An echo heard only from the weakest fields

The only number anyone has for the magnetic field in the emptiest space between galaxies is a lower limit made of three lengths and a missing halo. Teraelectronvolt gamma rays from a distant blazar are stopped by the background light they cross, the electron–positron pairs they make scatter the microwave background up to gigaelectronvolts, and if nothing turned the pairs the scattered light would form a glow around the blazar. The glow is not there. A void field strong enough to bend the pairs out of the beam would explain the absence, and so would a plasma that takes the beam’s energy before the pairs scatter anything. Either way, what has been measured is an absence.

A field turns a lepton through an angle, and a turned lepton’s light takes a longer path. That suggests a measurement with a value rather than a bound. When a blazar flares, the flare’s own teraelectronvolt photons arrive first; the gigaelectronvolt light made from the ones that were stopped arrives later, by an amount that grows as the square of the field. Time the echo and the field follows. The idea is old and simple, and this essay asks what it costs: how bright a flare has to be for its echo to be heard, and over which fields that is possible at all.

The same energy, spread over a longer time #

The figure’s first lesson is the one that decides everything after it. The energy in the echo does not depend on the field. A pair made by a 4-TeV photon gives its energy to the microwave background in about a megaparsec whatever the field is doing, and as long as the turning is less than the jet’s opening angle, the light it makes still reaches the observer. What the field controls is the detour, and so the arrival time. In a weak field the echo’s whole fluence arrives within hours; in a field a hundred times stronger the delay is ten thousand times longer and the same fluence is spread ten thousand times thinner.

The fluence in the echo is set by the flare and by distance. A flare is measured here by its fluence above a teraelectronvolt as seen from the Earth, in units of a day of the Crab Nebula’s steady flux — $4.8 \times 10^{-6}$ $\mathrm{erg\,cm^{-2}}$, a convenient yardstick because the Crab is the brightest steady source in the teraelectronvolt sky. Ten Crab-days is a flare ten times the Crab for a day, or a hundred times for a couple of hours. From a blazar 500 megaparsecs away, more than half the flare’s light above a few teraelectronvolts never arrives, so the flare actually emitted was much brighter than the one seen. Following that missing energy through the pairs into the gigaelectronvolt band gives the echo a fluence about twice the flare’s observed fluence above a teraelectronvolt, ten times $1.0 \times 10^{-5}$ $\mathrm{erg\,cm^{-2}}$ for the flare drawn.

Against that sits the telescope. A space gamma-ray telescope looking at a fixed point of sky detects a source by accumulating photons against a background, and the faintest steady flux it can detect falls as the square root of the time it looks, until the confusion of faint sources sets a floor. That is the dashed curve: $5 \times 10^{-11}$ $\mathrm{erg\,cm^{-2}\,s^{-1}}$ in a day, $10^{-12}$ after several years. It is the same square-root law that sets the error bar that comes from counting in any photometer, and it is what makes the arithmetic of the echo unforgiving. An echo of fixed fluence spread over a delay $\tau$ has a mean flux proportional to $1/\tau$; the flux needed to detect it falls only as $1/\sqrt{\tau}$. The ratio of the two falls as $1/\sqrt{\tau}$, and since $\tau$ grows as the square of the field, the echo’s detectability falls in proportion to the field. Every factor of ten in field costs a factor of ten in flare.

A trough with two walls #

There is a second wall on the other side. A blazar flare is itself bright in gigaelectronvolts — the same electrons that make the teraelectronvolt flare usually make gigaelectronvolt light too — so an echo that arrives during the flare cannot be told from it. Only what arrives after the flare has ended counts. For a field so weak that the delay is shorter than the flare, almost all of the echo is lost inside it, and the fraction that leaks out afterwards is the ratio of the delay to the flare’s length.

The two walls make a trough. Plotting, for each field, the flare fluence needed for the echo arriving after the flare to be detected gives a curve that falls steeply from the left, bottoms out where the delay is a few times the flare’s duration, and rises to the right as the field.

