Part III: The GLV Building Blocks as a Philosophical Framework. Layer 3 Non linear amplification

I think you only really begin to see how sophisticated the universe is in distorting our picture once you look not only at how much, and in which direction, light is deflected, but also at the scale at which that happens. Layer 1 was about an average, the winding road, summarised in δ. Layer 2 added that this winding is not the same everywhere, some directions are optically heavier than others. Layer 3 adds one more dimension to this. It says that the lens strength of the cosmic web itself depends on the structure you are looking at. Large, long wavelength patterns are affected differently from small, fine grained patterns. Scale matters.

An intuitive image helps. Imagine you are looking at a landscape through ribbed glass. The large lines, the horizon, the mountains, stay more or less where they were. But the details, the branches of a tree, the edges of a house, begin to wobble and dance. The lens in the glass treats large structures and small structures differently. It hardly smears the large pattern, but it makes the small patterns restless, and sometimes even more noticeable. Something similar happens in the cosmic web. The combination of many small lenses along the line of sight strengthens some spatial scales just a little more than others.

In the language of GLV this means that the optical correction from Layer 1 and Layer 2 is not neutral across all length scales. If you describe spacetime in terms of wavelengths of structure, long waves for large patterns, short waves for small details, then those short waves are more sensitive to the combined lens fields. Each small lens on its own has only a tiny influence, but the non linear stacking of those effects along a light path can nudge the amplitude of short scale structure upward. You could say that the web has a very slight preference for fine detail. That is Layer 3, non linear scale amplification.

In GLV I summarise that preference in an amplification factor, call it β, with a characteristic scale at which the effect starts to increase. For structures much larger than that scale, β is negligible, the lens stack hardly changes the relative strength of large patterns. For structures closer to that scale, and especially for even smaller patterns, the amplification becomes more noticeable. That does not mean the cosmic web suddenly becomes a microscope, the effect remains small. But in a field where we work with percent levels, and tenths of a percent, in precision measurements, a small preference for certain scales is already enough to shift the balance.

You see this very clearly in the cosmic microwave background. The spectrum of temperature fluctuations across the sky shows a series of peaks. The first peak corresponds roughly to the largest sound waves in the early plasma, the second, third, and higher peaks belong to progressively smaller structures at the moment of decoupling. In the standard model, dark matter helps to make the higher peaks relatively loud compared to the first. That extra strength on small scales is often presented as an elegant argument for an invisible matter component.

In the GLV picture I dare to place another layer underneath that. If the lens field strengthens small structures optically just a little more than large ones, then the intrinsic ratio between the first and the third peak does not have to be exactly what we now see in our flat reconstructions. A slightly favoured amplification of short wavelengths, visible as a slight lifting of the third peak relative to the first, can arise from the optical stacking of lens fields from the moment of decoupling all the way to our telescopes. You then need less intrinsic small scale driving built into the beginning of the universe, and therefore less additional matter as the carrier of that driving.

The same mechanism appears in baryon acoustic oscillations, the imprint of sound waves in the distribution of matter. The typical separation between clusters and large structures is often used as a standard ruler, a fixed scale in the cosmos. Observations show, however, that this ruler at low redshift seems to be stretched just a little compared to expectation. In classical analyses this is linked to details of the expansion history, and to dark energy. In GLV another possibility is added, the lens stack can stretch that ruler optically.

If short wavelengths in the matter distribution receive a slight optical amplification, then the ring scale where the BAO imprint shows up can appear, in hindsight, a little larger than the original sound horizon. It is like looking at a rigid pattern through ribbed glass. The ribs do not add new structure, but they do change the way you see the existing structures. In that way a five percent optical stretching of that ring can arise without the underlying geometric scale really changing. In terms of GLV this is exactly the kind of effect that belongs to Layer 3, a scale dependent amplification on top of the global distance bias from Layer 1, and the directional dependence from Layer 2.

You might ask why this non linear scale amplification occurs at all. The core is that lens fields are not a simple sum. If you hold two magnifying glasses on top of each other you do not get only a stronger lens, you also get distortions you do not see with a single glass. In the cosmic web it is even more complicated. You do not have two lenses, you have a whole stack of small, overlapping deviations in spacetime. Some lenses sit on roughly the same scale, others on very different scales. Light passing through such a stack gets not only an average deflection, it also picks up extra small scale modulations. It is precisely in those modulations that the information sits that Layer 3 is trying to capture.

From the quantum and field side, this is not strange. Interactions in a field theory are rarely completely scale blind. We often see that processes are more dominant at certain energies or length scales than at others. In the zero point field of this model it is no different. The way mass condensations and field ripples reinforce one another always produces a kind of preference pattern in k space, in the space of wavelengths. Layer 3 is nothing more than the optical shadow of that preference pattern. The relativistic lens equations are sensitive to it, because they effectively count how all the little mass peaks along a line of sight work together to deflect the light.

What does all this mean for our interpretation of dark matter, in connection with the earlier layers. Layer 1 reduces the need for extra mass by pulling distances back a little. Layer 2 spreads that correction across directions. Layer 3 mainly targets the question of why certain phenomena on small scales, such as the height of the third background peak, or the strength of small structures in lens maps, suggest there is more gravity than is visible. If part of that strengthening is optical, a consequence of scale dependent lens stacking, then the intrinsic matter distribution can be less sharp than we currently think. Part of what we interpret as dark matter can be compensation for the optical brightness of small details.

In everyday life we know this kind of misdirection all too well. A photo with high contrast and sharpened edges often feels sharper and richer than the reality was. You suddenly see every wrinkle and every smear. The underlying scene is the same, but the image that reaches you looks more dramatic and more detailed. In my view the universe does something similar to us. The non linear combinations of lens fields give the cosmic picture a very slight sharpening on certain scales. We take that as given, we pour it into a model, and we conclude there must be more structure than what is visible. Layer 3 invites you to ask first, does that extra contrast come from the scene itself, or from the optical system through which we see it.

There is also a subtle link with time in Layer 3, which anticipates what I develop in the next layer. The longer light travels, the more lens interactions it accumulates. Non linear scale amplification is therefore not only a function of scale, it is also a function of depth. Structures at the same physical scale can look optically a little different at different redshifts, because the lens stack in between is different. That means that when we follow the growth of structure through time, we always have to remember that we are not only looking at an evolving cosmic web, we are also looking at an accumulating optical distortion. Layer 3 becomes the bridge between spatial scale and depth.

In GLV all of this gets a formal translation into an extra scale dependent factor in the optical metric. In this book you do not need the formula. It is enough to hold on to the intuition that the web itself has a kind of characteristic grain. Structures smaller than that grain are optically pushed a little harder, structures much larger than it slide through more or less unaffected. δ tells you how much the whole map is stretched on average. α and the direction vector n tell you in which direction that stretching is a little stronger or weaker. β tells you for which details in the map the optical glass turns up the contrast and the brightness.

If you take Layer 1, Layer 2, and Layer 3 together, you get a very different feel for cosmic measurements. Distance, direction, and scale are not neutral concepts, they are optically loaded. The universe we reconstruct from light is therefore a little larger, a little more skewed, and a little sharper than the geometric reality in three spherical spacetime. In the next layer I will show how the time dimension is not free of these optical influences either. The farther light has travelled, the thicker the lens stack, and the stronger some of these effects become. But under all of that the common thread of GLV remains, as long as we treat light as an honest, straight messenger, we will attribute properties to the universe itself that in reality arise from the optical system through which we view it.