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Cosmography for a General Spacetime Centred at Arbitrary Redshift

This paper develops a formalism for general cosmographic expansions centered at arbitrary redshifts to translate observational data into model-independent geometric and dynamical information, demonstrating through Lemaître-Tolman-Bondi examples that in non-FLRW spacetimes, the resulting coefficients act as effective parameters dependent on the observational range rather than strict local quantities.

Original authors: Jonas Broe Bendtsen, Asta Heinesen, Sofie Marie Koksbang

Published 2026-08-10

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CC BY 4.0

Original authors: Jonas Broe Bendtsen, Asta Heinesen, Sofie Marie Koksbang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible ocean. For decades, scientists have been trying to map its currents and depths, but they've mostly assumed the water is perfectly smooth and uniform, like a calm, flat lake. This assumption, called the FLRW model, has been the standard map for everything from the birth of stars to the expansion of space itself. But what if the ocean isn't a calm lake? What if it's full of hidden whirlpools, massive underwater mountains, and deep, empty trenches that warp the water around them?

To navigate this, astronomers use "cosmography," which is essentially a mathematical way of describing the universe's shape and motion using only what we can see, without forcing it into a pre-made box. Think of it like trying to describe the shape of a bumpy hill by rolling a ball down it and measuring how fast it goes. If the hill is smooth, a simple formula works perfectly. But if the hill has a sudden cliff or a deep ditch, that simple formula breaks down, and the ball's path becomes unpredictable. The big question is: Can we still use our simple formulas if we stop assuming the hill is smooth? And if we do, does the math still tell us something real about the hill, or does it just become a confusing jumble of numbers?

This paper, written by Jonas Broe Bendtsen, Asta Heinesen, and Sofie Marie Koksbang, tackles exactly that problem. They introduce a new, more flexible way to do cosmography. Instead of only looking at the universe from our current spot (redshift zero), they show how to build these mathematical maps starting from any point in the universe's history, at any redshift. They tested this new method on two specific, wild scenarios: a giant cosmic "void" (a huge empty bubble) and a massive "overdensity" (a dense clump of matter).

Here is what they found. When they tried to use their new, flexible math to describe the distance to objects in these bumpy, lumpy universes, the results were a bit of a mixed bag. If they tried to build their map in a region where the density changed very sharply—like standing right on the edge of a giant cosmic cliff—their mathematical formulas started to wobble and fail. The "Taylor expansion" (the fancy name for their step-by-step math trick) needed so many steps to get it right that it became useless for simple, low-order approximations. It's like trying to draw a perfect circle using only a few straight lines; if the curve is too sharp, you need a million lines, and the drawing becomes a mess.

However, the paper suggests that if they moved their starting point to a smoother area of the universe, the math worked much better. The coefficients (the numbers in their formula) stayed stable and actually represented the local geometry of that smooth region. But in the chaotic, bumpy zones, the numbers they got didn't seem to represent a single, local truth. Instead, they became "effective parameters"—a fancy way of saying they were just a statistical average of the chaos they were looking at, rather than a clear description of a specific point.

The authors also tried a "piecewise" approach, which is like stitching together many small, local maps instead of trying to draw one giant map of the whole universe. They found that while this helped reduce errors in the smoother regions, it didn't magically fix the problems in the steep, bumpy areas. The math still struggled to converge when the density gradients were too steep.

Ultimately, this paper doesn't claim to have solved the mystery of the bumpy universe, nor does it say the old smooth-universe model is wrong. Instead, it acts as a cautionary guide. It suggests that when we look at real observational data, we need to be very careful about where we place our mathematical "magnifying glass." If we try to interpret the numbers from a chaotic, lumpy region as if they were simple, local facts, we might be misinterpreting the universe. The paper concludes that while these new, flexible tools are powerful for probing the universe at different redshifts, their ability to give us a clear, local picture of geometry and dynamics depends entirely on how smooth the universe happens to be at that specific spot. In short: the math works great on flat ground, but on a rocky cliff, you need a lot more than a simple formula to know what's going on.

Technical Summary: Cosmography for a General Spacetime Centred at Arbitrary Redshift

Problem Statement
Traditional cosmographic mapping of the Universe relies heavily on the assumption of a Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime, often restricting analysis to the $\Lambda$ CDM model. While cosmographic expansions (Taylor series of observable relations like redshift-distance) are standard tools for inferring parameters such as $H_0$ , they have historically been centred exclusively at $z=0$ . Recent studies indicate that in spacetimes containing large-scale structures, these expansions may suffer from limited radii of convergence or slow convergence, causing low-order truncations to deviate significantly from exact redshift-distance relations. Consequently, the physical interpretation of expansion coefficients as local geometric quantities becomes ambiguous. Furthermore, existing general cosmographic frameworks for inhomogeneous spacetimes have largely remained centred at $z=0$ . This paper addresses the need for a formalism that can translate observational data into geometric and dynamical information without predetermined cosmological models, specifically by extending general cosmographic expansions to arbitrary redshifts ( $z_*$ ).

