Cosmological Phase Transitions in Warped Space: Gravitational Waves and Collider Signatures

We study the electroweak phase transition within a 5D warped model including a scalar potential with an exponential behavior, and strong back-reaction over the metric, in the infrared. By means of a novel treatment of the superpotential formalism, we explore parameter regions that were previously inaccessible. We find that for large enough values of the t'Hooft parameter (e.g. $N\simeq 25$) the holographic phase transition occurs, and it can force the Higgs to undergo a first order electroweak phase transition, suitable for electroweak baryogenesis. The model exhibits gravitational waves and colliders signatures. It typically predicts a stochastic gravitational wave background observable both at the Laser Interferometer Space Antenna and at the Einstein Telescope. Moreover the radion tends to be heavy enough such that it evades current constraints, but may show up in future LHC runs.


Introduction
The Standard Model (SM) of ElectroWeak (EW) and strong interactions has been put on solid grounds by the past and current experimental data collected at e.g. the Large Hadron Collider (LHC) or the Large Electron Positron collider [1,2]. Still the model is unable to cope with some cosmological observables and suffers from theoretical drawbacks. For instance, it fails to explain a number of observational and consistency issues such as the baryon asymmetry of the universe, the strong CP problem, the origin of the flavor structure, the origin of inflation and the strong sensitivity to high scale physics. In particular the latter problem, a.k.a. the hierarchy problem, has motivated the introduction of Beyond the SM (BSM) physics which makes nowadays the subject of active experimental searches at the LHC.
One of the best motivated BSM frameworks was introduced years ago by Randall and Sundrum [3]. In this scenario the hierarchy between the Planck and EW scales is generated by the Anti de Sitter (AdS) warp factor involved in the extra dimension. An appealing feature of this framework is that the five-dimensional (5D) model is holographically dual to a non-perturbative four-dimensional (4D) Conformal Field Theory (CFT) and the dynamics of the strongly-coupled states of the 4D theory can be investigated perturbatively by means of the 5D theory.
Once the extra dimension is integrated out, the Randall-Sundrum theory contains towers of heavy states, the Kaluza-Klein (KK) modes of all SM particles, propagating in the bulk. It also contains a light state, the radion, dual to the dilaton, a Goldstone boson of the conformal invariance of the dual 4D theory. In the absence of a potential stabilizing the brane distance (see e.g. Ref. [4]), the radion (and equivalently the dilaton) is massless but, as soon as the extra dimension is stabilized, it acquires a mass. Still the radion typically remains the lightest BSM state and it can play a relevant role in the collider and early-universe phenomenology. In particular, it undergoes a phase transition during which it acquires a Vacuum Expectation Value (VEV) and which, in the dual language, corresponds to a (holographic) phase transition from the deconfined to the confined phase. In other words, the dilaton condenses.
The holographic phase transition has been studied by a number of authors and it has been concluded to be of first-order [5][6][7][8][9][10][11][12][13]. However, in models with small backreaction on the gravitational metric, in order to avoid the graceful exit problem, one has to consider scenarios where the number of degrees of freedom in the CFT phase (i.e. the number of "colors" N of the SU (N ) symmetry) is small, thus jeopardizing the perturbativity of the 5D gravitational theory. It is hence worth investigating models where the conformal symmetry is strongly broken in the infrared (IR), but the corresponding large back-reaction can be conveniently treated. In this way one expects to avoid the graceful exist problem even with N large, with clear benefits for the perturbativity of the 5D gravitational theory.
In the present paper we provide a method to deal with the large back-reaction issue. This method is a generalization of the superpotential procedure [14], and to show its capabilities, we apply it to analyze a class of theories where conformality is strongly broken at the IR brane. We dub these theories soft-wall models as they generate a naked singularity in the 5D metric beyond the location of the IR brane. Although the singularity is outside the physical interval, between the two branes, the distance of the singularity from the IR brane is important because it controls the breaking of conformality. This kind of models were introduced as minimal ultraviolet (UV) completions with no tension with EW precision data [15][16][17][18][19][20][21][22][23][24], as an alternative to models with extended (custodial) gauge symmetry [25]. Recently, the same models were also considered to accommodate the (g µ − 2) [26] and B-meson anomalies [27][28][29][30][31][32][33], in agreement with the quark mass and mixing angle spectra, and the natural generation of lepton flavor universality violation.
The outline of the paper is as follows. In Sec. 2 we introduce the general formalism for the 5D action, including the Gibbons-Hawking-York (GHY) boundary term. We also review the Equations of Motion (EoM) and Boundary Conditions (BCs) of the theory, and show that solving the EoM is equivalent to applying the superpotential procedure [14].
In Sec. 3 we develop a novel method to employ the superpotential formalism in the presence of mistuned BCs. This allows to calculate the effective potential between the two branes as a function of their distance without major problems with the backreaction. It hence opens up the possibility of studying warped models without imposing tight upper bounds on the amount of back-reaction.
In Sec. 4 we introduce the particular soft-wall model and we apply the generalized superpotential method to it. Since the method needs to be carried out numerically, we focus on some benchmark scenarios with different degrees of back-reaction (up to N 25). In all cases, the relevant parameters are set to solve the hierarchy problem.
The relation between the UV and IR brane distance and the canonically normalized radion field is analyzed in Sec. 5. The effective potential for the brane distance, previously obtained, can then be reinterpreted as a function of the physical radion field. This in particular allows to ensure that in our benchmark scenarios the KK gravitons are much heavier than the radion. For this reason the radion phase transition can be analysed within an Effective Field Theory (EFT) where the SM-like particles and the radion are the only dynamical fields.
The EFT at finite temperature of the soft-wall model is computed in Sec. 6. We obtain that, depending on the amount of back-reaction, the free energy difference be-tween the confined and deconfined phases can span several orders of magnitude. This of course has relevant effects on the value of the nucleation temperature and, in turn, on the phenomenology of the model.
In Sec. 7 we analyze the phase transition of the radion in detail. We find that, in agreement with precedent analyses [5][6][7], for tiny back-reaction the nucleation rate tends to be too small to overcome the Hubble expansion rate, and hence the universe is stuck in an eternal inflationary phase. Instead, for scenarios with large back-reaction, the universe inflates by (at most) a few e-folds and eventually completes the transition. In these cases the nucleation temperature is typically of the order of the EW scale, contrarily to what happens in most of the (small-back-reaction) frameworks considered in the literature [8][9][10][11][12]. Moreover, depending on the benchmark choice, the transition can end up with a reheating temperature smaller or larger than the nucleation temperature of the EW phase transition in the SM. We highlight the implication of this feature in Sec. 8, with some remarks about the feasibility of EW baryogenesis.
In Sec. 9 we discuss the prospects for detecting the stochastic gravitational wave (GW) background that the radion phase transition induces. Interestingly enough, the signal is so strong that both the Laser Interferometer Space Antenna (LISA) and Einstein Telescope (ET) will have very good chances to detect it.
We observe that the large-back-reaction regime favors the radion mass to be large, typically around the TeV scale. The corresponding collider phenomenology is studied in Sec. 10. No tension with present LHC data is found for the benchmark scenarios although, for future integrated luminosity, the radion decay into W + W − and ZZ might lead to detectable signatures.
Finally general conclusions are drawn in Sec. 11.

