We reverse the logic behind the apparent existence of H0-tension, to design diagnostics for cosmological models. The basic idea is that the non-constancy of H0 inferred from observations at different redshifts is a null hypothesis test for models within the FLRW paradigm -if H0 runs, the model is wrong. Depending on the kind of observational data, the most suitable form of the diagnostic can vary. As examples, we present two H0 diagnostics that are adapted to two different BAO observables. We use these and the corresponding BAO data to Gaussian reconstruct the running of H0 in flat ΛCDM with Planck values for the model parameters. For flat ΛCDM when the radiation contribution can be neglected, with comoving distance data, the diagnostic is a simple hypergeometric function. Possible late time deviations from the FLRW paradigm can also be accommodated, by simply keeping track of the (potentially anisotropic) sky variation of the diagnostic.

We introduce the dipole cosmological principle, the idea that the Universe is a maximally Copernican cosmology, compatible with a cosmic flow. It serves as the most symmetric paradigm that generalizes the FLRW ansatz, in light of the increasingly numerous (but still tentative) hints that have emerged in the last two decades for a non-kinematic component in the CMB dipole. Einstein equations in our “dipole cosmology” are still ordinary differential equations — but instead of the two Friedmann equations, now we have four. The two new functions can be viewed as an anisotropic scale factor that breaks the isotropy group from SO(3) to U(1), and a “tilt” that captures the cosmic flow velocity. The result is an axially isotropic, tilted Bianchi V/VII_h cosmology. We assess the possibility of model building within the dipole cosmology paradigm, and discuss the dynamics of expansion rate, anisotropic shear and tilt, in various examples. A key observation is that the cosmic flow (tilt) can grow even while the anisotropy (shear) dies down. Remarkably, this can happen even in an era of late time acceleration.

We present the dipole cosmological principle, i.e., the notion that the Universe is a Copernican cosmology that agrees with the cosmic flow. It suits the most symmetric paradigm that generalizes the Friedmann–Lemaître–Robertson–Walker ansatz in the context of numerous suggestions that have appeared in the literature for non-kinematic components in the cosmic microwave background dipole. Field equations in our “dipole cosmology” are still ODEs, but we now have four instead of two Friedmann equations. The two extra functions can be regarded as additional scale factors that break the isotropy group from SO(3) to U(1) and a “tilt” that denotes the cosmic flow. The result is an axially isotropic Universe. We examined the dynamics of the expansion rate, anisotropic shear, and tilt in some cases. One important observation is that the cosmic flow (tilt) can grow while the anisotropy (shear) dies down.