A null that stays put only if the sky cooled on schedule

Every statement about the microwave background at an earlier time rests on a one-line law: its temperature was higher in the past in proportion to one plus the redshift, $T(z) = T_0(1+z)$. The law is so familiar that it reads like a definition. It is a prediction. It holds if the expansion stretches every photon’s wavelength by the same factor, if photons are neither made nor destroyed along the way, and if nothing feeds the background energy after it was released. Each of those is an assumption about the universe, and an assumption can be checked.

Checking it needs a thermometer at a distance, which sounds like a contradiction. The background reaches the observer at one temperature, from one epoch, and it has no way to say what temperature it had when it passed some galaxy eight billion years ago. What can say so is something that was standing in the radiation at that time and was changed by it. Two very different objects do that. One is a cold molecule whose lowest rotational levels are populated by the radiation around it. The other is a cluster of galaxies, whose hot gas reshapes the spectrum of the photons passing through it, and whose shadow does not fade with distance however far away the cluster is.

The cluster’s version rests on a fact about that shadow which is usually stated as a convenience. The thermal distortion dims the sky at low frequency and brightens it at high frequency, and the crossing between the two lies at 217 gigahertz. Survey papers quote this one number for every cluster, near or far. It deserves a second look, because a cluster at redshift 1 saw the background at twice the temperature and at twice the frequency, and those two doublings have to cancel exactly for the number to be the same.

Why one curve serves every cluster #

Inverse Compton scattering moves photons upward in frequency while conserving their number. The shape of the resulting distortion, measured in thermodynamic temperature, depends on a single dimensionless ratio, the photon’s energy over the thermal energy of the radiation at the place where the scattering occurred:

$$\frac{\Delta T}{T} = y\left[x\coth\frac{x}{2} - 4\right], \qquad x = \frac{h\nu_{\rm rest}}{kT(z)}.$$

The quantity $y$ is the Compton parameter, and it carries everything about the gas. The bracket carries everything about the radiation, and it passes through zero at $x = 3.83$. Now suppose the law is written with room to fail, $T(z) = T_0(1+z)^{1-\beta}$. The rest frequency of a photon seen today at $\nu$ is $\nu(1+z)$, so

$$x = \frac{h\nu}{kT_0}\,(1+z)^{\beta}.$$

On the law, with $\beta = 0$, the redshift disappears. Every cluster at every distance draws the same curve against observed frequency, and the null falls where $h\nu/kT_0 = 3.83$, at 217.5 gigahertz. With any other value it does not disappear, and the null belonging to a cluster at redshift $z$ moves to $217.5\,(1+z)^{-\beta}$ gigahertz.

So the coincidence in the survey papers is a measurement waiting to be read. The null is one number that a cluster supplies without reference to the cluster’s mass, its pressure or its distance. Whether two clusters at different redshifts share it asks only whether the background cooled on schedule between them. That statement is the same as the one that the shadow does not fade, seen from the frequency axis instead of the brightness axis: the signal is independent of redshift because the source it is imprinted on redshifts along with it. If the background had cooled differently, the independence would fail, and it would fail in frequency before it failed anywhere else.

A thermometer read in 1941 #

The other thermometer is older than anyone’s knowledge of what it was reading. In 1941 Walter Adams, working from high-dispersion spectra at Mount Wilson, recorded absorption lines of the cyanogen radical CN in the light of stars behind interstellar clouds. Most came from the molecule’s lowest rotational level, as they should in gas that has had time to cool. A weaker line came from the next level up. Andrew McKellar used the ratio of the two to compute what temperature would populate that upper level, and he reported a figure of about 2.3 kelvin.

The measurement was correct to within what its errors allowed, and it was made twenty-four years before Arno Penzias and Robert Wilson found an unexplained excess in a radio horn at Holmdel. Nobody asked what could be exciting the molecule. Gerhard Herzberg, quoting the number in his 1950 textbook, remarked that it had only a very restricted meaning, and that judgement was reasonable: a temperature derived from one transition can be made by collisions with electrons, by local radiation, by many things. Only in 1966, after the Holmdel result, did George Field and others, and independently Iosif Shklovsky, realise that CN clouds in every direction were reading one number. The interpretation has an exacting test built into it, since the molecule’s first excited level sits at a wavelength of 2.6 millimetres, far closer to the blackbody peak than the centimetre bands of the first radio measurements. The CN value was among the first evidence that the radio excess had the spectrum of a blackbody at all.

The lesson that carried forward is that a molecule is a thermometer that can be moved. Its energy levels are fixed by quantum mechanics and do not care where it is. Put the same absorber in a cloud at high redshift, in front of a bright quasar, and the ratio of its populations records the radiation there, at the epoch when the quasar’s light passed through. John Bahcall and Richard Wolf pointed this out in 1968. Because the radiation was hotter then, levels too high to be occupied today become occupied, and the transition with the right energy for the job changes with the redshift. Neutral carbon’s fine-structure levels became the favoured probe in the 1990s, and then carbon monoxide, whose lines in the ultraviolet can be seen in absorption against a quasar and whose sequence of levels gives several ratios at once from one cloud.

