philosophy of physics

Philosophy of Cosmology If these arguments are c orrec t, then c osmology should be treated as a merely desc riptive or historic al sc ienc e that c annot disc over novel physic al laws. Both arguments rest on problematic assumptions regarding laws of nature and sc ientific method. Here I will sketc h an alternative ac c ount that allows for the possibility of testing c osmologic al laws despite the uniqueness of the universe. Before turning to that task, I should mention a different sourc e of skeptic ism regarding the possibility of sc ientific c osmology based on distinc tive laws. Kant argued that attempts at sc ientific c osmology inevitably lead to antinomies bec ause no objec t c orresponds to the idea of the “universe.” Relativistic c osmology c irc umvents this argument insofar as c osmologic al models have global properties that are well-defined, albeit empiric ally inac c essible. (This is disc ussed further in sec tion 5.) Yet c ontemporary worries resonant with Kant c onc erning how to arrive at the appropriate c onc epts for c osmologic al theoriz ing. Smolin (2000) c ritic iz es relativistic c osmology for admitting suc h global properties and proposes instead that: “Every quantity in a c osmologic al theory that is formally an observable should in fac t be measurable by some observer inside the universe.”4 1 A different question arises, for example, in extrapolating c onc epts to domains suc h as the early universe. Rugh and Zinkernagel (2009) argue that there is no physic al footing for spac etime c onc epts in the very early universe due to the lac k of physic al proc esses that c an be used to determine spac etime sc ales. Munitz 's formulation makes his assumptions about the relationship between laws and phenomena c lear: the phenomena are instanc es of the law, just as Fa ∧ Ga would be an instanc e of the “law” ∀x(Fx → Gx). Even if we grant this c onc eption of laws, Munitz 's argument would only apply to a spec ific kind of c osmologic al law. If we take EFE as an example of a “c osmologic al law,” then it has multiple instantiations in the straightforward sense that every subregion of a solution of EFE is also a solution.4 2 The same holds for other loc al dynamic al laws applic able in c osmology, suc h as those of QFT. A single universe has world enough for multiple instantiations of the loc al dynamic s. This is true as well of laws whose effec ts may have, c oinc identally, only been important within some finite subregion of the universe. For example, c onsider a theory, suc h as inflation (see sec tion 6 below), whose implic ations are only manifest in the early universe. The laws of this theory would be “instantiated” again if we were ever able to reac h suffic iently high energy levels in an experimental setting. Although the theory may in prac tic e only have testable implic ations “onc e,” it has further c ounterfac tual implic ations. Munitz 's argument would apply, however, to c osmologic al laws that are formulated direc tly in terms of global properties, as opposed to loc al dynamic al laws extrapolated to apply to the universe as a whole. Subregions of the universe would not c ount as instantiations of a “global law” in the same sense that they are instantiations of the loc al dynamic al laws. Penrose's Weyl c urvature hypothesis (proposed in Penrose 1979) is an example of suc h a law.4 3 This law is formulated as a c onstraint on initial c onditions and it does differ strikingly in c harac ter from loc al dynamic al laws. Phenomena are not, however, “instantiations” of laws of nature in Munitz 's straightforward logic al sense. Treating them as suc h attributes to the laws empiric al c ontent properly attributed only to equations derived from the laws with the help of supplementary c onditions.4 4 A simple example should help to make this c ontrast c lear. Newton's three laws of motion must be c ombined with other assumptions regarding the relevant forc es and distribution of matter to derive a set of equations of motion, desc ribing, say, the motion of Mars in response to the Sun's gravitational field. It is this derived equation desc ribing Mars's motion that is c ompared to the phenomena and used to c alc ulate the positions of Mars given some initial c onditions. The motion of Mars is not an “instanc e” of Newton's laws; rather, the motion of Mars is well approximated by a solution to an equation derived from Newton's laws along with a number of other assumptions. Ellis's argument does not explic itly rest on a c onc eption of phenomena as instantiations of laws. But he and Munitz both overlook a c ruc ial aspec t of testing laws. Continuing with the same example, there is no expec tation that at a given stage of inquiry one has c ompletely c aptured the motion of Mars with a partic ular derived equation, even as further physic al effec ts (suc h as the effec ts of other planets) are inc luded. The suc c ess of Newton's theory (in this c ase) c onsists in the ability to give more and more refined desc riptions of the motion of Mars, all based on the three laws of motion and the law of gravity. This assessment does not depend primarily on “multiplic ity of instanc es,” experimental manipulation, or observation of other members of an ensemble. Instead, the modal forc e of laws is reflec ted in their role in developing a ric her ac c ount of the motions. Due to this role they c an be subjec t to ongoing tests. The standard arguments that it is not possible to disc over laws in c osmology assume that the universe is not only unique, but in effec t “given” to us entirely, all at onc e—leaving c osmologists with nothing further to disc over, and Page 11 of 34

Philosophy of Cosmology no refinements to make and test. A novel law in c osmology c ould be supported by its suc c ess in providing suc c essively more refined desc riptions of some aspec t of the universe's history, just as Newtonian mec hanic s is supported (in part) by its suc c ess in underwriting researc h related to the solar system. This line of argument, if suc c essful, shows that c osmologic al laws are testable in muc h the same sense as Newton's laws. This suggests that “laws of the universe” should be just as amenable to an empiric ist treatment of the laws of nature as are other laws of physic s.4 5 None of this is to say that there are no distinc tive obstac les to assessing c osmo-logic al laws. But we need to disentangle obstac les that arise due to spec ific features of our universe from those that follow from the uniqueness of the objec t of study. Consider (c ontrary to the Standard Model) a universe that reac hed some finite maximum temperature as t → 0, and suppose (perhaps more absurdly) that physic ists in this universe had suffic ient funds to build ac c elerators to probe physic s at this energy sc ale. Many of the c hallenges fac ed in early universe c osmology in our universe would not arise for c osmologists in this other possible universe. They would have independent lines of evidence (from accelerator experiments and observations of the early universe) to aid in rec onstruc ting the history of the early universe, rather than basing their c ase in favor of novel physic s solely on its role in the rec onstruc tion. This suggests that obstac les fac ing c osmology have to do primarily with theoretic al and observational ac c essibility, whic h may be exac erbated by uniqueness, rather than with uniqueness of the universe per se. 5. Glo bal Structure The Standard Model takes the universe to be well-approximated by an FLRW model at suffic iently large sc ales. To what extent c an observations determine the spac etime geometry of the universe direc tly? The question c an be posed more prec isely in terms of the region visible to an observer at a loc ation in spac etime p—the causal past, J−(p), of that point. This set inc ludes all points from whic h signals traveling at or below the speed of light c an reac h p.4 6 What c an observations c onfined to J−(p) reveal about: (1) the spac etime geometry ofJ− (p) itself, and (2) the rest of spac etime outside of J−(p)? Here we will c onsider these questions on the assumption that GR and our other physic al theories apply universally, setting aside debates (suc h as those in sec tion 3) about whether these are the c orrec t theories. How muc h do these theories allow us to infer, granting their validity? Spac etime geometry is reflec ted in the motion of astronomic al objec ts and in effec ts on the radiation they emit, suc h as c osmologic al red-shift. To what extent would the spac etime geometry be fixed by observations of an “ideal data set,” c onsisting of c omprehensive observations of a c ollec tion of standard objec ts, with known intrinsic siz e, shape, mass, and luminosity, distributed throughout the universe? Of c ourse astronomers c annot avail themselves of suc h a data set. Converting the ac tual data rec orded by observatories into a map of the universe, filled with different kinds of astronomic al objec ts with spec ified loc ations and states of motion, is an enormously diffic ult task. The diffic ulty of c ompleting this task poses one kind of epistemic limitation to c osmology. Exploring this limitation would require delving into the detailed astrophysic s used to draw c onc lusions regarding the nature, loc ation, and motion of distant objec ts. This kind of limitation c ontrasts with one arising from a different sourc e, namely that we have an observational window on J−(p) rather than the entire spac etime. Even if we had ac c ess to an ideal data set, what we c an observe is not suffic ient to answer questions regarding global spac etime geometry unless we ac c ept further princ iples underwriting loc al-to-global inferenc es. The modest goal of pinning down the geometry of J−(p) observationally c an be realiz ed by observers with the ideal data set mentioned above (see Ellis 1980; Ellis et al. 1985). The relevant evidenc e c omes from two sourc es: the radiation emitted by distant objec ts reac hing us along our null c one, and evidenc e, suc h as geophysic al data, gathered from “along our world line,” so to speak. Ellis et al. (1985) prove that the ideal data set is nec essary and suffic ient, in c onjunc tion with EFE and a few other assumptions, to determine the spac etime geometry of J−(p). Considering the ideal data set helps to c larify the c ontrast between what we c an in princ iple determine loc ally, namely the spac etime geometry of J−(p), and what we c an determine globally. For points p, q with nonintersec ting c ausal pasts, we would not expec t the physic al state on J− (p) to fix that of J− (q).4 7 Does the spac etime geometry of J−(p), or of a c ollec tion of suc h sets, nonetheless c onstrain the large-sc ale or global properties of spac etime? Global properties of spac etime vary in general relativity, bec ause unlike earlier theories suc h as Newtonian mec hanic s, spac etime is treated as dynamic al rather than as a fixed bac kground. EFE Page 12 of 34

Philosophy of Cosmology impose a loc al c onstraint on the spac etime geometry, but this is c ompatible with a wide variety of global properties.4 8 Various global properties have been defined as part of stating and proving theorems suc h as the singularity theorems, inc luding “c ausality c onditions” that spec ify the extent to whic h a spac etime deviates from the c ausal struc ture of Minkowski spac etime (see Geroc h and Horowitz 1979 for a c lear introduc tion). For example, a globally hyperbolic spac etime possesses a Cauc hy surfac e, a null or spac elike surfac e ∑ intersec ted exac tly onc e by every inextendible timelike c urve. In a spac etime with a Cauc hy surfac e, EFE admit a well-posed initial value formulation: specifying appropriate initial data on a Cauchy surface ∑ determines a unique solution to the field equations (up to diffeomorphism). This is properly understood as a global property of the entire spac etime. Although submanifolds of a given spac etime may be c ompatible or inc ompatible with global hyperbolic ity, this property c annot be treated as a property asc ribed to loc al regions and then “added up” to deliver a global property. What does J−(p) reveal about the rest of spac etime? Suppose we do not impose any strong global assumptions suc h as isotropy and homogeneity. Fully spec ifying the physic al state in the region J− (p) plac es few c onstraints on the global properties of spac etime. This is c lear if we c onsider what is shared by all the spac etimes into whic h J−(p) c an be isometric ally embedded, where we allow p to be any point in a given spac etime.4 9 (That is, we shift from c onsidering the c ausal past of a single observer to the c ausal past of all possible observers in the spac etime.) Call this the set of spac etimes “observationally indistinguishable” (OI) from a given spac etime. Exc ept for the exc eptional c ase where there is a p′ suc h that, like Borges's Aleph, J−(p′) inc ludes the entire spac etime, there is a tec hnique (due to Malament 1977; Manc hak 2009) for c onstruc ting OI c ounterparts that do not share all the global properties of the original spac etime.50 The property of having a Cauc hy surfac e, for example, will not be shared by all the members of a set of OI spac etimes. More generally, the only properties that will be held in c ommon in all members of the set of OI spac etimes are those that c an be c onc lusively established by a single observer somewhere in the spac etime.51 The sc ope of underdetermination c an be reduc ed by imposing c onstraints that eliminate potential OI spac etimes. Consider, for example, restric ting c onsideration to spac etimes that are spatially homogeneous. The isometries on ∑ (implied by homogeneity), whic h c arry any point on ∑ into any other, apparently bloc k the c onstruc tion of an indistinguishable c ounterpart with different global properties.52 Homogeneity is just one example of a global property that c ould be imposed. Whatever property is imposed to eliminate underdetermination, it must be global to be effec tive given that the tec hnique for c onstruc ting indistinguishable c ounterparts preserves loc al properties. This line of argument c larifies the “c osmologic al princ iple.” The c osmologic al princ iple is the strongest of many possible “uniformity princ iples” or global stipulations that allow loc al-to-global inferenc es. If we require only that the J−(p) sets for all observers c an be embedded in a c osmologic al model, then the global properties of spac etime are radic ally underdetermined. Introduc ing different c onstraints on the c onstruc tion of the indistinguishable c ounterparts mitigates the degree of underdetermination. The c osmologic al princ iple is the strongest of these c onstraints—strong enough to eliminate the underdetermination: every observer c an take their limited view on the universe as ac c urately reflec ting its global properties. Figure 17.1 This figure contrasts the standard big bang model (a) and Ellis, Maartens, and Nel's (1978) model (b); in the latter, a c ylindric al timelikeD singularity surrounds an observer O loc ated near the axis of symmetry, and the constaDnt time surface t from which the CBR is emitted in the standard model is replaced with a surface r at fixed distance from O. However, this merely pushes the5 3original question bac k one step: What grounds do we have for imposing suc h a global constraint on spacetime? It is unappealing to simply assert that the cosmological principle holds a priori, or to treat it as a prec ondition for c osmologic al theoriz ing. But one may hope to justify the princ iple by appealing to a Page 13 of 34

Philosophy of Cosmology weaker general principle in conjunction with theorems relating homogeneity and isotropy Global isotropy around every point implies global homogeneity, and it is natural to seek a similar theorem with a weaker antec edent formulated in terms of observable quantities. The Ehlers-Geren-Sac hs (EGS) theorem (Ehlers, Geren, and Sac hs, 1968) shows that if all fundamental observers in an expanding model find that freely propagating bac kground radiation is exac tly isotropic , then their spac etime is an FLRW model.54 If our c ausal past is “typic al,” observations along our worldline will c onstrain what other observers should see. This assumption is often c alled the “Copernic an princ iple,” whic h requires that no point p is distinguished from other points q by any spac etime symmetries. This princ iple rules out models suc h as Ellis, Maartens and Nel's (1978) example of a “c ylindric al” c ounterpart to the observed universe (see figure 17.1).55 (This example illustrates the tension between the Copernic an princ iple and anthropic reasoning (see sec tion 7 below). Ellis, Maartens, and Nel point out that in their model one would only expec t to find observers near the axis of symmetry of the model, as that is the only region hospitable to life.) Combining the observed near isotropy of the CBR, the EGS theorem, and the Copernic an princ iple yields an argument in favor of the approximate validity of the FLRW models. Alternatively, one c ould dispense with the Copernic an princ iple and its ilk by showing that an early phase of the universe's evolution leads to an approximately FLRW universe. This was the aim of Misner's “c haotic c osmology” program launc hed shortly after the disc overy of the CBR, an aim taken up with greater ac c laim by inflationary c osmology (see sec tion 6 below). If this approac h suc c eeds, then homogeneity and isotropy over some length sc ale would be a c onsequenc e of underlying physic s, effec tively replac ing a priori princ iples regarding the uniformity of nature with factual claims about the universe's evolution. The warrant for an inductive inference regarding distant regions of the universe would then depend on the justification for this account. Note, however, that the ac c ount may not justify the c onc lusion that the universe is globally almost-FLRW. In the c ase of inflation, for example, homogeneity and isotropy hold in the interior of an inflationary bubble (whic h c ould be muc h larger than J−(p)), but the universe at muc h larger sc ales has dramatic nonuniformities (bubble walls, c olliding bubbles, regions between the bubbles, and so on). The Copernic an princ iple has c ome under inc reased sc rutiny rec ently due to its role in the c ase for dark energy. Departures from an FLRW geometry c ould simply indic ate the failure of the models rather than the presenc e of a new kind of matter. Rec ently there have been two suggestions for ways to test the Copernic an princ iple on sc ales c omparable to the observable universe. First, the Sunyaev-Zel'dovic h effec t56 c an be used to indirec tly measure the isotropy of the CBR as observed from distant points. Any anisotropies in the CBR as seen at a distant point q will be reflec ted in a temperature differenc e in the sc attered radiation; the distortion in the observed blac k-body spec trum in princ iple reveals the failure of isotropy from distant points not on our worldline (Caldwell and Stebbins, 2008). This allows one to prove that the loc al universe is almost-FLRW based on an EGS theorem and observations of the CBR without invoking the Copernic an princ iple (Clifton, Clarkson, and Bull, 2011). A sec ond test of the Copernic an princ iple is based on a c onsistenc y relation between several observables that holds in the FLRW models (Uz an, Clarkson, and Ellis, 2008). These disc ussions foc us on whether J−(p) c an be well approximated by an FLRW model. This question is c losely tied to assessing the c ase for dark energy and in determining the parameters of the Standard Model. What are the further implic ations if the universe is almost-FLRW on muc h larger sc ales, or if the c osmologic al princ iple holds globally throughout all of spac etime? More generally, what are the empiric al stakes of determining the global properties of spac etime? Some global spac etime properties are plausibly treated as prec onditions for the possibility of formulating loc al dynamic al laws.57 And the global properties are obvious c andidates for fundamental features of spac etime from a realist's point of view. Proofs of the singularity theorems require assumptions regarding global c ausal struc ture. Further, the origin and eventual fate of the universe are quite different in a globally almost-FLRW model and in an observationally indistinguishable c ounterpart to it. Yet despite all of this, there is a c lear c ontrast between c laiming that the observable universe is almost-FLRW and the extension of that to a global c laim regarding all of spac etime. The former plays a fundamental role in evidential reasoning in c ontemporary c osmology, whereas the latter is disc onnec ted from empiric al researc h by its very nature. Thus, the status of the c osmologic al princ iple seems to differ signific antly in prac tic e from that of other princ iples supporting induc tive generaliz ations—it does not lead, as in Newton's c ase of taking gravity to be truly universal, to a wide variety of further c laims that c an serve as the basis for a subsequent researc h program. 6. Early Universe Co smo lo gy Page 14 of 34

