F(R) gravity

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 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.^[3] 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.

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 ${\displaystyle \Gamma _{\alpha \beta }^{\mu }}$ 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: $${\displaystyle \delta {\sqrt {-g}}=-{\frac {1}{2}}{\sqrt {-g}}g_{\mu \nu }\delta g^{\mu \nu }}$$

The Ricci scalar is defined as $${\displaystyle R=g^{\mu \nu }R_{\mu \nu }.}$$

Therefore, its variation with respect to the inverse metric ${\displaystyle g^{\mu \nu }}$ is given by $${\displaystyle {\begin{aligned}\delta R&=R_{\mu \nu }\delta g^{\mu \nu }+g^{\mu \nu }\delta R_{\mu \nu }\\&=R_{\mu \nu }\delta g^{\mu \nu }+g^{\mu \nu }\left(\nabla _{\rho }\delta \Gamma _{\nu \mu }^{\rho }-\nabla _{\nu }\delta \Gamma _{\rho \mu }^{\rho }\right)\end{aligned}}}$$

For the second step see the article about the Einstein–Hilbert action. Since ${\displaystyle \delta \Gamma _{\mu \nu }^{\lambda }}$ is the difference of two connections, it should transform as a tensor. Therefore, it can be written as $${\displaystyle \delta \Gamma _{\mu \nu }^{\lambda }={\frac {1}{2}}g^{\lambda a}\left(\nabla _{\mu }\delta g_{a\nu }+\nabla _{\nu }\delta g_{a\mu }-\nabla _{a}\delta g_{\mu \nu }\right).}$$

Substituting into the equation above: $${\displaystyle \delta R=R_{\mu \nu }\delta g^{\mu \nu }+g_{\mu \nu }\Box \delta g^{\mu \nu }-\nabla _{\mu }\nabla _{\nu }\delta g^{\mu \nu }}$$ where ${\displaystyle \nabla _{\mu }}$ is the covariant derivative and ${\displaystyle \square =g^{\mu \nu }\nabla _{\mu }\nabla _{\nu }}$ is the d'Alembert operator.

Denoting ${\displaystyle F(R)={\frac {df}{dR}}}$, the variation in the action reads: $${\displaystyle {\begin{aligned}\delta S[g]&=\int {\frac {1}{2\kappa }}\left(\delta f(R){\sqrt {-g}}+f(R)\delta {\sqrt {-g}}\right)\,\mathrm {d} ^{4}x\\&=\int {\frac {1}{2\kappa }}\left(F(R)\delta R{\sqrt {-g}}-{\frac {1}{2}}{\sqrt {-g}}g_{\mu \nu }\delta g^{\mu \nu }f(R)\right)\,\mathrm {d} ^{4}x\\&=\int {\frac {1}{2\kappa }}{\sqrt {-g}}\left(F(R)(R_{\mu \nu }\delta g^{\mu \nu }+g_{\mu \nu }\Box \delta g^{\mu \nu }-\nabla _{\mu }\nabla _{\nu }\delta g^{\mu \nu })-{\frac {1}{2}}g_{\mu \nu }\delta g^{\mu \nu }f(R)\right)\,\mathrm {d} ^{4}x\end{aligned}}}$$

Doing integration by parts on the second and third terms (and neglected the boundary contributions), we get: $${\displaystyle \delta S[g]=\int {\frac {1}{2\kappa }}{\sqrt {-g}}\delta g^{\mu \nu }\left(F(R)R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }f(R)+[g_{\mu \nu }\Box -\nabla _{\mu }\nabla _{\nu }]F(R)\right)\,\mathrm {d} ^{4}x.}$$

By demanding that the action remains invariant under variations of the metric, ${\displaystyle {\frac {\delta S}{\delta g^{\mu \nu }}}=0}$, one obtains the field equations: $${\displaystyle F(R)R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }f(R)+\left[g_{\mu \nu }\Box -\nabla _{\mu }\nabla _{\nu }\right]F(R)=\kappa T_{\mu \nu },}$$ where ${\displaystyle T_{\mu \nu }}$ is the energy–momentum tensor defined as $${\displaystyle T_{\mu \nu }=-{\frac {2}{\sqrt {-g}}}{\frac {\delta ({\sqrt {-g}}{\mathcal {L}}_{\mathrm {m} })}{\delta g^{\mu \nu }}},}$$ where ${\displaystyle {\mathcal {L}}_{m}}$ is the matter Lagrangian.

