A Common Origin of the H₀ and S₈ Cosmological Tensions and a Resolution within a Modified ∧CDM Framework

The two most severe cosmological tensions in the Hubble constant \( H_0 \) and the matter clustering amplitude \( S_8 \) have the same relative discrepancy of 8.3%, which suggests that they may have a common origin. Modifications of gravity and exotic dark fields with numerous free parameters introduced in the Einstein field equations often struggle to simultaneously alleviate both tensions; thus, we need to look for a common cause within the standard \( \Lambda \)CDM framework. At the same time, linear perturbation analyses of matter in the expanding \( \Lambda \)CDM universe have always neglected the impact of comoving peculiar velocities \( \mathbf{v} \) (generally thought to be a second-order effect), the same velocities that in physical space cannot be fully accounted for in the observed late-time universe when the cosmic distance ladder is used to determine the local value of \( H_0 \). We have reworked the linear density perturbation equations in the conformal Newtonian gauge (sub-horizon limit) by introducing an additional drag force per unit mass \( -\Gamma(t)\mathbf{v} \) in the Euler equation with \( \Gamma \equiv \gamma(2 H) \), where \( \gamma\ll1 \) is a positive dimensionless constant and \( 2H(t) \) is the time-dependent Hubble friction. We find that a damping parameter of \( \gamma = 0.083 \) is sufficient to resolve the \( S_8 \) tension by suppressing the growth of structure at low redshifts, starting at \( z_\star\simeq 3.5-6.5 \) to achieve \( S_8\simeq 0.78-0.76 \), respectively. Furthermore, we argue that the physical source causing this additional friction (a tidal field generated by nonlinear structures in the late-time universe) is also responsible for a systematic error in the local determinations of \( H_0 \): the inability to subtract peculiar tidal velocities along the lines of sight when determining the Hubble flow via the cosmic distance ladder. Finally, the dual action of the tidal field on the expanding background—reducing both the matter and the dark-energy sources of the squared Hubble rate \( H^2 \), thereby holding back the cosmic acceleration \( \ddot a \)—is of fundamental importance in resolving cosmological tensions and can also substantially alleviate the density coincidence problem.