# Hearing without seeing: gravitational waves from hot and cold hidden sectors

- 18 Downloads

## Abstract

We study the spectrum of gravitational waves produced by a first order phase transition in a hidden sector that is colder than the visible sector. In this scenario, bubbles of the hidden sector vacuum can be nucleated through either thermal fluctuations or quantum tunnelling. If a cold hidden sector undergoes a thermally induced transition, the amplitude of the gravitational wave signal produced will be suppressed and its peak frequency shifted compared to if the hidden and visible sector temperatures were equal. This could lead to signals in a frequency range that would otherwise be ruled out by constraints from big bang nucleosynthesis. Alternatively, a sufficiently cold hidden sector could fail to undergo a thermal transition and subsequently transition through the nucleation of bubbles by quantum tunnelling. In this case the bubble walls might accelerate with completely negligible friction. The resulting gravitational wave spectrum has a characteristic frequency dependence, which may allow such cold hidden sectors to be distinguished from models in which the hidden and visible sector temperatures are similar. We compare our results to the sensitivity of the future gravitational wave experimental programme.

## Keywords

Beyond Standard Model Cosmology of Theories beyond the SM## Notes

### **Open Access**

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## References

- [1]B.S. Acharya, S.A.R. Ellis, G.L. Kane, B.D. Nelson and M. Perry,
*Categorisation and Detection of Dark Matter Candidates from String/M-theory Hidden Sectors*,*JHEP***09**(2018) 130 [arXiv:1707.04530] [INSPIRE].ADSCrossRefGoogle Scholar - [2]P. Schwaller,
*Gravitational Waves from a Dark Phase Transition*,*Phys. Rev. Lett.***115**(2015) 181101 [arXiv:1504.07263] [INSPIRE].ADSCrossRefGoogle Scholar - [3]M. Fairbairn, K. Kainulainen, T. Markkanen and S. Nurmi,
*Despicable Dark Relics: generated by gravity with unconstrained masses*,*JCAP***04**(2019) 005 [arXiv:1808.08236] [INSPIRE].ADSCrossRefGoogle Scholar - [4]M. Breitbach, J. Kopp, E. Madge, T. Opferkuch and P. Schwaller,
*Dark, Cold and Noisy: Constraining Secluded Hidden Sectors with Gravitational Waves*, arXiv:1811.11175 [INSPIRE]. - [5]K. Enqvist, J. Ignatius, K. Kajantie and K. Rummukainen,
*Nucleation and bubble growth in a first order cosmological electroweak phase transition*,*Phys. Rev.***D 45**(1992) 3415 [INSPIRE]. - [6]J.R. Espinosa, T. Konstandin, J.M. No and G. Servant,
*Energy Budget of Cosmological First-order Phase Transitions*,*JCAP***06**(2010) 028 [arXiv:1004.4187] [INSPIRE].ADSCrossRefGoogle Scholar - [7]L. Leitao and A. Megevand,
*Hydrodynamics of ultra-relativistic bubble walls*,*Nucl. Phys.***B 905**(2016) 45 [arXiv:1510.07747] [INSPIRE]. - [8]G. Nardini, M. Quirós and A. Wulzer,
*A Confining Strong First-Order Electroweak Phase Transition*,*JHEP***09**(2007) 077 [arXiv:0706.3388] [INSPIRE].ADSCrossRefGoogle Scholar - [9]J.R. Espinosa and M. Quirós,
*Novel Effects in Electroweak Breaking from a Hidden Sector*,*Phys. Rev.***D 76**(2007) 076004 [hep-ph/0701145] [INSPIRE]. - [10]J.R. Espinosa, T. Konstandin, J.M. No and M. Quirós,
