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On the Carroll–Chen Model

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Abstract

I argue that the Carroll–Chen cosmogonic model does not provide a plausible scientific explanation of the past hypothesis (the thesis that our universe began in an extremely low-entropy state). I suggest that this counts as a welcomed result for those who adopt a Mill–Ramsey–Lewis best systems account of laws and maintain that the past hypothesis is a brute fact that is a non-dynamical law.

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Notes

  1. Albert (2000, 96).

  2. Both Albert and Loewer would add another non-dynamical law that is “a probability distribution (or density) over possible initial conditions that assigns a value 1 to PH and is uniform over those micro-states that realize PH” Loewer (2008, 157), where by ‘PH’ Loewer means the past hypothesis.

  3. Lewis (1973b, 73).

  4. Callender (2004b, 249). For more on the Mill–Ramsey–Lewis best-systems account (BSA), see Cartwright et al. (2005, 797–799), Earman (1984, 196–199), Lewis (1973b, 72–77, 1983, 365–368, 1994, 478–482), Loewer (2001, 619, 2012) and Ramsey (1990, 150). For criticisms of the BSA see Armstrong (1983, 70–71), Belot (2011, 70–72), and van Fraassen (1989, 48–51).

    Penrose (1989a, 391–482) seemed to regard something like the past hypothesis as a law; it is just that he understood the initial low-entropy state in terms of Weyl curvature which vanishes as one approaches the beginning of the universe.

  5. See the comments in Sklar (1993, 311–312).

  6. See e.g., Carroll (2006, 1132, 2008a, 48, 50), Carroll and Chen (2004, 3–4), Cf. Carroll (2010, 288). For the implicit claim that the state is improbable see Penrose (2007, 729–731) and Price (2004, 230–231). The fact that a state is unnatural does not necessarily imply that that state is improbable.

  7. Even some of those who would insist that such a state is scientifically brute believe that it could be explained. See Callender (2004a, 199), though cf. his comments in (2004b, 241).

  8. Some of these alternative approaches assume different understandings of our universe’s low-entropy state.

  9. Two points about presentation: First, the discussion will be mostly informal. I delve into mathematical details only when absolutely necessary, and I discuss the physical significance of the underlying technical work, citing that work at every turn for those who might want to explore the underlying mathematical physics. I apologize for the abundance of footnotes, and note here that I stand by my interpretive work, and quote not a few theoretical physicists who agree with my interpretation of the underlying technical details.

    Second, while the analysis in general rests on results already in the literature, these results find novel applications in my critique of the CC-M. For example, no one has articulated just how especially problematic the measure problem and N-bound validity are for the CC-M. No one, so far as I’m aware, has used the EGS or BGV theorems to object to metauniverse nucleation and quantum tunneling. Moreover, no one has voiced the specific worries I will express about the incompleteness of the CC-M. My use of an argument from causation for the impossibility of the background space of the CC-M has heretofore never been supplemented in the way I justify several of the key premises, and some specific points about the internal inconsistency of the model are completely novel.

  10. As Roger Penrose stated, the “early spatial uniformity represents the universe’s extraordinarily low initial entropy”, Penrose (2012, 76). See also Penrose (2007, 706–707). Most writing on the subject agree with Penrose here. See the broader discussions in Albrecht (2004, 371–374), Greene (2004, 171–175), North (2011, 327), Penrose (1979, 611–617, 1989b, 251–260, 2012, 73–79), Price (1996, 79–85, 2004, 227–228) and Wald (1984, 416–418, 2006, 395). See also Callender (2010, 47–51). Earman (2006, 417–418, cf. the comments on 427) is very skeptical of the contemporary orthodoxy on these matters.

  11. Let me say here what I’m concerned with when I discuss or mention the arrow of time. First, I am not interested in the asymmetry of time itself. I am, however, concerned with the asymmetry of the contents of the cosmos (on this distinction see Price 1996, 16–17; North 2011, 312). There are, therefore, many arrows of time, though some maintain that these arrows can be reduced to the thermodynamic arrow. It is this supposed principal arrow with which I’m worried when I comment on the arrow of time below.

  12. “The low entropy starting point is the ultimate reason that the universe has an arrow of time, without which the second law would not make sense.” Dyson et al. (2002, 1). Cf. (Bousso (2012, 2–3; 26) for a different view. The discussion of these sorts of issues in North (2011) is first-rate.

