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Spectra of conditionalization and typicality in the multiverse ∗ Feraz Azhar Department of History and Philosophy of Science, University of Cambridge, Free School Lane, Cambridge, CB2 3RH, United Kingdom (Dated: January 25, 2016) An approach totesting theories describing amultiverse, that hasgained 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 unam- biguously, will assign low probabilities to any such experimental measurements. An alternative to thereby rejecting these theories, is to conditionalize the distributions involved by restricting atten- 6 1 tion to domains of the multiverse in which we might arise. In order to elicit a crisp prediction, 0 however,one needstomake afurtherassumption about how typicalwe are of thechosen domains. 2 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 n setting; namely, on predictions of the total number of species that contribute significantly to dark a matter. In particular, for each conditionalization scheme studied, we analyze how correlations be- J tween densities of different dark matter species affect the prediction, and explicate the effects of 2 assumptions regarding typicality. We find that the effects of correlations can depend on the con- 2 ditionalization 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” ] O consistingofconjunctionsoftheory,conditionalizationschemeandtypicalityassumption. Thiscon- clusionhighlightstheacutechallengesinvolvedinusingsuchteststoidentifyapreferredframework C that aims todescribe ourobservational situation in a multiverse. . h p I. INTRODUCTION attentiontodomainsthatshareallobservationalfeatures - o that we have thus far measured, except for the quantity tr A central concern regarding contemporary cosmolog- whosevalueweareaimingtopredict[12–16]. Intermedi- s ical theories that describe a multiverse, such as those ate approaches between these two ends of the spectrum a [ involving inflationary scenarios [1–5], possibly in combi- proposetoconditionalizeonsomecharacterizationofour nation with the string theory landscape [6–9], is: how observational situation without demanding that all (rel- 1 do we elicit testable predictions from these theories? A evant)knownfeaturesbeincluded. Thislastapproachis v particularlynaturalclassofpredictionsarethosederived termed “anthropic”, and can be thought of as according 8 3 from theory-generated probability distributions over ob- withCarter’s“weakanthropicprinciple”[17](see Hartle 9 servables, such as parameters of the standard models [18] for a clear discussion). 5 of particle physics and cosmology, or indeed observables There are, as one might expect, inherent difficulties 0 generatedfromthe outcomesofexperimentswehaveyet inimplementing either anthropic ortop-downcondition- 1. to perform. But the task of extracting such predictions alization schemes, arising from how best to characterize 0 has been elusive. “us”intheanthropiccase,orhowtocharacterizeourob- 6 It is expected, owing to the variety of conditions that servationalsituationinapracticablewayinthetop-down 1 are likely to obtain in any multiverse scenario, that a case. Evenifoneisabletoaddresstheseissues,therere- : v theory-generated probability of our observations in such mains a further assumption that needs to be made in i a scenario will turn out to be low. In this case, short of order to extract a crisp prediction. This amounts to an X disfavoring (all) such theories, one can restrict attention assumption regardinghow typicalwe are of the domains r to domains in the multiverse in which our observational that these conditionalization schemes explicitly restrict a situationmightobtain,andthencomparethenew(renor- attentionto. For a renormalizedprobabilitydistribution malized) probability distribution with our observations. function(and,indeed,fordistributionsexhibitingtheap- As described by Aguirre and Tegmark [10] (see propriate shape), this amounts to an assumption about also [11]), this process of conditionalization can occur howfarawayfromthepeakofadistributionwecanallow in a variety of different ways. One possibility, termed our observations to be, while still taking those observa- the “bottom-up”approach,is to not conditionalizeone’s tionstohavebeenpredictedbytheconjunctionoftheory, distribution at all, and correspondsto accepting the raw conditionalization scheme and typicality assumption—a theory-generatedprobability distribution as the primary conjunction we will refer to as a framework, in accord means of generating a prediction. At the opposite end with the terminology of Srednicki and Hartle [14]. of the spectrum, “top-down” conditionalization restricts What typicality assumptions one should support is controversial. There are essentially two camps: those who assert that we should always assume typicality in the context of an appropriately conditionalized the- ∗ Emailaddress: [email protected] ory, that is, those who support the “principle of medi- 2 ocrity”[13,19–22],andthosewhoseetypicalityasanas- shown to be η 5 [29]. obs ≈ sumptionthatcanbesubjecttoerror,andthattherefore The space in which ~η will take values will be referred