Astronomy & Celestial Mechanics
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Cosmology

A lower limit made of three lengths and a missing echo

The only measurement of a magnetic field in the empty space between galaxies is a lower limit, and it comes from light that did not arrive. Teraelectronvolt gamma rays from distant blazars make electron pairs in intergalactic space, the pairs should re-emit the energy at gigaelectronvolts in a halo around each source, and the halos are missing. Whether that absence weighs a field of a ten-thousandth of a billionth of a microgauss depends on three lengths, the field's own tangle, and how long the blazar has been shining.

Assumes Primordial fields, Olbers's paradox and Large-scale structure.

15 min read 6 figures Measuring the unreachableLight is the only messenger

Every magnetic field measured in the universe has been measured inside something. Planets, stars, galaxies and the gas in clusters of galaxies all carry fields, and in every case the field can be explained by a dynamo that amplified a much weaker one — which means, as the seed problem makes plain, that the field present today says nothing about the field the dynamo started from. The only place a primordial field would survive unamplified is the space where nothing has happened: the voids between the filaments of galaxies, tens of megaparsecs across and nearly empty.

Nothing in a void emits anything that a field would mark, and a void is emptier and expanding faster than the universe around it. There are no relativistic electrons to radiate synchrotron emission, too few thermal electrons for Faraday rotation to register, no dust to align. The one measurement that exists is indirect twice over. It uses gamma rays from sources outside the voids, electron–positron pairs made by those gamma rays inside the voids, and photons that those pairs should have sent toward the Earth and did not. The result, first published in 2010 and refined since, is a lower limit: a field of at least about 10−1610^{-16} gauss, filling a large fraction of intergalactic space.

That is ten-thousandth of a billionth of a microgauss, and the argument that yields it is a comparison of three lengths.

How far a void field turns the leptons that make each cascade photon. The deflection angle of a cascade lepton over its cooling length, in radians on a logarithmic scale, against the energy of the cascade photon it produces, for fields of 10⁻¹⁸ G, 10⁻¹⁷ G, 10⁻¹⁶ G, 10⁻¹⁵ G, 10⁻¹⁴ G coherent over at least that length. The angle falls as the square of the lepton energy, so as the inverse of the photon energy: the gigaelectronvolt halo is made by the slowest leptons, which are turned most. At 1 GeV the deflections are 1e-3, 0.01, 0.11, 1.06, 10.62 rad. The dashed line is the jet's half-opening angle, 0.1 rad; a lepton turned further than that sends its photons away from the observer, and the halo at that energy disappears.
Fig. 1 How far a void field turns a cascade electron over the distance it travels before it has given up its energy, against the energy of the photons that electron makes, for fields from 10−1810^{-18} to 10−1410^{-14} gauss. The deflection falls as the inverse of the photon energy. The dashed line is the half-opening angle of a blazar’s jet; electrons turned further than that send their photons away from the observer.

Light that is stopped by light

A blazar is an active galactic nucleus with a jet pointed nearly at the Earth, and the brightest of them emit gamma rays above a teraelectronvolt. Photons that energetic cannot cross the universe freely. When one meets an infrared or optical photon of the faint extragalactic background light — the accumulated starlight of all galaxies through cosmic history — the two can annihilate into an electron and a positron, provided the product of their energies exceeds the square of the electron’s rest energy. A ten-teraelectronvolt gamma ray finds a suitable partner in the near infrared, and the mean distance it travels before doing so is of order a hundred megaparsecs; a one-teraelectronvolt gamma ray goes several times further.

That absorption is measured. The spectra of distant blazars steepen above a teraelectronvolt exactly as the background light predicts, and the steepening is how the background light’s intensity is known best. So the first length in the argument, the pair-production length, is not in doubt, and it is long: the pairs are made far from the blazar, typically in the voids along the line of sight, because voids are most of the volume — a statistical statement about the universe’s structure that holds for any line of sight long enough.

