The Supercooling Window at Weak and Strong Coupling

Supercooled first order phase transitions are typical of theories where conformal symmetry is predominantly spontaneously broken. In these theories the fate of the flat scalar direction is highly sensitive to the size and the scaling dimension of the explicit breaking deformations. For a given deformation, the coupling must lie in a particular region to realize a supercooled first order phase transition. We identify the supercooling window in weakly coupled theories and derive a fully analytical understanding of its boundaries. Mapping these boundaries allows us to identify the deformations enlarging the supercooling window and to characterize their dynamics analytically. For completeness we also discuss strongly coupled conformal field theories with an holographic dual, where the complete characterization of the supercooling window is challenged by calculability issues.


I. INTRODUCTION
The detection of gravitational waves (GWs) from compact object mergers has reinvigorated the prospects of observing a stochastic GW background.Looking beyond the mergers observed at LIGO-VIRGO [1], fundamental physics can be directly responsible for numerous potentially observable stochastic sources, see for example Refs.[2,3].One such signal is the unique remnants of an early Universe first-order phase transition (PT), which generically produce GWs [4][5][6].However, the detectability of these signals is highly sensitive to the dynamics of the microscopic theory.For instance, the amplitude of the GW signal today depends on the energy released in the transition and its duration, see for example Ref. [7].
PTs liberating large amounts of energy stand out as the most promising candidates to produce detectable GW signals.If this energy originates from the vacuum energy of the meta-stable minimum, the PT is accompanied by a period of supercooling.This occurs when the meta-stable vacuum energy begins to dominate the energy density of the Universe, resulting in an additional period of inflation.It has long been understood that supercooled first order PTs are expected in theories with an approximate scale invariance at weak [8][9][10][11][12][13] and strong coupling [14][15][16], where the dilaton dynamics determines the vacuum tunneling rate.
The information regarding the supercooled transition is encoded in the zero temperature dilaton potential and its interactions with the thermal plasma.These must be carefully arranged to enter a supercooling phase and at the same time end it via a strongly first order transition.
In this paper we explore how generic a supercooled PT can be for a given dilaton potential.To quantify this statement, we identify the boundaries of the supercooling window in minimal setups and discuss how these can be extended/reduced in non-minimal constructions.
Many of the results presented here have appeared in some form within the extensive literature on supercooling.Our goal is to systematize the discussion and put forward a semi-analytical understanding of the behavior of the vacuum tunneling rate independently of any specific model.This will allow us to extract the general parametric dependence of the supercooling window.
Our approach exploits the re-parameterization invariance of the equations of motion and of the Euclidean bounce action, allowing the latter to be written in terms of a reduced number of parameters [17,18].The bounce action can then more easily be determined by a fit to the full solution, obtained numerically with standard codes [19,20].This simple observation allow us to obtain a detailed analytical characterization of the supercooling window for a large class of theories.
For weakly coupled theories, in the minimal setup, the dilaton potential is dominated by thermal corrections, as it is in the Coleman-Weinberg model [8,[21][22][23][24] or the Gildener-Weinberg model [9].Moreover, the dynamics of the PT can be fully captured in the high-temperature expansion.We show how cubic thermal corrections become dominant near the boundary of the supercooling window, where the expected signal is the strongest.This observation allow us to find an analytic approximation of the boundary of the supercooling window which we present in Eq. (20) and agrees astonishingly well with the full numerics as shown in Fig. 2.
The supercooling window can be enlarged with respect to the purely radiative case by adding a small temperature-independent deformation destabilizing the origin at low enough temperatures, hence ensuring the completion of the supercooled PT. 1 The scaling dimension of the relevant deformation (a negative mass squared or a negative cubic at weak coupling) controls the timescale of the PT which is strongly correlated with the strength of the GW signal.We show in Eq. (33) how the negative cubic favors slow first order PTs with respect 1 Explicit examples of this general mechanism were introduced in specific models in Ref. [12,25].

arXiv:2212.08085v3 [hep-ph] 17 Mar 2024
to the negative mass squared leading to a wider parameter space where a strong GW signal can be realized.Our analytical estimates match the numerical results shown in Fig. 4. Vice-versa, relevant deformations stabilizing the origin (a positive mass squared or a positive cubic at weak coupling) obstruct the completion of the PTs and must be suppressed compared to the cut-off scale, in order to ensure that the supercooling window does not shrink substantially compared to the purely radiative case.We derive a simple analytical parametric of these deformations in Eq. ( 40) which we compare against the full numerical solution in Fig. 6.
Our paper is organized as follows: Before turning to our results, we review in Sec.II the useful formulas necessary to describe the dynamics of PTs in the early Universe.Readers already familiar with the subject may wish to skip this summary.In Section III we define the supercooling window for weakly coupled theories in the simplest setup, where the breaking of conformal invariance is fully dominated by the interactions of the dilaton with the thermal bath, while in Section III A we discuss departures from this configuration.In Section IV, we define the supercooling window for strongly coupled CFTs admitting a Randall Sundrum description.We conclude in Section V.The details of our fitting procedure and the behavior of the bounce action are described in Appendix A. In Appendices B and C we provide a summary of the standard formulas used to compute the GW signal and the reach of present and future experiments.

