A redshift that changes while it is watched

A cosmological redshift is not a velocity. It is the ratio of the size of the universe when light arrives to its size when the light left, $1 + z = a_{\rm obs}/a_{\rm emit}$, and it is fixed by the two moments rather than by anything the source was doing. That definition has a consequence which sounds like a quibble and is in fact a measurement. The moment of arrival is not fixed. It is now, and now keeps moving.

Watch the same galaxy for ten years and the light arriving at the end left it a little later than the light arriving at the start. Both the numerator and the denominator of the ratio have changed. Whether the redshift has gone up or down depends on how fast the universe was expanding when the galaxy emitted, against how fast it is expanding today, and that comparison is exactly the question the discovery of the acceleration raised. A redshift drifts, and the sign of its drift is a direct reading of whether the expansion has accelerated since the light set out.

An idea older than the acceleration #

The effect was pointed out long before anyone needed it. In 1962 Allan Sandage showed that the redshifts of galaxies must change over time in a way that depends on the expansion history, and estimated that the change would be undetectable for the foreseeable future — a decade’s drift was far below anything a photographic spectrum could resolve. The idea waited for thirty-six years. In 1998 Abraham Loeb revisited it with the instruments and the sources that had appeared in between: spectrographs stable to metres per second and the dense forests of absorption lines in the spectra of distant quasars. The same year, and independently, the supernova teams announced that the expansion was accelerating, which turned a curiosity about the sign of a tiny number into the most direct test of a result that otherwise rested on how bright an exploding star is.

The name that has stuck, the Sandage–Loeb test, records both halves of that history. What it describes is a measurement nobody has yet made, and it is worth being clear about why it is still a proposal after six decades: not because the physics is uncertain, but because the signal is as small as the arithmetic says it is.

Where the formula comes from #

The derivation is short enough to give whole. Light leaving a galaxy at time $t_e$ and arriving at $t_0$ has redshift $1 + z = a(t_0)/a(t_e)$. A little later, at $t_0 + \Delta t_0$, the light arriving left at $t_e + \Delta t_e$, and because the galaxy’s comoving distance has not changed the two intervals are related by the expansion itself: $\Delta t_e = \Delta t_0 / (1 + z)$. A clock at the source runs slow by exactly the redshift, as seen from here. Differentiating the ratio then gives

$$\frac{dz}{dt_0} = (1+z)\,H_0 - H(z),$$

where $H(z)$ is the expansion rate at the moment of emission. It is more naturally quoted as the change in the apparent velocity, $c\,\dot z/(1+z)$.

The formula compares two numbers. $(1+z)H_0$ is what the expansion rate at the source’s epoch would have been if it had been exactly as fast, relative to the size of the universe, as it is now — the rate an empty, coasting universe would have had. $H(z)$ is what the rate actually was. If the universe has been accelerating since the light left, the rate then was lower than the coasting value, the difference is positive, and the redshift climbs. If it has only decelerated, the rate then was higher, and the redshift falls.

The three curves in the figure are the three possibilities. An empty universe coasts, so its drift is identically zero — a check the calculation makes at every redshift. A universe of matter alone has decelerated since the beginning, and every redshift in it is falling. The universe as measured has done both: it decelerated for most of its history and has accelerated for the last few billion years, so nearby galaxies, whose light left during the acceleration, drift upward, and distant ones, whose light left during the deceleration, drift down. The crossing, at redshift 1.91, is the redshift at which the two effects cancel over the whole of the light’s journey.

The crossing is the measurement #

The value of that crossing depends on what the universe contains, and it depends on it sensitively.

A shift of a few hundredths in the matter density moves the sign change by nearly a unit of redshift, and doubles or halves the largest drift. That is a steep lever, and it pulls in a different direction from every other measurement of the same parameters. The supernova measurement and every other distance-based method infer the expansion history from how bright or how large something appears, so they depend on the source being what it is assumed to be and on the chain of calibrations that turns a flux into a distance. The drift uses no distance at all. It needs only a spectral line, measured twice, and the fact that the source stayed where it was.

