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.
