Christopher Gregory Weaver 1
On the Carroll-Chen Model
On the Carroll-Chen Model
Christopher Gregory Weaver
Forthcoming in the Journal for General Philosophy of Science published by Springer
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.
1 Introduction
2 The Carroll-Chen Model
2.1 The Background de Sitter Spacetime and Unbounded Entropy
2.2 Nucleating Metauniverses and Unbounded Entropy
3 Philosophical Objections to the Carroll-Chen Model
3.1 Inconsistency, Ambiguity, and Admitted Incompleteness
3.2 Causation and the Carroll-Chen Model
4 Scientific Objections to the Carroll-Chen Model
4.1 Unbounded Entropy?
4.2 Nucleation and Metauniverse Creation
5 Conclusion and Philosophical Significance
1
Introduction
There is currently some controversy over whether or not the past hypothesis cries out for
a scientific explanation, where that hypothesis states that our universe began in an exceedingly
low-entropy state1 (see e.g., Price [2004]; and Callender [2004a], [2004b]). If the relevant state
does not cry out for a scientific explanation, then one would be well within one’s epistemic
rights in characterizing that hypothesis as scientifically brute in that it has no scientific
explanation. In fact, some philosophers maintain that we should resist looking for a scientific
explanation since on an independently reasonable account of laws of nature (i.e., the bestsystems account of laws (BSA)) it becomes plausible to regard the past hypothesis (or something
near enough) as physical law (see Albert [2015] pp. 1-30); Callender [2004a], pp. 207-213;
Loewer [2008], pp. 157-162).2 The BSA says that a contingent truth is a law if, and only if, it is
an axiom or theorem incorporated into the true deductive systems that achieve “a best
1
(Albert [2000], p. 96).
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], p. 157), where by ‘PH’ Loewer means the past hypothesis.
2
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Christopher Gregory Weaver 2
On the Carroll-Chen Model
combination of simplicity and strength.”3 There is clearly no principled barrier to the
incorporation of non-dynamical laws into the best systems. With respect to the past hypothesis,
Craig Callender recommended just such a course of action in his debate with Huw Price:
…according to one view of laws of nature, one attractive to some contemporary
empiricists, the Past Hypothesis would count as a law…The Ramsey-Lewis [BSA]
theory would seem to abruptly end our debate, [about explanation and PH] for it will
declare that the Past Hypothesis doesn’t call for explanation. Why was entropy low in
the past? ‘Because it’s physically impossible for it not to be,’ a real conversation
stopper, answers this question.4
Explanation stops somewhere. Non-dynamical laws of a form corresponding to the past
hypothesis are not easily conceived of as legitimate explananda. Many, standing in a tradition
that stretches back to Immanuel Kant would insist that the range of explanation-types that are
admissible require an appeal to some earlier state of the world. With respect to the past
hypothesis, such an earlier state may very well be unavailable, and so no proper scientific
explanans gets off the ground.5
Still, a veritable gaggle of philosophers and physicists find the initial low-entropy state to
be highly unnatural or improbable6 and on that basis maintain that the past-hypothesis cries out
for an explanation.7 In fact, most cosmologists working on the low-entropy initial condition vie
for a dynamical explanation of that condition. As Andreas Albrecht noted,
…most cosmologists would instinctively take a different perspective. They would
try and look further into the past and ask how could such strange ‘initial’
conditions possibly have been set up by whatever dynamical process went before.
(Albrecht [2004], pp. 374-375)
In the spirit of Albrecht’s instinctive perspective there have emerged a great many
attempts to scientifically explain the relevant state by appeal to inflation (e.g., Davies [1983] and
Guth [2004], p. 37), pre-big bang models (e.g., Steinhardt and Turok [2002a], [2002b], [2005]),
and other developments in cosmology and cosmogony (e.g., holographic cosmology for which
see (Banks [2007], pp. 27-34 and [2015]).8 Here I address just one of these potential scientific
explanations, specifically the explanation proposed by a cosmogonic model recently developed
3
(Lewis [1973b], p. 73).
(Callender [2004b], p. 249). For more on the Mill-Ramsey-Lewis best-systems account (BSA), see
(Cartwright et. al. [2005], pp. 797-799; Earman [1984], pp. 196-199; Lewis [1973b], pp. 72-77, [1983], pp. 365-368,
[1994], pp. 478-482; Loewer [2001], p. 619, [2012]; and Ramsey [1990], p. 150). For criticisms of the BSA see
(Armstrong [1983], pp. 70-71; Belot [2011], pp. 70-72; and van Fraassen [1989], pp. 48-51).
Penrose [1989a], pp. 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], pp. 311-312).
6
See e.g., (Carroll [2008a], p. 48, p. 50; [2006], p. 1132; Carroll and Chen [2004], pp. 3-4; Cf. Carroll
[2010], p. 288. For the implicit claim that the state is improbable see Penrose [2007], pp. 729-731; and Price [2004],
pp. 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], p. 199, though cf. his comments in [2004b], p. 241).
8
Some of these alternative approaches assume different understandings of our universe’s low-entropy state.
4
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Christopher Gregory Weaver 3
On the Carroll-Chen Model
by Sean Carroll and Jennifer Chen (hereafter I will refer to this model with the designation ‘CCM’). If my criticisms are successful, the current project would have important implications for
current debates on laws of nature. Most accounts of laws that are not of the BSA variety,
maintain that all laws are dynamical. To take just one example, Tim Maudlin ([2007], p. 15)
characterizes laws (with appropriate adjunct principles) as sui generis and primitive entities that
describe or perhaps even produce the temporal evolutions of physical systems. For Maudlin,
laws of nature are dynamical. His account cannot accommodate crowning the past hypothesis
with lawhood status. The BSA on the other hand can. A cumulative case for regarding the past
hypothesis as law and therefore also for regarding it as scientifically brute can be built up out of
well-reasoned objections to attempts to dynamically explain that hypothesis. If the best attempts
at dynamically accounting for the low entropy state repeatedly failed, evidence for the BSA
approach would amass, and the case against accounts like Maudlin’s would be all the more easier
to make. And so by criticizing the CC-M, I hope to add to the aforementioned case, and I hope to
motivate regarding the past hypothesis as scientifically brute. The case and motivation help force
our hand in the debate about natural laws in general philosophy of science.
My examination of the CC-M will proceed as follows: Sect. 2 provides an informal
explication of the CC-M. Sect. 3 subjects the CC-M to some philosophical criticism. I argue that
the model’s purported explanation of the arrow of time fails on account of the model’s
inconsistency and incompleteness. I then show that the well-foundedness of causation implies
that the model cannot be verisimilar. Sect. 4 suggests that Carroll and Chen (henceforth C&C)
cannot plausibly maintain that entropy is unbounded from above, and that the model’s
recommended mechanisms for metauniverse (or baby universe) nucleation are implausible.9
2 The Carroll-Chen Model
Our universe began in an extremely smooth, non-empty, homogeneous state. That initial
non-empty smoothness or homogeneity just is the initial low-entropy state of the cosmos.10 Our
best science suggests that our arrow of time points in the direction of entropic increase, since our
best metaphysics of science suggests that time’s arrow reduces to the arrow of entropic
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], p. 76). See also Penrose [2007], pp. 706-707). Most writing on the subject agree with
Penrose here. See the broader discussions in (Albrecht [2004], pp. 371-374; Greene [2004], pp. 171-175; North
[2011], p. 327; Penrose [1979], pp. 611-17, [1989b], pp. 251-260, [2012], pp. 73-79; Price [1996], pp. 79-85,
[2004], pp. 227-228; and Wald [1984], pp. 416-418, [2006], p. 395). See also Callender ([2010], pp. 47-51). Earman
([2006], pp. 417-418, cf. the comments on p. 427) is very skeptical of the contemporary orthodoxy on these matters.
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Christopher Gregory Weaver 4
On the Carroll-Chen Model
increase.11,12 C&C find these facts to be “unnatural” (q.v., n. 6). Their model attempts to advance
a promising strategy for understanding the arrow of time and initial smoothness naturally. The
strategy itself recommends a scientific explanation of the initial smoothness and so also the
arrow of time. This explanation has need of the conjecture that the initial low-entropy state was
produced by way of “dynamical evolution from a generic state” (C&C [2004], p. 6) (C&C
[2004], p. 29; cf. C&C [2005], p. 1671). The following theses are indispensable to the proposed
scientific explanation:
(Thesis 1): Our metauniverse was produced by a background Universe that is an
empty/pure (dS) or asymptotic (AsDS) de Sitter space-time.13
(Thesis 2): The Universe produced our metauniverse by means of a fluctuation.
Such a fluctuation gave birth to a proto-inflationary region. Within our
universe, the mechanisms of eternal inflation are responsible for the largescale structure of our metauniverse.
