On the Growth of Perturbations as a Test of Dark Energy and Gravity

The strongest evidence for dark energy comes presently from geometric techniques such as the supernova distance-redshift relation. By combining the measured expansion history with the Friedmann equation one determines the energy density and its time evolution, hence the equation of state of dark energy. Because these methods rely on the Friedmann equation which has not been independently tested it is desirable to find alternative methods that work for both general relativity and other theories of gravity. Assuming that sufficiently large patches of a perturbed Robertson-Walker spacetime evolve like separate Robertson-Walker universes and that shear stress is unimportant on large scales, we reduce the long-wavelength metric, density, and velocity potential perturbations to quadratures. The general solution is given including curvature perturbations, entropy perturbations (whose evolution generally requires solving additional equations of motion), and nonzero background curvature for any theory of gravity permitting a Robertson-Walker solution. When combined with the expansion history measured geometrically, the long-wavelength solution provide a test that may distinguish modified gravity from other explanations of dark energy. Alternative cosmological-scale tests of gravity are proposed in terms of the constraint equations of general relativity.