Three lengths

The pair then has a second fate. Each lepton carries about half the gamma ray’s energy and moves through the microwave background, which to a lepton of a teraelectronvolt is a dense bath of photons. It scatters them by inverse Compton scattering, boosting each by the square of its Lorentz factor, and loses its own energy in the process. The distance over which it loses most of its energy is the cooling length, a fraction of a megaparsec at a teraelectronvolt, shorter at higher energy.

The third length is the gyroradius: how tightly a void field would curl the lepton’s path. It grows in proportion to the lepton’s energy and in inverse proportion to the field.

Three lengths in a gamma-ray cascade. Three lengths against the energy of an electron or positron made in the cascade, both on logarithmic scales. The pair-production length of the parent gamma ray, twice the lepton's energy, is tens to hundreds of megaparsecs: pairs are made far from the blazar, out in intergalactic space and mostly in voids. The inverse-Compton cooling length is a fraction of a megaparsec at a teraelectronvolt, and shorter at higher energy. The gyroradius, in fields of 10⁻¹⁵ G and 10⁻¹⁷ G, grows with energy. Where the gyroradius is comparable to the cooling length, the lepton is turned through a large angle before it has given its energy away, and its cascade photons miss the observer. At 10⁻¹⁵ G the two cross at 0.58 TeV; at 10⁻¹⁷ G, at 0.058 TeV.
Fig. 2 The three lengths against the lepton’s energy. The parent gamma ray’s pair-production length is tens to thousands of megaparsecs; the cooling length a fraction of a megaparsec; the gyroradius, for fields of 10−1510^{-15} and 10−1710^{-17} gauss, grows with energy. Where the gyroradius falls to the cooling length the lepton is turned through a full radian before it has cooled: at 10−1510^{-15} gauss that happens below about 0.6 teraelectronvolts.

The deflection over the cooling length is the ratio of the two short lengths, δ≈DIC/RL\delta \approx D_{\rm IC}/R_L, as long as the field keeps one orientation over the whole cooling length. Because the cooling length falls as the inverse of the energy and the gyroradius rises in proportion to it, the deflection falls as the inverse square of the lepton’s energy. Low-energy leptons are turned far more than high-energy ones — and it is the low-energy leptons that make the gigaelectronvolt halo.

Why the halo is at a gigaelectronvolt

The energy a scattered microwave-background photon ends up with is set by the lepton’s Lorentz factor squared. A background photon’s mean energy is about six ten-thousandths of an electronvolt; a lepton of one teraelectronvolt has a Lorentz factor of two million; the product, with a factor of four-thirds, is a few gigaelectronvolts.

The energy a cascade lepton gives a background photon. The typical energy of a microwave background photon after one inverse-Compton scattering by a cascade lepton, against the lepton's energy, both on logarithmic scales: (4/3) γ² times the photon's mean energy, 3.2 GeV times the square of the lepton's energy in teraelectronvolts. The band marks the energies a space gamma-ray telescope measures well, 0.1 to 100 GeV; they are produced by leptons of 0.18 to 5.6 TeV, which come from primary gamma rays of twice that energy. A blazar observed at a few teraelectronvolts from the ground should therefore light up the gigaelectronvolt sky around itself, and the halo's spectrum is fixed by this square law.
Fig. 3 The energy of a microwave-background photon after inverse-Compton scattering by a cascade lepton, against the lepton’s energy: about 3.2 gigaelectronvolts times the square of the lepton’s energy in teraelectronvolts. The band is the 0.1 to 100 gigaelectronvolt range that a space gamma-ray telescope measures well. It is filled by leptons of 0.2 to 6 teraelectronvolts, made by primary gamma rays of twice that energy.