Philosophy of Cosmology Extrapolating the Standard Model bac kward in time leads to a singularity within a finite time, and as t → 0 the temperature and energy sc ales inc rease without bound. Even if the singularity itself is somehow avoided, the early universe is expec ted to have reac hed energy sc ales far higher than anything produc ed at Fermilab or CERN. The early universe is thus a fruitful testing ground for high-energy physic s, and sinc e the early 1980s there has been an explosion of researc h in this area. Yet it is not c lear whether observations of the early universe c an play anything like the role that ac c elerator experiments did in guiding an earlier phase of researc h in partic le physic s. Other aspec ts of the Standard Model are based on extrapolating well-established physic s, but the physic s applied to the early universe often c annot be tested by other means. Instead the c ase in favor of new physic al ideas is often based on their role in a plausible rec onstruc tion of the universe's history. Here I will assess a c ommon style of argument adopted in this literature, namely that a theory of early universe c osmology should be ac c epted bec ause it renders the observed history of the universe probable rather than merely possible. There is general agreement that the (c osmologic al) Standard Model should be supplemented with an ac c ount of physic al proc esses in the very early universe. The early universe falls within the domains of applic ability of both quantum field theory and general relativity, yet the two theories have yet to be c ombined suc c essfully. The framework of the Standard Model is thus not expec ted to apply to the very early universe. Although researc h in quantum gravity is often motivated by c alls for “theoretic al unific ation” and the like, it c an also be motivated by the more prosaic demand for a c onsistent theory applic able to phenomena suc h as the early universe and blac k holes (c f. Callender and Huggett, 2001). This “overlapping domains” argument does not imply anything in detail regarding what an early universe theory should look like, or how it would augment or c ontribute to the Standard Model. The overlapping domains argument should not be confused with the common claim that general relativity is inc omplete bec ause it “breaks down” as t → 0 and fails to provide a desc ription of what happens at (or before) the singularity.58 It is hard to see how general relativity c an be c onvic ted of inc ompleteness on its own terms. (Here I am following the line of argument in Earman (1995); Curiel (1999).) If general relativity proved to be the c orrec t final theory, then there is nothing more to be said regarding singularities; the laws of general relativity apply throughout the entire spac etime, and there is no obvious inc ompleteness. On the other hand, there are good reasons to doubt that general relativity is the c orrec t final theory, and further reasons to expec t that the suc c essor to general relativity will have novel implic ations for singularities. But then the argument for inc ompleteness is based on grounds other than the mere existence of singularities. Cosmologists often give a very different reason for supplementing the Standard Model: it is explanatorily deficient, bec ause it requires an “improbable” initial state. Guth (1981) gave an influential presentation of two aspec ts of the Standard Model as problematic: The standard model of hot big-bang c osmology requires initial c onditions whic h are problematic in two ways: (1) The early universe is assumed to be highly homogeneous, in spite of the fac t that separated regions were c ausally disc onnec ted (horiz on problem) and (2) the initial value of the Hubble c onstant must be fine tuned to extraordinary ac c urac y … (flatness problem). (Guth 1981, 347) Figure 17.2 This figure illustrates the horizon problem. Lightcones are at 45° but distances are distorted, muc h like a Merc ator projec tion. Two points p, q on the surfac e of last sc attering td, both falling within our past light c one, do not have overlapping light c ones. Horiz ons in c osmology measure the maximum distanc e light travels within a given time period; the horiz on delimits the spac etime region from whic h signals emitted at some time te traveling at or below the speed of light c ould reac h a given point. The existenc e of partic le horiz ons in the FLRW models indic ates that distant regions are not in c ausal Page 15 of 34

Philosophy of Cosmology c ontac t.59 There are observed points on the CBR separated by a distanc e greater than the partic le horiz on at that time (see figure 17.2). The Standard Model assumes that these regions have the same properties—e.g., the same temperature to within 1 part in 105 —even though they were not in c ausal c ontac t. In slightly different terms, if one expec ts no c orrelations between the c ausally disjoint regions it is mysterious how the observable universe c ould be so well approximated by an FLRW model. The flatness problem arises bec ause the energy density at early times has to be very c lose to the value of the c ritic al density Ω = 1.6 0 An FLRW model c lose to the “flat” k = 0 model, with nearly c ritic al density, at some spec ified early time is driven rapidly away from c ritic al density under FLRW dynamic s; the flat model is an unstable fixed point under dynamic al evolution.6 1 This aspec t of the dynamic s makes it extremely puz z ling to find that the universe is still c lose to the c ritic al density—this requires an extremely finely-tuned c hoic e of the energy density at the Planc k time Ω(tp), namely | Ω(tp) − 1| ≤ 10−59 . The horiz on and flatness problems both reflec t properties of the FLRW models. There are other similar “fine-tuning” problems related to other aspec ts of the Standard Model. The ac c ount of struc ture formation requires a set of “seed” perturbations that have two troubling features: first, the perturbations have to be c oherent on super- horiz on length sc ales, and, sec ond, the amplitude of the perturbations was muc h smaller than one would expec t for natural possibilities suc h as thermal fluc tuations.6 2 There are other puz z ling features not related to the seed perturbations. It is not c lear, for example, why the baryon-to-photon ratio, relevant to nuc leosynthesis c alc ulations, has the partic ular value it does. (This list c ould be extended.) The general c omplaint is that the Standard Model requires a variety of seemingly implausible assumptions regarding the initial state. Why did the universe start off with suc h a glorious pre-established harmony between c ausally disjoint regions? How was the initial energy density so delic ately c hosen that we are still c lose to the flat model? (And so on.) Although these features are all possible ac c ording to the Standard Model, the fac t that they obtain seems, intuitively, to be inc redibly improbable. The Standard Model treats these posits as brute fac ts not subjec t to further explanation. By c ontrast, Guth proposed to supplement the Standard Model by modifying the very early expansion history of the universe, drawing on ideas in partic le physic s. Guth proposed that the universe underwent a transient phase of λ- dominated, exponential expansion at roughly 10−3 5 s. Introduc ing this inflationary stage eases the c onflic t between a “natural” or “generic ” initial state and the observed universe, in the following sense. Imagine c hoosing a c osmologic al model at random from among the spac e of solutions of EFE. Even without a good understanding of this spac e of solutions or how one's c hoic e is to be “ac tualiz ed,” it seems c lear that one of the maximally symmetric FLRW models must be an inc redibly “improbable” c hoic e.6 3 New dynamic s in the form of inflation makes it possible for “generic ” pre-inflationary initial c onditions to evolve into the uniform, flat state required by the Standard Model.6 4 Ac c ording to the Standard Model alone, what we observe is inc redibly improbable; ac c ording to the Standard Model plus inflation, what we observe is to be expec ted. This is an example of a general strategy, whic h I will c all the “dynamic al approac h”: given a theory that apparently requires spec ial initial c onditions, augment the theory with new dynamic s suc h that the dependenc e on spec ial initial c onditions is reduc ed. Mc Mullin (1993) desc ribes a preferenc e for this approac h as ac c epting an “indifferenc e princ iple,” whic h states that a theory that is indifferent to the initial state, that is, robust under c hanges of it, is preferable to one that requires spec ial initial c onditions. Theorists who ac c ept the indifferenc e princ iple c an identify fruitful problems by c onsidering the c ontrast between a “natural” initial state and the observed universe and then seek new dynamics to reconcile the two. This line of reasoning is frequently endorsed as a motivation for inflation in the huge literature on the topic following Guth's paper. However, a number of skeptic s have c hallenged the dynamic al approac h as a general methodology and as a motivation for ac c epting inflation.6 5 One line of c ritic ism c onc erns whether inflation ac hieves the stated aim of eliminating the need for spec ial initial c onditions, as opposed to merely shifting it to a different aspec t of the physic s. In effec t inflation exc hanges the degrees of freedom assoc iated with the spac etime geometry of the initial state for the properties of a field (or fields) driving an inflationary stage. This exc hange has obvious advantages if physic s c an plac e tighter c onstraints on the relevant fields than on the initial state of the universe. What is gained, however, if the field (or fields) responsible for inflation has to be in a spec ial state to trigger inflationary expansion, or to have other finely tuned properties, to be c ompatible with observations? There are also direc t c hallenges to the dynamic al approac h itself, sometimes presented in c onc ert with advoc ac y Page 16 of 34

Philosophy of Cosmology of an alternative “theory of initial c onditions” approac h. First, why should we assume that the initial state of the universe is “generic ”? Penrose, in partic ular, has argued that this proposal is not c ompatible with a neo- Boltz mannian ac c ount of the sec ond law of thermodynamic s (c f. Albrec ht 2004). Penrose (1979) treats the sec ond law as arising from a lawlike c onstraint on the initial state of the universe, requiring that it has low entropy. Rather than introduc ing a subsequent stage of dynamic al evolution that erases the imprint of the initial state, we should aim to formulate a “theory of initial c onditions” that ac c ounts for its spec ial features. Sec ond, how should we make sense of the implic it probability judgments employed in these arguments? The assessment of an initial state as “generic ,” or, on the other hand, as “spec ial,” is based on a c hoic e of measure over the allowed initial states of the system. But on what grounds is one measure to be c hosen over another? Furthermore, how does a c hosen measure relate to the probability assigned to the ac tualiz ation of the initial state? It is c lear that the usual way of rationaliz ing measures in statistic al mec hanic s, suc h as appeals to ergodic ity, do not apply in this c ase bec ause the state of the universe does not “sample” the allowed phase spac e.6 6 Assessing the dynamical approach depends on a number of central issues in philosophy of science. Philosophers steeped in debates regarding sc ientific explanation may find it exc iting to disc over a major sc ientific researc h program motivated by explanatory intuitions. Proponents of inflation often sound as though their main concern is to make the early universe safe for Reic henbac h's princ iple of the c ommon c ause. Or, they emphasiz e the unific ation between partic le physic s and c osmology ac hieved in their models. While these c onnec tions are intriguing, they both must be treated with a grain of salt.67 A more general question is whether the explanatory intuitions betray an overly strong rationalistic tendenc y to demand explanations of everything. Callender (2004a, b) argues in favor of ac c epting a posited initial state as a brute fac t, in part by showing that purported “explanations” of it are mostly vac uous.6 8 A quite different approac h purports to explain various features of the universe as nec essary c onditions for our presenc e as observers, to whic h we now turn. 7. Anthro pic Reaso ning There has been a great deal of c ontroversy regarding anthropic reasoning in c osmology in the last few dec ades.6 9 Weinberg (2007) desc ribes the ac c eptanc e of anthropic reasoning as a radic al c hange for the better in how theories should be assessed, c omparable to the introduc tion of symmetry princ iples. In assessing c osmologic al theories we need, on this view, to ac c ount for selec tion effec ts due to our presenc e as observers and to c onsider fac tors suc h as the number of observers predic ted to exist by c ompeting theories. How exac tly this is to be done remains a matter of dispute. There is no widely ac c epted standard ac c ount of anthropic reasoning. Critic s of this line of thought argue that insofar as anthropic reasoning introduces new aspects of theory assessment, as opposed to merely putting an anthropic gloss on some ac c epted induc tive methodology, it is ill-motivated or even inc oherent. A methodology that is itself c ontroversial is not partic ularly useful in forging c onsensus, so the artic ulation and assessment of anthropic reasoning is c learly an essential task. Philosophers have already c ontributed to this effort and should c ontinue to do so. My aim here is to provide a brief overview of the debate, with an emphasis on connections with the philosophical literature. Two exemplary c ases should suffic e to introduc e anthropic reasoning. Dirac (1937) noted that various “large numbers” defined in terms of the fundamental constants have the same order of magnitude. This coincidence (and others) inspired his “Large Number Hypothesis”: dimensionless numbers constructed from the fundamental c onstants “are c onnec ted by a simple mathematic al relation, in whic h the c oeffic ients are of the order of magnitude unity” (Dirac , 1937, 323). Sinc e one of these numbers inc ludes the age of the universe t0 , so must they all. This implies time variation of the gravitational “c onstant” G. Dic ke (1961) argued that attention to selec tion effec ts undermined the evidential value of this surprising c oinc idenc e. Surprise at the c oinc idenc e might be warranted if t0 c ould be treated as “a random c hoic e from a wide range of possible values” (Dic ke, 1961, 440), but there c an only be observers to wonder at the c oinc idenc e for some small range of t. Dic ke (1961) argued that the value of t must fall within an interval suc h that Dirac 's c oinc idenc e automatic ally holds given two nec essary c onditions for the existenc e of observers like us.7 0 The evidenc e allegedly provided by the large number c oinc idenc e bears no relation to the truth or falsity of Dirac 's hypothesis or the Standard Model.7 1 Taking the c oinc idenc e as evidenc e for the large number hypothesis would be as misguided as c onc luding (rec yc ling Eddington's example) that there are no fish smaller than 6 inc hes in a pond based on the absenc e of suc h small Page 17 of 34

