Baryogenesis and gravity waves from a UV-completed electroweak phase transition
James M. Cline, Avi Friedlander, Dong-Ming He, Kimmo Kainulainen, Benoit Laurent, David Tucker-Smith
BBaryogenesis and gravity waves from a UV-completedelectroweak phase transition
Benoit Laurent ∗ and James M. Cline † McGill University, Department of Physics, 3600 University St., Montr´eal, QC H3A2T8 Canada
Avi Friedlander ‡ Queen’s University, Department of Physics & Engineering PhysicsAstronomy Kingston, Ontario, K7L 3N6 Kingston, Canada
Dong-Ming He § University of Science and Technology of China, Hefei, Anhui 230026 andUniversiteit van Amsterdam, Science Park 904, Amsterdam, 1098XH, Netherlands
Kimmo Kainulainen ¶ Department of Physics, P.O.Box 35 (YFL), FIN-40014 University of Jyv¨askyl¨a, Finland andHelsinki Institute of Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland
David Tucker-Smith ∗∗ Department of Physics, Williams College, Williamstown, MA 01267
We study gravity wave production and baryogenesis at the electroweak phase transition, in areal singlet scalar extension of the Standard Model, including vector-like top partners to generatethe CP violation needed for baryogenesis. The singlet makes the phase transition strongly first-order through its coupling to the Higgs boson, and it spontaneously breaks CP invariance througha dimension-5 contribution to the top quark mass term, generated by integrating out the heavytop quark partners. We improve on previous studies by incorporating updated transport equations,compatible with large bubble wall velocities v w , to determine the friction on the wall, and thereby v w and the wall thickness, rather than treating these as free parameters. The baryon asymmetry is alsocomputed with no assumptions, directly from the microphysical parameters. The size of the CP-violating dimension-5 operator is constrained by collider, electroweak precision, and renormalizationgroup running constraints. We identify regions of parameter space that can produce the observedbaryon asymmetry, and simultaneously produce gravitational waves that could be observed by futureexperiments. Contrary to standard lore, we find that for strong deflagrations, the efficiencies of largebaryon asymmetry production and strong GW-signal can be positively correlated. CONTENTS
I. Introduction 2II. Z -symmetric singlet model 3A. Laboratory constraints 4B. Explicit breaking of Z symmetry 6III. Phase Transition and Bubble Nucleation 7IV. Wall velocity and shape 8A. Transport equations for fluid perturbations 10V. Cosmological signatures 11A. Gravitational Waves 11B. Baryogenesis 12 ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ kimmo.kainulainen@jyu.fi ∗∗ [email protected] a r X i v : . [ h e p - ph ] F e b VI. Monte Carlo results 13A. Deflagration versus detonation solutions 14B. Baryogenesis and gravity wave production 15C. Dependence on λ s and Λ 17D. Theoretical uncertainties 17E. Comparison of the GW signal with previous studies 19VII. Conclusion 20A. Effective Potential 20B. Relativistic fluid equation 22C. Gravitational Wave Production 23References 24 I. INTRODUCTION
Phase transitions in the early universe provide an opportunity for probing physics at high scales through cos-mological observables, in particular, if the transition is first order. In that case, it may be possible to explain theorigin of baryonic matter through electroweak baryogenesis (EWBG) [1–4] or variants thereof [5]. Such transitionscan also produce relic gravitational waves (GWs) that may be detectable by future experiments like LISA [6, 7],BBO [8], DECIGO [9, 10] and AEDGE [11].It is remarkable that even though the electroweak phase transition (EWPT) is a smooth crossover in the stan-dard model (SM) [12, 13], it can become first order with the addition of modest new physics input, in particulara singlet scalar coupling to the Higgs [14–20], that can also be probed in collider experiments [21–29]. There havebeen many studies of such new physics models with respect to their potential to produce observable cosmologicalsignals [30–42]. However, it is challenging to make a first-principles connection between microphysical models andthe baryon asymmetry or GW production, since these can be sensitive to the velocity v w and thickness L w of thebubble walls in the phase transition, which are numerically demanding to compute [43–53]. Most previous studiesthat encompass EWBG and GW studies of the EWPT therefore leave v w and L w as free parameters. This limita-tion was addressed recently in Ref. [54], which undertook a comprehensive investigation of the EWPT enhancedby coupling the Higgs boson to a scalar singlet with Z symmetry. The simplicity of this model facilitates doingan exhaustive search of its parameter space.In the present work we continue the investigation started in Ref. [54], which determined v w and L w over much ofthe model parameter space, but did not try to predict the baryon asymmetry or GW production. Moreover, thatstudy was limited to subsonic wall speeds, due to a breakdown of the fluid equations that determine the frictionon the wall. Recently a set of improved fluid equations was postulated in Refs. [55, 56], that do not suffer fromthe subsonic limitation. We use these in the present work in order to fully explore the parameter space, wherehigh v w can be favorable to observable GWs, and also compatible with EWBG. It will be shown that for strongdeflagrations, the fluid velocity in front of the wall saturates and even decreases with increasing wall velocity v w .Since the walls become thinner at the same time, the baryon asymmetry is enhanced at larger wall velocities forthese transitions, becoming positively correlated with a strong GW signal.A further improvement in this work is to present an ultraviolet completion of the effective coupling that givesrise to the CP-violation needed for EWBG. We introduce heavy vectorlike top partners which when integratedout induce a CP-violating coupling of the singlet scalar s to top quarks, giving the source term for EWBG. We present the details in section II, including comprehensive laboratory constraints on the top partners and thesubsequent constraints on the effective theory. The finite-temperature effective potential of the theory is alsooutlined there. Hints of the presence of such a particle in LHC data were recently presented in Ref. [57]
The paper continues in Sect. III with a brief description of our methodology for finding the high-temperaturefirst-order phase transitions, and characterizing their strength. This is followed in Sect. IV by a detailed accountof how the bubble wall speed and shape are determined. The techniques for computing the baryon asymmetryand GW production are described in Sect. V. We present the results of a Monte Carlo exploration of the modelparameter space with respect to these observables in Sect. VI, with emphasis on the overlap between successfulEWBG and potentially observable GWs. Conclusions are given in Sect. VII, followed by several appendicescontaining details about construction of the finite-temperature effective potential, solving junction conditions forthe phase transition boundaries, and predicting GW production.
II. Z -SYMMETRIC SINGLET MODEL We study the Z -symmetric singlet scalar extension of the SM with a real singlet s coupled to the Higgs doublet H . The scalar potential is V ( H, s ) = µ h H † H + λ h (cid:0) H † H (cid:1) + λ hs (cid:0) H † H (cid:1) s + µ s s + λ s s . (1)We work in unitary gauge, which consists of taking H = h/ √
2; the Goldstone bosons still contribute to theone-loop and thermal corrections, but they are set to zero in the tree-level potential. We assume µ h < µ s <
0, which implies that the potential has non-trivial minimums at v ≡ h = ±| µ h | / √ λ h ≈
246 GeV, s = 0 and h = 0, s = ±| µ s | / √ λ s . The scalar fields’ mass in the vacuum can then be written in terms of the parameters ofthe potential as m h = − µ h ≈ (125 GeV) and m s = − λ hs µ h / (2 λ h ) + µ s .The other relevant interaction of s is a dimension-5 operator yielding an imaginary contribution to the topquark mass [58]: L BG = − y t √ ht L (cid:16) i s Λ (cid:17) t R + H . c . (2)This term will be ignored during the discussion on the phase transition; however it is essential for generating thebaryon asymmetry, since it gives the CP-violating source term when s temporarily gets a VEV in the bubble wallsof the electroweak phase transition. In Eq. (2) we have adopted a special limit of a more general model, in whichthe dimension-5 contribution is purely imaginary. This can be understood as a consequence of imposing CP inthe effective Lagrangian, with s coupling like a pseudoscalar, s → − s . Hence it is consistent to omit terms oddin s in the scalar potential (1), even though Eq. (2) is odd in s . The CP symmetry prevents a VEV from beinggenerated for s by loops.The effective operator is generated by integrating out a heavy singlet vectorlike top quark partner T , whosemass term and couplings to the third generation quarks q L = ( t L , b L ), Higgs and singlet fields are y t ¯ q L Ht R + η ¯ q L HT R + iη ¯ T L st R + M ¯ T L T R + H . c (3)including also the SM q L -Higgs coupling. This is invariant under CP if s → − s . Integrating out T leads to theeffective operator in (2) with scale Λ = y t Mη η (4)We consider experimental constraints on the scale Λ below.In previous literature, thermal corrections were frequently approximated by including just the first term of thehigh-temperature expansion of the thermal functions presented in the Appendix B. However, this approximationfails at temperatures below the mass of particles strongly coupled to the Higgs, as can happen in models witha high degree of supercooling. Therefore, we employ the full one-loop thermal functions. This will be shown tohave a large impact on the values of the tunneling action, and thus of the nucleation temperature. In addition tothe tree-level potential and the thermal corrections, we also include the one-loop correction and the thermal massParwani resummation [59]. The complete effective potential then becomes V eff = V tree + V CW + V T + δV. (5)The details are presented in Appendix A. The interaction term iη T L sT R also respects CP for real η . We neglect it to simplify our analysis. s t gg s tt W bb FIG. 1. Feynman diagrams for decay of the singlet s . A. Laboratory constraints
It is important to determine how low the scale Λ of the dimension-5 operator in Eq. (4) can be, since it has astrong impact on the baryon asymmetry η b ; in the limit of large Λ, η b scales as 1 / Λ. The relevant masses andcouplings are constrained by direct searches for the top partner and precision electroweak studies. Moreover theproperties of the singlet s are constrained by collider searches.After electroweak symmetry breaking, a Dirac mass term (¯ t L , ¯ T L )( m t µM ) (cid:16) t R T R (cid:17) is generated for t, T , with m t = y t v/ √ µ = η v/ √ t R , T R ) and ( t L , T L ), with mixingangles tan 2 θ L = 2 M µM − m t − µ , tan 2 θ R = 2 m t µM + µ − m t . (6)For example, consider a benchmark point with η = 0 .