That square law ties the whole argument together. A blazar whose teraelectronvolt spectrum is measured from the ground should, by energy conservation, reappear in the gigaelectronvolt band as a cascade — reprocessed light whose total power equals the absorbed power, arriving from a slightly larger angle and slightly later. For a handful of blazars with hard spectra reaching past ten teraelectronvolts, the expected cascade at a few gigaelectronvolts is larger than the gigaelectronvolt flux that the Fermi telescope actually measures from the direction of the source. Something has removed it. The clearest case is a blazar catalogued as 1ES 0229+200, at a redshift of 0.14, about six hundred megaparsecs away, whose spectrum rises in power all the way to ten teraelectronvolts and which is barely detected at a gigaelectronvolt at all. Its absorbed power alone, reprocessed, would make it one of the brighter gigaelectronvolt sources in its part of the sky. That source, and a few like it with steady teraelectronvolt emission over many years of monitoring, carry the argument; the rest of the blazar population, with softer spectra or more variable emission, adds little.

A field that sends the halo elsewhere

A void field removes it by deflection. A lepton turned through more than the jet’s opening angle before it cools emits its scattered photons in a direction that does not point at the Earth: a jet of half-opening angle about a tenth of a radian, as the Doppler boosting of blazars implies, beams its cascade toward the observer only if the leptons stay within that cone. Deflect them further and the reprocessed energy goes into the rest of the sky, a faint and isotropic contribution that nobody could attribute to any particular blazar.

Requiring the deflection of the leptons that make gigaelectronvolt photons to exceed a tenth of a radian gives the field. With the three lengths above, the answer for a field coherent over the cooling length is about 10−1610^{-16} gauss. But fields are not necessarily coherent over a megaparsec, and the bound depends on how tangled the field is.

The weakest void field that removes the gigaelectronvolt halo. The field strength needed to turn the leptons that make 1 GeV cascade photons through the jet's half-opening angle of 0.1 rad, against the field's coherence length, both on logarithmic scales. For a field coherent over more than the cooling length, 0.64 Mpc for these 0.56 TeV leptons, one orientation holds for the whole cooling and the bound is flat at 9.4·10⁻¹⁷ G. For a more tangled field the lepton random-walks through many cells, the deflections partly cancel, and the required strength rises as the inverse square root of the coherence length. A field above the curve would explain a missing halo; the published bounds, from full cascade simulations, lie between 10⁻¹⁷ and 10⁻¹⁵ G depending on the source and the assumptions about its history, in the same place.
Fig. 4 The weakest field that turns the leptons making 1 GeV photons through a tenth of a radian, against the field’s coherence length. For fields coherent over more than the cooling length, about 0.6 megaparsecs here, the requirement is flat at about 10−1610^{-16} gauss. For a more tangled field the lepton random-walks through many cells whose deflections partly cancel, and the required strength rises as the inverse square root of the coherence length — a hundred times stronger for a coherence length ten thousand times shorter.

The shape of that bound is a statement about what could have made the field. A field generated in the first second of the universe, during a phase transition, would have a coherence length set by the horizon at that time — tiny today, a fraction of a parsec — and would need to be stronger to satisfy the bound. A field from the inflationary epoch could be coherent on megaparsec scales and satisfy it at its weakest. The gamma rays are therefore sensitive to the product of a field’s strength and the square root of its coherence length, which is not a property of the field alone but of how it was made.

Which photons carry the bound

The bound depends on which part of the cascade is missing. Because the deflection falls as the square of the lepton’s energy, and the cascade photon’s energy rises as the square of the lepton’s energy, the deflection of the leptons that make photons of energy EE falls simply as 1/E1/E. The lowest-energy cascade photons come from the most deflected leptons and give the most stringent limit.

The weakest void field that removes the gigaelectronvolt halo. The field strength needed to turn the leptons that make 10 GeV cascade photons through the jet's half-opening angle of 0.1 rad, against the field's coherence length, both on logarithmic scales. For a field coherent over more than the cooling length, 0.20 Mpc for these 1.77 TeV leptons, one orientation holds for the whole cooling and the bound is flat at 9.4·10⁻¹⁶ G. For a more tangled field the lepton random-walks through many cells, the deflections partly cancel, and the required strength rises as the inverse square root of the coherence length. A field above the curve would explain a missing halo; the published bounds, from full cascade simulations, lie between 10⁻¹⁷ and 10⁻¹⁵ G depending on the source and the assumptions about its history, in the same place.
Fig. 5 The same bound derived from the absence of 10 GeV cascade photons instead of 1 GeV. The leptons that make them have about three times the energy, are turned ten times less by a given field, and so the field needed to remove this part of the halo is ten times stronger — about 10−1510^{-15} gauss for a field coherent over the cooling length, which is now shorter.