The direct points #

The measurements are few, because the clouds have to be just right: dense enough to contain molecules, thin enough to transmit the quasar’s light, and not so dense that collisions swamp the radiative excitation. Each result is really an upper limit, since anything else that excites the molecule only raises the temperature it seems to imply. Correcting for collisions requires a model of the cloud, which is built from its other lines.

These points are direct in a way nothing else in this subject is. They are temperatures at times far earlier than any supernova sample reaches, and the carbon monoxide results taken together hold $\beta$ to about three per cent. The furthest, published in 2022, used a different trick. Water molecules in a cold companion of a dusty galaxy at redshift 6.34 are seen absorbing the microwave background itself, because at that redshift the background is warm enough to pump the molecule’s levels out of balance with their surroundings. Its range is wide. Even so it reaches back to when the universe was less than a billion years old, further than any cluster will be found, and it agrees with the law.

What these points lack is number. A handful of sightlines is too few to tell a real trend from peculiarities of individual clouds. A bias of half a kelvin in the correction for collisions would be a large share of the signal at redshift 2. The two thermometers are also complementary in range: molecules work above redshift about 1.5, where the ultraviolet lines come within reach of a ground-based spectrograph, and clusters work below about 1, where they can be found in numbers and resolved in several bands. The gap between them is the part of cosmic history in which dark energy came to dominate.

Reading the crossing #

In practice nobody measures the null directly. There is little signal to be seen near a zero, so the experiment measures the distortion in the bands on either side of it and asks where the curve through those measurements must cross. That is why the method needed a satellite: at 143 and 353 gigahertz, a pair on opposite sides of the null, the atmosphere is hostile to one and opaque enough to spoil the other. The idea of using the ratio goes back to Fabbri, Melchiorri and Natale in 1978, and to Yoel Rephaeli in 1980, long before a telescope could perform it.

The attractive feature of a ratio is what cancels. The Compton parameter, the cluster’s mass and shape and distance, and the angular size of the patch being measured all cancel, leaving only the frequency dependence. A cluster is therefore a self-calibrating object: nothing has to be known about it except its redshift. Compare the length that the same effect gives when combined with X-rays, which needs a model of the gas’s whole structure and suffers from the unknown elongation of each cluster. For the temperature law, all of that is irrelevant. The same object measured for a distance is plagued by its own structure, and measured for a temperature it is not.

The price is a small lever. A departure of $\beta = 0.01$ moves the null by less than one per cent of its frequency at redshift 1, and the lever grows only logarithmically with redshift, since it is the exponent of one plus redshift that is being measured. The clusters most valuable for this test are distant ones, but those are hardest to resolve, and a cluster barely resolved by the beam leaks its signal between bands that have different beam sizes. Nearby clusters are bright and resolved, but they carry little information per cluster.

Those are the numbers an idealised calculation yields, and the actual results make plain how much they leave out. Analyses of the Planck clusters give $\beta = 0.012 \pm 0.016$ from one treatment and $0.009 \pm 0.017$ from another, and combining those with molecular absorbers brings the error to about a per cent. They needed many more clusters than the idealised line says, because the limiting noise at these frequencies is not in the receiver. It is the cosmic infrared background from unresolved galaxies, the dust of the Milky Way, and the infrared galaxies within the clusters themselves, all of which rise steeply above 217 gigahertz and all of which enter the high band and not the low.

The thing that moves the null for a duller reason #

There is a more serious contaminant, and it comes from the cluster’s own physics. The formula above is the non-relativistic one, and in a gas of 10 keV the electrons move at about a fifth of the speed of light. As the account of the relativistic correction showed, this flattens the curve and pushes the null upward. Its size is not a detail in this setting.

The null is therefore a thermometer with two bulbs. It responds to the temperature of the background and to the temperature of the gas, and a measurement at redshift 1 cannot separate them on its own. A shift of 1.5 gigahertz corresponds to the whole of the current limit on $\beta$, and an error of 3 keV in a cluster’s temperature produces that same shift. Hot clusters are also the brightest, so the sample with the best signal is the one with the largest correction. The two effects even have a partial correlation with redshift built into any catalogue: a survey selected on signal finds only massive, hot clusters at large distance, while at small distance it finds cooler ones as well, so the mean electron temperature rises with redshift along the very axis on which the departure is sought.

What saves the method is that temperatures can be measured separately. X-ray spectra give the gas temperature to a few per cent for the nearer clusters, and the correction can be applied. The dependence on gas temperature also differs from the dependence on the background’s temperature when a whole spectrum is available, because the relativistic correction changes the shape of the curve at high frequency in a way a simple shift of the axis would not. With two bands the two effects are mixed. With five or six they begin to separate, and that is the argument for the many-band instruments now being built, which could reach $\beta$ at a few parts in a thousand if the foregrounds cooperate.