(Thesis 3): Entropy is unbounded from above.
I will now informally discuss each claim, and in the process shed more light on less central
aspects of the CC-M.
2.1 The Background de Sitter Spacetime and Unbounded Entropy
C&C seek a scientific explanation of our metauniverse’s initial low-entropy state that
does not include finely-tuned boundary conditions or temporally asymmetric microscopic
dynamics (C&C [2004], p. 6, p. 27). In order to acquire such an explanation, C&C need a
background Universe. This background space-time has a “generic initial” Cauchy hypersurface
that is wholly natural. There is also a sense in which the entire background space-time is
admitted to be natural. For C&C, however, “natural means high-entropy” ([ibid., 7]), thus the
background space-time can be understood as a “middle moment” (to borrow Carroll’s wording)
with the highest amount of entropy that an individual interrelated cosmos with a positive vacuum
energy can have. Carroll ([2010], p. 362) wrote:
That middle moment was not finely tuned to some special very-low-entropy
initial condition, as in typical bouncing models. It was as high as we could get, for
a single connected universe in the presence of a positive vacuum energy. That's
the trick: allowing entropy to continue to rise in both directions of time, even
though it started out large to begin with. (Carroll ([2010], p. 362))
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], pp. 16-17; North [2011], p. 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, Kleban, and Susskind [2002, p. 1]). Cf. (Bousso [2012], pp.
2-3; p. 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.
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Christopher Gregory Weaver 5
On the Carroll-Chen Model
In their ([2004]) depiction of the CC-M, the background space-time evolves in two directions
away from some generic initial surface. There is then further evolution on both sides of the
surface into de Sitter phases with a positive cosmological constant. Details about the nature of
the initial surface are left to the imagination, though C&C suggest that such specifics are
irrelevant. One can define an initial condition over that initial surface since it is not a surface that
is “an equilibrium state with maximal entropy.”14 In fact, such a condition over the initial
Cauchy surface will be the surface “of minimum entropy” (C&C [2004], p. 5). Thus, entropy
increases away from the initial surface in two directions. Such dual entropic increase constitutes
the dependency base for two arrows of time. As the two sides of space-time approach their
respective de Sitter phases, each arrow of time will become in some sense ambiguous. This is
because empty de Sitter phases are in thermal equilibrium states. There is, therefore, no entropic
increase once either side of the ultra-large scale structure reaches respective de Sitter phases, and
this further implies that there are no arrows of time during the corresponding phases of the
cosmic evolution of the Universe (ignoring for now the introduction of metauniverse nucleation).
In subsequent work (e.g., Carroll [2006], p. 1134), Carroll seems to modify the CC-M
(this modified version of the account will be individuated via the locution ‘MCC-M’ for modified
CC-M). MCC-M’s background space-time shares some affinities with the space-time described
by Willem de Sitter’s solution to Einstein’s field equations. That solution’s line element is as
follows (using de Sitter’s coordinates):
(4): ds2 = −dr2 – R2 sin2 (r/R) (dϕ2 + sin2 ϕ dθ2) + cos2 (r/R) c2 dt2
(Eq. 1)15
(Eq.1) predicts that matter (what de Sitter called “world-matter”) is completely missing from the
space-time, and so de Sitter’s space-time is empty (de Sitter [1918], p. 229). The background
space-time of the MCC-M is likewise empty.
(Eq. 1) implies that the cosmological constant is positive in value (q.v. note 15). And in
contemporary cosmology and astrophysics, a positive cosmological constant is thought to
correspond to the real presence of dark vacuum energy. Thus, de Sitter’s space-time includes a
positive vacuum energy, and the same turns out to be true of the MCC-M’s background spacetime. The space-time geometry recommended by (Eq. 1) is such that the space-time described is
hyperbolical. More generally, de Sitter space-time is represented as a Minkowskian 5-space with
a Lorentzian 4-sphere inside it that can be described by the following metric ds2 = dt2 – dw2 – dx2
– dy2 – dz2.16 And lastly, because the Universe on the MCC-M is a pure de Sitter spacetime, it is
past-geodesically complete (see (Carroll [2010], pp. 361-362)).
14
(Carroll and Chen [2004], p. 27). I’m borrowing their wording here. The quotation in context is about
something different, viz., the fact that the background space 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], p. 230); but see also the discussion in (de Sitter [1917],
p. 7; and Earman [1995], p. 7).
16
(Penrose [2007], pp. 747-748); Misner, Thorne, and Wheeler [1973], p. 745; and for an extensive
treatment of de Sitter and anti-de Sitter space-times see Hawking and Ellis [1973], pp. 124-134; but see also the
discussions in Bousso [1998], [2000], pp. 19-21; and Ginsparg and Perry [1983], pp. 245-251). I should add here
that de Sitter space-time is also thought to have infinite volume. See (Carroll and Chen [2004], p. 27), and see the
nice illustration of the space-time in (Carroll [2006], p. 1134).
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Christopher Gregory Weaver 6
On the Carroll-Chen Model
2.2 Nucleating Metauniverses and Unbounded Entropy
de Sitter space-time is exceedingly cold, less than 10-28 Kelvin, though its temperature is
still above zero (Carroll ([2010], p. 313; Gibbons and Hawking [1977], p. 2739). The
temperature of de Sitter space-time is positive because it possesses “thermal radiation with a
characteristic wavelength of the order of the Hubble radius.” (quoting Gibbons and Hawking
[1977], p. 2739) The fact that de Sitter space-time has a positive temperature implies that that
space-time countenances fluctuations that result in the existence of “…new inflating patches,
which can eventually evolve into universes like ours…” (C&C [2005], p. 1673). With a positive
vacuum energy, and the positive temperature of the background spacetime, fluctuations can
cause an inflaton field to ascend its potential so as to produce the beginning stages of eternal
inflation, that is to say, the production of a sufficiently ample vacuum energy (C&C [2004, p.
27]; Carroll [2006], p. 1133, [2008b], p. 8). With respect to how this might all precisely work,
Carroll seems to rely heavily upon the tunneling story written down by Edward Farhi, Alan
Guth, and Jemal Guven ([1990]), he remarked:
…de Sitter space, the solution of Einstein's equation in the presence of a positive
cosmological constant, is unstable; there must be some way for it to undergo a
transition into a state with even more entropy. Chen and I imagined that the
mechanism was the quantum creation of baby universes, as suggested by
Farhi, Guth, and Guven [14]… (Carroll [2008b], p. 8 emphasis mine)
And while it is true that our metauniverse began in a very low-entropy state, that state exhibited
more entropy than the relevant “tiny comoving volume of de Sitter” spacetime “from which it
arose…” (C&C [2004], p. 26). This is because the entropy density per that tiny volume of de
Sitter spacetime is considerably low (C&C [2005], p. 1673; Carroll [2006], p. 1133; cf. Aguirre,
Carroll and Johnson [2011]). The fluctuations aren’t random. Rather, they fall out of the
obtaining of a certain condition that is itself produced by the space-time. C&C remarked,
“[b]ecause the entropy density of the background is so low, it is easier to fluctuate into a small
proto-inflationary patch than into a universe that looks like ours today” (C&C [2005], p. 1673
emphasis in the original). Thus, thermal fluctuations, in an empty de Sitter space-time in which
there is low entropy density in the background, yield a proto-inflationary patch out of which our
metauniverse can form via the mechanism of eternal inflation. Figure 1 below represents the
M
CC-M model well:
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Christopher Gregory Weaver 7
On the Carroll-Chen Model
[Figure 1: The CC-M, based, in part, on Carroll’s figure 87 in Carroll [2010], p. 363]
Because advanced stages of the Universe’s evolution are empty de Sitter on both the CCM and MCC-M, metauniverse nucleation conditions arise. The birth of metauniverse’s with
respective eternally inflating phases produces an avenue for unbounded entropic increase
(Carroll [2010], p. 360-362, p. 365). That entropy in the Universe is unbounded from above has
very clear implications. First, if (Thesis 3) is true, then the amount of energy in the background
space-time is infinite. Second, given (Thesis 3), there are infinitely many degrees of freedom.
And third, (Thesis 3) implies that with respect to the Universe, there is no such thing as an
entropic or thermodynamic equilibrium state. If any of these implications are proven false, it will
follow by modus tollens that (Thesis 3) is false as well.
3
Philosophical Objections to the Carroll-Chen Model
Science is not the sole arbiter of truth. In fact, scientists themselves appropriate various
philosophical tools for the purposes of evaluating and assessing scientific theories and models. It
is in the spirit of philosophical evaluation that I argue—in this section—that certain
philosophical considerations weigh heavily against the CC-M in that they show the model cannot
provide an explanation of our metauniverse’s initial low-entropy state, and that the model’s
background Universe cannot be as described.
