Exploring extra dimensions through inflationary tensor modes

Journal of High Energy Physics, Mar 2018

Sang Hui Im, Hans Peter Nilles, Andreas Trautner

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Exploring extra dimensions through inflationary tensor modes

HJE Exploring extra dimensions through in ationary tensor Sang Hui Im 0 1 Hans Peter Nilles 0 1 Andreas Trautner 0 1 Large Extra Dimensions, Spacetime Singularities 0 Nussallee 12 , 53115 Bonn , Germany 1 Bethe Center for Theoretical Physics and Physikalisches Institut der Universitat Bonn Predictions of in ationary schemes can be in uenced by the presence of extra dimensions. This could be of particular relevance for the spectrum of gravitational waves in models where the extra dimensions provide a brane-world solution to the hierarchy problem. Apart from models of large as well as exponentially warped extra dimensions, we analyze the size of tensor modes in the Linear Dilaton scheme recently revived in the discussion of the \clockwork mechanism". The results are model dependent, signi cantly enhanced tensor modes on one side and a suppression on the other. In some cases we are led to a scheme of \remote in ation", where the expansion is driven by energies at a hidden brane. In all cases where tensor modes are enhanced, the requirement of perturbativity of gravity leads to a stringent upper limit on the allowed Hubble rate during in ation. Brane Dynamics in Gauge Theories; Cosmology of Theories beyond the SM 1 Introduction 2 General considerations 3 Large extra dimensions and Randall-Sundrum scenario 2.1 2.2 2.3 3.1 3.2 4.1 4.2 4.3 4.4 4.5 Metric and expansion law E ective Planck mass during in ation Tensor modes in braneworld in ation General form of the metric warping Solution for the metric during in ation 3.2.1 3.2.2 Large extra dimensions Randall-Sundrum scenarios 4 Linear dilaton model Generalities The static case The expanding case with external stabilization The expanding case with dilaton stabilization Comparison to results in the literature 5 Conclusions A Agreement with earlier results on the RS1 case B Dictionary of di erent coordinate conventions the cosmic microwave background test the situation at highest cosmological energies and and assumes that radii of extra dimensions are xed by a separate mechanism that does not in uence the speci c prediction of the in ationary scheme. Even with this simpli ed assumption there is an impact of the presence of the extra dimensions: gravity could have a di erent strength in the bulk and in uence the size of tensor modes in the in ationary model under consideration [ 1 ] (cf. also [ 2, 3 ], and [4, ch. 5.1] for a review). This is particularly interesting in models that try to solve the weak scale hierarchy by large or warped extra dimensions. In this case, matter and in aton eld live on our visible infrared (IR) brane while gravity is stronger in the bulk and at a hidden ultraviolet (UV) brane.1 Work along these directions has been done in the framework of large extra dimensions (LED) [5, 6] and warped extra dimensions a la Randall-Sundrum (RS) [7]. The present work has its origin in the study of in ationary models within the so-called Linear Dilaton model (LD) [8{11] which regained popularity from a discussion of aligned axions [12, 13] and the clockwork scheme [14{18]. It can accommodate a solution of the weak-scale hierarchy problem in a braneworld scenario (with IR- and UV-brane) with power law warping (in contrast to exponential warping in the RS case). When studying the LD model within the framework of the simplifying IRB assumption we were led to some inconsistencies to be explained later. To achieve the standard in ationary picture on the visible brane some contributions from the invisible brane (or bulk) are needed. This observation leads us to reconsider a more general picture of extra-dimensional in ation beyond the simplest assumption (both in the LD and the RS model) and this is a main subject of this paper. Our discussion is organized as follows. In section 2 we will summarize the formulae relevant for the discussion. As a warm-up, we then repeat the discussion for the LED case with one extra dimension using the simplifying IRB assumption. In this case we nd an enhancement of tensor modes as the e ective Planck mass is reduced through extra dimensional e ects. There is an upper bound on the Hubble scale during in ation as well as on the reheating temperature after in ation. Still in some regions of parameter space one could nd a sizable tensor-to-scalar ratio due to the transdimensional enhancement. We then turn to the RS model and consider in ation under the simplifying IRB assumption. Again the Planck mass is reduced during in ation, implying that the strength of gravity is enhanced. The tensor-to-scalar ratio is enhanced and we obtain an upper bound on the Hubble parameter. This reproduces known results in the literature based on the simpli ed assumptions [ 1 ]. Next we turn our attention to a wider class of in ationary solutions. We assume a two-brane RS model with our matter on the IR brane. The exponential warping of the extra dimension could explain the weakness of gravity on the visible brane and thus solve the hierarchy problem. Even in the static case we see that the properties of the system strongly depend on the physics at both branes: IR and UV. The radius of the extra 1Concerning the terminology in this work, we will always refer to our brane - at which the Standard Model lives - as the visible brane and place it at the origin (z = 0) of any extra-dimensional coordinate. Irrespectively, we refer to the IR brane as one at which the weak scale hierarchy problem is solved, and to dimension can be tuned through a choice of the brane tensions on the visible and hidden brane. The implications of energy on the two branes are highly interdependent. A model where in ation is driven originally by in ationary dynamics at the visible brane could be made static by tuning the energy density of the hidden brane. Alternatively the physics at the hidden brane could be the only source of in ationary behavior, a phenomenon one might call \remote" in ation. In this case the Planck mass is enhanced during in ation implying that the strength of gravity is reduced compared to the static case. We also treat a speci c case in this general class of solutions discussed earlier by Nihei-Kaloper-Kim-Kim (NKKK) [19{21] as well as the single brane warped model RS2 [ 22 ]. In section 4 we consider the Linear Dilaton model. As in RS we have a negative cosmological constant in the bulk and two branes (with matter elds and in aton eld on the visible IR brane). A hierarchy of scales appears because of a power-law warping (in contrast to exponential warping in the RS-case). This situation is more complicated as there is an additional degree of freedom in the bulk (the dilaton). Despite of this we can derive the solution for the static case (H = 0) in full generality. Unfortunately this is no longer true in the expanding case H 6= 0. There we perform a perturbative expansion in H2. We again adopt the simplifying IRB assumption that we can ignore speci c properties of the stabilization process (imposed by an external mechanism). The study of this naive case leads to the amazing result that we are not able to recover the standard in ationary expansion on the visible brane (in contrast to the RS-case). The IRB assumption leads to a contradiction. In ationary behavior on the visible brane can only be obtained if there is some contribution from the hidden brane. This is similar to the discussion of \remote" in ation in the RS-case. But here we have no choice: some \remote" contribution is required (in contrast to the RS-case where such a contribution was optional). The origin of this particular behavior is the presence of the dilaton as an additional bulk degree of freedom. On the other hand, the presence of this additional bulk eld opens the possibility to stabilize the radius with the dilaton eld without the use of new external degrees of freedom. This situation is examined in section 4.4. We again have to do a perturbative expansion in H2 (completed with a full numerical solution). Surprisingly this situation allows the conventional in ationary scenario where in ation is driven from the visible brane (with no \remote" contribution needed). The tensor modes can be computed and are found to be suppressed compared to the four-dimensional case (in contrast to the RS case). We also compare these results with the analysis of ref. [23] done in a conceptually di erent setup. Section 5 is devoted to conclusions and outlook. 