Does string theory predict an open universe?

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by  R. Buniy

Does string theory predict an open universe?
Roman V. Buniy,1,∗ Stephen D. H. Hsu,1,† and A. Zee2,‡
1Institute of Theoretical Science, University of Oregon, Eugene, OR 97403
2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106
Ithasbeenclaimedthatthestringlandscapepredictsanopenuniverse,withnegativecurvature.
Thepredictionisaconsequenceofalarge numberofmetastablestringvacua,andthepropertiesof
the Coleman–De Luccia instanton which describes vacuum tunneling. We examine the robustness
ofthisclaim,whichisofparticularimportancesinceitseemstobeoneofstringtheory’sfewclaims
to falsifiability. We find that, due to subleading tunneling processes, the prediction is sensitive to
unknownpropertiesofthelandscape. Underplausibleassumptions,universeslikeoursareaslikely
tobe closed as open.
INTRODUCTION are described by the Coleman-De Luccia (CDL) instan-
8
0 ton [7], which exhibits an O(4) symmetry in Euclidean
0 space. For scalar fields, the instanton configurationwith
If, as suggested by recent results [1], string theory ex-
2 O(4) symmetry has the lowest action [8]. We assume
hibitsalandscapeofmorethan10500distinct,metastable
n that this is also the case when gravitational degrees of
vacua, its status as a conventional scientific theory is
a freedom are present. In that case, the Euclidean metric
J in jeopardy. Most physicists feel that scientific theories
must have the form
must make predictions which are falsifiable by experi-
7
2 meveennt.bSeuicnhfinaitlear–gemdiigvhetrsmityeaonftvhaactuaes–setnhteianlluymabneyr mloway- ds2 =dξ2+f(ξ)2dΩ2S3 , (1)
v energyphysicsisrealizablefromstringtheory. Optimisti- where dΩ2 denotes the distance element of a unit 3-
1 cally, future workmay revealsome testable propertiesof S3
3 sphere. The radialcoordinate ξ is orthogonalto families
2 thelandscape,suchascouplingconstantrelationsorcon- of3-spheressatisfying 4 x2 =ξ2. Whenthis solution
0 straintsonparticle content. However,atthe momentwe is analytically continuePdit=o1Miinkowski space, the corre-
1 are unaware of any such predictions, with the possible
sponding metric in the interior of the bubble is given by
6 exception of Weinberg’s anthropic determination [2] of
(1) for imaginary values of ξ, and with the 3-geometries
0
/ the value of the cosmological constant, and even that is H3 satisfying the constraint
h sensitive to assumptions about the spectrum of primor-
-t dial density perturbations [3]. Even ultra high-energy 3
p physics experiments may not yield additional informa- Xx2i −t2 =r2−t2 =ξ2
e
tion,sincescatteringattrans-Planckianenergiesleadsto i=1
h
: black holes [4] of ever increasing size, whose subsequent
v (t>r in the interior):
behavior (evaporation) is controlled by the low-energy
i
X physics of the ambient vacuum state. If recent results
ds2 =dτ2−f(iτ)2dΩ2 , (2)
r areanyguide,stringtheorywillbe extremelydifficult to H3
a
falsify.
whereξ =+iτ. HereH3isdefinedtohavesignature(−−
It is therefore important to carefully consider any ro- +). FollowingCDL,wehavemultiplied the metricbyan
bust implications of the string landscape, particularly overallsigninordertoobtaintheirMinkowskisignature.
thosethatmightbe testable inthe forseablefuture. One It is clear that the H3 geometries are hyperbolic, and
of these, recently elaboratedin [5], is the predictionthat hence the resulting bubble universe is open.
our universe must be open, with negative curvature. A Weseethataresultingopenuniversedependscrucially
recentanalysiscombiningWMAP andSloanDigitalSky ontheO(4)invarianceoftheEuclideansolution,andthe
SurveydatagivesΩtotal =1.003±0.010[6],butimproved subsequentanalytic continuation. Inthe nextsectionwe
future observations could yield a statistically significant will investigate the properties of bubbles which might
central value larger than unity, implying positive curva- resultfromtunneling which is not dominatedby a single
ture. Would this rule out string theory? Euclidean solution.
