Spin-dependence of Gravity-mediated Dark Matter in Warped Extra-Dimensions

We study the spin-dependence of Dark Matter (DM) particles which interact gravitationally with the Standard Model (SM) in an extra-dimensional Randall-Sundrum scenario. We assume that both the Dark Matter and the Standard Model are confined to the TeV (Infra-red) brane and only interact via gravitational mediators, namely Kaluza-Klein gravitons and the radion. We analyze the different DM annihilation channels and find that it is possible to achieve the presently observed relic abundance of Dark Matter, $\Omega_{\rm DM}$, within the freeze-out mechanism for DM particles of spin 0, 1/2 and 1. We study the region of the model parameter space for which $\Omega_{\rm DM}$ is achieved and compare it with the different experimental and theoretical bounds. We also consider the impact of the radion in the phenomenology. We find that, for DM particles mass $m_{\rm DM} \in [1,15]$ TeV, most of the parameter space is excluded by the current constraints or will be excluded by the LHC Run III or by the LHC upgrade, the HL-LHC. The presence of the radion does not modify significantly the non-excluded region. The observed DM relic abundance can still be achieved for DM masses $m_{\rm } \in [4,15]$ TeV and $m_{G_1}<10$ TeV for scalar and vector boson Dark Matter. On the other hand, for spin 1/2 fermion Dark Matter, only a tiny region with $m_{\rm DM } \in [4, 15]$ TeV, $m_{G_1} \in [5,10]$ TeV and $\Lambda>m_{G_1}$ is compatible with theoretical and experimental bounds.


Introduction
The Standard Model of Fundamental Interactions is a very powerful tool to understand electromagnetic, weak and strong interactions at least up to the energy scale tested at the LHC. After the discovery of the Higgs boson in 2012 [1] the model is complete and it may well be possible that a huge energy desert above the TeV scale should be crossed before finding some new phenomena. Accelerators much larger than the LHC [2] are currently under study in order to explore the energy landscape above the TeV. However, a reasonable hope can drive us in the future: the Standard Model on its own is incapable of explaining the observed baryon asymmetry in the Universe; it does not provide a unique mechanism to generate neutrino masses; and, more compellingly, it offers no clues at all to what Dark Matter and Dark Energy are. The Nature of Dark Matter (DM) is, indeed, one of the longest-standing puzzles to be explained in order to claim that we have a "complete" picture of the Universe. Astrophysical and cosmological data (see, e.g., Ref. [3] and refs. therein) point out that some kind of matter that gravitates but that does not interact with other particles by any other detectable mean exists. No candidate to fill the rôle of DM has yet been observed in high-energy experiments at colliders, though. For this reason, any meaningful extension of the Standard Model usually includes some DM candidate, a stable (or long-lived, with a lifetime as long as the age of the Universe) particle with very small or none interaction with Standard Model particles. These states are usually supposed to be heavy and are called "WIMP's", or "weakly interacting massive particles". Examples of these are the neutralino in supersymmetric extensions of the SM [4] or the lightest Kaluza-Klein particle in Universal Extra-Dimensions [5]. The typical range of masses for these particles was expected to be m DM ∈ [100, 1000] GeV. However, searches for heavy particles at the LHC have pushed bounds on the masses of the candidates into the multi-TeV region. Experiments searching for DM particles through their interactions with a fixed target, known as "Direct Detection" (DD) experiments (see, e.g., Ref. [6]) or through their annihilation into Standard Model particles, or "Indirect Detection" (ID) experiments (see, e.g., Ref. [7]) have thoroughly explored the m DM ∈ [100, 1000] GeV region, pushing constraints on the interaction cross-section between DM and SM particles to very small values. Notice that both DD and ID experiments have a limited sensitivity above the TeV, as they have been mostly designed to look for O(100) GeV particles. For all of this, it seems interesting to explore further the possibility that DM is indeed made of WIMPy-like particles with masses in the multi-TeV range and none or very small interaction with SM particles beside for their gravitational interaction. Four-dimensional gravitational interaction is, however, too weak to explain the observed DM abundance in the Universe for multi-TeV particles. A way out to this problem is to enhance the gravitational interaction by lowering the fundamental scale of gravity. This is easily done in any extra-dimensional setup: if gravity feels more than 4 dimensions, than the Planck mass M P is only an effective scale relevant for processes at too large distances (or too small energies) to test the fundamental scale M D . Several extra-dimensional models have been proposed in the last twenty years to solve the "Hierarchy Problem", i.e. the large hierarchy between the electro-weak scale, Λ EW ∼ 250 GeV, and the Planck scale, M P ∼ 10 19 GeV. Extra-dimensional models solve the hierarchy problem by either replacing the Planck scale M P with a fundamental gravitational scale M D (being D = 4 + n the number of dimensions and n the number of extra spatial dimensions) that could be as low as a few TeV (Large Extra-Dimensions models, or LED, see Refs. [8][9][10][11][12]), or by "warping" the space-time such that the effective Planck scale Λ felt by particles of the SM is indeed