f(R) gravity

f(R) gravity is a type of modified gravity theory first proposed in 1970 by Buchdahl as a generalisation of Einstein's General Relativity. Although it is an active field of research, there are known problems with the theory. It has the potential, in principle, to explain the accelerated expansion of the Universe without adding unknown forms of dark energy or dark matter.

Introduction

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

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

to

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

where $\kappa\equiv 8\pi G$, $g\,$ is the determinant of the metric tensor $g\equiv |g_{\mu\nu}|$ and $f(R)\,$ is some function of the Ricci Curvature.

Metric f(R) Gravity

Derivation of field equations

In metric f(R) gravity, one arrives at the field equations by varying 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:

$\delta \sqrt{-g}= -\frac{1}{2} \sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu}$

The Ricci scalar is defined as

$R = g^{\mu\nu} R_{\mu\nu}.\!$

Therefore, its variation with respect to the inverse metric $g^{\mu\nu}\,$ is given by

$\begin{align} \delta R &= R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu} \delta R_{\mu\nu}\\ &= R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu}(\nabla_\rho \delta \Gamma^\rho_{\nu\mu} - \nabla_\nu \delta \Gamma^\rho_{\rho\mu}) \end{align}$

For the second step see the article about the Einstein-Hilbert action. Now, since $\delta \Gamma^\lambda_{\mu\nu}\,$ is actually the difference of two connections, it should transform as a tensor. Therefore, it can be written as

$\delta \Gamma^\lambda_{\mu\nu}=\frac{1}{2}g^{\lambda a}\left(\nabla_\mu\delta g_{a\nu}+\nabla_\nu\delta g_{a\mu}-\nabla_a\delta g_{\mu\nu} \right)$

and substituting in the equation above one finds:

$\delta R= R_{\mu\nu} \delta g^{\mu\nu}+g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}$

where $\nabla_\mu$ is the covariant derivative and $\Box$ is the D'Alembert operator defined as $\Box=g^{\mu\nu}\nabla_\mu \nabla_\nu$.

Now the variation in the action reads:

$\begin{align} \delta S[g]&= \int {1 \over 2\kappa} \left(\delta f(R) \sqrt{-g}+f(R) \delta \sqrt{-g} \right)\, \mathrm{d}^4x \\ &= \int {1 \over 2\kappa} \left(F(R) \delta R \sqrt{-g}-\frac{1}{2} \sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu} f(R)\right) \, \mathrm{d}^4x \\ &= \int {1 \over 2\kappa} \sqrt{-g}\left(F(R)(R_{\mu\nu} \delta g^{\mu\nu}+g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) -\frac{1}{2} g_{\mu\nu} \delta g^{\mu\nu} f(R) \right)\, \mathrm{d}^4x \end{align}$

where $F(R)=\frac{\partial f(R)}{\partial R}$. Doing integration by parts on the second and third terms we get:

$\begin{align} \delta S[g]&= \int {1 \over 2\kappa} \sqrt{-g}\delta g^{\mu\nu} \left(F(R)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} f(R)+[g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu]F(R) \right)\, \mathrm{d}^4x \end{align}$

By demanding that the action remains invariant under variations of the metric, ie $\delta S[g]=0\,$, one obtains the field equations:

$F(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}+\left[g_{\mu\nu} \Box-\nabla_\mu \nabla_\nu \right]F(R) = \kappa T_{\mu\nu}$

where $T_{\mu\nu}\,$ is the energy-momentum tensor defined as

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g} L_m)}{\delta g^{\mu\nu}}$

where $L_m \,$ 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 $\kappa\equiv 8 \pi G=1$):

$3F H^{2} = \rho_{{\rm m}}+\rho_{{\rm rad}}+\frac{1}{2}(FR-f)-3H{\dot F}$

$-2F\dot{H} = \rho_{{\rm m}}+\frac{4}{3}\rho_{{\rm rad}}+\ddot{F}-H\dot{F}$

where $H=\frac{\dot{a}}{a}$, the dot is the derivative with respect to the cosmic time t and the terms ρ_m, ρ_rad represent the matter and radiation densities respectively and satisfy the continuity equations:

$\dot{\rho}_{{\rm m}}+3H\rho_{{\rm m}}=0$

$\dot{\rho}_{{\rm rad}}+4H\rho_{{\rm rad}}=0$

Modified Newton's constant

An interesting feature of these theories is the fact that the gravitational constant is time and scale dependent. To see this, add a small scalar perturbation to the metric (in the conformal Newtonian gauge):

$ds^2 = -(1+2\Phi)dt^2 +\alpha^2 (1-2\Psi)\delta_{ij}dx^i dx^j \,$

where Φ,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 G_eff. Doing so, we get the gravitational potential (valid in sub-horizon scales $k^2\gg a^2 H^2$):

$\Phi = -4 \pi G_{eff} \frac{a^2}{k^2} \delta\rho _m$ where δρ_m is a perturbation in the matter density and G_eff is:

$G_{eff}=\frac{1}{8\pi F}\frac{1+4\frac{k^2}{a^2R}m}{1+3\frac{k^2}{a^2R}m}$

and

$m\equiv\frac{RF_{,R}}{F}$

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 $f(R)\,$ becomes General Relativity plus a scalar field. To see this, identify

