seeding the electroweak phase transition, Phys. Rev. Lett. 129, 261303 (2022).
- P. Agrawal, S. Blasi, A. Mariotti, and M. Nee, Electroweak phase transition with a double well done doubly well, J. High Energy Phys. 06 (2024) 089.
- P. Amaro-Seoane et al. (LISA Collaboration), Laser interferometer space antenna, arXiv:1702.00786.
- X. Gong et al., Descope of the ALIA mission, J. Phys. Conf. Ser. 610, 012011 (2015).
- W.-R. Hu and Y.-L. Wu, The Taiji program in space for gravitational wave physics and the nature of gravity, Natl. Sci. Rev. 4, 685 (2017).
- V. Corbin and N. J. Cornish, Detecting the cosmic gravitational wave background with the big bang observer, Classical Quantum Gravity 23, 2435 (2006).
- H. Kudoh, A. Taruya, T. Hiramatsu, and Y. Himemoto, Detecting a gravitational-wave background with next-generation space interferometers, Phys. Rev. D 73, 064006 (2006).
- C. Caprini et al., Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions, J. Cosmol. Astropart. Phys. 04 (2016) 001.
- S. Babak, A. Petiteau, and M. Hewitson, LISA sensitivity and SNR calculations, arXiv:2108.01167.
- T. L. Smith, T. L. Smith, R. R. Caldwell, and R. Caldwell, LISA for cosmologists: Calculating the signal-to-noise ratio for stochastic and deterministic sources, Phys. Rev. D 100, 104055 (2019); 105, 029902(E) (2022).
- M. Cepeda et al., Report from Working Group 2: Higgs physics at the HL-LHC and HE-LHC, CERN Yellow Rep. Monogr. 7, 221 (2019).
- I. Agapov et al., Future circular lepton collider FCC-ee: Overview and status, in Snowmass 2021 (2022), 3; arXiv:2203.08310.
- M. Benedikt et al., Future circular hadron collider FCC-hh: Overview and status, arXiv:2203.07804.
- A. Aryshev et al. (ILC International Development Team Collaboration), The International Linear Collider: Report to snowmass 2021, arXiv:2203.07622.
- C. Accettura et al., Towards a muon collider, Eur. Phys. J. C 83, 864 (2023); 84, 36(E) (2024).
- D. Curtin et al., Exotic decays of the 125 GeV Higgs boson, Phys. Rev. D 90, 075004 (2014).
- G. Aad et al. (ATLAS Collaboration), Search for invisible Higgs-boson decays in events with vector-boson fusion signatures using of proton-proton data recorded by the ATLAS experiment, J. High Energy Phys. 08 (2022) 104.
- A. Tumasyan et al. (CMS Collaboration), A search for decays of the Higgs boson to invisible particles in events with a top-antitop quark pair or a vector boson in proton-proton collisions at , Eur. Phys. J. C 83, 933 (2023).
- G. Aad et al. (ATLAS Collaboration), Combination of searches for invisible decays of the Higgs boson using of proton-proton collision data at collected with the ATLAS experiment, Phys. Lett. B 842, 137963 (2023).
- L. Lee, C. Ohm, A. Soffer, and T.-T. Yu, Collider searches for long-lived particles beyond the standard model, Prog. Part. Nucl. Phys. 106, 210 (2019); 122, 103912(E) (2022).
- D. Curtin et al., Long-lived particles at the energy frontier: The MATHUSLA physics case, Rep. Prog. Phys. 82, 116201 (2019).
- M. Gon////çalves, M. Mühlleitner, R. Santos, and T. Trindade, Dark matter in multi-singlet extensions of the standard model, J. High Energy Phys. 03 (2026) 157.
- S. R. Coleman and E. J. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7, 1888 (1973).
- S. P. Martin, Taming the Goldstone contributions to the effective potential, Phys. Rev. D 90, 016013 (2014).
- J. Elias-Miro, J. R. Espinosa, and T. Konstandin, Taming infrared divergences in the effective potential, J. High Energy Phys. 08 (2014) 034.
- S. Baum, M. Carena, N. R. Shah, C. E. M. Wagner, and Y. Wang, Nucleation is more than critical: A case study of the electroweak phase transition in the NMSSM, J. High Energy Phys. 03 (2021) 055.
- A. Chatterjee, A. Datta, and S. Roy, Electroweak phase transition in the -invariant NMSSM: Implications of LHC and Dark matter searches and prospects of detecting the gravitational waves, J. High Energy Phys. 06 (2022) 108.
- P. Ghosh, T. Ghosh, and S. Roy, Interplay among gravitational waves, dark matter and collider signals in the singlet scalar extended type-II seesaw model, J. High Energy Phys. 10 (2023) 057.
- S. Roy, Dilution of dark matter relic abundance due to first order electroweak phase transition in the singlet scalar extended type-II seesaw model, Phys. Rev. D 111, 015037 (2025).
- P. Bittar, S. Roy, and C. E. M. Wagner, Self consistent thermal resummation: A case study of the phase transition in 2HDM, J. High Energy Phys. 12 (2025) 021.
- L. Dolan and R. Jackiw, Symmetry behavior at finite temperature, Phys. Rev. D 9, 3320 (1974).
- S. Weinberg, Gauge and global symmetries at high temperature, Phys. Rev. D 9, 3357 (1974).
- D. A. Kirzhnits and A. D. Linde, Symmetry behavior in gauge theories, Ann. Phys. (N.Y.) 101, 195 (1976).
- D. J. Gross, R. D. Pisarski, and L. G. Yaffe, QCD and instantons at finite temperature, Rev. Mod. Phys. 53, 43 (1981).
