Undergrad Gravitational Field with Dark Energy: Observable Effects?

[kodama]

TL;DR

how to calculate gravitational field with dark energy

since the cosmological constant observed is that there is a small amount of energy in empty space, and in general relativity anytime there is energy there is curvature and therefore gravity, how to calculate gravitational field with dark energy and does it have any observable effects on matter?

the cc is on the same order as mond ao so I wonder if somehow ao is the result of the gravity of dark energy acting on matter

 

[Ibix]

how to calculate gravitational field with dark energy and does it have any observable effects on matter?

You can easily add a cosmological constant term to the Schwarzschild solution - the result is Schwarzschild-de Sitter spacetime. However, in reality the term is so tiny that you could never hope to detect effects on the solar system scale.

the cc is on the same order as mond

Are you mixing up dark matter and dark energy? Dark matter and MOND are competing theories. Dark energy is only relevant on much larger scales where MOND doesn't work at all.