A unique gravitational wave signal from phase transition during inflation
AA unique gravitational wave signal from phase transition during inflation
Haipeng An,
1, 2
Kun-Feng Lyu,
3, 4
Lian-Tao Wang,
5, 6 and Siyi Zhou Department of Physics, Tsinghua University, Beijing 100084, China Center for High Energy Physics, Tsinghua University, Beijing 100084, China Department of Physics, the Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong S.A.R., P.R.C. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Department of Physics and Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA The Oskar Klein Centre for Cosmoparticle Physics & Department of Physics,Stockholm University, AlbaNova, 106 91 Stockholm, Sweden
We study the properties of the gravitational wave (GW) signals produced by first order phasetransitions during the inflation era. We show that the power spectrum of the GW oscillates withits wave number. This signal can be observed directly by future terrestrial and spatial gravitationalwave detectors and through the B-mode spectrum in CMB. This oscillatory feature of GW is genericfor any approximately instantaneous sources occurring during inflation, and is distinct from the GWfrom phase transitions after the inflation. The details of the GW spectrum contain information aboutthe scale of the phase transition and the later evolution of the universe.
Introduction
Gravitational waves (GWs), once pro-duced, propagate freely through the universe and canbring us the information of their origin and the historyof the universe. They can be detected in many proposed,either terrestrial or space based, detectors [1–16]. Pri-mordial GWs can also leave hints on the cosmologicalmicrowave background (CMB) and can be detected inthe B-mode power spectrum [17–19]. Possible sourcesof the primordial GWs are inflation [20–24], first orderphase transitions [25, 26], and cosmic strings [27–32].It is highly plausible that there was an inflationary erain early universe [33–35] (See Ref. [36]). The simplestinflation model is driven by a slow rolling inflaton. Toproduce enough inflation, the typical excursion of the in-flaton field must be large. As such, it may induce signifi-cant changes in the dynamics of the spectator fields. Thismay happen through a direct coupling between the infla-ton field to other spectator fields (see, e.g., Ref. [37]).The change in temperature during inflation is anotherpossibility (see Refs. [38, 39] as examples). Such changescan trigger dramatic events during inflation, such as afirst order phase transition [40, 41].In this letter, we show that the GWs produced by bub-ble collisions in first order phase transition during infla-tion can provide a unique oscillatory signal in its powerspectrum, which contains information of both inflationand the phase transition. It should be clear from the dis-cussion below that the signal is generic for approximatelyinstantaneous GW sources.
GWs from local and instantaneous sources.
Theequation of motion for the transverse and traceless GWperturbation h ij is h (cid:48)(cid:48) ij + 2 a (cid:48) a h (cid:48) ij − ∇ h ij = 16 πG N a σ ij , (1)where (cid:48) indicates derivatives with respect to the confor- mal time τ , G N is the Newton’s gravity constant, and σ ij is the transverse, traceless part of the energy mo-mentum tensor. There are several important time scalesin the problem: τ (cid:63) is the time of bubble collision andgeneration of the GW. The inflation ends at τ end . Wedenote the conformal time duration and the co-movingspatial spread of the bubble collision event to be ∆ τ,x .∆ τ,x (cid:28) | τ (cid:63) | by assumption for instantaneous and localsources that happened during inflation. The modes ofinterest to us are all outside the horizon at the time wheninflation ends, k | τ end | (cid:28)
1, where k is the co-moving mo-mentum. Spectral shape of the GW signal.
