The fifth image a galaxy hides

The lens with no focal length established what a single mass does to the light passing it. A deflection that falls as one over the distance from the axis has no focal plane, and for a source behind a point mass the lens equation is a quadratic, so there are two images, one on each side, with a ring when the alignment is perfect. That is the right description of a star lensing a more distant star. It is the wrong one for a galaxy, and the difference is not a matter of accuracy. A point mass always makes two images. A galaxy never does.

The reason is not in the physics of the deflection, which is the same factor-of-two deflection in both cases, summed over the mass. It is in the geometry of a map. A lens sends every direction on the sky to a direction in the source plane: the place the light would have come from if nothing had bent it. An image is a direction on the sky that the map sends to the source. Counting images is counting how many times a map covers a point, and there is a theorem about that count for maps that are smooth.

Where every line has to cross #

The drawing takes the simplest spread-out lens, a galaxy whose density falls as one over the square of the radius except in a small central core, and cuts through it along a line. The horizontal axis is where on the sky light arrives from; the vertical axis is where its source really is. An image of a source is a place where the curve reaches the source’s height.

Far from the centre the curve runs parallel to the diagonal, one Einstein radius below it on the right and one above it on the left: the deflection there is about one Einstein radius, pointing inward, as for any mass seen from far away. Near the centre the curve turns back, because the core’s high density bends light near the axis so strongly that the arrival direction and the source direction run the opposite way. The curve dips below zero on the right and rises above it on the left before it recovers.

The important property is not the dip but that the curve has no gaps. It starts far below at the left edge and ends far above at the right, and it passes through every height in between without jumping. A horizontal line at any height must therefore cross it an odd number of times: once if it misses the dip, three times if it passes through it. A source a third of an Einstein radius from the axis has three images; one well outside the dip has one. There is no height with two.

The dashed curve is the point mass, and it is the exception because it breaks the one condition. Its deflection grows without limit towards the centre, so its curve runs to minus infinity just right of the axis and to plus infinity just left of it — two separate branches, each crossing every height once. The odd count needs a curve that is continuous across the centre, and a point mass is not. Every real mass distribution has a finite central density, and the moment it does, the two branches join through the middle and the third crossing appears.

Five images of one quasar #

A real galaxy is not round, and it does not sit alone. Both effects add a stretch to the lens along one direction, and the simplest way to include them is an external shear: a deflection that pushes light outward along one axis and inward along the other. With a shear of 0.15, a typical value for a lensing galaxy, the line of the previous drawing becomes a surface, and the question of how many times a smooth surface covers a point has the same answer. The count is still odd.

The drawing places a source slightly off the centre and finds every image by brute force: every cell of a fine grid on the sky is mapped back to the source plane, the cells whose images cover the source are kept, and each is polished to machine precision. Five images come out. Four sit near the Einstein ring, arranged roughly in a cross, and are magnified between two and a half and five times. The fifth sits almost exactly at the centre and is magnified by about a hundredth — demagnified, dimmer than the source would be with no lens at all.

The images are not all the same way round. A lens map can preserve the handedness of a small patch of sky or reverse it, and the sign of the map’s local stretching, its Jacobian determinant, says which. Three of the five images are the right way round and two are mirror images. The parities sum to one, and that sum is the deeper form of the theorem. A continuous map of the sky that runs off to infinity in every direction covers each point of the source plane with a net count of exactly one, counting a mirrored copy as minus one. Images can be added only in pairs of opposite parity, which leave the net count unchanged, and that is why the total is always odd.

This is the configuration of the Einstein cross, the quadruply lensed quasar found behind a nearby spiral galaxy in 1985, and of a few dozen similar systems found since. They show four images because the fifth, at the lens’s centre, is faint and buried under the light of the galaxy’s own nucleus, where no telescope can pick out a demagnified point source. The count that the theorem requires is five, and the count observed is four, and the difference is a statement about how hard it is to see something rather than about what is there.

A map that folds #

The places on the sky where the map’s Jacobian is zero are called critical curves, and there are two of them in the image drawing: an outer one near the Einstein ring and a small inner one round the core. On a critical curve the map squashes a small patch of sky to a line, and the magnification — one over the Jacobian — is formally infinite. Mapped into the source plane, the critical curves become the caustics, and the caustics are what decide the count.

