# Is the inverse of the Hubble constant always approximately the age of the Universe?

So I know from class that $\frac{1}{H_0}$ is an approximation for the age of the Universe, where $H_0$ is Hubble's Constant. Now let's say that a large amount of time passes - this means that Hubble's constant must fall in order to consistently remain an approximation for the age of the Universe.

However, I also learnt that the universe is expanding at an increasing rate (the expansion of the universe is accelerating). Since the recessional velocity is given by $v = H_0 d$, I would expect that in order for this equation to fit with the idea that the universe is expanding at an increasing rate, $H_0$ would actually increase over time in order to yield a greater recessional velocity and hence show that the expansion of the universe occurs at an increasing rate.

So why is it that $H_0$ falls over time?

• Write down the full expression for $\dot{v}$. What condition must H satisfy for $\dot{v} > 0$? Feb 8, 2018 at 22:14
• Close to physics.stackexchange.com/questions/436985/… and others. Mar 14, 2019 at 7:22
• It's a coincidence. We are living in a very special cosmic era, with Lady Gaga nominated for the Academy Award for Best Actress, silicon-based live forms beating carbon-based ones at go and chess, and $t_0 =1/H_0$ Mar 14, 2019 at 13:46

The approximation of the age of the Universe as $t_H=1/H_0$ is valid only as long as the expansion history of the Universe is approximately linear. The black line in the figure below shows the expansion history so far (and $2\,{\rm Gyr}$ into the future) for our Universe, assuming the WMAP7 cosmology. The dotted lines mark today (which I've assigned $t=0$). Approximating the age as $t_H$ corresponds to taking the tangent to the curve today and extrapolating back to $a=0$, which I've shown with the red dashed line. As you can see it's pretty close, because the history is pretty close to linear.
If the expansion history is strongly non-linear, extrapolating the tangent backward can be a very poor estimate of the age. For instance, in the same cosmology as above, look what happens in a few tens of $\rm Gyr$. At $t=50$ (an age of about $63\,{\rm Gyr}$), $t_H$ as the age would be an error by about a factor of $2$!