The trough says the measurement exists, and says how narrow it is. A one-day flare of ten Crab-days opens a window more than two decades wide in field, from $4 \times 10^{-22}$ to $10^{-19}$ gauss. A flare of one Crab-day, which is already a large flare, opens nothing: its echo is undetectable at every field. A shorter flare moves the trough left and deepens it, because a shorter flare lets a shorter delay be separated from it; a ten-day flare moves the trough right and raises its floor, because only delays of weeks are separable from it and an echo spread over weeks is faint. All three curves join on the right, where the delay is so long that the flare’s length no longer matters and only the dilution counts.

The left-hand wall deserves a second look, because it is not a property of the field but of the instrument and the source. A flare is lost inside itself only because the gigaelectronvolt light the blazar makes directly during the flare cannot be separated from the echo by timing. If it could be separated another way — by its spectrum, which for the direct emission is usually a falling power law and for the echo rises towards higher energy within the band, or by modelling the direct emission from the teraelectronvolt light curve itself — the left wall would move left and the trough would deepen. The figure takes the cautious version: nothing arriving before the flare’s end is counted at all.

The right-hand wall is the one that limits what an echo can say about the fields the halo bounds are about. At $10^{-18}$ gauss a one-day flare needs about a hundred Crab-days, and at $10^{-17}$ gauss tens of thousands. The halo searches are argued about at $10^{-16}$. No flare from any blazar comes close.

A delay with a signature #

The trough would be a curiosity if the echo could not be recognised. A blazar varies all the time, at every energy, and a rise in gigaelectronvolt flux a month after a teraelectronvolt flare could be the blazar’s own business. The echo has to carry a signature no source can imitate, and it does: its delay depends on energy in a way fixed entirely by the cascade.

The delay is the pair-production length times the square of the deflection, divided by twice the speed of light. A cascade photon of energy $E_c$ was made by a lepton of energy proportional to $\sqrt{E_c}$, because inverse-Compton scattering of the microwave background multiplies a photon’s energy by the square of the lepton’s Lorentz factor. That lepton was made by a gamma ray whose pair-production length falls as the inverse of its energy, and it is deflected over its cooling length by an angle that falls as the inverse square of its energy. Putting the three together,

$$\tau \;\propto\; \frac{1}{E_e}\left(\frac{B}{E_e^2}\right)^2 \;\propto\; \frac{B^2}{E_e^5} \;\propto\; B^2\,E_c^{-5/2}.$$

A factor of a hundred in energy is a factor of a hundred thousand in delay. Photons at a hundred gigaelectronvolts arrive minutes after the flare, those at one gigaelectronvolt more than a year later, and the echo sweeps down the spectrum on a schedule whose exponent is fixed and whose normalisation is the field. Lags inside a source behave nothing like this. Synchrotron cooling makes low-energy emission lag high-energy emission as the inverse square root of the energy; light-travel effects across an emitting region give lags that hardly depend on energy at all. Either could produce a delay at one energy. Neither produces five decades of delay across two decades of energy.

The square in the field is also what makes a detected echo a precise measurement. A delay measured to a factor of two — which is all a light curve with a few dozen photons in its tail would give — fixes the field to forty per cent, because the field goes as the square root of the delay. Most of what is known about magnetic fields in astronomy is known to a factor of a few; an echo timed to within a factor of two at two energies would be one of the better-determined fields anywhere outside the solar system, and the only one ever measured in a void.

The scaling also means a detected echo would test itself. Its delay at two energies fixes the exponent independently of the field; if the exponent came out at −2.5 the cascade interpretation would be confirmed, and the normalisation would then be the field. That is the difference between an echo and a delay in a reverberating quasar, where a lag is turned into a radius by assuming a geometry: here the geometry is the thing the energy dependence checks.

