Cuscuton cosmology: Dark energy meets modified gravity

In a companion paper, we have introduced a model of scalar field dark energy, Cuscuton, which can be realized as the incompressible (or infinite speed of sound) limit of a k-essence fluid. In this paper, we study how Cuscuton modifies the constraint sector of Einstein gravity. In particular, we study Cuscuton cosmology and show that even though Cuscuton can have an arbitrary equation of state, or time dependence, and is thus inhomogeneous, its perturbations do not introduce any additional dynamical degree of freedom and only satisfy a constraint equation, amounting to an effective modification of gravity on large scales. Therefore, Cuscuton can be considered to be a minimal theory of evolving dark energy, or a minimal modification of a cosmological constant, as it has no internal dynamics. Moreover, this is the only modification of Einstein gravity to our knowledge, that does not introduce any additional degrees of freedom (and is not conformally equivalent to the Einstein gravity). We then study two simple Cuscuton models, with quadratic and exponential potentials. The quadratic model has the exact same expansion history as Lambda CDM, and yet contains an early dark energy component with constant energy fraction, which is constrained to Omega(Q)less than or similar to 2%, mainly from WMAP Cosmic Microwave Background and Sloan Digital Sky Survey Lyman-alpha forest observations. The exponential model has the same expansion history as the Dvali-Gabadadze-Poratti self-accelerating brane-world model, but generates a much smaller Integrated Sachs-Wolfe effect, and is thus consistent with the Cosmic Microwave Background observations. Finally, we show that the evolution is local on superhorizon scales, implying that there is no gross violation of causality, despite Cuscuton's infinite speed of sound.