II. PHASE TRANSITIONS TOOLKIT
In this section we summarize the essential conventions, notations and methodology for studying cosmological first order PTs, proceeding via the nucleation and percolation of true vacuum bubbles.We encourage expert readers to skip this section and proceed to Section III, although here and in Appendix B we provide a careful treatment of the approximations used to derive analytical results in the proceeding sections.
The nucleation rate of true vacuum bubbles is controlled by the tunneling rate between the scalar potential's false and true vacuum due to either thermal 2 or 2 We use a simplified expression for the pre-factor of the thermal tunnelings' exponent, T 4 instead of the usual approximation T 4 (S 3 /2πT ) 3/2 .This simplification does not lead to qualitative Under the assumption of spherically symmetric bubbles [32,33] and for a single scalar field controlling the tunneling rate S d is defined by the d-dimensional O(d)symmetric Euclidean action where r = √ τ 2 + x 2 with τ and x being the Euclidean time and position and Z ϕ is the wave function renormalization.To determine the initial bubble radius R appearing in Eq. ( 1), one must solve the field profile ϕ(r) that satisfies the so called "bounce" equation of motion with boundary conditions ϕ(r → ±∞) = ϕ − and dϕ dr r=0 = 0.Here the metastable vacuum is assumed to lie at the origin and ϕ − is the location of a deeper minimum in the potential.The final step requires inserting the solution for ϕ(r) into the action in Eq. ( 2) and minimizing with respect to the radius.
In what follows we make extensive use of the reparameterization invariance of Eq. (3), and the induced re-scaling of Eq. ( 2).More specifically an appropriate choice of field and coordinate transformations can be used to reduce the number of free parameters controlling the scalar potential.Starting from Eq. (2), we may write the action as utilizing r → Lρ and ϕ → ξφ.We then obtain the dimensionless action and scalar potentials (denoted with a tilde) As we will show the dimensionless action reduces weakly coupled renormalizable models to a single parameter system.
We now detail the micro-physics inputs required to describe the PT: i) the PT strength α, ii) the duration of the PT β H and how it connects to the bubble size at collision R ⋆ iii) the time T ⋆ at which the PT completes.The last parameter to predict the GW spectrum is the changes in the results, but does allow analytic results to be derived.
wall velocity v w which depends on the interactions of the expanding vacuum bubble with the surrounding plasma.We detail the exact treatment of these dynamics for our numerical results in Appendix B.
The strength of the transition α parameterizes the amount of energy available for GW production, in the form of latent heat ϵ(T ) = ∆V (T ) − T ∆V ′ (T )/4 normalized by the radiation energy density where ∆V (T ) is the positive potential difference between the false and true vacuum at a given temperature and the contribution of the temperature derivative of the effective potential can be easily neglected for supercooled PTs [34].The radiation energy density is ρ R (T ) ≡ π 2 g ⋆ tot (T )T 4 /30 with g ⋆ tot (T ) = g ⋆ SM (T ) + g ⋆ BSM (T ), where g ⋆ SM encodes the usual SM radiation degrees of freedom and g ⋆ BSM is model dependent.
Next we define the timescale of the PT.The nucleation rates in Eq. ( 1) are dominated by their exponents.Expanding these around T ⋆ , the transition timescale for a thermal PT can be defined as where H ⋆ ≡ H(T ⋆ ).For fast enough PT's (i.e.β H ≳ 10) the time of the PT can be easily related to the mean bubble size at where P false (T ⋆ ) = e −I(T⋆) is the probability of finding a point in the false vacuum [32,36,37].In Appendix B we detail the behavior I(T ⋆ ), but for fast enough PTs we can approximate Eq. ( 8) as R ⋆ H ⋆ ≈ 3v w /β H assuming that P false (T ⋆ ) ≈ 0. Interestingly one can show that the bubble size maximizing the energy distribution satisfies R * H * = 3v w /(β H + 1) which agrees with the standard formula as long as β H is large enough (see Appendix B for a detailed derivation).For thermal PTs, the critical temperature T c is defined as the point where true and false vacuum are degenerate: ∆V (T c ) = 0.As the temperature decreases, the tunneling rate grows quickly, leading to the nucleation temperature T n , which marks the onset of the PT.This is defined by the time-integrated probability of a single bubble being nucleated per Hubble volume reaching one.This can be approximated as Γ(T n )/H(T n ) 4 = 1 [38,39], where H(T ) is the usual Hubble rate H(T Pl .This is well approximated by Pl for supercooled PTs.If the PT is fast enough, the nucleation temperature is a good approximation of T ⋆ .Throughout the analytic section of this work we use T ⋆ = T n .In Appendix B we show that this approximation deviates at most by 20% by the percolation temperature T p defined as I(T p ) = 0.34.For numerical results we use T ⋆ = T p throughout as it gives a more accurate determination of R ⋆ .

III. THE SUPERCOOLING WINDOW AT WEAK COUPLING
At the renormalizable level, the most general potential for a single real scalar may be written as where the vacuum energy can always be set to zero, and any tadpole in ϕ reabsorbed via a linear redefinition.In general, m 2 (T ), δ(T ) and λ(T ) are complicated functions of the temperature T and of any other couplings or mass scales in the theory.The re-parameterization invariance introduced in Eq. ( 5) allows us to rewrite the potential as a function of a single parameter.Identifying yields where κ(T ) takes values between −∞ < κ ≤ κ c and the scalar kinetic term is canonically normalized (Z ϕ = 1).The equation κ(T c ) = κ c = 2/9 defines the critical temperature where the two minima of the potential in Eq. ( 11) are exactly degenerate. 3 The bounce solution Sd (κ) can be deduced once and for all by numerically computing the bounce for different values of κ and then performing a one-parameter fit (see Ref. [17,18] for similar results).For weakly coupled theories the tunneling rate is always dominated by thermal fluctuations (see Appendix A) and we can take Z ϕ ≃ 1 neglecting 1-loop correction to the wave function.Using this approach the O(3) symmetric bounce action can be written as 3 Strictly speaking a bounce can be defined for κc < κ < κmax, where the origin becomes the global minimum and the far away vacuum the false one.For κ larger than κmax = 1/4 the potential in Eq. ( 11) has only one global minimum at the origin.
where B3 (x) is given explicitly in Appendix A and it is defined such that B3 (0) = 1 and the two fitting functions match at κ = 0 where the bounce action admits a known analytical limit [40].For κ > 0 the functional dependence for κ → κ c is fixed to reproduce the thin-wall approximation at zeroth order in the thin-wall expansion [32].
The subleading terms computed in Ref. [41] do not impact significantly our results.For κ < 0 the solution is chosen such that for κ → −∞ we recover the solution of Ref. [42].

