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In this paper, we investigated the effect of dark matter on the weakdeflection angle by black holes at the galactic center. We consider three knowndark matter density profiles such as the Cold Dark Matter (CDM), Scalar FieldDark Matter (SFDM), and the Universal Rotation Curve (URC) from the Burkertprofile. To achieve this goal, we used how the positional angles are measuredby Ishihara et al. method based on Gauss-Bonnet theorem (GBT) on the opticalmetric. With the help of the non-asymptotic form of the GBT, the longitudinalangle difference is also calculated. First, we find the emergence of apparentdivergent terms on the said profiles, which indicates that the spacetimedescribing the black hole-dark matter combination is non-asymptotic. We showedthat these apparent divergent terms vanish when the distance of the source andreceiver are astronomically distant from the black hole. Using the currentobservational data in the Milky Way and M87 galaxies, we find interestingbehaviors of how the weak deflection angle varies with the impact parameter,which gives us some hint on how dark matter interacts with the null particlesfor each dark matter density profile. We conclude that since these deviationsare evident near the dark matter core radius, the weak deflection angle offersa better alternative for dark matter detection than using the deviation fromthe black hole shadow. With the DM profiles explored in this study, we findthat SFDM is the most difficult to detect, while the easiest is the URCprofile. With the dark matter profiles explored in this study, we find that thevariation of the values for weak deflection angle strongly depends on the darkmatter mass on a particular profile.

Dark matter effect on the weak deflection angle by black holes at the center of Milky Way and M87 galaxies