A metallicity term too small to close a gap

A Cepheid variable’s period of pulsation tells its luminosity, and its luminosity and its apparent brightness tell its distance. That is the relation Henrietta Leavitt found in the Small Magellanic Cloud in 1912, and it is still the foundation of the local measurement of the Hubble constant: Cepheids measured in the Milky Way, the Large Magellanic Cloud and the galaxy NGC 4258, all at geometrically known distances, calibrate the relation; Cepheids in a few dozen nearby galaxies that have hosted type Ia supernovae give those galaxies’ distances; and the supernovae, so calibrated, measure the expansion rate far beyond. A star that tells its distance by how slowly it blinks carries the whole chain.

The essay on the scatter that points the way dust does showed how the Wesenheit magnitude — a combination of two bands that cancels dust’s dimming — tightens the relation to a scatter of under a tenth of a magnitude. It ended on the smaller effect that tightness makes visible: whether Cepheids of the same period are brighter or fainter in galaxies with more metals. The local Hubble constant is now eight per cent higher than the value the microwave background predicts, at a significance of five standard deviations, and the metallicity dependence of the Cepheid relation is one of the places a systematic error could hide.

This essay asks how large an effect could hide there. The answer is a short calculation, and it rules the metallicity term out as the explanation by a wide margin, while showing exactly where the term still matters.

A relation that depends on what the star is made of #

A Cepheid’s luminosity at a given period depends on its mass, and the relation between mass and luminosity depends on the star’s chemical composition. A star richer in metals has a higher opacity, a slightly different structure on its loop through the instability strip, and a different distribution of its light across wavelengths. All of these shift its position at a given period by small amounts, and their net effect on the Wesenheit magnitude is the metallicity term, γ, in magnitudes per dex — per factor of ten in the abundance of iron.

A term of minus two tenths of a magnitude per dex makes the Milky Way’s Cepheids, whose iron abundance is about solar, nine hundredths of a magnitude brighter than the Large Magellanic Cloud’s, which have about forty per cent of the Sun’s iron, and fifteen hundredths brighter than the Small Cloud’s. Those offsets are comparable with the relation’s own scatter, so they are invisible in any single galaxy’s Cepheids and have to be measured from the offsets between galaxies of known distance.

Why metals touch a pulsation at all #

A Cepheid pulsates because a layer in its envelope acts as a valve. Where helium is being ionised for the second time, at about forty thousand kelvin, compression makes the gas more opaque rather than less, so the layer traps heat when squeezed and releases it on expansion, pushing the pulsation along — the valve that has to sit at the right depth for the star to pulsate at all. The depth of that layer below the surface depends on the star’s temperature, which is what confines Cepheids to a narrow instability strip in the diagram of colour and brightness.

Metals enter twice. They contribute to the envelope’s opacity directly, through the bound–bound absorption of iron and its neighbours, which changes how the star’s luminosity relates to its mass and so where it sits on the period–luminosity relation. And they change the star’s earlier evolution: the blue loop that carries a helium-burning giant back across the instability strip extends further or less far depending on composition, and a star crosses the strip at a different luminosity for its mass. The term in the relation is the net of these effects, and the effects partly cancel, which is why the net is small and why its sign could be argued about.

The chain the term travels along #

The Hubble constant measured locally is the last link of a chain in which every distance is measured with the last one. Geometric distances to the calibrators calibrate the Cepheids’ zero point; Cepheids in the hosts give the hosts’ distances; the hosts’ supernovae calibrate the supernovae’s peak luminosity; and supernovae in galaxies far enough away for their own motions to be negligible measure the expansion. Each link transfers a calibration to objects that differ from the ones it was set on, and the metallicity term is the correction for one such difference: the Cepheids at the second link are not made of the same material as the Cepheids at the first.

The chain’s strength is that each link has been checked by independent routes: four distances to the same galaxy agree within their errors, and the calibrated supernovae can be checked against those calibrated by other candles. The metallicity term is one place where the check can be made precisely, because the calibrators span a range of composition and the term can be measured on them directly.

Why the sign was argued about for decades #

For most of the twentieth century the metallicity term was not known to better than its sign, and the sign was disputed. Theoretical models of Cepheid pulsation predicted that metal-rich Cepheids should be fainter in visible light at a given period; empirical comparisons of Cepheids at different radii in the same galaxy, where the metallicity changes along the disc, found them brighter. The disagreement came partly from what was being compared: a metallicity effect on the visible magnitudes, on the colours and on the bolometric corrections all at once, so that a term measured in one band and one colour combination need not resemble the term in another.