That would argue for using the lowest energies possible, but two things push the other way. The gamma-ray telescope’s angular resolution is worst at its lowest energies, degrading to several degrees below a few hundred megaelectronvolts, so a halo there cannot be told from the point source it surrounds. And the blazar’s own gigaelectronvolt emission, which has nothing to do with the cascade, is brightest there and has to be modelled and subtracted. In practice the bounds come from the range of one to ten gigaelectronvolts, where the halo is expected to be bright, the resolution is good and the source’s own spectrum is measured well — and the difference between the two figures, a factor of ten in field, is part of why published bounds span an order of magnitude for the same sources.

The spectral shape is itself a check. A field that removes the low-energy cascade while leaving the high-energy part would steepen the expected halo spectrum at a predictable energy, and a halo removed by plasma instabilities, which act on all leptons more or less equally, would not. With enough sources, the two causes of absence leave different signatures in the gigaelectronvolt spectrum.

The halo that arrives late

A lepton turned through an angle also takes a longer path, and its photons arrive after the primary gamma rays that would have come straight. The delay is of order the pair-production length times the square of the deflection, divided by the speed of light.

How late the cascade arrives, against the void field. The delay of cascade photons behind the primary gamma rays, in years on a logarithmic scale, against the field strength, for cascade photons of 1, 10, 100 GeV: the detour of a lepton turned through an angle δ, about the pair-production length times δ²/2c. The delay grows as the square of the field. In a field of 10⁻¹⁹ G the echo of a teraelectronvolt flare arrives 13 years later at 1 GeV and 15 days later at 10 GeV, and can be looked for; in 10⁻¹⁷ G the lags are 10⁵ years and 416 years, and what is seen now is the echo of the blazar's distant past. The bound from a missing halo then depends on how long the blazar has been shining — a few years of observation cannot exclude a source that turned on a decade ago. The dashed lines mark a year and a thousand years.
Fig. 6 The delay of the cascade behind the primary gamma rays, against the field strength, for cascade photons of 1, 10 and 100 gigaelectronvolts. The delay grows as the square of the field. In a field of 10−1910^{-19} gauss the gigaelectronvolt echo of a flare arrives about a decade later; in 10−1710^{-17} gauss the lag is a hundred thousand years at 1 GeV and centuries at 10 GeV. For stronger fields the leptons are turned out of the beam altogether.

The delay changes what a missing halo means. The cascade seen today was launched by the gamma rays the blazar emitted long ago, by the amount of the delay. If the blazar has been bright for millions of years, the delay does not matter. If it switched on a decade ago — and blazars are notoriously variable — then the cascade simply has not had time to arrive, and its absence says only that the delay exceeds a decade. That requires a much weaker field: about 10−1810^{-18} gauss rather than 10−1610^{-16}.

The published bounds therefore come in two versions. Assuming the blazars have shone steadily for as long as the delay, the bound is about 10−1610^{-16} gauss or above. Assuming only that they have shone as long as they have been watched — a few years for the teraelectronvolt data — it drops by two orders of magnitude. The most careful joint analysis of ground-based and space gamma-ray data, published in 2023, assumed ten years of activity and found a field of at least about 7×10−167 \times 10^{-16} gauss for coherence lengths of a megaparsec and more. The history of a galaxy’s nucleus has entered the measurement of the emptiest space in the universe.

The delay also offers a way to see a field directly rather than bound it. A bright teraelectronvolt flare, if the void field were as weak as 10−1910^{-19} gauss, would be followed by a gigaelectronvolt echo weeks to years later, and its timing would give the field strength. Searches after the brightest flares have found no echoes, which excludes the weakest fields in the range where echoes would be seen, and closes the gap from below.

