  In this work, we calculate the ground state energy of pure neutron matter
using the renormalization group based low-momentum effective interaction
$V_{\text{low-}k}$ in Bogoliubov many-body perturbation theory (BMBPT), which
is a perturbative expansion around the Hartree-Fock-Bogoliubov (HFB) ground
state. In order to capture the low-density behavior of neutron matter, it turns
out to be better to use a density dependent cutoff in the $V_{\text{low-}k}$
interaction. Perturbative corrections to the HFB energy up to third order are
included. We find that at low densities corresponding to the inner crust of
neutron stars, the HFB state that includes pairing is a better starting point
for perturbation expansion. It is observed that including the higher order
perturbative corrections, the cutoff dependence of the ground state energy is
reduced.
