Anthropic Likelihood for the Cosmological Constant and the Primordial
  Density Perturbation Amplitude
  Weinberg et al. calculated the anthropic likelihood of the cosmological
constant 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 \rightarrow \infty. 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. Compared to Weinberg's model, this model gives a lower anthropic
likelihood of \Lambda_0 (T_+(\Lambda_0) ~ 5%). On the other hand, the anthropic
likelihood of the primordial density perturbation amplitude from this model is
high, while the likelihood from Weinberg's model is low. 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) ~ 10%, while the scale
factor cutoff measure and the causal patch measure give T_+(\Lambda_0) \geq
30%.
