Developing on the ideas of ( section 4 of ) [ 1 ] and [ 2 ] and using the formalisms of [ 3 , 4 ] , after inclusion of perturbative and non-perturbative \alpha ^ { \prime } corrections to the Kähler potential and ( D 1 - and D 3 - ) instanton generated superpotential , we show the possibility of getting finite values for the non-linear parameter f _ { NL } while looking for non-Gaussianities in type IIB compactifications on orientifolds of the Swiss Cheese Calabi-Yau { \bf WCP } ^ { 4 } [ 1 , 1 , 1 , 6 , 9 ] in the L ( arge ) V ( olume ) S ( cenarios ) limit .
First we show that in the context of multi-field slow-roll inflation , for the Calabi-Yau volume { \cal V } \sim { 10 } ^ { 5 } and D 3 - instanton number n ^ { s } \sim 10 along with N _ { e } \sim 18 , one can realize f _ { NL } \sim 0.03 , and for Calabi-Yau volume { \cal V } \sim { 10 } ^ { 6 } with D 3 - instanton number n ^ { s } \sim 10 resulting in number of e-foldings N _ { e } \sim { 60 } , one can realize f _ { NL } \sim 0.01 .
Further we show that with the slow-roll conditions violated and for the number of the D 3 - instanton wrappings n ^ { s } \sim { \cal O } ( 1 ) , one can realize f _ { NL } \sim { \cal O } ( 1 ) .
Using general considerations and some algebraic geometric assumptions , we show that with requiring a “ freezeout ” of curvature perturbations at super horizon scales , it is possible to get tensor-scalar ratio r \sim { \cal O } ( 10 ^ { -3 } ) with the loss of scale invariance |n _ { R } -1 | = 0.01 and one can obtain f _ { NL } \sim { \cal O } ( { 10 } ^ { -2 } ) as well in the context of slow-roll inflation scenarios in the same Swiss-Cheese setup .
For all our calculations of the world-sheet instanton contributions to the Kähler potential coming from the non-perturbative \alpha ^ { \prime } corrections , the degrees of genus-zero rational curves correspond to the largest value of the Gopakumar-Vafa invariants for the chosen compact projective variety , which is very large .
To our knowledge , such values of non-Gaussianities and tensor-scalar ratio in slow-roll inflationary and/or slow-roll violating scenarios , have been obtained for the first time from string theory .
We also make some observations pertaining to the possibility of the axionic inflaton also being a cold dark matter candidate as well as a quintessence field used for explaining dark energy .