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A Gaussian Covariance Matrix for Joint Pre- and Post-Reconstruction Full-Shape Power Spectrum Analysis

This paper presents and validates a semi-analytical Gaussian covariance model for the joint analysis of pre-reconstruction, post-reconstruction, and cross full-shape galaxy power spectra, demonstrating that it yields cosmological constraints consistent with mock-based numerical covariances while effectively capturing the dominant covariance structure.

Original authors: Yuting Wang, Ruiyang Zhao, Gong-Bo Zhao, Kazuya Koyama

Published 2026-08-07

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CC BY 4.0

Original authors: Yuting Wang, Ruiyang Zhao, Gong-Bo Zhao, Kazuya Koyama

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, cosmic ocean. For decades, astronomers have been trying to map the waves in this ocean, not water, but the invisible web of galaxies that stretches across billions of light-years. By studying how these galaxies cluster together, scientists can figure out the rules of the game: how fast the universe is expanding, what it's made of, and how it began. However, the universe is messy. Galaxies don't sit perfectly still; they drift, they swirl, and they interact in complicated ways. To get a clear picture, scientists use a clever trick called "reconstruction," which is like trying to rewind a movie of the universe to see where the galaxies started before they got tangled up. This helps reveal a faint, rhythmic pattern in the galaxy distribution, a cosmic "fingerprint" left over from the Big Bang.

But here's the catch: when you try to measure these patterns, you have to deal with "noise." It's like trying to hear a whisper in a crowded stadium. Sometimes the noise comes from the fact that we can't count every single galaxy perfectly (shot noise), and sometimes it comes from the fact that the universe itself is a bit lumpy and unpredictable. To trust their measurements, scientists need a "covariance matrix." Think of this as a giant rulebook that tells them how much their measurements might wiggle or mess up due to this noise. If they get this rulebook wrong, their conclusions about the universe could be off. The problem is that making this rulebook the old-fashioned way is incredibly slow and expensive, like trying to simulate the entire universe thousands of times just to check your math.

This paper introduces a new, faster way to write that rulebook. The authors, led by Yuting Wang and colleagues, have developed a "semi-analytical" method. Instead of running thousands of heavy computer simulations to guess how the noise behaves, they use a mathematical formula based on the idea that the universe's fluctuations are mostly "Gaussian" (a fancy word for a predictable, bell-curve shape). They specifically tackle the tricky part of the reconstruction process: the "cross shot noise." This is the specific type of noise that happens when you compare the galaxy map before and after the "rewind" trick. They also invented a new, quicker tool (an estimator) to measure this noise directly without having to split their data in half, which saves time and reduces errors.

The team tested their new method against a massive set of fake universes (called "mock catalogues") that they know the answers to. They found that their new, fast mathematical rulebook works almost perfectly for the scales they care about. When they used their new method to measure the universe's properties, the results matched the results from the slow, heavy computer simulations. They discovered that for their specific measurements, the new method is reliable up to a scale of $k_{max} = 0.18 \text{ h Mpc}^{-1}$ for the pre- and post-reconstruction maps, and $k_{max} = 0.12 \text{ h Mpc}^{-1}$ for the cross-map. While the method starts to get a little less accurate on very small scales where the universe gets truly chaotic, it captures the most important parts of the noise structure. This means astronomers can now analyze their data much faster and more efficiently, opening the door to studying the universe with greater precision without waiting years for computer simulations to finish.

Technical Summary: A Gaussian Covariance Matrix for Joint Pre- and Post-Reconstruction Full-Shape Power Spectrum Analysis

Problem Statement
Galaxy redshift surveys utilize full-shape analyses of the galaxy power spectrum to extract cosmological information from baryon acoustic oscillations (BAO), redshift-space distortions (RSD), and the broadband matter power spectrum. Density-field reconstruction is a standard technique used to enhance BAO signals by partially reversing nonlinear evolution. While traditional analyses combine pre-reconstruction full-shape data with post-reconstruction BAO measurements, recent work has proposed a joint analysis of pre-reconstruction, post-reconstruction, and cross power spectra ( $P_{pre}$ , $P_{post}$ , and $P_{cross}$ ) to leverage complementary information.

