f(R,G) Gravity: Extended Curvature Theories

• f(R,G) gravity is an extended theory that replaces the Ricci scalar in GR with a general function of curvature invariants R and G, enabling richer geometric and dynamic features.
• The weak-field analysis reveals two independent gravitational potentials with Yukawa-like corrections, highlighting differences from standard GR and f(R) gravity.
• These models offer practical scenarios for double inflation, wormhole solutions, and compact star configurations while satisfying effective energy conditions.

$f(R,G)$ gravity is an extended theory of gravity in which the Einstein–Hilbert Lagrangian $R$ of General Relativity is replaced by a general function of two curvature invariants, the Ricci scalar $R$ and the Gauss–Bonnet invariant $G$. In metric formalism, the action is written as

$$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$

with

$$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$

This framework contains GR, $f(R)$ gravity, $R+f(G)$ models, and pure $f(G)$ gravity as special cases, and is used to study higher-curvature quantum and string-inspired corrections, early-time inflation, late-time acceleration, and strong- and weak-field deviations from GR (Laurentis et al., 2013, Laurentis et al., 2015).

1. Action, geometric content, and field equations

The defining feature of $f(R,G)$ gravity is the promotion of the gravitational Lagrangian from $R$0 to an arbitrary function $R$1. The special cases stated in the literature are

$R$2

In four dimensions, $R$3 is a topological invariant and does not contribute to the equations of motion if the coefficient is constant, but a non-linear function $R$4 or a dynamical coupling does contribute (Shamir et al., 2017).

Varying the action with respect to the metric yields fourth-order field equations. Using

$R$5

the field equations contain the standard $R$6, $R$7, and $R$8 terms familiar from $R$9 gravity, together with curvature-squared and curvature-derivative couplings multiplying $R$0. The trace equation likewise contains both $R$1 and $R$2 sectors. Because $R$3 and $R$4 contain second derivatives of the metric, and because the equations involve derivatives of $R$5 and $R$6, the theory is generically fourth-order in the metric (Laurentis et al., 2013).

A recurrent interpretation is the effective-fluid rewriting

$R$7

in which the modified-curvature terms are shifted to the right-hand side and treated as an effective curvature fluid. This rewriting is used extensively in cosmology, wormhole physics, and compact-star modelling (Shamir et al., 2017, Shamir et al., 2017).

2. Weak-field regime, Newtonian limit, and post-Newtonian structure

A central structural result is that the weak-field expansion around Minkowski space,

$R$8

separates the Ricci and Gauss–Bonnet sectors by order. With

$R$9

the Ricci scalar starts at $G$0,

$G$1

whereas the Gauss–Bonnet invariant starts at $G$2. Consequently, the Newtonian limit of analytic $G$3 gravity depends only on $G$4 and $G$5; the $G$6-sector does not modify Newtonian gravity directly (Laurentis et al., 2013).

The Newtonian-order equations reduce to a coupled system for two independent gravitational potentials $G$7 and $G$8,

$G$9

and admit a general solution in terms of Green’s functions. Physically relevant choices give Yukawa-like corrections to the Newtonian potential with characteristic length determined by $$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$0. The key point is that there are two independent gravitational potentials $$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$1 and $$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$2, already richer than GR’s single potential (Laurentis et al., 2013).

For $$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$3 gravity, the Newtonian limit is exactly GR because $$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$4 is $$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$5; one finds $$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$6 and the usual Poisson equations. By contrast, generic $$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$7 or $$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$8 models with $$S=\int d^4x \sqrt{-g}\, f(R,G) + S_{\text{matter}},$$9 produce a scalaron-like Yukawa correction (Laurentis et al., 2013). The weak-field and slow-motion expansion up to $$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$0 in a spherically symmetric background confirms that the spatial behaviors at $$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$1 are the same for $$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$2 gravity and $$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$3 gravity, while the static corrections at $$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$4 depend on $$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$5 and $$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$6 (Wu et al., 2015).

