A field that would have moved the ruler

A galaxy’s magnetic field remembers nothing of the seed it grew from. A dynamo multiplies any seed above about $10^{-20}$ gauss up to the microgauss fields observed, so the field in a galaxy today is silent on whether the universe was born magnetised. The only places a primordial seed might survive unamplified are the voids between clusters, and the limits on it from the microwave background, from nucleosynthesis and from the missing gamma-ray haloes of blazars leave twelve orders of magnitude of field strength open. For the question of where galactic fields came from, a primordial field is both possible and unnecessary.

There is a different question on which a primordial field would not be silent at all, and it is one of the most argued-over numbers in cosmology. A field of about a tenth of a nanogauss, too weak to affect the acoustic peaks of the microwave background directly and far too weak to matter to any galaxy, would have left a mark on the moment the universe became transparent — and through it on the value of the Hubble constant that the microwave background implies.

Clumps too small for the light to see #

Before recombination, the baryons and photons were a single fluid, held together by Thomson scattering: the photons pushed on the electrons, the electrons dragged the protons, and any attempt to compress the baryons was resisted by the photons’ pressure. But the photons’ grip is limited by their mean free path. On scales smaller than the distance a photon travels between scatterings — kiloparsecs, in comoving terms, shortly before recombination — the photons stream freely through the gas and cannot hold it smooth. Something pushing on the baryons alone can compress them on those scales without the photons resisting.

A magnetic field is such a thing. It exerts a force on the charged plasma — the Lorentz force — but not on the photons, and in a plasma that conducts as well as the early universe’s it is frozen into the gas it pushes, so the field and the baryons move together while the photons slip through both. A field tangled on small scales pushes the plasma from regions of strong field into regions of weak field, piling up baryons in some places and evacuating others. Jedamzik and Abel showed in 2013 that a field of order a tenth of a nanogauss (in present-day comoving units) would produce density contrasts of order one on scales below the photon mean free path before recombination.

The photons cannot see those clumps. They are far smaller than anything the microwave background resolves — the finest detail in the background corresponds to tens of megaparsecs — and they do not disturb the acoustic oscillations of the photon–baryon fluid on the scales that make the peaks. What they change is the one thing that depends on the square of the density rather than the density itself: the rate of recombination.

Why the peaks do not notice #

It is worth being precise about why a field strong enough to clump the baryons is not already excluded by the acoustic peaks themselves. The peaks are a pattern in the photons’ temperature on scales of tens to hundreds of megaparsecs, and a field affects them directly only through its energy and stress on those scales — which, for a field whose power is concentrated on kiloparsec scales, is tiny. The damping of the peaks at the smallest angles the background resolves is set by photon diffusion, the process that erases acoustic structure below a few megaparsecs, and it operates on scales a thousand times larger than the clumps. A field tangled on kiloparsec scales is therefore invisible in the photon pattern at the level of the direct bound, and becomes visible only through a quantity that is sensitive to density fluctuations at every scale at once. The recombination rate, which is a local square, is exactly such a quantity: it averages the square of the density over all scales, however small, and passes the result up to the scale of the whole last-scattering surface.

That is the reason the clumping argument reaches field strengths the direct analyses cannot. It uses the one nonlinear process in the early plasma as an amplifier for fluctuations far below the resolution of any telescope.

Recombination goes as the square of the density #

Hydrogen recombines when a free electron and a proton meet, so the rate per unit volume goes as the product of their densities — as the density squared. In a smooth plasma every region recombines at the same rate. In a clumped one, the dense regions recombine faster and the rarefied ones slower, and because the rate is a square, the gain in the dense regions outweighs the loss in the rarefied ones. The average of the square is larger than the square of the average by a factor $1 + b$, where the clumping factor $b$ is the variance of the density contrast.

The figure computes this with the standard three-level treatment of the hydrogen atom — the one that captures the bottleneck that holds recombination back, the photons emitted when an electron reaches the ground state can immediately reionise a neighbour, so atoms can only settle by the slow two-photon decay of the 2s level or by Lyman-α photons redshifting out of resonance. The plasma is divided into two equal volumes, one at 1.71 and the other at 0.29 times the mean density, a clumping factor of 0.5, and each is followed separately; the free electrons are then averaged over the mass. The smooth plasma is half recombined at redshift 1273. The clumped one reaches the same point at 1292.

