# calculation of characteristic expansion time of the universe

Hubble's constant is estimated at 15km/s at a distance of million light years, without considering effects of mutual gravity, relativity, etc. I am trying to calculate the characteristic expansion time of the universe. Its formula is given as reciprocal of the Hubble constant, because time is distance by velocity. But:

$$v=kx$$

This implies that velocity and distance are exponential functions of time and so time cannot simply be calculated by dividing 1 million light years by 15km/s. So what am I missing? Why is the characteristic expansion time simply the reciprocal of Hubble's constant?

• Please edit for spelling. Put capital letters at the start of sentences, names of people and the word "I", and the correct spelling is "because" not "cuz" Apr 23 '18 at 20:13
• 15 km/s/Mlyr is somewhat off — 21 km/s/Mlyr is more correct. If you divide 1 Mlyr with 21 km/s, you get roughly 14.3 Gyr, which is pretty close to the 13.8 Gyr inferred from Planck measurements.
– pela
Apr 24 '18 at 18:37

If galaxies are currently moving away from each other, then it implies they were closer together in the past. Consider a pair of galaxies currently separated by a distance $r$, with a velocity $v = H_0r$ relative to each other. If there are no forces acting to accelerate or decelerate their relative motion, then their velocity is constant, adn the time that has elapsed since they were in contact is
$t_0 = \frac{r}{v} = \frac{r}{H_0r} = H_0^{-1}$,
independent of their current separation $r$.