we should not necessarily demand typicality, whatever to as “parameter space”. The variation of this vector of our specification of the conditionalized theory [14, 23– densities from one domain to the next is described by 26]. If one allows for the latter possibility, then one is a probability distribution P(~η ). The construction of |T facedwithaspectrumofpossibletypicalityassumptions. suchprobabilitydistributionsisadifficult,openproblem, Givensometheory,itisthenbysomeappropriatechoice andtomakeprogresswewillspecifysimple,exampledis- ineach ofthespectraofconditionalizationand typicality tributions as we proceed. We begin then by considering that one must extract predictions. the least restricted case: that of bottom-up conditional- In this paper, we take seriously the need to consider ization. the existence of these two spectra, and investigate the manner in which they interact, for a range of conditionalization schemes and typicality assumptions. This in- II. BOTTOM-UP CONDITIONALIZATION vestigation is carried out in the concrete context of an attempt to predict the total number of species that con- According to bottom-up conditionalization, one as- tribute significantly to dark matter. In particular, we sumesthat the rawprobabilitydistributionP(~η )con- extendtheworkofAguirreandTegmark[10]byconsider- |T stitutes the primary means of generating predictions. ingcaseswhereprobabilitydistributionsoverdensitiesof We will assume, following the general line of argument dark matter species can be correlated, and then analyze inAguirreandTegmark[10],thatinprinciple,the range the effects of bottom-up (Sec. II), top-down (Sec. III), in which each of the component densities η in ~η could i and anthropic (Sec. IV) conditionalization schemes, in take values is large (and is the same for each species additiontotheeffectsofatypicalityineachofthesecases. i). Assume also that the joint probability distribution We find that (i) atypicality can significantly change the P(~η )isunimodal,thatis,hasasinglepeakthatcould prediction in each case studied in such a way that (ii) |T fallanywhereinthe rangeoverwhichP(~η )couldpos- different frameworks can overlap, as regards their pre- |T sibly be significant [in this section, we make no assump- dictions; that is, different frameworks can lead to the tions regarding the nature of any correlations between same prediction for the number of dominant species of the component densities for any particular P(~η )]. In dark matter (Sec. V). These results leave open the chal- |T the absenceofanyfurtherinformation,weareinterested lenge,inmorerealisticsettings,ofconstructingthesesets in the following two questions: (i) how many of the N of equivalent frameworks (as judged by the equivalence components share the highest occurring density, and (ii) oftheirpredictions);while alsohighlightinghowdifficult howdoes this predictiondependupon the assumptionof it may be to use suchtests to identify a preferredframe- typicality? work. To make this problem tractable, let us discretize the range over which each of the densities could take values into M equal-sized bins, such that the central density of A. The general cosmological setting each bin is significantly different from its neighbors. We thus have an N-dimensional grid containing MN boxes, We begin by briefly outlining the generalcosmological where we assume the peak of P(~η ) is equally likely |T scenario within which we will be working. The current to fall into any box, and we are interested first [i.e., in favoredtheory regardingthe compositionofdark matter (i) above], in the probability that a total of j of the N does not rule out the possibility of multiple (new, non- components share the highest occupied density box. To baryonic) particle species contributing to the total dark be clear, a particular will indeed give rise to a single T matterdensity[10,27,28]. Thegeneralargumentofthis P(~η ) where the peak of this distribution will have a |T paper will build upon this possibility: we will assume single location in the grid—we are looking into the sit- that some theory describes a multiverse consisting of uation where we have no further information about the T distinct domains, in each of which a total of N distinct location of this peak, and are interested in predictions species of dark matter can exist, but where the relative aboutwherethepeakwilllie,assumingthatitisequally contributions of each of these species to the total dark likely to fall into any of the boxes we have constructed. matter density can vary from one domain to the next. It should be intuitively clear that for N 2 and high ≥ The densities of each of these components will be given enough M (i.e., M N), the chance of the peak falling ≫ by a dimensionless dark-matter-to-baryon ratio (we will along the equal density diagonal of the N-dimensional also assume that the density of baryons can vary from grid is small. We can formalize this intuition with one domain to the next), with the density of component thefollowingamendmenttothecorrespondingargument igivenbyηi Ωi/Ωb,sothatthedensitiesofallN com- in Aguirre and Tegmark [10, Sec. 3.2]. In this amended ≡ ponents are represented by ~η = (η1,η2,...,ηN). Note argument,thefinalresultweobtainfortheprobabilityof that our observations currently constrain the total dark j components sharing the highest occupied density box, N matter density η η . From results recently re- namely (j),isdifferentfromtheirresult[theirequation obs ≡ i=1 i P leased by the Planck collaboration, this quantity can be (1)],buttheoverallconclusionoftheanalysisofbottom- P 3 up conditionalization remains the same. this is indeed the case. We find for j : h i N j j (j) h i≡ P j=1 X N M−1 1 N The problem as stated in the previous paragraph can = j kN−j +δ MN j j,N be recast in the following (dimensionally-reduced) form j=1 (cid:18) (cid:19)"k=1 # X X where we consider M distinguishable bins, correspond- M−1 N 1 N N ing to the discretized densities in the range over which = j kN−j + any dark matter species can take values, and N distin- MN j MN k=1 j=1 (cid:18) (cid:19) guishableballs,wherethei’thballrepresentsthepeakof X X M−1 N the i’thmarginaldistributionPi(ηi|T). Ourassumption = N N −1 kN−j + N that the peak of P(~η ) is equally likely to fall into any MN j 1 MN |T k=1 j=1(cid:18) − (cid:19) boxisequivalenttothestatementthattheprobabilityof X X M−1 any ball falling into any bin is the same. Let (j) rep- N P = (1+k)N−1+1 resent the probability that exactly j of the N balls fall MN " # intothehighestoccupieddensitybin. Thenthefollowing kX=1 closed-formexpression,obtainedthroughasimplecount- N M = kN−1, (2) ing argument gives us the required probability (j): MN P k=1 X M−1 (j)= 1 N kN−j +δ , (1) where the binomial theorem has been used in obtaining P MN(cid:18)j(cid:19)"Xk=1 j,N# (tfhoermfifatlhlyl:infeo.rFfioxredMN≫, liNm,Mo→ne∞cjan=sho1w).thInatthhjeic∼as1e h i where δ is the Kronecker delta function. where N M, j can take values much greater than ≫ h i 1, that is, it is possible for multiple components to dom- inate (formally: for fixed M, limN→∞ j = ). The h i ∞ upshot is that as long as the range over which each of the dark matter densities can take values, namely M, is muchlargerthanthetotalnumberofdarkmatterspecies To understand where this result comes from, consider under consideration, namely N, the average number of the case where some j <N balls share the highest occu- species sharing the highest occurring density will be 1. pied density bin. If that bin is the k’th from the lowest densitybinoftheM possiblebins(where1 k M 1), ≤ ≤ − This result has been derived under the assumption of thenalltheremainingN j ballscanbe arrangedinthe k lower density bins in k−N−j ways. The sum in Eq. (1) typicality, in that the peak of the joint distribution dic- tates the prediction. To be clear, the average in Eq. (2) correspondstothesumoverallpossiblechoicesofk. The istakenoverallpossiblelocationsofthepeakofP(~η ), prefactor N just counts the number of ways of select- |T j andassumesthatwedonotinfactknowwherethismight ingexactlyj oftheN balls. Thisproductisthendivided (cid:0) (cid:1) be. Of course, for any fixed , the peak of the joint dis- by the total number of possible arrangements of balls in T tributionP(~η ) willbe locatedina singlebox,andthe bins, i.e., MN, giving the appropriate probability. The |T argument following Eq. (2) indicates that this box will Kronecker delta function keeps track of the particular probably correspond to a single dominant dark matter case where j = N, in which case there exists an extra component. arrangementin which the j balls sharing the highest occupied density bin (i.e., all N of them) can indeed be Indeed, irrespective of the location of the peak of any placed in the lowest density bin. It is straightforwardto particulardistributionP(~η ),therewillpresumablyex- show that this distribution is appropriately normalized: |T N ist directions in parameter space in which more than a (j)=1. j=1P single component would contribute significantly. There- P foreatypicalitycanleadtoarangeofdifferentpredictions for the total number of species that contribute significantly. Of course, we are constrained here in that only those theories (and their associated typicality assump- Our intuition that the chance is small of the peak of tions)thatpredictatotaldensitythatagreeswithourob- N P(~η )fallingalongtheequaldensitydiagonalintheN- served value, i.e., that satisfy η =η 5, would |T i=1 i obs ≈ dimensional grid (i.e., in the dimensionally-reduced de- be favored. Nevertheless, the combination of bottom- P scription in the paragraph above, of all N balls falling up conditionalization and atypicality presents us with a into the same bin), suggests that j 1. We will out- large amount of freedom regarding predictions that may h i ∼ line how for the most likely relative values of N and M, arise. 