3. Observational Protocol
Proposes two empirical tests:
a. Tensor-to-scalar ratio (r r): Prediction r0.0030.001 r0.0030.001 for string theory (KKLT) moduli-based inflation, different from the simple inflation model (r>0.01 r>0.01).
b. Statistical Anomalies in Large-Scale Structure: N-body simulations show that local variations of leave a signature on the galaxy correlation function ((r)x(r)) on the scale >100. >100 Mpc.
2. Mathematical Formalism
2.1 Inflaton Model from String Theory
2.1.1 Inflaton Potential of Stabilization Moduli (KKLT)
In the KKLT scenario (Kachru-Kallosh-Linde-Trivedi, 2003), the inflaton potential originates from the stabilization of the Khler moduli (\( \phi \)) and volume (\( \mathcal{V} \)) in the compactification of type IIB string theory. Potential effectiveness:
\[V(\phi) = V_0 + A e^{-a\phi} - B e^{-b\phi} + \sum_{i=1}^N \frac{q_i^2}{2} \phi_i^2,\]
where: \( V_0 \): Residual vacuum energy of the uplifting term (D3-brane).
- \( A e^{-a\phi} - B e^{-b\phi} \: Contribution of instanton and anti-D3 brane to moduli stabilization (Kachru et al., 2003).
- \( \sum_{i=1}^N \frac{q_i^2}{2} \phi_i^2 \): Flux field energy (\( G_{(3)} \)) in compacted dimensions, with \( q_i = \frac{1}{(2\pi)^4 \alpha'^2} \int_{\Sigma_i} G_{(3)} \).
Parameter Derivation:
