⚛️ general relativity

A theoretical prediction for the dipole in nearby distances using cosmography

This paper develops and validates a model-independent cosmographic method to predict the dipole in luminosity distances caused by nearby inhomogeneities, demonstrating through numerical relativity simulations that it significantly improves upon naive predictions for redshifts up to $z \approx 0.07$ in smooth fields and $z \approx 0.02$ in non-linear regimes.

Original authors: Hayley J. Macpherson, Asta Heinesen

Published 2026-07-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hayley J. Macpherson, Asta Heinesen

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, expanding balloon. For decades, scientists have studied this balloon by assuming it's perfectly smooth and round, like a beach ball. This smooth model, called the FLRW model, has been a fantastic tool for understanding how the universe grows. But if you look closely at a real beach ball, you might find a few bumps, dents, or even a patch of sand stuck to it. In our cosmic neighborhood, these "bumps" are massive clumps of matter—galaxies, clusters, and vast empty voids. Because the universe isn't perfectly smooth, light traveling through it doesn't follow a perfectly straight, predictable path. It gets nudged, stretched, and squeezed by these cosmic bumps.

When we look out into the universe, we see a "dipole." Think of this like standing in a crowd of people moving in a specific direction; you feel a breeze on one side of your face and still air on the other. In cosmology, this dipole is a difference in how things look depending on which way you look. Usually, we blame this on our own motion through space, like a car driving through the rain. But what if the "wind" isn't just from our motion, but from the uneven distribution of matter around us? If we ignore these cosmic bumps and assume the universe is a smooth beach ball, we might get the wrong answer about how fast the universe is expanding or how far away things are. This is the puzzle this paper tackles: how do we accurately measure the universe's expansion when we are surrounded by a messy, lumpy neighborhood?

The authors, Hayley J. Macpherson and Asta Heinesen, set out to create a new map for this messy neighborhood. Instead of pretending the universe is smooth, they developed a mathematical method to predict how the "bumps" in space create a dipole in the distances we measure to faraway stars. They call this method "cosmography," which is essentially a way of describing the shape and movement of the universe using only geometry, without needing to guess the specific laws of gravity that might be at play.

To test their new map, the team didn't just use math on paper; they built a virtual universe inside a supercomputer. They used a technique called "numerical relativity," which is like running a high-fidelity physics simulation where Einstein's equations of gravity are solved exactly, without any shortcuts. In this virtual world, they placed 20 different "observers" at random spots and sent out light rays in every direction to see how the virtual universe's lumps affected the distance measurements.

The paper finds that their new method works surprisingly well, but with some important limits. In their simulations, where the universe was relatively smooth (like a slightly bumpy beach ball), their prediction matched the "true" distance measured by the computer's light rays to within about 10% for redshifts up to $z \approx 0.07$ . Redshift is a measure of how much the universe has stretched the light from an object; a value of 0.07 corresponds to relatively nearby cosmic distances. However, when they made the virtual universe more clumpy and chaotic (with density contrasts up to 30%), their prediction remained accurate only for much closer distances, up to $z \approx 0.02$ .

Crucially, the authors show that their method is a significant improvement over older, simpler ways of guessing the dipole. Without their smoothing technique, the older methods were off by a factor of ten or more in the same simulations. By "smoothing" the data—averaging out the tiny, chaotic bumps to see the larger trends—they were able to capture the main dipole signature much more accurately.

The paper also clarifies what causes this dipole. It suggests that at these nearby distances, the dipole isn't just because we are moving; it's largely because the expansion rate and the density of matter are changing as you look in different directions. The authors note that if you look far enough away, the effect of these local bumps fades, and the dipole is once again dominated by our motion through space, just like the wind on a car.

In short, this paper doesn't claim to have solved the mystery of the universe's expansion or proven that our current models are wrong. Instead, it provides a much sharper tool for measuring the "bumpy" parts of our cosmic neighborhood. It suggests that to get precise measurements of the universe's expansion, especially in the nearby cosmos, we must account for the fact that space isn't a smooth sheet but a textured, uneven landscape. The authors conclude that while their method is a powerful step forward, it still needs refinement to handle the most extreme, non-linear clumps of matter, and future work will need to explore how to best smooth out these cosmic wrinkles to reveal the true shape of the universe.

Technical Summary: A Theoretical Prediction for the Dipole in Nearby Distances Using Cosmography

Problem Statement
Cosmography, a model-independent method for inferring the kinematics of the Universe via series expansions of observables (such as the distance-redshift relation), is typically constrained by the assumption of a Friedmann-Lemaître-Robertson-Walker (FLRW) background geometry. However, at low redshifts ( $z \lesssim 0.1$ ), the Universe has not yet converged to perfect homogeneity and isotropy. In this regime, neglecting inhomogeneities can bias parameter inferences, particularly regarding the "dipole tension," where observed dipoles in distance-redshift relations and galaxy counts often exceed those predicted by $\Lambda$ CDM based on Cosmic Microwave Background (CMB) data. Conventional cosmographic approaches either neglect these structures or apply model-dependent corrections (e.g., peculiar velocity corrections) to map data into a fictitious homogeneous space. The authors argue that as data precision approaches the sub-percent level, a model-independent, geometrically consistent treatment of anisotropies is required to accurately interpret the cosmic neighborhood.