4 III. TOP-DOWN CONDITIONALIZATION Multiplying through by and summing over k gives kj C N Our arguments thus far have been rather general, and η η⋆ = λ . (7) it will be instructive in whatfollows to restrictattention j − j − Ckj toparticulardistributionssoastoextractmoreconcrete kX=1 predictions. We turn now to the most restrictive type of By summing the N equations implicit in Eq. (7), rear- conditionalization scheme—that of top-down condition- ranging, and using the constraint [Eq. (4)], we can solve alization. for the Lagrange multiplier: Consider, again, the case where we have a total of N species of dark matter and we are interested in ascer- N η⋆ η taining the total number of species that contribute sig- λ= j=1 j − obs. (8) N nificantly tothe totalobserveddarkmatter density ηobs. P k,j=1Ckj Assume that the joint probability distribution function P Substituting Eq. (8) into Eq. (7) we find that the max- givensometheory islocallyGaussian(nearη ),with obs T imum of P(~η ) subject to the top-down constraint oc- |T curs at N 1 P(~η ) exp (η η⋆)( −1) (η η⋆) , |T ∝ −2 i− i C ij j − j  η N η⋆ N where each componentiX,jh=a1s substantial probabilityne(a3r) ηi =ηi⋆+ obs−Nk,lP=1jC=k1l j!mX=1Cmi. (9) η , i.e., for each i, η⋆ η (where we allow for some A judicious choice of tPhe η⋆’s and/or the sums of the tooblesrance in the precisie∼relaotbiosnshipbetween η⋆ and η columns of the covariance miatrix , therefore, can lead i obs C here), and is the covariance matrix (a symmetric, pos- to substantial contributions by less than all N species. C itive definite, N N matrix). This matrix, of course, × However, this conclusion is overturned, i.e., all N encodes potential correlations between each of the com- species contribute equally, in the case where (i) no sym- ponents. Notethatif δ thentherighthandsideof Cij ∝ ij metriesarebrokenwithregardtothelocationofthepeak Eq. (3) reduces to a product of independent Gaussians. of the joint probability distribution, namely, if for each Top-down conditionalization in this scenario amounts i, todemandingthatthepredictionextractedfromthisdis- tribution agrees with the totality of our data regarding η⋆ =η¯ (10) i dark matter (see Sec. IA); namely, that the sum over the densities of dark matter components agrees with the for some η¯ η , and (ii) we choose an appropriate obs total observed dark matter density, that is: functional fo∼rm for the covariance matrix. In particular, let us assume that the N N covariancematrix is given N by ˜where × ηi =ηobs. (4) C Xi=1 1 α α ··· α 1 α Gityenaemraotuinngtsatoprfienddicitnigon~ηusnudcehrtthhaetaEssqu.m(3p)tiiosnmoafxtiympiizceadl-, C˜=σ¯2... ... ·.·.·. ... (11) subjecttoEq.(4),anditistothistaskthatwenowturn.   α α 1  ···    with α ( (N 1)−1,1), so that ˜ is indeed positive A. Typicality ∈ − − C definite, and σ¯ is a free parameter. This choice fixes all variances to be the same, and all pairs of covariances to We proceedasinAguirreandTegmark[10],andfocus be the same, that is, ontheconstrainedoptimizationprobleminwhichweop- timize the logarithm of the distribution P(~η ), that is, (ηi η¯)2 =σ¯2 i, (12) |T h − i ∀ we aim to maximize (η η¯)(η η¯) =ασ¯2 i=j. (13) i j h − − i ∀ 6 N Then substituting Eq. (10) and Eq. (11) into Eq. (9) I(~η) lnP(~η ) λ η , (5) i gives, for each i, ≡ |T − i=1 X η Nη¯ where λ is a Lagrange multiplier. A quick calculation ηi =η¯+ Nσ¯2+oNbs(−N 1)ασ¯2 σ¯2+(N −1)ασ¯2 shows that for each k, ∂I(~η)/∂ηk =0 when (cid:20)1 − (cid:21)(cid:2) (cid:3) =η¯+ (η Nη¯) obs N N − (η η⋆)( −1) = λ. (6) 1 i− i C ik − = ηobs. (14) i=1 N X 5 So for a rather simple probability distribution [defined surface by demanding that by Eqs. (3), (10) and (11)] with (marginal) probability 1 distributions over distinct dark matter species that are, η′ =ǫ η , (17) 1 N obs in effect, the same: the most probable,i.e., most typical, ǫ 1 solution to the ensuing constrained maximization prob- η′ = 1 η j =1, (18) lem is that all N components contribute equally to the j − N N 1 obs ∀ 6 (cid:16) (cid:17) − total dark matter density. for0 ǫ N. Noticethat N η′ =η ,andspecies1 ≤ ≤ i=1 i obs This extends the argument of Aguirre and Tegmark dominates when ǫ N. The parameterizationis chosen → P ′ [10, Sec. 3.3] to the case where correlations are now insuchawaythatanyexcessinη overthemostprobable 1 explicitly built into the joint probability distribution value, η1′ = N1ηobs, is drawn equally from among the P(~η ). Note that the results above include the case of remaining N 1 components. We gauge the degree of |T − probabilistically independent dark matter species, that typicality (and thereby the degree of atypicality) by the ′ ′ ′ is, for the special case of α=0 we would againfind that ratio of the probability PE(~η ), with ~η ~η (ǫ) using Eq. (14) holds. the parameterization in Eqs.|(T17) and (18≡), to PMAX. A lengthy (but straightforward) calculation reveals that So how does atypicality affect the picture? We will this ratio is given by now show that under an assumptionof atypicality, there exists a region in parameter space where just a single P (~η′(ǫ) ) η 2 (ǫ 1)2 E obs |T =exp − , (19) dark matter component dominates. That is, atypicality PMAX −2σ¯2(1 α)N(N 1) can dramatically change the prediction. (cid:26) − − (cid:27) where as before, α describes correlations between different species [cf. Eq. (13)]. We point out two interesting features of the result in Eq. (19): (i) The dominance of species 1 indeed relies on an as- sumptionofatypicality. Toillustrate this,consider B. Atypicality the case where there exist two total dark matter components (N = 2) where the density of species We focus on the case where the underlying probabil- 1 is three times that of species 2 (ǫ=3N/4). If we ity distribution is given by Eq. (3), but is subject to the further assume that σ¯ = ηobs/5 and α = 1/2, the assumption that the peak of the distribution does not degreeoftypicalitythatachievesthis dominanceis privilegeanydarkmatterspecies,thatis,Eq.