A. Radiative breaking of conformal symmetry
Taking the theory to be classically scale invariant, and assuming the thermal corrections to be dominated by a single coupling g, we obtain simple expressions for the parameters of the scalar field potential in the high-T expansion: where N b is the number of the bosonic degrees of freedom in the thermal bath. 4n these scenarios, the classically flat ϕ direction is lifted by radiative corrections, which generate a stable vacuum at ⟨ϕ⟩ where conformal symmetry is radiatively broken.At the true minimum, the heavy states obtain a mass of order m b ≃ g⟨ϕ⟩, while the scalar flat direction mass is loop suppressed m 2 ϕ ≃ N b g 4 16π 2 ⟨ϕ⟩ 2 .At zero temperature, the energy difference between the stable minimum and the origin is therefore of order ∆V 0 ≃ N b g 4 16π 2 ⟨ϕ⟩ 4 .Concrete realizations of this scaling are the Coleman-Weinberg model (CW model) [8,[21][22][23][24], consisting of a complex scalar charged under a U (1) gauge symmetry with coupling strength g and the Gildener-Weinberg setup (GW model) [9] which consists of N b real scalars coupled through quartic interactions with strength g2 .For simplicity, we present our results in terms of the CW model where N b = 3 and define M ≡ e − 1 3 +γ E Λ/4π where γ E ≃ 0.577 and Λ is the renormalization scale in the MS scheme.The potential energy difference at zero temperature is proportional to M 4 as Our results easily generalize to the GW model by replacing g = g/ √ N b and M = e γ E Λ 4π to account for numerical factors coming from the different finite pieces in the 1loop potential between vector and scalar normalization.9), for varying temperatures.Solid curves are the full numerical temperature corrected potential and dashed curves are the high T expanded potential in Eq. ( 13).The high T approximation works well near the meta-stable vacuum and near ϕtop(T ), while not capturing the full behavior for large field values near the true vacuum ⟨ϕ⟩.Here, we show the potential for g = 1 and M = 1 GeV, for which Tc ≃ 3.8 GeV.
We now proceed to describing the thermal history of the CW model.At early times, the origin is the true vacuum of the theory.As the Universe cool down, the dilaton potential undergoes a PT whose dynamics can be fully captured within the high-T expansion of Eq. ( 13) as shown in Fig. 1.This can be verified by tracking the position of the top of the barrier between the two vacua ϕ top (T ), whose existence ensures that the PT is of the first order.We find that m b (ϕ top (T ))/T ≲ 0.5π, where 0.5π is the value of m b (ϕ top (T ))/T at the critical temperature T c ≃ e 4/3 M , with T c estimated in the high-T expansion.This ratio scales as m b (ϕ top (T ))/T ∼ 1/ log (M/T ) easily satisfying the high-T condition at the temperatures T < T c which are relevant for the PT dynamics.
Since the onset of nucleation is governed by S 3 /T the nucleation condition becomes In the parametrization of Eq. ( 11) the bounce action reads where we used the relations m 3 (T )/δ 2 (T ) = 2π 2 T /(9g 3 ) and κ(T ) = 1/6 log(T /M ).Equipped with Eq. ( 16), we can now study the parametric dependence of the first order PT on the gauge coupling g.  20) compares to the full numerical result.The black dotted line shows how the boundary gets modified neglecting the thermal cubic.Left: Fixing M = 1 TeV supercooled PTs require g to be in the range 0.51 < g < 0.87.For g < 0.58 the nucleation is controlled by the thermally generated cubic in Eq. ( 13) as detailed in Eq. ( 20).For g < 0.53 daisies corrections become important as shown in Eq. ( 22).For g < 0.51 the transition never completes and the universe remains in a state of eternal inflation.For g > 0.87 the transition completes in radiation domination.For g > 1 the perturbative control of the theory is lost.Right: Summary of the current probes (yellow) and the future reach of the many proposed experiments (shades of gray).See Appendix C for a review.The lower boundary of the supercooling window is showed as a function of the cut-off scale M .
We are particularly interested in understanding the boundaries of the supercooling region -defined as the regime where a first order PT completes with α(T n ) ≥ 1 (i.e. after a period of inflation).Our results are summarized in Fig. 2 where we observe the following: i) The lower bound on g separates the region where the PT completes from the region where inflation does not end.ii) The upper bound on g separates the supercooling region from the region where the first order PT completes during radiation domination.
Interestingly, from the right panel of Fig. 2 we see that the totality of the supercooling window can be probed at future GW detection experiments as long as the scale of the PT lies below 10 11 GeV (with the usual optimism in the expected reach of proposed future experiments as detailed in Appendix B).We checked that this conclusion is unaffected by possibly larger astrophysical background in the LIGO frequency band [43,44], essentially because of the enormous GW signal generated by the supercooled PTs.
As we see from Fig. 2, the lower bound on the gauge coupling is of crucial phenomenological relevance since it distinguishes a strong first order PT from a regime in which the period inflation is eternal and its fate de-pends on the behavior of quantum fluctuations [45][46][47].The upper bound of the supercooling window is instead only indicating where α(T n ) stops being larger than O(1).This does not have an immediate phenomenological impact since, depending on the experiment, first order PTs with α(T n ) < 1 can also lead to a detectable GW signal.
The number of e-foldings of inflation before the PT completes is defined as with T eq being the temperature where the energy density in radiation and vacuum energy is equal.In Fig. 2 and all subsequent figures we show N e as light green contours.The number of e-folds is bounded from above by requiring i) quantum fluctuations of the dilaton field to be negligible, ii) the CMB power spectrum to match the Planck observations [48].These two constraints require N e to be less than N max e = log(T eq /H V ) and N CMB e = 23.8+ log (T reh /TeV) respectively and ended up being unimportant in the phenomenologically interesting region of the CW model.
In the remainder of this section we derive analytic ex-pressions for the lower and upper boundaries of the supercooling window in the CW model.a.The lower bound on supercooling can be defined by studying the nucleation condition during vacuum domination (VD), which reads where we defined κ n ≡ κ(T n ) = 1/6 log(T n /M ).We have taken the κ < 0 expression for the bounce action, defined in Eq. ( 16), as T n < M is expected to hold in this regime.An approximate formula for the boundary of nucleation in the large supercooling limit can be found by expanding the above expression for where the constant term on the left-hand side is the leading-order contribution to the bounce due to the cubic scalar self-interaction introduced in Eq. ( 11).
Naively, one would like to approximate the nucleation temperature by ignoring the constant term and getting a simple analytical solution , which is often quoted in the literature.However, this approximation is only justified for |κ n | ≫ 1 which in practice is never realized in the relevant parameter space.The actual behavior of κ n is shown in Fig. 3 right where we can see that κ n ∼ O(1) at the boundary of the supercooling window.Therefore, the thermal cubic should be included in order to reliably describe the nucleation in the deep supercooling regime.We find that the solution of Eq. ( 19) approximate κ n up to 10% corrections which correspond to having neglected the higher orders in the |κ n | ≫ 1 expansion.Luckily these corrections have a negligible impact on the determination of lower bound on the supercooling window.
Studying the zeros of the discriminant of Eq. ( 19) we can find the boundary values of g that give an interception point between the bounce action and the nucleation curve.The discriminant is a cubic equation in g 3 with coefficients depending on ∆ 2 ≡ 3/ log M H V so that the boundary of the supercooling window corresponds to a single real solution if the discriminant is negative or the smallest of the three real solutions if the discriminant is positive. 5Series expanding the result to O(∆ 4 ) yields 5 For completeness we give the full equation describing the zeros of the discriminant here in terms of Xg ≡ g 3 : This corresponds to the minimal nucleation temperature at which the PT completes avoiding eternal inflation.The leading term in Eq. ( 20) is g CW min ≃ 2π/ log 2/3 (M/H V ), which corresponds to the lowest possible coupling neglecting the thermal cubic contribution.The new correction proportional to (1 − ∆) 1/3 is controlled by the thermal cubic whose role is to reduce the value of g CW min , enlarging the supercooling window.Fig. 2 shows that our analytic approximation (dashed black line) reproduces very well the boundary of the supercooling window obtained by a brute force numerical scan which is plotted in blue (see Ref. [13] for a similar numerical analysis of the CW model).
The value of g CW min is also modified by next-to-leading order corrections to the thermal potential.The inclusion of daisy diagrams [49][50][51][52], i.e. a resummation of the leading-order hard thermal loops, serve to reduce the thermal barrier and subsequently extend the parameter space where nucleation is viable.In the language of Eq. ( 13), we can write the shift in the mass and cubic induced by these corrections as Note that these closed form expressions require series expanding in small ϕ/T , which is a good assumption around the thermal barrier, as well as a field redefinition to remove the term that arises linear in ϕ.Here we see that the inclusion of Daisy diagrams decrease m 2 (T ) while simultaneously increasing the size of the negative cubic δ(T ) throughout the supercooling regime.The effects of the Daisy diagrams accounts for the difference between the analytic approximation in Eq. ( 20) and the full numerical result in Fig. 2. The parametrization in Eq. ( 16) allows us to extract a simple analytic approximation for the behavior of β H as defined in Eq. ( 7).In the deep supercooling regime we can expand S 3 /T to second order in κ n → ∞ to get where we neglected terms of order |κ| −2 inside the parenthesis.
Solving Eq. ( 19) for T n , we gain a good approximation for β H in the supercooling region for g ≲ 0.6.For larger g the k n → ∞ expansion breaks down as shown in Fig. 3 right.
b.The upper bound on supercooling can be defined imposing T n = T vac , where T vac is the temperature below which the Hubble rate shifts from radiation to vacuum  15).The blue line corresponds to the lower boundary of the supercooling window in the CW model (g = 0.51).For the other curves we fix g = 0.4.The violet (light green) curve shows for the effect of a non-thermal negative mass (cubic) with m0/M = 10 −11 (δ0/M = 10 −11 ) which tends to enlarge the supercooling window.The red (sea-green) curve shows the effect of a positive non-thermal mass (cubic) deformation with m0/M = 10 −11 (δ0/M = 10 −11 ) which instead tends to shrink the supercooling window.Right: The behavior of the κ parameter at nucleation (κn) as a function of the gauge coupling.The zoomed-in region shows how the lower boundary of the supercooling window indicated by red (sea-green) points shrinks for positive mass (cubic) deformations which are larger enough.We take as an example m0/M = 10 −5 (δ0/M = 10 −5 ).