Each patch on its own shows the effect directly. A patch at twice the mean density is half recombined at redshift 1311 instead of 1273; a patch at a fifth of the mean, much later. The dependence is weak — a logarithmic one, because the recombination is regulated by the balance between two exponentially steep rates, of which only one depends on the density — but it has a definite sign, and since most of the mass sits in the dense patches, the mass-weighted average recombines earlier than a smooth plasma of the same mean density.

A surface of last scattering moved earlier #

The microwave background is the light from the moment the plasma became transparent, and what matters for it is not when the hydrogen was half recombined but when the photons last scattered.

The visibility function is the probability that a photon of the background last scattered at a given redshift: it rises as the plasma becomes transparent and falls as the remaining free electrons become too few to scatter anything. Its peak is the redshift of last scattering, about 1080 in this hydrogen-only calculation for a smooth plasma, a little below the 1090 of the full codes that include helium and the finer atomic physics. With a clumping factor of 0.3 the peak moves to 1086; with 0.6, to 1096. The surface of last scattering is still a thin shell — the photons decouple within about a hundred units of redshift — but it sits at an earlier time.

An earlier surface of last scattering is a younger universe. The sound waves in the photon–baryon plasma, which had been travelling outward from every overdensity since the beginning, had less time to travel, and the distance they reached — the sound horizon, which sets the spacing of the acoustic peaks — is correspondingly shorter.

A shorter ruler, and a larger constant #

The microwave background measures the angle that the sound horizon subtends on the sky, to a precision of a few parts in ten thousand. The angle is the sound horizon divided by the distance to the surface of last scattering, and the Hubble constant is inferred by requiring the two to match: assume the physics of recombination, compute the sound horizon, and the measured angle then fixes the distance, which, with the other parameters, fixes the expansion rate today. The inferred Hubble constant is about 67.4 kilometres a second per megaparsec. The value measured locally from Cepheids and supernovae is about 73, a difference of five standard deviations that has survived a decade of scrutiny.

If recombination happened earlier than assumed, the true sound horizon is shorter than the one computed, the measured angle implies a shorter distance to last scattering, and a shorter distance means a faster expansion — a larger Hubble constant. The direction is exactly that of the discrepancy. The size, in the simplest calculation drawn here — clumping with every other parameter held fixed — is modest: a clumping factor of 0.5 shortens the sound horizon by 0.8 per cent and raises the inferred constant from 67.4 to 67.9, closing about a tenth of the gap.

The fixed-parameter calculation understates what a real fit does. The acoustic peaks constrain the matter density, the baryon density and the other parameters jointly with the sound horizon, and when the recombination physics changes, the best-fitting values of all of them move. Jedamzik and Pogosian, fitting the full microwave background and baryon acoustic oscillation data with clumped recombination in 2020, found that the inferred Hubble constant rose to about 70 to 71 kilometres a second per megaparsec for clumping factors near 0.5, easing the tension to under three standard deviations — enough to make the idea one of the few that moved the inference in the right direction without spoiling the fit to anything else. Later analyses that added the finest-scale measurements of the microwave background from ground-based telescopes, which are sensitive to the detailed shape of the visibility function, found the clumped models less favoured, and the question of whether clumping of that size is allowed is now being answered at those small scales.

The strength such a field would need #

The field strength involved is worth placing against the others this subject has.

At the bottom is the seed a dynamo needs, around $10^{-20}$ gauss, which is what makes the galactic field uninformative. Above it is the lower limit on the field in intergalactic voids, $10^{-17}$ to $10^{-15}$ gauss, inferred from the gigaelectronvolt haloes that blazar gamma rays should cascade into and do not. At the top is the upper limit from the microwave background’s acoustic peaks and polarisation, about a nanogauss. The clumping that would move recombination needs about a tenth of a nanogauss, tangled on small scales — twelve orders of magnitude above what galaxies need, a factor of ten below what the peaks exclude directly.