much smaller than the fundamental scale M D M P (see Refs. [13,14]), or by a mixture of the two options (see Refs. [15,16]). The possibility that Dark Matter particles, whatever they be, may have an enhanced gravitational interaction with SM particles has been studied mainly in the context of warped extra-dimensions. The idea was first advanced in Refs. [17,18] and subsequently studied in Refs. [19][20][21][22][23]. The generic conclusion of these papers was that when all the matter content is localized in the so-called TeV (or infrared brane), after taking into account current LHC bounds it was not possible to achieve the observed Dark Matter relic abundance in warped -2 -models for scalar DM particles (whereas this was not the case for fermion and vector Dark Matter). However, an important caveat was that these conclusions were drawn assuming the DM particle being lighter than the first Kaluza-Klein graviton mode. In this case, the only kinematically available channel to deplete the Dark Matter density in the Early Universe is the annihilation of two DM particles into two SM particles through virtual KK-graviton exchange. In Ref. [24], we studied the particular case of scalar DM in warped extra-dimensions allowing for DM particles to be heavier than the first KK-graviton mode. In this case, annihilation of two DM particles into two KK-gravitons becomes kinematically possible and, through this channel, the observed relic abundance can indeed be achieved in a significant region of the parameter space within the freeze-out scenario. Radion exchange and DM annihilation into radions (added as in the Goldberger-Wise mechanism [25], to stabilize the size of the extradimension) were also taken into acoount, showing in which part of the parameter space they may contribute or not to achieve the relic abundance. Recent papers studying different aspects of spin-2 mediation of the interaction between DM particles and the Standard Model have been published in Refs. [26][27][28].
A similar analysis was carried on in Ref. [29] in the framework of the Clockwork/Linear Dilaton extra-dimensional model. Also there it was shown that DM (made of scalar, fermion or vector boson particles) on the IR-brane coupled gravitationally with the SM may achieve the observed relic abundance through the freeze-out mechanism. In order to put on equal footing the Randall-Sundrum and the Clockwork/Linear Dilaton models, we extend in the present paper our work of Ref. [24] (where only the scalar DM case was studied) to the case in which DM particles can be either scalar, spin 1/2 fermions or vector bosons. The region of the parameter space for which the observed DM relic abundance is achieved in the freeze-out framework for scalar and vector boson DM particles corresponds to DM masses in the range m DM ∈ [1,15] TeV, with the first KK-graviton mass ranging from hundreds of GeV to tens of TeV. On the other hand, we found that it is very difficult to achieve the observed relic abundance for spin 1/2 fermion DM (only a tiny region of the parameter space with m DM ∼ m G 1 ∼ a few TeV and Λ ∼ 1 TeV survives after taking into account the LHC Run III bounds). In most part of the allowed parameter space, however, the effective gravitational scale Λ for which interactions between SM particles and KK-gravitons occur must be larger than 10 TeV, approximately. Therefore, in this scenario, the hierarchy problem cannot be completely solved and some hierarchy between Λ and Λ EW is still present. This is something, however, common to most proposals of new physics aiming at solving the hierarchy problem, as the LHC has found no hint whatsoever of new physics to date. As it was the case in our previous analysis for scalar DM in warped extra-dimensions, a large part of the allowed parameter space (almost all of it, in the case of spin 1/2 fermion DM) will be tested using the LHC Run III and the HL-LHC data. By the end of the next decade, therefore, the possibility that DM is indeed made of WIMPy particles that interact -3 -only gravitationally in an extra-dimensional framework can be fully explored.
Notice that a different approach to DM gravitationally coupled to the SM was followed in the recent Ref. [30], where it was studied the possibility that scalar DM in a Randall-Sundrum scenario is only feebly interacting with the SM and, thus, it never reaches thermal equilibrium. It was shown that the observed relic abundance may be achieved also in this case through the so-called freeze-in mechanism (see Ref. [31] for more details on this mechanism). The paper is organized as follows: in Sect. 2 we show our results for the annihilation cross-sections of DM particles into SM particles, KK-gravitons and radion/KKdilatons; in the first part of Sect. 3 we review the present experimental bounds on the parameters of the model (the effective Planck scale Λ, the mass of the first KKgraviton, m G 1 and the DM mass m DM ) from the LHC and from direct and indirect searches of Dark Matter, and recall the theoretical constraints (coming from unitarity violation and effective field theory consistency); in the second part of Sect. 3 we explore the allowed parameter space such that the correct relic abundance is achieved for DM particles; and, eventually, in Sect. 4 we conclude. In App. A we give the Feynman rules for the theory considered here. Complete expressions for KKgravitons and radion decay amplitudes and DM annihilation cross-sections into SM particles, KK-gravitons and/or radions in the small relative velocity approximation can be found in Ref. [29] and will not be repeated here.