Philosophy of Cosmology fish in a fisherman's basket, even though the fisherman's net has gaps too large to hold these fish. Attention has rec ently foc used on a different kind of anthropic reasoning exemplified by Weinberg's (1987) predic tion for λ.7 2 Just as in Dic ke's arguments regarding t, within the Standard Model the value of λ c annot be freely c hosen. Bec ause a λ term does not dilute with expansion, a c osmologic al model with λ 〉 0 will transition from matter-dominated to vacuum-dominated expansion. Weinberg showed that structure formation via gravitational enhanc ement stops in the vac uum-dominated stage. The existenc e of large gravitationally bound systems (large enough to lead to the formation of stars) then imposes an upper bound on possible values of λ, keeping other aspec ts of the Standard Model fixed.7 3 It is plausible to take the existenc e of gravitationally bound systems as a nec essary prec ondition for the existenc e of observers. There is also a lower bound: a negative λ term c ontributes to EFE like normal matter and energy, and adding a large negative λ term leads to a model that rec ollapses before there is time for observers to arise. So far the argument is similar to Dic ke's eluc idation of anthropic bounds on t. But Weinberg next predic ted that λ's observed value should be c lose to the mean of the values suitable for life. If we inhabit a “multiverse” in whic h the value of λ varies in different regions,7 4 the predic tion is obtained by using the presenc e of observers as a selec tion effec t. Weinberg assumed that the probability distribution for values of λ in the multiverse is uniform within the anthropic bounds and that we are typic al members of the referenc e c lass of observers in the universe Vilenkin (1995) c alls this the “princ iple of medioc rity” (PM). In Bayesian terms, an initially flat probability distribution for the value of λ is turned into a predic tion—a sharply peaked distribution around a preferred value—by c onditionaliz ing on the existenc e of large gravitationally bound systems, serving as a proxy for observers. Eac h of these assumptions is c ontroversial. I will postpone more detailed disc ussion of the multiverse until the next sec tion and take up the PM shortly. The first assumption is often justified by appeals to simplic ity or naturalness, but it is on unsure footing without further spec ific ation of how the multiverse is generated.7 5 Nonetheless, Weinberg's predic tion of a positive value of λ within two orders of magnitude of c urrently ac c epted values has been widely c ited as a striking suc c ess of anthropic reasoning.7 6 Different views regarding anthropic reasoning c an be c harac teriz ed in part by whether they take Weinberg's argument as a valid extension of Dic ke's. Many anthropic skeptic s ac c ept Dic ke's reasoning but see it as an illustration of how to take selec tion effec ts into ac c ount, without any truly anthropic elements (e.g., Earman, 1987; Smolin, 2007). Dic ke simply follows through the c onsequenc es of the existenc e of main sequenc e stars and heavy elements. The nature of “observers” and whether they are typical members of a given reference class play no role. Furthermore, as Roush (2003) emphasiz es, Dic ke's argument devalues a partic ular body of evidenc e. The apparent c oinc idenc es that troubled Dirac reflec t deep biases in the evidenc e available to us, and as a result have no value in assessing his hypothesis. Weinberg's argument, by c ontrast, takes the suc c essful “predic tion” of a surprising value for a partic ular parameter as evidenc e in favor of a multiverse. Thus it is more in line with Dirac 's idea that suc h c oinc idenc es c an be revealing rather than with Dic ke's response. It also depends on assumptions regarding our “typic ality” among members of a referenc e c lass, raising a number of issues that Dic ke's argument avoids. Proponents of anthropic reasoning argue that these issues have to be dealt with in order to assess c osmologic al theories. Some have argued that the PM must be assumed in order to extrac t any predic tions at all from c osmologic al theories that desc ribe an infinite universe.7 7 Consider an observation O, for example that the CBR has an average temperature within the observer's Hubble volume of T = 3.14159…K, in agreement with the dec imal expansion of π to some spec ified number of digits. Suppose we have a c osmologic al theory T that predic ts the existenc e of an open FLRW model with infinite spatial slic es ∑ and also assigns a nonz ero probability to O. Then there is an observer for whom O is true somewhere in the vast reac hes of the infinite universe. The point generaliz es to other observations and threatens to undermine the use of any observations to assess c osmologic al theories.7 8 (This c hallenge arises even in the Standard Model, provided that the universe is not c losed, and does not depend on more spec ulative multiverse proposals.) This skeptic al c onc lusion c an only be evaded by ac c epting the princ iple of medioc rity, ac c ording to this line of thought: we are interested not in the reports of suc h improbable “freak observers,” but rather in our observations— where we regard ourselves as randomly selec ted from an appropriate referenc e c lass. Even “infinite universe” theories c an make predic tions by employing the PM, onc e the appropriate referenc e c lass has been spec ified. The PM leads, unfortunately, to absurd results in other c ases. These problems are arguably due to the explic it Page 18 of 34

Philosophy of Cosmology relianc e on the c hoic e of a referenc e c lass. This c hoic e does not reflec t a fac tual c laim about the world, yet it c an lead direc tly to striking empiric al results, as illustrated in the Doomsday argument (e.g., Leslie 1992; Gott 1993; Bostrom 2002). The argument follows from applying the PM to one's plac e in human history, in partic ular by asserting that one should occupy a “typical” birth rank among the reference class consisting of all humans who have ever lived. This implies that there are roughly as many humans born before and after one's own birth. For this to be true, given the c urrent rate of population growth, “doomsday”—a rapid drop in the growth rate of the human population—must be just around the c orner.7 9 The c onc lusion of the argument depends c ritic ally on the referenc e c lass. Starkman and Trotta (2006) argue that Weinberg's predic tion of λ is similarly sensitive to the referenc e c lass used in applying the PM. Philosophers have discussed a number of other cases, from Sleeping Beauties to Presumptuous Philosophers, meant to test princ iples proposed for anthropic reasoning.8 0 Stated more generally, these proposals regard how to inc orporate indexic al information (about, e.g., one's loc ation in the history of mankind) in evidential reasoning. Straightforward modific ations of the PM to avoid the Doomsday argument lead to c ounterintuitive results in these other c ases. Bostrom (2002) advoc ates responding to the Doomsday argument by c onsidering a different referenc e c lass when applying the PM, but his arguments that there is a unique referenc e c lass that resolves the problems are unc onvinc ing. An alternative response is to take the number of observers in the referenc e c lass into ac c ount, by weighting the prior probability by this number.8 1 For example, if a theory predic ts that there will be 10 more observers (in the appropriate referenc e c lass) than a c ompeting theory, then the prior probabilities should have this same ratio. This effec tively bloc ks the Doomsday argument. It has unpalatable c onsequenc es of its own, however, if it is taken as a general methodologic al princ iple: it implies nearly unshakeable c onfidenc e in theories that predic t large numbers of observers.8 2 The c ombined effec t of ac c epting PM and adjusting the priors to take ac c ount of the number of observers is to eliminate the dependenc e on a c hoic e of a partic ular referenc e c lass, as Neal (2006) shows. Rather than introduc e the referenc e c lass only to eliminate its impac t, why not simply apply Bayesian c onditionaliz ation? Neal (2006) argues that standard Bayesian c onditionaliz ation on all nonindexic al evidenc e available resolves the various puz z les assoc iated with anthropic reasoning, with one c aveat. On this approac h anthropic reasoning is just a spec ies of Bayesian c onditionaliz ation, and there is no need to introduc e further methodologic al princ iples.8 3 (It is c ruc ial to c onditionaliz e on everything bec ause, as analyses of selec tion effec ts like Dic ke's show, it is not always transparent whic h aspec ts of our evidenc e are relevant.) This approac h leads to the following assessment of anthropic predic tions, suc h as Weinberg's predic tion of λ. Consider a multiverse theory TM in whic h the value of λ (and perhaps other parameters) takes on different values in different regions, c ontrasted with a theory T1 in whic h the value of λ is not fixed by theoretic al princ iples, but does not vary in different regions. Suppose that ∆ is the range of values of λ c ompatible with all available evidenc e (inc luding, for example, the existenc e of galaxies at high redshifts), and that ac c ording to TM the frac tion of regions with a value of λ within ∆ is given by f,84 whereas T1 assigns a probability of g to ∆. If one assigns equal priors to the two theories, the odds ratio for TM to T1 upon c onditionaliz ation will be given by f/g. The evaluation of the two theories depends on the probability they assign to a value of λ within ∆. Whether the theory involves a “multiverse” with λ varying in different regions is irrelevant to the c omparison. The assessment also does not depend on c onsidering how ∆ c ompares to ∆′, the range of parameter values of λ c ompatible with “intelligent life” (or “advanc ed c iviliz ations,” etc .). The c aveat is that this analysis applies to universes in whic h the evidenc e is suffic iently ric h to single out a unique observer. Neal acknowledges that in an infinite universe the argument above regarding “freak observers” poses a threat, given that there will be multiple observers with the same total body of evidenc e. He goes on to argue, however, that it is implausible that our evidential reasoning should depend on whether the universe is large enough to c ontain observers with exac tly the same evidenc e. (This is, of c ourse, exac tly the c ontext in whic h c osmologists feel the need to invoke the PM—see, e.g., Garriga and Vilenkin 2007.) Philosophers have rejec ted the use of PM on other grounds. Norton (2010) has c hallenged the employment of probability distributions as a way of representing neutrality of evidential support, as part of a more general c ritic ism of Bayesian-ism. He argues that the ability to get something from nothing—a striking empiric al result from innoc uous assumptions, as in the Doomsday argument—reflec ts the extra representational baggage assoc iated with desc ribing ignoranc e using a probability measure. Probability measures are assumed to be c ountably additive, and Page 19 of 34

Philosophy of Cosmology this prevents them from expressing c omplete evidential neutrality. Assigning a uniform prior probability over the values of some parameter suc h as λ implies that a value in a finite interval is disfavored by the evidenc e, rather than treating all of these values neutrally. One might hope that invoking a “random” c hoic e among members of a referenc e c lass c an underwrite asc riptions of probability. Norton c ounters that invoc ations of indifferenc e princ iples suc h as PM ac tually support the asc ription of neutral evidential warrant rather than uniform probability. This brief survey has sketched three different lines of thought regarding anthropic reasoning. The most c onservative option is to apply standard Bayesian methodology to c ases where anthropic issues arise. The hope is that these c ases c an be treated by c arefully attending to details without introduc ing new princ iples of general sc ope, and without invoking referenc e c lasses. One advantage of the c onservative position is the availability of arguments in favor of the basic tenets of Bayesianism. It would be surprising if the validity of these methodologic al princ iples were in fac t sensitive to whether we live in a vast, finite universe or a truly infinite universe. Against the c onservatives, Norton direc tly attac ks the use of probability to represent degrees of belief in c ases of neutral support, suc h as undetermined parameters. This general c ritic ism of Bayesianism has implic ations muc h broader than anthropic reasoning, but the c onc lusions it leads to in this c ase are similar to those of the c onservative Bayesian: a rejec tion of the need to provide anthropic explanations of partic ular parameter values. Finally, a third position is that there are important and new methodologic al princ iples required to handle indexic al information and selec tion effec ts. One goal of suc h an ac c ount would be to c larify this style of reasoning, whic h is widely employed within c ontemporary c osmology. What is lac king so far, in my view, is a c ompelling ac c ount of what these princ iples are and a motivation for ac c epting them. 8. MULTIVERSE Anthropic reasoning is often disc ussed in tandem with the multiverse (c f. Zinkernagel 2011). Weinberg's anthropic predic tion for λ is based on applying a selec tion effec t to a multiverse in whic h the value of λ varies in different regions. The multiverse idea has gained trac tion in part bec ause Weinberg's approac h is widely regarded as the only viable solution to the c osmologic al c onstant problem, and other similar problems may also admit only anthropic solutions. Two different lines of thought in physic s also support the introduc tion of the multiverse. First, within inflationary c osmology the same mec hanism that produc es a uniform, homogeneous universe on sc ales on the order of the Hubble radius leads to a dramatic ally different global struc ture of the universe. Inflation is said to be “generic ally eternal” in the sense that inflationary expansion continues in different regions of the universe, constantly creating bubbles suc h as our own universe, in whic h inflation is followed by reheating and a muc h slower expansion.8 5 The individual bubbles are effec tively c ausally isolated from other bubbles and are often c alled “poc ket universes.” The sec ond line of thought relates to the proliferation of vac ua in string theory. Many string theorists now expec t that there will be a vast landsc ape of allowed vac ua, with no way to fulfill the original hope of finding a unique c ompac tific ation of extra dimensions to yield low-energy physic s. Both of these developments suggest treating the low-energy physic s of the observed universe as partially fixed by paroc hial c ontingenc ies related to the history of a partic ular poc ket universe. Other regions of the multiverse may have drastic ally different low-energy physic s bec ause, for example, the inflaton field tunneled into a loc al minima with different properties.8 6 Here my main foc us will be on a philosophic al issue that is relatively independent of the details of implementation: In what sense does the multiverse offer satisfying explanations? But, first, what do we mean by a “multiverse” in this setting?8 7 These lines of thought lead to a multiverse with two important features. First, it c onsists of c ausally isolated poc ket universes, and sec ond, there is signific ant variation from one poc ket universe to another. There are other ideas of a multiverse, suc h as an ensemble of distinc t possible worlds, eac h in its own right a topologic ally c onnec ted, maximal spac etime, c ompletely isolated from other elements of the ensemble. But in contemporary cosmology, the pocket universes are all taken to be effectively c ausally isolated parts of a single, topologic ally c onnec ted spac etime—the multiverse. Suc h regions also oc c ur in some c osmologic al spac etimes in c lassic al GR. In De Sitter spac etime, for example, there are inextendible timelike geodesic s γ 1,γ 2 suc h that J−(γ 1) does not intersec t J−(γ 2 ). In c ases like this the definition of “effec tively c ausally isolated” c an be c ashed out in terms of relativistic c ausal struc ture, but for a quantum multiverse the definition needs to be amended. Page 20 of 34

Philosophy of Cosmology The example of poc ket universes within De Sitter spac etime lac ks the sec ond feature, variation from one poc ket universe to another. This c an take several forms, from variation in the c onstants appearing in the Standard Models of c osmology and partic le physic s to variation of the laws themselves. Within the c ontext of eternal inflation or the string theory landsc ape, what were previously regarded as “c onstants” may instead be fixed by the dynamic s. For example, λ is often treated as the c onsequenc e of the vac uum energy of a sc alar field displac ed from the minimum of its effec tive potential. The variation of λ throughout the multiverse may then result from the sc alar field settling into different minima. Greater diversity is suggested by the string theory landsc ape, ac c ording to whic h the details of how extra dimensions are c ompac tified and stabiliz ed are reflec ted in different low-energy physic s. In the multiverse some laws will be demoted from universal to paroc hial regularities. But presumably there are still universal laws that govern the mechanism that generates pocket universes. This mechanism for generating a multiverse with varying features may be a direc t c onsequenc e of an aspec t of a theory that is independently well- tested. Rather than treating the nature of the ensemble as spec ulative or c onjec tural, one might then have a suffic iently c lear view of the multiverse to c alc ulate probability distributions of different observables, for example. In this c ase, there is a direc t reply to multiverse c ritic s who objec t that the idea is “unsc ientific ” bec ause it is “untestable”: other regions of the multiverse would then have much the same status as other unobservable entities proposed by empiric ally suc c essful theories.8 8 Unfortunately for fans of the multiverse, the c urrent state of affairs does not seem so straightforward. Although multiverse proposals are motivated by trends in fundamental physic s, the detailed ac c ounts of how the multiverse arises are typic ally beyond theoretic al c ontrol. As long as this is the c ase, there is a risk that the c laimed multiverse explanations are just-so stories where the mec hanism of generating the multiverse is c ontrived to do the job. This strikes me as a legitimate worry regarding c urrent multiverse proposals, but I will set this aside for the sake of disc ussion. Suppose, then, that we are given a multiverse theory with an independently motivated dynamic al ac c ount of the mec hanism c hurning out poc ket universes. What explanatory questions might this theory answer, and what is the relevanc e of the existenc e of the multiverse itself to its answers? 8 9 Here we c an distinguish between two different kinds of questions. First, should we be surprised to measure a value of a partic ular parameter X (suc h as λ) to fall within a partic ular range? Our surprise ought to be mitigated by a disc ussion of anthropic bounds on X, revealing various unsuspec ted c onnec tions between our presenc e and the range of allowed values for the parameter in question. But, as with Dic ke's approac h disc ussed above, this explanation c an be taken to demystify the value of X without also providing evidenc e for a multiverse. The value of this disc ussion lies in trac ing the c onnec tions between, e.g., the time-sc ale needed to produc e c arbon in the universe or the c onstraints on expansion rate imposed by the need to form galaxies. The existenc e of a multiverse is irrelevant to this line of reasoning. A sec ond question pertains to X, without referenc e to our observation of it: Why does the value of X fall within some range in a partic ular poc ket universe? The answer to this question offered by a multiverse theory will apparently depend on contingent details regarding the mechanism that produced the pocket universe. This explanation will be historic al in the sense that the observed values of the parameter will ultimately be trac ed bac k to the mec hanism that produc ed the poc ket universe.9 0 It may be surprising that various features of the universe are given this type of explanation rather than following as necessary consequences of fundamental laws. However, the suc c ess of historic al explanations does not support the c laim that other poc ket universes must exist. Analogously, the suc c ess of historic al explanations in evolutionary biology does not imply the existenc e of other worlds where pandas have more elegant thumbs. To put the point in a slightly different form, the value of c onverting questions about modalities in c osmology into questions about loc ation within a vastly enlarged ontology is not c lear. Both types of questions c an apparently be answered adequately without making the further ontological commitment to the actual existence of other pocket universes. 9. Co nclusio n One theme running through the disc ussion above is the attempt to identify distinc tive evidential c hallenges fac ed in c osmology. There is an ec ho of skeptic ism regarding the possibility of knowledge of the universe-as-a-whole in the disc ussion of global properties of the universe (sec tion 5). Loc al observations are not suffic ient to warrant c onc lusions regarding global properties without help from general princ iples like the c osmologic al princ iple, whic h Page 21 of 34

