Radion dynamics in the Multibrane Randall-Sundrum Model

The radion equilibrium in the Randall-Sundrum (RS) model is guaranteed by the back-reaction of a bulk scalar field. In this paper we study an extended scenario, where an intermediate brane exists in addition to the two branes at the fixed points, due to the discontinuity of bulk cosmology constants in two spatial regions. We conducted a complete analysis of the linearized Einstein's equations after applying the Goldberger-Wise mechanism. Our result elucidates that in the presence of nonfixed point branes under the rigid assumption, a unique radion field is conjectured as legitimate in the RS metric perturbation. The cosmological expansion in this setup is briefly discussed.

Extra dimension theories have been constructed to resolve the gauge hierarchy problem without invoking the supersymmetry [1][2][3].One theoretical appealing proposal is a slice of anti-de Sitter (AdS 5 ) space as a solution to Einstein's equation with a negative bulk cosmology constant plus opposite brane tensions [4,5].This RS model can naturally explain the weakness of graviton coupling to SM matters and the TeV scale emerges from a warped geometry factor.To achieve the expected scale at the infrared brane, a dimensionless parameter should be kL ∼ 35, with the curvature k = − Λ 12M 3 , in terms of the 5D Planck scale and bulk cosmology constant.Thus as an attempt to address the hierarchy problem, the brane separation needs to be stabilized at L ∼ 35/k.An observation in light of the late-time cosmology in brane models also calls for stabilizing the flat direction.For the RS1, the Einstein equation G 55 = κ 2 T 55 evaluated at the brane predicts H 2 ∼ ρ (2Λ w + ρ), with the quadratic term Λ 2 w cancelled by the negative Λ in the bulk [6].Without stabilizing the radius, in the IR brane, it seems that the matter and radiation density is forced to be negative (ρ < 0) for Λ w < 0 [7].While this constraint can be eliminated after including the effect of radion stabilization [8,9].As a consequence, the radion acquires a mass and the conventional Friedmann-Robertson-Walker (FRW) equations are recovered.
One elegant way to stablise the radius was the Goldberger-Wise (GW) mechanism [10,11], by introducing a massive bulk scalar minimally coupled to the graviton.With appropriate brane terms, the bulk scalar can develop a y-dependent vacuum expectation value (VEV), so that the effective potential of radion after integration over the fifth dimension will gain an extrema at a fixed brane separation.In this letter, we are going to investigate the radion dynamics after imposing GW mechanism in a RS model with multibranes [12].The authors in that paper worked out 2 spin-0 radion fields in the metric perturbation, with one of them representing the relative moving of the intermediate brane.However, that claimant contradicts the degrees of freedom counting in a 5D extra-dimension theory [13,14].We will illustrate below that assuming the intermediate brane is rigid for a static configuration, only one radion field is the legiti-mate solution to the linearized Einstein's equations with the jump conditions matched at all branes.Following that proof, the radion mass will be derived by including the backreaction of the GW bulk scalar.
We start with the metric ansatz g MN dx M dx N perturbed by the transverse graviton h µν (x, y) and one radion field f (x) on an S 1 /Z 2 orbifold: where the subscripts are µ, ν = 0, 1, 2, 3 and M, N ∈ (µ, 5).Compared with the metric in [15], the fluctuation of 2ǫ(y)∂ µ ∂ ν f (x) is assumed to be permitted.Now we consider the five dimensional action of graviton coupling to a single bulk scalar field: with κ 2 = 1/(2M 3 ) and y i = {0, ±r, L} designating the location of branes in the y-coordinate.Note that the action integration spans over the entire S 1 circle with L > r > 0. The bulk scalar can be expanded around a y-dependent vacuum expectation value (VEV): φ(x, y) = φ 0 (y) + ϕ(x, y).Note that the brane terms are crucial to compensate the discontinuity caused by the orbifold compactification.By varying the action Eq.( 2) with respect to the 5d metric tensor g MN , one can derive the Einstein's equations in terms of Ricci tensor: R MN = κ 2 TMN ≡ κ 2 T MN − 1 3 g MN T a a , with the energy-momentum tensor given by It is advantage to work in this approach because at most the linear order perturbation is involved.We have calculated the Ricci tensor R MN from the metric, and the source term TMN can be written in a compact form: By decomposing R MN and TMN till the linear order of metric perturbations, we listed their exact expressions (Eq.(44-51)) in Appendix .