We focus onthree regimes with qualitatively different features. • | τ (cid:63) | − < k < ∆ − τ,x . In this regime, we can ignore thespatial inhomogeneity caused by the bubbles and treatthe bubble collisions as instantaneous sources. There-fore the bubble collisions can be approximated as deltafunction sources,˜ σ ij ≈ ˜ T (0) ij a − ( τ (cid:63) ) δ ( τ − τ (cid:63) ) , (2)where ˜ T (0) ij is a constant depending on the strength of thesource.During inflation and after the bubble collision, | τ (cid:63) | (cid:29)| τ | > | τ end | , we have (after Fourier transformation, andsuppressed i, j indices )˜ h k ( τ ) ≈ − πG N H inf ˜ T (0) τk (cid:20)(cid:18) kτ − kτ (cid:63) (cid:19) cos k ( τ − τ (cid:63) )+ (cid:18) k ( τ τ (cid:63) ) (cid:19) sin k ( τ − τ (cid:63) ) (cid:21) , (3)where H inf is the Hubble expansion rate. In posi-tion space, when k τ τ (cid:63) (cid:28) h ( τ, x ) is approximately a r X i v : . [ a s t r o - ph . C O ] S e p FIG. 1: Left: configuration of bound state of GW in theinflation era. Right: Fourier modes of GW during inflationfrom τ (cid:63) to τ end . - - k | τ ★ | d ρ G W / d l og k ( a r b it r a r yun it ) FIG. 2: Illustration of the shape of the GW spectrum pro-duced by instantaneous source during inflation. G N H inf ˜ T (0) | τ (cid:63) | − Θ( τ − τ (cid:63) − | x − x (cid:63) | ), which is a uni-form over density ball with radius | τ (cid:63) | , as shown in theleft panel of Fig. 1. At the end of inflation, the universeis filled with such GW balls. In this regime, we can ig-nore the terms suppressed by ( kτ (cid:63) ) − in Eq. (3). Afterthe production of the GW at τ ≈ τ (cid:63) , it will continue tooscillate until it exits the horizon at k | τ | ≈ kτ end →
0, its value is frozento ˜ h fk = − πG N H inf ˜ T (0) k (cid:18) cos kτ (cid:63) − sin kτ (cid:63) kτ (cid:63) (cid:19) . (4)Since k | τ (cid:63) | >
1, we can neglect the term proportional tosin kτ (cid:63) /kτ (cid:63) . Hence, ˜ h fk ∼ cos kτ (cid:63) , as shown in the rightpanel of Fig. 1.The GW starts to oscillate again after re-entering thehorizon, with ˜ h fk as the initial condition. For example, ifwe assume the universe evolves into the radiation domi-nation (RD) immediately after inflation,˜ h k ( τ ) = ˜ h fk × sin kτkτ . (5)Hence, ˜ h k ( τ ) ∝ cos( kτ (cid:63) ). The energy density of the GWhas the form ρ GW ∼ a ( τ ) (cid:90) d k (2 π ) | ˜ h (cid:48) k ( τ ) | . (6)As a result, deeply inside the horizon ( kτ (cid:29) dρ GW d log k ∼ k (cid:18) cos kτ (cid:63) − sin kτ (cid:63) kτ (cid:63) (cid:19) ≈ k cos kτ (cid:63) . (7) We see that the GW has a distinct oscillatory featurein the frequency space, with a period of π/τ (cid:63) . This fea-ture stems from the instantaneous nature of the GW pro-duction, which sets up a GW spectrum proportional tocos k | τ (cid:63) | at the end of the inflation. The GW energydensity also has an overall factor of k − [41], since themodes with longer wavelength redshift less before exit-ing the horizon. An illustration of the GW spectrum isshown in Fig. 2. • k < τ − (cid:63) . In this regime, we can ignore the detailsof the GW source, and treat it as a delta function inspace-time. Hence, Eq. (3) still applies. In the limit k | τ end | (cid:28) k | τ (cid:63) | (cid:28)
1, from Eq. (4), h kf is independent of k at leading order. From Eq. (7) we have dρ GW /d log k ∝ k , also shown in Fig. 2, which is similar to the case of apoint-like source in Minkowski space. • k (cid:38) ∆ − τ,x . In this regime, we have k | τ (cid:63) | (cid:29)
