Abstract
Large excursion of the inflaton field can trigger interesting dynamics. One important example is a first-order phase transition in a spectator sector which couples to the inflaton. Gravitational waves (GWs) from such a first-order phase transition during inflation, an example of an instantaneous source, have an oscillatory feature. In this work, we show that this feature is generic for a source in an era of accelerated expansion. We also demonstrate that the shape of the GW signal contains information about the evolution of the early universe following the phase transition. In particular, the slope of the infrared part of the GW spectrum is sensitive to the evolution of the Hubble parameter when the GW modes reenter the horizon after inflation. The slope of the profile of the intermediate oscillatory part and the ultraviolet part of the GW spectrum depend on the evolution of the Hubble parameter when the modes exit horizon during the inflation and when they reenter the horizon during the reheating. The ultraviolet spectrum also depends on the details of the dynamics of the phase transition. We consider the GW signal in several models of evolution during and after inflation, and compare them with the minimal scenario of quasi- de Sitter inflation followed by radiation domination after a fast reheating, and demonstrate that the shape of the GW can be used to distinguish them. In this way, the GW signal considered in this paper offers a powerful probe to the dynamics of the early universe which is otherwise difficult to explore directly through CMB, large scale structure, big bang nucleosynthesis (BBN), and other well-studied cosmological observables.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys. Rev. D 23 (1981) 347 [INSPIRE].
A.D. Linde, A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys. Lett. B 108 (1982) 389 [INSPIRE].
A. Albrecht and P.J. Steinhardt, Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking, Phys. Rev. Lett. 48 (1982) 1220 [INSPIRE].
Planck collaboration, Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641 (2020) A10 [arXiv:1807.06211] [INSPIRE].
H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B 766 (2007) 21 [hep-th/0605264] [INSPIRE].
H. Jiang, T. Liu, S. Sun and Y. Wang, Echoes of Inflationary First-Order Phase Transitions in the CMB, Phys. Lett. B 765 (2017) 339 [arXiv:1512.07538] [INSPIRE].
K. Sugimura, D. Yamauchi and M. Sasaki, Multi-field open inflation model and multi-field dynamics in tunneling, JCAP 01 (2012) 027 [arXiv:1110.4773] [INSPIRE].
A. Ashoorioon, Exit from Inflation with a First-Order Phase Transition and a Gravitational Wave Blast, Phys. Lett. B 747 (2015) 446 [arXiv:1502.00556] [INSPIRE].
A. Ashoorioon, A. Rostami and J.T. Firouzjaee, Examining the end of inflation with primordial black holes mass distribution and gravitational waves, Phys. Rev. D 103 (2021) 123512 [arXiv:2012.02817] [INSPIRE].
A. Ashoorioon, K. Rezazadeh and A. Rostami, NANOGrav Signal from the End of Inflation and the LIGO Mass and Heavier Primordial Black Holes, arXiv:2202.01131 [INSPIRE].
Y.-T. Wang, Y. Cai and Y.-S. Piao, Phase-transition sound of inflation at gravitational waves detectors, Phys. Lett. B 789 (2019) 191 [arXiv:1801.03639] [INSPIRE].
H. An, K.-F. Lyu, L.-T. Wang and S. Zhou, A unique gravitational wave signal from phase transition during inflation, arXiv:2009.12381 [INSPIRE].
eLISA collaboration, The Gravitational Universe, arXiv:1305.5720 [INSPIRE].
LISA collaboration, Laser Interferometer Space Antenna, arXiv:1702.00786 [INSPIRE].
S. Kawamura et al., The Japanese space gravitational wave antenna: DECIGO, Class. Quant. Grav. 28 (2011) 094011 [INSPIRE].
TianQin collaboration, TianQin: a space-borne gravitational wave detector, Class. Quant. Grav. 33 (2016) 035010 [arXiv:1512.02076] [INSPIRE].
W.-H. Ruan, Z.-K. Guo, R.-G. Cai and Y.-Z. Zhang, Taiji program: Gravitational-wave sources, Int. J. Mod. Phys. A 35 (2020) 2050075 [arXiv:1807.09495] [INSPIRE].
J. Crowder and N.J. Cornish, Beyond LISA: Exploring future gravitational wave missions, Phys. Rev. D 72 (2005) 083005 [gr-qc/0506015] [INSPIRE].
