Dark Energy and Doubly Coupled Bigravity
  We analyse the late time cosmology and the gravitational properties of doubly
coupled bigravity in the constrained vielbein formalism (equivalent to the
metric formalism) when the mass of the massive graviton is of the order of the
present Hubble rate. We focus on one of the two branches of background
cosmology where the ratio between the scale factors of the two metrics is
algebraically determined. The Universe evolves from a matter dominated epoch to
a dark energy dominated era where the equation of state of dark energy can
always be made close to -1 now by appropriately tuning the graviton mass. We
also analyse the perturbative spectrum of the theory in the quasi static
approximation well below the strong coupling scale where no instability is
present and we show that there are five scalar degrees of freedom, two vectors
and two gravitons. In a cosmological FRW background for both metrics, four of
the five scalars are Newtonian potentials which lead to a modification of
gravity on large scales. In this scalar sector, gravity is modified with
effects on both the growth of structure and the lensing potential. In
particular, we find that the $\Sigma$ parameter governing the Poisson equation
of the weak lensing potential can differ from one in the recent past of the
Universe. Overall, the nature of the modification of gravity at low energy,
which reveals itself in the growth of structure and the lensing potential, is
intrinsically dependent on the couplings to matter and the potential term of
the vielbeins. We also find that the time variation of Newton's constant in the
Jordan frame can easily satisfy the bound from solar system tests of gravity.
Finally we show that the two gravitons present in the spectrum have a
non-trivial mass matrix whose origin follows from the potential term of
bigravity. This mixing leads to gravitational birefringence.
