  Massive bigravity, a theoretically consistent modification of general
relativity with an additional dynamical rank two tensor, successfully describes
the observed accelerated expansion of the Universe without a cosmological
constant. Recent analyses of perturbations around a cosmological background
have revealed power law instabilities in both scalar and tensor perturbations,
motivating an analysis of the initial conditions, evolution, and cosmological
observables to determine the viability of these theories. In this paper we
focus on the tensor sector, and study a primordial stochastic gravitational
wave background in massive bigravity. The phenomenology can differ from
standard General Relativity due to non-trivial mixing between the two
linearized tensor fluctuations in the theory, only one of which couples to
matter. We study perturbations about two classes of cosmological solutions in
bigravity, computing the tensor contribution to the temperature anisotropies in
the Cosmic Microwave Background radiation and the present stochastic
gravitational wave background. The result is strongly dependent on the choice
of cosmological background and initial conditions. One class of background
solution generically displaying tremendous growth in the amplitude of
large-wavelength gravitational waves, while the other remains observationally
indistinguishable from standard General Relativity for a wide variety of
initial conditions. We analyze the initial conditions for tensor modes expected
in an inflationary cosmology, finding again that there is a strong dependence
on the assumed background. For one choice of background, the semi-classical
theory is beyond the perturbative regime. For the other choice, inflation
generically yields initial conditions that, when evolved, give rise to a
stochastic background observationally indistinguishable from standard General
Relativity.
