Astronomy & Celestial Mechanics
the sky, computed
Cosmology

An edge that needs no pulsation

The local measurement of the expansion rate rests on a calibration step, and for decades that step was the Cepheid variables. The tip of the red giant branch replaces it with an edge — the brightness at which old stars ignite helium, nearly the same in every galaxy — found by counting stars rather than modelling them. It gives a slightly lower expansion rate, and whether the difference is a flaw in one calibration or in both is the local half of the Hubble tension.

Assumes Hubble constant, Distance ladder and Stellar evolution.

14 min read 5 figures Measuring the unreachableLight is the only messenger

The expansion rate measured from the local universe and the one inferred from the microwave background disagree by about eight per cent, and each side has been tested by rebuilding it. On the early side, the inference passes through a computed length — the sound horizon — that an energy component lost before recombination could shorten. On the local side, the measurement passes through a chain: every distance is measured with the last one, and the middle link of the chain, connecting geometric distances in the nearest galaxies to the supernovae that reach far enough to see the expansion, was for decades the Cepheid variable stars.

The tension invites the question of whether that middle link carries a systematic error of a few per cent. The way to answer it is to replace the link with one built on different physics and see whether the expansion rate changes. The tip of the red giant branch is that replacement, and its physics could hardly be more different from a pulsating star’s.

The end of the red giant branch, found as an edge. The luminosity function of a synthetic galaxy halo — the number of red giants against apparent magnitude in the I band, smoothed — and its derivative, which is what the edge detector computes, searched over the magnitudes brighter than the branch's crowded lower part, as every application of the method does. 1582 stars are drawn: red giants rising in number towards fainter magnitudes as ten to the 0.3 times the magnitude, ending at the tip at magnitude 24, with a sixth as many asymptotic giants thinning out above it, all measured with errors of 0.08 magnitudes. The derivative peaks sharply where the counts jump, here at 24.09 — 0.09 from the true tip. The red giant branch ends where helium ignites in a degenerate core of almost fixed mass, so the tip's absolute magnitude is close to −4.05 for old, metal-poor stars, and the edge's apparent magnitude is a distance.
Fig. 1 The luminosity function of a synthetic galaxy halo — red giants against I-band magnitude, smoothed — and its derivative, which is what the edge detector computes, searched over the magnitudes brighter than the branch’s crowded lower part. 1,582 giants are drawn, rising in number towards fainter magnitudes and ending at the tip at magnitude 24, with a sixth as many asymptotic giants above it, measured with errors of 0.08 mag. The derivative peaks at 24.09.

Why the branch ends where it does

A star like the Sun, once its central hydrogen is exhausted, swells into a red giant because its centre shrank. The helium core it leaves behind contracts and heats, and in a star of less than about two solar masses it becomes degenerate: held up by the rule that no two electrons can share a state, rather than by its temperature. Hydrogen burning in a shell around the core keeps adding helium to it. As the core grows its gravity strengthens, the shell above it burns faster, and the star brightens and climbs the red giant branch.

The climb ends abruptly. When the degenerate core reaches about 0.47 solar masses its centre reaches about a hundred million kelvin and helium ignites. In a degenerate gas the pressure does not respond to temperature, so the ignition runs away — the helium flash — until the degeneracy is lifted, the core expands, and the star drops onto a fainter, stable helium-burning sequence. The flash reaches ten billion solar luminosities inside the star for a few seconds and never reaches the surface; what an observer sees is the star getting fainter, and what a population shows is a giant branch that stops. The brightest a red giant can become is therefore set by the core mass at ignition, and that mass is almost the same for every low-mass star, because it is fixed by the physics of degenerate matter rather than by the star’s total mass. The red giant branch ends at nearly the same luminosity in every old population: an absolute magnitude of about −4.05 in the I band for metal-poor stars.

That gives a galaxy’s old halo a sharp feature. Count its brightest stars, magnitude by magnitude: the counts rise towards fainter magnitudes along the giant branch and then, at the tip, fall almost to nothing — only the sparser asymptotic giants, in a later phase of evolution, are brighter. The apparent magnitude of that edge, compared with −4.05, is the distance.

A brightness set by neutrinos

The core mass at ignition is fixed by a competition, and one side of it is particle physics. A degenerate helium core is cooled by neutrinos produced in its dense plasma, which escape directly; the cooling is strongest at the centre, where the density is highest, so the hottest point is not the centre but a shell part-way out, and the flash ignites off-centre. How much mass the core has to accumulate before ignition depends on how fast the neutrinos carry heat away. Faster cooling means a heavier core at ignition, and a heavier core means a brighter tip.

