Universal gravitational-wave signatures from heavy new physics in the electroweak sector
Astrid Eichhorn, Johannes Lumma, Jan M. Pawlowski, Manuel Reichert, Masatoshi Yamada
PPrepared for submission to JCAP
Universal gravitational-wavesignatures from heavy new physics inthe electroweak sector
Astrid Eichhorn, a Johannes Lumma, b Jan M. Pawlowski, b ManuelReichert, a,c
Masatoshi Yamada b a CP3-Origins, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark b Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidel-berg, Germany c Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, U.K.E-mail: [email protected], [email protected],[email protected], [email protected],[email protected]
Abstract.
We calculate the gravitational-wave spectra produced by the electroweak phasetransition with TeV-scale Beyond-Standard-Model physics in the early universe. Our studycaptures the effect of quantum and thermal fluctuations within a non-perturbative frame-work. We discover a universal relation between the mean bubble separation and the strengthparameter of the phase transition, which holds for a wide range of new-physics contributions.The ramifications of this result are three-fold: First, they constrain the gravitational-wave spectra resulting from heavy (TeV-scale) new physics. Second, they provide a wayto distinguish heavy from light new physics directly from the gravitational-wave signature.Third, they suggest that a concerted effort of gravitational-wave observations together withcollider experiments could be required to distinguish between different models of heavy newphysics. a r X i v : . [ h e p - ph ] S e p ontents B.1 Latent heat and duration time in the thin-wall approximation 29B.2 A simple model case: φ model 31 C Result tables 34
Gravitational-wave (GW) signals from a first-order phase transition [1–5] constitute an in-triguing link between particle physics and gravitational physics. If measured, these signalscan be used to unveil the details of the evolution of the early universe during these phasetransitions. Accordingly, the investigation of such GW- signals has attracted considerableattention, see [6–9] for reviews.The evolution of the early universe with a Standard Model (SM) high-energy sectorfeatures two phase transitions that could potentially trigger GW signals: The QCD phasetransition has been proposed as a source of gravitational waves in [10], but the transition isnot of first-order, at least at low density. Similarly, the electroweak phase transition wouldonly be of first order for a very low Higgs mass excluded by experiments, see [11–15]. We– 1 –onclude that a close-to-equilibrium evolution within the SM through these phase transitionsdoes not trigger gravitational waves.Consequently, GW signals, that can be linked to a first-order phase transition, offeran exciting imprint of and evidence for Beyond-Standard-Model (BSM) physics. There areseveral potential sources for such a phase transition, most of them linked to yet unsolvedhigh-energy-physics questions. For example, phase transitions in the dark sector could resultin detectable GW signals, see, e.g., [16–25].In the present work, we concentrate on a potentially first-order electroweak phase transi-tion triggered by BSM-contributions to the Higgs potential. It is well-known that new-physicsmodels impact the Higgs potential and in particular, may change the order of the electroweakphase transition. Investigations have been done in supersymmetric models, [26], and morerecently, more generic new-physics models, e.g., models with one [27–34] or several additionalSM singlets [35], two-Higgs models [36–38] and composite Higgs models [39–41]. At the LHC,new physics in the electroweak sector could be found by direct and indirect searches. Even ifthe mass scale of new physics is above the direct reach of the LHC, imprints of the new physicsexist in the Higgs self-coupling [30, 42–48] that can be measured at a hadron collider [49–51].Future GW interferometers like LISA and DECIGO offer an exciting complementary testof BSM physics. They will be sensitive to gravitational waves in the frequency range that isof interest for the electroweak phase transition [16]; pulsar timing arrays are sensitive to lowerfrequencies, and could therefore find imprints of the QCD phase transition [52]. In particular,a GW signature can provide information on the shape of the underlying potential [53].By now, there is a plethora of works that explore the impact of BSM fields on theGW signature explicitly, e.g., [31–35, 37, 39–41, 54–59]. Alternatively, the contributions ofBSM-physics to the Higgs potential can be introduced in a phenomenological approach asmean-field contributions to the finite-temperature effective potential by hand, see, e.g., [60]for a φ correction, as well as [61] for a logarithmic correction. The zero-temperature one-loop contribution to the potential, arising from φ corrections has been investigated in [62].Taking a step beyond, the one-loop quantum and thermal corrections have been explored for φ contributions to the microscopic potential [54, 58, 61, 63, 64].In this paper, we provide a fully non-perturbative calculation of the parameters de-termining GW signals from a first-order electroweak phase transition. We consider rathergeneric new-physics models, see Fig. , and include their low-energy quantum and thermalfluctuations below the new-physics scale M NP non-perturbatively. In these models, the Higgspotential at the scale M NP is not well-approximated by a φ -potential due to integrating outnew massive particles with masses beyond M NP . To model the impact of such heavy newphysics, we include several distinct classes of modification for the potential at M NP that weexpect to cover generic cases of new physics. Starting from these rather generic potentialsat M NP , we integrated out quantum and thermal fluctuations below M NP in [48], and usethis quantum effective potential here to evaluate the GW signals from the electroweak phasetransition in these models.The physics properties of the first-order transition, and hence that of the resulting GWsignal, are characterized by two parameters, the strength parameter α (energy budget) andthe mean bubble separation R , typically measured in units of the Hubble parameter H ( T p ) atthe percolation temperature T p . Importantly, we find that these two parameters are relatedby an almost universal curve over the whole parameter range. To highlight its importancethis novel result is displayed already in Fig. , more details can be found in Sec. 4. As aconsequence, the properties of the first-order phase transition which are important for the– 2 – t r o n g s up e r c oo li n g A s y m p t o t i c fi t − − . − − . α R H ( T p ) φ φ ln( φ ) φ exp( − /φ ) Figure 1 . We display the strength parameter α versus the mean bubble separation R , multiplied bythe Hubble parameter H ( T p ) at the percolation temperature T p , as measured for the rather genericclasses of new-physics models. We find an almost universal relation between RH and α for allmodifications. The red area marks the strong supercooling regime as defined in Sec. 3.6. At small α ,we find an asymptotic power-law behavior with RH = 10 α . GW signal, are not sensitive to the specific form of the new-physics contribution at theultraviolet scale M NP , but rather on ’integrated’ information that is well described by eitherthe transition strength α or the mean bubble separation. Quite surprisingly, this universalityencompasses both non-perturbative as well as perturbative heavy new physics. This hasramifications both for GW searches as well as for a “multi-messenger” approach which usesGW detectors and particle colliders concertedly.This paper is structured as follows: In Sec. 2, we introduce the theoretical framework wework in and recall the main results from [48] for the electroweak phase transition that we buildon here. In Sec.3, we review how to obtain the spectrum of gravitational waves from the finite-temperature effective Higgs potential. We present our results in Sec. 4, where we discover anintriguing universality between the strength parameter and the mean bubble separation, seeSec. 4.1. Furthermore, we provide GW spectra for all modifications of the Higgs potential,see Sec. 4.3, and we link the LISA signal-to-noise ratio with the high-luminosity LHC resultson the Higgs self-couplings in Sec. 4.4. We conclude in Sec. 5. We build on the results of [48], where the impact of higher-order contributions in the Higgspotential to the Higgs self-couplings was studied and briefly review them here. We includemodifications of the Higgs potential at a new-physics scale M NP . The new-physics scale istypically chosen in the TeV range, such that it impacts the physics of the electroweak phasetransition. Lower new-physics scales are typically in tension with LHC results, whereashigher new-physics scales decouple the new physics from the electroweak phase transition.Starting from a given potential at M NP , we successively integrate out quantum fluctuationsto obtain the quantum effective potential of the Higgs field for a given finite temperature T . Following the evolution of the full quantum effective potential as a function of T allowsus to determine the order of the electroweak phase transition that results from a given new-– 3 – φ ln( φ ) φ exp( − /φ ) SM . . . . . φ c /T c λ H / λ H , . . . φ c /T c λ H / λ H , Figure 2 . Three- and four-Higgs self coupling normalized to its tree-level value, λ H /λ H , and λ H /λ H , , as a function of φ c /T c for all three modifications of the Higgs potential. The Higgs self-coupling increases with increasing strength of the phase transition. The results are taken from [48]. physics modification at M NP . We use the functional renormalization group (FRG) [65] asa non-perturbative tool, for reviews of the FRG, see, e.g., [66–71]. In the context of Higgsphysics, this method is particularly well suited to explore various questions related to theHiggs potential, such as its (non-perturbative) stability [72–77], as well as the impact of aportal to dark matter [78, 79].We do not work with a fully-fledged dynamical implementation of the whole SM, instead,we work in the framework of [74]. We account for the effects of weak gauge bosons through afiducial coupling and a thermal mass. The use of a fiducial coupling instead of a dynamicalweak gauge sector has proven to reproduce the running of the SM up to the Planck scale[74]. Accordingly, we also neglect the would-be Goldstone bosons and focus on a real scalar φ with Z reflection symmetry. In the spontaneously symmetry-broken phase, this scalarfield relates directly to the physical Higgs field φ = H + v , where v denotes the vacuumexpectation value (vev). In [48], the electroweak phase transition has been explored in thisframework.Heavy new physics beyond the mass threshold M NP can be parameterized by higher-order operators in the Higgs potential at and below M NP . It is well known that the inclusionof such higher-order operators, e.g., a φ interaction, can lead to a strong first-order phasetransition [80, 81]. In [48], generic BSM contributions were investigated, all leading to a strongfirst-order phase transition. The BSM contributions, ∆ V , are added to the perturbativelyrenormalizable form of the potential at the new-physics scale, i.e., V k = M NP ( φ ) = µ φ + λ φ + ∆ V ( φ ) . (2.1)The new physics contributions considered in [48] were parameterized in three classes, • Polynomial contributions: A ∆ V ( φ ) = λ M φ interaction is the leading-order effectof new physics in a standard effective-field theory expansion. • Logarithmic contributions: The modification to the effective action is logarithmic, as inthe Coleman-Weinberg potential. This motivates modifications of the form ∆ V ( φ ) = λ ln φ ln φ M . – 4 – Exponential contributions: Non-perturbative effects inspire an exponential dependenceon M /φ , i.e., a contribution of the form ∆ V ( φ ) = λ exp φ exp (cid:16) − M φ (cid:17) .These three classes rather generically cover different types of contributions that may arisefrom integrating out new physics above the new-physics scale M NP , which serves as an ultra-violet cutoff scale. They serve as initial potentials at the cutoff scale M NP before integratingout quantum fluctuations below M NP . We numerically evaluate the running of the scalarpotential V k ( φ ) at finite temperature T on a grid in the scalar field φ , for more details see[48]. Let us highlight that in [48], it was confirmed explicitly that – as expected – a φ contri-bution is well captured by a φ -modification, and similarly, a φ ln φ contribution resemblesa φ ln φ one, and finally, a φ exp( − φ − ) contribution can be encoded in a φ exp( − φ − )one. Therefore, we restrict ourselves to the three generic choices listed above for the presentwork.All three types of modifications can change the electroweak phase transition from across over in the SM to a first-order phase transition. The strength of the phase transitionis parameterized by the ratio φ c /T c , with φ c = (cid:104) φ (cid:105) T c the expectation value of the quantumfield at the nontrivial minimum at the critical temperature T c . A strong first-order phasetransition with φ c /T c > T c . This in turn implies that the new-physics scale M NP cannot be too far above the TeVrange, since all three types of modifications ∆ V are suppressed under the RG flow to the IRdue to their higher-order nature. Despite the non-perturbative nature of the second and thirdclass of modifications, the logarithmic and exponential ones, this perturbative suppression ofhigher-order terms holds for the set of parameters we explore. In [48], it was estimated that M NP ≈