1. Thedetails of the bubble collision become essential, and wewill need numerical simulations to obtain the shape ofthe signal. At such small scales, the curvature of thespace-time is not important when the GW is produced.However, the inflation effect distorts the GW spectrum.As a result, the energy density behaves as dρ GW d log k ∼ k − dρ flatGW d log k p , (8)where k p = k/a , is the physical momentum. dρ flatGW /d log k p is the GW spectrum produced from thesame source in the Minkowski space-time. The distor-tion factor k − stems from the k factor in the denomi-nator of Eq. (4). dρ flatGW /d log k p usually decreases as k − rp ,with r = 1 for bubble collisions [42]. Therefore, forGW produced by approximately instantaneous sourcesduring inflation, the UV part of the spectrum decreasesas k − . At the same time, due to the uncertaintiesof both the finite size and the duration, the oscillatorypattern would be smeared out. This finite size effectshould also blunt the oscillation pattern in the regime | τ (cid:63) | − < k < ∆ − τ,x . Detailed simulation is required todetermine how the spectrum is smeared. We mimic thiseffect by replacing the factor (sin kτ (cid:63) /kτ (cid:63) − cos kτ (cid:63) ) inEq. (7) with (2∆) − (cid:82) τ (cid:63) +∆ τ (cid:63) − ∆ dτ (cid:48) (cid:63) (sin kτ (cid:48) (cid:63) /kτ (cid:48) (cid:63) − cos kτ (cid:48) (cid:63) ) ,where ∆ ∼ ∆ τ,x embodies the size of the smearing effect.Combining the above analysis, the general form of theGW spectrum when it is back into the horizon in RD canbe written as dρ GW d log k = a ( τ end ) a ( τ ) S ( k p ) dρ flatGW d log k p , (9)where S ( k p ) is S ( k p ) = H k p (cid:26)
12 + cos(2 k p /H inf ) sin(2 k p ∆ p )4 k p ∆ p + 14 k p ∆ p (cid:18) − cos(2 k p /H inf − k p ∆ p ) k p /H inf − k p ∆ p − − cos(2 k p /H inf +2 k p ∆ p ) k p /H inf + k p ∆ p (cid:19)(cid:27) . (10)∆ p = a − ( τ (cid:63) )∆ is the physical size of the source. Detectability of the GW signal.
To get the strengthand the frequency of the GW today, we need to studya specific model. We assume that the backreaction fromthe spectator sector that underwent the phase transitionto the evolution of the inflaton field is negligible. Tohave a detectable signal, the latent heat density releasedduring the phase transition should be larger than H .Hence, the plasma, with energy density that can be es-timated as T ∼ H / (2 π ) , is negligible. As a result,the production of the GW is dominated by bubble col-lisions. A comprehensive description of bubble collisionand GW production can be found in Ref. [42]. Duringbubble collision the parameter β ≡ − dS /dt determinesthe size of the bubble and the wavelength of the GW,where S is the action of the bounce at the end of thephase transition. Here we use S since the phase tran-sition rate is dominated by the quantum tunneling. Forthe phase transition to complete during inflation, we as-sume β (cid:29) H inf . When β ≈ H inf , numerical simulation isneeded which is beyond the scope of this work.In the case of an instantaneous reheating and followedby RD, all the energy of the inflaton field converts intothe radiation energy. Hence, today’s relative abundanceof GW can be written asΩ GW ( k today ) = Ω R × S ( k ) × ∆ ρ vac ρ inf dρ flatGW ∆ ρ vac d log k p , (11)where Ω R is today’s abundance of radiation, ∆ ρ vac is thelatent energy density of the phase transition sector. Thelast factor is the flat space-time spectrum