A clock that flickers without a tick

A redshift stretches everything that arrives from a distant source, not only its wavelengths. If successive wave crests leave a galaxy one period apart and arrive $(1+z)$ periods apart, so do successive photons, successive pulses, and successive events of any kind. The shift that is not a Doppler shift is the stretching of the scale between emission and reception, and it applies to time intervals exactly as it applies to wavelengths. A process that takes a month at redshift one should be seen to take two.

For type Ia supernovae this has been measured directly and with great clarity. A supernova’s light curve has a rise and fall whose width at the source is calibrated, to a few per cent, by its peak brightness and colour, and at redshift one the light curves are seen to last twice as long, which is exactly the expansion’s prediction and exactly what models in which redshift is caused by photons losing energy on the way cannot produce. The test is now part of the case that the dimming of distant objects is the expansion’s rather than a property of the light.

Quasars are the obvious second test, because they are bright to much higher redshift — beyond seven, against two for supernovae — and because they vary. For a while they appeared to fail it: an analysis of several hundred quasar light curves in 2010 found no sign that the distant ones varied more slowly. The claim was taken seriously, because a failure of time dilation would be a failure of the expanding universe itself. In 2023 an analysis of a hundred and ninety quasars observed for two decades found the dilation after all, at about the expected size. The difference between the two results is the subject of this essay, and it is not a difference in data quality. It is that a quasar is a clock with no standard tick.

A wander with no period #

A quasar’s optical brightness changes by ten or twenty per cent over months to years, in a way that is well described statistically as a damped random walk: at each moment the brightness takes a small random step, and it is also pulled gently back towards its mean, so that it wanders but does not drift away. The process has no period and no preferred shape. It has one characteristic time, the damping timescale $\tau$, over which the brightness forgets where it was. On intervals much shorter than $\tau$ the light curve behaves like a random walk, its changes growing as the square root of the interval; on intervals much longer, its changes saturate at a fixed amplitude.

At redshift two, every interval at the source arrives three times longer, and so does $\tau$. The distant curve in the figure is the first third of the nearby one, stretched; it wanders more slowly and completes fewer excursions in the same observing time. That stretch is the whole of the signal. But the figure flatters the test in a way no real survey can: the two curves share their random sequence, so the stretch can be seen by comparing their shapes. Real quasars share nothing. Each has its own sequence and its own timescale, and the only thing that can be compared is the timescale itself, measured on each and averaged over many.

How a random process can keep time at all #

It may seem odd that something with no period can be a clock. The resolution is that a random process has a statistical shape even when it has no repeating one. The structure function — the mean squared difference in brightness between two observations, as a function of the time between them — rises with the interval and then flattens, and the bend between the two regimes sits at $\tau$. Measure the structure function for one quasar and the bend can be located, roughly; stretch time by three and the bend moves by three. A clock of this kind ticks only in an average sense, and its reading for any one quasar is uncertain by tens of per cent because a few timescales of light curve contain only a few independent excursions.

The same randomness is what makes quasar light curves treacherous for other purposes. A damped random walk has more power at long periods than at short ones, and a curve of it, searched for periodicity, throws up peaks that look like cycles and are not — the tallest peak in a periodogram of red noise is routinely mistaken for a binary black hole. For the dilation test that randomness is the signal rather than a nuisance: what is wanted is not a period but the place where the wandering stops remembering.

The same expansion also predicts a slower, stranger clock. The redshift of any distant source changes by a few centimetres a second each decade as the expansion accelerates, and measuring that drift directly is a test of the expansion history that uses quasars as backlights rather than as clocks. The dilation test is the first-order version of the same idea: it asks only whether time at the source runs slow by the factor the redshift says, not how that factor is changing.

The timescale is the clock, and a clock is useful only if its rate at the source is known. For a supernova the source-frame width is calibrated from nearby examples. For a quasar the source-frame $\tau$ depends on the quasar, and the dependence is not random.

What sets a quasar’s timescale at the source #

The variability is thought to arise in the accretion disc, from fluctuations in the heating of the gas that propagate through it on a thermal or a dynamical timescale at the radius where the observed light is emitted. Both of those depend on the black hole’s mass. A disc round a heavier hole is larger, the radius that emits a given wavelength sits further out, and the timescales at that radius are longer. Measurements across samples of quasars with masses from reverberation and from the widths of their emission lines find $\tau$ growing roughly as the black hole’s mass to the power 0.38 in one study and with somewhat different powers in others.

