  In this paper, we investigated the effect of dark matter on the weak
deflection angle by black holes at the galactic center. We consider three known
dark matter density profiles such as the Cold Dark Matter (CDM), Scalar Field
Dark Matter (SFDM), and the Universal Rotation Curve (URC) from the Burkert
profile. To achieve this goal, we used how the positional angles are measured
by Ishihara et al. method based on Gauss-Bonnet theorem (GBT) on the optical
metric. With the help of the non-asymptotic form of the GBT, the longitudinal
angle difference is also calculated. First, we find the emergence of apparent
divergent terms on the said profiles, which indicates that the spacetime
describing the black hole-dark matter combination is non-asymptotic. We showed
that these apparent divergent terms vanish when the distance of the source and
receiver are astronomically distant from the black hole. Using the current
observational data in the Milky Way and M87 galaxies, we find interesting
behaviors of how the weak deflection angle varies with the impact parameter,
which gives us some hint on how dark matter interacts with the null particles
for each dark matter density profile. We conclude that since these deviations
are evident near the dark matter core radius, the weak deflection angle offers
a better alternative for dark matter detection than using the deviation from
the black hole shadow. With the DM profiles explored in this study, we find
that SFDM is the most difficult to detect, while the easiest is the URC
profile. With the dark matter profiles explored in this study, we find that the
variation of the values for weak deflection angle strongly depends on the dark
matter mass on a particular profile.