Philosophy of Cosmology is itself on unsure footing. This does not, however, support a general skeptic ism about c osmology. Most c ontemporary researc h in c osmology is c ompatible with agnostic ism regarding the global properties of the universe. The c hallenges arise, not from the limits imposed by the c ausal struc ture of GR, but from the diffic ulty in gaining ac c ess to the relevant phenomena via independent routes. As the disc ussion in sec tion 3 illustrates, assuming that the Standard Model is basic ally c orrec t makes it possible to infer the presenc e of dark matter and dark energy. It is diffic ult to rule out the possibility that the same observations used as the basis for this inferenc e instead reveal flaws in the Standard Model. Yet this does not mean that all the responses to the observations should be given equal c redenc e. Philosophers of sc ienc e ought to offer an ac c ount of empiric al support that c larifies the assessment of different responses. Regarding early universe c osmology (sec tion 6), the theory being used to desc ribe the underlying physic s is tested through its role in providing a rec onstruc tion of the universe's history. The field has been partially driven by strong explanatory intuitions favoring a theory that renders the observed history probable or expec ted, although it is unc lear how to move beyond intuitive disc ussions of probability. Cosmologists have to fac e the possibility that the data they use to assess theories is subjec t to unexpec ted anthropic selec tion effec ts (sec tion 7). Whether these selec tion effec ts c an be treated within standard approac hes to c onfirmation theory or require new princ iples of anthropic reasoning is c urrently being debated. Finally, c osmologists may also see their explanatory aims c hange, with various features of the universe trac ed to environmental features of our poc ket universe rather than being derived from dynamic al laws (sec tion 8). References Aguirre, A. (2007). Making predic tions in a multiverse: Conundrums, dangers, c oinc idenc es. In Universe or multiverse? ed. B. Carr, 367–386. Albrec ht, A. (2004). Cosmic inflation and the arrow of time. In Science and Ultimate Reality, ed. J. D. Barrow, P. C. W. Davies, C. L. Harper. Cambridge: Cambridge University Press, 363–401. Alpher, R. A., H. Bethe, and G. Gamow (1948). The origin of c hemic al elements. Physical Review 73: 803– 804. Balashov, Y. (2002). Laws of physic s and the universe. In Einstein Studies in Russia, 107– 148. Boston, MA/Basel/Berlin: Birkhäuser. Barrow, J. D., and F. J. Tipler (1986). The anthropic cosmological principle. Oxford: Oxford University Press. Beisbart, C. (2009). Can we justifiably assume the c osmologic al princ iple in order to break model underdetermination in cosmology? Journal for General Philosophy of Science 40: 175–205. Bekenstein, J. D. (2010). Alternatives to dark matter: Modified gravity as an alternative to dark matter. ArXiv e- prints 1001.3876. Bertone, G., D. Hooper, and J. Silk (2005). Partic le dark matter: Evidenc e, c andidates and c onstraints. Physics Report 405: 279–390. hep-ph/0404175. Bianc hi, E., and C. Rovelli (2010). Why all these prejudic es against a c onstant? ArXiv preprint arXiv: 1002.3966. Blau, S. K., and A. Guth (1987). Inflationary c osmology. In 300 years of gravitation, ed. S. W. Hawking and W. Israel. Cambridge: Cambridge University Press, 524–603. Bostrom, N. (2002). Anthropic bias: Observation selection effects in science and philosophy. New York: Routledge. Buc hert, T. (2008). Dark energy from struc ture: a status report. General Relativity and Gravitation 40(2): 467– 527. Caldwell, R. R., and A. Stebbins (2008). A test of the Copernic an princ iple. Physical Review Letters 100: 191302. Callender, C. (2004a). Measures, explanations and the past: Should spec ial initial c onditions be explained? British Journal for the Philosophy of Science 55: 195–217. Page 22 of 34

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Philosophy of Cosmology general relativity. Philosophy of Science 70: 812–832. ———. (2005). The methodologic al value of c oinc idenc es: Further remarks on dark matter and the astrophysic al warrant for general relativity. Philosophy of Science 72: 1324–1335. Vilenkin, A. (1995). Predic tions from quantum c osmology. Physical Review Letters 74: 846– 849. Wald, R. (1984). General relativity. Chic ago: University of Chic ago Press. ———. (2006). The arrow of time and the initial c onditions of the universe. Studies in History and Philosophy of Modern Physics 37: 394–398. Wallac e, D. (2011). The logic of the past hypothesis. Unpublished. Available at http://philsci- archive.pitt.edu/id/eprint/8894. Weinberg, S. (1972). Gravitation and cosmology. New York: John Wiley & Sons. ———. (1987). Anthropic bound on the c osmologic al c onstant. Physical Review Letters 59: 2607. ———. (1989). The c osmologic al c onstant problem. Reviews of Modern Physics 61: 1– 23. ———. (2007). Living in the multiverse. In Universe or multiverse, ed. B. Carr. Cambridge: Cambridge University Press, 29–42. ———. (2008). Cosmology. New York: Oxford University Press. Wolf, J. A. (2011). Spaces of constant curvature. 6th ed. Providenc e, RI: Americ an Mathematic al Soc iety. Zinkernagel, H. (2002). Cosmology, partic les, and the unity of sc ienc e. Studies in the History and Philosophy of Modern Physics 33: 493–516. ———. (2011). Some trends in the philosophy of physic s. Theoria 71: 215– 241. Notes: (1) A “standard c andle” is an objec t whose intrinsic luminosity c an be determined; the observed apparent magnitude then provides an accurate measurement of the distance to the object. (2) This is not to say that there is no literature on the topic , and muc h of it will be c ited below. For more systematic reviews of the literature by someone whose c ontributions have shaped the field, and whic h I draw on in the following, see Ellis (1999, 2007). (3) Of the several textbooks that c over this territory, see in partic ular Peebles (1993); Dodelson (2003); Weinberg (2008); see Longair (2006) for a masterful historic al survey of the development of c osmology and astrophysic s. (4) At the time, Einstein formulated Mac h's princ iple as the requirement that inertia derives from interac tions with other bodies rather than from a fixed bac kground spac etime. His model eliminated the need for anti-Mac hian boundary c onditions by eliminating boundaries: it desc ribes a universe with spatial sec tions of finite volume, without edges. See Smeenk (2012) for further disc ussion. (5) An isometry is a transformation that preserves the spac etime geometry; more prec isely, a diffeomorphism ϕ that leaves the spac etime metric invariant, i.e., (ϕ*g)a b = gab. (6) A topologic al spac e is simply connected if, roughly speaking, every c losed loop c an be smoothly c ontrac ted to a point. For example, the surfac e of a bagel is multiply c onnec ted, as there are two different types of loops that c annot be c ontinuous deformed to a point. There is another possibility for a globally isotropic spac e with c onstant positive c urvature that is multiply c onnec ted, namely projec tive spac e (with the same metric as spheric al spac e but a different topology). These three possibilities are unique up to isometry See, e.g., Wolf (2011), for a detailed disc ussion. Page 27 of 34

Philosophy of Cosmology (7) See Ellis (1971) for a pioneering study of this kind of model, and Lac hiez e-Rey and Luminet (1995) for a more rec ent review. (8) EFE are: Gab + ʌgab = 8π Tab, where Gab is the Einstein tensor, Tab is the stress-energy tensor, gab is the metric , and ʌ is the c osmologic al c onstant. Equation (1) follows from the “time-time” c omponent of EFE, and equation (2) is the differenc e between it and the “spac e-spac e” c omponent. (All other c omponents vanish due to the symmetries.) The Raychaudhuri equation is a fundamental equation that describes the evolution of a cluster of nearby worldlines, e.g., for the partic les making up a small ball of dust, in response to c urvature. It takes on the simple form given here due to the symmetries we have assumed: in the FLRW models the small ball of dust c an c hange only its volume as a func tion of time, but in general there c an be a volume-preserving distortion (shear) and torsion (rotation) of the ball as well. (9) The stress energy tensor for a perfec t fluid is given by Tab = (ρ + p)ζaζb + (p)gab, where ζa is the tangent vec tor to the trajec tories of the fluid elements. (10) These are both c lasses of solutions, where members of the c lass have spatial sec tions with c urvature of the same sign but different values of the spatial c urvature at a given c osmic time. (11) One c an treat the c osmologic al c onstant as a distinc tive type of matter, in effec t moving it from the left to the right side of EFE and treating it as a c omponent of the stress-energy tensor. It c an be viewed instead as properly inc luded on the left-hand side as part of the spac etime geometry. This issue of interpretation does not, however, make a differenc e with regard to the behavior of the solution. (12) Hubble's distanc e estimates have sinc e been rejec ted, leading to a drastic dec rease in the estimate of the c urrent rate of expansion (the Hubble parameter, H0 ). However, the linear redshift-distanc e relation has withstood sc rutiny as the sample siz e has inc reased from 24 bright galaxies (in Hubble 1929) to hundreds of galaxies at distanc es 100 times greater than Hubble's, and as astrophysic ists have developed other observational methods for testing the relation (see Peebles 1993, 82–93 for an overview). (13) The problem is underspec ified without some stipulation regarding the worldlines traversed by the observers emitting and rec eiving the signal. Assuming that both observers are fundamental observers, a photon with frequenc y ω emitted at a c osmic time t1 will be measured to have a frequenc y ω′   = R(t1) at a later time t2 . (For an R(t2) expanding universe, this leads to a red-shift of the light emitted.) Given a partic ular solution one c an c alc ulate the exac t relationship between spec tral shift and distanc e. (14) Quantitatively estimating the dynamic al effec ts of the expansion on loc al systems is remarkably diffic ult. One approac h is, sc hematic ally, to imbed a solution for a loc al system (suc h as a Sc hwarz sc hild solution) into an FLRW spac etime, taking c are to impose appropriate junc tion c onditions on the boundary. One c an then c alc ulate an upper bound on the effec t of the c osmologic al expansion; the effec t will presumably be smaller in a more realistic model, whic h inc ludes a hierarc hy of imbedded solutions representing struc tures at larger length sc ales suc h as the galaxy and the Loc al Group of galaxies. Bec ause of the nonlinearity of EFE it is surprisingly subtle to make the idea of a “quasi-isolated” system immersed in a bac kground c osmologic al model prec ise, and to differentiate effec ts due to the expansion from those due to c hanges within the loc al system (suc h as growing inhomogeneity). See Carrera and Giulini (2010) for a rec ent systematic treatment of these issues. (15) This assumption of loc al thermal equilibrium as an “initial state” at a given time presumes that the interac tion timesc ales are muc h less than the expansion timesc ale at earlier times. (16) The departures from equilibrium are desc ribed using the Boltz mann equation. The Boltz mann equation formulated in an FLRW spac etime inc ludes an expansion term. As long as the c ollision term (for some c ollec tion of interac ting partic les) dominates over the expansion term then the interac tions are suffic ient to maintain equilibrium, but as the universe c ools, the c ollision term bec omes subdominant to the expansion term, and the partic les dec ouple from the plasma and fall out of equilibrium. To find the number density at the end of this freez ing out proc ess, one typic ally has to solve a differential equation (or a c oupled set of differential equations for multiple partic le spec ies) derived from the Boltz mann equation. (17) See Olive, Steigman, and Walker (2000) for a review of big bang nuc leosynthesis. Page 28 of 34

Philosophy of Cosmology (18) These are c alled “primordial” or “relic ” abundanc es to emphasiz e that they are the abundanc es c alc ulated to hold at t ≍ 20 minutes. Inferring the values of these primordial abundanc es from observations requires an understanding of the impac t of subsequent physic al proc esses, and the details differ substantially for the various light elements. (19) The term “re-c ombination” is misleading, as the elec trons were not previously bound in stable atoms. See Weinberg (2008) and sec tion 2.3 for a desc ription of the intric ate physic s of rec ombination. (20) The blac k-body nature of the spec trum was firmly established by the COBE (Cosmic Bac kground Explorer) mission in 1992. The diffic ulty in finding an alternative stems from the fac t that the present universe is almost entirely transparent to the CBR photons, and the matter that does absorb and emit radiation is not distributed uniformly. To produc e a uniform sea of photons with a blac k-body spec trum, one would need to introduc e an almost uniformly distributed type of matter that thermaliz es radiation from other proc esses to produc e the observed mic rowave bac kground, yet is nearly transparent at other frequenc ies. Advoc ates of the quasi-steady state c osmology have argued that whiskers of iron ejec ted from supernovae c ould serve as just suc h a thermaliz er of radiation in the far infrared. See, e.g., Li (2003) for a disc ussion of this proposal and persuasive objec tions to it. (21) More prec isely, the different perturbation modes have the same density c ontrast when their wavelength equals the Hubble radius, H−1. (22) Cosmologists use “c onc ordanc e model” to refer to the Standard Model of c osmology with the spec ified c ontributions of different types of matter. The c ase in favor of a model with roughly these c ontributions to the overall energy density was made well before the disc overy of c osmic ac c eleration (see, e.g., Ostriker and Steinhardt (1995); Krauss and Turner (1999)). Coles and Ellis (1997) give a useful summary of the opposing arguments (in favor of a model without a dark energy c omponent) as of 1997, and see Frieman, Turner, and Huterer (2008) for a more recent review. (23) See Trimble (1987) for a disc ussion of the history of the subjec t and a systematic review of various lines of evidenc e for dark matter. (24) “Hot” vs. “c old” refers to the thermal veloc ities of relic partic les for different types of dark matter. Hot dark matter dec ouples while still “relativistic ,” in the sense that the momentum is muc h greater than the rest mass, and relic s at late times would still have large quasi-thermal veloc ities. Cold dark matter is “non-relativistic ” when it dec ouples, meaning that the momentum is negligible c ompared to the rest mass, and relic s have effec tively z ero thermal veloc ities. (25) Type Ia supernovae do not have the same intrinsic luminosity, but the shape of the light c urve (the luminosity as a func tion of time after the initial explosion) is c orrelated with intrinsic luminosity. See Kirshner (2009) for an overview of the use of supernovae in cosmology. (26) These brief remarks are not exhaustive; there are further lines of evidenc e for dark matter and dark energy; see, e.g., Bertone, Hooper, and Silk (2005) for a review of evidenc e for dark matter and Huterer (2010) on dark energy. (27) See Vanderburgh (2003, 2005) for a philosopher's take on these debates. (28) See Sotiriou and Faraoni (2010) for a review of one approac h to modifying GR, namely by adding higher-order c urvature invariants to the Einstein-Hilbert ac tion. These so-c alled “ f (R) theories” (the Ric c i sc alar R appearing in the ac tion is replac ed by a func tion f (R)) have been explored extensively within the last five years, but it has proven to be diffic ult to satisfy a number of seemingly reasonable c on straints. Uz an (2010) gives a brief overview of other ways of modifying GR in light of the observed ac c eleration. (29) The mass estimates differ both in total amount of mass present and its spatial distribution. Estimating the mass based on the amount of electromagnetic radiation received (photometric observations) requires a number of further assumptions regarding the nature of the objects emitting the radiation and the effects of intervening matter, suc h as sc attering and absorption (extinc tion). (30) This behavior is usually desc ribed using the rotation c urve, a plot of orbital veloc ity as a func tion of the Page 29 of 34

