Abstract
A precise modelling of the dynamics of bubbles nucleated during first-order phase transitions in the early Universe is pivotal for a quantitative determination of various cosmic relics, including the stochastic background of gravitational waves. The equation of motion of the bubble front is affected by the out-of-equilibrium distributions of particle species in the plasma which, in turn, are described by the corresponding Boltzmann equations. In this work we provide a solution to these equations by thoroughly incorporating the non-linearities arising from the population factors. Moreover, our methodology relies on a spectral decomposition that leverages the rotational properties of the collision integral within the Boltzmann equations. This novel approach allows for an efficient and robust computation of both the bubble speed and profile. We also refine our analysis by including the contributions from the electroweak gauge bosons. We find that their impact is dominated by the infrared modes and proves to be non-negligible, contrary to the naive expectations.
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References
B.-H. Liu, L.D. McLerran and N. Turok, Bubble nucleation and growth at a baryon number producing electroweak phase transition, Phys. Rev. D 46 (1992) 2668 [INSPIRE].
S.Y. Khlebnikov, Fluctuation-dissipation formula for bubble wall velocity, Phys. Rev. D 46 (1992) R3223 [INSPIRE].
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].
P.B. Arnold, One loop fluctuation-dissipation formula for bubble wall velocity, Phys. Rev. D 48 (1993) 1539 [hep-ph/9302258] [INSPIRE].
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].
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].
P. John and M.G. Schmidt, Do stops slow down electroweak bubble walls?, Nucl. Phys. B 598 (2001) 291 [hep-ph/0002050] [INSPIRE].
G.D. Moore, Electroweak bubble wall friction: Analytic results, JHEP 03 (2000) 006 [hep-ph/0001274] [INSPIRE].
J.M. Cline, M. Joyce and K. Kainulainen, Supersymmetric electroweak baryogenesis, JHEP 07 (2000) 018 [hep-ph/0006119] [INSPIRE].
A. Mégevand and A.D. Sanchez, Velocity of electroweak bubble walls, Nucl. Phys. B 825 (2010) 151 [arXiv:0908.3663] [INSPIRE].
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].
L. Leitao and A. Mégevand, Spherical and non-spherical bubbles in cosmological phase transitions, Nucl. Phys. B 844 (2011) 450 [arXiv:1010.2134] [INSPIRE].
A. Mégevand, Friction forces on phase transition fronts, JCAP 07 (2013) 045 [arXiv:1303.4233] [INSPIRE].
S.J. Huber and M. Sopena, An efficient approach to electroweak bubble velocities, arXiv:1302.1044 [INSPIRE].
A. Mégevand and F.A. Membiela, Stability of cosmological deflagration fronts, Phys. Rev. D 89 (2014) 103507 [arXiv:1311.2453] [INSPIRE].
L. Leitao and A. Mégevand, Hydrodynamics of phase transition fronts and the speed of sound in the plasma, Nucl. Phys. B 891 (2015) 159 [arXiv:1410.3875] [INSPIRE].
A. Mégevand and F.A. Membiela, Stability of cosmological detonation fronts, Phys. Rev. D 89 (2014) 103503 [arXiv:1402.5791] [INSPIRE].
A. Mégevand, F.A. Membiela and A.D. Sanchez, Lower bound on the electroweak wall velocity from hydrodynamic instability, JCAP 03 (2015) 051 [arXiv:1412.8064] [INSPIRE].
T. Konstandin, G. Nardini and I. Rues, From Boltzmann equations to steady wall velocities, JCAP 09 (2014) 028 [arXiv:1407.3132] [INSPIRE].
J. Kozaczuk, Bubble Expansion and the Viability of Singlet-Driven Electroweak Baryogenesis, JHEP 10 (2015) 135 [arXiv:1506.04741] [INSPIRE].