dominated and reads
where we used the energy difference between the false and true vacuum at zero temperature in the CW model, see Eq. ( 14).The nucleation condition defining T n can be written as where for T ≥ T vac , the quartic λ(T ) > 0 implies κ > 0 as follows from Eq. ( 11) allowing for a simple parameterization of S 3 .The equation above is an algebraic equation which defines κ n and can be solved in general.This is shown as a dashed red curve in the right-hand panel of Fig. 2.

B. Additional sources of explicit breaking
We now wish to study the consequences of incorporating additional scales breaking explicitly the conformal symmetry in the zero temperature potential.In the high temperature expansion, these deformations can be parameterized as shifts of the temperature dependent terms in Eq. (13).In what follows we study the set of possible deformations, examining how their presence changes the behavior of S 3 /T and ultimately the possibility of realizing supercooled PTs.A schematic view of how these deformations affect S 3 /T is presented in Fig. 3 left.In Sec.III B 1 we describe deformations destabilizing the origin with either a negative squared mass or a cubic. 6hese will make S 3 /T decreasing at low temperature as shown by the violet and green lines in Fig. 3, hence enlarging the supercooling window compared to the CW case as shown in Fig. 4. In Sec.III B 2 we describe deformations stabilizing the fake vacuum at origin with either a positive squared mass or a positive cubic.These will make S 3 /T increasing at low temperature as shown by the red and orange lines in Fig. 3, hence shrinking the supercooling window compared to the CW case as shown in Fig. 6.

Enlarging the supercooling window
Here, we would like to explore how the boundary of the supercooling window is extended as explicit breaking contributions destabilizing the origin are added to Eq. ( 13).This prospect can be studied by introducing non-thermal relevant deformations, which act to eliminate the thermally induced barrier at some finite temperature T flat , implying that the PT necessarily completes, even outside of the classically invariant supercooling window.The obvious candidates are a negative mass −  33).The light blue shading indicates the region where a GW signal is detectable.In the white region to the left βH has grown so that no detectable signal can be within the reach of any planned GW experiment.
The light green curves indicate the number of e-folds during the inflationary period induced by the PT transition as defined in Eq. ( 17).In the white region below the N max e the period of inflation is incompatible with CMB observations.comparable to the scale of the relevant deformations, where it quickly drops to zero corresponding to lowering the thermal barrier.
If nucleation only occurs just before (or after) the thermal barrier disappears the PT is effectively second order and no strong GW signal is expected.The relevant question to quantify here is then how large is the parameter space where the PT completes with a large enough GW signal.As we will see this depends very much on the scaling of β H , which is controlled by the slope of the bounce action drop and very sensitive to the scaling dimension of the temperature-independent deformation.
The effects of introducing small temperatureindependent deformations can easily encoded in the high-T expansion as negative mass: negative cubic: Importantly, the negative non-thermal cubic also affects the thermal mass at one loop, as where we expanded the loop contribution at leading order for small m 0 /M, δ 0 /M ≪ 1.The mass shift arises from the tadpole for ϕ, induced at one-loop by the presence of the cubic.Performing a field redefinition to remove this term gives rise to the shift in Eq. ( 28).
As can be seen from Eqs. ( 26) and ( 28), for both deformations there is a temperature T flat such that the effective thermal mass vanishes: m 2 (T flat ) = 0.In the region of interest the nucleation temperature should be very close to T flat .Approaching T flat the κ parameter defined in Eq. ( 11) tends to zero as explicitly shown in the right panel of Fig. 3.These two features characterize the dynamics of PTs that complete for g ≪ g CW min , where the nucleation is triggered mostly by the temperatureindependent deformations.
To describe this region we can then work under two simplifying assumptions: i) we take ∆T n = T n − T flat to be a small parameter keeping only the leading order term in the ∆T n /T flat expansion, ii) we expand the bounce action for κ n ≪ 1.In this limit the bounce action can be written in general as Setting log(T n /H V ) ≃ log (T flat /H V ) we can use Eq. ( 29) to get a simple expression for the nucleation temperature around the limit of a vanishing barrier: Lastly, we can estimate the timescale of the PT by expressing β H in the same limit where we already substituted the value of T n obtained in Eq. (30).
The above formulas can be used to get simple parameterics for the two deformations at hand.Finding the ).Here we clearly see the transition from the scaling behavior at small g, c.f. Eq. (33) (dashed), back to the radiative CW model (shown in blue) at large gauge coupling using the fully numerical results shown as solid curves.For completeness we also show the departure from the conformal behavior in red.The boundary of the supercooling window for the CW model is indicated by the gold dashed vertical line.
zeroes of Eqs. ( 26) and ( 28) we get T flat at first order in g ≪ 1 to be from which one can easily derive the behavior of dm 2 (T )/ dT and δ(T ) in the two cases.Putting all these together we get the asymptotic behavior of β H for g ≪ g CW min for the two deformations: The different scaling of β H with the gauge coupling can be simply understood from Eq. ( 31) by remembering that for the for the mass deformation: T flat ∼ m 0 /g, δ ∼ g 2 m 0 and dm 2 (T )/ dT ∼ gm 0 ; while for the cubic deformation: T flat ∼ δ 0 /g 2 , δ ∼ δ 0 and dm 2 (T )/ dT ∼ δ 0 .The asymptotic behavior of Eq. ( 33) agrees extremely well with the full numerical result for g ≪ g CW min as shown in Fig. 5.In the same figure we show the departure from CW behavior.This behavior is universal in both cases and can be easily derived as β H ∼ 1/g 3 (dashed red line).
As a result, our parametric can easily explain why the cubic deformation enlarges the supercooling parameter space so much more than the mass one as shown in Fig. 4. The scaling dimension of the deformation controls the dependence of β H on g which ultimately sets the strength of GW signal.
The left boundary for small g in Fig. 4 is determined purely by β H which controls both the strength and the peak frequency of the PT in the limit where α ≫ 1 (see Appendix C for details).In particular if we focus on the sound wave contribution, that typically dominates the GW production in our setup, the peak frequency scales as f peak ∼ β while the signal strength scales as Ω GW ∼ 1/β 2 H .The precise boundary of the parameter space will depend on the details of the signal and the experimental reach and we do not find it particularly enlightening to quantify.Our shading in Fig. 4 indicates that β H ∼ 500−1000 are the maximal allowed to obtain a detectable GW signal at any frequency, although for particular frequency windows the amazing expected reach of future GW interferometers could probe even larger values of β H .
The fact that the size of the deformation cannot be too small compared to the cut-off can be easily understood from the fact that the smaller the deformation, the longer the bounce will track the CW solution before nucleation, resulting in a larger number of e-folds of inflation as defined in Eq. ( 17).The upper bound on the number of e-folds of inflation gives a lower bound on T n and hence a lower bound on the size of the deformation which is shown in Fig. 4.
Before concluding this section, we briefly comment on explicit models where the cubic deformation dominates over the mass and the quartic.One example is the linear coupling of the CW dilaton ϕ to an operator which dynamically develops a vacuum expectation value.The dilaton potential at zero temperature can be schematically written as where λ(ϕ) ∼ 3g 4 /8π 2 in the CW model and we added the VEV of an operator ⟨ Ôϵ ⟩ of dimension ≥ 3 with ϵ ≪ λ.Shifting the tadpole term by using the field re- induces a mass and negative cubic terms for ϕ scaling as λ 1/3 ϵ 2/3 , λ 2/3 ϵ 1/3 .In the limit ϵ ≪ λ the induced mass becomes sub-dominant compared to the cubic, which is the leading deformation from classical scale invariance.This model can then be mapped to our parametrization above by identifying . Explicit examples were discussed in Refs.[12,25].
In realistic scenarios both a positive mass squared and a cubic deformation will be generated at tree level, so we briefly discuss how our result is modified when both m 0 and δ 0 are present.As shown in Fig. 3 if a large positive non-thermal mass squared dominates the dynamics, it will make the bounce growing to infinity at low temperatures before meeting the nucleation condition.The existence of a solution to m 2 (T flat ) = 0 requires then an upper bound on m 0 , which however does not seem to require any additional fine-tuning to be fulfilled.

