Inferring the Hubble Constant Using Simulated Strongly Lensed Supernovae and Neural Network Ensembles

TL;DR

This work presents a pipeline that uses simulated strongly lensed SNe Ia observed by the Roman Space Telescope to infer the Hubble constant $H_0$ via time-delay cosmography. An ensemble of five 3D CNNs processes image time-series and, through a simulation-based inference framework, yields full posteriors for the time-delay distance $D_{ m\Delta t}$ and $H_0$. On a test set of 100 glSNe Ia, the joint analysis achieves $H_0=(69.20 \pm 3.03)\ \mathrm{km\ s^{-1}\ Mpc^{-1}}$, a $4.4\%$ precision that aligns with the ground-truth value within uncertainties, illustrating the potential of ML+SBI for fast, automated cosmology with glSNe. The study highlights which parameters most drive uncertainty and outlines clear pathways for improvement, such as multi-band data and more realistic lens models, to further tighten constraints as glSNe samples grow.

Abstract

Strongly lensed supernovae are a promising new probe to obtain independent measurements of the Hubble constant (${H_0}$). In this work, we employ simulated gravitationally lensed Type Ia supernovae (glSNe Ia) to train our machine learning (ML) pipeline to constrain $H_0$. We simulate image time-series of glSNIa, as observed with the upcoming Nancy Grace Roman Space Telescope, that we employ for training an ensemble of five convolutional neural networks (CNNs). The outputs of this ensemble network are combined with a simulation-based inference (SBI) framework to quantify the uncertainties on the network predictions and infer full posteriors for the $H_0$ estimates. We illustrate that the combination of multiple glSN systems enhances constraint precision, providing a $4.4\%$ estimate of $H_0$ based on 100 simulated systems, which is in agreement with the ground truth. This research highlights the potential of leveraging the capabilities of ML with glSNe systems to obtain a pipeline capable of fast and automated $H_0$ measurements.

Figures (9)

• Figure 1: Distributions of the properties of the simulated glSNe Ia, resulting from the simulated doubly imaged dataset.
• Figure 2: Example of an element in our doubly imaged dataset. Here a simulated glSN Ia system is shown, where: (a) the images' light curves are shown, with vertical grey lines indicating the time at which the images were simulated; (b) the time-series (i.e. the element), comprised of the images simulated at all the aforementioned times, is shown. Each element holds the evolution of a glSN, where observations are taken every five days. The glSN Ia system shown here has: $\theta_{\textrm{E}}$ = 1.157, $z_l$ = 0.232, $z_s$ = 0.527, $\Delta t$ = 78.48 days, $\mu_{\textrm{img}~0}$ = 2.688 and $\mu_{\textrm{img}~1}$ = 0.854.
• Figure 3: Joint 2D PDF of predicted and true $D_{\Delta t}$ values, computed using a set of 24,245 doubly imaged supernovae. We obtain the approximate posterior $\mathcal{P}(D_{\Delta t}^{\textrm{true}} | D_{\Delta t}^{\textrm{pred}})$ for a particular lens system by making a vertical slice at the predicted value of $D_{\Delta t}$.
• Figure 4: Illustration of the ensemble architecture used in this study. The ensemble processes a sequence of images through five independent 3D CNNs, each designed to infer a specific parameter: image positions, source position, ellipticities, Einstein radius, and time delay. Each layer in the CNNs is followed by a tanh activation function, ensuring bounded outputs suitable for parameter estimation. The shared architecture of the CNNs is detailed at the bottom of the figure, with the primary distinction being the dimensionality of the final layer, which is tailored to each output parameter. These individual outputs are subsequently utilized to compute the time-delay distance, which is combined with redshift measurements to derive an estimate of the Hubble constant.
• Figure 5: Comparison between predicted (y-axis) versus ground truth (x-axis) for all parameters predicted by the 3D CNN ensemble model. The colored boxes indicate parameters predicted by the same model and they are as follows: (i) Inside the red square are the predictions for all the image position coordinates; (ii) Inside the green square are the source position coordinates; (iii) Inside the blue square are the Ellipticities predictions, $e_1$ and $e$; (iv) Inside the yellow square are the predictions for the Einstein Radius, $\theta_E$; and, finally, (v) Inside the purple square are the time delay, $\Delta t$, predictions.