The Big Bang, Hubble's Law and the Age of the Universe

1Everything is rushing away

In the 1910s Vesto Slipher measured the spectra of spiral "nebulae" and found that most were redshifted: they were moving away from us, some at over 1000 km s⁻¹. In 1927 Georges Lemaître, and in 1929 Edwin Hubble with distances from Cepheids (Lesson 3), showed that the farther a galaxy is, the faster it recedes. Run that film backwards and every galaxy converges: the universe must once have been far smaller, denser and hotter. This lesson weighs the evidence for that conclusion, the Big Bang theory, and uses it to find the age of the universe. It also meets a puzzle that is still open today.

2Redshift

Every element absorbs light at its own set of wavelengths (Lesson 2). In the spectrum of a distant galaxy the same patterns appear, but every line is moved towards the red end by the same fraction (Figure 5.1). The redshift is

$$z=\frac{\mathrm{\Delta }\lambda }{{\lambda }_0}=\frac{{\lambda }_{\text{observed}}-{\lambda }_0}{{\lambda }_0}\text{and for }v\ll c:z\approx \frac{v}{c}z\; =\; \backslash frac\{\backslash Delta\backslash lambda\}\{\backslash lambda\_0\}\; =\; \backslash frac\{\backslash lambda\_\{\backslash text\{observed\}\}\; -\; \backslash lambda\_0\}\{\backslash lambda\_0\}\; \backslash qquad\backslash text\{and\; for\; \}\; v\; \backslash ll\; c:\; \backslash quad\; z\; \backslash approx\; \backslash frac\{v\}\{c\}$$

Figure 5.1: Redshift of the hydrogen lines (schematic).

For nearby galaxies the redshift is just a Doppler shift: the source is moving away. For distant galaxies the better picture is that space itself is expanding and stretching the light on its way, so that $1+z=\frac{\text{size of the universe now}}{\text{size when the light was emitted}}1\; +\; z\; =\; \backslash frac\{\backslash text\{size\; of\; the\; universe\; now\}\}\{\backslash text\{size\; when\; the\; light\; was\; emitted\}\}$. The formula $v=czv\; =\; cz$ then holds only for small $zz$ (say below 0.1).

3Hubble's law

Figure 5.2: A Hubble diagram (illustrative data).

Measured distances (from Cepheids and supernovae) and speeds (from redshifts) lie close to a straight line through the origin: $v=H_{0}dv\; =\; H\_0d$. This is exactly what uniform expansion predicts (Figure 5.3): if every distance grows by the same factor in the same time, then more distant galaxies must move apart faster, in proportion to their distance, and an observer in any galaxy sees the same law. There is no centre: the Big Bang was not an explosion at a point in space, but the expansion of space everywhere.

Figure 5.3: Why uniform expansion gives speed proportional to distance.

4The value of H₀, and a tension

Two kinds of measurement give $H_{0}H\_0$, and they disagree.

The difference, about 5.6 km s⁻¹ Mpc⁻¹, is about five times the combined uncertainty. This is the Hubble tension. Either one method has an error that nobody has yet found, or the standard model of the universe is missing something. Other teams, including some using the James Webb Space Telescope, have found values in between, and the question is still being actively researched.

5The age of the universe

If every galaxy had always moved at its present speed, the time since all were together would be the same for every galaxy: $t=\frac{d}{v}=\frac{d}{H_{0}d}=\frac{1}{H_0}t\; =\; \backslash frac\{d\}\{v\}\; =\; \backslash frac\{d\}\{H\_0d\}\; =\; \backslash frac\{1\}\{H\_0\}$, the Hubble time. In reality the expansion first slowed down, because the gravity of matter pulled against it, and over the last few billion years it has been speeding up, driven by the mysterious dark energy. That acceleration was discovered in 1998 from distant Type Ia supernovae (Nobel Prize in Physics 2011). The full model fitted to the Planck data gives an age of 13.8 billion years (13.787 ± 0.020 Gyr), and a universe made of about 5% ordinary matter, 27% dark matter and 68% dark energy.

6The evidence for the Big Bang

1. The expansion described by Hubble's law.
2. The cosmic microwave background (CMB). In 1964 Arno Penzias and Robert Wilson, testing a large horn antenna at Bell Laboratories, found a faint microwave signal coming equally from every direction. It is the afterglow of the hot early universe. About 380 000 years after the Big Bang the universe had cooled to about 3000 K, cool enough for electrons and nuclei to combine into neutral atoms, and the light then travelled freely. Expansion since then ($z\approx 1100z\; \backslash approx\; 1100$) has stretched it into microwaves: today it is an almost perfect black-body spectrum at 2.725 K, the same in every direction to about one part in ${10}^{5}10^\{5\}$. The tiny ripples in it are the seeds of today's galaxies.
3. The abundances of the light elements. In the first few minutes, the universe was hot enough for fusion. Theory predicts that it ended up about 75% hydrogen and 25% helium by mass, with traces of deuterium and lithium. This is what is found in the oldest stars and in gas clouds that have never been inside stars. Stars alone could not have made so much helium.
4. Galaxies change with distance. Looking far away is looking back in time, and very distant galaxies are smaller, more irregular and forming stars faster: the universe has evolved. A universe that has always looked the same, as the rival "steady state" theory claimed, would not show this, and could not explain the CMB.