Shrinking the supercooling window
In this section we study how the boundary of the supercooling window shrinks once deformations stabilizing the origin are added to Eq. ( 13).In contrast to the CW case, the action in these cases does not continue to decrease indefinitely.From both the red and orange curves in Fig. 3 we observe that the action reaches a minimum at a temperature T min comparable to the size of the explicit breaking.Hence, it is expected that for sufficiently large values of the explicit breaking parameter nucleation will be prevented.
We will examine the case of a positive mass-squared m 2 0 and that of a positive cubic term δ 0 which can be easily captured in the high-T expansion by shifting the thermal mass and cubic in Eq. ( 13) respectively: positive mass: positive cubic: In both cases, when the deformations are small, the supercooling boundary can be determined by expanding the action to leading order in |κ n | → ∞, keeping the first non-trivial correction due to m 0 or δ 0 .The right panel of Fig. 3 confirms that this approximation is justified.In the large κ n limit the bounce action admits the following simple form where ε(T ) ≪ 1 can be easily found for the two deforma-tions of interest: , positive δ 0 .
At zeroth order in the expansion of ε(T ) we can express the new supercooling window boundary, g min,ε as where g CW min is the CW result of Eq. ( 20) and E(T min ) encodes the effects of the explicit breaking.Solving the nucleation condition we get The shrinking of the supercooling window can then be understood by studying the properties of E(T min ) given in Eq. (40).Namely, E(T min ) is minimized for E( √ M H V ) = 1 (corresponding to the zeroth order conformal result) and otherwise it is an increasing function of T min .Since for the deformations in question we always find that T min > √ M H V it is clear that g CW min increases with T min , reducing the viable parameter space for supercooling.
The only remaining step to obtain the supercooling boundary is determining T min which amounts to find the minimum of Eq. (37).For the two deformations under consideration we find , positive δ 0 . ( with T = 2m 0 /g for the mass case and T = 4πδ 0 /3g 3 in the cubic case.Plugging the expressions from Eq. ( 41) into Eq.( 40) we explicitly see that as m 0 , δ 0 get larger, the value of g min,ε increases until the supercooling window closes completely.Since the T min for the cubic deformation increases parameterically faster as a function of g than in the mass case, the corresponding supercooling parameter space shrinks faster than in the mass case as shown in Fig. 6.
The full behavior of the deformation dependent shift E(T min ) for both deformations is shown in Fig. 6, where we also plot the behavior of the nucleation temperature T n and of β H .Our crude approximation in Eq. ( 40) (shown as the black dashed line in Fig. 6) captures only the qualitative behavior of the boundary but fails completely as soon as the deformation become sufficiently large.
From Fig. 6 we clearly see that the deformation term should be at least loop suppressed in order for the supercooling window to not shrink completely.