Philosophy of Cosmology distanc e from the galac tic c enter. The “expec ted” behavior (dropping as r−1/2 after an initial maximum) follows from Newtonian gravity with the assumption that all the mass is c onc entrated in the c entral region, like the luminous matter. The disc repanc y c annot be evaded by adding dark matter with the same distribution as the luminous matter; in order to produc e the observed rotation c urves, the dark matter has to be distributed as a halo around the galaxy. (31) In a seminal paper, Ostriker and Peebles (1973) argued in favor of a dark matter halo based on an N body simulation, extending earlier results regarding the stability of rotating systems in Newtonian gravity to galaxies. These earlier results established a c riterion for the stability of rotating systems: if the rotational energy in the system is above a c ritic al value, c ompared to the kinetic energy in random motions, then the system is unstable. The instability arises, roughly speaking, bec ause the formation of an elongated bar shape leads to a larger moment of inertia and a lower rotational energy. Considering the luminous matter alone, spiral galaxies appear to satisfy this c riterion for instability; Ostriker and Peebles (1973) argued that the addition of a large, spheroidal dark matter halo would stabiliz e the luminous matter. (32) This assumption has been c hallenged; see Cooperstoc k and Tieu (2007) for a review of their c ontroversial proposal that a relativistic effec t important in galac tic dynamic s, yet absent from the Newtonian limit, eliminates the need for dark matter. (33) Gravitational lensing oc c urs when light from a bac kground objec t suc h as a quasar is deflec ted due to the spac etime c urvature produc ed, ac c ording to GR, by a foreground objec t, leading to multiple images of a single objec t. The detailed pattern of these multiple images and their relative luminosity c an be used to c onstrain the distribution of mass in the foreground object. (34) See, in partic ular, Weinberg (1989) for an influential review of the c osmologic al c onstant problem prior to the disc overy of dark energy, and, e.g., Polc hinski (2006) for a more rec ent disc ussion. (35) Energy c onditions plac e restric tions on the stress-energy tensor appearing in EFE. They are useful in proving theorems for a range of different types of matter with some c ommon properties, suc h as “having positive energy density” or “having energy-momentum flow on or within the light cone.” In this case the strong energy condition is violated; for the c ase of an ideal fluid disc ussed above, the strong energy c ondition holds iff ρ +3p ≥ 0. Cf, for example, c hapter 9 of Wald (1984) for definitions of other energy c onditions. (36) In more detail, the relevant integral is For a Planc k sc ale c utoff, ℓp ≍ 1.6 × 10−3 5 m, the resulting vac uum energy density is given by ρv ≍ 2 × 10110 erg/cm3 , c ompared to observational c onstraints on the c osmologic al c onstant—ρʌ ≍2× 10−10 erg/cm−3 . Choosing a muc h lower c utoff sc ale, suc h as the elec troweak sc ale ℓew ≍ 10−18 m, is not enough to eliminate the huge disc repanc y (still 55 orders of magnitude). Reformulated in terms of the effec tive field theory approac h, the c osmologic al c onstant violates the tec hnic al c ondition of “naturalness.” Defining an effec tive theory for a given domain requires integrating out higher energy modes, leading to a resc aling of the c onstants appearing in the theory This resc aling would be expec ted to drive the value of terms like the c osmologic al c onstant up to the sc ale of the c utoff; a smaller value, suc h as what is observed, apparently requires an exquisitely fine-tuned c hoic e of the bare value to c ompensate for this sc aling behavior, given that there are no symmetry princ iples or other mec hanisms to preserve a low value. (37) See, in partic ular, Rugh and Zinkernagel (2002) for a thorough c ritic al evaluation of the c osmologic al c onstant problem, as well as Earman (2001), Saunders (2002), and Bianc hi and Rovelli (2010). (38) See ? for an overview of the use of inhomogeneous models as an alternative to dark energy. (39) The pronounc ements of the steady state theorists drew a number of philosophers into debates regarding c osmology in the 1960s. See Kragh (1996) for a historic al ac c ount of the steady-state theory, and the rejec tion of it in favor of the big bang theory by the sc ientific c ommunity, and Balashov (2002) for a disc ussion of their views regarding laws. Page 30 of 34

Philosophy of Cosmology (40) See Pauri (1991); Sc heibe (1991); Torretti (2000) for disc ussions of the implic ations of uniqueness and the status of laws in cosmology. (41) This is the first of two princ iples Smolin advoc ates as nec essary to resolve the problem of time, and he further argues that they bring c osmologic al theoriz ing more in line with sc ientific prac tic e. (42) That is, for any open set O of the spac etime manifold M, if 〈M,gab, Tab〉 is a solution of EFE, then so is 〈O, gab | O, Tab| O〉 taken as a spac etime in its own right. (43) The Weyl tensor represents, roughly speaking, the gravitational degrees of freedom in GR with the degrees of freedom for the sourc e terms removed. Penrose's hypothesis holds that this tensor vanishes in the limit as one approac hes the initial singularity. (44) This mistake also underlies muc h of the disc ussion of “c eteris paribus” laws, and here I draw on the line of argument due to Smith (2002); Earman and Roberts (1999). (45) There may be other philosophic al requirements on an ac c ount of laws of nature that do draw a distinc tion between laws of physic s and laws of the universe. (46) In Minkowski spac etime, this set is the past lobe of the light c one at p, inc luding interior points and the point p itself. A point p causally precedes q (p 〈 q), if there is a future-direc ted c urve from p to q with tangent vec tors that are timelike or null at every point. The sets J ± (p) are defined in terms of this relation: J−(p) = q : q 〈 p , J+(p) = q : p 〈 q , the causal past and future of the point p, and the definition generalizes immediately to spacetime regions. (47) The Gauss-Codac c i c onstraint equations do impose some restric tions on spac elike separated regions, although these would not make it possible to determine the state of one region from the other; see Ellis and Sc iama (1972). (48) A local property of a spac etime is one that is shared by loc ally isometric spac etimes, whereas global properties are not. (Two spac etimes are loc ally isometric iff for any point p in the first spac etime, there is an open neighborhood of the point suc h that it c an be mapped to an isometric open neighborhood of the sec ond spac etime (and vic e versa).) (49) The underdetermination problem still arises if we consider the past of future-inextendible curves; see Glymour (1977); Malament (1977) for discussion. (50) Malament (1977) reviews several different definitions of observational indistinguishability and gives a series of c onstruc tions of OI spac etimes lac king spec ific global properties. Note that Malament defines OI in terms of the chronological rather than causal sets, whic h inc lude the interior of the light c one but not the c one itself (The definition follows the one given in footnote 46, dropping the phrase “or null.”) Manc hak (2009) proves that Malament's tec hnique for c onstruc ting suc h spac etimes fails only in the exc eptional c ase noted in the text. Cf Norton (2011), who argues that the induc tive generaliz ations from J− (p) to other regions of spac etime lac k c lear justific ation. (51) As Malament emphasiz es, this inc ludes the failure of the c ausality c onditions to hold. (52) Pic k a point in p ∈ M suc h that p lies in ∑ and its image ϕ (p) ∈ M′ under the isometric imbedding map ϕ . If homogeneity holds, then M′ must inc lude an isometric “c opy” ∑′ of the entire Cauchy surface ∑ along with its entire c ausal past. Take ξ to be an isometry of the spatial metric defined on ∑, and ξ′ an isometry on ∑′. Sinc e ϕ ◦ξ(p) = ξ′ ◦ϕ (p), and any point q ∈ ∑ c an be reac hed via ∑, it follows that ∑′ is isometric to E′. Mapping points along an inextendible timelike c urve from M into M eventually leads to an isometric c opy of our original spac etime, assuming that both spac etimes are inextendible. Turning this into a proof that OI c ounterparts are c ompletely eliminated requires further assumptions about the topology of the solutions. (de Sitter spac time and its unrolled c overing spac e are both inextendible and homogeneous yet have distinc t global topology, as John Manc hak reminded me.) (53) See Beisbart (2009) for a thorough disc ussion of different attempts to justify the c osmologic al princ iple. (54) Rec ent work has c larified the extent to whic h this result depends on the various exac t c laims made in the antec edent. The fundamental observers do not need to measure exac t isotropy for a version of the theorem to go Page 31 of 34

Philosophy of Cosmology through: Stoeger, Maarten, and Ellis have further shown that almost isotropic CMBR measurements imply that the spac etime is an almost FLRW model, in a sense that c an be made prec ise; see Clarkson and Maartens (2010) for a review. (55) Their model replac es temporal evolution in the Standard Model with spatial variation, with spheric al sym metry around a preferred axis. They c onstruc t the model to rec apture the observational results of the Standard Model for observers situated near the axis of symmetry. Suc h a preferred loc ation is exac tly what the Copernic an princ iple rules out. (56) The Sunyaev-Zel'dovic h effec t refers to the distortion of the spec trum of CBR photons that results from sc attering by hot gases in galaxy c lusters. Due to the sc attering by the hot gas the CBR spec trum will have an exc ess of high-energy photons and a defic it of low-energy photons; measurements of this distortion c an in princ iple be used to measure the temperature and mass of the gas in the c luster. (57) For example, topologic al properties suc h as temporal orientability, whic h allows for a globally c onsistent c hoic e of the direc tion of time, seem to be presupposed in formulating loc al dynamic al laws. (58) Here I am adopting the usual way of desc ribing the objec tion, although this language c an be quite misleading as it implic itly assumes that the singularity c an be “loc aliz ed” in some sense. There are c onvinc ing arguments in favor of taking singular as an adjec tive desc ribing spac etime as a whole; see Curiel (1999), Geroc h, Can-bin, and Wald (1982). (59) Following Rindler (1956), a horiz on is the surfac e in a time slic e t0 separating partic les moving along geodesic s that c ould have been observed from a worldline γ by t0 from those whic h c ould not. The distanc e to this surfac e, for signals emitted at a time te, is given by: ((3)) Different “horiz ons” c orrespond to different c hoic es of limits of integration, with the “partic le horiz on” defined as the limit te → 0. The integral c onverges for R(t) ∝ tn with n 〈 1, whic h holds for matter or radiation-dominated expansion, leading to a finite horiz on distanc e. See Ellis and Rothman (1993) for a c lear introduc tion to horiz ons. (60) Ω  =:   ρ , where the c ritic al density is the value of ρ for the flat FLRW model, ρc  =   3  (H2 − Λ ). ρc 8π 3 (61) It follows from the FLRW dynamic s that ∣Ω−1∣  α R3γ−2  (t). γ 〉 2/3 if the strong energy c ondition holds, and in Ω that c ase an initial value of Ω not equal to 1 is driven rapidly away from 1. (62) One c an evolve observed fluc tuations bac kward to determine the amplitude of the fluc tuation spec trum at a given “initial” time ti. For ti on the order of the Planc k time, for example, Blau and Guth (1987) c alc ulate that the fluc tuations obtained by evolving bac kward from the time of rec ombination imply a density c ontrast of ≍ 10−4 9 at ti, nine orders of magnitude smaller than thermal fluc tuations. The c omparison depends on the c hoic e of the time ti: if this is treated as a free variable, then there will be some time at whic h the fluc tuations are c omparable to thermal fluc tuations. (63) For any reasonable c hoic e of measure over the spac e of solutions, these models are presumably a measure- z ero subset. (64) Inflation solves the horiz on problem bec ause the horiz on distanc e inc reases exponentially during inflation; for a suffic iently long period of inflation, all the points on the surfac e of last sc attering will have overlapping past light c ones. The inflationary phase also reverses the dynamic al feature of the FLRW models responsible for the flatness problem. Bec ause γ = 0 (in the equation in f n. 61) for most models of inflation, inflationary expansion drives Ω toward 1, enlarging the range of c hoic es Ω (tp) c ompatible with observations. (65) One of the main lines of c ritic ism of inflation is due to Roger Penrose; see Penrose (2004, c h. 28) for a rec ent exposition. See Earman and Mosterin (1999) for a philosopher's take on inflation, Linde (2007), for example, for a rec ent review and Turok (2002) for a c ritic al assessment. Page 32 of 34

Philosophy of Cosmology (66) For further disc ussion, see, e.g., Callender (2004a); ?); Wald (2006); Wallac e (2011). (67) For further disc ussion of c ausality in relation to the horiz on problem, see Earman (1995), and for a c ritic al assessment of unific ation c laims, see Zinkernagel (2002). (68) See Pric e (2004) for a defense of the opposing point of view, in an exc hange with Callender (2004a,b). (69) Barrow and Tipler (1986) is an influential early survey of the field; see Carr (2007) for a rec ent c ollec tion of essays. (70) These nec essary c onditions are: (1) that main sequenc e stars are still burning, and (2) that an earlier generation of red giants had time to produc e c arbon in supernovae. (71) Bayesians c an ac c ount for this by explic itly c onditionaliz ing on some c laim c harac teriz ing the selec tion effec t A: Ps(·) = P(· \\A). The selec tion effec t may render an originally “informative” piec e of evidenc e E useless, in that Ps(E\\H) ≍ Ps(E\\ H). In these terms, Dic ke's argument shows that Ps(LN\\HD) Ps(LN\\HSM ) 1, where LN is the large number c oinc idenc e, HD is Dirac 's c osmologic al theory, and HS M is the Standard Model. (72) This is not to say that Weinberg's paper is the first appearanc e of this kind of anthropic reasoning in c ontemporary c osmology; Collins and Hawking (1973) is an earlier influential example, in whic h similar reasoning is used to ac c ount for the isotropy of the universe. (73) More prec isely, the upper bound relates the λ term to the total energy density in matter at the time when most galaxies formed; the upper bound on λ is 200 times the present matter density Considering variation of multiple parameters may undermine this bound; larger values of λ c an be tolerated if one inc reases the amplitude of the initial spec trum of density perturbations, for example. See Aguirre (2007) for a disc ussion of the problems assoc iated with c onsidering a single parameter. (74) Weinberg (1987) did not base his suggestion on a partic ular multiverse proposal, instead listing four proposals that would provide a suitable setting for his argument. (75) There have been c alc ulations for the prior probability distribution over λ in different proposed multiverses; the assumption holds in some but not all of them (see, e.g., Garriga and Vilenkin 2000). (76) This is a vast improvement on the estimates produc ed by partic le physic s, whic h are off by up to 120 orders of magnitude. In a later treatment, Weinberg argues for a lower anthropic bound, suc h that the probability assigned to c urrent observations is either 5 or 12% (depending on other assumptions); see Weinberg (2007) for an overview and referenc es. (77) See, e.g., Vilenkin (1995); Bostrom (2002). (78) Obviously this argument requires some assumptions regarding methodology; it is typic ally formulated within a Bayesian approac h, and the c onc lusion need not follow on other ac c ounts of induc tive method. Shortly I will return to the question of whether this is a good argument even on a Bayesian approac h. (79) There are various different formulations of the argument (see Bostrom 2002 for an entry point into this literature). One formulation starts with the assumption that the probability of one's own birth rank being r is given by Pr(r \\ N) = 1/N, where N is the total number of humans ever born (assuming that N ≥ r). If one further assigns a prior probability Pr(N) = k/N (with a c onstant k), then the posterior probability obtained using Bayes's theorem is Pr(N\\r) = k/N. It follows that there is a less than 5% probability that the total number of humans ever born will exc eed 20r. The argument is entirely general and results from invoking the PM in c hoosing a time within a proc ess that extends over some finite duration. (80) See Bostrom (2002) and Neal (2006) for disc ussions of the different versions of “anthropic reasoning” and the various puz z les they are meant to address. (81) This was proposed by Dieks (1992) in response to the Doomsday argument; see Bostrom (2002) and Dieks (2007) for further disc ussion. The idea has also been disc ussed in light of Elga's (2000) Sleeping Beauty problem. Page 33 of 34

