How do we know that the energy density of dark matter in the Universe scales as that of ordinary matter in the expanding Universe? Is it a guess?
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There are two types of Dark Matter, Hot, and Cold.
The density fluctuations in the early universe, galaxy formation, BAO oscillations seem to favor the cold dark matter (CDM) over the hot dark matter (HDM) (e.g., neutrinos).
Let us take the WIMPS (weakly interacting massive particles) as our CDM candidate.
Since they are massive ($\approx 50 - 100$ GeV) they must have been decoupled in the early universe after they become non-relativistic.
In other words, if it was non-relativistic ($w_{CDM} = 0$) in the very early times, and it must have remained like that until now. So they scale as the baryonic matter.
Mathematically we say that,
if, $kT \gg m_xc^2$ the particle is relativistic.
if, $kT \ll m_xc^2$ the particle is non-relativistic
So, if we say that the mass of the WIMP is between $50-100$ GeV, we can calculate when it become non-relativistic, which is around $T_{dec} \approx 10^{15} K$ (this corresponds to $t \approx 10^{-9}$ second).
So after this point, they just basically scaled as baryonic matter.
Note:
The calculate this,
take the boltzmann constant $k = 1.380649 \times 10^{-23} JK^{-1}$ and $100GeV = 1.60218 \times 10^{-8} J$
so
$$T = m_x/k = \frac{1.60218 \times 10^{-8}}{1.380649 \times 10^{-23}} \approx 10^{15}K$$
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$\begingroup$ While everything you write is correct, this only seems to answer the main but not the sub-questions, those seem to ask rather about the observational evidence. I'm thinking we can measure the density in the CMB and today and it matches if you scale by a^-3... $\endgroup$– rflCommented Oct 29, 2020 at 20:11
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$\begingroup$ hmmm... structure formation should yield a measurement... gotta do some digging... $\endgroup$– rflCommented Oct 29, 2020 at 20:26
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$\begingroup$ @rfl how can you measure the current CDM density, in order to compare it with the CDM measurement from the CMB ? $\endgroup$ Commented Oct 29, 2020 at 20:26
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$\begingroup$ oh, you can't. but you can measure the amount of dark matter at the time of decoupling, that's what the cmb does, and find Omega_darkmatter_cmb=25%. Now you can look at clusters and galaxies today, to try to find the Omega_darkmatter_today. They match if you take into account scaling by scale factor a^(-3). $\endgroup$– rflCommented Oct 29, 2020 at 20:33
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$\begingroup$ So I didn't find direct measurement on the scale-dependence of the dark matter density. But there are limits on the decay of dark matter. I guess one could re-cast those limits on limits on the deviation from the simple a^-3 scale dependence... $\endgroup$– rflCommented Nov 10, 2020 at 12:27