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Reconstructing the largest scales of the Universe with field-level inference applied to the Quaia Quasar Catalogue

This study applies the BORG field-level inference algorithm to the all-sky Quaia quasar catalogue to generate the largest 3D reconstruction of the Universe's initial conditions and matter distribution to date, validating the results with a ~4σ cross-correlation against Planck CMB lensing.

Original authors: Adam Andrews, Arthur Loureiro, Jens Jasche, Stuart McAlpine, Guilhem Lavaux, Florent Leclercq

Published 2026-02-03

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CC BY 4.0

Original authors: Adam Andrews, Arthur Loureiro, Jens Jasche, Stuart McAlpine, Guilhem Lavaux, Florent Leclercq

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

The Big Picture: Rebuilding a 3D Puzzle from a Few Dots

Imagine you are trying to reconstruct a massive, invisible 3D sculpture of the entire universe. You can't see the sculpture itself, but you have a box filled with a few thousand glowing marbles (quasars) that are scattered inside it. These marbles are the only clues you have. They are sparse, some are blurry, and they are spread out over a huge distance.

The goal of this paper is to figure out what the invisible sculpture (the distribution of dark matter) looks like based only on where these glowing marbles are sitting.

The Tool: "BORG" (The Cosmic Time Machine)

The authors used a sophisticated computer algorithm called BORG. Think of BORG as a "reverse-engineering time machine."

The Forward Model: Usually, scientists start with a theory of how the universe began and run a simulation forward in time to see what it looks like today.
The Reverse Model: BORG does the opposite. It starts with the messy, real-world data we have today (the quasars) and works backward to figure out what the universe looked like at the very beginning (the "initial conditions"). It then runs that forward again to see if it matches what we see today.

It's like looking at a pile of rubble and a few scattered bricks, then using physics to deduce exactly what the original castle looked like, where every stone was placed, and how the wind blew it apart.

The Data: The "Quaia" Catalogue

The researchers used a new, massive list of quasars called the Quaia catalogue.

What are quasars? They are the brightest beacons in the universe, powered by supermassive black holes. Because they are so bright, we can see them from incredibly far away, acting as "streetlights" that illuminate the dark, empty spaces between them.
The Challenge: There aren't many of them (low density), and they are hard to measure precisely (some are blurry). It's like trying to map a city at night using only a few flickering streetlamps that are sometimes obscured by fog.

The team analyzed two versions of this data:

Quaia Clean: A stricter list of the brightest, most reliable quasars.
Quaia Deep: A list that includes fainter quasars, giving them more data points but with a bit more "noise" or uncertainty.

The Process: Filling in the Blanks

The team didn't just draw dots where the quasars were. They used the laws of physics (gravity) to fill in the gaps.

The Analogy: Imagine you see a few ripples in a pond. Even if you can't see the whole wave, you know how water moves. If you see a few ripples, you can mathematically predict where the rest of the wave is, even in the dark parts of the pond.
The Result: They created a 3D map of the dark matter (the invisible stuff that holds galaxies together) and the velocity fields (how fast and in what direction the universe is expanding and flowing).

They reconstructed a volume of space so huge it would take light 10 billion years to cross it. This is the largest 3D reconstruction of the universe ever made using this specific method.

The Proof: Did They Get It Right?

How do you know your reconstruction is real and not just a computer hallucination? The authors performed a "sanity check" using the Cosmic Microwave Background (CMB).

The Analogy: Imagine you reconstructed a 3D map of a forest based on a few trees. To prove you did it right, you look at the shadows cast by the forest on the ground from a distant light source (the Sun). If your 3D map matches the shadows, your map is likely correct.
The Test: The CMB is the "afterglow" of the Big Bang. As light from the Big Bang travels to us, the gravity of the matter in the universe bends it (lensing). The team compared their reconstructed 3D map of matter against the actual bending of light measured by the Planck satellite.
The Result: The two matched with high confidence (about a 4-sigma significance). This means there is a very high probability that their reconstructed map of the universe is real and not a random guess.

