Question

The visible section of the Universe is a sphere centered on the bridge of your nose, with radius $13.7$ billion light-years.

(a) Explain why the visible Universe is getting larger, with its radius increasing by one lightyear in every year.

(b) Find the rate at which the volume of the visible section of the Universe is increasing.

Answer 1

(a) What we can see is limited by the finite age of the Universe and by the finite speed of light. We can see out only to a look-back time equal to a bit less than the age of the Universe. Every year on your birthday the Universe also gets a year older, and light now in transit arrives at Earth from still more distant objects. So the radius of the visible Universe expands at the speed of light, which is
$\frac{dr}{dt}=c=1ly/yr$
(b) The volume of the visible section of the Universe is $\frac{4}{3}\pi r^3$, where $r=13.7$ billion light-years. The rate of volume increase is
$\frac{dV}{dt}=\frac{d}{dt}(\frac{4}{3}\pi r^3)=\frac{4}{3}\pi 3r^2\frac{dr}{dt}=4\pi r^{2}c$
$=4\pi [(13.7\times {10}^{9}ly)(\frac{9.4605\times {10}^{15}m}{1ly}){]}^2(3.00\times {10}^{8}m/s)$
$=6.34\times {10}^{61}{m}^3/s$