A null test to probe the scale-dependence of the growth of structure as a test of General Relativity

The main science driver for the coming generation of cosmological surveys is understanding dark energy which relies on testing General Relativity on the largest scales. Once we move beyond the simplest explanation for dark energy of a cosmological constant, the space of possible theories becomes both vast and extremely hard to compute realistic observables. A key discriminator of a cosmological constant, however, is that the growth of structure is scale-invariant on large scales. By carefully weighting observables derived from distributions of numbers of galaxies and a dipole pattern in their apparent sizes, we construct a null test which vanishes for any model of gravity or dark energy where the growth of structure is scale-independent. It relies only on very few assumptions about cosmology, and does not require any modelling of the growth of structure at late times. We show that with a survey like the Dark Energy Spectroscopic Instrument a scale-dependence of the order of 10-20 percent can be detected with 3 standard deviations with the null test, which will drop by a factor of 2 for a survey with a larger number of galaxies and volume, like the Square Kilometre Array will be capable of. We also show that the null test is very insensitive to typical uncertainties in other cosmological parameters including massive neutrinos and scale-dependent bias, making this a key null test for the cosmological constant.