Key Takeaways

First of its Kind: This is the first time this specific "field-level inference" method has been applied to quasars.
Massive Scale: They mapped a volume of the universe that is 10 billion light-years across, creating the largest 3D map of its kind to date.
Robustness: Even though the data was sparse and noisy, the physics-based method successfully recovered the large-scale structure of the universe.
Validation: The map was cross-checked against independent data (CMB lensing) and found to be accurate, proving that quasars are powerful tools for mapping the invisible universe.

In short, the authors took a sparse, noisy list of distant cosmic beacons and used the laws of physics to build a detailed, 3D hologram of the invisible web of matter that makes up our universe, and they proved it works by comparing it to the "shadows" cast on the oldest light in existence.

Technical Summary: Reconstructing the Largest Scales of the Universe with Field-Level Inference Applied to the Quaia Quasar Catalogue

1. Problem Statement

Mapping the large-scale structure (LSS) of the Universe is essential for extracting fundamental physics, yet traditional analyses relying on summary statistics (e.g., power spectra, bispectra) suffer from suboptimal data compression, failing to fully exploit high-dimensional information. While quasars are unique tracers capable of probing the Universe up to redshifts $z \sim 7$ and covering ultra-large scales, their analysis is complicated by low number densities (high shot noise), foreground contamination, and redshift uncertainties (particularly for photometric samples).

The recently released Quaia quasar catalogue, derived from Gaia and unWISE data, offers an all-sky coverage ( $f_{sky} \approx 0.71$ ) and a broad redshift range ( $0.084 < z \lesssim 4.0$ ). However, leveraging this dataset for a full three-dimensional reconstruction of the matter distribution requires a method that can simultaneously handle survey selection effects, redshift-space distortions, quasar bias, and the inherent sparsity of the tracer population. Existing methods often struggle to integrate these systematics into a coherent physical model without losing information.

2. Methodology

This work applies the BORG (Bayesian Origin Reconstruction from Galaxies) algorithm, a field-level inference (FLI) framework, to the Quaia quasar catalogues. Unlike traditional approaches that compress data into summary statistics, BORG performs a Bayesian forward-modeling of the entire observed galaxy field.

2.1 Data Samples

The analysis utilizes two distinct samples from the Quaia catalogue (Storey-Fisher et al. 2024):

Quaia Clean: A magnitude-limited sample ( $G < 20.0$ ) with $\sim 755,850$ sources.
Quaia Deep Cut: An extended sample ( $G < 20.5$ ) with $\sim 1,295,502$ sources. This sample undergoes stricter angular selection, excluding regions near the Large and Small Magellanic Clouds (LMC/SMC) and imposing a completeness threshold $> 0.5$ to mitigate foreground contamination.

Both samples are partitioned into eight radial bins to probe different cosmic epochs while maintaining statistical power. Redshifts are transformed into the CMB rest frame to remove the dipole induced by the Sun's peculiar motion.

2.2 Forward Model

The BORG algorithm employs a physics-based forward model to evolve initial conditions into present-day observables:

Initial Conditions: A Gaussian white-noise field is modulated by a primordial power spectrum (fixed to Planck 2018 cosmology) to generate the initial gravitational potential.
Gravitational Evolution: The density field is evolved using Lagrangian Perturbation Theory (LPT) (first-order) to model structure formation from the initial conditions to the present day. This approach is computationally efficient for the large scales and coarse resolution ( $39.1 \, h^{-1}\text{Mpc}$ ) considered.
Observational Effects: The model incorporates redshift-space distortions, light-cone effects, and the survey window function (including radial selection functions and angular completeness masks).
Bias Modeling: A Poisson power-law bias model is used to relate the underlying dark matter density to the observed quasar distribution.

2.3 Inference Engine

The algorithm explores the high-dimensional posterior distribution of the initial conditions and late-time fields using Hamiltonian Monte Carlo (HMC) combined with slice samplers. This generates an ensemble of physically plausible realizations of:

The primordial initial conditions.
The present-day dark matter density field.
The large-scale velocity fields (radial and absolute).
The predicted quasar distribution.