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Christopher Gregory Weaver 8
On the Carroll-Chen Model
3.1 Inconsistency, Ambiguity, and Admitted Incompleteness
Formulations of the CC-M are inconsistent.17 The CC-M is ambiguously described. And
the scientific explanation of our initial non-empty and smooth state provided by the CC-M is
admittedly incomplete. Given such inconsistency and incompleteness, C&C’s explanation fails.18
On the CC-M, our metauniverse is a closed and “essentially autonomous” system, “free
from outside influences” (Carroll [2010], p. 335 emphasis in the original). One might wonder
how our metauniverse achieved such independence on the CC-M. According to some of
Carroll’s work, such independence was achieved by means of the mechanism of metauniverse
nucleation developed by Edward Farhi, Alan Guth, and Jemal Guven ([1990], I will refer to their
tunneling story with the locution ‘FGG’). On the FGG, when there is successful nucleation,
metauniverses completely separate from their mother Universe. Here is Carroll’s description of
the process:
What we see is simultaneous fluctuation of the inflaton field, creating a bubble of
false vacuum, and of space itself, creating a region that pinches off from the rest
of the universe. The tiny throat that connects the two is a wormhole…But this
wormhole is unstable and will quickly collapse to nothing, leaving us with two
disconnected spacetimes: the original parent universe and the tiny baby. (Carroll
[2010], pp. 357-358 emphasis mine; cf. Carroll [2008a], p. 56)
Importantly though, the background de Sitter space-time (or the regions of that space-time that
are empty de Sitter) have no respective arrows of time. This is because empty de Sitter spacetime is in a state of thermal equilibrium. Ontologically prior to metauniverse nucleation, there is
no entropic increase. Such a fact (noted by Carroll himself [2010], p. 355) makes interpreting
Carroll’s comments regarding the relationships between the arrows of time per metauniverses,
and the direction of time in the background space-time difficult to interpret, for he stated that
“…local direction of time [i.e., the direction of time in our metauniverse] may not be related to
that of the background space-time” (Carroll [2006], p. 1134). But again, with respect to the
background space-time, or at least the appropriate regions thereof, there just is no direction of
time. Something is awry.
There is a second inconsistency in the model (and here I lean on Nikolić [2004], p. 2),
though this second charge applies only to the CC-M (and so not the MCC-M). The initial Cauchy
hypersurface in the background space-time is thought to be generic. But this is not so. At every
Cauchy hypersurface of the background space-time, save the initial Cauchy hypersurface,
entropy increases away from that hypersurface out along a single direction in time. Only at the
initial Cauchy hypersurface does entropy increase in two directions. And so I agree with Nikolić,
17
Unless otherwise indicated, in this section, just about everything I say about the CC-M holds for the
CC-M. Therefore, (again, unless I indicate otherwise) wherever one sees ‘CC-M’, read ‘MCC-M’ as well.
18
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.
M
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Christopher Gregory Weaver 9
On the Carroll-Chen Model
“…the initial hypersurface having two directions of time is not typical at all” (Nikolić [2004], p.
2 emphasis in the original).
Although I will discuss scientific issues relevant to (Thesis 2) below, I want to
immediately point out a perceived ambiguity and incompleteness in Carroll’s discussion of
metauniverse nucleation. First, I have already noted above, that Carroll interprets his work with
Chen in such a way that it is committed to the quantum tunneling mechanism of Farhi, Guth, and
Guven ([1990]). But something is amiss. In their original ([2004]) paper, C&C seem to deny that
their mechanism of nucleation involves any such quantum tunneling process. They stated:
In our discussion is that we [sic] examine the case of an harmonic oscillator
potential without any false vacua; in such a potential we can simply fluctuate up
without any tunneling. The resulting period of inflation can then end via
conventional slow-roll, which is more phenomenologically acceptable than
tunneling from a false vacuum (as in “old inflation” [7]). Thus, the emptying-out
of the universe under typical evolution of a generic state can actually provide
appropriate initial conditions for the onset of inflation, which then leads to regions
that look like our universe. (Carroll and Chen [2004], p. 21 emphasis mine)
But C&C ([2004], pp. 22-23; pp. 25-26; cf. n. 4 on p. 26) concede that the fluctuation route to
metauniverse nucleation and large-scale structure formation is incredibly improbable.
I described the incompleteness of the model as “admitted incompleteness” because
Carroll has said that the nucleation process is part of an “extremely speculative” proposal and
that it specifically “lies beyond the realm of established physics.” (Carroll [2006], p. 1133).
In other work (see particularly, Carroll [2006], p. 1133 and the sources cited in notes 3741 on p. 1135; cf., Carroll [2012], p. 191; Carroll [2010], pp. 284-286), Carroll indicated that the
multiverse maybe a prediction of string theory and inflation. We should not be optimistic about
string theory in this context since “...there is presently no fully satisfactory embedding of de
Sitter space into string theory” (Bousso, DeWolfe, and Myers [2003], pp. 297-298). In fact, there
are no-go theorems that seek to establish that certain supergravity theories (IIA SUGRA string
theories) are incompatible with de Sitter space-time (see Maldacena and Nuñez [2001], pp. 845847). Moreover, “[a]ll explicit and fully trustworthy solutions that have ever been constructed in
string theory have a non-positive cosmological constant” (quoting Van Riet [2012], p. 1; cf.
Stominger [2001], p. 2). Carroll (with Johnson, and Randall) seems to agree, “…string
theory…seems to favor Minkowski or anti-de Sitter vacua” (Carroll, Johnson, and Randall
[2009], p. 2).
There are further problems with injecting string theory into the model, for that theory
requires a great many dimensions which must somehow be compactified into any pure or
asymptotically de Sitter space-time if one or the other is your space-time of choice. The problem
is that there are no-go theorems proving that compactified theories that abide by the null energy
condition (along with several other plausible conditions for string theoretic models) cannot be
wed to inflationary theory.19 It has also been shown that compactified theories that violate the
null energy condition, but which otherwise satisfy other very plausible conditions (for string
theoretic models) cannot be united with inflationary theory or theories. (Steinhardt and Wesley
[2009], pp. 104026-6 to 104026-8). So I’m not sure what to make of Carroll’s claim that a
19
I have in mind the results of Steinhardt and Wesley ([2009], pp. 104026-4 to 104026-6). Though cf.
Koster and Postma ([2011]).
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Christopher Gregory Weaver 10
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multiverse is a prediction of inflation coupled with string theory. The two are not necessarily
agreeable partners.
The foregoing reasoning indicates that FGG nucleation out of a de Sitter space-time is
merely speculative and that Carroll’s discussion of it should be thought of as exploratory. I
believe it is therefore safe to conclude that a central piece of the model is missing, and so the
CC-M is incomplete in that it does not have a clear recommended dynamical path from the
background Universe to the birth of metauniverses like ours.
The incompleteness of the CC-M has a bearing on the question of whether or not the
model provides a legitimate scientific explanation of our initial low-entropy state. Assuming
some robust version of scientific realism, explanations, when they successfully explain, are at
least approximately true. It is not clear how an explanans can be verisimilar, if it is unclear
which proposition, if any, is expressed by that explanans on account of the kind of
incompleteness the CC-M displays. Thus, I find this gap in the model to be severely delimiting.
We cannot, in my opinion, justifiably claim that the CC-M proffers an actual scientific
explanation of the initial non-empty smoothness of our metauniverse, since it is altogether
unclear what the explanation is on the CC-M.
3.2 Causation and the Carroll-Chen Model
Considerations in this section so far establish that the CC-M does not provide an actual
scientific explanation of our metauniverse’s arrow of time since the model is inconsistent and
woefully incomplete. I will now provide a demonstration of the falsity of (Thesis 1) on the basis
of philosophical considerations having to do with the fundamental nature of causation.
3.2.1
Preliminaries
For the purposes of the main argument in this section, I will assume that purely
contingent facts are the relata of obtaining causal relations. Such facts are particular kinds of
concrete states of affairs involving contingent substances or substrates exemplifying properties
or standing in relations. The nexuses of exemplification within purely contingent facts tie
together contingent substrates (and only contingent substrates) with respective properties.20 The
relations within purely contingent facts relate contingent substrates and only contingent
substrates to one another. Moreover, the improper parts of all purely contingent facts must
themselves be contingent. An improper part of an object o just is o. Thus; no purely contingent
fact exists in all worlds.
I allow for complex purely contingent facts, as would many other theoreticians since
most who work on mereology go in for an unrestricted view of composition (see (Rea [1998] and
van Cleve [2008] for arguments). Call such complexes higher-order purely contingent facts.
Higher-order purely contingent facts are themselves purely contingent in the sense that only
contingent substrates fuse together and help comprise the respective high-order sums. This holds
only if the following principle is true:
20
Reminiscent of Chisholm’s understanding of events in his ([1990], p. 419 see definition D11). See also
(Koons [2000], pp. 31-43).