*Some Cosmological Implications of Hidden Sectors*,*Phys. Rev.***D 78**(2008) 123528 [arXiv:0809.3215] [INSPIRE]. - [11]A. Addazi,
*Limiting First Order Phase Transitions in Dark Gauge Sectors from Gravitational Waves experiments*,*Mod. Phys. Lett.***A 32**(2017) 1750049 [arXiv:1607.08057] [INSPIRE]. - [12]J. Jaeckel, V.V. Khoze and M. Spannowsky,
*Hearing the signal of dark sectors with gravitational wave detectors*,*Phys. Rev.***D 94**(2016) 103519 [arXiv:1602.03901] [INSPIRE]. - [13]P.S.B. Dev and A. Mazumdar,
*Probing the Scale of New Physics by Advanced LIGO/VIRGO*,*Phys. Rev.***D 93**(2016) 104001 [arXiv:1602.04203] [INSPIRE]. - [14]A. Addazi and A. Marciano,
*Gravitational waves from dark first order phase transitions and dark photons*,*Chin. Phys.***C 42**(2018) 023107 [arXiv:1703.03248] [INSPIRE]. - [15]M. Aoki, H. Goto and J. Kubo,
*Gravitational Waves from Hidden QCD Phase Transition*,*Phys. Rev.***D 96**(2017) 075045 [arXiv:1709.07572] [INSPIRE]. - [16]I. Baldes,
*Gravitational waves from the asymmetric-dark-matter generating phase transition*,*JCAP***05**(2017) 028 [arXiv:1702.02117] [INSPIRE].ADSGoogle Scholar - [17]J. Ellis, M. Lewicki and J.M. No,
*On the Maximal Strength of a First-Order Electroweak Phase Transition and its Gravitational Wave Signal*, arXiv:1809.08242 [INSPIRE]. - [18]I. Baldes and C. Garcia-Cely,
*Strong gravitational radiation from a simple dark matter model*,*JHEP***05**(2019) 190 [arXiv:1809.01198] [INSPIRE].ADSCrossRefGoogle Scholar - [19]V. Brdar, A.J. Helmboldt and J. Kubo,
*Gravitational Waves from First-Order Phase Transitions: LIGO as a Window to Unexplored Seesaw Scales*,*JCAP***02**(2019) 021 [arXiv:1810.12306] [INSPIRE].ADSCrossRefGoogle Scholar - [20]R. Jinno, S. Lee, H. Seong and M. Takimoto,
*Gravitational waves from first-order phase transitions: Towards model separation by bubble nucleation rate*,*JCAP***11**(2017) 050 [arXiv:1708.01253] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [21]D. Croon, V. Sanz and G. White,
*Model Discrimination in Gravitational Wave spectra from Dark Phase Transitions*,*JHEP***08**(2018) 203 [arXiv:1806.02332] [INSPIRE].ADSCrossRefGoogle Scholar - [22]D. Bödeker and G.D. Moore,
*Can electroweak bubble walls run away?*,*JCAP***05**(2009) 009 [arXiv:0903.4099] [INSPIRE].CrossRefGoogle Scholar - [23]D. Bödeker and G.D. Moore,
*Electroweak Bubble Wall Speed Limit*,*JCAP***05**(2017) 025 [arXiv:1703.08215] [INSPIRE].MathSciNetCrossRefGoogle Scholar - [24]G.D. Moore and T. Prokopec,
*How fast can the wall move? A Study of the electroweak phase transition dynamics*,*Phys. Rev.***D 52**(1995) 7182 [hep-ph/9506475] [INSPIRE]. - [25]G.D. Moore and T. Prokopec,
*Bubble wall velocity in a first order electroweak phase transition*,*Phys. Rev. Lett.***75**(1995) 777 [hep-ph/9503296] [INSPIRE]. - [26]G.D. Moore and K. Rummukainen,
*Electroweak bubble nucleation, nonperturbatively*,*Phys. Rev.***D 63**(2001) 045002 [hep-ph/0009132] [INSPIRE]. - [27]P. John and M.G. Schmidt,
*Do stops slow down electroweak bubble walls?*,*Nucl. Phys.***B 598**(2001) 291 [*Erratum ibid.***B 648**(2003) 449] [hep-ph/0002050] [INSPIRE]. - [28]J. Kozaczuk,
