$\chi\rm{SB}$ of cascading gauge theory in de Sitter

${\cal N}=1$ supersymmetric $SU(N)\times SU(N+M)$ cascading gauge theory of Klebanov et.al spontaneously breaks chiral symmetry in Minkowski space-time. We demonstrate that in de Sitter space-time the chiral symmetry breaking occurs for the values of the Hubble constant $H\lesssim 0.7 \Lambda $, as well as in a narrow window $0.92(1) \Lambda \le H\le 0.92(5) \Lambda $. We give a precise definition of the strong coupling scale $\Lambda $ of cascading gauge theory, which is related to the glueball mass scale in the theory $m_{glueball}$ and the asymptotic string coupling $g_s$ as $\Lambda \sim g_s^{1/2} m_{glueball}$.

This theory has two gauge couplings g 1 , g 2 associated with the two gauge group factors, and a quartic superpotential while the sum of the gauge coupling remains exactly marginal [2], where g s is the asymptotic string coupling of the gravitational dual [4], perturbative β-function of the difference of couplings is nonzero [4]: Λ is the strong coupling scale of the theory. Given (1.4) and (1.5), the effective weakly coupled description of SU(N + M) × SU(N) gauge theory exists only in a finite-width energy band centered about Λ -one encounters Landau poles both in the IR 6) and the ultraviolet (UV), to leading order in M 2 /N 2 . As explained in [2], to extend the theory past the strong coupling regions one must perform self-similar transformations (Seiberg dualities [5]): IR properties of the cascading gauge theories were reviewed in [4] (see also [9]); an important feature of the theory is the characteristic scale in the glueball mass spectrum: where ǫ is a conifold deformation parameter of the holographic dual [2], and α ′ = ℓ 2 s is the string scale.
Notice that confinement triggers spontaneous breaking of the chiral symmetry [2]: there is no spatially homogeneous and isotropic phase which is confined with U(1) chiral symmetry. It will be instructive to have a geometrical classification of these phases, in the warped-deformed conifold holographic dual of the theory [2,13,15]. To this end, consider analytical continuation along the time direction t → t E ≡ it. Euclidean time t E is then periodically identified as where T is the equilibrium temperature of the phase. Topologically, the compact directions of the holographic dual are unbroken chiral symmetry : ; broken chiral symmetry : (1.12) We can thus geometrically characterize different phases depending on which cycle shrinks to zero size in the interior of the ten-dimensional Euclidean type IIB supergravity dual: (1. 13) According to [12] there is the first-order confinement/deconfinement phase transition between PhaseA s and PhaseB at 1 T c = 0.614(1) Λ thermal P g 1/2 s = 0.614(1) 3 1/2 e 1/3 2 7/12 ǫ 2/3 P g 1/2 s = 0.220(2) g 1/2 s m glueball , (1.14) where the relation between P and M is given by (2.7) and m glueball is defined as in (1.10).
At temperature T < T c the phase PhaseA s is metastable -it becomes perturbatively unstable below T χSB < T c [13], where G 5 is given by (2.8). PhaseA b has larger thermal free energy density than that of the chirally symmetric deconfined phase PhaseA s at the corresponding temperature, and thus it does not dominate the canonical ensemble. On the other hand, PhaseA b is entropically favored over PhaseA s at the corresponding energy density, and thus is the dominant phase in the microcanonical ensemble. According to [14] the phase PhaseA b is thermodynamically unstable, and thus it is dynamically (perturbatively) unstable towards developing spatial inhomogeneities [17].
In this paper we would like to understand vacua of cascading gauge theories in de Thermodynamics can be studied in canonical or microcanonical ensembles 3 . The latter one is suitable to study the dynamics of the equilibration process. de Sitter evolution of gauge theory states is eternally sourced by the space-time accelerated 2 There is no difference between the them at late times as the curvature effects are diluted as ∝ exp(−2Ht). 3 As we emphasized above the thermal equilibrium phase structure is different in the two ensembles of the cascading gauge theory. expansion and thus is (loosely) equivalent to the microcanonical ensemble; there is no correspondence to the canonical ensemble.
Insisting on spatial homogeneity and isotropy, an initial state typically 4 relaxes to thermal equilibrium configuration, which can be assigned a thermal (time-independent) entropy density. Holographic dynamics of conformal gauge theories with a simple scale transformation can be mapped to an evolution in Minkowski space-time [19] -here the late-time de Sitter vacua are conformally equivalent to equilibrium states of the microcanonical ensemble. There is no equilibration of non-conformal gauge theories at late-times in de Sitter [19] 5 : the comoving entropy density production rate is nonzero.