The argumentfor anopen universe is as follows. Con- Inthe remainderofthis letter wewillassume apoten-
sider anenergysurface withmany localminima,mostof tial of the form given in Figure 1. Tunneling takes place
whichhavemuchmore energydensity thanthe observed betweenconfigurations1and2,withenergydensitiesρ
1,2
value of the dark energy density Λ ∼ 10−10eV4. Given respectively. An extended flat region is assumed beyond
genericinitialconditions,itislikelythatouruniversear- the tunneling point 2, in order that the bubble interior
rivedatitscurrentstateviatunnelingfromamuchmore experience an inflationary epoch after nucleation. In [5]
energetic metastable vacuum. Such tunneling processes itwas deducedthat roughly60e-foldings ofinflationare
2
ρ
where the subscript k on dΩ2, denotes the usual FRW
openandclosedgeometriesfornegativeandpositiveval-
ues, respectively. The bubble state |Bi, including metric
and field theory degrees of freedom, is characterized by
ρ1 parameters such as R,R˙,ρ2,σ,r, with r the bubble ra-
ρ2 dius.
Λ Given these assumptions, it is the initial conditions of
1 2 the bubble that determine the curvature of the interior
universe. Unlike in the CDL case, the evolution of the
FIG. 1: A possible potential in thestring theory landscape. interior is not the analytic continuation of a particular
Euclidean solution. Instead, we examine the Einstein
equation for the interior
necessaryinorderthatthenegativecurvatureoftheCDL
bubble not suppress galaxy formation. H2 =κρ−k/R2 , (4)
For simplicity, we assume ρ ≃ ρ = ∆4, and a thin-
walled bubble with surface ten1sion σ2∼∆3. We define where H = R˙/R, and κ = 8πG/3. The sign of k is
determined by whether the initial interior density ρ is
2
ρ1−ρ2 greater or less than the critical density ρ =H2/κ:
ǫ≡ ≪1 , c
∆4
k =R˙2(ρ /ρ −1) . (5)
so a critical bubble has radius r ∼(∆ǫ)−1. This radius 2 c
∗
can be of order the de Sitter horizon size rdS ∼ M/∆2 It is easy to arrange for positive k without violating any
(whereM isthe Planckscale)ifǫis smalland∆nottoo conservationlaws. Forexample,thebubblemightresem-
small relative to M. This last condition is important for ble the CDL bubble, except with a smaller ρ resulting
c
the non-Euclidean inflationary evolution of the interior from smaller initial R˙. The total energy of the bubble
of the bubble [9], as we will discuss further below. can still be made equal to its volume times ρ , thereby
1
conserving energy.
Eq. (4) can be rewrittenin the form of an energy con-
BEYOND SEMICLASSICAL DOMINANCE
servation equation
It is easy to imagine situations in which the tunneling R˙2−κρR2 =−k. (6)
amplitude is not dominated by a single Euclidean solu-
tion. For example, if some of the fields involved in the A region with k > 0 can simply be regarded as one in
tunneling are strongly coupled, the usual semiclassical which the sum of negative gravitational binding energy
expansionbreaksdown,andmanypathsinthefunctional andpositiveenergydensityinR˙ islessthanzero. Sucha
integral play a role. Since there are so many field the- region is gravitationally bound: it decouples from the
ory degrees of freedom on the landscape, it seems quite any expansion (inflation) of the surrounding universe,
plausible that some of them will be strongly coupled. and eventually undergoes collapse. To an outside ob-
In addition, even if the leading amplitude is given by server it ultimately appears as a black hole. Note that
the CDL instanton, there will still exist subleading pro- any region that is isotropic and homogeneous can be lo-
cesses, albeit with exponentially smaller amplitudes. As cally described by the FRW metric, and the value of k
explained below and in the following section, even ex- is simply determinedby the rate of expansionrelativeto
ponentially suppressed bubbles are of interest, as they the matter energy density.
