Physics · Gravitation and the universe · Year 1

The expanding universe

Why are galaxies moving away from us, and how old is the Universe?

The situation

In 1929 Edwin Hubble found that the redshift of the lines in the spectra of galaxies (measured by V. Slipher) is proportional to distance — more distant galaxies recede faster. In 1965 A. Penzias and R. Wilson detected with a horn antenna a faint radio noise coming from every direction of the sky — radiation with a temperature of only 2.7 K. Both discoveries lead to one conclusion: the universe is expanding and was once very hot and dense. When a raisin loaf rises in the oven, each raisin also moves away from the others. When did the universe “begin”, and what holds galaxies together?

Topic

Cosmology: the expanding universe. Hubble's law, the cosmic microwave background and the rotation of galaxies — dark matter

Key ideas

Hubble's law
Cosmic microwave background
Dark matter

Formulas

v = H_{0} d, t_{H} = \frac{1}{H_{0}} \approx 14 billion years (H_{0} = 70 km/(s·Mpc))

Hubble's law: the speed of recession is proportional to distance. $1 / H_{0}$ is the Hubble time, an estimate of the age of the universe.

H^{2} = H_{0}^{2} (\frac{Ω_{r}}{a^{4}} + \frac{Ω_{m}}{a^{3}} + Ω_{Λ}), a = \frac{1}{1 + z}

The Friedmann equation (flat universe): radiation, matter and dark energy determine the rate of expansion.

T = T_{0} (1 + z), λ_{m a x} = \frac{b}{T} = \frac{2.898 mm·K}{2.7255 K} = 1.06 mm

The relic radiation cools as the universe expands; Wien's law gives its maximum.

v (r) = \sqrt{\frac{G M (r)}{r}} \Rightarrow v = const \Rightarrow M (r) = \frac{v^{2} r}{G} \propto r

Flat rotation curve: even beyond where the visible stars end, the mass keeps increasing — dark matter.

f = \frac{f_{0}}{1 + v / c}, f_{0} = 1420.406 MHz (λ = 21 sm)

The 21 cm line of neutral hydrogen: the speed of the gas is found from the Doppler shift.

Theory

Hubble's law and expansion

If a loaf in the oven doubles in size in 60 minutes, then from any raisin the others recede at the speed $v = H d$ — whichever raisin you choose, the graph is the same. So the expansion has no centre. The raisins themselves do not grow: galaxies, stars and atoms are also bound by gravitational and electric forces and do not expand. The cosmological redshift $z$ is the stretching of a light wave as space stretches: $1 + z = 1 / a$ . For nearby galaxies $v \approx c z$ . $1 / H_{0}$ ≈ 14 billion years — this would be the age of the universe if the expansion had always been at the same rate.

The Big Bang and the first minutes

Going back in time, the universe becomes hotter and denser. Until about 10 µs it was a quark–gluon plasma; then quarks combined into protons and neutrons. There was slightly more matter than antimatter (the baryon asymmetry: about one extra particle per 10⁹ pairs) — everything else annihilated and turned into photons. At about 3 minutes (T ≈ 10⁹ K) came primordial nucleosynthesis: almost all neutrons combined into helium-4. At that time n/p ≈ 1/7, so the helium mass fraction is $Y = 2 (n / p) / (1 + n / p) \approx 0.25$ — about this value (≈ 0.245) is measured in gas clouds poorest in heavy elements.

Cosmic microwave background

Until z ≈ 1100 (≈ 380 thousand years) the universe was a hot plasma: photons scattered off free electrons and could not travel far. At T ≈ 3000 K electrons combined with nuclei to form atoms (recombination) — the universe became transparent. Since then the wavelength of the photons has stretched ≈ 1100 times: today they are blackbody radiation at 2.7255 K, with a maximum at 1.06 mm. This radiation is the same over the whole sky, the differences being only ≈ 1/100 000 — it was exactly these density “patches” that grew through gravity and formed galaxies and their clusters (the theory of galaxy formation).

The Milky Way and galaxies

Our Galaxy is a barred spiral: in the centre there is a bulge (old, yellowish stars), a thin disc and spiral arms (young, bluish stars, gas and dust, open clusters), and around it a spherical halo with about 150 old globular clusters. The Sun is ≈ 8 kpc (26 000 light-years) from the centre and orbits at ≈ 230 km/s; one revolution takes ≈ 230 million years. Hubble classes: elliptical (E), spiral (S), barred spiral (SB), irregular (Irr). The Milky Way, Andromeda and the Triangulum Galaxy, together with several dozen dwarf galaxies, make up the Local Group. Galaxies gather into clusters (Virgo, Coma). In the nuclei of some galaxies a supermassive black hole swallows gas and shines brighter than the whole galaxy — active nuclei, quasars (for example 3C 273, z = 0.158) and radio galaxies.

The interstellar medium and the 21 cm line

The space between the stars is not empty: it holds tenuous gas (mostly hydrogen), dust and partly ionised cosmic plasma. In an atom of neutral hydrogen (HI), when the electron and proton spins flip from parallel to antiparallel, a wave of 21 cm (1420.4 MHz) is emitted. This is a forbidden transition: for a single atom it happens on average once in ≈ 11 million years, but there is so much hydrogen that the signal is strong and dust does not block it. Gas around hot young stars is ionised — HII regions (pink from Hα light). In dense cold molecular clouds (H₂, CO) new stars are born; in them OH and H₂O molecules emit strong radio waves as cosmic masers. Forbidden lines such as [O III] in nebulae show that the gas is extremely tenuous.

Dark matter, dark energy and the model

In 1933 F. Zwicky found that galaxies in the Coma cluster move at speeds that the visible mass cannot hold; in the 1970s V. Rubin and others measured the flat rotation curves of spiral galaxies. Mass also bends light — gravitational lensing: the images of distant galaxies around a cluster turn into arcs and rings, which makes it possible to “weigh” dark matter directly. In 1998 type Ia supernovae showed that the expansion is accelerating — dark energy. Today's composition: ≈ 5 % ordinary matter, ≈ 27 % dark matter, ≈ 68 % dark energy. Model: the dough rises exponentially; the sizes of the galaxies are enlarged; the M31 model is approximate (bulge + disc in a spherical approximation + isothermal halo); the relic map is schematic, not real data; the temperature before e⁺e⁻ annihilation is approximate. Stellar evolution, white dwarfs and neutron stars are in the “Physics of stars” experiment, black holes in the “Black holes” experiment.

Did you know?

Penzias and Wilson at first thought that the persistent noise in the antenna came from the droppings of pigeons that had nested in it. The noise remained even after cleaning — it was the relic radiation; they received the Nobel Prize in 1978.
The Andromeda Galaxy (M31) is one of the most distant objects visible to the naked eye: its light takes about 2.5 million years to reach us.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Special relativity — Grade 10
Law of universal gravitation — Grade 10
Relativistic dynamics — Grade 10
Kepler's laws — Grade 10
Physics of stars — Year 1
Black holes — Year 4