55 and a physical T mass M T = 800 GeV. These parameterscorrespond to M = 794 GeV and mixing angles θ L = 0 .
126 and θ R = 0 . y t and thephysical top mass differs from the SM one by less than 1%, which is allowed by current LHC constraints [60, 61].For sufficiently large η , decays of T to ht/Zt/W b induced by mixing are highly subdominant to T → st , andsearches for vector-like top partners that focus on the former channels are evaded. Near the Goldstone-equivalentlimit (which should apply reasonably well for M T = 800 GeV and relatively small s masses, m s ∼
100 GeV), thebranching ratio of T → st is B ( T → st ) (cid:39) η η + 2 η . (7)We roughly estimate from Refs. [62, 63] that for M T = 800 GeV, vector-like quark searches that target SM finalstates are evaded provided B ( T → st ) > ∼ η > ∼ . m s , M T )for models in which T → st dominates, finding that top partner masses above ∼
750 GeV are allowed in the casewhere s decays 100% into two gluons. This is true in our model, where the dominant s decays are induced bythe loop diagrams shown in Fig. 1. One can estimate that the gluon final state dominates over that of b quarksby a factor of ( g s m s /g w m b ) (cid:38) , and over decays into photons by ( g s /e ) ∼ T , which is corrected by [66]∆ T = T sm s L (cid:18) − (1 + c L ) + s L r + 2 c L rr − r (cid:19) (cid:46) . , (8)where T sm = 1 .
19 is the SM value, c L = cos θ L , s L = sin θ L , and r = ( M T /m t ) ; the upper limit is from section10 of [67]. The benchmark point chosen above almost saturates this constraint, giving ∆ T (cid:39) . s to t in the mass eigenstate basis is y st = η cos θ R sin θ L ∼ η θ L , while that to T is y sT = − η cos θ L sin θ R ∼ − η θ R .The squared matrix element for the decays s → gg is [68] |M| = (cid:16) α s π (cid:17) m s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i = t,T y si m i τ i (cid:104) sin − (cid:16) τ − / i (cid:17)(cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (9)where τ i = 4 m i /m s . The parton-level production cross section for gg → s is ˆ σ = π |M| δ (ˆ s − m s ) / (256 ˆ s )where the 256 comes from averaging over gluon colors and spins. Integrating this over the gluon PDFs gives the �������� Λ = ��� ��� Λ = ��� ��� Λ = ���� ��� ��� ��� ��� ��� ������������ � � ( ��� ) σ × � ( � → γγ )( � � ) �������� ��� ��� ��� ��� ������������������������������� � � ( ��� ) � � � � � ������� Λ ( � � � ) (a) (b)FIG. 2. Left (a): experimental limits from ATLAS [69, 70] and CMS [71] for resonant production of s by gg fusion followedby decays into photons (solid lines), versus predictions at different values of of Λ. Right (b): corresponding lower boundson Λ. hadron-level cross section σ ( pp → s ) = π m s |M| L g ≡ π m s |M| (cid:90) m s /s dxx [ xf g ]( x )[ xf g ]( m s /sx ) (10)in which dependence on m s drops out except in the parton luminosity factor L g . This production is probedvia decays s → γγ , whose branching ratio is approximately B ( s → γγ ) = (8 / α /α s [68]. For the dominant s → gg decay into gluons, in principle LHC dijet resonance searches could be constraining, but these exist only for m s (cid:38)
500 GeV which is beyond the range of interest for the present study. To a good approximation, σ ( pp → s )is determined by m s and Λ. In Fig. 2(a) we show limits from ATLAS [69, 70] and CMS [71] on σB ( s → γγ )as a function of m s , along with the predictions for various Λ, and in Fig. 2(b) we show the associated lowerbounds on Λ. In the low-mass region (65 GeV < m s <
110 GeV), lower bounds on Λ range roughly from 400GeV to 650 GeV; in the intermediate-mass region (110 GeV < m s <
160 GeV), Λ is not bounded by diphotonresonance searches, and for much of the high-mass region ( m s >
160 GeV), Λ is bounded to be above 1 TeV. Forour subsequent scans of parameter space, we adopt a fixed reference value for Λ,Λ ref = 540 GeV , (11)which is large enough to be consistent with much of the low- m s region. Because Λ ref is well below the lower-boundson Λ in the high-mass region, we confine our scans to m s <
160 GeV for consistency. We show the constraints from precision electroweak data, diphoton resonance searches, and vector-like quarksearches in the η - η plane in Fig. 3, for M T = 800 GeV, where we approximate the T search constraints by therequirement B ( T → st ) > .
9, and for M T = 1300 GeV, heavy enough to evade T searches for any B ( T → st ).For the chosen m s , it is apparent that the reference value Λ = 540 GeV is attainable for η > ∼ . M T = 800GeV and η > ∼ M T = 1300 GeV. For slightly heavier s in the window 110 GeV < m s <
160 GeV, diphotonresonance searches are evaded and the red contours disappear. In this case even lower values of Λ are allowedprovided one is willing to consider larger values of η .Because the baryon asymmetry η b scales roughly as 1 / Λ, it is straightforward to reinterpret our final results forlarger (or smaller) Λ. For example, from Fig. 5(c) one can infer that a significant fraction of points remain viablefor baryogenesis for Λ = 2Λ ref (or for even larger Λ), a scale consistent with more modest couplings, η ∼ . η could invalidate the effective theory above the heavy top partner threshold M at scales only slightly larger than M , which would require us to specify additional new physics in order to have Although we do not pursue this point here, lower values of Λ are consistent with m s >
160 GeV if B ( s → γγ ) is suppressed dueto, for example, a dominant invisible decay channel; LHC constraints on tt plus missing energy [72, 73] are in that case evaded for M T > ∼ ����������������������������������� ��� ��� ��� ��� ������������������������ η � η � � � = ��� ��� � � = ��� ��� ����������������������������������� ��� ��� ��� ��� ������������������������ η � η � � � = ���� ��� � � = ��� ��� FIG. 3. For selected T and s masses, constraints on η and η from precision electroweak data (green), diphoton resonancesearches [70, 71](red), and searches for vector-like quarks [62] (blue), along with contours of Λ in GeV. The allowed regionis unshaded. a complete description. There are two principal challenges arising from the running of the couplings, dη d ln µ ∼ = η π (12) dλ s d ln µ ∼ = 9 λ s π − η π + λ s η π (13)where µ denotes the renormalization scale. The most serious problem is that for large values of η , the self-coupling λ s is quickly driven to zero, and the scalar potential becomes unstable. The second is that η reaches aLandau pole at somewhat higher scales. The first problem could be ameliorated by coupling additional scalars to s , without impacting our results for EWBG or GWs. For this reason, we do not limit the scope of our investigationbased on the running of λ s . Regarding the second problem, we note that even for η = 3, the Landau pole isnearly an order of magnitude above M , which we consider to be an acceptably large range of validity for theeffective theory. B. Explicit breaking of Z symmetry Since we are considering a scenario where the Z symmetry s → − s is spontaneously broken during the earlyuniverse and restored at the EWPT, domain walls form before the EWPT, and the universe will consist of domainswith random signs of the s condensate. The source term for EWBG that arises from Eq. (2) is linear in s , resultingin baryon asymmetries of opposite signs, that could average to zero after completion of the EWPT. To avoid thisoutcome, the Z symmetry should be explicitly broken, by potential terms V b = µ b s ( h − v ) + µ (cid:48) b s (14)with small coefficients µ b , µ (cid:48) b . We have used the freedom of shifting s by a constant to remove a possible tadpoleof s at the true vacuum ( h, s ) = ( v, V b can ameliorate the situation in several ways. First, if the transition tothe broken s -phase is of second order, even a small tilt can suffice to make the lower-energy vacuum dominate.Second, in a first order transition, symmetry breaking terms can bias the bubble nucleation rates to prefer thelower-energy vacuum. Indeed, the number of bubbles nucleated during the transition is n ∼ (cid:82) t ∗ t c d t Γ( t ), where t ∗ is the time when transition completes, and Γ( t ) ∼ exp( − S /T ). Writing the action as S ± = ¯ S ∓ δS in thetwo respective vacua, the relative number density of bubbles in each phase at the end of the transition becomes n + /n − ≈ exp(2 δS ∗ /T ∗ ). In general [74] S ∝ E , where E is the coefficient of the cubic term in the potential. Usingthis scaling we may write δS ∗ = ( δE/E ) ¯ S ∗ , where typically S ∗ /T ∗ ≈ E ≈ (3 λ s ) / T / π ,so taking V b = µ b (cid:48) s , corresponding to δE = µ b (cid:48) , and T ∗ ≈
100 GeV, the condition for single-phase vacuumdominance becomes µ b (cid:48) > ∼ . λ / s GeV. Barring very large λ s , this condition is easily met with no limitations onour analysis.Even if a domain wall network forms, the higher-energy domains will collapse due to pressure gradients, and weshould ensure that this process completes before the EWPT. The collapse starts with the acceleration of a wall atrelative position R according to ¨ R = − ∆ V /τ , where τ ∼ √ λ s w is the surface tension (distinct from the tension σ used above in the nucleation estimate), ∆ V ∼ V b (0 , w ) ∼ mu (cid:48) b w is the difference in the vacuum energies, and w ∼ µ s / √ λ s is the singlet VEV. Using H = 1 / t and T ≈
