Spectral Chebyshev Approximation of Cosmic Expansion in $f(R)$ Gravity

Abstract

We present a numerical framework to study the cosmological background evolution in $f(R)$ gravity by employing a \textit{spectral Chebyshev collocation approach}. Unlike standard integration methods such as Runge--Kutta that often encounter stiffness and accuracy issues, this formulation expands the normalized Hubble function $E(z) = H(z)/H_0$ as a finite Chebyshev series. The modified Friedmann equation is then enforced at selected Chebyshev--Gauss--Lobatto points, converting the original nonlinear differential equation into a system of algebraic relations for the series coefficients. This transformation yields exponentially convergent and numerically stable solutions over the entire redshift domain, $0<z<z_{max}$, eliminating the need for adaptive step-size control. We apply the method to two widely studied $f(R)$ models, Hu--Sawicki and Starobinsky, and perform a combined analysis using cosmic chronometer $H(z)$ data and the Union~3.0 supernova compilation. The reconstructed expansion histories match observations to within $2\sigma$ over $0 < z < 2$, producing best-fit parameters of approximately $(\Omega_{m0}, H_0, \Lambda_{\mathrm{eff}}) \simeq (0.29, 68, 1.2\text{--}2.5\,H_0^2)$. These results indicates that both models reproduce the observed late-time acceleration while permitting small geometric corrections to $\Lambda$CDM. Overall, the spectral Chebyshev method provides a precise and computationally efficient framework for probing modified-gravity cosmologies in the precision-data era.