The shift that is not a Doppler shift

Take a spectrum of a galaxy a hundred megaparsecs away and every absorption line in it sits at a longer wavelength than the same line does in a laboratory. Take a spectrum of one twice as far and the displacement is twice as large. The relation is linear, it holds in every direction that has been looked in, and it has been checked over four decades of distance.

The obvious reading is that the galaxies are moving away, and that reading has a problem which is not subtle: if everything is receding from here, and the recession is faster the further away it is, then this is the one place in the universe where nothing is running away from anything. That is a conclusion so unlikely that it is worth asking what else could produce the same measurement.

What a Doppler shift is, and what this is not #

A Doppler shift is a statement about relative motion at the moment of emission. A source approaching along the line of sight emits crests that are bunched together, one receding emits them spread apart, and the fractional change in wavelength is the ratio of the line-of-sight speed to the speed of light. The shift is fixed the instant the light leaves. Nothing that happens afterwards changes it. The cosmological redshift is a different quantity that produces a similar-looking spectrum. It is not fixed at emission and it is not a property of the source’s motion. It accumulates over the journey, and what it records is the ratio of the size of the universe now to its size when the light left:

$$1 + z = \frac{a(t_{\rm obs})}{a(t_{\rm emit})}.$$

The scale factor $a$ is a single number that describes how far apart things are at a given time, with $a=1$ chosen to mean now. A wavelength travelling through an expanding universe is stretched in the same proportion as everything else, so a photon emitted when the universe was half its present size arrives with twice its emitted wavelength, and $z = 1$. The redshift is a measurement of a ratio of epochs, not of a speed.

One immediate consequence is that the stretching applies to a thermal spectrum as much as to a line, and applies to it in a way that leaves the spectrum thermal.

The distinction is not pedantry, and there is a clean test that separates the two. A Doppler shift depends on the velocity at emission and nothing else. A cosmological redshift depends on the expansion history over the whole light path — so two galaxies at the same present distance, in universes with different expansion histories, show different redshifts. That difference is measurable, and it is what the supernova cosmology of the 1990s measured.

Why the law comes out linear #

The velocity–distance relation follows from the assumption of uniformity and from nothing else, and the derivation is two lines. Let every object sit at a fixed comoving coordinate $\chi$, so its physical separation from any chosen origin is $d = a(t)\chi$. Differentiate:

$$\dot d = \dot a \chi = \frac{\dot a}{a}\,(a\chi) = H(t)\, d,$$

with $H \equiv \dot a / a$. The recession rate is proportional to distance because the derivative of a product with a fixed factor is proportional to that factor. There is no dynamics in this at all — it is true of any uniform expansion whatever, of a rising loaf of bread as much as of a universe, and it holds from every point because $\chi$ was measured from an arbitrary origin.

That last clause is the argument the hero figure makes with pictures. The linearity is not a coincidence to be explained; it is the only velocity field consistent with an expansion that looks the same everywhere. Any other function of distance would single out a point, because a non-linear law measured from one origin becomes a different law measured from another.

Peculiar velocities are worth dwelling on, because they are the one part of a galaxy’s redshift that really is a Doppler shift. A galaxy in a cluster is orbiting in the cluster’s potential at several hundred kilometres per second — the same dispersion that weighs the cluster — and that motion adds an ordinary Doppler term on top of the cosmological one. For the Andromeda galaxy the peculiar term wins outright: it is approaching at 110 km/s and its spectrum is blueshifted. Nothing about the expansion prevents that. Expansion is a statement about the average, and a bound system has already decoupled from it — the Local Group is not expanding, the Milky Way is not expanding, and neither is a metre rule.

What was actually measured #

The linear law is a relation between two quantities, and only one of them is easy.

The redshift is easy, and unusually so. It is a ratio of wavelengths, measured on a pattern of lines rather than on any single one, and it needs no calibration beyond a laboratory comparison spectrum. Modern surveys measure redshifts for millions of galaxies to a precision of a few parts in $10^5$, and the dominant error is not in the instrument but in deciding which lines are which.

