The Doppler effect and Hubble's law

Sound first

You already know the Doppler effect by ear. The pitch of a siren drops the moment it passes you. While the source approaches, each wavefront is emitted a little closer to you than the one before, so the fronts arrive bunched together and the frequency you hear is raised. Once it recedes they arrive stretched apart, and the pitch falls. For a source moving at speed v_s through sound travelling at v, the observed frequency is

for approach, and a receding source flips the sign to v + v_s so the pitch drops. Nothing about the source itself has changed. The motion sets the size of the shift, so the shift measures the motion, and that single idea carries the rest of this lesson from ambulances to the edge of the observable universe.

Motion, measured from light

Light shows the same effect, with no medium needed. A source moving away stretches every wavelength it emits towards the red; a source approaching squeezes them towards the blue. For speeds much less than c, the size of the fractional shift equals the speed as a fraction of light's:

Those two are useless until you say what Δ means, so fix it now. Take Δf as the observed frequency minus the rest frequency, and Δλ as the observed wavelength minus the rest wavelength. A source receding at speed v then gives Δλ/λ = +v/c and Δf/f = −v/c, because receding light arrives stretched and therefore lower in frequency. Approach reverses both signs. The data booklet prints Δf/f = −Δλ/λ = v/c, which agrees with all of that provided its v counts as positive for a source coming towards you.

Safer still, work with magnitudes and settle the direction physically. Away means red, towards means blue, and the size of the shift gives the speed. The quantity z is the red shift, positive for recession by construction. Both equations hold only while v is far below c, so treat anything past z of about 0.1 as an estimate and say so.

The measurement is possible at all because spectra carry absorption lines at exactly known laboratory wavelengths. The whole line pattern slides together as one block, and that shift is what is measured.

Binary stars: the wobble in the lines

Many stars come in orbiting pairs too close for any telescope to split. The spectrum reveals them. Watch a binary in the plane of its orbit and each star alternately approaches and recedes along the line of sight, so its spectral lines swing blue, then red, then blue again, once per orbit. The result is a wavelength that oscillates about the rest value.

Two readings follow directly. The period of the oscillation is the orbital period of the pair, and the maximum shift gives the orbital speed through z = Δλ/λ. Tilt the orbit out of the line of sight and the measured shifts shrink, so the clean textbook case specifies the plane of the orbit.

Hubble's law, and what it implies

Apply the red shift measurement to galaxies and a clear pattern appears. Apart from a handful of close neighbours, every galaxy is receding, and the further away it is, the faster it goes. Speed is proportional to distance:

with H the Hubble constant, taken throughout as 65 km s⁻¹ Mpc⁻¹, the value the AQA booklet adopts for calculations; published measurements sit nearer 70. On the adopted value, every megaparsec of extra distance adds 65 kilometres per second of recession. The law reads naturally as evidence that the universe is expanding, and here the shrapnel picture needs dismantling carefully. The galaxies are not flying through space away from one privileged spot. Space-time itself is stretching, carrying the galaxies apart, and an observer in any galaxy sees the same law with themselves at the apparent centre. There is no shrapnel and no crater.

That last claim has a name, and OCR asks for it by name while the other boards only lean on it. The Cosmological principle is the assumption that, viewed on a large enough scale, the universe is homogeneous, of the same composition and density everywhere, so no place is special; isotropic, looking the same in every direction, so no direction is special; and that the laws of physics are universal, the same here as anywhere else. Learn those three together.

Two clauses in that are worth pressing on. The scale clause is the fine print, since close up the universe is plainly lumpy, all stars and voids and empty gaps, and the principle only claims smoothness once you average over hundreds of megaparsecs. The third part is the one students skip and the one every measurement in this lesson depends on, because reading a galaxy's red shift means trusting that hydrogen ten billion light years away absorbs at the same wavelengths as hydrogen in a laboratory here.

What the expansion will do next depends on what the universe is made of, and most of that mass and energy is not directly observed. Galaxies rotate as though they hold far more mass than shines, and the unseen extra is dark matter. The expansion, measured through distant supernovae, is also accelerating where gravity alone would demand a slowdown, driven by something labelled dark energy.

Between them they determine the universe's fate, but only within a model. Feed a cosmological model enough matter and little dark energy and it predicts an expansion that slows and one day reverses. Feed it the mix the current measurements favour, with dark energy dominant, and the standard model predicts an expansion that runs away, ending cold, dark and dilute. These are outputs of a model fitted to data, not things anyone has watched happen, and an answer that says so is the stronger answer.

Run the expansion backwards and everything was once together, in the Big Bang. Read that carefully: it was not an explosion at some address in space. Run the model back and it reaches an early state of space-time that was extremely hot and dense everywhere at once, and the expansion measured today is that same space-time still stretching; what, if anything, preceded that state lies beyond what present physics establishes.

Two further observations support the Big Bang, and both are wanted by name. The cosmic microwave background is a faint glow arriving from every direction, a black-body spectrum at 2.7 K, the cooled and red-shifted afterglow of the hot early universe. Feed 2.7 K into Wien's law and the peak lands near one millimetre. Second, the universe's relative abundance of hydrogen and helium, about three to one by mass, matches what fusion in the first few minutes of a hot dense universe would cook up. No ordinary stellar history accounts for that much helium.

What happened next, an OCR section

OCR asks for the evolution of the universe from the Big Bang to the present. No other board sets it, so read this section only if OCR A is your specification. It is one story with a handful of dated chapters, and you have already met two of them: the hydrogen to helium ratio and the microwave background are both events in this chronology rather than loose facts about it.

One process drives all of it. The universe expands, expansion cools it, and each stage below happens at the moment the temperature falls far enough to permit it. Learn the chapters in order and the reasons come with them.

The first two rows are quick. For a fraction of the first second nothing composite can hold together, and matter is a soup of quarks, leptons and photons; by about a microsecond the quarks have bound into protons and neutrons.

The third row sets a number that has survived ever since. Primordial nucleosynthesis runs from roughly one second to twenty minutes, fusing those protons and neutrons into helium nuclei. That window is short because the expansion is cooling the universe out of the fusion range, and what it leaves behind is the three to one hydrogen to helium ratio by mass quoted above. Stars have processed only a small share of the universe's matter since, which is why the ratio is still close to what those twenty minutes set.

The universe is nonetheless still opaque, because free electrons scatter photons before they can cross. At about 380 000 years the temperature has fallen to some 3000 K, cool enough for nuclei to hold on to electrons. Neutral atoms form, the scattering stops, and light travels freely for the first time. That released light is the microwave background, and the sum ties the two facts together: expansion has stretched its wavelengths by a factor of about 1100 since, and 3000 K divided by 1100 is the 2.7 K measured today.

A long dark age follows. Gravity works on the slight density variations printed on the young universe, pulling matter into clumps that collapse and heat until the first stars ignite, a few hundred million years in. Those stars fuse elements heavier than helium and scatter them when they die, so each generation forms from richer material, and clumps merge into galaxies. The Sun is a late arrival, forming from that enriched debris about 9 billion years in, which is why the Earth has iron in it at all.

For most of its history the expansion was slowing, gravity pulling back on it. The type Ia supernova measurements of the earlier lesson show that over the last few billion years it has instead been accelerating, with dark energy now the dominant term in the budget. The present age is 13.8 billion years, comfortably near the crude 1/H estimate above, and the chronology closes on the dark matter and dark energy of the section before this one.