(10)holds, small: PE(~η′(ǫ) )/PMAX =exp( 25/8) 0.04; |T − ≈ and the covariance matrix is given by Eq. (11)—these (ii) Correlationsaffectthedegreeoftypicalityrequired lattertwoassumptionsaremadeinordertomitigateob- toachievethesamedominance. Henceintheexam- vious biases in the search for a single dominant species. ple in(i)above: settingα=0[i.e.,nocorrelations, We will label the resulting distribution P (~η ) (‘E’ for E |T cf. Eq.(13)],whilekeepingallotherparametersthe ‘equal’); so that, from Eqs. (3), (10) and (11), same, gives P (~η′(ǫ) )/PMAX = exp( 25/16) E |T − ≈ 0.21. N 1 PE(~η ) exp (ηi η¯)(˜−1)ij(ηj η¯) . In this way, an assumption of atypicality can change the |T ∝ −2 − C −   iX,j=1  prediction from N equally dominant components to a (15) single dominant component. To highlight the effects of atypicality, we will lookfor a regioninparameterspaceawayfromthemaximumofthe probabilityonthe constraintsurface[whichoccurswhen C. Non-Gaussianities and typicality assumptions Eq. (14) is satisfied], while remaining on the constraint surface [i.e., respecting Eq. (4)]. So, from Eq. (14), the Thus far, we have focussed on the Gaussian case. But maximum of the probability on the constraint surface, it is interesting to explore the situation where we explic- denoted by PMAX, is given by itlybreakthis Gaussianity—foraswewillshow,this can change predictions under various assumptions regarding N PMAX P η = 1 η . (16) typicality. In particular, we will show that our earlier E i obs ≡ (cid:26) N (cid:27)i=1(cid:12)(cid:12)T! conclusion, which established equal contributions to the (cid:12) total dark matter density under typicality, as expressed Our excursion on the constraint surface w(cid:12)ill explore the in Eq. (14), can be overturnedwhen we break Gaussian- (cid:12) possibilityofjustasinglespeciesdominating,andwewill ity. takethatspeciestobethefirstspecies(thoughofcourse, We will establish this result numerically and by con- nothing physical depends on this choice). Hence, follow- struction, in the case where N =2. We will assume that ing Azhar [25, Sec. 3.2.2], we remain on the constraint the underlying distribution Q(η ,η ) is non-Gaussian, 1 2 |T 6 where the non-Gaussianityis controlledby a single positive parameter µ, such that µ=0 recovers the Gaussian case. Assume, then, that 2 2 1 Q(η ,η ) 1+µ (η η⋆)4 exp (η η⋆)(˜−1) (η η⋆) , (20) 1 2|T ∝" k− k # −2 i− i C ij j − j  Xk=1  iX,j=1    where ˜isthetwo-dimensionalversionofEq.(11),andso a probability that is significantly greater than the single C α ( 1,1). We want to maximize Q(η ,η ) subject (local)minimum(byafactorof&2),andthedominance 1 2 ∈ − |T to the constraint that the sum of the densities is the ofonecomponentisalsogreaterthanintheµ=0.1case, observed dark matter density: with a roughly 3.5:1.5 split in Fig. 1c, and a roughly 3.7:1.3 split in Fig. 1f. η1+η2 =ηobs. (21) Atypicality therefore predicts either the existence of two equally contributing components [corresponding to So proceeding as in section IIIA, we wish to maximize the local minima (red squares) in Fig. 1c, 1f] or indeed just a single dominant component (that is more domi- 2 nant than the prediction under typicality, that is, as one J(η ,η ) lnQ(η ,η ) λ η (22) 1 2 1 2 i ≡ |T − movesalongthe constraintsurfacetowardseither axisin i=1 X Fig. 1c, 1f, say1). where λ is a Lagrange multiplier. Setting Thus we have exhibited a scenario in the top-down ∂J(η ,η )/∂η = 0, gives the following two equa- approach where: atypicality corresponds to equal con- 1 2 k tions (for k =1,2): tributions, and typicality to unequal contributions (i.e., one dominant species), to the total dark matter density. 4µ(η η⋆)3 2 Non-Gaussianities can change the nature of the predic- 1+µ 2k−(ηk η⋆)4 − (ηj −ηj⋆)(C˜−1)jk =λ. (23) tion. i=1 i− i Xj=1 P The setof equationsgivenby Eq.(23) andEq.(21) con- IV. ANTHROPIC CONDITIONALIZATION stitutethreeequationsforthethreeunknowns η ,η ,λ . 1 2 { } We proceed to solve these numerically, and plot the resulting solutions in Fig. 1. The free parameters in the Anthropic conditionalization represents an intermedi- problem are µ,η⋆,η⋆,α,σ¯,η . Figure 1 displays re- ate point between bottom-up and top-down approaches sults such th{at fo1r ea2ch of twobos}values of σ¯ (each value inthatsomemultiversedomainsareindeedexcisedinthe correspondstoarowofthe plot), the parametercontrol- computation of probabilities but the restrictionis not as ling the non-Gaussianity, namely µ, is varied over three stringent as in the case of top-down conditionalization. possible values from 0 to 0.1 to 1 (from left to right). From a calculational point of view, as discussed Each value corresponds to a column of the plot. We in Aguirre and Tegmark [10], one can implement an- choose η = 5 (which is, recall, approximately the ex- thropic conditionalization by adopting a weighting fac- obs perimentally observed value), η⋆ =3.6=η⋆ (so that the tor W that multiplies the raw probability distribution 1 2 distributionhasweightneartheexperimentallyobserved P(~η )andexpressesthe probabilityoffinding domains |T value), and α= 0.5 (for illustrative purposes). in which we might exist, as a function of the relevant − parameterweareinvestigating. InAguirreandTegmark We see that in the Gaussian case (µ = 0, Fig. 1a, [10], and in what follows, the assumption is made that 1d), there is a single maximum (blue circle) which oc- W W(η) is afunction ofthe totaldarkmatter density curs at η = η /2 = η in accord with the general re- 1 obs 2 ≡ N sultderivedearlier[Eq.