100 GeV, one finds that walls reach light speed in time δt/t = τ H/δV ∼ − √ λ s (eV /µ b (cid:48) ), which is practically instantaneous on the timescales of interest, for reasonablevalues of µ b (cid:48) . The higher energy domains subsequently collapse at the speed of light, since there is no appreciablefriction. The time required for this process to complete is determined by R ∗ = 2 a ( t ) (cid:90) t t dta ( t ) , (15)where R ∗ is the comoving size of the domain wall separation. By the Kibble mechanism one expects that R ∗ = AH − ∗ with A (cid:46)
1, leading to the ratio of domain wall collapse to formation times t /t = (1 + A/ .The temperature interval corresponding to this time interval is ∆ T /T ≈ A , assuming that the growth phase alsoproceeded at the speed of light.The temperature of the first phase transition, T can be estimated as that when ∂ V /∂s becomes negative. Inthe approximation of neglecting V b and keeping only leading terms in the high-T expansion one finds T − T c ∼ λ h w c /c s where T c is the critical temperature of the EWPT, and c s = (3 λ s + 2 λ hs ) /
12. Thus the temperaturedifference between transitions is of order ∆ T c ∼ λ h w / ( c s T c ). Requiring that ∆ T c /T c > A then gives A < λ h λ s + 2 λ hs w c T c . (16)Given that A ∼ ( T ∗ /S ∗ )(∆ T /T ) ∗ ∼ − − − [43] this is a very weak constraint. We conclude that it is easyto avoid cosmological problems associated with the domain walls by small symmetry breaking terms which do notaffect the rest of our analysis. III. PHASE TRANSITION AND BUBBLE NUCLEATION
In the examples of interest for this work, the phase transition in the Z -symmetric singlet model proceeds intwo steps: starting from the high-temperature global minimum h = s = 0, a transition first occurs to nonzero s ,while the Higgs field remains at h = 0. This is followed by the EWPT, in which s returns to zero and h developsits VEV. The h s interaction provides the potential barrier to make this a first order transition.As usual, the first order transition occurs at the bubble nucleation temperature T n , which is below the criticaltemperature T c , where the two potential minima become degenerate, V eff ( h, s, T c ) | h =0 ,s = wc = V eff ( h, s, T c ) | h = vc,s =0 (17)Bubble nucleation occurs when the vacuum decay rate per unit volume Γ d becomes comparable to H , the Hubblerate per Hubble volume. The decay rate is [75]Γ d ∼ = T (cid:18) S πT (cid:19) / exp (cid:18) − S T (cid:19) , (18)where S is the O(3) symmetric action, S = 4 π (cid:90) r dr (cid:32) (cid:18) dhdr (cid:19) + 12 (cid:18) dsdr (cid:19) + V eff (cid:33) . (19)The precise criterion that we use for nucleation isexp ( − S /T n ) = 34 π (cid:18) H ( T n ) T n (cid:19) (cid:18) πT n S (cid:19) / , (20)which is satisfied when S /T n ∼ = 140 [76]. We used the package CosmoTransitions [77] to calculate S . The actionobtained with the full potential can differ significantly from the commonly used thin wall approximation [78, 79]or the approximation of evaluating it along the minimal integration path for the potential [37]. We compare thepredictions for nucleation of these approximations to the full one-loop result, for several exemplary models, inTable III. The approximate methods tend to underestimate the action, giving a higher nucleation temperature;hence we use the values derived from the full one-loop action in the following. λ hs m s (GeV) S /T | T =100 GeV T n (GeV)Thin wall MPP 1-loop Thin wall MPP 1-loop1 120 234 277 427 93.5 92.6 89.81.7 200 68.7 101 151 115.6 109.8 100.13.2 300 37.9 36.8 54.3 134.3 133.8 121.6TABLE I. Examples of the dimensionless tunneling action S /T , evaluated at T = 100 GeV, and ensuing nucleationtemperatures, computed within the thin wall and minimal potential path (MPP) approximations, compared with the valueobtained using the resummed one-loop potential. In there example, λ s = 1 and Λ = 540 GeV. There are two complementary parameters for characterizing the strength of the first order transition. One isthe ratio of the Higgs VEV to the temperature at the time of nucleation, v n /T n , which is especially relevant forEWBG, as we will discuss in Sect. V B. The other, which is more important for GW production, is the ratio ofreleased vacuum energy density to the radiation energy density [80, 81]: α = 1 ρ γ (cid:18) ∆ V − T n dVdT (cid:19) , (21)where ρ γ = g ∗ π T n / g ∗ is the effective number of degrees of freedom in the plasma (we use g ∗ = 106 . α quantifies the amount of supercool-ing that occurs prior to nucleation, which determines how much free energy is available for the production of GWs. IV. WALL VELOCITY AND SHAPE
The derivation of the wall velocity and field profiles is a technically demanding problem [43], that was firstaddressed in the context of Higgs plus singlet models in Refs. [47, 49, 82], in various approximations. One mustsolve the equations of motion (EOM) for the scalar sector coupled to a perfect fluid, E h ( z ) ≡ − h (cid:48)(cid:48) ( z ) + dV eff ( h, s ; T + ) dh + (cid:88) i N i dm i dh (cid:90) d p (2 π ) E δf i ( (cid:126)p, z ) = 0 ,E s ( z ) ≡ − s (cid:48)(cid:48) ( z ) + dV eff ( h, s ; T + ) ds + (cid:88) i N i dm i ds (cid:90) d p (2 π ) E δf i ( (cid:126)p, z ) = 0 , (22)where z is the direction normal to the wall, that is to a good approximation planar by the time it has reachedits terminal velocity. We use a sign convention where the wall is moving to the left, so that z > h or s in the plasma, with N i and m i respectively denoting the number of degrees of freedom and the field-dependent mass of the corresponding species,and δf i the deviation from equilibrium of its distribution function. All the temperature-dependent quantitiesappearing in these equations are evaluated at T + , which is the plasma’s temperature just in front of the wall. Wecalculate T + in Appendix B using the method described in Ref. [81], and δf i will be computed in Sect. IV A.The terms in Eqs. (22) with δf i represent the friction of the plasma on the wall, that leads to a terminal wallspeed v w <
1, unless the friction is too small and the wall runs away to speeds close to that of light. Following The term “friction” is strictly speaking not correct, but we adopt this commonly used terminology. More accurately, the last termsin (22) represent the additional pressure created by the out-of-equilibrium perturbations, which modify the effective action in thesame way as the usual thermal excitations. previous work, we take the dominant sources of friction to be from the top quark ( i = t ) and electroweak gaugebosons ( i = W ), neglecting the contributions to friction from the Higgs itself and from the singlet. This approxi-mation is bolstered by the smaller number of degrees of freedom N h = N s = 1 compared to N t = 12 and N W = 9,as well as the smallness of the Higgs self-coupling λ h and the not-too-large values of the cross-coupling λ hs thatwill be favored in the subsequent analysis. Then the friction term for the s equation of motion vanishes, since s couples only to itself and to the Higgs, apart from its suppressed dimension-5 coupling to t . This allows for somesimplification in the following procedure.In Ref. [54], a similar study of the present model was done, where no a priori restriction of the wall shape wasassumed, but it was found that the actual shapes conform to a very good approximation to the tanh profiles h ( z ) = h z/L h )] ,s ( z ) = s − tanh( z/L s + δ )] , (23)where h and s are respectively the vacuum expectation values (VEV) of the h and s fields in the broken andunbroken phases. Hence we adopt the ansatz (23), which allows the singlet and Higgs wall profiles to have differentwidths, and to be offset from each other by a distance L s δ . The s field’s VEV is taken to be the usual one evaluatedat T + , which solves the equation dV eff (0 , s ; T + ) /ds (cid:12)(cid:12) s = s = 0. The situation is more complicated for the h field, forwhich the Higgs VEV should be evaluated at T − , the plasma’s temperature behind the wall. Since we are fixinga constant temperature T + in the potential, the change in the effective action due to the shift in the backgroundtemperature must be accounted for by the perturbation in the broken phase. As a consequence we are choosing h so that it solves the equation (cid:32) dV eff ( h, T + ) dh + (cid:88) i N i dm i dh (cid:90) d p (2 π ) E δf i ( (cid:126)p, z ) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h = h ,z →∞ = 0 . (24)This choice guarantees that the Higgs EOM is satisfied far behind the wall. We will estimate the uncertainty ofour results due to this approximation in Sect. VI D.To approximately solve the Higgs EOM, one can define two independent moments M , of E h ( z ), and assumethat they both vanish at the optimal values of v w and L h . A convenient choice is [49] M ≡ (cid:90) dz E h ( z ) h (cid:48) ( z ) = 0 , (25) M ≡ (cid:90) dz E h ( z )[2 h ( z ) − h ] h (cid:48) ( z ) = 0 . (26)These also have nice physical interpretations that naturally distinguish them as good predictors of the wall speedand thickness, respectively. M is a measure of the net pressure on the wall, so that Eq. (25) can be interpretedas the requirement that a stationary wall should have a vanishing total pressure; nonvanishing M would causeit to accelerate. Therefore one expects that Eq. (25) principally determines the wall speed v w , while dependingonly weakly on the thickness L h . With that sign convention, M can be interpreted as the pressure in front ofthe wall minus the pressure behind it, so that M > M is a measure of the pressure gradient in the wall. If nonvanishing, it would lead to compressionor stretching of the wall, causing L h to change. Hence Eq. (26) mainly determines L h , and depends only weaklyon v w . The two equations are approximately decoupled, facilitating their numerical solution. This is illustratedin Fig. 4, which shows the dependence of M and M on v w and L h .We chose a different approach to determine the singlet wall parameters L s and δ . Instead of solving momentequations analogous to (25,26), one can determine their values by minimizing the s field action S ( L s , δ ) = (cid:90) dz (cid:26)