General formalism
We follow the notation and conventions of Ref. [10] 1 . We consider a slice of 5D spacetime between two branes at values r = r 0 , the UV brane, and r = r 1 , the IR brane. The 5D action of the model, including the stabilizing bulk scalar φ(x, r), reads as where V (φ) and Λ α (φ) are the bulk and brane potentials of the scalar field φ, and the index α = 0 (α = 1) refers to the UV (IR) brane. The parameter κ 2 = 1/(2M 3 ), with M being the 5D Planck scale, can be traded by the parameter N in the holographic theory by the relation [34] where is a constant parameter of the order of the Planck length, which determines the value of the 5D curvature. The metric g M N is defined in proper coordinates by The EoM read then as 2

6)
and, assuming a Z 2 symmetry across the branes, the localized terms impose the constraints The EoM can then be written in terms of the superpotential W (φ) as [14] φ = 1 2 and while the BCs read as (2.14) Note that the EoM in Eqs. (2.6)-(2.10) and Eqs. (2.11)-(2.14) are completely equivalent, having both sets three integration constants. In particular one of the integration constants appears in Eq. (2.12).
Starting from a potential V and integrating Eq. (2.12) is usually a very complicated task which normally cannot be accomplished analytically. On the other hand starting from a superpotential function W , and computing the potential V from Eq. (2.12), amounts to fixing the corresponding integration constant to zero, and no radion potential can be generated using this method. To circumvent this problem (for details see the next section) we propose an alternative procedure: we determine the effective potential by integrating the action over the solutions of the EoM with the scalar BC (2.10) (or equivalently (2.14)) imposed at both branes, but we mistune the BC (2.9) [or equivalently (2.13)] while finely adjusting the potential Λ 0 3 . In this way, by means of the mistuning we break the flatness of the radion potential, and by means of the Λ 0 adjustment we achieve a zero cosmological constant at the minimum of the potential.
For concreteness we consider for the brane potentials the form where Λ α is a constant, hereafter considered as a free parameter as it does not enter in Eqs. (2.10) and (2.14), and γ α is a dimensionful parameter. Using Eq. (2.15) for the brane potentials, the BCs in Eq. (2.14) can be written as (2.17) 3 See e.g. the thorough discussion in Ref. [37].
As we will see in Sec. 3 the effective potential will depend through Λ α (φ(r α )) on the γ α parameters. In the simple (stiff wall) limit where γ α → ∞, the BCs (2.16) and the potential (2.17) simplify to measures the mistuning we are doing, while the parameters γ α have introduced a dynamical mechanism by which φ(r α ) = v α . In fact the condition φ(0) = v 0 is enforced by fixing the integration constant of the first equality in Eq. (2.11), while the condition φ(r 1 ) = v 1 is enforced by fixing the integration constant appearing in Eq. (2.12) as we will see in Sec. 3. In the generic case of finite γ α , an analytic solution to the BCs (2.14) does in general not exist but still numerical solutions can be worked out, as we will see in Sec. 3. In the following, and unless explicit mention, we work in the limit γ α → ∞.

The effective potential
By using Eqs. (2.6)-(2.8), the action (2.1) can be written as where we have included a factor of 2 in S bulk and S GHY from orbifolding, as we are integrating over S 1 /Z 2 . By joining all these terms together we get with where we are using the EoM degrees of freedom to fix r 0 = 0 and A(0) = 0. The variable r 1 is thus the branes distance and establishes the relationship between κ 2 and the 4D rationalized Planck mass, M P = 2.4 × 10 18 GeV, via the expression wherer ≡ r/ is dimensionless. In particular, for some given N andr 1 , Eqs. (2.2) and (3.7) fix the value of .
In the limiting case γ α → ∞, using the superpotential formalism, the first equation in (2.11) has just one integration constant and thus only the value of the field at, say the UV brane, is fixed (thus v 0 is fixed). Therefore within the superpotential formalism, if we start from a superpotential W 0 from which the bulk potential V is deduced, we fix to zero the integration constant that should have appeared in Eq. (2.12). We have then lost the freedom to choose the value of φ at the IR brane (v 1 ), in particular we cannot set v 1 at the value for which r 1 solves the hierarchy problem [cf. Eq. (2.18)]. However, as we now explain, there exists a way of reintroducing such a freedom. Let us call the "lost" integration constant s.
We consider a potential V that is expressed in terms of a zero-order superpotential being solution of Eq. (2.12) to all orders. This means that Eq. (2.12) does not fix the integration constant s, which should then be fixed from the BC φ(r 1 ) = v 1 . An explicit solution is given for n = 1 by [38] (see also discussion in [39,40]) while for n > 1 it can be iteratively defined as From now on we assume sW 1 W 0 , so that we can keep the expansion in Eq. (3.8) to linear order, which corresponds to use W = W 0 + sW 1 + O(s 2 ), an approximation that should be verified a posteriori. We can similarly expand the field φ and metric A as 4 φ(r) = φ 0 (r) + sφ 1 (r) + O(s 2 ) , (3.12) As we are solving Eq. (2.11) order by order perturbatively, condition sW 1 W 0 also implies sφ 1 φ 0 and sA 1 A 0 . The corresponding expansion of W then reads (3.14) Using now the first expression in Eq. (2.11) we get corresponding to the values of φ(r) in the UV and IR branes, respectively. In particular one can fix C 1 = v 0 such that φ 0 (0) = v 0 and φ 1 (0) = 0. Then the condition φ(r 1 ) = v 1 leads to fixing the integration constant s as 5 Therefore the superpotential in Eq. (3.14) gets an explicit dependence on the brane distance, W (r 1 ), which in turn creates a non-trivial dependence on r 1 of the effective potential of Eq. (3.6). As the latter only gets contributions from the branes, one can then expand the superpotential on the branes as so that the effective potential can be expanded to first order in s(r 1 ): Eq. (3.20) involves several key parameters that play a relevant role in our analysis. The second line, and in particular the function s(r 1 ), provides a non-trivial dependence on the brane distance r 1 . We anticipate that r 1 can be interpreted as the constant background value of the (canonically unnormalized) radion/dilaton field. Consequently, the cosmological constant at the minimum of the radion potential can be set to zero by an accurate choice of the terms in the first line, which are independent of r 1 . We fine-tune Λ 0 for such a scope 6 .
Similarly, from Eq. (3.14) and the second expression in Eq. (2.11) one finds where φ 1 (φ) is given by Eq. (3.16) with the substitution φ 0 →φ. The integration constants in Eqs. (3.22) and (3.23) have been chosen to fix A(0) = A 0 (0) = 0. Given that φ 0 = φ + O(sφ 1 ), and since sA 1 A 0 , we can keep the zero order φ 0 φ in the definition of A 1 in Eq. (3.23). This, together with the BC φ(r 1 ) = v 1 , leads to As we see, A 1 (r 1 ) does not explicitly depend on r 1 , it only depends on v α and the superpotential parameters.

