F(R) gravity explained

is a type of modified gravity theory which generalizes Einstein's general relativity. gravity is actually a family of theories, each one defined by a different function,, of the Ricci scalar, . 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. gravity was first proposed in 1970 by Hans Adolph Buchdahl^[1] (although was used rather than 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.

Introduction

In gravity, one seeks to generalize the Lagrangian of the Einstein–Hilbert action:$$S[g]=\; \backslash int\; R\; \backslash sqrt\; \backslash ,\; \backslash mathrm^4x$$to$$S[g]=\; \backslash int\; f(R)\; \backslash sqrt\; \backslash ,\; \backslash mathrm^4x$$where

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

There are two ways to track the effect of changing

R to 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. While the two formalisms lead to the same field equations for General Relativity, i.e., when f(R) = R , the field equations may differ when f(R) ≠ R .

Metric gravity

Derivation of field equations

\Gamma ^{\mu} _{\alpha\beta} 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:$$\backslash delta\; \backslash sqrt=\; -\backslash frac\; \backslash sqrt\; g\_\; \backslash delta\; g^$$

The Ricci scalar is defined as$$R\; =\; g^\; R\_.$$

Therefore, its variation with respect to the inverse metric

g ^{\mu\nu} is given by

For the second step see the article about the Einstein–Hilbert action. Since

\delta\Gamma ^{λ} _{\mu\nu} is the difference of two connections, it should transform as a tensor. Therefore, it can be written as $$\backslash delta\; \backslash Gamma^\backslash lambda\_=\backslash fracg^\backslash left(\backslash nabla\_\backslash mu\backslash delta\; g\_+\backslash nabla\_\backslash nu\backslash delta\; g\_-\backslash nabla\_a\backslash delta\; g\_\; \backslash right).$$

Substituting into the equation above:$$\backslash delta\; R=\; R\_\; \backslash delta\; g^+g\_\backslash Box\; \backslash delta\; g^-\backslash nabla\_\backslash mu\; \backslash nabla\_\backslash nu\; \backslash delta\; g^$$

where

\nabla _{\mu} is the covariant derivative and \square = g ^{\mu\nu} \nabla _{\mu\nabla} _{\nu} is the d'Alembert operator.

Denoting

F(R) = (df)/(d R) , the variation in the action reads:

Doing integration by parts on the second and third terms (and neglected the boundary contributions), we get:$$\backslash delta\; S[g]=\; \backslash int\; \backslash frac\; \backslash sqrt\backslash delta\; g^\; \backslash left(F(R)R\_-\backslash fracg\_\; f(R)+[g\_\{\backslash mu\backslash nu\}\backslash Box\; -\backslash nabla\_\backslash mu\; \backslash nabla\_\backslash nu]F(R)\; \backslash right)\backslash ,\; \backslash mathrm^4x.$$

By demanding that the action remains invariant under variations of the metric,

(\delta S)/(\delta g ^{\mu\nu}) =0 , one obtains the field equations:$$F(R)R\_-\backslash fracg\_f(R)+\backslash left[g\_\{\backslash mu\backslash nu\}\; \backslash Box-\backslash nabla\_\backslash mu\; \backslash nabla\_\backslash nu\; \backslash right]F(R)\; =\; \backslash kappa\; T\_,$$where T _{\mu\nu} is the energy–momentum tensor defined as$$T\_=-\backslash frac\backslash frac,$$where l L _{m} is the matter Lagrangian.

The generalized Friedmann equations

Assuming a Robertson–Walker metric with scale factor

a(t) we can find the generalized Friedmann equations to be (in units where \kappa = 1 ):$$3F\; H^\; =\; \backslash rho\_+\backslash rho\_+\backslash frac(FR-f)-3H$$$$-2F\backslash dot\; =\; \backslash rho\_+\backslash frac\backslash rho\_+\backslash ddot-H\backslash dot,$$where$$H\; =\; \backslash frac$$ is the Hubble parameter, the dot is the derivative with respect to the cosmic time, and the terms _m and _rad represent the matter and radiation densities respectively; these satisfy the continuity equations:$$\backslash dot\_+3H\backslash rho\_=0;$$$$\backslash dot\_+4H\backslash rho\_=0.$$

