Astronomy & Celestial Mechanics
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Cosmology

A dimming short by what young stars repay

In an expanding universe a galaxy's surface brightness should fall as the fourth power of one plus its redshift; if light merely tired on its way through static space, as the first power. Measured on elliptical galaxies at redshift 0.85, it falls as about the two-and-a-half power — neither. The verdict on expansion rests on what the galaxies did in the meantime, and the decisive fact is not the size of the shortfall but its sign: expansion needs the galaxies to have been brighter when their stars were younger, and tired light needs them to have been fainter.

Assumes Expansion, Surface brightness and Stellar evolution.

16 min read 6 figures Light is the only messengerMeasuring the unreachable

A redshift on its own says that light arrives with longer wavelengths than it left with. It does not say why. Read as expansion, it is the stretching of space between emission and reception, and it brings a set of other effects with it: clocks at the source appear slowed, sources appear larger than their distance implies, and the rate at which photons arrive drops. Read as something that happens to the photon alone — a gradual loss of energy on a long journey through a static universe, the idea Fritz Zwicky proposed in 1929 and called tired light — it brings none of them.

Richard Tolman pointed out in 1930 that one measurement would decide between the two without any knowledge of distances: the surface brightness of a galaxy, its light per unit area of sky. In a static Euclidean space surface brightness does not depend on distance at all, because the flux and the solid angle both fall as the inverse square and their ratio stays fixed. It is the brightness that distance cannot touch. In an expanding space it falls steeply with redshift. Compare the surface brightnesses of the same kind of galaxy near and far, and the redshift’s cause is read off the slope.

Four ways for a galaxy to fade with redshift. The dimming of a galaxy's surface brightness, in magnitudes per square arcsecond, against redshift, for the same galaxy placed at greater and greater distance. In an expanding universe the bolometric surface brightness falls as (1+z)⁴: 3.01 magnitudes by z = 1 and 7.8 by z = 5. Measured per unit frequency, as magnitudes in a filter are, one power is absorbed by the stretching of the band and the dimming is (1+z)³. If photons lost energy on a static journey ("tired light"), only one power would remain — 0.75 magnitudes at z = 1. In a static space with no loss there is no dimming at all, because flux and solid angle both fall as the inverse square of distance. At z = 1 expansion and tired light differ by 2.3 magnitudes, a factor of 8 in brightness.
Fig. 1 The same galaxy’s surface-brightness dimming against redshift under four readings of the redshift. Expansion, counting all the light, takes 3.01 magnitudes by redshift 1 and 7.8 by redshift 5; through a filter it takes one power fewer. Tired light keeps 0.75 magnitudes at redshift 1, and a static space with no loss keeps none. At redshift 1 the first and the third differ by 2.3 magnitudes, a factor of 8 in brightness — the gap the test is built to see.

The figure draws the four candidates. In an expanding universe the bolometric surface brightness falls as (1+z)4(1+z)^4: three magnitudes by redshift one, nearly eight by redshift five. Measured through a filter, per unit frequency, one power is absorbed by the stretching of the passband and the fall is (1+z)3(1+z)^3. Tired light keeps only (1+z)1(1+z)^1 — three-quarters of a magnitude at redshift one. And a static universe with no loss at all keeps nothing.

Where four powers come from

The four powers are worth separating, because the test’s force lies in the three that tired light lacks.

Where the four powers come from, at redshift 1. The 3.01 magnitudes by which a galaxy's bolometric surface brightness has fallen at redshift 1, split into its factors of (1+z). Two come from area: the light left when the galaxy was closer, so it subtends a larger angle than its present distance implies, spreading its light over (1+z)² more sky. One comes from each photon's energy, reduced by the stretch. One comes from the arrival rate, because the interval between photons is stretched as well. Each power is 0.75 magnitudes. A static universe in which light merely tires keeps the second bar alone (lower row); the other three are the signature of expansion: 2.26 magnitudes, a factor of 8.
Fig. 2 The 3.01 magnitudes by which a galaxy’s bolometric surface brightness has fallen at redshift 1, split into its factors of (1+z)(1+z), each worth 0.75 magnitudes. Two come from area: the galaxy subtends a larger angle than its present distance implies, spreading its light over (1+z)2(1+z)^2 more sky. One comes from each photon’s energy, one from the stretched interval between photons. Tired light in a static space keeps only the energy factor (lower row); the other three — 2.26 magnitudes, a factor of 8 — are the signature of expansion.

