This paper derives an upper limit on the density \rho _ { \scriptscriptstyle \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 _ { 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 _ { \scriptscriptstyle \Lambda } / M _ { pl } ^ { 4 } < 10 ^ { -90 } , which is \sim 30 orders of magnitude larger than the value in our universe \rho _ { \scriptscriptstyle \Lambda } / M _ { pl } ^ { 4 } \sim 10 ^ { -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 _ { 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 .