IV. THE SUPERCOOLING WINDOW AT STRONG COUPLING
In this section, we derive the supercooling window for a class of strongly coupled theories which have a known holographic dual.We focus on large N CFT with spontaneously broken conformal symmetry where the dilaton potential and the holographic principle [53,54] can be used to trace the PT between broken and unbroken conformal symmetry as first shown in Ref. [27].
If conformal invariance is mainly spontaneously broken, the confined phase can be well described by the effective dilaton potential, radiatively generated by the coupling of the dilaton to a marginally irrelevant operator of dimension [O] = 4 + ϵ with coupling strength g ≪ 1.The zero temperature dilaton potential describing the confined phase can be written as where ⟨ϕ⟩ is the dilaton VEV which is defined in terms of . Here λ 0 and g UV are the values of the dilaton quartic and the coupling strength g at the UV scale Λ UV .Moreover, we have expanded the running dilaton quartic λ(g(ϕ)) at the leading order in small g, defining λ ′ 0 ≡ dλ/ dg| g=0 .The above construction can be viewed as the holographic dual of a 5-dimensional theory of gravity in antide Sitter space with IR and UV branes [54] stabilized by the Goldberger-Wise mechanism [55,56].The dilaton is interpreted as the radion, describing the position of the IR brane.A non-flat radion potential is therefore associated with the breaking of conformal symmetry.The dilaton potential in Eq. ( 42) can be shown to match the radion potential, for small ϵ ≪ 1.
We take λ 0 < 0 as the bare dilaton quartic and 0 < ϵ ≪ 1 parametrizes the small positive anomalous dimension.The normalization of the dilaton potential can be obtained via the AdS/CFT correspondence or directly by considering the contribution of the irrelevant operator to the trace anomaly [26,57].The smallness of ϵ, determines the hierarchy of scales between the dilaton and the other CFT states so that the vacuum structure can be studied in terms of the dilaton potential alone.This is analogous to the loop suppression of the dilaton mass in the CW model of Sec.III A. We also assume the number of degrees of freedom contributing thermally to the dilaton potential after confinement to be small which requires g * ,light ≪ 45N 2 /4.
At high temperatures, the system is in its deconfined phase, consisting of a strongly coupled large N CFT.The contribution of the thermal CFT plasma to the free energy is F deconfined ∼ −N 2 T 4 as supported by holographic results [58].The details of the full potential in this phase depends on the strongly coupled dynamics and are incalculable.This introduces a certain arbitrariness in matching the dilaton potential in the confined phase with the value of the free energy in the deconfined phase (see Refs. [28][29][30] for an extensive discussion).The simplest option is to take where the two sides of the potential are then glued together at the origin of field space and the dilaton potential in the deconfined/confined phases is denoted by negative/positive field values for ϕ, respectively.In what follows, we will show the region of parameter space which is insensitive to the choice of deconfined potential.
The potential energy difference between the false vacuum at the origin and the dilaton VEV ⟨ϕ⟩ at zero temperature is simply where Λ ≡ ⟨ϕ⟩|λ0ϵ| 1/4 2 3/2 π 1/2 is the effective confinement scale, defined similarly to [31].The free energies of both phases equilibrate at T c = Λ, so that a (de)confining PT which completes at T n < T c must always be supercooled, as T 4 n < ∆V 0 .This is in sharp contrast with the weak coupling case where the same model could describe a PT both in vacuum and radiation domination.
The full potential describing the PT dynamics is then given by the sum of the confined and the deconfined dilaton potential and can be written as where we used the reparametrization invariance defined in Eq. ( 5) with to get a lagrangian that up to an overall rescaling by N 2 T 4 , depends only on a single parameter Here, κ ranges between 4e < κ < ∞, where its critical value is κ c = 4e, at which the two phases have the same vacuum energy.The d dimensional bounce action is given by where Bd (κ) is a fitting function regularized over the TW solution, admitted near the critical value κ c = 4e.Unlike in the weakly coupled case, the thermally driven bounce does not always dominate, hence the tunneling rate is dictated by the min[S 3 /T, S 4 ].Due to their different scaling with λ 0 , it is expected that for sufficiently low T and |λ 0 | ≫ 1 the quantum contribution may dominate, i.e. S 4 < S 3 /T .Explicit forms for the O(3), O(4) actions, as well as a comparison between the two, are given in Appendix A. In Appendix A 2 we find that for the majority of the relevant parameter space, quantum tunneling dominates and ultimately controls the boundary of nucleation.The supercooling boundary can now be found by solving the induced nucleation condition Eq. ( 15) in the limit T nuc → 0 (i.e.κ → ∞).S 4 admits a simple solution for the nucleation temperature, given explicitly in Eq. (A15), with the lowest possible temperature obtained at Requiring T > T min nuc sets a lower bound on the explicit breaking of conformal symmetry ϵ as In deriving Eq. ( 49), we completely ignored the contribution to the action due to the φ motion along the deconfined region of the potential, i.e. φ < 0. We can estimate the contribution of this motion to the bounce in the TW approximation [28,30] neglecting friction, as where T is the critical bubble size, resulting in Note that the surface tension in Eq. ( 51) is integrated between ϕ = −T and ϕ = 0, implicitly taking the dilaton The supercooling window for a large N strongly coupled CFT with confining scale Λ = 1 TeV.The light blue region marks the supercooling window for the maximal dilaton quartic λ0 = 16π 2 .Dashed and dotted lines indicate how the window shrinks for smaller |λ0|.In white, the region where nucleation is inefficient and a first order transition fails to complete.In the light gray (dark gray) region higher order terms in N (in ϵ) are non-negligible.In the hatched region the incalculable CFT contribution to the bounce action dominates the tunneling.The discontinuities in all the blue curves indicate the transition to thermal tunneling driven by the O(3) action which typically happens at low values of N .
potential on the CFT side to be minimized at ϕ = −T .While this assumption is well motivated in the holographic picture by the existence of the AdS black hole solution [27], relaxing it can drastically change the values of the bounce actions in Eq. ( 52), which scale linearly with the CFT vacuum position as S CFT ∼ |ϕ|/T .
The truly calculable region of parameter space is then defined by requiring S dilaton > S CFT , which sets an upper boundary on the explicit breaking of conformal symmetry ϵ ≲ 0.18 In Fig. 7 we show the supercooling window for a strongly coupled CFT as a function of N and ϵ, fixing the confinement scale Λ = 1 TeV and varying the bare quartic λ 0 .The only calculable boundary of the window is the one for smaller ϵ which is the analogous of the small g boundary in the weakly coupled case.As already noticed in Ref. [31] requiring a non-empty Universe imposes a strong upper bound on N which can be large enough to justify the large N expansion only for very large values of the dilaton quartic λ 0 ≳ 1.

V. CONCLUSIONS
Supercooled PTs offer one of the best possibilities to produce sizeable GW stochastic backgrounds in the early Universe.This motivated us to understand their dynamics systematically, characterizing the available parameter space with the goal of understanding how generic a supercooled PT can be.This is critical as it is well known that supercooled PTs live at the boundary of eternal inflation [31,[59][60][61].In practice this implies for minimal models that the coupling constant controlling the closeto-marginal operator breaking conformal symmetry has to be judiciously chosen to allow the PT to complete.As a starting point of our analysis we quantify precisely the allowed range for this coupling which we call the supercooling window for weakly coupled and strongly coupled theories.
The boundary of the supercooling window at weak coupling where conformal symmetry is radiatively broken is well described by a simple formula we derived in Eq. ( 20) up to the daisy resummation of the thermal loops (see Fig. 2).Our simple formula highlights the importance of the thermally generated cubic which was neglected in previous analytical approximations.This contribution dominates the dynamics close to the boundary of the supercooling window because it reduces the thermal barrier hence favoring bubble nucleation.At the same time, the small range of couplings where supercooling can be realized suggests looking beyond the minimal model.
Depending on their relative sign with respect to the thermal contributions, small zero-temperature deformations of the potential can either increase or decrease the size of the thermal barrier between the false vacuum and the true vacuum.While in the former case the supercooling window obviously shrinks, it is interesting to ask what happens in the latter case where at some sufficiently low temperature the barrier completely disappears and the PT behaves as a second order or cross-over transition.
We explicitly study the cases of mass and cubic deformation destabilizing the origin, deriving analytically the parametric scaling of the time scale of the PT (often called β H in the literature).We show that in both cases the supercooling window is enlarged with respect to the minimal case.Compared to the mass case, the cubic deformation gives a wider region where a strong first order PT completes before the barrier disappears.This can be understood analytically from Eq. ( 33) comparing how fast the the barrier disappears with respect to the time scale of nucleation.Adding a cubic deformation makes the supercooling window wide enough at the price of realizing a large hierarchy between the dynamics generating the cubic deformation and the one responsible for the spontaneous breaking of conformal symmetry.Examples of concrete setups were given in Ref. [12,25].
For completeness we define the supercooling window in strongly coupled gauge theories with a holographic dual.Analogously to previous studies [31,62] we find that a successful nucleation generically challenges the large N expansion and the small-backreaction limit.Our analysis reinforces the need for constructing fully calculable strongly coupled setups at large N where supercooled PTs can occur.Non minimal models addressing this issue were put forward in Ref. [14-16, 25, 28, 29, 63-72] and more recently in Ref. [73].We hope to return to these issues in the future.
actions is given to leading order by STW The fitting functions are then obtained as We note that in order to derive the nucleation temperature given in Eq. ( 49), one must expand the full actions in the limit of small temperature (i.e.κ → ∞), and solve the nucleation condition using the following actions The nucleation temperature admits a simple form when driven by quantum fluctuations, given by This temperature is minimized at N 2 ≃ 6.44 c 4/3 log 2 M pl ΛN , where c ≡ 1.8 |λ0ϵ| 3/4 as in [31], resulting in Eqs. ( 49) and (50).A similar derivation for thermally driven transitions does not lead to an analytic expression for the nucleation temperature due to the log 3/4 κ factor which appears in Eq. (A14), requiring the solution of a seventh order equation for log Λ/T n .However, an upper bound on N 2 can still be obtained by considering the discriminant of the nucleation condition, leading to an upper bound given by N 2 ≤ 3.4 c log 7 4 M pl ΛN .