  13. Below, I call the universes that help compose the multiverse “metauniverses”. These are spawned by the ‘Universe’ (capital-U), the background de Sitter spacetime.

  14. Carroll and Chen (2004, 27). I’m borrowing their wording here. The quotation in context is about something different, viz., the fact that the background space–time is never in an equilibrium state because metauniverses can always be generated resulting in the further increase of entropy.

  15. Given that r0 = 0 and that Λ = 3/R2; where R corresponds to a positive constant, and r is the Schwarzschild radius. The equation is from de Sitter (1918, 230); but see also the discussion in de Sitter (1917, 7) and Earman (1995, 7).

  16. Penrose (2007, 747–748), Misner et al. (1973, 745); and for an extensive treatment of de Sitter and anti-de Sitter space–times see Hawking and Ellis (1973, 124–134); but see also the discussions in Bousso (1998, 2000, 19–21) and Ginsparg and Perry (1983, 245–251). I should add here that de Sitter space is also thought to have infinite volume. See Carroll and Chen (2004, 27), and see the nice illustration of the space in Carroll (2006, 1134).

  17. Figure 1 is based, in part, on Carroll’s own Fig. 87 in Carroll (2010, 363).

  18. Unless otherwise indicated, in this section just about everything I say about the CC-M holds for the MCC-M. Therefore, (again, unless I indicate otherwise) wherever one sees ‘CC-M’, read ‘MCC-M’ as well.

  19. Before I proceed, I should provide a bit of an apologetic for what I’m up to in this section. First, C&C are completely honest and humble about the CC-M’s incompleteness. I do not mean to mercilessly pile on their worries about how to complete the model. My contention below will be that given scientific realism and the fact that substantive portions of the CC-M are inconsistent and admittedly not well understood, one cannot plausibly maintain that the CC-M provides a legitimate explanation of the low-entropy state. That is an important academic and philosophical point. Second, subsequent sections of this paper criticize the model on the assumption that there are ways of providing the details. So even if one does not agree with the aforementioned contention, one will still have to respond to some damaging criticism.

  20. I have in mind the results of Steinhardt and Wesley (2009, 104026-4 ff.). Though cf. Koster and Postma (2011).

  21. Reminiscent of Chisholm’s understanding of events in his (1990, 419 see definition D11). See also Koons (2000, 31–43).

  22. See Bunzl (1979, 145 and 150).

  23. Kim defined “causal closure of the physical domain” as the thesis that “…any physical event that has a cause at time t has a physical cause at t.” Kim (1989b, 43, emphasis in the original). He says that explanatory exclusion is the principle that “[n]o event can be given more than one complete and independent explanation.” Kim (1989a, 79, emphasis in the original).

  24. De Muijnck does try to rescue a counterfactual account of causation from cases of overdetermination.

  25. Mackie’s term, from Mackie (1974, 43), cf. De Muijnck (2003, 65–66), Schaffer (2003, 28). Schaffer tells us that, “[q]uantitative overdetermination occurs whenever the cause is decomposable into distinct and independently sufficient parts” Schaffer (2003, 28).

  26. Paul’s term from Paul (2007).

  27. I should point out that Paul believes that the consequence of there being such prevalent constitutive overdetermination is “mysterious and problematic” Paul (2007, 277). Paul attempts to rid the world of such prevalence by arguing that fundamental causal relata are property instances shared by overlapping entities involved in obtaining causal relations. See Paul (2007, 282).

  28. Schaffer (2003, 42. n. 9) emphasis in the original, though I’ve reversed the order. The objection is tied to Kim (1989a). Some interpret Kim as suggesting that overdetermination just doesn’t involve the causation of an event by an object and the parts that compose it. See Sider (2003, 719. n. 2).

  29. In the discussion above, I assume a simple counterfactual analysis of causation, not unlike the one defended in Lewis (1973a) or the later more sophisticated analysis in Lewis (2004). My point could easily be adjusted so as to accommodate other, (perhaps) more plausible accounts of causation.

  30. Clarkson, Ghezelbash, and Mann stated, “…one implication of the N-bound (and the maximal mass conjecture) is that a quantum gravity theory with an infinite number of degrees of freedom (such as M-theory) cannot describe spacetimes with Λ = 0…” Clarkson et al. (2003, 360).

