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A cosmographic analysis using DESI-DR2 and strong lensing: II. Distance Ratio measurements

This paper presents a model-independent cosmographic analysis combining DESI-DR2 baryon acoustic oscillation data, strong-lensing distance ratios, and multiple Type Ia supernova samples to constrain cosmic expansion parameters and spatial curvature, finding results consistent with a flat $\Lambda$ CDM universe.

Original authors: Darshan Kumar (HNAS), Deepak Jain (DU), Shobhit Mahajan (DU)

Published 2026-06-05

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CC BY 4.0

Original authors: Darshan Kumar (HNAS), Deepak Jain (DU), Shobhit Mahajan (DU)

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 been trying to figure out exactly how fast this balloon is inflating, whether it's perfectly round (flat), or if it's slightly squashed or stretched (curved). Usually, to do this, they have to guess the "recipe" of the universe first—assuming it's made of specific ingredients like dark energy and dark matter.

This paper is like a group of detectives who decided to solve the mystery without guessing the recipe first. They wanted to measure the shape and speed of the universe's expansion using only the raw evidence they could see, a method called "cosmography."

Here is how they did it, using some creative analogies:

1. The Three Clues (The Datasets)

The researchers combined three different types of cosmic clues to get a clearer picture:

The Cosmic Ruler (Strong Gravitational Lensing): Imagine a massive galaxy sitting between us and a distant star. Its gravity acts like a giant, cosmic magnifying glass, bending the light from the star and creating a ring or multiple images. By measuring how big that ring looks, the scientists can calculate a specific "distance ratio." Think of this as measuring the ratio of two shadows to figure out the distance between two objects without needing to know exactly how bright the light source is.
The Cosmic Yardsticks (Type Ia Supernovae): These are exploding stars that all burn with roughly the same brightness. Because we know how bright they should be, if they look dimmer, we know they are farther away. The team used three different catalogs of these "standard candles" (PantheonPlus, Union3, and DESY5) to map out the universe's history.
The Cosmic Echo (DESI-DR2): This is a massive new survey that maps millions of galaxies. It looks for "Baryon Acoustic Oscillations"—essentially, frozen sound waves from the early universe that left a specific pattern in how galaxies are spaced. Think of this as finding a specific, repeating pattern in a crowd of people that tells you exactly how far apart they are standing.

2. The Method: Measuring Without a Map

Usually, scientists try to fit all these clues into a pre-made map (a specific cosmological model). If the clues don't fit, they might blame the map.

Instead, this team used a Distance Sum Rule. Imagine you are standing at point A, looking at a tree (point B) and a mountain (point C). You know the distance from you to the tree and from the tree to the mountain. In a perfectly flat world, the distance from you to the mountain should be exactly the sum of the other two. If the world is curved (like the surface of a sphere), that simple math changes.

By using the "distance ratios" from the gravitational lenses (the magnifying glasses) combined with the supernova and DESI data, they could test if the universe is flat, curved, or open without assuming a specific model of dark energy or gravity first. They treated the expansion of the universe like a story, reading it chapter by chapter (using a mathematical expansion) to see how the speed of expansion is changing over time.

3. The Results: What They Found

The team ran their analysis in two ways: first using just the lensing and supernova data, and then adding the new, high-precision DESI data.

Without the new data (DESI): The picture was a bit fuzzy. The clues suggested the universe is likely flat, but the margins of error were wide. They could tell the universe is accelerating (speeding up its expansion), but they couldn't pin down the finer details of how it's accelerating. It was like looking at a distant landscape through a slightly foggy window; you can see the general shape, but the details are blurry.
With the new data (DESI-DR2): Adding the DESI data was like clearing the fog. Suddenly, the measurements became much sharper.
- Shape: The universe is confirmed to be flat (or very, very close to it) with high confidence.
- Speed: The universe is definitely accelerating, and the measurements of this acceleration match the standard "Lambda-CDM" model (the current leading theory of the universe) very well.
- The "Snap": They tried to measure even higher-order changes in the expansion (called "jerk" and "snap," which describe how the acceleration itself is changing). While these are still hard to measure precisely, the new data tightened the constraints, making the numbers more reliable.

4. The Bottom Line

The paper concludes that by combining the "magnifying glass" effect of gravitational lenses with the "yardsticks" of supernovae and the "echoes" of the DESI survey, they have built a robust, model-independent way to measure the universe.

They found that:

The universe is consistent with being flat.
The expansion is accelerating, just as standard theories predict.
The new DESI data is crucial; without it, the measurements were too loose to be definitive. With it, the results are tight and reliable.

Essentially, they proved that you don't need to assume the "recipe" of the universe to measure its shape and speed; you just need to look at the right combination of cosmic shadows, exploding stars, and galaxy patterns. The universe, it seems, is playing by the rules of the standard model, at least as far as this specific set of measurements can tell.

Technical Summary: Cosmographic Analysis Using DESI-DR2 and Strong Lensing: II. Distance Ratio Measurements

Problem Statement
The standard spatially flat $\Lambda$ CDM model faces unresolved tensions, including the Hubble constant discrepancy, inconsistencies in spatial curvature parameters, and theoretical issues regarding fine-tuning and coincidence. These challenges necessitate model-independent approaches that avoid assumptions tied to specific cosmological dynamics. While previous studies have utilized the Distance Sum Rule (DSR) with various probes, there is a need to rigorously test cosmic expansion and curvature using strong gravitational lensing (SGL) distance ratios combined with the latest large-scale structure and supernova data, without relying on a fixed background cosmological model or traditional distance ladder calibrations.