Modified Newton's constant

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):$$\backslash mathrms^2\; =\; -(1+2\backslash Phi)\backslash mathrmt^2\; +\backslash alpha^2\; (1-2\backslash Psi)\backslash delta\_\backslash mathrmx^i\; \backslash mathrmx^j$$where and 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 _eff. Doing so, we get the gravitational potential (valid on sub-horizon scales):$$\backslash Phi\; =\; -4\backslash pi\; G\_\backslash mathrm\; \backslash frac\; \backslash delta\backslash rho\_\backslash mathrm$$where _m is a perturbation in the matter density, is the Fourier scale and _eff is:$$G\_\backslash mathrm=\backslash frac\backslash frac,$$with$$m\backslash equiv\backslash frac.$$

Massive gravitational waves

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 becomes general relativity plus a scalar field. To see this, identify$$\backslash Phi\; \backslash to\; f\text{'}(R)\; \backslash quad\; \backslash textrm\; \backslash quad\; \backslash frac\backslash to\backslash frac,$$and use the field equations above to get$$\backslash Box\; \backslash Phi=\backslash frac$$

Working to first order of perturbation theory:$$g\_=\backslash eta\_+h\_$$$$\backslash Phi=\backslash Phi\_0+\backslash delta\; \backslash Phi$$and after some tedious algebra, one can solve for the metric perturbation, which corresponds to the gravitational waves. A particular frequency component, for a wave propagating in the -direction, may be written as$$h\_(t,z;\backslash omega)=A^(\backslash omega)\backslash exp(-i\backslash omega(t-z))e^\_+A^(\backslash omega)\backslash exp(-i\backslash omega(t-z))e^\_\; +h\_f(v\_\backslash mathrm\; t-z;\backslash omega)\; \backslash eta\_$$where$$h\_f\backslash equiv\; \backslash frac,$$

and _g = d/d is the group velocity of a wave packet centred on wave-vector . The first two terms correspond to the usual transverse polarizations from general relativity, while the third corresponds to the new massive polarization mode of 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 _G < 1 (in units where  = 1), this mode is dispersive. However, in gravity metric formalism, for the model

f(R) = \alpha R ^{2} (also known as pure R ^{2} model), the third polarization mode is a pure breathing mode and propagate with the speed of light through the spacetime. ^[7]

Equivalent formalism

Under certain additional conditions^[8] we can simplify the analysis of theories by introducing an auxiliary field . Assuming

f''(R) ≠ 0 for all, let be the Legendre transformation of so that \Phi = f'(R) and R = V'(\Phi) . Then, one obtains the O'Hanlon (1972) action:$$S\; =\; \backslash int\; d^4x\; \backslash sqrt\; \backslash left[\backslash frac\{1\}\{2\backslash kappa\}\backslash left(\backslash Phi\; R\; -\; V(\backslash Phi)\backslash right)\; +\; \backslash mathcal\{L\}\_\{\backslash text\{m\}\}\backslash right].$$

We have the Euler–Lagrange equations:$$V\text{'}(\backslash Phi)=R$$$$\backslash Phi\; \backslash left(R\_\; -\; \backslash fracg\_\; R\; \backslash right)\; +\; \backslash left(g\_\backslash Box\; -\backslash nabla\_\backslash mu\; \backslash nabla\_\backslash nu\; \backslash right)\; \backslash Phi\; +\; \backslash frac\; g\_V(\backslash Phi)\; =\; \backslash kappa\; T\_$$

Eliminating, 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:$$\backslash tilde\_=\backslash Phi\; g\_,$$we transform to the Einstein frame:$$R\; =\; \backslash Phi\; \backslash left[\backslash tilde\{R\}\; +\; \backslash frac\{3\backslash tilde\{\backslash Box\}\; \backslash Phi\}\{\backslash Phi\}\; -\backslash frac\{9\}\{2\}\backslash left(\backslash frac\{\backslash tilde\{\backslash nabla\}\; \backslash Phi\}\{\backslash Phi\}\backslash right)^2\; \backslash right]$$$$S\; =\; \backslash int\; d^4x\; \backslash sqrt\backslash frac\backslash left[\backslash tilde\{R\}\; -\; \backslash frac\{3\}\{2\}\backslash left(\backslash frac\{\backslash tilde\{\backslash nabla\}\backslash Phi\}\{\backslash Phi\}\; \backslash right)^2\; -\; \backslash frac\{V(\backslash Phi)\}\{\backslash Phi^2\}\; \backslash right]$$after integrating by parts.