*Bubble Expansion and the Viability of Singlet-Driven Electroweak Baryogenesis*,*JHEP***10**(2015) 135 [arXiv:1506.04741] [INSPIRE].ADSCrossRefGoogle Scholar - [29]S.J. Huber and T. Konstandin,
*Gravitational Wave Production by Collisions: More Bubbles*,*JCAP***09**(2008) 022 [arXiv:0806.1828] [INSPIRE].ADSCrossRefGoogle Scholar - [30]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**(2014) 041301 [arXiv:1304.2433] [INSPIRE]. - [31]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**(2015) 123009 [arXiv:1504.03291] [INSPIRE]. - [32]D.J. Weir,
*Revisiting the envelope approximation: gravitational waves from bubble collisions*,*Phys. Rev.***D 93**(2016) 124037 [arXiv:1604.08429] [INSPIRE]. - [33]T. Konstandin,
*Gravitational radiation from a bulk flow model*,*JCAP***03**(2018) 047 [arXiv:1712.06869] [INSPIRE].ADSCrossRefGoogle Scholar - [34]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**(2017) 103520 [arXiv:1704.05871] [INSPIRE]. - [35]D. Cutting, M. Hindmarsh and D.J. Weir,
*Gravitational waves from vacuum first-order phase transitions: from the envelope to the lattice*,*Phys. Rev.***D 97**(2018) 123513 [arXiv:1802.05712] [INSPIRE]. - [36]A. Kosowsky, M.S. Turner and R. Watkins,
*Gravitational waves from first order cosmological phase transitions*,*Phys. Rev. Lett.***69**(1992) 2026 [INSPIRE].ADSCrossRefGoogle Scholar - [37]M. Kamionkowski, A. Kosowsky and M.S. Turner,
*Gravitational radiation from first order phase transitions*,*Phys. Rev.***D 49**(1994) 2837 [astro-ph/9310044] [INSPIRE]. - [38]J. Ignatius, K. Kajantie, H. Kurki-Suonio and M. Laine,
*The growth of bubbles in cosmological phase transitions*,*Phys. Rev.***D 49**(1994) 3854 [astro-ph/9309059] [INSPIRE]. - [39]A. Kosowsky, A. Mack and T. Kahniashvili,
*Gravitational radiation from cosmological turbulence*,*Phys. Rev.***D 66**(2002) 024030 [astro-ph/0111483] [INSPIRE]. - [40]M. Hindmarsh,
*Sound shell model for acoustic gravitational wave production at a first-order phase transition in the early Universe*,*Phys. Rev. Lett.***120**(2018) 071301 [arXiv:1608.04735] [INSPIRE]. - [41]R. Jinno and M. Takimoto,
*Gravitational waves from bubble collisions: An analytic derivation*,*Phys. Rev.***D 95**(2017) 024009 [arXiv:1605.01403] [INSPIRE]. - [42]R. Jinno and M. Takimoto,
*Gravitational waves from bubble dynamics: Beyond the Envelope*,*JCAP***01**(2019) 060 [arXiv:1707.03111] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [43]P. Binetruy, A. Bohe, C. Caprini and J.-F. Dufaux,
*Cosmological Backgrounds of Gravitational Waves and eLISA/NGO: Phase Transitions, Cosmic Strings and Other Sources*,*JCAP***06**(2012) 027 [arXiv:1201.0983] [INSPIRE].ADSCrossRefGoogle Scholar - [44]A. Katz and A. Riotto,
*Baryogenesis and Gravitational Waves from Runaway Bubble Collisions*,*JCAP***11**(2016) 011 [arXiv:1608.00583] [INSPIRE].ADSCrossRefGoogle Scholar - [45]T. Hambye,
*Hidden vector dark matter*,*JHEP***01**(2009) 028 [arXiv:0811.0172] [INSPIRE].ADSCrossRefGoogle Scholar - [46]T. Hambye and A. Strumia,
*Dynamical generation of the weak and Dark Matter scale*,*Phys. Rev.***D 88**(2013) 055022 [arXiv:1306.2329] [INSPIRE]. - [47]V.V. Khoze, C. McCabe and G. Ro,