To the zeroth order, one can obtain the background (BG) equations for the VEV φ 0 and metric A: where the prime denotes the partial derivative with respect to y and the last equation originates from G 55 = R 55 − 1 2 g 55 R = κ 2 T 55 .The analytic solutions for these nonlinear equations can be found using the superpotential method [16,17], with the backreaction effect automatically accounted.Provided the bulk potential V (φ) can be written in the form of: then a solution to the BG equations is given by: To reproduce the usual exponential metric in multibrane RS model [12], the superpotential is derived as: with the brane potentials: where the subscript ± denotes the y = (0, L) branes and the jump for a general quantity W is defined as Note that λ ±r does not depend on the φ field since there is no jump for φ ′ (y) at y = ±r.We also remark that this method can be generalized to the scenario with several bulk scalars given that the superpotential is of the special class W = n i=1 W i (φ i ), where each W i (φ i ) only depends on a single scalar field.Now we are ready to investigate the coupled equations for the excitations using the linearized Einstein Equation δR MN = κ 2 δ TMN .First of all, we need to figure out the conditions for decoupling the transverse graviton from the scalar excitation.At the linear order, the (µ5)-component gives the first orthogonal condition: While the (µν)-component is more complicated, we can extract out the ∂ µ ∂ ν f (x) term from R µν and Tµν and match them (ref Eq.(45-46), Eq.(49) in Appendix): The rationale is that except for gravitons and the above ones, the other terms are all proportional to η µν .Then applying the background equation ( 5), we derived the second orthogonal condition: We would like to mention that Eq.( 12) and Eq.( 14) are equivalent to the transverse and traceless gauging fixing for the graviton.In particular, Eq.( 14) indicates that, since there is no singular term to match here.In analogy to the bulk graviton in the RS model that is illustrated in Appendix [18], under the orbifold symmetry ǫ ′ (y) = −ǫ ′ (−y), this translates into the continuous boundary condition ǫ ′ (0) = ǫ ′ (L) = 0 at the fixed points.While the remaining junction condition [ǫ ′ (y)]| y=±r = 0 will constrain the integration constants of the 5d profile F (y) in the two spatial regions.
To obtain the equation of motion (EOM) for the radion field, one can construct the quantity e 2A Rµν ηµν + R 55 to remove the term of V ′ (φ 0 )ϕ in the Einstein's equations.Then substituting into that with Eq.( 5) and Eq.( 14) for a further simplification, one will arrive at the following ansatz: The discontinuity conditions of F ′ at the branes can be obtained by matching the singular term in the above equation: After identifying the ones of A ′ and φ ′ 0 at the junctions: we can see that the jump equation ( 17) is consistent with the first orthogonal condition (12).Equipped with the EOM and BC, one can find out the independent degrees of freedoms in the multibrane setup.Note that it would be necessary to put all the permitted fluctuations into the metric.
(1) We should first examine the massless modes f (x) = 0 (without the GW field φ = 0) given by Eq.( 16) and Eq.( 12): Taking a second differentiation of Eq.( 20), one can see that only the BC [A ′ ] | i = κ 2 3 λ i is required to make the above two equations agree.Most importantly, one can immediately infer from Eq.( 20) that G(y) is continuous in the y-coordinate.
