### Decoupling and mean free path

Before the photons [decoupled](https://en.wikipedia.org/wiki/Decoupling_(cosmology)) from matter, they scattered frequently on free electrons, so their mean free path was small compared to the size of the (observable) Universe. It is often said thad the Universe was "foggy" until decoupling, but I think people often overestimate this fogginess. The mean free path is $\ell = 1 / n_e \sigma_\mathrm{T}$,
where $n_e$ is the number density of free electrons and $\sigma_\mathrm{T}\simeq6.65\times 10^{-25}\,\mathrm{cm}^2$ is the Thomson cross section of the electron. Calculating $n_e$ from the ionization state of the gas, this works out$^\dagger$ to roughly 2000 light-years just before recombination begins to kick in at a time $t\simeq200\,000\,\mathrm{yr}$ after the Big Bang (BB), 20 light-years at $t\simeq50\,000\,\mathrm{yr}$ when matter started dominating over radiation, 16 kilometers at $t\simeq15\,\mathrm{m}$ when nucleosynthesis ended, and 20 meters at $t\simeq10\,\mathrm{s}$, when leptons and antileptons annihilated and the photon epoch began.

But this scattering is not really important for how *bright* the Universe were. Photons arrive at your eye all the time, and if they have scattered multiple times from their point of origin, you won't know *where* they were created, but you will still see them.

### Planck's law

Until stars were formed a few hundred million years after BB, the brightness of the Universe was extremely homogeneous and given by a near-perfect blackbody Planck spectrum with a temperature of $T = T_0(1+z)$, where $T_0=2.725\,\mathrm{K}$ is the current temperature of the [CMB](https://en.wikipedia.org/wiki/Cosmic_microwave_background), and $z$ is the redshift corresponding to the time $t$. That is, the brightness at wavelength $\lambda$ is:
$$
B_\lambda(\lambda,T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc/\lambda k_\mathrm{B} T} - 1},
$$
where $h$, $c$, $k_\mathrm{B}$ are Planck's constant, the speed of light, and Boltzmann's constant, respectively.

### Percieved brightness

I'm going to assume that you're human, and hence that the brightness you're interested in is the optical wavelength region, i.e. around $\lambda\sim550\,\mathrm{nm}$. The peak of a Planck spectrum shifts toward higher frequencies the higher the temperature is, and hence the ratio of optical-to-UV/X-ray/gamma radiation will decrease. But regardless, the *absolute* brightness will always increase at any wavelength for larger temperatures.

At $t\simeq10\,\mathrm{s}$, what later becomes our observable Universe was already $\sim30$ light-years in radius (although the observable Universe of *that* epoch was only 8000 km). The scale factor (ratio of the size at that time to the correct size) was thus $a\sim7\times10^{-10}$, the corresponding redshift $z\sim1.4\times10^9$, and hence the temperature of the Universe was $T \sim 3.7\times10^9\,\mathrm{K}$.

Plugging into the equation above, I get that the brightness in the optical was
$$
B_\lambda(550\,\mathrm{nm},3.7\times10^9\,\mathrm{K}) \simeq 3\times 10^{20}\,\mathrm{W}\,\mathrm{m}^{-3}\,\mathrm{sr}^{-1}.
$$
So, what does this number mean? To get a feeling of how it would look, we can compare the amount of light received from a human field of view, to the light received when looking directly at the Sun. [Andersen et al. (2018)](https://arxiv.org/abs/1801.03278v1) did exactly this, in order to calculate the period of time that a human could see anything in the early Universe. They found that while the Universe became too dim for a human to sense any light around $t=5.7$ million years after BB, it was as bright as looking at the Sun when the Universe was around $T\simeq1600\,\mathrm{K}$, just over 1 million years after BB, and so had a brightness in the optical of around $B_\lambda(550\,\mathrm{nm},1600\,\mathrm{K}) \simeq 2\times 10^{8}\,\mathrm{W}\,\mathrm{m}^{-3}\,\mathrm{sr}^{-1}$, or a factor of $10^{12}$ times smaller than at $t\simeq10\,\mathrm{s}$.

In other words, ten seconds after the Big Bang **the Universe was a trillion times brighter than looking at the Sun**.

---
$^\dagger$<sub>I wrote a Python code called `timeline` to calculate $\ell$ and other quantities of the Universe as a , available on GitHub [here](https://github.com/anisotropela/universe-timeline).</sub>