2 2.1 General considerations Metric and expansion law We are interested in ve-dimensional \braneworld" scenarios where gravity is propagating in ve dimensions, while the Standard Model is con ned to a four-dimensional slice of spacetime. The weak scale hierarchy problem can then be solved by an apparent, large { 3 { four-dimensional Planck mass which is caused by the tiny overlap of the (massless) graviton zero mode with the visible brane. In a very general manner, the action is given by S = Z for the metric can be written as (i) b_(z; t) = 0 (the size of the extra dimension is static), and (ii) A(z; t) = f (z) a(t) (A is a separable function). The rst assumption (i) is certainly ful lled if there is a mechanism to stabilize the size of the extra dimension, for example via a stabilizing potential for the radion mode [24]. Alternatively, if cosmological constants are the only form of energy density on the branes, one can achieve a consistent solution for a static extra dimension also by ne-tuning of the brane energy densities (cf. e.g. [19{21]). In either case, a stabilization will involve a contribution to the (55) component of the energy-momentum tensor TMN .2 The (55) component of the Einstein equations, thus, serves to determine the size of the 5th dimension independently of the details of the stabilization mechanism [25{27]. This means that, as long as the stabilization mechanism decouples from all other equations, we can simply put aside the (55) equation while assuming that the radius is stabilized at some value (see e.g. [ 1 ]). We will later see that this is not always the case, and we will then also take into account the (55) equation. The remaining choice b(z) = 1 then corresponds to a choice of coordinate system. One can show that under the assumption (i), point (ii) is ful lled if and only if (n=A) is independent of z. Considering matching conditions on the four-dimensional branes, this requires that L0; is time independent [28], i.e. the energy densities are dominated by cosmological constants. This is a good assumption here, because we are interested in in ationary solutions of the scale factor a(t). The Einstein's equations are of the form GMN = RMN 2 R gMN = 2 TMN ; (2.4) with 2 M 3 . With the ansatz (2.3) the following features arise: 2Our conventions are: M; N; : : : = 0; 1; 2; 3; 5; ; ; : : : = 0; 1; 2; 3; i; j; : : : = 1; 2; 3; metric signature (+1; 1; 1; 1; 1); dots and primes denote the derivatives with respect to t or z, respectively. { 4 { is independent of z [ 26 ]. Due to the assumption (ii) of our ansatz above, however, this relation is automatically ful lled here. The only relevant Einstein equation, hence, is the (00) component of (2.4) which is given by ansatz (2.2), the (05) equation gives rise to the insight that is the approximately constant energy density that drives in ation. In the picture of the IRB assumption, this energy density is due to a (su ciently at) in aton potential on the visible brane. The crucial point is that MPl;e and MPl;exp will, in general, not coincide during in ation, thus, giving rise to a change of the tensor mode power spectrum as compared to in ation in the purely four dimensional case [ 1 ]. { 5 { and we realize that H corresponds to the physical expansion rate of a three-dimensional slice of space at the ve dimensional point z0 where f (z0)=1. The proper physical Hubble rate at a di erent slice of four-dimensional space time, say at z = z1, can be obtained from H by a rede nition of the time coordinate f 2(z1)dt2 ! d 2, and, therefore, is given by 2.2 E ective Planck mass during in ation The relevant quantity for the actual strength of 4D gravity is the prefactor of the fourdimensional Ricci scalar (the normalization factor of the zero-mode graviton) which arises upon integrating out the 5th dimension in (2.1). The e ective 4D Planck mass MPl;e obtained in this way is given by MP2l;e = M 3 Z 5D dz f (z)2 : H2 = As gravitational tensor modes are intrinsically bulk degrees of freedom, they are susceptible to the ve dimensional geometry during in ation. For the treatment of tensor mode perturbations (primordial gravitational waves) we follow [ 1 ] (see also [ 2, 3 ]). Even though the discussion about the tensor modes in [1] is based on the IRB assumption that the in aton eld is con ned to the visible brane, we remark that the result is applicable to general braneworld in ation including \remote in ation" where in ation is driven by an in aton eld located at the hidden brane. This is because the form of the zero mode graviton solution is independent of the speci c dynamics responsible for in ation. How the spectrum of gravitational waves is modi ed then depends only on the underlying geometry and not on the microscopic details of the four-dimensional model of in ation. The power spectrum of primordial tensor modes PT(`) = 2 2 H(`) MPl;e 2 ; generally deviates from its four-dimensional value. This can be attributed to a change of the e ective reduced Planck mass MPl;e during in ation [ 1 ]. Kaluza-Klein modes other than the massless tensor mode are not relevant because they are separated by a su ciently broad mass gap [ 3 ]. In sharp contrast, scalar mode (density) perturbations originating from quantum uctuations of the in aton eld do depend on the speci c scenario of braneworld in ation. If one adopts the IRB assumption, the scalar perturbation is a purely four-dimensional degree of freedom con ned to the visible brane. Then, given the ordinary four-dimensional Hubble law (2.11) on the brane, the presence of extra dimensions does (to leading order in slow-roll) not a ect the power spectrum of scalar metric perturbations which is given by [ 1 ] PS(`) = still predict the altered behavior of primordial tensor modes based on the knowledge of the e ective Planck mass during in ation. 3 3.1 Large extra dimensions and Randall-Sundrum scenario General form of the metric warping The actions of the LED model [ 5, 6 ] and of the Randall-Sundrum model [7] are given by simpli cations of (2.1). Let us consider the case of a bulk cosmological constant in addition { 6 { to the unspeci ed radion stabilization mechanism in the bulk Lb(x; z) = Then, the (00) Einstein equation (2.6) which determines the metric warping f is given by the (00) equation in the bulk reads Depending on the sign of , this equation has the general solutions (cf. also [29]) f 2(z) = < >> H2 z2 + c1 z + c2 A e > 0 without loss of generality. Each solution has two constants which are to be determined from the boundary conditions. Depending on the setting, the boundary conditions are set by symmetry constraints or the placement of branes. For example, if there is an in nitely thin brane with constant energy density (\brane tension") at a position z = z0, the four-dimensional \brane" Lagrangian takes the form This gives rise to a discontinuity of the rst derivative of f across the brane [30, 31] which Lz0 (x) (z0) = z0 (z0) : between solutions for f in the di erent regions can be obtained, for example, by requiring that the functions are related by the orbifold transformation z ! z, or similar relations. is given by 3.2 3.2.1 (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) c2 = H2 6 R Solution for the metric during in ation Large extra dimensions The LED case is characterized by = 0 and the introduction of a single brane. The extra dimension is taken to be compact, with a size denoted by 2 R. We follow [ 1 ] and place the visible brane with tension 0 = at z = z0 = 0. The boundary condition f (0) = f (2 R) xes c1, while the jump