DM annihilation cross-section in RS model
Experimental data from astrophysical and cosmological measurements clearly show that a significant fraction of the Universe energy density manifests itself in the form of a non-baryonic (i.e. electromagnetically inert) matter. This component is called Dark Matter and, in the cosmological ΛCDM [32] "standard model", is usually assumed to consist of stable (or long-lived) heavy particles, i.e. non-relativistic (or "Cold") Dark Matter. In the freeze-out scenario, the DM component is supposed to be in thermal equilibrium with the rest of particles in the Early Universe (differently from the case of the freeze-in scenario, in which the DM has never been in equilibrium with the Standard Model). The evolution of the Dark Matter density n DM follows the following Boltzmann equation [33]: where T is the temperature, H(T ) is the Hubble parameter as a function of the temperature, and n eq DM is the DM number density at equilibrium (see Ref. [33] for an explicit expression for n eq DM ). Eq. (2.1) depends on two factors: the first proportional to the Hubble expansion rate at temperature T , and the second to the thermally-averaged cross-section, σv .
In order for n DM (T ) to freeze-out, as the Universe expanded and cooled down the thermally-averaged annihilation cross-section times the number density should fall below the Hubble expansion rate, σv × n 2 DM < H(T ). At that moment, the DM decouples from the SM particles bath and its density in the co-moving frame freezes to a constant density called DM relic abundance. The experimental value of the relic abundance can be computed starting from the DM density in the ΛCDM model, Ω CDM h 2 = 0.1198 ± 0.0012, where h parametrizes the present Hubble parameter (see Ref. [34]). Solving eq. (2.1) we may find, then, the thermally-averaged cross-section at freeze-out 1 σ FO v = 2.2 × 10 −26 cm 3 /s [35].
In order to obtain this quantity, we first compute the total annihilation crosssection of the DM particles: where in the first term, σ ve , the DM particles annihilate through virtual exchange (thus the subscript ve) through KK-graviton, radion or the Higgs boson 2 . In this cross-section we sum over all SM particles in the final state and in the KK-graviton modes tower when needed. We computed the analytical value of σv using the exact expression from Ref. [38]: where K 1 and K 2 are the modified Bessel functions and v M øl is the Møller velocity. The second term, σ rr , corresponds to DM annihilation into radions. The third term, σ Gr , corresponds to DM annihilation into one radion and one KK-graviton G n . Eventually, the fourth term, σ GG , corresponds to DM annihilation into a pair of KK-gravitons G n and G m .
If the DM mass m DM is smaller than the mass of the first KK-graviton G 1 and of the radion, only the first channel is possible. After that, depending on the mass of the radion with respect to G 1 , the other channels open. For a radion mass smaller than m G 1 (as is usually the case in phenomenological models using the Goldberger-Wise mechanism to stabilize the size of the extra-dimension), we will take into account in sequence the second, the third and, eventually, the fourth term in eq. (2.2).
A common approximation in the freeze-out paradigm is to consider a small relative velocity v between the DM particles when the freeze-out occurs. Therefore, the c.o.m. energy s is usually replace by s ∼ 4m 2 DM and only leading order terms in v are kept. Formulae for the DM annihilation into SM particles in the so-called velocity expansion were given in Ref. [29] and will be not repeated here. We address the interested reader to that reference. Notice that the DM annihilation cross-section into SM particles via virtual exchange of KK gravitons is velocity suppressed (d wave), due to the spin 2 of the mediators, while the corresponding one through virtual radion is s wave. DM annihilation channels into two radions, two KK-gravitons or one radion and one KK-graviton are also in s wave. In Fig. 1 we present the different contributions from σ ve , σ rr , σ Gr and σ GG to the thermally-averaged DM annihilation cross-section as a function of the DM mass m DM for scalar (left panel), spin 1/2 fermion (middle panel) and vector boson (right panel) Dark Matter particles, respectively. The parameters for which the Figure has been obtained are m r = 1 GeV, m G 1 = 1 TeV and Λ = 10 TeV. These values have been chosen so as to give a general feeling of the typical results that can be obtained. In all plots, the freeze-out thermally-averaged cross-section σ FO v is depicted by a dotted horizontal (red) line. The virtual KK-graviton exchange is represented by (purple) dot-dashed lines, and it shows the characteristic spaced multiple-resonances behaviour of the warped scenarios (differently from the case of CW/LD model [29], where the spacing between one KK-graviton mode and the next one is rather small, and a huge number of KK-modes must be coherently summed). We can see in the left panel that, as it was already found in Refs. [19][20][21][22][23], for scalar DM the virtual exchange channel is insufficient to reach σ FO v . This is not the case for fermion and vector boson DM, for which the resonant channel dominates the cross-section for DM masses between 1 and 10 TeV. The direct production of