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The Back-in-time Void Finder: dynamical identification of cosmic voids through optimal transport reconstruction

This paper introduces the Back-in-Time Void Finder (BitVF), a novel algorithm that utilizes optimal transport reconstruction to identify cosmic voids via Lagrangian displacement fields, demonstrating superior robustness against shot noise and redshift-space distortions compared to traditional topological methods.

Original authors: Simone Sartori, Sofia Contarini, Elena Sarpa, Giulia Degni, Federico Marulli, Stephanie Escoffier, Lauro Moscardini

Published 2026-01-23

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CC BY 4.0

Original authors: Simone Sartori, Sofia Contarini, Elena Sarpa, Giulia Degni, Federico Marulli, Stephanie Escoffier, Lauro Moscardini

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 not as a static picture, but as a giant, slow-motion movie of matter moving around. For decades, astronomers have tried to map the "cosmic web"—the vast network of galaxy clusters, filaments, and the huge, empty spaces in between called cosmic voids.

This paper introduces a new tool called BitVF (Back-in-Time Void Finder) to find these empty spaces more accurately than ever before. Here is how it works, explained simply:

The Problem: Trying to Map a Crowd in a Fog

Currently, finding cosmic voids is like trying to map the empty spaces in a crowded room by only looking at a few people scattered around.

The "Shot Noise" Issue: Because galaxies are sparse (like a few people in a huge stadium), traditional methods often get confused. They mistake random gaps between people for giant empty rooms, or they miss real empty rooms because the "people" (galaxies) are too far apart to define the edges clearly.
The "Redshift" Distortion: When we look at galaxies, we see them based on how fast they are moving away from us (redshift), not just where they actually are. This is like looking at a car speeding toward you; it looks squashed. This makes the shapes of cosmic voids look weird and distorted, like a funhouse mirror.

The Solution: Rewinding the Movie

The authors propose a clever trick: rewind the movie.

Instead of just looking at where galaxies are now, BitVF asks: "If we ran the movie backward, where did these galaxies come from?"

The Analogy: Imagine a crowd of people at a party who have all drifted apart over time. If you could magically rewind time, you would see everyone flowing out of the empty rooms and into the crowded rooms.
The Physics: In the universe, matter flows away from empty spaces (voids) and piles up in dense areas. By mathematically "rewinding" the positions of galaxies back to a time when the universe was smooth and uniform, BitVF can see the flow lines.
The Discovery: If you trace these flow lines backward, the empty spaces (voids) are the places where the flow lines are all pointing away from a center. BitVF identifies these "outflow centers" as the true voids.

How It Works (The "Optimal Transport" Magic)

To do this rewinding, the paper uses a mathematical concept called Optimal Transport.

The Metaphor: Imagine you have a pile of sand (the current galaxies) and you want to move it to form a perfectly flat, smooth sheet (the early universe). The "Optimal Transport" algorithm is the most efficient way to move every grain of sand to its starting spot with the least amount of energy.
Why it helps: Unlike old methods that just count how many galaxies are in a specific box (which is noisy and shaky), this method looks at the entire flow of the universe. It's like looking at the wind pattern across a whole country rather than just checking if a single tree is blowing. This makes the map much smoother and less sensitive to missing data.

What They Found

The team tested this new tool against the best existing methods using supercomputer simulations:

Smoother Maps: The voids found by BitVF have cleaner, more natural shapes. They aren't jagged or broken up by random noise.
Resilient to Missing Data: Even if you remove 90% of the galaxies from the simulation (making the map very sparse), BitVF still finds the big voids correctly. Old methods fall apart when data is sparse.
Fixing the Distortion: BitVF is better at handling the "funhouse mirror" effect of redshift. It can reconstruct the voids so they look like they do in reality, not just how they look through the distortion of motion.

The Bottom Line

The authors have built a new "void finder" that works by understanding the dynamics (the movement) of the universe rather than just its static shape. It's like switching from taking a blurry, shaky photo of a crowd to analyzing the flow of the crowd to find the empty spaces.

This tool is ready to be used with upcoming massive galaxy surveys (like DESI) to help us understand the universe's structure, and the code is being made public for other scientists to use.

Technical Summary: The Back-in-time Void Finder (BitVF)

Problem Statement
Cosmic voids are powerful cosmological probes, yet their identification is hindered by significant observational and methodological challenges. Traditional topological and density-based void finders are highly sensitive to shot noise, particularly in underdense regions where tracer sampling is sparse and galaxy bias is elevated. Furthermore, identification in redshift space is compromised by Redshift-Space Distortions (RSD) and the Alcock–Paczynski (AP) effect, which distort void shapes, shift centers along the line-of-sight (LOS), and break the one-to-one mapping between real and redshift-space voids. These limitations necessitate a void-finding approach that is dynamically motivated, robust to sampling fluctuations, and capable of handling complex survey geometries and RSD.

