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Mujic{\Lambda}: Reconstructing Initial Conditions from Incomplete Redshift Surveys with Projected Optimization

This paper introduces Mujic{\Lambda}, an optimization-based framework that reconstructs initial conditions from incomplete redshift surveys by augmenting the L-BFGS algorithm with projection operators and rank-order matching to enforce Gaussianity, thereby enabling robust recovery of the cosmic web and density fields for next-generation constrained simulations.

Original authors: Chenze Dong, Benjamin Horowitz, Adrian E. Bayer, Khee-Gan Lee

Published 2026-04-30

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CC BY 4.0

Original authors: Chenze Dong, Benjamin Horowitz, Adrian E. Bayer, Khee-Gan Lee

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: Reversing the Universe's "Shuffle"

Imagine the universe as a giant, complex jigsaw puzzle. When the universe began (the Big Bang), the pieces were scattered almost randomly, but with a very specific, smooth pattern (like a calm, flat ocean with tiny ripples). Over billions of years, gravity acted like a strong wind, blowing the pieces together to form clusters, long chains, and empty spaces. This is what we see today: the "Cosmic Web."

The problem for astronomers is that we only see the final picture (the galaxies as they are now). We want to know what the puzzle looked like at the very beginning. This is called reconstructing the initial conditions.

However, there are two big hurdles:

We don't see the whole puzzle: Our telescopes can only look at a specific slice of the sky, and some parts are missing or blurry (incomplete data).
The math is tricky: If you try to reverse-engineer the puzzle using standard math, the missing pieces often cause the computer to "hallucinate" or invent fake patterns that look real but aren't.

The Solution: MujicΛ

The authors introduce a new tool called MujicΛ (pronounced "Moo-jik-Lambda"). Think of it as a super-smart detective that tries to solve the puzzle backwards, but with a special set of rules to keep it honest.

Here is how it works, step-by-step:

1. The "Forward Model" (The Simulation Engine)

Imagine you have a video game that can simulate how gravity pulls dust together to form stars and galaxies.

MujicΛ starts by guessing what the universe looked like at the beginning.
It runs its "video game" forward in time to see what the universe would look like today based on that guess.
It then compares its simulated universe to the actual telescope data we have.

2. The Problem with Standard Guessing

If you just let a computer guess and adjust to minimize the error, it often gets "desperate." Because our telescope data has holes (missing galaxies), the computer might try to cheat. It might invent weird, jagged patterns in the empty areas just to make the math work, even though those patterns violate the laws of physics (specifically, the rule that the early universe was smooth and random, or "Gaussian").

3. The "Projection" (The Reality Check)

This is where MujicΛ gets clever. It uses a technique called Projected Optimization.

The Analogy: Imagine you are trying to walk a straight line through a foggy forest. Every time you take a step, you might drift off course. A normal walker might keep drifting until they are lost.
MujicΛ's approach: After every step, it has a "guardian" (the projection operator) that gently nudges you back onto the straight line. It forces the solution to stay within the "rules" of the early universe (keeping it smooth and random).
It also uses Rank-Order Matching. Imagine you have a deck of cards. If the computer's guess has the right types of cards but in the wrong order, this step shuffles the deck so the distribution of values matches the perfect, random pattern expected from the Big Bang, without changing the overall structure too much.

What Did They Test?

The authors didn't test this on real telescopes yet. Instead, they built a fake universe (a "mock catalog") using a supercomputer simulation called the Millennium Simulation.

They created a fake galaxy survey with gaps and missing data, just like a real telescope would see.
They fed this fake data into MujicΛ.
The Result: MujicΛ successfully reconstructed the "beginning" of this fake universe.
- It got the big structures (massive clusters and long filaments) right.
- It kept the "beginning" smooth and random, exactly as physics predicts.
- It correctly identified the "Cosmic Web" (voids, sheets, filaments, and knots).

Why Does This Matter?