Note that the second term in the above equation is only half in the coefficient compared with [12].We can calculate the related field using G = F ′ /A ′ .By imposing the continuity conditions F (r − ε) = F (r+ε) and G(r−ε) = G(r+ε), one can determine the junction condition: as implicated by the second orthogonal equation ( 14) for [A ′ ]| y=r = k 2 − k 1 .According to Eq.( 22), one gets a class of solutions and it is the property of [ǫ ′′ (y)]| y=r that determines whether ǫ ′ (r) is nonzero.However since ǫ ′ (y) can be arbitrary away from the branes, one can always tune [ǫ ′′ (y)]| y=r = 0 to achieve ǫ ′ (±r) = 0 (equivalent to c 1 = c 2 ), same as ǫ ′ (0) = ǫ ′ (L) = 0 at the fixed points.For completeness, another EOM can be derived from R 55 = κ 2 T55 .In the massless limit, the radion wave-function obeys: with After simple algebra, the above equation is simplified to be −2A ′ (F ′ − A ′ G) = 0, trivially satisfied due to Eq.( 20).This confirms there is no extra constraint for ǫ ′ (y) in the bulk but its boundary values ǫ ′ (y)| y={0,±r,L} are gauge invariant.Thus we deduce that ǫ ′ (y) is a redundant degree of freedom unless its bulk value is fully irrelevant and a unique radion associated with IR brane is the legitimate perturbation in the AdS 5 metric.For c 1 = c 2 and ǫ ′ (y) = 0, Eq.( 21) reproduces the familiar radion solution G = 2F = c e 2A derived in the paper [19].
(2) We will look into the EOM ( 16) for a massive radion by applying the GW stabilization.From the orthogonal equation ( 14), the gauge fixing G = 2F leads to ǫ ′ (y) = 0 or ǫ ′ (y) ∼ e 4A .Supplemented with the BC ǫ ′ (0) = ǫ ′ (L) = ǫ ′ (±r) = 0, one can see that ǫ ′ (y) must be identically zero in the two regions.Hence the EOM is simplified to be [15]: A few properties related to the Lagrangian expansion in Eq.( 2) are commented in order.After applying the EOM in the bulk, the tadpole term in the 4D effective Lagrangian with φ = 0 is calculated to be (see Appendix): Using Eq.( 20), it is easy to verify that the tadpole term vanishes with the brane potentials in Eq. (10)(11).Note that the brane term at y = −r due to the Z 2 orbifold symmetry is necessary for such cancellation.
The 4D Lagrangian at the leading order are the kinetic terms.For the radion one gets: where the second term in the bracket drops out due to Eq.( 12).Before stabilization (φ 0 = 0), using F ′ = A ′ G, the coefficient of the kinetic term can be rewritten as: For A ′ = Constant, this only depends on the boundary values of F (y).But after stabilization, because of the fact F − A ′ ǫ ′ e 2A ∝ e 2A and A ′ = Constant in each region, the normalization from Eq.(26) becomes dependent on the bulk value of ǫ ′ (y).This indicates that ǫ ′ (y) must 1 Due to the conformally flat property of AdS 5 , the first term of the radion kinetic term in Eq.( 26) can be derived from the Fierz-Pauli Lagrangian in a straightforward manner, by replacing The detail to derive the second term can be found in the paper [20].
be zero after the corresponding symmetry is broken.It is not suitable to use ǫ ′ (r) to create another degree of freedom.
For clarity, we remarks on the second solution in the paper [12], obtained by imposing the nonmixing condition for the kinetic terms of 2 radions in Eq.(26 For φ 0 = 0 and F 1 = c e 2A , the second term equals ′ that precisely vanishes.Evaluated with F 2 in the general expression of Eq.( 21), this directly fixes the ratio: indicating another zero mode with a spike at y = r generated by a nonzero ǫ ′ 2 (r) (for ). However after stabilization, the second term in Eq.( 28) cannot be organized into a total differentiation anymore, as a result the arbitrary bulk value of ǫ ′ 2 (y) will enter.This is also true if F 1 holds a ǫ ′ 1 part.Thus this unusual solution has to be abandoned.