*Higgs vacuum stability from the dark matter portal*,*JHEP***08**(2014) 026 [arXiv:1403.4953] [INSPIRE].ADSCrossRefGoogle Scholar - [48]A. Karam and K. Tamvakis,
*Dark matter and neutrino masses from a scale-invariant multi-Higgs portal*,*Phys. Rev.***D 92**(2015) 075010 [arXiv:1508.03031] [INSPIRE]. - [49]T. Hambye, A. Strumia and D. Teresi,
*Super-cool Dark Matter*,*JHEP***08**(2018) 188 [arXiv:1805.01473] [INSPIRE].ADSCrossRefGoogle Scholar - [50]S.R. Coleman and E.J. Weinberg,
*Radiative Corrections as the Origin of Spontaneous Symmetry Breaking*,*Phys. Rev.***D 7**(1973) 1888 [INSPIRE]. - [51]J.I. Kapusta and C. Gale,
*Finite-temperature field theory: Principles and applications*,*Cambridge Monographs on Mathematical Physics*, Cambridge University Press, Cambridge U.K. (2011).Google Scholar - [52]D. Curtin, P. Meade and H. Ramani,
*Thermal Resummation and Phase Transitions*,*Eur. Phys. J.***C 78**(2018) 787 [arXiv:1612.00466] [INSPIRE]. - [53]M.E. Carrington,
*The Effective potential at finite temperature in the Standard Model*,*Phys. Rev.***D 45**(1992) 2933 [INSPIRE]. - [54]P. Fendley,
*The Effective Potential and the Coupling Constant at High Temperature*,*Phys. Lett.***B 196**(1987) 175 [INSPIRE]. - [55]J.R. Espinosa and M. Quirós,
*Improved metastability bounds on the standard model Higgs mass*,*Phys. Lett.***B 353**(1995) 257 [hep-ph/9504241] [INSPIRE]. - [56]P. Adshead, Y. Cui and J. Shelton,
*Chilly Dark Sectors and Asymmetric Reheating*,*JHEP***06**(2016) 016 [arXiv:1604.02458] [INSPIRE].ADSCrossRefGoogle Scholar - [57]E. Hardy and J. Unwin,
*Symmetric and Asymmetric Reheating*,*JHEP***09**(2017) 113 [arXiv:1703.07642] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [58]C. Pitrou, A. Coc, J.-P. Uzan and E. Vangioni,
*Precision big bang nucleosynthesis with improved Helium-4 predictions*,*Phys. Rept.***754**(2018) 1 [arXiv:1801.08023] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [59]J.L. Feng, H. Tu and H.-B. Yu,
*Thermal Relics in Hidden Sectors*,*JCAP***10**(2008) 043 [arXiv:0808.2318] [INSPIRE].ADSCrossRefGoogle Scholar - [60]A. Fradette and M. Pospelov,
*BBN for the LHC: constraints on lifetimes of the Higgs portal scalars*,*Phys. Rev.***D 96**(2017) 075033 [arXiv:1706.01920] [INSPIRE]. - [61]
- [62]T. Flacke, C. Frugiuele, E. Fuchs, R.S. Gupta and G. Perez,
*Phenomenology of relaxion-Higgs mixing*,*JHEP***06**(2017) 050 [arXiv:1610.02025] [INSPIRE].ADSCrossRefGoogle Scholar - [63]A.D. Linde,
*Decay of the False Vacuum at Finite Temperature*,*Nucl. Phys.***B 216**(1983) 421 [*Erratum ibid.***B 223**(1983) 544] [INSPIRE]. - [64]I. Affleck,
*Quantum Statistical Metastability*,*Phys. Rev. Lett.***46**(1981) 388 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [65]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**(1992) 2384 [INSPIRE]. - [66]G.W. Anderson and L.J. Hall,
*The Electroweak phase transition and baryogenesis*,*Phys. Rev.***D 45**(1992) 2685 [INSPIRE]. - [67]D. Huang and B.-Q. Lu,
*Comment on*“*Hearing the signal of dark sectors with gravitational wave detectors*”,*Phys. Rev.***D 97**(2018) 068303 [arXiv:1803.03180] [INSPIRE]. - [68]C.L. Wainwright,
