The Expansion of the Universe

We cannot infer the distance from our galaxy to another unless we know the history of the expansion of the universe. It takes a finite time for light to reach us, during which the universe has changed. Our observations of wavelength, angular size, and luminosity are geometrically affected by the change in scale in a predictable way.

The metric for a flat ($k=0$) universe is

$$ds^2 = - c^2 dt^2 + dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$ (1)

with conventional units for space and time. The spatial expansion of the universe at a time $t$ in its history will be given by a scale factor $a(t)$ which applies throughout all space (there is no dependence on $r$, $\theta$, or $\phi$). To do this we introduce $\chi$, a dimensionless ``conformal'' coordinate that preserves shapes during the expansion. It is given by

$$dr = a(t) d\chi$$ (2)

The metric with $a(t)$ becomes

$$ds^2 = - c^2 dt^2 + a^2(t)(d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta d\phi^2))$$ (3)

The present time is $t=t_0$ and the present scale factor is $a(t_0)=a_0$. At any other time, $t$, the ratio $a(t)/a_0$ would give the scale of the universe relative to the present epoch. Apart from local motions, two galaxies will maintain the same dimensionless conformal separation for all time, while the actual physical separation will increase or decrease as $a(t)$.