f(R) is a type of modified gravity theory which generalizes Einstein's general relativity. f(R) gravity is actually a family of theories, each one defined by a different function, f, of the Ricci scalar, R. The simplest case is just the function being equal to the scalar; this is general relativity. 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 Starobinsky on cosmic inflation.^[2] A wide range of phenomena can be produced from this theory by adopting different functions; however, many functional forms can now be ruled out on observational grounds, or because of pathological theoretical problems.

This article may be too technical for most readers to understand. (July 2019)

In f(R) gravity, one seeks to generalize the Lagrangian of the Einstein–Hilbert action:

$${\displaystyle S[g]=\int {1 \over 2\kappa }R{\sqrt {-g}}\,\mathrm {d} ^{4}x}$$

to

$${\displaystyle S[g]=\int {1 \over 2\kappa }f(R){\sqrt {-g}}\,\mathrm {d} ^{4}x}$$

where ${\displaystyle \kappa ={\tfrac {8\pi G}{c^{4}}},g=\det g_{\mu \nu }}$ is the determinant of the metric tensor, and ${\displaystyle f(R)}$ is some function of the Ricci scalar.^[3]

There are two ways to track the effect of changing ${\displaystyle R}$ to ${\displaystyle f(R)}$, i.e., to obtain the theory field equations. The first is to use metric formalism and the second is to use the Palatini formalism.^[3] While the two formalisms lead to the same field equations for General Relativity, i.e., when ${\displaystyle f(R)=R}$, the field equations may differ when ${\displaystyle f(R)\neq R}$.

Under certain additional conditions^[8] we can simplify the analysis of f(R) theories by introducing an auxiliary field Φ. Assuming ${\displaystyle f''(R)\neq 0}$ for all R, let V(Φ) be the Legendre transformation of f(R) so that ${\displaystyle \Phi =f'(R)}$ and ${\displaystyle R=V'(\Phi )}$. Then, one obtains the O'Hanlon (1972) action:

$${\displaystyle S=\int d^{4}x{\sqrt {-g}}\left[{\frac {1}{2\kappa }}\left(\Phi R-V(\Phi )\right)+{\mathcal {L}}_{\text{m}}\right].}$$

We have the Euler–Lagrange equations

$${\displaystyle V'(\Phi )=R}$$

$${\displaystyle \Phi \left(R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }R\right)+\left(g_{\mu \nu }\Box -\nabla _{\mu }\nabla _{\nu }\right)\Phi +{\frac {1}{2}}g_{\mu \nu }V(\Phi )=\kappa T_{\mu \nu }}$$

Eliminating Φ, we obtain exactly the same equations as before. However, the equations are only second order in the derivatives, instead of fourth order.

We are currently working with the Jordan frame. By performing a conformal rescaling

$${\displaystyle {\tilde {g}}_{\mu \nu }=\Phi g_{\mu \nu },}$$

we transform to the Einstein frame:

$${\displaystyle R=\Phi \left[{\tilde {R}}+{\frac {3{\tilde {\Box }}\Phi }{\Phi }}-{\frac {9}{2}}\left({\frac {{\tilde {\nabla }}\Phi }{\Phi }}\right)^{2}\right]}$$

$${\displaystyle S=\int d^{4}x{\sqrt {-{\tilde {g}}}}{\frac {1}{2\kappa }}\left[{\tilde {R}}-{\frac {3}{2}}\left({\frac {{\tilde {\nabla }}\Phi }{\Phi }}\right)^{2}-{\frac {V(\Phi )}{\Phi ^{2}}}\right]}$$

after integrating by parts.

Defining ${\displaystyle {\tilde {\Phi }}={\sqrt {3}}\ln {\Phi }}$, and substituting

$${\displaystyle S=\int \mathrm {d} ^{4}x{\sqrt {-{\tilde {g}}}}{\frac {1}{2\kappa }}\left[{\tilde {R}}-{\frac {1}{2}}\left({\tilde {\nabla }}{\tilde {\Phi }}\right)^{2}-{\tilde {V}}({\tilde {\Phi }})\right]}$$

$${\displaystyle {\tilde {V}}({\tilde {\Phi }})=e^{-{\frac {2}{\sqrt {3}}}{\tilde {\Phi }}}V\left(e^{{\tilde {\Phi }}/{\sqrt {3}}}\right).}$$

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. The matter Lagrangian is assumed to be independent of the connection. These theories have been shown to be equivalent to Brans–Dicke theory with ω = −3⁄2.^[9][10] Due to the structure of the theory, however, Palatini f(R) theories appear to be in conflict with the Standard Model,^[9][11] may violate Solar system experiments,^[10] and seem to create unwanted singularities.^[12]

In metric-affine f(R) gravity, one generalizes things even further, treating both the metric and connection independently, and assuming the matter Lagrangian depends on the connection as well.

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

$${\displaystyle f(R)=a_{0}+a_{1}R+a_{2}R^{2}+\cdots }$$

The first term is like the cosmological constant and must be small. The next coefficient a₁ can be set to one as in general relativity. For metric f(R) gravity (as opposed to Palatini or metric-affine f(R) gravity), the quadratic term is best constrained by fifth force measurements, since it leads to a Yukawa correction to the gravitational potential. The best current bounds are |a₂| < 4×10⁻⁹ m² or equivalently |a₂| < 2.3×10²² GeV⁻².^[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.^[13]

Starobinsky gravity has the following form

$${\displaystyle f(R)=R+{\frac {R^{2}}{6M^{2}}}}$$

where ${\displaystyle M}$ has the dimensions of mass.^[16]

Starobinsky gravity provides a mechanism for the cosmic inflation, just after the Big Bang when ${\displaystyle R}$ was still large. However, it is not suited to describe the present universe acceleration since at present ${\displaystyle R}$ is very small.^[17][18][19] This implies that the quadratic term in ${\displaystyle f(R)=R+{\frac {R^{2}}{6M^{2}}}}$ is negligible, i.e., one tends to ${\displaystyle f(R)=R}$ which is General Relativity with a null cosmological constant.

Gogoi-Goswami gravity has the following form

$${\displaystyle f(R)=R-{\frac {\alpha }{\pi }}R_{c}\cot ^{-1}\left({\frac {R_{c}^{2}}{R^{2}}}\right)-\beta R_{c}\left[1-\exp \left(-{\frac {R}{R_{c}}}\right)\right]}$$

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are two dimensionless positive constants and ${\displaystyle R_{c}}$ is a characteristic curvature constant. ^[20]

f(R) gravity as presented in the previous sections is a scalar modification of general relativity. More generally, we can have a

$${\displaystyle \int \mathrm {d} ^{D}x{\sqrt {-g}}\,f(R,R^{\mu \nu }R_{\mu \nu },R^{\mu \nu \rho \sigma }R_{\mu \nu \rho \sigma })}$$

coupling involving invariants of the Ricci tensor and the Weyl tensor. Special cases are f(R) gravity, conformal gravity, Gauss–Bonnet gravity and Lovelock gravity. Notice that with any nontrivial tensorial dependence, we typically have additional massive spin-2 degrees of freedom, in addition to the massless graviton and a massive scalar. An exception is Gauss–Bonnet gravity where the fourth order terms for the spin-2 components cancel out.

• Extended theories of gravity
• Gauss–Bonnet gravity
• Lovelock gravity