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For cold dark matter, density profiles are well known and easy to find information about - eg. NFW, Burkert, Einasto, and others.

But for some reason I couldn't find explicit expressions for the density profiles for Hot and Warm dark matter.

I need to know what are $\rho_{_{HDM}}(r)$,$\rho_{_{WDM}}(r)$.

I'm interested in the differences between cold/hot/warm dark matter and why do we use Cold dark matter ($\Lambda$-CMD model) and not Hot or Cold dark matter. I need to back my claims with formal quantitative analysis and not just a qualitative explanation.

If you can recommend about other aspects which differentiate the models, or a good review paper on the subject, I would welcome it.

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I think you're going to have a difficult time finding this. The reason being, nobody has really bothered studying non-CDM density profiles because it was pretty clear from the onset that CDM was the (most promising) answer. Why spend time further analyzing a model that looks wrong? I will say though, that there are issues with CDM models (e.g., where are all the "satellite" halos that CDM predicts), and because of that, some people have turned to WDM or a Cold-Warm dark matter model.

But how do we know these non-CDM models are wrong, you may ask. The answer for that comes from N-body simulations. There have been a number of continually impressive simulations over the decades that model the formation of large structure in our universe (see this paper for example). These simulations have included various forms of dark matter in an attempt to discover how different types of dark matter affect the formation of structure and content in our universe. The resounding answer seems to always be that hot dark matter just produces an incompatible universe from what we actually observe in ours. There's no point in thoroughly modeling a hot dark matter density profile when it is clear that the HDM universe is drastically different from our own.

I would suggest that if you want to mathematically compare the various dark matter models, that you look at the power spectra from them. A good place to start might be this paper.

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There are indeed models for halos made of warm dark matter. The profiles will depend on exactly what the dark matter is and what its temperature is. I am no expert, but a starting point could be a paper by Vinas et al. (2012), who claim that the profiles are much flatter in the centre than those of CDM halos.

http://arxiv.org/abs/1202.2860

Hot dark matter is defined as such during the early universe and will be nearly uniformly distributed at the epochs of early structure formation. However at later epochs, hot dark matter with rest mass can cool down. For example, the cosmic neutrino background would have been highly uniform when it decoupled and when most galaxies formed. But because of the non-zero neutrino rest mass and the expansion of the universe, these neutrinos are now non-relativistic and will be affected by cosmic structures and become "clumpy". The same will be true for any dark matter particle with a rest mass greater than about 0.1 eV.

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