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1. See Smeenk 2012 and O’Raifeartaigh et al. forthcoming forfurther discussion of Einstein’s contributions.

2. There are several excellent textbook treatments of cosmology,including in particular Dodelson 2003, Mukhanov 2005, Durrer 2008, andPeter & Uzan 2013.

3. A geodesic is an analog of a straight line in two senses: thegeodesic between two points extremizes the length, and the tangentvector to the curve remains parallel to itself as it is paralleltransported along the curve.

4. Represented compactly using the metric field, \(g_{ab}\). Theequations are non-linear both in terms of involving products ofderivatives \(g_{ab,c}\) of \(g_{ab}\) with other derivative and alsowith its inverse \(g^{ab}.\)

5. These solutions arehomogeneous andisotropic:homogeneity requires that at a given moment of cosmic time everyspatial point “looks the same,” and isotropy holds ifthere are no geometrically preferred spatial directions. Thus thespatial geometry of \(\Sigma\) is such that there is an isometrycarrying any point \(p \in \Sigma\) to any other point lying in thesame surface (homogeneity), and at any point \(p\) the three spatialdirections are equivalent to each other (isotropy). An isometry is atransformation that preserves the spacetime geometry; more precisely,a diffeomorphism \(\phi\) that leaves the spacetime metric invariant,i.e., \((\phi^*g)_{ab} = g_{ab}\). It is an isotropy about a point\(p\) if it leaves \(p\) fixed: \((\phi^*p) =p\).

6. These are the three possibilities if \(\Sigma\) is assumed to besimply connected, which holds if every closed loop can besmoothly contracted to a point. There are many possibilities for aglobally isotropic space with constant curvature that is multiplyconnected (Ellis 1971b), for example a toroidal topology (a closedversion of flat space) or projective space (with the same metric asspherical space but a different topology).

7. \(\dot{R}\) is the derivative of \(R\) with respect to cosmic time\(t\), \(\Lambda\) is the cosmological constant, and \(G\) isNewton’s gravitational constant. The Raychaudhuri equation is afundamental equation that describes the evolution of a cluster ofnearby worldlines, e.g., for the particles making up a small ball ofdust, in response to spacetime curvature (Ellis 1971a). It takes onthe simple form given here due to the symmetries we have assumed: inthe FLRW models the small ball of dust can change only its volume as afunction of time, but in general there can be a volume-preservingdistortion (shear) and rotation of the ball as well.

8. The stress energy tensor for a perfect fluid is given by \(T_{ab} =(\rho + p) u_a u_b + (p) g_{ab}\), where \(u_a\) is the unit tangentvector to the trajectories of the fluid elements (\(u^a u_a =-1\)).

9. These models typically treat the evolution of perturbations usingNewtonian gravity, over a fixed background cosmological model; theassumptions include specification of the initial spectrum of densityperturbations and cosmological parameters. There are enormouscomputational challenges to giving a fully relativistic account ofstructure formation of comparable dynamical range, and it isfurthermore unclear to what extent a relativistic account divergesfrom Newtonian N-body simulations.

10. See Ellis et al. 2012: §11.1, 13.1 for further discussion andreferences.

11. See Wainwright & Ellis 1997 for discussion.

12. See Clarkson & Maartens 2010, February et al. 2010, and Zhang& Stebbins 2011.

13. There could in principle be a curvature dominated phase before thishappens, but in practice this seems to not be the case. See Ehlers& Rindler 1989 for an illuminating phase space representation ofFLRW solutions with pressure and a cosmological constant that showsall the possibilities.

14. See Beringer et al. 2012 and Ade et al. 2016 for example, for reviewsof evidence bearing on the cosmological parameters. The total numberof parameters used to specify a cosmological model varies in differentstudies, but typically 5–10 fundamental parameters are used todetermine the best fit to a given data set.

15. See, e.g., Particle Data Group 2016 for a recent review ofobservational constraints on the cosmological models.

16. See Norton (1993, 1994) for a discussion of this contrast betweenscientific and philosophical attitudes toward underdetermination.

17. See, for example, Laudan & Leplin 1991 for a sharp criticism ofthe treatment of empirical content by Quine, van Fraassen, and others,along these lines.

18. The light cone is the boundary of the causal past; EFE can be used todetermine the spacetime geometry in the causal past from this dataset. On this approach the ideal data set includes variousastrophysical assumptions about the nature of sources, and theirhistorical evolution, used to measure spacetime geometry, withoutassumptions regarding background geometry. There is an importantlimitation: there is no way to register the impact of dark matter ordark energy directly on the ideal observational set withoutsubstantive modeling assumptions.

19. The Gauss-Codacci constraint equations do impose some restrictions onspacelike separated regions, although these would not make it possibleto determine the state of one region from the other (see Ellis &Sciama 1972). Events to the future of \(J^-(p)\) will in general beinfluenced by regions of spacetime that did not register on\(J^-(p)\), and extrapolations to the future are only valid if ano-interference condition holds.

