# CMBR temperature over time?

How has CMBR temperature dropped as function of time? A graph would be nice, but I'd be happy with times (age of universe) when it cooled enough to not be visible to human eye, became room temperature equivalent, or reached some interesting temperatures regarding matter in the universe.

If there is a nice formula giving the temperature as function of time, that would be great too.

In terms of the redshift, the background temperature is $$T(z) = T_0\left(1+z\right)$$ where $T_0\sim2.725$ K is today's CMB temperature. For simplicity, one can invoke a uniformly-expanding universe to get the relation between $z$ and $t$ as $$1+z\propto\frac{1}{t^{2/3}}$$ So as $t\to0$, $T(t)\to\infty$ and as $t\infty$, we see $T(t)\to0$. This $1/t^{2/3}$ relationship can be plotted by Wolfram Alpha.
Note that the relationship between $z$ and $t$ is a bit more complex when considering a non-uniformly expanding universe, but the $T$ & $z$ relationship should still be valid.
The exact formula in the Standard ΛCDM-model is $$T(t) = T_0\big(1+z(t)\big) = \frac{T_0}{a(t)},$$ where $a(t)$ is the cosmological scale factor, which can be calculated by numerically inverting the formula $$t(a) = \frac{1}{H_0}\int_0^a\frac{a'\,\text{d}a'}{\sqrt{\Omega_{R,0} + \Omega_{M,0}\,a' + \Omega_{K,0}\,a'^2 + \Omega_{\Lambda,0}\,a'^4}},$$ with $H_0$ the present-day Hubble constant, $\Omega_{R,0}, \Omega_{M,0}, \Omega_{\Lambda,0}$ the relative present-day radiation, matter and dark energy density, and $\Omega_{K,0}=1-\Omega_{R,0}- \Omega_{M,0}- \Omega_{\Lambda,0}$. See this post for more details.