of GW dρ flatGW ∆ ρ vac d log k p = κ (cid:18) H inf β (cid:19) ∆( k p /β ) , (12)where κ = 1 if the energy density of the plasma is negli-gible, and the simulation result shows∆( k p /β ) = ˜∆ × . k p k . p ˜ k . p + 2 . k . p , (13)where ˜ k p = 1 . β and ˜∆ = 0 . β − to estimate ∆ p in S .Finally, the observed GW frequency is f today f (cid:63) = a ( τ (cid:63) ) a (cid:32) g (0) ∗ S g ( R ) ∗ S (cid:33) / T CMB (cid:104)(cid:16) g ( R ) ∗ π (cid:17) (cid:16) H πG N (cid:17)(cid:105) / , (14) where the superscript ( R ) indicates that the values of pa-rameters at reheating temperature. Due to the distortioninduced by inflation, the position of the highest peak ofthe spectrum corresponds to k p ≈ H inf . As a result, as-suming g ( R ) ∗ S = g ( R ) ∗ ≈ f today = 1 . × Hz (cid:18) H inf m pl (cid:19) / (cid:18) a (cid:63) a (cid:19) . (15)Take high scale inflation as an instance, ( H inf /m pl ) / ∼ − − − . Detectors based on the pulsar timing tech-nology, such as EPTA [9], IPTA [10], and SKA [11] aresensitive to GWs with frequencies around 10 − Hz. FromEq. (15), they can probe the GWs produced at the eraof about 40 e-folds before the end of inflation as shownby the blue curves in Fig. 3. The space-based detectors,such as LISA [2], eLISA [1], DECIGO [3], BBO [7, 8],ALIA [6], TianQin [4] and Taiji [5], is sensitive to fre-quencies around 10 − Hz, corresponding to about 20 e-folds before the end of inflation as shown by the magentaand red curves in Fig. 3. The proposed ground-baseddetectors (e.g. the Einstein Telescope [15] and the Cos-mic Explorer [16]) are sensitive to GWs with frequenciesaround 10 − Hz, which correspond to about 15 e-foldsfrom the end of inflation. They can detect the signal ofthe phase transition if β ≈ H inf . However, as shown bythe purple dotted curve in Fig. 3, the oscillatory featureis expected to be smeared out since ∆ p ∼ H − . Signals on CMB
If the phase transition happenedabout 60 e-folds before the end of inflation, it would leavean imprint on the CMB B-mode power spectrum [40].Since the strength of the GWs depends on H inf onlythrough the ratio H inf /β , as shown in Eqs. (11) and (12),it is possible to see sizable B-mode spectrum from CMBeven in low scale inflation models.We simulate the B-mode power spectra induced by firstorder phase transitions using the class package [46]. Theresult is shown in Fig. 4, where β/H inf and ∆ ρ vac /ρ inf are fixed to be 30 and 0 .
01. The frequency of GW de-pends on H inf , which we have chosen to be H inf = 10 GeV. The solid, dashed and dot-dashed curves are thespectrum for N e = 59, 58 and 57, respectively. Thereare small wiggles induced by the oscillatory pattern inthe GW power spectrum. Since the spherical harmonicsare not orthogonal to the Fourier modes, the oscillatorypattern is smeared. The amplitude of the oscillation isonly about 10% of the total. As a comparison, the blackdotted curve in Fig. 4 shows the B-mode power spectrum FIG. 3: Ω GW as a function of f . The blue, magenta, red and purple curves are for N e = 38, 25, 21 and 15, respectively. Thevalue of β/H inf for each curve are shown in the legend. For all the curves H inf = 10 GeV and ∆ ρ vac /ρ inf = 0 .
1. The curvefor the sensitivity of BBO phase 2 (BBO2) is from [43]. Curves for sensitivities of other detectors are from [44].FIG. 4: B-mode power spectra from the bubble collision hap-pened at N e = 59 , ,
57 during inflation with β/H inf = 30, H inf = 10 GeV. ∆ ρ vac /ρ inf = 0 .