Based on the analysis of degree of freedom, it is valid to take the intermediate brane as rigid in a static solution, with its location fixed by the boundary of unequal cosmology constants.In this setup, the r parameter is traded with the scalar VEV at the y = r brane.We now pursue to stabilize the radion associated with the separation between the UV and IR branes.The effective potential of radion is given by integrating over the fifth dimension: where the VEV of the GW bulk scalar is φ 0 = φ P e −uy with φ P denoting the UV brane value.Then the first derivative of the potential is: and the extrema gives The mass of radion in RS1 was calculated in [15,21].Following the procedure, we expand the background metric A(y) and the radion wave function Q(y) in a measure of the backreaction: with l = κφ P / √ 2 and the mass parametrized as m 2 = m2 l 2 .The ϕ ′ in the EOM (24) can be eliminated using Eq.(12).At the zeroth order the EOM yields the massless case without back-reaction.Expanding to the l 2 order, one derives: In the limit of a stiff brane potential, namely γ ± → ∞, the BC reduces to be ( At the y = r brane we impose the continuous BC from the jump matching 2 : After a lengthy calculation, the mass of radion is determined by BC at the y = L brane: with the constant fixed by the other two BCs, Thus the last term in the radion mass ( 35) is negligible due to the large suppression from a warped factor.Finally we briefly discuss the cosmological expansion rate by taking the metric to be of a time evolution form: The perturbations (δa, δn, δb) are caused by adding the matter densities.We will inspect the G 55 equation using the following ansatz: 2 In general, the mass parameter in the superpotential may be set to be unequal (u 1 = u 2 ) in two spatial regions.Firstly, the brane tension gets a shift 1 2 (u 1 − u 2 )φ 2 0 and the BC of radion profile at y = r alters to be: = 0 for such case.In addition, to be compatible with the scalar φ BC, i.e. [ϕ ′ − 2F φ ′ 0 ]|y=r = 0, one needs to impose the constraint (u 2 −u 1 ) φ 0 ϕ − 3 Taking the jump of G 55 = κ 2 T 55 at the y = r brane where the reflection symmetry is not operative, we obtain at the leading order: where the first quantity on the right side vanishes due to Eq.( 6) and δa ′ (r) represents the average of The jump equations for δa ′ and δn ′ are given in [6,8]: with ρ and p being the matter density and pressure at the y = r brane.By replacing Eq.(40) in Eq.( 39), one arrives that, Note that there is no singularity at k 1 = k 2 in the above equation since the density ρ along with the intermediate brane will disappear in that limit.Then we will average the G 55 = κ 2 T 55 with respect to the y = r brane by keeping only the linear term of ρ: Substituting the 3δa ′ +δn ′ term in Eq(42) with Eq.(41), we derive the FRW equation near the intermediate brane to be: For consistency with late time cosmology, one expects where (ρ i , p i ) are the physically measured quantities on each brane.Although, without knowing δb, it is not possible to make a robust prediction, nonetheless, Eq.( 43) is a constraint that the multibrane system needs to satisfy.
In summary, in this letter we derived two orthogonal conditions to decouple the transverse graviton from the modulus field in a multibrane RS model.By solving the linearized Einstein's equation we find out that the perturbation ǫ(y)∂ µ ∂ ν f (x) merely plays the role of gauge fixing for the radion field and can not be used to create a new excitation in the presence of radius stabilization.
The intermediate brane originates from the necessity to match the jump condition of the background metric.To get a GW-like minima from the effective potential, it is reasonable to assume that the nonfixed point brane is (quasi) rigid, with a time evolution due to matter densities.Under such rigid assumption, the radius of IR brane can actually be promoted to be a dynamic field.Instead the location of intermediate brane is purely a parameter that signals the discontinuity of bulk cosmology constants.Hence one can anticipate that the stabilization of the IR radius in the multibrane model is similar to the RS1, with the minima affected by the unequal curvatures.After applying the GW mechanism, we show that with a small back-reaction, the radion mass is well below the cut off scale of IR brane, that is consistent with the NDA argument from the AdS/CFT correspondence [22][23][24].A favorable property is that the radion and its Kaluza-Klein towers after the proposed stabilization are orthogonal in the limit of stiff brane potentials.The Einstein's equation R MN = κ 2 TMN ≡ κ 2 T MN − 1 3 g MN T a a determines the dynamics of metric fields.We derived the components for the Ricci tensor and the energy-momentum tensor in a RS-like model.