Methodology
This work employs a cosmographic framework, expanding observables as a Taylor series in the variable $y = z/(1+z)$ to ensure convergence at higher redshifts. The analysis focuses on the kinematics of the universe's expansion up to the fourth order, constraining the Hubble constant ( $H_0$ ), spatial curvature ( $\Omega_{k0}$ ), deceleration ( $q_0$ ), jerk ( $j_0$ ), and snap ( $s_0$ ) parameters.

The study integrates three distinct observational datasets:

Strong Gravitational Lensing (SGL): A sample of 161 galaxy-scale lenses (compiled from surveys including SLACS, BELLS, and SL2S) is analyzed using the Singular Isothermal Sphere (SIS) model. The analysis utilizes the distance ratio $d_{ls}^A / d_{os}^A$ derived from the Einstein radius and velocity dispersion measurements. The DSR is applied to these ratios to determine spatial curvature independently of $H_0$ .
Type Ia Supernovae (SNIa): Three independent compilations are analyzed: PantheonPlus, Union3, and DES Y5. Theoretical apparent magnitudes are calculated, and nuisance parameters are analytically marginalized to remove dependence on calibration.
Baryon Acoustic Oscillations (BAO): The DESI-DR2 dataset (three years of observations) is included, utilizing distance ratios normalized by the sound horizon ( $r_d$ ). A Gaussian prior on $r_d$ is applied to break degeneracies between $H_0$ and $r_d$ .

Statistical inference is performed using Bayesian Markov Chain Monte Carlo (MCMC) methods (via the emcee sampler). The total $\chi^2$ is constructed by summing the contributions from SGL, SNIa, and DESI-DR2 datasets. Flat priors are assumed for all parameters within broad ranges to avoid boundary bias.

Key Contributions

Model-Independent Curvature Constraint: The paper applies the DSR directly to SGL distance ratio measurements without assuming a fiducial cosmological model, providing an independent geometric probe of spatial curvature ( $\Omega_{k0}$ ).
Integration of DESI-DR2: This work represents the first application of the latest DESI-DR2 BAO data in conjunction with SGL distance ratios and multiple SNIa compilations within a cosmographic framework.
Fourth-Order Cosmography: The analysis extends the cosmographic expansion to the "snap" parameter ( $s_0$ ), offering constraints on higher-order derivatives of the scale factor that are typically poorly constrained.
Robustness Validation: The study explicitly tests the sensitivity of results to the SIS mass profile assumption by marginalizing over a velocity bias parameter ( $f_E$ ) and evaluates the impact of prior boundary choices on the Bayesian inference.

Results

Without DESI-DR2: When combining SGL with individual SNIa datasets (PantheonPlus, Union3, or DESY5), the constraints on cosmographic parameters remain broad. The curvature parameter $\Omega_{k0}$ is consistent with a flat universe at the 68% confidence level but shows hints of deviation. The deceleration parameter $q_0$ confirms acceleration but with varying precision depending on the SNIa sample. Higher-order parameters ( $j_0$ and $s_0$ ) are largely unconstrained.
With DESI-DR2: The inclusion of DESI-DR2 data significantly tightens parameter intervals across all combinations.
- Curvature: $\Omega_{k0}$ is constrained to be consistent with a flat universe at the 95% confidence level (e.g., $0.139^{+0.061}_{-0.058}$ for SGL+PantheonPlus+DESI).
- Expansion Parameters: The deceleration parameter $q_0$ is determined with improved precision (e.g., $-0.481^{+0.057}_{-0.054}$ ), confirming cosmic acceleration. The jerk parameter $j_0$ becomes consistent with the standard $\Lambda$ CDM prediction of unity within the 68% confidence level.
- Higher-Order Dynamics: While the snap parameter $s_0$ remains weakly constrained, the inclusion of DESI-DR2 yields modest improvements in its constraint intervals compared to analyses without this data.
Consistency: The inferred values for $q_0$ and $j_0$ are compatible with $\Lambda$ CDM predictions for all dataset combinations. The results show that the tensions observed without DESI-DR2 are primarily due to limited statistical precision rather than fundamental inconsistencies in the cosmological model.
Systematics: The analysis confirms that the results are robust against the choice of SIS mass profile (marginalizing over $f_E$ ) and prior boundaries for $H_0$ .

Significance and Claims
The authors claim that this work provides a unique and robust method to investigate cosmic curvature and expansion by combining SGL distance ratios with independent supernova and BAO measurements. By utilizing the DSR framework, the study avoids reliance on the distance ladder or specific dark energy parameterizations.

The primary significance lies in demonstrating that the integration of DESI-DR2 data with SGL distance ratios produces tighter, more consistent cosmographic constraints that align with standard cosmology. The paper establishes that while SGL distance ratios alone stabilize parameter estimation, they cannot fully overcome statistical limitations without the high-precision BAO data provided by DESI. This study serves as the final component of a two-part cosmographic series, complementing the time-delay analysis of Paper I by probing relative geometry rather than absolute distance scales, thereby offering a more complete view of late-time cosmology. The authors conclude that their methodology lays the groundwork for future precision cosmology using upcoming observational datasets.

1. The Three Clues (The Datasets)

2. The Method: Measuring Without a Map

3. The Results: What They Found

4. The Bottom Line

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