Philosophy of Cosmology (82) Henc e the Presumptuous Philosopher (see Bostrom 2002), whose posterior probability in the theory with more observers remains high despite receiving disconfirming evidence. (83) This is not to say that various c onsiderations emphasiz ed in the anthropic literature, suc h as the number of observers predic ted to exist in a partic ular situation, are irrelevant. Rather, suc h fac tors c an be ac c ounted for in a Bayesian approach by paying careful attention to the details without adding further general principles. (84) How to c alc ulate this frac tion depends upon the measure assigned over the multiverse, so that one c an c ount regions. Here for the sake of illustration I will simply assume that suc h a frac tion is well defined and that it yields a finite result. (85) Note that arguments to this effect usually involve a lot of hand-waving. (86) The Everett interpretation of quantum mechanics attributes a branching structure to the universal wave func tion of the universe, and the individual branc hes c an be regarded as something akin to poc ket universes (see Wallac e, this volume, for a disc ussion of the Everett interpretation). However, unlike the other ac c ounts the laws of physic s do not vary in the different branc hes. There is a c lear distinc tion between the two c ases, although rec ently there has been interest in exploring connections between these two lines of thought. (87) See Tegmark (2009) for an influential classification of four different types or levels of the multiverse. (88) This line of argument has appeared numerous times in the literature; see, e.g., Livio and Rees (2005) for a c lear formulation. (89) Here I am indebted to discussions with John Earman. (90) The explanation may also be path-dependent in the sense of depending not just on an initial state, but on various stoc hastic proc esses leading to the formation of the poc ket universe. Chris Smeenk Chris Sm eenk is Associate Professor of Philosophy at the University of Western Ontario. He received a B.A. degree in Physics and Philosophy from Yale University in 1995, and pursued graduate studies at the University of Pittsburgh leading to a PhD in History and Philosophy of Science in 2003. Prior to arriving at UWO, he held a post‐doctoral fellowship at the Dibner Institute for History of Science and Technology (MIT) and was an assistant professor in the Departm ent of Philosophy at UCLA (2003–2007). His m ain research interests are history and philosophy of physics, general issues in philosophy of science, and seventeenth‐century natural philosophy.

Index The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Index Ab initio strategy, the tyranny of sc ales, 263– 64 Abraham-Lorentz equation, 119, 131, 137 ACDM model, 617, 618 Action-at-a-distance forces, 45 rigid body mechanics, 73 Ac tion princ iple, unitary equivalenc e, 491 Additivity, phase transitions, 201n2 Advanc ed Green's func tions, c lassic al mec hanic s, 109 bounded spatial domain, wave equation in, 121–23 Cauc hy problem, 120 Dirac delta func tion, 111– 12 Fourier transformations, 111 overview, 109–13 point source, 110–11 spatial propagation, 120–21 undamped harmonic osc illator, 110 wave equation, 120–23 Airplane wings, boundary layers, 26 Airy, George Biddell, 17 “Algebraic Imperialism,” unitary equivalenc e, 513 Analytic mec hanic s, 44 Anderson, Edward, 562 Anderson, Hans, 163 Anderson, Philip, 2, 142 Andrews, Thomas, 142, 147, 191 Anthropic reasoning, 638–43 Bayesian c onditionaliz ation, 642, 643 c osmic bac kground radiation (CBR), 640 Page 1 of 112

Index Doomsday argument, 641, 643 “Large Number Hypothesis,” 638–39 multiverse, 639 “princ iple of medioc rity” (PM), 639– 42 Anti-atomism, 55 Antifoundationalism, effec tive field theory (EFT), 240 Anti-partic les, symmetry, 294n10 Antireduc tionism, effec tive field theory (EFT), 240 Approximations effective field theory (EFT), 243 hydrodynamic s, philosophy of, 31 Arc length along the slot, rigid body mec hanic s, 71– 72, 72 Asymmetry, indistinguishability, 365n33 Averaging and homogeniz ation, differenc es, 277–78 Axiomatic presentation, 47– 58 c hoic e of sc ale length, 50, 50 c onc eptual system, 57 decompositional programs, 53, 53 degeneration, 53 “escape hatches,” 56 forc e of rolling fric tion, 54, 54 foundational point of view, 48–49 freez ing to a sc ale level, 55 homogeniz ation theory, 51, 53, 55 point-mass swarm, 51 points, 49–50 reduc ed variables, 51, 52, 55n14, 56 Riemann-Hugoniot approac h to shoc k waves, 53n12 rigid c rystalline forms, 51 rolling on a rigid trac k, 54– 55 rotation, 48 sc ale siz es, relationships between, 51–52 shoc k waves, Riemann-Hugoniot approac h, 53n12 Stieltjes-Lesbeque integration, 53n12 theory fac ades, 48, 57 visc osity of fluid, 54 Bac h, Alexander, 355– 56 Bac k-bending, infinite idealiz ation, 206–8, 207 Balanc e princ iples, c ontinuum mec hanic s, 94 Barbour, Julian Hole Argument (Einstein), 578 Mach's Principle: From Newton's Bucket to Quantum Gravity, 557 rationality, failure of, 539–40 Batterman, R.,emergenc e of EFT, 248– 51 Bayesian c onditionaliz ation, 642, 643 Page 2 of 112

Index Bead sliding on rigid wire, 71 Bélanger, Jean Baptiste, 22 Bell, John S., 481–82, 569 Bernoulli, Daniel. See Bernoulli-Euler beam; Bernoulli-Euler equation; Bernoulli's Law Bernoulli-Euler beam, 99, 100 Bernoulli-Euler equation, 90, 90, 91 Bernoulli's Law, 15–16, 27 Best c lassic al approximation, 46 Best matc hing, Mac hian relationalism, 559, 559– 60 Best Systems presc ription, have-it-all relationalism, 566 Big Bang, 609, 615 elec troweak theory, 303, 403 global struc ture, c osmology, 631 singularity, 600 “Big dessert” assumption, 404 Billiard c ollisions, point-mass mec hanic s, 68– 69, 69 Bit c ommitment problem, quantum mec hanic s, 429– 30 Bjerknes, Vilhelm, 19 Blac k-body spec trum of radiation, 615 Blac k hole evaporation, 518 Blac k, Max, 374 Bloc h sphere, 423 Bloc ks-and-c ords c onstruc tion, 91, 92 Block transforms, 170–72, 172 Blowup, c lassic al mec hanic s, 45n3 Bohm-Aharonov effec t, 507 Bohr, Niels, 9, 463 Boiling, 149, 163–64 fluctuations, 151 phase transitions, 147 Boltz mann equation, 614n16 Boltz mann-Gibbs formulation Ising model, 154 matter, infinities and renormaliz ation, 180 Boltz mann, Ludwig Boltz mann equation, 142 indistinguishability entropy, 346–47 haecceitism, 356 Maxwell-Boltz mann statistic s, 371n42 quantum, indistinguishability and, 342 thermodynamic s, 343 mec hanic s, princ iples of, 53 statistic al physic s, theory of, 142 Born, Max, 431, 434 Page 3 of 112

Index Born rule c lassic regime, 8, 417, 419, 431 c onsistenc y, 442 general phenomenology of measurements, 446, 448 measurement, 451–52, 454 proper mixtures, 424 unsharp spin measurements, 445 Bose-Einstein, 206, 208 Bottom-up approac h effec tive field theory (EFT), 224, 229– 30 the tyranny of sc ales, 257 Boundary conditions Diric hlet variety of, 122 global spac etime struc ture, 600 radiation theory, 123–34 Boundary layers, 13, 23–27, 53n12 airplane wings, 26 disc ontinuity surfac e, 24– 25 eddy resistanc e, 24, 25 and modules, 37 ships, 24 skin resistance, 24 wave resistanc e, 24 Bounded spatial domain, wave equation in, 121–23 Boussinesq, Joseph, 17 Branch count, Everett interpretation, 477–78 Broglie-Bohm theory, 431, 440, 482 Broken symmetries matter, infinities and renormaliz ation, 144–45 Brown, Harvey R., 541–43, 544n46, 569, 570n92 Brownian partic le, 437 BSW ac tion, Mac hian relationalism, 562– 64 Bubbles within bubbles, the tyranny of sc ales, 267, 284 Bulls-eyes, rigid body mec hanic s, 75– 76 Burgess, C. P., 234–35 Butterfield, Jeremy, 219, 260–64 Cabibbo-Kobayashi-Maskawa framework, 402 Callen-Symanz ik equation, 180 Caloric c urve, bac k-bending of, 207 Canonic al antic ommutation relations (CARS), 489, 490, 501– 2, 504 “uniqueness results,” 494, 497 Canonical ensemble, 147 Cao, T., 243, 244, 249–41, 407, 411 Carnap, Rudolf, 369, 372 CARS. See Canonic al antic ommutation relations (CARS) Page 4 of 112

Index Cartan distribution, 324 Cartesian coordinates, 536 Cartesian locations, 63 Cartesian motion, 524–26 Cassini spac e probe to Saturn, 61, 61 Cat example Everett interpretation, 464–65, 467 Sc hrödinger, quantum mec hanic s, 435 Cauchy, Augustin c ontinuum mec hanic s, “Cauc hy's Law,” 98– 99 point-mass mec hanic s, 67 rigid body mec hanic s, “Cauc hy's Law,” 78n32 surfac e waves, 17 the tyranny of sc ales, Euler's rec ipes, 273– 75 vortex motion, 17 Cauc hy integral, dispersion theory, 133 Cauchy problem advanc ed Green's func tions, 120 retarded Green's functions, 120 Cauc hy's equation, the tyranny of sc ales, 270– 75 “Cauchy's Law,” 78n32, 98–99 Cauc hy surfac e global struc ture, c osmology, 629 spacetime properties, 597 Causal c ontinuity c ondition, spac etime properties, 597 Causal c urve, relativistic spac etime, 591 Causal direc tionality, retarded Green's func tions, 116 Causal future, relativistic spac etime, 590– 91 “Causality” as initial value problem, retarded Green's func tions, 116– 20 “Causality c onditions,” global struc ture, 629 Causal simplic ity c ondition, spac etime properties, 597 Causal struc ture, spac etime properties, 596 Causation in c lassic al mec hanic s, 107– 40 Abraham-Lorentz equation, 119, 131, 137 advanc ed Green's func tions, 4, 109 bounded spatial domain, wave equation in, 121–23 Cauc hy problem, 120 Dirac delta func tion, 111– 12 Fourier transformations, 111 overview, 109–13 point source, 110–11 spatial propagation, 120–21 undamped harmonic osc illator, 110 wave equation, 120–23 backward causation, 108n6 Page 5 of 112

Index dispersion theory, 132–37 Cauchy integral, 133 description, 109 Fourier transformations, 133, 135 Hilbert transform pairs, 134, 136 Kramers-Kronig relations, 135–36 passivity and c ausality, 137 Eliez er's theorem, 132 Lorentz -Dirac equation description, 109 point-partic le elec trodynamic s, 132 overview, 4 point-partic le elec trodynamic s, 129– 32 radiation theory, 123–29 boundary conditions, 123–34 description, 109 finiteness c ondition, 124 Fourier transformation tec hnique, 126– 27 Helmholtz equation, 123, 125–27, 129–30 inc oming waves, 127 Laplac e transform tec hnique, 125 outgoing waves, 127 Princ iple of Limiting Amplitude, 128 reasons for, 123–29 reduc ed wave equation, 125. See also Helmholtz equation Sommerfeld radiation c ondition. See Sommerfeld radiation c ondition time dependent boundary condition, 124 time-harmonic waves, 127–29 waveguide, 124 retarded Green's functions Abraham-Lorentz equation, 119 bounded spatial domain, wave equation in, 121–23 Cauc hy problem, 120 c ausal direc tionality, 116 “c ausality” as initial value problem, 116– 20 c ontours for damped osc illator, 115 damping, added, 113–16, 135–36 Dirac delta function, 111–12, 119 “final c ondition,” 119 “final value problem,” 119 Fourier transformations, 111, 113, 114 initial value problem, 116–20 Lebesgue measure z ero, 118 overview, 4, 109 physic al motivation, 113 Page 6 of 112

Index point source, 110–11 privileging, 113–20 Residue Theorem, 114 spatial propagation, 120–21 undamped harmonic osc illator, 110 wave equation, 120–23 time-reversal invarianc e, 136n37 CBR. See Cosmic bac kground radiation (CBR) CCCRs. See Circ ular Canonic al Commutation Relations (CCCRs) CERN's Large Hadron Collider, 305 “Ceteris paribus” laws, 626n44 Chain reac tion, unific ation in physic s, 387– 88 Chandler, David, 163 “Chaotic cosmology” program, 631–32 Charge c onjugation, symmetry, 292– 94 Charleton, Walter, 526 Chia Chiao Lin, 20 Choic e of sc ale length, axiomatic presentation, 50, 50 Christenson, Cronin, Fitc h, and Turlay (CPT theorem), 294– 95 Chronology c ondition, spac etime properties, 596 Chronology violating region of spac etime, relativistic spac etime, 591 Circ ular Canonic al Commutation Relations (CCCRs), 506 Circ ular motion, 525 CKM. See Cabibbo-Kobayashi-Maskawa framework Classic al distillations of quantum proc esses, rigid body mec hanic s, 74– 75 Classic al dynamic al behavior, quantum mec hanic s, 438 Classic al elec trodynamic s, Everett interpretation, 466n6 Classic al mec hanic s, 43– 106 ac tion-at-a-distanc e forc es, 45 analytic mec hanic s, 44 axiomatic presentation, 47–58 c hoic e of sc ale length, 50, 50 c onc eptual system, 57 decompositional programs, 53, 53 degeneration, 53 “escape hatches,” 56 forc e of rolling fric tion, 54, 54 foundational point of view, 48–49 freez ing to a sc ale level, 55 homogeniz ation theory, 51, 53, 55 point-mass swarm, 51 points, 49–50 reduc ed variables, 51, 52, 55n14, 56 Riemann-Hugoniot approac h to shoc k waves, 53n12 rigid c rystalline forms, 51 Page 7 of 112

