The Cosmological Principle (CP) -- the notion that the Universe is spatially isotropic and homogeneous on large scales -- underlies a century of progress in cosmology. It is conventionally formulated through the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) cosmologies as the spacetime metric, and culminates in the successful and highly predictive $\Lambda$-Cold-Dark-Matter ($\Lambda$CDM) model. Yet, tensions have emerged within the $\Lambda$CDM model, most notably a statistically significant discrepancy in the value of the Hubble constant, $H_0$. Since the notion of cosmic expansion determined by a single parameter is intimately tied to the CP, implications of the $H_0$ tension may extend beyond $\Lambda$CDM to the CP itself. This review surveys current observational hints for deviations from the expectations of the CP, highlighting synergies and disagreements that warrant further study. Setting aside the debate about individual large structures, potential deviations from the CP include variations of cosmological parameters on the sky, discrepancies in the cosmic dipoles, and mysterious alignments in quasar polarizations and galaxy spins. While it is possible that a host of observational systematics are impacting results, it is equally plausible that precision cosmology may have outgrown the FLRW paradigm, an extremely pragmatic but non-fundamental symmetry assumption.

We present the ensemble expectation values for the translation invariant, rank-2 Minkowski tensors in three-dimensions, for a linearly redshift space distorted Gaussian random field. The Minkowski tensors W 0,2 1 , W 0,2 2 are sensitive to global anisotropic signals present within a field, and by extracting these statistics from the low redshift matter density one can place constraints on the redshift space distortion parameter β = f /b. We begin by reviewing the calculation of the ensemble expectation values W 0,2 1 , W 0,2 2 for isotropic, Gaussian random fields, then consider how these results are modified by the presence of a linearly anisotropic signal. Under the assumption that all fields remain Gaussian, we calculate the anisotropic correction due to redshift space distortion in a coordinate system aligned with the line of sight, finding inequality between the diagonal elements of W 0,2 1 , W 0,2 2 . The ratio of diagonal elements of these matrices provides a set of statistics that are sensitive only to the redshift space distortion parameter β. We estimate the Fisher information that can be extracted from the Minkowski tensors, and find W 0,2 1 is more sensitive to β than W 0,2 2 , and a measurement of W 0,2 1 accurate to ∼ 1% can yield a ∼ 4% constraint on β. Finally, we discuss the difference between using the matrix elements of the Minkowski tensors directly against measuring the eigenvalues. For the purposes of cosmological parameter estimation we advocate the use of the matrix elements, to avoid spurious anisotropic signals that can be generated by the eigenvalue decomposition.

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The Minkowski tensors (MTs) can be used to probe anisotropic signals in a field, and are well suited for measuring the redshift-space distortion (RSD) signal in large-scale structure catalogs. We consider how the linear RSD signal can be extracted from a field without resorting to the plane-parallel approximation. A spherically redshift-space distorted field is both anisotropic and inhomogeneous. We derive expressions for the two-point correlation functions that elucidate the inhomogeneity, and then explain how the breakdown of homogeneity impacts the volume and ensemble averages of the tensor Minkowski functionals. We construct the ensemble average of these quantities in curvilinear coordinates and show that the ensemble and volume averages can be approximately equated, but this depends on our choice of definition of the volume average of a tensor and the radial distance between the observer and field. We then extract the tensor Minkowski functionals from spherically redshift-space distorted, Gaussian random fields and gravitationally evolved dark matter density fields at z = 0 to test if we can successfully measure the Kaiser RSD signal. For the dark matter field, we find a significant, ∼10% anomalous signal in the MT component parallel to the line of sight that is present even on large scales R G ≳ 15 Mpc, in addition to the Kaiser effect. This is due to the line-of-sight component of the MT being significantly contaminated by the Finger of God effect, which can be approximately modeled by an additional damping term in the cumulants.