condition for the derivatives (3.6) evaluated at z 0+ = 0 and z0 = 2 R xes the relation Recall that H corresponds to the usual four-dimensional Hubble rate at the slice where f (z) = 1. Since we are interested in the visible brane at z = 0 we chose the four-dimensional coordinates such as to normalize f (0) = 1 corresponding to c2 = 1. Altogether, the solution then is given by f 2(z) = H2 z 2 2 R z + 1 ; with H2 = 2 6 R Finally we can obtain the e ective Planck mass in 4D from the metric warping and compare it to the e ective Planck mass appearing in the expansion law. From the metric four dimensions as compared to the static case. In contrast, from the four-dimensional Hubble expansion law one nds (3.8) (3.9) H2 = 2 6 R relevant scale for the tensor mode perturbations. Consequently the tensor-to-scalar ratio is modi ed due to transdimensional e ects [ 1 ]. The fact that the e ective Planck mass in (3.9) is reduced shows that gravity is stronger during in ation, i.e. the tensor-to-scalar ratio is enhanced. In fact, if the Hubble scale is too large the metric warp function (3.8) crosses zero at which point there appear curvature singularities in the bulk.3 This signals the breakdown of perturbative gravity. Avoiding this situation results in an upper bound 2 R2 H2 < 1 : (3.11) This result generalizes to n compact extra dimensions in the form cn R H < 1, with a factor cn scales to be M one nds R O(1) that depends on the details of the compacti cation [ 1 ]. Taking all 5D TeV and requiring a successful solution to the hierarchy problem 1030=n 19 m resulting in an approximate upper bound on H < 10 30=n TeV corresponding to a maximal reheating temperature of TRH < 1021=2 15=n GeV. For the speci c case of n = 2 this corresponds to H < 10 12 GeV and TRH < 103 GeV. Depending on the assumed microscopic model of in ation there will eventually be even stronger bounds imposed on the product HR by the (non-)observation of CMB B-mode polarization. 3We thank the referee for drawing our attention to this point. { 8 { The Randall-Sundrum scenario [7] is characterized by taking < 0 while introducing two branes. Without loss of generality, the branes are placed at z = 0 and z = R. The respective brane tensions are denoted by 0 and , respectively. We will solve for f (z) in the region 0 z R, while the solution in the region R z 0 can be obtained by the orbifold transformation z ! The general solution for f (z)2 in the bulk is given by (cf. (3.4)) f 2(z) = A e four-dimensional Planck mass can be computed in a general fashion, MP2l;e = 2 M 3 Z 0 R where we have introduced the warp factor of the static case ! := e R Let us compute A, B, and H for general (time-independent) boundary conditions. The junction conditions (3.6) take the form f 2 = A ! + B ! 1 f 2 = A ! B ! 1 ; where we use the abbreviations f0 := f (0) ; f := f ( R) : The rst line of (3.17) are simple identities, while the second line arises from the junction conditions (3.15). Taking , 0 , , R together with a normalization condition for f (e.g. f0 = 1) as input, this can be viewed as a system of four equations with four unknowns (A, B, H, f ). In particular, the expansion rate H is a function of the input parameters { 9 { and It is useful to de ne the dimensionless quantities 2 3 ; and := = p compare the brane tensions to the bulk cosmological constant. In particular, we will shortly see that for 0 = = 1 the originally considered static case [7] is obtained. Altogether the constraints from the boundary conditions can be written as 0 := 3 2 0 = p 0 and completely determined by the requirement that the jumps in the derivative of f (z) at z = 0 and z = R are consistent with the imposed brane tensions. Let us take the visible brane to be located at z = 0 and, therefore, adopt the normalization f0 = 1. A general expression for the Hubble rate (at z = 0) then is given by while the parameters A and B can be expressed as and 0 := Let us reproduce some known results, while pointing out novel insights. Static RS1. For an arbitrary but xed radius R (i.e. a xed warp factor !) we see that the expansion rate (3.19) vanishes if 0 = = 1. This corresponds to the cases (A = 1; B = H = 0) as well as (A = H = 0; B = 1) implying that the metric takes the standard form ds2 = e z dt2 ij dxidxj dz2 : Clearly, this is the originally considered RS1 case [7], which is static. For the case 0 = = 1 we reproduce the well-known result MP2l;e RHS=10 = nding that MPl;e appears exponentially enhanced over the fundamental scale M . Note that tuning 0 and to equal but opposite values 1 is not the only possibility in order to obtain a static (H = 0) case. Alternatively, there can be a non-trivial interplay of the size of the fth dimension and the expansion rate of the four-dimensional slices. For a large number of combinations of 0 and , see gure 1, it is possible to tune R to the very speci c value RRS = 2 1 ln which gives rise to a vanishing expansion rate of all four-dimensional slices. Most notably, for any given non-trivial value of 0 6= the tension of the other brane such as to stop in ation. 1 and any value of the radius one can always tune This clearly demonstrates that for in ationary solutions the evolution of the IR and UV branes are highly interdependent. In the most extreme case, for example, one could drive in ation of the visible brane by dynamics located solely on the hidden brane (\remote" in ation). Expanding RS1. Using the general results (3.19) and (3.20), we can also reproduce the in ationary case considered by Giudice et al. [ 1 ]. Therefore, we take 0 = = 1. This corresponds to the usual ne tuned brane tensions of the static case plus an extra energy density 0 on the visible brane, characterized by the dimensionless parameter (3.21) (3.22) (3.23) (3.24) H2 2 ! 1 e The e ective Planck mass during in ation then is The Hubble rate comes out as and the metric warping is given by the value of R = RRS (darker means bigger) such that H = 0. At the solid boundary lines R asymptotes the values indicated in the plot. Values RRS 70, for which the desired hierarchy between MPl;e and M is obtained, lie very close to the vertical and horizontal boundary lines. Empty circles show the choice of brane tensions in the original Randall-Sundrum scenario, where H = 0 while R at these points can take any value. This result is exact. As shown in appendix A, our results for the expansion rate, metric warping and Planck mass agree with the results of [ 1 ] after taking into account the di erent conventions. Clearly, the Planck mass is reduced during in ation, implying that the strength of gravity is enhanced. If the Hubble rate becomes too large, f 2(z) crosses zero and there appears a curvature singularity in the bulk. Avoiding the onset of strongly coupled gravity thus imposes an upper bound on the Hubble rate -4 -2 0 2 4 H2 < 2 (1 ! !)2 2 ! ; (3.25) (3.26) (3.27) (3.28) The Hubble rate of the visible brane comes out as and we have to restrict < 2! 