The Wesenheit magnitude reduces the problem by combining bands, and the near-infrared reduces it further, because the infrared light of a Cepheid is less sensitive to the absorption lines of metals than the blue. What remains is smaller and has settled, in the last decade, on a negative value: metal-rich Cepheids are brighter in the Wesenheit magnitudes used for the Hubble constant, by between a tenth and three tenths of a magnitude per dex depending on the analysis.

Four calibrators, one slope #

The term is measured from the calibrators themselves.

Each calibrator whose distance is known geometrically gives one point: the offset of its Cepheids’ relation from a reference, at its Cepheids’ metallicity. The Milky Way’s Cepheids have parallaxes from the Gaia satellite; the Large Magellanic Cloud’s distance comes from eclipsing binaries to about a per cent; NGC 4258’s from the orbits of water masers round its central black hole, to about one and a half; the Small Magellanic Cloud’s, again from eclipsing binaries. The slope of the line through the four is the metallicity term.

Each point’s error bar is its distance’s uncertainty, a few hundredths of a magnitude. With four points spread over eight tenths of a dex in metallicity, the slope is fixed to about four hundredths of a magnitude per dex — precise enough to say the term is negative and to put it near two tenths.

The lever #

The precision depends on the calibrators’ spread in metallicity much more than on their number.

The Milky Way and NGC 4258 have Cepheids of similar metallicity, and two points close together fix a slope poorly: at two hundredths of a magnitude each, they leave the term uncertain by nearly two tenths of a magnitude per dex, as large as the term itself. Adding the metal-poor Magellanic Clouds stretches the lever arm to eight tenths of a dex and shrinks the uncertainty five-fold. The slope is measured best between calibrators unlike the hosts it is applied to — which means it is measured mostly on metal-poor Cepheids and applied to metal-rich ones, on the assumption that the relation is linear in metallicity across the whole range.

How far a wrong term moves the Hubble constant #

The term enters the Hubble constant through the difference between the calibrators’ metallicities and the hosts’.

If the hosts’ Cepheids were exactly as metal-rich as the calibrators’, the term would cancel entirely: the same correction would apply to both ends of the chain. They are not quite. The supernova hosts are large spiral galaxies whose Cepheids are about as rich as the Milky Way’s, while the calibrators, weighted by how many Cepheids and how precise a distance each contributes, include the metal-poor Magellanic Clouds. The effective difference is about a tenth to two tenths of a dex.

An error δγ in the adopted term then shifts every host’s distance modulus by δγ times that difference, the calibrated supernovae’s absolute magnitude by the opposite amount, and the Hubble constant by a factor of ten to the minus one-fifth of their product. A term wrong by two tenths of a magnitude per dex — as large as the term itself — with hosts two tenths of a dex richer moves the Hubble constant by under two per cent. The direction depends on the sign: a term too weak in magnitude inflates the local Hubble constant, a term too strong deflates it.

What the tension would need #

Turn the calculation round and ask what error would explain the whole difference.

To move the Hubble constant by eight per cent with a two-tenths-of-a-dex difference, the term would have to be wrong by almost nine tenths of a magnitude per dex — four times larger than the term, and twenty times its present uncertainty. With a one-tenth difference, by 1.7. No measurement of the term has come close: the analyses that disagree with each other do so by a tenth or two, and they agree on the sign.

That is a clean exclusion, and it is the kind this measurement allows: not a proof that the local Hubble constant is right, but a demonstration that this particular systematic error cannot be the reason it is high. The essay on the same constant measured twice, five sigma apart listed the systematic errors the local measurement has been checked against; the metallicity term belongs among those that are well understood and too small. The same arithmetic disposes of it as an explanation in either direction: a term wrong the other way, too strong rather than too weak, would lower the local value and widen the tension rather than close it. Whatever the tension’s origin — a statistical fluctuation, a systematic error elsewhere in the chain, or new physics that changes the early universe’s prediction — the composition of Cepheids accounts for at most a quarter of it, and only if every estimate of the term so far has been wrong by its own size in the same direction.

A steeper term #

The calculation is linear, so it is easy to see what a steeper term would change.