The outer critical curve maps to a small diamond with four cusps, reaching a quarter and a third of an Einstein radius along the two axes of the shear. The inner one maps to a large oval, reaching about seven tenths. A source outside the oval has one image. A source between the oval and the diamond has three. A source inside the diamond has five. The drawing checks this by counting images at four test positions rather than taking it on trust.

Crossing a caustic changes the count by exactly two, and there is a simple picture of why. Along a caustic, the lens map folds the sky over itself like a sheet of paper creased and laid back. A source just on the folded side of the crease has two images of opposite parity, lying on either side of the critical curve, one from each layer of the fold. As the source approaches the crease the two images approach each other, grow brighter, and at the crease merge on the critical curve and vanish. The magnification of each image near a fold grows as one over the square root of the source’s distance from it, which is finite in total brightness integrated over any source of finite size and infinite only for a point.

The same fold is responsible for the rainbow’s sharp edge, where the deviation of light through a drop passes through a minimum and two rays of different impact parameter merge into one direction, and for the envelope of every possible throw, where two trajectories to a target merge into one and beyond which there are none. A caustic is always the edge of a region reached twice, and the count on the far side is always two fewer. A mirror that cannot focus draws the bright cusped curve in a coffee cup for the same reason. Gravity adds nothing to the geometry, only a particular map to fold.

The count along a path #

The drawing moves a source in a straight line across the lens, just off its long axis, and counts at each position. The count runs one, three, five, three, one. It never takes the values two or four, and it never changes by one. Each change comes at a caustic, where the total brightness jumps: coming in, the source crosses the oval and gains a pair, crosses the diamond and gains another; going out, it loses them in reverse order.

The same folding happens on much smaller scales inside the lensing galaxy. Each of its stars adds a tiny caustic network of its own on top of the galaxy’s, and the relative motion of source, lens and observer carries the source across those small folds over months to years, brightening one image of a quasar without the others. How sharp each brightening is depends on how large the source is compared with the fold, so a crossing converts a size far below any telescope’s resolution into a duration. The fold turns a size into a time.

Why the fifth image is so faint #

The fifth image lives inside the inner critical curve, where the convergence of the lens — its surface density in units of the critical density — is far above one. There the lens is not bending light by a small correction; it is turning the arrival direction round entirely, and the magnification, which goes as one over the product of two factors of roughly one minus the convergence, becomes tiny.

The drawing keeps the source and the shear fixed and shrinks the core from three tenths of an Einstein radius to three thousandths. Every lens in the series gives exactly five images, so the theorem survives. But the central image’s magnification falls from about a fifth to ten millionths, roughly as the square of the core, while the four outer images barely change. A real galaxy’s inner density profile is steep and its core, if it has one, is small, so its central image is expected to be tens of thousands of times fainter than the others — and it would be seen, if at all, against the galaxy’s own bright nucleus.

That prediction has been tested where it can be. At radio wavelengths the lensing galaxy’s own light does not compete, and in 2004 a faint third image was identified at the centre of a radio-loud system that had been known to show two, with a brightness that required the galaxy’s inner density to be steep. The central image is therefore not only a requirement of a theorem but a measurement: its brightness depends on the density of the galaxy’s innermost few hundred parsecs, a region that is otherwise almost impossible to weigh.

In the limit of no core at all the central image does not just become faint; it disappears, and the lens drops to four images, or two, and the odd-number rule fails. That is not a contradiction. A density that rises without limit at the centre makes the map discontinuous there, exactly as the point mass does, and the theorem was only ever a statement about continuous maps. The count depends on whether the lens is smooth, and the field outside a body cannot tell how its mass is arranged inside — but the fifth image can, because it is made by the light that passed through the middle.

Peaks, passes and a pit #

The theorem has a second derivation that is worth having, because it uses a different picture and gives something the first does not. Light from the source reaches the observer by every path, and the images are the paths for which the travel time is stationary — Fermat’s principle applied to a region where the metric slows light down, which is the delay the Sun imposes on radio signals turned into a map. Plotted over the sky, the arrival time is a surface. It has a geometric part, a bowl centred on the source’s true position that grows as the square of the angle away from it, and a gravitational part, a mound where the lensing mass slows the light.