1 to ensure that H2 > 0. In the case ! one nds. which also ensures that MP2l;e > 0. Taking all 5D scales to be M TeV and requiring a solution to the hierarchy problem due to a warping R 70 this bound restricts the Hubble rate to H < 10 12 GeV. Assuming maximally e cient reheating, this corresponds to a bound TRH < 103 GeV. The fact that gravity is stronger during in ation generically leads to an enhancement of tensor mode perturbations. As an interesting alternative, note that one could, in principle, also drive in ation from a completely remote sector that gives rise to an approximately constant additional energy density on the hidden brane. To model this we take 0 = 1, = 1 + , where (3.29) This shows that the physical Hubble rate of the visible brane is highly susceptible to even smallest energy densities on the hidden brane. For example, the currently observed Hubble rate of 10 32 eV on the visible brane can be caused by an additional energy density of only 10 70 eV4 on the hidden brane. The necessary ne-tuning of energy density on the hidden brane demonstrates that the cosmological constant problem of our visible brane is not a local but in fact a global ne-tuning problem. Nevertheless, the necessary degree of netuning on the hidden brane is the same as the usual 4D cosmological constant problem, as the natural mass scale on the hidden brane (for canonically normalized elds) is given by The e ective Planck mass during such a \remote" in ation caused by hidden-brane For this scenario the Planck mass is enhanced during in ation, implying that the strength of gravity is reduced. Gravity is weakly coupled throughout, meaning that there are no constraints on the possible values of H. Despite the fact that the expansion law looks standard in terms of the canonical hiddenbrane energy density, we stress that it may not be possible here to directly interpret the e ect of the altered e ective Planck mass on the tensor-to-scalar ratio. In particular, in ation is driven by an energy density located on the hidden brane which sharply contradicts the IRB assumption that the in aton dynamics should be con ned to the visible brane. The results of [ 1 ] do not simply generalize to cases that violate this assumption. A dedicated study would be required to see how density and tensor mode perturbations on the visible brane can be a ected or even seeded in other cases. Given that there is no direct coupling of the in aton sector to the visible sector, reheating could occur via gravitational particle production [32] (cf. also [33{35]). The low e ciency of this reheating mechanism requires the in ation scale to be rather high, de nitely well above the BBN scale. This is no problem here because H is not bounded as discussed above. Exploring the observational consequences of such a \remote" in ation scenario is beyond the scope of this work. Nihei-Kaloper-Kim-Kim special case. So far, we have not speci ed the mechanism which stabilizes the size of the fth dimension. One possibility to obtain a xed size of the extra dimension R without any bulk dynamics is by ne-tuning the brane tensions against each other [19{21]. This is a very speci c variant of the expanding RS1 case, in the sense that the four-dimensional slices may expand but the UV and IR brane tensions are xed relative to each other in order to warrant that the fth dimension is static. With an empty bulk, the general solutions (3.4) receive an additional constraint from the (55) Einstein equation, which is then given by This restricts the coe cients of the general solution to the form Using the parametrization ce0 e2 c0 the metric warping results as size R of the fth dimension and the brane tensions4 It immediately follows from eqs. (3.37) that j 0; j > 1 and there is a relation between the R = R1 1 ln 1)( + 1) 1) : The metric warping together with the boundary condition as a function of the chosen radius for a xed value of 0 is displayed in gure 2. Furthermore, the Hubble rate of the 4As a curiosity, note that this is precisely 2RRS (3.23). This is in full agreement with [21]. Due to the additional constraint, there is one less parameter than in the general solution. Furthermore, normalizing f (0) = 1 xes and there is no free parameter left. form The novel constraint is also manifest in the boundary conditions (3.15) which take the such that f (0) = 1. The dashed line shows the normalized brane tension (R1) which the hidden brane must carry if it would be located at a distance R1 away from the visible brane. There is a special radius z = R0 for which the metric is zero and f 0=f does not exist. If a brane is placed at this special radius it decouples and can have any brane tension. visible brane (as always in our convention at z = 0) comes out as while the Hubble rate at the hidden brane (located at z = R1) is given by 2 4 H2 = 2 0 1 ; H2 H2 f 2 = 2 4 2 1 : (3.41) HJEP03(218)4 Our solutions fully agree with eqs. (16) and (20) of [21] after noting that their k =2, L5 R, and k1;2=k 0; . However, the discussion in [21] was limited to the parameter region 1 < 0 < . This limitation makes sense if the hierarchy problem is to be solved at the z = R1 brane and if one requires that the metric should not have a zero at any point in the extra dimension. On the other hand, if we do not impose these requirements we nd that there are additional regions in parameter space for which a consistent solution can be found, cf. gure 3 (see also [19, 20]). In particular, we nd that there is another solution for the size of the extra dimension R = R0 1 ln ( 0 + 1) 1) ; for which the general solution for H2 (3.19) is reconciled with (3.39). This solution makes sense when 0 > 1 (i.e. c0 > 0), irrespective of the value of . For this solution, the ne-tuned value of the radius is independent of the energy density on the hidden brane! In order to understand why this is the case, it is important to note that z = R0 corresponds to the zero of the metric (3.35) (cf. gure 2). Despite the fact that f 0( R0)=f ( R0) does not exist, this second solution for the radius is consistent with the boundary conditions for HJEP03(218)4 ; for the Nihei-KaloperKim-Kim special case. The contours illustrate the value of R1 where it is allowed and otherwise the value of R0 (darker means bigger) such that a consistent solution is achieved. The Hubble rate of the visible brane is everywhere given by (3.39). Contour lines with constant R can also be viewed as parametric curves on which the general expression for H2, Equation (3.19), is reconciled with (3.39). any value of . This may appear surprising in view of the condition (3.37). However we note that in deriving (3.6), which is the general origin of (3.37), it was tacitly assumed that the metric has no zero f (z0) 6= 0. Without imposing this requirement the actual boundary conditions for a brane at z = z0 read 3 f (z0) f 0(z0+) f 0(z0 ) = f (z0)2 2 z0 : (3.42) It follows that at metric zeros f (z0) = 0 the boundary conditions are trivially ful lled and the parameter z0 will not enter the solution. The expansion rate of the hidden brane, Hz0 , becomes meaningless in this case. We conclude that putting the hidden brane precisely at the zero of the metric is a valid solution for which the hidden brane decouples and does not play any role for physics on the visible brane. Let us discuss how the hierarchy problem can be addressed in the NKKK setup, while pointing out the e ect on the in ationary tensor modes. The (orbifolded) extra dimension extends between the visible brane at z = 0 and a hidden brane at z = R1 with the general solution for the metric given by (3.35). Integrating out the extra dimension, we nd the strength of e ective four-dimensional gravity to be The two signs correspond to the two cases c0 7 0. There are no restrictions on the possible values of H and R from this since MP2l;e is strictly positive. In all cases where the hierarchy f (z)2 = e z with an e ective Planck mass problem is solved, the Planck mass is enhanced during in ation implying that gravity is weakened and the tensor mode perturbations are reduced. For c0 7 0 and taking the limit H ! 0 (corresponding to j 0j ! 1) the metric approaches the standard RS warped form The hierarchy problem at the visible brane (without loss of generality placed at z = 0) can be reliably addressed only in the case c0 < 0 where the metric warping increases away from the brane. In the case c0 > 0 and R1 > R0 the zero of f (z)2 is present in the bulk. In contrast to the cases discussed in the previous sections, the zero of f (z)2 does not correspond to a true (e.g. curvature) singularity but is instead related to the presence of a causal horizon in the extra dimension [ 2, 20 ]. Following, for example, [ 2, 36 ] one can restrict the size of the extra dimension to a causally connected region, i.e. let the extra dimension end on the horizon at R0 = 2c0= . In this case the integral (2.10) results in MP2l;e = 2 2 M 3 2 H2 hsinh 2 arcsinh This case cannot address the hierarchy problem. The e ective Planck mass only implicitly depends on the size of the extra dimension via H which is in a one-to-one relationship with R. The extra dimension can become in nite in size only if H ! 