A term of minus four tenths, twice the usual value, would put the Small Cloud’s Cepheids a third of a magnitude faint of the Milky Way’s at the same period — an offset the calibrators’ distances, good to a few hundredths, would reveal immediately. The calibrators constrain the term directly, so it cannot be large without the calibrators disagreeing with each other. The one way to hide a large effect would be a term that is not linear: flat across the metal-poor range where the calibrators measure it, and steep in the metal-rich range where the hosts live. No physical model predicts that, and the Milky Way’s own Cepheids, which span a few tenths of a dex across the Galaxy’s disc and have Gaia parallaxes, can test it directly in the metal-rich range.

Which Cepheids a host gives up #

The hosts’ effective metallicity depends on which of their Cepheids are measured, and that is decided by what can be seen. In a galaxy twenty or thirty megaparsecs away, Cepheids near the crowded centre are blended with their neighbours and hard to measure; those in the outer disc are cleaner. Disc galaxies are richer in metals towards their centres, by a few hundredths of a dex per kiloparsec, so a sample drawn preferentially from the outskirts is more metal-poor than the galaxy’s Cepheids as a whole. The selection is not a flaw in the analysis — it is the same principle that makes a sample brighter than the population it came from — and it is handled by measuring each Cepheid’s metallicity from its position on the host’s gradient. But it means the hosts’ metallicity is a property of the survey as well as of the galaxies, and it changes as the survey improves: near-infrared imaging from space, which resolves crowded fields better, reaches further into the metal-rich centres, and moves the effective difference between hosts and calibrators.

Where else the metals hide #

The metallicity term is not the only place composition enters the chain. The supernovae themselves may depend on their hosts’ metallicity or age, through the mass of nickel their explosions make; the tip of the red giant branch, which offers an independent route to the same host galaxies without Cepheids, depends on metallicity through the giants’ colour, and is measured in the old, metal-poor haloes of the hosts rather than in their discs. Each route has its own composition dependence, and they differ.

That difference is useful. The two routes disagree with each other by a few per cent in some analyses and agree in others, and the disagreement, where it exists, is the right size to come from the composition dependences of one or both — and much too small to account for the tension. The Hubble tension is eight per cent between the local measurements and the early universe; the disagreement between local measurements is a fraction of that. Whatever composition does to the local chain, it does it at the level of the local measurements’ disagreement with each other.

What the arithmetic assumes #

The term is linear. Every number here assumes the offset is proportional to the metallicity. A curved dependence would make the term measured on metal-poor calibrators inapplicable to metal-rich hosts, and is the one loophole the calibrators cannot close without more metal-rich calibrators.

Metallicity is one number per galaxy. Each galaxy’s Cepheids are given a single metallicity. Real discs have gradients of a few hundredths of a dex per kiloparsec, and the Cepheids observed in a host are drawn from a range of radii that depends on crowding and on the survey’s field; their mean metallicity is an estimate with its own error.

The calibrators’ metallicities are approximate. The values used are representative; published values differ by a tenth of a dex depending on whether iron or oxygen abundances are used and how they are converted.

The calibrators’ distances are taken as given. The Milky Way’s Cepheids rest on Gaia parallaxes with a zero-point correction of their own, and other geometric methods for Cepheids, such as the one that rests on one unmeasured number — the projection factor converting a line’s velocity into the surface’s — carry systematic errors comparable with the term’s effect. An error in one calibrator’s distance tilts the fitted slope, and the Small Cloud, at the end of the lever, has the most leverage.

The hosts’ difference is an estimate. The effective metallicity difference between hosts and calibrators depends on how the calibrators are weighted, and is between a tenth and two tenths of a dex in published analyses.

Still open: whether the relation stays straight where the hosts are #

The metallicity term of the Cepheid relation is measured, negative, near a fifth of a magnitude per dex, and fixed by calibrators that span eight tenths of a dex. Within any plausible error it moves the local Hubble constant by a per cent or two, and the tension with the early universe would need an error several times larger than the term. The term is not the explanation.

What remains open is the shape of the dependence in the metal-rich range, where the hosts are and where the calibrators are fewest. The Milky Way’s Cepheids, with parallaxes now good to a few per cent for hundreds of stars across the Galaxy’s metallicity gradient, can measure the term within the metal-rich range alone; space-based near-infrared observations of Cepheids in the hosts themselves, less crowded and less sensitive to metals, reduce the term’s leverage further. If the term stays straight, it leaves the local measurement where it is, and the tension, if it is not a statistical accident, needs physics in the early universe or an error somewhere else in the chain.