12 ( s (cid:48) ) + [ V eff ( h, s, T + ) − V eff ( h, s ∗ , T + )] (cid:27) = s L s + (cid:90) dz [ V eff ( h, s, T + ) − V eff ( h, s ∗ , T + )] , (27)with respect to L s and δ . Here s ∗ is a field configuration with arbitrary fixed parameters L ∗ s and δ ∗ , that wechoose to be L ∗ s = L h and δ ∗ = 0. The second term is just a constant, but it allows for the convergence of the0 v w L h T n M / T n v w L h T n M / T n (a) (b)FIG. 4. Moments of the Higgs EOM (a) M and (b) M as a function of the wall velocity v w and the Higgs wall width L h for a model with parameters λ hs = 1, λ s = 1 and m s = 130 GeV. The red dot is the solution of Eqs. (25,26). Asexpected, M is roughly independent of L h while M depend mainly on L h . The moments are discontinuous at v w ≈ . v + and T + are discontinuous. integral by canceling the contributions of V eff at z → ±∞ . This method has the advantage that it does notdepend on any arbitrary choice of moments, and it is more efficient to numerically minimize the function of twovariables than to solve the system of equations for the moments of the EOMs. A. Transport equations for fluid perturbations
The final step toward the complete determination of the velocity and the shape of the wall is to computethe distribution functions’ deviations from equilibrium δf i , by solving the Boltzmann equation for each relevantspecies in the plasma. The method of approximating the full Boltzmann equation by a truncated set of coupledfluid equations was originally carried out in Ref. [43], for the regime of slowly-moving walls (see also Ref. [49]).This approach was recently improved in Ref. [56] in order to be able to treat wall speeds close to or exceedingthe speed of sound consistently. We briefly summarize the formalism, which has been adopted for the present study.The out-of-equilibrium distribution function can be parametrized in the wall frame as f = 1exp[ βγ ( E − v + p z )(1 − δτ ) − µ ] ± δf u , (28)where β = 1 /T + and the ± is + for fermions and − for bosons. δτ and µ are the dimensionless temperature andchemical potential perturbations from equilibrium, and δf u is a velocity perturbation whose form is unspecified,but is constrained by (cid:82) d p δf u = 0. By assuming that the perturbations are small, one can expand f to linearorder in µ , δτ and the velocity perturbation δf u to obtain δf ≈ δf u − f (cid:48) [ µ + βγδτ ( E − v + p z )] , (29)with f (cid:48) = ddX e X ± (cid:12)(cid:12)(cid:12)(cid:12) X = βγ ( E − v + p z ) . (30)To simplify the problem, one models the plasma as being made of three different species: the top quark, the W bosons (shorthand for W ± and Z ) and a background fluid, which includes all the remaining degrees of freedom.It is convenient to write the velocity perturbation as u ∝ (cid:82) d p ( p z /E ) δf u when constructing the moments of thelinearized Boltzmann equation. By taking three such moments, using the weighting factors 1, E and p z /E , theperturbations are determined by transport equations Aq (cid:48) + Γ q = S, (31) q (cid:48) bg = − ˜ A − (Γ bg ,t q t + Γ bg , W q W ) , (32)1where prime denotes d/dz , q i = ( µ i , δτ i , u i ) (cid:124) , q = ( q (cid:124) W , q (cid:124) t ) (cid:124) , the Γ matrices are collision terms, and S is the sourceterm, whose definitions, as well as those of the the matrices A , Γ, ˜ A − , Γ bg ,t , Γ bg , W , can be found in Ref. [56].If A and Γ were independent of z , one could use the Green’s function method to solve Eq. (31); however, A is afunction of m i ( z ) /T . To deal with this dependence on z , we discretize space, z → z + n ∆ z with n = 0 , · · · , N − πi ∆ z (cid:18) kN − (cid:22) kN (cid:23)(cid:19) ˜ q k + 1 N N − (cid:88) l =0 (cid:94) ( A − Γ) ( k − l ) mod N ˜ q l = (cid:94) ( A − S ) k , k = 0 , · · · , N − , (33)where the tilde denotes the discrete Fourier transform. This is a linear system that is straightforward to numeri-cally solve for ˜ q k . Once ˜ q k is known, it can be transformed back and interpolated to obtain q ( z ). Eq. (32) canthen be integrated using a Runge-Kutta algorithm.Finally, one can substitute Eq. (29) into the Higgs EOM (22) to express the friction in terms of the fluidperturbations µ i , δτ i and u i . This leads to the result (cid:90) d p (2 π ) E δf i = T (cid:104) C , µ i + C , ( δτ i + δτ bg ) + D , − v ( u i + u bg ) (cid:105) , (34)where the functions C m,nv and D m,nv can be found in Ref. [56]. V. COSMOLOGICAL SIGNATURES
We have now established the machinery needed to compute all the relevant properties of the first order phasetransition bubbles, starting from the fundamental parameters of the microscopic Lagrangian. In this section wedescribe how to apply these results for the estimation of GW spectra and the baryon asymmetry.
A. Gravitational Waves
We follow the methodology of Refs. [7, 81, 83] to estimate future gravitational wave detectors’ sensitivity to theGW signals that can be produced by a first-order electroweak phase transition in the models under consideration.The GW spectrum Ω gw ( f ) is the contribution per frequency octave to the energy density in gravitational waves, i.e., (cid:82) Ω gw d ln f is the fraction of energy density compared to the critical density of the universe. The spectrum getsseparate contributions from the scalar fields, sound waves in the plasma and magnetohydrodynamical turbulencecreated by the phase transition: Ω gw ( f ) = Ω φ ( f ) + Ω sw ( f ) + Ω m ( f ) , (35)Each of these contributions depends on the wall velocity v w , the supercooling parameter α (Eq. (21)), and theinverse duration of the phase transition, defined as β = H ( T n ) T n ddT S T (cid:12)(cid:12)(cid:12)(cid:12) T = T n . (36)Another useful quantity is the mean bubble separation, which can be written in terms of v w and β as [7] R = (8 π ) / β max[ c s , v w ] . (37)It has been shown in Ref. [50] that interactions with gauge bosons prevent the wall from running away indefinitelytowards γ → ∞ . In that case, the contribution from the scalar fields has been shown to be negligible. Further-more, the estimates for the magnetohydrodynamical turbulence are very uncertain and sensitive on the details ofthe phase transition dynamics [84] and are expected to be much smaller than the contribution from sound waves.Hence, we consider only the effects from the latter and set Ω m ( f ) = Ω φ ( f ) = 0. For convenience, we reproducethe numerical fits of the GW spectrums derived in Refs. [7, 81, 83] in appendix C.2We will use these predictions with respect to four proposed space-based GW detectors: LISA [85], AEDGE [11],BBO [86] and DECIGO [9]. A successful GW detection depends upon having a large enough signal-to-noise ratio[87], SNR = (cid:115) T (cid:90) f max f min df (cid:20) Ω gw ( f )Ω sens ( f ) (cid:21) (38)where Ω sens ( f ) denotes the sensitivity of the detector and T is the duration of the mission. The sensitivity curvesfor the detector LISA, BBO and DECIGO were obtained from Ref. [88]. Whenever SNR is greater than a giventhreshold SNR thr , we conclude that the signal can be detected. In general, this threshold can depend upon theconfiguration of the detector. For all the experiments, we take SNR thr = 10 and T = 1 . × s. In the following,SNR max will designate the maximum signal-to-noise ratio detected by one of the detectors:SNR max ≡ max[SNR LISA , SNR
AEDGE , SNR
BBO , SNR
DECIGO ] . (39)While Ω sens ( f ) can be obtained from the noise spectrum of a detector, it is not practical to compare it to the GWspectrum directly; one needs to compute the SNR to determine if signal is detectable. A useful tool for visualizingthe sensitivity of a detector is the power-law sensitivity curve (PLS) [89], defined as the envelope of the set ofpower-law curves, Ω gw = af b , that generate a SNR of SNR thr when observed over a duration T . By construction,every power-law curve above the PLS has SNR > SNR thr and can therefore be detected.
