DBI Inflation in N=1 Supergravity
  It was recently demonstrated that, when coupled to N=1 supergravity, the Dirac-Born-Infeld (DBI) action constructed from a single chiral superfield has the property that when the higher-derivative terms become important, the potential becomes negative. Thus, DBI inflation cannot occur in its most interesting, relativistic regime. In this paper, it is shown how to overcome this problem by coupling the model to one or more additional chiral supermultiplets. In this way, one obtains effective single real scalar field DBI models with arbitrary positive potentials, as well as multiple real scalar field DBI inflation models with hybrid potentials. 
Michael Koehn, Jean-Luc Lehners and Burt A. Ovrut
hep-th astro-ph.CO