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We investigate the possibility of testing the Cosmological Principle within a specific axisymmetric metric. Starting from the Einstein field equations, we derive a perturbed form of the Friedmann equation based on this metric and obtain a generalized form for matter density perturbations. By combining these, we establish the relationship between the scale factor a and the terms for matter density and dark energy over time. We then use the Union2.1 and PantheonPlus Type Ia supernovae datasets and CMB shift parameter R to constrain the perturbation parameters introduced in the model. This analysis allows us to assess whether matter perturbation could lead to observable inhomogeneity in cosmic expansion. Finally, we summarize the key parameters used in the calculation and discuss their physical significance. Our results indicate that within this axisymmetric metric, the contribution of matter perturbations to cosmic anisotropy and inhomogeneity is negligible, posing no challenge to the Cosmological Principle.
Chen et al. (Thu,) studied this question.