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Wanda,
Thanks for the lead. I did a Cornell Archive search and came up with many papers. The most recent may be found at http://arxiv.org/PS_cache/hep-th/pdf/0509/0509012.pdf
Here is the abstract:
Kahler Moduli Inflation
Authors: Joseph P. Conlon, Fernando Quevedo
Comments: 17 pages, 1 figure; v2. references added
Report-no: DAMTP-2005-78
We show that under general conditions there is at least one natural inflationary direction for the Kahler moduli of type IIB flux compactifications. This requires a Calabi-Yau which has h^{2,1}>h^{1,1}>2 and for which the structure of the scalar potential is as in the recently found exponentially large volume compactifications. We also need - although these conditions may be relaxed - at least one Kahler modulus whose only non-vanishing triple-intersection is with itself and which appears by itself in the non-perturbative superpotential. Slow-roll inflation then occurs without a fine tuning of parameters, evading the eta problem of F-term inflation. In order to obtain COBE-normalised density perturbations, the stabilised volume of the Calabi-Yau must be O(10^5-10^7) in string units, and the inflationary scale M_{infl} ~ 10^{13} GeV. We find a robust model independent prediction for the spectral index of 1 - 2/N_e = 0.960 - 0.967, depending on the number of efoldings.
Here is an earlier discussion by the leaders of the field of dtring cosmology
http://arxiv.org/abs/hep-th/0308055
Towards Inflation in String Theory
Authors: Shamit Kachru, Renata Kallosh, Andrei Linde, Juan Maldacena, Liam McAllister, Sandip P. Trivedi
Comments: 41 pages, harvmac; v2: results of appendix A extended to include branes at angles, typos corrected, refs added
Report-no: SLAC-PUB-9669, SU-ITP-03/18, TIFR/TH/03-06
Journal-ref: JCAP 0310 (2003) 013
We investigate the embedding of brane inflation into stable compactifications of string theory. At first sight a warped compactification geometry seems to produce a naturally flat inflaton potential, evading one well-known difficulty of brane-antibrane scenarios. Careful consideration of the closed string moduli reveals a further obstacle: superpotential stabilization of the compactification volume typically modifies the inflaton potential and renders it too steep for inflation. We discuss the non-generic conditions under which this problem does not arise. We conclude that brane inflation models can only work if restrictive assumptions about the method of volume stabilization, the warping of the internal space, and the source of inflationary energy are satisfied. We argue that this may not be a real problem, given the large range of available fluxes and background geometries in string theory.
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