The $\mathrm{\Lambda }$ cold dark matter cosmological model faces increasingly significant and robust tensions among independent cosmological probes, prompting renewed scrutiny of its foundational assumptions. While general relativity and the nature of darkenergy are now routinely tested with cosmological surveys, less progress has been made testing the spacetime geometry at the largest scales, and in particular testing the assumption that observables (distances, redshifts, expansion of space, etc.) on the largest scales are described by a single Friedmann-Lemaître-Robertson-Walker (FLRW) metric. In order to enable such tests, we introduce a model-independent framework that combines successive derivatives of the angular diameter distance, $d_A(z)$, with the line-of-sight expansion rate, $\mathcal{H}(z)$, to expose the physical content of well-known FLRW consistency relations. This allows us to perform diagnostic tests of the large-scale geometry that are free of assumptions about dark energy and the theory of gravity on large scales. In addition, we derive a new nonparametric estimator for the cosmic density field that is independent of the Friedmann equations. This enables qualitatively new, observationally accessible tests of the FLRW framework and provides a stringent, model-independent diagnostic for departures from standard cosmology using current and forthcoming distance and expansion rate measurements.

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• Evolution of the Universe
• Large scale structure of the Universe
• Relativistic aspects of cosmology