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[Interpretation: ∇x: x is purely contingent (where what it means to be purely
contingent is to only have substrates that are contingent as proper parts or
constituents, though the entity in question may have properties or relations as
constituents); Cxy: x is substrate-composed solely of y (where what it means for x
to be substrate-composed solely of y, is for x to have y as its only proper part that
is a substrate, though x may also have as proper parts or constituents properties or
relations); Domain: unrestricted]
(Purely Contingent Parts Principle, PCPP): ■∀x(Ǝys((Cxys & ∇ys) ⊃ ∇x))
That is to say, necessarily, for any entity x, if there are some ys, such that x is substratecomposed solely of the ys, and the ys are purely contingent, then x is purely contingent. The
above principle makes use of “plural referring expressions” or plural quantification, but as Peter
van Inwagen has said, that idea “has sufficient currency” contemporarily (van Inwagen [1990], p.
23). PCPP seems to me to be highly intuitive. I do not have an argument for it, and so I should
perhaps adopt a generally good dialectical strategy at this point and place a defeasibility operator
‘Ð’ in front of the prejacent of PCPP:
(PCPP*): ■Ð[∀x(Ǝys((Cxys & ∇ys) ⊃ ∇x))]
That is to say, necessarily, normally, for any entity x, if there are some ys, such that x is
substrate-composed solely of the ys, and the ys are purely contingent, then x is purely contingent.
Christopher Gregory Weaver has argued in more than one place, that all purely
contingent facts have causes (Weaver [2012], [2013]). I will not revisit his reasoning for such a
conclusion here. I will simply assume the universality of causation, and proceed in
demonstrating that the causal relation is well-founded with respect to purely contingent facts.
3.2.2
The Well-Foundedness of Causation
In his very important work, Realism Regained: An Exact Theory of Causation, Teleology,
and the Mind ([2000]), Robert Koons proffered a very interesting argument for the wellfoundedness of the causal relation (see p. 113) which depended upon the universality of
causation. A close cousin of that argument proceeds as follows.
(Premise 1): All purely contingent facts have causes and there is a purely
contingent fact m that is the sum of all purely contingent facts.
(Premise 2): If all purely contingent facts have causes and there is a purely
contingent fact m that is the sum of all purely contingent facts, then m has
a cause, call it c (i.e., c causes m).
(Premise 3): If for any obtaining causal relation which composes m, the cause in
such a relation is preempted by c, then it is not the case that there is a
purely contingent fact m that is the sum of all purely contingent facts.
(Premise 4): If c causes m, then either, for any obtaining causal relation which
composes m, the cause in such a relation will be preempted by c, or c
causes m by indirectly (through the transitivity of causation) causing all of
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its constituent purely contingent facts by being the initial cause of m’s
earliest obtaining purely contingent fact or facts.
(Premise 5): If m is an infinitely long causal chain whose links involve only
purely contingent facts, then it is not the case that c causes m by indirectly
(through the transitivity of causation) causing all of its constituent purely
contingent facts by being the initial cause of m’s earliest obtaining purely
contingent fact or facts.
(Conclusion): Therefore, it is not the case that m is an infinitely long causal chain
whose links involve only purely contingent facts.
Premise (1) follows from mereological universalism, the universality of causation, and
from (PCPP*). Premise (2) is an obvious truth. With respect to justification for premise (3), one
should first pick out any obtaining causal relation which helps compose m. Let’s say the relation
is one involving C* and E*, where C* is the would-be cause of E* (the relation is an arbitrary
constituent of m). C*, if preempted by c, will be barred from causally producing E*. Since this
would hold for any obtaining causal relation, there simply could not be an m given that a cause c
brings about m, since every would-be cause featured in causal relations building up m would fail
to actually bring about the relevant effects. Premise (5) is certainly true. An infinitely long causal
chain has no initial cause and no earliest obtaining purely contingent fact or facts. That leaves
premise (4). Why would one affirm it? Why could we not uphold the claim that c causes m by
causing—via overdetermination—all of m’s constituent purely contingent facts? Koons ([2000],
p. 113) failed to (at least) explicitly defend the supposition that we are dealing here with a case of
preemption instead of a case of symmetric overdetermination, or even joint causation.
There are several ways to supplement Koons’ argumentation. We might follow Martin
Bunzl who argued that symmetric overdetermination is impossible (see Bunzl [1979]), though he
allowed for explanatory overdetermination see p. 145), but Bunzl’s reasoning required a specific
analysis of events, particularly the analysis of Donald Davidson ([1967]).21 Davidson’s analysis
includes identity or individuation conditions for events (see Davidson [1969], [2001], p. 179),
where some event e1 is identical to event e2, just in case, for any event x, x directly causes e1 just
in case, x directly causes e2, and for any event y, e1 directly causes y just in case, e2 directly
causes y (see [ibid.]; and the discussion in Brand [1977], p. 331; Simons [2003], p. 374). This
view of the identity conditions of events is flawed, for as Myles Brand ([1977], p. 332) pointed
out, it equates all events which do not have direct causes, and which do not directly cause other
events. It is an undesirable consequence that all ineffectual events are identical. So we should
abandon Davidson’s analysis of events because of its view of the identity
conditions of causal relata.
There is a different path for defending a related but weaker thesis: With respect to the
actual world, there are no cases of overdetermination. One might have good reasons for believing
in explanatory exclusion, and causal closure.22 Some believe that if those two dogmas hold, then
there are no actual cases of overdetermination. Wim De Muijnck suggests that the intuition could
be stronger, “[m]etaphysically speaking, no such thing as overdetermination seems possible; this
21
See (Bunzl [1979], p. 145, and p. 150).
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], p. 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], p. 79 emphasis in the original).
22
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is a consequence…of causal closure and explanatory exclusion” (De Muijnck [2003], p. 65).23 I,
however, find causal closure to be objectionable, and so it is best to look for proper
substantiation of premise (4) elsewhere.
Some philosophers have suggested that symmetric overdetermination is improbable. One
response to this charge that draws from (Schaffer [2003], pp. 27-29; and Paul [2007]) is that
“quantitative”24, and/or “constitutive overdetermination”25 is prevalent. If one is a nonreductionist about the structure of material objects, then a great many cases of macroscopic
causation will involve Paul’s constitutive overdetermination, in that the parts which compose
such wholes (any macro-level material entity) will contribute to causally producing that which
the macro-level entity (the whole) brings about (cf. Paul26 [2007], pp. 276-277). Schaffer
recommends that one fend off the objection that parts of the causally efficacious object are not
metaphysically distinct enough to breed overdetermination, by noting that the parts are
“nomologically correlated” while still “metaphysically distinct”.27 I agree with Schaffer’s
response.
Let me recommend a different strategy. Give attention to a standard explication of
symmetric overdetermination as articulated by L.A. Paul ([2007], pp. 269-270 emphasis mine):
In contemporary discussions of causation, standard cases of symmetric causal
overdetermination are defined (roughly) as cases involving multiple distinct
causes of an effect where the causation is neither joint, additive, nor preemptive
(and it is assumed the overdetermining causes do not cause each other)…Each
cause makes exactly the same causal contribution as the other causes to the effect
(so the causal overdetermination is symmetric); each cause without the others is
sufficient for the effect; and for each cause the causal process from cause to effect
is not interrupted. (Paul [2007], pp. 269-270)
Now pick out an arbitrary obtaining causal relation that composes m. Label the cause D, and the
effect E. If we understand symmetric overdetermination in the way Paul recommends, we cannot
say that c is a direct overdetermining cause with D of E, precisely because c is a cause of D in
that it is the cause of m. Perhaps a charitable reading of (Koons [2000], p. 113) would suggest
that by his admission that (in the case as I have adjusted it) c is “causally prior” to D in that c is
also the cause of D, c and D cannot be overdetermining causes of E. On the charitable reading,
my attempted improvement on (Koons [2000], p. 113) per this paragraph only involves further
23
De Muijnck does try to rescue a counterfactual account of causation from cases of overdetermination.
Mackie’s term, from Mackie ([1974], p. 43); cf. De Muijnck ([2003], pp. 65-66); Schaffer ([2003], p. 28).
Schaffer tells us that, “[q]uantitative overdetermination occurs whenever the cause is decomposable into distinct and
independently sufficient parts”. Schaffer ([2003], p. 28).
25
Paul’s term from (Paul [2007]).
26
I should point out that Paul believes that the consequence of there being such prevalent constitutive
overdetermination is “mysterious and problematic”. (Paul [2007], p. 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], p. 282).
27
(Schaffer [2003], p. 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], p. 719. n. 2).
24
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elaboration upon just how causal priority precludes a symmetric overdetermination reading of
the relevant case.