B. Baryogenesis
The mechanism of electroweak baryogenesis is sensitive to the speed and shape of the bubble wall during thephase transition. In most previous studies, these quantities were treated as free parameters to be varied, but inthis work we have already derived them, as was discussed in Section IV.An important requirement for EWBG is to avoid the washout, by baryon-violating sphaleron interactions, ofthe generated asymmetry inside the bubbles of broken phase, once they have formed. This leads to the well-knownconstraint [90] v n T n > . ∼
5% of viable models in the scan over parameter space to be described below.Near the bubble wall, CP-violating processes associated with the effective interaction in Eq. (2) give rise toperturbations of the plasma, that result in a local chemical potential µ B L for left-handed baryons, which byimposing the chemical equilibrium of strong-sphaleron interactions, is related to those of the t L , t cR and b L quarksby µ B L = 12 (cid:0) K t (cid:1) µ t + 12 (cid:0) K b (cid:1) µ b − K t µ t c , (41)where K a -functions were defined in [92] ( K a = D a in the notation of [55]). The µ B L potential biases sphalerons,leading to baryon number violation, whose associated Boltzmann equation can be integrated to obtain the baryonto photon ratio η b = 405 Γ sph π v w γ w g ∗ T (cid:90) dz µ B L f sph e − sph | z | / v w (42)where f sph quantifies the diminution of the sphaleron rate in the broken phase [93]. For AEDGE, we use the envelope of minimal strain that can be achieved by each resonance, with its width scaled to approximateΩ sens ( f ). This curve is expected to reproduce the correct SNR up to about 10%. The extra factor of γ w = 1 / (cid:112) − v w in the denominator was pointed out by Ref. [55]. µ t L , µ t cR and µ b L appearing in Eq. (41). They satisfy fluid equations resembling the network (31,32), except thatthe potentials relevant for EWBG are CP-odd, whereas those determining the wall profiles are CP-even.The CP-odd transport equations have been discussed extensively in the literature, leading to two schools ofthought as to how best to compute the source term for the CP asymmetries. These are commonly known asthe VEV-insertion [94, 95] or WKB (semiclassical) [96–101] methods, respectively. A detailed discussion andcomparison of the two approaches was recently given in Ref. [55], which quantified the well-known fact that theVEV-insertion source tends to predict a larger baryon asymmetry than the WKB source, by a factor of ∼
10. Inthe present work we adopt the WKB approach, which was updated in Ref. [55] to allow for consistently treatingwalls moving near or above the sound speed. In addition, that reference computed the source term arising fromthe same effective interaction (2) as in the present model, so we can directly adopt the CP-odd fluid equationsstudied there.
VI. MONTE CARLO RESULTS
In order to study the properties of the phase transition, we performed a scan over the parameter space of themodels, imposing several constraints. We found that variations in λ s do not qualitatively change the results,prompting us to initially fix its value at λ s = 1, leaving λ hs and m s as the free scalar potential parameters. Wewill first discuss this slice of parameter space, and later consider the quantitative dependence on λ s . We also choseΛ = 540 GeV, which is conservative since there are no collider constraints on its value for singlet masses in theregion m s = [110 , η b , whichis expected to scale roughly as 1 / Λ. Finally, in order to prevent Higgs invisible decays, we imposed m s > m h / λ i by small increments δ i . The trial model isadded to the chain using a conditional probability P = min (cid:20) v n /T n . , (cid:21) (43)that favors models having strong first order phase transitions, and for which a solution to the nucleation condition(20) can be found. We adjust the δ i so that roughly half of the models are kept, with larger values of δ i beingmore likely to result in a rejection.
75 100 125 150 m s (GeV) h s v w
75 100 125 150 m s (GeV) h s S N R m a x
75 100 125 150 m s (GeV) h s b / o b s (a) (b) (c)FIG. 5. Scan of the parameter space with λ s = 1 and Λ = 540 GeV. The colors represent (a) the terminal wall velocity v w , (b) the maximum signal-to-noise ratio of gravitational waves that could be detected by either LISA, AEDGE, BBO orDECIGO and (c) the baryon asymmetry (in units of the observed value) produced by the phase transition. The red dotsin (a) correspond to detonation solutions with v w ≈
1, and the latter are not included in (c) since they are expected toproduce a negligible baryon asymmetry. v w M / T n M ( v + , T + ) M ( v w , T n ) v w v + / v w T + / T n (a) (b)FIG. 6. Left (a): Pressure on the wall M as a function of the wall velocity v w . The solid (dashed) line corresponds to thepressure evaluated at the velocity v + ( v w ) and the temperature T + ( T n ). Right (b): Relation between the naive variables v w , T n and the ones relevant for evaluating M , namely v + and T + . Both plots were obtained using the parameters m s = 130 GeV, λ hs = λ s = 1 and L h = 5 /T n . The shaded region corresponds to hybrid wall solutions characterized by c s < v w < ξ J . This procedure yielded 833 models with strong phase transitions, of which 708 were amenable to finding so-lutions for the moment equations (25-26). Our analysis is expected to hold for γ (cid:46)
10. Above that bound, ouralgorithm becomes numerically unstable which prevents it from giving any trustworthy result. It is thereforeimpossible to determine the type of solution of the 125 remaining models using our method alone: they couldeither stabilize at an ultra-relativistic speed satisfying 10 (cid:46) γ < ∞ , or run away indefinitely towards γ → ∞ . Thevalue of the baryon asymmetry should not be affected by this ambiguity since it is negligible for v w ≈
1. However,the GW spectrum produced during the phase transition is sensitive to the type of solution since runaway wallshave a non-negligible fraction of their energy stored in the wall, while for non-runaway walls, the energy getsdissipated into the plasma, so the fraction of energy in the wall becomes negligible. This ambiguity can be liftedusing the result of Ref. [50], which found that in the limit γ → ∞ , interactions between gauge bosons and thewall create a pressure proportional to γ , preventing it from running away . We can therefore assume that the 125models without a solution to the moment equations (25-26) correspond to non-runaway walls with v w ≈ λ hs versus m s . A. Deflagration versus detonation solutions
A striking feature of these results is that all the detonation solutions have v w ≈ We have tested that thisis not specific to our choice of fixed parameter values, but also holds for all models having 0 . < λ s < >
110 GeV; hence it seems to be a general property of phase transitions in the Z -symmetric singlet framework.One can understand this behavior by considering the net pressure opposing the wall’s expansion, M (recall Eq.(25-26)), as a function of the wall velocity, as illustrated in Fig. 6. It shows how M differs when evaluatedwith the appropriate quantities v + , T + rather than the incorrect ones v w , T n . Using the latter, we would find nosolution to the equation M = 0 for the exemplary model used in Fig. 6, and would then incorrectly conclude thatit satisfies v w ≈
1. The relevant quantities are those measured right in front of the wall, v + and T + . The speed v + is smaller than v w for v w < ξ J , which would lower the pressure against the wall ( ξ J is the Jouguet velocity,defined as the smallest velocity a detonation solution can have). However, in the same region, the temperature T + is larger than T n , which causes the pressure to increase. The latter effect turns out to dominate over theformer. Indeed, the actual pressure, represented by the solid blue line in Fig. 6, increases much more rapidly than More recently, the authors of Ref. [51] have carried out an all-orders resummation at leading-log acuracy, finding that the pressureis in fact proportional to γ for fast-moving walls. Strictly speaking there are models with v w < v w = 0 and accelerates until it reaches the solution with the lowest velocity. v w v + v w L h T n L h T n L s T n (a) (b) (c)FIG. 7. Shape and velocity of the deflagration solutions. (a) Correlation between the wall velocity v w and the fluid velocityin front of the wall, v + ; (b) dimensionless wall width L h × T n versus v w ; and (c) correlation of the s and h wall widths.Colors indicate the supercooling parameter α (Eq. (21)) in (a,b), or the wall offset δ (Eq. (23)) in (c). M ( v w , T n ) close to the speed of sound. This qualitative difference allows for a solution to M = 0, which wouldhave been missed if we had used the naive quantities v w and T n .We find that the previous statements apply quite generally: for all models, T + > T n when v w < ξ J , and thisalways leads to a much higher pressure on the wall, even if the difference between T + and T n is quite small;the pressure barrier at v w = ξ J is always greater than the maximum possible value for a detonation solution.Therefore, if the phase transition is strong enough to overcome the pressure barrier at ξ J , the solution becomes adetonation, but the pressure in the region v w > ξ J is never enough to prevent it from accelerating towards v w ≈ solutions are shown in Fig. 7, which demonstratesthat the behaviors for subsonic (deflagration) and supersonic (hybrid) walls are rather different. Subsonic wallsgenerally have v + ≈ v w , which is expected since the fluid should not be strongly perturbed by a slowly movingwall. The wall width is not uniquely determined by v w , but there exists a clear correlation, with slower wallsbeing thicker. For supersonic cases, the correlation between v + and v w gets inverted: higher wall velocity leadsto lower v + . The wall width becomes uniquely determined by v w and the relation between these two variablesis to a good approximation linear. One observes that stronger phase transitions, quantified by higher values of α , generally produce faster and thinner walls. Even for the strongest transitions our walls still have thickness LT > ∼