In [21] it was pointed out that the comoving entropy production rate R can be attribute entirely to the spatial expansion In holography, the non-equilibrium entropy density s = s(t) is associated with the Bekenstein entropy of the dynamical apparent horizon (AH) [22,23]. In [24] an example of fully nonlinear holographic evolution from initially homogeneous and isotropic state in de Sitter was presented where the late-time dynamics approaches de Sitter vacuum with entanglement entropy (1.18).
Implementing de Sitter holographic dynamics as in [24] for cascading gauge theories is outside the scope of this paper. Rather, as in [19] and [20], we assume that we specify a well-defined spatially homogeneous and isotropic initial state 6 (well-defined initial condition for the gravitation evolution) in a holographic dual. This would correspond to some coarse grained state in the gauge theory specified with the density matrix ρ.
We identify the von Neumann entropy S S = − Tr(ρ ln ρ) , 4 Not all strongly interacting systems equilibrate. See [18] for a holographic example. 5 See also [20] for a detailed recent analysis. 6 We believe that restriction to homogeneity and isotropy is not relevant for the late-time dynamics, given the accelerated background space-time expansion.
with the Bekenstein entropy of the AH in the holographic dual 7 . Partial differential equation of the gravitational dual at late times reduce to system of ordinary differential equations [24] which we analyze in details here. Inequivalent  Parallel to classification of the thermal equilibrium states, we now explain topological/symmetry considerations to classify de Sitter vacua of cascading gauge theorythe discussion is more intuitive for the closed spatial slicing in (1.17). To access AH (and thus to evaluate s ent ), the dual gravitational bulk must be described in Eddington-Finkelstein (EF) coordinates. Fefferman-Graham (FG) coordinates cover only a patch of the former, which is outside of the EF frame AH [24], and thus is not suitable for the computation of the vacuum entanglement entropy. Still, FG frame is useful to implement analytical continuation to Euclidean (Bunch-Davies) vacuum  (1.21) Parallel to (1.13), we can geometrically characterize different de Sitter vacua of the cascading gauge theory depending on which cycle shrinks to zero size in the interior of 7 This procedure is implicit in all examples of holographic evolutions in Chesler-Yaffe framework [25]. Besides 'holographic quenches' of background space-time [26] (similar to de Sitter 'quenches' of interest here) it was successfully applied to quenches of the coupling constants of relevant operators in [27,28]. the ten-dimensional Euclidean FG frame type IIB supergravity dual: To evaluate s ent we proceed in two steps 8 : • first, we construct FG frame vacua, subject to the 'boundary conditions' (1.22) (see appendix B.1 for technical details); • second, we use coordinate transformation to EF frame for each of these vacua (see [24] and appendix B.2 for technical details), and access the corresponding AH.
We summarize now our results: • TypeA s de Sitter vacua were studied previously in [29][30][31]. These vacua share resemblance with thermal deconfined chirally symmetric states of cascading gauge theory, i.e., PhaseA s . We find here that As H 2 Λ 2 increases, the Kretschmann scalar at the AH in the holographic dual increases, making supergravity approximation less reliable.
• We find that while • TypeB de Sitter vacua were studied previously in [31]. These vacua share resemblance with thermal confined states of cascading gauge theory with spontaneously broken chiral symmetry, i.e., PhaseB. We find here that (1.31) 9 We introduce novel technique used to identify phases/vacua with spontaneously broken symmetry. The rest of the paper is organized as follows. In section 2 we discuss holographic dual effective action of cascading gauge theory. Section 2 contains a guide to set of Appendices with technical details. Cascading gauge theory de Sitter vacuum entanglement entropy is identified with the Bekenstein entropy of the AH in the holographic dual at late times, see section 3. In section 3.1 we identify AH in ten dimensional holographic dual and compute its area density. In section 3.2 we establish that both the 10 While this is likely to be true in general, the statement is strictly precise for de Sitter evolution of spatially homogeneous and isotropic states of cascading gauge theory. 11 This should be understood in the same sense as existence of TypeA s vacua: the supergravity approximation used to construct TypeB vacua is robust against higher-derivative α ′ corrections from full string theory.