The soft-wall metric
We consider the exponential superpotential used in soft-wall phenomenological models [15]: This function W 0 (φ) is an exact solution of the EoM involving the scalar potential Following the general procedure described in Sec. 3, we find The scalar field φ = φ 0 + sφ 1 results where the location of the naked singularity, r S , is given by Note that integration constants have been fixed such that in which the integrand is (4.8) The integrals in Eqs. (4.5) and (4.7) cannot be computed analytically in general and therefore all calculations of the effective potential will be performed numerically. For the warp factor A = A 0 + sA 1 , we can determine A 0 as Instead A 1 cannot be given in terms of an analytic solution and we have thus to determined it numerically. In particular for A 1 (r 1 ) we use the general expression provided in Eq. (3.24).
In order to solve the hierarchy problem we have to fix A(r 1 ) 35. This can be done by conveniently choosing the brane parameters v α and γ in the superpotential, as well as κ 2 , which provides the physical KK scale ρ 1 ≡ −1 exp[−A(r 1 )] 7 . Moreover, by fixing the parameter κ 2 and the metric A(r), the value of is established from the 4D Planck mass value as in Eq. (3.7). Since s(r 1 )A 1 (r 1 ) A 0 (r 1 ), to solve the hierarchy problem it is enough to work to zero order in the s expansion, which means A 0 (r 1 ) 35. Then, from A 0 (r 1 ) 35 and assigning some values to γ and r S (i.e. v 0 ), one can find r 1 . Moreover using the approximation φ 0 (r 1 ) v 1 one can roughly estimate v 1 from This simple approximation is useful to guide the eye although the correct value of r 1 has to eventually be computed numerically. Eq. (4.10) also highlights that the IR brane is shielding the singularity since r 1 < r S . The amount of back-reaction in our solution can be read off from comparing the size of the two terms in the right hand side of the approximation Two extreme possibilities arise for a fixed value of κ 2 : i) For −3/2 γ 1 and 3/2 |v 1 − v 0 | 1, the second term is small compared with the first one and the hierarchy problem is mainly solved by the first term. In this case there is little back-reaction on the metric and the length of the extra dimension is comparable to the AdS case, i.e. r 1 A 0 (r 1 ) . ii) For −3/2 γ 1 and/or 3/2 |v 1 − v 0 | 1 the second term can be comparable to the first term and the length of the extra dimension is smaller than in the AdS case, i.e. r 1 < A(r 1 ) . This case is also characterized by the fact that the IR brane is close enough to the singularity, i.e. (r S −r 1 ) r 1 . For different values of κ 2 (N 2 ) the amount of back-reaction decreases with decreasing (increasing) values of κ 2 (N 2 ).
Since the superpotential formalism does not permit an analytic approach, in the present paper we carry out our investigation by concentrating in a few concrete benchmark scenarios. They cover parameter configurations with large or small back-reactions on the metric 8 , and are expected to give some qualitative insight on a vast class of plausible models where the hierarchy problem is solved. Our benchmark scenarios belong to the following classes: 7 The scale ρ 1 is O(TeV) for −1 M P = 2.4 × 10 18 GeV and A(r 1 ) 35. In the numerical calculations we will work in units where = 1.
8 Models with bulk potentials quadratic in φ as those originally considered by Goldberger and Wise [4], would fall in our formalism into the small back-reaction class. -Large back-reaction (class B) (4.14) -Large back-reaction & smaller N (class D) For concreteness, in all of them we choose the remaining free parameters to obtain A(r 1 ) 35. We also rescale the brane tensions Λ α as where λ α are dimensionless constants and the value of λ 0 , which is negative, is used to fine-tune the cosmological constant to zero at the minimum of the potential. At this point, λ 1 is a free parameter, which, as we will see, will determine the shape of the effective potential.
In the left panels of Fig. 1 we show some numerical result of V eff for several values of λ 1 in classes A and B scenarios (upper and lower panels, respectively). The potentials are normalized to zero at r 1 → ∞, where there is a minimum in all cases. The plots highlight how the parameter λ 1 controls the shape of the potential. For |λ 1 | 1 (with λ 1 < 0), the absolute minimum at r 1 is very deep, and the maximum between the absolute minimum and the local minimum at r 1 → ∞ is tiny. Moreover there is a critical value of |λ 1 | for which the absolute minimum becomes degenerate with the minimum at r 1 → ∞, and even disappears (becomes a saddle point) for smaller values of |λ 1 |.
In the right panels of Fig. 1 we also show the relative size of the O(s) terms in the superpotential expansion, displayed as sW 1 (v α )/W 0 (v α ). In the upper panel we present the results for the class A scenarios (small back-reaction) while in lower panel we do it for the class B scenarios (large back-reaction). Notice that within one given class the ratio sW 1 (v α )/W 0 (v α ) does not depend on the particular λ 1 value. As we can see, in the region r 1 > r 1 relevant for the study of the phase transition, the ratio is small enough to guarantee the validity of the s-expansion, as assumed in the analysis.
In view of this behavior of V eff , in the rest of the paper we restrict ourselves to configurations with potentials having two minima (and reliable s expansion). Specifically, for each class we take some generic set of values for λ 1 . Such values are provided in Tab. 1 (for the color code of λ 1 in the table, see Section 7). Within each class, the choice of λ 1 unequivocally define benchmark scenarios. The scenarios A 1 , B 1 , . . . , B 11 , C 1 , C 2 , D 1 and E 1 are those we investigate numerically in the next sections. Tab. 1 also includes the value of in units of M P that we obtain via Eq. (3.7). As expected, −1 results very close to M P .

The radion field
We now introduce the radion field as a perturbation of the metric whose definition is The Einstein EoM can be solved with the radion ansatz F (x, r) = F (r)R(x) such that the excitation of the field φ, ϕ(x, r), can be reparametrized as [41] ϕ(x, r) = 3 so that the only remaining degree of freedom is the radion field R(x). In particular we adopt the ansatz F (r) e 2A which is appropriate for a light radion/dilaton 9 Figure 1: Left panels: Effective potential for different values of λ 1 in units of . Only the relevant regime r 1 > r 1 is considered. Right panels: The relative correction to the superpotential sW 1 (v 1 )/W 0 (v 1 ) as a function of r 1 . The panels on the top correspond to 'small back-reaction scenario (class A)', while the panels on the bottom correspond to 'large back-reaction scenario (class B)'.
case Eq. (5.3) leads to ϕ(x, r) 0. Moreover the geodesic distance between the branes can be written as [10] r(x) ≡ ds = by which R(x) can be interpreted as the excitation of the (unnormalized) radion field r(x) with background value r 1 . This provides the functional dependence of the effective potential V ef f (r 1 ) we consider in Eq. (3.20) and subsequently.
Scen.  The kinetic term of the action is given by [24] 2 d 4 x from where we can see that the field r(x) is not canonically normalized. One uses to define the canonically normalized 10 field µ(x) with kinetic and mass terms as with m rad being the mass of the normalized radion. The field µ(x) is related to r(x) by where in the last step the background field r 1 is approximated by the whole field configuration r(x). The formal solution to Eq. (5.8) is In this case the effective potential is given by the function where r(µ) is the inverse function provided by Eq. (5.9). In general the relationship between the fields µ and r can only be obtain numerically. However the relation can be easily solved analytically in the particular regime of no back-reaction, e.g. in the AdS scenario. In that case it turns out that A(r) = r/ and X F (r) = exp(2r/ ) so that also Eq. (5.8) can be solved analytically leading to µ(r) = −1 e −r/ or, equivalently, r = log(1/µ ), which is the usual expression obtained in the Randall-Sundrum theory.
The effective potential for the cases of small and large back-reaction (and thus N large) are shown in Fig. 2. We observe that the shape of the potential in every case, i.e. the depth and location of the minimum, has important consequences for the dilaton phase transition. The flatter the potential, the slower the way to the false minimum, the bigger the euclidean action (as we will see) and the more difficult (if not impossible) the phase transition. The flatness of the potential is associated with the amount of back-reaction 12 . This happens for the potentials in classes A, B, C and E in the left panel of Fig. 2, as we can see. In fact we will see that in class A, unlike in classes B, C and E, the euclidean action is so large that the transition rate never overcomes the expansion rate of the universe. Moreover, the location of the minimum is also important for the phase transition. In fact the smaller the value of µ , the shorter the road along the potential to the false minimum, and thus the smaller the euclidean action. This fact is exemplified in the right panel of Fig. 2 where the potential for the class D scenario is shown. Even if the potential is flatter than in case A, the value of µ for case D is one order of magnitude smaller than in case A, and then the euclidean action is also smaller and allows the phase transition, as we will see.
For the validity of the 4D treatment of the radion field it is necessary that the KK graviton modes are significantly heavier than the radion and thus can be integrated out in the EFT. In that case, it is energetically expensive for the KK fields to move and the transition can be studied in an effective theory where the only extra (with respect to the SM) dynamical degree of freedom is the radion. To check such a hierarchy, the following analytical approximate formulas turn out to be useful [24]: , (5.12) in which the last term is negligible for strict stiff wall boundary potentials (γ α → ∞).
Similarly, the mass of graviton KK modes can be approximated as [24] . (5.14) Therefore the validity of the EFT requires the ratio to be small. Using Eqs. (5.12), (5.14) and (5.15), for our benchmark scenarios we obtain although the more precise values depend on the specific value of λ 1 of each scenario 13 . It then turns out that the radion is not very light in the scenarios of the class B and C because of the large back-reaction and the strong departure from conformality near the IR brane. Still there is enough hierarchy between the radion and the KK graviton masses to justify the use of the EFT effective potential for the analysis of the phase transition in most of the benchmark scenarios although class C might be borderline 14 . However this does not happen for scenarios A, B, D and E where the radion is lighter, as compared with the corresponding KK graviton mass. The precise values of the mass ratios for the different benchmarks are shown in the Tab. 1 which also includes the scale ρ 1 , defined in Eq. (5.11), and the radion VEV µ , corresponding to the minima of the potentials. 6 The effective potential at finite temperature At finite temperature the system allows for an additional gravitational solution with a Black Hole (BH) singularity located in the bulk. In the AdS/CFT correspondence this BH metric describes the high temperature phase of the system where the dilaton is sent to the symmetric phase µ = 0 and thus the condensate evaporates [5].
Let us consider a BH metric of the form where h(r) is a blackening factor which vanishes at the position of the event horizon, Eq. (6.5) can be eliminated in favor of (6.2)-(6.4) by means of the identity so that we have three differential equations with five integration constants which can be fixed by imposing BCs at the UV brane r = 0, and regularity conditions at the singularity r = r h . Four of these integration constants are then set as while the fifth one is A(r h ) and is traded for the physical parameter T h representing the Hawking temperature of the system. Indeed, from Eq. (6.1) it can be derived that the temperature T h and the entropy S of the BH can be expressed as [36,42] 15 The quantity T h has a key role in the phase transition. To appreciate this, it is useful to consider the thermodynamics relations for the internal and free energies with U and F = U − T S being the internal energy and the free energy, respectively. In fact Eq. (6.10) makes manifest that F (T h ) has a minimum at T h = T that amounts to where we have employed Eq. (6.8) and the definition (6.12) In particular, under the assumption of a h being a smooth function of T , we can approximate the free energy as