two radions, depicted by a dashed (green) line, is relevant for m DM below 1 TeV in the case of scalar DM, whereas it is much smaller than the resonant channel for fermion and vector bosons. The same happens for the virtual radion exchange cross-section, depicted by a dashed (blue) line, mostly irrelevant 3 in all cases. This is not the case for the direct production of one KK-graviton and one radion (represented by a dashed brown line), kinematically possible for m DM ≥ 1/2m G 1 . In the scalar case this channel is strongly suppressed. For vector bosons, σ Gr is much smaller than the virtual KKgraviton exchange but much larger than σ rr and the virtual radion exchange. On the other hand, in the fermion case, this cross-section is in the same ballpark of the virtual KK-graviton exchange one and may play a role for m DM < 1 TeV. The last contribution, depicted by a solid (orange) line, represents the contribution of direct production of two KK-gravitons, kinematically allowed for m DM ≥ m G 1 (for larger values of m DM , new channels open as long as 2m DM ≥ m Gm + m Gn ). For scalar DM, this channel is the driving force to achieve σ v FO for m DM > 1 TeV, as it was found in Ref. [24]. On the other hand, both for fermion and vector DM, this channel is of the same order of the virtual KK-graviton exchange and contributes to the total cross-section but is not changing the general behaviour of the latter. Eventually, the red-shaded area in the upper-right corner represents the region of the parameter space for which the effective field theory we are using here is no longer valid, as the cross-section is trespassing the unitarity bound σv ≥ 1/s. As a useful tool to understand the difference between the cross-sections for scalar, fermion and vector DM particles, we remind in Tab. 1 the dependence of the thermally-averaged annihilation cross-section σv on the relative velocity v (see Ref. [29]). Recall that v acts as a suppression factor and, therefore, the larger the power to which it appears, the smaller the cross-section.  . In all cases, the radion mass has been kept fixed to m r = 500 GeV. (notice that the actual value -8 -of the radion mass has no real impact onto the DM total annihilation cross-section, though). In all panels we represent scalar, fermion and vector DM particles by dashed (blue), dot-dased (orange) and solid (green) lines, respectively. As in Fig. 1, the horizontal (red) dashed line and the red-shaded area represent the freeze-out thermal cross-section σv FO and the region for which the effective field theory is not valid.
We can see some generic features: (1) for vector boson DM, virtual KK-graviton exchange always dominates the cross-section; (2) for scalar DM, the freeze-out crosssection is achieved only after the opening of the direct KK-graviton production channel; (3) fermion DM has a much softer dependence on m DM than scalar and vector boson DM (as it was already discussed in Ref. [29]); (4) the lower (the higher) Λ, the lower (the higher) the DM mass for which the freeze-out cross-section is achieved. In order to understand the dependence of the DM annihilation cross-section on the three free paramers of the model m DM , m G 1 and Λ, we show in Fig. 3 the region of the (m DM , m G 1 ) plane for which σ v FO is achievable, drawing the corresponding value of Λ for which σ v th = σ v FO . The upper panels represent our results in the case of an unstabilized extra-dimension, i.e. in the absence of the radion. On the other hand, in the lower panels we have included a radion accordingly to the Goldberger-Wise stabilization mechanism. Both in the upper and lower cases, from left to right the three panels show the scalar, fermion and vector boson cases, respectively. The main difference between the unstabilized and stabilized cases is the gray region in the upper left corner present for DM of any spin. This region represents the portion of the parameter space for which the observed DM relic abundance cannot be achieved. We can see that, when no radion is present in the physical spectrum, the region at low DM mass and large m G 1 is not able to reproduce σ v FO for any value of Λ. On the other hand, when a radion is included, this region becomes accessible as the direct radion production channel σ rr opens for relatively low values of the radion mass, m DM ≥ m r . Apart from this difference, the two rows are rather similar. The typical range of Λ for which achieving σ v FO is possible is Λ ∈ [10 −1 , 10 5 ] TeV. A periodic pattern in Λ can be clearly seen for low m DM for any spin of the DM particle, a consequence of the fact that for these values of m DM the freeze-out cross-section is achieved through the virtual KK-graviton exchange diagram (see Fig. 1). We can also see that the scalar and vector boson cases are extremely similar for m DM ≥ 1 TeV (as it can also be seen in Fig. 2, whenever σ v FO is achieved through direct KKgravitons production). On the other hand, the range of Λ for which the freeze-out cross-section is achievable in the fermion DM case is smaller, Λ ∈ [10 −1 , 10 3 ] TeV, as a consequence of the milder m DM dependence of the fermion DM annihilation crosssection. This points out that the fermion DM case will be more easily falsified by resonant searches at the LHC Run-III and its high-luminosity upgrade, the HL-LHC.

