Closed-form approximations of fundamental quantities of Lemaitre-Tolman-Bondi cosmologies from Symbolic Regression: I. Results on the Garcia-Bellido-Haugbølle 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.

Figures (5)

• Figure 1: 2D Distribution of the cosmological parameters generated from bubble as described in Sec. \ref{['sec:ltb_data']}, showing the relation between the input parameters (rows) and the target output quantities (columns). In here, the most darker color corresponds to regions that are denser in points, whereas the lighter represents regions with sparser points, where the white color indicates lack of population. Additionally, the histograms for all quantities here represented are shown in the first row for the target functions and the last column for the input cosmological parameters of our Symbolic Regression methodology.
• Figure 2: Representation of the SR found expressions of the transverse scale factor $a_1$ and the radial scale factor $a_2$. Left panel: Comparison between the theoretical values obtained via bubble software and the estimated values from the Orchestrator, where a yellow color represents the most dense regions. In the bottom panels of each left panel, we showcase the relative error between these two values. Right panel: Redshift distribution of each parameter for the models labeled in Tab \ref{['tab:LTB_models']}. The dashed line represents the theoretical distribution from bubble and the solid line represents the redshift distribution obtained from the estimated equation.
• Figure 3: Representation of the SR found expressions of the transverse Hubble rate $H_T$, the radial Hubble rate $H_R$, and the angular diameter distance $D_A$. Left panel: Comparison between the theoretical values obtained via bubble software and the estimated values from the Orchestrator, where a yellow color represents the most dense regions. In the bottom space of each left panel, we showcase the relative error between these two values. Right panel: Redshift distribution of each parameter for the models labeled in Tab. \ref{['tab:LTB_models']}. The dashed line represents the theoretical distribution from bubble and the solid line represents the redshift distribution obtained from the estimated equation.
• Figure 4: Scalability of TuringBot on the Expanse system. The figure shows the number of symbolic expressions generated per minute as a function of the number of threads.
• Figure 5: Simplified flowchart illustrating the Orchestrator workflow used to automatically generate input files for the Symbolic Regression engine using multiple parallel batch process dispatches.