That makes the tip’s brightness a measurement of the neutrino cooling rate — and so of any physics that would add to it. A neutrino with a small magnetic moment would interact with the plasma’s electromagnetic fields and be produced more copiously; a hypothetical light particle coupling to electrons, the axion being the standard example, would carry energy out as well. Either would delay the flash and brighten the tip. The tips of globular clusters, whose distances are known independently, are measured to a few hundredths of a magnitude and agree with the standard cooling, which sets some of the tightest limits there are on the neutrino’s magnetic moment and on the axion’s coupling to electrons. A distance indicator is, at the same time, a laboratory for the physics of matter at densities nothing on the Earth reaches.

The two uses pull in opposite directions. For the expansion rate the cooling has to be standard, so that the calibrated −4.05 transfers from one galaxy to another; for the particle limits the distance has to be known, so that a measured tip can be compared with the predicted one. Each relies on the other’s assumption, and the consistency of the two is part of why both are trusted.

An edge found by counting

The edge is found by an edge detector, the same operation that finds edges in an image: smooth the luminosity function and differentiate it, and the tip is where the derivative peaks. The figure is a synthetic halo drawn from a known population so that the method can be checked against the answer, and it recovers the tip to within a tenth of a magnitude.

How it is computed matters more than it looks. The raw counts rise steadily along the giant branch and are noisiest where they are largest, at its faint end, and differentiating them there produces gradients as large as the tip’s own. Practitioners therefore search only the magnitudes brighter than the crowded lower branch. Weighting the derivative to suppress the noise — dividing by the counts, or by their square root — moves the peak towards the sparse, bright side of the blurred step, by more than a tenth of a magnitude in the synthetic halos used here. The choice of detector is a choice about which bias to accept, and the published analyses test several.

How precisely the tip can be found, against how many giants sit below it. The robust scatter — 1.48 times the median absolute deviation — of the tip measured by the edge detector across 60 synthetic halos each, in magnitudes, against the number of red giants in the first magnitude below the tip, for photometric errors of 0.05 and 0.15 magnitudes. With 25 giants the tip is uncertain by 0.33 magnitudes at the better photometry; with 1600, by 0.030. With few giants the detector also fails outright in 62 per cent of halos, latching onto a noise spike far from the tip; with 1600, in 2 per cent. A tenth of a magnitude is five per cent in distance and so five per cent in the expansion rate, so a precise tip needs hundreds of giants in a halo field — a matter of counting stars, with an error that can be measured by repeating the count, rather than of modelling the stars' physics.
Fig. 2 The robust scatter of the tip found by the edge detector across 60 synthetic halos each, against the number of red giants in the first magnitude below the tip, for photometric errors of 0.05 and 0.15 mag. With 25 giants the tip is uncertain by about a third of a magnitude; with 1,600 by 0.03. With few giants the detector fails outright in most halos, latching onto noise; with 1,600 in two per cent.

The precision is a matter of counting. With a few dozen giants in the first magnitude below the tip the edge is drowned in Poisson noise and the detector often finds a fluctuation instead. With hundreds it locks on, and with over a thousand the scatter falls to a few hundredths of a magnitude — a little over one per cent in distance. Because the precision comes from the number of stars, it can be estimated from the data themselves, by resampling the stars actually measured, rather than from a model of the stars’ physics. That is the method’s main virtue against the Cepheids, whose period–luminosity relation carries corrections for metal content and for crowding by unresolved neighbours that have to be modelled.

Why the I band

The tip’s brightness is constant in one band and not in others, and the reason is temperature.

A tip that is flat in one band and tilted in the others. The absolute magnitude of the tip of the red giant branch against the giants' colour, which tracks their metal content, in three bands, from approximate calibrations. In the I band the tip is nearly flat — it brightens by only 0.22 magnitudes across the colour range drawn — because a metal-rich giant's extra luminosity is cancelled by its redder light falling outside the band. In the near-infrared J and K bands the tip is brighter, which lets it be seen further, but tilted by 0.9 and 1.8 magnitudes across the same range, so the colour must be measured and corrected for. The I band's flatness is why the method was built on it, and the infrared's brightness is why space telescopes are moving it there anyway.
Fig. 3 The absolute magnitude of the tip against the giants’ colour, which tracks their metal content, in the I, J and K bands, from approximate calibrations. In I the tip brightens by only 0.22 mag across the range drawn; in J by 0.9 and in K by 1.8, though it is brighter in both.