Parameter space analysis
In this Section we search the different regions of the parameter space (m DM , m G 1 , Λ) for which is possible to achieve the correct relic abundance, σ v th = σ v FO . We will first review briefly present experimental bounds on the mass of the first KK-graviton and the effective gravitational scale Λ and remind theoretical unitarity bounds on m DM . Eventually, in Fig. 4 we show the region of the (m DM , m G 1 ) plane for which the observed DM relic abundance is achieved for scalar, fermion and vector boson DM, extending our previous results of Ref. [24].

Experimental Bounds
There are two kinds of experimental bounds to be imposed in the model parameter space: resonance searches at the LHC; and Direct and Indirect Dark Matter searches. We will review both kinds of bounds in Sects. 3.1.1, 3.1.2 and 3.1.3.

LHC bounds
The strongest constraints come from resonant searches at LHC Run II at √ s = 13 TeV. In the RS model, two kinds of particles can be resonantly produced at the LHC: the radion and the KK-graviton tower. Out of the latter, bounds are usually imposed over the first KK-graviton mode, G 1 , as in the absence of a signal we can only conclude that the mass of the corresponding resonance is larger than the maximum avaiable energy to produce it. In the case a positive signal were to be found in the LHC Run III or at the HL-LHC, we should clearly look for more, heavier, resonances and check if the spacing between them is compatible with the values of m Gn expected in the model.
In order to estimate the impact of the LHC Run II, it is necessary to analyse the production cross-section of these two kind of particles. The bound is over the production of bulk particles and it is independent of the DM mass and spin. The analysis realised in Ref. [24], therefore, is totally valid and it can be used in the three cases of scalar, fermion and vector boson DM particles. The conclusion of the study of the production was that the bounds on the resonant production of the radion are much weaker than those corresponding to KK-graviton production. Indeed, theq q r vertex is proportional to the corresponding quark mass and, then, resonant radion production is dominated by gluon-fusion at the considered energy. However, the interaction between gluons/photons and the radion arises through quarks and W boson loops via the trace anomaly [39]. Eventually, detection of resonant particles at the LHC occurs dominantly in two possible ways, X → γγ and X → ll. However, radion decay to γγ and l l is much smaller than the corresponding decay of a KKgraviton. As a consequence, the overall bounds over m r are weaker than those over m G 1 , as anticipated above. Bounds over m G 1 and Λ from Refs. [40][41][42] are given in Fig. 7 of Ref. [24].

Direct Dark Matter Detection
Another possible source of experimental constraints is given by the DM searches at direct detection experiments. Taking a zero momentum transfer for the DM-nucleon scattering, the total cross-section for spin-independent elastic scattering between Dark Matter and nuclei is [23]: where m p is the proton mass, f p and f n are the nucleon form factors and, eventually, Z and A are the number of protons and the atomic number, respectively. In the zero momentum transfer approximation eq. (3.1) is independent of the DM particle spin. The strongest bounds from Direct Detection (DD) Dark Matter searches are found at XENON1T, which uses as target mass 129 Xe, (Z = 54 and A − Z = 75). In -11 -order to compute the possible bounds over the three cases studied in the present work we use the exclusion curve of XENON1T [43] to set constraints in the (m DM , m G 1 , Λ) parameter space.

Indirect Dark Matter Detection
Regarding DM indirect searches, there are several astrophysical experiments analysing different signals. The Fermi-LAT collaboration, for example, studied the gammaray flux reaching Earth coming from Dwarf spheroidal galaxies [44] and the galactic center [45,46], while AMS-02 has reported data about the positrons [47] and antiprotons [48] arriving at Earth from the center of the galaxy. These results are relevant for DM models that generate a continuum spectrum of different SM particles, such as the RS scenario we are considering. For the scalar and fermion DM cases we have a d-wave and p-wave suppression, respectively, in the virtual annihilation exchange into SM particles. For these two case, only DM annihilation into KK-gravitons and radions lead to observable signals. On the other hand, in the vector boson DM case we have s-wave in all DM annihilation channels. This channel, therefore, is the most constrained by these class of experiments. Current experimental data for indirect detection DM searches, however, allows to constrain DM only below ∼ 100 GeV. In the case considered here, i.e. for DM particles with a mass above ∼ 1 TeV, the limits on the cross-section are well above the required value σ FO v . Thus, indirect searches have no impact on the viable parameter space in our case.