Defining

\tilde{\Phi} = \sqrt{ 3 } ln {\Phi} , and substituting$$S\; =\; \backslash int\; \backslash mathrm^4x\; \backslash sqrt\backslash frac\backslash left[\backslash tilde\{R\}\; -\; \backslash frac\{1\}\{2\}\backslash left(\backslash tilde\{\backslash nabla\}\backslash tilde\{\backslash Phi\}\backslash right)^2\; -\; \backslash tilde\{V\}(\backslash tilde\{\backslash Phi\})\; \backslash right]$$$$\backslash tilde(\backslash tilde)\; =\; e^\; V\; \backslash left\; (e^\; \backslash right).$$

This is general relativity coupled to a real scalar field: using 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) 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.)

Palatini gravity

In Palatini 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 .^{[9] [10]} Due to the structure of the theory, however, Palatini 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]

Metric-affine gravity

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

Observational tests

As there are many potential forms of 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 by Taylor expanding$$f(R)\; =\; a\_0\; +\; a\_1\; R\; +\; a\_2\; R^2\; +\; \backslash cdots$$

The first term is like the cosmological constant and must be small. The next coefficient ₁ can be set to one as in general relativity. For metric gravity (as opposed to Palatini or metric-affine 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 or equivalently ^{[13] [14]}

The parameterized post-Newtonian formalism is designed to be able to constrain generic modified theories of gravity. However, 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 gravity, like General Relativity, is entirely consistent with the bounds from Cassini tracking.

Starobinsky gravity

See main article: Starobinsky inflation. Starobinsky gravity has the following form$$f(R)\; =\; R\; +\; \backslash frac$$where

M has the dimensions of mass.^[16]

Starobinsky gravity provides a mechanism for the cosmic inflation, just after the Big Bang when

R was still large. However, it is not suited to describe the present universe acceleration since at present R is very small.^{[17] [18] [19]} This implies that the quadratic term in f(R) = R + (R ^{2})/(6 M ^{2}) is negligible, i.e., one tends to f(R) = R which is General Relativity with a null cosmological constant.

Gogoi-Goswami gravity

Gogoi-Goswami gravity has the following form$$f(R)\; =\; R\; -\; \backslash frac\; R\_c\; \backslash cot^\; \backslash left(\backslash frac\; \backslash right)\; -\; \backslash beta\; R\_c\; \backslash left[1\; -\; \backslash exp\backslash left(-\; \backslash frac\{R\}\{R\_c\}\; \backslash right)\; \backslash right]$$where

\alpha and \beta are two dimensionless positive constants and R _{c} is a characteristic curvature constant. ^[20]

Tensorial generalization

gravity as presented in the previous sections is a scalar modification of general relativity. More generally, we can have a $$\backslash int\; \backslash mathrm^Dx\; \backslash sqrt\backslash ,\; f(R,\; R^R\_,\; R^R\_)$$coupling involving invariants of the Ricci tensor and the Weyl tensor. Special cases are 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.