Metal-rich giants are cooler at the tip than metal-poor ones, because metals make their envelopes more opaque; their bolometric luminosity at the tip is slightly higher. In the I band, near a micron, those two effects nearly cancel — the extra luminosity of a cooler, metal-rich giant is emitted increasingly at longer wavelengths, outside the band — and the tip is flat to a fifth of a magnitude across the colours of real halos. In the near infrared the tip is a magnitude or more brighter, which lets it be seen further, but it tilts steeply with colour, so a tip measured there needs each galaxy’s giants’ colours measured and corrected for. Halos are chosen for their old, metal-poor stars partly for this reason, and the outer halo fields observed are placed far enough from the disc to avoid young stars and dust.

How far the edge reaches

The tip’s practical limit is how faint a star can be measured in a crowded field.

How far the tip can be seen. The apparent magnitude of the tip of the red giant branch against distance, in the I band (blue) and in the near infrared, where it is about 1.3 magnitudes brighter (green), with the depths at which the tip can be measured precisely by the Hubble and James Webb space telescopes. The tip reaches about 14 Mpc with Hubble and 47 Mpc with Webb in the infrared. The galaxies near enough to calibrate type Ia supernovae — a few dozen hosts of well-observed supernovae within forty megaparsecs — sit across this range, which is why the tip could calibrate only some of them until Webb, and why the number of hosts a method can reach is as important as its precision per galaxy.
Fig. 4 The apparent magnitude of the tip against distance, in the I band and in the near infrared, where it is about 1.3 mag brighter, with the depths at which it can be measured precisely by the Hubble and James Webb space telescopes: about 14 Mpc with Hubble and 47 Mpc with Webb in the infrared.

To calibrate the supernovae that measure the expansion rate, the tip has to be measured in the galaxies that hosted well-observed type Ia supernovae, and those are rare enough nearby that the sample is a few dozen galaxies within forty megaparsecs. With the Hubble Space Telescope the tip reaches about fourteen megaparsecs, which included only some of the hosts; the Cepheids, bright young stars in the discs, reach further and calibrated more. Webb’s infrared sensitivity and sharper images reach three times further and bring most of the hosts within range, at the price of the colour correction the infrared requires.

What the Cepheids have to model, and the tip does not

The comparison with the Cepheids is clearer for knowing what each has to assume. A Cepheid is a young, massive star crossing an instability strip where a valve in its envelope sits at the right depth to drive pulsation, and its period tells its brightness through a relation calibrated on nearby examples. To use it in a distant galaxy the relation needs a correction for the star’s metal content, which shifts it by a few hundredths of a magnitude per factor of two in metals; a correction for dust, since Cepheids live in the dusty discs where stars are forming; and, most contentiously, a correction for crowding, because at tens of megaparsecs each Cepheid’s image includes the light of unresolved neighbours that make it look brighter.

The tip avoids all three by construction. Its stars are old and in the halo, where there is little dust and little crowding; its brightness is nearly independent of metals in the I band; and its measurement is a count rather than a period and a brightness. What it gives up is reach — the halo’s giants are fainter than the disc’s Cepheids — and a single number per galaxy rather than dozens of stars, each with its own period. The two methods’ errors are nearly independent, which is exactly why they are compared.

A third edge

A third method has joined them, and it is an edge too. Among the asymptotic giants above the tip, the ones that have dredged enough carbon to their surfaces become carbon stars, and in the near infrared their luminosity function has a narrow peak at a nearly fixed brightness. The J-region asymptotic giant branch method, as it is called, uses that peak as the standard candle, with a population intermediate in age between the tip’s old stars and the Cepheids’ young ones. Its expansion rates have come out near the tip’s, around 68 to 70, which is either support for a lower local value or a sign that the two old-population methods share a systematic the Cepheids do not.

Where its expansion rate sits

The first full measurement of the expansion rate through the tip was published by Freedman and collaborators in 2019, calibrated on the Large Magellanic Cloud and on the masers in the galaxy NGC 4258, and applied to the same supernova sample the Cepheid measurements use.

