One possible way to explain the observed acceleration in the expansion of the universe is to change Einstein's theory of General Relativity. The simplest modification is to introduce a cosmological constant, which can be explained by energy that exists in the vacuum. In f(R) gravity and other modified gravity models, scientists go beyond this simple modification. In the f(R) gravity model, spacetime reacts differently to the matter in the universe than it does in General Relativity.
In General Relativity, gravity is a manifestation of the curvature of space and time, where the source of this curvature are all of the forms of mass and energy in the universe. In the absence of any mass or energy spacetime can become completely flat. What f(R) gravity does is allow spacetime to act as a source of its own curvature, so there can still be some curvature even if spacetime is completely empty and the energy is zero. So, as the universe expands and empties out, some curvature remains, resulting in cosmic acceleration.
By making this modification to gravity an additional ("5th") force is introduced. By comparing observations of the masses of galaxy clusters with the predictions of f(R) gravity, the range of this 5th force can be estimated. On distance scales smaller than this range, gravity is stronger than predicted by Einstein's equations. The smaller that this range is, the less effect that this modification to gravity has on the growth of galaxy clusters.
If the cosmological constant is the explanation for cosmic acceleration then the acceleration will continue forever and all galaxies outside the Local Group should eventually disappear from view, resulting in a lonely universe. If f(R) gravity applies then the 5th force will die away in the far future, cosmic expansion will slowly decelerate and a lonely universe will be avoided. It will be many billions of years before either one of these future scenarios can play out.
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