The fractal bubble model with a cosmological constant

Abstract

We generalize the fractal bubble (FB) model, recently proposed in the literature as an alternative to the standard ΛCDM cosmology, to include a non-zero cosmological constant. We retain the same volume partition of voids and walls as the original FB model, and the same matching conditions for null geodesics, but do not include effects associated with a nonuniform time flow arising from differences of quasilocal gravitational energy that may arise in the coarse-graining process. The Buchert equations are written and partially integrated and the asymptotic behavior of the solutions is given. For a universe with Λ = 0, as is the case in the FB model, an initial void fraction with hyperbolic curvature evolves in such a way that it asymptotically fills completely our particle horizon. Conversely, in the presence of a nonvanishing Λ, we show that this does not happen and the voids fill a finite fraction , where the value of is expected to depend on Λ and the initial fraction fvi and also to be small. For its determination, a numerical integration of the equations is necessary. Finally, an interesting prediction of our model is a formula giving a minimum allowed value of present day dark energy as a function of the age of the universe and of the matter and curvature density parameters at our time.To my mother Cristina Pergolini