$\Phi \rightarrow f'(R)~~~~~\textrm{and}~~~~ \frac{dV}{d\Phi}\rightarrow\frac{2f(R)-R f'(R)}{3}$

and use the field equations mentioned above to get

$\Box \Phi=\frac{dV}{d\Phi}$

Working to first order of perturbation theory:

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\,$

$\Phi=\Phi_0+\delta \Phi \,$

and after some tedious algebra, one can solve for the metric perturbation, which corresponds to the gravitational waves:

$h_{\mu\nu}(t,z)=A^{+}(t-z)e^{+}_{\mu\nu}+A^{\times}(t-z)e^{\times}_{\mu\nu} +h_f(t-\upsilon_G z) \eta_{\mu\nu}$

where

$h_f\equiv \frac{\delta \Phi}{\Phi_0}$

and $\upsilon_G \,$ is the group velocity of a wave packet $h_f\,$ centered in momentum $p \,$ as $\upsilon_G =\frac{p}{\omega}$. The first two terms correspond to the usual from General Relativity gravitational waves while the third corresponds to the new massive polarization mode of $f(R)\,$ theories, which moves at a speed $\upsilon_G <1 \,$ (in units where $c \equiv 1 \,$.

Equivalent formalism

We can simplify the analysis of f(R) theories by introducing an auxiliary field Φ. Assuming $f''(R) \neq 0$ for all R, let V(Φ) be the Legendre transform of f(R) so that Φ = f'(R) and R = V'(Φ). Then,

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

We get the Euler-Lagrange equations

V'(Φ) = R

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

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

We are currently working with the Jordan frame. By performing a conformal rescaling $\tilde{g}_{\mu\nu}=\Phi g_{\mu\nu}$, we get to the Einstein frame.

$R=\Phi^{-1} \left[ \tilde{R} + \frac{3\tilde{\Box} \Phi}{\Phi} -\frac{9}{2}\left(\frac{\tilde{\nabla} \Phi}{\Phi}\right)^2 \right]$

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

after integrating by parts.

Define $\tilde{\Phi} = \sqrt{3} \ln{\Phi}$.

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

$\tilde{V}(\tilde{\Phi}) = e^{-2/\sqrt{3}\;\tilde{\Phi}}V(e^{\tilde{\Phi}/\sqrt{3}})$

This is none other an ordinary general relativity coupled to a real scalar field! Trying to use f(R) theories to describe the accelerating universe is practically equivalent to using quintessence.

Palatini f(R) Gravity

In Palatini f(R) gravity, one treats the metric and connection independently and varies the action with respect to each of them separately. Also important, one assumes the matter Lagrangian does not depend on the connection. These theories have been shown to be equivalent to Brans–Dicke theory with ω = − 3 / 2. Due to the structure of the theory, however, Palatini f(R) theories appear to be in conflict with the Standard Model, may violate Solar system experiments, and seem to create unwanted singularities.

Metric-Affine f(R) Gravity

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.

Tensorial generalization

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

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

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. It is suggested to consider dependency to the covariant derivative of the Riemann tensor in order to resolve more problems. 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

• Gauss-Bonnet gravity
• Lovelock gravity

Further reading

• A.A. Starobinsky, Phys. Lett. B, 91, 99 (1980), doi:10.1016/0370-2693(80)90670-X
• S.M. Carroll, V. Duvvuri, M. Trodden, and M.S. Turner, Phys. Rev. D 70, 043528 (2004), arXiv:astro-ph/0306438, June 2003: Is Cosmic Speed-Up Due to New Gravitational Physics?
• N.J. Poplawski, Phys. Rev. D 74, 084032 (2006), arXiv:gr-qc/0607124, Jul 2007: Interacting dark energy in f(R) gravity
• S. Capozziello, V.F. Cardone, A. Troisi, Feb 2006. 5pp. JCAP 0608:001,2006. [e-Print: astro-ph/0602349 Dark energy and dark matter as curvature effects.]
• S. Tsujikawa, arXiv:0705.1032, May 2007: Matter density perturbations and effective gravitational constant in modified gravity models of dark energy
• V. Faraoni, arXiv:0810.2602, Oct. 2008: f(R) gravity: successes and challenges
• E.E. Flanagan, arXiv:astro-ph/0308111, Aug. 2003: Palatini form of 1/R gravity
• S. Capozziello, Mauro Francaviglia, Gen.Rel.Grav.40:357 (2008) [ e-Print: arXiv:0706.1146 [astro-ph]Extended Theories of Gravity and their Cosmological and Astrophysical Applications]
• A. De Felice and S. Tsujikawa, "f(R) Theories", Living Rev. Relativity 13 (2010) 3, http://www.livingreviews.org/lrr-2010-3
• T.P. Sotiriou and V. Faraoni, arXiv:0805.1726, May 2008: f(R) theories of gravity
• T.P. Sotiriou, arXiv:0810.5594, Oct. 2008: 6+1 lessons from f(R) gravity
• S. Capozziello et al. ,Phys.Lett.B 669, 255 (2008) arXiv:0812.2272, Dec 2008: Massive gravitational waves from f(R) theories of gravity: Potential detection with LISA

External links

• F(R) gravity on arxiv.org