Constraints on Vacuum Energy from Structure Formation and Nucleosynthesis

This paper derives an upper limit on the density $\rho_{\scriptstyle\Lambda}$ of dark energy based on the requirement that cosmological structure forms before being frozen out by the eventual acceleration of the universe. By allowing for variations in both the cosmological parameters and the strength of gravity, the resulting constraint is a generalization of previous limits. The specific parameters under consideration include the amplitude $Q$ of the primordial density fluctuations, the Planck mass $M_{\rm pl}$, the baryon-to-photon ratio $\eta$, and the density ratio $\Omega_M/\Omega_b$. In addition to structure formation, we use considerations from stellar structure and Big Bang Nucleosynthesis (BBN) to constrain these quantities. The resulting upper limit on the dimensionless density of dark energy becomes $\rho_{\scriptstyle\Lambda}/M_{\rm pl}^4<10^{-90}$, which is $\sim30$ orders of magnitude larger than the value in our universe $\rho_{\scriptstyle\Lambda}/M_{\rm pl}^4\sim10^{-120}$. This new limit is much less restrictive than previous constraints because additional parameters are allowed to vary. With these generalizations, a much wider range of universes can develop cosmic structure and support observers. To constrain the constituent parameters, new BBN calculations are carried out in the regime where $\eta$ and $G=M_{\rm pl}^{-2}$ are much larger than in our universe. If the BBN epoch were to process all of the protons into heavier elements, no hydrogen would be left behind to make water, and the universe would not be viable. However, our results show that some hydrogen is always left over, even under conditions of extremely large $\eta$ and $G$, so that a wide range of alternate universes are potentially habitable.