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It is shown that, in supergravity models of inflation where the gauge kineticfunction of a gauge field is modulated by the inflaton, we can obtain a newinflationary attractor solution, in which the roll-over of the inflaton suffersadditional impedance due to the vector field backreaction. As a result,directions of the scalar potential which, due to strong Kaehler corrections,become too steep and curved to normally support slow-roll inflation can nownaturally do so. This solves the infamous eta-problem of inflation insupergravity and also keeps the spectral index of the curvature perturbationmildly red despite eta of order unity. This mechanism is applied to a model ofhybrid inflation in supergravity with a generic Kaehler potential. The spectralindex of the curvature perturbation is found to be 0.97 - 0.98, in excellentagreement with data. The gauge field can act as vector curvaton generatingstatistical anisotropy in the curvature perturbation. However, this anisotropycould be possibly observable only if the gauge coupling constant is unnaturallysmall.

How to remedy the eta-problem of SUSY GUT hybrid inflation via vector backreaction