Methodology
The paper develops a theoretical framework to predict the dipole component of luminosity distances arising from nearby inhomogeneities without assuming a specific background cosmology.

Theoretical Framework: The authors utilize a generalized cosmographic expansion of the luminosity distance $d_L(z)$ truncated at third order in redshift. Within the "quiet universe" approximation (assuming pressureless dust, vanishing magnetic Weyl tensor, and General Relativity), they derive expressions for the dipole coefficients ( $d^{(2)}_{L,\mu}$ and $d^{(3)}_{L,\mu}$ ) in terms of gradients of the expansion rate ( $\theta$ ) and matter density ( $\rho$ ).
Smoothing Procedure: To address convergence issues caused by high-density contrasts in non-linear regimes, the authors introduce a smoothing operation. Instead of using local values of $\theta$ and $\rho$ , they replace them with volume-averaged values ( $\langle\theta\rangle$ and $\langle\rho\rangle$ ) smoothed over a spatial scale corresponding to the redshift of observation using a Gaussian kernel. This yields a robust prediction for the dipole amplitude and direction applicable over a broader redshift range than local cosmography.
Numerical Validation: The prediction is tested against "exact" solutions using fully relativistic numerical relativity (NR) simulations performed with the Einstein Toolkit. The simulations employ improved initial data methods ("exact" perturbation option) to minimize constraint violations, evolving matter-dominated (Einstein-de Sitter) universes on cubic grids.
Ray Tracing: To obtain ground-truth data, the authors perform fully relativistic ray tracing through the simulated space-times. They calculate exact luminosity distances and redshifts for 20 random observers across 12,288 geodesic directions, extracting the dipole component from the resulting sky maps.

Key Contributions

Derivation of a Dipole Prediction: The paper provides a closed-form theoretical prediction (Equation 8) for the luminosity distance dipole in terms of smoothed gradients of expansion and density, free from any underlying FLRW background assumption.
Smoothing Implementation: It demonstrates that smoothing the cosmographic coefficients over a scale matching the observation redshift significantly improves the accuracy of dipole predictions in inhomogeneous universes compared to unsmoothed, local cosmography.
Validation in Relativistic Simulations: The study offers the first quantification of this prediction's accuracy using fully relativistic NR simulations that do not enforce an FLRW background, bridging the gap between theoretical anisotropic cosmography and realistic non-linear structure formation.

Results

Accuracy in Quasi-Linear Regimes: In simulations with quasi-linear density contrasts (typical fluctuations $\sigma \approx 0.01$ ), the smoothed prediction accurately captures the dipole signature to within $\sim 10\%$ for redshifts $z \lesssim 0.07$ . The direction of the dipole is predicted with even higher accuracy ( $< 1\%$ mismatch for $z < 0.07$ ).
Accuracy in Non-Linear Regimes: In simulations with larger density contrasts (typical fluctuations $\sigma \approx 0.05$ , max $|\delta| \sim 0.3$ ), the accuracy range reduces to $z \lesssim 0.02$ for a $10\%$ error margin.
Improvement over Local Cosmography: The inclusion of the smoothing procedure yields an improvement of up to an order of magnitude in prediction accuracy compared to naive, local third-order cosmography (which does not smooth the coefficients).
Role of Higher-Order Terms: Including the third-order term in the prediction is crucial for accurately capturing the dipole amplitude across the tested redshift range, whereas the dipole direction is well-captured by the second-order term alone.
Convergence Behavior: At higher redshifts ( $z \gtrsim 0.25$ ), the ray-traced dipole converges toward a dipole induced by the observer's peculiar motion relative to the large-scale average frame (a special-relativistic boost), indicating that the smoothed cosmographic prediction (based on gradients) becomes sub-dominant to kinematic effects at these scales.

Significance
The paper claims that its primary significance lies in addressing the convergence properties of anisotropic cosmographic series expansions, which otherwise limit their applicability to very narrow redshift ranges in the presence of inhomogeneities. By providing a model-independent prediction for the dipole that accounts for gradients in expansion and density, the method offers a tool to interpret nearby distance measurements without relying on the $\Lambda$ CDM framework. This is particularly relevant for resolving tensions between local dipole measurements and CMB predictions. The authors emphasize that while their results are robust in quasi-linear regimes, the application to deeply non-linear structures requires further development in smoothing techniques and potentially the inclusion of particle dynamics to model bound structures, which are currently beyond the scope of their continuous fluid approximation.

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