*CosmoTransitions: Computing Cosmological Phase Transition Temperatures and Bubble Profiles with Multiple Fields*,*Comput. Phys. Commun.***183**(2012) 2006 [arXiv:1109.4189] [INSPIRE].ADSCrossRefGoogle Scholar - [69]A. Strumia, N. Tetradis and C. Wetterich,
*The Region of validity of homogeneous nucleation theory*,*Phys. Lett.***B 467**(1999) 279 [hep-ph/9808263] [INSPIRE]. - [70]A. Strumia and N. Tetradis,
*Bubble nucleation rates for cosmological phase transitions*,*JHEP***11**(1999) 023 [hep-ph/9904357] [INSPIRE]. - [71]A. Strumia and N. Tetradis,
*A Consistent calculation of bubble nucleation rates*,*Nucl Phys.***B 542**(1999) 719 [hep-ph/9806453] [INSPIRE]. - [72]A. Strumia and N. Tetradis,
*Bubble nucleation rates for radiatively induced first order phase transitions*,*Nucl. Phys.***B 554**(1999) 697 [hep-ph/9811438] [INSPIRE]. - [73]B.W. Mintz, A. Bessa and E.S. Fraga,
*On the nucleation of hadronic domains in the quark-hadron transition*,*Nucl. Phys.***A 820**(2009) 291C [arXiv:0810.2798] [INSPIRE]. - [74]A. Bessa, E.S. Fraga and B.W. Mintz,
*Phase conversion in a weakly first-order quark-hadron transition*,*Phys. Rev.***D 79**(2009) 034012 [arXiv:0811.4385] [INSPIRE]. - [75]A. Megevand and S. Ramirez,
*Bubble nucleation and growth in very strong cosmological phase transitions*,*Nucl. Phys.***B 919**(2017) 74 [arXiv:1611.05853] [INSPIRE]. - [76]A.H. Guth and E.J. Weinberg,
*Could the Universe Have Recovered from a Slow First Order Phase Transition?*,*Nucl. Phys.***B 212**(1983) 321 [INSPIRE]. - [77]T. Konstandin, G. Nardini and M. Quirós,
*Gravitational Backreaction Effects on the Holographic Phase Transition*,*Phys. Rev.***D 82**(2010) 083513 [arXiv:1007.1468] [INSPIRE]. - [78]T. Konstandin and G. Servant,
*Cosmological Consequences of Nearly Conformal Dynamics at the TeV scale*,*JCAP***12**(2011) 009 [arXiv:1104.4791] [INSPIRE].ADSCrossRefGoogle Scholar - [79]C. Caprini et al.,
*Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions*,*JCAP***04**(2016) 001 [arXiv:1512.06239] [INSPIRE]. - [80]
- [81]D.J. Weir,
*Gravitational waves from a first order electroweak phase transition: a brief review*,*Phil. Trans. Roy. Soc. Lond.***A 376**(2018) 20170126 [arXiv:1705.01783] [INSPIRE]. - [82]C. Caprini, R. Durrer, T. Konstandin and G. Servant,
*General Properties of the Gravitational Wave Spectrum from Phase Transitions*,*Phys. Rev.***D 79**(2009) 083519 [arXiv:0901.1661] [INSPIRE]. - [83]C. Caprini and D.G. Figueroa,
*Cosmological Backgrounds of Gravitational Waves*,*Class. Quant. Grav.***35**(2018) 163001 [arXiv:1801.04268] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [84]D. Croon and G. White,
*Exotic Gravitational Wave Signatures from Simultaneous Phase Transitions*,*JHEP***05**(2018) 210 [arXiv:1803.05438] [INSPIRE].ADSCrossRefGoogle Scholar - [85]
- [86]C.J. Hogan,
*Gravitational radiation from cosmological phase transitions*,*Mon. Not. Roy. Astron. Soc.***218**(1986) 629.ADSCrossRefGoogle Scholar - [87]A. Kosowsky, M.S. Turner and R. Watkins,
*Gravitational radiation from colliding vacuum bubbles*,*Phys. Rev.***D 45**(1992) 4514 [INSPIRE]. - [88]A. Kosowsky and M.S. Turner,