H₀: nine determinations in two families. Published determinations of H₀, each with its quoted one-sigma interval, sorted into two families — measured locally, in the late universe, against inferred from z ≈ 1100 through a model. The shaded band behind each family is that family's inverse-variance weighted mean: 72.66 ± 0.75 across 5 of them, against 67.40 ± 0.41 across 4. The difference is 5.26 ± 0.85 km/s/Mpc, which is 6.2 standard deviations, computed here from the quoted errors alone. That number is an upper bound on the significance rather than the significance: the determinations within each family share calibrations, samples and in two cases the same supernovae, so they are not independent, and a correlated pair combines to something wider than the formula used here gives. What the figure does establish is that the split is not one discrepant measurement against a consensus — it is two internally consistent groups, and the grouping is by method rather than by result.
Fig. 5 Published determinations of the expansion rate with their quoted intervals, grouped into those measured locally and those inferred from the early universe through a model. The tip of the red giant branch’s 69.8 km/s/Mpc sits between the Cepheids’ 73.0 and the microwave background’s 67.4.

It gave 69.8 kilometres a second per megaparsec, with an uncertainty of about two and a half per cent — between the Cepheid value of 73 and the microwave background’s 67.4, and consistent with both. Later analyses with Webb data have moved it between about 70 and 72 depending on the sample of galaxies, the edge detector and the treatment of the infrared colour correction, and the teams that built the Cepheid calibration and the teams that built the tip calibration have analysed each other’s data with different results. The disagreement within the local measurements is smaller than the tension with the early universe, and its size is about the size of the effects in the second figure: a few hundredths of a magnitude in how the edge is found and calibrated.

The tip’s place in the argument is therefore not that it settles the tension but that it narrows where the problem can be. If the tip and the Cepheids agree, the middle step of the local distance scale is not the explanation, and the problem lies in the supernovae, in the geometric calibrators both methods share, or in the early-universe inference. If they disagree, one of them carries a systematic error of a few per cent, and the tension partly dissolves. The current measurements sit uncomfortably between those cases.

What the calibrators share

Both methods need their absolute brightness calibrated somewhere with a geometric distance, and the calibrators are few: the Large Magellanic Cloud, whose distance comes from eclipsing binaries to about one per cent; the masers orbiting the black hole in NGC 4258, to about one and a half per cent; and the parallaxes of stars in the Milky Way, which for the tip means globular-cluster giants whose distances Gaia has measured. An error in a calibrator moves every distance calibrated on it, which is why a distance that rests on one unmeasured number is the vulnerability the whole distance scale shares. The tip and the Cepheids use the same calibrators, so they test each other’s middle step but not the calibrators underneath, and they inherit the parallax corrections in the Milky Way part of the calibration together.

The calibrators also differ in what kind of tip they offer. The Large Magellanic Cloud’s halo is near and well populated, so its tip is measured to a few hundredths of a magnitude, but its giants are a mixture of ages and metal contents unlike the old halos of distant spirals. The halo of NGC 4258 is old and metal-poor, like the targets, but seven megaparsecs away, so its tip is measured with the same precision as theirs and adds that uncertainty to every distance calibrated on it. The calibration therefore has to choose between a calibrator that is precise and one that is representative, and the published values differ by a few hundredths of a magnitude depending on the choice — about one per cent in the expansion rate.

The stars that have passed the tip

The stars that ignited helium do not vanish; they settle a few magnitudes fainter, burning helium in their cores for about a hundred million years. In a metal-rich population they collect in a tight clump on the colour–magnitude diagram, the red clump, whose brightness is also nearly fixed and is itself used as a distance indicator in the Milky Way and nearby galaxies. The clump and the tip are the before and after of the same event, and their brightness difference, about three and a half magnitudes in the I band, is set by the core mass at the flash. Measured in the same galaxy, the difference is a check that the population behaves as the calibration assumes, independent of the distance — one of the few internal consistency tests a single photometric data set can provide.

What the synthetic halo leaves out

The population is idealised: a single giant branch of one metal content, a power-law luminosity function with one slope, a fixed share of asymptotic giants, Gaussian photometric errors. Real halos have spreads in age and metal content, contamination by foreground stars and background galaxies, crowding that makes errors larger and non-Gaussian towards faint magnitudes, and in some galaxies a young population whose bright asymptotic giants can mimic an edge. The tip’s absolute magnitude also depends weakly on age for populations younger than a few billion years. Each enters the error budget of the real measurements, and the published analyses carry them; what the synthetic halo shows is the method’s skeleton — that the edge is a counting statistic, how its precision grows with the counts, and what choosing a detector costs.

