Part III: The GLV Building Blocks as a Philosophical Framework. Layer 2 Direction Sensitive Lens Distortion

Layer 2 begins with a question we rarely ask out loud, whether the universe is equally honest to us in every direction. In Layer 1 it was about an average, we discovered that light paths wind, and that we therefore systematically overestimate radial distances. We captured that in one number, δ, an average shortening of about eleven percent when you redraw the winding road as a straight line. But nothing in nature guarantees that this effect is exactly the same in every direction. On the contrary, as soon as you look at the actual mass distribution in the universe, it becomes clear that some directions are heavier than others. Layer 2 is about those preferred directions.

The cosmic web is not a neat, homogeneous mist. Matter does not sit in space like an evenly distributed cloud, it sits like beads on threads. Massive clusters sit at the nodes of long filaments, in between lie relatively empty voids. If you take an imaginary cube of the universe and look at it from all sides, you see the same statistical patterns everywhere. But from one point, within one light cone, reality can feel very different. On one side of the sky there may hang a thick filament strand, on the other side perhaps a deep void. The result is that light paths from different directions, on average, run through a different lens stack.

Layer 2 says exactly that, the radial distance bias from Layer 1 is not isotropic. The overestimation of distances, and therefore of mass and scale, depends on the direction in which you look. In some sectors of the sky δ is a little larger than the average, because more mass has clumped along the line of sight there. In other sectors δ is smaller than average, because light there travels mainly through quieter regions. Averaged over all directions you still end up with δ around 0.11, but reality is δ plus a directional deviation.

You can imagine it as a kind of wind direction in lens strength. There is a preferred vector in space, an n, which indicates in which direction the lens stack is thickest. If you look roughly in that direction, light paths wind a little more, and the effective radial shortening is larger. If you look in the opposite direction, paths wind a little less and the shortening is a little smaller. In the language of GLV I capture this extra direction sensitivity in one amplitude, α, and a preferred direction, n. Together they describe a dipole like variation in δ across the sky. Without α Layer 2 is silent, then δ is the same everywhere. As soon as α is larger than zero, the universe in our measurements starts to pull a little more on one side than on the other.

It is important that this is not a violation of the cosmological principle on the largest scales. Statistically the universe can still be homogeneous and isotropic. But we do not look at the whole statistic, we look from one place, through one light cone, with one specific history. Layer 2 takes that dead seriously. It says, even in a statistically well distributed universe our measurements can show a preferred direction, simply because our own environment and our lines of sight run through the web in an uneven way.

What do you notice of that in concrete terms. One of the clearest examples is the kind of system called the Bullet Cluster, a colliding double lens of clusters, where the hot gas clouds and the gravitational mass have partly overlapped. In standard cosmology the centre of lensing strength is usually equated with the centre of the underlying dark matter. In the Bullet Cluster we see that the lensing mass map has a clear offset relative to the bright gas. That is often seen as a signature of colliding dark matter halos that slide through each other while the gas lags behind.

In the GLV story Layer 2 adds another reading. If the lens stack is not symmetric everywhere, but in one direction a little heavier, then the optical centre of the lens is shifted relative to the baryonic centre. The photons we observe are not evenly distributed over all directions, they come through that lens cone where, by chance, more or less foreground mass lies. The centre of lensing strength that we reconstruct from the light field can therefore shift in a dipole pattern, slightly toward the heavier side of the web. The offset between gas and lensing mass is then not by definition proof of a separate dark component, it can be a signal of direction sensitive shortening in the light paths that run through that cluster.

Another area where Layer 2 can be felt is closer to home, in the distribution of satellite galaxies around the Milky Way and similar galaxies. For a long time there has been a tension between simulations and observations. Simulations with cold dark matter predict a substantial population of massive satellites, while in practice we see fewer of those heavy neighbours than expected. This is sometimes called the missing satellites problem. GLV suggests that part of that tension does not have to sit in the true population, it can sit in the way we look at that population.