Methodology
The authors develop a formalism for general cosmographic expansions in 4-dimensional Lorentzian spacetimes centred at an arbitrary redshift $z_*$ .

Formal Derivation: The angular diameter distance, $d_A(z)$ , is expanded as a Taylor series around $z_*$ . The authors derive explicit expressions for the first three derivatives ( $d'_A, d''_A, d'''_A$ ) in terms of geometric and dynamical spacetime quantities, including the isotropic expansion of the image ( $\hat{\theta}$ ), shear ( $\hat{\sigma}$ ), Ricci curvature ( $R_{\mu\nu}$ ), and the line-of-sight expansion rate ( $H$ ).
Mapping: The framework includes a mapping between the angular diameter distance expansion and the luminosity distance ( $d_L$ ) expansion via the Etherington reciprocity relation, providing transformation rules for the Taylor coefficients.
Numerical Application: To test the formalism, the authors apply it to two Lemaître–Tolman–Bondi (LTB) models, which represent spherically symmetric inhomogeneous dust+ $\Lambda$ $Λ$ spacetimes:
- LTB1: A central void (underdensity) surrounded by a compensating overdensity.
- LTB2: A central overdensity.
  Both models are embedded in a flat $\Lambda$ CDM background.
Simulation: Exact redshift-distance relations are obtained by solving geodesic equations and transport equations for the tidal matrix. The authors generate skymaps of the expansion coefficients using 49,152 light rays distributed via HEALpix. They evaluate the accuracy of third-order expansions centred at various redshifts ( $z_*$ ) and test "piecewise" cosmographic expansions where multiple local expansions are stitched together.

Key Results

Convergence and Density Gradients: The accuracy of the third-order Taylor expansion is highly dependent on the local density gradient at the expansion centre $z_*$ . In regions with steep density gradients (e.g., the boundary of the void in LTB1), the expansion accuracy drops dramatically even for small deviations from $z_*$ . Conversely, expansions centred in regions with smoother density profiles (or the FLRW background) yield significantly higher accuracy.
Coefficients as Effective Parameters: In general spacetimes with structure, the expansion coefficients do not strictly retain their interpretation as local geometric quantities (as they do in standard FLRW cosmology). Instead, they act as "effective parameters" reflecting the finite observational range probed. The paper finds that in regions of strong inhomogeneity, the radius of convergence may be reduced, or convergence may be slow, requiring higher-order terms for stability.
Piecewise Expansions: Stitching together multiple local expansions (piecewise cosmography) improves the mean relative error, particularly in smoother models like LTB2. However, in models with sharp gradients (LTB1), increasing the number of expansion points does not consistently reduce the maximum error, as the local convergence properties remain the limiting factor.
Multipole Structure: Unlike expansions centred at $z=0$ , which admit a finite multipole representation (truncating after a finite number of multipoles), expansions centred at arbitrary $z_*$ lose this finite structure. In the general case, every multipole can contribute, and the spherical harmonic hierarchy generally does not truncate.

Significance and Claims
The paper claims that extending cosmography to arbitrary redshifts provides a flexible framework for constructing local approximations of the redshift-distance relation anywhere along the past light cone, rather than relying on extrapolations from $z=0$ .

Model Independence: The formalism allows for mapping the Universe with minimal assumptions: that spacetime is a 4-dimensional Lorentzian manifold, light follows null-geodesics, and redshift is an invertible function of the affine parameter.
Interpretation of Coefficients: The study highlights that the physical interpretation of fitted coefficients is not guaranteed solely by the quality of the fit. In inhomogeneous spacetimes, coefficients must be interpreted as averages over larger regions of spacetime or along the lightcone, rather than as strictly local quantities.
Future Utility: While the loss of the finite multipole structure makes the arbitrary- $z$ expansion less tractable than the $z=0$ case, the authors suggest it enables new applications such as localized cosmographic reconstructions, consistency checks at specific redshifts, and model-independent cross-checks against probes naturally centred at $z>0$ (e.g., BAO).
Limitations: The authors modestly note that their study cannot disentangle a low radius of convergence from slow convergence, nor can it definitively determine the radius of convergence. However, it quantifies the accuracy of third-order truncations, which is practically relevant for data analysis. They conclude that for density contrasts modest on scales probed by supernovae, the coefficients likely provide meaningful descriptions of local expansion and geometry on appropriately smoothed scales.

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