3. Since the semiclassical force mostly affects particles with momenta (cid:104) k z (cid:105) ∼ T , we find L (cid:104) k z (cid:105) > ∼
3, so thatthe semiclassical approximation is still valid. In fact the semiclassical picture has been shown to remain valid forsurprisingly narrow walls [102], working very well for L (cid:104) k z (cid:105) ≈ L (cid:104) k z (cid:105) ≈
2. There is alinear correlation between the h and s wall widths, but the slope is not 1; in all cases, we find that L h > L s . Thedistribution of wall offset values δ is also indicated in Fig. 7(c). B. Baryogenesis and gravity wave production
Of the 833 sampled models, 513 are able to generate the baryon asymmetry at a level large enough to agreewith observations, and 321 can produce observable gravitational waves. More detailed results are presented inTable II. The complementarity of the experiments considered here, with respect to the present model, can beappreciated by considering the relation between the maximum GW amplitude max[Ω gw h ] and the frequency ofthis peak amplitude f max , as shown in Fig. 8 (a). While a large fraction of the models are above BBO’s andDECIGO’s PLS, the peak frequency of the strongest detonation walls are positioned exactly in LISA’s region of Henceforth we take “deflagration” to also include hybrid solutions f max (Hz) m a x [ g w h ] LISAAEDGEBBODECIGODeflagrationDetonation f (Hz) g w h l o g ( S N R m a x ) f (Hz) g w h l o g ( S N R m a x ) (a) (b) (c)FIG. 8. (a): Maximum amplitude of GW as a function of the peak frequency f max with the PLS (solid line) and thesensitivity Ω sens h (dashed line) of the four considered detectors. (b) and (c): Spectrum of GW produced by the 10 modelswith the highest SNR max for (b) deflagration and (c) detonation solutions. maximal sensitivity, and the same is true for the deflagration walls and three other detectors. Consideration ofthe complete spectrum’s shape, shown in Fig. 8 (b,c) for deflagration and detonation solutions respectively, showsthat even if the detonation spectrum peaks in LISA’s, BBO’s and DECIGO’s PLS, the tail of the distributioncan be above that of AEDGE. This implies that detonation walls can be probed by all the detectors, while onlyAEDGE, BBO and DECIGO can efficiently observe deflagration walls.In previous studies, where the wall velocity was considered as a free parameter, there was an expectation thatbaryogenesis would be less efficient with increasing v w , whereas gravity waves would become more so. In thepresent study, where v w is not adjustable but is a derived parameter, we surprisingly find that rather than EWBGand observable GWs being anticorrelated, instead they are positively correlated, as is illustrated in Fig. 9 (a).This can be understood from the fact (see Fig. 7 (b)) that L h is a decreasing function of v w , which enhancesEWBG. Moreover, the relevant velocity for EWBG is v + , which is a decreasing function of v w for supersonicwalls, and is bounded by v + < c s ; this effect also enhances EWBG for fast-moving walls. The actual relationbetween η b and v w is shown in Fig. 9 (b) and, at least for supersonic walls, there is a positive correlation betweenthese two variables, which results in models having both high GW production and baryon asymmetry. Fig. 9also indicates that supercooling parameter α is positively correlated with both η b and SNR max : stronger phasetransitions generally lead to both higher GW and baryon production.Detailed predictions for EWBG in the Z symmetric model were previously made in Refs. [37] and [29], asopposed to merely requiring the sphaleron bound (40) to be satisfied. Comparisons with the present work arehindered by the fact that different source terms for the CP asymmetry were assumed. In Ref. [37], the dimension-6 coupling i ( y t / √ s/ Λ) ¯ ht L t R was used, rather than the dimension-5 coupling in Eq. (2). Moreover, a value v w = 0 . L h = v n / √ V b was made for the wall width, where v n isthe Higgs VEV at the nucleation temperature, and V b is the potential barrier between the two minima. For thesame potential parameters ( λ s = 0 .
1) as in [37], we find no values of v w below 0.43, and our determination of L h is two to three times larger than the estimate in [37]. Both of these discrepancies would lead to overestimatingthe efficiency of EWBG, helping to explain why Ref. [37] obtains a high frequency of successful models, despitethe extra suppression that should result from using a dimension-6 source term.In Ref. [29], the dimension-5 coupling to leptons rather than the top quark was studied, and a different for-malism (the VEV insertion approximation) for computing the CP asymmetry was employed, which tends to givesignificantly larger estimates for the baryon asymmetry than the WKB method that we adopt [55]. Ref. [29] alsoused a different prescription for resumming thermal masses, which relies upon a high-temperature expansion forthe thermal self-energies, and different renormalization conditions. For the same parameters as in the benchmarkmodels given there, we do not find the right pattern of symmetry breaking for the phase transitions, furtherimpeding meaningful comparisons of our respective results.7 b / obs S N R m a x v w b / o b s (a) (b)FIG. 9. (a): Relation between the SNR max and the baryon asymmetry produced by the phase transition. (b): Baryonasymmetry as a function of the wall velocity. Both plots only show the deflagration models. C. Dependence on λ s and Λ To study the quantitative dependence on the singlet self-coupling λ s , we performed 3 other scans similar tothe one previously described, taking λ s = 0 . , . λ s (cid:38) .
01 (0 . λ s (cid:38) .
1. These results confirm that LISA would only be suitable for probing detonation walls , which are notgood candidates for EWBG. The three other detectors would be sensitive to both types of solutions. Increasing λ s generally leads to stronger phase transitions, resulting in more models with successful EWBG and detectable GWs.The value of Λ (recall Eq. (4)) can in principle also have an effect on the strength of the phase transition,through the effective potential’s dependence on the top quark mass. The leading thermal term added to thepotential varies like h s T / Λ , which becomes negligible at high Λ, but could significantly modify the behaviorof the phase transition for Λ ∼ T n , resulting in a larger baryon asymmetry and GW production. We have verifiedthat this term is already subdominant when Λ = 540 GeV. However, for m s >
110 GeV, the weaker constraintsallow for values of Λ as low as 300 GeV, which could have an important effect on the phase transition.To test the sensitivity to lower values of Λ, we repeated the previous scans using Λ = Λ min ( m s ), where Λ min isgiven by Λ min ( m s ) =
540 GeV , m s <
110 GeV300 GeV ,
110 GeV < m s <
160 GeV (44)The results are shown in Table II . As one could anticipate from the relation η b ∼ / Λ, EWBG is more efficientat lower values of Λ. One can also see that the number of detonation walls or walls generating detectable GWdoes not change substantially, which indicates that the lower values of Λ do not change the character of the phasetransition.
D. Theoretical uncertainties
In Ref. [56], the integrals that determine the collision rates Γ appearing in the Boltzmann equation network(31-32) were reevaluated, and it was noticed that the leading log approximation that was used in their derivationleads to theoretical uncertainties of O (1) in the fractional error. To study the impact of these uncertainties onour results, we recomputed the wall velocity with uniformly rescaled collision rates, Γ →
2Γ and Γ → Γ /
2. Theensuing variations of velocity ∆ v and wall width ∆ L are shown in Figs. 10 (a) and (b) respectively. The effect No deflagration solutions with SNR
LISA >
10 were found in the scans presented in Table II. The λ s = 0 .
01 scan is omitted since all accepted models satisfy m s <
110 GeV, making the results identical to those of the previousscan. Λ λ s η b /η obs > AEDGE >
10 SNR
BBO >
10 SNR
DECIGO >
10 Total SNR
LISA >
10 SNR
AEDGE >
10 SNR
BBO >
10 SNR
DECIGO > . − . . − . . . − . . . +0 . − . . +0 . . +2 . − . . . . − . . − . min TABLE II. Statistics from the scans performed with λ s = 0 . , . , , min . Each entry correspondsto the percentage of models satisfying the indicated constraint. In the row for λ s = 1 and Λ = 540 GeV, the exponents(indices) correspond to the error obtained by substituting the collision matrix Γ for 2Γ (Γ / min is the minimum valueof Λ allowed by laboratory constraints. on v w can be significant for slow walls, leading to a ±
40 % change when v w ∼ .