  31. His argument is explanatory: “It is hard to see what, other than the Λ–N correspondence, would offer a compelling explanation [of] why such disparate elements appear to join seamlessly to imply a simple and general result” (Bousso (2000, 18).

  32. Bousso (2000, 3). Even if the type of entropy in play is information-theoretic or Von Neumann entropy, that fact would be irrelevant. The main argument of this section still runs.

  33. Bousso (2000, 2). In subsequent discussion, I will sometimes speak of N-bound validity for a space–time. What I mean by such a judgment is that Eq. 2 (Proposition 5) holds for those space–times.

  34. Add to these assumptions the further two conditions that the system feature elements that are “spatially bounded”, and that the system has constant energy; Sklar (1993, 36).

  35. Throughout the remainder of the paper, one may read ‘MCC-M’ wherever one sees ‘CC-M’. All subsequent argumentation will be applicable to both.

  36. For some the following nagging objection will remain: Fields in QFT admit infinitely many degrees of freedom, therefore something is wrong with the above argumentation. The reasoning is out of touch with the contemporary state of the art in quantum cosmology. Numerous considerations suggest that QFT breaks down and most cosmologists (it seems) no longer believe that QFT will reside prominently in the endgame quantum theory of gravity. In fact, David J. Gross has said that “[t]he longstanding problem of quantizing gravity is probably impossible within the framework of quantum field theory… QFT has proved useless in incorporating quantum gravity into a consistent theory at the Planck scale. We need to go beyond QFT, to a theory of strings or to something else, to describe quantum gravity” Gross (1997, 10).

  37. The Copernican principle says, roughly, that our causal past and position in space–time is not unique or distinctive. See Stoeger et al. (1995, 1).

  38. Borrowing some wording from Smeenk (2013, 630–631). See also Stoeger et al. (1995, 1). There is a interesting discussion of these matters in Clarkson and Maartens (2010), Maartens (2011). It is important to add that the result from Ehlers, Geren, and Sachs does not extend to times prior to the decoupling era. They remarked, “the result presented cannot be taken to mean that the universe in its earliest stages was necessarily a Friedmann model…” Ehlers, Geren, and Sachs (1968, 1349, emphasis mine). The beginning of the era in question is the moment(s) at or during which radiation and matter decoupled (about 240,000 or so years after the initial singularity).

  39. (Clifton et al. (2012, 051303-1 to 051303-2). For more on the Sunyaev–Zel’dovich effect see Weinberg (2008, 132–135).

  40. It may be that in order to alleviate worries about fine-tuning and the cosmological constant, one should appropriate a scalar field model of dark energy. In addition, it seems that the best way of understanding dark energy via quintessence is to posit a scalar field model of dark energy. As Weinberg remarked, “[t]he natural way to introduce a varying vacuum energy is to assume the existence of one or more scalar fields, on which the vacuum energy depends, and whose cosmic expectation values change with time” (Weinberg 2008, 89). For more on dark energy and scalar field models of such energy, see Sahni (2002, 3439–3441).

  41. Clarkson and Maartens (2010, 2) stated,

    “Isotropy is directly observable in principle, and indeed we have excellent data to show that the CMB is isotropic about us to within one part in ~105 (once the dipole is interpreted as due to our motion relative to the cosmic frame, and removed by a boost).”

    Weinberg (2008, 129) confesses that treating the CMBR as perfectly isotropic and homogeneous is “a good approximation”. Weinberg went on to affirm that “the one thing that enabled Penzias and Wilson to distinguish the background radiation from radiation emitted by earth’s atmosphere was that the microwave background did not seem to vary with direction in the sky.” ([ibid.]).

  42. Clifton, Clarkson, and Bull embrace their idealized assumptions in Clifton et al. (2012, 051303-4).

  43. See Hawking and Ellis (1973, 353-354). For a discussion of the CMBR anisotropies, see Lyth and Liddle (2009, 152–169) and Weinberg (2008, 129–148).