The paper claims that MujicΛ is a robust way to reverse-engineer the universe's history, even when our data is messy or incomplete.

It's a "Safety Net": Unlike other methods that might get confused by missing data and invent fake physics, MujicΛ forces the answer to stay physically realistic.
It's a "Head Start": The authors suggest this tool can be used to create a very good "first guess" for more complex, slower computer programs. This would speed up future studies of how galaxies form and evolve.
Future Use: The paper specifically mentions that this tool is well-suited for upcoming massive sky surveys (like the Subaru Prime Focus Spectrograph) that will map millions of galaxies.

Summary

MujicΛ is a new mathematical tool that helps astronomers look at a messy, incomplete map of galaxies today and accurately figure out what the smooth, random universe looked like billions of years ago. It does this by constantly checking its own work against the rules of physics, ensuring it doesn't invent fake history just to fill in the blanks.

1. Problem Statement

The reconstruction of initial conditions (ICs) and the underlying density field from realistic galaxy redshift surveys is a critical challenge in cosmology. While Bayesian sampling methods (e.g., BORG, ELUCID) offer robust uncertainty quantification, they are computationally expensive. Conversely, optimization-based methods (e.g., MAP estimation) are faster but suffer from significant instability when applied to realistic data due to:

Survey Incompleteness: Real surveys have complex selection functions and incomplete sky coverage, rendering the inverse problem highly under-constrained.
Model Misspecification: Simplifying assumptions in forward models (e.g., galaxy bias) can lead optimizers to "reward hacking," where the solution minimizes the loss function by deviating from physical reality (e.g., producing non-Gaussian fields) rather than recovering the true linear field.
Gaussianity Violation: Standard optimizers often fail to maintain the Gaussian prior of the initial conditions, leading to unphysical reconstructions that diverge from $\Lambda$ CDM cosmology.

2. Methodology: MujicΛ Framework

The authors introduce MujicΛ (Mapping the Universe with Jax-based Initial Condition Reconstruction), an optimization-based framework designed to reconstruct ICs while strictly enforcing Gaussianity and robustness to survey geometry.

A. Forward Model

MujicΛ utilizes a differentiable forward model ( $M_\theta$ ) that maps initial conditions to observed galaxy counts:

Initial Conditions: Reparameterized as complex Gaussian variables $\tilde{\delta}_{ic} \sim \mathcal{CN}(0, 1)$ .
Structure Formation: Uses JaxPM, a differentiable particle-mesh N-body solver, to evolve the field from $z=9$ to the present day ( $z=0$ ) using 2LPT and FastPM steps.
Observational Effects: Incorporates lightcone effects, Redshift-Space Distortions (RSD), and a galaxy bias model (power-law bias with a soft cutoff).
Survey Response: Explicitly models survey incompleteness via a response function $R$ , which acts as a mask on the predicted galaxy density.
Likelihood: Assumes a Poisson distribution for galaxy counts in mesh cells, optimizing the log-likelihood $\mathcal{L}$ .

B. Augmented Optimization Strategy

Unlike standard Maximum A Posteriori (MAP) approaches, MujicΛ optimizes the likelihood subject to constraints that enforce the Gaussian prior. It employs two key mechanisms:

Projected L-BFGS (PGD):
- Uses the Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm for efficient large-scale optimization.
- Projection Operator: After every gradient step, the solution is projected back onto the feasible set of Gaussian initial conditions. This involves:
  - Z-score Normalization: Enforcing zero mean and unit standard deviation.
  - Clipping/Resetting: Resetting extreme values (outliers) that exceed a threshold ( $s_{th}=10$ ) to random Gaussian draws, preventing non-physical divergences.
Rank-Order Matching:
- Applied periodically during optimization in both real space and Fourier space (within specific $k$ -bins).
- Mechanism: The algorithm sorts the current field values and replaces them with values drawn from the target Gaussian distribution, preserving the original rank order.
- Purpose: This acts as a "generalized projection" that strictly enforces the full probability density function (PDF) of the prior, correcting deviations that the projection operator might miss. It also helps the optimizer escape local minima.

