Current Approach: Anthropic selection within causal patches (Bousso, 2011).
Alternatives:
1. Geometric Cutoff: Restrict to observers within a Hubble volume (Vilenkin, 2007).
2. Bayesian Weighting: Apply entropy-based priors P()eSdS P(L)andSdS
Derivation of de Sitter Entropy
The entropy of a de Sitter horizon with cosmological constant LL is derived from the Gibbons-Hawking formula:SdS=A4G=RH2G=3MPl2, SdS=4GA=GRH2=L3MPl2
where RH=3/L RH=3/L is the horizon radius. For string theory compactifications, corrections arise from higher-dimensional effects: SdSstring=22VMPl2+O(gs), SdSstring=L2p2VMPl2
+O(gs), with Vls6 Vls6 the compactification volume.
Slow-Roll Consistency Check:
For the KKLT potential V()=V0Aea V()=V 0 But and , the slow-roll parameters must satisfy:MPl216(VV)2a2A2e2a16V021, 16pMPl2(VV)216V02a2A2and2and 1,
MPl28VVa2Aea8V01. or8pMP2VV8V0a2Butand 1.
For 10MP 10MPl, A105MPl4 A105MPl4, and a0.1 a0.1, we find 106 106, 0.01 or0.01, validating slow-roll.