  44. Ehlers et al. (1968) also ignored the cosmological constant.

  45. Maartens and Matravers (1994, 2701). These galactic observations correspond to propositions (O1)–(O4) in Maartens and Matravers (1994, 2694). They are not observations of isotropic background blackbody radiation. See also (Maartens (2011) and Hasse and Perlick (1999).

  46. Maartens’ discussion of the specific result I am interested in is an expansion on earlier work with Chris Clarkson in Clarkson and Maartens (2010).

  47. Maartens (2011, 5131).

  48. Such that Fμ = Fμν  = Fμνα = 0 holds (from Eq. 3.24 of Maartens (2011, 5125).

  49. See Maartens (2011, 5125, 5131).

  50. “FLRW models with ordinary matter have a singularity at a finite time in the past.” (Smeenk 2013, 612). Hawking and Ellis stated, “…there are singularities in any Robertson–Walker space–time in which μ > 0, p ≥ 0 and Λ is not too large.” Hawking and Ellis (1973, 142). See also Wald (1984, 213–214); and the discussion of FLRW models in Penrose (2007, 717–723).

  51. See the review of many of these theorems in Hawking and Ellis (1973, 261–275).

  52. (Hawking and Penrose 1970, 529). This paper also provides an excellent review of both Hawking and Penrose’s previous work on singularity theorems, see especially (ibid., 529–533).

  53. Hawking (1996, 19).

  54. The quoted bit is from (Hawking and Penrose 1970, 531). Of course, they were not concerned with inflationary cosmology in 1970. Here is the broader context of the quote, “…we shall require the slightly stronger energy condition given in (3.4), than that used in I. This means that our theorem cannot be directly applied when a positive cosmological constant λ is present.” (Hawking and Penrose 1970, 531, emphasis in the original). Many authors have noted that inflationary cosmological models violate the strong energy condition of the Hawking–Penrose theorem. See, for example Wall (2013, 20. n. 14); and Borde and Vilenkin (1996, 824. n. 17), inter alios.

  55. The three conditions are stated in Borde and Vilenkin (1996, 819).

  56. See Borde and Vilenkin (1997, 718–719).

  57. Borde et al. (2003).

  58. “The result depends on just one assumption: The Hubble parameter H has a positive value when averaged over the affine parameter of a past-directed null or noncomoving timelike geodesic.” (Borde et al. 2003, 151301-4). See also Mithani and Vilenkin (2012, 1) “…it [the BGV] states simply that past geodesics are incomplete provided that the expansion rate averaged along the geodesic is positive: H av  > 0.” (ibid.); and Vilenkin (2013, 043516-1).

  59. Vilenkin (2013, 043516-1).

  60. Borde et al. (2003, 151301-4), cf. Guth (2004, 49). See also Mithani and Vilenkin (2012, 1–2).

  61. Farhi et al. (1990, 419).

  62. Carroll and Chen (2004, 27. n. 6).

  63. Vilenkin stated, “[e]ven though the BGV theorem is sometimes called a ‘singularity theorem’, it does not imply the existence of spacetime singularities. All it says is that an expanding region of spacetime cannot be extended to the past beyond some boundary \(\mathcal{B}\)” Vilenkin (2013, 043516-1).

  64. See Carroll (2008b, 4, 2010, 50, 349–350, particularly 408. n. 277), but cf. Penrose (1996, 36) for a different view.

  65. Vilenkin (2006, 175) emphasis mine. Ashtekar (2009, 9) acknowledged that the BGV does not rely on Einstein’s field equations.

  66. It seems that C&C go in for a generalized second law. In their discussion of black hole entropy and Hawking radiation, they stated that “[o]ne can prove [69], [70], [71], [72] certain versions of the Generalized Second Law, which guarantees that the radiation itself, free to escape to infinity, does have a larger entropy than the original black hole.” Carroll and Chen (2004, 18).

  67. See Wall (2013). Cf. Jacobson (1994).

  68. Carroll and Chen (2004, 25). There is also the separate question of how likely or natural it is that our metauniverse’s large scale structure is due to some prior inflationary era. Carroll and Tam address this question to some degree in their (2010).

  69. That work also required a quantum theory with an infinitely dimensional Hilbert space.

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    Christopher Gregory Weaver

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Weaver, C.G. On the Carroll–Chen Model. J Gen Philos Sci 48, 97–124 (2017). https://doi.org/10.1007/s10838-016-9337-9

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