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** f(R)** is a type of modified gravity theory which generalizes Einstein's general relativity.

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

There are two ways to track the effect of changing to , 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 , the field equations may differ when .

In metric `f`(`R`) gravity, one arrives at the field equations by varying the action with respect to the metric and not treating the connection independently. 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.

The variation of the determinant is as always:

The Ricci scalar is defined as

Therefore, its variation with respect to the inverse metric is given by

For the second step see the article about the Einstein–Hilbert action. Since is the difference of two connections, it should transform as a tensor. Therefore, it can be written as

Substituting into the equation above:

where is the covariant derivative and is the d'Alembert operator.

Denoting , the variation in the action reads:

Doing integration by parts on the second and third terms (and neglected the boundary contributions), we get:

By demanding that the action remains invariant under variations of the metric, , one obtains the field equations:

Assuming a Robertson–Walker metric with scale factor we can find the generalized Friedmann equations to be (in units where ):

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):

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. To see this, identify

Working to first order of perturbation theory:

and `v`_{g}(`ω`) = d`ω`/d`k` is the group velocity of a wave packet `h`_{f} 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. This mode is a mixture of massless transverse breathing mode (but not traceless) and massive longitudinal scalar mode. ^{[5]} ^{[6]} The transverse and traceless modes (also known as tensor modes) propagate at the speed of light, but the massive scalar mode moves at a speed `v`_{G} < 1 (in units where `c` = 1), this mode is dispersive. However, in `f`(`R`) gravity metric formalism, for the model (also known as pure model), the third polarization mode is a pure breathing mode and propagate with the speed of light through the spacetime. ^{[7]}

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

We have the Euler–Lagrange equations

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

Defining , and substituting

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

The first term is like the cosmological constant and must be small. The next coefficient `a`_{1} 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 |`a`_{2}| < 4×10^{−9} m^{2} or equivalently |`a`_{2}| < 2.3×10^{22} 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

Starobinsky gravity provides a mechanism for the cosmic inflation, just after the big bang when was still large. However, it is not suited to describe the present universe acceleration since at present is very small.^{[17]}^{[18]}^{[19]} This implies that the quadratic term in is negligible, i.e., one tends to which is General Relativity with a null cosmological constant.

Gogoi-Goswami gravity has the following form

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

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- See Chapter 29 in the textbook on "Particles and Quantum Fields" by Kleinert, H. (2016), World Scientific (Singapore, 2016) (also available online)
- Sotiriou, T. P.; Faraoni, V. (2010). "f(R) Theories of Gravity".
*Reviews of Modern Physics*.**82**(1): 451–497. arXiv:0805.1726. Bibcode:2010RvMP...82..451S. doi:10.1103/RevModPhys.82.451. S2CID 15024691. - Sotiriou, T. P. (2009). "6+1 lessons from f(R) gravity".
*Journal of Physics: Conference Series*.**189**(9): 012039. arXiv:0810.5594. Bibcode:2009JPhCS.189a2039S. doi:10.1088/1742-6596/189/1/012039. S2CID 14820388. - Capozziello, S.; De Laurentis, M. (2011). "Extended Theories of Gravity".
*Physics Reports*.**509**(4–5): 167–321. arXiv:1108.6266. Bibcode:2011PhR...509..167C. doi:10.1016/j.physrep.2011.09.003. S2CID 119296243. - Salvatore Capozziello and Mariafelicia De Laurentis, (2015) "F(R) theories of gravitation". Scholarpedia, doi:10.4249/scholarpedia.31422
- Kalvakota, Vaibhav R., (2021) "Investigating f(R)" gravity and cosmologies". Mathematical physics preprint archive, https://web.ma.utexas.edu/mp_arc/c/21/21-38.pdf

*f*(*R*) gravity on arxiv.org- Extended Theories of Gravity