Exploring Modified Gravity: Constraints on the $\mu$ and $\Sigma$ Parametrization with WMAP, ACT, and SPT

The cosmic acceleration problem remains one of the most significant challenges in cosmology. One of the proposed solutions to this problem is the modification of gravity on large scales. In this paper, we explore the well-known $\mu$-$\Sigma$ parametrization scenarios and confront them with observational data, including the cosmic microwave background (CMB) radiation from the Wilkinson Microwave Anisotropy Probe (WMAP), Atacama Cosmology Telescope (ACT), and South Pole Telescope (SPT), as well as large-scale structure data from the Sloan Digital Sky Survey (SDSS: BAO+RSD) and Pantheon Supernovae (SN) catalog. We employ a Bayesian framework to constrain the model parameters and discuss the implications of our results on the viability of modified gravity theories. Our analysis reveals the strengths and limitations of the $\mu$-$\Sigma$ parametrization and provides valuable insights into the nature of gravity on cosmological scales. From the joint analysis of the ACT + WMAP + SDSS + SN, we find $\mu_0 -1 = 0.02 \pm 0.19$ and $\Sigma_0 -1 = 0.021 \pm 0.068$ at 68% CL. In light of the SPT + WMAP + SDSS + SN, we find $\mu_0 -1 = 0.07 \pm 0.18$ and $\Sigma_0 -1 = -0.009^{+0.078}_{-0.11}$ at 68% CL. In all the analyses carried out, we do not find any deviations from the theory of general relativity. Our results represent an observational update on the well-known $\mu$-$\Sigma$ parameterization in view of current CMB data, independent and competitive with the constraints obtained with the Planck data.