Fayet-Iliopoulos terms in supergravity and D-term inflation
  We analyse the consequences of a new gauge invariant Fayet-Iliopoulos (FI)
term proposed recently to a class of inflation models driven by supersymmetry
breaking with the inflaton being the superpartner of the goldstino. We first
show that charged matter fields can be consistently added with the new term, as
well as the standard FI term in supergravity in a K\"ahler frame where the
$U(1)$ is not an R-symmetry. We then show that the slow-roll conditions can be
easily satisfied with inflation driven by a D-term depending on the two FI
parameters. Inflation starts at initial conditions around the maximum of the
potential where the $U(1)$ symmetry is restored and stops when the inflaton
rolls down to the minimum describing the present phase of our Universe. The
resulting tensor-to-scalar ratio of primordial perturbations can be even at
observable values in the presence of higher order terms in the K\"ahler
potential.