20. For example, there are models which are locally (rather thanglobally) isotropic and homogeneous, such that the surfaces \(\Sigma\)have finite volume but are multiply connected, consisting of, roughlyspeaking, cells pasted together. Ellis 1971b explores this kind ofmodel; see Lachieze-Rey & Luminet 1995 for a more recent review.Although isotropy and homogeneity hold locally at each point, abovesome length scale there would be geometrically preferred directionsreflecting how the cells are connected.

21. Alocal spacetime property is a property such that for anypair of locally isometric spacetimes, they either both have theproperty or neither does. The property of being a solution to EFE is alocal property in this sense. Global properties, by contrast, can varybetween locally isometric spacetimes. There are a hierarchy ofconditions that characterize the global causal structure ofspacetimes. See Manchak 2013 for further discussion andreferences.

22. Roughly, the counterpart can be constructed by stringing together acollection of “copies” of the causal pasts, like laundryhanging on a clothesline. The properties of this counterpart spacetimeare not constrained in regions outside the copies of the causal pasts.(For example, a closed timelike curve could exist in a region outsidethe copies of the causal pasts, violating global hyperbolicity.) SeeMalament 1977 and Manchak 2009 for further discussion. Malament (1977)reviews several different definitions of observationalindistinguishability; Manchak (2009) establishes the generality ofMalament’s “clothesline construction.”

23. See, for example, Baker et al. (2015) regarding the regimes ofgravitational theory probed by solar system tests, observations ofgravitational waves, and etc. Obviously, the physics horizon is basedin part on an assessment of what is technologically and economicallyfeasible, and will shift over time as new avenues of testing openup.

24. The CMB indicates that baryonic matter was very smooth at the time ofdecoupling because it was strongly coupled to radiation. Dark matterdecouples from radiation earlier than baryonic matter, and can be muchlumpier at the time the CMB is emitted; these lumps then generateperturbations in baryonic matter.

25. At the time of writing, there are no generally accepted candidatesfor successful detection of dark matter particles; instead, ongoingexperimental searches have ruled out parts of the parameter space ofcandidate particles.

26. See Frieman et al. 2008 for a summary.

27. Synge’s G-method (Synge 1961) states that you can always runthe EFE backwards from any given geometry \(g_{ab}\) to determine whatenergy-momentum stress tensor \(T_{ab}\) would solve the EFE exactlyfor that geometry. This does not provide a physical basis for the thusdetermined nature of such ‘matter’, which will for examplegenerically violate the energy conditions.

28. See Ellis and Madsen (1991)for the general procedure, and Lidsey et al. 1997 for itsuse as regarding reconstructing the inflaton potential; this isessentially a version of Synge’s G-method.

29. In the case of say geology, even though each one is different, thereare many mountains, rivers, and continents to observe and compare.

30. The first singularity theorem showed that these divergences followfrom the Raychaudhuri equation (Ellis 1971a). One might have hopedthat pressure would counteract these divergences, but this is not thecase since it enters into the equation with the same sign as theenergy density. Equation (2) is a special case of the generalRaychaudhuri equation, in which shear, vorticity, and accelerationvanish due to the symmetries of the FLRW models.)

31. Refocusing leads to the “onion” shape of the past lightcone: it reaches a maximum radius at some finite time, and decreasesat earlier times (Ellis 1971a). See Ellis & Rothman 1993 forfurther discussion.

32. Cf. Earman (1995)’s defense, following Charles Misner, of“tolerance for spacetime singularities,” based on acritical assessment of the various reasons given in the physicsliterature for regarding singularities as a shortcoming of GR. SeeCuriel and Bokulich (2009 [2012]) for further discussion andreferences.

33. Bergmann argues that

a theory that involves singularities and involves them unavoidably,moreover, carries within itself the seeds of its own destruction;(1980: 186)

similarly, textbook discussions of the singularity theorems (such asWald 1984: Chapter 9) emphasize that the singularity theorems showthat the GR’s classical description of gravity breaks down insome regimes. See Curiel and Bokulich (2009 [2012]) for furtherdiscussion.

34. Penrose has emphasized this point; see Chapter 3 of Penrose 2016 fora recent discussion. Wikipedia has a thorough discussion of the BKLsingularity (seeOther Internet Resources).

35. These features are usually taken to be puzzling because theyapparently require “fine-tuning,” discussed further in§4.2.

36. A particle horizon at time \(t_0\) is the timelike 3-surface inspace-time separating world lines of particles moving alongfundamental geodesics that could have interacted with a worldline\(\gamma\) at the time \(t_0\) (their world lines would haveintersected the past light cone of that event) from those which couldnot. On the time slice \(t_0\), this defines a 2-sphere comprising themost distant matter that could have interacted with \(\gamma\) at thattime. For a radiation-dominated FLRW model, the expression for horizondistance \(d_h\) is finite; the horizon distance at decouplingcorresponds to an angular separation of \(\approx 1^{\circ}\) on thesurface of last scattering.

