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The relationship between the metric and nonrelativistic matter distribution depends on the theory of gravity and additional fields, providing a possible way of distinguishing competing theories. With the assumption that the geometry and kinematics of the homogeneous universe have been measured to sufficient accuracy, we present a procedure for understanding and testing the relationship between the cosmological matter distribution and metric perturbations (along with their respective evolution) using the ratio of the physical size of the perturbation to the size of the horizon as our small expansion parameter. We expand around Newtonian gravity on linear, subhorizon scales with coefficient functions in front of the expansion parameter. Our framework relies on an ansatz which ensures that (i) the Poisson equation is recovered on small scales (ii) the metric variables (and any additional fields) are generated and supported by the nonrelativistic matter overdensity. The scales for which our framework is intended are small enough so that cosmic variance does not significantly limit the accuracy of the measurements and large enough to avoid complications from nonlinear effects and baryon cooling. The coefficient functions provide a general framework for contrasting the consequences of Lambda CDM and its alternatives. We calculate the coefficient functions for general relativity with a cosmological constant and dark matter, GR with dark matter and quintessence, scalar-tensor theories, f(R) gravity and braneworld models. We identify a possibly unique signature of braneworld models. Constraining the coefficient functions provides a streamlined approach for testing gravity in a scale dependent manner. We briefly discuss the observations best suited for an application of our framework.
Amin et al. (Wed,) studied this question.