We present a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or longitudinal) gauge. We first derive the transformation law relating the two gauges. We then write down in parallel in both gauges the coupled, linearized Boltzmann, Einstein and fluid equations that govern the evolution of the metric perturbations and the density fluctuations of the particle species. The particle species considered include cold dark matter (CDM), baryons, photons, massless neutrinos, and massive neutrinos (a hot dark matter or HDM candidate), where the CDM and baryon components are treated as fluids while a detailed phase-space description is given to the photons and neutrinos. The linear evolution equations presented are applicable to any Ω = 1 model with CDM or a mixture of CDM and HDM. Isentropic initial conditions on super-horizon scales are derived. The equations are solved numerically in both gauges for a CDM+HDM model with Ω cold = 0.65, Ω hot = 0.3, and Ω baryon = 0.05. We discuss the evolution of the metric and the density perturbations and compare their different behaviors outside the horizon in the two gauges. In a companion paper we integrate the geodesic equations for the neutrino particles in the perturbed conformal Newtonian background metric computed here. The purpose is to obtain an accurate sampling of the neutrino phase space for the HDM initial conditions in N -body simulations of the CDM+HDM models.

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The acceleration of the universe can be explained either through dark energy or through the modification of gravity on large scales. In this paper we investigate modified gravity models and compare their observable predictions with dark energy models. Modifications of general relativity are expected to be scale-independent on super-horizon scales and scale-dependent on sub-horizon scales. For scale-independent modifications, utilizing the conservation of the curvature scalar and a parameterized post-Newtonian formulation of cosmological perturbations, we derive results for large scale structure growth, weak gravitational lensing, and cosmic microwave background anisotropy. For scale-dependent modifications, inspired by recent f (R) theories we introduce a parameterization for the gravitational coupling G and the post-Newtonian parameter γ. These parameterizations provide a convenient formalism for testing general relativity. However, we find that if dark energy is generalized to include both entropy and shear stress perturbations, and the dynamics of dark energy is unknown a priori, then modified gravity cannot in general be distinguished from dark energy using cosmological linear perturbations.