Warm and Hot dark matter density profiles

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.