What is Equation of state for the Dark Matter?

w = 0 for dark matter. It's pressureless.

 

It doesn't interact with itself either. At least not much.

 

Yep. Regular matter has w ~ 0 as well - it's because it has in essence no kinetic energy compared to its total energy. It doesn't really matter whether it interacts or not. Relativistic matter would have an equation of state parameter such that 0 < w < 1/3, with the exact value depending on the kinetic energy. It approaches 1/3 as its' kinetic energy becomes much greater than its rest energy.

 

That comes about from its relativistic motion, though. If the typical velocity of dark matter particles was also near the speed of light, it too would behave like that.

 

Well, it's also because it's currently at low density. Normal matter experiences quite a lot of pressure at higher densities. But on cosmic scales this effect is completely negligible.

 

Er, well, no. The velocities of these particles within galaxies and galaxy clusters is actually irrelevant as far as this determination is concerned. The dark matter particles get heated by their fall into those gravitational potential wells. Dark matter particles that haven't fallen into such wells are actually probably much cooler than the CMB, because the photons get lots of energy dumped into them by various processes, but as the dark matter loses its ability to interact early-on, it doesn't get this extra energy.

What happens is that the earlier the dark matter decouples from the normal matter, the lower the temperature. Usually this means that higher-mass dark matter particles end up with lower temperatures, but it is rather model-dependent.