where Ω 0 is a constant in the definition of the 5-form flux 12 , see (2.5), R 10 is given by and R 5 is the five-dimensional Ricci scalar of the metric that forms part of the ten dimensional full metric One-forms {g i } (for i = 1, · · · , 5) are the usual forms defined in the warp-squashed T 1,1 and are given as in [13], for coordinates 0 ≤ ψ ≤ 4π, 0 ≤ θ a ≤ π and 0 ≤ φ a ≤ 2π (a = 1, 2). All the covariant derivatives ∇ λ are with respect to the metric (2.3). Fluxes (and dilaton Φ) are parameterized in such a way that functions h 1 (y), h 2 (y), h 3 (y) appear as Parameter P must be appropriately quantized [4,12]: corresponding to the number M of fractional branes (the difference of ranks of cascading gauge theory gauge group factors) on the conifold. Finally, G 5 is the five dimensional effective gravitational constant where 16πG 10 = (2π) 7 (α ′ ) 4 is 10-dimensional gravitational constant of type IIB supergravity.
Chirally symmetric states of the cascading gauge theory correspond to enhancement of the global symmetry 13 SU(2) × SU(2) × Z 2 → SU(2) × SU(2) × U(1), and are described by the gravitational configurations of (2.1) subject to constraints 14 , 9) or in the boundary QFT language [13], We find it convenient to introduce The ultimate goal is to compute the entanglement entropy of cascading gauge theory -using the dual holographic picture with the effective gravitational action (2.1)in distinct vacua (see (1.22)) in four dimensional de Sitter space-time. As explained in the introduction, this is does in two steps: constructing de Sitter vacua in Fefferman-Graham coordinate frame , a = a(r) , σ = σ(r) , ω a2,b2,c2 = ω a2,b2,c2 (r) .
(2.13) 13 In the planar limit. 14 This is a consistent truncation of the cascading gauge theory to U (1) symmetric sector constructed in [15].
It is important to keep in mind that EF frame vacua (2.13) are the late-time limits of the evolution in EF frame: , Ω 1,2,3 = Ω 1,2,3 (t, r) . (2.14) We now summarize technical details delegated to various Appendices. 15 Developing the precise holographic dictionary between these normalizable coefficients and the corresponding expectation values, while interesting, is not important for the results presented, and thus is outside the scope of the paper.
• appendix B.2 establishes the map between EF and FG frame description for each type of the vacua: TypeA s , TypeA b and TypeB.
• In the limit H → 0, TypeB vacuum in FG frame represents the extremal KS solution [2]. We use this limit in appendix B.3 to related the strong coupling scale Λ of the cascading gauge theory to the complex structure conifold deformation parameter ǫ used in [2], see (B.80).
• appendix C covers numerical procedures for construction of FG frame dual backgrounds (see C.1) and EF frame dual backgrounds (see C.2). We introduce three different computational schemes -SchemeI, SchemeII and SchemeIII (C.6)explain how they are related and outline their computational advantages in accessing different regions of the parameter space of the model. We introduce the AH location function L AH (C.8), used to identify the apparent horizon.
• appendix D presents technical details for construction of TypeA s de Sitter vacua in computational scheme SchemeII in the conformal limit, i.e., b → 0.
• appendix E collects the expression for the Kretschmann scalar (E.1) of the background geometry (2.13). It is used to test the validity of the supergravity approximation.
• appendix F contains equations of motion and the asymptotic expansions for the chiral symmetry breaking perturbations about FG frame TypeA s de Sitter vacua with explicit symmetry breaking parameter -the gaugino mass term. These perturbations are used to identify TypeA b vacua "close" to TypeA s vacua.
3 Apparent horizon in de Sitter evolution of cascading gauge theory Apparent horizon 16 in holographic dual is crucial for identifying the attractor vacuum for the evolution of generic homogeneous and isotropic states of cascading gauge theory 16 In general AH is observer dependent. It is natural to define AH with respect to an observer reflecting the symmetries of the spatial slices -homogeneity and isotropy in x in (2.14), see [25].
Such an identification correctly reproduces the hydrodynamic limit [32] and can be proven to comply with the second law of thermodynamics [19,24], thus serving as a useful definition of the dynamical (nonequilibrium) entropy.
in de Sitter: given competing trajectories for the evolution, dynamics proceeds along trajectory resulting in the maximum entropy at late times. We identify AH directly in ten-dimensional EF frame gravitational dual in section 3.1. We reproduce the same result in EF gravitational dual of the effective five-dimensional description in section 3.2. Both in ten-dimensions and upon Kaluza-Klein reduction to five dimensions the area of the AH stays the same. In section 3.3 we use equations of motion (A.3)-(A. 13) to prove that the area of the AH is nondecreasing upon evolution. We identify the (dynamical) area density of the AH A 10 (t) with the dynamical entropy density s of the boundary gauge theory as where a = e Ht is the boundary spatial metric scale factor, see (1.17). The entanglement entropy s ent is related to the late-time limit of s as where R is the comoving entropy production rate in de Sitter vacuum first introduced in [19]. Finally, in section 3.4 we show that