The distance is the whole difficulty, and it is the distance ladder that supplies it — a chain in which each rung is calibrated against the one below, so that the fractional errors accumulate all the way up. Its bottom rung is a parallax, which is the only geometric measurement in the chain, and the rung that reaches a galaxy far enough away for the Hubble flow to dominate is a variable star whose period gives its luminosity. So the honest statement of what the observation delivers is: a redshift measured to five digits, plotted against a distance whose uncertainty is a few per cent at best and dominated by calibration rather than by noise. The scatter on a Hubble diagram is almost entirely horizontal. That asymmetry shapes everything about the subject, including which of the two determinations of the expansion rate the field trusts, and it is why so much effort has gone into distance indicators that skip rungs.

There is one further thing the observation does not deliver, and it needs saying because the axis label hides it. A “recession velocity” quoted for a distant galaxy is not measured. What is measured is $z$; the velocity is $cz$ only in the limit of small $z$, and at $z=1$ the naive conversion gives $c$ exactly, which is a statement about the arithmetic rather than about the galaxy.

Recession faster than light #

The linear law extrapolates without limit, so at a distance of about 14 billion light years the recession rate reaches $c$, and beyond that it exceeds it. This is not a breakdown of the theory and nothing is being violated — but it is worth being precise about why not, because the usual reassurance is the wrong one.

The rule that nothing travels faster than light is a statement about local motion: no object passes another at more than $c$. Recession velocity is not a local quantity. It is the rate of change of a distance between two points that are nowhere near each other, computed by adding up a great many local separations, and there is no theorem bounding that sum. Two galaxies each moving slowly with respect to their own surroundings can have any recession velocity whatever, given enough surroundings between them. The consequence a reader is most likely to have met is that galaxies at redshifts above about 1.5 are receding faster than light, and are nevertheless observed. Both halves are true and neither is remarkable once the recession velocity is understood as an accounting quantity rather than a speed.

The test that expansion has to pass #

A redshift on its own is compatible with more than one explanation, and the alternatives were taken seriously for decades. What distinguishes expansion from a photon simply losing energy on its way here is that expansion does other things as well, and one of them is measurable.

Consider the surface brightness of a galaxy — its flux per unit solid angle. In a static universe that quantity is independent of distance: a galaxy twice as far away delivers a quarter of the flux over a quarter of the solid angle, and the ratio does not change. That is why a nebula does not fade as it recedes and why surface brightness is not a distance indicator.

In an expanding universe it does change, and by a large factor. Each photon arrives with its energy reduced by $(1+z)$; photons arrive less often, by another factor of $(1+z)$; and the solid angle is enlarged relative to the static case by two further factors, from the way angular diameter distance and luminosity distance differ. The result is that surface brightness falls as

$$\Sigma \propto (1+z)^{-4},$$

which at $z = 1$ is a factor of sixteen and at $z = 2$ a factor of eighty-one.

A tired-light model predicts $(1+z)^{-1}$ — one factor, from the energy loss alone, with no time dilation and no geometric enlargement. The two predictions differ by three powers, and the comparison is the Tolman test.

It has been done, and it favours expansion decisively. The difficulty is that galaxies evolve: a galaxy at $z = 1$ is younger and brighter intrinsically than one nearby, so the raw comparison mixes the geometry with the population. Using elliptical galaxies, whose evolution is comparatively simple, and correcting for it with stellar population models, the measured exponent comes out near four rather than near one.

The same expansion factor shows up in a cleaner place. A distant supernova’s light curve is stretched in time by exactly $(1+z)$ — a fifteen-day rise becomes thirty at $z = 1$ — and that has been measured directly, on dozens of objects, with no population modelling required at all.

The two tests are worth keeping together because they bracket the argument. One measures a geometric consequence of expansion and needs a model of how galaxies evolve; the other measures a kinematic consequence and needs nothing but a clock. Between them they leave a static interpretation of the redshift with nowhere to stand.

What the picture cannot show #

The hero figure draws a lattice with an edge, and the universe does not have one. The lattice is a window onto something that continues, and the whole argument depends on that: if it had a boundary, the boundary would be a preferred place, and the law would stop looking the same from every point. Nothing in the observation says the universe is infinite; what it says is that the region observed shows no sign of an edge or a centre, which is a much weaker claim and the only one available from inside.