Theoretical limits
Besides the experimental limits, there are two relevant theoretical assumptions to be fulfilled in order to ensure the validity of the approach used in this paper. First, we have been performing a tree-level computation of the DM annihilation crosssections, only. We must, therefore, worry about unitarity issues. In particular, the t-channel annihilation cross-section into a pair of KK-gravitons, σ GG , diverges as m 8 DM /(m 4 Gn m 4 Gm ) for scalar and vectorial DM particles and m 4 DM /(m 2 Gn m 2 Gm ) for spin 1/2 particles in the non-relativistic limit s m 2 DM . It is, therefore, mandatory to check that the effective theory is still unitary. We will take as unitarity bound that σ < 1/s 1/m 2 DM . This bound is shown in Fig. 4 as a green-meshed area. Second, we should concern about the consistency of the effective theory framework. In a Randall-Sundrum framework, the effective scale of the theory is represented by Λ. At energies much above this scale, KK-gravitons become stronglycoupled and the theory inherits the intrinsic non-renormalizability of the Einstein action, independently on the number of space-time dimensions. In this region, therefore, the effective field theory approach is no longer valid. We will force, then, m G 1 to be less than Λ in order to trust our results. As we are including the first KKgravitons in the low-energy spectrum, they should be lighter than the effective field -12 -theory scale to be dynamical degrees of freedom of the theory. Notice that, in the allowed region, also the relation m DM ≤ Λ is automatically fulfilled.

Results
We present our final results in the (m DM , m G 1 ) plane in Fig. .4. The different panels represent the region of the three-dimensional parameter space (m DM , m G 1 , Λ) for which the DM annihilation cross-section can achieve the freeze-out value. From left to right, the panels represent our results for scalar, fermion and vector boson Dark Matter. On the other hand, the difference between upper and lower plots stands in that in upper plots the size of the extra-dimension is unstabilized, whereas in the lower ones we add the radion to the spectrum and implement the Goldberger-Wise mechanism to stabilize r c .
In each of the panels, we depict by a white area the allowed region: this means that for each pair of values in the (m DM , m G 1 ) plane, it exists a specific value of Λ for which σ v th = σ v FO . The grey-shaded area, on the other hand, represent the region for which, for a particular choice in the (m DM , m G 1 ) plane, no value of Λ fulfills the freeze-out condition. We can see that a grey-shaded area exists in all of the three upper plots. This means that, in the absence of the radion, it always exists a region of the parameter space for which it is impossible to achieve σ v FO , independently of the spin of the Dark Matter particle. On the other hand, in all of the three lower panels the grey-shaded region is absent: it is always possible to reach σ v FO in the presence of a radion. This happens as the radion mass is not fixed: by choosing a conveniently light radion mass, the direct radion production channel σ rr gives an extra component to the total cross-section such that the observed relic abundance can be achieved. In all of the lower panels, we fix the radion mass to m r = 1 GeV. Notice that bounds on the radion are much weaker than those on the first KK-graviton, as it was explained in Sect. 3.1.1.
On top of the allowed or disallowed regions, we draw the experimental bounds from Sects. 3.1.1, 3.1.2 and 3.1.3. The red-shaded area is the region of the parameter space incompatible with Direct Detection experiments. The peculiar periodic structure arises as for a fixed value of m DM the correct relic abundance can be achieved with multiple choices of the two other free parameters of the model, m G 1 and Λ (see Fig. 2 for a similar situation in a different plane). We see that this bound only constrains very low values of the Dark Matter mass, independently from the Dark Matter spin. On the other hand, the light blue-shaded region is much more constraining: this corresponds to resonance searches at the LHC Run II, with a luminosity of 36 fb −1 at √ s = 13 TeV [40][41][42]. In all cases, this bound is much stronger than those from DD and excludes Dark Matter masses below 1 TeV (or more, depending on the DM spin). The LHC bound saturates in m G 1 around 5 TeV. Above this value, the LHC is no longer able to push its bounds, independently from the luminosity, as the c.o.m. energy is not enough to produce the resonance. This is not the case Figure 4. Region of the (m DM , m G 1 ) plane for which σv th = σ F O v . Upper panels represent our results in the unstabiliteze case, i.e. when no radion is considered; lower panels depict the stabilized case, where the size of the extra-dimension is fixed by the Goldberger-Wise mechanism and a (light) radion is added to the spectrum. The radion mass in this case is m r = 1 GeV. From left to right we present our results for scalar, fermion and vector boson DM particles. In all panels, the white (grey-shaded) area represents the region of the parameter space for which it is possible (impossible) to achieve the correct relic abundance. Over these regions, we have superimposed theoretical and experimental bounds. In particular, the pink-meshed area is the region for which the low-energy Randall-Sundrum effective theory is untrustable as m G 1 < Λ; the vertical green-meshed area on the right of all panels is the region where the unitarity constraint is not fulfilled, the red-shaded area is the region of the parameter space excluded by Direct Dark Matter Detection searches; eventually, the three blue-shaded areas represent the region of the parameter space excluded by resonance searches at the LHC Run II with 36 fb −1 (light blue) and foreseeably excluded by the LHC Run III with 300 fb −1 (blue) and the HL-LHC with 3000 fb −1 (dark blue).