01. The spectrum generatedfrom quantum fluctuations for tensor-scalar ratio r = 0 . produced by quantum fluctuations during inflation withthe tensor-scalar ratio r = 0 . σ level [19]. Of course, the infla-tionary history in this era will also be probed and poten-tially constrained by other CMB and large scale struc-ture observables. Search for the GW signgal discussedhere will provide complementary information. We willleave a more detailed discussion to a separate work. Summary and outlook
The GW spectrum producedfrom instantaneous sources during inflation has an oscil-latory feature, as shown in Figs. 3 and 4, and can bedetected by future GW detectors. This feature allowsus to distinguish it from GW generated by sources af-ter the inflation. From the frequency of the oscillationin the spectrum, we can learn the energy scale of the
FIG. 5: Reaches of the proposed DECIGO [3], BBO [7],SKA [11] and CMB-S4 [19] projects. For CMB-S4, we re-quire the sum of values of D BBl of our spectrum to be smallerthan the 0.003 quantum fluctuation spectrum for l from 50to 100. For other projects, the reach is set by requiring oursignal to be below their sensitivity curves as shown in Fig. 3. phase transition in the unit of the Hubble expansion rateduring inflation. The information of the time the phasetransition happened are encoded in the frequency of theGW. Fig. 5 shows the future reaches of LISA, DECIGO,BBO, SKA and CMB-S4 projects for ∆ ρ vac /ρ inf = 0 . H inf = 10 GeV.For the inflationary history outside the ten e-foldsaround the CMB era, there is no direct measurementof the power spectrum. Hence, the evolution there couldbe very different from the simple form assumed in thispaper. At the same time, if there is a first order phasetransition happened at around N e ≥
10, the details ofthe oscillatory spectrum can help us map out this part of“missing history”. A detailed study of this subject willbe presented in a separate work.If the phase transition happened in the regime thatcan be detected in CMB, the mass of the fields in thespectator sector must be larger than H inf so that theirperturbations induced by the phase transition will decayquickly after evolving out of the horizon. On the otherhand, if the phase transition happens in the missing his-tory and light degrees of freedom exist in the spectatorsector, the perturbations may induce primordial blackholes or dark blobs, leading to additional signals in thefuture. Acknowledgement
We thank Yi Wang, HongliangJiang, Zhong-Zhi Xianyu and Junwu Huang for usefuldiscussions. HA is supported by NSFC under Grant No.11975134, the National Key Research and DevelopmentProgram of China under Grant No.2017YFA0402204 andthe Tsinghua University Initiative Scientific ResearchProgram. KFL was supported in part by the NationalScience Foundation under Grant No. NSF PHY-1748958and by the Heising-Simons Foundation and acknowledgesthe hospitality of Kavli Institute for Theoretical Physicswhile this work was in progress. LTW is supported bythe DOE grant DE-SC0013642. The work of SZ was sup-ported in part by the Swedish Research Council undergrants number 2015-05333 and 2018-03803. [1] P. A. Seoane et al. (eLISA) (2013), 1305.5720.[2] P. Amaro-Seoane et al. (LISA) (2017), 1702.00786.