0, in which case we approach the RS2 limit. A naive second possibility to address the hierarchy problem would be c0 > 0 with R1 > R0, corresponding to the upper right region of gure 3 where 0 ; > 1 and the metric is cusped upwards on both branes (cf. gure 2). However, in this case the causal horizon is present in the bulk and the extra dimension consists of two causally disconnected regions without interaction. The graviton zero mode then would have to be normalized in causally connected regions only, corresponding to the previously discussed case that the extra dimension ends on the horizon (3.45). A di erent possibility could arise if the NKKK case is only the late time limit to a situation in which there was initially no horizon but it formed dynamically. Just before the time of horizon formation the graviton zero mode would have to be normalized to the full extra dimension and it remains to be investigated in a fully dynamical setting how this normalization would change after the horizon is formed. Exploring this possibility would require a fully dynamical treatment of the extra dimension and the formation of the horizon, which is beyond the scope of this work and so we will not further discuss this here. Expanding RS2. Let us also discuss the RS2 model [ 22 ]. Here, the brane with positive tension is the visible brane centered at z = 0, and the radius of the extra dimension is eventually taken to in nity. This model does not solve the hierarchy problem, but it is of interest simply due to the fact that an in nite extra dimension is allowed in consistency with observation. The original (static) RS2 setup is obtained by choosing 0 = = 1. In this case the e ective four-dimensional Planck mass is MP2l;e RHS=20 = There is no exponential hierarchy generated between the 5D and 4D scales, and there is no obstruction in taking R ! 1. For the expanding case, we assume that the in aton sector is con ned to the visible brane. Therefore, we keep = 1 but add a surplus energy density to the visible brane given by 0 = 1 + 0, with 0 as given in (3.24). The general results (3.4) and (3.19) apply and one nds a Hubble rate which is consistent with the usual four-dimensional expansion law. The metric warping is given by f 2(z) = 1 + H2 and the resulting e ective Planck mass is RS2 MP2l;e = MP2l;e H=0 1 H2 2 ! 3 + 2 1 R ! : The correction factor always reduces the e ective Planck mass, i.e. it increases the strength of gravity during in ation. Since the discussion of [ 1 ] is fully applicable, we conclude that the tensor modes during in ation are enhanced in the RS2 setup. Analogous to the LED and RS1 cases the metric can cross zero at which point a curvature singularity appears in the bulk. Avoiding this situation imposes the bound which also ensures MP2l;e > 0. Obeying this bound enforces H ! 0 as R ! 1 implying that the in ating RS2 case is inconsistent with taking the size of the extra dimension to in nity. In principle this is nothing new, as it also happens in the LED case, cf. Equation (3.9) and the related discussion in [ 1 ]. The fact that we cannot take the size of the extra dimension to in nity without letting H ! 0 is independent of whether we take into account the (55) equation or not, as is clear from the previous section. This conclusion does not change in case one allows for other values of (for example, = 0 or = 1 + 0). H2 < 2 ! ; (3.46) (3.47) (3.48) (3.49) (3.50) 4.1 Generalities Let us consider the linear dilaton con gurations [8{11, 37] of little string theory (see [38] for more references on LST). This case is akin to the RS case in the sense that there will be a negative bulk cosmological constant and two branes. The crucial di erence to the RS case is the presence of an additional scalar eld, the dilaton. Ultimately, this gives rise to power law warping in contrast to the exponential warping of the RS case. The action in the Einstein frame is given by [9, 10, 16, 23] HJEP03(218)4 S = Z [L0 (z) + L R)] + Lb(x; z) ; ) (4.1) where 2k2 2 normalized eld is = M 3=2S=p3. and we have allowed for the possibility of having branes and extra contributions in the bulk. S is the (dimensionless) dilaton and the corresponding canonically The dilaton eld itself can be used to stabilize the radius of the fth dimension. Alternatively, one could also introduce a Goldberger-Wise (GW) type scalar eld [24] ful lling the same purpose. In analogy to the LED and RS case, we will rst be agnostic about the details of the radius stabilization and simply ignore the corresponding (55) Einstein equation, assuming that it gives rise to a stable radius. We are again looking for solutions to Einstein's equations that satisfy the metric ansatz The Einstein tensor is the same as above, while the energy-momentum tensor now is given by 1 3 2 TMN = e 2 S=3 gMN : The (00) equation in the bulk then can be written as Here we are assuming that the dilaton is homogeneous in four dimensions S = S(z). It follows that H = const: and the (ij) and (00) equations are degenerate. Additionally, there appears the equation of motion of the dilaton, which is in the bulk given by S00 + 4 S0 + 2 2 e 2 S=3 = 0 : The Bianchi identity is identical to the dilaton equation of motion and does not give an independent constraint. Due to the exponential dilaton factors in the action, the boundary conditions are modi ed in comparison to the previous cases. For a brane at position z = z0 carrying a (4.2) (4.3) (4.4) (4.5) f 0 f constant four-dimensional energy density Lz0 = the brane have to ful ll z0 the discontinuities of f 0 and S0 across as well as We were not able to nd a closed form solution for (4.4) and (4.5) for a general H 6= 0. We will, thus, discuss the exactly solved static case (H = 0) rst, after which we present a perturbative solution for the expanding case. It should be mentioned here that a closed form solution for the LD model in the Jordan frame has been presented in [39]. Despite the di culties in transforming the solution to the Einstein frame [39] their derivation makes use of the unperturbed (55) equation. This is why we cannot directly adopt their solution here. Nevertheless, let us emphasize that it would be eminently useful to have an exact solution also for the LD case, just as for the LED and RS cases above. The static case given by Setting H = 0, the most general simultaneous solution to (4.4) and (4.5) in the bulk is ; f (z) = c0 + cS 3 z : 2 k Here, c0 and cS are arbitrary dimensionless constants and = 1 is an undetermined sign. By rescaling of the four-dimensional coordinates it is always possible to normalize f (0) = 1, thus, xing the constant c0 = 1. By contrast, cS corresponds to the normalization of S(0), whose value, however, can also be chosen without loss of generality. This can be understood by noting that cS can be absorbed into k~ := cSk and it thereby disappears completely from the bulk action which now contains k~ and S~(z) = 3 ln(1 + 2k~z=3) which is automatically normalized to S~(0) = 0. Reformulating the brane Lagrangians in terms of S~ one nds that they have to be globally rescaled by cS, for example L0 ! cSL0. To maintain canonically normalized kinetic terms on the branes one then has to rescale the elds in L0 which likewise leads to an unphysical rescaling of couplings. Thereby cS can be completely absorbed from the theory without loss of generality. Correspondingly, a normalization of S(0) = 0 can always be chosen without physical consequences [16]. We stress this point here because in the expanding (H 6= 0) case below this conclusion will not hold and the physical results change if the boundary condition S(0) = 0 is changed. The bulk solution of the static case is consistent with the boundary conditions on the two branes (4.6), (4.7) only for the ne-tuned brane tensions5 = 4 k M 3 : (4.9) 5Note that this corresponds to values = 2=p3 in our above notation for the RS case, showing that the ne-tuned brane tensions in the CW case are di erent from the ones required in the RS case. 0 = 0 = (4.6) (4.7) (4.8) Here, it makes sense to de ne the dimensionless quantities and := 4 k M 3 1 and assuming the usual orbifold symmetry z ! z for solutions in the