Part III: The GLV Building Blocks as a Philosophical Framework. Layer 4 Time and Depth in the Light Cone

Until now we have acted as if δ were a kind of fixed knob. In Layer 1, δ appeared as the average shortening of radial distances, because light winds. In Layer 2, a direction dependence was added on top of that. In Layer 3, we saw that not all scales are treated equally, small structures are optically emphasised just a little more than large ones. In all those layers, however, there was still a silent assumption underneath, that δ is the same everywhere along the light cone. Layer 4 turns precisely on that. Not only distance, direction, and scale matter, time and depth matter too.

We never see the universe at a single moment. Every measurement we make sits on a different step of the time ladder. We see a nearby galaxy as it looked tens of millions of years ago, a distant system as it looked billions of years ago, and the cosmic background even from a time when there were no stars at all. Our light cone is not a photograph, but a stack of slides with different ages. At the same time, the lensing cosmic web itself is also changing through time. Dense structures grow, voids become emptier, clusters form and merge. Layer 4 is about the overlap of those two facts. The light we see travels through a lens stack that itself evolves while the photon is on its way.

You can feel this clearly with a simple image. Imagine a long mountain route, not only with hairpin turns, but also with villages and cities along the way that grow over time. A cyclist leaves a small village, first rides through a quiet landscape, later passes through increasingly busy places, and ends in a large city. If you only know the total travel time and the final distance, you can never calculate how much heavy traffic the cyclist encountered along the way. The traffic was not the same everywhere, it increased as he came closer to the city. In the cosmos it is the same. A photon that comes from far away first passes through a young, relatively smooth universal environment, but later, closer to us, through a much more clumped structure. The lens stack is therefore not homogeneous over the travel time, it is different in the second half of the route than in the first.

Layer 4 translates this into δ as a function of depth. For very nearby sources, the route is short, the light has passed only through a thin layer of the cosmic web. The optical route and the geometric distance are then close together, δ is small there. For sources at middle distances, say around redshift one, the lens stack is thicker and the web is already fairly developed. That is where the bulk of the path shortening lies, that is where δ is close to the value we derive with GLV from the five “gearwheel” observations. For extremely distant sources, such as the CMB, the number of lenses passed through increases further, but the extra gain per gigaparsec becomes smaller, because in the far youth of the universe the web was still less clumpy. δ grows with depth, but it saturates.

That is important, because in many analyses we do exactly the opposite. We treat the whole light path as if it passed in one go through a static, fully developed web. In the lensing kernels of standard cosmology this is corrected for, but often it stays in the background of the interpretation. Layer 4 brings it forward with a simple message, δ is in fact a function δ(z). At small z the effective δ is a fraction of the asymptotic value, at z of order one you are around the calibration value, beyond that the growth levels off.

That z dependence has direct consequences for how we think the universe changes through time. Take the supernova story. Distant supernovae appear fainter than expected, which is interpreted as an acceleration of expansion. The standard plot, the Hubble diagram, assumes that the relationship between brightness and distance is translated everywhere into a straight scale with the same δ. Through a GLV lens you say, be careful, supernovae at different redshifts have light that has travelled through different effective δ layers. Nearby, δ is smaller, farther away you are closer to the full δ of the cosmic web. If you do not include that, it looks as if the Hubble curve bends upward with z, while part of that bend is the work of Layer 4. The light of distant supernovae has simply accumulated more “optical kilometres” in the clumped, later half of the universe than the light of nearby ones.

You see the same thing with the growth of structure. We measure how strongly density contrast increases through time, often summarised in a growth parameter per redshift. In the standard model, that is tightly linked to the contents, to how much ordinary matter and how much dark matter there is, and to the expansion history. If the measured growth at some depths lags behind expectation, that is quickly linked to missing dark components, or to dark components coupling in a different way. When you take Layer 4 seriously, an extra factor enters, the lens stack that optically amplifies small structures, Layer 3, does not do so with the same strength everywhere. Nearby structures benefit more from the full lens stack than structures in the far youth, where the web is still under construction.

