Are my assumptions about universe thermodynamics right?

Question

Let's find the temperature of universe for times between $$10^{-30}s-1000\;years$$ These times are bigger than inflation era but smaller than nonrelativistic matter dominance (50000 years) and smaller than light transparency era (400000 years). So, I can just use ultra relativistic assumptions: $$p=\dfrac{1}{3}\rho$$ $$\dot{\rho}+3\dfrac{\dot{a}}{a}(\rho+p)=0\Rightarrow \dfrac{d\rho}{p+\rho}=-3dln(a)\Rightarrow\rho=\dfrac{c}{a^4}$$ where a is scale factor $$\left(\dfrac{\dot{a}}{a}\right)^2=\dfrac{8\pi}{3}G\rho \Rightarrow a(t)=const\cdot t^{1/2}\Rightarrow \rho\sim 1/t^2$$ From the other side, for ultra relativistic matter I can write: $$ \rho_i=\frac{g_i}{2 \pi^2} \int \frac{E^3}{\mathrm{e}^{E / T} \mp 1} d E= \begin{cases}g_i \frac{\pi^2}{30} T^4 & - \text { bosons; } \\ \frac{7}{8} g_i \frac{\pi^2}{30} T^4 & -\text { fermions }\end{cases} $$ so $$\rho\sim 1/t^2\sim T^4\Rightarrow t\cdot T^2=known\;constant$$ is that true?

Answer 1

That works as long as the relativistic content of the universe remains constant.

More generally we can say that $$\rho(T) = \frac{\pi^2}{30}g_*(T)\, T^4,$$ where $$g_*(T) = \sum_{\text{bosons } i} \left(\frac{T_i}{T}\right)^4 g_i + \sum_{\text{fermions } i} \frac{7}{8}\left(\frac{T_i}{T}\right)^4 g_i.$$

See figure 1 of these notes for how $g_*$ evolves as a function of temperature, assuming only Standard Model relativistic content.

Moreover, if the relativistic content changes, then $a\propto T^{-1}$ (and hence $\rho\propto a^{-4}$) does not exactly hold. Instead we can say that entropy is conserved, so the entropy density, $$s=\frac{2\pi^2}{45}g_{*s}(T)\, T^3$$ with $$g_{*s}(T) = \sum_{\text{bosons } i} \left(\frac{T_i}{T}\right)^3 g_i + \sum_{\text{fermions } i} \frac{7}{8}\left(\frac{T_i}{T}\right)^3 g_i,$$ scales as $s\propto a^{-3}$. So for example, $$a(T) = \left(\frac{g_{*s}(T_0)}{g_{*s}(T)}\right)^{1/3}\frac{T_0}{T}a(T_0),$$ where $T_0$ is the temperature today (or at any other reference time). Figure 1 of the same notes also shows how $g_{*s}$ evolves as a function of temperature. You can also see this earlier question for further discussion of these points.

Now we can say that the time is $$t=\int_0^a \frac{da^\prime}{a^\prime H(a^\prime)}$$ (which follows from the definition of $H$) with $$H^2 = \frac{8\pi G}{3}\rho$$ and the relationships between $\rho$, $T$, and $a$ given above.