Constraints on the mass and self-coupling of Ultra-Light Scalar Field Dark Matter using observational limits on galactic central mass

It is well known that Ultra-Light Dark Matter (ULDM), usually scalar fields of mass $m \sim 10^{-22}~{\rm eV}$, can solve some of the outstanding problems of the Cold Dark Matter (CDM) paradigm. Such a scalar field could have non-negligible self-coupling $\lambda$. In this work, using the known observational upper limit on the amount of centrally concentrated dark matter in a galaxy, we arrive at the observational constraints in the $\lambda - m$ (self coupling $-$ mass) parameter space. It is found that the observational limit on the mass $m$ of the ULDM depends upon the sign and strength of the self-interactions. We demonstrate that, for $m \sim 10^{-22}~{\rm eV}$, self-coupling values of ${\cal O}(10^{-96})$ (corresponding to a scattering length of $a_s \sim 10^{-82}~{\rm m}$) can be probed using limits on the dark matter mass within 10 pc of the centre of M87 galaxy. Our analysis suggests that if Ultra Light Axions (ULAs) form all of dark matter, its mass has to be less than $\sim 6 \times 10^{-23}$ eV.