AH in ten dimensions
The apparent horizon of the bulk gravitational dual to cascading gauge theory dynamics in de Sitter is located at the radius r = r AH where the expansion θ of a congruence of outward pointing null vectors vanishes (i.e., it stops expanding outwards). Working in the coordinates of equation (2.14), we characterize such a congruence with the null vector k = ∂ t + A∂ r . The null vector k points toward the boundary of the space-time outside of the initial black hole, and points inward inside the initial horizon.
Following [33], the expansion of a congruence of affine parameterized null vectors n is given by However, it turns out that k β ∇ β k α = ∂ r A k α , i.e., k is not affine. To remedy this, we rescale k by exp{´∂ r A dλ}, where λ is the parameter along which the congruence k evolves. This ensures that the rescaled null vector satisfies the geodesic equation with λ as an affine parameter. Reference [33] then gives the expansion of k to be Substituting in for ∇ α k α computed in the metric (2.14) We see that θ = 0, when Eq. (3.7) determines the location of the AH, i.e., r AH = r AH (t). The area density of the AH A 10 is (3.9)

AH in Kaluza-Klein reduction to five dimensions
We would like to reproduce (3.7) and (3.9) from the five-dimensional perspective.
The five dimensional area density A 5 of the AH in (3.14) is given by leading to the dynamical entropy density reproducing (3.9).

Area theorem for the AH
Following [19] and using the equations of motion (A.3)-(A.13) we prove now that the dynamical entropy density s defined as in (3.21) grows with time t, i.e., Note that the AH location is determined from (see (3.19)) (3.23) which is used to algebraically solve for dr AH dt r=r AH . The latter expression is then sub- We use equations of motion (A.3)-(A.13) to eliminate all second order derivative in (3.25); we further eliminate ∂ t Σ using (3.23) to arrive at where F 2 is manifestly positive (3.27) Constraint (A.12) can be integrated (once) to obtain which implies that provided the integral in (3.28) is convergent and since the quantity d + (Σ 3 Ω 1 Ω 2 2 Ω 2 3 ) changes sign at r = r AH , see (3.23). Combining (3.26), (3.29) and (3.31) we arrive at (3.22).
For future reference we present the expressions for the location of the AH and the entanglement entropy density in de Sitter vacua. Using (A.14) and (A.15) we find from (3.23) and (3.9) AH location : vacum entanglement entropy : (3.32)

