  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.
