A Quantitative Framework for Testing the Hubble Tension in a Bianchi Type I Cosmological Background

The Hubble tension is usually formulated as a disagreement between two determinations of a single scalar parameter, H0, within an exactly isotropic FLRW model. We develop a quantitative framework in which the tension is instead treated as a consistency test of the scalar FLRW compression of cosmological data in a homogeneous but anisotropically expanding Bianchi type I background. Beyond synthesizing established results on Bianchi I kinematics, null geodesics and optical propagation, the original contribution is a worked weak-shear, axisymmetric calculation that maps a specified shear history into a low-redshift luminosity-distance quadrupole. The calculation explicitly separates the direction-dependent redshift–affine-parameter mapping from the Jacobi-focusing contribution and then propagates the resulting distance quadrupole through an analytic polar-cap toy window. For freely decaying shear we obtain AD(z)=−BH0+(2q0−1)BH0z/2+(5−q0−18q02+6j0)BH0z2/12+O(z3,BH02), where BH0=(H‖0−H⊥0)/H0 and j0 is the mean jerk parameter. A representative BBN limit, Ωσ0≲10−23, implies |BH0|≲9.5×10−12 and a distance-modulus quadrupole below approximately 2.4×10−11 mag at z=0.15. The early-Universe bound used in this comparison is adopted from previous work and is not itself a new result of the present analysis; the novelty is its propagation through the derived direction-dependent redshift and Sachs–Jacobi mapping into quantitative limits on the luminosity-distance quadrupole and on the catalog-window bias of an isotropic H0 fit. By contrast, even a maximally aligned one-percent directional shift requires Ωσ0≃2.5×10−5, while a shift comparable with the Planck 2018–SH0ES 2022 benchmark separation requires Ωσ0≃1.8×10−3. Thus the minimal shear-only model cannot resolve the tension, although the framework supplies a falsifiable program for testing sustained late-time anisotropy with supernovae, BAO, distance-ladder measurements and future standard sirens.