Entanglement entropy of TypeB de Sitter vacua
We demonstrate here that entanglement entropy of TypeB de Sitter vacuum vanishes -this implies that the corresponding comoving entropy production rate vanishes.
de Sitter comoving entropy production rate vanishes in conformal field theories as well [20]. In CFTs the reason is simple: de Sitter vacuum is a conformal transformation of a thermal equilibrium state and entropy production is invariant under conformal transformations [19]. We do not understand the physical reason why the same is true for a de Sitter vacuum in nonconformal gauge theory (TypeB vacuum in cascading gauge theory).

TypeA s de Sitter vacua
TypeA s vacua in FG frame were discussed in details in [31]. As emphasized in [19] and [20] this is not enough to access vacuum entanglement entropy -one needs the holographic construction in EF frame. In section 4.1 we present numerical results for TypeA s vacua for generic values of H 2 Λ 2 , in particular the results for the entanglement entropy, see fig. 6. We discuss TypeA s in the conformal limit Λ ≪ H in section 4.2.
In section 4.3 we estimate H s min (see (1.25)) below which TypeA s vacua construction in type IIB supergravity becomes unreliable. We identify the source of breaking of the supergravity approximation. involves (linearly) f ′ 2 and can be used instead of one of the second order equations (namely, the one involving f ′′ 2 ). Thus, altogether we have a coupled system of 4 second order ODEs (linear in {f ′′ 3 , h ′′ , K ′′ , g ′′ }) and a single first order equation (linear in f ′ 2 ). As a result, a unique solution must be characterized by 9 = 2 × 4 + 1 parameters; these are the UV/IR parameters UV : tational scheme is adopted. We illustrate now that this is indeed the case using the IR parameters in (4.1) as an example 17 . Comparison of the different computational schemes is done using dimensionless and rescaled quantities: ln H 2 Λ 2 (as a vacuum label) (4.2) 17 The same is true for the UV parameters as well. PSfrag replacements PSfrag replacements PSfrag replacements Next, FG frame TypeA s de Sitter vacua have to be reinterpreted in EF frame, see appendix B.2. The diffeomorphism transformation is performed at the radial location FG : Details of numerical construction of EF frame vacua from FG frame vacua are collected in appendix C.2. An important quantity is the parameter s h 0 , see (2.13), PSfrag replacements  As with FG frame UV/IR parameters (4.1), results for s h 0 should not depend on the choice of the computational scheme, provided we compare properly dimensionless and rescaled quantities, i.e., ln H 2 Λ 2 andŝ h 0 (C.15), PSfrag replacements PSfrag replacements where z AH = −r AH is the location of the apparent horizon at asymptotically late times, see (3.32). To determine the location of the apparent horizon, along with integrating the gravitational background functions {a, σ, w c2 , w a2 , K 1 , g} (remember that w b2 = w c2 , K 3 = K 1 and K 2 = 1 when the chiral symmetry is unbroken), we evaluate the AH location function L AH (z), see (C.8). AH is located at the first zero of this function for z > 0. A typical profile of the AH location function is shown in fig. 5. Once the AH is identified, TypeA s vacua entanglement entropy is computed following (3.32): where following (C.1) we introduced dimensionless and rescaled functions and the radial coordinate: {z , a , σ , w c2 , w a2 , K 1 , g} =⇒ {ẑ ,â ,σ ,ω c2 ,ω a2 ,K 1 ,ĝ} ; In the last equality in (4.7) we used expressions for G 5 (2.8) and P (2.7). We compute entanglement entropy in different computational schemes; results must agree, provided we compare dimensionless and rescaled quantities, Explicitly, SchemeI : ln