The alternative that needs no field

The argument has one serious rival, and it has been argued for more than a decade. The pair beam produced by a blazar is a stream of relativistic electrons and positrons moving through the tenuous plasma of intergalactic space, and a beam through a plasma can drive instabilities — electrostatic waves that grow by drawing on the beam’s energy. If the beam–plasma instabilities grow fast enough, the pairs lose their energy heating the intergalactic gas rather than by scattering the microwave background, and the cascade never forms. The halo is missing, and no field is needed.

Whether the instabilities are fast enough depends on how they saturate, which is set by nonlinear plasma physics in conditions no laboratory reaches: a beam a million times less dense than a very tenuous plasma, over megaparsecs. Some calculations find the instabilities drain the beam faster than inverse-Compton cooling; others find them quenched by the inhomogeneity of the intergalactic gas or by nonlinear damping long before they matter. If they do operate, they would heat the voids measurably, and the temperature of the lowest-density gas, read from the Lyman-alpha forest, is one test; so far it neither requires nor forbids the heating.

The situation is a familiar one in this subject. An absence is being asked to carry a quantitative conclusion, and the conclusion is exactly as strong as the exclusion of the alternatives.

What the arithmetic leaves out

The lengths used in the figures are the order-of-magnitude relations every analysis begins with, not the analyses themselves. The published bounds come from Monte Carlo simulations that follow each lepton through many scatterings, track the evolving energy spectrum of the cascade, use detailed models of the extragalactic background light as a function of redshift, and fold in the angular resolution of the telescopes. Those simulations change the numbers by factors of a few and do not change the scalings drawn here.

The jet’s opening angle is taken as a single number, a tenth of a radian. Real jets have structure, and the cascade photons within the cone arrive with an angular spread that, for fields around 10−1510^{-15} gauss, would make each blazar look slightly extended. Searches for that extended emission, stacking many blazars, have found none, which excludes the intermediate range of fields where the halo would be both present and resolvable; a field above 10−1610^{-16} gauss that does not produce a visible halo must be strong enough to spread it too thin to see.

And the argument assumes that the voids the pairs are made in are typical of the voids in general. The pair-production length is long enough that most pairs are made in low-density regions, but the lowest-density regions along a particular line of sight need not be representative. A field that filled only the denser parts of intergalactic space — produced by outflows from galaxies rather than by the early universe — could satisfy the bound along some lines of sight and not others, and the handful of sources with the hardest spectra do not yet sample enough sky to tell.

A field that would be the oldest thing measured

If the lower limit is a measurement, it is a remarkable one. A field of 10−1610^{-16} gauss filling the voids is a field no galaxy made, because galactic outflows are not believed to reach the centres of the largest voids, and so it would be the first magnetic field measured that is plausibly primordial. Its strength would be large enough, compressed into a protogalaxy, to seed a galactic dynamo without difficulty — at the flux conservation that carries a field from a cloud into a star, a void field of 10−1610^{-16} gauss becomes 10−1210^{-12} in a galaxy before any dynamo acts. And it would be far weaker than the fields that could have shortened the cosmic ruler by clumping the early plasma, which need a thousand times more, so the two ideas are not in conflict and the gamma rays cannot settle the other.

The limit also has a bearing on another kind of particle. The highest-energy cosmic rays are deflected by intergalactic fields on their way from their sources, and whether their arrival directions point back to anything depends on fields between the galaxies. A void field of 10−1610^{-16} gauss is too weak to matter to a proton of 102010^{20} electronvolts; the fields that do scatter them are those of the filaments and the Galaxy. The gamma-ray bound constrains the emptiest parts of the web, and the cosmic rays the denser parts.

Still open: an echo, or a heated void

The two routes that would turn a lower limit into a detection are both being tried. One is the echo: a sufficiently bright and brief teraelectronvolt flare followed, days to years later, by gigaelectronvolt emission with the spectrum the cascade predicts, whose delay would measure the field directly. The next generation of ground-based gamma-ray telescopes, more sensitive by an order of magnitude, will record flares bright enough for the echo to be detectable if the field lies in the lower part of the allowed range. The other is the instability: if beam–plasma instabilities are draining the pairs, the voids are being heated by the blazars, and the heating would show as a temperature–density relation in the intergalactic gas that rises toward low densities rather than falling. Whether the emptiest space in the universe is threaded by a field left from its first moments, or warmed by the jets of distant black holes, is being decided by those two measurements.