10 TeV is the maximal new-physics scale that can still impact the electroweak phasetransition at fixed Higgs mass and vev.In [48], the modifications were connected to two LHC observables, the effective three-Higgs coupling λ H and four-Higgs coupling λ H . A similar connection has been made inthe literature in [42–47]. Due to the modifications of the scalar potential at M NP , the three-and four-Higgs couplings are enhanced, compared to their values in the SM without suchmodifications. We display this in Fig. . With the high-luminosity LHC run, a modificationof more than λ H /λ H , > . indirect evidence for a first-order phase transition. One of our main aims inthis work is to connect this LHC observable with GW signatures of a first-order electroweakphase transition that will become detectable at future, space-based GW detectors, such asthe planned observatory LISA and the proposed observatory DECIGO. Similar links betweenGW detectors and collider searches have been established within other new-physics settings,e.g., in [32, 34, 56, 83, 84].As mentioned before, the above three classes span, rather generically, a wide range ofpossible modifications to the potential. Exploring these three classes allows us to map out keyparts of the parameter space relevant for GWs. Ultimately, this provides us with an estimateof achievable signal-to-noise-ratio, peak frequency, and peak amplitude for GW signals froman electroweak first-order phase transition driven by heavy new physics.– 5 – Review: Calculation of gravitational-wave spectra from 1st-order phasetransition
In this section, we review the calculation of the GW spectrum from a first-order phasetransition and explain the assumptions and approximations we make to obtain GW spectra.We aim at providing a self-contained discussion of this topic; reviews can be found in [8, 9,60, 85–87]. At the end of this section, we will summarize which equations we are using forthe computation of the GW spectra.During a first-order phase transition, bubbles of the true vacuum are produced in thefalse vacuum. These bubbles expand and eventually collide with each other [1–4, 88–93].After the initial collision, sound shells continue to propagate in the plasma [94–98]. More-over, a first-order transition can generate magneto-hydrodynamical turbulence in the cosmicplasma [99–106]. All three processes trigger tensor fluctuations in the energy-momentumtensor that describes the primordial plasma, and therefore they source GWs. The processthat typically adds the largest contribution to the GW signal is the propagation of soundwaves in the plasma [94, 97].For all three components, fits for the resulting spectrum of GWs are available, see, e.g.,[86]. The most important information is the peak frequency f peak and the peak amplitudeˆ h Ω peak . The peak frequency depends on the inverse duration time of the phase transition, asexpected on dimensional grounds. The amplitude of the GW signal depends on the amountof energy that is released. Both quantities can be calculated from the finite-temperatureeffective potential. To calculate GW spectra, several different temperatures that characterize the phase transitionin an expanding universe are relevant: • T c is the critical temperature, at which both minima of the effective potential aredegenerate, i.e., V eff ( φ = 0 , T c ) = V eff ( φ = φ c , T c ) , (3.1)where φ c = (cid:104) φ (cid:105) T c is the vacuum expectation value in the broken phase at T c . • T n is the nucleation temperature, which is determined by comparing the decay rateΓ( t ) of the false vacuum to the expansion rate of the universe, described by the Hubbleparameter H ( t ). At nucleation temperature, one bubble nucleation per causal Hubblevolume takes place on average, i.e., N ( T n ) = (cid:90) t n t c d t Γ( t ) H ( t ) = 1 . (3.2)Here, t c is the time at which the temperature equals the critical temperature. This isthe time at which bubble nucleation can in principle start. • The percolation temperature T p is defined as the temperature at which the probabilityto have the false vacuum is about 0.7; i.e., 34 % of the false vacuum has been convertedinto the true vacuum. – 6 –dditionally, two further temperatures can play a role, namely the reheating temperatureand the minimization temperature, both of which will be defined later.The order of the temperatures is always T p ≤ T n ≤ T c . In a cosmological context, thephase transition does not occur at T c , but the cosmological phase transition temperature is thepercolation temperature T p . Supercooling can occur when the expansion rate of the universeis large enough to strongly suppress the tunneling probability, even if the true vacuum liesat a significantly lower value of the potential than the false vacuum. Strong supercoolingcan therefore enhance the strength of the phase transition and the amount of energy that isreleased. When a phase transition is strongly supercooled, it becomes important to accuratelydefine the criterion for the phase transition and to distinguish the characteristic temperatures,which is explained in the following sections. Last, if the expansion rate of the universebecomes too large, the criteria for the nucleation and percolation temperature might neverbe fulfilled and the phase transition might not complete. In first-order phase transitions, expanding bubbles of the broken phase are stochasticallycreated through tunneling from the false vacuum. The nucleation rate for bubbles of thetrue vacuum, or equivalently the decay rate of the false vacuum, takes the following form asa function of time t [107]: Γ( t ) = A ( t ) e − S ( t ) . (3.3)At much lower temperatures than the typical energy scale of the system, S ( t ) is given bythe four-dimensional Euclidean action, denoted here by S ( t ), and the pre-factor is A ( t ) = r − [ S ( t ) / (2 π )] where r is the initial radius of the nucleated bubble. At finite temperature,the bubbles of true vacuum are induced by thermal fluctuations. The nucleation rate (3.3) isdetermined by A ( T ) = T [ S ( T ) / (2 πT )] / and S ( T ) = S ( T ) /T [108, 109]. Here S is thethree-dimensional Euclidean action, which reads S ( T ) = (cid:90) d x (cid:20)
12 ( ∇ φ ) + V eff ( φ, T ) (cid:21) = 4 π (cid:90) ∞ d r r (cid:34) (cid:18) d φ d r (cid:19) + V eff ( φ, T ) (cid:35) , (3.4)where r = (cid:112) x + y + z is the three-dimensional distance. To evaluate S ( T ), we first needto evaluate φ ( r ) by solving d φ ( r )d r + 2 r d φ ( r )d r = ∂V eff ∂φ , (3.5)with the boundary conditions, φ ( r → ∞ ) = 0 and d φ ( r = 0) / d r = 0. The solution to (3.5) isthe so-called bounce solution, which provides the radius of a bubble at temperature T . Onecan solve (3.5) using the overshooting/undershooting method. In this work, we employ thePython code CosmoTransitions [110] to find the bounce solution.The expansion history of the universe can be mapped onto its thermal history throughthe relation d T d t = − H ( T ) T, (3.6)– 7 –here H ( T ) is the Hubble parameter, given by H ( T ) = (cid:115) ρ rad + ρ vac M , (3.7)with the reduced Planck mass squared, M pl = 2 . · GeV. The radiation and vacuumenergy densities are given by ρ rad = g ∗ π T , ρ vac = ∆ V eff ( φ, T ) = − [ V eff ( φ = (cid:104) φ (cid:105) T , T ) − V eff ( φ = 0 , T )] , (3.8)where (cid:104) φ (cid:105) T denotes the true vacuum at temperature T . The factor g ∗ denotes the effectivenumber of relativistic degrees of freedom, given by g ∗ = (cid:88) i =boson g i (cid:18) T i T (cid:19) + 78 (cid:88) i =fermion g i (cid:18) T i T (cid:19) , (3.9)where g i and T i are the number of degrees of freedom and the decoupling temperature ofparticle species i , respectively. We treat the number of relativistic degrees of freedom as aconstant value, g ∗ = 106 .
75 in the SM, although in general it depends on the temperature.The new degrees of freedom that generate the modification of the Higgs potential at M NP are heavy and thus do not contribute to g ∗ at the temperature of the electroweak phasetransition. Using (3.6), we can rewrite dependences on time in terms of temperature, so thathereafter we freely exchange between dependences on time and temperature. In particular,we can use it to rewrite (3.2), the defining equation for the nucleation temperature, i.e., thenumber of bubbles per causal Hubble volume at nucleation temperature, N ( T n ) in terms ofthe nucleation temperature itself: N ( T n ) = (cid:90) t n t c d t Γ( t ) H ( t ) = (cid:90) T c T n d TT Γ( T ) H ( T ) = 1 . (3.10)For fast phase transitions the integral in (3.10) is dominated by T ≈ T n and we arrive at theapproximation Γ( T n ) H ( T n ) ≈ , (3.11)which is commonly used. Using the definitions of Γ( T ) and H ( T ) given above, the relation(3.11) approximately yields S ( T n ) T n (cid:39) M pl /T n ) . (3.12)Assuming that the nucleation temperature is located in the range 10 GeV < ∼ T n < ∼ GeV,one has 140 < ∼ S ( T n ) /T n < ∼ T n in the case of a fast phase transition,– 8 –hereas, in general cases, one should give a more accurate prescription. To that end, wedefine the false vacuum probability, in order to describe the percolation of bubbles [111, 112], P ( T ) = e − I ( T ) . (3.13)Here, the weight function is given by [64] I ( t ) = (cid:90) tt c d t (cid:48) Γ( t (cid:48) ) a ( t (cid:48) ) a ( t ) (cid:20) π r ( t (cid:48) , t ) (cid:21) , (3.14)where a ( t ) is the scale factor in the Friedmann-Robertson-Walker metric and r ( t (cid:48) , t ) is thecomoving radius of a bubble that is nucleated at t (cid:48)(cid:48) ( T (cid:48)(cid:48) ) and evolutes until t ( T ). In terms ofthe wall velocity v w ( T ), the comoving bubble radius is given by r ( t (cid:48) , t ) = (cid:90) tt (cid:48) d t (cid:48)(cid:48) v w ( t (cid:48)(cid:48) ) a ( t ) a ( t (cid:48)(cid:48) ) . (3.15)Note that the scale factors in (3.14) play a role in the dilution of nucleated bubbles. Using(3.6), the weight function (3.14) can be rewritten in terms of temperature I ( T ) = 4 π (cid:90) T c T d T (cid:48) Γ( T (cid:48) ) H ( T (cid:48) ) T (cid:48) (cid:32)(cid:90) T (cid:48) T d T (cid:48)(cid:48) v w ( T (cid:48)(cid:48) ) H ( T (cid:48)(cid:48) ) (cid:33) . (3.16)The percolation of bubbles starts when I ( T ) > ∼ .
