A horizon drawn by light at three kelvin

The cosmic-ray spectrum runs as a nearly straight power law across eleven decades of energy, bends at the knee and the ankle, and then, above about fifty exaelectronvolts — $5\times10^{19}$ electronvolts, the kinetic energy of a well-struck tennis ball carried by one nucleus — falls away. The fall was predicted in 1966, within a year of the discovery of the microwave background, by Greisen and independently by Zatsepin and Kuzmin, and it was observed forty years later by detectors covering thousands of square kilometres. It is usually told as a triumph: a prediction about the highest-energy particles in nature, made from the coldest light in nature, confirmed.

The prediction is right. The question is whether it is the reason for what is seen. A cut-off at fifty exaelectronvolts has two possible causes, and they happen to sit at nearly the same energy. One is a horizon made by the microwave background. The other is the sources running out of accelerating power. Telling them apart turned out to need not the shape of the spectrum but the identity of the particles.

Cold light that is a gamma ray to a proton #

The microwave background fills space with about 410 photons per cubic centimetre — the light that makes the night sky dark rather than bright is outnumbered by it a hundredfold — each carrying on average $6\times10^{-4}$ electronvolts. To a proton at rest they are nothing. To a proton with a Lorentz factor $\Gamma$ they are blueshifted by a factor of about $2\Gamma$, and for a proton of $10^{20}$ eV, whose Lorentz factor is $10^{11}$, a typical background photon arrives in the proton’s frame with an energy of about a hundred million electronvolts — a gamma ray.

Above a photon energy of 145 MeV in the proton’s frame, the collision can make a pion: $p + \gamma \to p + \pi^0$ or $n + \pi^+$. The cross-section rises steeply above that threshold and peaks at 340 MeV, where the proton and photon briefly form the Δ(1232) resonance, and each such collision costs the proton about a fifth of its energy. So the rate at which a proton loses energy depends on how many background photons are energetic enough, in its frame, to cross the threshold. The background is a blackbody, whose photons fall off exponentially above a few times $kT$, and the fraction of them that can do this rises steeply with the proton’s energy.

The figure is computed from exactly that: the blackbody photon density at 2.725 kelvin, integrated against a resonance cross-section and the fraction of energy lost per collision. Nothing about the sources, the galaxy or the history of the universe goes in. Below about $6\times10^{19}$ eV hardly any background photons reach the threshold, and the only energy loss is the stretching of the proton’s momentum by cosmic expansion, which takes a Hubble length of about four gigaparsecs. Above it, the loss length collapses by two orders of magnitude in less than a factor of two in energy. At $10^{20}$ eV it is 155 megaparsecs; at $10^{21}$, where every photon in the peak of the blackbody is above threshold, it levels off at about 19. A proton seen above $10^{20}$ eV has come from within a hundred megaparsecs or so — a horizon drawn by light at three kelvin.

The same arithmetic, applied to gamma rays rather than protons, weighs the faint infrared background by the light it removes from distant blazars. There the target is a photon field whose density is uncertain and the projectile’s energy is known; here the target is the best-measured blackbody in nature and the projectile’s origin is the unknown. The CMB horizon is one of the few distances in astrophysics that is set entirely by a laboratory cross-section and a temperature.

Every proton ends near the threshold #

A horizon for particles that lose energy continuously has an odd property: it forgets where they started.

A proton far above threshold loses energy fastest, because its loss length is shortest; as it drops towards the threshold, its losses slow. So protons that start at very different energies converge. A proton made at $10^{22}$ eV and one made at $10^{20}$ eV are within a factor of two of each other after a hundred megaparsecs, and after three hundred they are within twelve per cent, both near $5.6\times10^{19}$ eV. The arriving energy of a proton that has travelled a few loss lengths says almost nothing about the energy at which it was made. What it says is that the source could make it and that the source was close enough for it to arrive — and the second constraint gets tighter as the energy rises.

That has a consequence for any spectrum of sources spread through space. Particles from distant sources pile up just below the threshold, having lost their excess on the way, and particles from nearby sources arrive with their original energies. A smooth injection spectrum becomes, at the Earth, a spectrum with a pile-up near the threshold and a steep fall above it. The fall is the prediction; the pile-up, smeared by the randomness of individual collisions and by the energy the protons also lose to making electron–positron pairs on the same photons below the pion threshold, is much less distinct in any real calculation than in a continuous-loss one, and the figures here do not try to draw it.

The distance to the sources closes #

The horizon becomes more useful when it is turned round and asked where the particles above a given energy come from.

For sources spread uniformly through space, nine-tenths of the protons arriving above $2\times10^{19}$ eV were made within about three gigaparsecs — most of the observable universe, in this simplified calculation that ignores the pair losses and the cosmic evolution of sources that matter at such distances. Above $4\times10^{19}$ eV that shrinks to about 940 megaparsecs. Above $6\times10^{19}$, to 220; above $10^{20}$, to 60. The horizon closes by a factor of fifty across a factor of five in energy.