G.M. Harry, P. Fritschel, D.A. Shaddock, W. Folkner and E.S. Phinney, Laser interferometry for the big bang observer, Class. Quant. Grav. 23 (2006) 4887 [Erratum ibid. 23 (2006) 7361] [INSPIRE].
V. Corbin and N.J. Cornish, Detecting the cosmic gravitational wave background with the big bang observer, Class. Quant. Grav. 23 (2006) 2435 [gr-qc/0512039] [INSPIRE].
M. Krämer and D.J. Champion, The European Pulsar Timing Array and the Large European Array for Pulsars, Class. Quant. Grav. 30 (2013) 224009 [INSPIRE].
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].
G. Janssen et al., Gravitational wave astronomy with the SKA, PoS AASKA14 (2015) 037 [arXiv:1501.00127] [INSPIRE].
LIGO Scientific collaboration, Advanced LIGO, Class. Quant. Grav. 32 (2015) 074001 [arXiv:1411.4547] [INSPIRE].
A. Abramovici et al., LIGO: The Laser interferometer gravitational wave observatory, Science 256 (1992) 325 [INSPIRE].
VIRGO collaboration, Advanced Virgo: a second-generation interferometric gravitational wave detector, Class. Quant. Grav. 32 (2015) 024001 [arXiv:1408.3978] [INSPIRE].
M. Punturo et al., The Einstein Telescope: A third-generation gravitational wave observatory, Class. Quant. Grav. 27 (2010) 194002 [INSPIRE].
D. Reitze et al., Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO, Bull. Am. Astron. Soc. 51 (2019) 035 [arXiv:1907.04833] [INSPIRE].
A. Hook, G. Marques-Tavares and D. Racco, Causal gravitational waves as a probe of free streaming particles and the expansion of the Universe, JHEP 02 (2021) 117 [arXiv:2010.03568] [INSPIRE].
J. Fumagalli, G.A. Palma, S. Renaux-Petel, S. Sypsas, L.T. Witkowski and C. Zenteno, Primordial gravitational waves from excited states, JHEP 03 (2022) 196 [arXiv:2111.14664] [INSPIRE].
V. Mandic, S. Bird and I. Cholis, Stochastic Gravitational-Wave Background due to Primordial Binary Black Hole Mergers, Phys. Rev. Lett. 117 (2016) 201102 [arXiv:1608.06699] [INSPIRE].
S. Clesse and J. García-Bellido, Detecting the gravitational wave background from primordial black hole dark matter, Phys. Dark Univ. 18 (2017) 105 [arXiv:1610.08479] [INSPIRE].
S. Wang, Y.-F. Wang, Q.-G. Huang and T.G.F. Li, Constraints on the Primordial Black Hole Abundance from the First Advanced LIGO Observation Run Using the Stochastic Gravitational-Wave Background, Phys. Rev. Lett. 120 (2018) 191102 [arXiv:1610.08725] [INSPIRE].
M. Raidal, V. Vaskonen and H. Veermäe, Gravitational Waves from Primordial Black Hole Mergers, JCAP 09 (2017) 037 [arXiv:1707.01480] [INSPIRE].
J. García-Bellido, M. Peloso and C. Unal, Gravitational Wave signatures of inflationary models from Primordial Black Hole Dark Matter, JCAP 09 (2017) 013 [arXiv:1707.02441] [INSPIRE].
H.-K. Guo, J. Shu and Y. Zhao, Using LISA-like Gravitational Wave Detectors to Search for Primordial Black Holes, Phys. Rev. D 99 (2019) 023001 [arXiv:1709.03500] [INSPIRE].
S.Y. Khlebnikov and I.I. Tkachev, Relic gravitational waves produced after preheating, Phys. Rev. D 56 (1997) 653 [hep-ph/9701423] [INSPIRE].
R. Easther, J.T. Giblin Jr. and E.A. Lim, Gravitational Wave Production At The End Of Inflation, Phys. Rev. Lett. 99 (2007) 221301 [astro-ph/0612294] [INSPIRE].
J. García-Bellido and D.G. Figueroa, A stochastic background of gravitational waves from hybrid preheating, Phys. Rev. Lett. 98 (2007) 061302 [astro-ph/0701014] [INSPIRE].
J. García-Bellido, D.G. Figueroa and A. Sastre, A Gravitational Wave Background from Reheating after Hybrid Inflation, Phys. Rev. D 77 (2008) 043517 [arXiv:0707.0839] [INSPIRE].
