Our understanding of the Universe relies on two pillars, gravity and matter. These are involved in compelling open questions: is the accelerated cosmic expansion due to gravity modifications? And what are the properties of dark matter? In this thesis, we present new methods to investigate these questions through the effect of gravitational redshift, which arises from the distortion of time in a gravitational potential. We first focus on linear cosmological scales, where this effect will become measurable with upcoming galaxy surveys. We show that this novel observable contains crucial information to distinguish non-standard dark matter scenarios from gravity modifications and to assess whether dark matter experiences gravity in the same way as ordinary matter. We then provide the first relativistic modelling for the non-linear gravitational redshift signal that has already been measured from galaxy clusters. Therefore, shedding light on the pillars of cosmology might literally be a matter of time.