A lag too short for the light

The broad emission lines of a quasar come from gas orbiting its black hole at thousands of kilometres a second, too close to resolve in any image. Its distance from the black hole is measured instead by a delay: the lines respond to changes in the continuum light after the time the light takes to reach the gas, and the delay, in days, is the radius in light-days. With a radius and the line’s width, the black hole’s mass follows from the virial theorem, up to a factor that multiplies every quasar mass.

Measuring a delay takes months of monitoring, and has been done for about two hundred nuclei. Every other black-hole mass — the hundreds of thousands in large spectroscopic surveys, and every quasar beyond a redshift of about one — comes from a shortcut. The measured delays scale with the luminosity: brighter nuclei have larger broad-line regions, roughly as the square root of the luminosity, as if the gas sits wherever the ionising light has a fixed intensity. So a single spectrum gives the luminosity, the luminosity gives the radius through that relation, and the radius and line width give the mass.

The relation was calibrated on the nuclei that had been monitored, and those were mostly nearby, modestly luminous and accreting slowly. This essay is about what happens to the shortcut when it is applied to nuclei that are not like that — the ones accreting fastest — and why the error it makes there is unusually hard to see.

A population below the line #

From the mid-2010s, reverberation campaigns deliberately targeted nuclei thought to be accreting rapidly, and found that their lags were shorter than the relation predicted — sometimes by factors of three to eight. The shortening correlated with the accretion rate: the faster the nucleus was accreting relative to the Eddington rate, the further below the relation its lag fell.

The model’s shortening is a parameterisation of the size the campaigns reported, not a derivation. The proposed physical cause is the accretion disc itself. At low accretion rates the disc is thin and the ionising radiation from its inner parts reaches the broad-line gas freely. At high rates the inner disc puffs up — radiation pressure inflates it into a thick, slim disc — and the thickened inner rim shadows the gas near the disc plane from part of the ionising light. The gas that responds sees a fainter source than the observer does, and sits closer in.

What the shortcut does to a mass #

A single-epoch mass uses the relation’s radius. For a rapidly accreting nucleus that radius is too large, by the shortening factor, and the mass, proportional to the radius, is too large by the same factor.

That would be a correctable bias if the accretion rate were known independently. It is not. The dimensionless accretion rate of an active nucleus is computed from its luminosity and its mass, through the physics of a standard accretion disc: the optical luminosity of a thin disc depends on the accretion rate and the black-hole mass together, and solving for the rate gives one proportional to the luminosity to the three-halves power divided by the square of the mass. So a mass that is too large makes the accretion rate too small — by the square of the factor.

That movement is what makes the error self-concealing. A survey that looked for nuclei to which the relation might not apply — rapid accretors — would select them by their computed accretion rates, and the computed rates of the most rapid accretors have been pushed down, below the threshold where anyone would suspect a problem. The relation, applied to them, produces numbers that say the relation applies.

A correction that has to be solved #

The fix seems straightforward: measure how the lag depends on the accretion rate, and correct each single-epoch radius by the shortening appropriate to its accretion rate. But the accretion rate used for the correction must itself come from a mass, and the mass is what is being corrected.

The single pass, which is what an analysis does if it corrects with the uncorrected accretion rate, recovers only a small part of the error. The fixed point — where the corrected mass and the mass used for the accretion rate agree — is the self-consistent answer, and it lies much further away.

How much further depends on the steepness of the shortening, through a simple piece of algebra. The mass after correction scales as the accretion rate to the power $-k$; the accretion rate scales as the mass to the power $-2$; so the fixed point satisfies $M \propto M^{2k}$ relative to the single-epoch value, and the correction to the logarithm of the mass is amplified by $1/(1 - 2k)$. For $k = 0.4$ that is a factor of five. For a nucleus of the same luminosity with a narrower line, 1500 km/s, the model’s self-consistent mass is nine times smaller than the single-epoch one, at an accretion rate in the hundreds.

And for $k \geq 0.5$ there is no fixed point at all: each correction lowers the mass enough to raise the accretion rate enough to require a larger correction, without limit. A shortening that steep would make the radius–luminosity method unusable for rapid accretors — not merely biased, but unable to give a self-consistent answer from a single spectrum. Measured shortenings are near the boundary, which is why the slope has become one of the more consequential numbers in the subject.