It is also the most direct answer to the question of whether the dark energy is a constant: a change in the equation of state moves the curve, and in particular moves the crossing, without any of the degeneracy with the matter density that a distance measurement suffers at a single redshift, because the drift is measured across a range of redshifts at once and its shape is a function of the history rather than of an integral over it.

One galaxy, watched for fifty billion years #

The drift is the slope of a curve that is only ever sampled at one point. The curve itself is worth drawing, because it shows what the slope is the slope of.

Early on, an observer would have seen this galaxy at a much higher redshift, because the light arriving then had left it when the universe was very young and very small. As the observer’s clock advanced, the light arriving came from later and later epochs of the galaxy, the ratio of sizes fell, and the redshift dropped. It bottomed out when the universe was 11.3 billion years old. Since then the acceleration has been stretching every light path faster than the light can cross it, and the redshift has turned round and is climbing, as it will for ever.

The same galaxy’s redshift being at a minimum a few billion years ago, in the past, is why its drift today is positive. A galaxy further away is at a different point on its own curve.

Put side by side, the three curves are one statement read at three distances. Every galaxy’s redshift falls and then rises; what differs is when the minimum comes. For a nearby galaxy the light comes from the recent, accelerating universe, and the minimum was long ago. For a distant one the light left in the decelerating era, and the minimum is still to come. The redshift at which the minimum is happening now is the crossing in the first figures — the one redshift whose drift today is zero.

A signal a hundred million times smaller than the Earth’s motion #

The numbers are small in a way that needs saying plainly. A drift of 2.5 centimetres per second in ten years, set against the apparent recession velocity of a galaxy at a redshift of a half — about a hundred and fifty thousand kilometres per second — is a change of about two parts in ten billion, accumulated over a decade. The Earth’s orbit carries every spectrometer on it at thirty kilometres per second in a direction that reverses every six months, and its rotation adds a few hundred metres per second more.

Those motions are known and removed. What makes the measurement conceivable at all is the precision spectroscopy developed for a different purpose: the search for planets by the reflex velocity of their stars, which pushed the stability of spectrographs from hundreds of metres per second to below one, with wavelength references now calibrated against laser frequency combs. The drift asks for another factor of a hundred beyond that, held stable over decades, on sources thousands of times fainter than nearby stars.

The sources proposed are not galaxies. They are the absorption lines that intergalactic hydrogen clouds imprint on the spectra of distant quasars — the forest of lines between a quasar and the observer — because there are hundreds of them in each spectrum, spread across the range of redshift where the drift is largest and changes sign, and because a diffuse cloud of gas in the space between galaxies is about as close to a comoving marker as nature provides. Averaging over many lines in many quasars, observed across twenty years or more with an extremely large telescope, is the only route anyone has found to the required precision.

The cost of that route is set by counting photons. A velocity measured from a spectral line is uncertain in proportion to the line’s width divided by the signal-to-noise ratio, and the noise falls only as the square root of the light collected. To reach centimetres per second from lines whose natural widths are kilometres per second, each epoch has to gather a very large number of photons from quasars that are faint, and it has to do so twice, decades apart, with an instrument whose wavelength scale has not moved by a part in ten billion in between. The forecasts that have been made for such a programme run to thousands of hours of time on the largest telescopes planned, spread over two decades, for a detection at a few standard deviations. It is an experiment whose duration is fixed by the universe, not by the instrument: no improvement in the spectrograph makes the drift over ten years any larger than it is.

The same clock ticks in every cosmological observable, and most of its ticks are even quieter. The temperature of the microwave background falls in proportion to the scale factor, so it drops at a fractional rate equal to the present expansion rate — about two hundred-millionths of a kelvin a century. The Hubble constant is a rate, and like every rate in an expanding universe it is itself slowly changing. The redshift drift is the one of these effects large enough, and in sources numerous enough, that a measurement of it is conceivable at all.

Things that also change a redshift #

The formula assumes that the source and the observer are both at rest relative to the expansion, and neither quite is.

The observer is accelerating. The Sun is in orbit round the centre of the Galaxy, and a body in orbit is always accelerating towards the centre of its orbit. Gaia has measured that acceleration from the apparent motions of distant quasars: about 0.23 nanometres per second squared. Over a decade that is a change of about seven centimetres per second in the Sun’s velocity — more than the entire cosmological drift of any galaxy in the first figure. It changes every observed velocity by its projection along the line of sight, so it appears as a pattern across the sky, strongest towards and away from the Galactic centre, and it has to be modelled and removed before a drift can be seen at all.

The source is accelerating too. A galaxy moving through the potential of a group or cluster carries a peculiar velocity that is not the expansion, and the velocity changes as the galaxy moves. For a galaxy falling through a massive cluster that change can be of the same order as the cosmological signal. Intergalactic absorbers, far from any cluster, are expected to be much quieter, which is another reason they are the sources of choice; how much quieter is itself a prediction of the models of structure formation that the measurement would test.

The path changes as well. Light from a quasar crosses thousands of megaparsecs of growing structure on its way, falling into potential wells and climbing back out of them, and in a universe with a cosmological constant the largest of those wells are slowly growing shallower while the light is inside them — so a photon regains a little more energy leaving a well than it gained entering it, and the amount depends on when the light made the crossing. The same effect leaves a faint correlation between the microwave background and the distribution of galaxies on the sky. Along a single sightline its contribution to the drift is expected to be well below the cosmological signal, and it averages down across many sightlines; it is one of the floors a forecast has to show it can argue past.

And the clocks disagree. Both redshifts are compared against atomic transitions that have to be the same at both ends of the decade. They are, to any precision anyone can test — which is to say the measurement also assumes that the fine-structure constant is not changing by a part in ten billion per decade, an assumption that has its own experimental programme.

What the four histories have to do with it #

The drift is the local derivative of the whole expansion history, and it is worth seeing that history once more from the point of view of what the derivative is sensitive to.

Four histories that agree exactly now are four different drift curves, because the drift compares the rate now with the rate then. The empty universe is a straight line on this figure and its drift is zero; the matter-only and closed universes bend over and every redshift in them falls; ΛCDM bends over and then straightens and bends up, and its drift changes sign. A measurement of distance integrates over the bend, which is why four definitions of distance disagree and none of them determines the history uniquely from one redshift. The drift reads the bend’s slope directly, at every redshift where there are lines to measure.

What this is not #

It is not a detection. No redshift drift has been measured. Every number here is a prediction, and the figures are forecasts of the size and sign of a signal for the parameters that the other measurements favour.

It is not the same as the acceleration being inferred elsewhere. The supernovae measure the integrated expansion through the distance-redshift relation, and the microwave background and baryon oscillations measure it through the angles standard rulers subtend. The drift measures the rate at two epochs through the time dependence of one quantity. They are consistent with each other within the model, and the value of the drift is precisely that it could disagree with all of them.

And it does not care what the expansion is made of until the formula is fed a model. The same line measured twice returns $\dot z$ whatever the universe contains. Only turning $\dot z$ into $H(z)$ uses $H_0$, and only turning $H(z)$ into densities uses the Friedmann equation.

The galaxies that are already leaving #

There is a connection with the galaxies that are already out of reach that the watched-galaxy curves make visible. A redshift that rises for ever is the signature of an event horizon: the light arriving from the galaxy comes from ever later moments of its history, compressed into ever longer stretches of the observer’s time, and the galaxy’s image slowly freezes and fades. The minimum on each curve is the moment at which the acceleration began to win for that galaxy’s light, and every curve has one. Given enough time every galaxy’s redshift is climbing, and the drift measured today is the first derivative of that farewell.

Still open: whether an expanding universe expands a bound system at all #

The drift is a statement about galaxies that are carried apart by the expansion. It raises the opposite question for things that are not: whether a planet’s orbit, a galaxy, or a group of galaxies feels the expansion rate $H$ at all, and if not, what of the expansion does reach inside a bound system — which turns out to be the acceleration and not the rate, and to have a precise size.