Constraining Power of Wavelet vs. Power Spectrum Statistics for CMB Lensing and Weak Lensing with Learned Binning

This paper introduces a novel learned binning method to compare wavelet-based statistics (WST and WPH) against traditional angular power spectra ($C_\ell$) for CMB and weak lensing, finding that while wavelet methods perform similarly to power spectra for CMB auto-correlations, they significantly outperform them in cross-correlations with galaxy weak lensing, particularly when using the new binning approach.

The Gist

Imagine the universe as a giant, invisible ocean of matter. We can't see this matter directly, but we can see how it bends the light from distant galaxies and the ancient glow of the Big Bang (the Cosmic Microwave Background, or CMB). This bending is called gravitational lensing.

Think of the universe's matter distribution like a bumpy, wrinkled sheet. When light travels over this sheet, it gets distorted. By measuring these distortions, astronomers try to figure out the recipe of the universe: how much "stuff" (matter) is in it, and how clumpy that stuff is.

This paper is about finding the best way to read the wrinkles on that sheet.

The Old Way: Counting Waves (The Power Spectrum)

For decades, astronomers have used a method called the Power Spectrum. Imagine you have a piece of fabric with a pattern on it. The Power Spectrum is like taking a ruler and measuring how big the waves are in the pattern. It tells you, "Okay, there are a lot of small ripples and a few big waves."

This method works great if the fabric is perfectly smooth and random (like static on an old TV). But the universe isn't perfectly random. As gravity pulls matter together over billions of years, it creates complex, lumpy structures (like galaxies and clusters) that look more like a crumpled piece of paper than a smooth wave. The old "ruler" method misses a lot of the interesting details in these crumples.

The New Way: Wavelets (The "Zoom Lens" and "Phase Harmony")

The authors of this paper tried a more sophisticated tool called Wavelets.

• Wavelet Scattering Transform (WST): Imagine instead of just measuring wave sizes, you have a magical zoom lens. You can zoom in on a tiny part of the fabric to see the texture, then zoom out to see how that texture fits into the bigger picture. This method captures the "shape" of the wrinkles, not just their size.
• Wavelet Phase Harmonics (WPH): This is like listening to a duet. Instead of just listening to one instrument (one map), you listen to two instruments playing together (the CMB map and the galaxy map). It measures how the timing and phase of the wrinkles in one map match up with the other.

The Problem: Too Much Data

Here's the catch: These new methods generate massive amounts of data. It's like trying to describe a 4K movie by listing the color of every single pixel. It's too much information to handle, and if you try to analyze it all at once, you might get confused by the noise (like trying to hear a whisper in a hurricane).

Usually, scientists have to throw away most of this data to make it manageable, which risks throwing away the good stuff too.

The Solution: "Learned Binning" (The Smart Sorter)

The authors invented a clever new trick called Learned Binning.
Imagine you have a huge pile of mixed-up Lego bricks (the data).

• The Old Way: You just grab a handful and hope you get the right pieces. Or, you sort them strictly by color (linear binning), which might miss the fact that a red 2x4 brick is more important than a red 1x1 brick.
• The New Way (Learned Binning): You have a smart robot that learns which bricks matter most for building a specific castle (the cosmological parameters). It doesn't just sort by color; it learns to group the bricks in the most efficient way to build the strongest castle. It compresses the pile into a few neat boxes without losing the essential pieces needed to solve the puzzle.

What They Found

The team tested these methods using computer simulations of the universe, comparing different telescopes (like Planck, ACT, SPT, and the upcoming Euclid mission).

1. For the CMB alone (looking at the "baby picture" of the universe): The fancy new wavelet methods were about the same as the old ruler method. The universe was still too smooth at that early stage for the extra complexity to help much.
2. For the "Cross-Correlation" (looking at the CMB and galaxies together): This is where the magic happened. When they looked at how the ancient light (CMB) and the modern galaxies interact, the Wavelet Phase Harmonics (WPH) method was a game-changer.
   • The Result: The new method was 2 to 3.5 times better at pinning down the universe's recipe than the old method.
   • Why? The interaction between the ancient light and modern galaxies happens in a part of the universe's history where matter has become very "clumpy" and non-random. The old ruler method missed these complex shapes, but the new "smart sorter" (WPH) caught them perfectly.