The energy factor is shared: in both pictures a photon arrives with less energy than it left with, by the ratio of wavelengths. The arrival-rate factor is a consequence of the same stretching applied to time. If successive wave crests are separated by (1+z)(1+z) times their original interval, so are successive photons, and so is every other interval at the source — which is why the light curves of distant supernovae are measured to be stretched by exactly 1+z1+z, a test of the same factor by a different route. A photon that tires on its journey has no reason to be delayed relative to the photon behind it.

The two area factors are the geometric heart of the matter. In an expanding universe the light from a distant galaxy left when the galaxy was closer to the point that would become the observer, so its angular size is set by the distance at emission while its flux is set by the distance light has travelled and the stretching since. The two distances are related by the factor (1+z)2(1+z)^2 — the distance-duality relation — and the ratio of flux to angle inherits that factor. In a static universe the two distances are one distance and the factor is absent.

So the test does not ask whether light is redshifted. It asks whether the geometry of the light’s journey and the timing of its arrival also changed, which tired light says they did not.

A surface brightness compared without a distance

The obvious difficulty is that “the surface brightness of a galaxy” is not one number. Galaxies are brightest at their centres and fade outward, and the surface brightness at a fixed angular radius samples different parts of galaxies at different distances. To compare like with like, the comparison has to be made at the same physical place in each galaxy — at a fixed number of kiloparsecs from the centre, say — and that requires converting angles to kiloparsecs, which requires a distance, which requires a cosmology. The test seems to need the answer before it starts.

The way out was to let each galaxy’s own profile choose the radius.

A radius chosen by the galaxy's own profile. The Petrosian function η of an elliptical galaxy's light profile — the difference, in magnitudes, between the mean surface brightness inside a radius and the surface brightness at that radius — against radius in units of the half-light radius, for a de Vaucouleurs profile. η rises from zero at the centre, and the radius where it reaches a chosen value is fixed by the profile's shape alone: η = 1 at 0.40, 1.5 at 1.23, 2 at 2.56 half-light radii. Because η is a ratio of two surface brightnesses, the radius it picks out is the same part of the galaxy at any redshift and under any cosmology, so the mean surface brightness inside it can be compared across redshift without first assuming the distance the test is meant to check.
Fig. 3 The Petrosian function η\eta of a de Vaucouleurs profile — the difference, in magnitudes, between the mean surface brightness inside a radius and the surface brightness at that radius — against radius in units of the half-light radius. It reaches 1 at 0.40 half-light radii, 1.5 at 1.23 and 2 at 2.56. Being a ratio of two surface brightnesses, it picks out the same part of a galaxy at any redshift and under any cosmology.

The Petrosian radius, introduced by Vahe Petrosian in 1976 for exactly this purpose, is where the local surface brightness has fallen a chosen amount below the mean inside it. Both quantities dim by the same factor with redshift, whatever that factor is, so their ratio is unaffected; the radius at which the ratio reaches a given value is fixed by the shape of the profile alone. The mean surface brightness inside that radius can then be compared between a galaxy nearby and a galaxy at redshift one with no assumption about how far away either is.

That made the test feasible. It did not make it easy. The profile has to be measured out to where the surface brightness is well below the sky in a distant galaxy whose half-light radius is less than an arcsecond, which needed the Hubble Space Telescope’s resolution; the galaxies chosen have to be of a kind whose profiles are similar near and far, which pointed to giant ellipticals in clusters; and the light has to be compared in the same rest-frame band, which requires a correction for the shift of the spectrum through the filter. Each of those steps carries a modest uncertainty. None of them is the large one.

