A twist that decides where the field ends

The emptiest space between galaxies has a magnetic field, or the gamma-ray sky has been misread. That is the conclusion of a lower limit made of three lengths and a missing halo: teraelectronvolt photons from distant blazars make electron–positron pairs in the voids, the pairs ought to scatter the microwave background into a gigaelectronvolt glow around each source, and the glow is missing. A void field of about $10^{-16}$ gauss coherent over a megaparsec would turn the pairs out of the beam and explain the absence. Nothing a galaxy does is obviously able to put a field there, which is why the bound is read, cautiously, as the first sign of a field older than any galaxy.

The obvious place to make such a field is a phase transition of the early universe. When the universe cooled through the electroweak scale, at about a hundred gigaelectronvolts and a hundred-billionth of a second, and again through the quark–hadron transition at a hundred and fifty megaelectronvolts and some tens of microseconds, it may have changed state by nucleating bubbles of the new phase that expanded and collided. Colliding bubble walls stir the plasma, stirred plasma carries currents, and currents make magnetic fields. The fields so made can be enormous — up to a few per cent of the energy in radiation — and they are small. They cannot be coherent over more than the distance light had travelled since the beginning, and at the electroweak transition that distance, stretched to today’s scale, is about a hundred astronomical units.

A field coherent over a hundred astronomical units, or over the astronomical unit that a bubble a hundredth of that size would give, is useless for the voids. A pair travels a megaparsec while it cools; a field that reverses its direction ten billion times along the way turns the pair by a random walk and hardly turns it at all. So the question is not whether a phase transition could make a strong field. It is what three hundred and eighty thousand years of turbulence does to that field before the universe becomes transparent and the plasma stops moving — and the answer turns out to hinge on one number that the field’s energy does not set.

A field that cannot stay where it was made #

A magnetic field embedded in a highly conducting plasma moves with the plasma, and in the early universe the plasma is turbulent: the bubble collisions that made the field also stirred the fluid, and the field itself, once strong, stirs it further. Freezing a field into a fluid means that the field’s structure is carried by the eddies, folded and stretched by them, and the energy of the small-scale structure is passed down to the dissipation scale and lost there as heat. Turbulence left to itself decays, and a magnetised turbulence decays by losing its smallest eddies first.

That loss has a consequence for the field’s coherence length. If the small-scale structure goes and the large-scale structure stays, the typical scale of what remains grows. A decaying field gets weaker and larger at the same time, and the figure’s two coloured tracks are the two ways it can do so. Each starts at the black dot — the point the transition fixes, a field of a microgauss on a scale of an astronomical unit, both measured in comoving units, so that the field’s dilution by expansion is already divided out. Each moves down and to the right. And each stops at the dashed line.

The stopping line is where the turbulence runs out of time. An eddy of size $\lambda$ decays once a magnetic disturbance — an Alfvén wave — has crossed it, and an Alfvén wave in a field of strength $B$ moves at a speed proportional to $B$. By recombination the universe has had a finite time, and the largest eddy that has been processed is the one an Alfvén wave can cross in that time. A stronger field processes larger eddies; the line of processed eddies is therefore a straight line of slope one on the logarithmic plane, close to $10^{-8}$ gauss at a megaparsec. Once the field’s coherence length reaches that line it stops decaying in any important way. The photons decouple, the plasma’s viscosity falls, the turbulence freezes into the gas, and the field that reaches the voids today is, up to the slow dilution the comoving units have already accounted for, the field that sat on the line at recombination.

The line comes from simulations of decaying magnetised turbulence and from the Alfvén-crossing argument, and both are uncertain by factors of a few in its normalisation. The figures here use it as given, because the argument does not depend on its exact position; it depends on the difference between the two tracks, which is a factor of nearly a hundred thousand in final coherence length.

What the decay is allowed to keep #

The two tracks differ because the decay of a turbulent field is constrained by what it conserves, and a field can conserve two different things.

A field with no net twist — whose field lines are as often linked one way as the other — loses energy and coherence together under one constraint that simulations have identified in the last decade: a quantity that behaves like $B^2\lambda^5$, the square of the field times the fifth power of its length, which the turbulence keeps nearly fixed. If $B^2\lambda^5$ is conserved then $B \propto \lambda^{-5/2}$: each tenfold growth in coherence length costs the field a factor of three hundred in strength. That is the steep track. It is steep enough that the field’s energy, which goes as $B^2$, falls by a factor of a hundred thousand for every decade of growth. Very little energy reaches large scales; nearly all of it is dissipated with the small eddies.

A field whose lines are linked has magnetic helicity, a measure of that linkage. Helicity is one of the best-conserved quantities in a highly conducting plasma — better conserved than the energy, because it is dissipated only at the very smallest scales while the energy cascades down to them — and its density scales as $B^2\lambda$. If the turbulence must keep $B^2\lambda$ fixed while it loses energy, the only way to lose energy is to grow the length, and $B \propto \lambda^{-1/2}$: a tenfold growth costs only a factor of three in field. That is the shallow track, and it is what is called an inverse cascade. The energy that remains is pushed up to ever larger scales rather than being dissipated with the small ones, because the helicity it carries has nowhere else to go.