The Takeaway

This paper is like upgrading from a black-and-white TV to a 3D IMAX experience.

• The Old Method gave us a flat, blurry picture of the universe's ingredients.
• The New Method (Learned Binning + Wavelets) allows us to see the texture, depth, and complex relationships between different parts of the universe.

By using this "smart sorter" to handle the data, astronomers can now extract much more precise information about the universe's density and structure, especially when combining data from different cosmic eras. It's a major step forward in understanding the invisible architecture of our cosmos.

Technical Summary

1. Problem Statement

Cosmological parameter estimation from gravitational lensing (both Cosmic Microwave Background lensing, $\kappa_{CMB}$, and galaxy weak lensing, $\kappa_{WL}$) traditionally relies on the angular power spectrum ($C_\ell$). However, $C_\ell$ captures only Gaussian information. Due to non-linear gravitational evolution, the large-scale structure of the universe exhibits significant non-Gaussianities, particularly at low redshifts, which $C_\ell$ fails to capture.

While higher-order summary statistics (such as Wavelet Scattering Transform (WST) and Wavelet Phase Harmonics (WPH)) exist to extract this non-Gaussian information, their application faces two major hurdles:

1. Dimensionality: These statistics generate high-dimensional data vectors, making covariance matrix inversion difficult or impossible with a limited number of simulations.
2. Hyperparameter Sensitivity: Constraints derived from these statistics are highly dependent on arbitrary choices regarding scale ranges and binning schemes, leading to potential overfitting and instability.

The paper aims to compare the constraining power of standard power spectra against wavelet-based statistics for $\kappa_{CMB}$ and cross-correlated $\kappa_{CMB} \times \kappa_{WL}$, while introducing a robust method to handle the dimensionality and hyperparameter issues.

2. Methodology

A. Simulations and Data

• Simulations: The authors use the ULAGAM suite of $N$-body simulations (2000 fiducial realizations + 100 variations per parameter shift for $\Omega_m$ and $\sigma_8$).
• Observables:
  • $\kappa_{CMB}$: Full-sky convergence maps integrated to the surface of last scattering ($z \approx 1089$). Mock maps are generated for Planck, ACT, SPT, and Simons Observatory (SO), including Gaussian noise and survey masks.
  • $\kappa_{WL}$: Convergence maps integrated to galaxy survey redshifts, specifically tailored to resemble Euclid Data Release 2 (DR2) with two tomographic bins (low and high redshift).
• Cross-Correlation: The study analyzes both auto-correlations ($\kappa_{CMB}$) and cross-correlations ($\kappa_{CMB} \times \kappa_{WL}$).

B. Summary Statistics

1. Angular Power Spectrum ($C_\ell$): The standard two-point statistic.
2. Wavelet Scattering Transform (WST): Applied to $\kappa_{CMB}$. It involves convolving maps with wavelets, taking the modulus, and averaging. The authors use $S_1$ and $S_2$ coefficients with Morlet wavelets.
3. Wavelet Phase Harmonics (WPH): Applied to $\kappa_{CMB} \times \kappa_{WL}$. This generalizes WST to two maps, capturing phase correlations between them using coefficients like $S_{00}, S_{01}, C_{01}$, etc., with "bump-steerable" wavelets.

C. Novel Approach: Learned Binning

To address the high dimensionality and hyperparameter dependence, the authors propose a Learned Binning approach:

• Concept: Instead of fixed binning or complex linear compression (like MOPED), they define a binning matrix $B$ via a function $f(x)$ parameterized by a small number of "pivots."
• Optimization: The pivot positions are optimized to maximize the determinant of the Fisher matrix (information content) while compressing the data to a manageable size (15 bins).
• Robustness: To prevent overfitting, the authors use a cross-validation strategy: they split simulations into training and validation sets to learn pivots, then apply them to the full dataset. They also use a "combined Fisher matrix" (geometric mean of standard and compressed Fisher matrices) to correct for biases in derivative estimates.

