Current constraints on Dark Matter self-interaction from galactic profiles The self-interaction of dark matter may be small but it cannot be negligible if it is able to dissipate energy to relax into galactic clumps (necessary to explain galaxy rotation curves).
According to some answers in this old question: How Does Dark Matter Form Lumps?, the gravitational self-interaction alone is enough to allow dark matter clumping (via n-body interactions). Although two answers suggest something other than gravity is needed (one states considering the weak force is necessary, while another answer argues for why gravity alone doesn't explain how in cosmology dark matter could clump first).
I am curious about:


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*Have the measurements of dark matter profiles of galaxies become good enough to provide indirect measurements of dark matter self-interactions?  

*Can this self-interaction be used to say anything about the mass of the dark matter particles? At the very least, can we say with certainty they have mass above some threshold (ruling out very light particles such as axions or neutrinos, and ruling out some kind of unseen massless particles)?

*Since the strength and radial distribution of the gravitational force vs the weak force differ so strongly, is it possible to determine from the self interaction whether dark matter interacts via the weak force?
 A: First of all, let me clarify that a non-gravitational interaction is absolutely not required to allow dark matter to clump.
There are many attempts to constrain the self-interaction cross section of dark matter. Qualitatively speaking, if dark matter scatters off of itself, this would tend to put an upper limit on the local density of dark matter. Measurements sensitive to this density (e.g. rotation curves, gravitational lensing) can therefore constrain the cross section.
There are several examples in the literature of such constraints (this being only a small selection). As it stands, there is still debate over whether a claim of a self-interaction detection (or really, an interestingly constraining limit) can be made, so learning much about the particle mass or the force mediating the interaction is still rather premature.
A: There is currently no strong evidence that dark matter interacts with ordinary matter via anything besides gravity. There have been proposals for non-gravitational interactions to explain discrepancies with small scale observations. But these discrepancies can also be explained by the effects of ordinary baryonic physics.
A: Dark matter is still hypothetical - name given to excess gravity that can not be explained in terms of known bayionic matter.
Your question assumes proven existence of dark matter, which is not necessarily true.
Properties attributed to dark matter are - transparent, cold, and non-interactive (except via gravity). These properties resemble those of empty space more than anything else.