-14 -in the (horizontal) Dark Matter mass axis, as for this parameter increasing the LHC luminosity does make the bound stronger: this is depicted by increasingly darker blue-shaded areas, corresponding to the LHC Run III (with an expected luminosity of 300 fb −1 ) and to the foreseen LHC luminosity upgrade, the HL-LHC (with a goal luminosity of 3000 fb −1 ). Eventually, the green-and pink-meshed areas represent theoretical consistency and unitarity bounds from Sect. 3.2. In particular, the pink-meshed area is the region of the parameter space for which the value of Λ needed to achieve σ v FO for a given point in the (m DM , m G 1 ) plane is lower than the first KK-graviton mass, Λ < m G 1 . In an OPE approach this condition is unviable, as we should integrate out particles heavier than the effective theory scale, in this case the whole tower of KK-gravitons. Notice that this constraint excludes most of the parameter space for which the observed relic abundance is achieved through direct radion production (the region that opens in the upper left corner for DM of any spin in the lower panels, absent in the upper row). The vertical green-meshed area in the rightmost side of each plot represents, on the other hand, the unitarity bound m DM ≤ 1/σ. This constraint puts an upper bound to the value of the Dark Matter mass for which the Randall-Sundrum model is able to explain the observed relic abundance within the freeze-out scenario. Notice that, incidentally, in all of the allowed (white) region the Dark Matter mass is also smaller than the value of Λ needed to achieve σ v FO , m DM < Λ.
Once we have described what is common to all panels, we may now particularize to each DM spin case. The two leftmost plots correspond, as explained above, to the scalar DM case without (above) and with (below) a Goldberger-Wise radion. This case was already shown in Ref. [24] and we get pretty similar results to those presented there (the only difference being that in this case we have taken into account the DM DM → r G n channel, previously overlooked). The region of the (m DM , m G 1 ) plane where it is possible to obtain the correct relic abundance and is not excluded by the theoretical and experimental bounds is dominated by direct graviton production. The virtual KK-graviton (and radion) exchange is always subdominant in this area. The difference between the unstabilized (above) and stabilized (below) cases is that in the latter it would be possible to reach the observed DM abundance for lower DM masses: this region, however, is excluded by the LHC Run II bounds for m G 1 < 5 TeV and by consistency of the effective theory for m G 1 > 5 TeV.
The two plots in the middle represent the spin 1/2 DM case. This case is the most constrained one between the three options studied here, as a consequence of the softer dependence of the cross-section on the DM mass (see Fig. 2). The direct KK-graviton production channel in the fermion DM case diverges as m 4 DM /m 2 Gn m 2 Gm instead than as m 8 DM /m 4 Gn m 4 Gm , as in the scalar and vector boson cases. The observed relic abundance is, therefore, reached later than for integer spin, closer to the region excluded by the unitarity limit, m DM < 1/σ. For spin 1/2 Dark Matter particles, the LHC bounds are extremely effective for m G 1 < 5 TeV, excluding all of the allowed -15 -region after taking into account the unitarity bound. Both in the upper and lower panels we can see that only a tiny triangular region survives, for which m G 1 > 5 TeV, m ∈ [4,15] TeV and Λ > m G 1 .
Eventually, the vector boson DM case is depicted in the two rightmost panels. This is the only one for which the virtual KK-graviton and radion exchange channels have some effect in the phenomenology in the allowed region. The periodic pattern caused by the dominance of these channels in some part of the parameter space induces the peculiar wiggled behaviour in the upper right corner of the LHC experimental bounds. The surviving allowed (white) region is very similar to what we got in the scalar DM case, as the cross-section dependence on the DM mass is analytically the same.

Conclusions
In this paper we have completed the analysis, presented in Ref. [24], of the possibility that the observed Dark Matter relic abundance can be explained by gravitationallyinteracting scalar Dark Matter (in agreement with all experimental probes regarding its existence) within the freeze-out mechanism in the Randall-Sundrum [13] extradimensional model. The otherwise exceedingly small gravitational interaction is known to be enhanced in extra-dimensional models either by the volume of the extra-dimension or by their curvature (being this latter option the one at work in our case). In a following paper [29], we studied the same possibility in a different extra-dimensional model, the more recent Clockwork/Linear Dilaton one. In that case, we analysed not only the scalar DM case, but only spin 1/2 and vector boson DM particles, comparing the differences of the three possibilities and finding the region of the parameter space for which achieving the observed DM relic abundance is compatible with present and future experimental and theoretical constraints. In this paper, therefore, we decided to complete our effort in the study of Dark Matter in the framework of extra-dimensional models studying the spin 1/2 and vector boson DM cases in the Randall-Sundrum model. At this point, we have eventually a completely symmetrical picture for both models.
