7.3 Anthropic predictions

The determination of the anthropic region for a set of parameters is in no way a prediction but simply a characterization of our understanding of a physical phenomenon P that we think is important for the emergence of observers. It reflects that, the condition C stating that the constants are in some interval, C =⇒ P, is equivalent to !P =⇒!C.

The anthropic principle [82] states that “what we can expect to observe must be restricted by the conditions necessary for our presence as observers”. It has received many interpretations among which the weak anthropic principle stating that “we must be prepared to take account of the fact that our location in the universe in necessarily privileged to the extent of being compatible with our existence as observers”, which is a restriction of the Copernican principle often used in cosmology, and the strong anthropic principle according to which “the universe (and hence the fundamental parameters on which it depends) must be such as to admit the creation of observers within it at some stage.” (see [35] for further discussions and a large bibliography on the subject).

One can then try to determine the probability that an observer measure the value x of the constant X (that is a random variable fluctuating in the multiverse and the density of observers depend on the local value of X). According to Bayes theorem,

P (X = x|obs) ∝ P (obs|X = x)P(X = x ).
P (X = x) is the prior distribution, which is related to the volume of those parts of the universe in which X = x at dx. P (obs|X = x) is proportional to the density of observers that are going to evolve when X = x. P (X = x|obs) then gives the probability that a randomly selected observer is located in a region where X = x ± dx. It is usually rewritten as [519Jump To The Next Citation Point]
P (x)dx = nobs(x)Ppriordx.
This higlights the difficulty in making a prediction. First, one has no idea of how to compute nobs(x ). When restricting to the cosmological constant, one can argue [519] that Λ does not affect microphysics and chemistry and then estimate nobs(x ) by the fraction of matter clustered in giant galaxies and that can be computed from a model of structure formation. This may not be a good approximation when other constants are allowed to vary and it needs to be defined properly. Second, P prior requires an explicit model of multiverse that would generate sub-universes with different values xi (continuous or discrete) for x. A general argument [528] states that if the range over which X varies in the multiverse is large compared to the anthropic region X ∈ [Xmin, Xmax ] one can postulate that Pprior is flat on [Xmin,Xmax ]. Indeed, such a statement requires a measure in the space of the constants (or of the theories) that are allowed to vary. This is a strong hypothesis, which is difficult to control. In particular if Pprior peaks outside of the anthropic domain, it would predict that the constants should lie on the boundary of the anthropic domain [443]. It also requires that there are sufficiently enough values of xi in the anthropic domain, i.e., δxi ≪ Xmax − Xmin. Garriga and Vilenkin [228] stressed that the hypothesis of a flat Pprior for the cosmological constant may not hold in various Higgs models, and that the weight can lower the mean viable value. To finish, one want to consider P (x) as the probability that a random observer measures the value x. This relies on the fact that we are a typical observer and we are implicitly making a self sampling hypothesis. It requires to state in which class of observers we are supposed to be typical (and the final result may depend on this choice [383Jump To The Next Citation Point]) and this hypothesis leads to conclusions such as the doomsday argument that have be debated actively [64, 383].

This approach to the understanding of the observed values of the fundamental constants (but also of the initial conditions of our universe) by resorting to the actual existence of a multiverse populated by a different “low-energy” theory of some “mother” microscopic theory allows us to explain the observed fine-tuning by an observational selection effect. It also sets a limit to the Copernican principle stating that we do not live in a particular position in space since we have to live in a region of the multiverse where the constants are inside the anthropic bound. Such an approach is indeed not widely accepted and has been criticized in many ways [7, 182, 480, 402, 479, 511, 475].

Among the issues to be answered before such an approach becomes more rigorous, let us note: (1) what is the shape of the string landscape; (2) what constants should we scan. It is indeed important to distinguish the parameters that are actually fine-tuned in order to determine those that we should hope to explain in this way [537, 538]. Here theoretical physics is indeed important since it should determine which of the numerical coincidences are coincidences and which are expected for some unification or symmetry reasons; (3) how is the landscape populated; (4) what is the measure to be used in order and what is the correct way to compute anthropically-conditioned probabilities.

While considered as not following the standard scientific approach, this is the only existing window on some understanding of the value of the fundamental constants.

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