[3] S. Kawamura et al., Class. Quant. Grav. , 094011(2011).[4] J. Luo et al. (TianQin), Class. Quant. Grav. , 035010(2016), 1512.02076.[5] W.-H. Ruan, Z.-K. Guo, R.-G. Cai, and Y.-Z. Zhang, Int.J. Mod. Phys. A , 2050075 (2020), 1807.09495.[6] J. Crowder and N. J. Cornish, Phys. Rev. D , 083005(2005), gr-qc/0506015.[7] G. Harry, P. Fritschel, D. Shaddock, W. Folkner, andE. Phinney, Class. Quant. Grav. , 4887 (2006), [Erra-tum: Class.Quant.Grav. 23, 7361 (2006)].[8] V. Corbin and N. J. Cornish, Class. Quant. Grav. ,2435 (2006), gr-qc/0512039.[9] M. Kramer and D. J. Champion, Class. Quant. Grav. ,224009 (2013).[10] G. Hobbs et al., Class. Quant. Grav. , 084013 (2010),0911.5206.[11] G. Janssen et al., PoS AASKA14 , 037 (2015),1501.00127.[12] J. Aasi et al. (LIGO Scientific), Class. Quant. Grav. ,074001 (2015), 1411.4547.[13] A. Abramovici et al., Science , 325 (1992).[14] F. Acernese et al. (VIRGO), Class. Quant. Grav. ,024001 (2015), 1408.3978. [15] M. Punturo et al., Class. Quant. Grav. , 194002 (2010).[16] D. Reitze et al., Bull. Am. Astron. Soc. , 035 (2019),1907.04833.[17] H. Hui et al., Proc. SPIE Int. Soc. Opt. Eng. ,1070807 (2018), 1808.00568.[18] H. Li et al., Natl. Sci. Rev. , 145 (2019), 1710.03047.[19] K. Abazajian et al. (2019), 1907.04473.[20] L. Grishchuk, Sov. Phys. JETP , 409 (1975).[21] A. A. Starobinsky, JETP Lett. , 682 (1979).[22] V. Rubakov, M. Sazhin, and A. Veryaskin, Phys. Lett. B , 189 (1982).[23] R. Fabbri and M. Pollock, Phys. Lett. B , 445 (1983).[24] L. Abbott and M. B. Wise, Nucl. Phys. B , 541(1984).[25] E. Witten, Phys. Rev. D , 272 (1984).[26] M. Kamionkowski, A. Kosowsky, and M. S. Turner, Phys.Rev. D , 2837 (1994), astro-ph/9310044.[27] T. Vachaspati and A. Vilenkin, Phys. Rev. D , 3052(1985).[28] R. H. Brandenberger, A. Albrecht, and N. Turok, Nucl.Phys. B , 605 (1986).[29] M. Hindmarsh, Phys. Lett. B , 28 (1990).[30] T. Damour and A. Vilenkin, Phys. Rev. D , 064008(2001), gr-qc/0104026.[31] X. Siemens and K. D. Olum, Nucl. Phys. B , 125(2001), [Erratum: Nucl.Phys.B 645, 367–367 (2002)], gr-qc/0104085.[32] M. Hindmarsh and T. Kibble, Rept. Prog. Phys. , 477(1995), hep-ph/9411342.[33] A. H. Guth, Adv. Ser. Astrophys. Cosmol. , 139 (1987).[34] A. D. Linde, Adv. Ser. Astrophys. Cosmol. , 149 (1987).[35] A. Albrecht and P. J. Steinhardt, Adv. Ser. Astrophys.Cosmol. , 158 (1987).[36] D. Baumann, in Theoretical Advanced Study Institute inElementary Particle Physics: Physics of the Large andthe Small (2011), pp. 523–686, 0907.5424.[37] X. Chen and Y. Wang, JCAP , 027 (2010), 0911.3380.[38] A. Berera and L.-Z. Fang, Phys. Rev. Lett. , 1912(1995), astro-ph/9501024.[39] A. Berera, Phys. Rev. Lett. , 3218 (1995), astro-ph/9509049.[40] H. Jiang, T. Liu, S. Sun, and Y. Wang, Phys. Lett. B , 339 (2017), 1512.07538.[41] Y.-T. Wang, Y. Cai, and Y.-S. Piao, Phys. Lett. B ,191 (2019), 1801.03639.[42] S. J. Huber and T. Konstandin, JCAP , 022 (2008),0806.1828.[43] G. Harry, https://dcc.ligo.org/public/0002/G0900426/001/G0900426-v1.pdf (2009).[44] C. Moore, R. Cole, and C. Berry, Class. Quant. Grav. , 015014 (2015), 1408.0740.[45] P. Ade et al. (BICEP2, Keck Array), Phys. Rev. Lett. , 031302 (2016), 1510.09217.[46] D. Blas, J. Lesgourgues, and T. Tram, JCAP07