The exponent that came out

The definitive version of the test, by Lori Lubin and Allan Sandage in 2001, compared elliptical galaxies in three clusters near redshift 0.85 with nearby ellipticals measured the same way. In the rest-frame red, the surface brightness had fallen by an amount corresponding to (1+z)2.6(1+z)^{2.6}; in the near-infrared, closer to (1+z)3.4(1+z)^{3.4}.

Neither is four and neither is one. Taken at face value, the red-band exponent sits nearer to expansion than to tired light, but by less than it seems: 2.6 is about halfway between in exponent and further from both in magnitudes. The raw measurement, by itself, favours neither picture decisively. What decides it is the third ingredient, and the third ingredient is the galaxies.

An elliptical galaxy at redshift 0.85 is seen about seven billion years earlier than one nearby. Its stars are that much younger. A population of stars that formed in a single early burst and has done nothing since grows fainter as it ages, because its most luminous members — the most massive stars still burning hydrogen, and the giants they become — die off in order of mass, and the light of the population is carried by ever smaller, ever fainter stars. So the distant galaxies were intrinsically brighter than their nearby counterparts, by an amount a model of stellar evolution can compute. The measured dimming is the cosmological dimming plus that evolution, and the evolution has the same size as the difference being tested.

What the galaxies would have to have done

Turn the comparison around. Given the measured exponent, each hypothesis requires a particular history of the galaxies themselves, and a stellar population either can supply that history or cannot.

What the galaxies would have to have done. If the surface brightness of comparable elliptical galaxies is measured to fall as (1+z) to the power 2.6 — the kind of exponent measured in the red near z = 0.85 — the shortfall from each hypothesis has to be supplied by the galaxies themselves. In an expanding universe they must have been brighter in the past, by 0.94 magnitudes at z = 0.85 (the curve that falls below zero). In a static universe with tiring light they must have been fainter, by 1.07 magnitudes. The lower curves are the brightening of a population of stars that formed at z = 2 or 5 and has simply aged since, with luminosity falling as its age to the power 0.9: 1.15 and 0.84 magnitudes brighter at z = 0.85. The expanding universe asks for the brightening ageing stars supply. The static one asks for galaxies that were fainter when their stars were younger, which no stellar population does.
Fig. 4 If surface brightness is measured to fall as (1+z)2.6(1+z)^{2.6}, each hypothesis needs the galaxies to make up the difference. In an expanding universe they must have been 0.94 magnitudes brighter at z=0.85z=0.85 than today; in a static universe with tiring light, 1.07 magnitudes fainter. The dashed curves are a population of stars formed at z=2z = 2 or 55 and simply ageing since, its luminosity falling as its age to the power 0.9: 1.15 and 0.84 magnitudes brighter at z=0.85z = 0.85. Expansion asks for the brightening ageing stars supply; tired light asks for galaxies that were fainter when their stars were younger.

The figure carries the whole verdict. If space expands, the galaxies at redshift 0.85 must have been about 0.9 magnitudes brighter than the same galaxies today. Ageing stars supply between 0.8 and 1.2 magnitudes, depending on when they formed. The required evolution and the computed evolution agree, which is the statement usually summarised as “corrected for evolution, the exponent is consistent with four”.

If light tires, the galaxies at redshift 0.85 must have been about a magnitude fainter than today. That is not a matter of degree. A stellar population fades as it ages; it does not brighten. For the static picture to work, galaxies with younger stars would have to be less luminous than the same galaxies seven billion years later with the same stars grown old, which runs against everything stellar evolution says about how a population’s light changes with time — the same physics that dates star clusters by their turn-off and is calibrated on them.

The same galaxies in a redder band

The near-infrared measurement makes the same point from a different angle, and it is worth drawing because its numbers are not the same. In light longer than a micron the stars that dominate an old population are the cool giants, whose number changes more slowly as the population ages; the brightening with lookback time is shallower, closer to the age to the power 0.6 than 0.9. The measured exponent was correspondingly closer to four.

