Testing spatial curvature and anisotropic expansion on top of the
  $\Lambda$CDM model
  We explore the possible advantages of extending the standard $\Lambda$CDM
model by more realistic backgrounds compared to its spatially flat
Robertson-Walker spacetime assumption, while preserving the underpinning
physics; in particular, by simultaneously allowing non-zero spatial curvature
and anisotropic expansion on top of $\Lambda$CDM, viz., the An-$o\Lambda$CDM
model. This is to test whether the latest observational data still support
spatial flatness and/or isotropic expansion in this case, 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 current cosmological
tensions associated with $\Lambda$CDM. We first present the theoretical
background and explicit mathematical construction of An-$o\Lambda$CDM. Then we
constrain the parameters of this model and its particular cases, namely,
An-$\Lambda$CDM, $o\Lambda$CDM, and $\Lambda$CDM, by using the data sets from
different observational probes, viz., Planck CMB(+Lens), BAO, SnIa Pantheon,
and CC data, and discuss the results in detail. Ultimately, we conclude that
(i) the observational data confirm the spatial flatness and isotropic expansion
assumptions of $\Lambda$CDM, though a very small amount of expansion anisotropy
cannot be excluded, e.g., $\Omega_{\sigma0}\lesssim10^{-18}$ (95% C.L.) for
An-$\Lambda$CDM from CMB+Lens data, (ii) the introduction of spatial curvature
or anisotropic expansion, or both, on top $\Lambda$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 the
CMB(+Lens) data nor does it improve the drastically reduced value of $H_0$ led
by the closed space.