Assuming a Robertson–Walker metric with scale factor ${\displaystyle a(t)}$ we can find the generalized Friedmann equations to be (in units where ${\displaystyle \kappa =1}$): $${\displaystyle 3FH^{2}=\rho _{\rm {m}}+\rho _{\rm {rad}}+{\frac {1}{2}}(FR-f)-3H{\dot {F}}}$$ $${\displaystyle -2F{\dot {H}}=\rho _{\rm {m}}+{\frac {4}{3}}\rho _{\rm {rad}}+{\ddot {F}}-H{\dot {F}},}$$ where $${\displaystyle H={\frac {\dot {a}}{a}}}$$ is the Hubble parameter, the dot is the derivative with respect to the cosmic time t, and the terms ρ_m and ρ_rad represent the matter and radiation densities respectively; these satisfy the continuity equations: $${\displaystyle {\dot {\rho }}_{\rm {m}}+3H\rho _{\rm {m}}=0;}$$ $${\displaystyle {\dot {\rho }}_{\rm {rad}}+4H\rho _{\rm {rad}}=0.}$$

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): $${\displaystyle \mathrm {d} s^{2}=-(1+2\Phi )\mathrm {d} t^{2}+\alpha ^{2}(1-2\Psi )\delta _{ij}\mathrm {d} x^{i}\mathrm {d} x^{j}}$$ 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 G_eff. Doing so, we get the gravitational potential (valid on sub-horizon scales k² ≫ a²H²): $${\displaystyle \Phi =-4\pi G_{\mathrm {eff} }{\frac {a^{2}}{k^{2}}}\delta \rho _{\mathrm {m} }}$$ where δρ_m is a perturbation in the matter density, k is the Fourier scale and G_eff is: $${\displaystyle G_{\mathrm {eff} }={\frac {1}{8\pi F}}{\frac {1+4{\frac {k^{2}}{a^{2}R}}m}{1+3{\frac {k^{2}}{a^{2}R}}m}},}$$ with $${\displaystyle m\equiv {\frac {RF_{,R}}{F}}.}$$

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 $${\displaystyle \Phi \to f'(R)\quad {\textrm {and}}\quad {\frac {dV}{d\Phi }}\to {\frac {2f(R)-Rf'(R)}{3}},}$$ and use the field equations above to get $${\displaystyle \Box \Phi ={\frac {\mathrm {d} V}{\mathrm {d} \Phi }}}$$

Working to first order of perturbation theory: $${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+h_{\mu \nu }}$$ $${\displaystyle \Phi =\Phi _{0}+\delta \Phi }$$ 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 z-direction, may be written as $${\displaystyle h_{\mu \nu }(t,z;\omega )=A^{+}(\omega )\exp(-i\omega (t-z))e_{\mu \nu }^{+}+A^{\times }(\omega )\exp(-i\omega (t-z))e_{\mu \nu }^{\times }+h_{f}(v_{\mathrm {g} }t-z;\omega )\eta _{\mu \nu }}$$ where $${\displaystyle h_{f}\equiv {\frac {\delta \Phi }{\Phi _{0}}},}$$ and v_g(ω) = dω/dk 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 ${\displaystyle f(R)=\alpha R^{2}}$ (also known as pure ${\displaystyle R^{2}}$ 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 ${\displaystyle f''(R)\neq 0}$ 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 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 ${\displaystyle f(R)=R+{\frac {R^{2}}{6M^{2}}}}$ is negligible, i.e., one tends to f(R) = R which is general relativity with a zero cosmological constant.

Gogoi–Goswami gravity (named after Dhruba Jyoti Gogoi and Umananda Dev Goswami) 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 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.

• Alternatives to general relativity
• Gauss–Bonnet gravity
• Lovelock gravity