The energy plot makes the asymmetry visible in a way the field plot does not. Both fields lose energy. The non-helical field loses it faster and also gets less for it: its coherence length grows by less than four decades before the stopping line catches it, and by then it has kept one part in $10^{19}$ of the energy it started with. The helical field loses energy more slowly and travels further along the length axis, nine decades, ending with ten orders of magnitude more energy than the non-helical one and on a scale a hundred thousand times larger. Helicity does not create energy. It decides where the surviving energy sits.

The floor, and where each field lands against it #

The shaded region in the opening figure is the floor set by the missing halos, drawn here as the field that turns a pair making one-gigaelectronvolt light by more than the jet’s opening angle over its cooling length. For a field coherent over more than that cooling length, about 0.6 megaparsecs, the floor is flat near $10^{-16}$ gauss. For a field tangled on smaller scales the turning becomes a random walk and is weaker by the square root of the number of reversals, so the floor rises as the inverse square root of the coherence length. A field coherent over a parsec must be about a thousand times stronger than one coherent over a megaparsec to turn the pairs equally.

That rising floor is what makes the comparison sharp. The non-helical electroweak field ends at 0.038 parsecs with $4 \times 10^{-16}$ gauss. That would clear the floor if it were coherent over a megaparsec, but at 0.038 parsecs the floor is about $4 \times 10^{-13}$ gauss, three decades higher. The pairs cross some seventeen million reversals of this field while they cool, and the random walk turns them through about a four-thousandth of the angle a coherent field of the same strength would. The helical field ends at 4.7 kiloparsecs with $5 \times 10^{-11}$ gauss — a field so strong that even tangled on kiloparsec scales it clears the floor by four and a half decades.

The difference is not marginal in either direction. An electroweak field without helicity cannot be the field the missing halos ask for; an electroweak field with helicity overshoots the floor so far that it is near the region the microwave background excludes. The voids’ field, if it is primordial and electroweak, is helical. If it is not helical, it is either not electroweak or not primordial.

How little twist is enough #

A field with no helicity at all and a field with all the helicity it can carry are the two extremes. A real field made by colliding bubbles would carry some helicity of one sign or the other by chance, and the processes proposed to make helicity deliberately — mechanisms tied to the violation of parity in the electroweak interaction, or to the same events that are thought to have made the excess of matter over antimatter — give amounts that are not known to within many orders of magnitude. What matters is how much is enough, and the answer is much less than intuition suggests.

The mechanism is the same conservation law applied to a partly twisted field. Its helicity, which is its energy times its length times the helical fraction, is conserved; its energy, while the decay is non-helical, falls as the fifth power of its length. So the helical fraction grows as the fourth power of the length. A field that starts with a helical share of one part in ten billion grows its coherence length by two and a half decades, multiplies its helical share by ten billion, and becomes fully helical. From then on it follows the shallow track. The steep phase still costs it — it enters the shallow phase weaker and later than a field that was helical from the start — but the shallow phase does the rest.

That is the figure’s reading. The final field against the floor rises steadily with the initial helicity across twenty decades, at about half a decade of field for every decade of twist, and it crosses the floor at a helical fraction of $6 \times 10^{-10}$. A primordial field made with a twist of a part in a billion ends in the voids as a field the missing halos would notice. That number happens to sit near the ratio of baryons to photons, which is the coincidence some of the helicity mechanisms invoke, but nothing in the decay ties the two together, and the figure says only that the threshold is low, not that any particular mechanism reaches it.

What a bigger bubble buys #

The two other free parameters of the transition are the size of the bubbles and the strength of the field they make, and either could, in principle, rescue a non-helical field without needing helicity. Larger bubbles start the field on a larger scale; a stronger field moves the processing line’s crossing point further out. Both effects are real, and both are weak.

The weakness has a simple cause. The decay forgets the starting point, and it forgets it faster the steeper the track. For a non-helical field the end point’s coherence length goes as the starting length to the five-sevenths power, because the field has to travel along a slope of minus five halves until it meets a line of slope one, and the arithmetic of the intersection compresses whatever range the start had. Four decades of bubble size become under three decades of final length, and with the floor’s own slope included, a bit over four decades of margin. Even bubbles as large as the Hubble length itself — the largest size causality allows — leave the non-helical field almost a decade short.

The helical curve is shallower still, because its track is shallower: a starting point further along a slope of minus a half moves the intersection less. That is the opposite of what one might expect, and it means the helical field is robust. Its end point barely cares where it started.

What a stronger start buys #

The energy argument is similar, and the figure shows the same compression from the other direction.