Thermal/Quantum Tunneling at Weak/Strong Coupling
The vacuum decay rate can be dominated by either thermal or quantum fluctuations.Due to the negative exponential dependence of the tunneling rate on the bounce action, as seen from Eq. ( 1), it is sufficient to compare S 4 (quantum) against S 3 /T (thermal) to determine which one drives the PT.The lesser of the two actions at a given temperature is therefore the dominant contribution to the rate.
The two actions have different scaling with the model parameters, and potentially, either one may dominate at a different regime of coupling strengths.Here, we discuss whether nucleation proceeds via thermal or quantum tunneling for the models discussed in the main text.
Weak coupling: We begin with the weakly coupled models considered in Section III.The two relevant actions are then given by Eq. (A1).Since these actions must converge to the TW approximation at sufficiently high temperature, and knowing that S TW 3 /T has a double pole in (T − T c ), while S TW 4 has a triple pole in (T − T c ), we conclude that S 3 /T < S 4 (T ) at T → T c , implying that high-T transitions are induced by thermal fluctuations.Lowering the temperature, the condition for continued thermal dominance over quantum is simply S 4 (T ) < S 3 (T )/T , translated via Eq.(A1) to m(T )/T < S4 (κ)/ S3 , where the most stringent condition can be found by requiring that there exist no κ for which this condition is met.By inspection, the left panel of Fig. 8 demonstrates that the functions S4 (κ), S3 (κ) are monotonically increasing with κ, with their minimal value obtained at κ → −∞.In this limit, both actions admit simple forms given by Eq. (A3), scaling as 1/κ, rendering their ratio constant.The aforementioned condition for κ → −∞ is then translated to the simple constraint and Te (red) vary with respect to the nucleation temperature as a function of the PT timescale.We observe that Tp is an accurate measure of the PT completing except for extremely small values of βH .Right: we show the resulting determination of R⋆ as a function of βH , for a number of different approximations.The main message is that for βH ≳ 10, the PT is sufficiently fast that the background expansion can be ignored while the maximum of the volume weighted distribution also coincides with the mean bubble size in this regime.
where V false (T ) = a(T ) 3 e −I(T ) .For the case of exponential nucleation it can be shown that this translates to the requirement that I(T e ) = 3/β H [35], i.e this condition is more stringent than the percolation requirement at values of β H ≲ 8.8.This behavior is shown in the left-hand panel of Fig. 9, where the temperature ratios with respect to the nucleation temperature are shown as a function of β H in the context of the Coleman-Weinberg model.
To summarize the discussion of T ⋆ , for fast PTs T n or equivalently T p signals the completion of the transition as T e ≫ T n ∼ T p while for supercooled transitions T n > T p ≳ T e , that is T p suffices except for extremely small values of β H .
With the appropriate temperature for a given β H identified, we must now identify the typical bubbles size R ⋆ at the point of their collisions.This is crucial as not only does the energy density in the bubble scale with its volume, but the majority of bubble simulations rely crucially on this measure to determine numerically the resulting GW signal.Following Ref. [35] we can determine the size of a bubble, R, at time t that was originally nucleated at time t R as R(t, t R ) = a(t)    I. Two benchmark points to illustrate bubble wall behavior.BP1 exhibits a small amount of supercooling with a large gauge coupling, hence sound-waves dominate the GW signal, while BP2 is in the opposite regime with an extreme amount of supercooling and small gauge coupling.Hence the GW signal is dominated by collisions of the bubble walls themselves.

GW Simulation Parameters
To assess the detectability of a given PT, its parameters (α, β H , T ⋆ and v w ) must be mapped to the spectrum of sourced gravitational waves.This mapping requires a combination of both lattice and magnetohydrodynamic simulations.This leads to sizeable theoretical errors, potentially larger than those of the above parameters [74].As a result we make conservative assumptions when utilizing the simulation results and include only two of the possible GW generation mechanisms; bubble-wall collisions and sound waves in the plasma. 7We will use the short-hand but often confusing notation h 2 Ω GW for the total signal: These two fitting functions and the additional derived quantities are given in the following subsections.Note that due to the values of γ eq in the models under consideration, the sound-wave contribution dominates over the signal generated from bubble wall collisions.

a. Bubble-wall Collisions
The latest simulations of bubble-wall sourced GWs give the following fitting functions [75] [note the addition of the redshifting factor compared to Eq. ( 52 The redshifting factor for the GW amplitude is given by g ⋆S (T eq ) g ⋆S (T reh ) The last piece is the energy fraction that the bubble wall carries κ ϕ .

FIG. 1 .
FIG. 1. Heuristic view of the Coleman-Weinberg potentialV (ϕ, T ) from Eq. (9), for varying temperatures.Solid curves are the full numerical temperature corrected potential and dashed curves are the high T expanded potential in Eq. (13).The high T approximation works well near the meta-stable vacuum and near ϕtop(T ), while not capturing the full behavior for large field values near the true vacuum ⟨ϕ⟩.Here, we show the potential for g = 1 and M = 1 GeV, for which Tc ≃ 3.8 GeV.