The near-parallel curve is the geometry behind the amplification. Where the corrected mass falls almost as fast as the mass used to compute it, each step of the staircase is nearly as long as the last, and the fixed point is far away and sensitive to everything that sets the curve’s position — the threshold, the slope, the calibration of the relation itself. A mass obtained this way for a nucleus deep in the rapid-accretion regime carries an uncertainty dominated not by the measurement but by the parameters of the correction.

Why the reverberation campaigns had missed it #

The original calibrators were chosen for practical reasons. Monitoring a nucleus for months needs a bright, nearby object with strong variability and a lag of a few weeks — short enough to measure in one observing season. Those requirements select moderately luminous nuclei in the local universe, and those are mostly slow accretors: the rapid accretors, which tend to have narrow lines and weaker variability, were under-represented.

The lag measured across the line, resolved in velocity, adds a second clue. In several rapid accretors the velocity-resolved lags show signs of outflow — the gas on the approaching side responding before the receding side — where slow accretors more often show orbits or inflow. If the gas in rapid accretors is partly outflowing, the virial assumption that converts a radius and a width into a mass is also less secure, and the factor multiplying every mass may itself differ for them.

Where the light is measured #

The relation uses the optical continuum at 5100 ångströms because it is easy to measure and lies near the Hβ line. For quasars at high redshift, Hβ moves into the infrared, and masses are computed instead from the magnesium line in the near ultraviolet or the carbon line further into it, with radius–luminosity relations of their own, cross-calibrated against Hβ in nuclei where both are seen.

Each cross-calibration inherits the Hβ relation’s calibration sample, and therefore its bias towards slow accretors. The carbon line, in particular, is often blueshifted and asymmetric in luminous quasars — signs of a wind rather than orbiting gas — and its width correlates poorly with the Hβ width in the same objects. The highest-redshift quasars, whose masses are most in demand, have their masses from the lines least tested against reverberation, by relations calibrated on objects least like them.

Narrow lines, fast growth #

The rapid accretors have a recognisable spectral signature, which is part of why they could be targeted. Their broad lines are relatively narrow — a couple of thousand kilometres a second rather than five or ten thousand — their spectra are rich in iron emission, and their X-ray spectra are steep. The class was named for the first property: narrow-line Seyfert galaxies, nearby nuclei whose lines are narrow for their luminosity.

The narrow lines are themselves a consequence of a small mass at a given luminosity: gas at the radius set by the luminosity orbits a lighter black hole more slowly. So the nuclei most affected by the shortening are the ones whose line widths already say they are small black holes accreting fast, and the shortcut — taking a radius too large — gives them more mass than their lines alone would suggest, moving them towards ordinary Seyfert galaxies in every diagram of mass against luminosity. Corrected, they become what their spectra suggested in the first place: small black holes in a phase of rapid growth.

The earliest quasars #

The consequence that has drawn most attention concerns the first billion years. Quasars at redshifts of six and seven, when the universe was under 800 million years old, have single-epoch masses of a billion solar masses and more, and their computed Eddington ratios are often modest — a few tenths. A black hole growing at a few tenths of the Eddington rate e-folds its mass every few hundred million years, and cannot grow from a stellar-mass seed to a billion solar masses in the time available.

The figure does not resolve the problem, and the problem has other proposed solutions — heavy seeds, formed by the direct collapse of gas clouds into black holes of ten thousand to a million solar masses; episodes of growth above the Eddington rate, which the Eddington limit caps in light but not in mass. What it shows is that the problem is stated in terms of masses and accretion rates that the radius–luminosity bias pushes in exactly the direction that makes it harder: masses up, rates down. A population of early quasars that were actually accreting at super-Eddington rates, with smaller black holes than assigned, would look, through the shortcut, like a population of overweight black holes growing slowly.

Eddington’s limit, and what it limits #

The accretion rate in this argument is measured against the Eddington rate, the rate at which the radiation from accretion would push on the infalling gas as hard as gravity pulls. The limit is written for free electrons scattering light, and it is a limit on luminosity, not on the rate at which mass can fall in: at rates far above it, the disc traps its own radiation and carries much of it into the black hole, and the luminosity rises only logarithmically with the inflow.