D. Forecasting

The authors use the Fisher Information Matrix formalism to forecast constraints on the matter density ($\Omega_m$) and the amplitude of matter fluctuations ($\sigma_8$). They verify that the binned statistics follow a Gaussian distribution (justifying the Fisher forecast) and test the convergence of their derivative estimates.

3. Key Contributions

1. First Application of WST/WPH to Cross-Correlations: This is the first study to apply Wavelet Scattering Transform and Wavelet Phase Harmonics specifically to $\kappa_{CMB} \times \kappa_{WL}$ cross-correlations.
2. Learned Binning Framework: A novel, interpretable method for compressing high-dimensional summary statistics that maximizes information retention while remaining robust to overfitting and noise.
3. Comparative Analysis: A systematic comparison of Gaussian ($C_\ell$) vs. non-Gaussian (WST/WPH) statistics across multiple survey configurations (Planck, ACT, SPT, SO $\times$ Euclid).

4. Results

A. CMB Lensing Auto-Correlations ($\kappa_{CMB}$)

• Finding: For $\kappa_{CMB}$ alone, WST and $C_\ell$ yield nearly identical constraints (within 5-15%) for all surveys considered.
• Implication: The non-Gaussian information in the high-redshift CMB lensing field is minimal or not effectively captured by the specific WST implementation used. The standard power spectrum is sufficient for CMB lensing auto-correlations.

B. Cross-Correlations ($\kappa_{CMB} \times \kappa_{WL}$)

• Finding: When crossing CMB lensing with Euclid DR2 weak lensing, WPH significantly outperforms cross-$C_\ell$.
• Quantitative Improvement:
  • Constraints on $\Omega_m$ improve by factors of 2.2 to 3.4.
  • Constraints on $\sigma_8$ improve by factors of 2.2 to 3.4.
  • Example: For SPT $\times$ Euclid DR2, WPH improves constraints by a factor of 3.4 compared to cross-$C_\ell$.
• Reasoning: The cross-correlation peaks at lower redshifts ($z \sim 1$) compared to CMB auto-correlations ($z \sim 2-6$). At these lower redshifts, the matter distribution is more evolved and highly non-Gaussian, providing WPH with rich information that power spectra miss.

C. Impact of Learned Binning

• For Wavelet Statistics: Learned binning significantly improves constraints compared to naive linear binning (e.g., improving $\Omega_m$ constraints by a factor of 2.7 for ACT $\times$ Euclid WPH).
• For Power Spectra: Learned binning provides negligible gains (<10%) for $C_\ell$, as the power spectrum is already a compact, optimal summary for Gaussian fields.

5. Significance and Conclusion

• Cosmological Insight: The study confirms that while non-Gaussian statistics offer little advantage for high-redshift CMB lensing alone, they are crucial for extracting maximum information from cross-correlations between CMB and low-redshift galaxy surveys.
• Methodological Advance: The "Learned Binning" approach provides a practical, interpretable solution to the dimensionality curse in simulation-based inference, allowing for the comparison of complex summary statistics without sacrificing interpretability or overfitting.
• Future Outlook: The authors suggest that future work should explore spherical wavelets to avoid patch-based projection artifacts and investigate other WPH coefficient combinations. The results strongly motivate the inclusion of non-Gaussian statistics in the analysis pipelines for upcoming surveys like Euclid and the Simons Observatory.

In summary, the paper demonstrates that combining CMB and Weak Lensing data using Wavelet Phase Harmonics with a learned binning compression scheme can tighten cosmological parameter constraints by up to a factor of 3.4 compared to traditional power spectrum methods, primarily by capturing the non-Gaussian structure of the low-redshift universe.