Calculating the age of the empty universe from the Hubble's constant

This is from p.120 of Ta-Pei Cheng's Relativity, Gravitation and Cosmology. I cannot understand the way the book calculated the age of the empty universe. The velocity $v$ of expansion is proportional to the distance $d$. That means the expansion velocity has not been constant. But why does the book conclude the age of universe from simply dividing the distance $d$ from the velocity of the current moment? Is there something that I am missing? I need help.

• It is a limit estimation Jan 13 '18 at 10:53
• What does that mean? Jan 13 '18 at 14:27
• That it is just an estimate that indeed does not fit to observations. They clearly state that in an empty universe H is supposed to be constant in time Jan 13 '18 at 14:29

An empty universe never changes. Also, its value of $$H_0$$ never changes, and using the Friedmann equation

https://en.wikipedia.org/wiki/Friedmann_equations#Detailed_derivation

$$H(t) = H_0 = 1.$$

Therefore

$$(1/a) (da/dt) = 1,$$ and $$da/dt = a.$$

Therefore

$$dt = da/a$$, and $$t = ln(a) + const.$$

Therefore

$$a = const$$ x $$e^t.$$

Therefore the age is not determinable, since there is no way to choose one value for the constant in preference to another.