For a star the same balance sets a hard ceiling — a brightness above it would blow the star apart, because the radiation acts on the star’s whole outer envelope. An accretion disc escapes that fate by geometry: radiation leaves along the axis, where there is little gas, while the inflow arrives in the plane, where the radiation is trapped and dragged inward. The limit applies to a spherical problem, and a disc is not spherical.

That is why the rapid accretors can exist at all, and also why their luminosities are poor guides to their accretion rates. A nucleus accreting at a hundred times the Eddington rate shines at a few times the Eddington luminosity. Its broad-line region is set by the ionising light it actually emits, and the shadowing by its thick inner disc reduces that further in the directions where the gas lies. The radius–luminosity relation fails for it because the light the observer measures, along the disc’s axis, is not the light the broad-line gas receives near the disc plane.

Independent checks #

Masses that do not come from the broad lines provide the test. A handful of nearby nuclei have masses from the motions of stars or gas resolved in the galaxy’s centre; the brightest nuclei have broad-line regions resolved directly by near-infrared interferometry, which measures the angular size and, with a distance, the physical size without any lag; and for some quasars the mass can be estimated from the accretion disc’s spectrum, whose shape depends on the mass and the accretion rate together.

Interferometric sizes for the brightest nuclei have come out smaller than the radius–luminosity relation predicts for their luminosities — consistent with shortening at high luminosity, and with masses lower than the single-epoch ones. The disc-spectrum masses for high-redshift quasars are similarly lower, on average, than their single-epoch masses. Neither check is precise enough yet to fix the slope of the shortening, but both point the same way.

A third check is statistical. The masses of nearby black holes correlate tightly with the velocity dispersions of the stars in their host galaxies’ bulges, and the mass at the centre that is not stars has been weighed that way in dozens of galaxies. Active nuclei with reverberation masses follow the same correlation once the virial factor is calibrated on it. Rapid accretors, with single-epoch masses from the uncorrected relation, tend to lie above it — too massive for their hosts — and the offset shrinks when their masses are corrected for shortened lags. The offset was once taken as evidence that rapid accretors are young systems whose black holes have outgrown their bulges; it may instead be the radius–luminosity bias, seen against a relation calibrated another way.

Where the model stops #

A parameterised shortening. The relation between lag and accretion rate is drawn as a power law above a threshold. The measured dependence is noisier and may depend on other properties — the strength of the iron emission, the shape of the line — that correlate with the accretion rate; the figures use a single parameter to show the consequences, not to fit the data.

A standard disc for the accretion rate. The accretion rate is computed from the luminosity and mass with the formula for a thin disc seen at an average inclination. At the highest rates the disc is not thin, and the formula’s dependence on mass may change, which would change the amplification of the correction.

A virial factor of one. Masses are computed with the line width as the velocity and a virial factor of one. The factor varies by several between objects, as the essay on it argued, and may differ systematically for rapid accretors.

Synthetic populations. The nuclei in the first figure are drawn at random in luminosity and accretion rate with a fixed scatter; real samples are selected by their own flux limits and variability, which shape where they fall.

The surprising connection #

The trap in this correction — an input computed from the output — is the same one that made the black holes that can be weighed the heavy ones: a measurement whose selection depends on the quantity measured. There, the sample was biased because only massive black holes had resolvable spheres of influence. Here, the bias is subtler: the quantity used to decide whether a correction is needed is itself distorted by the absence of the correction. Both are versions of a problem every calibrated relation faces when applied beyond its calibration sample — but this one has the additional property that its own output hides the evidence that it has been applied out of range.

Still open: how steep the shortening is #

The slope of the lag’s dependence on accretion rate decides whether the single-epoch method can be corrected for rapid accretors or fails for them outright, and the measured slopes lie near the value at which the correction stops converging. Reverberation campaigns of rapidly accreting nuclei at higher luminosities and redshifts, now under way with multi-object spectroscopic monitoring that follows hundreds of quasars at once, will measure lags for the kinds of object the shortcut is most used on. Whether they find a slope safely below a half — a correctable bias — or one that makes the masses of the most rapidly growing black holes impossible to determine from a single spectrum is the question on which every black-hole mass beyond the nearby universe now partly depends.