PHYSICAL REVIEW D, vol.85, no.8, 2012 (SCI-Expanded)
Weinberg et al. calculated the anthropic likelihood of the cosmological constant Lambda using a model assuming that the number of observers is proportional to the total mass of gravitationally collapsed objects, with mass greater than a certain threshold, at t -> infinity. We argue that Weinberg's model is biased toward small Lambda, and to try to avoid this bias we modify his model in a way that the number of observers is proportional to the number of collapsed objects, with mass and time equal to certain preferred mass and time scales. The Press-Schechter formalism, which we use to count the collapsed objects, identifies our collapsed object at the present time as the Local Group, making it inconsistent to choose the preferred mass scale as that of the Milky Way at the present time. Instead, we choose an earlier time before the formation of the Local Group and this makes it consistent to choose the mass scale as that of the Milky Way. Compared to Weinberg's model (T+(Lambda(0)) similar to 23%), this model gives a lower anthropic likelihood of Lambda(0) (T+(Lambda(0)) similar to 5%). On the other hand, the anthropic likelihood of the primordial density perturbation amplitude Q(0) from this model is high (T+(Q(0)) similar to 63%), while the likelihood from Weinberg's model is low (T+(Q(0)) << 0.1%). Furthermore, observers will be affected by the history of the collapsed object, and we introduce a method to calculate the anthropic likelihoods of Lambda and Q from the mass history using the extended Press-Schechter formalism. The anthropic likelihoods for Lambda and Q from this method are similar to those from our single mass constraint model, but, unlike models using the single mass constraint which always have degeneracies between Lambda and Q the results from models using the mass history are robust even if we allow both Lambda and Q to vary. In the case of Weinberg's flat prior distribution of Lambda (pocket-based multiverse measure), our mass history model gives T+(Lambda(0)) similar to 10%, while the scale factor cutoff measure and the causal patch measure give T+(Lambda(0)) greater than or similar to 30%.