2.4 Validation Strategy

To validate the reconstructions, the authors perform:

Internal Consistency Checks: Comparison of recovered power spectra and bispectra against theoretical predictions (CLASS) and checks for non-Gaussianities in the initial conditions.
External Cross-Correlation: A critical validation step involves cross-correlating the BORG-inferred matter density field with the Planck 2018 CMB lensing convergence map ( $\kappa_{Planck}$ ). This tests whether the reconstructed large-scale structures align with the integrated matter distribution observed via gravitational lensing of the CMB.

3. Key Contributions and Results

3.1 Unprecedented Reconstruction Volume

The study produces the largest field-level reconstruction of the observable Universe to date in terms of comoving volume.

Volume: $(10 \, h^{-1}\text{Gpc})^3$ .
Resolution: Maximum spatial resolution of $39.1 \, h^{-1}\text{Mpc}$ (voxel size).
Scope: The reconstruction covers the full Quaia survey volume, providing posterior maps of initial conditions, present-day dark matter, and velocity fields.

3.2 Inferred Fields

Density Field: The reconstructed dark matter density field shows strong qualitative consistency with the observed quasar distribution. It captures large-scale coherent structures (filaments and voids) and extends beyond survey boundaries through gravitational coupling, with uncertainties naturally increasing in regions of low tracer density.
Initial Conditions: The inferred primordial fluctuations are consistent with Gaussian statistics (mean $\approx 0$ , skewness and kurtosis consistent with zero), confirming the validity of the forward model and the absence of artificial non-Gaussianities introduced by the inference process.
Velocity Fields: The reconstructed velocity fields reveal coherent large-scale bulk flows, with inflows and outflows tracing the underlying density distribution. The signal-to-noise ratio follows the survey window function, highlighting the algorithm's ability to account for observational limitations.

3.3 Statistical Validation

Power Spectrum & Bispectrum: The inferred matter power spectrum agrees with theoretical predictions from CLASS across a wide range of scales. The reduced bispectrum is consistent with zero, indicating no spurious non-linear contributions.
Sub-volume Consistency: Power spectra computed in sub-regions of the reconstruction volume show statistical consistency, ruling out significant spatial model mismatches or localized contamination.

3.4 CMB Lensing Cross-Correlation

The most significant external validation is the detection of a cross-correlation signal between the BORG-reconstructed density field and the Planck CMB lensing map.

Significance: For the Quaia Deep Cut sample at $N=256$ resolution, the detection significance reaches $\sim 4\sigma$ (specifically $3.96\sigma$ via amplitude fitting and $3.0\sigma$ via phase alignment).
Amplitude: The best-fit amplitude parameter $A$ is $0.70 \pm 0.17$ , consistent with the theoretical expectation ( $A=1$ ) within uncertainties.
Implication: This detection confirms that the BORG framework successfully recovers the true large-scale matter distribution traced by quasars, validating the reconstruction against an independent cosmological probe.

4. Significance and Claims

The paper claims to establish a robust framework for exploiting quasar surveys in field-level cosmology. Its primary significance lies in:

First Application to Quasars: This is the first application of field-level inference to a quasar dataset, demonstrating that sparse, noisy tracers can be used to reconstruct the 3D matter distribution and initial conditions.
Handling Systematics: The work demonstrates that a fully forward-modeled, hierarchical Bayesian approach can effectively account for complex observational systematics (selection effects, redshift errors, bias) without losing information to data compression.
Validation of Large-Scale Physics: The $\sim 4\sigma$ detection of the CMB lensing signal provides independent confirmation that the reconstructed fields accurately reflect the large-scale structure of the Universe.
Foundation for Future Cosmology: The authors posit that this framework enables precision studies of ultra-large-scale physics, including constraints on primordial non-Gaussianity ( $f_{NL}$ ), tests of general relativity, and investigations of the Integrated Sachs-Wolfe (ISW) effect. The reconstructed fields serve as a "digital twin" of the local Universe, providing data products (posterior maps) for constrained simulations and further cosmological analysis.

The authors note that the reconstruction resolution is currently limited by shot noise and spectrophotometric redshift uncertainties, which prevent reliable inference on smaller scales. However, the results confirm that field-level inference is a viable and powerful tool for transforming sparse quasar observations into statistically robust, three-dimensional maps of the Universe's large-scale structure.