Index rolling on a rigid trac k, 54– 55 rotation, 48 sc ale siz es, relationships between, 51–52 shoc k waves, Riemann-Hugoniot approac h, 53n12 Stieltjes-Lesbeque integration, 53n12 theory fac ades, 48, 57 visc osity of fluid, 54 best c lassic al approximation, 46 blowup, 45n3 c ausation in. See Causation in c lassic al mec hanic s c omposite, 44 “c onc eptually simple surrogate, 44 c onnec ted rigid parts, 44 c ontac t forc es, 45 continua, 44 c ontinuum mec hanic s, 83– 104 applied force, 84 balanc e princ iples, 94 Bernoulli-Euler beam, 99, 100 Bernoulli-Euler equation, 90, 90, 91 bloc ks-and-c ords c onstruc tion, 91, 92 “Cauc hy's Law,” 98– 99 c ompatibility, 85, 85 c omposite c ontinua, 88 c onstitutive assumptions, 98 decorated points, 84, 84–85 dimensionless point c ube, 95n42 drumheads, 92 Hooke's law, 99 inertial forc e, 97n44 infinitesimal c ubes, 96 infinitesimal trac tion vec tors, 85 labyrinth of the c ontinuum, 90, 101 lattic e defec ts, 98 loaded beam, 91n40 material derivatives, 87 mechanical elements, 91 “method of extensive abstrac tion,” 96n43 physic al infinitesimals, 84– 85, 85 point-mass setting, 98 Problem of the Physic al Infinitesimal, 86, 90, 93 reference plane, 96 response planes, 95 shearing pattern, 85 short-range forc es, 97, 97, 98 Page 8 of 112

Index spec ial forc e laws, 99 strain tensors, 85, 95, 95 stress and strain tensors, 96n43 stress-energy tensors, 84 stress tensors, 85, 95, 95, 96 surfac e forc es, 87 tensor fields, 98 toothpaste, 89, 89 trac tion vec tors, 83, 83– 84, 93 vibrating strings, 92, 92, 101 wave movements, 88, 88 enric hed relationalism, 545– 53 diffeomorphism group (DPM), 547, 550 embedding of relational history, 546 Galilei group, 549–50 kinematic shift argument, 546–47 Maxwell group, 550–51 Maxwellian relationalist, 552 Newton-Cartan theory, 551, 553 nonsimultaneous events, 549 Sklarations, 548–49 “false fac ts,” 47 fields, 44 flexible beam, 44 flexible bodies, 44 mass point lattic e, 44 mathematic al c omplexity, 45 new interest in, 2 Newton's laws. See Newton, Isaac ontologic ally mixed c irc umstanc es, 44n2 ordinary differential equations (ODEs) c ontinuous variables, lift from, 55n14 and foundational princ iples, 45n4 PDE's distinguished, 45 and Schrödinger equation, 46 and spin, 45n5 tasks governed by, 49 overview, 3–4 partial differential equations (PDEs) and foundational princ iples, 45n4 ODE's distinguished, 45 point-mass mechanics, 44–46, 57–69 billiard c ollisions, 68– 69, 69 Cartesian locations, 63 Cassini spac e probe to Saturn, 61, 61 Page 9 of 112

Index c oeffic ient of restitution, 69 c onstitutive modeling c onditions, 66, 82 inertial reac tion, 65 Lennard-Jones potential, 60 magnitude F, 65, 65 matc hed asymptotic s, 69 methodologies of avoidanc e, 62 “natural coordinates,” 63 purely elastic c ollision, 69n26 representative center, 59 rotating rigid objec ts, 58 spec ial forc e laws, 63 steel ball pendulums, 64–65, 64–66 quantum mec hanic s, 45 rigid body mec hanic s, 44, 67, 70– 83 ac tion-at-a-distanc e forc es, 73 arc length along the slot, 71– 72, 72 bulls-eyes, 75–76 c lassic al distillations of quantum proc esses, 74– 75 c onstraint relationships, 70 c ontac t forc es, 73 d'Alembert's principle, 79 dead load, 73 dimensional inharmonious quantities, 76 dynamic loading, 73 Eulerian c uts, 73, 74 forc ed c losed, 70n27 forces, 73–74 free body diagrams, 73, 74 generaliz ed c oordinates, 71 Greenwood's proofs, 80–82 “higher or lower” pairs, 70n28 independently variable, 72 isolated partic le, 81 kinematic s of mec hanisms, 79 Lagrange's principle, 78–82, 81, 82 meaningfully c ombined, 77 mobility spac e, 79 “pinned constraint,” 71 point-mass perspective, 71 pressures, 74 princ iple of virtual work, 79n33 probability differences, 76 punc tiform point of view, 79, 80 redirec tion of thrust, 71 Page 10 of 112

Index representative points, 81, 93 static load, 73 stress, 74 theory of measure, 76 torque r, 77, 78 trac tion forc es, 73 turning moment, 77, 78 virtual displac ement, 81 virtual variations, 79n33 virtual-work reasoning, 72, 72 Schrödinger equation, 46 small siz e sales, 46 surrogate for c lassic al doc trine, 44 Classical Particle Indistinguishability (Bac h), 355– 56 Classic al regime, quantum mec hanic s. See Quantum mec hanic s Clausius, Rudolf, 142, 191 Cloc ks, relativistic spac etimes, 536 Closed c ausal c urve, relativistic spac etime, 591 Closing uniform topology, unitary equivalenc e, 510n23 Coarse-graining indistinguishability equilibrium entropy, 348 statistic al mec hanic s, 346 renormaliz ation group theory, phase transitions, 197 Coeffic ient of restitution, point-mass mec hanic s, 69 Cohen, E. G. D., 163 Coherent states, quantum mec hanic s, 432– 34 Cold dark matter, 617 Collapse postulate, quantum mec hanic s, 8 density operators, 424 measurement, 417, 418, 454 Compac tly generated Cauc hy horiz on, global spac etime struc ture, 602 Compac t manifold, 588 Compatibility, 85, 85 Complex inverse temperature, infinite idealiz ation phase transitions, 209 Composite, c lassic al mec hanic s, 44 Composite c ontinua, c ontinuum mec hanic s, 88 Compton wavelength, 411, 413 Computational templates, hydrodynamic s, 32– 33n41 Conc eptually simple surrogate, 44 Conceptual novelty, phase transitions, 199–200 infinite idealiz ation, 204–10 Conc eptual system, axiomatic presentation, 57 Conc iliatory approac h, Everett interpretation, 475– 77 “Conc ordanc e model,” dark matter and dark energy, 617n22 Page 11 of 112

Index Condensed matter physic s, 3– 4, 383– 84 Conformal fac tor, relativistic spac etime, 592 Connected rigid parts, 44 Constitutive assumptions, c ontinuum mec hanic s, 98 Constitutive modeling c onditions, point-mass mec hanic s, 66, 82 Constraint relationships, rigid body mec hanic s, 70 Constraint solutions, spac etime properties, 595 Contac t forc es, 45 rigid body mechanics, 73 Continua, c lassic al mec hanic s, 44 Continuous models, quantum mec hanic s, 438– 39 Continuous phase transition, 147 Continuous symmetry, 295–97 gauge theories, 299 Continuum EFTs, 237–39 Continuum mec hanic s, 83– 104 applied force, 84 balanc e princ iples, 94 Bernoulli-Euler beam, 99, 100 Bernoulli-Euler equation, 90, 91 bloc ks-and-c ords c onstruc tion, 91, 92 “Cauc hy's Law,” 98– 99 c ompatibility, 85, 85 c omposite c ontinua, 88 c onstitutive assumptions, 98 decorated points, 84, 84–85 dimensionless point c ube, 95n42 drumheads, 92 “Euler's c ontinuum rec ipe,” 257 Hooke's law, 99 inertial forc e, 97n44 infinitesimal c ubes, 96 infinitesimal trac tion vec tors, 85 labyrinth of the c ontinuum, 90, 101 lattic e defec ts, 98 loaded beam, 91n40 material derivatives, 87 mechanical elements, 91 “method of extensive abstrac tion,” 96n43 point-mass setting, 98 Problem of the Physic al Infinitesimal, 86, 90, 93 reference plane, 96 response planes, 95 shearing pattern, 85 short-range forc es, 97, 97, 98 Page 12 of 112

Index spec ial forc e laws, 99 strain tensors, 85, 95, 95 stress-energy tensors, 84 stress tensors, 85, 95, 95, 96 surfac e forc es, 87 tensor fields, 98 toothpaste, 89, 89 trac tion vec tors, 83, 83– 84, 93 the tyranny of sc ales Euler's recipes, 273–74 “material partic les,” 270n18 vibrating strings, 92, 92, 101 wave movements, 88, 88 Continuum model equations, 256–57 Continuum versions, effec tive field theory (EFT), 251– 52 Contours for damped osc illator, retarded Green's func tions, 115 Conventions, system of and indistinguishability, 246 Convex normal, spac etime properties, 593 Coordinate independent transformations,symmetries, 533 Copenhagen interpretation Everett interpretation, 463 quantum mec hanic s, 420 Copernic an princ iple, 631– 32 Coriolis, Gaspard, 22 Correlation func tion c alc ulations, 168– 69 Correlation length, mean field theory, 162 Corresponding states, princ iple of, 162 Cosmic bac kground radiation (CBR), 614– 16, 632 anthropic reasoning, 640 dark matter and dark energy, 618 early universe c osmology, 634 Cosmologic al princ iple, 610, 630 Cosmology, philosophy of, 10–11, 607–52 anthropic reasoning, 638–43 Bayesian c onditionaliz ation, 642, 643 c osmic bac kground radiation (CBR), 640 Doomsday argument, 641, 643 “Large Number Hypothesis,” 638–39 multiverse, 639 “princ iple of medioc rity” (PM), 639– 42 dark matter and dark energy, 617–24, 646 ACDM model, 618 “c onc ordanc e model,” 617n22 c osmic bac kground radiation (CBR), 618 dark energy, c ase for, 620– 21 Page 13 of 112

Index Einstein-Hilbert ac tions, 619n28 general relativity (GR), 619 “hot” versus “cold,” 618n24 lensing effect, 621 light-bending, 620–21 modific ation of Newtonian dynamic s (MOND), 623– 24 Newtonian gravity, 619–20 “old” c osmologic al c onstant problem, 621– 22 Planc k sc ale, 621 quantum field theory (QFT), 621–22 rotation curves, 620n30 “standard candle,” 618 stress-energy tensors, 621n35 systematic error, 620 Type Ia supernovae, 618n25 “z odiac al masses,” 619 early universe c osmology, 633– 38 c osmic bac kground radiation (CBR), 634 “dynamic al approac h,” 636– 37 Einstein Field Equations (EFEs), 636 flatness problem, 635 Friedman-Lemaitre-Robertson-Walker (FLRW) models, 634–36 horiz on problem, 634–35, 635 inflation, 636n64, 637 “overlapping domains” argument, 633 Standard Model, 633–36 “theoretic al unific ation,” 633 “theory of initial c onditions,” 637 Friedman-Lemaitre-Robertson-Walker (FLRW) models, 623 global structure, 628–33 Big Bang model, 631 Cauc hy surfac e, 629 “c ausality c onditions,” 629 “c haotic c osmology” program, 631– 32 Copernic an princ iple, 631– 32 c osmic bac kground radiation (CBR), 632 “c osmologic al princ iple,” 630 Ehlers-Geren-Sachs (EGS) theorem, 630–32 Einstein Field Equations (EFEs), 628–29 Friedman-Lemaitre-Robertson-Walker (FLRW) models, 631–33 Gauss-Codac c i c onstraint equations, 629n47 global hyperbolic spac etime, 10, 629 homogeneity, 630 loc al property of spac etime, 629n48 Minkowski spacetime, 628n46, 629 Page 14 of 112

Index observationally indistinguishable (OI) spac etime, 629– 30 spacetime geometry, 628 Sunyaey-Zel'dovich effect, 632 multiverse, 643–47 De Sitter universe, 644 Everett interpretation, 644n86 inflation, 644 and string theory, 644 “standard candle,” 609n1 Standard Model, 10–11 ACDM model, 617 assumption of, 628 barriers, 615 Big Bang, 609, 615 black-body spectrum of radiation, 615 Boltz mann equation, 614n16 c old dark matter, 617 c osmic bac kground radiation (CBR), 614– 16 c osmologic al princ iple, 610 Einstein Field Equations (EFEs), 610–11, 613 expanding universe models, 609–14 freez e out of partic les, 614 Friedman-Lemaitre-Robertson-Walker (FLRW) models, 610, 612–13, 614n16, 617, 623 galaxies and c lusters of galaxies, length sc ale, 613 global isotropy, reduc tion of, 611 Mac h's princ iple, 610 overview of, 609–17 parameters of, determining, 632 Raychaudhuri equation, 611 “re-combination,” 615 structure formation, 616–17 thermal history, 614–16 Type Ia supernovae, 609 uniqueness of universe, 624–27 “c eteris paribus” laws, 626n44 and “Laws of Physic s,” 625 Mars's motion, 626–27 and quantum field theory (QFT), 626 and relativistic c osmology, 625– 26 Sun's gravitational field, 626–27 Coulomb forc e, infinite idealiz ation, 205 Coulomb's law, 60, 401 Coulumb barriers, 615 CP symmetry, elec troweak theory, 402– 3 CPT theorem (Christenson, Cronin, Fitc h, and Turlay), 294– 95 Page 15 of 112

Index Critic al fixed points, 179, 180 Critic al opalesc enc e, fluc tuations, 151– 52 Critic al point, 147 Crossover theory, infinite idealiz ation, 220–21 Crystalline materials, 143–44 Crystals, Defects, and Microstructures (Phillips), 284–85 Curie, Pierre, 157, 290 “Curie Princ iple,” 290 Cutoff, effec tive field theory (EFT), 241– 43, 250n26 Cylindric al Minkowski spac etime, relativistic spac etime, 591 Damping, retarded Green's func tions, 113– 16, 135– 36 Dark matter and dark energy, 617–24, 646 ACDM model, 618 “c onc ordanc e model,” 617n22 c osmic bac kground radiation (CBR), 618 dark energy, c ase for, 620– 21 Einstein-Hilbert ac tions, 619n28 general relativity (GR), 619 “hot” versus “cold,” 618n24 lensing effect, 621 light-bending, 620–21 modific ation of Newtonian dynamic s (MOND), 623– 24 Newtonian gravity, 619–20 “old” c osmologic al c onstant problem, 621– 22 Planc k sc ale, 621 quantum field theory (QFT), 621–22 rotation curves, 620n30 “standard candle,” 618 stress-energy tensors, 621n35 systematic error, 620 Type Ia supernovae, 618n25 “z odiac al masses,” 619 Darwin, Charles, 47 d'Alembert, Jean le Rond boundary layers, 23 hydrodynamic s, 13 point-mass mec hanic s, 66 rigid body mechanics, 79 unific ation in physic s, 389– 90 vortex motion, 19 Dead load, rigid body mec hanic s, 73 De Boer, Reint, 272 Debye, Peter, 163, 340 Dec ision-theoretic framework, c onstruc tion of, 480 Decoherence Page 16 of 112

Index Everett interpretation, 461, 468–70 system-environment split, 468–69 quantum mec hanic s, 437– 41 c ontinuous models of, 438– 39 Everett interpretation. See Everett interpretation Newtonian behavior, 439 Dec ompositional programs, axiomatic presentation, 53, 53 Decorated points, 84 Dec oupling, effec tive field theory (EFT), 238, 240– 41 Degeneration, axiomatic presentation, 53 De Grav (Newton), 526 Dennett, Daniel, 472–73 Density operators, quantum mec hanic s, 420– 23 bit c ommitment problem, 429– 30 entangled states, 427 Hilbert spac e, 425– 26 Hilbert-space vectors, 419n7, 420–21 improper mixtures, 427 no-go theorem for safe bit c ommitment protoc ols, 430 no-hidden variables theorem, 422 nontrivial spin properties, 428 normaliz ation, 422n10 no-signaling theorem, 425–26 proper mixtures, 423–28 reduced states, 425 simplex, 423–24 spin-1/2 systems, 422–23, 425, 443 Dependence, relativistic spacetime, 590–93 Derivationally independent EFT, 246 Desc artes, René, 70n27, 524, 525 De Sitter universe, 541n37 Deutsch, David, 478, 480–81 Diamond, Jared, 141 Dic ke, G., 638, 640 Diffeomorphism group (DPM) enric hed relationalism, c lassic al mec hanic s, 547, 550 Hole diffeomorphism, 575 Mac hian relationalism, 557– 58, 561 Differential equations, symmetries of, 323–26 Cartan distribution, 324 c lassic al symmetries, 323– 24, 334– 37 generaliz ed symmetries, 323–26 Kepler problem, 325 Korteweg-de Vries vector, 325 Lenz -Runge vec tor, 325 Page 17 of 112

