Projective Origins of the Inflaton

We detail how the recently developed projectively-invariant gravitational model, Thomas-Whitehead Gravity (TW Gravity), naturally gives rise to a field acting as the inflaton. In the formulation of TW Gravity, a field $\mathcal{D}_{ab}$ is introduced into the components of a projective connection. This field, $\mathcal{D}_{ab}$, is related to a rank-two tensor field $\mathcal{P}_{ab}$. Through the action of TW Gravity, in terms of projective curvature invariants, the tensor field $\mathcal{P}_{ab}$ acquires dynamics. Decomposing $\mathcal{P}_{ab}$ into its trace and traceless degrees of freedom, we show that the action can describe a type of non-minimally coupled inflaton action. We calculate values for the spectral index $n_s$, tensor-to-scalar ratio $r$, and scalar amplitude $A_s$. These calculated quantities are shown to lie within bounds from the most recent Planck analysis, for a discrete range of free parameters.