34 (corresponding to P ( T ) < ∼ .
7) [113].Hence, the percolation temperature T p is defined such that I ( T p ) (cid:39) . The mean bubble separation R ( t ) is a relevant length scale for the generation of gravitationalwaves produced by the phase transition. The number of bubbles per horizon is estimatedfrom the averaged nucleation rate ¯Γ( t ) obtained from the false vacuum probability P ( T ),cf. (3.13), and the decay rate of the false vacuum Γ( t ), cf. (3.3),¯Γ( t ) = P ( t )Γ( t ) . (3.17)As the universe gradually fills up with bubbles of the true vacuum, the bubble nucleation issuppressed. Thus the number density of bubbles at time t is given by [114, 115], n B ( t ) = (cid:90) tt c d t (cid:48) a ( t (cid:48) ) a ( t ) ¯Γ( t (cid:48) ) , (3.18)from which the mean bubble separation R ( t ) at t is defined as R ( t ) = ( n B ( t )) − / . (3.19)It corresponds to the average distance between centers of nucleation points. We estimate R ( t ) for specific cases below. – 9 – .4 Energy released by the first-order phase transition The amplitude of the GW signal strongly depends on the energy budget of the phase transi-tion, which is commonly described by the strength parameter α . Most analyses employ thebag model, where the bag constant (cid:15) describes the jump in energy and pressure across thephase boundary. The strength parameter is then given by α = (cid:15)/ ( a + T ) evaluated at thepercolation temperature, where a + = π g eff /
30 relates to the relativistic degrees of freedomin the symmetric phase. The remaining question is how to relate a concrete particle-physicsmodel to the bag model. The currently most typically employed link uses that the bag con-stant (cid:15) is related to the trace of the energy-momentum tensor θ = ( e + 3 p ) /
4, where theenergy density is given by the (0,0) component of the stress-energy tensor, e = T , andthe pressure corresponds to the spatial entries, p = T ii , with i = 1 , , α is α θ = 34 θ + − θ − a + T (cid:12)(cid:12)(cid:12)(cid:12) T = T p , (3.20)where + labels the symmetric vacuum and − the symmetry-broken vacuum. Within thefield-theoretic description, the energy density and pressure can be obtained from the effectivepotential, e + / − = V eff − T ∂V eff /∂T and p + / − = − V eff . This leads us to α θ = 30 π g eff T (cid:18) ∆ V eff − T ∂ ∆ V eff ∂T (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) T = T p = 1 ρ rad (cid:18) ρ vac − T ∂ρ vac ∂T (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) T = T p . (3.21)Here, ρ vac and ρ rad are defined in (3.8). The parameter α θ is the ratio between the latentheat and the radiation energy and measures the strength of the phase transition in a waythat is directly relevant for the GW production.For a weakly supercooled phase transition, i.e., T p < ∼ T c , the vacuum-energy density ρ vac can be ignored, while for a strongly supercooled case, T p (cid:28) T c , the vacuum-energy densitybecomes much larger than the radiation-energy density, such that (3.21) can be approximatedby α θ (cid:39) ρ vac ρ rad (cid:12)(cid:12)(cid:12)(cid:12) T = T p . (3.22)An alternative way to match the strength parameter to the bag model is via the energydifference of the two phases α e = 34 e + − e − a + T (cid:12)(cid:12)(cid:12)(cid:12) T = T p = 1 ρ rad (cid:18) ρ vac − T ∂ ∆ V eff ∂T (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) T = T p , (3.23)which differs by a factor of four in the second term compared to (3.21). The difference isdue to the fact that the pressure-contribution to θ , which is present in (3.20), is ignored inthe above definition based on the energy densities alone. This change of definition can bepartially compensated, if we properly adjust the corresponding efficiency parameter, whichwe define in Sec. 3.8. Nonetheless, α θ and α e are similar for strong phase transitions, i.e.,where α is large and the vacuum energy dominates, while α e overestimates the strength ofweak phase transitions by a factor of four. For a discussion of the strength parameter beyondthe bag model, see [116]. – 10 – .5 First-order phase transitions without strong supercooling For sufficiently fast phase transitions, the bubble nucleation rate at finite temperature canbe approximated as Γ ≈ Γ( t ) e β ( t − t ) + ... , (3.24)for some t , typically taken as the percolation time t p . Herein β can be understood as theinverse duration of the phase transition. The nucleation rate grows exponentially. From aTaylor expansion of (3.3) with S = S ( T ) /T , it follows that the timescale is set by β = − dd t S ( T ) T (cid:12)(cid:12)(cid:12)(cid:12) t = t . (3.25)Equivalently, using (3.6), the inverse duration of the phase transition can be written as˜ β := βH (cid:12)(cid:12)(cid:12)(cid:12) T = T = T d S d T (cid:12)(cid:12)(cid:12)(cid:12) T = T , (3.26)where the expression is evaluated at the temperature corresponding to t . Note that (3.25)and (3.26) only hold if the duration of the phase transition is short enough such that thelinearization of the action is a valid approximation.In fast phase transitions, for which the bubble-wall velocity and the scale factor areapproximately constant, the comoving radius of a bubble (3.15) is given by r ( t, t (cid:48) ) = v w · ( t − t (cid:48) ).Then, with (3.24), we obtain the weight function I ( t ) in (3.14) and the averaged nucleationrate ¯Γ( t ) in (3.17) in terms of β , I ( t ) = I e β ( t − t ) , ¯Γ( t ) = Γ e β ( t − t ) e − I e β ( t − t . (3.27)Here, we use that t c (cid:28) t p and thus we can take the limit t c → −∞ as a simplification. Inthe above expressions, we have defined Γ = Γ( t ) and I = 8 πv w Γ β − . Thus, the numberdensity (3.18) can be expressed in terms of β as n B ( t ) = β πv w [1 − P ( t )] . (3.28)For t > t p , the phase transition completes very fast and hence P ( t > t p ) (cid:39)
0. Thus the meanbubble separation (3.19) at t = t p ( T = T p ) is given by [114] R = (8 π ) v w β − , (3.29)To calculate the GW spectra for phase transitions without strong supercooling, the inverseduration β and the mean bubble separation R can therefore be used interchangeably. For sufficiently strong phase transitions, α ∼ O (1), the weight S ( T ) /T has a minimum at T m and increases for T < T m . This defines the minimization temperature T m , which is onlyimportant in the situation of strong supercooling.In this case, β as defined above goes to zero at the minimization temperature T m and caneven become negative for T < T m , such that the approximation in (3.24) breaks down [117–119]. For a more robust determination of the duration of the phase transition, one has to– 11 –onsider the quadratic term in ( t − t m ) in the Taylor expansion of the nucleation rate in(3.24), i.e., Γ ∝ e − S T = e β V ( t − t m ) + ... , (3.30)where we used that for sufficiently strong phase transitions, β →
0. The duration of thephase transition is now set by the second derivative of S /T with respect to t , i.e., β V ≡ (cid:115) − d d t (cid:18) S ( T ) T (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = t m = H ( T ) T (cid:115) d d T (cid:18) S ( T ) T (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = T m . (3.31)In the next sections, we will use a common parameter ˜ β to denote β/H in cases withoutstrong supercooling, and β V /H in cases with strong supercooling.Next, we evaluate the mean bubble separation in the case of a strongly supercooledfirst-order phase transition. For the initial nucleation rate (3.30), when the false vacuumprobability satisfies P (cid:39)
1, the number density is given by [117] n B ( t ) = n max (cid:2) β V ( t − t m ) / √ (cid:3) (cid:34) − H ( t − t m ) + (cid:18) √ Hβ V (cid:19) (cid:35) , (3.32)where n max = √ π Γ( t m ) β − V and erf denotes the error function. In the derivation of (3.32),a constant Hubble parameter is assumed. This approximation holds in the case of a stronglysupercooled first-order phase transition for which the vacuum energy dominates, see (3.7)and (3.8), where ρ rad is negligible and ρ vac is roughly temperature independent. For t − t m ∼√ β − V , we obtain the mean bubble separation [119] R = ( n max ) − / = (cid:104) √ π Γ( T m ) β − V (cid:105) − . (3.33)The relation between mean bubble separation and the inverse duration of the phase transitioncontains the decay rate evaluated at the minimization temperature, requiring knowledgeabout the effective scalar potential. In comparison, in the case without strong supercooling,the relation does not require this piece of information, see (3.29). The requirement that the phase transition needs to be completed within an expanding uni-verse results in a maximum possible strength of the electroweak phase transition. A na¨ıvecriterion for the completion of the phase transition is that the percolation of bubbles takesplace as defined below (3.16). A stronger criterion is that the physical volume of the falsevacuum needs to decrease. This criterion is stronger than the percolation condition since thephysical volume of the true vacuum does not only have to increase, it also has to outgrowthe expansion of the universe. This condition reads [64, 115]1 V false d V false d t = 3 H ( t ) − d I ( t )d t = H ( T ) (cid:18) T d I ( T )d T (cid:19) < . (3.34)The phase transition completes if this criterion is fulfilled at the percolation temperature. In[64], a regime of even stronger phase transitions was explored. There, this criterion does nothold at the percolation temperature, but instead holds at lower temperatures. Here, we donot make that distinction and only check if (3.34) is fulfilled at T p .– 12 – .8 Calculation of gravitational-wave spectra From the mean bubble separation R (or equivalently the inverse duration ˜ β in cases withoutstrong supercooling), the wall speed v w and the strength parameter α , the GW spectrum atemission can be calculated. Due to the cosmic expansion, the peak frequency at emission, f ∗ peak , is red-shifted to the peak-frequency today, f peak , according to f peak = a p a f ∗ peak , (3.35)where a p is the scale-factor at the time of emission, i.e., at percolation time, and a is thescale factor today. Given an expansion history of the universe, the relation between f peak and f ∗ peak can be calculated. For instance, assuming that the universe transitioned into radiation-dominated right after the phase transition and has expanded adiabatically ever since, oneobtains [4] f peak = H H ( T p ) f ∗ peak = 1 . · − Hz (cid:16) g ∗ (cid:17) (cid:18) T reh
