\[n_i = \frac{1}{(2\pi)^2 \alpha'} \int_{\Sigma_i} G_{(3)} \in \mathbb{Z}.\]
2.2.2 Anthropic Probability
The relative probability of observing a particular \( \Lambda \) is: \[P(\Lambda) \approx \exp\left( \frac{24\pi^2 M_{\text{Pl}}^4}{V(\phi)} \right) \times \Theta(\Lambda_{\text{max}} - \Lambda),\]
with:
- Exponential: De Sitter entropy of a bubble with vacuum energy \( \Lambda \) (Gibbons & Hawking, 1977).
- \( \Theta(\Lambda_{\text{max}} - \Lambda) \): Heaviside ladder function representing the anthropic constraint (Weinberg, 1987), with \( \Lambda_{\text{max}} \sim 10^{-121} M_{\text{Pl}}^4 \).
Exponential Derivation:
The de Sitter entropy is given by \( S = \frac{3M_{\text{Pl}}^2}{8\Lambda} \), so the number of microstates: \[\mathcal{N}(\Lambda) \sim e^{S} = \exp\left(\frac{3M_{\text{Pl}}^2}{8\Lambda} \right).\]
The factor \( 24\pi^2 \) arises from the inflaton phase volume correction (Garriga & Vilenkin, 2001).
2.3. Derivation P(L) from the Fokker-Planck Equation
1. Fokker-Planck Equation for Stochastic Inflaton