We present DeepSLIDE: a Strong Lens based Inference for Dark Energy. We use Simulation-Based Inference (SBI) with Neural Ratio Estimation (NRE) to constrain Dark Energy parameter from a population of strong galaxy-galaxy lenses.

Introduction

Strong gravitational lensing offers crucial insights into cosmology, but traditional Monte Carlo methods for cosmological inference are computationally prohibitive and inadequate for processing the thousands of lenses anticipated from future cosmic surveys. New tools for inference, such as SBI using NRE, address this challenge effectively. NRE is a classifier neural network to differentiate between two probability distributions:

$(x,w) \sim\ p(x,w)$ with class label y=1

$(x,w) \sim\ p(x)p(w)$ with class label y=0

where $x$ is the strong lens image and $w$ is the dark energy equation-of-state parameter that generated the image.

By training a machine learning model on simulated data of strong lenses, we can learn the likelihood-to-evidence ratio $\frac{p(x|w)}{p(x)}$. This is used for robust inference of the posterior $p(w|{x_{0}})$ from a population of strong lens images ${x_{0}}$.

$\textbf{Analysis Workflow}$

The following figure summarizes the workflow of the analysis.

The strong lens images are generated using a simulator where the parameters are sampled from a prior distribution. The training data for the NRE network (classifier) includes the image and the parameter of interest. The network outputs the likelihood-to-evidence ratio. The trained model is implemented on the observations to estimate the posterior.

Getting started

Data

The data used for this analysis can be found on zenodo (link will be provided shortly).

The images are generated using Deeplenstronomy package.

This data can be generated using the yaml files in /deeplenstornomy_templates as inputs to $\texttt{Deeplenstronomy}$.

The simulation outputs the data into a folder which includes images (/CONFIGURATION_1_images.npy) and the metadata (/CONFIGURATION_1_metadata.csv) associated with the image generation.

$\textbf{Training data}$: We train, validate, and test the NRE model on simulated data of 1M strong lens images.

$\textbf{Test data for population-level analysis}$: We generate three datasets of 3000 images each by fixing w = -1.2, -1.0, and -0.8 respectively.

Notebooks for analysis

The notebooks in /notebooks can be run to reproduce the results of the paper. The python scripts are currently being developed to be run from the terminal.

$\textbf{Model Training}$

train_model.ipynb This notebook includes reading in the data, preprocessing of the images, and training the model. Three models with random weight initializations (seed input to the model) are run in our analysis to check robustness. One of the models working_model_1M-2-034_seed128_v2.keras is available on zenodo (link will be provided shortly).

This trained model can be directly loaded (without having to re-train the model) using

model = tf.keras.models.load_model(model_name)

$\textbf{Model Evaluation}$

compare_random_seeds.ipynb code is for checking the performance of the model on test data of 2000 images. The code includes plotting the Receiver Operating Curve (ROC) for the three models. It also includes calculating and plotting the analytical posteriors of a few randomly selected strong lenses.

plot_image_posterior.ipynb code is for plotting the training data and show the correlation between the Einstein radius and $w$. This code also plots the image of strong lens from the training data and the corresponding analytical posterior.

plot_residuals is to plot the predicted mean $w$ of the analytical posterior with 1 $\sigma$ error bar Vs the true $w$ of the 2000 test images. We also compute the posterior coverage plot to check the model uncertainty.

$\textbf{Population-level Analysis}$

NRE_varyastro_w12.ipynb, NRE_varyastro_w1.ipynb, and NRE_varyastro_w08.ipynb include functions to compute the joint posterior from 3000 images with $w$ fixed to -1.2, -1.0, and -0.8 respectively. We show both MCMC and analytical methods to calculate the posterior.

Compare_mcmc_analytical.ipynb compares the posteriors from the MCMC and analytical methods.

Authors

Sreevani Jarugula, Brian Nord, Abhijith Gandrakota, and Aleksandra Ćiprijanović

References

If you use this code, please cite our paper (Link to be posted shortly)