One might think that the charitable reading does not help preclude a joint causation
understanding of the case of concern. But this is not right. We were supposing in the Koons-style
reasoning above that D is a sufficient cause of E. So D is enough to bring about E, and since we
are not dealing here with symmetric overdetermination, the relationship between c, D,
and E must be understood in terms of asymmetric overdetermination, i.e., preemption.
The above Koons-style argument will generalize to any infinitely long causal chain
composed of obtaining causal relations that feature only purely contingent facts.
3.2.3
Well-Foundedness and the CC-M
The background space-time of the CC-M contains two actual infinities on each side of the
“initial” Cauchy hypersurface. Each infinity spawns a sea of infinite metauniverses. There
therefore seems to be a sense in which the CC-M implies the existence of an infinitely long
causal chain given the falsity of causal eliminativism.
The actual infinities in the background space-time involve an ongoing infinite process of
cosmic evolution via the evolution of the de Sitter space-time. We can choose a random
arbitrarily large surface of the space-time, sum it up and judge that such a surface will serve as a
causal dependency base for some subsequent exceedingly large surface of the same space-time.
That surface will also serve as a causal dependency base for some succeeding surface and so on
ad infinitum. Thus, the CC-M implies that there is an infinitely long causal dependency chain.28
Such dependency will suffice as insurance for respective obtaining causal relations, and so the
CC-M likewise implies that there is an infinitely long causal chain (again, given the falsity of
causal eliminativism).
If the above successfully connects the CC-M to the implication that there is an infinitely
long causal chain, then the Koons-style argument for well-foundedness will run, and (Thesis 1)
will come out false.
But doesn’t the well-foundedness result generalize to other infinite spacetimes? Perhaps.
My objection would seem to constitute a metaphysical constraint on solutions to the low entropy
problem given that causation cannot be eliminated and surfaces (or events on such surfaces) of
the relevant spacetimes can be understood as dependency bases for one another. As I stated when
I began this section, science is not the sole arbiter of truth. It must give ear to other knowledge
conducive pursuits. The case of causation and well-foundedness suggests that the explication of
an approximately true physical model must succumb to restrictions imposed by metaphysical
law, the law in this case being that causation is by its very metaphysical nature well-founded.
4
Scientific Objections to the Carroll-Chen Model
The CC-M does not pass philosophical muster. I will now argue that even given the
failure of preceding philosophical argumentation, the CC-M suffers from insurmountable
scientific problems and so cannot actually explain our metauniverse’s initial low-entropy state.
28
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.
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4.1 Unbounded Entropy?
I will now take up (Thesis 3). I maintain that the N-bound confirms the Tom Banks/Willy
Fischler Λ-N correspondence thesis, at least when the background space-time of the MCC-M is in
view, and that such confirmation means that (Thesis 3) is false. I also argue that while it is
unclear if the N-bound holds for the background space-time of the CC-M, there are arguments to
which one can turn for the purposes of establishing Λ-N correspondence for that space-time, and
so (Thesis 3) is false given the CC-M as well.
4.1.1
Λ-N Correspondence
Tom Banks ([2000], p. 5) has argued that the value of Λ, the cosmological constant, is
inversely proportional to the value of N. N is the logarithm of the dimension of the Hilbert space
belonging to quantum theory describing space-time. By consequence, if one’s quantum theory
conceives of N as finite, then that quantum theory (its Hilbert space) will contain finitely many
dimensions (Bousso [2000], p. 2. n. 2). The correspondence of Λ to N entails that there is a large
(though finite) number of degrees of freedom. If, however, N really is finite, then quantum
theories of gravity featuring infinitely many degrees of freedom will be implausible.29
Raphael Bousso has noted that proofs of what he calls the “N-bound” constitute evidence
for Λ-N correspondence.30 The N-bound states that every space-time with Λ > 0 is a space-time
whose total observable entropy is bounded by:
(5): N = 3π/Λ
(Eq. 2)31
Or, any space-time with a positive cosmological constant is one which cannot feature an
observable entropy whose value is greater than N = 3π/Λ.32 The N-bound trivially holds for
empty de Sitter space-times like the background space of the MCC-M. In addition, Bousso at one
time believed that one could show that the N-bound is valid for asymptotically de Sitter spacetimes—such as our metauniverse—on the basis of the generalized second law. He remarked:
It is not difficult to see that the N-bound is true for vacuum solutions like de Sitter
space (a trivial case). Moreover, one can argue that it is satisfied for all spacetimes which are asymptotically de Sitter at late times, by the generalized second
law of thermodynamics. (Bousso [2000], p. 2)
29
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, Ghezelbash, and Mann [2003], p. 360)
30
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], p. 18).
31
(Bousso [2000], p. 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.
32
(Bousso [2000], p. 2). In subsequent discussion, I will sometimes speak of N-bound validity for a spacetime. What I mean by such a judgment is that Eq.2 (proposition 5) holds for those space-times.
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But my use of Bousso’s work will not require such a generalization. That I can ignore the
stronger argumentation for the more general point is advantageous, for Bousso himself (with
collaborators) provided counter-examples to the N-bound (see Bousso, DeWolfe, and Myers
[2003]). These counter-examples involved space-times with dimensionality greater than four.
Moreover, Clarkson, Ghezelbash, and Mann ([2003]) attempted to show that the N-bound is
invalid for a four-dimensional Taub-Bolt space-time that is locally asymptotically de Sitter, and
which features NUT charge and (unfortunately) closed timelike curves ([ibid.], pp. 360-361).
With respect to N-bound validity, the only point that my argumentation requires is that the Nbound is valid for dS or “empty” de Sitter space-time, and both Bousso ([2000], cf. the remark in
[2012], p. 123509-15.) and Smolin ([2002], pp. 45-46) have acknowledged its validity in that
context.
How does all of this relate to the MCC-M? Recall (Thesis 3) above, and remember that if
(Thesis 3) holds, then there are infinitely many degrees of freedom (Carroll and Chen [2004], p.
7; cf. pp. 14-15, and p. 30). The N-bound, which is trivially valid for dS space-time (the very
background space-time of MCC-M) is strong confirming evidence for the Banks/Fischler Λ-N
correspondence thesis (as Bousso suggested). But N comports to the logarithm of the dimension
of the Hilbert space in quantum theory. If the correspondence thesis is right, then N is probably
finite. Therefore, there should be finitely many dimensions of the Hilbert space in the correct
quantum theory, and so there are also only finitely many degrees of freedom. This conclusion
ensures that (Thesis 3) is false. Entropy is not unbounded from above. The argument in play can
be summarized as follows:
(Premise 1): If the Λ/N correspondence thesis holds for dS space-time, then the
correct quantum theory describing dS space-time features a finitely
dimensional Hilbert space.
(Premise 2): If the N-bound is valid for dS space-time, and the best explanation of Nbound validity for dS space-time is the Λ/N correspondence thesis, then
the Λ/N correspondence thesis holds for dS space-time.
(Premise 3): The N-bound is valid for dS space-time, and the best explanation of Nbound validity for dS space-time is the Λ/N correspondence thesis.
(Premise 4): If the correct quantum theory for dS space-time features a finitely
dimensional Hilbert space, then dS space-time features only finitely
many degrees of freedom.
(Premise 5): If dS space-time features only finitely many degrees of freedom, then it
is not the case that the global entropy of dS space-time is unbounded
from above.
(Conclusion): Therefore, it is not the case that the global entropy of dS space-time is
unbounded from above.
The first premise is true by virtue of the meaning of the correspondence thesis. The
second premise holds on account of the cogency of inference to the best explanation reasoning.
In the absence of defeaters and underdetermination, such reasoning provides cognizers with
epistemic justification for their belief that the purported best explanation holds. The first
conjunct of premise three follows from points already made above. The second conjunct follows
from the fact that there is simply no competing explanation of the relating of the two seemingly
incommensurable parameters, viz. Λ and N (Bousso [2000], p. 18). It seems that the
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correspondence thesis wins by default. Premises four and five seem straightforward enough, and
our conclusion follows from elementary moves in logic.
In an attempt to defend the MCC-M, one might respond by emphasizing that the means by
which the Universe increases its entropy is by giving birth to metauniverses (Carroll [2010], pp.
359-360). Appeals to the N-bound do nothing to subvert that possibility. This response is flawed.
According to C&C, if it is not the case that there are infinitely many degrees of freedom, then
their story regarding metauniverse nucleation and unbounded entropy cannot run. Entropy is
unbounded from above only if there are infinitely many degrees of freedom. The above
argumentation cuts down this necessary condition, and so results in a bound on entropy.