D. Bodeker and G.D. Moore, Electroweak Bubble Wall Speed Limit, JCAP 05 (2017) 025 [arXiv:1703.08215] [INSPIRE].
J.M. Cline and K. Kainulainen, Electroweak baryogenesis at high bubble wall velocities, Phys. Rev. D 101 (2020) 063525 [arXiv:2001.00568] [INSPIRE].
B. Laurent and J.M. Cline, Fluid equations for fast-moving electroweak bubble walls, Phys. Rev. D 102 (2020) 063516 [arXiv:2007.10935] [INSPIRE].
M. Barroso Mancha, T. Prokopec and B. Swiezewska, Field-theoretic derivation of bubble-wall force, JHEP 01 (2021) 070 [arXiv:2005.10875] [INSPIRE].
S. Höche et al., Towards an all-orders calculation of the electroweak bubble wall velocity, JCAP 03 (2021) 009 [arXiv:2007.10343] [INSPIRE].
A. Azatov and M. Vanvlasselaer, Bubble wall velocity: heavy physics effects, JCAP 01 (2021) 058 [arXiv:2010.02590] [INSPIRE].
S. Balaji, M. Spannowsky and C. Tamarit, Cosmological bubble friction in local equilibrium, JCAP 03 (2021) 051 [arXiv:2010.08013] [INSPIRE].
R.-G. Cai and S.-J. Wang, Effective picture of bubble expansion, JCAP 03 (2021) 096 [arXiv:2011.11451] [INSPIRE].
X. Wang, F.P. Huang and X. Zhang, Bubble wall velocity beyond leading-log approximation in electroweak phase transition, arXiv:2011.12903 [INSPIRE].
A. Friedlander, I. Banta, J.M. Cline and D. Tucker-Smith, Wall speed and shape in singlet-assisted strong electroweak phase transitions, Phys. Rev. D 103 (2021) 055020 [arXiv:2009.14295] [INSPIRE].
J.M. Cline et al., Baryogenesis and gravity waves from a UV-completed electroweak phase transition, Phys. Rev. D 103 (2021) 123529 [arXiv:2102.12490] [INSPIRE].
J.M. Cline and B. Laurent, Electroweak baryogenesis from light fermion sources: A critical study, Phys. Rev. D 104 (2021) 083507 [arXiv:2108.04249] [INSPIRE].
F. Bigazzi, A. Caddeo, T. Canneti and A.L. Cotrone, Bubble wall velocity at strong coupling, JHEP 08 (2021) 090 [arXiv:2104.12817] [INSPIRE].
W.-Y. Ai, B. Garbrecht and C. Tamarit, Bubble wall velocities in local equilibrium, JCAP 03 (2022) 015 [arXiv:2109.13710] [INSPIRE].
M. Lewicki, M. Merchand and M. Zych, Electroweak bubble wall expansion: gravitational waves and baryogenesis in Standard Model-like thermal plasma, JHEP 02 (2022) 017 [arXiv:2111.02393] [INSPIRE].
Y. Gouttenoire, R. Jinno and F. Sala, Friction pressure on relativistic bubble walls, JHEP 05 (2022) 004 [arXiv:2112.07686] [INSPIRE].
G.C. Dorsch, S.J. Huber and T. Konstandin, On the wall velocity dependence of electroweak baryogenesis, JCAP 08 (2021) 020 [arXiv:2106.06547] [INSPIRE].
G.C. Dorsch, S.J. Huber and T. Konstandin, A sonic boom in bubble wall friction, JCAP 04 (2022) 010 [arXiv:2112.12548] [INSPIRE].
S. De Curtis et al., Bubble wall dynamics at the electroweak phase transition, JHEP 03 (2022) 163 [arXiv:2201.08220] [INSPIRE].
S. De Curtis et al., Collision integrals for cosmological phase transitions, JHEP 05 (2023) 194 [arXiv:2303.05846] [INSPIRE].
S. De Curtis et al., Bubble wall dynamics at the electroweak scale, PoS ICHEP2022 (2022) 080 [INSPIRE].
S. De Curtis et al., Dynamics of bubble walls at the electroweak phase transition, EPJ Web Conf. 270 (2022) 00035 [arXiv:2209.06509] [INSPIRE].
