The coefficients \( a, b \) are determined by the Euler numbers characteristic of the Calabi-Yau manifold (\( \chi \)): \[a = \frac{2\pi}{g_s^{1/4} \mathcal{V}^{1/6}}, \quad b = \frac{4\pi}{g_s^{1/2} \mathcal{V}^{1/3}},\]
with \( g_s \): Coupling string, \( \mathcal{V} \): Volume compactification.
2.1.2 Inflaton Motion Equation with Quantum Fluctuations
The inflaton dynamics satisfies the stochastic Langevin equation:
\[\ddot{\phi} + 3H\dot{\phi} + V'(\phi) = \xi(t), \quad \langle \xi(t)\xi(t') \rangle = \frac{H^3}{4\pi^2} \delta(t-t'),\]
where:
- \( 3H\dot{\phi} \): Hubble damping (universe expansion effect).
- \( \xi(t) \): Quantum noise from inflaton vacuum fluctuations (Starobinsky, 1986).
Modified Slow-Roll Terms:
For inflation to occur, the potential must meet:
\[\epsilon = \frac{M_{\text{Pl}}^2}{16\pi} \left( \frac{V'}{V} \right)^2 \ll 1, \quad \eta = \frac{M_{\text{Pl}}^2}{8\pi} \frac{V''}{V} \ll 1 + \frac{H^2}{4\pi^2 V'}.\]