(14)]. When we breakGaussian- η ≡ i=1ηi, and we will look into the effects of assum- ity we overturn this result. For smaller deviations from ing an η-dependent Gaussian fall-off for this weighting P Gaussianity (µ = 0.1, Fig. 1b, 1e), the maxima on the factor. That is, we will assume, following [10] that constraintsurfacecorrespondto(twosymmetriccasesin 1 which) one component slightly dominates over the other W(η) exp η2 , (24) (with a roughly 3:2 split in Fig. 1b and a roughly3.4:1.6 ∝ (cid:26)−2η02 (cid:27) split in Fig. 1e). There is also a single (local) minimum on the constraint surface between these maxima, whose probability is close to theirs. This result is amplified in the case where the non-Gaussianity is stronger (µ = 1, 1 Note that one would need to worry about boundary conditions Fig. 1c, 1f). The maxima on the constraint surface have ofthedistributionspresentedtogiveprecisedetailsinthiscase. 7 a b c 8 8 8 7 7 7 6 6 6 5 5 5 2 4 2 4 2 4 η η η 3 3 3 2 2 2 1 1 1 0 0 0 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 η η η 1 1 1 d e f 8 8 8 7 7 7 6 6 6 5 5 5 2 4 2 4 2 4 η η η 3 3 3 2 2 2 1 1 1 0 0 0 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 η η η 1 1 1 FIG.1. ContourplotsofthedistributionQ(η1,η2|T)[seeEq.(20)]. Ineachpanel,theblacklinedenotestheconstraintsurface, andtheredlinedenotesthelineofequaldensity. Alongtheconstraintsurface, bluecirclescorrespondto(global)maximaand red squares to (local) minima. Note that in (b,c,e,f), pairs of maxima have the same probability in each panel. Parameters ⋆ ⋆ have been set as follows: ηobs = 5, η1 = 3.6 = η2, and α = −0.5. (a–c) σ¯ = ηobs/6 with µ = 0,0.1,and1 respectively. (d–f) σ¯ = ηobs/5 with µ = 0,0.1,and1 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 section IIIA). The equality of contribution is overturned in a significant way for µ=1, that is (c,f), where in each case, the maxima correspond to unequal contributions and thelocal minimum corresponds toequal contributions. where we also assume that we have a way of calculating tion P (~η ,W), which takes this anthropic weighting tot the standard deviation η .2 The optimization problem factor into |aTccount, where 0 that implements the assumption of typicality now de- mands that we maximize the total probability distribu- Ptot(~η ,W) P(~η )W(η). (25) |T ∝ |T We will investigate the result of doing this for the correlated Gaussian case discussed in Sec. III, beginning first 2 Note that for the sake of calculational simplicity, we extend with the case where we do not restrict the covariance the domain of validity of the Gaussian fall-offfor the anthropic matrix. In addition, in contrast to our discussion so far, weightingfactorW(η), beyondthat exploredin[10],wherethis we will focus less on the equality of contribution of dif- domaincorrespondedtoη>η0. Thisraisesasubtletyregarding ferent components to the total dark matter density; and the value(s) of η that can appropriately be considered to max- more on a new feature that arises exclusively in the an- imize W(η). This is a debate that lies outside the scope of the thropic approach: that of the determination of precisely problemconsideredinthispaper,butwouldneedtobeaddressed inalessstylizedsetting. how many componentsN contribute equallyto the total 8 dark matter density. such that P (~η ,W) is maximized, where, substitut- tot |T ing Eq. (3) and Eq. (24) into Eq. (25), we have A. The optimization routine We assume again, that there are N possible species of dark matter and that we need to find the value of ~η N 1 1 P (~η ,W) exp (η η⋆)( −1) (η η⋆) exp η2 . (26) tot |T ∝ −2 i− i C ij j − j  −2η2  iX,j=1  (cid:26) 0 (cid:27)   For each k, setting ∂lnP (~η ,W)/∂η =0 gives In addition, the optimal total density η is given by tot k opt |T i=N1(ηi−ηi⋆)(C−1)ik =−η102η. (27) ηopt =Nγ =Nη02+Nσ¯2[η¯1η+02(N −1)α]. (31) X Multiplying through by , summing over k and rear- kj C ranging, we find N 1 η =η⋆ η . (28) j j − η2 Ckj There is another way one can derive this last result 0 kX=1 [Eq.