100 GeV (cid:19) f ∗ peak H ( T p ) . (3.36)Here, g ∗ is the effective number of relativistic degrees of freedom at the time of produc-tion and T reh is the temperature at reheating. Using energy conservation a ρ rad ( T reh ) = a p ( ρ rad ( T p ) + ∆ V ), the reheating temperature can be calculated [64], T reh = (cid:18) a p a reh (cid:19) (cid:18) ρ rad ( T p ) g ∗ π (cid:18) Vρ rad ( T p ) (cid:19)(cid:19) (cid:39) T p (cid:16) α ( T p ) (cid:17) . (3.37)Here, a reh is the scale factor at reheating and is assumed to be a reh (cid:39) a p . It holds that T reh ≤ T c , with the critical temperature T c . Further, H ( T p ) is the Hubble rate at the timeof production and H = a p a H ( T p ) is the Hubble rate redshifted until the present time. Forphase transitions that last about a Hubble time or less at a temperature in the 100 GeVrange, the peak frequency falls into the LISA and/or DECIGO sensitivity region.Once released, the gravitational waves are described in the linearized approximation,such that the total spectrum of gravitational waves is simply the sum of the spectra fromthree processes, ˆ h Ω GW = ˆ h Ω coll + ˆ h Ω sw + ˆ h Ω turb . (3.38)Herein, ˆ h = H/ (100 km/s/Mpc) is the dimensionless Hubble parameter, and the three differ-ent Ω denote the GW amplitudes produced by the i) collisions of bubble walls [1–4, 88–93](Ω coll ), ii) sound waves in the plasma after bubble collision [94–98] (Ω sw ) and iii) magneto-hydrodynamic turbulence in the plasma [99–106] (Ω turb ). We follow [6, 9, 60] and use thefollowing expressions to obtain the respective GW spectra from α and R :ˆ h Ω coll ( f ) = ˆ h Ω peakcoll . f /f collpeak ) . . f /f collpeak ) . , ˆ h Ω sw ( f ) = ˆ h Ω peaksw (cid:32) ff swpeak (cid:33)
47 + 37 (cid:32) ff swpeak (cid:33) − , – 13 – h Ω turb ( f ) = ˆ h Ω peakturb (1 + 8 πf /H ) (cid:32) ff turbpeak (cid:33) (cid:34) (cid:32) ff turbpeak (cid:33)(cid:35) − . (3.39)The peak frequencies read f collpeak (cid:39) . · − Hz (cid:16) g ∗ (cid:17) (cid:18) T reh
100 GeV (cid:19) (cid:32) (8 π ) H ( T p ) R (cid:33) (cid:18) . v w . − . v w + v w (cid:19) ,f swpeak (cid:39) . · − Hz (cid:16) g ∗ (cid:17) (cid:18) T reh
100 GeV (cid:19) (cid:32) (8 π ) H ( T p ) R (cid:33) ,f turbpeak (cid:39) . · − Hz (cid:16) g ∗ (cid:17) (cid:18) T reh
100 GeV (cid:19) (cid:32) (8 π ) H ( T p ) R (cid:33) . (3.40)For fast phase transitions without strong supercooling, the mean bubble separation R canbe approximated by (3.29) for which we obtain the well-known formulae [6] written in termsof the inverse duration time β . In a situation with strong supercooling, R is instead givenby (3.33). The peak amplitudes for each spectrum are given byˆ h Ω peakcoll (cid:39) . · − (cid:32) H ( T p ) R (8 π ) (cid:33) (cid:18) κ coll α α (cid:19) (cid:18) g ∗ (cid:19) . v w .
42 + v w , ˆ h Ω peaksw (cid:39) . · − (cid:32) H ( T p ) R (8 π ) (cid:33) (cid:18) κ sw α α (cid:19) (cid:18) g ∗ (cid:19) , ˆ h Ω peakturb (cid:39) . · − (cid:32) H ( T p ) R (8 π ) (cid:33) (cid:18) κ turb α α (cid:19) (cid:18) g ∗ (cid:19) , (3.41)where κ coll , κ sw , and κ turb are the efficiency factors that denote how much α is convertedinto the energy of the wall (scalar field) and the bulk motion of bubbles. These factorsdepend on v w and α . We review the detailed formulae in App. A. Within the total α , theeffective quantity κ coll α is transferred to the strength of gravitational waves produced by thecollision of bubbles, and then the remnant α eff = α (1 − κ coll ) becomes an energy source forthe dynamics of bubbles generating gravitational waves from sound waves and turbulence.The efficiency factors κ sw and κ turb are given by κ sw = ( H ( T p ) τ sw ) κ v , κ turb = (cid:2) − ( H ( T p ) τ sw ) (cid:3) κ v . (3.42)Here, τ sw is the length of the sound-wave period [58, 61] and the efficiency factor κ v is thefraction of the energy transferred into the bulk motion of bubbles. The longer the sound-wave period lasts, the less energy remains available for GW production through turbulence,accordingly κ turb decreases with increasing τ sw . The length of the sound-wave period isdefined by τ sw = min (cid:20) H ( T p ) , R ∗ ¯ U f (cid:21) . (3.43)Thus, H ( T p ) τ sw takes into account the reduction of GW spectrum from sound waves when thesound-wave period, in which ˆ h Ω sw is actively produced, is shortened [64]. The length of the– 14 –ound-wave period (3.43) depends on the root-mean-square fluid velocity of the plasma [58,97], ¯ U f = 3 v w (1 + α ) (cid:90) v w c s d ξ ξ v ( ξ ) − v ( ξ ) (cid:39) α eff α eff κ v , (3.44)where c s = 1 / √ v ( ξ ) is the solution to (A.17), and we assumed v w (cid:39) κ v depends on the velocity of the bubble wall,see (A.19) in App. A. For v w = 1, we have κ v = α eff α α eff .
73 + 0 . √ α eff + α eff . (3.45)Several comments are in order: Firstly, the formulae for the GW amplitudes depend onunderlying assumptions, e.g., for the bubble dynamics. In particular, the formulae (3.39)have been derived in the envelope approximation in which colliding bubble walls immediatelylose their energy, see, e.g., [92, 93]. The validity of this approximation, however, stronglydepends on the features of both the phase transition and the bubble dynamics, especially onthe magnitude of α . One can distinguish two cases for the bubble dynamics before collisions.One is the runaway bubble case, in which the phase transition produces a lot of energy, i.e., alarge α , most of which is transferred to the bubble wall. This case, in which incidentally theenvelope approximation breaks down, requires a huge α of the order of 10 [120], which isfar beyond the range of α achievable in our setup. The other is the non-runaway bubble case.The bubble wall reaches a constant terminal velocity due to the pressure and the friction withthe plasma. In this case, one typically relies on the envelope approximation, as we will dohere, although it has recently been questioned whether the envelope approximation is validfor α (cid:29) κ coll is negligible in comparisonwith κ sw and κ turb , [58, 122], which we have checked numerically. Therefore, κ coll ≈ v w = 1, and use the approximation κ coll ≈
0. We caution that for smallerbubble-wall velocities, a significant suppression (of the order of 10 − ) of the GW amplitudefrom sound waves has been suggested to occur in comparison with that predicted in earlierworks [123]. In summary, we are using the following equations to determine the GW spectra from theeffective scalar potential • We use the strength parameter, where the bag constant is related to the trace of theenergy-momentum tensor, see (3.21), evaluated at the percolation temperature. • For the mean bubble separation, we distinguish between the cases with and withoutstrong supercooling. For strong supercooling, a minimization temperature exists andthen the inverse duration is given by (3.31) and the mean bubble separation followsfrom (3.33). If no minimization temperature exits, we are in a regime without strongsupercooling and the defining equations are (3.25) and (3.29). • For the efficiency parameters, we use κ coll ≈ v w = 1.– 15 – t r o n g s up e r c oo li n g T h i n - w a ll a s y m p t o t i c s A s y m p t o t i c fi t − − . α ˜ β φ φ ln( φ ) φ exp( − /φ ) Figure 3 . We display the α - ˜ β –values that can be achieved from the three different modifications ofthe Higgs potential that we consider here. We find an almost universal relation between ˜ β and α forall modifications. The red area marks the strong supercooling regime as defined in Sec. 3.6. At small α , we find an asymptotic power-law behavior with ˜ β = 0 . · α − . . It does not match the asymptoticpower-law behavior expected from the thin-wall approximation, see App. B. • The GW spectra are finally determined by (3.39) – (3.41).