A common misconception is that the Gauss–Bonnet sector must immediately alter weak-field observables. The detailed perturbative analysis shows a more specific statement: the Gauss–Bonnet sector is hidden at Newtonian order but enters at PN and PPN orders through $$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$7, $$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$8, and higher derivatives (Laurentis et al., 2013). An analogous statement holds in linearized radiation theory: the linearized field equations of $$G = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$$9 gravity are the same as those of linearized $f(R)$0 gravity, and the Gauss–Bonnet curvature scalar $f(R)$1 does not contribute to the effective stress-energy tensor of gravitational waves, nor to the energy, momentum, and angular momentum carried by gravitational waves, in the linearized regime (Wu et al., 2018).

3. Cosmological dynamics and inflationary applications

In spatially flat FLRW spacetime,

$f(R)$2

the curvature invariants are

$f(R)$3

so both background expansion and its derivatives feed directly into the higher-curvature sector (Laurentis et al., 2015).

A widely studied inflationary toy model is

$f(R)$4

In this setting the $f(R)$5 term produces the standard Starobinsky-like scalaron, while the $f(R)$6 term introduces a second effective scalaron. The resulting dynamics are driven by two effective masses (lengths) related to the $f(R)$7 scalaron and the $f(R)$8 scalaron working respectively at early and very early epochs of cosmic evolution. In this sense, a double inflationary scenario naturally emerges (Laurentis et al., 2015).

For the model $f(R)$9, the effective masses are

$R+f(G)$0

At very high curvature, the $R+f(G)$1-sector dominates; at somewhat later but still early times, the $R+f(G)$2 sector dominates; and at low curvature, the linear $R+f(G)$3 term recovers GR (Laurentis et al., 2015).

The same literature stresses that $R+f(G)$4 can exhaust all the curvature budget related to curvature invariants without considering derivatives of $R+f(G)$5, $R+f(G)$6, or $R+f(G)$7 in the action. This is one reason why $R+f(G)$8 models are often presented as a compact higher-curvature alternative to more general effective actions (Laurentis et al., 2015).

4. Viability, effective energy conditions, and observational consistency

In cosmological applications, the modified Friedmann equations are commonly rewritten in terms of an effective energy density and effective pressure, and the null, weak, strong, and dominant energy conditions are imposed on the effective fluid. For $R+f(G)$9 gravity on a flat FRW background, this yields explicit inequalities involving $f(G)$0, $f(G)$1, $f(G)$2, and cosmographic parameters such as the Hubble, deceleration, jerk, and snap parameters (Atazadeh et al., 2013).

For the model

$f(G)$3

the weak energy condition was found to hold for

$f(G)$4

when recent estimated values of the Hubble, deceleration, jerk and snap parameters are used. For

$f(G)$5

the weak energy condition is satisfied for

$f(G)$6

in the parameter ranges examined (Atazadeh et al., 2013).

Weak-field viability imposes a different but complementary set of constraints. To recover the correct Newtonian strength one requires $f(G)$7 and $f(G)$8. If $f(G)$9, Solar-System tests require the scalaron to be sufficiently massive, so that Yukawa corrections are short-ranged. Since $f(R,G)$0 starts at PN order, Gauss–Bonnet couplings must be small enough not to spoil PN and PPN bounds (Laurentis et al., 2013).

The literature also identifies a misconception that all higher-curvature terms are equally visible in all observables. In $f(R,G)$1 gravity this is not the case: some couplings are tested already at Newtonian order through Yukawa corrections, while others are hidden until PN, PPN, or nonlinear regimes (Laurentis et al., 2013, Wu et al., 2018).

5. Compact objects, wormholes, and gravastars

The strong-field sector of $f(R,G)$2 gravity has been explored with static spherically symmetric geometries, often using separable models such as

$f(R,G)$3

For traversable wormholes, a non-constant redshift function is essential because a constant redshift function makes the Gauss–Bonnet term vanish in the model employed. Using

$f(R,G)$4

together with anisotropic, isotropic, and barotropic matter, static, traversable, asymptotically flat wormhole geometries were constructed. The null energy and weak energy conditions for the effective energy-momentum tensor are usually violated for the ordinary matter content, but some small feasible regions for the existence of wormhole solutions were found where the energy conditions are not violated (Shamir et al., 2017).

