F(R) gravity

Derivation of field equations

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.

Generalized Friedmann equations

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: $${\displaystyle {\dot {\rho }}_{\rm {m}}+3H\rho _{\rm {m}}=0;}$$ $${\displaystyle {\dot {\rho }}_{\rm {rad}}+4H\rho _{\rm {rad}}=0.}$$

Modified gravitational 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): $${\displaystyle \mathrm {d} s^{2}=-(1+2\Phi )\mathrm {d} t^{2}+\alpha ^{2}(1-2\Psi )\delta _{ij}\mathrm {d} x^{i}\mathrm {d} x^{j}}$$ 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 $${\displaystyle m\equiv {\frac {RF_{,R}}{F}}.}$$

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 $${\displaystyle \Phi \to f'(R)\quad {\textrm {and}}\quad {\frac {dV}{d\Phi }}\to {\frac {2f(R)-Rf'(R)}{3}},}$$ 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]