Abstract
We compute the production rate of the energy density carried by gravitational waves emitted by a Standard Model plasma in thermal equilibrium, consistently to leading order in coupling constants for momenta k ∼ πT. Summing up the contributions from the full history of the universe, the highest temperature of the radiation epoch can be constrained by the so-called Neff parameter. The current theoretical uncertainty ∆Neff ≤ 10−3 corresponds to Tmax ≤ 2 × 1017 GeV. In the course of the computation, we show how a subpart of the production rate can be determined with the help of standard packages, even if subsequently an IR subtraction and thermal resummation need to be implemented.
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References
S. Weinberg, Gravitation and Cosmology John Wiley & Sons, New York (1971).
J. Ghiglieri and M. Laine, Gravitational wave background from Standard Model physics: Qualitative features, JCAP 07 (2015) 022 [arXiv:1504.02569] [INSPIRE].
A.A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. 30 (1979) 682 [INSPIRE].
C. Caprini and D.G. Figueroa, Cosmological backgrounds of gravitational waves, Class. Quant. Grav. 35 (2018) 163001 [arXiv:1801.04268] [INSPIRE].
D. Bödeker, M. Sangel and M. Wörmann, Equilibration, particle production and self-energy, Phys. Rev. D 93 (2016) 045028 [arXiv:1510.06742] [INSPIRE].
G. Baym, S.P. Patil and C.J. Pethick, Damping of gravitational waves by matter, Phys. Rev. D 96 (2017) 084033 [arXiv:1707.05192] [INSPIRE].
R. Flauger and S. Weinberg, Absorption of gravitational waves from distant sources, Phys. Rev. D 99 (2019) 123030 [arXiv:1906.04853] [INSPIRE].
A. Abbas, Anomalies and charge quantization in the standard model with arbitrary number of colours, Phys. Lett. B 238 (1990) 344 [INSPIRE].
Y. Schröder, M. Vepsäläinen, A. Vuorinen and Y. Zhu, The ultraviolet limit and sum rule for the shear correlator in hot Yang-Mills theory, JHEP 12 (2011) 035 [arXiv:1109.6548] [INSPIRE].
J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].
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 (2014) 2250 [arXiv:1310.1921] [INSPIRE].
N.D. Christensen et al., A comprehensive approach to new physics simulations, Eur. Phys. J. C 71 (2011) 1541 [arXiv:0906.2474] [INSPIRE].
T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].
T. Hahn and M. Pérez-Victoria, Automated one-loop calculations in four and D dimensions, Comput. Phys. Commun. 118 (1999) 153 [hep-ph/9807565] [INSPIRE].
B.R. Holstein, Graviton physics, Am. J. Phys. 74 (2006) 1002 [gr-qc/0607045] [INSPIRE].
N.E.J. Bjerrum-Bohr, B.R. Holstein, L. Planté and P. Vanhove, Graviton-photon scattering, Phys. Rev. D 91 (2015) 064008 [arXiv:1410.4148] [INSPIRE].
D. Besak and D. Bödeker, Thermal production of ultrarelativistic right-handed neutrinos: complete leading-order results, JCAP 03 (2012) 029 [arXiv:1202.1288] [INSPIRE].
M. Laine, Thermal 2-loop master spectral function at finite momentum, JHEP 05 (2013) 083 [arXiv:1304.0202] [INSPIRE].
G. Jackson, Two-loop thermal spectral functions with general kinematics, Phys. Rev. D 100 (2019) 116019 [arXiv:1910.07552] [INSPIRE].
E. Braaten and R.D. Pisarski, Soft amplitudes in hot gauge theories: a general analysis, Nucl. Phys. B 337 (1990) 569 [INSPIRE].
J.C. Taylor and S.M.H. Wong, The effective action of hard thermal loops in QCD, Nucl. Phys. B 346 (1990) 115 [INSPIRE].
