Role of internal space correlations in the dynamics of a higher-dimensional Bianchi type-I universe: Shear scalar and Hubble parameter perspectives

We investigate exact solutions of the Einstein field equations in higher-dimensional, spatially homogeneous Bianchi type-I spacetimes, introducing a real parameter A that correlates the expansion rates of external and internal spaces. Extending beyond Robertson-Walker spacetime, our approach includes positive and negative correlations, suggesting a broader and isotropic/anisotropic cosmological model space. Positively correlated dimensions manifest as a cosmological constant at late times, while at early times, they mimic stiff-fluid-like dark energy that dilutes faster than radiation, paralleling early dark energy models. This suggests a pathway for alleviating the Hubble tension by tailoring higher-dimensional dynamics to reduce the sound horizon. When anisotropic expansion is allowed, these models achieve isotropization more efficiently than predicted by Wald's cosmic no-hair theorem. Negative correlations, in contrast, yield a higher-dimensional steady-state universe where the shear scalar remains constant, effectively emulating a negative cosmological constant. These distinct behaviors arise from a simple signature change: positive correlation accelerates shear scalar decay, while negative correlation stabilizes it. We demonstrate that the solutions admit analytic continuation from the Lorentzian to Euclidean regime (t- -ii), revealing a wormhole-like topology that connects two asymptotic regions via a throat, with A- -A.