In both the RS and the CW/LD models two branes are considered, the so-called UV (or Planck) and IR (or TeV) branes. Standard Model matter is traditionally constrained to the IR-brane in both cases. We also choose to constrain the Dark Matter particle, whichever its spin, to the IR-brane. In this particular scenario the interaction between two particles located in the IR-brane via gravity is proportional to 1/M 2 P when the interaction occurs thanks to the Kaluza-Klein zero-mode (i.e. the standard graviton), whereas the interaction with higher Kaluza-Klein modes is suppressed only by two powers of the effective scale Λ. Since Λ can be as low as a few TeV (so as to solve the so-called hierarchy problem, the original motivation for the existence of extra-dimensions), a huge enhancement in the cross-section is possible -16 -with respect to standard linearized General Relativity. In addition to the KK-tower of gravitons, we also consider a radion field, added in such a way so as to stabilize the size of the extra-dimension taking advantage of the Goldberger-Wise mechanims. Other possibilties could be (and have been) considered, such as allowing for the Dark Matter to freely explore the bulk. However, we have found that also in our restrictive case the freeze-out mechanism is efficient enough to explain the observed DM relic abundance.
Once fixed the setup, we have computed the different contributions to the thermallyaveraged DM annihilation cross-section σv for each of the three DM particles studied here with spin 0, 1/2 and 1. The channels considered for the analysis are the virtual KK-gravitons and radion exchange and the direct production of two "gravitational" modes (either two KK-gravitons, or one KK-graviton and one radion, or two radions). As a consequence of the polarization of the spin-2 KK-gravitons, the dominant channel for any of the considered DM spins is the direct production of two KK-gravitons, when the DM mass is larger than 1 TeV, approximately. In the scalar and vector cases the corresponding cross-section is enhanced at large DM masses by a term proportional to m 8 DM /(m 4 Gn m 4 Gm ). In contrast with the spin 0 and 1 cases, the cross-section for direct KK-gravitons production in the spin 1/2 case is enhanced by a softer factor, m 4 DM /(m 2 Gn m 2 Gm ). As a consequence, the observed relic abundance for spin 1/2 DM particles is achieved at larger values of the DM mass where, however, the unitarity bound on the DM mass takes over.
We have scanned the three-dimensional parameter space of the model, (m DM , m G 1 , Λ), looking for the regions for which σv th = σ F O v whilst being compatible with present and foreseeable theoretical and experimental bounds. Our results were eventually shown in Fig. 4. We have found that the most relevant experimental constraint comes from LHC Run II resonance searches, whereas Direct and Indirect Dark Matter Detection experiments are mostly irrelevant for DM masses above 1 TeV. The theoretical requirements that m DM < 1/σ and that Λ be larger than m DM , m G 1 constrain significantly the parameter space, also.
Our main result is that a significant portion of the (m DM , m G 1 ) plane is able to reproduce the current data about the DM relic abundance for any of the considered DM spins. Most part of the allowed region is, however, excluded by theoretical and experimental bounds. This is particularly true in the case of spin 1/2 Dark Matter, for which only a tiny triangular region survives for m G 1 > 5 TeV, m DM ∈ [4,15] TeV and Λ > m G 1 . This region can only be explored by accelerators with more c.o.m. energy than the LHC. On the other hand, both for scalar and vector boson DM particles, the LHC and its upgrades cannot exclude a region with m DM ∈ [4,15] TeV and m G 1 < 10 TeV. In this region, Λ ranges from a few TeV to 10 4 TeV, approximately. In most of this region, therefore, the hierarchy problem cannot be solved and a (softer) hierarchy is still present between Λ and the electro-weak scale Λ EW . We have found that the presence or absence of the radion is mostly irrelevant -17 -and our results do not depend on it.
The Feynman rules for the n = 0 KK-graviton can be obtained by the previous ones by replacing Λ with M P . We do not give here the triple KK-graviton vertex, as it is irrelevant for the phenomenological applications of this paper.

A.2 Radion Feynman rules
The radion, r, couple with particles localized in the IR-brane with the trace of the energy-momentum tensor, T = g µν T µν . The only exception are photons and gluons that, being massless, do not contribute to T at tree-level. However, effective couplings of these fields to the radion are generated through quarks and W loops, and the trace anomaly.
The interaction between one radion and two scalar fields S of mass m S is given by: The vertex that involves one radion and two Dirac fermions ψ of mass m ψ takes the form: The interaction between two massive vector bosons V of mass m V and one radion is given by: whereas the vertex corresponding to the interaction between two massless SM gauge bosons and one radion is: where α i = α EM , α s for the case of the photons or gluons, respectively, and [39]: U V + 1 2 q F 1/2 (x q ) , U V = −11 + 2n/3, where n is the number of quarks whose mass is smaller than m r /2. The explicit form of F 1/2 and F 1 is given by: Eventually, the 4-legs diagrams are given by: .20) and