Again the argument from the N-bound shows that with respect to the background de
Sitter space-time of the MCC-M, there are finitely many degrees of freedom. Carroll ([2008b],
pp. 6-7) believes that the MCC-M would in that case have a fundamental problem with Poincaré
recurrence. Recall that on the basis of Newtonian mechanical laws of motion, and with respect to
an energetically isolated system whose volume is finite, Poincaré proved an important theorem.
The result is this: given the aforementioned assumptions33, a relevant system which starts off in
state s at t, will, given enough time, evolve back into a state arbitrarily close to s, and it will do
this infinitely many times (paraphrasing Sklar [1993], p. 36). There are quantum analogs of this
theorem, and Carroll ([2008b], pp. 6-7) believes he can escape these analogs by appeal to an
infinitely dimensional Hilbert space. But you will remember that the argument from the N-bound
cuts down the dimensions of Hilbert space to only a finite amount due to the Banks/Fischler Λ-N
correspondence thesis. Thus, by Carroll’s own lights, the problem of Poincaré recurrence
remains.
4.1.2
The N-Bound and the CC-M
Does the argument from the N-bound apply equally well to the background space-time of
the CC-M? I am not sure. C&C’s description of that space is fragmented. We do not know the
dimensionality of the space-time, nor what precise generic conditions the space-time evolves
away from. In addition, we do not know what precise kinds of matter occupy the space-time in
its non-de Sitter regions. Ignorance of these matters makes it difficult to determine N-bound
validity. For although Bousso ([2000]) originally argued that the N-bound is valid for all spacetimes with a positive cosmological constant (as I have already pointed out) he would later (with
collaborators) reverse his opinion on the matter by proffering counter-examples to his original
proof (Bousso, DeWolfe, and Myers [2003], p. 299). But let us suppose that the N-bound is not
valid for the background space-time of the CC-M. Tom Banks ([2000], pp. 5-6) provided three
convincing arguments that all motivate the Λ-N correspondence thesis in the context of AsDS
space-times. From the little we can discern about the nature of the background space-time of the
CC-M, we can somewhat safely infer that that space-time is AsDS. Hence, the Hilbert space of
the appropriate quantum theory describing that space-time is finitely dimensional. (Thesis 3) is
therefore false when either the CC-M or MCC-M is in view.
33
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], p. 36).
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I will now continue to assume that the CC-M/MCC-M34 is complete, and move on and
reflect, in the next sub-section, on (Thesis 2), evaluating the proposed mechanisms for
metauniverse nucleation in the work of C&C.35
4.2 Nucleation and Metauniverse Creation
As I have already pointed out, Carroll seems to commit himself to the quantum tunneling
process of metauniverse nucleation as articulated by Farhi, Guth, and Guven ([1990]). That
process will not serve as a proper mechanism for the nucleation of our metauniverse, if our
metauniverse has an initial singularity. On this point Farhi, Guth, and Guven ([1990], p. 419)
stated, “…any plausible scheme to create a universe in the laboratory must avoid an initial
singularity.” As a result, Farhi, Guth, and Guven try to articulate a theory of quantum tunneling
which avoids the Penrose singularity theorem of ([1965]). I will argue that while the FGG
mechanism may escape the Penrose theorem, it does not escape other theorems which entail that
our metauniverse has an initial singularity, and that our metauniverse is past-geodesically
incomplete.
4.2.1
The EGS Theorem and Related Results
According to the EGS theorem (proven in (Ehlers, Geren, and Sachs [1968])), given the
Copernican principle36, and that observers situated in some expanding model discern (via
observations) that free and unrestrained “propagating background radiation is” isotropic, the
space-time in which such observers are situated must be FLRW.37 Clifton, Clarkson, and Bull
([2012]) (CCB) showed that space-time geometry is, for an observer, FLRW “using the CMB
alone” without the Copernican principle ([ibid.], p. 051303-4). Given certain conditions, their
work may also indicate “our entire causal past must…be FLRW” ([ibid.], p. 051303-3). One
acquires their results by assuming that an observer beholds isotropic CMBR while the Sunyaev34
Throughout the remainder of the paper, one may read ‘MCC-M’ wherever one sees ‘CC-M’. All
subsequent argumentation will be applicable to both.
35
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], p. 10).
36
The Copernican principle says, roughly, that our causal past and position in space-time is not unique or
distinctive. (Stoeger, Maartens, and Ellis [1995], p. 1).
37
Borrowing some wording from (Smeenk [2013], pp. 630-631). See also (Stoeger, Maartens, and Ellis
[1995], p. 1). There is a nice 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], p. 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).
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Zel’dovich effect ((SZ) which involves baryonic matter scattering the photons of the CMBR
(Clarkson [2012], p. 700)) is present in that beholding.38 The idea is that if a single onlooker
perceives blackbody CMBR that is isotropic, and that CMBR is accompanied by particular SZrelated scattering events, then that observer can infer that her universe is FLRW, given that the
necessary assumptions of the EGS theorem (save the Copernican principle) hold, and that either
(a) the observations are over a prolonged period of time, or (b) the SZ-related effects involve
double scattering (paraphrasing (Clarkson [2012], p. 700)). I should add that the CCB results
hold even given the presence of dark energy, it is just that such dark energy must be susceptible
to a scalar field description.39
Both the EGS and CCB results are significant since our observations regarding the
cosmic microwave background radiation suggest that that radiation is nearly isotropic.40 The
qualifier ‘nearly’ is important since it seems that both EGS and CCB reasoning require highly
idealized propagating radiation in so far as that radiation must be exactly isotropic.41 Our
metauniverse’s CMBR exhibits certain anisotropies42, and so it is unclear what work these
theorems can do for me.43
There are related results that do not rely on a condition of perfectly isotropic CMBR. For
example, Stoeger, Maartens, and Ellis ([1995], p. 1) argued that our cosmos is approximately or
nearly FLRW given the Copernican principle, the fact that background blackbody radiation is
freely propagating everywhere and that such radiation is perceived, by observers, to be
approximately or nearly isotropic (plus a few additional technical assumptions). Maartens and
Matravers ([1994]) have articulated a matter analog of the EGS theorem. Their result establishes
that our universe is FLRW given the Copernican principle, and that a class of galactic
observations along a postulated observer’s world line is isotropic.44
38
(Clifton, Clarkson, and Bull [2012], pp. 051303-1 to 051303-2). For more on the Sunyaev-Zel’dovich
effect see (Weinberg [2008], pp. 132-135).
39
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], p. 89) For more on
dark energy and scalar field models of such energy, see (Sahni [2002], pp. 3439-3441).
40
Clarkson and Maartens ([2010], p. 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], p. 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.])
41
Clifton, Clarkson, and Bull embrace their idealized assumptions in (Clifton, Clarkson, and Bull [2012], p.
051303-4]).
42
See (Hawking and Ellis [1973], pp. 353-354). For a discussion of the CMBR anisotropies, see (Lyth and
Liddle [2009], pp. 152-169; and Weinberg [2008], pp. 129-148).
43
(Ehlers, Geren, and Sachs [1968]) also ignored the cosmological constant.
44
(Maartens and Matravers [1994]: p. 2701). These galactic observations correspond to propositions (O1)(O4) in (Maartens and Matravers [1994], p. 2694). They are not observations of isotropic background blackbody
radiation. See also (Maartens [2011]; and cf. Hasse and Perlick [1999]).
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The most formidable EGS-like result was recently discussed by Roy Maartens ([2011],
pp. 5121-5125) in his excellent review of much of the associated literature.45 The theorem has it
that with respect to a region of a space-time featuring dark energy (whether understood in terms
of a perfect fluid, quintessence, or cosmological constant) and dust matter, if (a) every postulated
observer in the region beholds partially isotropic CMBR, (b) the observed CMBR “rest frame is
geodesic”46 with an expanding four-velocity, and (c) the self-same radiation is collisionless with
a vanishing octupole, quadrupole and dipole47, then the region in question is FLRW.48 The
assumptions of this theorem are quite weak. The theorem constitutes indirect evidence for a
general FLRW cosmos given that the conditions are satisfied in the appropriate region, and that
the postulated observers do not occupy a privileged region of spacetime. I therefore agree with
Maartens “[t]his is the most powerful observational basis that we have for background
homogeneity and thus an FLRW background model” (Maartens [2011], p. 5125).
What is the relevance of all of this to the CC-M? It turns out that every FLRW model
(with matter like ours) features an initial singularity.49 And since the assumptions of several of
the EGS-like results are quite weak, and since it is plausible to regard some of those less
idealized results together with the satisfaction of their antecedents as evidence that our entire
spacetime is FLRW, we are justified in maintaining that our metauniverse is best described by an
FLRW model. Thus, the FGG mechanism for metauniverse nucleation cannot be the mechanism
responsible for our universe’s nucleation out of a background de Sitter spacetime. Some other
theory of nucleation that is not impeded by the singular nature of our metauniverse is required.