two separate domains, one obtains 2 k 3 j j z f (z) = fs(z) := 1 + and S(z) = Ss(z) := 3 ln 1 + 2 k This is the standard linear dilaton solution (see e.g. [9{11]) which appears here as \logarithmic dilaton" due to our euclidean coordinate choice for the fth dimension. Compared to the exponential warping in the static RS metric (3.21) we nd here a power-law warping. Taking the fth dimension to be of size z 2 [ Planck mass is given by R; R] the e ective four-dimensional MP2l;e H=0 = 2 M 3 Z LD 0 R Taking the fundamental scale to be M k TeV and requiring the observed value for MPl;e we nd that k R & 1011, corresponding to an extra dimension of size 10 nm. Note that it is crucial to choose the same sign for and the possible values of z. Choosing = 1 (or equivalently allowing for negative values of z) the metric vanishes and the dilaton pro le diverges at zsing = ( )3=2k corresponding to a physical singularity. This would give rise to a natural cuto size of the extra dimension R 3=2k. The presence of such singularities has already been noted in [39]. We are interested in cases where the presence of the extra dimension solves the hierarchy problem. Therefore, we limit ourselves to parameters which allow for an arbitrary size of the extra dimension and avoid the singularity in the dilaton pro le. 4.3 The expanding case with external stabilization Let us now generalize the solution to the case H 6= 0. Since we were not able to nd an exact solution for f (z) and S(z) for the general case, we will assume that the dimensionless quantity := H2=k2 is small and nd a perturbative solution in . In the limit linear dilaton solution should be recovered. Therefore, we adopt the ansatz ! 0 the f (z) = fs(z) [1 + df (z)] ; and S(z) = Ss(z) + dS(z) ; (4.13) where fs(z) and Ss(z) are the solutions of the static case given in (4.11). Plugging the ansatz into equations (4.4) and (4.5) we expand in 1 and use that fs and Ss are solutions of the static case. At linear order in we nd that df (z) and dS(z) have to ful ll 2 fs2 dkf200 + 16 3 df 0 fs k dS00 fs k2 + + 4 9 fs k dS0 8 dS0 3 k + 8 9 + 8 dS 3 fs dS + 8 df 0 k 2 = 0 ; = 0 : (4.10) (4.11) (4.12) (4.14) (4.15) These two equations can be decoupled, thereby giving rise to a third oder equation for dS which can be solved. Subsequently the solution for dS can be used in order to solve also for df . The corrections to the bulk solutions are then given by dS(z) = df (z) = ln fs(z) + 9 4 9 2 c1 c2 + ; c2 c1 162 fs(z)3 + c4 as well as c3, respectively. The junction conditions on the branes are As our bulk solution is only valid up to order we can only require that the boundary conditions are solved up to that order. This implies that deviations of the brane tensions should be small compared to the static case, i.e. 0 = 1. The boundary conditions are then solved by (4.16a) (4.16b) (4.17a) (4.17b) (4.18) (4.19) displayed in gure 4. Given the metric warping f (z), the e ective Planck mass during in ation can be com1 3 fs2; + fs; + 1 ; c1 = c3 = fs2; + fs; ; fs2; + fs; + 5 ; together with the relations 243 2 27 4 = 1 ; and 0 = fs2; fs; fs2; fs2; + fs; 1 1 : Here we have used the abbreviation fs; warping is fully speci ed and given by fs( R). Finally, our solution for the metric where h:o: denotes terms of higher order in H2=k2 pleteness, we also state the leading order correction to the dilaton pro le which is given by Our perturbative solutions agree well with a numerical solution of (4.4) and (4.5), as (a) 10 0.10 0.01 (black) as compared to the static solution (gray). Next to our perturbative results (black) we also show a numerical solution (black, dashed). For the chosen parameters, the resulting Hubble rate is H 4 The inferred bound on HR from the perturbativity requirement on gravity MP2l;e > 0 during in ation is H2 2 R2 . 3 ; corresponding to H < 10 eV (i.e. TRH < 105GeV) for R 10 nm. Note that it is not possible here to straightforwardly interpret the decrease of the Planck mass during in ation in terms of the tensor-to-scalar ratio, as we will discuss in the following. The rst relation in (4.19) can be written as H2 = 4 k2 0 1 3 0 2 R2 = 2 3 k R 0 ; where 0 denotes the surplus energy density on the visible brane and we have expanded in k R 1 to simplify the result. Note that we do not recover the standard expansion law on the visible brane unlike in the LED or RS case with in ation. Just like in the cases of \remote" in ation in the RS model and the special case of Nihei-Kaloper-Kim-Kim, the expansion law of the visible brane is non-standard. The origin of the non-standard expansion law here is the necessary relation between the surplus energy densities on the IR and UV branes, manifest in the second equation of (4.19). Such a relation is, of course, inconsistent with the picture of having the in aton sector con ned to one of the branes. Restoring the individual contributions of the two brane energy densities to the expansion one can write H2 0 + 8 27 k This shows that if one would ignore the required interrelation of energy densities and simply set ! 0 the standard expansion law on the visible brane would be recovered. However, such an ad-hoc prescription is inconsistent with our solution, in particular with (4.23) (4.24) (4.25) the relation (4.19). In this sense the common wisdom, that a stabilized radius leads to a standard in ation law on the brane, does not hold for the Linear Dilaton model. Clearly, the requirements to apply the analysis of [ 1 ] are not ful lled here. In particular, it is not possible to assume that the energy density that drives in ation | i.e. the in aton and its potential | is con ned to the visible brane. A dedicated study would be required to track the impact of the non-standard expansion law on the observable scalar and tensor mode perturbations after in ation. Let us remark that the physical origin of the relation between the brane energy densities in (4.19), which leads to the non-standard expansion law, is the dilaton degree of freedom. In the present case, the dilaton dynamics does not decouple from the system even in low energy limit so that the correlation of the two brane energy densities still holds. In a sense, the situation is similar to the NKKK case in the RS model. In that case, the radion is massless so that the (55) Einstein equation does not decouple from the system. This in turn requires the brane energy density relation (3.38) for a given radius R, which leads to the non-standard expansion law (3.39). Just as the correlation (3.38) becomes irrelevant when the radion is heavy in the RS model, also the relation (4.19) in the Linear Dilaton model will be broken if the dilaton gets massive. We will see this to be the case in the next subsection where we introduce additional brane localized dilaton potentials, which make the dilaton uctuations over the background solution heavy enough to decouple them from the system in the low energy limit. So far we have been agnostic about the details of the stabilization mechanism. The most economic way to stabilize the extra dimension in the Linear Dilaton model is to invoke the dilaton eld itself [10]. If S experiences strong boundary potentials, the eld values Ss(0) = S0 and Ss( R) = S on the branes are xed, corresponding to two additional boundary conditions. In the static case, the size of the extra dimension then is determined by the relation [10] k R = exp 3 2 S S0 3 1 ; (4.26) completely analogous to the usual Goldberger-Wise mechanism [24]. An analogous scheme has been adopted for the expanding case in [23]. However, we nd that one has to be very careful in applying the stabilization scheme of the static case for the expanding case. The reason is that the boundary conditions (4.17) are modi ed by the brane potentials of S. This does not a ect the solution of the static case, as the modi ed boundary conditions are automatically ful lled. In the dynamical case, however, the change is important as we will see in the following. 