The case of small back-reaction
The regime of small back-reaction has been broadly studied [5][6][7]. In this case the constant part of the bulk potential V (φ) −6/(κ 2 2 ) dominates, and neglecting the back-reaction of the scalar field on the metric A is a good approximation. Thus the solutions to Eqs. (6.2)-(6.5) read (6.14) Moreover, from Eqs. (6.8) and (6.14) one recovers the usual expressions leading to the standard expression for the free energy in AdS space [5]:

The case of large back-reaction
In the case of large back-reaction, the blackening factor h(r) has to be obtained by solving Eqs. (6.2)-(6.5) numerically, from where one can easily deduce a h (T ) and F min . We show the result of this procedure in Fig. 3 whose left and right panels deal, respectively, with the class A (i.e. small back-reaction) and B (i.e. large back-reaction) scenarios. The resulting function a h (T ) is marked in (blue) solid, while the quantity κ 2 F min /(π 4 3 T 4 ) is marked in (red) dashed. We see that, as anticipated, for small-back reaction a h (T ) basically reproduces the case of pure AdS (i.e. a h (T ) = 1), whereas for large back-reaction it results a h (T ) 1. This effect has important phenomenological implications since it strongly influences the nucleation temperature of the phase transition, as we discuss in Sec. 7. The comparison between a h and κ 2 F min /(π 4 3 T 4 ) highlights the fact that Eq. (6.13) is a very good approximation of F min for all practical purposes.
We have checked that these features do not depend on the specific benchmark scenarios we have considered. In particular the behavior of a h (T ) is generic and only depends, in all cases, on the amount of back-reaction on the gravitational metric.

The dilaton phase transition
The phase transition can start when the free energy of the BH deconfined phase, F d , becomes smaller than the free energy in the soft-wall confined phase, F c . Those free energies are defined by where g ef f c (g ef f d ) is the number of SM-like degrees of freedom in the confined (deconfined) phase, F min is given in (6.11) and finally E 0 is defined as 16 . In this way the critical temperature T c at which the phase 16 For numerical purposes we need to focus on a given particle setup: we assume that at low energy the confined phase does not contain BSM fields besides the radion. In this phase, at T much below the mass scale of the n = 1 modes, g B(F ) (T ) matches the SM number of bosonic (fermionic) degrees of freedom. It follows that g ef f = g B (T ) + 7 8 g F (T ) = 106.75 at 172 GeV T m G . On the other hand, at very high temperatures, in the deconfined phase only the elementary degrees of freedom will contribute to the free-energy F d , which we will then assume to be contributed by most of the SM degrees of freedom, as we will only consider, as we will see later on in this section, a few (composite) states (as the right-handed top quark and the Higgs scalar) living in the IR brane. Under this transition starts being allowed (the nucleation temperature T n is indeed below it) is given by The values of T c for the different considered benchmark scenarios are shown in Tab. 1.
To study in detail the dilaton/radion phase transition we have to consider the bounce solution of the Euclidean action, as described in Refs. [43][44][45]. For the canonically normalized field µ, the Euclidean action driven by thermal fluctuation is O(3) symmetric and given by [45,46] The corresponding bounce equation is with ρ = √ x 2 and BCs 17 Thermal fluctuations are not the only way of overcoming the barrier between the false and true vacua. At low enough temperatures it can also occur via quantum fluctuations. In this case the bounce solution is O(4) symmetric, with Euclidean action S 4 provided by [43,46] S 4 = 2π 2 dρρ 3 6 3 κ 2 where ρ = √ x 2 + τ 2 (with τ being the Euclidean time), and satisfies the differential equation reasonable assumption, the contribution to the free energies coming from the SM degrees of freedom is balanced between the confined and deconfined phases, and can be neglected. This approximation will be justified in Secs. 7.3 and 8. 17 Notice that the BC at µ = 0 is not the standard one which fixes the behaviour of the solution at ρ → ∞. Here we exploit the fact that µ(ρ) reaches µ = 0 at very large values of ρ, so that at even larger ρ the friction term in Eq. (7.5) is negligible. Our BC at µ = 0 is thus equivalent to the standard one due to approximate energy conservation (i.e. approximate lack of friction) in the subsequent evolution of the bounce.
As we are normalizing the potential to zero at the origin µ = 0, instead of normalizing it at the (fake) BH minimum as in the original calculations [43,45], we have to add the omitted contribution to the Euclidean action. In a suitable approximation this is given for the O(3) solution by where, in practice, we can take c 3 c 4 1 and A T 4 c (changing these values has negligible impact on the results of this paper). Then Γ/V is dominated by the least action such that in non-pathological regimes we can assume S E = (S 3 + ∆S 3 )/T for O(3) bubbles and S E = S 4 + ∆S 4 for O(4) bubbles. Only when the first and second terms in the right hand side of Eq. (7.13) are very close to each other, one should take care of the full expression of Eq. (7.13).
The onset of nucleation then happens at the temperature T n such that the probability for a single bubble to be nucleated within one horizon volume is O(1). A simple estimate translates into the upper bound on the Euclidean action [10,11] S E 4 log (M P / µ ) ≈ 140 , (7.14) which will be considered throughout the forthcoming numerical analysis.   3)) solution dominates, as expected. Remarkably, neither S 4 nor S 3 /T , and therefore S E , ever reach the threshold 140. This happens because the free-energy in the BH solution is large, i.e. a h O(1), and the system tries to cool down as much as possible to minimize the energy barrier between the confined and deconfined phases. Nevertheless, due to the flatness of the potential, the barrier is still too big even at zero temperature. As a consequence the bubble nucleation is always too suppressed to compete with the Hubble expansion of the universe, and the bubbles of the confined phase µ never percolate. This leads to a universe where a (huge) portion of the space remains in an inflationary phase (see Sec. 7.3). The viability of the scenario A 1 is then quite debatable and we do not further investigate it hereafter.

