Why Anthropic Reasoning Cannot Predict $\Lambda$

We revisit anthropic arguments purporting to explain the measured value of the cosmological constant. We argue that different ways of assigning probabilities to candidate universes lead to totally different anthropic predictions. As an explicit example, we show that weighting different universes by the total number of possible observations leads to an extremely small probability for observing a value of $\Lambda$ equal to or greater than what we now measure. We conclude that anthropic reasoning within the framework of probability as frequency is ill-defined and that in the absence of a fundamental motivation for selecting one weighting scheme over another the anthropic principle cannot be used to explain the value of $\Lambda$, nor, likely, any other physical parameters.

Figure 1

Anthropically predicted probability density distribution as a function of $R$, the ratio of the cosmological constant in another part of the multiverse to the value it takes in our Universe, in our MANO scheme and for different values of the parameter $\tau$ controlling the cosmic time when intelligent life emerges (in units of 13.7 Gyr, with $\tau =1$ being the case of our Universe). The probability of observing a value of $\Lambda$ equal to or greater than what we measure (dashed vertical line) is very small as all the weight lies close the minimum value.