If δ is a little larger in some directions than in others, then the apparent distances and brightnesses of satellites shift in a subtle, but systematic way. Satellites that lie in a direction where the lens stack is thick are placed optically a little farther away than they are geometrically. Their dynamics is translated to too large a scale, which makes them seem too massive or too fast. In a direction with a relatively thin lens stack the reverse happens. If you then choose a single threshold for which satellites you count as massive and which you do not, you quietly create a direction dependent selection. From within our light cone it can then look as if in some sectors of the sky there are too few massive satellites. Through the GLV lens you see in that a contribution from the dipole in δ, the fact that our own lens environment is different on one side of the Milky Way than on the other.

Perhaps the most refined signal of Layer 2 sits in the rotation curves of disks. In Layer 1 I looked at radial profiles of rotation speed, speed as a function of distance from the centre. But a disk does not only have radius, it also has angle. If the lens stack is not isotropic, then the optical image of the speed distribution will not be the same everywhere at equal radius. In the outer rings of a galaxy you then expect small oscillations in the measured speed as you move around the disk, a kind of sine like wiggle in the azimuthal direction.

You can see it like this, the radial distance bias from Layer 1 gives an average radius for a ring, but Layer 2 lays a very slight wave on top of it, where some angles look optically a little longer or shorter. That angle dependent variation in δ translates into small, periodic differences in the inferred speed. In the GLV analysis of real rotation curves we do indeed see indications of such wiggles in the outer disks. The signal is weak and it asks for very precise measurements, but that is exactly what makes it a beautiful test for Layer 2, a test that has nothing to do with exotic particles, and everything to do with subtle lens geometry.

Philosophically, Layer 2 is a reminder of something we often forget, all cosmic knowledge comes in through one light cone. It is like a museum where you are allowed to walk only down one corridor. You see what hangs on your wall, but you have no idea how the rooms on the other side are arranged. Anisotropy in lens distortion means that even a universe that is globally neat can look skewed locally in your corridor. On one side of your field of view there can hang a heavy mass strand that increases δ, while the other side is relatively quiet. If you then use the same straight rulers and spherical models everywhere, you build a map with a built in preference, a map that places a little more weight in the direction of the heavy wall.

Layer 2 does not try to polish that away, it tries to make it explicit. The parameters δ, α, and n together describe how strong the radial distance bias is on average, and how that bias differs between directions. δ tells the average story of the winding route, α tells how much extra contrast there is in that story between different sky directions, and n indicates where the preferred direction lies. In technical GLV work you can pour that into formulas. In this book it is enough to hold on to the intuition, in some viewing directions the optical shortening is a bit larger, in others a bit smaller, and you can summarise that pattern as a dipole, or in some cases a quadrupole.

What does that buy us. If Layer 1 and Layer 2 are both taken into account, you have to reach less quickly for a separate dark sector for a number of puzzles. The offset between mass and gas in colliding clusters gains a measurement dependent component, the suppression of massive satellites around Milky Way like galaxies can partly arise from a direction dependent δ, the small waves in rotation curves fit better within a lens geometry with a preferred direction than within a perfectly spherical halo. You are not replacing hard data with interpretation, you are replacing an overly simple grid with a grid that takes the real structure of the zero point field more seriously.

It remains true, of course, that Layer 2 has to be tested. A dipole in lens distortion is only credible if you can actually see it in data. In the more technical GLV documents it is described how surveys such as Euclid and SKA can look for precisely those offsets and wiggles. For this book I mainly want to make clear that there is a consistent idea behind it. If the universe is a cosmic web and we have only one light cone, then it is unlikely that all directions are optically exactly the same. Layer 2 takes that unlikelihood seriously, and tries to capture it in a simple, testable form.

Together with Layer 1, Layer 2 forms the skeleton of the optical part of GLV. Layer 1 says, never underestimate the difference between an optical and a geometric light path, δ is not zero. Layer 2 adds, never underestimate that we look into that web from one specific direction, the bias is not the same everywhere. In the next layer the refinement becomes even greater. There it is no longer only about average distance and direction, it is also about scale, about the fact that not all wavelengths and not all structure sizes are affected in the same way by the lens stack. But even there the basis remains the same, in a universe that is not empty and that truly has a web structure, light never tells us a fully neutral story. And the more we acknowledge that, the less we have to fill the universe with invisible substances to compensate for our misinterpretations.