W.-Y. Ai, B. Laurent and J. van de Vis, Model-independent bubble wall velocities in local thermal equilibrium, JCAP 07 (2023) 002 [arXiv:2303.10171] [INSPIRE].
P. Athron et al., Cosmological phase transitions: From perturbative particle physics to gravitational waves, Prog. Part. Nucl. Phys. 135 (2024) 104094 [arXiv:2305.02357] [INSPIRE].
I. Baldes, M. Dichtl, Y. Gouttenoire and F. Sala, Bubbletrons, arXiv:2306.15555 [INSPIRE].
A. Azatov, G. Barni, R. Petrossian-Byrne and M. Vanvlasselaer, Quantisation Across Bubble Walls and Friction, arXiv:2310.06972 [INSPIRE].
G.C. Dorsch and D.A. Pinto, Bubble wall velocities with an extended fluid Ansatz, arXiv:2312.02354 [INSPIRE].
W.-Y. Ai, Logarithmically divergent friction on ultrarelativistic bubble walls, JCAP 10 (2023) 052 [arXiv:2308.10679] [INSPIRE].
W.-Y. Ai, X. Nagels and M. Vanvlasselaer, Criterion for ultra-fast bubble walls: the impact of hydrodynamic obstruction, JCAP 03 (2024) 037 [arXiv:2401.05911] [INSPIRE].
S. De Curtis, L. Delle Rose and G. Panico, Composite Dynamics in the Early Universe, JHEP 12 (2019) 149 [arXiv:1909.07894] [INSPIRE].
P.B. Arnold, G.D. Moore and L.G. Yaffe, Transport coefficients in high temperature gauge theories. 2. Beyond leading log, JHEP 05 (2003) 051 [hep-ph/0302165] [INSPIRE].
P.B. Arnold, G.D. Moore and L.G. Yaffe, Effective kinetic theory for high temperature gauge theories, JHEP 01 (2003) 030 [hep-ph/0209353] [INSPIRE].
P.B. Arnold, G.D. Moore and L.G. Yaffe, Transport coefficients in high temperature gauge theories. 1. Leading log results, JHEP 11 (2000) 001 [hep-ph/0010177] [INSPIRE].
U.W. Heinz and S.M.H. Wong, Elliptic flow from a transversally thermalized fireball, Phys. Rev. C 66 (2002) 014907 [hep-ph/0205058] [INSPIRE].
P.F. Kolb et al., Centrality dependence of multiplicity, transverse energy, and elliptic flow from hydrodynamics, Nucl. Phys. A 696 (2001) 197 [hep-ph/0103234] [INSPIRE].
P.F. Kolb, J. Sollfrank and U.W. Heinz, Anisotropic transverse flow and the quark hadron phase transition, Phys. Rev. C 62 (2000) 054909 [hep-ph/0006129] [INSPIRE].
D. Teaney and E.V. Shuryak, An Unusual space-time evolution for heavy ion collisions at high-energies due to the QCD phase transition, Phys. Rev. Lett. 83 (1999) 4951 [nucl-th/9904006] [INSPIRE].
D.H. Rischke, S. Bernard and J.A. Maruhn, Relativistic hydrodynamics for heavy ion collisions. 1. General aspects and expansion into vacuum, Nucl. Phys. A 595 (1995) 346 [nucl-th/9504018] [INSPIRE].
D.H. Rischke, Y. Pursun and J.A. Maruhn, Relativistic hydrodynamics for heavy ion collisions. II. Compression of nuclear matter and the phase transition to the quark-gluon plasma, Nucl. Phys. A 595 (1995) 383 [nucl-th/9504021] [INSPIRE].
S. Bernard, J.A. Maruhn, W. Greiner and D.H. Rischke, Relativistic hydrodynamics for heavy ion collisions: Freezeout and particle spectra, Nucl. Phys. A 605 (1996) 566 [nucl-th/9602011] [INSPIRE].
