least gravitationally) to moduli fields, originating from the string theory compactification. A volume modulus can be stabilized using the following form of the potentials [666, 674, 675]: where T is a volume modulus, the modulus, χ , is a matter field introduced to allow for an additional contribution to the superpotential which is charged under a new U (1) abelian symmetry, with V$_{2}$ its gauge superfield, and δ ′ GS is the Green-Schwarz parameter. The moduli potential derived contains the D -terms of the form: where f$_{mod}$ ( T ) is the gauge kinetic function for that sector. Interestingly, the above stabi- lization potential has a non-vanishing vacuum which can generate an effective FI term for inflation, if T and χ are also charged under the U (1)$_{ξ}$, giving with δ ξ GS the Green-Schwarz parameter of U (1)$_{ξ}$. The form of the K¨ ahler potential can be invariant under the shift symmetry of the inflaton field as the minimal potential would induce a SUGRA- η problem, due to non-vanishing F -terms of the moduli sector during inflation. However, if we assume that the total K¨ ahler is simply the sum, K = K$_{mod}$ + K$_{infl}$ , the contribution from that sector to the (effective mass) 2 of the waterfall field would spoil the graceful waterfall exit from inflation. A total K¨ ahler potential of the form: would preserve the general behavior of the D -term inflationary model, as long as the am- plitude for the moduli are small. The current observations constrain the parameter space of this model, leaving 3 classes of models, all predicting a spectral index below or close to unity. One last modification due to the moduli sector is the nature of the cosmic strings formed at the end of inflation, which are not being BPS anymore and potentially containing massive fermionic currents. 126 W$_{mod}$ ( T, χ ) = W$_{0}$ + Ae − aT χ b , K$_{mod}$ = − 3 ln( T + ¯ T − | χ | 2 + δ$_{GS}$V$_{2}$ ) , (270) V D $_{mod}$( T, χ ) = { 3 δ ′ $_{GS}$[1 + ( a/b ) | χ | $^{2}$] } 2 8Re( f ( T )) exp $^{2}$( − K/ 3) , (271) ξ = 3 δ ξ $_{GS}$(1 + a/b | χ | $^{2}$) 4Re( T ) − 2 χ 2 , K$_{infl}$ = | S − ¯ S | $^{2}$/ 2 + | Φ$_{+}$ | 2 + | $^{Φ}$− | $^{2}$, (272) K = − 3 ln( T + ¯ T − | χ | 2 − K$_{infl}$/ 3 + $_{δ}$′ $_{GS}$V$_{2}$ + δ ξ $_{GS}$V$_{1}$ ) , (273)