4.2.2
The BGV Theorem
On the standard hot big bang model, implications of proper solutions to Einstein’s field
equations suggest that our metauniverse is past geodesically incomplete in that our metauniverse
features an initial singularity. Attempts to avoid this result were blocked by work on singularity
theorems in the 1960s and 1970s (see some of the related literature in ((Geroch [1966]),
(Hawking [1965], [1966a], [1966b], [1967]), (Penrose [1965]), and (Hawking and Penrose
[1970]).50 In ([1970]) Hawking and Penrose attempted to generalize on this work by advancing
“[a] new theorem on space-time singularities”.51 Hawking would later describe this newer
theorem as one that predicts that there is a singularity in the future, and that there is a singularity
in the past “at the beginning of the present expansion of the universe.”52 The theorem had need
of four seemingly modest conditions, one of which demanded that space-time be described by
Einstein’s field equations along with a cosmological constant that is negative or equal to zero in
45
Maartens’ discussion of the specific result I am interested in is an expansion on earlier work with Chris
Clarkson in (Clarkson and Maartens [2010]).
46
(Maartens [2011], p. 5131]).
47
Such that Fµ = Fµν = Fµνα = 0 holds (from equation 3.24 of Maartens [2011], p. 5125).
48
See (Maartens [2011], p. 5125, p. 5131).
49
“FLRW models with ordinary matter have a singularity at a finite time in the past.” (Smeenk [2013], p.
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], p. 142). See also (Wald [1984], pp. 213-214); and the discussion
of FLRW models in (Penrose [2007], pp. 717-723).
50
See the review of many of these theorems in (Hawking and Ellis [1973], pp. 261-275).
51
(Hawking and Penrose [1970], p. 529). This paper also provides an excellent review of both Hawking
and Penrose’s previous work on singularity theorems (see especially [ibid.], pp. 529-533).
52
(Hawking [1996], p. 19).
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value. It turned out that this modest condition was not modest enough. When inflationary stages
of cosmic evolution are added to the standard model, a positive cosmological constant is
required, thus, the Hawking-Penrose theorem “cannot be directly applied” to such models.53
Later theorems were proven. One of these was a result of the work of Arvind Borde and
Alexander Vilenkin ([1996], pp. 819-822). They showed that a space-time is past-null
geodesically incomplete if that space-time satisfies what were perceived to be even more modest
conditions than those used to deliver erstwhile singularity theorems.54 One such condition (viz.,
the null convergence condition which is implied by the weak energy condition) was shown to be
problematic in light of diffusion regions, and so that condition was not mild enough.55
Borde and Vilenkin would later return, this time with Alan Guth, to prove a newer
theorem.56 The Borde-Guth-Vilenkin (BGV) theorem entails that all space-times whose Hubble
parameters are on average greater than zero, are past-geodesically incomplete.57 Notice that the
theorem does not necessarily suggest that the relevant space-times feature an initial singularity.
This is because the theorem is not actually a singularity theorem. The theorem only implies that
every past-null or past-timelike geodesic is such that it cannot extend further than a pastboundary ℬ.58
The BGV has broad application potential since it only relies on a single, model
independent assumption. For example, Borde, Guth, and Vilenkin apply the theorem to the early
cyclic cosmogonic model of Steinhardt and Turok ([2002a]).59 They also apply the theorem to
the higher-dimensional model of Martin Bucher ([2002]).
Our space-time or metauniverse is such that it can be accurately described with a Hubble
constant whose value is on average greater than zero. Hence, the BGV theorem can be easily
applied to our metauniverse. This point is underscored by the fact that the BGV was originally
developed for the purposes of demonstrating that inflationary models are past-incomplete.
Carroll and Chen are fans of inflation (a fortiori eternal inflation). They at least believe that in
the past our metauniverse exhibited an extraordinary inflationary stage of cosmic evolution. And
so the theorem should be easily applicable to our metauniverse as understood by the CC-M.
Is the presence of a past-boundary indicative of an initial singularity? For my present
intents and purposes, it is. Farhi, Guth, and Guven define “[a]n initial singularity” as “a point on
the boundary of space-time at which at least one backward-going (maximally extended) null
53
The quoted bit is from (Hawking and Penrose [1970], p. 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], p. 531 emphasis in the original). Many
authors have noted that inflationary cosmological models violate the strong energy condition of the HawkingPenrose theorem. See, for example, (Wall [2013], p. 20. n. 14; and Borde and Vilenkin [1996], p. 824. n. 17), inter
alios.
54
The three conditions are stated in (Borde and Vilenkin [1996, p. 819]).
55
See (Borde and Vilenkin [1997], pp. 718-719).
56
(Borde, Guth, and Vilenkin [2003]).
57
“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, Guth, and
Vilenkin [2003], p. 151301-4). See also (Mithani and Vilenkin [2012], p. 1 “…it [the BGV] states simply that past
geodesics are incomplete provided that the expansion rate averaged along the geodesic is positive: Hav > 0.” (ibid.));
and (Vilenkin [2013], p. 043516-1).
58
(Vilenkin [2013], p. 043516-1).
59
(Borde, Guth, and Vilenkin [2003], p. 151301-4; cf. Guth [2004], p. 49) See also (Mithani and Vilenkin
[2012], pp. 1-2).
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Christopher Gregory Weaver 22
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geodesic terminates.”60 The BGV entails such geodesic incompleteness given that our
metauniverse satisfies the Hubble parameter condition (which on the CC-M it does).
C&C discuss the BGV theorem, citing (Borde, Guth, and Vilenkin [2003]) and
interpreting the result in such a way that it suggests that eternal inflationary models have
singularities.61 This reading of the theorem is multiply flawed.62 C&C seem to imagine that
because singularities “occur all the time at the center of black holes, and eventually disappear as
the black hole evaporates” the BGV is unproblematic for their model (Carroll and Chen [2004],
p. 27. n. 6). They go on to remark that the fact that the theorem entails the presence of
singularities does not itself entail that there is a “spacelike” boundary “for the entire spacetime.”
(ibid.) This is a misstatement of the result. The theorem implies the existence of just such a
boundary (as Vilenkin noted). An interesting, separate question is whether or not the BGV
applies to the Universe, or to our metauniverse given the CC-M. I have argued that it at least
applies to our metauniverse.
4.2.3
Evasion by the Quantum?
What about escaping the singularity and geodesic incompleteness via the quantum?
Surely there is some hope that a more complete cosmogonic model outfitted with a full-blooded
quantum understanding of gravity will consign our metauniverse’s initial singularity and past
boundary to the trash bin of physical cosmology. McInnes reports that “[i]t has been
argued…that quantum-mechanical effects allow the singularity in the Farhi-Guth ‘wormhole’ to
be evaded...” (McInnes [2007], p. 20, who cites Fischler, Morgan, and Polchinski [1990]; though
cf. Vachaspati [2007]). Carroll has expressed similar optimism.63 Sadly however, quantum
cosmogony does not justify such optimism. There is no piece of classical cosmology on which
the BGV theorem essentially relies, and for which we have sufficient evidence that that piece will
be completely done away with in the quantum regime. In other words, the BGV theorem does
not assume a classical theory of gravity. Vilenkin made this point clear:
A remarkable thing about this theorem is its sweeping generality. We made no
assumptions about the material content of the universe. We did not even assume
that gravity is described by Einstein’s equations. So, if Einstein’s gravity requires
some modification, our conclusion will still hold. The only assumption that we
made was that the expansion rate of the universe never gets below some nonzero
value, no matter how small. This assumption should certainly be satisfied in the
inflating false vacuum. The conclusion is that past-eternal inflation without a
beginning is impossible.64
60
(Farhi, Guth, and Guven [1990], p. 419).
(Carroll and Chen [2004], p. 27. n. 6).
62
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 ℬ”. (Vilenkin [2013], p. 043516-1)
63
See (Carroll [2008b], p. 4, [2010], p. 50, pp. 349-350, particularly p. 408. n. 277), but cf. (Penrose
[1996], p. 36) for a different view.
64
(Vilenkin [2006], p. 175 emphasis mine). Ashtekar ([2009], p. 9) acknowledged that the BGV does not
rely on Einstein’s field equations.
61
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But what about my use of results which capitalize on the EGS theorem and related reasoning?
Are not those results classical? Yes, the results are classical. They depend upon the assumption
that Einstein’s field equations describe the cosmos. However, we have no conclusive evidence
that these results will be overturned by a complete quantum cosmology.
Perhaps you are still dissatisfied with my argumentation. The question, “how can we be
sure that there is an initial space-time singularity at ℬ in a full quantum physical context?” may
still strike you as a deep worry. I believe I can mollify the force of such a worry, since Aron C.