*Gravitational radiation from colliding vacuum bubbles: envelope approximation to many bubble collisions*,*Phys. Rev.***D 47**(1993) 4372 [astro-ph/9211004] [INSPIRE]. - [89]LIGO Scientific, Virgo collaboration,
*Upper Limits on the Stochastic Gravitational-Wave Background from Advanced LIGO*’*s First Observing Run*,*Phys. Rev. Lett.***118**(2017) 121101 [*Erratum ibid.***119**(2017) 029901] [arXiv:1612.02029] [INSPIRE]. - [90]LIGO Scientific collaboration,
*Advanced LIGO*,*Class. Quant. Grav.***32**(2015) 074001 [arXiv:1411.4547] [INSPIRE]. - [91]B. Sathyaprakash et al.,
*Scientific Objectives of Einstein Telescope*,*Class. Quant. Grav.***29**(2012) 124013 [*Erratum ibid.***30**(2013) 079501] [arXiv:1206.0331] [INSPIRE]. - [92]C.L. Carilli and S. Rawlings,
*Science with the Square Kilometer Array: Motivation, key science projects, standards and assumptions*,*New Astron. Rev.***48**(2004) 979 [astro-ph/0409274] [INSPIRE]. - [93]G. Hobbs et al.,
*The international pulsar timing array project: using pulsars as a gravitational wave detector*,*Class. Quant. Grav.***27**(2010) 084013 [arXiv:0911.5206] [INSPIRE]. - [94]E. Thrane and J.D. Romano,
*Sensitivity curves for searches for gravitational-wave backgrounds*,*Phys. Rev.***D 88**(2013) 124032 [arXiv:1310.5300] [INSPIRE]. - [95]I. Garcia Garcia, S. Krippendorf and J. March-Russell,
*The String Soundscape at Gravitational Wave Detectors*,*Phys. Lett.***B 779**(2018) 348 [arXiv:1607.06813] [INSPIRE]. - [96]M.C. Bento, O. Bertolami and R. Rosenfeld,
*Cosmological constraints on an invisibly decaying Higgs boson*,*Phys. Lett.***B 518**(2001) 276 [hep-ph/0103340] [INSPIRE]. - [97]G.F. Giudice, E.W. Kolb and A. Riotto,
*Largest temperature of the radiation era and its cosmological implications*,*Phys. Rev.***D 64**(2001) 023508 [hep-ph/0005123] [INSPIRE]. - [98]H.L. Child and J.T. Giblin, Jr.,
*Gravitational Radiation from First-Order Phase Transitions*,*JCAP***10**(2012) 001 [arXiv:1207.6408] [INSPIRE].ADSCrossRefGoogle Scholar - [99]T. Kahniashvili, A. Kosowsky, G. Gogoberidze and Y. Maravin,
*Detectability of Gravitational Waves from Phase Transitions*,*Phys. Rev.***D 78**(2008) 043003 [arXiv:0806.0293] [INSPIRE]. - [100]T. Kahniashvili, L. Campanelli, G. Gogoberidze, Y. Maravin and B. Ratra,
*Gravitational Radiation from Primordial Helical Inverse Cascade MHD Turbulence*,*Phys. Rev.***D 78**(2008) 123006 [*Erratum ibid.***D 79**(2009) 109901] [arXiv:0809.1899] [INSPIRE]. - [101]T. Kalaydzhyan and E. Shuryak,
*Gravity waves generated by sounds from big bang phase transitions*,*Phys. Rev.***D 91**(2015) 083502 [arXiv:1412.5147] [INSPIRE]. - [102]C. Caprini and R. Durrer,
*Gravitational waves from stochastic relativistic sources: Primordial turbulence and magnetic fields*,*Phys. Rev.***D 74**(2006) 063521 [astro-ph/0603476] [INSPIRE]. - [103]C. Caprini, R. Durrer and G. Servant,
*The stochastic gravitational wave background from turbulence and magnetic fields generated by a first-order phase transition*,*JCAP***12**(2009) 024 [arXiv:0909.0622] [INSPIRE].ADSCrossRefGoogle Scholar - [104]T. Kahniashvili, L. Kisslinger and T. Stevens,
*Gravitational Radiation Generated by Magnetic Fields in Cosmological Phase Transitions*,*Phys. Rev.***D 81**(2010) 023004 [arXiv:0905.0643] [INSPIRE].