The same physics makes $\tau$ depend on wavelength. Bluer light comes from hotter gas closer to the hole, where everything happens faster, so at fixed mass $\tau$ shortens towards the blue, roughly as the wavelength to the power 0.17.

Neither dependence is large on its own. Doubling a black hole’s mass lengthens its timescale by thirty per cent; halving the wavelength shortens it by eleven per cent. The trouble is that a survey that spans a range of redshift automatically spans a range of both.

A survey that weighs its quasars by distance #

A survey sees the quasars brighter than its flux limit, and the luminosity that corresponds to that limit rises as the square of the luminosity distance. Quasars shine at a range of fractions of their black hole’s Eddington limit, but if a typical fraction is taken, the faintest quasar a survey can see at each redshift is also the lightest black hole it can see there.

That is a factor of a hundred and fifty in the mass of the faintest members between redshift half and redshift four. Not every quasar in a sample sits at the floor — there are brighter ones at every redshift — but the floor moves the whole distribution, and the population at redshift three is a population of much heavier black holes than the population at redshift a half. It is the ordinary bias of a sample brighter than the population it came from, and here it acts on the clock.

The luminosity distance that sets the floor is itself one of the several distances an expanding universe defines, and it grows faster than the others because it carries two factors of $(1+z)$ — one for the redshift of each photon’s energy and one for the dilation of their arrival rate. So the same dilation that the test is looking for is already inside the selection that confounds it: part of the reason distant quasars must be intrinsically luminous to be seen at all is that their photons arrive slowly.

Three things that slow the flicker #

Put the three dependences together for the faintest quasars a survey can see, and the observed timescale’s growth with redshift separates into its causes.

Time dilation contributes a factor of five between redshift 0.2 and redshift four. The bluer rest-frame wavelength contributes a shortening of about a fifth, because a fixed observing band at redshift four samples light emitted at a fifth of its wavelength. The heavier black holes contribute a lengthening by fifteen, three times larger than the dilation itself. A test that fitted the observed timescales of such a sample to a power of $(1+z)$ would find an exponent near 2.7 rather than one, and would be measuring the survey’s selection rather than the universe’s expansion.

The real samples are not this extreme, and the selection term in the figure is an upper bound — a sample built to have similar luminosities at all redshifts can suppress it. But the figure makes the structure of the problem plain. Dilation is the only one of the three effects that does not depend on the quasar, and it is not the largest. Any measurement has to estimate the other two from the same data and remove them, which is why the quasar test is a fit with several free parameters rather than a comparison of two numbers.

The 2023 analysis did exactly that. It fitted, simultaneously, the dependence of the timescale on luminosity, on rest-frame wavelength, and on $(1+z)$, using quasars with spectroscopic masses observed in several bands. The dilation exponent came out at $1.28^{+0.28}_{-0.29}$ — consistent with one, and inconsistent with zero.

A baseline that hides what it measures #

There is a second trap, and it works against the expansion rather than for it. A damped random walk’s timescale can be recovered only from a light curve that covers several timescales, because the estimate depends on seeing the curve forget its past. Too short a light curve looks like a pure random walk with no saturation, and its timescale is not merely uncertain but biased.

The direction of the bias comes from subtracting the curve’s mean. A short curve’s own mean sits closer to its points than the true mean does, so the variability looks smaller and its correlations decay faster than they really do, and the recovered timescale comes out short. The simulations show how slowly the bias goes away: even ten timescales return two-thirds of the truth by this estimator, and a reliable estimate needs dozens. Better estimators, which fit the whole process rather than one correlation, reduce the bias but do not remove it; the rule of thumb from simulation studies of quasar data is that a light curve should cover at least ten timescales.

Now apply that to a survey of fixed length. A quasar at redshift three with a rest-frame timescale of two hundred days has an observed timescale of eight hundred, and a ten-year survey covers it four and a half times. The same quasar nearby is covered eighteen times. The distant quasar’s timescale is underestimated more than the nearby one’s, and the error grows with redshift — precisely in the direction that would hide a $(1+z)$ dependence. A survey too short for its distant members will find too little dilation, and the earlier null result used light curves for which this mattered. The 2023 measurement used a baseline of about two decades, which brought most of its quasars out of the regime where the bias dominates.

