Cosmology is the science of large things: galaxies, clusters of galaxies, the Universe itself. Gravity propagates freely at very large distances (actually, infinite distances) and the details of planets and stars, for example, don’t make a big difference when it comes to the expansion of the Universe. Almost all the information we have about the general properties of the Universe has been obtained averaging over huge scales, ten or hundred times larger that the typical distance between a galaxy and its nearest neighbors. Normally, what happens at smaller scales is either not important for as concern the global evolution, or too difficult to analyse because many non-gravitational effects come into play. It’s only because we can forget about the many little complications of small scales that we can build a solid map of the entire cosmic landscape.
Since matter in the Universe is distributed rather homogeneously when averaged over large scales, the fluctuations on top of this homogeneous distribution are very small and consequently one can linearize the equations, that is, consider only small deviations from homogeneity. This simple fact allows for an enormous simplification in the mathematical structure of our problem. Moving to small scales means one has to give up this simplicity and deal with non-linear equations. However, one can make just a little step in the direction of hopeless complication and stop at next order: not linear, not fully non-linear, but just second order. This is what one calls a perturbative approach, and almost all of physics is built this way, one step at a time.
In a recent paper, we studied how one can use second-order perturbation theory to test whether gravity is standard or requires corrections. In a previous paper, we did the same at linear level, and we found a combination of observables from galaxy clustering, gravitational lensing and supernovae that directly probes an important property of gravity, called anisotropic stress, that can discriminate between standard and modified gravity. Importantly, this test of gravity is very much model-independent, that is, does not depend on other assumptions like, for instance, what is the relation between the distribution of visible galaxies and of dark matter. In the new paper we extend this to smaller scales, finding another relation between various observables that probes directly the growth of fluctuations and therefore another property of gravity. Together with the linear test, one can in principle completely determine the phenomenological correction to gravity, if any.
We could not yet apply this to real data because they are not sufficient for the time being. But at least we have been able to show that small scales can teach us something about gravity, that large scales can’t.