A critical challenge in such joint analyses is the construction of an accurate covariance matrix that consistently accounts for correlations among all three power-spectrum measurements. While numerical covariance matrices estimated from large suites of mock catalogues are accurate, they are computationally expensive, particularly for high-dimensional data vectors. Analytical approximations offer a more efficient alternative but require rigorous validation, especially regarding the complex shot-noise contributions introduced by the reconstruction process.

Methodology
The authors develop a semi-analytical Gaussian covariance framework for the joint analysis of $P_{pre}$ , $P_{post}$ , and $P_{cross}$ . The methodology comprises three main components:

Gaussian Covariance Formalism: The authors adopt the Gaussian covariance formalism, assuming Gaussian density fluctuations. Using Wick's theorem, they derive the covariance matrix for the joint data vector. The matrix includes all auto- and cross-correlations among the multipoles ( $\ell=0, 2, 4$ ) of the three power spectra. The 3D power spectra are reconstructed from their multipoles using Legendre projections, incorporating both the signal and shot-noise terms.
Cross Shot-Noise Modelling: A key theoretical contribution is the modeling of the reconstruction-induced cross shot noise. Unlike auto-correlation shot noise, the cross shot noise between the pre- and post-reconstruction fields is scale-dependent due to the displacement field applied during reconstruction. The authors model this using displacement-field statistics, deriving an analytical expression for the cross shot noise ( $N_{cross}$ ) based on the cumulant expansion of the displacement field. This model accounts for the Gaussian smoothing kernel and the fiducial bias and growth rate parameters used in reconstruction.
New Cross Shot-Noise Estimator: To validate the analytical model and improve measurement precision, the authors propose a new estimator for the cross shot noise. Unlike the traditional half-sum and half-difference (HS-HD) method, which requires splitting the galaxy catalogue into disjoint subsets (thereby increasing statistical noise), the new estimator directly measures the cross shot noise by computing the Fourier transform of a shifted noise catalog derived from the displacement vectors ( $x_{rec} - x$ ). This approach avoids catalogue splitting and reduces numerical statistical uncertainty on small scales.

Validation and Results
The semi-analytical Gaussian covariance is validated against numerical covariance matrices derived from 986 independent GLAM mock catalogues.

Shot-Noise Validation: The new estimator for cross shot noise yields mean measurements consistent with the HS-HD method but exhibits lower variance on small scales. The analytical model for cross shot noise accurately reproduces the scale dependence measured in the mocks, confirming that the Gaussian displacement approximation effectively describes the reconstruction-induced noise suppression.
Covariance Structure: The semi-analytical Gaussian covariance accurately reproduces the diagonal structure of individual covariance blocks (auto-correlations) and the cross-correlations between different power-spectrum multipoles.
Parameter Inference: Using emulator-based parameter inference, the authors compare cosmological constraints derived using the semi-analytical Gaussian covariance against those using the numerical mock-based covariance.
- For the joint fit of $P_{pre}$ , $P_{post}$ , and $P_{cross}$ , the constraints obtained with the Gaussian covariance are consistent with the numerical results over the adopted fitting ranges: $k_{max} = 0.18 \, h \, \text{Mpc}^{-1}$ for $P_{pre}$ and $P_{post}$ , and $k_{max} = 0.12 \, h \, \text{Mpc}^{-1}$ for $P_{cross}$ .
- As the fitting range extends to smaller scales (higher $k$ ), deviations emerge, particularly for the cross power spectrum, indicating that non-Gaussian contributions (such as mode coupling) become increasingly important. However, within the conservative fitting ranges, the Gaussian approximation captures the dominant contribution to the covariance structure.

Significance and Claims
The paper claims that the proposed semi-analytical Gaussian covariance framework provides a computationally efficient alternative to mock-based covariance estimation for joint full-shape analyses. By explicitly accounting for correlations between pre- and post-reconstruction density fields and modeling the scale-dependent cross shot noise, the framework yields cosmological constraints consistent with numerical estimates over the tested scales.

The authors emphasize that while the Gaussian approximation is adequate for the dominant covariance structure within the specified $k$ -ranges, non-Gaussian effects become significant on smaller scales, particularly for the cross power spectrum. Consequently, the framework serves as a robust tool for current analyses but suggests that future improvements should incorporate non-Gaussian effects (e.g., trispectrum contributions and super-sample covariance) and survey window functions to handle irregular geometries in real survey data. The introduction of the new cross shot-noise estimator further offers a slight reduction in numerical uncertainty without the computational overhead of catalogue splitting.

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