Multiverse proponents are often vague about how the parameter values are chosen across the defined ensemble. If there is a 'law of laws' describing how parameter values are assigned as one slips from one universe to the next, then we have only shifted the problem of cosmic biophilicity up one level. Why? First, because we need to explain where the law of laws comes from. But there is a second problem. Each law of laws specifies a different version of the multiverse, and not all multiverses are bound to contain at least one biophilic universe. In fact, on the face of it, most multiverses would not contain even one component universe in which all the parameter values were suitable for life. To see this, note that each parameter will have a small range of values - envisage it as a highlighted segment on a line - consistent with biology. Only in universes where all the relevant highlighted segments intersect in a single patch (i.e. all biophilic values are instantiated together) will biology be possible. If the several parameters vary independently between universes, each according to some rule, then for most sets of rules the highlighted segments will not concur. So we must not only explain why there is any law of laws; we must also explain why the actual law of laws (i.e. the actual multiverse) happens to be one that intersects the requisite patch of parameter space that permits life.
Often it is asserted that there is no law of laws, only randomness. Thus in Smolin's version of the multiverse, gravitational collapse events 'reprocess' the existing laws with small random variations . In this case, given an infinite multiverse, randomness would ensure that at least one biophilic universe exists with a finite (albeit minute) probability. (That is, there will always be a patch of parameter space somewhere with all highlighted segments intersecting.) Plausible though this is, the assumption of randomness is not without its problems. Without a proper measure over the parameter space, probabilities cannot be properly defined. There is a danger of predicting meaningless or paradoxical results. There is also a danger in some multiverse models that the biophilic target universes may form only a set of measure zero in the parameter space, and thus be only infinites-imally probable. Furthermore, in some models, various randomness measures may be inconsistent with the underlying physics. For example, in the model of a single spatially infinite Universe in which different supra-Hubble regions possess different total matter densities, it is inconsistent to apply the rule that any value of the density may be chosen randomly in the interval [0, p], where p is some arbitrarily large density (e.g. the Planck density). The reason is that for all densities above a critical value (very low compared with the Planck density), the Universe is spatially finite, and so inconsistent with the assumption of an infinite number of finite spatial regions .
The need to rule out these 'no-go' zones of the parameter space imposes restrictions on the properties of the multiverse that are tantamount to the application of an additional overarching biophilic principle. There would seem to be little point in invoking an infinity of universes only then to impose biophilic restrictions at the multiverse level. It would be simpler to postulate a single universe with a biophilic principle.
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