That window is not arbitrary. A field produced in a phase transition in the first fraction of a second — the electroweak or quark–hadron transitions, the candidates the first essay listed — would naturally be tangled on small scales, since it could be coherent only over the horizon at the time, and would have been processed by the plasma’s turbulence into roughly the strengths and scales that clumping needs. The same field, surviving in voids today, would sit above the blazar lower limit. A single primordial field of about a tenth of a nanogauss is therefore consistent with the missing blazar haloes, with every direct limit, and with a partial easing of the Hubble tension — which is an economy that makes the idea attractive and not yet any evidence for it.

The heat a decaying field leaves behind #

A field strong enough to clump the baryons does work on them, and the work ends up as heat. Before recombination, energy injected into the plasma is shared with the photons, and if it arrives late enough — after about two months of cosmic time, when the processes that keep the photons in a perfect blackbody have become too slow — it cannot be fully thermalised. It leaves a small distortion of the background’s spectrum: a chemical potential, or, later, a Compton distortion of the kind that hot gas in clusters produces. The microwave background’s spectrum is a blackbody to better than one part in ten thousand, and that limit constrains how much magnetic energy could have decayed in the relevant epoch.

For a field of a tenth of a nanogauss the predicted distortion is below the existing limit, which is why the spectrum has not already excluded the idea, but only by a factor of a few. A future measurement of the spectrum a hundred times more sensitive than the one made in the 1990s would detect the distortion from such a field or rule it out, independently of both the Hubble constant and the recombination argument. Three separate consequences — clumped recombination, a spectral distortion and a void field deflecting blazar cascades — follow from one primordial field, and each can be tested on its own.

Another way to shorten the same ruler #

Clumping is one of a family of proposals with a common logic. The Hubble constant inferred from the background is proportional to the inverse of the sound horizon, so anything that shortens the sound horizon raises it, and there are two ways to shorten a distance travelled by sound: stop the sound earlier, or make the universe expand faster while it travels. Clumping does the first. Early dark energy — a component that briefly adds to the energy density just before recombination and then dilutes away — does the second, speeding the expansion and reducing the time the sound had. Whether dark energy is a constant is a question about late times; early dark energy is a proposal about a moment four hundred thousand years after the Big Bang.

The two leave different fingerprints. Early dark energy changes the expansion rate and therefore the relative heights of the acoustic peaks through the way the gravitational potentials decay; clumping changes the ionisation history and therefore the shape of the damping tail and the polarisation. A sound horizon about seven per cent shorter is what the local measurement would need, and no proposal has yet produced that much without disturbing something else the background measures precisely. The clumping idea stands out for achieving part of the shift with no new particle and no new field of force — only a magnetic field of a strength nothing yet excludes.

What the calculation leaves out #

The model is deliberately minimal. It follows hydrogen only, where the full recombination codes also follow helium, which recombines earlier and supplies part of the electron density; it uses the three-level atom with a single fudge factor, where the modern codes follow hundreds of atomic levels and the radiative transfer of the Lyman lines in detail. Those refinements move the smooth last-scattering redshift from 1080 to 1090 and the sound horizon by about half a per cent, which is why the figures quote shifts rather than absolute values.

It represents clumping as two zones of equal volume, which is the simplest of the models that were fitted to data. The real density distribution produced by a tangled field is continuous and evolving: the field’s own energy decays as it drives the plasma, and the clumps themselves are stirred and smoothed as the photons’ mean free path grows during recombination. Different assumptions about the distribution change the shift in the last-scattering redshift for a given clumping factor by tens of per cent. And the calculation does not model how the field produces the clumps; it takes the clumping factor as given and asks what it does. Connecting a clumping factor to a field strength requires magnetohydrodynamic simulations of the plasma before recombination, which exist but depend on the field’s spectrum.

Still open: clumps at the finest scales the background shows #

The clumping idea makes a prediction beyond the Hubble constant. An earlier, slightly broader recombination changes the damping tail of the microwave background — the finest-scale anisotropies, where the photons’ diffusion erases the acoustic oscillations — in a specific way that no change of the standard parameters can mimic. The ground-based telescopes now mapping the background at arcminute resolution measure exactly that tail, and their data, combined with the satellite’s, are what have begun to disfavour the largest clumping factors. Whether a primordial field left a measurable imprint on recombination, and so whether the ruler of cosmology has been read slightly wrong, is being decided in the smallest detail of the oldest light — while the field itself, if it exists, is waiting in the emptiest parts of the universe to be found by the gamma rays it would deflect.