Closed-form approximations of fundamental quantities of Lemaitre-Tolman-Bondi cosmologies from Symbolic Regression: I. Results on the Garcia-Bellido-Haugb{\o}lle parameterization

Abstract

We introduce a novel set of analytic approximations for five fundamental functions in spherically symmetric, inhomogeneous Lemaitre-Tolman-Bondi (LTB) cosmologies, derived via Symbolic Regression (SR). Focusing on the constrained Garcia-Bellido-Haugboelle (GBH) parameterization, we sample the four-dimensional LTB parameter space using the bubble LTB numerical code and apply SR to reconstruct closed-form expressions for the radial and transverse scale factors A_parallel(r,t) and A_perp(r,t), the corresponding Hubble functions H_parallel(r,t) and H_perp(r,t), and the angular diameter distance D_A(z). Our best-fit formulas reproduce the numerical data with high precision: the relative mean error across all quantities remains below 0.3 percent, except for the radial Hubble function, where it reaches 1.4 percent. These compact expressions enable rapid evaluation of LTB predictions, supporting fast parameter scans, likelihood analyses, and model comparisons without time-consuming integrations. We provide explicit coefficients and discuss the domain of validity, demonstrating that SR-driven approximations can serve as robust surrogates for exact LTB solutions in both theoretical investigations and observational analyses.