Wall ([2013]) has recently proven a quantum singularity theorem that relies only upon the
generalized second law (GSL),65 which states (roughly) that generalized entropy never decreases
as time marches forward (Wall, [2012], p. 104049-1). Or, with respect to any causal horizon, the
sum of the horizon entropy, plus the field entropy external to any such horizon will necessarily
increase as time marches forward unless already at equilibrium (ibid.). Interestingly, the GSL
implies that the thermodynamic behavior of certain open systems (e.g., a causal horizon’s
exterior) is akin to that of certain closed systems (ibid.).
There is a strong case to be made for the claim that the GSL is true even in contexts in
which quantum gravitational degrees of freedom would be relevant.66 Given that it does indeed
hold, Wall has shown (with the help of several theorems) that multifarious important results for
theoretical cosmology fall out. For example, with respect to an application of his work to our
FLRW metauniverse, Wall stated:
Putting all these considerations together, if the GSL is a valid law of nature, it
strongly suggests that either the universe had a finite beginning in time, or else it
is spatially finite and the arrow of time was reversed previous to the Big Bang. In
the latter case, it could still be said that the universe had a beginning in a
thermodynamic sense, because both branches of the cosmology would be to the
thermodynamic future of the Big Bang. (Wall [2013], p. 22)
Of course, the CC-M posits an eternally inflating FLRW sub-model of our metauniverse. Thus,
the reversed arrow of time idea cannot be added to that sub-model.
You might maintain that C&C need not appropriate the FGG proposal. There are, after
all, suggested improvements of the tunneling story told there. Why then cannot C&C simply
side-step the objections in this section by appropriating one of these ameliorations. The problem
is that other mechanisms like the one in (Fischler, Morgan and Polchinski [1990]) fail if our
metauniverse features an initial singularity. That is why Fischler, Morgan, and Polchinski
diligently seek to rub initial singularities out (see [ibid.] pp. 4046-4047).
We can conclude then, that Wall provides us with yet another reason for why we ought to
believe that our metauniverse is past-null geodesically incomplete. This, I believe, serves as a
significant defeater for the claim that our metauniverse nucleated by means of the FGG
mechanism from a background de Sitter space-time.
65
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], p. 18)
66
See Wall (2013). Cf. Jacobson (1994).
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4.2.4
Fluctuation
The means by which our metauniverse came forth out of a background space-time need
not have involved a quantum tunneling process like the one recommended by Farhi, Guth, and
Guven. In fact, C&C’s original paper ([2004]) did not use the FGG mechanism. Rather, it urged
that a suitable proto-inflationary patch could have—via the harmonic oscillation of a potential—
fluctuated into existence from the background de Sitter space-time. But C&C believe that the
entire process is immensely improbable.67 In fact, the probability is so small that C&C describe it
as possibly the “smallest positive number in the history of physics…” (Carroll and Chen [2004],
p. 26. n. 4). C&C can acknowledge wholeheartedly such a small probability without fear or
trepidation because their model is very much a “wait and see” model (cf. McInnes [2007] p. 8).
Because the background space-time is eternal, and geodesically complete, fluctuations of just the
right sort will inevitably occur, a fortiori, they will occur an infinite amount of times. On this
“wait and see” feature of the model, C&C ([2004], p. 27 emphasis mine) stated:
…the physical volume of spacetime in the de Sitter phase will continue to
increase, just as in eternal inflation. The total spacetime volume of the de Sitter
phase is therefore infinite, and the transition into our proto-inflationary universe is
guaranteed eventually to occur. Indeed, it will eventually occur an infinite number
of times. (Carroll and Chen [2004], p. 27 emphasis mine)
The more general idea seems to be that because the de Sitter vacuum is both unstable and eternal,
anything that can physically occur, will occur, and it will occur an infinite amount of times.
One can see how the infinities are in some sense compounded on the CC-M once one
realizes that the mechanism for producing the large-scale structure of our metauniverse is eternal
inflation. According to Alan Guth, on such a sub-model, “anything that can happen will happen:
in fact, it will happen an infinite number of times” (Guth [2004], p. 49). The latter implication of
eternal inflation is relevant since—you will remember—the means by which entropy increases
without bound is through the birth of metauniverses. Because our metauniverse will evolve into a
de Sitter space-time, it will eventually start to behave like the background space-time, and spawn
proto-inflationary patches that eternally inflate into even more metauniverses. But you see,
Guth’s point is that eternal inflation also implies the inevitable birth of other metauniverses
without the extra thesis that our metauniverse is an asymptotically de Sitter space-time. For on
eternal inflation, certain regions of space-time never stop inflating. Some of these inflating
regions will give birth to other universes with varying features (see Linde [2004], pp. 431-432;
and Steinhardt [2011], p. 42).
So the background universe yields an infinite amount of metauniverses, and an infinite
amount of these will, through eternal inflation, yield an infinite amount of metauniverses as well.
What’s the problem? The problem is that this wreaks havoc on probability judgments. If your
sample space is infinite, it does not appear possible to have a well-defined probability measure to
underwrite your probability and likelihood judgments. This problem of infinities and
probabilities in eternal inflation-based cosmologies is well-known (see Page [2008], p. 063536-1
and the literature cited therein). However, it is also well-known that there is no current
67
(Carroll and Chen [2004], p. 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]).
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Christopher Gregory Weaver 25
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satisfactory solution to the problem. In fact, Paul Steinhardt noted that “[m]any remain hopeful
even though they have been wrestling with this issue for the past 25 years and have yet to come
up with a plausible solution” (Steinhardt [2011], p. 42 emphasis mine).
Notice that my criticism here would run even if C&C dispensed with eternal inflating
sub-models. The problem of infinities appears when theorizing about ultra-large-scale structure
(i.e., the Universe). The problem is compounded when eternal inflating sub-models of
metauniverses such as our own are added in. I conclude then, that while C&C’s original paper
does not invoke the FGG mechanism (despite judgments from Carroll to the contrary), a
heretofore-unresolved theoretical problem remains, the problem of infinity and likelihood.
Kimberly K. Boddy, Sean Carroll, and Jason Pollack (2014) have recently attempted to
avoid the infinities pregnant within inflationary theory by arguing that quantum fluctuations
needed during the inflationary era to induce eternally inflating regions never occur due to the
phenomenon of quantum decoherence as understood by the many-worlds or Everettian
interpretation of QM (see ibid., pp. 3-4, p. 28). The proposal will not help Carroll escape the
clutches of the measure problem as articulated here since the existence of infinitely many
universes is guaranteed by the existence of the Universe or multiverse itself. The background
space-time that is empty dS produces an infinite amount of metauniverses without the
mechanism of eternal inflation. Second, many-worlds interpretations of QM face a deeply
perplexing problem. How does one justify the Born rule of QM given such an interpretation?
(See on this Albert (2010)). I find the attempts to derive the Born rule from considerations
having to do with decision theory to be dependent upon questionable assumptions (e.g.,
Wallace’s (2012, pp. 178-179) diachronic consistency and state supervenience principles). I also
find the reasoning to be a bit circular (Baker (2007)). Thus, this newer work provides no escape
from the measure problem.68
5
Conclusion and Philosophical Significance
The CC-M is admittedly though still woefully incomplete. This incompleteness transfers
to its proposed scientific explanation of our initial low-entropy state. Even if we grant that the
model and its accompanying explanation are in some sense complete, all of its essential theses
fail. We should therefore refrain from looking to the CC-M for a dynamical explanation of the
arrow of time. This result should be welcomed by those who affirm a best systems account of
laws of nature, for as was suggested in sect. 1 above, the demonstrable failure of models like the
CC-M suggests that the low-entropy initial condition is scientifically brute (it has no scientific
explanation). One can readily accommodate such a fact by insisting that not all laws are
dynamical. The past hypothesis is itself a law of nature, and the best account of laws that allows
for non-dynamical versions of them is the Mill-Ramsey-Lewis best systems approach.
68
That work also required a quantum theory with an infinitely dimensional Hilbert space.
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Acknowledgments: I would like to thank David Albert, Tom Banks, Barry Loewer, Don N.
Page, Quayshawn Spencer, and Aron C. Wall for helpful comments on earlier drafts of the paper.
I would like to thank the referees and editors with the Journal for General Philosophy of Science
for helpful comments and suggestions. Special thanks to Anthony Aguirre, Robin Collins, and
Matthew Johnson for valuable conversations about some of the arguments presented here. And
let me express gratitude and thanks to Sean Carroll for kindly and patiently answering several of
my questions about the Carroll-Chen model during the UC Santa Cruz Summer Institute for the
Philosophy of Cosmology in the summer of 2013, and the Philosophy of Cosmology Tenerife
(Spain) Conference of 2014.
Christopher Gregory Weaver
PhD in Philosophy, Rutgers University
Assistant Professor of Philosophy
University of Illinois at Urbana-Champaign
christophergweaver@gmail.com
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