Large back-reaction
To describe the behavior of the radion phase transition in the regime of large backreaction, we first focus on classes B and C, and then comment on the remaining parameter configurations.  Fig. 4 but for scenario B 8 . Lower panels: As in Fig. 4 but for scenario B 2 . The findings are qualitatively similar to those arising in the most common parameter scenarios where the back-reaction is large (cf. Fig. 6).
The upper (lower) panel of Fig. 5 shows the numerical results for the bounce in scenario B 8 (B 2 ). Similarly, the upper (lower) panel of Fig. 6 deals with scenario C 2 (C 1 ). The plots illustrate that for large values of |λ 1 | (lower panels) the phase transition is dominated by the O(3) bounce, while for lower values of |λ 1 | it is dominated by the O(4) bounce (upper panels). The plots moreover highlight that µ 0 and T n , which is the largest temperature where S 4 or S 3 /T crosses the horizontal dashed line, are of the same order of µ . This happens due to the fact that a h 1: the temperature in the free energy has to be substantially increased to compensate the smallness of the prefactor a h , in comparison to what happens in a h ≈ 1 configurations (remember that T appears only in F d once the SM-like plasma is neglected, as previously stressed).
As expected, Figs. 5 and 6 also show that the nucleation temperature provided by the O(3) ansatz, if it exists, is higher than the one arising in the O(4) case. In particular  Fig. 4 but for scenario C 2 . Lower panels: As in Fig. 4 but for scenario C 1 . The findings are qualitatively similar to those arising in the most common parameter scenarios where the back-reaction is large (cf. Fig. 5).
the nucleation temperature of the latter case is small enough not to jeopardize the correctness of the O(4) action calculation. In fact, the O(4) ansatz assumes a space topology that is a good approximation of the (compactified) finite-temperature space only whenρ T n 1 [5,8]. We have checked that our solutions fulfill such a condition. The numerical results obtained for other benchmark scenarios with large backreaction are qualitatively similar to those just described. We then simply report our findings in Tab. 1, together with those above. Besides quoting the results, we display the value of λ 1 in blue (red) when the phase transition occurs via the O(4) (O(3)) solution. Overall, all the considered benchmark configurations hint at the fact that in the ballpark of the large back-reaction parameter space, the transition is possible and occurs with T n / µ of the order of between one or one tenth. Much smaller values of T n are of course feasible by tuning the parameters.

Inflation and reheating
As Tab. 1 shows, when the radion phase transition happens, T n is smaller than the value of µ inside the nucleated bubble, µ 0 , or the value of µ at the radion potential minimum, µ (of course µ 0 < µ(T n ) with µ 0 µ(T n ) in our scenarios). The considered scenarios thus exhibit a quite large order parameter µ /T n , namely 2 µ /T n 25, signaling the presence of a strong first order phase transition. This is a consequence of the cooling in the initial (BH) phase, which also triggers a (very brief) inflationary stage just before the onset of the phase transition.
The energy density ρ = F − T dF/dT in the two phases is given by Inflation in the deconfined phase happens provided that E 0 dominates the value of ρ d over the thermal corrections. So inflation in the deconfined phase starts at the temperature and finishes everywhere when bubbles percolate, which is expected to occur at a temperature very closed to T n (for details, see e.g. Ref. [8]). So, the amount of e-folds of inflation occurring just before the radion phase transition is N e ≈ log(T i /T n ).
The precise values of T i and N e for the different benchmark scenarios depend on the matter content in the different confinement/deconfinement phases, i.e. the values of g ef f c and g ef f d . As previously stated, we assume that in the confined phase, at low energy, the dynamical degrees of freedom are the SM fields plus the massive radion, i.e. g ef f c = 106.75 18 . Among these, only the Higgs and the right-handed top quark are localized towards the IR brane, so that g ef f d = 97.5. The consequent values of T i and N e in the considered benchmark scenarios are shown in Tab. 2. Notice that the scenario D 1 , Eq. (7.17), yields T i < T n and thus there is no inflationary period before nucleation. In the the other scenarios, instead, a brief inflationary epoch exists, so that the plasma contribution due to the SM-like degrees of freedom is subdominant at the onset of the radion phase transition. This proves a posteriori that our calculation of T n by disregarding the thermal contribution proportional to g ef f d is fully justified. 18 We are not counting here the radion/dilaton, which is highly localized towards the IR brane and thus composite in the dual theory, whose mass in the confined phase is larger than the nucleation temperature. Its contribution ∆g ef f c ∝ exp(−m rad /T n ) is Boltzmann suppressed, as it decouples from the thermal plasma.  Under the approximation that the percolation temperature is very similar to T n , during the phase transition the energy density is approximately conserved. At the end of the phase transition the universe then ends up in the confined phase at the reheating temperature T R given by ρ c (T R ) = ρ d (T n ) , (7.18) or, equivalently, The value of T R for the different benchmark scenarios is shown in Tab. 2. It turn out that in most of the cases T R is quite close to the TeV scale, nevertheless a parameter window with T R at the EW scale exists (e.g. scenario D 1 ). We will comment on the consequences of this observation in the next section.

The electroweak phase transition
Depending on the particle setup and the embedding of the Higgs field in the model, the confinement/deconfinement phase transition can be tightly connected to the EW phase transition. This is the case in our setup (specified in Sec. 7.3) where the Higgs, the radion, and the right-handed top are localized towards the IR brane and hence only exist in the confined phase.
All SM-like fields propagating in the bulk, as well as those localized at the branes, are present in thermal plasma of the confined phase. Their contribution to the free energy is ∆F c = −π 2 g ef f c T 4 /90, with g ef f c 106.75 at 100 GeV T m rad . Instead, the fields localized near the IR brane are beyond the BH horizon and, being outside the physical space, they are not present in the deconfined phase. Within our particle setup, the thermal plasma before the radion phase transition contributes to the free energy as ∆F d = −π 2 g ef f d T 4 /90, with g ef f which corresponds to |∆F min /F min | 0.01. Therefore the nucleation temperature of the radion phase transition is essentially unaffected by the presence of the SM-like degrees of freedom in the plasma. Disregarding them in the calculation of T n is hence fully justified, even when the phase transition does not start in an inflationary epoch, as in our scenario D 1 .
On the other hand the SM-like particles do not contribute to the free energy only via the plasma term: when the BH horizon moves beyond the IR brane during the phase transition, the Higgs field (H) appears and there is an extra dynamical field besides the radion. The effective potential becomes a function of both fields and can be written as [8] while the SM potential V SM in the effective theory, after integrating the extra dimension, is given by where the Higgs mass is m 2 H = 2λv 2 (125 GeV) 2 with λ = v 2 /m 2 0.123 and v = 246 GeV, and the term ∆V SM (H, T ) contains the Higgs field dependent loop corrections both at zero and at finite temperature. V SM (H, T ) has its absolute minimum at H(T ) = v(T ) whose value, in the first (leading) approximation for the thermal corrections, turns out to be [46,47] v where T EW , the temperature at which the SM minimum at the origin turns into a maximum, is given by In principle, the analysis of the radion phase transition should also take into account the H degree of freedom. However, in practice, this is not necessary. In fact, V SM provides a contribution O(λv 4 (T )/4), so that it effectively shifts the µ 4 term in V ef f (µ) by the amount O(λv 4 (T )/(4 µ 4 )), which is vanishing for T n > T EW and is +O(10 −4 ) otherwise. Such a correction is therefore too small to substantially affect the results of the radion phase transition, obtained without including V SM (cf. Fig. 2). The calculations in Sec. 7 turn out to be justified a posteriori.
We can see from from Tab. 1 that some scenarios lead to T n < T EW , so that the EW symmetry is broken at the same time that the confinement/deconfinement phase transition, while other scenarios yield T n > T EW and the EW symmetry remains unbroken during the radion phase transition 19 . Nevertheless, it ultimately depends on T R whether the universe really ends up in the EW broken phase after the deconfined/confined bubble percolation or, in other words, whether the dilaton and the EW phase transitions are sequential or simultaneous. This has consequences for electroweak baryogenesis [47,48], as we now discuss.

Sequential phase transitions: T R > T EW .
Models with T R > T EW are exhibited by the scenarios of classes B, C and E (see Tab. 2). In those cases, even when T n < T EW , at the end of the reheating process the Higgs field is in its symmetric phase and the universe evolves along a radiation dominated era. Within the particle setup we have assumed so far, the EW symmetry breaking would occur as in the SM, that is, via a crossover that prevents the phenomenon of electroweak baryogenesis [49,50]. Had we chosen a low energy particle content rich of new BSM degrees of freedom, the dynamics of the EW symmetry breaking would have been the one corresponding to the chosen low energy setup (while the radion phase transition would have been basically unchanged). In this sense, when T R > T EW , the implementation of electroweak baryogenesis remains a puzzle for which the UV soft-wall framework is not helpful.