C. Training Strategy

To accelerate convergence, the authors use an annealing strategy:

First, a low-resolution reconstruction ( $4 \text{ cMpc}/h$ ) is performed.
The resulting field serves as the initial guess for the full-resolution run ( $1.95 \text{ cMpc}/h$ ).
Rank-order matching is applied every 5 iterations (low-res) and every 20 iterations (high-res).

3. Key Contributions

Novel Optimization Framework: MujicΛ is one of the few optimization-based codes that explicitly maintains Gaussianity of initial conditions while modeling realistic survey effects (incompleteness, selection functions) up to megaparsec scales.
Robustness to Incompleteness: By combining projection operators with rank-order matching, the method prevents the optimizer from diverging in regions with poor data coverage (outside the survey mask).
Differentiable Pipeline: The integration of JaxPM allows for end-to-end differentiability, enabling efficient gradient-based optimization with complex forward models.
Cosmic Web Recovery: The framework successfully recovers the topology of the cosmic web (voids, sheets, filaments, knots) despite the under-constrained nature of the problem.

4. Results

The method was validated using a mock lightcone catalog derived from the Millennium simulation (semi-analytic model H15) with a survey geometry mimicking a wide-field spectroscopic survey ( $0.07 < z < 0.22$ , $20^\circ$ angular cut).

Gaussianity Preservation: Histograms of the reconstructed initial conditions in both real and Fourier space align perfectly with a Gaussian distribution, unlike vanilla L-BFGS which produces non-Gaussian outliers.
Density Field Reconstruction:
- Visual: The reconstructed density field accurately recovers massive protoclusters and filamentary structures within the survey window.
- Quantitative:
  - Correlation coefficient ( $r_c$ ) between truth and reconstruction is 0.54 at $2 \text{ cMpc}/h$ smoothing and 0.78 at $8 \text{ cMpc}/h$ .
  - In the survey window, $r_c$ remains high ( $>0.8$ ) up to $k \approx 0.1 \text{ h/cMpc}$ .
  - Outside the survey window, the field remains uncorrelated with the truth (as expected), demonstrating that the method does not hallucinate structures in unobserved regions.
Power Spectra: The reconstructed matter power spectrum ( $P_{mm}$ ) matches the truth within 0.3 dex down to $k = 1 \text{ h/cMpc}$ .
Cosmic Web Classification:
- Void/Sheet Recovery: $\sim60\%$ of voids and sheets are correctly classified.
- Filament/Knot Recovery: Lower accuracy; filaments are often misclassified as sheets due to imperfect recovery of high-density peaks.
- Eigenvector Recovery: The eigenvectors of the deformation tensor (crucial for intrinsic alignment studies) show high recovery rates, with $>70\%$ of voxels having $|\cos \theta| > 0.6$ for the largest eigenvalue.

5. Significance and Future Applications

Next-Generation Surveys: MujicΛ is specifically tailored for upcoming large-scale surveys like Subaru PFS, DESI, Euclid, and Roman, which feature complex geometries and incomplete sky coverage.
Constrained Simulations: The reconstructed ICs can serve as high-quality initial guesses for Bayesian samplers (e.g., MaiGO), significantly accelerating convergence for field-level inference.
Galaxy Evolution Studies: By providing a 3D density field and cosmic web classification, MujicΛ enables environmental studies of galaxy evolution and the intergalactic medium (IGM).
Scalability: The framework is modular and can incorporate more sophisticated forward models (e.g., Lagrangian bias, neural network gradients) and multi-tracer data (e.g., Lyman- $\alpha$ forest, FRBs) in future iterations.

In summary, MujicΛ bridges the gap between the speed of optimization-based methods and the physical fidelity required for realistic cosmological analysis, offering a robust tool for decoding the cosmic web from incomplete observational data.