Question 26.10: Age of the Universe Use the current value of the Hubble cons...
Age of the Universe
Use the current value of the Hubble constant, H_{0}=72 km / s / Mpc, to estimate the age of the universe in years. Assume that the galaxies have always been moving apart with their current speeds.
ORGANIZE AND PLAN In the Big Bang theory, the universe begins with today’s observable universe at a single point. We can use the Hubble relation to find out when that was. If their speeds haven t changed, two galaxies that today are a distance R apart have been moving at speed v=H_{0} R (Hubble relation, Equation 26.4, with H=H_{0} ) since the Big Bang. So they’ve been moving for time .T = R/v Substituting from the Hubble relation gives T=1 / H_{0} for the time since the galaxies were together. Since R doesn’t enter this equation, it holds for any pair of galaxies. Therefore, the entire observed universe would have been at the same point at this time, so T is our estimate for the age of the universe H_{0}. is in mixed units, so we’ll need to convert to all SI.
v = HR (Hubble relation) (26.4).
Known: H_{0}=72 km / s / Mpc , 1 Mpc =3.09 \times 10^{22} m (from text).
T_{\text {universe }}=\frac{1}{H_{0}}=\frac{1 s \cdot Mpc }{72 km } \times \frac{1 km }{1000 m } \times \frac{3.09 \times 10^{22} m }{1 Mpc }.
=4.29 \times 10^{17} s.
\text { With } 1 \operatorname{year}(y)=3.16 \times 10^{7} s \text {, this is }T_{\text {universe }}=4.29 \times 10^{17} s \times \frac{1 y }{3.16 \times 10^{7} s }=1.36 \times 10^{10} y.
or 13.6 billion years.
REFLECT That’s a good estimate, but don’t consider it exact. The precise relation between the Hubble constant and age of the universe depends on how cosmic expansion varies with time.