What is dark energy in the fabric of space-time?

[Daine]

I know that according to Einstein's theory of relativity, space-time is like a fabric which can be pliable. Gravity is the shape, or the warping of that fabric. In this analogy, what would dark energy (the unknown form of energy that is causing the universe to expand) be?

 

[mfb]

space-time is like a fabric which can be pliable

That is an attempt to explain the theory to laymen.

Gravity is the shape, or the warping of that fabric.

And that is taking the analogy too far already.

 

[Mordred]

I know that according to Einstein's theory of relativity, space-time is like a fabric which can be pliable. Gravity is the shape, or the warping of that fabric. In this analogy, what would dark energy (the unknown form of energy that is causing the universe to expand) be?

As mentioned these desciptives are heuristic attempts to describe spacetime.

First let's start off with the proper definitions, space is the volume under geometry. That volume requires 3 dimensions to describe. x,y,z which are your spatial dimensions.

Spacetime is any coordinate system that describes time via its interval between measurement points under a coordinate basis. We describe the interval via a new coordinate ct where c is not the speed of light but a constant of proportionality. Giving us (ct,x,z) this gives us a relation to the length element with which we can now graph into a spacetime diagram.

Now that we have our coordinate basis, we can assign a value or function at each of those coordinates. The value assigned being freefall motion. ( constant velocity ie no change in direction or speed).

Now under Euclidean flat geometry with time for 4d, with time given the dimension of length. Galilean relativity applies. This is our everyday experience (Newtonian physics) with no time variation.
Our geometry is preserved and will follow all vector summation rules. Pythagorous theorem of lengths and angles are preserved.
However due to how signals propogate between separation points at higher potentials and as the velocity approaches c.
This no longer holds accurate, we start to deviate in the signal intervals. This is defined by the simultaneous changes on two of our coordinate axis. ct and x. To which we can assign a new constant of proportionality to restore our Euclidean geometry. That being $$\gamma$$
Galilean relativity transforms,
$$\acute{x}={x}-vt$$
$$\acute{y}=y$$
$$\acute{z}=z$$

Lorentz Transforms

$$\acute{x}=\gamma x-vt$$
$$\acute{y}=y$$
$$\acute{z}=z$$
$$\acute{t}=\gamma (t-\sqrt{\frac{vx}{c^2}})$$

lets make sure you understand this first before going into the $$ds^2$$ separation distances between two points under polar coordinates. The above is Cartesian, and you have a learning curve to understand the cosmological constant which can be shortened by reading the following.
http://cosmology101.wikidot.com/universe-geometry
page 2 which has the geometric relations of interest with regards to cosmology applications of the above.
http://cosmology101.wikidot.com/geometry-flrw-metric/