- R. R. Parwani, Resummation in a hot scalar field theory, Phys. Rev. D 45, 4695 (1992); 48, 5965(E) (1993).
- P. B. Arnold and O. Espinosa, The effective potential and first order phase transitions: Beyond leading-order, Phys. Rev. D 47, 3546 (1993); 50, 6662(E) (1994).
- C. G. Boyd, D. E. Brahm, and S. D. H. Hsu, Resummation methods at finite temperature: The tadpole way, Phys. Rev. D 48, 4963 (1993).
- B. Allen and J. D. Romano, Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities, Phys. Rev. D 59, 102001 (1999).
- R.-G. Cai, Z. Cao, Z.-K. Guo, S.-J. Wang, and T. Yang, The gravitational-wave physics, Natl. Sci. Rev. 4, 687 (2017).
- C. Caprini and D. G. Figueroa, Cosmological backgrounds of gravitational waves, Classical Quantum Gravity 35, 163001 (2018).
- J. D. Romano and N. J. Cornish, Detection methods for stochastic gravitational-wave backgrounds: A unified treatment, Living Rev. Relativity 20, 2 (2017).
- N. Christensen, Stochastic gravitational wave backgrounds, Rep. Prog. Phys. 82, 016903 (2019).
- J. R. Espinosa, T. Konstandin, J. M. No, and G. Servant, Energy budget of cosmological first-order phase transitions, J. Cosmol. Astropart. Phys. 06 (2010) 028.
- M. S. Turner, E. J. Weinberg, and L. M. Widrow, Bubble nucleation in first order inflation and other cosmological phase transitions, Phys. Rev. D 46, 2384 (1992).
- R. Apreda, M. Maggiore, A. Nicolis, and A. Riotto, Gravitational waves from electroweak phase transitions, Nucl. Phys. B631, 342 (2002).
- M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Numerical simulations of acoustically generated gravitational waves at a first order phase transition, Phys. Rev. D 92, 123009 (2015).
- A. Kosowsky, M. S. Turner, and R. Watkins, Gravitational waves from first order cosmological phase transitions, Phys. Rev. Lett. 69, 2026 (1992).
- A. Kosowsky, M. S. Turner, and R. Watkins, Gravitational radiation from colliding vacuum bubbles, Phys. Rev. D 45, 4514 (1992).
- A. Kosowsky and M. S. Turner, Gravitational radiation from colliding vacuum bubbles: Envelope approximation to many bubble collisions, Phys. Rev. D 47, 4372 (1993).
- C. Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: An update, J. Cosmol. Astropart. Phys. 03 (2020) 024.
- D. Bodeker and G. D. Moore, Electroweak bubble wall speed limit, J. Cosmol. Astropart. Phys. 05 (2017) 025.
- M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Gravitational waves from the sound of a first order phase transition, Phys. Rev. Lett. 112, 041301 (2014).
- J. T. Giblin, Jr. and J. B. Mertens, Vacuum bubbles in the presence of a relativistic fluid, J. High Energy Phys. 12 (2013) 042.
- J. T. Giblin and J. B. Mertens, Gravitional radiation from first-order phase transitions in the presence of a fluid, Phys. Rev. D 90, 023532 (2014).
- K. Schmitz, New sensitivity curves for gravitational-wave signals from cosmological phase transitions, J. High Energy Phys. 01 (2021) 097.
- C. Caprini and R. Durrer, Gravitational waves from stochastic relativistic sources: Primordial turbulence and magnetic fields, Phys. Rev. D 74, 063521 (2006).
- T. Kahniashvili, A. Kosowsky, G. Gogoberidze, and Y. Maravin, Detectability of gravitational waves from phase transitions, Phys. Rev. D 78, 043003 (2008).
- T. Kahniashvili, L. Campanelli, G. Gogoberidze, Y. Maravin, and B. Ratra, Gravitational radiation from primordial helical inverse cascade MHD turbulence, Phys. Rev. D 78, 123006 (2008); 79, 109901(E) (2009).
- T. Kahniashvili, L. Kisslinger, and T. Stevens, Gravitational radiation generated by magnetic fields in cosmological phase transitions, Phys. Rev. D 81, 023004 (2010).
- C. Caprini, R. Durrer, and G. Servant, The stochastic gravitational wave background from turbulence and magnetic fields generated by a first-order phase transition, J. Cosmol. Astropart. Phys. 12 (2009) 024.
- L. Kisslinger and T. Kahniashvili, Polarized gravitational waves from cosmological phase transitions, Phys. Rev. D 92, 043006 (2015).
- M. Hindmarsh and M. Hijazi, Gravitational waves from first order cosmological phase transitions in the sound shell model, J. Cosmol. Astropart. Phys. 12 (2019) 062.
- H.-K. Guo, K. Sinha, D. Vagie, and G. White, Phase transitions in an expanding universe: Stochastic gravitational waves in standard and non-standard histories, J. Cosmol. Astropart. Phys. 01 (2021) 001.
- M. B. Hindmarsh, M. Lüben, J. Lumma, and M. Pauly, Phase transitions in the early universe, SciPost Phys. Lect. Notes 24, 1 (2021).
- U.-L. Pen and N. Turok, Shocks in the early universe, Phys. Rev. Lett. 117, 131301 (2016).
- M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Shape of the acoustic gravitational wave power spectrum from a first order phase transition, Phys. Rev. D 96, 103520 (2017); 101, 089902(E) (2020).
- D. J. Weir, Gravitational waves from a first order electroweak phase transition: A brief review, Phil. Trans. R. Soc. A 376, 20170126 (2018); 381, 20230212(E) (2023).