Spectra of conditionalization and typicality in the multiverse

An approach to testing theories describing a multiverse, that has gained interest of late, involves comparing theory-generated probability distributions over observables with their experimentally measured values. It is likely that such distributions, were we indeed able to calculate them unambiguously, will assign low probabilities to any such experimental measurements. An alternative to thereby rejecting these theories, is to conditionalize the distributions involved by restricting attention to domains of the multiverse in which we might arise. In order to elicit a crisp prediction, however, one needs to make a further assumption about how typical we are of the chosen domains. In this paper, we investigate interactions between the spectra of available assumptions regarding both conditionalization and typicality, and draw out the effects of these interactions in a concrete setting; namely, on predictions of the total number of species that contribute significantly to dark matter. In particular, for each conditionalization scheme studied, we analyze how correlations between densities of different dark matter species affect the prediction, and explicate the effects of assumptions regarding typicality. We find that the effects of correlations can depend on the conditionalization scheme, and that in each case atypicality can significantly change the prediction. In doing so, we demonstrate the existence of overlaps in the predictions of different “frameworks” consisting of conjunctions of theory, conditionalization scheme and typicality assumption. This conclusion highlights the acute challenges involved in using such tests to identify a preferred framework that aims to describe our observational situation in a multiverse.

Figure 1

Contour plots of the distribution $Q({\eta }_1,{\eta }_2|\mathcal{T})$ [see Eq. (20)]. In each panel, the black line denotes the constraint surface, and the red line denotes the line of equal density. Along the constraint surface, blue circles correspond to (global) maxima and red squares to (local) minima. Note that in (b), (c), (e), and (f), pairs of maxima have the same probability in each panel. Parameters have been set as follows: ${\eta }_{\mathrm{obs}}=5$, ${\eta }_1^{\star }=3.6={\eta }_2^{\star }$, and $\alpha =-0.5$. (a)–(c) $\overline{\sigma }={\eta }_{\mathrm{obs}}/6$ with $\mu =0,0.1$, and 1 respectively. (d)–(f) $\overline{\sigma }={\eta }_{\mathrm{obs}}/5$ with $\mu =0,0.1$, and 1 respectively. The Gaussian cases (a), (d) exhibit a single maximum corresponding to equal contributions to the total dark matter density from the two components (as discussed in Sec. 3a). The equality of contribution is overturned in a significant way for $\mu =1$, that is (c), (f), where in each case, the maxima correspond to unequal contributions and the local minimum corresponds to equal contributions.

Figure 2

Change in the prediction for $N$ in the presence of correlations under the assumption of typicality. (a), (b), and (c) exhibit solutions for the choices $X=2.35$, 2.5, and 2.7 respectively. The red circles correspond to $N_0$, the prediction for independent species of dark matter [see Eq. (44)—each of these solutions is not finely tuned under the choice of parameters described herein]. The parabolas correspond to the left-hand side of Eq. (45), whose $x$-axis intercepts are the predictions for $N_{\alpha }$, shown in blue squares, where $\alpha =0.25$ in each case [a prediction of $N_{\alpha }<2$, as for the smaller of the two predictions in (c), is discounted as “unphysical”]. The gray segments overlapping the $x$ axes correspond to the range of solutions that are not finely tuned for the correlated case (namely, the region of the $x$ axis where the constraint given by Eq. (41) is satisfied—recall that $F=1$ under the assumption of typicality, and we have set $\overline{\eta }={\eta }_{\mathrm{obs}}$, and ${\eta }_0^2=X{\overline{\sigma }}^2$). We have set ${\eta }_{\mathrm{obs}}=5$ in accord with the experimentally observed value, and for the sake of illustration, we have set ${\overline{\sigma }}^2=2.8$. We note that in (a) and (b), there exist two, distinct, physically acceptable predictions for $N_{\alpha }$, the greater of which is also significantly different from the case where there are no correlations.

Figure 3

The effects of atypicality on the prediction of $N_{\alpha }$, where $\alpha =0.25$. (a) $X=2.5$, $F=0.875$; (b) $X=2.5$, $F=1.05$; (c) $X=2.7$, $F=0.875$; (d) $X=2.7$, $F=1.05$. In each case, we have also set ${\eta }_{\mathrm{obs}}=5$ and ${\overline{\sigma }}^2=2.8$. For each panel, the intersection of the gray parabola with the $x$ axis corresponds to the prediction under typicality, whereas the intersection of the black parabola with the $x$ axis, marked by blue squares, corresponds to the prediction under the assumption of atypicality (recall that we accept only those solutions for which $N_{\alpha }\ge 2$). The gray segment overlapping the $x$ axis corresponds to the range of solutions for the atypical scenario that are not finely tuned. We see in each case that predictions for $N_{\alpha }$ can shift significantly.