How many quasars a test needs #

Once the selection and the baseline are dealt with, what remains is the scatter of individual timescales about the relation with mass and wavelength, which is large — about a factor of two, or 0.3 in the logarithm, for quasars of the same mass seen at the same wavelength. That sets how many quasars a test needs.

The exponent’s error falls as the square root of the number of quasars and in proportion to the spread of $\ln(1+z)$ the sample covers, which is why the range in redshift matters as much as the number. A hundred and ninety quasars over redshift 0.2 to 4 give an error of about 0.13 from the scatter alone. The published error is about twice that, because the luminosity and wavelength terms are fitted from the same data and their uncertainties propagate into the exponent. Either way, a sample of that size can tell dilation from no dilation at several standard deviations, and cannot yet test it at the level of ten per cent.

What a ragged clock is worth #

It is fair to ask why a test so beset by systematics is worth making when supernovae already pass it cleanly. The comparison of the two clocks shows both the cost and the reason.

A supernova is a precise clock: a couple of dozen of them over a modest range of redshift fix the exponent to a tenth. A quasar is a ragged one and needs ten times as many objects even over a range of redshift four times larger. But the ragged clock reaches further. Supernovae are found reliably only to redshift two or so; quasars are seen at redshift seven. Time dilation is not a test that a single redshift settles: alternatives to expansion that mimic it at low redshift but not at high ones are what a long baseline in redshift exists to exclude, and only quasars, and gamma-ray bursts with their own and still worse-calibrated clocks, extend it to the first billion years. The quasar test is also made with an entirely different kind of object — a random process round a black hole, not a thermonuclear explosion of a white dwarf — so it shares none of the supernova test’s assumptions.

And there is a reason to take the earlier null result seriously in the first place: it was not careless. A test of time dilation that fails is the single most consequential null result cosmology could produce, and the reason it failed was that the dilation is entangled with the population in a way that the supernova test, with its calibrated clock, never has to confront. Learning to separate them is what made the quasar test possible, and it is the same separation every population-based test of the expansion has to make.

Why tired light fails the quasar test too #

The tests discussed in this collection constrain the same thing from different directions. The surface brightness of galaxies dims as the fourth power of $(1+z)$ in an expanding universe, and one of those four powers is time dilation: the photons arrive at a reduced rate. The supernova light curves isolate that power directly. The quasar timescales isolate it again with an independent clock and to much higher redshift. A redshift caused by photons losing energy in transit, with the universe static, predicts no dilation at all, and both clocks exclude that. A redshift caused by expansion predicts exactly one power of $(1+z)$, and both clocks find it, the quasar one within its larger error.

What neither test can do on its own is distinguish the dilation from something that changes the source’s clock with cosmic time — quasars at redshift four that are genuinely slower because the young universe made different accretion discs. The two tests rule that out together: supernovae have no reason to evolve in the same way as accretion discs, and they agree.

Simplifications that flatter the quasar clock #

The figures use simplified versions of each dependence: a single power of mass and of wavelength, a single Eddington fraction for every quasar, a light curve sampled evenly with no seasonal gaps, and a scatter that does not depend on anything. Real quasar light curves come in seasons, with gaps of several months each year that themselves alias against timescales of a year — the same problem that sampling patterns pose for any period search. The masses come from emission-line widths calibrated against reverberation, which have their own scatter of a factor of two or more. And the damped random walk is itself an approximation: on timescales of days some quasars vary more steeply than it allows, and whether its single timescale is the whole description is still argued. Each simplification makes the test look cleaner than it is.

The essential point survives them: the dilation is one of three effects that change quasar timescales with redshift, and not the largest, and the only way to see it is to model the other two.

Still open: whether the flicker evolves #

The measured exponent is consistent with one, at an error of about thirty per cent. To reach ten per cent, the test needs a couple of thousand quasars with masses and with light curves long enough for their dilated timescales, which a decade-long survey with several hundred visits per field, now beginning, will supply for tens of thousands. At that precision the question changes. A precise exponent of exactly one would confirm what the supernovae already say, to higher redshift. An exponent that came out slightly different would not threaten the expansion, which has passed too many tests for that; it would say that quasar discs in the young universe were not the same as nearby ones at the same mass — that the clock itself evolves — and the quasar test would become a measurement of accretion physics across cosmic time rather than of cosmology. Which it turns out to be depends on whether the residual scatter of the timescales hides structure that the mass and the wavelength do not explain.