Specifically, an increase in the expansion rate would be due to dark energy (which isn’t dark, it’s transparent, nor is it energy, but rather energy density). There are a couple of terms that influence the expansion rate of the universe and the specific relationship of each to the expansion rate of the universe is given by the Friedmann equation. There are a lot of ways to write this equation, but one that is helpful to see what each term does is this one:

The left hand side is one over the Hubble constant squared times the term that contains the expansion rate. Specifically a-dot is the first time derivative of this variable ‘a’ (can write this as da/dt). So what is the variable a? It is the scale factor of the universe. That is roughly the distance between points in space where today we call this number = 1 to have some reference point. In the future, this term will be larger than 1 as the universe continues to expand and in the past, this term was less than 1 as the distance between points in space was smaller. So the term a-dot divided by a is how fast is the scale factor changing over time compared to its present value.

Now there are four terms on the right-hand side…

- The first term is the fractional energy density of the curvature of the universe. That sounds like a mouthful, but that term is basically equal to zero since the universe itself has no curvature on average/large scales.
- The second term is the fractional energy density of matter. This term includes regular matter and dark matter (which isn’t dark, it’s actually transparent though it is at least “matter”). While dark matter isn’t made up of any standard model particles, it isn’t so spooky and there are many different observations that point to its existence. However, notice that this term is a constant divided by a^3. If you think about how this term changes over time, notice that it starts off pretty large (when a is very small) and then decreases in importance over time.
- Compare this to the third term which is the fractional energy density of radiation. This includes all the photons every produced in stars plus the cosmic microwave background photons. The latter makes up the bulk of this term (about 99%+ of it), but notice how this term depends on a^4. That means when the universe is very small and young, this term is VERY important but quickly becomes less important faster than the matter term since it is a^4 which decreases faster than a^3.
- Finally there is the final term, the fractional energy density of dark energy. This term is not so spooky either and is a natural term to include in the equations of general relativity. It is related to the energy density of empty space itself. From quantum mechanical considerations, it makes a lot of sense to think that empty space isn’t truly empty, but we presently don’t have any underlying theories to derive this dark energy density term, it’s just a property of space that we measure, similar to how we can’t derive a constant like G, the universal gravitational constant, we just measure it and it is what it is. This term does NOT decrease over time and is just… constant. Interestingly, this means that while the first three terms decay to zero, the right-hand side of this equation eventually equals a constant. This term was not important early in the universe’s history (when the scale factor ‘a’ was much less than 1), but is presently the dominant term (see next figure). Since the right-hand side equals a constant, this implies the left-hand side is also a constant and you get a pretty simple differential equation (relatively speaking) to solve. Basically, as the universe expands, it has to keep expanding faster to keep the left-hand side constant, i.e. an accelerating expansion.