J.I. Kapusta, P. Lichard and D. Seibert, High-energy photons from quark-gluon plasma versus hot hadronic gas, Phys. Rev. D 44 (1991) 2774 [Erratum ibid. 47 (1993) 4171] [INSPIRE].
R. Baier, H. Nakkagawa, A. Niégawa and K. Redlich, Production rate of hard thermal photons and screening of quark mass singularity, Z. Phys. C 53 (1992) 433 [INSPIRE].
T. Altherr and P.V. Ruuskanen, Low-mass dileptons at high momenta in ultra-relativistic heavy-ion collisions, Nucl. Phys. B 380 (1992) 377 [INSPIRE].
P.B. Arnold, G.D. Moore and L.G. Yaffe, Photon emission from quark gluon plasma: complete leading order results, JHEP 12 (2001) 009 [hep-ph/0111107] [INSPIRE].
P. Aurenche, F. Gelis and H. Zaraket, A Simple sum rule for the thermal gluon spectral function and applications, JHEP 05 (2002) 043 [hep-ph/0204146] [INSPIRE].
S. Caron-Huot, O(g) plasma effects in jet quenching, Phys. Rev. D 79 (2009) 065039 [arXiv:0811.1603] [INSPIRE].
M. Laine, P. Schicho and Y. Schröder, A QCD Debye mass in a broad temperature range, Phys. Rev. D 101 (2020) 023532 [arXiv:1911.09123] [INSPIRE].
M. Bolz, A. Brandenburg and W. Buchmüller, Thermal production of gravitinos, Nucl. Phys. B 606 (2001) 518 [Erratum ibid. 790 (2008) 336] [hep-ph/0012052] [INSPIRE].
J. Pradler and F.D. Steffen, Thermal gravitino production and collider tests of leptogenesis, Phys. Rev. D 75 (2007) 023509 [hep-ph/0608344] [INSPIRE].
V.S. Rychkov and A. Strumia, Thermal production of gravitinos, Phys. Rev. D 75 (2007) 075011 [hep-ph/0701104] [INSPIRE].
P. Graf and F.D. Steffen, Thermal axion production in the primordial quark-gluon plasma, Phys. Rev. D 83 (2011) 075011 [arXiv:1008.4528] [INSPIRE].
A. Salvio, A. Strumia and W. Xue, Thermal axion production, JCAP 01 (2014) 011 [arXiv:1310.6982] [INSPIRE].
A. Brandenburg and F.D. Steffen, Axino dark matter from thermal production, JCAP 08 (2004) 008 [hep-ph/0405158] [INSPIRE].
E. Braaten and T.C. Yuan, Calculation of screening in a hot plasma, Phys. Rev. Lett. 66 (1991) 2183 [INSPIRE].
T.L. Smith, E. Pierpaoli and M. Kamionkowski, New Cosmic Microwave Background Constraint to Primordial Gravitational Waves, Phys. Rev. Lett. 97 (2006) 021301 [astro-ph/0603144] [INSPIRE].
S. Henrot-Versillé et al., Improved constraint on the primordial gravitational-wave density using recent cosmological data and its impact on cosmic string models, Class. Quant. Grav. 32 (2015) 045003 [arXiv:1408.5299] [INSPIRE].
B.X. Hu and A. Loeb, An Upper Limit on the Initial Temperature of the Radiation-Dominated Universe, arXiv:2004.02895 [INSPIRE].
P.F. de Salas and S. Pastor, Relic neutrino decoupling with flavour oscillations revisited, JCAP 07 (2016) 051 [arXiv:1606.06986] [INSPIRE].
J.J. Bennett, G. Buldgen, M. Drewes and Y.Y.Y. Wong, Towards a precision calculation of the effective number of neutrinos Neff in the Standard Model: the QED equation of state, JCAP 03 (2020) 003 [arXiv:1911.04504] [INSPIRE].
M. Escudero Abenza, Precision early universe thermodynamics made simple: Neff and neutrino decoupling in the Standard Model and beyond, JCAP 05 (2020) 048 [arXiv:2001.04466] [INSPIRE].
