These two possibilities send us back to the large number hypothesis by Dirac  that has been used as an early motivation to investigate theories with varying constants. The main concern was the existence of some large ratios between some combinations of constants. As we have seen in Section 5.3.1, the running of coupling constants with energy, dimensional transmutation or relations such as Equation (185) have opened a way to a rational explanation of very small (or very large) dimensional numbers. This follows the ideas developed by Eddington [178, 179] aiming at deriving the values of the constants from consistency relations, e.g., he proposed to link the fine-structure constant to some algebraic structure of spacetime. Dicke  pointed out another possible explanation to the origin of Dirac large numbers: the density of the universe is determined by its age, this age being related to the time needed to form galaxies, stars, heavy nuclei…. This led Carter  to argue that these numerical coincidence should not be a surprise and that conventional physics and cosmology could have been used to predict them, at the expense of using the anthropic principle.
The idea of such a structure called the multiverse has attracted a lot of attention in the past years and we refer to  for a more exhaustive account of this debate. While many versions of what such a multiverse could be, one of them finds its root in string theory. In 2000, it was realized  that vast numbers of discrete choices, called flux vacua, can be obtained in compactifying superstring theory. The number of possibilities is estimated to range between 10100 and 10500, or maybe more. No principle is yet known to fix which of these vacua is chosen. Eternal inflation offers a possibility to populate these vacua and to generate an infinite number of regions in which the parameters, initial conditions but also the laws of nature or the number of spacetime dimensions can vary from one universe to another, hence being completely contingent. It was later suggested by Susskind  that the anthropic principle may actually constrain our possible locations in this vast string landscape. This is a shift from the early hopes  that M-theory may conceivably predict all the fundamental constants uniquely.
Indeed such a possibility radically changes the way we approach the question of the relation of these parameters to the underlying fundamental theory since we now expect them to be distributed randomly in some range. Among this range of parameters lies a subset, that we shall call the anthropic range, which allow for universe to support the existence of observers. This range can be determined by asking ourselves how the world would change if the values of the constants were changed, hence doing counterfactual cosmology. However, this is very restrictive since the mathematical form of the law of physics managed as well and we are restricting to a local analysis in the neighborhood of our observed universe. The determination of the anthropic region is not a prediction but just a characterization of the sensitivity of “our” universe to a change of the fundamental constants ceteris paribus. Once this range is determined, one can ask the general question of quantifying the probability that we observe a universe as ours, hence providing a probabilistic prediction. This involves the use of the anthropic principle, which expresses the fact that we observe are not just observations but observations made by us, and requires us to state what an observer actually is .
Living Rev. Relativity 14, (2011), 2
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