4.4 The expanding case with dilaton stabilization Let us consider the case that the dilaton itself is used as a stabilizer. The dilaton-stabilized solution is somewhat more elaborate than the stabilization by an additional GoldbergerWise scalar, simply due to the fact that the back reaction of the stabilizing eld is fully accounted for in the computation of the metric. Assuming an otherwise empty bulk, the bulk Lagrangian is fully speci ed and the (55) Einstein equation should be taken into account. In addition, the boundary conditions (4.17) are modi ed by the brane potentials of S. (4.27) (4.28) (4.29) (4.30a) S(z0)) : i (4.30b) potential. f0; Rg to S0(z0 ) = Here, 0; are dimensionless parameters that characterize the strength of the respective This modi es the boundary conditions (4.6) and (4.7) at the respective position z0 = 2 3 e S(z0)=3 h 2 e S(z0)=3 h z0 + z0 M 4 (Sz0 S(z0))2i ; z0 + z0 M 4 (Sz0 S(z0))2 + 6 z0 M 4 (Sz0 The (55) equation in the bulk is given by Stabilization can be achieved by imposing strong boundary potentials for S which can be modeled in (4.1) by the choice L0 = L 0 0 M 4 (S0 M 4 (S S(z))2 ; S(z))2 : Here, we take the boundary potential parameters S0 and S to be the same for the static and the dynamic case. In the static case, the solutions (4.11) are compatible with the modi ed boundary conditions if the additional constraints Ss(0) = S0 and Ss( R) = S are ful lled. Furthermore, the (55) Einstein equation is automatically ful lled. Thus, the original static case solution is completely consistent also with the assumption of dilaton stabilization and one eventually arrives at (4.26) which determines the stabilized size of the extra dimension. In the dynamical case, by contrast, the assumption of dilaton stabilization a ects the nal form of the solution of f and S. The bulk solution is still given perturbatively by the ansatz (4.13) with the general solution (4.16). As the bulk Lagrangian is fully speci ed we require that the (55) equation is solved to linear order in = H2=k2, which is the case only if c2 = 0. As above, the boundary conditions are required to be ful lled to leading order in . While the boundary conditions for f are unchanged at leading order in , it is evident that the boundary conditions for S in (4.30b) are modi ed as compared to (4.17). Deviations of the brane tensions compared to the static case should again be small, i.e. 0 = 1. Finally, we assume that the boundary k,6 and require again the physical condition S(0) = 0. potentials are strong, 0; M Altogether, a consistent solution is given by c1 = c3 = 243 3 fs3; ; 0 0 + fs3; ; and c2 = 0 ; = 4 0 + 3 fs3; fs3; : 1 (4.31) 6Since both and k are dilaton shift symmetry breaking parameters, one can control the relative size of k compared to M . 10 0.10 0.01 the case where the dilaton itself serves as a stabilizer. Next to our perturbative results (black) we also show a numerical solution (black, dashed) and the static case solution (gray). For the chosen parameters, the resulting Hubble rate is H 8:5 10 18k 8:5 10 15 GeV corresponding to a reheating temperature around the electroweak scale (assuming maximally e cient reheating). 0 and The solution for the metric warping and dilaton pro le is fully speci ed by this. Note that there is no constraint on the relative tensions of the two branes in this case, meaning that can be varied independently. We note that the last relation of (4.31) can be H2 = 0 fs3; For the case = 0 corresponding to = 0 the standard expansion law on the visible brane is recovered. In the following we limit ourselves to this case, i.e. we assume that in ation is driven from a surplus energy density located solely on the visible brane. The solution for the metric warping then is while the dilaton pro le at leading order is given by (4.33) As before h:o: denotes terms of higher order in H2=k2 The solutions are displayed in gure 5 together with a numerical solution. Given the metric warping f (z), the e ective Planck mass during in ation can be com LD MP2l;e = MP2l;e H=0 4 9 1 + H2 k 3 R3 + h:o: : We note that the Planck mass is enhanced during in ation meaning that the strength of gravity is reduced. Since in ation here is solely driven from the visible brane, the simplifying assumptions of [ 1 ] hold and we can interpret our result in terms of the in ationary tensor-to-scalar ratio. Since gravity is weakened during in ation the amplitude of tensor modes is reduced. There is no upper bound on H from perturbativity requirement on gravity. It is noteworthy, however, that the corrections to the static case are proportional to (H=k)2(kR)3 in contrast to all other models above, where the k dependence cancels and the corrections were proportional to powers of (HR) only. This does not modify the conclusion of [ 1 ], that the 4D consistency relation PT(`)=PS(`) = 4 nT also holds in the 5D case. The extra dimension is stabilized by the dilaton at a size R, which is determined by the transcendental relation S S0 = 3 ln 1 + k R 2 3 6 0 3 + 2 k R 2 : (4.36) HJEP03(218)4 Clearly, this corresponds to (4.26) of the static case. For our particular case of interest = 0) the corrections to the radius with respect to the static case result (4.26) are O(kR) 1 and, thus, completely negligible. Comparison to results in the literature In ation in the Linear Dilaton model has recently also been studied by Kehagias and Riotto (KR) [23]. They have investigated in ation in the LD model under the assumption that the Standard Model and the in aton both reside on the UV brane and have found that the tensor modes are suppressed. Our setting is conceptually di erent because we consider the case that the visible brane, which hosts the Standard Model and in aton elds, is the IR brane such that the gauge-hierarchy problem can also be solved. We also discuss the di erence between stabilizing the extra dimension by the dilaton compared to the conventional Goldberger-Wise stabilization. Investigating the setup considered by KR we nd qualitative agreement (i.e. suppression of the tensor modes) but we were not able to quantitatively reproduce their results on the metric warping, dilaton pro le, and the e ective Planck mass. 5 We have considered the scheme of in ation in theories with extra space dimensions. In this framework some novel questions arise: why do some dimensions in ate while others are frozen? In the framework of UV-complete theories (as e.g. string theory) this question is related to the mechanism of moduli stabilization. Another question concerns the location of the in aton eld (is it a brane- or a bulk- eld) and whether the predictions of the in ationary scenario are in uenced by the presence of extra dimensions. This is the question discussed in the present paper. We have concentrated our analysis on those situations where the extra dimensions explain the hierarchy between the weak-scale and the Planck scale. In this case the relative strength of gravity varies in the bulk between visible and hidden brane and this can have consequences for the size of in ationary tensor modes, discussed here in detail. Examples under consideration are large extra dimensions (LED), the Randall-Sundrum scenario (RS) and the linear dilaton model (LD). Up to now the discussion concentrated mainly on simpli ed cases that satisfy the IRB assumption where the in aton sits on the visible brane and where the mechanism of stabilizing the extra dimensions is assumed not to in uence the predictions of in ation. In a rst step we have reexamined the IRB case for LED and RS where exact solutions could be obtained. We observed enhanced tensor modes compared to in ationary prediction in four space-time dimensions. We also stress that in these cases we obtain an upper limit on the Hubble scale H. The LD case is more complicated due to the presence of an additional bulk eld (the dilaton). We are not able to nd exact solutions here but can derive a perturbative expansion in H2. Within the LD framework we nd that the naive IRB assumption leads to inconsistencies. Contributions from the hidden brane (or bulk) are needed to obtain the conventional in ationary scenario. This leads us to a scheme of \remote" in ation, where in ation is (partially) driven by energies on the hidden