Methodology
The authors introduce the Back-in-time Void Finder (BitVF), a dynamical algorithm that identifies voids as regions of negative divergence in a Lagrangian displacement field reconstructed from the present-day tracer distribution.

Optimal Transport (OT) Reconstruction:
- The method employs a discrete optimal transport algorithm to map observed tracers (Eulerian positions) back to an initially homogeneous Lagrangian distribution.
- Unlike direct density estimates, this approach minimizes a global transport cost (Monge cost function) to find the most efficient mapping between the current inhomogeneous distribution and a uniform primordial state.
- The implementation utilizes an efficient, iterative quartet-swap scheme (an improvement over the pairwise-swapping PIZA method) to minimize the sum of squared distances between matched tracer-random pairs.
- The reconstruction naturally incorporates tracer bias without requiring prior cosmological assumptions; the bias is accounted for via a linear scaling of the displacement field.
Divergence Computation and Smoothing:
- The displacement field $\Psi$ is reconstructed backward in time. Voids are defined as regions where $\nabla_q \cdot \Psi < 0$ , corresponding to local mass outflow when evolved forward.
- The divergence field is computed on a discrete grid (Cartesian for periodic boxes, curvilinear adaptive for light-cones) using Gauss's theorem.
- To mitigate shot noise, the method averages over multiple random realizations (typically 10–50) and applies optional Gaussian smoothing to the divergence field.
Watershed Segmentation:
- Voids are identified via a watershed algorithm applied to the smoothed divergence field. Cells are iteratively assigned to the local minimum of the divergence field, defining connected basins of coherent outflow.
- Void centers are refined using a weighted correction based on neighboring divergence values.
Redshift-Space Correction:
- For redshift-space data, the method can model RSD using a bias-corrected Kaiser framework. By assuming a fiducial cosmology and tracer bias, the reconstructed displacement field allows for the recovery of real-space positions, effectively mitigating RSD-induced systematic shifts.

Key Contributions

Novel Algorithm: BitVF is the first framework specifically designed for Stage IV galaxy surveys (e.g., DESI, Euclid) that defines voids dynamically via Lagrangian displacement rather than topological or density thresholds.
Robustness to Noise: By relying on the non-local, coherent large-scale mass flow encoded in the displacement field, BitVF is intrinsically more robust to shot noise and tracer sparseness than density-based methods.
RSD Mitigation: The method demonstrates an intrinsic ability to preserve real-space void size functions in redshift space and offers a pathway to reconstruct real-space catalogs using a fiducial cosmology.
Implementation: The algorithm is implemented in the CosmoBolognaLib package, featuring parallelized execution (OpenMP) with a computational scaling of approximately $O(N^{1.1})$ , making it tractable for millions of tracers.

Results
The authors validated BitVF using high-resolution N-body simulations (Aletheia suite) and realistic light-cone mocks (Buzzard/DESI-like).

Comparison with REVOLVER: When compared to the topological void finder REVOLVER (based on Voronoi tessellation), BitVF produces void catalogs with:
- Smoother Density Profiles: BitVF voids exhibit smoother transitions between interiors and boundaries, reflecting coherent mass flows rather than the "top-hat" structures often seen in topological methods.
- Stable Abundances: BitVF void counts and size distributions remain stable under significant tracer subsampling (down to 2.5% of the particle population) and shot noise, whereas REVOLVER shows significant instability and spurious structure generation in sparse samples.
- Size Differences: BitVF identifies voids with systematically smaller effective radii than REVOLVER because it stops expansion at the boundary where divergence changes sign, excluding compensating overdense shells.
Velocity Reconstruction: The reconstructed velocities derived from the OT displacement field show a Pearson correlation coefficient of $R \approx 0.77$ with true halo velocities, validating the dynamical reconstruction.
Light-Cone Performance: Applied to DESI-like mock catalogs, BitVF successfully mitigates RSD systematic effects, preserving real-space void statistics more faithfully than topological methods.

Significance and Claims
The paper claims that BitVF provides a physically motivated, robust alternative to existing void finders, particularly for the era of Stage IV surveys where tracer densities vary and RSD effects are prominent. By grounding void identification in the dynamics of mass outflow rather than static density thresholds, the method offers:

Reduced Sensitivity to Systematics: It inherently handles shot noise and tracer sparseness better than topological approaches.
Improved Interpretability: The dynamical definition aligns with the physical evolution of cosmic structures, potentially reducing the identification of spurious voids.
Versatility: The framework is applicable to both idealized simulation boxes and complex, masked light-cone survey data.

The authors conclude that BitVF represents a significant step forward in void cosmology, offering a tool that balances accuracy, computational efficiency, and applicability to the large datasets expected from future surveys. The code and tools are made publicly available to facilitate further research.