  We consider a modified gravity framework for inflation by adding to the
Einstein-Hilbert action a direct $f(\phi)T$ term, where $\phi$ is identified as
the inflaton and $T$ is the trace of the energy-momentum tensor. The framework
goes to Einstein gravity naturally when the inflaton decays out. We investigate
inflation dynamics in this $f(\phi)T$ gravity (not to be confused with
torsion-scalar coupled theories) on a general basis and then apply it to three
well-motivated inflationary models. We find that the predictions for the
spectral tilt and the tensor-to-scalar ratio are sensitive to this new
$f(\phi)T$ term. This $f(\phi)T$ gravity brings chaotic and natural inflation
into better agreement with data and allows a larger tensor-to-scalar ratio in
the Starobinsky model.