Simultaneous phase transitions: T R < T EW .
For T R < T EW the reheating does not restore the EW symmetry and eventually the Higgs lies at the minimum of V(H, T R ). The value of its minimum, v(T R ), can be considered as the upper bound of the Higgs VEV during the (simultaneous) EW and deconfined/confined phase transitions. Taking this upper bound, it results that the EW baryogenesis condition 20 v(T R ) is fulfilled in the presence of a SM-like low energy particle content (and m H 125 GeV) when T R satisfies the bound [8] (see also Ref. [51]) To summarize, in scenarios with T R < T EW the nature of the EW phase transition is then entirely dependent on the radion reheating temperature. More specifically: • If the reheating temperature is T EW T R > T H , the EW phase transition is too weak (i.e. it does not satisfy Eq. (8.6)) and the sphalerons inside the bubble wipe out any previously created baryon asymmetry.
• If the reheating temperature is below T H , then the sphalerons inside the bubble do not erase the possible baryon asymmetry accumulated inside the bubble during their expansion. Therefore EW baryogenesis can take place if there is a strong enough source of CP violation in the theory. However, the radion phase transition in the generic scenarios leading to T R T H should be studied paying particular attention to the bounce procedure. In fact the vacuum energy E 0 might not have dominated the energy density prior to the transition (see Eq. (7.19)), as the dilaton and Higgs potentials might be of the same order of magnitude. The precise bounce solution would then need to be solved in the two-field space (µ, H), as in Ref. [52] 21 .
A parameter configuration leading to T R < T H is provided by scenario D 1 . In this case the dilaton and EW phase transitions happen simultaneously at T = T n 112 20 The SM at finite temperature has an IR singularity at the origin such that perturbative calculations in this region are unreliable. In fact lattice calculations point toward an extremely weak phase transition, or cross-over, for Higgs masses around the experimental value. However for temperatures low enough condition (8.6) is fulfilled, and the perturbative potential near the minimum can be approximately trusted. 21 The precise evaluation of such bounce solutions goes beyond the scope of the present paper whose main aim is more to stress new possibilities than providing refined results.
GeV, ending up with T = T R = 133.7 GeV < T EW , so that both the radion and the Higgs acquire a VEV. Before and after the reheating, the bound of Eq. (8.7) is fulfilled, and the condition of strong-enough first order phase transition for EW baryogenesis is satisfied 22 .
A key quantity is the velocity v w at which the bubble walls are expanding at the moment of their collisions. In standard cases this would be determined as the asymptotic solution of the EoM of the field driving the phase transition [75][76][77]: where δf j is the small deviation from the Boltzmann distribution of the species j with mass m j . However in our case, whereμ = µ forμ ≥ 0 andμ = −T h forμ < 0, not all δf j are small 23 . Thus Eq. (9.1) does not capture the complex dynamics of the confined/deconfined phase transition, and strongly-coupled techniques, still under development, should be applied; see e.g. Refs. [78,79]. In any case, it seems reasonable to expect supersonic walls, even reaching v w ≈ 1 in the extremely supercooled scenarios (i.e. very strong phase transitions, in practice). For concreteness we thus discuss in detail two reasonable options for v w , namely v w = v 1 ≡ 0.70 and v w = v 2 ≡ 0.95. A further critical feature is the behavior of the plasma during, and after, the bubble collisions. Besides the energy stored in the bubble walls, the turbulent or coherent motions of the plasma, excited by the bubble expansion, can contribute to the SGWB spectrum too. Including them would enhance not only the amplitude of 22 For a recent analysis see Ref. [52,53]. 23 For instance, fields exactly localized on the IR brane are degrees of freedom that do not exist in the deconfined phase and suddenly appear when the BH horizon crosses the IR brane (atμ = 0). This abrupt change implies δf j to be of the same order of the Boltzmann distribution f j , i.e. the species j is far away from the thermal equilibrium. By continuity, large deviations are also expected for fields non-exactly localized. For these, it is manifest that their non-trivial prefactor ∂m 2 j /∂μ is not sufficient to enforce the sum in Eq. (9.1) to be a small perturbation. the GW frequency spectrum but even the shape of the spectrum at high frequencies.
Unfortunately, no robust result on the plasma effects exists for the ambiguous case of a deconfined/confined phase transition, in particular when T n is so high that most of the SM-like degrees of freedom in the plasma are not Boltzmann suppressed. We thus refrain ourselves from including plasma effects in the subsequent analysis.
In view of the above considerations, in our analysis we employ the envelope approximation results [56,60,61,72,73,80]. In such a regime, the frequency power spectrum of the SGWB is given by [74] In particular for the chosen velocities v 1 and v 2 it turns out that ξ(v 1 ) 0.04, ξ(v 2 ) 0.07,ξ(v 1 ) 0.28,ξ(v 2 ) 0.24. The size of the peak of the power spectrum, f p , can span many orders of magnitudes, and strongly depends on β/H and T R . The latter is basically set by 4 √ E 0 (see Eq. (7.15)). Had we not bothered about the solution to the hierarchy problem, values of T R differing from the TeV scale by orders of magnitude (in particular for 4 √ E 0 much larger than TeV) would have been consistent with the theoretical framework 24 . Also β/H can span many order of magnitude and radically modify f p . Its lower bound is set by Big Bang Nucleosynthesis (BBN), which provides an upper bound on the number of relativistic species during nucleosynthesis that can be converted into the constraint [81,82]. For the spectrum in Eq. (9.2) this constraint implies h 2 Ω GW 5.6 × 10 −7 , corresponding to log 10 (β/H ) 0.045 for v w = v 1 and log 10 (β/H ) 0.16 for v w = v 2 .
24 Even though 4 √ E 0 much below the TeV scale might be in tension with LHC data; see Sec. 10.
In the next two decades several GW observatories will have the potential to observe, or constrain, the SGWB produced in our benchmark models. Fig. 7 (left panel) highlights the sensitivity curves of the main existing and forthcoming GW experiments. The dashed-dotted lines at f ∼ 1 nHz and ∼ 10 Hz are the power-law sensitivity curves h 2 Ω pls,NANO and h 2 Ω pls,LIGO O1 [83,84] reached by the NANOGRAV and aLIGO collaborations, respectively [85,86]. These collaborations do not find any SGWB in their data and consequently rule out any spectrum h 2 Ω GW (f ) that intersects one of the two dashed-dotted curves and behaves as a power law inside them (EPTA and PPTA also achieve a bound similar to the NANOGRAV's one [87,88]). The solid lines correspond to the future sensitivity curves of SKA, LISA, ET and aLIGO at its design sensitivity. Since for SKA, LISA and ET there exists no official and/or updated power-law sensitivity curve, for all future detectors we perform our analysis starting from the "standard" sensitivity curves. Specifically, for SKA we determine h 2 Ω sens,SKA(100) and h 2 Ω sens,SKA(2000) from Ref. [89,90], assuming observation of respectively 100 and 2000 milli-second pulsars (light and dark red lines respectively) during 20 years with 14 days of cadence and 3 × 10 −8 timing precision. For LISA (orange line) we take the sensitivity curve h 2 Ω sens,LISA from Ref. [91], while for aLIGO at its design sensitivity (green line) we obtain h 2 Ω sens,LIGOdesign by joining the sensitivity curves of Virgo, LIGO and KAGRA of Ref. [92]. For ET (yellow line) we use the "ET-D" sensitivity curve presented in Ref. [93]. The dashed lines display the SGWBs h 2 Ω GW corresponding to the benchmark scenarios B 1 , B 2 and B 11 summarized in Tab. 2 (the values of α and β/H from Eqs. (9.7) and (9.6) are also quoted in the table). The SGWB spectra touching the blue area are ruled out by the BBN bound previously discussed.
The right panel of Fig. 7 sketches the parameter region (hatched areas) of the plane β/H -T R that NANOGRAV, EPTA, PPTA and aLIGO O1 rules out, assuming the spectrum in Eq. (9.2) with v ω = v 2 . The exclusion is based on the criterion that a spectrum touching the power-low sensitivity curves of these experiments would have already been detected 25 . The blue area is the BBN bound above mentioned. The remaining areas sketch the β/H -T R parameter regions for which Ω GW will yield a Signal-to-Noise Ratio (SNR) larger than 10 in the data which will be collected in the next two decades. For concreteness, for each experiment we check the condition with T i = 20, 3, 7 and 8 years, respectively, for i ="SKA","LISA", "ET" and "aLIGO design" (these numbers are very indicative estimates of the amount of data that each experiment may take by 2040 including duty cycles). The parameter reach that we obtain for LISA does not substantially differ from the one previously calculated in Ref. [74]. We remind that Fig. 7 assumes v w = v 2 . The forecast for a different bubble velocity, v ω , can be obtained from the right panel of the figure by shifting the coloured regions by 0.5 log 10 [0.07/ξ(v ω )] and log 10 [0.06/(ξ(v ω ) ξ(v ω ))] along the log 10 (β/H ) and log 10 (T R /GeV) axes, respectively. Thus for the case v w = v 1 the shifts are around 10% in log 10 (β/H ), and 1% in log 10 (T R /GeV), which are negligible with respect to the approximations on the spectrum we are making. Notice also that this rescaling proves that subsonic velocities, suitable for EW baryogenesis, are not incompatible with detection. For instance, within our approximations (which might not be reliable for small velocities), the "simultaneous phase transition" of the scenario D 1 would be detectable at LISA, even with v w 0.02 (fully consistent with the scenario of EW baryogenesis [94]). Unfortunately this would not hold for the ET detector, whose detection region would stay completely on the right of the point representing D 1 .
In conclusion, for both v w = v 1 and v w = v 2 , all our benchmark scenarios are promising for detection at both LISA and ET, whereas SKA and aLIGO, as well as present GW constraints, do not reach them. Out of our benchmarks, only scenario B 11 is ruled out due to the BBN bound. In general, measuring the SGWB at two experiments, sensitive to very different frequencies, will allow to better understand the nature of the SGWB. We further comment on the possible implications of this result in the conclusions.

