When did the radiation domination end and matter domination start?

Question

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.

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$bullet$ 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.

Cham · Answer

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}

ProfRob · Answer

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.

michael nettleton · Answer

The thesis implies that the big bang originates in the creation of matter preceding that of radiation, in conflict with the interpretation in Genesis. Why not collisions between energetic photons leading to the creation of matter?

Answered by michael nettleton on January 20, 2021

When did the radiation domination end and matter domination start?

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