TypeA s de Sitter vacua in the conformal limit
To study the conformal limit it is convenient to use the computational scheme SchemeII (see (C.6)), i.e., , we use the symmetry transformations SFG2-SFG4 of (B.13)-(B.15) to set H = g s = K 0 = 1 and allow b ≡ P 2 to vary. The FG frame equations of motion  Explicit equations for {f 2n , f 3n , h n , k n , g n } for n = 1, 2 along with the UV/IR asymptotics are presented in appendix D.1. Numerically solving these equations we find perturbative in b predictions for the UV/IR parameters (4.1). As explained in appendix C.2 we also need the FG frame parameter s h 0 , see (B.68). Given (4.11) we find from (C.15) (4.12) Using results of appendix 4.2 we evaluate the integrals in (4.12) to find PSfrag replacements Following appendix B.2 we convert perturbative FG frame construction (4.11) to EF frame: (4.14) Explicit equations for {a n , s n , v n ≡ w c2n + 4w a2n , w a2n , k n , g n } for n = 1, 2 along with the initial conditions are presented in appendix D.  PSfrag replacements We will show now that the location of the AH z AH , as determined from the zero of the AH location function L AH (C.8), is 18 Subleading terms depend on coefficients that have to be determined numerically. green curve: perturbative approximation to order O(b 2 ); see (4.12) with (4.13). (4.20) In fact, from the general structure of the perturbative equations we expect rendering successive higher order perturbative corrections in (4.14) at z = z AH small despite the singular behavior of {a n , s n , w c2n , w a2n , k n , g n } in this limit 19 .  .24) and (4.25).
Given (4.19) and (4.20) we find from (C.8): so that the first zero of the apparent horizon location function occurs at From (3.32) we find perturbative predictions in the conformal limit for the TypeA s de Sitter vacua entanglement entropy: In fig. 10 we compare numerical results for z AH and s ent in computational scheme SchemeII (blue curves) with the perturbative predictions (4.24) and (4.25) at leading (red curves) and next-to-leading (green curves) orders in the conformal limit: b → 0.
Restoring dimensional parameters, from (4.25), PSfrag replacements Besides numerical (technical) difficulties associated with construction of these vacua, there are conceptual ones, associated with the breakdown of the supergravity approximation -the effective action (2.1) becomes less reliable as the background space-time curvature of (2.13) grows. In fig. 11 (left panel) we present the Kretschmann scalar of (2.13) evaluated at the apparent horizon in different computations schemes, see appendix E: PSfrag replacements PSfrag replacements polynomials. The fits suggest that the curvature is divergent at We take (4.29) as an indication that TypeA s vacua do not exist 20 for In fig. 12 (left panel) we identify the rapid curvature growth with the fact that the size of (deformed) T 1,1 , R 2 T 1,1 , evaluated at the apparent horizon becomes vanishingly small in string units, P ∝ Mα ′ = M ℓ 2 s . Note that in the limit R 2 T 1,1 → 0 TypeA s vacua entanglement entropy vanishes, see (4.7). Right panel shows the deformation parameter δ T 1,1 of the T 1,1 : the size of the U(1) fiber compare to the (4.32) 20 It would be interesting to rigorously establish this.     , see table 1:

TypeA b de Sitter vacua
with the remaining metric functions and the string coupling as in TypeA s vacua, i.e., {f c = f 2 , h, g}. It is straightforward to verify that truncation to {δf, δk 1,2 } is consistent (at the linearized level). Equations of motion for the fluctuations and their asymptotic expansions in the UV (ρ → 0) and the IR (y = 1 ρ ) are collected in appendix F. Once the non-normalizable coefficient (the explicit chiral symmetry breaking parameter, i.e., the gaugino mass term) is fixed to δf 1,0 = 1, the expansions are characterized by 6 UV/IR parameters UV : which is the correct number of parameters to find a unique solution of 3 second-order i.e., the non-normalizable parameter δf 1,0 , is kept fixed. This occurs at To use the critical fluctuations as a seed for TypeA b vacua, we need to know the 'susceptibilities'  Given (5.6), fully nonlinear TypeA b vacua, with k s close to k crit s , can be constructed following the same procedure as the one employed in construction of Klebanov-Strassler black hole in [14]. We highlight the main steps: • Let's denote the amplitude of the symmetry breaking condensate (see (5.1)) Then,

