How can space expand in accelerating speed when dark energy is constant?

Question

I came to know that dark energy is constant.As mentioned in the Friedmann equations

How can a constant energy cause an accelerating rate of expansion in the universe?

Answer 1

This is a very good question and it comes down to what is meant by "accelerating expansion".

In cosmology we define something called the scale factor $a$. By definition at the present time $a=1$, billions of years ago when linear distances on a cosmic scale (e.g. the distance to a faraway galaxy) were half what they are now $a$ was equal to $0.5$ and many years in the future when cosmic linear distances have increased to twice their current size the $a$ will equal $2$.

$a$ is a function of cosmological time and we also define its 1st derivative and 2nd derivatives, with respect to cosmological time, $\dot{a}$ and $\ddot{a}$.

$\dot{a}$ is a measure of the Universe's rate of expansion and so when $\dot{a}$ is increasing with time we say that the expansion of the Universe is accelerating. $\ddot{a}$ is a measure of the change of the rate of the Universe's expansion so equivalently we say if $\ddot{a}$ is positive then the expansion of the Universe is accelerating.

However note we have defined $a$ at the present time to be $1$, but the present time is entirely arbitrary. This means that $a$, $\dot{a}$ and $\ddot{a}$ depend on when we define $a$ to be equal to $1$. To get round this can instead think of the Hubble Parameter $H = \frac{\dot{a}}{a}$ as a better measure of the rate of expansion as it does not depend on when $a=1$.

In a de Sitter Universe (spatially flat, empty, positive cosmological constant) the Friedmann equations reduce to:

$$ \frac{\ddot{a}}{a} = H^2 = \frac{\Lambda c^2}{3}$$

From this we can see that $H$ and $\ddot{a}$ are both positive constants. So in one sense (the sense of $H$) the rate of expansion of the Universe is constant, but in another sense (in the sense of $\dot{a}$) the rate of expansion is increasing. So we can see a link between constant energy desnity (the energy density of the cosmological constant is constant) and a constant rate of expansion as long as we define the rate of expansion in a certain way.

In the currently-favoured cosmological model dark energy is a cosmological constant and we are currently in an era when dark energy dominates the dynamics of the Universe. (Looking at the full Friedmann equations) the contribution of matter reduces $\ddot{a}$, but as the cosmological constant dominates it is still positive and we say that the Universe's acceleration is expanding. Looking at $H$ however we say that, because the energy density of matter decreases with time asymptotically to zero, $H$ is decreasing asymptotically to the value given above for the de Sitter Universe.