The brightest flares come from the wrong blazars #

Everything so far has been for a blazar 500 megaparsecs away, about a redshift of 0.12. The distance enters twice, and the two entries pull in opposite directions.

A teraelectronvolt photon’s pair-production length on the extragalactic background light is hundreds of megaparsecs at one teraelectronvolt and tens at ten. From a nearby blazar most of the flare’s teraelectronvolt light arrives — that is why the nearby blazars are the bright ones in the teraelectronvolt sky — and correspondingly little was converted into pairs. From a distant blazar most of it was stopped, and the flare seen is the remnant of a much brighter flare that went into pairs.

The ratio is fifteen between the nearest and the farthest. Per unit of observed flare, a blazar a gigaparsec away makes an echo fifteen times brighter than one at 150 megaparsecs, because the observer sees only the fraction of the flare that the background light let through and the echo is made from the rest.

The flares that have been seen are the other way round. The brightest teraelectronvolt flares on record come from the nearest blazars: Markarian 501 and Markarian 421, both within about 150 megaparsecs, have been seen at several times the Crab’s flux for weeks. Their echoes are the weakest per Crab-day, and their flares are long, which moves them up the right-hand side of the trough. The most violent flare from a blazar at a useful distance, PKS 2155−304’s in July 2006 at a redshift of 0.116, reached about ten times the Crab above a few hundred gigaelectronvolts — but for less than an hour at a time, giving a fluence above a teraelectronvolt of well under one Crab-day. On the curves drawn here, no single flare yet recorded clears the floor of the trough for its distance with much room to spare.

That does not make the echo impossible. It makes it a waiting game, with the odds set by the luminosity function of blazar flares at redshifts near a tenth to a quarter, and by how many such blazars a ground-based array can monitor continuously. A flare of ten Crab-days from a blazar at five hundred megaparsecs would be about ten times brighter than any yet recorded at that distance. The next generation of Cherenkov arrays will see far more flares, but the bottleneck is the gigaelectronvolt side: the echo has to be detected by a space telescope, and the one now flying is the only one.

Below the halo, and asking nothing about the past #

The echo and the halo measure the same field through the same physics, and the question is how their reaches compare. The halo bound has a weakness the echo does not. A halo delayed by more than a blazar’s age has not yet arrived, so a missing halo says only that either the field is strong enough to turn the pairs out of the jet, or the delay is longer than the time the blazar has been bright. A blazar observed for ten years and assumed to have shone only that long gives a much weaker bound than one assumed to have shone for millions.

The two methods are complementary rather than redundant, and the figure shows exactly how. The halo works where fields are strong; its reach depends on an assumption about the blazar’s history that cannot be checked from the halo itself. The echo works where fields are weak, and its reach depends on nothing about the past: the flare is observed, and the echo either follows on the predicted schedule or does not. The ten-year halo bound at one gigaelectronvolt sits just at the top of the echo’s window, which is why the published conservative bounds — those that assume only the years of observation — and the echo searches together cover a continuous range rather than leaving a gap.

They do not, however, cover the range the halo bound is usually quoted at. If the field is $10^{-16}$ gauss, as the long-history bounds favour, no echo will ever be seen from any flare, and the echo method’s contribution is to close off the weak end that the conservative bound leaves open.

A tangled field moves the window, not its width #

All of the above assumes a field coherent over more than the distance a lepton travels while it cools, about a third of a megaparsec at a teraelectronvolt. A field tangled on smaller scales turns a lepton by a random walk through many cells, and the net deflection grows only as the square root of the number of cells crossed. The same delay then requires a stronger field, by the square root of the ratio of the cooling length to the coherence length.