PSfrag replacements
It is rather challenging to find the solutions of the corresponding system of ODEs in 15-dimensional parameter space by brute force -fortunately, we already know some solutions which are close to k crit s , see section 5.1. As for the construction of TypeA s we use three different computation schemes, see appendix C.1. There are some differences though: both in SchemeII and SchemeIII we use as a pivot value 22 Numerical results must not depend on which computational scheme is adopted. We illustrate now that this is indeed the case using a sample of IR parameters in (5.12) as an example 23 . Comparison of the different computational schemes is done using dimensionless and rescaled quantities: ln H 2 Λ 2 (as a vacuum label) (C.2) and {f h a,b,c,0 ,K h 1,2,3,0 ,ĝ h 0 } (C.4). Explicitly: SchemeII : ln SchemeIII : ln Following (5.14), we collect results of {f h a,0 −f h b,0 ,K h 1,0 } as functions of ln H 2 Λ 2 in different computational schemes in fig. 17: SchemeI (blue curves), SchemeII (red curves) 22 As will be clear from the presented results this is a convenient value. 23 The same is true for the rest of IR parameters and the UV parameters as well.  PSfrag replacements where z AH = −r AH is the location of the apparent horizon at asymptotically late times, see (3.32). To determine the location of the apparent horizon, along with integrating the gravitational background functions {a, σ, w a,b,c,2 , K 1,2,3 , g}, we evaluate the AH location function L AH (z), see (C.8). AH is located at the first zero of this function for z > 0. 1.5 × 10 -9 2. × 10 -9 2.5 × 10 -9 PSfrag replacements  Once the AH is identified, TypeA b vacua entanglement entropy is computed following (3.32): where following (C.1) we introduced dimensionless and rescaled functions and the radial coordinate: {z , a , σ , w a2,b2,c2 , K 1,2,3 , g} =⇒ {ẑ ,â ,σ ,ω a2,b2,c2 ,K 1,2,3 ,ĝ} ; z = HP g 1/2 sẑ , a = H 2 P g 1/2 sâ , σ = HP 1/2 g 1/4 sσ , w a2,b2,c2 = P g 1/2 sω a2,b2,c2 , K 1,3 = P 2 g sK1,3 , K 2 =K 2 , g = g sĝ .

(5.18)
In the last equality in (5.17) we used expressions for G 5 (2.8) and P (2.7). We compute entanglement entropy in different computational schemes; results must agree, provided we compare dimensionless and rescaled quantities, see (4.9). Explicitly, SchemeI : ln

Validity of supergravity approximation for TypeA b vacua
In this section we briefly comment on the validity of the supergravity approximation in construction of TypeA b vacua. In fig. 23 we present the Kretschmann scalar of (2.13) evaluated at the apparent horizon in different computations schemes for the TypeA b vacua, see appendix E:

TypeB de Sitter vacua
TypeB de Sitter vacua were studied previously in [31]. We showed in section 3.4 that the entanglement entropy of these vacua vanishes. Thus, these vacua can arise as latetime dynamical attractors of cascading gauge theory in de Sitter only when neither We identify the source of breaking of the supergravity approximation.