What the galaxies would have to have done. If the surface brightness of comparable elliptical galaxies is measured to fall as (1+z) to the power 3.4 — the kind of exponent measured in the near-infrared near z = 0.85 — the shortfall from each hypothesis has to be supplied by the galaxies themselves. In an expanding universe they must have been brighter in the past, by 0.40 magnitudes at z = 0.85 (the curve that falls below zero). In a static universe with tiring light they must have been fainter, by 1.60 magnitudes. The lower curves are the brightening of a population of stars that formed at z = 2 or 5 and has simply aged since, with luminosity falling as its age to the power 0.6: 0.77 and 0.56 magnitudes brighter at z = 0.85. The expanding universe asks for the brightening ageing stars supply. The static one asks for galaxies that were fainter when their stars were younger, which no stellar population does.
Fig. 5 The same accounting for an exponent of 3.4, the kind measured in the near-infrared near z=0.85z = 0.85, with the ageing population’s luminosity falling as its age to the power 0.6, as it does in redder light. Expansion now needs the galaxies 0.40 magnitudes brighter at z=0.85z = 0.85; ageing stars formed at z=2z = 2 or 5 supply 0.77 and 0.56. Tired light needs them 1.60 magnitudes fainter.

Here the agreement is less neat. Expansion asks for four-tenths of a magnitude of brightening, and the simple ageing models give between a half and three-quarters — a disagreement of two or three tenths, the size of the differences between stellar population models in this band and of the uncertainty in the rest-frame correction. It is a disagreement about how much the galaxies evolved, not about which way. In the near-infrared the static picture needs the galaxies to have been more than a magnitude and a half fainter when younger, which is further from anything a stellar population can do than in the red.

So two bands with different exponents give two different required evolutions for the expanding universe, each roughly what ageing stars provide in that band, and two different required evolutions for the static one, each with the wrong sign. That the required evolution in an expanding universe tracks the band in the way stellar models say it should is itself evidence: an error in the cosmology would not know which band it was being measured in.

That is the subtle strength of the test. Its precision on the exponent is poor, because the evolution correction is as large as the effect and is known to perhaps a few tenths of a magnitude. But its power to discriminate does not depend on that precision, because the two hypotheses require corrections of opposite sign. One requires what stars do; the other requires what they cannot.

A test that runs through two populations

The test compares two populations of galaxies seven billion years apart and assumes they are the same population at two ages. For giant ellipticals in rich clusters that is the best assumption available, because their stars are old, their star formation ended early, and their light evolves in the simplest way. It is still an assumption, and the size of the galaxies is where it is most exposed.

A size that changes is a dimming that changes. Bolometric surface-brightness dimming against redshift in an expanding universe when galaxies of a given luminosity were smaller in the past, with radius scaling as (1+z) to the power −0.5 or −1, compared with tired light in a static space (lower). Halving a galaxy's radius at fixed luminosity quadruples its surface brightness, so size evolution of the kind measured for galaxies at fixed mass offsets the expansion dimming by 5β log(1+z) magnitudes. Galaxies whose radii shrank as (1+z) to the power −1.5 would reproduce tired light exactly. The Tolman test compares surface brightnesses, and a surface brightness is a luminosity divided by an area: it tests the expansion only as well as the evolution of both is known.
Fig. 6 Bolometric surface-brightness dimming in an expanding universe when galaxies of a given luminosity were smaller in the past, with radius scaling as (1+z)−0.5(1+z)^{-0.5} or (1+z)−1(1+z)^{-1}, against tired light in a static space (dashed). Halving a galaxy’s radius at fixed luminosity quadruples its surface brightness, so size evolution offsets the expansion dimming by 5βlog⁡10(1+z)5\beta\log_{10}(1+z) magnitudes; shrinking as (1+z)−1.5(1+z)^{-1.5} would reproduce tired light exactly.

