(a) Calculate the approximate age of the universe from the average value of the Hubble constant,\({{\rm{H}}_{\rm{0}}}{\rm{ = 20km/s}} \cdot {\rm{Mly}}\). To do this, calculate the time it would take to travel\({\rm{1 Mly}}\)at a constant expansion rate of\({\rm{20 km/s}}\). (b) If deceleration is taken into account, would the actual age of the universe be greater or less than that found here? Explain.

The recession velocity for a galaxy is given by,

\(v = {H_o}d\)

Here\({H_o}\)is the Hubble constant and\(d\)is the distance to the galaxy.

Taking the edge of the universe having \({\rm{10 Gly}}\) away from us, we then obtain:

\(\begin{array}{c}t = \frac{D}{v}\\ = \frac{{{\rm{1}}{{\rm{0}}^{{\rm{10}}}}{\rm{ \times 3 \times 1}}{{\rm{0}}^{\rm{8}}}{\rm{m/s}}}}{{{\rm{20}}\,{\rm{km/s}}}}y\\ = {\rm{1}}{\rm{.5 \times 1}}{{\rm{0}}^{{\rm{14}}}}{\rm{ y}}\end{array}\)

Therefore, the age of the universe is \({\rm{1}}{\rm{.5 \times 1}}{{\rm{0}}^{{\rm{14}}}}{\rm{ y}}\).

As, we haven't factored in deceleration, this time is really \({\rm{1}}{{\rm{0}}^{\rm{4}}}\) times shorter, therefore it's an overestimate.

Therefore, the age of the universe is less.