What is the Force Formulation of Dark Energy?

[particlemania]

Well what is the apparent force applied on two bodies of mass m1 and m2 due to Cosmological Constant?

First of all, what is the meaning of a constantly accelerating expansion of universe? Is the second time derivative of volume constant or that of the diametric size of universe?

Well in a lecture by Leonard Susskind, I remember, he said that the force is of a form similar to elastic spring force, just repulsive in nature. That is, for a universal constant k, the force is kr, in repulsive sense.

But in History Channel Documentary featuring Michio Kaku and Sean Caroll, it said that Expansion preserves proportion, and a mass-independent force will mean a mass-dependent acceleration, which wouldn't preserve proportions.

So what is the force formulation of dark energy?

 

[bapowell]

The effect of dark energy on the expansion of space is a result of gravitational phenomena. This behavior is described by Einstein's Equations. What's important to understand is that objects in the universe are not undergoing acceleration relative to their local frames of reference, the way, say, a car accelerates down a roadway. Rather, objects in the universe (like galaxies and clusters of galaxies) are at rest relative to their surroundings -- it is the space itself that is expanding, and they are carried along with it. What we mean by accelerated expansion is this:

$$\ddot{a} > 0$$

where a(t) is the scale factor of the universe. This function determines the distance between points in spacetime. As it grows larger, points on the spacetime manifold grow further apart. The physical manifestation of this is an expanding universe.