2. On the other hand for nearlysupersonic walls, v w (cid:38) c s , the wall speed is quite insensitive to Γ. The variation of L h is generally below 5%,much smaller than the corresponding variation in Γ.This behavior is not surprising since, near the speed of sound, the pressure on the wall is mainly determinedby the variation of T + , which does not depend on Γ. Likewise, the results for the baryon asymmetry and GWproduction turn out to be relatively robust against variations in Γ. This is demonstrated by the error intervals inthe λ s = 1 row of Table II. The error on the ratio of models satisfying η b /η obs > i >
10 is of order 10%,which is much smaller than the range of variation in Γ.Another source of uncertainty is the discrepancy between the temperatures computed with the Boltzmannequation (see Section IV A) and the conservation of the energy-momentum tensor (see Appendix B). Ideallyone should obtain T + = T BE ( z → −∞ ) and T − = T BE ( z → ∞ ), where T BE ( z ) = T + (1 + δτ bg ( z )) is the localtemperature calculated with the Boltzmann equation. The first condition is always satisfied since we impose theboundary condition δτ bg ( −∞ ) = 0, but we fail to recover the second one due to the different approximationsmade in the two methods. The discrepancy becomes larger as v w approaches the Jouguet velocity ξ J , where T + increases compared to T − ≈ T n (see Fig. 6 (b)). On the other hand, δτ bg does not change significantly in thesame region. Hence, we observe an error in the temperature of order ∆ T = T − − T BE ( ∞ ) ≈ T − − T + .Since the temperature is not accurate in the broken phase, the Higgs EOM is not automatically satisfiedasymptotically. To solve that problem, we shift the actual Higgs VEV h − evaluated in the broken phase by anamount − ∆ h , so that the adjusted VEV h = h − − ∆ h asymptotically solves the EOM (see Eq. (24)). This givesan additional source of uncertainty for v w and L h .We estimate the errors induced on v w and L h by ∆ T and ∆ h , assuming they are small enough to justifykeeping just the first order terms. Assuming that v w is completely determined by the solution of M = 0 and L h by M = 0, the error on these solutions can be obtained by expanding around the estimated values. For example,for the error in the wall velocity is estimated by0 = M ( v w + ∆ v, h + ∆ h, T ( z ) + ∆ T ( z )) ≈ M ( v w , h , T ( z )) + ∂M ∂v w ∆ v + (cid:90) dz δM δT ( z ) ∆ T ( z ) + ∆ h M , (45)where ∆ h M = M ( v w , h + ∆ h, T ) − M ( v w , h , T ), and we integrate over the temperature variation because M is a functional of T ( z ). Since v w is the solution of M ( v w , h , T ( z )) = 0, the absolute errors on v w and L h areestimated as | ∆ v | ≈ ( | ∆ T M | + | ∆ h M | ) (cid:12)(cid:12)(cid:12)(cid:12) ∂M ∂v w (cid:12)(cid:12)(cid:12)(cid:12) − , | ∆ L | ≈ ( | ∆ T M | + | ∆ h M | ) (cid:12)(cid:12)(cid:12)(cid:12) ∂M ∂L (cid:12)(cid:12)(cid:12)(cid:12) − , (46)where ∆ T M i = (cid:82) dz ( δM i /δT ( z ))∆ T ( z ). Notice that Eq. (46) overestimates the errors since ∆ T M i and ∆ h M i haveopposite signs. From Eqs. (22,25,26), one can see that the functional derivative δM i /δT ( z ) can be approximated9 v w v / v w v w L / L h v w | v |/ v w | L |/ L h (a) (b) (c)FIG. 10. (a) and (b): Relative changes ∆ v/v w and ∆ L/L h in the wall velocities and widths obtained by substitutingΓ →
2Γ or Γ / v w and L h due to the discrepancy between the temperatures computedwith the Boltzmann equation and the conservation of the energy-momentum tensor (see Eq. (49)). by ddT ( ∂V eff /∂h ), so that ∆ T M i ≈ (cid:90) dz ddT (cid:18) ∂V eff ∂h (cid:19) F i ( z )∆ T ( z ) , (47)where F = h (cid:48) and F = h (cid:48) (2 h − h ). We can simplify this integral with the approximation ∆ T ( z ) ≈ ( T − − T + )[1 +tanh( z/L h )] /
2. Furthermore, we approximate ddT (cid:0) ∂V∂h (cid:1) as being constant and half of its maximal value, occurringnear z = 0. Then ∆ T M i ≈
12 ( T − − T + ) C i ddT (cid:18) ∂V eff ∂h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) z =0 , (48)where C = (cid:82) dzF ( z )[1 + tanh( z/L h )] / h / C = h /
6. Substituting this expression in Eq. (46), wefinally obtain that the errors on v w and L h are given by | ∆ v | ≈ (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)
14 ( T − − T + ) h ddT (cid:18) ∂V eff ∂h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) z =0 + | ∆ h M | (cid:27) (cid:12)(cid:12)(cid:12)(cid:12) ∂M ∂v w (cid:12)(cid:12)(cid:12)(cid:12) − , | ∆ L | ≈ (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)
112 ( T − − T + ) h ddT (cid:18) ∂V eff ∂h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) z =0 + | ∆ h M | (cid:27) (cid:12)(cid:12)(cid:12)(cid:12) ∂M ∂L h (cid:12)(cid:12)(cid:12)(cid:12) − . (49)The relative errors are presented in Fig. 10 (c) for the scan with λ s = 1 and Λ = 540 GeV. The error on v w isbelow 7% for 97% of the models, and exhibits no strong correlation with v w . This happens because ∆ T = T − − T + and dM /dv w are roughly proportional (see Fig. 6), and therefore cancel each others’ contributions. The relativeerror on L h is small at low velocity (or large L h ), but becomes more significant near the speed of sound, howeverwithout ever exceeding 10%. E. Comparison of the GW signal with previous studies
We end this section with a brief comparison with recent studies of the GW produced during a first-orderelectroweak phase transition. With the prospect of the upcoming LISA experiment, numerous forecasts of theGW spectrum have been made for various extensions of the Standard Model [40, 42, 103–105]. Most of these findregions of model parameter space that would produce detectable GWs. Here we focus on studies of the singletscalar extensions [37, 38, 41, 106, 107].Our results agree qualitatively with the conclusions of previous work, in the prediction of GWs detectable byLISA, DECIGO and BBO. However there are distinctions stemming from differences in methodology. To compute0the GW contribution from the sound waves, previous authors used the numerical fit presented in Ref. [6], whilewe used the updated formula of Ref. [7, 83]. This leads to a smaller peak frequency, decreasing the number ofdetectable models.A more significant difference arises from our determination of the wall velocity, which was treated as a freeparameter in previous work, whereas we have computed it from the microphysics. The GW spectrum and hencesignal-to-noise ratio and ultimately the detectability are strongly dependent on the wall speed. For example, Ref.[106] assumed v w = 0 .
95 for all models, which considerably enhanced GW production and led to more optimisticpredictions. Moreover, using a fixed value for v w hides the discontinuous transition between the deflagration anddetonation solutions shown in Fig. 8 and in Table II. One could therefore incorrectly conclude that some modelsboth yield successful EWBG and GWs detectable by LISA. Our more detailed analysis indicates that LISA willonly probe detonation solutions with v w ≈
1, which are incompatible with EWBG.
VII. CONCLUSION
In this work we have taken a first step toward making complete predictions for baryogenesis and gravity wavesfrom a first order electroweak phase transition, starting from a renormalizable Lagrangian that gives rise to theeffective operator needed for CP-violation. This is in contrast to previous studies in which quantities like thebubble wall velocity or thickness were treated as free parameters, instead of being derived from the microphysicalinput parameters as we have done here. This is a necessary step for properly assessing the chances of havingsuccessful EWBG and potentially observable GWs, since the two observables are correlated in a nontrivial way,when they are both computed from first principles.We have incorporated improved fluid equations, both for the CP-even perturbations that determine the frictionacting on the bubble wall [56], and for the CP-odd ones that are necessary for baryogenesis [55], that can properlyaccount for wall speeds close to the sound barrier. Earlier versions of these equations were singular at the soundspeed, making reliable predictions impossible for fast-moving walls. Contrary to previous lore, we find that EWBGcan be more efficient for faster walls, due in part to the tendency for fast walls to be thinner.The Z -symmetric singlet model with vector-like top partners, analyzed in this work, was chosen for its sim-plicity, but the methods we used can be applied to other particle physics models that could enhance the EWPT.For example, singlets with no Z symmetry have additional parameters, and would thus be likely to have morefreedom to simultaneously yield large GW production and sufficient baryogenesis. It would be interesting toidentify other UV-completed models with these properties. A limitation we identified with the Z -symmetricmodel is that for the large values of the η coupling that are desired for EWBG, the singlet self-coupling is rapidlydriven toward zero by renormalization group running, above the top partner threshold. Acknowledgments.
We thank T. Flacke, M. Lewicki and G. Servant for helpful correspondence. The workof JC and BL was supported by the Natural Sciences and Engineering Research Council (Canada). The work ofKK was supported by the Academy of Finland grant 31831.
Appendix A: Effective Potential
We describe here the full effective potential used to describe the phase transition in the Z -symmetric singletmodel. It takes the general form V eff ( h, s, T ) = V tree ( h, s ) + V CW ( h, s, T ) + V T ( h, s, T ) + δV ( h, s ) . (A1) V tree is the scalar degrees of freedom’s tree-level potential obtained in the unitary gauge by setting in Eq. (1) H → h/ √ V BG term: V tree ( h, s ) = µ h h + λ h h + λ hs h s + µ s s + λ s s . (A2)1 V CW is the Coleman-Weinberg potential in the MS renormalization scheme that incorporates the vacuum one-loopcorrections and V T is the thermal potential: V CW ( h, s, T ) = 164 π (cid:88) i = W,Z,γ L , , ,χ,t n i ˜ M i ( h, s, T ) (cid:34) log ˜ M i ( h, s, T ) µ − C i (cid:35) ,V T ( h, s, T ) = (cid:88) i = W,Z,γ L , , ,χ,t n i T π (cid:90) ∞ dy y log (cid:104) ± e − √ y + M i ( h,s,T ) /T (cid:105) − ˜ gπ T , (A3)where the sums go over all the massive particles, including the thermal mass. Here, we include the contributionfrom the W and Z gauge bosons, the photon’s longitudinal polarization γ L , the Goldstone bosons χ , the topquark and the eigenvalues of the mass matrix of the Higgs boson and singlet scalar m and m . We impose therenormalization energy scale as µ = v , where v = 246 GeV is the Higgs vacuum expectation value. The ± in thethermal integral is + for fermion and − for bosons and ˜ g = (cid:80) B N B + (cid:80) F N F = 85 .