Index Lie-Bäcklund transformations, 324–26 local symmetries, 324–25 Maxwell's theory, 325n21 nonloc al symmetries, 323, 325 and physic al equivalenc e, 329– 30 Dimensional inharmonious quantities, rigid body mec hanic s, 76 Dimensionless point c ube, c ontinuum mec hanic s, 95n42 Dirac delta func tion advanc ed Green's func tions, 111– 12 retarded Green's func tions, 113, 119 Dirac , Paul A.M., 640. See also Lorentz -Dirac equation on indistinguishability, 340 “Large Number Hypothesis, 638–39 Dirac-von Neumann interpretation, 419 Diric hlet variety of boundary c onditions, 122 Discrete symmetries, 292–94 Disc retiz ed position measurements, quantum mec hanic s, 443 Dispersion theory, 132–37 Cauchy integral, 133 description, 109 Fourier transformations, 133, 135 Hilbert transform pairs, 134, 136 Kramers-Kronig relations, 135–36 Displac ement, Maxwell's elec trodynamic s, 386– 89 Distinguishability c onditions, spac etime properties, 596 Distribution of z eros, infinite idealiz ation, 208–10 Divergence symmetries, 327nn33 and 34 Domb, Cyril, 166 Dominant energy c onditions, spac etime properties, 595 Donald, Matthew, 436 D1 and D2, symmetry and equivalenc e, 319– 20 Doomsday argument, 641, 643 DPM. See Diffeomorphism group (DPM) Drumheads, c ontinuum mec hanic s, 92 D2, symmetry and equivalenc e, 329– 30 Duhem, Pierre, 55, 67 Dynamic al ac c ount, symmetry, 302n22 Dynamic al approac h early universe c osmology, 636– 37 have-it-all relationalism, 569–74 Dynamic al-c ollapse theories, Everett interpretation, 463 Dynamic al laws, have-it-all relationalism, 565 Dynamic al symmetries Hole Argument (Einstein), 576 spac etime and, 527– 29 Page 18 of 112

Index spacetime, substantivalist and relationalist approaches to, 529 Dynamic al systems theory, 180 “A Dynamic al Theory of the Elec tromagnetic Field” (Maxwell), 390 Dynamic loading, rigid body mec hanic s, 73 Dynamics, 145–47 Dyson, Freeman, 288 Early universe c osmology, 633– 38 c osmic bac kground radiation (CBR), 634 “dynamic al approac h,” 636– 37 Einstein Field Equations (EFEs), 636 flatness problem, 635 Friedman-Lemaitre-Robertson-Walker (FLRW) models, 634–36 horiz on problem, 634–35, 635 inflation, 636n64, 637 “overlapping domains” argument, 633 Standard Model, 633–36 “theoretic al unific ation,” 633 “theory of initial c onditions,” 637 Earman, John have-it-all relationalism, 565 Hole Argument (Einstein), 523, 575, 577 Mac hian spac etime, defined, 557n75 A Primer on Determinism, 1 relationalism, 552 enric hed relationalism, 555n2 symmetry and equivalence, 318–19 Eddy resistanc e, 24, 25 Edinburgh Festschrift, 431, 434 E-expansion, renormaliz ation group theory, 176–77 EFEs. See Einstein Field Equations (EFEs) Effec tive field theory (EFT), 224– 54 antifoundationalism, 240 antireductionism, 240 approximations, EFTs and, 243 autonomy, sense of, 247–48 c ausal autonomy, 248 explanatory autonomy, 248 predic tive autonomy, 247 reductive autonomy, 247 bottom-up approach, 224, 229–30 “c oarse” theory, 248 c ontinuum EFTs, 237– 39 c ontinuum versions, 251– 52 c utoff, realistic interpretations of, 241– 43, 250n26 Wilsonian approac h, 242 Page 19 of 112

Index decoupling, 238, 240–41 definitional extension of T, 247 derivationally independent EFT, 246 duality transformation, 232 emergence, EFTs, 243–51 autonomy, sense of, 247–48 Batterman's notion of emergence, 248–51 c ausal autonomy, 248 “c oarse” theory, 248 c ontinuum versions, 251– 52 definitional extension of T, 247 derivationally independent EFT, 246 explanatory autonomy, 248 Fermi EFT of weak forc e, 245 intertheoretic relation, 245– 47 Lagrangian formalism, 247 Lagrangian, initial and Lagrangian for superfluid Helium, 246 limiting relations, emergenc e and, 248– 51 predic tive autonomy, 247 reductive autonomy, 247 statistic al mec hanic s, 249– 50 superfluid, EFT of, 245–46, 246 thermodynamics, 249–50 top-down EFTs, 244–45 Fermi EFT of weak forc e, 245 Fourier transformations, 239 Gauss's Law, 231–32 Green's func tions, 235, 251 hierarchy problem, 228n6 Higgs term, 228n6 intertheoretic relation, 245– 47 Lagrangian density, 226, 235, 251 Lagrangian formalism, 247 Lagrangian, initial and Lagrangian for superfluid Helium, 246 limiting relations, emergenc e and, 248– 51 low-energy superfluid helium-4 film, 230–32 Lagrangian density, 230–32 mass-dependent schemes, 236–37 mass-independent schemes, 237–39 Minkowski spac etime, 231 nature of EFTs, 225–32 non-relativistic QCD, 229n7 ontologic al implic ations, 239– 43 antifoundationalism, 240 antireductionism, 240 Page 20 of 112

Index approximations, EFTs and, 243 c utoff, realistic interpretations of, 241– 43 decoupling, 240–41 quantum field theory (QFT), 240–43 quasi-autonomous domains, 240–41 Wilsonian approac h, 240 overview, 5 predic tability, 232– 35 quantum field theory (QFT), 240–43, 251 quasi-autonomous domains, 240–41, 243 and emergenc e, 244 Wilsonian approac h, 241 relativistic quantum field theories (RQFTs) emergence, EFTs, 250–51 overview, 224–25 renormaliz ation group (RG), 232–35 nonrenormaliz able EFTs, 233 renormaliz ation sc hemes, 235–39 c ontinuum EFTs, 237– 39 Green's function, 235 Lagrangian density, 235 mass-dependent schemes, 236–37 mass-independent schemes, 237–39 Wilsonian approach, 236–37 sc alar field theory, 227n3 Standard Model, 228 statistic al mec hanic s, 249– 50 superfluid, EFT of, 245–46, 246 thermodynamics, 249–50 top-down approach emergence of EFTs, 244–45 naturalness, hypothesis of, 228 overview, 225–29 quantum c hromodynamic s (QCD), 228– 29 symmetry considerations, 228 types of EFTs, 235–39 unification in physics, 384, 406–13 Gell-Mann/Low formulation, 408–9 Hamiltonians, 410 quantum elec trodynamic s (QED), 411 reduc tionism, problems of, 411n25 Wilson-Kadanoff model, 412 Wilsonian approac h, 252 c utoff, realistic interpretations of, 242 mass-independent schemes, 237–39 Page 21 of 112

Index ontologic al implic ations, 240 and renormaliz ation group (RG) tec hniques, 232 renormaliz ation sc hemes, 236–37 top-down approach, 226, 228–29 Ehlers-Geren-Sachs (EGS) theorem, 630–32 Ehrenfest, P., 191, 350–51 Ehrenfest's theorem, quantum mec hanic s, 432– 34 Ehrenfest-Trkal-van Kampen approac h, indistinguishability, 350– 51, 366 Eiffel, Gustav, 26 Eigenvec tors, quantum mec hanic s, 418 “Eightfold Way,” 302n22 Einstein, Albert c orresponding states, princ iple of, 162 c ritic al opalesc enc e, 151 general relativity, 325, 537–39, 541, 543–44 Hole Argument, 300n17, 523, 574–79 indistinguishability quantum, indistinguishability and, 342–43 quantum partic les, 363 on indistinguishability, 340 light-bending, 620–21 mean field theory, 158 Noether, Emmy, obituary, 297 Principle of Equivalence, 537 quantum mec hanic s, 440 c lassic al regime, 431 c oherent states, 433– 34 substantivalist-relationalist debate, 522n1 symmetry GTR princ iple, 292, 297 STR, formulation of, 311–12 theories of spec ial and general relativity, 290 unific ation in physic s, 392 Einstein Field Equations (EFEs), 537, 543 early universe c osmology, 636 global struc ture, c osmology, 628– 29 relativistic spac etimes, 537 Standard Model, 610–11, 613 Einstein-Hilbert ac tions, dark matter and dark energy, 619n28 Einstein's equation, spacetime properties, 595 Einstein tensor, 537, 594 Electroweak theory, 393–401 Big Bang, 303, 403 “big dessert” assumption, 404 Cabibbo-Kobayashi-Maskawa framework, 402 Page 22 of 112

Index CP symmetry, 402–3 gauge hierarchy problem, 404 Glashow model, 397 Higgs boson, 394–95, 403–6 Higgs field, 398–401 isospin, 396n14 Lagrangian, 395 Lagrangian invarianc e, 397– 99 Large Hadron Collider (LHC), 405–6 Lie group, 395 loop quantum gravity (LQG), 405n22 from mathematic s to physic s, 397– 401 and Maxwell's theory, 396, 399 multiplets, 396 naturalness, 404n19 Noether's theorem, 396–97 non-Abelian c ase, 396, 401 phase transformation, 396n13 Planc k sc ale, 403 problems with, 403–6 QCD vac uum, 394 quantum c hromodynamic s (QCD), 402– 3 Schrödinger equation, 395 Standard Model, 394, 402–6 string theory, 405 SU(2) and SU(3) color groups, 394, 396–97 supersymmetry (SUSY), 405 symmetry as tool for unific ation, 395– 97 Theory of Everything (TOE), 405 Yukawa couplings, 402 Yukawa interac tion, 402n17 Elec troweak unific ation, symmetry, 303– 6 Elementary partic les, renormaliz ation group theory, 176 Elementary Principles in Statistical Mechanics (Gibbs), 352–53 Eliez er's theorem, 132 Eliminativism and Gibbs paradox, 378 indistinguishability, 376–78 “preferred basis problem,” 377 quantum fields, 377–78 Ellis, G. F. R., 625, 627 Embedding of relational history, enric hed relationalism, 546 Emergence EFTs. See Effec tive field theory (EFT) of phase transitions, 197–204 Page 23 of 112

Index Energy-momentum tensor, spacetime properties, 594 Enric hed relationalism c lassic al mec hanic s, 545– 53 diffeomorphism group (DPM), 547, 550 embedding of relational history, 546 Galilei group, 549–50 kinematic shift argument, 546–47 Maxwell group, 550–51 Maxwellian relationalist, 552 Newton-Cartan theory, 551, 553 nonsimultaneous events, 549 Sklarations, 548–49 relativity, 553–57 four-forc e, 554 gravitational wave, 556 kinematic ally possible models (KPMs), 555 Minkowski distanc es, 553– 56 Ensemble equivalent, 207 Ensemble generates averages, 180 Ensemble generates ensemble, 180 Entangled states, quantum mec hanic s, 427 Entropy, indistinguishability, 343–45 Boltz mann definition, 346–47 Environment, entanglement with, 434–37 Epistemic puz z les, Everett interpretation, 479–81 Equal areas law, symmetry, 288n2 Equilibrium, 145–47 Equilibrium entropy, indistinguishability, 348–49 c oarse-graining, 348 Equivalenc e, symmetry and. See Symmetry and equivalenc e Euclidean coordinate systems Mac hian relationalism, 557 spac etime, substantivalist and relationalist approac hes to, 528, 529, 557 Euc lidean spac e, relativistic spac etimes, 536 Euc lidean symmetries, Hole Argument (Einstein), 577 Eulerian flow, 26 Euler-Lagrange equations, symmetry, 296 Euler, Leonhard c ontinuum mec hanic s, 257 balanc e princ iples, 94 Bernoulli-Euler beam, 99, 100 Bernoulli-Euler equation, 90, 91, 91n40 laws of motion, 96 Eulerian flow, 26 explanatory progress, 29 Page 24 of 112

Index fluid motion, 13–14 instabilities, 19 point-mass mec hanic s, 59– 60, 69 rigid body mechanics Eulerian c uts, 73, 74 “First Law,” 78n32 vortex motion, 19 vortex filaments, defined, 18 Euler's rec ipes, the tyranny of sc ales, 273– 75 and Cauchy, Augustin, 273–75 c ontinuum, 273– 74, 278 c ontroversy, 274– 75 discrete, 273–74 and Navier-Stokes equations, 275 and Young's modulus, 275 Everett, Hugh I. See also Everett interpretation insight of, 463–66 Everett interpretation, 8–9, 460–88 Bell's inequalities, 481–82 branch count, 477–78 “c at” example, 464– 65, 467 c lassic al elec trodynamic s, 466n6 c onc iliatory approac h, 475– 77 Copenhagen interpretation, 463 dec ision-theoretic framework, c onstruc tion of, 480 decoherence, 461, 468–70 system-environment split, 468–69 Dennett's criterion, 472–73 dynamic al-c ollapse theories, 463 epistemic puz z les, 479–81 higher-order ontology, 470–74 Hilbert space, 464, 467n8 indexic al unc ertainty, 476 and many-exac t worlds theories, 467– 68 and many-minds theories, 467–68 measurement problem, 462–63 mod-squared amplitude, 479n17, 480 multiverse, 644n86 non-interac ting partic les, 463n5 overview, 460–61 parallel tradition, 478n16 pilot wave theory, 482 “preferred basis problem,” 466–68 and many-exac t worlds theories, 467– 68 and many-minds theories, 467–68 Page 25 of 112

Index overview, 461 probability problem, 466, 474–77 objec tive-probability role, 479n17 parallel tradition, 478n16 philosophic al aspec ts of probability, 474– 75 and possibility, 475–77 probability simplic iter, 475 and uncertainty, 475–77 quantitative problem, 477–79 quantum mechanics, 440, 644n86 quasi particles, 473–74 semantics, 476 Standard Model, 471 struc ture, role of, 470– 74 “superposition state,” 462 wave pac kets, 461 Expanding universe models, 609–14 Explanatory irreduc ibility, phase transitions, 210– 14 Explanatory progress, hydrodynamics, 27–30 c omponents of explanation, 28– 29 heterogeneous spec ializ ations, 30 homogeneous spec ializ ations, 30 pragmatic definition of, 29– 30 sourc es of, 27– 28 Extended singularities Ising model, 152 matter, infinities and renormaliz ation, 183 Extended singularity theorem, Ising model, 154–55 Extensivity, phase transitions, 201n2 “False fac ts,” c lassic al mec hanic s, 47 Fecundity and generality, trade-off between, 335n13 Fermi EFT of weak forc e effec tive field theory (EFT), 245 Fermilab's Tevatron, 305 Fermions, indistinguishability, 364–65 Ferromagnetic phase transitions, 192 Ferromagnets, 145 Ising model, 156 the tyranny of sc ales, 265– 66, 278 spontaneous magnetiz ation, 265 unitary inequivalenc e, as example of, 508, 509 Feynman, R., 119 anti-particles, 294n10 diagrams, indistinguishability, 367, 368 the tyranny of sc ales, 256 unitary equivalenc e, 492 Page 26 of 112