(31)],whichishelpfulinunderstandingthenatureof As in section IIIA, choosing the η⋆’s and/orthe sums of theoptimizationbeingcarriedout,andsoweoutlinethe j results of this alternate derivation here. Namely, when the columns of the covariance matrix appropriately, one P(~η ) is Gaussian with mean vector (η⋆,η⋆,...,η⋆ ) can find contributions to the total dark matter density |T 1 2 N and covariance matrix [as in Eq. (3)], then the prob- such that all species do not contribute equally. C [i.eH.,oEwqe.v(e1r,0)u],nadnerdtthheatatshsuemcopvtaiorinantcheatmηaj⋆tri=xiη¯sgfoivreanllbyj abbyilRit(yη)d,isitsriablusotioGnauosvseiranthweitshummeanNi=1ηNii=1≡ηi⋆η,anddenvoaterid- N P Eq. (11) say, we recover the case of equal contributions ance (see for example [30, chapter II, Sec. i,j=1Cij P discussed above. In particular, we obtain, for each j, 13]). Under the simplifying assumptions introduced ear- P lier, namely, if we set η⋆ = η¯ for all i, and ˜ as ηj =η¯− η12η[1+(N −1)α]σ¯2, (29) in Eq. (11), we have: iNi=1ηi⋆ = Nη¯ and NiC,j=→1CCij = 0 Nσ¯2[1+(N 1)α]. Thus − P P where the right-hand side does not depend on j. The 1 solution to these equations is just ηj = γ, say, so that R(η) exp (η Nη¯)2 . (32) η ≡ Ni=1ηi =Nγ. Thus we find ∝ (cid:26)−2Nσ¯2[1+(N −1)α] − (cid:27) P η¯η2 Theresultingprobabilitydistributionoverthesumofthe η =γ = 0 . (30) densitiesη,denotedbyP (η ,W),canbeshowntobe: j η2+Nσ¯2[1+(N 1)α] tot |T 0 − P (η ,W) R(η)W(η) tot |T ∝ 1 1 =exp (η Nη¯)2 exp η2 −2Nσ¯2[1+(N 1)α] − −2η2 (cid:26) − (cid:27) (cid:26) 0 (cid:27) 1 =exp (η Φ)2 , (33) −2Σ2 − (cid:26) (cid:27) where Thus P (η ,W) is also Gaussian with its maximum tot |T occurringatΦ;inagreementwiththe optimalvalue η η2Nσ¯2[1+(N 1)α] opt Σ2 = 0 − , (34) η2+Nσ¯2[1+(N 1)α] 0 − η¯η2 Φ=N 0 . (35) η2+Nσ¯2[1+(N 1)α] 0 − 9 displayed in Eq. (31). for N, with Fη η2 N = obs 0 . (38) η2η¯ Fη σ¯2 B. Prediction and fine tuning 0 − obs The demand that the original prediction is not finely tuned, namely, that 1η 2η [Eq. (37)], bounds N How then, in light of the above discussion, do we pro- F opt ≤ 0 such that pose to extract a prediction from P (η ,W) while al- tot |T lowingforvariationsinassumptionsregardingtypicality? 2Fη2 We know that the probability distribution P (η ,W) N 0 , (39) tot |T ≤ η0η¯ 2Fσ¯2 isGaussian[seeEqs.(33),(34),and(35)],andsoithasa − single maximum; moving sufficiently far away from this for η η¯ > 2Fσ¯2; otherwise no such upper bound exists. 0 maximum takes us into regions of atypicality. Hence, in- Equations (38) and (39) imply that if 0 < F < 1, both troducing a factor F > 0 that measures deviations from thevalueofN thatispredictiveandtheupperboundon the maximum, we propose that the framework specified N such that the prediction is not finely tuned, decrease bythetheory ,togetherwiththeanthropicconditional- relative to the case of typicality (i.e., relative to F =1). T izationfactor W(η), and the typicality assumptionchar- WhenF >1,these valuesincreaserelativeto the caseof acterized by F, is typicality. In this way,atypicality canchangethe nature of the prediction. 1 predictive if η =η , (36) opt obs F D. Correlated species and typicality assumptions where F = 1 corresponds to the assumption of ‘maximum’ typicality, and deviations from F = 1 correspond to some degree of atypicality. For nonzero α, again using Eq. (31), we find that the In addition, following Aguirre and Tegmark [10], we framework we are examining is predictive, i.e., satisfies do not want this prediction to be too finely tuned, in Eq. (36), when the sense that increasing the value of the prediction t(hi.ee.,anF1tηhorpotp)icshcoounldditnioontatlaizkaetiuons tfoaoctofarrWin(tηo)th[geivteanilboyf αηobsσ¯2N2+ (1−α)ηobsσ¯2− F1η¯η02 N +ηobsη02 =0, (cid:20) (cid:21) Eq. (24)]. In this way, we will assume that the value of (40) the predictionforthetotalobserveddarkmatter density and this prediction is not finely tuned, i.e., satisfies is Eq. (37), when 1 1 not finely tuned if Fηopt ≤2η0, (37) −2ασ¯2N2+ Fη¯η0−2σ¯2(1−α) N −2η02 ≤0. (41) (cid:20) (cid:21) namely,withintwostandarddeviations,2η ,ofthemean 0 We note that now, depending on the balance of the pa- oftheGaussianconditionalizationfactorW(η). Thepre- rameters in the problem, it is possible that under the cisetolerancehereislessimportantthanthegeneralcon- inclusionofnonzerocorrelations,thereexisttwo distinct clusions we will develop below. solutionstothepredictionforthetotalnumberofspecies We will show (as indeed mentioned by Aguirre and that contribute equally to the total dark matter density. Tegmark [10]) that the criteria expressed in Eqs. (36) We focus firston this effect in more detail, before study- and (37) can be used to predict the total number of ing the effects ofatypicality onthe resulting predictions. equally contributing components to the total dark mat- In particular, we will be interested in two different ques- ter density, and it is within the context of this type of tions: (1) under the assumption of typicality, how does prediction that we will analyze the effects of atypicality. a nonzero α change predictions for N (relative to the To gain intuition about how these two criteria operate, α=0case,andassumingthesepredictionsarenotfinely we begin by analyzingthe caseof independent species of tuned), and (2) for a fixed nonzero α, how does atypi- dark matter. calitychangethe predictionforN (againassumingthese predictions are not finely tuned)? C. Independent species and typicality assumptions 1. The effect of correlations, α6=0, on N, under typicality In the case of independent species