# Age of a dark energy dominated universe

In a flat universe that is dominated by dark energy (or cosmological constant), the Friedmann equation can be written as:

$$H^2 = (\frac{\dot a}{a})^2 = \frac{8\pi G\varepsilon_{\Lambda}}{3c^2}$$

Where $$H$$ is the Hubble parameter, $$a$$ is scale factor, $$\dot a$$ is scale factor's time derivative, $$\varepsilon_\Lambda$$ is the energy density of cosmological constant. Currently, our belief is that $$\varepsilon_\Lambda$$ is a constant, so the right side can be replaced by Hubble constant $$H_0$$

Rearrange Friedmann equation and integrating each side:

$$a(t) = e^{H_0(t-t_0)}$$

$$t_0$$ is the time right now.

From here, my textbook just says that a flat universe containing nothing but a cosmological constant is infinitely old, but I certainly do not see why this is the case, it is because cosmological constant will drive the universe to expand forever?

In non-dark-energy-dominated phases, $$a\propto (t-t_1)^\gamma$$ for some $$\gamma>0$$ and some reference time $$t_1$$. This means that $$a=0$$ when $$t=t_1$$, so that time $$t_1$$ is the beginning of the Universe. (Normally we take $$t_1=0$$.)
$$a(t) = e^{H_0(t-t_0)}$$
which tells you that there is no finite time at which $$a=0$$. Instead, $$a=0$$ is only reached in the limit that $$t\to-\infty$$. That's the sense in which the Universe would be infinitely old (assuming it was always dark energy-dominated).