  The rotation curves of spiral galaxies exhibit a great diversity that
challenge our understanding of galaxy formation and the nature of dark matter.
Previous studies showed that in self-interacting dark matter (SIDM) models with
a cross section per unit mass of $\sigma/m\approx{\cal O}(1)~{\rm cm^2/g}$, the
predicted dark matter central densities are a good match to the observed
densities in galaxies. In this work, we explore a regime with a larger cross
section of $\sigma/m\approx20-40~{\rm cm^2/g}$ in dwarf galactic halos. We will
show that such strong dark matter self-interactions can further amplify the
diversity of halo densities inherited from their assembly history. High
concentration halos can enter the gravothermal collapse phase within $10~{\rm
Gyr}$, resulting in a high density, while low concentration ones remain in the
expansion phase and have a low density. We fit the rotation curves of $14$
representative low surface brightness galaxies and demonstrate how the large
range of observed central densities are naturally accommodated in the strong
SIDM regime of $\sigma/m\approx20-40~{\rm cm^2/g}$. Galaxies that are outliers
in the previous studies due to their high halo central densities, are no longer
outliers in this SIDM regime as their halos would be in the collapse phase. For
galaxies with a low density, the SIDM fits are robust to the variation of the
cross section. Our findings open up a new window for testing gravothermal
collapse, the unique signature of strong dark matter self-interactions, and
exploring broad SIDM model space.
