Thank you for this reply! I read about asymmetric DM but as you said this is more of a parallel between baryons and DM, assuming DM has an antiparticle. I was wondering about a more direct implication (like the heavy neutrinos effect) in which DM doesn't necessary have an antiparticle (it is it's own anti particle), but it decays more to particles than antiparticles. In this case DM doesn't have to be asymmetric, or be subject to the same mechanism as the baryons, instead DM itself is the mechanism (or at least one of them) which leads to baryon asymmetry? Is there any such model out there?You may be interested in the concept of asymmetric dark matter in general. The idea behind it is to make dark matter more similar to matter in the sense that the DM abundance is due to an asymmetry in the dark sector similar to the baryon asymmetry in the visible sector. Within this class of models there are several different ways in which the dark and baryon asymmetries can be generated together.
Also, another possible explanation of the baryon asymmetry is in terms of very heavy neutrinos (typically introduced to explain the lightness of the standard model neutrinos) decaying preferentially to matter over antimatter as you describe. However, the very same heavy neutrinos would typically not be dark matter.
Ah I see. However, can't you have some sort of custodial symmetry? Like at high energies in the past the decay was allowed, but in the effective Lagrangian nowadays the decay is extremely suppressed (the higher terms that would allow it goes like ##\Lambda^{-n}##) so you can have stable DM now and yet lots of decays in the past?Typically, if you use the asymmetry in the decay to produce baryons, then your decay rate must be too short for the particle to be dark matter (for which it needs to be stable on cosmological time scales).