Stochastic Selection of Cosmological Constant in Chaotic Inflation: A String Theory Landscape Approach

3. Probability Distribution:  

   Total micro state: \[\mathcal{N}(\Lambda) \sim e^{S_{\text{dS}}} = \exp\left( \frac{24\pi^2 M_{\text{Pl}}^4}{V(\phi)} \right), \]  where \( V(\phi) \) is the inflaton potential when inflation ends.  

4. Numerical Example:  

   For \( \Lambda \sim 10^{-122} M_{\text{Pl}}^4 \): \[S_{\text{dS}} \approx 2.9 \times 10^{122}, \quad \mathcal{N}(\Lambda) \sim e^{10^{122}}. \]  

Implementation Notes

1. Simulation:  

   - For high accuracy, use **4th-order Runge-Kutta** for stochastic integration.  

   - Parallelization with **MPI** for multiverse scale (\( >10^6 \) bubbles).  

2. Parameter:  

   - The values \( q_i \) and \( n_i \) must be validated with **moduli stabilization** (example: KKLT).  

3. Entropy