FIG. 2 .
FIG.2.The supercooling window for a classically scale invariant theory at weak coupling.As discussed in Sec.III A we take the CW model as a reference.Below the solid blue no nucleation is possible while above the dashed red line the transition occurs in radiation domination.The green lines indicate the number of of e-fold of inflation.The black dashed line shows how the approximation in Eq. (20) compares to the full numerical result.The black dotted line shows how the boundary gets modified neglecting the thermal cubic.Left: Fixing M = 1 TeV supercooled PTs require g to be in the range 0.51 < g < 0.87.For g < 0.58 the nucleation is controlled by the thermally generated cubic in Eq. (13) as detailed in Eq. (20).For g < 0.53 daisies corrections become important as shown in Eq. (22).For g < 0.51 the transition never completes and the universe remains in a state of eternal inflation.For g > 0.87 the transition completes in radiation domination.For g > 1 the perturbative control of the theory is lost.Right: Summary of the current probes (yellow) and the future reach of the many proposed experiments (shades of gray).See Appendix C for a review.The lower boundary of the supercooling window is showed as a function of the cut-off scale M .

10 - 11 10 FIG. 3 .
FIG.3.The behavior of the nucleation condition in the different scenarios presented here complemented by the behavior of the κ parameter defined in Eq. 11 at nucleation.For both panels, we fix M = 1 TeV.Left: The bounce action as a function of temperature in the different scenarios discussed here.The thick gray line is the R.H.S. of the nucleation condition in Eq. (15).The blue line corresponds to the lower boundary of the supercooling window in the CW model (g = 0.51).For the other curves we fix g = 0.4.The violet (light green) curve shows for the effect of a non-thermal negative mass (cubic) with m0/M = 10 −11 (δ0/M = 10 −11 ) which tends to enlarge the supercooling window.The red (sea-green) curve shows the effect of a positive non-thermal mass (cubic) deformation with m0/M = 10 −11 (δ0/M = 10 −11 ) which instead tends to shrink the supercooling window.Right: The behavior of the κ parameter at nucleation (κn) as a function of the gauge coupling.The zoomed-in region shows how the lower boundary of the supercooling window indicated by red (sea-green) points shrinks for positive mass (cubic) deformations which are larger enough.We take as an example m0/M = 10 −5 (δ0/M = 10 −5 ).

m 2 02 ϕ 2
FIG. 4. Enlarging the supercooling window as a result of a temperature-independent deformation breaking conformal symmetry with a negative mass-squared (left) or a negative cubic (right), for a fixed scale of M = 1 TeV.The gold dashed vertical line denotes the lower boundary of the supercooling window for the CW model.The hatched purple regions indicate where the deformations are no longer a small perturbation of the original CW model.Dashed gray contours are growing values of βH whose scaling is fully capture by Eq. (33).The light blue shading indicates the region where a GW signal is detectable.In the white region to the left βH has grown so that no detectable signal can be within the reach of any planned GW experiment.The light green curves indicate the number of e-folds during the inflationary period induced by the PT transition as defined in Eq. (17).In the white region below the N max

FIG. 5 .
FIG.5.Evolution of βH as a function of the gauge coupling g for a temperature-independent negative mass-squared (violet) or cubic (light green) with m0/M = 10 −4 (δ0/M = 10 −4 ).Here we clearly see the transition from the scaling behavior at small g, c.f. Eq. (33) (dashed), back to the radiative CW model (shown in blue) at large gauge coupling using the fully numerical results shown as solid curves.For completeness we also show the departure from the conformal behavior in red.The boundary of the supercooling window for the CW model is indicated by the gold dashed vertical line.

10 FIG. 6 .
FIG. 6. Shrinking the supercooling window as a result of explicit conformal symmetry breaking with a positive mass-squared (left) and a positive cubic (right), for a fixed scale of M = 1 TeV.Light blue shaded regions indicate a supercooled PT, while the light red regions indicate nucleation during radiation domination.The white region does not exhibit bubble nucleation, while the transparent blue line indicates the boundary of the supercooling window.Lastly, we show both the numerically determined values of both βH (grey dashed contours) and the number of e-folds of vacuum domination (light green contours).The gold dashed vertical line denotes the lower boundary of the supercooling window for the scale-invariant model.The balc dashed line in both plots indicate our analytical approximation in Eq. (39).
FIG. 7.The supercooling window for a large N strongly coupled CFT with confining scale Λ = 1 TeV.The light blue region marks the supercooling window for the maximal dilaton quartic λ0 = 16π 2 .Dashed and dotted lines indicate how the window shrinks for smaller |λ0|.In white, the region where nucleation is inefficient and a first order transition fails to complete.In the light gray (dark gray) region higher order terms in N (in ϵ) are non-negligible.In the hatched region the incalculable CFT contribution to the bounce action dominates the tunneling.The discontinuities in all the blue curves indicate the transition to thermal tunneling driven by the O(3) action which typically happens at low values of N .

10 FIG. 9 .
FIG.9.Temperature and typical bubble size at collisions as a function of the PT timescale.Left: here we show how Tp (black) and Te (red) vary with respect to the nucleation temperature as a function of the PT timescale.We observe that Tp is an accurate measure of the PT completing except for extremely small values of βH .Right: we show the resulting determination of R⋆ as a function of βH , for a number of different approximations.The main message is that for βH ≳ 10, the PT is sufficiently fast that the background expansion can be ignored while the maximum of the volume weighted distribution also coincides with the mean bubble size in this regime.

FIG. 10 .
FIG.10.Contours of α as well as the bubble wall boost factor γ⋆ normalized to the equilibrium value γeq from Eq. (B12).Note that α is evaluated at the percolation temperature Tp, whereas the nucleation line between vacuum and radiation domination (red, thick-dashed line) depends on the nucleation temperature.

4 / 3 = 1 . 64 ×GW κ ϕ α 1 + α 2 (H reh R ⋆ ) 2 (= 5 × 10 −8 κ ϕ α 1 + α 2 (H reh R ⋆ ) 2 ( 90 1
10 −5 100 g ⋆ (T reh ) energy density in photons today being Ω γ h 2 = 2.473 × 10 −5[76], the relativistic dofs in the photon bath g ⋆ (T 0 ) = 2, and the entropy degrees of freedom at matter-radiation equality g ⋆S (T eq ) = 3.909.Here we have also assumed that g ⋆ (T reh ) = g ⋆S (T reh ).The above can be re-written in the following formdΩ ϕ h 2 d(ln f ) = 3.22 × 10 −3 F 0 a + b) c f b ϕ f a b f (a+b)/c ϕ + af (a+b)/c c ,(B19)with a = 3, b = 1.51 and c = 2.18.More sophisticated results depending on the initial bubble wall profile are can be found in Ref.[77].The scalar field generated peak frequency today is f 0 g ⋆S (T eq ) g ⋆S (T reh )1/3 T γ0 T reh M P g ⋆ (T reh ) with corresponding χ 2 = 0.9993(1.0015)for B 3 (B 4 ) respectively.The full bounce actions are then given by Note, that the above is written as a function of time rather than temperature, which is significantly simpler for an inflating background.Assuming perfect de Sitter with scale factor a(t) = a 0 e Ht we can firstly evaluate I(t) Then using the relation between a bubble size at time t given that it was nucleated at time t R 4P false (t R ) .