Restrictions on theories which describe particle which is the dark matter candidate Lets have theory which describes (cold) dark matter candidate. I know two cosmological (not astrophysical) restrictions for particle: its lifetime has to be larger than the lifetime of the Universe, and its energy density has to correspond CDM density. What are the other restrictions?
 A: Good question! 
I agree with the two restrictions on dark matter (DM) that you mentioned. In total I would mention four main restrictions: 


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*It must be non-luminous:


In practice this means no coupling (or extremely weak) to $U(1)_{em}$ and no coupling to $SU(3)_c$. We know it cannot interact with the strong force because e.g. radiation of gluons would give rise, among other things, to neural pions that decay to photons. 


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*It must have a very weak self interaction:


Many observations constrain the self interaction properties of DM. A velocity dependent interaction however could get around the strongest constraints on some scales to give a significant self interaction on other scales. 


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*It must be cold:


DM has to be non-relativistic during structure formation, this means it must have a mass larger than $m_\chi \gtrsim 1$ keV.  


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*It must be stable:


If DM had a decay rate comparable to the age of the universe it would affect cosmology significantly, something we do not see. 
In addition to these restrictions there are a lot of other restrictions that depends on assumptions of your theory. An important example is that of thermally produced dark matter. If you assume that DM was, at some point, in thermal equilibrium with the standard model particles, then we can use the fact that we know the current DM density to find the annihilation rate of DM. The result you end up with looks something like this: 
$$ \langle \sigma v \rangle_{\text{ann}} \approx 2.5\cdot10^{-9} \text{GeV}^{-2} \approx 10^{-36} \text{cm}^2 \approx 3 \cdot10^{-26} \text{cm}^3/\text{s} \approx 1 \, \text{pb},$$
where $\langle \sigma v \rangle_{\text{ann}}$ is the thermally averaged annihilation cross section. The exact numerical value depends somewhat on the model you are looking at, but always gives about the same order of magnitude as this. 
The constraint from thermal production is more useful than the others because it is a specific result that your model must reproduce. That means that it doesn't just eliminate some parts of parameter space, but it (usually) lets you fix one of the parameters of your theory completely in therms of the other parameters. 
A: One astrophysical constraint is that dark matter particles must not be produced in too large numbers in stars. Suppose that the reaction $\gamma + Ze^+ \to \gamma + Ze^+ + D$ or something similar, is allowed where $D$ is a dark matter particle. Since a star is opaque to photons, that energy stays in the star, but the dark matter particle escapes, like neutrinos, and carries away energy.