Compact-star studies use the same higher-curvature framework as a strong-gravity test bed. With a Starobinsky-like $f(R,G)$5 sector and a power-law $f(R,G)$6 sector,

$f(R,G)$7

anisotropic compact-star models for Her X‑1, SAX J1808‑3658, and 4U 1820‑30 were constructed using the Krori–Barua ansatz. For positive values of the $f(R,G)$8 model parameter $f(R,G)$9, all three stars behave as usual; the models satisfy the standard energy conditions, the generalized TOV balance, causality bounds on sound speeds, and Herrera’s cracking criterion in the ranges studied (Shamir et al., 2017).

A related analysis with a $R$00 hyperbolic $R$01 model and three different $R$02 models extended this conclusion to six stars—Her X‑1, SAX J1808.4‑3658, 4U1820‑30, PSR J1614 2230, VELA X‑1, and Cen X‑3—and again reported that, for the positive value of parameter $R$03 of the model $R$04, all the six stars behave normally (Rahman et al., 2022). Charged anisotropic compact objects built from a Hu–Sawicki-like $R$05 and three viable $R$06 choices were also found to satisfy NEC, WEC, SEC, and DEC, with causal sound speeds and stability under Herrera’s criterion for VELA X‑1, SAXJ1808.4‑3658, and 4U1820‑30 (Ilyas et al., 2023).

Ultra-compact alternatives to black holes have also been studied. A Mazur–Mottola–type gravastar in generic $R$07 gravity was constructed by matching a de Sitter interior to a Schwarzschild exterior through a stiff thin shell. In that setting, shell length, entropy, and energy acquire explicit dependence on the effective dark source generated by the $R$08 sector (Bhatti et al., 2020).

6. Quasilocal energy, horizon thermodynamics, and theoretical outlook

A recent structural development is the construction of a generalized Misner–Sharp energy in $R$09 gravity for static spherically symmetric spacetimes and FLRW cosmology. Two independent derivations were given: the integration method and the conserved charge method based on the Kodama vector. The two constructions coincide, and the result reduces to standard $R$10 gravity when the Gauss–Bonnet term is absent (Akbarieh et al., 4 Jun 2025).

On FLRW backgrounds, evaluating the generalized Misner–Sharp energy at the apparent horizon reveals a non-equilibrium thermodynamic structure. For the model

$R$11

the apparent-horizon energy, pressure, enthalpy, and specific heats contain explicit $R$12, $R$13, and $R$14 terms, showing that horizon thermodynamics is inherently non-equilibrium in $R$15 gravity (Akbarieh et al., 4 Jun 2025).

Several theoretical caveats remain standard. The weak-field expansion developed for analytic $R$16 theories relies on analyticity around $R$17 and $R$18; non-analytic functions require a separate treatment (Laurentis et al., 2013). Stability and ghost-freedom impose additional constraints not fully analyzed in the weak-field papers (Laurentis et al., 2013). Linearized gravitational-wave calculations show that the Gauss–Bonnet sector does not contribute to the effective stress-energy tensor of gravitational waves in the Minkowski-based linear regime, though $R$19 plays an important role in nonlinear effects in general (Wu et al., 2018).

Taken together, these results define a characteristic profile for $R$20 gravity. Conceptually, it is a generalized Lovelock-inspired modification of GR in which the action is a general analytic function of $R$21 and $R$22. Dynamically, it leads to fourth-order field equations and, in the weak-field limit around Minkowski, to two gravitational potentials governed by higher-order Poisson-type equations. Phenomenologically, the Ricci sector controls the Newtonian limit, while the Gauss–Bonnet sector is hidden at Newtonian order and reappears at PN, PPN, strong-field, and thermodynamic levels. Cosmologically, $R$23- and $R$24-scalarons can generate a double inflationary scenario. Astrophysically, the theory accommodates wormholes, gravastars, and compact stars in representative models, while quasilocal energy and horizon thermodynamics acquire curvature-dependent corrections (Laurentis et al., 2013, Laurentis et al., 2015, Shamir et al., 2017, Akbarieh et al., 4 Jun 2025).