Images sit at the stationary points of that surface, and a smooth surface that rises to infinity at its edges has stationary points of three kinds: minima, saddles and maxima. A theorem from the topology of landscapes, which every hill-walker implicitly knows, says that on such a surface the number of pits minus the number of passes plus the number of peaks is one. Minima and maxima are images the right way round; saddles are mirror images. So the parity sum is one and the count is odd for the same reason that an island with two summits has a col between them.

This picture also says what each image is. The four bright images of the cross are two minima and two saddles, lying on the rim of the gravitational mound, and the faint central image is a maximum — the top of the mound itself. It is a path that takes the longest time among its neighbours, which is why light following it arrives last, after the others, by an interval that is days to weeks for a galaxy lens. The time delays between images of a variable quasar have been measured for dozens of systems, and they depend on the distances to lens and source; the fifth image’s delay would be the longest of all.

Where the smooth lens stops being the right model #

The theorem assumes the lens is transparent: that light passing through the centre of the galaxy reaches the observer. Dust in the lensing galaxy can absorb the central image entirely, which is one more reason it is rarely seen at optical wavelengths. It also assumes a lens with no hole in its mapping. A supermassive black hole at the galaxy’s centre is a point mass inside a smooth distribution, and it changes the count near the centre: depending on its mass and the core’s, it can swallow the central image, or split it into two faint images of opposite parity, leaving four or six in total. That is another instance of the rule failing exactly where continuity does, and it means that a measured central image, or its absence, also limits the mass of the central black hole.

The images drawn here are points. A real source has a size, and one near a caustic becomes an arc, stretched along the critical curve with a length that depends on the magnification. Giant arcs in galaxy clusters are sources lying across cluster caustics, and their shapes carry the same information as the image counts, but spread out rather than concentrated.

The thin-lens assumption also matters. Everything here treats the galaxy as a sheet at one distance, which is excellent when the galaxy’s depth along the line of sight is small compared with the distances to it and beyond it, as it always is in practice. Two lenses at different distances along one line of sight are not a thin lens, and their combined caustics can make seven or more images of one source; the parity sum is still one.

The images a smooth model cannot account for #

Four images near the Einstein ring should have brightnesses related to one another in a way the smooth lens predicts. When a source sits near a cusp of the diamond, three of the images lie close together, and the sum of their signed magnifications — counting the mirrored one as negative — should be close to zero. Many observed quadruple systems violate that relation, one image being too faint or too bright by tens of per cent.

The violations are usually not errors in the models. They are the signature of small-scale structure in the lensing galaxy: satellite galaxies, clumps of dark matter, or stars, each adding a small perturbation to the map near one image and not the others. How much of the anomaly is due to dark matter clumps of a particular mass, and whether their abundance matches what models of dark matter predict, is an active question, and the answer constrains what dark matter is. The pictures, which draw a lens with no substructure at all, show what the images would be if the galaxy were smooth, and the discrepancies with real systems are measurements of its roughness.

Still open: how dense the centre of a galaxy is #

The central image is the only direct probe of a galaxy’s innermost mass that works at cosmological distances, and it has been used in only a handful of systems, because it is faint and the galaxy is bright. Radio interferometers with enough sensitivity to find central images routinely could measure the inner density profiles of dozens of galaxies and settle whether their cores are flat or cusped at scales below a hundred parsecs — the same question dark-matter models disagree about in nearby dwarf galaxies. Whether the central images, once found in numbers, will match the density profiles inferred from the stars’ motions in nearby galaxies has not been determined.

The habit worth carrying away is to count before computing. A smooth map that runs off to infinity at its edges covers each point an odd number of times, counting orientation, and it can only gain or lose coverings in pairs on the curves where it folds. That statement needs no knowledge of what the map is, only that it has no holes, and it says at once that a lens galaxy makes five images where four are seen, that a source drifting behind it changes its image count by two, and that a lens which appears to make an even number has something sharp hidden in its middle.