Standard Einsteinian gravity contains a wave/particle called graviton that propagates the force in space, just like the photon propagates electrodynamics in space. The graviton is supposed to be massless, again like the photon, but since so far nobody has ever seen a graviton, its mass has never been measured. A few years ago, some physicists found the way to generalize Einsteinian gravity by introducing a massive graviton. This is not as easy as it seems and it actually requires adding a second copy of the gravitational theory, almost identical to the usual one but not directly coupled to matter. In other words, we still feel directly only one form of gravity (or metric) but a second one is lurking there and has some indirect observable effect too. The combination of the two gravities creates a massive graviton in addition to the massless graviton.
The idea is simple and quite appealing. Moreover, soon it was discovered that it could also explain the cosmic acceleration without necessarily adding a cosmological constant, as long as the graviton mass is extremely small. After the initial excitement, however, many particular models have been found to be actually unstable and had to be discarded. For a while, only one special case was thought to offer a viable cosmology. But even this model was shown later to generate unacceptably large fluctuations on the cosmic microwave background spectrum in the form of too large gravitational waves. There was still a loophole open though: the gravitational waves were so large that they should have become non-linear by the time we observe them. This in turn implies that the simplified linear treatment cannot be carried through and should be somehow reconsidered.
This is what we did in a recent paper. Here we assumed that the linear growth stops when the waves become non linear. This is definitely a very rough approximation and actually a full study should require estimating the non linear growth. It is however very plausible that when the waves become too strong they just freeze; moreover, since the dangerous waves are those associated with the second metric, which then transmit their growth to the ordinary metric (the one we observe), one could hope that freezing the former would save the latter. The paper contains mixed results. On one hand, we find that even the freezing is not enough to save the model at face value. On the other, we restrict the problem to the still badly observed very large scale sector of the CMB polarization. Moreover, we find that playing with initial conditions or with time-dependent coupling parameters one can produce acceptable results.
So, although the freezing mechanism is not actually as good as we hoped for, the jury is still out on whether bigravity cosmology works or not.