Analytic auto-differentiable $\Lambda$CDM cosmography

TL;DR

This paper derives analytic expressions for key cosmological distances in the standard $\\Lambda$CDM model, enabling fast, differentiable computations suitable for Bayesian inference with large datasets.

Contribution

It introduces a novel analytic approach using elliptic integrals for distance calculations in $\\Lambda$CDM cosmology, implemented in a PyTorch package for efficient automatic differentiation.

Findings

Analytic solutions are accurate and faster than numerical methods.

The implementation enables scalable Bayesian analysis with millions of supernovae.

Demonstrates practical application in high-dimensional cosmological parameter inference.

Abstract

I present general analytic expressions for distance calculations (comoving distance, time coordinate, and absorption distance) in the standard $\Lambda$CDM cosmology, allowing for the presence of radiation and for non-zero curvature. The solutions utilise the symmetric Carlson basis of elliptic integrals, which can be evaluated with fast numerical algorithms that allow trivial parallelisation on GPUs and automatic differentiation without the need for additional special functions. I introduce a PyTorch-based implementation in the phytorch.cosmology package and briefly examine its accuracy and speed in comparison with numerical integration and other known expressions (for special cases). Finally, I demonstrate an application to high-dimensional Bayesian analysis that utilises automatic differentiation through the distance calculations to efficiently derive posteriors for cosmological parameters from up to $10^6$ mock type Ia supernovae using variational inference.