Final Report Summary - COSMOGRAV (Cosmological tests of gravity) Cosmology studies the expansion of Universe on the largest scales and its effect on the formation of structures, such as galaxies. The study of the cosmic expansion requires full knowledge of the constituents of matter that fill the universe as well as the theory of gravity. Our best candidate theory of gravity to date is Einstein's general relativity while all known forms of stable matter are either radiation (photons and neutrinos), or baryons (protons and other nuclei). However when confronted with observations this paradigm fails in two important respects.The observed dynamics of galaxies leads us to posit a missing component of matter invisible to photons, which goes by the name of cold dark matter (CDM). This simple and attractive amendment is highly successful in explaining the dynamics of our universe, but the agreement with observations is not without tension. For instance CDM predicts more structure than we observe. Even when augmented with CDM, the aforementioned paradigm fails in a profound way. It predicts that the expansion of the universe must be slowing down while on the contrary it has been observed that it is actually speeding up! This requires a second missing component called the dark energy.Unlike CDM, for which many compelling proposals exist, the nature of dark energy is a complete mystery and one of the greatest puzzles of modern physics. Both of these components have yet to be detected directly which leads to the possibility that their inference based on general relativity is incorrect. This project focuses on critically questioning the nature of these two problems. To determine whether we need a new theory of gravity rather than new types of matter a framework of parametric deviations from general relativity on cosmological scales is proposed. The proposed framework consists of a set of numbers tied to different observations over a wide range of scales. These numbers can be used to decide unequivocally whether a particular aspect of the missing mass/energy is due to a different gravitational law or the presence of unseen matter. So far, the framework has been successfully been formulated on the largest scales, where matter densities and gravitational potentials are small. This allows one to assume certain approximations valid in this limit to simplify the equations used. Our research has shown that a very wide class of theories of gravity can be described using this framework.Given that the law of gravity plays a fundamental role in both the missing mass and the missing energy problem, it is very conceivable that we are merely seeing two sides of the same coin. Alternative gravity theories offer the possibility to solve both problems using the same mechanism. Part of the project was to investigate a number of alternative theories of gravity in the light of cosmological observations. This was achieved with the help of the parameterized framework and also with the study of specific examples of such theories.We also applied our framework to create techniques for measuring departures from general relativity and to test dark energy with cosmological observations. One particular focus was the growth of structure which is encoded in a parameter called the growth index. We used current observations to constrain the parameters that measure departures from general relativity. To begin with, we used the simplest theory that falls under this framework, namely, the Brans-Dicke theory. We found that the Brans-Dicke theory can be considered as an approximation to more complicated theories of gravity involving scalar fields. Our results place strong constraints on the Brans-Dicke and by extrapolation to all theories of gravity based on scalar fields.A further possibility is that dark matter and dark energy are coupled together. We have investigated this possibility and have generalized previous attempts under a single framework. In the process we uncovered new was to couple these two substances together. Our results opened up new ways for testing dark matter, dark energy and gravity with cosmology. Our methods are now being applied to new experiments and observations. We now have a tremendous flow of data and the computational power to analyse them. These combined with answers to unsolved problems mentioned above, make cosmology one of the most exciting fields of study, and will likely change our view of the cosmos.