Dear Shantanu,

thanks a lot for your ensemble of questions!

Some of the answers would have taken too long in the talk, so let me get back to them here:

1) what is a PIEMD:

Take a look at Eq.(6) of this paper:

https://ui.adsabs.harvard.edu/abs/2005M ... L/abstract2) do you assume substructure in your halo model:

Yes and no, depends on the viewpoint, since we could consider the n_p particles that are interacting as "substructure" given that, in simulations, their mass is usually several thousand solar masses. On the other hand, when I want to describe the entire dark matter halo, I am (so far) only interested in the final state of the halo, which would correspond to a thermal equilibrium state if that existed.

3) which of these cases corresponds to dwarf spheroidals:

In principle, any case can be applied to them, it just depends on how you sort the dwarves into the hierarchy of masses you want to model. The most important lesson that I learnt from setting up a halo model is that halos are an emergent quantity and their behaviour decisively depends on how we define them (e.g. where we put r_max). If your dwarf spheroidals are some satellites of a galaxy in a large galaxy cluster and you are interested in the halo of the overall galaxy cluster, the dwarves would be microscopic particles and averaged over. If you are interested in the halo of such a dwarf spheroidal you can measure the half-light radius and define this as the outer boundary, then you arrive at case number 4.

4) can this model explain the observations made by Donato or Kormendy & Freeman:

This is a point that I didn't address in the talk: on the slides, I also had a minimum radius r_min, which, in the end, was defined to be the radial position of the innermost particle. This value has to be larger than zero in order to keep all quantities finite. Extrapolating to 0 means that the PDF becomes infinite, hence, the formalism breaks down. This is to be expected because, the simplified assumptions may not be valid on these scales anymore. At least, we do not know what happens below the scales we observe. So r_min makes sure that we are only using the model down to scales we can resolve/observe. The cusp-core debate is -- to some extend -- related to this issue: for simulations, we cannot know the behaviour of the mass density below the radius of convergence (=r_min for the simulations). For observations the same applies due to the finite resolution of the spectrograph (to determine the rotation curves). Anything else is an extrapolation into a regime that my model does not provide, on purpose.

The other point is that Donato et al include the luminous matter content to explain the cores, while my approach is currently only considering dark matter. However, I hope to reproduce the effect that luminous matter flattens the inner mass density profiles, as also found in simulations.

5) how easy it is to extend this formalism to non-Newtonian gravity theories:

So far, the model is based on a probability density function of a power-law that is motivated by a scale-free Newtonian gravity. If your modified gravity theory provides you with another PDF, then it is quite easy to transfer the formalism. Yet, so far, I am still working on a mathematically sound derivation from Newtonian gravity to the PDF that I showed, so it may be a bit more difficult than previously assumed.

6) what about self-interacting DM:

Self-interactions break the assumptions that the particles are collisionless (=independent). Hence, we cannot simply multiply the individual PDFs for all particles, but the joint p_E of the entire ensemble is more complicated since it has to account for correlations among the particles. Despite that, it is straightforward to include this feature into the formalism, given the collisional model.

I hope that these answers are helpful. If not, don't hesitate to ask further.

Best regards and thanks a lot for the great discussion!

Jenny Wagner