Nothing in this essay establishes that the expansion is of space rather than through it. The two descriptions — galaxies receding through a static space, and a scale factor multiplying a fixed grid — agree on every observable in the local neighbourhood, and the phrase “space itself expands” carries a suggestion of substance that the mathematics does not. The scale factor is a coordinate convention that makes the equations simple. What is physical is the ratio of separations at two times, and that is what $1+z$ measures.

And a single redshift dates nothing. Converting $z$ into a time, a distance, or an age requires the whole expansion history, which means it requires a model with its parameters already fixed by other measurements. A quasar at $z = 6$ is at “12.9 billion light years” only in the ΛCDM cosmology; in a matter-only universe with the same $H_0$ the same redshift corresponds to a different age, a different distance and a different everything. The redshift is the datum. The rest is inference, and this field’s standing obligation is to keep saying which is which.

How it was worked out #

Vesto Slipher measured the first spiral-nebula velocities at Lowell Observatory from 1912, on exposures running to tens of hours, and by 1917 had twenty-five: twenty-one of them receding, four approaching, and speeds far larger than anything then known for a star. He had no distances and made no cosmological claim. What he had was the observation that the pattern was overwhelmingly one-sided, which is odd for random motion and was recognised as odd.

The distances came from Edwin Hubble’s Cepheids, and the 1929 paper that combined them used twenty-four galaxies and got a slope of about 500 km/s/Mpc — seven times the modern value, because the period–luminosity relation he calibrated against had not yet been split into two populations and his distances were correspondingly too small. The linear relation survived the correction; the constant did not.

Georges Lemaître had derived the relation from general relativity two years earlier, in 1927, and had estimated the coefficient from Slipher’s velocities and Hubble’s own earlier distances. The paper was in French, in an obscure journal, and the paragraph containing the estimate was absent from the 1931 English translation — omitted, it now appears, by Lemaître himself, who thought the numbers superseded. The relation is called the Hubble–Lemaître law by the International Astronomical Union since 2018 for that reason.

What is worth taking from the history is not the priority dispute but the order of events. The theory produced the linear law before the observation confirmed it, and the observation that confirmed it was wrong by a factor of seven in its constant while being right in its form. The form was the discovery. The constant took another seventy years.

The generalisation #

The argument in the hero figure is not about astronomy. It says that a quantity which grows in proportion to separation, measured from any origin, is the signature of a similarity transformation — and that such a field is the unique one with no preferred point.

The same reasoning appears whenever a system is examined for a centre. A rotation curve does the reverse and finds one: a galaxy’s velocity field is not linear in radius, so a galaxy does have a centre, and the shape of the departure from linearity is what weighs it. The barycentre of a binary is found the same way, by noticing that both stars’ displacements are proportional to their distances from one point and to no other. In each case the question is whether a velocity field can be written as a scaling about some point, and the answer distinguishes a bound system from an unbound one.

The surprising direction is the reverse one. Because the linear law is a kinematic identity rather than a dynamical result, it contains no information about the contents of the universe at all. Every cosmology in the family — empty, matter-filled, closed, accelerating — obeys $v = Hd$ exactly at every instant. Measuring the slope measures the expansion rate now and says nothing whatever about what is causing it or what it will do next. Getting at that requires measuring the departures from linearity at large redshift, which is a far harder observation and is the subject of the next several essays.

Where the ladder goes next #

The immediate question is what the constant of proportionality means, given that it has the units of inverse time and can be read as an age, a length or a density. The next essay takes that up, and finds that all three readings are wrong by amounts that themselves carry the physics.

Later rungs on this anchor: the departures from linearity, and how a second-order term becomes a measurement of acceleration; the relation between redshift and the scale factor when the expansion is not monotonic; peculiar velocities as a signal rather than a nuisance, and the bulk flows measured in them; whether the expansion can be detected inside a bound system, and why the answer is effectively no; the cosmological principle as an assumption and as a measurement; and the distinction between an expanding universe and a static one with tired light, which is a genuinely testable alternative and one that the surface brightness of distant galaxies rules out.