In the algebraic branch of the bigravity FRW solution, the late time cosmology is determined. By tuning the graviton mass, the asymptotic dark energy equation of state can be made arbitarily close to -1. The full theory contains 5+2+2 SVT modes, the dof of two gravitons (one massive and one massless). The scalar sectors leads to modifications of structure formation and lensing on large scales - which are senstive to how the matter is coupled to both metrics. The mass mixing of the two gravitons leads to a gravitational birefringence.

Studying the modified phenomenology of multi-metric theories of gravity provides valuable cosmological predictions which can be tested as alternatives to General Relativity. In addition to the dRGT mass terms which can be included in a multi- metric theory, there is also more freedom in how matter couples to gravity - it may do so through a combination of the two metrics. This paper outlines some of the different cosmological effects due to different matter couplings.

While generalized Galileon theories can stably violate the null energy condition in flat space, when coupling them to gravity (giving the Horndeski theory) any such violation gives rise to instabilities. This no-go theorem extends previous work of 1605.05992 to all Horndeski theories, and suggests that Beyond Horndeski will be necessary in order to have a stable violation of the null energy condition.

In an FRW universe, the null energy condition keeps the Hubble rate decreasing with time. As we extrapolate backwards in time to the early Universe, one of two things must happen: (i) Hubble rate increases to Planck scales and our understanding of gravity breaks down, or (ii) the null energy condition is violated. Interest in the latter case goes under the name 'non-singular (bouncing) cosmologies', and may provide an alternative scenario to the initial Big Bang singularity - for example there could be a `bounce' from a contracting to an expanding universe during a null-energy-condition-violating phase of the early universe.

Massive bigravity with doubly coupled matter fields admits novel cosmological solutions. While previous work had investigated the stability of SVT perturbations around particular branches of solution, this latest paper carries out a perturbative analysis about completely general equations of motion, and provides the full quadratic action for the scalar perturbations (which is shown to contain a ghost if one chooses the non-algebraic branch).

Providing stability criteria for a whole class of spin-2 field theories (namely two metrics interacting via dRGT mass terms and both coupled to external matter) plays a crucial role in guiding future phenomenological investigation, allowing the community to focus their resources on analysing only the stable solutions.