J.F. Dufaux, A. Bergman, G.N. Felder, L. Kofman and J.-P. Uzan, Theory and Numerics of Gravitational Waves from Preheating after Inflation, Phys. Rev. D 76 (2007) 123517 [arXiv:0707.0875] [INSPIRE].
T. Vachaspati and A. Vilenkin, Gravitational Radiation from Cosmic Strings, Phys. Rev. D 31 (1985) 3052 [INSPIRE].
R.H. Brandenberger, A. Albrecht and N. Turok, Gravitational Radiation From Cosmic Strings and the Microwave Background, Nucl. Phys. B 277 (1986) 605 [INSPIRE].
M. Hindmarsh, Gravitational radiation from kinky infinite strings, Phys. Lett. B 251 (1990) 28 [INSPIRE].
T. Damour and A. Vilenkin, Gravitational wave bursts from cusps and kinks on cosmic strings, Phys. Rev. D 64 (2001) 064008 [gr-qc/0104026] [INSPIRE].
X. Siemens and K.D. Olum, Gravitational radiation and the small-scale structure of cosmic strings, Nucl. Phys. B 611 (2001) 125 [Erratum ibid. 645 (2002) 367] [gr-qc/0104085] [INSPIRE].
M.B. Hindmarsh and T.W.B. Kibble, Cosmic strings, Rept. Prog. Phys. 58 (1995) 477 [hep-ph/9411342] [INSPIRE].
R. Durrer, P.G. Ferreira and T. Kahniashvili, Tensor microwave anisotropies from a stochastic magnetic field, Phys. Rev. D 61 (2000) 043001 [astro-ph/9911040] [INSPIRE].
C. Caprini and R. Durrer, Gravitational wave production: A Strong constraint on primordial magnetic fields, Phys. Rev. D 65 (2001) 023517 [astro-ph/0106244] [INSPIRE].
L. Pogosian, T. Vachaspati and S. Winitzki, Signatures of kinetic and magnetic helicity in the CMBR, Phys. Rev. D 65 (2002) 083502 [astro-ph/0112536] [INSPIRE].
C. Caprini, R. Durrer and T. Kahniashvili, The Cosmic microwave background and helical magnetic fields: The Tensor mode, Phys. Rev. D 69 (2004) 063006 [astro-ph/0304556] [INSPIRE].
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].
C. Caprini, R. Durrer and E. Fenu, Can the observed large scale magnetic fields be seeded by helical primordial fields?, JCAP 11 (2009) 001 [arXiv:0906.4976] [INSPIRE].
J.R. Shaw and A. Lewis, Massive Neutrinos and Magnetic Fields in the Early Universe, Phys. Rev. D 81 (2010) 043517 [arXiv:0911.2714] [INSPIRE].
S. Saga, H. Tashiro and S. Yokoyama, Limits on primordial magnetic fields from direct detection experiments of gravitational wave background, Phys. Rev. D 98 (2018) 083518 [arXiv:1807.00561] [INSPIRE].
L.P. Grishchuk, Amplification of gravitational waves in an istropic universe, Sov. Phys. JETP 40 (1975) 409 [Zh. Eksp. Teor. Fiz. 67 (1974) 825] [INSPIRE].
A.A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. 30 (1979) 682 [INSPIRE].
V.A. Rubakov, M.V. Sazhin and A.V. Veryaskin, Graviton Creation in the Inflationary Universe and the Grand Unification Scale, Phys. Lett. B 115 (1982) 189 [INSPIRE].
R. Fabbri and M.d. Pollock, The Effect of Primordially Produced Gravitons upon the Anisotropy of the Cosmological Microwave Background Radiation, Phys. Lett. B 125 (1983) 445 [INSPIRE].
L.F. Abbott and M.B. Wise, Constraints on Generalized Inflationary Cosmologies, Nucl. Phys. B 244 (1984) 541 [INSPIRE].
G. Ballesteros, J. Rey, M. Taoso and A. Urbano, Primordial black holes as dark matter and gravitational waves from single-field polynomial inflation, JCAP 07 (2020) 025 [arXiv:2001.08220] [INSPIRE].
N. Bhaumik and R.K. Jain, Primordial black holes dark matter from inflection point models of inflation and the effects of reheating, JCAP 01 (2020) 037 [arXiv:1907.04125] [INSPIRE].
