Cosmology in Eddington-inspired Born-Infeld gravity is investigated using a scalar Born-Infeld field (e.g. tachyon condensate) as matter. In this way, both in the gravity and matter sectors we have Born-Infeld-like structures characterized by their actions and via two separate constants, lambda and alpha(2)(T), respectively. With a particular choice of the form of (phi) over dot (the time derivative of the Born-Infeld scalar), analytical cosmological solutions are found. Thereafter, we explore some of the unique features of the corresponding cosmological spacetimes. For kappa > 0, our solution has a de Sitter-like expansion both at early and late times, with an intermediate deceleration sandwiched between the accelerating phases. On the other hand, when kappa < 0, the initial de Sitter phase is replaced by a bounce. Our solutions, at late time, fit well with available supernova data-a fact we demonstrate explicitly. The estimated properties of the Universe obtained from the fitting of the kappa > 0 solution are as good as in Lambda CDM cosmology. However, the kappa < 0 solution has to be discarded due to the occurrence of a bounce at an unacceptably low redshift.
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