The Expanding Universe and Relativistic Cosmology
This unit derives the Friedmann equation and uses it to compute how fast the universe expanded at any epoch, how old it is and how far away its galaxies are. Students learn why distance has several meanings at high redshift, what the horizons of the universe are, and how the Hubble constant is measured rung by rung with Cepheids, red giants and supernovae.
1 The cosmological principle and the scale factor
Unit 8 ended with the Hubble-Lemaître law: galaxies recede from us at speeds proportional to their distances, \(v = H_0 d\). This unit builds the theory behind that law and uses it to compute the age of the universe, its expansion history, and the distances to objects whose light has been traveling for most of cosmic time.
The theory rests on one assumption, the cosmological principle: averaged over large enough volumes, the universe is homogeneous (the same at every place) and isotropic (the same in every direction). On small scales this is plainly false. The solar neighborhood is about two million times denser than the cosmic average, and galaxies sit in clusters and filaments around voids tens of megaparsecs wide. Above roughly 100 Mpc the lumps average out, and the cosmic microwave background (Unit 14) has the same temperature in every direction to one part in \(10^5\).
Such a universe has no center and no edge, and it has very little freedom to move. Rotation or shear would single out a direction, so the only motion allowed is a uniform expansion or contraction in which every distance between galaxies changes by the same factor.
1.1 The scale factor and the Hubble parameter
One function of time describes such a motion, the scale factor \(a(t)\). If two galaxies are a distance \(x\) apart today, at time \(t_0\), their distance at any time \(t\) is \begin{equation} r(t) = a(t)\,x, \qquad a(t_0) = 1. \label{eq:scale} \end{equation} The fixed label \(x\) is the comoving distance. The changing length \(r(t)\) is the proper distance, which a chain of rulers would measure if all were read at the same cosmic time. Take the time derivative of Equation \eqref{eq:scale} and write \(\dot{a}\) for \(\dd a/\dd t\): \[ v = \dot{r} = \dot{a}\,x = \frac{\dot{a}}{a}\,r = H(t)\,r, \qquad H(t) \equiv \frac{\dot{a}}{a}. \] The Hubble-Lemaître law is a direct consequence of uniform expansion. It holds for an observer in any galaxy, so the fact that everything recedes from us says nothing special about our position. The Hubble parameter \(H(t)\) changes with time, and its present value is the Hubble constant \(H_0\). Measurements lie between 67 and 73\(\unit{km\,s^{-1}\,Mpc^{-1}}\) (Section 6); this unit uses \(H_0 = 70\unit{km\,s^{-1}\,Mpc^{-1}}\), which is \(2.27\times10^{-18}\unit{s^{-1}}\) in SI units. Two natural scales follow. The Hubble time \(t_H = 1/H_0 = 14.0\) Gyr is the age the universe would have if it had always expanded at today’s speed, and the Hubble distance \(d_H = c/H_0 = 4280\) Mpc is the distance at which \(v = H_0 d\) reaches the speed of light. Atoms, stars and galaxies are held together by electric forces or by their own gravity and do not expand.
1.2 Cosmological redshift
Light that crosses an expanding universe arrives with a longer wavelength than it had at the source. As in Unit 3, the redshift \(z\) is defined by \(1 + z = \lambda_{\mathrm{obs}}/\lambda_{\mathrm{emit}}\).
For \(z \ll 1\) this agrees with the Doppler formula \(z \approx v/c\) of Unit 3. At large \(z\) the redshift is better read as a label for an epoch. JWST spectra place the galaxy GN-z11 at \(z = 10.60\). Its light left when distances between galaxies were 11.6 times smaller than now and the mean density of matter was \(11.6^3 \approx 1560\) times higher, and its ultraviolet Lyman \(\alpha\) line, emitted at 121.6 nm, arrives at 1410 nm in the infrared. The same stretching applies to any time interval at the source: Type Ia supernovae at \(z = 1\) take twice as long to brighten and fade as nearby ones.