37. It follows from the FLRW dynamics that \(\frac{|\Omega - 1|}{\Omega}\propto R^{3\gamma - 2}(t)\). \(\gamma > 2/3\) if the strong energycondition holds, and in that case an initial value of \(\Omega\) notequal to 1 is driven rapidly away from 1. Observational constraints on\(\Omega (t_0)\) can be extrapolated back to a constraint on the totalenergy density of the Planck time, namely \(|\Omega(t_p) - 1| \leq10^{-59}\).

38. The Hubble radius \(d(H_0)\) is defined in terms of the instantaneousexpansion rate \(\dot{R}(t)\), by contrast with the particle horizondistance \(d_h\), which depends upon the expansion history since thestart of the universe. For radiation or matter-dominated solutions,the two quantities have the same order of magnitude.

39. Brout et al. 1978, Vilenkin 1983, and Hartle & Hawking 1983 areearly papers pursuing a quantum account of the creation from differenttheoretical perspectives; see also Butterfield & Isham 2000.

40. Here we are in agreement with Albert 2012, which initiated apolemical exchange with Lawrence Krauss (whose book was the subject ofAlbert’s review).

41. Carter (1974) introduceda terminology that has been widely used: the “weak anthropicprinciple” refers to a selection effect, as illustrated in thecase of Dicke’s response to Dirac, whereas the “stronganthropic principle” holds that the universemust (insome sense) allow for the existence of life. These two claims strikeus as too different to be weak and strong versions of a singleunderlying principle. Rather than adopting these terms, below we willdraw a related distinction between selection effects and the use oftypicality assumptions to make anthropic predictions.

42. See Titelbaum (2013) for a recent proposal and references to relatedwork.

43. Although we will not pursue the topic here, Weinberg’s argumentis a special case that avoids some of the questions that arise ingiving a general account of “anthropic prediction.” Forexample, the argument concerns variation of a single parameter,whereas the general case requires considering the variation of severalparameters. See Aguirre (2007) for an account of the challenges andcomplications involved in carrying out anthropic predictions for avariety of parameters, and Starkman & Trotta 2006 for furtherproblems with these methods.

44. This is closely related to Vilenkin (1995)’s “Principleof Mediocrity,” and Bostrom (2002)’s “Self-SamplingAssumption” (although he eventually argues for a principleapplied to “observer-moments” rather than observers); seealso Dorr and Arntzenius (2017).

45. As Aguirre et al. 2007 notes, it is possible to choose some otherobject to conditionalize on in a Weinberg-style argument; but thisleads to similar problems regarding the choice of reference class andappeal to indifference.

46. More precisely, the assignment of probabilities depends on algebraicstructure—the event algebra—defined on the sample space.Many different event algebras can be assigned over the same samplespace.

47. See Norton (2010), who is also critical of Bostrom’s approach,but advocates a non-Bayesian approach to inductive reasoning.

48. There are different ways of enumerating the fundamental constants inboth cases, but the arguments below do not depend upon choosing aspecific list.

49. Barrow & Tipler 1986 is an influential early discussion offine-tuning; for more recent discussions, see Barnes 2012 and Lewis& Barnes 2016.

50. See McGrew et al. 2001 and Colyvan et al. 2005 for challenges tojustifying probabilities in this case, and Manson 2009 for a responseand general discussion of fine-tuning.

51. A somewhat similar view is taken to follow from the existence of manydistinct vacua in string theory (the “landscape”). Bothviews are distinct from the Everettian multiverse, although there havebeen speculative attempts to link these distinct multiverses.

52. See Ade et al. 2016 and Martin et al. 2014 regarding the most recentconstraints from the CMB.

53. See Freivogel et al. 2006 for multiverses based in Coleman de Lucciatunnelling, and Aguirre 2007 for a an overview of EI.

54. A third one is that it is claimed by some that a multiversedefinitively requires universes with spatial sections of negativecurvature (Freivogel et al. 2006), so if if this is not observed thehypothesis is disproved. However that claim is disputed (Buniy et al.2008).

55. See, for example, Vilenkin 2007.

56. There are active debates in the philosophy of physics regarding thestatus of infinite idealizations in other areas of physics; see, e.g.,Batterman 2005 and Menon & Callender 2013.

57. Despite the prevalence of multiverse debates in popular science,there is in fact only a small minority of cosmologists activelyworking on this topic: see Hossenfelder’s comments in a blogpost (2013, inOther Internet Resources), and the lack of mention of the topic in the standard textbooksreferred to above and the data analysis by the Planck team (Ade et al.2016).

58. It is also assumed that the consequence does not itself entail thehypothesis.

59. For further discussion along these lines, see Norton 2000, Harper2012, Smith 2014, and Stein 1994; Smeenk (2017) considers theimplications of this line of thought for inflation.

60. Although most of the anthropic literature is concerned only withnecessary conditions for life, rather thansufficient. In other words, they do not in fact deal withbiology, only with conditions for stars, planes, and elements likeCarbon and Oxygen to exist. To make the link to life proper one needsto extend this discussion to the possibility spaces for life tofunction, as discussed for example by Andreas Wagner in hisilluminating bookArrival of the Fittest (2014) describingvarious spaces of genotype to phenotype maps.