Where this goes next

Thirty e-foldings erase the memory of a seed. Field strength against time for three seed fields 8 orders of magnitude apart, amplified at one e-folding every 3·10⁸ years — a galactic dynamo's measured turnover rate — and stopped at the 3·10⁻⁶ gauss the disc actually has. In 10 billion years the budget is 33 e-foldings, which is a factor of 3·10¹⁴. That is the finding: the three tracks reach the same ceiling within 5.5 billion years of one another, so the field a galaxy has today carries essentially no information about the field it started with. Any seed above about 10⁻²⁰ gauss will do, and mechanisms that produce far less than that are the only ones ruled out. The measurement that does constrain a seed has to be made where no dynamo ever ran, which means the voids between clusters — and the limit there comes from gamma rays that never arrived. The seed that cannot be remembered A galactic dynamo affords something like thirty e-foldings over the age of a galaxy, which multiplies any seed field above a ten-thousandth of a billionth of a microgauss up to the microgauss actually observed. The field a galaxy has today therefore says nothing about the field it started with, and the only place a seed survives unamplified is the emptiness between clusters. The sky's darkness, measured as an opacity. Optical depth to electron–positron pair production against gamma-ray energy, for sources at redshifts 0.03, 0.1, 0.3, 1, both axes logarithmic. A gamma ray of energy E is absorbed most readily by background photons near twice the square of the electron rest energy divided by E: at 1 TeV that is 2.37 µm and at 100 GeV it is 0.24 µm, so the energy axis is a wavelength axis for the background light, running backwards — and the background it is evaluated against is the same two-component spectrum the star formation history produced, not a flat number. Above the marked τ = 1 the universe is opaque. The depth is computed in the delta-function approximation, the cross-section replaced by 0.2 of the Thomson value over a bandwidth of order the energy, with the background's comoving density evolving as (1+z)^1.2. That is a factor-of-two calculation and the shape is what it gets right: the horizon closes from z = 0.59 at 100 GeV to z = 0.16 at 1 TeV. The measurement runs the other way. A blazar's spectrum is observed, the absorbed part is the difference between it and the spectrum the source is believed to have emitted, and that difference gives the background — in the near infrared, where no direct measurement can subtract the zodiacal light well enough to compete. A background weighed by what it stops The faintest light in the universe cannot be photographed from inside the Solar System, because the zodiacal foreground is a hundred times brighter. It can be weighed instead, by the bite it takes out of a blazar at a trillion electronvolts. A void: a deficit of matter, and the outflow it drives. The density contrast of a typical cosmic void against distance from its centre in units of its radius (solid), from the profile fitted to stacked voids in simulations, with a central density 15 per cent of the mean and a compensating overdense ridge just outside the edge. The dashed curve is the excess expansion rate the deficit drives, v/(Hr) = −(f/3)Δ(r), for a growth rate f = 0.75: the centre expands 21 per cent faster than the universe around it, and the outflow speed, which is this rate times the radius, peaks near 0.7 void radii. Beyond the ridge the flow reverses: matter at the ridge is pulled back towards the denser wall. A void is a small open universe expanding inside a flat one, and every galaxy in it is moving outward. An emptiness that expands faster A cosmic void is not a hole in the universe but a region with a fifth of the average density, and gravity makes it behave like a small open universe: it expands faster than its surroundings, every galaxy in it streams outward, and in a redshift survey it is stretched along the line of sight — the opposite of what the same gravity does around a cluster. The stretch measures how fast structure grows, and the average void's roundness measures the geometry of the universe.

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BlazarCoherence lengthExtragalactic background lightGyroradiusInverse Compton scatteringPair cascadePrimordial fieldsVoids