On the growth of perturbations as a test of dark energy and gravity

The strongest evidence for dark energy at present comes 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 and 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, that shear stress is unimportant on large scales, and that energy and momentum are locally conserved, we derive several relations between long-wavelength metric and matter perturbations. These relations include generalizations of the initial-value constraints of general relativity. For a class of theories including general relativity we reduce the long-wavelength metric, density, and velocity potential perturbations to quadratures including curvature perturbations, entropy perturbations, and the effects of nonzero background curvature. When combined with the expansion history measured geometrically, the long-wavelength solution provides a test that could distinguish modified gravity from other explanations of dark energy.