Revealing the 3D Cosmic Web through Gravitationally Constrained Neural Fields

TL;DR

The paper tackles 3D dark matter mass mapping from weak gravitational lensing by formulating a coordinate-based neural-field representation constrained by a differentiable forward model of lensing. It optimizes the 3D overdensity via an MLP over 3D coordinates to reproduce observed shear, incorporating a lens-plane power-spectrum regularizer and an ensemble-based uncertainty framework. On simulations with survey-like realism, the approach achieves improved localization along the line of sight and faithful recovery of non-Gaussian structures, outperforming Wiener-filter baselines. This flexible, scalable framework is poised to enhance interpretation of upcoming weak-lensing surveys and tighten tests of structure formation and dark matter properties.

Abstract

Weak gravitational lensing is the slight distortion of galaxy shapes caused primarily by the gravitational effects of dark matter in the universe. In our work, we seek to invert the weak lensing signal from 2D telescope images to reconstruct a 3D map of the universe's dark matter field. While inversion typically yields a 2D projection of the dark matter field, accurate 3D maps of the dark matter distribution are essential for localizing structures of interest and testing theories of our universe. However, 3D inversion poses significant challenges. First, unlike standard 3D reconstruction that relies on multiple viewpoints, in this case, images are only observed from a single viewpoint. This challenge can be partially addressed by observing how galaxy emitters throughout the volume are lensed. However, this leads to the second challenge: the shapes and exact locations of unlensed galaxies are unknown, and can only be estimated with a very large degree of uncertainty. This introduces an overwhelming amount of noise which nearly drowns out the lensing signal completely. Previous approaches tackle this by imposing strong assumptions about the structures in the volume. We instead propose a methodology using a gravitationally-constrained neural field to flexibly model the continuous matter distribution. We take an analysis-by-synthesis approach, optimizing the weights of the neural network through a fully differentiable physical forward model to reproduce the lensing signal present in image measurements. We showcase our method on simulations, including realistic simulated measurements of dark matter distributions that mimic data from upcoming telescope surveys. Our results show that our method can not only outperform previous methods, but importantly is also able to recover potentially surprising dark matter structures.

Figures (9)

• Figure 1: Weak Lensing Measurements. (Left) As light travels through the universe, it is lensed by intervening matter structures, causing slight changes between the unlensed intrinsic shape $e_\text{int}$ and the observed shape $e_\text{obs}$ of galaxies. Measurements of these faint shape changes is a useful probe of the 3D structure of dark matter. (Right) To quantify the effects of cosmic shearing, it is useful to consider galaxy shapes in the complex elliptical domain, where the components of the complex ellipticity describe the axis ratio and orientation angle of a given ellipse (Figure adapted from schneider2006gravitational). In kinematic weak lensing, the detectable cosmic shear signal is outweighed by our uncertainty in a galaxy's intrinsic shape by more than an order of magnitude. In traditional weak lensing the uncertainty is more than two orders of magnitude greater than the shear signal. The lensing effects of intervening matter can be described by a shear $\gamma$, which is approximately additive in the ellipticity domain. In practice, we combine many shear measurements from a dense field of galaxies to obtain a coherent signal from the underlying matter distribution.
• Figure 2: Proposed 3D Mass Mapping Pipeline. We model a 3D matter overdensity field $\hat{\delta}$ as a continuous function using a fully-connected neural network. We then differentiably compute cosmic shear measurements from a given galaxy catalog through this overdensity field with a physics-based forward model to produce a set of predicted shear measurements. Next, we solve for the weights of the neural network by minimizing a data loss between the model and observed shear measurements plus a physically-motivated power spectrum regularization loss. We obtain a final reconstruction by taking the median from a deep ensemble of 100 independently initialized neural fields.
• Figure 3: Kinematic weak lensing result: We present 3D reconstruction results of a realistic lensing volume derived from an N-body simulation. Results includes 12 lens planes from redshift $z = 0$ to $z = 1$. To concisely visualize reconstructions, we average every 3 adjacent lens planes to form 4 lensing regions, increasing in distance away from the sensor. Ground truth (GT) lensing regions are shown on the far left. For visual comparison, we blur the ground truth volume with a Gaussian filter in the line-of-sight direction that maximizes the cross-correlation with each reconstructed slice. We then visualize this optimally blurred ground truth volume for each reconstruction, along with the average line-of-sight standard deviation used for each region (top right corner). Our reconstruction corresponds well with a GT volume with significantly lower radial blurring, especially at lower redshift, showing that our method is less susceptible to smearing along the line of sight.
• Figure 4: Traditional weak lensing result: We perform 3D reconstruction on a simulated traditional weak lensing survey assuming no information on the intrinsic shapes of galaxies. As in Fig. \ref{['fig:lsskine']}, we use the same ground truth volume and setup, dividing our volume into four redshift regions and generating blurred ground truth volumes for visual comparison. The optimal blur level for each region and method is indicated in the top right corner of each blurred ground truth image. Our reconstruction has less radial blurring than the Wiener filter baseline used in previous weak lensing surveys, and has higher correlation with the ground truth volumes.
• Figure 5: Sensitivity to Non-Gaussianity: In this experiment, we perform reconstruction on a toy lensing field of 4 lens planes, each with an MNIST digit 0 through 3 in one if its corners. Due to the non-Gaussian nature of these images, the Wiener filter which uses a Gaussian prior struggles to reconstruct the lens planes accurately. We blur the ground truth volume to match each reconstructed lensplane as described in Sec. \ref{['sec:lss']} and report the optimal $z$ blur (px) in each corner; the Wiener filter exhibits significantly more blurring in low redshifts.