What a nonzero β would mean #

The test would be of limited interest if a failure could be absorbed by a small change in some parameter, but that is not how it would go. The law follows from general principles. If photons travel on null geodesics in an expanding metric and their number is conserved, a blackbody remains a blackbody with temperature falling as the scale factor. To break the law without destroying the spectrum, something must trade energy with the photons, or photon number must change. The spectrum measured by the satellite in 1990 constrains exactly such a trade, at very early times and very tightly. Later, at low redshift, the limit is much weaker, and that is the epoch these tests probe.

Models with such a trade exist, and they are mostly motivated by dark energy. If a cosmological term that behaves so awkwardly were not a constant but were decaying, it might decay into something, and photons are one candidate. That would feed energy to the background at late times and make it cool more slowly than scheduled, which would be a positive $\beta$. A coupling between photons and a light scalar field or axion in intergalactic magnetic fields could work in the opposite direction, converting photons away.

Which connects this to a second test that seems to have nothing to do with it. Luminosity distance and angular-diameter distance must obey $D_L = (1+z)^2 D_A$ if photons are conserved. A process that destroys or creates photons breaks that relation, since a source becomes fainter or brighter than its angular size implies. So the same physics would appear in two places: the temperature of the background at redshift $z$, read from clusters and molecules, and the dimming of supernovae relative to the sizes of standard rulers.

The tie was worked out by Anastasios Avgoustidis and collaborators, and it turns a pair of independent measurements into a consistency test. The relation holds only for adiabatic processes, those that leave the spectrum thermal, so a measurement off the line would itself be informative: it would mean that whatever changes the photon number also distorts the spectrum, or that the supernovae are being dimmed by something that has nothing to do with the background. The temperature test also has an advantage over the supernova version, in that it depends on no standard candle at all. Its limit along the line is among the most stringent on $\varepsilon$ that exists, from a method whose only calibration is the frequency response of a radiometer.

The check that nobody needed, until it might be needed #

So far the test has found nothing. Every measurement agrees with $\beta = 0$, and the combined limit of about a per cent is a real achievement given what it had to contend with. It would be natural to ask why anyone would continue, and the answer is that the law is an input to almost everything else in this subject. The conversion between the velocity a cluster’s map records and a microkelvin signal assumes the background’s temperature at that redshift. The redshifted 21-centimetre trough is a contrast between the temperature of the gas and that of the radiation behind it. The ratio of thermal populations in a cloud’s fine-structure lines is corrected for pumping by the background. An error in the law would be folded into all of them, quietly, and would be mistaken for something about the objects.

There is a further point in the fact that a null can be measured at all. The same independence of redshift that makes the effect so useful for finding clusters is what also makes it a clean check of the assumption behind that independence. An experiment designed for one purpose thus tests the foundation of its own method. That is unusual, and in this case it worked out in the method’s favour.

What the picture cannot show #

The figures treat each cluster as a pure thermal distortion plus a stated relativistic correction. A real cluster carries also the kinematic signal of its bulk motion, which has a different spectrum and does not vanish at the null: a cluster moving at 300 kilometres a second adds a signal that displaces the apparent crossing by more than the shifts discussed here, for any one object, and only the average over a large sample removes it. The dust within clusters, and the unresolved infrared background behind them, have been left out of the count of clusters altogether.

The power law is also only a convenient form. A real physical departure would have a shape determined by when and how the energy was exchanged, perhaps nothing at low redshift and a rise beyond it, and a test with a single exponent would average it away. The molecular points are shown at face value, with the published allowance for collisions and pumping, which is itself model-dependent. And the relation between β and ε holds only for the class of adiabatic processes.

What the test cannot do at all is reach back past the first clusters and the first quasars with an equivalent thermometer. Between the furthest molecular point and the surface the background itself was released from there is a very long stretch with almost no direct temperature measurement. The law’s validity there is inferred from the shape of the spectrum today and the acoustic peaks at the far end. It is an argument at both ends and a gap in the middle.

Still open: whether the law holds in the stretch where dark energy took over #

The stretch from redshift 2 down to 0.5 is the epoch in which the expansion began to accelerate, and so it is the epoch in which a decaying dark energy feeding the background would leave its largest mark. It is also where the two thermometers are weakest. Molecular absorbers begin to fail below redshift 1.5, and cluster samples thin out above redshift 1, and they become hard to resolve.

The microwave surveys now coming into operation will find tens of thousands of clusters out to redshift 2, in five or six bands and at arcminute resolution. Ground-based extremely large telescopes will extend molecular work to fainter quasars and perhaps to lower redshift with space-based ultraviolet instruments behind them. If the two methods then meet in that interval and each holds $\beta$ to a few parts in a thousand, the law will have been verified across the epoch where it is most in question. Whether foregrounds in the high-frequency bands permit that, or whether the limit stalls at a per cent for reasons unrelated to the receivers, is a question about dust as much as about cosmology, and the cluster counts being used to measure the growth of structure will have to settle the same question on the way.