In Sec. 3, we have reviewed the calculation of GW spectra from a first-order electroweakphase transition with a given effective potential. In the present work, we utilize rathergeneric effective potentials obtained from integrating-out quantum and thermal effects belowa given high-energy (TeV) cutoff scale M NP , see Sec. 2. The initial effective potentials atthe ultraviolet scale M NP stem from integrating out BSM physics above this scale, and thedifferent classes of potentials parameterize a wide range of BSM models. The GW spectra are computed from the parameters α and R (or ˜ β ), that relate to theenergy released during the phase transition and the mean bubble separation (or the inversetime duration), respectively. For the evaluation of potential new physics, it is important toknow which combinations of pairs ( α, ˜ β ) can be achieved. The values of ( α, ˜ β ) depend on thespecifics of the effective potential around the phase-transition temperature. Varying thesepotentials freely leads to a scatter plot in the α - ˜ β –plane, see, e.g., [9].In our study, this scatter plot reduces to a rather small band of parameters in the α - R and α - ˜ β –plain, see Fig. and Fig. , respectively. This is a remarkable result in viewof the wide range of BSM physics our study is expected to cover, and the correspondingdiverse shapes of the effective potential. This universal curve arises from correspondingcurves for α and ˜ β as functions of the vacuum expectation value φ c /T c , see Fig. : withincreasing strength of the phase transition encoded in φ c /T c , α grows, while ˜ β decreases.The φ c /T c -dependence of α, ˜ β already shows a roughly universal behavior for the differenttypes of BSM-modifications. Moreover, for small φ c /T c (cid:46)
1, both log α and log ˜ β show a– 16 – trong supercooling A s y m p t o t i c fi t T h i n - w a l l a s y m p t o t i c s . − − . φ c /T c α φ φ ln( φ ) φ exp( − /φ ) A s y m p t o t i c fi t T h i n - w a ll a s y m p t o t i c s . φ c /T c ˜ β φ φ ln( φ ) φ exp( − /φ ) Figure 4 . We display α (left panel) and ˜ β (right panel) as a function of φ c /T c for all potentials. Wefind an almost universal relation between α , ˜ β and φ c /T c for all modifications. The red area marksthe strong supercooling regime as defined in Sec. 3.6. At small φ c /T c , we find asymptotic power-lawbehaviors with α = 0 . · ( φ c /T c ) . and ˜ β = 48800 · ( φ c /T c ) − . . The thin-wall approximationmatches the power-law behavior of α perfectly, while it does not match that for β , see App. B. linear dependence on log φ c /T c , indicating a universal power law. In turn, for φ c /T c (cid:38)
1, nosimple power law is present but universality still holds true.The above observations entail that the generic BSM physics encoded in the differentpotential classes introduced in Sec. 2 lead to a universal relation between the energy releasedduring the phase transition and its inverse duration time. For the universality seen in Fig. ,Fig. , and Fig. , one would expect the specifics of the initial effective potentials at the UVscale M NP be washed-out by quantum and thermal fluctuations below the UV scale. Indeed,in the perturbative regime, the RG flow typically washes out the “memory” of the initialconditions at the cutoff scale M NP over just a few orders of magnitude in scales. Specifically,despite being non-polynomial around the origin in field space, all potentials can be expandedabout a given finite field value or background ¯ φ . The expansion coefficients of ( φ − ¯ φ ) n with n ≥ k , driving them towards zero with k n − . Based on this observation, one might conclude that the underlying reason for ourdiscovery of a universal curve in the α - R -plane (or α - ˜ β -plane) is the underlying universalityof the effective potential that emerges from different classes of microphysics.We confront this expectation with our numerical data on the finite-temperature effectivepotential at the critical temperature, which we display in Fig. . All potentials displayed inFig. lead to similar values of α and R (or ˜ β ) and they also fulfill φ c /T c ≈
1. Nevertheless,the effective potentials exhibit clear differences between the three classes of modifications.These differences between the classes exist at all temperatures. At vanishing temperature,they give rise to differences in the three- and four-Higgs coupling, which we have discussed inSec. 2 and displayed in Fig. . Furthermore, each potential displayed in Fig. has a differentvalue of φ c , T c , T p , etc., although they have the value of φ c /T c ≈ α - R and α - ˜ β -relation is not due to a universal formof the effective potential. Instead, it is a form of universality that arises in the integratedinformation from the effective potential, which enters the GW spectra. Therefore, we expectthat there is a universal potential that could be modeled such as to provide the α ( φ c /T c ), R ( φ c /T c ), and ˜ β ( φ c /T c ) curves. Such a universal potential would encode only the coarse-grained information reflected in these two parameters. On the other hand, it would notencode the LHC observables, as these distinguish between the three classes of microphysics.– 17 – . . . . . . . φ/φ c V e ff ( φ ) i n G e V a t T = T c φ φ ln( φ ) φ exp( − /φ ) Figure 5 . Displayed are the effective potentials for all three modifications of the Higgs potentialat T = T c with φ c /T c ≈
1. Despite their differences, the potentials lead to very similar values of α and R and thus also to similar GW spectra. This makes the observed universality in Fig. , Fig. ,and Fig. , highly non-trivial. Importantly, the potentials differ in their explicit values of φ c and T c : φ c,φ = 116 . φ c, ln = 116 . φ c, exp = 110 . Finally, to investigate the origin of the power-law dependence of α ( φ c /T c ), R ( φ c /T c ),and ˜ β ( φ c /T c ), we discuss the regime of φ c /T c (cid:46)
1. For very small φ c /T c , we expect to beable to rely on the thin-wall approximation with (cid:15) ( T p ) = V eff (0 , T p ) − V eff ( (cid:104) φ (cid:105) T p , T p ) →
0. Inthis limit, close to the second-order phase transition, analytic computations are accessible,see App. B: the percolation temperature T p is close to the critical temperature and we canexpand all quantities in the reduced temperature δ c ( T ) = ( T c − T ) /T c . Concentrating on theleading coefficients, this leads to scaling relations for α and ˜ β as a function of φ c /T c , which wedisplayed in Fig. . The power-law for α matches the observed asymptotic fit very well, whilewe note a clear difference in the power-law for ˜ β . We conjecture that the reason is that wehave not yet reached small enough values of φ c /T c , such that the thin-wall approximation issufficiently fulfilled. Furthermore, the leading-order coefficient of the expansion in ˜ β may bestrongly numerically suppressed, which is why we instead observe the next-to-leading orderbehavior. Importantly, the α relation does not depend on the thin-wall approximation, buton the φ approximation, which explains the excellent matching in the α relation, see App. Bfor more details. We leave a more detailed study that could explain the quantitative valuesof both power laws to future studies.In summary, we make the tentative discovery that, unlike light new physics, heavy newphysics leads to a universal α ( ˜ β ) curve. This has ramifications for both GW detections aswell as collider searches, that we will spell out in detail below. To map perturbative new-physics models directly to GW-parameters, we provide Fig. ,which links the φ coupling at the new-physics scale to GW parameters. For this class ofnew physics, this coupling is sufficient to capture the salient effect of the new physics. In termsof GW physics, all modifications at the new-physics scale effectively reduce to one universalpotential with the universal relations between α and R ( ˜ β ) as shown in Fig. and Fig. .The universality of our results indicates that a family of universal effective potentials shouldexist, which captures the GW parameters correctly. This family of potentials depends on– 18 – − − . λ (2 TeV) α φ
60 80 100 120 1401010 λ (2 TeV) ˜ β φ Figure 6 . We display α (left panel) and ˜ β (right panel) as a function of λ at the initial scale M NP = 2 TeV. These relations can be used to directly match a given new-physics model to the GWsignature. the parameters φ c , T c , and T p . For these potentials, we relate the parameters of the effectivefield theory description at the cutoff scale M NP to the parameters α and ˜ β , cf. Fig. . This isuseful in terms of model-building, as it provides a more direct map from new-physics modelsto the GW signature: In a given perturbative new-physics model, one can calculate thesize of the φ coupling at the cutoff scale M NP . Within a perturbative new-physics setting,the φ coupling is expected to capture the leading-order contribution. In [83] it has beenshown that already the next-to-leading order φ contribution has a strongly subleading effecton the phase transition. The size of λ in turn translates directly into corresponding GWparameters.In summary, our results here provide a direct map from perturbative new physics, whichcan be captured in terms of a φ coupling, to the GW parameters. This provides a directway to estimate, for a given perturbative new-physics model, whether or not it is likely tobe detectable by GW detectors. We compare the GW spectra for the three classes of potentials to the LISA and DECIGOsensitivity curves, taken from [124–127]. Our first result, in agreement with the literature, isthe detectability of the GW signal at both of these planned detectors for electroweak phasetransitions which are strong enough, cf. Fig. . Phase transitions with φ c /T c ≈ − . φ c /T c wouldrequire an instrument at higher frequencies and increased sensitivity, such as DECIGO. Thealternative technology underlying AION and AEDGE [128], using atom interferometry, mightbe able to reach sensitivities relevant for an electroweak phase transition of lower φ c /T c . Inparticular, our results indicate that LISA is mostly sensitive to phase transitions with strongsupercooling. In contrast, DECIGO is also sensitive to those phase transitions with moderatesupercooling.As a general trend, increasing strength of the phase transition, i.e., larger φ c /T c results ina shift of the peak-frequency towards lower frequencies, as well as a growth of the amplitude.The same trend has already been observed, e.g., for the class of φ potentials, see, e.g., [62].Due to the universality of α ( ˜ β ), this feature is shared by all three classes of potentials, cf. thethree panels in Fig. . Due to the shift in frequency with increasing φ c /T c , the peak frequency– 19 – - - - - - - - - - - - - - - - - - - - - - - - - Figure 7 . We display the GW spectra for all modifications of the Higgs potential for different valuesof φ c /T c . The top-left panel displays the polynomial modifications, the top-right panel the logarithmicmodifications, and the bottom panel the exponential modifications. The dashed GW spectra are inthe strong supercooling regime as defined in Sec. 3.6. of the GW spectra lies in the outer areas of the LISA , as well as DECIGO sensitivity bands,for those cases where the maximum amplitude lies above the sensitivity curves, respectively.The GW spectra depend on several parameters, such as the efficiency factors that de-termine how much energy is converted into gravitational waves, or the wall speed of theexpanding bubbles. Our assumptions and choices for these parameters are summarized inSec. 3.9. As is well-known in the literature, the sound-wave contribution is typically thedominant component of the spectrum, see [97] for results from numerical simulations. As aself-consistency check of our treatment, our results pose no exception, as is exemplified in theleft panel of Fig. . We have confirmed that the same is true for all other cases we analyze.Note that our use of (3.42) for the efficiency factors is the most conservative choice, as it– 20 – - - - - - - - - - - × - × - × - × - Figure 8 . Left panel:
We show the contributions of sound waves and turbulence to the GWspectrum. The sound-wave contribution is the dominant contribution for all investigated cases.
Rightpanel:
We display the effect of the wall speed on the GW spectrum. With increasing wall speed thespectra are shifted towards larger amplitudes and peak frequencies. The effect is small comparedto changes in φ c /T c . For both panels, the displayed example spectrum stems from the polynomialmodification of the Higgs potential with φ c /T c = 2 . leads to a lower amplitude for the dominant sound-wave contribution than other choices thatcan also be found in the literature.Further, we test the robustness of our results under variations of the wall speed, whichwe set to v w = 1 for the main part of this work. Decreasing it generically results in a loweramplitude and slightly larger peak frequencies, and the difference between v w = 0 . v w = 1 can easily be a factor of 2 in peak amplitude, cf. the right panel of Fig. . Our choiceof the maximum wall speed is accordingly a less conservative one.For GW detectors, the relevant quantity that informs about the detectability of astochastic GW signal is the signal-to-noise-ratio (SNR) [129, 130], which can be obtainedfrom the GW spectrum, the sensitivity curve of the detector, ˆ h Ω det , and the observationtime T , obtained from the duration of the mission times the duty cycle, as in [131]SNR = (cid:118)(cid:117)(cid:117)(cid:116) Ts (cid:90) f max f min d f (cid:32) ˆ h Ω GW ˆ h Ω det (cid:33) . (4.1)Typically, an observation time of roughly four years is assumed, with a duty cycle of 75% [124]i.e., T ≈ π · s. To determine a threshold SNR is not straightforward, as the detectabilityof a signal is influenced by various aspects, such as, e.g., whether matched filtering techniquesare applicable which in general improves the detectability of a signal significantly [124]. Inour case, as the spectrum is known, these techniques are applicable. Additionally, galacticbinaries constitute an expected stochastic background-signal that needs to be accounted for[132, 133]. We obtain SNRs very significantly above 1 for cases where the peak amplitude atpeak frequency lies above the LISA sensitivity curve. Let us note that we used f min = 10 − Hzas a lower cutoff for the sensitivity of LISA, in agreement with [124]. For the phase transitionswith strong supercooling, the SNR would be significantly increased by an extension to, e.g., f min = 10 − Hz, see Fig. . – 21 – NR > . . . . . . − − − λ H /λ H , S N R φ φ ln( φ ) φ exp( − /φ ) SNR > − − − λ H /λ H , S N R φ φ ln( φ ) φ exp( − /φ ) Figure 9 . We display the normalized three-Higgs (left panel) and four-Higgs self coupling (rightpanel) as a function of the LISA SNR for the three different modifications of the Higgs potential. Thegreen area indicates a SNR above 1 and the grey areas display the confidence level with which thehigh-luminosity run of the LHC can exclude the modification of the Higgs self-coupling, see (4.2). Acombined effort of GW detectors and particle collider is needed to distinguish between the differentmodels of NP.
