# Why does the photon density decrease more than baryon density when the universe expands?

I read that most photons today are from the cosmic background radiation, so the number must have stayed roughly the same since CMB was released. Also, the baryon density has been roughly constant since the first fraction of a second. But why did photons dominate early in the universe, and now they are like 0.001%?

## 2 Answers

The rate at which energy density dilutes is different. Think of the energy of particles as $E^2=m^2c^4 + p^2c^2$.

We will consider how the energy density of those particles dilutes as the universe expands. Let's write $a$ as the characteristic size of the universe. You may think of a typical volume of universe as a cube of side length $a$ (e.g. the total volume of that characteristic amount of universe is $a^3$).

For a heavy, slow moving particle (e.g. a baryon, sometimes also referred to as dust or cold matter), almost all the energy of these particles can be written as $E\approx mc^2$. (There is a little bit of kinetic energy also, but $mc^2>>pc$, so we may ignore the contribution from kinetic energy for this discussion). As the universe expands, the number density of particles dilutes $\propto a^{-3}$ which means the energy density of these particles dilutes as $\rho =\frac{E}{V}\propto a^{-3}$.

Now let's consider what would happen to a fast-moving particle (like a photon, or some other particle where the $pc>>mc^2$ (which includes the case where $m=0$)). As the universe expands, the number density of photons dilutes as $n \propto a^{-3}$. However, the photon frequency decreases with the expansion $\nu\propto a^{-1}$. Because frequency is proportional to energy, the energy of each particle dilutes with the expansion as $E\propto a^{-1}$. Therefore, the energy density of photons in the universe decreases with time as $\rho = \frac{E}{V} \propto a^{-4}$.

To add a few things to Bob's answer:

in those regions of the universe, between galaxy clusters, where matter is not dominant, but dark energy is, in those voids of space, space itself expands.

Now as the photons travel through those regions, they travel in expanding space, so their wavelength will be stretched too.

Thus, their frequencies will decrease, their energy will decrease. Now if you take the number of photons in the universe as constant, and the number of baryons as constant, and take that expanding space does not affect the energy density of baryons, you see that photons' energy density will decrease more then baryons'.

Now as the universe expands, and in this case the expansion accelerates, the voids of inter galaxy cluster space are going to dominate the universe, and after a while even inter galaxy spaces will dominate, so even galaxies will be so far apart, that matter will not have a gravitational effect there and dark energy will dominate in these regions between galaxies.

Now as dark energy will dominate more parts of the universe, photons will more likely travel in these regions, so their energy density will decrease even more as the expansion accelerates.