N. Bhaumik and R.K. Jain, Small scale induced gravitational waves from primordial black holes, a stringent lower mass bound, and the imprints of an early matter to radiation transition, Phys. Rev. D 104 (2021) 023531 [arXiv:2009.10424] [INSPIRE].
H.V. Ragavendra, P. Saha, L. Sriramkumar and J. Silk, Primordial black holes and secondary gravitational waves from ultraslow roll and punctuated inflation, Phys. Rev. D 103 (2021) 083510 [arXiv:2008.12202] [INSPIRE].
J. Lin, Q. Gao, Y. Gong, Y. Lu, C. Zhang and F. Zhang, Primordial black holes and secondary gravitational waves from k and G inflation, Phys. Rev. D 101 (2020) 103515 [arXiv:2001.05909] [INSPIRE].
Z. Yi, Q. Gao, Y. Gong and Z.-h. Zhu, Primordial black holes and scalar-induced secondary gravitational waves from inflationary models with a noncanonical kinetic term, Phys. Rev. D 103 (2021) 063534 [arXiv:2011.10606] [INSPIRE].
F. Zhang, Y. Gong, J. Lin, Y. Lu and Z. Yi, Primordial non-Gaussianity from G-inflation, JCAP 04 (2021) 045 [arXiv:2012.06960] [INSPIRE].
J.L. Cook and L. Sorbo, Particle production during inflation and gravitational waves detectable by ground-based interferometers, Phys. Rev. D 85 (2012) 023534 [Erratum ibid. 86 (2012) 069901] [arXiv:1109.0022] [INSPIRE].
N. Barnaby, E. Pajer and M. Peloso, Gauge Field Production in Axion Inflation: Consequences for Monodromy, non-Gaussianity in the CMB, and Gravitational Waves at Interferometers, Phys. Rev. D 85 (2012) 023525 [arXiv:1110.3327] [INSPIRE].
R. Namba, M. Peloso, M. Shiraishi, L. Sorbo and C. Unal, Scale-dependent gravitational waves from a rolling axion, JCAP 01 (2016) 041 [arXiv:1509.07521] [INSPIRE].
J. García-Bellido, M. Peloso and C. Unal, Gravitational waves at interferometer scales and primordial black holes in axion inflation, JCAP 12 (2016) 031 [arXiv:1610.03763] [INSPIRE].
O. Özsoy, Synthetic Gravitational Waves from a Rolling Axion Monodromy, JCAP 04 (2021) 040 [arXiv:2005.10280] [INSPIRE].
O. Özsoy and Z. Lalak, Primordial black holes as dark matter and gravitational waves from bumpy axion inflation, JCAP 01 (2021) 040 [arXiv:2008.07549] [INSPIRE].
L.T. Witkowski, G. Domènech, J. Fumagalli and S. Renaux-Petel, Expansion history-dependent oscillations in the scalar-induced gravitational wave background, JCAP 05 (2022) 028 [arXiv:2110.09480] [INSPIRE].
J. Fumagalli, M. Pieroni, S. Renaux-Petel and L.T. Witkowski, Detecting primordial features with LISA, arXiv:2112.06903 [INSPIRE].
J. Fumagalli, S. Renaux-Petel and L.T. Witkowski, Oscillations in the stochastic gravitational wave background from sharp features and particle production during inflation, JCAP 08 (2021) 030 [arXiv:2012.02761] [INSPIRE].
M. Braglia, X. Chen and D.K. Hazra, Probing Primordial Features with the Stochastic Gravitational Wave Background, JCAP 03 (2021) 005 [arXiv:2012.05821] [INSPIRE].
I. Dalianis and C. Kouvaris, Gravitational waves from density perturbations in an early matter domination era, JCAP 07 (2021) 046 [arXiv:2012.09255] [INSPIRE].
R.-G. Cai, S. Pi and M. Sasaki, Universal infrared scaling of gravitational wave background spectra, Phys. Rev. D 102 (2020) 083528 [arXiv:1909.13728] [INSPIRE].
S.J. Huber and T. Konstandin, Gravitational Wave Production by Collisions: More Bubbles, JCAP 09 (2008) 022 [arXiv:0806.1828] [INSPIRE].
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].
O. Gould, J. Kozaczuk, L. Niemi, M.J. Ramsey-Musolf, T.V.I. Tenkanen and D.J. Weir, Nonperturbative analysis of the gravitational waves from a first-order electroweak phase transition, Phys. Rev. D 100 (2019) 115024 [arXiv:1903.11604] [INSPIRE].
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].
