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How do we know that dark matter energy density scales as $\rho\propto a^{-3}$?

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  • $\begingroup$ It is just an hypothesis. The simplest one. It is possible that DM interact with itself, so the DM pressure isn't necessarily 0 : $p \ne 0$ and $\rho_{\mathrm{DM}}$ may be more complicated than simple pressureless matter. Currently, it is the simplest model of "cold" DM. Some guys are postulating a global scalar field (which have pressure even if it isn't interacting with itself!). $\endgroup$ – Cham May 9 at 17:49
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Dark matter is assumed to be similar to ordinary matter, but without electromagnetic interactions. (The density of ordinary non-relativistic matter scales as the inverse cube of the scale factor, so non-relativistic or “cold” dark matter is assumed to as well.) This assumption seems to be validated by the observational success of the standard model of cosmology, called Lambda-Cold-Dark-Matter, particularly regarding the details of the cosmic microwave background.

So we “know” this in the same way that we “know” many things in physics: it is one part of an accepted and successful theory that makes other predictions which are observed. That leads us to have reasonable confidence that it is true.

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  • $\begingroup$ "so non-relativistic or “cold” dark matter is assumed to as well." At high enough temperatures, everything should be relativistic including dark matter! Right? Is this therefore incorrect to infer that dark matter should not always have scaled like $\sim a^{-3}$? @G.Smith $\endgroup$ – SRS May 9 at 18:07
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    $\begingroup$ When they were highly relativistic in the early universe, both ordinary matter and dark matter scaled like radiation did, as the inverse fourth power. $\endgroup$ – G. Smith May 9 at 18:16
  • $\begingroup$ So is this scaling feature is valid after DM became nonrelativistic? @G.Smith $\endgroup$ – SRS May 9 at 18:19
  • $\begingroup$ The transition to this scaling occurred as both regular matter and dark matter became non-relativistic. $\endgroup$ – G. Smith May 9 at 18:21
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The expansion of the universe is described by a solution to Einstein's equation called the FLRW metric and two related equations called the Friedmann equations. In these we have an energy density that is normally divided into three parts:

  1. relativistic matter and radiation

  2. non-relativistic matter

  3. dark energy/cosmological constant

For more on this see my answer to Is Dark Matter called "Matter" only because of gravity?

The three parts scale as $a^{-4}$, $a^{-3}$ and $a^0$ (i.e. constant) respectively. The densities of these three components were measured by the Planck experiment and found to be:

$$ \begin{align} \Omega_{R,0} &= 9.24\times 10^{-5},\\ \Omega_{M,0} &= 0.315,\\ \Omega_{\Lambda,0} &= 0.685, \end{align}$$

So we know that just under a third of the stuff in the universe scales in the way we expect non-relativistic matter to scale, and we therefore assume that it is indeed non-relativistic matter. However visible matter makes up only about 4-5% of the matter/energy in the universe, so we are left with 26% or so of stuff that is measured by Planck to scale in the same way as non-relativistic matter but that we cannot see. This is what we call dark matter.

So the answer to your question:

How do we know that dark matter energy density scales as $\rho\propto a^{-3}$?

is simply that the Planck measurements tell us that there is around 26% of the stuff in the universe that scales as $a^{-3}$ but that we can't see. And for want of a better name this we have called dark matter.

So it isn't that we have determined that dark matter scales as $a^{-3}$, but rather that we have observed something scaling as $a^{-3}$ and called it dark matter.

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