Klebanov-Strassler warp-throat conifold-background will be the basis for our explicit analysis of warped D-brane inflationary cosmology

For a visual treat of the mathematical 'picture', scroll to the bottom of this post. In part two of this series on M-theoretic world-brane cosmology, I showed that a Klebanov-Strassler geometry naturally arises by considering string theory compactification on $Ad{S_5} \times {X_5}$ where ${X_5}$ is the Einstein manifold in five dimensions, with the interaction-Lagrangian of the massless Klebanov-Strassler field and the brane fields fermions is

then I showed that after integrating over the extra dimensional part, the effective 4-D Lagrangian reduces to

with the fundamental Planck scale $M$ and the 4-D Planck scale ${M_{pl}}$ related as

Moreover, I demonstrated that the moduli spaces of compact Calabi-Yau spaces naturally contain conifold singularities and that the local description of these singularities is a conifold, a noncompact Calabi-Yau three-fold whose geometry is given by a cone, and that the orbifolded conifold equation

Loosely put: Klebanov-Strassler spacetime geometry is the warped product of 4-D Minkowski spacetime with 6-D Calabi-Yau orientifold

In part one, I showed, in the context of 4-D low-energy effective description of the KKLT string flux compactifications proposal, that in the limit of N = 1 supergravity, where the moduli potential ${V_F}$ is characterized by a superpotential $W$ and a Kähler potential ${\rm K}$

where $W$ is defined by

yields a standard Calabi-Yau compactification containing 3-form flux ${G_3} \equiv {F_3} - \tau {H_3}$ that contributes to the superpotential via the Gukov-Vafa-Witten 4-fold term

with $\Omega$ the holomorphic 3-form on the Calabi-Yau three-fold and