Heavy radion phenomenology
As concluded in the previous sections, a considerable amount of back-reaction on the metric facilitates the confined/deconfined phase transition. It also typically implies that the radion is lighter than any KK resonances and has a mass around the TeV scale, at least for the parameter choices solving the hierarchy problem. Due to this mass hierarchy, in our particle setup with only SM-like fields at the EW scale the radion can decay only into SM-like fields. In particular, since the radion couples to the trace of the energy momentum tensor, its production and decay channels are those of the SM Higgs, although with different strengths. We can thus estimate the detection prospects for the radion at the LHC by rescaling the cross sections and branching rations valid for a generic SM-like Higgs, H [95] 26 with mass equal to the radion mass.

Radion couplings
As in our particle setup the 125-GeV Higgs boson is localized towards the IR brane to solve the hierarchy problem, hereafter we make the simplifying hypothesis that the Higgs is exactly localized at the IR brane. This allows to avoid technicalities that would affect the final result only marginally. The relevant 4D action for the radion, the generic Higgs H and the SM fields is then where all 5D fields have already been rescaled with the corresponding power of the warp factor and the 5D Dirac mass is M (φ) = ∓c L,R W (φ). Moreover V (H) has the form of V SM (H, T = 0) in Eq. (8.2) but with a generic λ (λ 0.123 only when H matches the 125-GeV SM Higgs H). In addition the zero modes are defined, in terms of the 4D fields, as 2) and the 5D (g 5 ) and 4D (g 4 ) gauge couplings are correspondingly related by g 5 = g 4 √ 2r 1 .
Using the radion ansatz F (x, r) ≡ e 2A R(x) and expanding Eq. (10.1) to first order in R(x), we obtain the reparametrization (see Eq. (5.9))

Couplings to massless gauge bosons
To compare the loop-induced couplings of the radion with those of the heavy Higgs H, it is useful to calculate the loop-induced couplings of both scalar fields. In the case of the heavy Higgs, the interactions to photons and gluons are given by the Lagrangians where τ i = m 2 H /4m 2 i and H = v + h. For the functions A 1/2 (τ ) and A 1 (τ ) we use their generic expressions defined e.g. in Ref. [96] although, in our regime of heavy Higgs with τ = m 2 H /(4m 2 i ) 1, they can be well approximated as It follows that 9) which implies that L Hγγ and L Hgg are respectively dominated by diagrams with Wboson exchange and top exchange.
For the radion interactions with the massless gauge bosons we take the results from Ref. [24]. The Lagrangian relevant for photons is given by and similarly for gluons. For our aim it is convenient to rexpress such Lagrangians in terms of the canonically normalized radion R. We find where c γ (c g ) measures the departure of the γγ (gg) coupling from the value that the hypothetical SM Higgs H has when m H = m rad . If the radion had couplings exactly equal to those of the SM Higgs, then c γ and c g would be equal to one, but in general they are given by  Table 3: Masses of the radion and the n = 1 graviton mode, and coupling coefficients of the radion interactions with the SM fields, for the scenario B 2 , B 8 , C 1 , C 2 , D 1 and E 1 .

Couplings to fermions
After canonically normalizing the fermions, the fermion masses are given by 14) and their couplings to the radion are manifest in the Lagrangian interaction As before, the coupling coefficient c f would be equal to one for a radion coupled to fermions exactly like the SM Higgs.
The coefficient c f is universal, i.e. equal for all fermions. However the full radion couplings to fermions depend on the fermion masses, as it happens for the Higgs; see Eq. (10.15). The values of c f in the considered benchmark scenarios are listed in Tab. 3.

Couplings to massive gauge bosons
In the Lagrangian involving the radion interactions with the massive gauge bosons, the couplings can be again normalized as with Were these couplings of the same size of those of the SM Higgs, we would have obtained c W = c Z = 1. The values of the coefficients c W and c Z in our selected scenarios are shown in Tab. 3.

Coupling to the Higgs boson
The coupling of the radion to Higgs bosons can be deduced from the interaction The interaction would have the same size of the SM trilinear interaction for c H = 1. For a generic radion it instead results The numerical values of c H for the considered models are exhibited in Tab. 3.

LHC constraints on the radion signal strengths
The production cross section and decays of the radion at the LHC can be calculated by manipulating the results on the productions and decays of a (heavy) SM Higgs. We concentrate on the scenarios B 2 , B 8 and D 1 since they well represent the collider phenomenology of our scenarios.

Radion production
At the LHC we can produce the heavy radion by the following main production mechanisms: • Gluon fusion, with a cross-section σ ggF (gg → R) related to the corresponding heavy SM Higgs prediction σ ggF SM (gg → H) by  (10.22) in the three considered benchmark scenarios.
• Vector-boson fusion, with a cross-section Likewise there exists the associated production with V , σ(pp → V * → RV ), which is proportional to |c V | 2 , and the associated production with tt, σ(gg → ttR). However they are tiny at the considered values of the radion mass so that they can be neglected as compared to the aforementioned production processes. In conclusion our benchmark scenarios highlight that at the LHC the TeV-scale radion is mainly produced via gluon fusion, and to some extent via vector-boson fusion.

Radion decay
The radion decays, mimicking the (heavy) SM Higgs, have the partial widths with X = γ, g, W, Z, f . On top of these channels, the radion can also decay into a pair of 125-GeV Higgses with partial width from which it turns out that the radion branching fraction into an X pair is    and their corresponding 95% C.L. upper bounds in the scenarios B 2 , B 8 and D 1 . The bound on the γγ channel does not distinguish between gluon and vector fusion production and then has to be compared to the sum of the two processes. No specific bound on S V BF ττ is considered. All quantities are in fb units. The bounds are taken from Refs. [97][98][99][100] .

