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Modified gravity changes the gravitational dynamics described by general relativity, for example through additional gravitational fields or curvature terms beyond the Einstein-Hilbert action. A proposed theory must specify both its equations and how matter couples to the geometry before trajectories, light propagation or cosmological evolution can be predicted.

The regime of a test matters. Linearized f(R) gravity about Minkowski spacetime can coincide with the Linearized Einstein equations for suitable derivatives of its defining function, while nonlinear terms still differ. Agreement in one weak-field calculation therefore need not identify the full theory. Comparisons involving orbital motion, gravitational lensing and gravitational waves constrain different combinations of dynamics, matter assumptions and observational geometry.

Because $R=O(ϵ)$ andthe Taylor series gives $f(R)=R+O({ϵ}^3)$ and $f^{\prime }(R)=1+O({ϵ}^2)$. The extra derivative terms in the f(R) equation are therefore at least second order. At first order the theory reduces to the Linearized Einstein equationsIn harmonic coordinates, equivalently the Lorenz gauge in linearized gravity ${\partial }_{\mu }{\bar{h}}^{\mu }{}_{\nu }=0$, the trace-reversed metric perturbationsatisfiesThus Linearized f(R) gravity about Minkowski spacetime has the same weak-field predictions as general relativity under these assumptions. Differences can first appear through nonlinear curvature terms or in backgrounds about which $f^{\prime \prime }(R)$ is nonzero.