In this post we’ll look at one of strongest pillars of evidence for the hot Big Bang. (For an introduction to the Big Bang, see this post.)

The decisive evidence for this extremely hot and dense origin of the universe as we know it is **the thermal distribution of the cosmic microwave background** (CMB). In a nutshell, the background radiation (light particles, or photons) filling the universe have exactly the same distribution of energies as particles in thermal equilibrium – that is, particles scattering off each other frequently enough to share a common temperature. This suggests that although they are not now in equilibrium, they were long ago – and this could only be the case in a much smaller, denser, and hotter universe.

If you took a large number of particles (say photons, electrons, and protons) and put them in a box, particles would scatter off each other, with more energetic particles sharing energy with other particles. Eventually the system would come to a state of thermal equilibrium, in which the distribution of energies clustered around an average energy, determining the temperature of the gas of particles. (Only a system in a state of equilibrium can be described with a temperature.) This distribution of energies is a **thermal distribution**:

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is the probability for a particle to have energy *E*.* T* is the temperature (a hotter gas means typical particles have more energy). The sign means “proportional to” – we need to scale everything so probabilities add up to one, but that’s not important here. (The “-1” in the denominator is a quantum mechanical effect that appears for photons, but is different for electrons or protons). For large energies, though, the amount of photons drops exponentially with the energy per photon.

No matter how you initially position the particles in the box – whether you cluster them all in one corner and give half the particles all the energy, or you position them randomly but with the exactly the same energy for each – given enough time they will come to equilibrium: their distribution of energies will approach a thermal distribution.

(For nonrelativistic particles, with speeds much less than the speed of light, the equilibrium energy distribution translates into a distribution of speeds. For photons, on the other hand, which are massless and travel at the speed of light, the distribution of energies translates into a distribution of frequency or wavelength, instead of speed.)

Well, guess how the energies (or frequencies) of CMB photons are distributed…in *exactly* this way. The CMB energy distribution – first measured by the COBE satellite – falls exactly on the thermal distribution described above (see the plot below), and is measured so precisely that the error bars shown are – this means that the odds of the actual value falling outside the experimental error bars is an unfathomably tiny number.

Two points to note about the plot above: First, this shows the distribution of intensity or, roughly speaking, the distribution of *energy*, which is proportional to the probability distribution given above multiplied by the photon energy. (More energetic photons obviously contribute more to the total energy, giving more weight to the high-energy part of the thermal distribution.) On the *x*-axis we have frequency, or equivalently the inverse of wavelength. Second, the theoretical curve is matched to the data points by adjusting the overall temperature; it is precisely through the CMB that we measure the temperature of the universe. But the *shape* of the distribution is the same for all temperatures, and it is this very definite prediction that the data matches precisely.

So CMB photons look like they’re in thermal equilibrium, but they can’t be in equilibrium today. Photons only affect each other by exchanging energy via other particles, such as electrons or protons. But all of these particles are clustered in galaxies and other bound objects. Empty space is far too dilute for particles to indirectly “pass energy” between photons scattering off them.

The natural explanation is that CMB photons *were* in equilibrium long ago, and have carried their equilibrium distribution of energies till today. (After the expansion of the universe made space too dilute for photons to continue colliding with other particles, they simply free-streamed through space, virtually untouched for billions of years. It’s true that all photons have lost energy due to redshift, but this just changes the overall temperature, leaving the thermal distribution intact.) But what conditions would put photons in equilibrium? Well, equilibrium is maintained through particles interacting and sharing energy, and the rate of interactions is determined by the density of particles and their speed. Faster particles forced into a smaller volume will scatter and share energy with each other more quickly. Photons, of course, travel at the speed of light, so they can’t have been moving faster in the early universe (although they did have more energy). So the only way they could have been in equilibrium would be by occupying a *smaller volume*, along with electrons and other particles via which they could share energy.

How much smaller a volume? It turns out, a little more than a thousand times smaller, with a temperature of over 5000 degrees Fahrenheit, over 13 billion years ago. This is hot, but not quite the hot Big Bang itself. If we rewind the clock even further, using relativity, we reach a moment a few hundreds of thousands of years beforehand where the universe is so hot and dense that atoms cannot even hold together! They would be bombarded apart into protons and neutrons. As the universe expands and cools after this point, atoms and elements start to form. When we apply what we know about atomic and nuclear physics, we’re able to predict how much Deuterium, Helium, and other elements should be created in this primordial epoch. And astronomical observations confirm the expectation, providing strong evidence for a history of the universe that extends back to nuclear temperatures and densities! But that is a story for another blog post.