25 with the sums running overall the lighter degrees of freedom not included in the first term of V T . The C i ’s are constants given by C , ,χ,t = 3 / C W,Z,γ L = 5 / , (A4)and the n i ’s are the particle’s number of degrees of freedom: n W T = 4 , n W L = n Z T = 2 , n Z L = n γ L = 1 , n , = 1 , n χ = 3 , n t = − . (A5)We adopt the method developed by Parwani [59] to resum the Matsubara zero-modes for the bosonic degrees offreedom. It consists of replacing the bosons’ vacuum mass m i ( h, s ) by the thermal-corrected one M i ( h, s, T ) = m i ( h, s ) + Π i ( T ), with the self-energy given byΠ s ( T ) = (cid:18) λ s + 16 λ sh (cid:19) T , Π h ( T ) = Π χ ( T ) = (cid:20) (cid:0) g + g (cid:1) + 12 λ h + 14 y t + 124 λ hs (cid:21) T , Π W L ( T ) = 116 g T , Π W T ( T ) = Π Z T ( T ) = Π γ T ( T ) = 0 . (A6)The thermal masses for the longitudinal mode of the photon and Z boson are M Z L ( s, h, T ) = 12 (cid:20) m Z ( s, h ) + 116 g cos θ w T + ∆( s, h, T ) (cid:21) and M γ L ( s, h, T ) = 12 (cid:20) m Z ( s, h ) + 116 g cos θ w T − ∆( s, h, T ) (cid:21) , (A7)with ∆( s, h, T ) = (cid:20) m Z ( s, h ) + 113 g cos θ w cos θ w (cid:18) m Z ( s, h ) + 1112 g cos θ w T (cid:19) T (cid:21) / . (A8)At low temperature ( m i /T (cid:29) i becomes essentially absent from the plasma. This is manifestly the case for V T , since the thermalintegrals decay exponentially in the limit M i /T ≈ m i /T (cid:29)
1. However, in the same limit, V CW would dependquadratically on T if we used the thermal masses defined above. This would spoil the potential’s low- T behaviour.Therefore, we define a regulated thermal mass ˜ M i = m i + R ( m i /T )Π i , that should only be used in V CW . R ( x ) is a regulator chosen to recover the right behaviour in the low and high- T limit. In order to do so, it should For the photon and Z boson’s longitudinal mode, we define Π i = M i − m i , which should reproduce the desired behaviour. R ( x = 0) = 1 and R ( x ) ∼ e − √ | x | when | x | (cid:29)
1. We choose here the integratedBoltzmann number density function given by R ( x ) = 12 [ x ] K (cid:16)(cid:112) [ x ] (cid:17) , (A9)where K is the modified Bessel function of the second kind and [ x ] = x tanh( x ) is a smoothed absolute value.The last term of Eq. (A1) contains the following counterterms: δV ( h, s ) = Ah + Bh + Cs + D, (A10)which are fixed by requiring the renormalization conditions0 = ∂V eff ∂h (cid:12)(cid:12)(cid:12)(cid:12) h = v,s =0 ,T =0 m h = ∂ V eff ∂h (cid:12)(cid:12)(cid:12)(cid:12) h = v,s =0 ,T =0 m s = ∂ V eff ∂s (cid:12)(cid:12)(cid:12)(cid:12) h = v,s =0 ,T =0 V eff | h = v,s =0 ,T =0 . (A11) Appendix B: Relativistic fluid equation
We here calculate the hydrodynamical properties of the plasma close to the wall using the method described inRef. [81]. The quantities of interest are the temperatures T ± and the velocities of the plasma measured in the wallframe v ± . The subscript + and − indicate that the quantity is measured in front or behind the wall respectively.By integrating the conservation of the energy-momentum tensor equation across the wall, one can show thatthe quantities T ± and v ± are related by the equations v + v − = 1 − (1 − α + ) r − α + ) r ,v + v − = 3 + (1 − α + ) r α + ) r , (B1)where α + and r are defined as α + ≡ (cid:15) + − (cid:15) − a + T ,r ≡ a + T a − T − ,a ± ≡ − T ± ∂V eff ∂T (cid:12)(cid:12)(cid:12)(cid:12) ± ,(cid:15) ± ≡ (cid:18) − T ± ∂V eff ∂T + V eff (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ± . (B2)These quantities are often approximated by the so-called bag equation of state, which is given in Ref. [81]. Thisapproximation is expected to hold when the masses of the plasma’s degrees of freedom are very different from T ,which is not necessarily true in the broken phase. Therefore, we keep the full relations (B2) in our calculations.Subsonic walls always come with a shock wave in front of the phase transition front. The Eqs. B1 can be usedto relate T ± and v ± at the wall and the shock wave, but we need to understand how the temperature and fluidvelocity evolve between these two regions. Assuming a spherical bubble and a thin wall, one can derive from theconservation of the energy-momentum tensor the following differential equations2 vξ = γ (1 − vξ ) (cid:18) µ c s − (cid:19) ∂ ξ v,∂ ξ T = T γ µ∂ ξ v, (B3)3where v is the fluid velocity in the frame of the bubble’s center and ξ = r/t is the independent variable, with r the distance from the bubble center t the time since the bubble nucleation. With that choice of coordinates, thewall is positioned at ξ = v w . µ is the Lorentz-transformed fluid velocity µ ( ξ, v ) = ξ − v − ξv , (B4)and c s is the speed of sound in the plasma c s = ∂V eff /∂TT ∂ V eff /∂T ≈ . (B5)The last approximation is valid for relativistic fluids, which models well the unbroken phase. In the broken phase,the particles get a mass that can be of the same order as the temperature, and it causes the speed of sound tobecome slightly smaller.One can find three different types of solutions for the fluid’s velocity profile: deflagration walls ( v w < c − s ) havea shock wave propagating in front of the wall, detonation walls ( v w > ξ J ) have a rarefaction wave behind it andhybrid walls ( c − s < v w < ξ J ) have both shock and rarefaction waves. ξ J is the model-dependent Jouguet velocity,which is defined as the smallest velocity a detonation solution can have. Each type of wall have different boundaryconditions that determine the characteristics of the solution. Detonation walls are supersonic solutions where thefluid in front of the wall is unperturbed. Therefore, it satisfies the boundary conditions v + = v w and T + = T n .For that type of solution, Eqs. (B1) can be solved directly for v − and T − .Subsonic walls always have a deflagration solution with a shock wave at a position ξ sh that solves the equation v − sh ξ sh = ( c + s ) , where v − sh is the fluid’s velocity just behind the shock wave measured in the shock wave’s frame.It satisfies the boundary conditions v − = v w and T + sh = T n . Because these boundary conditions are given at twodifferent points, the solution of this system can be somewhat more involved than for the detonation case. Indeed,one has to use a shooting method which consists of choosing an arbitrary value for T − , solving Eqs. (B1) for T + and v + , integrating Eqs. (B3) with the initial values T ( v w ) = T + and v ( v w ) = µ ( v w , v + ) until the equation µ ( ξ, v ( ξ )) ξ = ( c + s ) gets satisfied. One can then restart this procedure with a different value of T − until the Eqs.(B1) are satisfied at the shock wave. Hybrid walls satisfy v + < c − s < v w and they have the boundary conditions v − = c − s and T + sh = T n , which make them very similar to the deflagration walls. Appendix C: Gravitational Wave Production
For the convenience of the reader, we here reproduce the formulae from Refs. [7, 81, 83] that determine the GWspectrum from sound waves and turbulence in a first order phase transition. They are given respectively byΩ sw ( f ) = 8 . × − K (cid:18) HRc s (cid:19) min (cid:20) , HR √ K sw (cid:21) (cid:18) g ∗ (cid:19) / S sw ( f ) , (C1)where K sw = κ sw α/ (1 + α ), with κ sw the efficiency coefficient of the sound wave, and h = 0 .
678 is the reducedHubble constant defined by H = 100 h km s − Mpc − [108]. As previously stated, we assume that all the wallshave non-runaway solutions and that the contribution from turbulence is negligible; hence we set Ω sw = Ω φ ( f ) = 0.The function parametrizing the shape of the GW spectrum is S sw ( f ) = (cid:18) ff sw (cid:19) (cid:32)
74 + 3 ( f /f sw ) (cid:33) , (C2)and the peak frequency f sw is f sw = 2 . × − Hz (cid:18) HR (cid:19) (cid:18) T n
100 GeV (cid:19) (cid:16) g ∗ (cid:17) . (C3)4Numerical fits for the efficiency coefficient κ sw (the fractions of the available vacuum energy that go into kineticenergy) were presented in [81]. For non-runaway walls, these fits depend on the wall velocity and are given by κ sw = c / s κ a κ b ( c / s − v / w ) κ b + v w c / s κ a , v w (cid:46) c s κ b + ( v w − c s ) δκ + ( v w − c s ) ( ξ J − c s ) [ κ c − κ b − ( ξ J − c s ) δκ ] , c s < v w < ξ J ( ξ J − ξ / J v − / w κ c κ d [( ξ J − − ( v w − ] ξ / J κ c +( v w − κ d , v w (cid:38) ξ J (C4)where c s = 1 / √ ξ J = (cid:112) α/ α + c s α δκ = − . √ α √ ακ a = 6 . v / w α . − . √ α + α κ b = α / .
017 + (0 .
997 + α ) / κ c = √ α .
135 + √ .
98 + α κ d = α .
73 + 0 . √ α + α (C5)We caution that while these fits, when used as input for a signal-to-noise estimate, are useful to get an overallestimate for the Gravitational wave signal in a given model, their precise predictions should be interpreted withcare. The fit for the sound wave production is rather reliable for relatively weak transitions α < .
1, which is therange where most of our models luckily fall into. For stronger transitions the fit can overestimate the GW-signalby as much as a factor of thousand (strong deflagrations) [109]. In addition to the strength of the transition, fitparameters have also shown be sensitive on the shape of the effective potential [110].
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