ULDM self-interactions, tidal effects and tunnelling out of satellite galaxies

It is well-known that Dark Matter (DM) inside a satellite galaxy orbiting a host halo experiences a tidal potential. If DM is ultra-light, given its wave-like nature, one expects it to tunnel out of the satellite - if this happens sufficiently quickly, then the satellite will not survive over cosmological timescales, severely constraining this dark matter model. In this paper, we study the effects of the inevitable quartic self-interaction of scalar Ultra-Light Dark Matter (ULDM) on the lifetimes of satellite galaxies by looking for quasi-stationary solutions with outgoing wave boundary conditions. For a satellite with some known core mass and orbital period, we find that, attractive (repulsive) self-interactions decrease (increase) the rate of tunnelling of DM out of it. In particular, for satellite galaxies with core mass $\sim \mathcal{O}(10^7-10^8)\ M_\odot$ and orbital period $\sim \mathcal{O}(1)\ \text{Gyr}$, one can impose constraints on the strength of self-interactions as small as $\lambda\sim \mathcal{O}(10^{-92})$. For instance, for ULDM mass $m = 10^{-22}\ \text{eV}$, the existence of the Fornax dwarf galaxy necessitates attractive self-interactions with $\lambda \lesssim -2.12 \times 10^{-91}$.