  Nonlinear, ghost-free massive gravity has two tensor fields; when both are
dynamical, the mass of the graviton can lead to cosmic acceleration that agrees
with background data, even in the absence of a cosmological constant. Here the
question of the stability of linear perturbations in this bimetric theory is
examined. Instabilities are presented for several classes of models, and simple
criteria for the cosmological stability of massive bigravity are derived. In
this way, we identify a particular self-accelerating bigravity model,
infinite-branch bigravity (IBB), which exhibits both viable background
evolution and stable linear perturbations. We discuss the modified gravity
parameters for IBB, which do not reduce to the standard $\Lambda$CDM result at
early times, and compute the combined likelihood from measured growth data and
type Ia supernovae. IBB predicts a present matter density $\Omega_{m0}=0.18$
and an equation of state $w(z)=-0.79+0.21z/(1+z)$. The growth rate of structure
is well-approximated at late times by
$f(z)\approx\Omega_{m}^{0.47}[1+0.21z/(1+z)]$. The implications of the linear
instability for other bigravity models are discussed: the instability does not
necessarily rule these models out, but rather presents interesting questions
about how to extract observables from them when linear perturbation theory does
not hold.