Surface brightness is luminosity divided by area, and area goes as the square of the radius. If galaxies of a given luminosity were more compact in the past, their surface brightness was higher for that reason alone, and the test reads the compactness as a shortfall in dimming. Massive quiescent galaxies are indeed measured to have been smaller at earlier times — at fixed stellar mass, two to four times smaller at redshift two than today, growing since by accreting smaller galaxies onto their outskirts. At redshift 0.85 the growth is smaller, a few tens of per cent, but in the direction that brightens the distant galaxies.

The Petrosian construction blunts this: it compares galaxies at the same point in their own profiles, not at the same physical radius, and selects the distant galaxies to match the nearby ones in Petrosian size. But it cannot remove it. If the population at redshift 0.85 is not the same population as the nearby one, only younger — if some of its members merged or grew or were selected differently — then the surface brightness difference contains a structural term the stellar models do not compute. In the extreme, galaxies shrinking with redshift as (1+z)−1.5(1+z)^{-1.5} in an expanding universe would produce exactly the dimming tired light predicts.

This is how the test has been contested since. Studies of ultraviolet surface brightnesses of disc galaxies out to high redshift have found them nearly constant, and that result has been read as support for a static universe. Read with the measured growth of galaxy sizes, it is what an expanding universe with evolving galaxies predicts. The disagreement is not about the photometry. It is about whether the distant galaxies and the nearby ones are the same objects at two ages, and that is a question about galaxy evolution rather than about the redshift.

Why the test is kept, though other tests are cleaner

The time dilation of supernova light curves tests the arrival-rate factor with no population model at all: a type Ia supernova’s rise and fall take a calibrated number of days at the source, and at redshift one they are measured to take twice as long, exactly as expansion predicts and tired light does not. The redshift drift — the slow change of a source’s redshift over decades — would test the expansion history directly. The microwave background’s spectrum, a near-perfect blackbody at 2.7 kelvin, is itself an argument: a tired-light process that shifted every photon by the same factor would have to do so without broadening the spectrum, which no proposed mechanism does, and the darkness of the night sky is another.

The Tolman test is the one that tests the geometry, and its history shows how long that took. Tolman and Hubble set out the test together in 1935, and Hubble’s own photographic surface brightnesses could not distinguish the cases; Sandage returned to it repeatedly over the following six decades, as the telescopes and the stellar models each improved enough to make the next attempt worth making. The 2001 measurement was the fourth paper of a series, and it was possible only once the Hubble Space Telescope could resolve a Petrosian radius at redshift one and the ages of old stellar populations were known well enough to compute their fading. Time dilation tests clocks; the background’s spectrum tests thermodynamics. Surface brightness tests the two area factors that come from the relation between angular size and flux, and those are the factors that depend on space itself changing size rather than on anything happening to light in transit. It is the most direct of the tests of what expansion means, and the least precise.

Where the comparison stops

The evolution model is a single burst. Real ellipticals formed their stars over an extended period, and a younger tail of star formation makes the past brightening larger at a given redshift. The range drawn, from formation at redshift two to five, spans most published models but not the extremes.

The power law in age is approximate. A luminosity falling as age to the power 0.9 describes the red light of an old population well; in bluer bands the slope is steeper, and in the near-infrared shallower, which is part of why the measured exponent differs between bands.

The size evolution is drawn as a power of (1+z)(1+z) at fixed luminosity. Measured growth is usually expressed at fixed stellar mass, and converting between the two requires the same luminosity evolution as the main test.

The rest-frame correction is a model. Comparing a galaxy at redshift 0.85 observed through one filter with a nearby galaxy observed through another needs a correction for the band the source never had, computed from a template spectrum. For old ellipticals the template is well known, and the correction’s uncertainty is a few hundredths of a magnitude — small next to the evolution, but in the same direction in every distant galaxy.

Dust is ignored. Ellipticals have little, but any difference between the near and far samples in dust content would enter as a colour-dependent offset.

The exponent 2.6 stands in for a measurement. The measured values differ by band and by sample, and the conclusion drawn — that the required evolution has the sign and size stellar ageing provides — holds across them.