brane. Motivated by this observation we reconsidered also the RS case beyond the IRB assumption and the properties of (partially) remote in ation. We provide a general class of in ationary solutions for remote in ation that include some speci c cases (as NKKK) discussed earlier. Depending on the speci c situations, tensor modes could be enhanced or reduced. The consequences for the tensor-to-scalar ratio are not known yet as this would require more calculations beyond the ones given in this paper. In some cases we nd an upper limit on the scale of H similar to that found in the IRB case. The analysis of remote in ation in the LD case leads to similar results. The calculation is performed perturbatively in H2 and supported by a full numerical solution. Still it would be desirable to extend this to an exact solution as we had derived in the LED and RS case. The complications in the LD case come from the presence of the additional dilaton eld, and this opens new possibilities. One could use the dilaton to stabilize the size of the extra dimension within the scheme itself (without the need for additional stabilizer elds). We have discussed this situation in detail and found the surprising result that this scheme can be made consistent with the IRB assumption (with the in aton eld on the IR brane). In this scheme one obtains a reduced tensor-to-scalar ratio while there is no upper limit on H. The presence of extra dimensions can have strong e ects on the prediction of in ationary models, especially in those cases where extra dimensions provide a solution to the weak-scale hierarchy problem. This is an exciting situation in view of new observations concerning the tensor-to-scalar ratio of uctuations of the cosmic microwave background. Acknowledgments We thank Stefan Forste for useful discussions. AT also thanks Zhongyi Zhang for useful discussions. This work has been supported by the German Science Foundation (DFG) within the SFB-Transregio TR33 \The Dark Universe". A Agreement with earlier results on the RS1 case We show that our results in (3.25){(3.27) on the Hubble rate, metric warping, and e ective Planck mass for the RS1 case are in agreement with the results obtained by Giudice et al. [ 1 ]. written as M B MPl as the fundamental scale. Dictionary of di erent coordinate conventions 1 # 4 2 K2 e 2 K z 2 H2 2 K2 ; (A.2) in perfect agreement with [ 1 ]. Using the same identities the Hubble rate (3.25) can be H2 = [ 1 ] K 0 3 M 3 1 2 2 : This agrees with eq. (95) of [ 1 ] up to a factor of 2 which is simply due to their choice of In order to make the connection with the results of [ 1 ] one has to recall that they work in the \ -frame" while we have chosen to put the visible brane at z = 0 throughout this note. In order to reproduce the metric warping in eq. (94) of [ 1 ] one performs the coordinate transformation z ! z + R in (3.26) to obtain f 2(z) j frame = 2 e z + 1 + H2 H2 2 2 ! 1 2 : Using the identities K =2 and 2 ! one then nds (A.1) (A.3) (B.1) (B.2) (B.3) In the discussion of the Linear Dilaton model multiple coordinate conventions have been used in the literature. The coordinates in this work are chosen such that the extra dimension is at implying that z denotes the proper length of the extra dimension. A di erent natural choice of coordinates other than ours is given by 3 2 k 2 3 y(z) = ln 1 + k z ; z(y) = e2 k y=3 1 : In this basis the metric of the static case is given by ds2 = e4 k y=3 dx dx dy2 ; while the dilaton pro le is given by S(y) (y) = 2 k y : The translation of all of our results to this basis is straightforward. In particular, we emphasize that our solutions (4.16) as well as the speci cally determined coe cients (4.18) and (4.31) are stated in a form which is independent of the chosen basis. Depending on the desired basis the explicit form of the solutions can be obtained by using fs(y) = e2 k y=3 instead of fs(z) = 1 + 2 k z=3. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. hep-th/0209133 [INSPIRE]. cosmology of theories with submillimeter dimensions and TeV scale quantum gravity, Phys. Rev. D 59 (1999) 086004 [hep-ph/9807344] [INSPIRE]. [7] L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [hep-ph/9905221] [INSPIRE]. [8] I. Antoniadis, S. Dimopoulos and A. Giveon, Little string theory at a TeV, JHEP 05 (2001) 055 [hep-th/0103033] [INSPIRE]. [9] I. Antoniadis, A. Arvanitaki, S. Dimopoulos and A. Giveon, Phenomenology of TeV little string theory from holography, Phys. Rev. Lett. 108 (2012) 081602 [arXiv:1102.4043] [10] P. Cox and T. Gherghetta, Radion dynamics and phenomenology in the linear dilaton model, JHEP 05 (2012) 149 [arXiv:1203.5870] [INSPIRE]. (2012) 125019 [arXiv:1202.6674] [INSPIRE]. [hep-ph/0409138] [INSPIRE]. [11] M. Baryakhtar, Graviton phenomenology of linear dilaton geometries, Phys. Rev. D 85 [12] J.E. Kim, H.P. Nilles and M. Peloso, Completing natural in ation, JCAP 01 (2005) 005 [13] K. Choi, H. Kim and S. Yun, Natural in ation with multiple sub-Planckian axions, Phys. Rev. D 90 (2014) 023545 [arXiv:1404.6209] [INSPIRE]. [14] K. Choi and S.H. Im, Realizing the relaxion from multiple axions and its UV completion with high scale supersymmetry, JHEP 01 (2016) 149 [arXiv:1511.00132] [INSPIRE]. [15] D.E. Kaplan and R. Rattazzi, Large eld excursions and approximate discrete symmetries from a clockwork axion, Phys. Rev. D 93 (2016) 085007 [arXiv:1511.01827] [INSPIRE]. [16] G.F. Giudice and M. McCullough, A clockwork theory, JHEP 02 (2017) 036 [17] N. Craig, I. Garcia Garcia and D. Sutherland, Disassembling the clockwork mechanism, [arXiv:1610.07962] [INSPIRE]. JHEP 10 (2017) 018 [arXiv:1704.07831] [INSPIRE]. arXiv:1705.10162 [INSPIRE]. Lett. B 465 (1999) 81 [hep-ph/9905487] [INSPIRE]. [18] G.F. Giudice and M. McCullough, Comment on \disassembling the clockwork mechanism", [19] T. Nihei, In ation in the ve-dimensional universe with an orbifold extra dimension, Phys. 4690 [hep-th/9906064] [INSPIRE]. (1999) 4922 [hep-ph/9907447] [INSPIRE]. [hep-th/9905210] [INSPIRE]. HJEP03(218)4 eld, Phys. Lett. B 481 (2000) 386 [hep-ph/0002229] [INSPIRE]. [hep-th/9503149] [INSPIRE]. [1] G.F. Giudice , E.W. Kolb , J. Lesgourgues and A. Riotto , Transdimensional physics and in ation, Phys. Rev. D 66 ( 2002 ) 083512 [ hep -ph/0207145] [INSPIRE]. [2] D. Langlois , R. Maartens and D. Wands , Gravitational waves from in ation on the brane , Phys. Lett. B 489 ( 2000 ) 259 [ hep -th/0006007] [INSPIRE]. [3] A.V. Frolov and L. Kofman , Gravitational waves from brane world in ation, [4] R. Maartens and K. Koyama , Brane-world gravity, Living Rev. Rel . 13 ( 2010 ) 5 [5] N. Arkani-Hamed , S. Dimopoulos and G.R. Dvali , The hierarchy problem and new dimensions at a millimeter , Phys. Lett. B 429 ( 1998 ) 263 [ hep -ph/9803315] [INSPIRE]. [6] N. Arkani-Hamed , S. Dimopoulos and G.R. Dvali , Phenomenology, astrophysics and [20] N. Kaloper , Bent domain walls as brane worlds , Phys. Rev. D 60 ( 1999 ) 123506 [21] H.B. Kim and H.D. Kim , In ation and gauge hierarchy in Randall-Sundrum compacti cation , Phys. Rev. D 61 ( 2000 ) 064003 [ hep -th/9909053] [INSPIRE]. [22] L. Randall and R. Sundrum , An alternative to compacti cation , Phys. Rev. Lett . 83 ( 1999 ) [23] A. Kehagias and A. Riotto , Clockwork in ation, Phys. Lett. B 767 ( 2017 ) 73 [24] W.D. Goldberger and M.B. Wise , Modulus stabilization with bulk elds , Phys. Rev. Lett . 83 [25] C. Csaki , M. Graesser , L. Randall and J. Terning , Cosmology of brane models with radion stabilization , Phys. Rev. D 62 ( 2000 ) 045015 [ hep -ph/9911406] [INSPIRE]. [26] P. Kanti , I.I. Kogan , K.A. Olive and M. Pospelov , Cosmological three-brane solutions , Phys. [28] J. Lesgourgues , S. Pastor , M. Peloso and L. Sorbo , Cosmology of the Randall-Sundrum model uctuations , Phys. Rev. D 42 ( 1990 ) 3413 [INSPIRE]. [34] T. Damour and A. Vilenkin , String theory and in ation, Phys. Rev. D 53 ( 1996 ) 2981

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Sang Hui Im, Hans Peter Nilles, Andreas Trautner. Exploring extra dimensions through inflationary tensor modes, Journal of High Energy Physics, 2018, 4, DOI: 10.1007/JHEP03(2018)004