of dark matter, In addressing the first question, we are interested in namely, when α = 0, a quick calculation reveals that comparing, for F = 1, Eq. (38) and the solution(s) to the prediction that 1η = η [Eq. (36)], in combina- Eq. (40). To make the comparison more tractable, we F opt obs tion with Eq. (31), can be translated into a prediction make two simplifying assumptions: namely, (i) that the 10 original probability distribution P(~η ) has significant section IVD1 as expressed in Eqs. (42) and (43). The |T probability near the observed value; more precisely, we equation expressing predictivity, Eq. (40), reduces to will assume 1 αN2+(1 α X)N +X =0, (47) η¯ ηobs, (42) α − − F α ≡ and the bound on N such that the prediction is not and(ii)thatthevarianceoftheconditionalizationfactor α W(η) is related in a simple way to the variances of the finely tuned, namely Eq. (41), reduces to dark matter components, 1 2ασ¯N2+ η √X 2σ¯(1 α) N 2Xσ¯ 0. η2 =Xσ¯2, (43) − α F obs − − α− ≤ 0 (cid:20) (cid:21) (48) forsomepositiveX whoserangewillbespecifiedshortly. For illustrative values of the parameters, the effects of Under these assumptions then, the prediction for the these equations on the prediction of the total number of numberofuncorrelatedspecies,asderivedfromEq.(38), species of dark matter contributing equally to the total and which we will now refer to as N , is dark matter density are explored in Fig. 3. We see there 0 that the prediction under the assumption of atypicality X [as determined by the appropriate x-axis-intercept(s) of N = . (44) 0 X 1 the black parabola in each panel; recall that only those − solutions where N 2 are considered physical] changes α InorderthatN0 isaphysicallyrealizableprediction(and significantly from th≥e prediction under the assumption so that we do not, at this stage, discount the framework of typicality [corresponding to the appropriate x-axis- that gave rise to this prediction), we choose X > 1, so intercept(s) of the gray parabola in each panel]. that N >1. 0 Similarly, these assumptions imply that Eq. (40) reduces to V. DIFFERENT FRAMEWORKS, SAME PREDICTION αN2+(1 α X)N +X =0, (45) α − − α where we now denote N by N ; the solution of which is So let us recapitulate what we have found thus far, α in order to better understand the nature of the overlaps X +α 1 (1 α X)2 4αX that exist between different frameworks as regards their Nα = − ± − − − . (46) predictionsforthetotalnumberofspeciesofdarkmatter 2α p that we should expect to observe. Let us note a couple of cases of interest here. Firstly, In the case of bottom-up conditionalization (Sec. II), if(1 α X)2 =4αX,thenthereisjustasinglesolution where we assumed the underlying probability distribu- − − Nα =(X+α 1)/2α. Settingα=0.25forexample,gives tion P(~η ) was unimodal and could, in principle, take − |T X =2.25 and so Nα =3 whereas N0 2 (one can show nonzero values within some N-dimensional cube in pa- ≈ that both of these solutions are not necessarily finely rameter space, the expected number of dominant dark tuned). Inthecasethat(1 α X)2 >4αX,N0 willex- matter components, under the assumption of typicality, − − hibit just a single solution, whereas Nα may exhibit two wasshownto be 1. Inthe terminologyof Sec.II, j 1 h i∼ (physical) solutions. Figure 2 displays some illustrative for M N. However, atypicality can change this pre- ≫ examples of what these solutions look like. There we ex- diction to a range of other possibilities, including equal hibitsolutionsN0 (redcircles)andNα (bluesquares)for contributions from all N components. α=0.25,undertheassumptionthatX =2.35,2.5,or2.7 Inthecaseoftop-downconditionalization(Sec.III),we (corresponding to Fig. 2a, 2b, or 2c respectively). We foundtheoppositeprediction,thatforacorrelatedGaus- see that in Figs. 2a and 2b, the introduction of correla- sian distribution P(~η ), all N components, under typ- |T tions leads to two distinct solutions for Nα, the greater icality, contribute significantly to the total dark matter of which is significantly different from N0. In the case density [Eq. (14)]. This result holds for probabilistically of Fig. 2c, the smaller of the two solutions for Nα is independent (α = 0) dark matter species as well; and it discounted as unphysical (as only those solutions in the generalizes the previous result of Aguirre and Tegmark correlated case make sense where Nα 2). [10,Sec.3.3]foraparticularP(~η ). However,there ex- ≥ |T ist regions of parameter space such that assumptions of atypicality lead to the prediction of just a single domi- 2. How does atypicality change the prediction when α6=0? nant dark matter component [generalizing the argument inAzhar[25,Sec.3.2],again,foraparticularP(~η )]. In |T The secondquestion we are interested in is the nature this casealso,we found that non-Gaussianitiescanover- of the change in the prediction as a result of atypical- turn the equality of contribution, such that typicality ity in the correlated Gaussian case. To investigate this corresponds to a single dominant species (for the N =2 in a simple setting, we again invoke the assumptions of case studied there), and atypicalitycorrespondsto equal