cosmology - Can the proportion of dark energy change?

This post generalizes the posts:
- Proportion of dark matter/energy to other matters/energy at the beginning of the universe?
- How do people calculate proportions of dark matter, dark energy and baryonic matter of the universe?

Can the proportion of dark matter/energy (to other matters/energy) change with time?

If so, is there a measurement of this evolution up to now, and a prediction for future?
How measure such a change?

Answer

The propotion of dark matter and dark energy can definitely change with time. This is because the universe expands. We call, $a(t)$, the scalefactor of the universe, it is proportional to all the distances in the universe (on a large scale). That is, if the universe doubles in size, $a(t)$ doubles in size.

The various density components in the universe change differently as the universe expands.

Regular (non-relativistic) matter density (like cold dark matter) just dilutes with the volume:

$$ \rho_m \propto \frac{1}{a(t)^3}.$$

Relativistic matter (radiation) density dilutes with volume, but also redshifts, this gives:

$$ \rho_r \propto \frac{1}{a(t)^4}.$$

Dark energy (or vacuum energy) density does not dilute, it has a constant density:

$$\rho_{de} \propto \Lambda = \text{constant}.$$

This means that as the universe expands the proportion of dark energy will increase as compared to the other formes of matter.

Since the big bang we have thus first had a radiation dominated period, followed by a period dominated by non-relativistic matter, and now we are in the period dominated by dark energy.

At this time the density of dark enery is roughly 70% of the total energy density, and non-relativistic matter (mostly dark matter) accounts for the last 30%.

The general equation that governs the time evolution of the density components is:

$$ \dot \rho = -3\frac{\dot a}{a}\left(\rho + \frac{p}{c^2}\right).$$

We can solve this equation if we are given a relation between the density and pressure, called the equation of state:

$$ p = w\rho c^2. $$

We get a general solution:

$$\rho = \rho_0 \left(\frac{a_0}{a}\right)^{3(1+w)}.$$