In summary, we conclude that heavy new physics could leave detectable imprints atfuture GW detectors. For LISA, cases without strong supercooling are more challengingto access, calling for a GW observatory with increased sensitivity. In contrast to light newphysics, where a stronger GW signal has been predicted in many cases, the characteristic shiftin peak-frequency towards lower frequencies that heavy-new-physics models exhibit, makesit more challenging to detect their imprints.
In this section, we explore how GW and collider signatures can be used concertedly to learnabout new physics. In particular, we focus on aspects of universality and how to distinguishdifferent types of new-physics contributions.The new physics that triggers a strong first-order phase transition and thereby a GWsignal, at the same time affects observable properties of the Higgs potential, namely theeffective three-Higgs and four-Higgs coupling, see Fig. . Both are enhanced over their valuesin the SM without new physics, as has been investigated in [48]. The high-luminosity runof the LHC will be able to test deviations in the Higgs self-couplings. The standard channelto measure the three-Higgs coupling at the LHC is Higgs pair production in gluon fusion[49, 50, 134–142], see, e.g., the diagrams in Fig. 1 of [48]. An exact cancellation occurs in theSM in the low-energy kinetic regime, leading to a distinct signal, should the SM relation beviolated. The optimal reach of the high-luminosity LHC run with 3 ab − is given by [82], λ H λ H , = 0 . ... . λ H λ H , = − . ... . λ H /λ H , outside the given range would detectably violate the cancellation ofdi-Higgs production in gluon fusion low-energy kinetic regime as discussed above.The cosmological constraint that the electroweak phase transition must complete, asdiscussed in Sec. 3.7, has an intriguing implication for the Higgs self-couplings observed at– 22 –he LHC: it yields a maximal possible value of the three and four-Higgs coupling, assumingthat no other mechanism, i.e., light degrees of freedom, modify the Higgs self-coupling. Wefind that these maximal values are given by λ H , max /λ H , = 2 .
69 and λ H , max /λ H , = 13 . , we compare the LHC observables against the SNR for the GW signal forall three classes of potentials. As expected, strong enough phase transitions are expectedto lead to detectable GW signals at LISA as well as a detectable LHC signal. Therefore,the two observational signatures can be used as cross-checks of each other: For instance, anenhancement of the triple-Higgs and quartic-Higgs coupling might come from other types ofnew physics, which leave the cross-over intact, and therefore do not provide the conditions forelectroweak baryogenesis. To strengthen the case for new physics that changes the cross-overto a strong first-order phase transition requires the observation of a GW signal. Conversely,GW signals could also arise, e.g., from first-order phase transitions in a dark sector [17]. TheLHC constraints on the three- and four-Higgs coupling could distinguish this from a phasetransition in the electroweak sector.Moreover, even amongst those scenarios with electroweak baryogenesis, a combinationof the GW signal with the LHC signature might allow learning more about the details of thenew physics. Fig. highlights that at a given GW SNR, the different classes of potentialslead to different signal strengths at the LHC. In some cases, these might potentially even bedistinguishable. In this paper, we have explored the GW signal sourced by a first-order electroweak phasetransition arising from BSM physics. As in [48], the new physics is parameterized by anaddition to the effective potential at the new-physics scale M NP . We focus on “heavy” newphysics, where the new-physics scale M NP ∼ TeV such that there are no additional lightBSM degrees of freedom. We have performed a non-perturbative analysis, by integratingout quantum and thermal fluctuations below the scale of new physics to obtain the finite-temperature effective potential. From the latter, we extract key parameters that determinethe GW spectrum.The key novel result of our study is an unexpected emergence of universality: Weobserve that the energy released by the phase transition, α , and the mean bubble separation, R , as well as the inverse duration of the phase transition, β , show a qualitatively universaldependence on φ c /T c for the distinct types of new-physics contributions we have explored, seeFig. . This results in a clustering of R ( α ) and β ( α ) around a common curve, see Fig. andFig. . Such a behavior is quite surprising because we investigated rather distinct possibilitiesfor the new physics by exploring a wide range of new-physics contributions at the new-physics scale. Specifically, we investigate an effective-field-theory inspired φ contribution,a Coleman-Weinberg-inspired logarithmic contribution, and a non-perturbative exponentialcontribution. We believe these to cover most generic cases of new-physics contributions. Infact, not only the UV potentials but also the corresponding effective potentials at and belowthe phase-transition temperature differ. Consequently, LHC observables, such as the three-Higgs self-coupling, do not exhibit universality. Hence the emergence of universality for the R ( α ) curve is nontrivial. The universality holds for new-physics scales significantly above theelectroweak scale, i.e., heavy new physics. Our result should be contrasted with the situationwhere the new-physics contribution is associated with a lower mass scale, and more of the α − R plane is accessible when model parameters are varied. Indeed, many examples in the– 23 –iterature illustrate that models with light new degrees of freedom generically deviate fromour newly-discovered universal curve, see, e.g., the examples in [9].It is an intriguing future goal to understand the roots of this qualitative universality inmore detail, including a classification of potential, more exotic new-physics classes that arenot captured by our universal result. Our results suggest that a universal effective potentialexists that is determined by a single free parameter in addition to depending on the phase-transition temperature T p and critical field value φ c , which encodes the impact of heavy newdegrees of freedom on the GW parameters. We stress that such a universal effective potentialmay be used to extract the GW parameters, but cannot account for the particle-physicsobservables, where non-universal features of the distinct classes of new physics matter.We expect this result to have ramifications for GW searches as it severely constrains theavailable parameter space for GW signatures – a property that was not previously noticedin the literature to the best of our knowledge. Indeed, observations that do not lie on theuniversal curve are a strong indication for the presence of light new degrees of freedom.Our results have direct relevance for the phenomenology of GW observations. Firstly,our results severely constrain the available parameter space for GW signatures – a propertythat was not previously noticed in the literature to the best of our knowledge. Secondly,observational results that do not lie on the universal curve are a strong indication for thepresence of light new degrees of freedom. Thirdly, under the hypothesis that our findings arerepresentative of many (if not all) new-physics contributions with a high new-physics scale,the maximum achievable SNR of a GW signal would follow from our study. We find an SNRat LISA that suggests that some new-physics cases should be detectable. At DECIGO, aneven larger range of detectable phase transitions opens up.Besides GW observatories, collider experiments, e.g., the LHC, could be sensitive to suchnew-physics contributions, through a deviation of the effective three-Higgs and four-Higgscouplings from their SM- values. As the expected signal could be challenging to detect bothat the LHC as well as at most GW observatories; a convincing detection could be more withinreach with the use of both types of instruments. Our discovery of universality strengthens thecase for a combination of both signatures: As a consequence of universality for the heavy newphysics, GW signals provide a way to generically distinguish heavy from light new physics.In contrast, distinguishing information on the form of the heavy new physics is not encodedin the GW signal, i.e., the GW signals for different models are degenerate. Interestingly, asshown in [48], the strength of the LHC-signal differs for the different classes of new-physicscontributions. The LHC might be able to lift the degeneracy of GW signatures of distinct newphysics, as the same value of α ( R ) (and therefore the GW signal) is associated with differentvalues of the effective Higgs couplings in different models. A concerted effort, involving bothGW observatories as well as the LHC, therefore seems indicated.GWs are already used together with electromagnetic and neutrino signals as part ofan ongoing effort to constrain fundamental physics with multi-messenger astronomy. Here,we highlight that another promising, multi-instrument campaign, using GW observatoriestogether with the LHC, might be yet another way in which fundamental physics beyond theSM might be constrained or even discovered. Acknowledgements
We acknowledge helpful discussions with Ryusuke Jinno. A.E. and J.L. acknoweldge supportby an Emmy-Noether grant of the DFG under grant number Ei/1037-1. A.E. is supported bya Villum Young Investigator grant of VILLUM FONDEN under grant no. 29405. J.M.P. is– 24 –upported by the DFG Collaborative Research Centre SFB 1225 (ISOQUANT) and the DFGunder Germany’s Excellence Strategy EXC - 2181/1 - 390900948 (the Heidelberg ExcellenceCluster STRUCTURES). M.R. is supported by the Science and Technology Research Council(STFC) under the Consolidated Grant ST/T00102X/1. The work of M.Y. is supported byan Alexander von Humboldt Fellowship.