Numerical results: TypeB
To establish the existence of TypeB vacua it is sufficient to construct them in FG frame  To validate our results, we use two different computation schemes: SchemeI and SchemeIII, see (C.6). Numerical results must not depend on which computational scheme is adopted. We illustrate now that this is indeed the case using a sample of IR parameters in (6.1) as an example 24 . Comparison of the different computational schemes is done using dimensionless and rescaled quantities: ln H 2 Λ 2 (as a vacuum label) (C.2) and {f h a ,ĥ h 0 ,k h 1,3 ,k h 2,2 ,k h 2,4 ,k h 3,1 ,ĝ h 0 } (C.5). Explicitly: SchemeIII : ln  We find remarkable agreements, e.g., The remaining parameters are validated at ∼ 10 −6 level or better.
Following (6.2), we collect results ofk h 1,3 as functions of ln H 2 Λ 2 in different computational schemes in fig. 25: SchemeI (blue curves) and Scheme III (green curves) (left panel); the accuracy of the collapsed results in different schemes is highlighted in 24 The same is true for the rest of IR parameters and the UV parameters as well.
where in the second equality we used (C.5).

Conclusion
In this paper we presented a comprehensive analysis of the vacua structure of cas- We mentioned that TypeA b vacua resemble thermal states of deconfined cascading gauge theory with Z 2 chiral symmetry. The holographic dual of these states is a Klebanov-Strassler black hole [14], which is unstable to local energy density perturbations -the sound waves in cascading gauge theory plasma. It would be interesting to study the fate of spatial inhomogeneities in TypeA b de Sitter vacua.
Ideally, we would like to develop numerical simulations of the cascading gauge theory in de Sitter, akin to the model studied in [24]. As a first step, it would be interesting Innovation. This work was further supported by NSERC through the Discovery Grants program.

B FG frame equations of motion, asymptotics, relation to EF frame and extremal Klebanov-Strassler solution
Fefferman-Graham frame can be used to describe only (the patch of) the gravitational dual to the cascading gauge theory de Sitter vacua. It is useful to setup the asymptotic boundary conditions, analytical continuation to Euclidean (Bunch-Davies) vacua, and study the H → 0 limit in which one recovers the KS solution [2].
Within the metric ansatz where we used the FG frame time τ and the radial coordinate ρ to distinguish them from the EF frame time t and the radial coordinate r in (A.1), we find the following equations of motion (independent of whether we use the flat boundary spatial slicing dM f 4 2 or the closed boundary spatial slicing (dM c 4 ) 2 ) describing de Sitter vacuum of cascading gauge theory [31]: (B.10) Additionally, we have the first order constraint symmetry SFG2: symmetry SFG3:

B.1 Asymptotics
The general UV (as ρ → 0) asymptotic solution of (B.3)-(B.11) describing the phase of cascading gauge theory with spontaneously broken chiral symmetry takes the form f a,n,k ρ n ln k ρ , (B.20) It is characterized by 11 parameters: The IR asymptotic expansion h h n y n , is characterized by 7 parameters: Note that given (B.29), i.e., S 4 indeed smoothly shrinks to zero size as y → 0. It is important to emphasize that TypeA vacua defined by (B.29) have either U(1) or Z 2 chiral symmetry -chiral symmetry is unbroken in the former (TypeA s ), and spontaneously broken in the latter (TypeA b ).

B.1.1 TypeA s vacua asymptotics
We provide here connection with the extensive earlier studies of TypeA s vacua in [31].
Chirally symmetric de Sitter vacua of cascading gauge theory (TypeA s ) correspond to a consistent truncation We find: in the UV, i.e., as ρ → 0,
We do not present the relations between all the UV/IR parameters stemming from (C.1) and (C.2) -they are straightforward to work out, but too long to be illuminating -and instead focus on the few ones for which we are reporting the numerical results: can produce different data sets fixing three of the four parameters {H, P, g s , K 0 }. As we demonstrate, with appropriate rescaling, the distinct data sets must collapse. We find it useful to implement three different computational schemes: SchemeI : H = P = g s = 1 , k s is varied ; SchemeII : Note that: SchemeI is equivalent to performing computations in the hatted variables in (C.1),
Numerical computations are done adopting the algorithms developed in [12].

E.1 Kretschmann scalar at AH of TypeB de Sitter vacua
In section 3.4 we showed that the AH horizon of the bulk gravitational dual to TypeB de Sitter vacua of cascading gauge theory is located at r AH = −z AH = 0, see (3.33).