See also

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

References

1. Non-linear Lagrangians and cosmological theory. Buchdahl . H. A.. Monthly Notices of the Royal Astronomical Society. 150. 1–8. 1970. 1970MNRAS.150....1B. 10.1093/mnras/150.1.1. free.
2. A new type of isotropic cosmological models without singularity. Starobinsky . A. A.. . 91. 99–102. 1980. 1 . 10.1016/0370-2693(80)90670-X. 1980PhLB...91...99S .
3. https://www.cambridge.org/core/books/dark-energy/EC55E8BF946C34D61B758273D8286618 L. Amendola and S. Tsujikawa (2013) “Dark Energy, Theory and Observations”
4. Tsujikawa . Shinji. Matter density perturbations and effective gravitational constant in modified gravity models of dark energy. Physical Review D. 2007. 76. 2. 023514. 10.1103/PhysRevD.76.023514 . 0705.1032. 2007PhRvD..76b3514T. 119324187.
5. 10.1103/PhysRevD.95.104034 . Polarizations of gravitational waves in f(R) gravity . Phys. Rev. D . 95 . 104034 . 2017 . Liang . Dicong . Gong . Yungui . Hou . Shaoqi . Liu . Yunqi . 10 . 1701.05998 . 2017PhRvD..95j4034L . 119005163 .
6. 10.1140/epjc/s10052-020-08684-3 . A new f(R) gravity model and properties of gravitational waves in it . The European Physical Journal C . 80 . 1101 . 2020 . Gogoi . Dhruba Jyoti . Dev Goswami . Umananda . 12 . 2006.04011 . 2020EPJC...80.1101G . 219530929 .
7. 10.1007/s12648-020-01998-8 . Gravitational Waves in f(R) Gravity Power Law Model . Indian Journal of Physics . 2022 . Gogoi . Dhruba Jyoti . Dev Goswami . Umananda . 96 . 2 . 637 . 1901.11277 . 2022InJPh..96..637G . 231655238 .
8. De Felice. Antonio. Tsujikawa. Shinji. f(R) Theories. Living Reviews in Relativity. 2010. 13. 1. 3. 10.12942/lrr-2010-3. free . 28179828. 5255939. 1002.4928. 2010LRR....13....3D.
9. The conformal frame freedom in theories of gravitation. Flanagan . E. E.. . 21. 3817–3829. 2004. 10.1088/0264-9381/21/15/N02 . 2004CQGra..21.3817F . gr-qc/0403063. 15 . 117619981 .
10. The Gravity Lagrangian According to Solar System Experiments. Olmo . G. J.. . 95. 261102. 2005. 10.1103/PhysRevLett.95.261102 . 2005PhRvL..95z1102O . gr-qc/0505101. 26. 16486333 . 27440524 .
11. How (not) to use the Palatini formulation of scalar-tensor gravity. Iglesias . A. . Kaloper . N. . Padilla . A. . Park . M.. . 76. 104001. 2007. 10.1103/PhysRevD.76.104001 . 2007PhRvD..76j4001I . 0708.1163. 10 .
12. A no-go theorem for polytropic spheres in Palatini f(R) gravity. Barausse . E. . Sotiriou . T. P. . Miller . J. C.. . 25. 062001. 2008. 10.1088/0264-9381/25/6/062001 . 2008CQGra..25f2001B . gr-qc/0703132. 6 . 119370540 .
13. Linearized f(R) gravity: Gravitational radiation and Solar System tests. Berry . C. P. L. . Gair . J. R.. . 83. 104022. 2011. 10.1103/PhysRevD.83.104022. 2011PhRvD..83j4022B . 1104.0819. 10 . 119202399 .
14. Dark Matter from R² Gravity. Cembranos . J. A. R.. . 102. 141301. 2009. 10.1103/PhysRevLett.102.141301. 2009PhRvL.102n1301C . 0809.1653. 14. 19392422 . 33042847 .
15. Parametrized post-Newtonian limit of fourth-order theories of gravity. Clifton . T.. . 77. 024041 . 2008. 10.1103/PhysRevD.77.024041. 2008PhRvD..77b4041C . 0801.0983. 2 . 54174617 .
16. 10.1016/0370-2693(80)90670-X . A new type of isotropic cosmological models without singularity . Physics Letters B . 91 . 99–102 . 1980 . Starobinsky . A.A . 1 . 1980PhLB...91...99S .
17. Web site: Will the Universe expand forever? . . 24 January 2014 . 16 March 2015.
18. Web site: Our universe is Flat. FermiLab/SLAC . 7 April 2015 . Lauren. Biron. symmetrymagazine.org.
19. Unexpected connections. Marcus Y. Yoo. Engineering & Science. LXXIV1. 2011. 30.
20. 10.1140/epjc/s10052-020-08684-3 . A new f(R) gravity model and properties of gravitational waves in it . The European Physical Journal C . 80 . 1101 . 2020 . Gogoi . Dhruba Jyoti . Dev Goswami . Umananda . 12 . 2006.04011 . 2020EPJC...80.1101G . 219530929 .

Further reading

• 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 . . 82 . 1 . 451–497 . 0805.1726 . 2010RvMP...82..451S . 10.1103/RevModPhys.82.451. 15024691 .
• Sotiriou . T. P. . 2009 . 6+1 lessons from f(R) gravity . . 189 . 9 . 012039 . 0810.5594 . 2009JPhCS.189a2039S . 10.1088/1742-6596/189/1/012039. 14820388 .
• Capozziello . S. . De Laurentis . M. . 2011 . Extended Theories of Gravity . . 509 . 45 . 167–321 . 1108.6266 . 2011PhR...509..167C . 10.1016/j.physrep.2011.09.003. 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

External links

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