Planck collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 [INSPIRE].
K. Abazajian et al., CMB-S4 Science Case, Reference Design and Project Plan, arXiv:1907.04473 [INSPIRE].
M. Laine and M. Meyer, Standard Model thermodynamics across the electroweak crossover, JCAP 07 (2015) 035 [arXiv:1503.04935] [INSPIRE] [http://www.laine.itp.unibe.ch/eos15/].
A.M. Cruise and R.M.J. Ingley, A prototype gravitational wave detector for 100 MHz, Class. Quant. Grav. 23 (2006) 6185 [INSPIRE].
T. Akutsu et al., Search for a Stochastic Background of 100-MHz Gravitational Waves with Laser Interferometers, Phys. Rev. Lett. 101 (2008) 101101 [arXiv:0803.4094] [INSPIRE].
F. Li, J. Baker, Robert M. L., Z. Fang, G.V. Stephenson and Z. Chen, Perturbative photon fluxes generated by high-frequency gravitational waves and their physical effects, Eur. Phys. J. C 56 (2008) 407 [arXiv:0806.1989] [INSPIRE].
M.-l. Tong, Y. Zhang and F.-Y. Li, Using a polarized maser to detect high-frequency relic gravitational waves, Phys. Rev. D 78 (2008) 024041 [arXiv:0807.0885] [INSPIRE].
C. Sabín, D.E. Bruschi, M. Ahmadi and I. Fuentes, Phonon creation by gravitational waves, New J. Phys. 16 (2014) 085003 [arXiv:1402.7009] [INSPIRE].
M. Goryachev and M.E. Tobar, Gravitational wave detection with high frequency phonon trapping acoustic cavities, Phys. Rev. D 90 (2014) 102005 [arXiv:1410.2334] [INSPIRE].
D. Hartley, T. Bravo, D. Rätzel, R. Howl and I. Fuentes, Analogue simulation of gravitational waves in a 3+1 dimensional Bose-Einstein condensate, Phys. Rev. D 98 (2018) 025011 [arXiv:1712.01140] [INSPIRE].
R. Schützhold, Interaction of a Bose-Einstein condensate with a gravitational wave, Phys. Rev. D 98 (2018) 105019 [arXiv:1807.07046] [INSPIRE].
M.P.G. Robbins, N. Afshordi and R.B. Mann, Bose-Einstein condensates as gravitational wave detectors, JCAP 07 (2019) 032 [arXiv:1811.04468] [INSPIRE].
H. Weldon, Effective fermion masses of order gT in high-temperature gauge theories with exact chiral invariance, Phys. Rev. D 26 (1982) 2789 [INSPIRE].
P. Aurenche, F. Gelis and H. Zaraket, Landau-Pomeranchuk-Migdal effect in thermal field theory, Phys. Rev. D 62 (2000) 096012 [hep-ph/0003326] [INSPIRE].
P.B. Arnold, G.D. Moore and L.G. Yaffe, Photon emission from ultrarelativistic plasmas, JHEP 11 (2001) 057 [hep-ph/0109064] [INSPIRE].
A. Anisimov, D. Besak and D. Bödeker, Thermal production of relativistic Majorana neutrinos: strong enhancement by multiple soft scattering, JCAP 03 (2011) 042 [arXiv:1012.3784] [INSPIRE].
P.B. Arnold, G.D. Moore and L.G. Yaffe, Photon and gluon emission in relativistic plasmas, JHEP 06 (2002) 030 [hep-ph/0204343] [INSPIRE].
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ArXiv ePrint: 2004.11392
Previous address: Physik-Department, TU München, James-Franck-Strasse 1, 85748 Garching, Germany. (Y. Zhu)
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Ghiglieri, J., Jackson, G., Laine, M. et al. Gravitational wave background from Standard Model physics: complete leading order. J. High Energ. Phys. 2020, 92 (2020). https://doi.org/10.1007/JHEP07(2020)092
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DOI: https://doi.org/10.1007/JHEP07(2020)092