Still open: whether the middle step of the local distance scale is right

The tip of the red giant branch and the Cepheids are two independent ways of crossing the same gap, and the expansion rate depends on which one is right to within a few per cent. The next decisive measurements are the ones that apply both, and a third method — the brightness of carbon-rich asymptotic giants, another edge in a luminosity function — to the same galaxies with the same telescope, analysed blind by teams that do not know the answer. If all three agree near 73, the local side is secure and the tension is with the early universe’s model. If they spread between 68 and 73, the local measurement’s uncertainty was underestimated, and part of the tension was a disagreement about how to find an edge.

Where this goes next

The Hubble constant the microwave background implies, with an early component. The Hubble constant that keeps the microwave background's measured angular sound horizon fixed, against the peak fraction of early dark energy at z = 3500, with the baryon density held at its standard value and the matter fraction at the 0.315 that late-time distances measure, so that the dark-matter density rises with the Hubble constant. With none, it is 67.4 km/s/Mpc and the horizon 144.3 Mpc. At 5 per cent it is 68.7; at 10 per cent 70.2, with the horizon at 138.5 Mpc; the local value of 73 is reached at 19 per cent. The shrinkage of the horizon is what raises the constant: a shorter ruler at the same angle must be nearer, and nearer at a fixed matter fraction means faster expansion. The extra dark matter this requires is the cost the published fits also find: at the value that reaches 73, the physical matter density is 18 per cent higher than in the standard model. A ruler shortened by something already gone The microwave background gives the Hubble constant only through a computed length, the distance sound travelled before the universe became transparent. Add a component of energy that exists for a few thousand years before recombination and then vanishes faster than radiation, and the universe expands a little faster while the sound is travelling, the ruler comes out a few per cent shorter, and the inferred expansion rate rises toward the local value. The price is paid in dark matter and in the finest details of the oldest light. H₀: nine determinations in two families. Published determinations of H₀, each with its quoted one-sigma interval, sorted into two families — measured locally, in the late universe, against inferred from z ≈ 1100 through a model. The shaded band behind each family is that family's inverse-variance weighted mean: 72.66 ± 0.75 across 5 of them, against 67.40 ± 0.41 across 4. The difference is 5.26 ± 0.85 km/s/Mpc, which is 6.2 standard deviations, computed here from the quoted errors alone. That number is an upper bound on the significance rather than the significance: the determinations within each family share calibrations, samples and in two cases the same supernovae, so they are not independent, and a correlated pair combines to something wider than the formula used here gives. What the figure does establish is that the split is not one discrepant measurement against a consensus — it is two internally consistent groups, and the grouping is by method rather than by result. The same constant, measured twice, five sigma apart Cepheids and supernovae give an expansion rate of about 73 kilometres per second per megaparsec. The microwave background gives 67.4. Both quote errors near one per cent, both have been rebuilt from scratch by rival teams, and the gap between them has grown as the measurements have improved. A Cepheid's velocity, radius and angular size through two pulsation cycles. Three curves for a model Cepheid with δ Cephei's period of 5.37 days and a mean radius of 44 solar radii, at 272 parsecs, through two cycles. Top: the radial velocity of its spectral lines, in km/s, oscillating about the star's own velocity of -16.8 km/s — negative is towards the observer, so the surface is expanding when the curve is lowest. Middle: the change in radius, from integrating that velocity over time and multiplying by the projection factor p = 1.3: a swing of 4.9 solar radii, 11 per cent of the star. Bottom: the angular diameter the same star presents, in milliarcseconds, swinging by 168 microarcseconds about 1.505. The middle and bottom curves have the same shape because they are the same motion — one in kilometres, one in angle — and their ratio is the distance. A distance that rests on one unmeasured number A pulsating star's swelling can be measured twice — in kilometres from the Doppler shift of its lines, and in milliarcseconds by an interferometer — and the ratio is its distance, with no calibration against any other star. Except one: the factor that turns a line's velocity into the speed of the surface, which geometry puts at 1.33 to 1.5, measurement puts at 1.2 to 1.35, and every distance carries in proportion.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Cepheid variableDegeneracy pressureHelium flashHubble constantLuminosity functionRed giantTip of the red giant branchA type Ia supernova