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 \to \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 .
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 ( \mathinner { \mathcal { T } _ { + } \mathopen { \left ( \Lambda _ { 0 } \right ) } } \sim 23 \% ) , this model gives a lower anthropic likelihood of \Lambda _ { 0 } ( \mathinner { \mathcal { T } _ { + } \mathopen { \left ( \Lambda _ { 0 } \right ) } } \sim 5 \% ) .
On the other hand , the anthropic likelihood of the primordial density perturbation amplitude Q _ { 0 } from this model is high ( \mathinner { \mathcal { T } _ { + } \mathopen { \left ( Q _ { 0 } \right ) } } \sim 63 \% ) , while the likelihood from Weinberg ’ s model is low ( \mathinner { \mathcal { T } _ { + } \mathopen { \left ( Q _ { 0 } \right ) } } \ll 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 \mathinner { \mathcal { T } _ { + } \mathopen { \left ( \Lambda _ { 0 } \right ) } } \sim 10 \% , while the scale factor cutoff measure and the causal patch measure give \mathinner { \mathcal { T } _ { + } \mathopen { \left ( \Lambda _ { 0 } \right ) } } \gtrsim 30 \% .