Still open: whether the distant galaxies are the near ones, younger

The test’s remaining weakness is the identification of the two samples. Large spectroscopic surveys now measure stellar masses, ages and star-formation histories for thousands of quiescent galaxies out to redshift one and beyond, and deep space-based imaging resolves their profiles. That allows the comparison to be made between galaxies matched by stellar mass and by the age of their stars rather than by appearance — and allows the evolution correction to be measured from the spectra of the same galaxies rather than modelled.

If the expansion picture is right, the surface-brightness dimming corrected with each galaxy’s own measured stellar age and size should come out at the fourth power with an uncertainty set by the photometry alone. That has not been done at the precision the data now permit. Until it is, the Tolman test remains what it has been since 2001: a measurement that cannot fix the exponent to better than a power, and that does not need to, because the static universe it was designed to test needs galaxies to grow brighter as they age.

Where this goes next

The same recession law from two different galaxies. Twenty galaxies at fixed comoving positions after the whole picture has been multiplied by 1.34, with each galaxy's displacement drawn from where it was to where it is. Every arrow's tail sits at the old separation and its tip at the new one, so the arrow is exactly 0.34 times its tail's distance from the highlighted galaxy — proportional to separation for one reason and no other: a uniform scaling moves everything in proportion to its distance from whatever point the scaling is measured about. The right-hand panel measures from a different galaxy and gets the identical law with the identical constant. That is the content of a linear velocity–distance relation. It is the signature of an expansion with no centre, and the observation that every galaxy recedes is therefore not evidence that this one is the centre — it is evidence that none of them is. The shift that is not a Doppler shift Every galaxy beyond the Local Group has its lines shifted to the red, by an amount proportional to its distance. Read as a velocity that looks like a confession that everything is fleeing from here; read as a change of scale it says the opposite, because a uniform expansion produces exactly the same law measured from any galaxy in it. What distance does, and does not, do to a galaxy. Three quantities against distance, each relative to its value at 5 Mpc, on logarithmic axes so that a power law is a straight line and its exponent is the slope. Flux falls with slope −2 and angular size with slope −1, both of which are ordinary. Their ratio has slope zero: a galaxy of surface brightness 23.5 magnitudes per square arcsecond has that surface brightness at every distance, and a sky of 22 is brighter than it at every distance too. The contrast against the sky — the quantity that decides whether the thing is detectable at all — is -1.5 magnitudes wherever it is put. The brightness distance cannot touch Flux falls as the inverse square of distance and so does solid angle, so their ratio does not fall at all. A galaxy's surface brightness is the same number wherever it is put, which means whole populations can be undetectable at any distance whatever. The ratio of two distances, for a universe with dark energy and one with dust. The distance-duality ratio η — the luminosity distance divided by (1 + z)² times the angular-diameter distance — against redshift. In a universe with a cosmological constant, or any universe in which light travels on the geometry and is conserved, it is exactly one (solid). In a universe of matter alone, decelerating, with grey intergalactic dust dimming its supernovae just enough to make their distances match the accelerating one's, it rises with redshift: 1.19 at redshift 0.5, 1.30 at 1, because the dust makes supernovae look farther than the geometry puts them while leaving the acoustic-scale distances untouched. The shaded band is η = 1 ± 0.03z, about what supernova and acoustic-scale distances together allow. The dusty universe's η rises at the start with a slope of about 0.45 per unit redshift, more than ten times the band's, and leaves it at once. Two distances that must agree if light is conserved A supernova's distance comes from how faint it is; an acoustic ruler's distance comes from how small it looks. In any universe whose light follows the geometry and is neither created nor destroyed, the two are tied by an exact factor of (1 + z)², whatever the universe contains. Dust that dimmed distant supernovae without reddening them could have faked the accelerating expansion — and would have broken that tie by a fifth by redshift a half. The tie holds to a few per cent.

About the same objects

Not linked from either essay — found by the objects both name.

The objects this essay names

Each one links to every other essay that touches it.

Angular-diameter distanceCosmological redshiftDistance dualityElliptical galaxyExpansion of spaceK-correctionLuminosity distanceStellar populationSurface brightnessTired light