The band and the halo line move up together as the field becomes more tangled, because both depend on the field only through the deflection it produces. That is a useful fact about what either method measures. Neither measures a field strength: both measure the product of a strength and the square root of a coherence length, wherever the coherence length is shorter than the cooling length. A field produced in the first instants after the Big Bang by a phase transition would be tangled on scales set by the horizon at the time, far smaller than a kiloparsec today; a seed blown out of galaxies would be coherent over the scales of the outflows that carried it. An echo would measure the combination, and the distinction between the two origins would have to come from elsewhere — from the imprint a strong enough field leaves on the microwave background, or from the field’s helicity, which is beyond the reach of anything drawn here.

An absence that measures a product #

There is one more assumption hidden in every curve, and it is the one the halo argument is most vulnerable to. The echo is made by pairs that scatter the microwave background. If beam–plasma instabilities drain the pair beam’s energy into the intergalactic gas first, the scattering never happens, the echo never forms, and its absence says nothing about the field.

The asymmetry is the most important thing the echo offers. A missing echo, like a missing halo, constrains a product — the field’s effect times the fraction of the beam that survives to scatter — and the product can be small because the field is outside the window or because the beam was drained. The searches that have looked for echoes after bright flares and found none have excluded the weakest fields only on the assumption that nothing drained the beam. A detected echo is different in kind. It would show that the beam survives, because the light came from scattering; its amplitude would measure how much survived; and its delay, with the energy dependence checked, would measure the field. One detection would settle both the field and the plasma question that the halo searches have argued about for a decade.

What the calculation leaves out #

The model is built to be traceable rather than complete, and several of its simplifications push the numbers by factors that matter.

Only the first generation of the cascade is followed. Cascade photons above about a hundred gigaelectronvolts are themselves absorbed over the larger distances and make a second generation of pairs at lower energy, which adds echo at gigaelectronvolts with longer delays. The correction is modest for the band drawn and grows for the most distant blazars.

The delay of each photon is taken as an exponential with a mean set by the pair-production length of the lepton that made it. A full treatment follows each lepton from its creation, through a deflection that accumulates as it cools, to the energy at which it emits; the delay distribution then has a longer tail. The mean and its scaling with energy and field are robust; the shape of the light curve at late times is not.

Every lepton is assumed to stay within the jet’s opening angle, and the observer to sit on the jet’s axis. An observer off-axis sees the direct flare dimmer and the echo relatively brighter. Real blazars are viewed within a few degrees of the jet axis, which is why they are blazars, but the correction is not zero.

The extragalactic background light is described by one pair-production length that scales inversely with energy. Measured spectra of that background have structure, and the absorbed fraction at each energy differs from the scaling by tens of per cent between models.

The telescope is a square-root law with a floor. Real sensitivity depends on the source’s position, on the spectrum assumed, on the background model and on the point-spread function, which at one gigaelectronvolt is about a degree. For the fields in the window, the echo’s angular extent is well under that, so it looks like a point source; for stronger fields, outside the window, it would be extended and harder still.

And the blazar’s own gigaelectronvolt light after a flare is treated as if it returned instantly to a steady level. In practice blazars flicker after flares, and the quiet level that an echo must stand out against is uncertain by more than the echo in the right half of the trough.

Still open: a flare bright enough, far enough away #

The picture is of a measurement that exists, has a clean signature, and has not yet been given the flare it needs. The window it opens lies below the halo bounds and touches the conservative ones, so a detection would mean the field is weaker than the long-history halo arguments allow — those arguments would then be wrong about blazar histories or about the plasma — and a non-detection after a large enough flare would close the weak end only if the beam is not being drained.

What would decide it is a flare of ten Crab-days or more, lasting a day or less, from a blazar at a redshift of a tenth or beyond, with a gigaelectronvolt telescope watching for the following months. The teraelectronvolt arrays now being built will monitor more blazars more continuously than any before, and if such flares occur at all, they will be seen. Whether the space telescope that would hear the echo is still flying when that happens, and whether a successor with better sensitivity at a few gigaelectronvolts is built, decides whether the oldest field in the universe is ever measured rather than bounded — or whether the voids keep returning only the silence that the plasma would also explain.