  We explore some aspects of $Sp(4)$ gauge theory with two fundamental fermions
in the context of composite Goldstone Dark Matter. We present preliminary
lattice results for the mass of the pseudoscalar iso-singlet meson $\eta'$
using unimproved Wilson fermions and the standard plaquette action. We find
that the $\eta'$ is slightly heavier than the pseudoscalar non-singlets $\pi$
and lighter than the vector mesons $\rho$ for multiple ensembles with
$m_\pi/m_\rho \geq 0.77$. This pattern is potentially relevant for Beyond the
Standard Model physics model building. Furthermore, we show that for $N_f=1+1$
flavours the disconnected contributions to the unflavoured pseudoscalar $\pi^0$
are small. We supplement this measurement by a calculation of the $\pi\pi$
scattering length $a_0$ which shows that the ensemble studied might be
phenomenologically relevant for Dark Matter models such as the Strongly
Interacting Massive Particles paradigm.