That is the reason the arrival directions of the highest-energy particles were expected to become anisotropic even though the sky at lower energies is almost perfectly smooth. Below the threshold the sources are averaged over a gigaparsec or more, far beyond the scale at which the universe becomes homogeneous, and they should be spread as evenly as galaxies on that scale. Above it they are confined to the local few hundred megaparsecs, where matter is distributed very unevenly — in the supergalactic plane, in a few nearby clusters and voids — and the flux should follow that structure. The largest observatory found exactly such a signal at lower energy than expected: a dipole of about six per cent amplitude above $8\times10^{18}$ eV, significant at more than five standard deviations, pointing about 125 degrees from the galactic centre. It is extragalactic — a dipole in the arrival directions of charged particles, as the microwave background’s own dipole is a record of the observer’s motion, though here the cause is the uneven distribution of nearby sources smeared by magnetic fields — and it is the first thing the ultra-high-energy sky has shown. Above $4\times10^{19}$ eV there are warmer patches, near the direction of a nearby starburst galaxy and a region of the sky in the northern hemisphere, at the three- to four-sigma level.

A limit that draws the same line #

The spectrum’s fall is measured with great precision by now. The energy at which the flux drops to half of what the lower-energy power law would give, for the largest detector, is near $5\times10^{19}$ eV — close to where the computed horizon sets in. That was taken for a decade as confirmation.

The difficulty is that a source population with a finite maximum energy produces a cut-off too, and there is no reason for that maximum to be far above the horizon. The acceleration of a charged particle by a shock or any magnetic structure is limited by the requirement that the particle stay confined while it gains energy: its Larmor radius, which grows with its energy and shrinks with its charge and the field, must be smaller than the accelerating region. The maximum energy therefore scales with the particle’s charge, and the natural parameter is not energy but rigidity, energy divided by charge. A population of accelerators that all reach roughly the same maximum rigidity cuts off each species at its own energy: protons first, then helium at twice the energy, nitrogen at seven times, iron at twenty-six. Where the proton cut-off sits depends on the accelerators, and nothing prevents it from sitting near the pion threshold.

If it does, the spectrum cannot distinguish the two. A horizon removes particles after they are made; a limit prevents them being made; either produces a steep fall at roughly the same energy. The shapes differ in detail — a horizon produces a pile-up and, for hard injection, a recovery at higher energies; a limit produces neither — but those details are in the least populated part of the spectrum, where each energy bin holds a few dozen events.

The short list of accelerators that could #

The confinement requirement can be turned into a census of candidate accelerators, and it is short. A region of size $R$ and field $B$, moving or shocking at speed $\beta c$, can accelerate a particle of charge $Z$ to at most about $Z\beta\,(B/1\,\mu{\rm G})(R/1\,{\rm kpc})\times10^{18}$ eV — Hillas’s criterion, which says only that the particle’s orbit must fit inside the accelerator. Reaching a rigidity of $5\times10^{18}$ volts with shocks moving at a tenth of the speed of light needs a product of field and size of about fifty microgauss-kiloparsecs.

Supernova remnants, at a few hundred microgauss over a few parsecs, fall short by more than an order of magnitude; they are the accelerators of the knee, not of the cut-off. The objects that pass are few. The lobes of radio galaxies, with fields of ten microgauss over a hundred kiloparsecs, pass easily. The relativistic jets whose brightness says they are moving at nearly the speed of light pass, with $\beta$ close to one. The galactic winds of starburst galaxies, with fields of a hundred microgauss over a kiloparsec, pass marginally, and the warm patch above $4\times10^{19}$ eV lies near the direction of the nearest bright starburst. Gamma-ray bursts and newborn magnetars pass by virtue of enormous fields in small regions. The criterion is necessary and far from sufficient, and a maximum rigidity of a few times $10^{18}$ volts, if that is what the composition says, is a strikingly modest number for accelerators this capable — which is itself a clue that the particles may be escaping before they reach the limit the criterion allows.

What the particles are made of #

The difference between the two pictures is in the composition. A horizon made by pion production acts on protons, and a proton spectrum cut off by the CMB stays a proton spectrum to the end. A population of accelerators limited in rigidity, on the other hand, runs out of each species in turn, and the mixture arriving at the Earth grows heavier with energy.

