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Testing spatial curvature and anisotropic expansion on top of the $Λ$CDM model

We explore the possible advantages of extending the $Λ$CDM model by more realistic backgrounds compared to its spatially flat RW spacetime assumption, while preserving the underpinning physics; in particular, by simultaneously allowing non-zero spatial curvature and anisotropic expansion on top of it, viz., the An-$oΛ$CDM model. This is to test whether the latest data support spatial flatness and/or isotropic expansion, and, if not, to explore the roles of spatial curvature and expansion anisotropy (due to its stiff fluid-like behavior) in addressing some of the cosmological tensions. We first present the theoretical background and explicit mathematical construction of An-$oΛ$CDM; combining the simplest anisotropic generalizations of the RW spacetime, viz., the Bianchi type I, V, and IX spacetimes. Then we constrain this model and its particular cases, viz., An-$Λ$CDM, $oΛ$CDM, and $Λ$CDM, by using the data sets from different probes, viz., Planck CMB(+Lens), BAO, SnIa Pantheon, and CC data, and discuss the results. Ultimately, we conclude that, within the setup under consideration, (i) the data confirm the spatial flatness and isotropic expansion, though a very small amount of present-day expansion anisotropy cannot be excluded, e.g., $Ω_{\sigma0}\lesssim10^{-18}$ (95\% C.L.) for An-$Λ$CDM from CMB+Lens, (ii) the introduction of spatial curvature or anisotropic expansion, or both, on top $Λ$CDM does not offer a possible relaxation to the $H_0$ tension, and (iii) the introduction of anisotropic expansion neither affects the closed space prediction from CMB(+Lens) nor does it improve the drastically reduced value of $H_0$ led by the closed space. We discuss why it is important and indispensable to maintain the geometric generalization work program, especially in models that offer solutions to cosmological tensions. [abridged]