After the inflationary era the Universe became radiation dominated. The era of radiation domination is defined as the phase during which the temperature of the Universe was so high that the kinetic energy of the massive particles in the Universe were too large compared to their rest masses. If I understand it correct, as the temperature dropped, the radiation dominated phase ended when ultrarelativistic particles became nonrelativistic. Now, here is my problem. Clearly, with the decrease in temperature, the class of heavier particles (e.g., some heavy dark matter) will become nonrelativistic much earlier than the class of lighter particles (e.g. electrons, quarks etc). On the other hand, photons and neutrinos will (almost) always remain relativistic.

$\bullet$ How do we pinpoint the time at which radiation domination ended and matter domination started?

Response after @Cham's answer:

$\bullet$ If particles with different masses become nonrelativistic at different times, how can there be a unique time for the onset of matter domination? I would like to have a physical understanding of when this transition happens.

  • $\begingroup$ There seem to be two effects contributing to matter domination over radiation: the fact that matter dilutes slower than radiation and the fact that over time some sources counted as radiation switch to being counted as matter. But that doesn't stop there being a unique time at which the influences are equal, it just makes it hard to calculate. $\endgroup$ – jacob1729 May 9 '19 at 18:35
  • $\begingroup$ The situation is even more complicated and There is room for empirical observation to change these figures. In the early universe, even such theoretical mechanisms as inflation and spontaneous symmetry breaking (yielding massive particles) would have a large impact on the answer to your question. $\endgroup$ – R. Rankin May 10 '19 at 0:58

The non-relativistic matter (all of the massive particles) is described by energy density \begin{equation}\tag{1} \rho_{\mathrm{mat}}(t) = \rho_{\mathrm{mat \, 0}} \, \frac{a_0^3}{a^3(t)}, \end{equation} where $a(t)$ is the universal scale factor and $\rho_{\mathrm{mat} \, 0}$ is the density today (i.e. at time $t_0$). We can define $a_0 \equiv a(t_0) = 1$ if we whish, but it is unnecesary. Radiation (all ultra-relativistic particles) behaves as \begin{equation}\tag{2} \rho_{\mathrm{rad}}(t) = \rho_{\mathrm{rad \, 0}} \, \frac{a_0^4}{a^4(t)}. \end{equation} The universe enters the matter domination era when $\rho_{\mathrm{mat}}(t) \approx \rho_{\mathrm{rad}}(t)$, i.e. when \begin{equation}\tag{3} \frac{a(t)}{a(t_0)} = \frac{\rho_{\text{rad} \, 0}}{\rho_{\text{mat} \, 0}}. \end{equation}

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  • $\begingroup$ +1 While your answer is fine it doesn't address my concern, namely, how does this transition happen physically? With the decrease in temperature, some heavier particles (e.g., some heavy dark matter) will become nonrelativistic much earlier than the class of lighter particles (e.g. electrons, quarks etc) which become nonrelativistic a little later. Which one would signal the onset of matter domination? @Cham $\endgroup$ – SRS May 9 '19 at 18:16
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    $\begingroup$ @SRS, as far as I can tell, the matter density includes all the particles (massive and less massive). We define the matter era as the period for which the total matter density dominates the radiation density. Since you can split that matter density into several contributions, I guess you could say that it is the more massive particles that defines the matter era, since $\rho_{\mathrm{mat} \, 0} = \rho_1 + \rho_2 + \ldots \approx \rho_1$. $\endgroup$ – Cham May 9 '19 at 18:23
  • $\begingroup$ For that to be true, the abundance of the most massive particles must dominate the energy density of nonrelativistic matter. @Cham $\endgroup$ – SRS May 9 '19 at 18:45

The definition of the radiation-/matter-dominated era is not what you say in the question. The boundary is simply defined as the transition point where the energy density of matter exceeds that of radiation (and the vacuum energy density).

This does indeed have a unique value.

It can be approximately calculated in the way described by @Cham and then by inverting an equation for $a(t)$ to find the corresponding time. The approximation does assume that all the matter is non-relativistic.

The changeover point turns out to be around 50,000 years after the big bang when the temperature was $\sim 10^4$ K ($\sim 0.8$ eV). At these temperatures only neutrinos could (probably) be considered relativistic but they contribute only a very small amount to the total matter density.

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  • $\begingroup$ I think the last sentence here is really important: OP is at least partly concerned that the standard calculation assumes particles don't change from 'radiation like' to 'matter like' which is false (things cool as the universe expands). The data though suggests the change over is well after everything cooled and became matter like, so all is well. $\endgroup$ – jacob1729 May 9 '19 at 21:33
  • $\begingroup$ @jacob1729 Yes; apart from the neutrinos. $\endgroup$ – Rob Jeffries May 9 '19 at 21:39
  • $\begingroup$ "The definition of the radiation-/matter-dominated era is not what you say in the question." What definition do you have in mind? @RobJeffries $\endgroup$ – SRS May 10 '19 at 3:57
  • $\begingroup$ The conventional one is that stated in my answer @SRS ? $\endgroup$ – Rob Jeffries May 10 '19 at 9:28

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