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

A2. Mathematical Derivations

de Sitter Entropy from First Principles

For a universe with cosmological constant :

1. Horizon Area: A=4RH2=12,RH=3/  A=4RH2=12,RH=3/

2. Gibbons-Hawking Entropy:SdS=A4G=3MPl2   SdS=4GA=3MP2

3. String Theory Correction:S=2Vgs24  S=gs242V

Slow-Roll Consistency Test

For KKLT potential V()=V0Aea  V()=V0Aea

:=MPl216(aAeaV0)2106,=a2MPl28AeaV00.01  =16MPl2(V0aAea)2106,=8a2MP2V0Aea0.01

Condition: <1 <  requires >1aln(a2AMPl28V0)  >a1ln(8V0a2AMPl2).

A3. Numerical Simulations