A Efficiencies of gravitational-wave production processes
The GW spectra depend on efficiency factors that describe the fractions of vacuum energytransferred into the dynamics of bubbles, see (3.41). The efficiency factor κ coll denotes thefraction stored in the bubble wall, which is released upon collisions of the bubble walls,while the energy converted into bulk motion of the fluid is characterized by κ sw and κ turb ,respectively.For κ coll , we need to understand the dynamics of the bubble wall, which we brieflyreview now. Imposing the thin-wall approximation, the Lagrangian of the bubble radius r = r ( t ) is given by [143] L = − M wall ( r ) γ + 4 π r p . (A.1)Here M wall ( r ) = 4 πr σ can be regarded as the “mass” of the bubble wall, γ = √ − ˙ r isthe Lorentz factor and p is the pressure on the bubble wall. The mass of the bubble wallis proportional to the bubble-wall tension σ , which is identified with the one-dimensionalEuclidean action S , see (B.8) in App. B. The dot on r denotes the time derivative, i.e.,˙ r = d r/ d t . The equation of motion for r is given byd γ d r + 2 γr = pσ . (A.2)With the initial condition γ ( r ) = 1, the solution is γ = p σ r + r r − p σ r r . (A.3)From (A.1), the total energy of the bubble wall is given by E tot = (4 πr σ ) γ − π r p . (A.4)The critical radius r c at which the total energy is minimized, i.e., d E tot / d r = 0, is givenby r c = 2 σ/p under the assumption of γ = 1 and constant p and σ under variations of r .In order that the bubble can expand, its initial radius has to be larger than r c . Assumingthat the initial radius is slightly larger than the critical one, i.e., r (cid:38) r c , the Lorentz factorcan be approximated by γ ≈ rr + r r . As the bubble radius increases, the second term issuppressed, whereas the first term dominates. Therefore, the approximated relation between γ and r reads γ ≈ rr . (A.5)One can see from the Lagrangian (A.1) that the pressure on the bubble wall is a key quantityfor the bubble dynamics. The relevant contribution that causes the expansion of the bubble– 25 –omes from the vacuum energy difference between the two vacua as given in (3.8). The frictionforce between the bubble wall and plasma can contribute to a reduction of the pressure. Ingeneral, it is complicated to evaluate the friction force. The highly relativistic case allows usto approximately obtain the pressure on the bubble wall: p (cid:39) ∆ V eff − ∆ P LO − γ ∆ P NLO , (A.6)with ∆ P LO = ∆ m T , ∆ P NLO = g ∆ m V T . (A.7)Here ∆ P LO and ∆ P NLO are the leading-order and next to leading-order pressures due tothe friction induced by 1 → → P LO describes the 1 → m = (cid:80) i c i N i ∆ m i , weighted by the number ofdegrees of freedom for particle species i , N i , and factors c i = 1 for a boson and c i = 1 / P NLO describes the 1 → e − → e − Z [120]. ∆ P LO is frame independent, i.e., it does not depends on the Lorentz factor, while the contributionfrom 1 → γ . The 1 → g ∆ m V = (cid:80) i g i N i ∆ m i , weighted by the gauge coupling constant g i . Althoughone can consider, in general, the 1 → φ → V L φ where φ and V L denote a SM particle and the longitudinal mode of a massive vectorboson ( W ± or Z ), respectively.The treatment (A.6) is only valid when ∆ V eff > ∆ P LO . As γ increases, we expect thatthe pressure (A.6) vanishes, p = 0, when the Lorentz factor reaches γ eq = ∆ V eff − ∆ P LO ∆ P NLO , (A.8)at which a bubble has the radius r eq = (3 γ eq / r from (A.5). The Lorentz factor γ becomesconstant once it reaches γ eq . Thus, the acceleration of the bubble wall stops and it expandswith a constant velocity. In other words, the frame-independent net pressure (vacuum energy)∆ V eff − ∆ P LO , which accelerates the bubble wall, balances with the next to leading-orderfriction γ ∆ P NLO at γ = γ eq .The efficiency factor κ coll describes how much of the vacuum energy is used to acceleratethe bubble wall. The total vacuum energy is simply given by ∆ V eff times the volume ofthe bubble at percolation, i.e., E V = ∆ V eff 4 π r ∗ , where we denoted the bubble radius atpercolation by r ∗ . For the bubble-wall energy, we distinguish whether the bubble radiusreaches r eq before percolation or not. As long as r < r eq , the bubble wall accelerates and thusall vacuum energy, except for the leading-order friction, is transferred into the accelerationof the bubble wall. Thus, if r ∗ < r eq (or equivalently if γ ∗ < γ eq ) then the bubble-wall energyis given by E wall = (∆ V eff − ∆ P LO ) π r ∗ . Then, the efficiency factor κ coll is given by κ coll = E wall E V = ∆ V eff − ∆ P LO ∆ V eff ( r ∗ < r eq ) . (A.9)For r > r eq , the bubble wall does not accelerate anymore and the bubble radius grows witha constant rate. Thus, in the case of r ∗ > r eq the efficiency factor reads [58] κ coll = E wall E V = (∆ V eff − ∆ P LO ) π r ∆ V eff 4 π r ∗ + ∆ A ∆ V eff 4 π r ∗ , ( r ∗ > r eq ) . (A.10)– 26 –here the first term accounts for the transferred energy before r eq and the second term forthe transferred energy after r eq . The numerator in the second term is the kinetic energy in(A.4) between r eq ≤ r ≤ r ∗ , given by∆ A = 4 π ( r ∗ − r ) σ eq γ eq = 4 π V eff ( r ∗ − r ) r eq , (A.11)with σ eq = r eq ∆ V eff / (3 γ eq ) the bubble wall tension at r eq which is derived from the first termin (A.3) only, as the last two terms are suppressed in this case. The quantity (A.11) entailsthe increase in the bubble-wall area when the bubble expands with a constant velocity.We rewrite these results in more convenient dimensionless quantities given by α ∞ = ∆ P LO ρ rad , α eq = ∆ P NLO ρ rad , (A.12)where α ∞ denotes the weakest phase transition for which the vacuum pressure exceeds theleading-order friction. α eq is defined such that we can rewrite the terminal Lorentz factorwith γ eq = α − α ∞ α eq . Furthermore, we define γ ∗ = r ∗ r , which is the Lorentz factor that thebubble wall would reach if the next-to-leading-order friction was neglected. The condition r ∗ ≶ r eq is then equivalent to γ ∗ ≶ γ eq . The efficiency factor for the collision of bubbles,(A.9) and (A.10), can be written as κ coll = γ eq γ ∗ (cid:20) − α ∞ α (cid:16) γ eq γ ∗ (cid:17) (cid:21) , γ ∗ > γ eq , − α ∞ α , γ ∗ ≤ γ eq . (A.13)The remnant α eff = α (1 − κ coll ) is transferred into the bulk motion of bubbles which producesGW from sound wave and turbulence. Note that if the vacuum energy dominates over thefriction, κ coll tends to be unity and then α eff (cid:39)
0, so that GW from the bulk motion ofbubbles are suppressed.The efficiency factors for sound waves and turbulence, κ sw and κ turb , are determined interms of the efficiency factor κ v , cf. (3.42). The efficiency factor κ v is defined by [91] κ v ( α, v w ) = 3 ρ vac v w (cid:90) v w c s w ( ξ ) ξ v ( ξ ) − v ( ξ ) d ξ, (A.14)where c s = 1 / √ c s > v w or c s < v w are possibly, see below. w ( ξ ) is the plasmaenthalpy profile, w ( ξ ) = w exp (cid:34) (1 − c − s ) (cid:90) v ( ξ ) v d v (cid:48) µ ( ξ, v (cid:48) )1 − v (cid:48) (cid:35) . (A.15)Here, we defined the Lorentz transformed fluid velocity µ ( ξ, v ) = ξ − v − ξv , (A.16)and the plasma velocity profile v ( ξ ) which is obtained from the differential equation2 vξ = 1 − ξv − v (cid:2) c − s µ ( ξ, v ) − (cid:3) ∂v∂ξ . (A.17)– 27 –epending on the initial conditions, the solutions to (A.17) are classified into three differenttypes [91]: (i) deflagrations, (ii) detonations and (iii) hybrids. (i): The bubble-wall velocityis subsonic, i.e., v w < c s . Therefore, collisions between the bubble wall and the fluid outsideare mostly avoided and little energy goes into the generation of sound waves and turbulence.In this case, the GW signal from sound waves tends to be suppressed [123]. (ii): The bubblewall velocity is supersonic ( v w > c s ). The active fluid is inside the wall, while the wall hitsthe fluid at rest. Hence, the strong gravitational waves from sound waves and turbulencecould be produced. (iii): The bubble wall moves at a supersonic speed, but smaller than theChapman-Jouguet detonation velocity v J which is defined by v J = (cid:113) α eff / α + (cid:112) /
31 + α eff . (A.18)This case is a combination of the two cases (i) and (ii), namely the wall is inside the activefluid.The fitting analysis to the numerically evaluated efficiency factor tells us that for thethree different cases, (A.14) can be well described by [91] κ v ( α eff , v w ) = α eff α × c / s κ A κ B ( c / s − v / w ) κ B + v w c / s κ A , for v w < ∼ c s ,κ B + ( v w − c s ) δκ + ( v w − c s ) ( v J − c s ) [ κ C − κ B − ( v J − c s ) δκ ] , for c s < ∼ v w < ∼ v J , ( v J − v / J v − / w κ C κ D [( v J − − ( v w − ] v / J κ C +( v w − κ D , for v J < ∼ v w , (A.19)where the efficiency factors for different wall-velocity regions read κ A (cid:39) . α eff . − . √ α eff + α eff v / w , ( v w (cid:28) c s ) ,κ B (cid:39) α / .
017 + (0 .
997 + α eff ) / , ( v w = c s ) ,κ C (cid:39) √ α eff .
135 + √ .
98 + α eff , ( v w = v J ) ,κ D (cid:39) α eff .
73 + 0 . √ α eff + α eff , ( v w = 1) . (A.20)In (A.19) for c s < ∼ v w < ∼ v J , the efficiency factor κ v depends on δκ , which is the derivative of κ v with respect to v w at v w = c s , approximately given by δκ (cid:39) − . (cid:18) √ α eff √ α eff (cid:19) . (A.21)Then the efficiency factors κ sw and κ turb are defined by (3.42).In many previous works, the setup κ sw = κ v and κ turb = (cid:15)κ v has been commonlyemployed, where (cid:15) is the fraction of turbulent bulk motion and is typically set to (cid:15) (cid:39) .
05– 0 .
