Testing typicality in multiverse cosmology

Feraz Azhar

Phys. Rev. D 91, 103534 – Published 27 May 2015

Abstract

In extracting predictions from theories that describe a multiverse, we face the difficulty that we must assess probability distributions over possible observations prescribed not just by an underlying theory, but by a theory together with a conditionalization scheme that allows for (anthropic) selection effects. This means we usually need to compare distributions that are consistent with a broad range of possible observations with actual experimental data. One controversial means of making this comparison is by invoking the “principle of mediocrity”: that is, the principle that we are typical of the reference class implicit in the conjunction of the theory and the conditionalization scheme. In this paper, we quantitatively assess the principle of mediocrity in a range of cosmological settings, employing “xerographic distributions” to impose a variety of assumptions regarding typicality. We find that for a fixed theory, the assumption that we are typical gives rise to higher likelihoods for our observations. If, however, one allows both the underlying theory and the assumption of typicality to vary, then the assumption of typicality does not always provide the highest likelihoods. Interpreted from a Bayesian perspective, these results support the claim that when one has the freedom to consider different combinations of theories and xerographic distributions (or different “frameworks”), one should favor the framework that has the highest posterior probability; and then from this framework one can infer, in particular, how typical we are. In this way, the invocation of the principle of mediocrity is more questionable than has been recently claimed.

Received 19 February 2015

DOI:https://doi.org/10.1103/PhysRevD.91.103534

Authors & Affiliations

Feraz Azhar^*

Department of History and Philosophy of Science, University of Cambridge, Free School Lane, Cambridge CB2 3RH, United Kingdom

^*feraz.azhar@alumni.physics.ucsb.edu

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 91, Iss. 10 — 15 May 2015

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(Left) First-person likelihoods for our data $D_{0}$ , given a theory $T$ and a xerographic distribution $ξ_{c}^{typD}$ , for a multiverse of cycles with a total of 3 cycles (see Sec. 3d). The theories considered include all possible theories assigning the observable red/blue to each cycle (for a total of $2^{3}$ theories), and the xerographic distributions can be nonzero only up to (and including) some terminal cycle (corresponding to the final element of $c$ ). We implement the assumption that we are typical with respect to other instances of our data up to (and including) that terminal cycle (for a total of 3 possible xerographic distributions; see C1 of Sec. 3a). Likelihoods are shown as a function of $p$ , the probability of the existence of observers in each cycle, which is here assumed to be the same for each cycle. Likelihoods for all possible frameworks (found by varying $T$ and $c$ —a total of 24 frameworks) have been superimposed on the plot. In this case, the existence of equivalent functional forms for likelihoods for different frameworks gives rise to 4 distinct traces (including likelihoods which are zero for all values of $p$ ). The uppermost trace corresponds only to the framework $(T_{all red}, ξ^{PM} ≔ ξ_{V}^{typD})$ , and is the supremum of the likelihoods for all possible values of $p$ . (Right) This plot displays similar information as in the left panel, but for the xerographic distribution that assumes we are typical of all observers, regardless of what data they see (C2 of Sec. 3a). Here, the existence of equivalent functional forms for likelihoods for different frameworks gives rise to 7 distinct traces (again, including likelihoods which are zero for all values of $p$ ). The uppermost trace is the same as the uppermost trace for the left panel, and corresponds only to the framework $(T_{all red}, ξ_{V}^{typO})$ .
Reuse & Permissions
Figure 2
The multiverse of cycles introduced in Sec. 3d with $N = 3$ cycles. Each subplot shows $[P^{(1 p)} (D_{0} | T, ξ^{PM}) - P^{(1 p)} (D_{0} | T, ξ_{c}^{typX})]$ vs $p$ (where X is either D or O). The term in square brackets is the difference between the first-person likelihood for the framework that implements the principle of mediocrity, namely ( $T, ξ^{PM} ≔ ξ_{V}^{typD}$ ), and the likelihood for another framework specified by varying only the xerographic distribution in accord with the cutoff procedure described in the last paragraph of Sec. 4a. The variable $p$ (labeling the $x$ axis of each subplot) is the probability of the occurrence of observers in any cycle (and is assumed to be the same in each cycle). The theory is shown on the left of each row, with 1 or 0 respectively denoting the prediction of red or blue in the corresponding cycle. The xerographic distribution against which we are comparing the principle of mediocrity is shown on the top of each column (traces referring to $ξ_{c}^{typO}$ are shown in grey to guide the eye). We see that for each row, that is, for a fixed theory, the difference is non-negative for all possible values of $p$ . It is in this sense that we claim the best xerographic distribution for a fixed theory is that corresponding to the principle of mediocrity.
Reuse & Permissions
Figure 3
(Left) First-person likelihoods for our data $D_{0}$ , given a particular theory $T$ and xerographic distributions $ξ_{c}^{typX}$ (where X is either D or O), vs $p$ , for the multiverse of cycles introduced in Sec. 3d for $N = 3$ . The likelihoods for three separate frameworks are shown. The black line shows the likelihood for the framework ( $T_{one red}, ξ^{PM} ≔ ξ_{V}^{typD}$ ), where the theory predicts a single red cycle and the xerographic distribution corresponds to the principle of mediocrity [ $P^{(1 p)} (D_{0} | T_{one red}, ξ^{PM}) = p$ ]. The uppermost grey trace corresponds to the framework $(T_{R R B}, ξ_{c = {1, 2}}^{typO})$ , where the theory predicts only the first two cycles are red [ $T_{R R B} \equiv (1, 1, 0)$ ], and the xerographic distribution corresponds to typicality with respect to observers considered among only the first two cycles [ $P^{(1 p)} (D_{0} | T_{R R B}, ξ_{c = {1, 2}}^{typO}) = p (2 - p)$ ]. We note that this framework does better than that containing a xerographic distribution implementing the principle of mediocrity. The lower grey trace corresponds to the likelihood for the framework ( $T_{two red}, ξ_{V}^{typO}$ ), whose theory predicts that some two of the cycles are red, and that we are typical with respect to observers considered over the entire multiverse [ $P^{(1 p)} (D_{0} | T_{two red}, ξ_{V}^{typO}) = \frac{2}{3} p (p^{2} - 3 p + 3)$ ]. This trace crosses the likelihood for ( $T_{one red}, ξ^{PM}$ ), implying a parametric dependence of the dominance of the framework implementing the principle of mediocrity. The crossover point of this dependence occurs for $p \sim 0.63$ . (Right) Similar results for the case $N = 4$ . Here, the black trace shows the likelihood for the framework implementing the principle of mediocrity with $P^{(1 p)} (D_{0} | T_{two red}, ξ^{PM}) = p (2 - p)$ . The uppermost trace corresponds to $P^{(1 p)} (D_{0} | T_{R R R B}, ξ_{c = {1, 2, 3}}^{typO}) = p (p^{2} - 3 p + 3)$ [where $T_{R R R B} \equiv (1, 1, 1, 0)$ ]. The lower grey trace corresponds to the likelihood $P^{(1 p)} (D_{0} | T_{three red}, ξ_{V}^{typO}) = 3 p (4 - 6 p + 4 p^{2} - p^{3}) / 4$ . In this latter case, the principle of mediocrity is only dominant for $p > \sim 0.42$ .
Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology