The measurements of the cosmic microwave background (CMB) have determined the cosmological parameters with high accuracy, and the observation of the flatness of space has contributed to the status of the concordance Λ cold dark matter (CDM) model. However, the cosmological constant Λ , necessary to close the model to critical density, remains an open conundrum. The Einstein equations and the Friedmann–Lemaître–Robertson–Walker (FLRW) metric are the foundation of modern cosmology. While the geometric interpretation of the Einstein equations describes the action of gravity as the dynamical curvature of space by matter, the FLRW metric is built on Milne’s concept of a kinematically determined Universe. In a preceding companion article, we considered that the Friedmann equation describes the expansion history of FLRW universes in the local reference frame of freely falling comoving observers, who perceive flat, homogeneous, and isotropic space in their local inertial frame. The observed late-time accelerated expansion is then attributed to a kinematic effect akin to a dark energy component. Our approach displayed an expansion history very similar to that of Λ CDM. Now we extend our approach to nonlinear structure formation. We include the impact on the expansion history caused by the cosmic web of the late Universe, once voids dominate its volume, and find that the initially constant w d e becomes time-dependent, evolving to a value of w d e ≃ − 0.9 at the present. While this impact of voids is minor, it could provide a possible explanation for the Hubble tension. We use the Cosmic Linear Anisotropy Solving System (CLASS) to calculate the expansion history and power spectra of our extension and compare our results to concordance Λ CDM and to observations. We find that our model agrees well with current data, in particular with the final data release PR4 of the Planck mission.
Foidl et al. (2026) studied this question.