massive gravity with a flat reference metric. Open solutions and solutions with general reference metrics suffer from instabilities. Therefore, viable cosmologies can only be found in massive gravity if one abandons the cosmological principle that the Universe is uniform on large scales, or otherwise generalizes dRGT. For instance, cosmological solutions are better behaved in bigravity, the theory which extends dRGT by giving dynamics. While these tend to possess instabilities as well, those instabilities might find a resolution in the nonlinear dynamics (through a Vainshtein-like mechanism) or by pushing the era of instability to the very early Universe. 3D massive gravity A special case exists in three dimensions, where a massless graviton does not propagate any degrees of freedom. Here several ghost-free theories of a massive graviton, propagating two degrees of freedom, can be defined. In the case of topologically massive gravity one has the action with the three-dimensional Planck mass. This is three-dimensional general relativity supplemented by a Chern-Simons-like term built out of the Christoffel symbols. More recently, a theory referred to as new massive gravity has been developed, which is described by the action Relation to gravitational waves The 2016 discovery of gravitational waves and subsequent observations have yielded constraints on the maximum mass of gravitons, if they are massive at all. Following the GW170104 event, the graviton's Compton wavelength was found to be at least , or about 1.6 light-years, corresponding to a graviton mass of no more than . This relation between wavelength and energy is calculated with the same formula (the Planck–Einstein relation) that relates electromagnetic wavelength to photon energy. However, photons, which have only energy and no mass, are fundamentally different from massive gravitons in this respect, since the Compton wavelength of the graviton is not equal to the gravitational wavelength. Instead, the lower-bound graviton Compton wavelength is about times greater than the gravitational wavelength for the GW170104 event, which was ~ 1,700 km. This is because the Compton wavelength is defined by the rest mass of the graviton and is an invariant scalar quantity. See also Horndeski's theory Dual graviton Further reading Review articles References Theories of gravity