[Paper Review] Wavelet Moments for Cosmological Parameter Estimation

Michael Eickenberg, Erwan Allys+11 more|arXiv (Cornell University)|Apr 15, 2022

Geophysics and Gravity Measurements73 references25 citations

TL;DR

The paper introduces 3D wavelet moment statistics computed from wavelet transforms of matter fields, and shows they yield state-of-the-art cosmological parameter constraints (including sum of neutrino masses) in Fisher forecasts, outperforming the power spectrum by factors of 5–10.

ABSTRACT

Extracting non-Gaussian information from the non-linear regime of structure formation is key to fully exploiting the rich data from upcoming cosmological surveys probing the large-scale structure of the universe. However, due to theoretical and computational complexities, this remains one of the main challenges in analyzing observational data. We present a set of summary statistics for cosmological matter fields based on 3D wavelets to tackle this challenge. These statistics are computed as the spatial average of the complex modulus of the 3D wavelet transform raised to a power $q$ and are therefore known as invariant wavelet moments. The 3D wavelets are constructed to be radially band-limited and separable on a spherical polar grid and come in three types: isotropic, oriented, and harmonic. In the Fisher forecast framework, we evaluate the performance of these summary statistics on matter fields from the Quijote suite, where they are shown to reach state-of-the-art parameter constraints on the base $Λ$CDM parameters, as well as the sum of neutrino masses. We show that we can improve constraints by a factor 5 to 10 in all parameters with respect to the power spectrum baseline.

Motivation & Objective

• Motivate the need to extract non-Gaussian information from nonlinear large-scale structure data.
• Propose a set of 3D wavelet-based summary statistics for matter fields.
• Assess information content of these statistics for cosmological parameters using Fisher forecasts.

Proposed method

• Construct 3D band-limited, radially separable wavelets of isotropic, oriented, and harmonic types on a spherical polar grid.
• Define wavelet moments S1 for q-parameterized powers of the complex wavelet modulus, including isotropic S1^I, oriented S1^O, and harmonic S1^H variants.
• Average over space and, for rotation-invariant descriptors, over orientations or spherical harmonic channels to obtain rotationally invariant statistics.
• Analyze Gaussian and non-Gaussian behavior of wavelet moments, and relate q=2 moments to the power spectrum via Parseval’s identity.
• Use Quijote N-body simulations to compute the statistics and evaluate information content via the Fisher matrix framework.

Experimental results

Research questions

• RQ1 Do 3D wavelet moments capture non-Gaussian information beyond the power spectrum in nonlinear structure formation?
• RQ2 How do isotropic, oriented, and harmonic wavelet families compare in constraining cosmological parameters?
• RQ3 What is the impact of the wavelet moment exponent q on parameter constraints and information content?
• RQ4 Can wavelet moments improve constraints on the sum of neutrino masses relative to baseline statistics?

Key findings

• Wavelet moments with q ≠ 2 contain additional information beyond the power spectrum.
• Using band-limited, polar-separable 3D wavelets (isotropic, oriented, harmonic) yields strong parameter constraints.
• In Fisher forecasts on Quijote simulations, wavelet moments improve constraints by a factor of 5–10 over the power spectrum baseline.
• The approach demonstrates state-of-the-art constraints on base ΛCDM parameters and the sum of neutrino masses.
• Gaussian-field expectations show q=2 moments reproduce power-spectrum information, while other q values reveal non-Gaussian features at small scales.