P.L. Bhatnagar, E.P. Gross and M. Krook, A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems, Phys. Rev. 94 (1954) 511 [INSPIRE].
G.W. Anderson and L.J. Hall, The electroweak phase transition and baryogenesis, Phys. Rev. D 45 (1992) 2685 [INSPIRE].
B. Laurent and J.M. Cline, First principles determination of bubble wall velocity, Phys. Rev. D 106 (2022) 023501 [arXiv:2204.13120] [INSPIRE].
M.A. York and G.D. Moore, Second order hydrodynamic coefficients from kinetic theory, Phys. Rev. D 79 (2009) 054011 [arXiv:0811.0729] [INSPIRE].
A. Rebhan, P. Romatschke and M. Strickland, Hard-loop dynamics of non-Abelian plasma instabilities, Phys. Rev. Lett. 94 (2005) 102303 [hep-ph/0412016] [INSPIRE].
A. Guiggiani, Bubble dynamics at the electroweak scale, [arXiv:2401.18043] [INSPIRE].
S. Caron-Huot and G.D. Moore, Heavy quark diffusion in perturbative QCD at next-to-leading order, Phys. Rev. Lett. 100 (2008) 052301 [arXiv:0708.4232] [INSPIRE].
P.B. Arnold, D. Son and L.G. Yaffe, The hot baryon violation rate is O(\( {\alpha}_w^5 \)T4), Phys. Rev. D 55 (1997) 6264 [hep-ph/9609481] [INSPIRE].
P. Huet and D.T. Son, Long range physics in a hot nonAbelian plasma, Phys. Lett. B 393 (1997) 94 [hep-ph/9610259] [INSPIRE].
D.T. Son, Effective nonperturbative real time dynamics of soft modes in hot gauge theories, hep-ph/9707351 [INSPIRE].
D. Bodeker, On the effective dynamics of soft nonAbelian gauge fields at finite temperature, Phys. Lett. B 426 (1998) 351 [hep-ph/9801430] [INSPIRE].
D. Bodeker, From hard thermal loops to Langevin dynamics, Nucl. Phys. B 559 (1999) 502 [hep-ph/9905239] [INSPIRE].
D. Bodeker, Diagrammatic approach to soft nonAbelian dynamics at high temperature, Nucl. Phys. B 566 (2000) 402 [hep-ph/9903478] [INSPIRE].
P.B. Arnold, D.T. Son and L.G. Yaffe, Longitudinal subtleties in diffusive Langevin equations for nonAbelian plasmas, Phys. Rev. D 60 (1999) 025007 [hep-ph/9901304] [INSPIRE].
P.B. Arnold and L.G. Yaffe, Nonperturbative dynamics of hot nonAbelian gauge fields: Beyond leading log approximation, Phys. Rev. D 62 (2000) 125013 [hep-ph/9912305] [INSPIRE].
Acknowledgments
The work of L.D.R. has been partly funded by the European Union — Next Generation EU through the research grant number P2022Z4P4B “SOPHYA - Sustainable Optimised PHYsics Algorithms: fundamental physics to build an advanced society” under the program PRIN 2022 PNRR of the Italian Ministero dell’Università e Ricerca (MUR) and partially supported by ICSC — Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing. Á.G.M. has been supported by the Secretariat for Universities and Research of the Ministry of Business and Knowledge of the Government of Catalonia and the European Social Fund. Á.G.M. acknowledges the support from the Departament de Recerca i Universitats from Generalitat de Catalunya to the Grup de Recerca ‘Grup de Física Teòrica UAB/IFAE’ (Codi: 2021 SGR 00649). IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.
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De Curtis, S., Rose, L.D., Guiggiani, A. et al. Non-linearities in cosmological bubble wall dynamics. J. High Energ. Phys. 2024, 9 (2024). https://doi.org/10.1007/JHEP05(2024)009
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DOI: https://doi.org/10.1007/JHEP05(2024)009