On the scalar graviton in n-DBI gravity
  n-DBI gravity is a gravitational theory which yields near de Sitter inflation
spontaneously at the cost of breaking Lorentz invariance by a preferred choice
of foliation. We show that this breakdown endows n-DBI gravity with one extra
physical gravitational degree of freedom: a scalar graviton. Its existence is
established by Dirac's theory of constrained systems. Firstly, studying scalar
perturbations around Minkowski space-time, we show that there exists one scalar
degree of freedom and identify it in terms of the metric perturbations. Then, a
general analysis is made in the canonical formalism, using ADM variables. It is
useful to introduce an auxiliary scalar field, which allows recasting n-DBI
gravity in an Einstein-Hilbert form but in a Jordan frame. Identifying the
constraints and their classes we confirm the existence of an extra degree of
freedom in the full theory, besides the two usual tensorial modes of the
graviton. We then argue that, unlike the case of (the original proposal for)
Horava-Lifschitz gravity, there is no evidence that the extra degree of freedom
originates pathologies, such as vanishing lapse, instabilities and strong
self-coupling at low energy scales.
