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

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