As a consequence of introducing an arbitrary function, there may be freedom to explain the accelerated expansion and structure formation of the Universe without adding unknown forms of dark energy or dark matter.
Some functional forms may be inspired by corrections arising from a quantum theory of gravity.
f(R) gravity was first proposed in 1970 by Hans Adolph Buchdahl[1] (although ϕ was used rather than f for the name of the arbitrary function).
It has become an active field of research following work by Alexei Starobinsky on cosmic inflation.
[3] There are two ways to track the effect of changing R to f(R), i.e., to obtain the theory field equations.
For completeness we will now briefly mention the basic steps of the variation of the action.
The main steps are the same as in the case of the variation of the Einstein–Hilbert action (see the article for more details) but there are also some important differences.
is the Hubble parameter, the dot is the derivative with respect to the cosmic time t, and the terms ρm and ρrad represent the matter and radiation densities respectively; these satisfy the continuity equations:
An interesting feature of these theories is the fact that the gravitational constant is time and scale dependent.
[4] To see this, add a small scalar perturbation to the metric (in the Newtonian gauge):
where Φ and Ψ are the Newtonian potentials and use the field equations to first order.
After some lengthy calculations, one can define a Poisson equation in the Fourier space and attribute the extra terms that appear on the right-hand side to an effective gravitational constant Geff.
where δρm is a perturbation in the matter density, k is the Fourier scale and Geff is:
This class of theories when linearized exhibits three polarization modes for the gravitational waves, of which two correspond to the massless graviton (helicities ±2) and the third (scalar) is coming from the fact that if we take into account a conformal transformation, the fourth order theory f(R) becomes general relativity plus a scalar field.
and after some tedious algebra, one can solve for the metric perturbation, which corresponds to the gravitational waves.
and vg(ω) = dω/dk is the group velocity of a wave packet hf centred on wave-vector k. The first two terms correspond to the usual transverse polarizations from general relativity, while the third corresponds to the new massive polarization mode of f(R) theories.
[7] Under certain additional conditions[8] we can simplify the analysis of f(R) theories by introducing an auxiliary field Φ.
This is general relativity coupled to a real scalar field: using f(R) theories to describe the accelerating universe is practically equivalent to using quintessence.
(At least, equivalent up to the caveat that we have not yet specified matter couplings, so (for example) f(R) gravity in which matter is minimally coupled to the metric (i.e., in Jordan frame) is equivalent to a quintessence theory in which the scalar field mediates a fifth force with gravitational strength.)
In Palatini f(R) gravity, one treats the metric and connection independently and varies the action with respect to each of them separately.
As there are many potential forms of f(R) gravity, it is difficult to find generic tests.
Additionally, since deviations away from General Relativity can be made arbitrarily small in some cases, it is impossible to conclusively exclude some modifications.
Some progress can be made, without assuming a concrete form for the function f(R) by Taylor expanding
[13][14] The parameterized post-Newtonian formalism is designed to be able to constrain generic modified theories of gravity.
However, f(R) gravity shares many of the same values as General Relativity, and is therefore indistinguishable using these tests.
[15] In particular light deflection is unchanged, so f(R) gravity, like General Relativity, is entirely consistent with the bounds from Cassini tracking.
[16] Starobinsky gravity provides a mechanism for the cosmic inflation, just after the Big Bang when R was still large.
Gogoi–Goswami gravity (named after Dhruba Jyoti Gogoi and Umananda Dev Goswami) has the following form
[20] f(R) gravity as presented in the previous sections is a scalar modification of general relativity.
An exception is Gauss–Bonnet gravity where the fourth order terms for the spin-2 components cancel out.