Question

Imagine that you live in a geometrically flat universe which is dominated by radiation. Hubble's constant in this universe is measured to be 70km s−1 Mpc−1 . A Mpc (Mega-parsec) is a million parsecs, and a parsec is 3.086×1016m .

How old is this universe (in billions of years)?

Answer 1

The age of this (and our) universe can be found from the following simple equation:

$$t=\frac{1}{H_0}.$$

The units of Hubble constant are km/(s⋅Mpc).

1) Convert km to m: 1 km/(s⋅Mpc) is 1000 m/(s⋅Mpc).

2) Now convert Mpc^-1 to pc^-1: 1000 m/(s⋅Mpc)=1000/1000000=0.001 m/(s⋅pc).

3) 1000 m/(s⋅pc) is, therefore, equivalent to 0.001/(3.086×10¹⁶) = 3.240×10^-20 s^-1.

Thus, the age of the universe is

$$t=\frac{1}{70\cdot3.240\cdot10^{-20}}=4.41\cdot10^{17}\text{ s}.$$

This is

$$4.41\cdot10^{17}/3600/24/365.22={13.976\cdot10^{9}}^\text{ y}.$$

The age is about 14 billion years.