That means you must always see the observed growth of contrast as a combination of real dynamics and a z dependent optical amplification. What we call “too little growth” at some depths may partly be that we are including the optical equaliser of Layer 3 and Layer 4 in the reconstruction in the wrong way. You are then not looking at a pure film of structure growth, but at a film played through a variable lens.

Layer 4 is also hidden inside the cosmic background radiation itself. The CMB we measure today is light that has travelled since decoupling through a universe that has been changing the whole time. The lensing signal in the CMB, the extra graininess in the temperature and polarisation map, is the integrated effect of the web that grew during the long intervening time. The lensing kernel for the CMB peaks roughly halfway through our visible three sphere, where the combination of lens mass and distance is largest. That is exactly the z zone where GLV calibrates δ, the zone where the five gearwheels, rotation curves, third peak, BAO, lensing, and growth parameter, are all sensitive at once.

Layer 4 says here, we must recognise that the δ we derive from the CMB and the other large datasets is mainly a δ that applies around that lensing peak, not automatically a δ that literally applies at every z. Within the nearby cosmos the effective δ is smaller, far beyond the lensing peak little more changes. As long as we do not distinguish between those regimes, we run the risk of confusing shifts in optical effectiveness with fundamental changes in the contents of the universe.

Another face of Layer 4 is pure travel time. Light that “takes a detour” through the lens stack arrives at a different time than light that would have followed an almost straight geodesic. With supernovae and other transients, that can be visible as a time delay between multiple images. On cosmic scales, it translates into a subtle difference between the time we think a signal has taken and the real cosmic time along the geodesic. In GLV this is not a separate effect, but another way of looking at δ, δ is in fact also a statement about travel time relative to the shortest possible route. Layer 4 connects that travel time directly to the evolution of the lens environment during the intervening time.

From the philosophical side, Layer 4 touches a nervous chord. We like to talk about “what the universe is like at time t”, as if you could freeze the universe somewhere and look at it from the outside. In reality, we only have access to one light cone, in which every redshift layer has a different age and a different optical filter. Depth and time never come apart in our observation. Layer 4 recognises that explicitly. The δ we measure, the direction vector we recognise in Layer 2, and the scale amplification from Layer 3, are in fact functions on the light cone, not on an abstract spatial carpet at one cosmic moment.

That also means that some tensions in current cosmology may be less deep than they appear. The H0 tension, the difference between the locally measured expansion rate and the value derived from CMB analyses, is one of them. Local measurements use nearby light paths, with a small effective δ. CMB measurements combine a long light path with a lens weight that peaks at middle depths. If in both cases you assume the same geometric δ, a discrepancy appears. If Layer 4 is right that δ(z) develops with depth, then it is not strange that a difference in derived H0 appears. Then it is partly a comparison between two different depth regimes through one lens, rather than a purely internal inconsistency of the universe.

In GLV, Layer 4 ultimately receives a formal translation as a z dependent correction factor in the optical metric. Instead of one δ, in the more technical pieces you write δ(z), and you take into account that δ makes its main contribution at a few characteristic depths, exactly where the web weighs most heavily in the lensing kernels. For this book the reader does not need to know that formula. It is enough to hold on to the intuition that path shortening and optical amplification are not built up in one stroke, but layer by layer, while the photon travels through time.

Layer 4 therefore makes the GLV picture richer again. Layer 1 said, light does not take a highway, the map becomes radially too large. Layer 2 said, that map is also slightly skewed, some directions are optically heavier than others. Layer 3 said, certain scales, especially small structures, receive extra optical emphasis. Layer 4 adds, and all those effects are built up through time, depending on how deep into the light cone you look and how far the cosmic web had grown at that moment.

In the next layer, another nuance is added, the cosmos is not only a quiet, gradual lens stack, but also has outbursts, bursts, temporary extra lens contributions that leave their own noise and patterns in the light cone. With that we shift from the average, calm evolution to the more erratic sides of the universe. But there too, the lesson of Layer 4 remains standing, every conclusion about “how the universe changes” must first pass through the sieve of time and depth in the light cone.