Experimental bounds
Since the radion is a narrow resonance, the cross section S To determine whether such collider features are experimentally allowed, we consider the ATLAS searches of Refs. [97][98][99][100] constraining the W W , ZZ, τ τ and γγ channels 27 . These furnish 95% C.L. bounds on S ggF W W , S ggF ZZ , S ggF ττ , S V BF ττ , S V BF W W , S V BF ZZ and S ggF γγ + S V BF γγ as functions of the scalar mass. Tab. 6 reports the pertinent limits and the respective predictions of S ggF XX and S V BF XX in each of the considered scenarios. Notice that the constraint on the γγ channel does not distinguish between the gluon and the vector-boson fusion productions, and for this reason it has to be compared with the sum of the two production processes.
We conclude that the scenarios B 2 , B 8 and D 1 are in full agreement with the current bounds 28 and, given the values collected in Tabs. 4 and 5, we expect the same conclusion to hold for all previously investigated benchmark configurations, as D 1 is the scenario with the smallest radion mass and largest coupling coefficients. In particular, among scenarios B 2 , B 8 and D 1 , only D 1 has some channels (i.e. the ZZ and W W ones) that are not far below the experimental constraints. In this sense, future LHC data with much larger integrated luminosity, will be able to probe some of the decay channels here investigated, but likely only future colliders [103] will have the capabilities to put strong constraints on the model or to discover the soft-wall radion. This will probably happen in conjunction with the LISA and ET measurements, given the time schedule of future collider and GW facilities

Conclusions
The hierarchy problem has motivated several ultraviolet completions of the Standard Model. Among these, the frameworks of warped extra dimensions have gained popularity in the last decade. The interest in these frameworks is two-fold: on the one side, they may be the correct description of nature if the latter has a five dimensional spacetime; on the other, they may be a useful tool for understanding a strongly-coupled sector in a four dimensional nature. The most investigated warped model is the Randall-Sundrum 27 The equivalent CMS searches (see e.g. Ref. [101]) tend to provide weaker bounds and therefore we do not take them into account. On the other hand, since we eventually find that our scenarios are well within the current limits, we do not expect our conclusions to depend on the particular analyses we consider. 28 In principle also the searches for the SM-like and graviton KK modes might be relevant. Under some model assumptions, the bounds in Ref. [102], for instance, require the KK gluons to be above 4 TeV, approximatively, and thus should not be in tension with most of our scenarios. Moreover such bounds are extremely model dependent and can thus be circumvented by adjusting our particle setups. For instance, assuming the first and second generation of quarks localized towards the UV brane could relax the bounds from Drell-Yan production, as the KK modes are extremely localized towards the IR brane, without major changing on our main results. one, followed by scenarios where the metric is less trivial, which can show phenomenological advantages related to the description of precision electroweak observables. In the present paper we have explored technical challenges and phenomenological issues of one of these setups, the soft-wall models, with special emphasis on the so-called holographic phase transition.
Concerning the technical achievements, we have extended the application of the superpotential formalism to configurations where the mechanism stabilizing the extra dimension can have a strong back-reaction on the metric. This formal result is remarkable because, in principle, it can be applied to any warped model, with clear advantages on the parameter space that can be investigated without losing control on the back-reaction effects. (We remind that the correct treatment of the back-reaction has strongly limited the parameter space that some of the previous studies could explore [5][6][7][8]).
As a concrete application, we have applied the proposed formalism to the soft-wall model, where the potential in the bulk behaves exponentially near the IR brane. The radion phase transition is controlled, on the one hand by the free energy in the confined phase (i.e. essentially the depth of the effective potential at its minimum), and on the other hand by the free energy in the deconfined phase. Concerning the confined phase, the depth of the effective potential is essentially controlled by the radion mass, which in turn is controlled by the amount of back-reaction on the gravitational metric 29 . The heavier the radion, the deeper (and steeper) the effective potential, the smaller the Euclidean action (as the Euclidean "time" to get to the true minimum is shorter), and consequently the higher the nucleation temperature. In this way, for the cases of large back-reaction considered in this paper, there is no supercooling and the nucleation temperature is usually above the electroweak temperature. On the other hand the free energy in the deconfined phase, as given by Eq. (6.13), depends on the factor a h (T ) which, for the cases of small back-reaction is a h (T ) 1, while for the cases of large backreaction is a h (T ) 1, a fact which makes easier the phase transition and increases the nucleation temperature by a factor O(a −1/4 h ). Therefore the nucleation temperature is not necessarily much below the electroweak scale, which implies that the SM-like particles in the plasma are not Boltzmann suppressed during the bubble expansion and collision. Clearly, the presence of this rich plasma could have relevant effects on the dynamics of the conformal symmetry breaking and, in turn, on the phenomenology of the model.
Our method has allowed to determine the radion potential even in the regime of large t'Hooft coupling, e.g. as large as N 25, when the back-reaction goes away if all the other parameters are fixed, and the phase transition meets more difficulties to happen. The reason the phase transition can take place in those cases is because we can still compensate the sizable back-reaction by changing the values of the other parameters, in particular the values of the field φ at the UV and IR branes, v 0 and v 1 . However we have found that the radion mass increases parametrically in the cases with large values of N , the low energy effective theory describing the SM degrees of freedom and the radion field should not be trustable, and one instead should consider the whole set of 5D Kaluza-Klein modes in the thermal plasma, a task outside the scope of the present paper.
In summary, in the class of models we consider in this paper, where conformality is strongly broken in the IR brane, and we can keep track of the back-reaction, the nucleation temperature is higher than the electroweak temperature and thus the dilaton phase transition naturally occurs (sequentially) before (at higher temperatures than) the electroweak phase transition 30 . However (less natural) solutions where both phase transitions are simultaneous can be implemented in our class of models only at the price of decreasing the value of N (see the benchmark scenario D 1 above). In all the cases there is no supercooling in the deconfined phase and the amount of inflation which takes place is very marginal and does not affect at all the dynamics of the phase transition.
Together with other quantities, the reheating temperature plays a key role in the signatures of the radion phase transition. In most of the considered benchmark scenarios (cf. scenarios of classes B, C and E) the reheating temperature is much above 150 GeV; thus the electroweak phase transition is subsequent to the holographic one and resembles the one of the SM. On the contrary, when this does not happen (see e.g. benchmark scenario D 1 ) and the Higgs is localized at the infrared brane, the electroweak phase transition turns out to be supercooled and then of first order. Electroweak baryogenesis in soft-wall models thus looks possible, although further studies would be required to better understand this issue.
We also have investigated the detectability prospects of the model at the forthcoming gravitational wave observatories. Present and future pulsar time array experiments, and the current generation of ground based interferometers, are not sensitive to the stochastic gravitational signals our benchmark scenarios lead to. The LISA and ET interferometers can instead measure all of them with a signal to noise ratio of about 10 or larger (assuming the absence of further astrophysical [104] or cosmological [82] sources). Simultaneous detection of the signal at both experiments is possible due to the broadband of the predicted power spectrum and the large signal amplitude that emerges when the bubbles are supersonic. Curiously, in a corner of parameter space, the signal is so powerful that the big bang nucleosynthesis constraint rules it out. On the other hand, for subsonic bubble velocities (v w 0.02), which are those favored by electroweak baryogenesis, the signal is weaker and redshifted, and only LISA can detect (most of) the benchmark scenarios.
We have moreover noticed that in the large-back-reaction regime the soft-wall scenario tends to provide a radion mass that is only slightly suppressed with respect to the radion vacuum expectation value. For this reason, once such a vacuum expectation value is fixed at the electroweak scale to alleviate the hierarchy problem, the radion mass is not necessarily of the order of the electroweak scale or below. Thanks to this feature, the radion is not in tension with present LHC searches. Its observation may be however feasible at the future LHC runs.
In conclusion, by means of the aforementioned superpotential formalism, we have determined some interesting features of the soft-wall models in the presence of large back-reaction. A heavy radion and a large nucleation temperature look to be the main smoking guns. Whereas measuring the former at colliders would be suitable by standard techniques, inferring the latter would need improvements in the prediction and detection of stochastic gravitational wave backgrounds. In fact, based on the envelope approximation we have followed, two phase transitions having the same reheating temperature but different nucleation temperatures would provide the same stochastic signal. Only going beyond the envelope approximation, and having well under control the plasma effects during the phase transition, would allow to disentangle scenarios with tiny nucleation temperature -where most of the SM degrees of freedom are Boltzmann suppressed, as it typically happens in the Randall-Sundrum model -from those with large nucleation temperature. More detailed theoretical predictions, as well as more refined phase transition simulations, are thus required in order to break this degeneracy. We look forward to knowing them in order to understand how to possibly disentangle a given warped framework from another one.
Note added: Before submission, the LISA CosWG preview of this paper unveiled the existence of Refs. [105,106], which partially overlap with Section 9, but all done independently of this paper.