  The distribution of the dark matter (DM) in DM-admixed-neutron stars (DANSs)
is supposed to be either a dense dark core or an extended dark halo, which is
subject to the DM fraction of DANS ($f_{\chi}$) and the DM properties, such as
the mass ($m_{\chi}$) and the strength of the self-interaction ($y$). In this
paper, we perform an in-depth analysis of the formation criterion for dark
core/dark halo and point out that the relative distribution of these two
components is essentially determined by the ratio of the central enthalpy of
the DM component to that of the baryonic matter component inside DANSs. For the
critical case where the radii of DM and baryonic matter are the same, we
further derive an analytical formula to describe the dependence of $f^{\rm
crit}_{\chi}$ on $m_{\chi}$ and $y$ for given DANS mass. The relative
distribution of the two components in DANSs can lead to different observational
effects. We here focus on the modification of the pulsar pulse profile due to
the extra light-bending effect in the case of a dark-halo existence and conduct
the first investigation of the dark-halo effects on the pulse profile. We find
that the peak flux deviation is strongly dependent on the ratio of the halo
mass to the radius of the DM component. Lastly, we perform Bayesian parameter
estimation on the DM particle properties based on the recent X-ray observations
of PSR J0030+0451 and PSR J0740+6620 by the Neutron Star Interior Composition
Explorer.
