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R. Kallosh and A. Linde (private communication): Throughout the paper we have assumed that the volume of internal space is frozen as in KKLT model. When inflaton field is around zero, it was shown in 23 that the non-perturbative correction (17) to the tree level superpotential makes it possible to have a positive minimum at finite volume. This is in part due to the fact that the anti-D3 brane correction (19) is small. During the inflation the anti-D3 brane correction is again like (19) in which (Formula presented) should be replaced by (Formula presented) With this correction, however, one may not get anymore a local minimum for the supergravity potential during inflation in which φ is very large. In this case one may need to consider also α corrections to the Kahler potential 26 to study stabilization of the volume of internal space
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R. Kallosh and A. Linde (private communication): Throughout the paper we have assumed that the volume of internal space is frozen as in KKLT model. When inflaton field is around zero, it was shown in 23 that the non-perturbative correction (17) to the tree level superpotential makes it possible to have a positive minimum at finite volume. This is in part due to the fact that the anti-D3 brane correction (19) is small. During the inflation the anti-D3 brane correction is again like (19) in which (Formula presented) should be replaced by (Formula presented) With this correction, however, one may not get anymore a local minimum for the supergravity potential during inflation in which φ is very large. In this case one may need to consider also α corrections to the Kahler potential 26 to study stabilization of the volume of internal space.
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Note that in some other cosmological models (Formula presented) can be negative. See, e.g., Ref. 33
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