Homogeneity and the causal boundary

Abstract

A Universe with finite age also has a finite causal scale $\chi_\S$, so the metric can not be homogeneous for $\chi>\chi_\S$, as it is usually assumed. To account for this, we propose a new causal boundary condition, that can be fulfil by fixing the cosmological constant $\Lambda$ (a free parameter for gravity). The resulting Universe is inhomogeneous, with possible variation of cosmological parameters on scales $\chi \simeq \chi_\S$. The size of $\chi_\S$ depends on the details of inflation, but regardless of its size, the boundary condition forces $\Lambda/8\pi G $ to cancel the contribution of a constant vacuum energy $\rho_{vac}$ to the measured $\rho_\Lambda \equiv \Lambda/8\pi G + \rho_{vac}$. To reproduce the observed $\rho_{\Lambda} \simeq 2 \rho_m$ today with $\chi_\S \rightarrow \infty$ we then need a universe filled with evolving dark energy (DE) with pressure $p_{DE}> - \rho_{DE}$ and a fine tuned value of $\rho_{DE} \simeq 2 \rho_m$ today. This seems very odd, but there is another solution to this puzzle. We can have a finite value of $\chi_\S \simeq 3 c/H_0$ without the need of DE. This scale corresponds to half the sky at $z \sim 1$ and 60deg at $z \sim 1000$, which is consistent with the anomalous lack of correlations observed in the CMB.