1, which stems from numerical simulations [97]. The recent investigations [58, 61, 64],– 28 –owever, have shown that when the period of the active production from sound waves isshorter than the Hubble time, the sound-wave GW amplitude is reduced and the GW fromturbulence instead becomes stronger. This could be taken into account by introducing theparameter H ( T p ) τ sw as given in (3.42). In this paper, we employ the recent treatment byfollowing Ref. [58, 61]. B Thin-wall approximation
In this appendix, we discuss the thin-wall approximation that allows us to derive analyticapproximations of α defined in (3.20), the strength parameter of the first-order phase transi-tion, and ˜ β defined in (3.26), the inverse duration time (divided by the Hubble parameter),as functions of φ c /T c . The thin-wall approximation corresponds to the limit ε →
0, where ε ( T p ) = V eff (0 , T p ) − V eff ( (cid:104) φ (cid:105) T p , T p ) is the depth of the effective potential at the true vacuumat the percolation temperature as shown in Fig. . This approximation holds in the weaksupercooling regime where the percolation temperature is close to the critical temperature, T p (cid:39) T c . We normalize the effective potential at φ = 0 so that V eff (0 , T ) = 0 and thus ε ( T p ) = − V eff ( (cid:104) φ (cid:105) T p , T p ). B.1 Latent heat and duration time in the thin-wall approximation
The thin-wall approximation holds for weak supercooling, where the vacuum-energy densityis negligible and (3.21) reduces to α (cid:39) L c ρ rad = 152 π L c g ∗ ( T c ) T c , (B.1)with the latent heat L c at the critical temperature T c , L c ≡ − T ∂ ∆ V eff ( (cid:104) φ (cid:105) T , T ) ∂T (cid:12)(cid:12)(cid:12)(cid:12) T = T c = T ∂V eff ( (cid:104) φ (cid:105) T , T ) ∂T (cid:12)(cid:12)(cid:12)(cid:12) T = T c , (B.2)where ∆ V eff ( (cid:104) φ (cid:105) T , T ) = ε ( T ) = − V eff ( (cid:104) φ (cid:105) T , T ). Next, we evaluate the duration time (3.26)from the bounce solution to (3.5). Within the thin-wall approximation, the bounce solutionis approximately given by the case where the two vacua are degenerate. This is realized whenthe “friction term”, the second term on the left-hand side in (3.5), is negligible, to witd φ d r = ∂V eff ∂φ . (B.3)This equation can be rewritten as d φ d r = − (cid:112) V eff ( φ, T ) . (B.4)Its solution reads r = (cid:90) φ c φ d ϕ (cid:112) V eff ( ϕ, T ) . (B.5) The thin-wall approximation to the evaluation of the Euclidean action at zero temperature was discussedby Coleman [107, 145, 146] and it was extended to the finite temperature case by Linde [109]. The discussionin this appendix follows the latter. – 29 – igure 10 . Schematic figures of the effective potential at T c and T p , and the bounce solution to theequation of motion (B.3). For the evaluation of this integral we split the integration domain in three parts, see the rightpanel of Fig. , φ ( r ) = φ ∗ (0 < r < r ∗ − ∆ /
2) (Region 1) φ w ( r ) ( r ∗ − ∆ / < r < r ∗ + ∆ /
2) (Region 2)0 ( r > r ∗ + ∆ /
2) (Region 3) . (B.6)Here φ ∗ and r ∗ are the field value and the radius of a bubble at a certain temperature T around T c , respectively, and ∆ is the width of the bubble wall. Within the approximation(B.6), the spatial integral of the three-dimensional Euclidean action (3.4) reads, S ( T ) = 4 π (cid:90) ∞ d r r (cid:34) (cid:18) d φ d r (cid:19) + V eff ( φ, T ) (cid:35) = − π r ∗ ε ( T ) (cid:124) (cid:123)(cid:122) (cid:125) Region 1 + 4 π (cid:90) r ∗ +∆ / r ∗ − ∆ / d r r (cid:34) (cid:18) d φ w d r (cid:19) + V eff ( φ w , T ) (cid:35)(cid:124) (cid:123)(cid:122) (cid:125) Region 2 + 0 (cid:124)(cid:123)(cid:122)(cid:125)
Region 3 . (B.7)The region 2 is evaluated as(Region 2) = 4 πr ∗ (cid:90) r ∗ +∆ / r ∗ − ∆ / d r (cid:34) (cid:18) d φ w d r (cid:19) + V eff ( φ w , T ) (cid:35) = 4 πr ∗ (cid:90) φ ∗ d ϕ (cid:112) V eff ( ϕ, T ) ≡ πr ∗ S ( T ) , (B.8)where we used (B.4) in the second equality. Here, S corresponds to the bubble-wall tension σ introduced in (A.1). From the stationary conditiond S ( T )d r ∗ = − πr ∗ ε ( T ) + 8 πr ∗ S ( T ) = 0 , (B.9)one finds r ∗ = 2 S ( T ) ε ( T ) , S ( T ) = 16 π S ( T ) ε ( T ) . (B.10)– 30 –ote that from these relations, one obtains r ∗ = (cid:20) S πε (cid:21) . (B.11)This result is compatible with the critical radius r c = 2 σ/p in App. A by inserting p = ε and σ = S . For weak supercooling the percolation temperature, T p < T c , is close to the criticaltemperature. For temperatures T p ≤ T ≤ T c , this entails a small reduced temperature( T − T c ) /T c (cid:28)
1. This allows us to use a linear approximation for the expansion of theeffective potential about T c or rather in powers of the reduced temperature. The linearexpansion coefficient in an expansion in powers of the reduced temperature is the latentheat, (B.2), and we arrive at V eff ( φ, T ) = V eff ( φ c , T c ) + T ∂V eff ∂T (cid:12)(cid:12)(cid:12)(cid:12) T = T c T − T c T c + O (cid:32)(cid:20) T − T c T c (cid:21) (cid:33) (cid:39) L c T − T c T c . (B.12)In (B.12) we have used that V eff ( φ c , T c ) = 0 at T c . For T = T p this leads us to V eff ( φ ∗ , T p ) = ε ( T p ) (cid:39) − L c δ c ( T p ) , with δ c ( T ) = T c − TT c . (B.13)Their insertion into the three dimensional Euclidean action (B.10) yields, at a certain T around T c , S ( T ) T (cid:39) π S ( T c ) T c L c δ c ( T ) − − δ c ( T ) (cid:32) − δ c ( T ) T c S ( T c ) ∂S ( T ) ∂T (cid:12)(cid:12)(cid:12)(cid:12) T = T c (cid:33) (cid:39) π S ( T c ) T c L c δ c ( T ) − , (B.14)from which one finds δ c ( T ) = (cid:18) π S ( T c ) T c L c (cid:19) / (cid:18) S ( T ) T (cid:19) − / . (B.15)The duration time (3.26) at the percolation temperature reads˜ β = T dd T S ( T ) T (cid:12)(cid:12)(cid:12)(cid:12) T = T p (cid:39) π S ( T c ) T c L c dd T δ c ( T ) − (cid:12)(cid:12)(cid:12)(cid:12) T = T p = 2 δ c ( T p ) S ( T p ) T p = (cid:18) π T c L c S ( T c ) (cid:19) / (cid:18) S ( T p ) T p (cid:19) / . (B.16) B.2 A simple model case: φ model We consider the effective potential V eff ( φ, T ) = A ( T − T ) φ − BT φ + λ T φ , (B.17)where A , B , and λ T are positive constant and T is the temperature at which the symmetricphase φ = 0 becomes metastable. At T c or for ε →
0, the potential values at φ = 0 and– 31 – = (cid:104) φ (cid:105) T take the same value V eff = 0, so that in such a case one can parametrize the effectivepotential as V eff ( φ, T c ) = λ T φ ( φ − φ c ) . (B.18)Comparing between (B.17) and (B.18) at T c , we obtain the relations A ( T c − T ) = λ T φ c , BT c = λ T φ c , T c (cid:18) − B Aλ T (cid:19) = T . (B.19)Here, T has to be positive, which implies that A > B /λ T . For the effective potential (B.18),the bounce solution to the equation of motion (B.3) is found to be φ ( r ) = φ c (cid:20) − tanh (cid:18) r − r ∗ ∆ (cid:19)(cid:21) , (B.20)with r ∗ the bubble wall radius and ∆ = (2 /φ c ) (cid:112) /λ T the bubble wall width. Inserting thisbounce solution into (B.8), one obtains S ( T p ) (cid:39) (cid:90) ∞ d r (cid:34) (cid:18) d φ d r (cid:19) + λ T φ ( φ − φ c ) (cid:35) = λ T φ c ∆32 (cid:90) ∞− r ∗ / ∆ d x cosh x (cid:39) √ λ / T φ c , (B.21)where we have employed the thin-wall approximation, namely − r ∗ / ∆ → −∞ in the lastequality and have used (cid:90) ∞−∞ d x cosh x = 43 . (B.22)Note that instead of the use of the bounce solution (B.20), we can obtain the same result byevaluating directly (B.8) with the potential (B.18), S ( T p ) = (cid:90) φ c d ϕ (cid:113) V eff ( ϕ, T p ) = √ λ / T φ c . (B.23)Thus, the latent heat L c , α and ˜ β in this simple model are evaluated at T p (cid:39) T c , respectivelyas L c = T ∂V eff ∂T (cid:12)(cid:12)(cid:12)(cid:12) T = T c = 2 (cid:18) A − B λ T (cid:19) T c (cid:18) φ c T c (cid:19) ,α (cid:39) π g ∗ ( T c ) (cid:18) A − B λ T (cid:19) (cid:18) φ c T c (cid:19) ≡ C α (cid:18) φ c T c (cid:19) , ˜ β (cid:39) · / π / λ / T (cid:18) A − B λ T (cid:19) (cid:18) S ( T p ) T p (cid:19) (cid:18) φ c T c (cid:19) − / ≡ C ˜ β (cid:18) φ c T c (cid:19) − / , (B.24)where we have used (B.19). The positivity of the latent heat implies that A > B /λ T , whichis the same condition as the positivity of T . As discussed in (3.12), the factor S ( T p ) /T p – 32 –akes a constant value between 140–150 when the weak supercooling occurs at T p ≈ T n . Wesee now that α behaves like ( φ c /T c ) , while ˜ β behaves like ( φ c /T c ) − / . From this fact, ˜ β behaves as a function of α such that ˜ β = Cα − / , (B.25)where C = C ˜ β /C − / α . – 33 – Result tables λ φ c /T c α ˜ β R · GeV H ( T p ) / GeV T c / GeV T n / GeV T p / GeV T reh / GeV SNR λ H /λ H , λ H /λ H ,
58 1.02 0.00274 46400 3 . · . · −
116 115 115 115 1 . · − . · . · −
112 111 110 111 6 . · − . · . · −
107 105 105 105 2 . · − . · . · − . · − . · . · − . · − . · . · − . · − . · . · − . · . · − . · . · − . · . · − . · . · − . · . · − . · . · − . · . · − Table 1 . The results for the φ modification of the Higgs potential are summarized. The double line indicates where the strong supercoolingregime, i.e., the regime where a minimization temperature exists as described in Sec. 3.6. –34– ln φ c /T c α ˜ β R · GeV H ( T p ) / GeV T c / GeV T n / GeV T p / GeV T reh / GeV SNR λ H /λ H , λ H /λ H , . · . · −
117 116 116 116 2 . · − . · . · −
113 111 111 111 5 . · − . · . · −
103 98.3 97.8 98.1 6 . · − . · . · − . · − . · . · − . . · . · − . . · . · − . · . · − . · . · − . · . · − . · . · − . · . · − . · . · − Table 2 . The results for the φ log( φ ) modification of the Higgs potential are summarized. The double line indicates where the strong supercoolingregime, i.e., the regime where a minimization temperature exists as described in Sec. 3.6. –35– exp / φ c /T c α ˜ β R · GeV H ( T p ) / GeV T c / GeV T n / GeV T p / GeV T reh / GeV SNR λ H /λ H , λ H /λ H , . . · . · −
110 110 109 110 1 . · − . . · . · −
109 108 108 108 2 . · − . . · . · −
108 107 107 107 9 . · − . . · . · −
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