A field with a hundred thousand times less energy starts three hundred times weaker, and both tracks carry that deficit to the line. The non-helical track carries a smaller share of it — the end point moves only 1.1 decades against the floor — because its steep decay was going to throw almost everything away anyway. The helical track carries more, 2.5 decades, because it throws away less: a helical field remembers its starting energy better because it keeps more of it. But even at the physical maximum, with the field holding all the energy the radiation has, the non-helical field ends almost three decades short.

So the three parameters of a transition are not interchangeable. Bubble size and initial energy move the end point by fractions of their own range; helicity moves it by decades for a twist that is, in energy terms, nothing. Of the three things a phase transition might have done, the one that decides whether its field can be the voids’ field is the one that carries no energy at all.

A later transition does not need the twist #

The quark–hadron transition happened when the universe was about a million times larger in comoving terms than at the electroweak one — its Hubble length, stretched to today, is half a parsec rather than a hundred astronomical units. A field made then starts a million times further along the length axis, and the steep decay’s five-sevenths compression turns that into four decades of final length.

That is enough. A non-helical QCD field on small bubbles grazes the floor and on large bubbles clears it by two decades. Helicity still helps — the helical versions sit six or seven decades above, uncomfortably close to the region the microwave background excludes — but it is no longer necessary. The comparison between the two transitions is the cleanest statement of what helicity does: it buys an electroweak field roughly the coherence length that a thousandfold later start buys a non-helical one.

The QCD transition has its own difficulty, which the figure does not show. In the standard model of particle physics it is not a phase transition at all but a smooth crossover, with no bubbles and no walls to collide, and the field-making mechanisms that need violent collisions would have nothing to work with. A field made at that epoch needs physics beyond the standard model just as surely as a helical electroweak field does. The figure says which would work, not which happened.

What the plane cannot show #

Every number in these figures rests on three idealisations, and each is a place the argument could fail.

The first is the decay law itself. That a non-helical magnetised turbulence conserves something like $B^2\lambda^5$ is the conclusion of simulations run over a few decades of scale, and the extrapolation across four decades of growth assumes the same law holds throughout. Earlier work used other exponents for the non-helical decay, and a shallower law would close part of the gap. It would not close three decades unless the law were nearly as shallow as the helical one, and no simulation has found that.

The second is the stopping line. Its normalisation depends on the plasma’s viscosity in the era before recombination, when photons diffuse over the scales in question and damp the motions — the same damping that smooths the smallest structure in the microwave background — and the turbulence in that era may be processed more slowly than the Alfvén argument says. A line a factor of ten lower moves every end point down the plane and the conclusion with it, but it moves both tracks together, and the comparison between them survives.

The third is the floor. It is the long-shining version of the halo argument, and if the blazars have shone for only a few years, the floor drops by more than a decade; if the plasma drains the pair beam before it can scatter, there is no floor at all. A void field that is three decades below the long-shining floor and just above the short-shining one is not excluded by any measurement — only by the reading of the halos as a field.

The test the twist leaves behind #

The case for helicity in the voids would be stronger if helicity could be seen there directly, and in principle it can. A helical field turns the pairs in a cascade preferentially one way, and the scattered gigaelectronvolt photons, if they arrive at all, should then carry a handedness: the arrival directions around a blazar would spiral with a sign set by the field’s twist. Searches for exactly this correlation in the diffuse gigaelectronvolt sky have been made and have reported hints of either sign at modest significance, none surviving.

A second route is the microwave background itself. A helical field would source parity-violating correlations between the background’s temperature and its polarisation, of a kind that ordinary scattering cannot make, and the amplitude would scale with the field’s strength at recombination — which, for the helical electroweak field drawn here, is at the strong end of what the plane allows. Current maps constrain such correlations to fields a little above a nanogauss, two decades above where the helical field ends.

A third is the echo. A flare’s gigaelectronvolt echo measures a field rather than bounding it, but only for fields below about $10^{-19}$ gauss — a window entirely below the floor, where neither electroweak field drawn here lands and where only a non-helical field made with almost no energy could. An echo detected in that window would argue against the helical electroweak picture; one that is not detected argues nothing.

Still open: whether anything made the twist #

The plane turns the question of the voids’ field from one about energy into one about topology. A field made at the electroweak transition, decaying freely, ends three decades below the floor or four and a half above it, and the only property of its creation that moves it from one to the other is a twist of a part in a billion. The bubble size cannot do it and neither can the field’s strength. A later transition can, but in the standard model the later transition makes no bubbles to begin with.

So the primordial reading of the missing halos now rests on a question particle physics has not answered: whether the early universe made fields with net helicity, and how much. The candidate mechanisms tie helicity to the violation of parity in the weak interaction, to the same sphaleron transitions that may have made the excess of matter the universe is built from, or to the axion-like fields that some models of inflation include, and their predictions differ by more than the twenty decades the third figure spans. If any of them reaches a part in a billion, the voids’ field could be the electroweak transition’s, carried there by an inverse cascade. If none does, the field the halos imply is younger than the first second — or the plasma took the beam, and there is no field to explain.