The composition of cosmic rays at these energies cannot be measured particle by particle; the particles are never caught. What is measured is the air shower each one makes, and in particular the depth in the atmosphere at which the shower reaches its maximum number of particles. A nucleus of mass number $A$ and energy $E$ behaves roughly like $A$ separate protons of energy $E/A$, each starting its own cascade; lower-energy cascades develop sooner, so heavier nuclei make showers that peak higher in the atmosphere, by an amount proportional to $\ln A$. The fluorescence telescopes that watch showers develop on dark nights measure that depth for thousands of events, and its mean and spread give the mean and spread of $\ln A$ — with a systematic uncertainty from the models of hadronic interactions at energies far beyond any accelerator.

What they show is the pattern in the figure. The composition is lightest, closest to pure protons, at a few times $10^{18}$ eV. Above that it grows steadily heavier, and by $3\times10^{19}$ eV it is dominated by nuclei of intermediate mass. The spread of shower depths narrows as the energy rises, which a mixture of protons and iron would not do and a mixture of neighbouring species would. The fits that best match both the spectrum and the composition have the sources accelerating to a maximum rigidity of a few times $10^{18}$ volts, with a very hard injection spectrum — so hard that the ankle and the cut-off become the proton and heavier-species ends of one population. In that picture the fall at fifty exaelectronvolts is mostly the sources running out, and the microwave background adds to it rather than causing it.

Heavy nuclei have their own horizon, and it happens to fall near the same energy again. A nucleus is not destroyed by the background as a proton is slowed by it, but broken up: photons at a few to tens of MeV in its frame — from the microwave and the cosmic infrared backgrounds — knock nucleons out of it, a process called photodisintegration, and iron nuclei above a few times $10^{20}$ eV and lighter nuclei at lower energies have loss lengths of tens to hundreds of megaparsecs. So even for heavy nuclei the highest-energy sky is local, and the conclusion that the particles above the cut-off come from within a few hundred megaparsecs survives whichever picture is right.

Nuclei that cannot point #

The composition has a second consequence, and it is the one that most changes what the highest-energy sky can be used for. A charged particle is bent by magnetic fields on its way in, and the bending is set by its rigidity.

The Larmor radius of a particle is its energy divided by its charge and the field, so a proton and an iron nucleus of the same energy follow orbits whose radii differ by a factor of twenty-six. A proton at $10^{20}$ eV crossing a few kiloparsecs of the galaxy’s regular field is deflected by a few degrees — enough to blur, not enough to hide, the direction of its source. An iron nucleus of the same energy is deflected by nearly ninety degrees in the same field, and the turbulent component of the field adds a random scattering on top. If the particles above the cut-off are dominated by intermediate and heavy nuclei, as the air showers indicate, they arrive from directions that have been scrambled by tens of degrees at least.

That is why the ultra-high-energy sky has yielded a dipole and a few warm patches rather than a catalogue of sources, and why it probably always will. The hope of the 1990s — that the highest-energy particles, being few, local and rigid, would point back at their sources one by one — rested on their being protons. With a heavy composition the sources can be identified only statistically, by correlating the smeared arrival directions with maps of nearby galaxies, and the correlation is limited by the uncertainty in the galaxy’s own magnetic field as much as by the number of particles collected.

What the drawing leaves out #

The loss lengths in these figures include pion production and cosmic expansion and leave out pair production, which costs a proton about a gigaparsec near $10^{19}$ eV and shapes the ankle. They treat the losses as continuous, which is right for pair production and approximate for pion production, where each collision costs a fifth of the energy and a proton that happens to escape collisions for a while arrives higher than the average. They place sources uniformly out to four gigaparsecs with no evolution in their density, and ignore redshift in the background itself, which was hotter in the past. None of these changes the location of the horizon by more than tens of per cent, and all are included in the propagation codes that fit the measured spectrum and composition.

The composition itself is the least secure input. The depth of shower maximum is measured well, but converting it to $\ln A$ requires a model of how particles interact at centre-of-mass energies ten times higher than any collider reaches, and the models disagree with each other by about half a unit of $\ln A$ and with the measured numbers of muons in showers by tens of per cent. The heavier trend is seen with every model; its size is not agreed.

Still open: whether the horizon or the sources set the end #

Both of the explanations for the cut-off are physics, and both are presumably at work: the microwave background removes protons above the threshold whether or not the sources can make them. The question is which one dominates, and the answer bears on what the sources are. A cut-off made by the horizon says nothing about the accelerators except that they reach beyond it; a cut-off made by the sources measures their maximum rigidity, a few times $10^{18}$ volts, and rules out any accelerator capable of much more. The observatories are being upgraded to measure the muon and electromagnetic parts of each shower separately, which should give the composition event by event rather than on average. If the highest-energy events turn out to include a light component — a few per cent of protons would do — the horizon will be doing the cutting and point-source astronomy with them becomes possible again. If they are uniformly heavy, the end of the cosmic-ray spectrum is a measurement of the accelerators, and the three-kelvin horizon, correct as it is, merely draws a second line in almost the same place.