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].
C. Caprini, R. Durrer and G. Servant, Gravitational wave generation from bubble collisions in first-order phase transitions: An analytic approach, Phys. Rev. D 77 (2008) 124015 [arXiv:0711.2593] [INSPIRE].
S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley and Sons, New York, NY, U.S.A. (1972).
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].
A. Kosowsky, M.S. Turner and R. Watkins, Gravitational waves from first order cosmological phase transitions, Phys. Rev. Lett. 69 (1992) 2026 [INSPIRE].
D.J. Weir, Revisiting the envelope approximation: gravitational waves from bubble collisions, Phys. Rev. D 93 (2016) 124037 [arXiv:1604.08429] [INSPIRE].
D. Cutting, E.G. Escartin, M. Hindmarsh and D.J. Weir, Gravitational waves from vacuum first order phase transitions II: from thin to thick walls, Phys. Rev. D 103 (2021) 023531 [arXiv:2005.13537] [INSPIRE].
H. Zhong, B. Gong and T. Qiu, Gravitational waves from bubble collisions in FLRW spacetime, arXiv:2107.01845 [INSPIRE].
T. Konstandin, Gravitational radiation from a bulk flow model, JCAP 03 (2018) 047 [arXiv:1712.06869] [INSPIRE].
J. Ellis, M. Lewicki and V. Vaskonen, Updated predictions for gravitational waves produced in a strongly supercooled phase transition, JCAP 11 (2020) 020 [arXiv:2007.15586] [INSPIRE].
M. Lewicki and V. Vaskonen, Gravitational wave spectra from strongly supercooled phase transitions, Eur. Phys. J. C 80 (2020) 1003 [arXiv:2007.04967] [INSPIRE].
M. Lewicki and V. Vaskonen, Gravitational waves from colliding vacuum bubbles in gauge theories, Eur. Phys. J. C 81 (2021) 437 [Erratum ibid. 81 (2021) 1077] [arXiv:2012.07826] [INSPIRE].
F. Lucchin and S. Matarrese, Power Law Inflation, Phys. Rev. D 32 (1985) 1316 [INSPIRE].
T.W.B. Kibble, Topology of Cosmic Domains and Strings, J. Phys. A 9 (1976) 1387 [INSPIRE].
A. Vilenkin, String Dominated Universe, Phys. Rev. Lett. 53 (1984) 1016 [INSPIRE].
A. Vilenkin, Gravitational Field of Vacuum Domain Walls and Strings, Phys. Rev. D 23 (1981) 852 [INSPIRE].
J. Preskill, S.P. Trivedi, F. Wilczek and M.B. Wise, Cosmology and broken discrete symmetry, Nucl. Phys. B 363 (1991) 207 [INSPIRE].
M. Gleiser and R. Roberts, Gravitational waves from collapsing vacuum domains, Phys. Rev. Lett. 81 (1998) 5497 [astro-ph/9807260] [INSPIRE].
B. Spokoiny, Deflationary universe scenario, Phys. Lett. B 315 (1993) 40 [gr-qc/9306008] [INSPIRE].
P.J.E. Peebles and A. Vilenkin, Quintessential inflation, Phys. Rev. D 59 (1999) 063505 [astro-ph/9810509] [INSPIRE].
G. Harry, The Big Bang Observer, https://dcc.ligo.org/public/0002/G0900426/001/G0900426-v1.pdf (2009).
C.J. Moore, R.H. Cole and C.P.L. Berry, Gravitational-wave sensitivity curves, Class. Quant. Grav. 32 (2015) 015014 [arXiv:1408.0740] [INSPIRE].
Y. Gouttenoire, G. Servant and P. Simakachorn, Kination cosmology from scalar fields and gravitational-wave signatures, arXiv:2111.01150 [INSPIRE].
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].
S. Pi, M. Sasaki and Y.-l. Zhang, Primordial Tensor Perturbation in Double Inflationary Scenario with a Break, JCAP 06 (2019) 049 [arXiv:1904.06304] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2201.05171
Rights and permissions
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.
About this article
Cite this article
An, H., Lyu, KF., Wang, LT. et al. Gravitational waves from an inflation triggered first-order phase transition. J. High Energ. Phys. 2022, 50 (2022). https://doi.org/10.1007/JHEP06(2022)050
Received:
Revised:
Accepted:
Published:
Version of record:
DOI: https://doi.org/10.1007/JHEP06(2022)050