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.
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.