may play an important role in the anthropic calculation Wementionanimportantcaveat. We assumethatthe
of probabilities. fateofthebubbleinteriorcanbededucedfromFRWini-
We nowrelaxthe conditionthatthe tunneling process tialdatadescribingitdeepinside. Indoingso,weneglect
be described by a Euclidean solution. We consider any the boundary interactions of the expanding bubble wall
amplitudes that are not explicitly forbidden by conser- with the false vacuum 2. In [9] it was shown that infla-
vation laws, such as energy conservation. That is, any tionary(non-Euclidean)internalevolutionofashrinking
non-zero transition hf|U|ii where the initial state |ii is false vacuumbubble is determinedby a naiveanalysisof
the vacuum 1 and the final state is that of a critical or the interior properties if the bubble is sufficiently large,
supercritical(expanding)bubbleofvacuum2: |fi≡|Bi. roughly the size of the de Sitter horizon determined by
We assume the bubble interior is homogeneous, so that its interior energy density. In the previous section we
deep inside its spacetime is described by the Friedman- notedthatbubbles ofthis size arepossibledepending on
Robertson-Walker (FRW) metric the parameters ǫ and ∆. In our case the bubble wall
is expanding away from the interior at relativistic speed
ds2 =dt2−R(t)2dΩ2 , (3) (i.e., the false vacuum is on the outside), whereas in the
k
3
case studied in [9] it is collapsing. In [9], when interior probabilities is
inflation is curtailed it is due to the impinging collapse,
P(CDL|us) Γ(CDL) N(60+)
which is not an issue for our case. Our neglect of the ∼ · , (7)
P(B∗|us) Γ(B∗) N(B∗)
boundaryinteractionislikelytobejustified,particularly
in the case of large bubbles. where Γ(CDL) and Γ(B∗) denote tunneling rates,
Intheremainderofthepaper,wefocusonthesubsetof N(60+) is the number of regions in the landscape of
bubbles |B∗i whose interiors are homogeneous and have the type in Figure 1 which can produce at least 60 e-
ρ very close to ρc (i.e., k very close to zero, with either foldings, and N(B∗) is the number that can produce a
sign). Suchbubblesareparticularlyfavorableforproduc- much smaller number of e-foldings sufficient to produce
ing universes like ours, as first noted by Linde [10]. Ho- a universe like ours from B∗ initial conditions. (For ex-
mogeneity ameliorates the horizon problem, while small ample, a homogeneous B∗ bubble with k close to zero
initial k lessens the flatness problem. The number of e- mightonly require40 e-foldings of inflation, andN(B∗),
foldings of subsequent inflation required for such bubble the number of potentials that produce 40 e-folds of in-
universes to resemble ours is much less than the 60 or so flation might be exponentially larger than N(60+).) We
required for the CDL bubble [5]. This opens the inter- have assumed a large but finite number of vacua, so the
esting possibility that, even if nucleation of B∗ bubbles N are finite numbers. Even if Γ(CDL) >> Γ(B∗), it
is highly suppressedrelativeto CDL bubbles, the overall could be that N(B∗) is sufficiently larger than N(60+)
conditional probability that a universe resembling ours thatP(B∗|us)dominatesP(CDL|us). Inthatcase,there
came fromB∗ tunneling may be greaterthan orof order couldberoughlyequalprobabilitiesofk slightlypositive
the probability that it came from CDL tunneling. The and negative, since B∗ bubbles could arise with either
relative tunneling suppression might be compensated by signof the curvature. Eventhatconclusionis dependent
the scarcity of flat potentials that can produce many e- onfurther assumptions, as the precise minimum amount
foldings of slow roll inflation. of inflation necessary for k negative and positive could
Note that the background spacetime in which the B∗ be slightly different.
bubble is nucleated has presumablybeen inflating for an We mightexpect that if inflationary e-foldings areim-
extended period, and hence has almost exactly flat ge- probable, then the amount of curvature should be near
ometry. The curvature parameter k of the B∗ bubble an anthropic boundary. What is the anthropic limit on
is assumed to be very close to zero, so we are essentially flatness? Let us consider k > 0 since that is the case of
gluingaveryslightlycurvedFRWregionontoanexactly mostinteresthere. Weleaveasidethedarkenergy,which
flatbackground. Itseemsunlikelythatanyboundaryef- further complicates the issue. It is not clear whether
fects could cause a change in the sign of k. the dark energy will persist forever, e.g., as in the case
of a cosmological constant, or is only a temporary phe-
nomenon (e.g., quintessence).
Positive curvature causes less rapid expansion than in
ANTHROPISM AND MODEL DEPENDENCE
ak =0universewithsimilarmattercontent,sostructure
formation is not the issue. Rather, the issue is whether
Let us adopt the anthropic assumption that the land- the universe recollapses before life can evolve. Universes
scape leads to the realization of many causally discon- whichsurviveforalongtimebeforerecollapsemightlead
nected universes with varying properties. Further, let us to more sentient beings (per unit volume) and be an-
assume that it is via bubble nucleation depicted in Fig- thropically favored. The number of beings might even
ure1thatuniverseswithsmallcosmologicalconstantare grow exponentially with time, depending on population
produced. Now,considertheconditionalprobabilitythat dynamics. Therefore, even if inflationary e-foldings are
a universe resembling ours (i.e., large, nearly flat, with improbable, one cannot exclude k > 0 universes which
structure formation) has k either negative or positive. are very flat. (Note, it would be quite surprising for the
Whetherpositiveornegativek ismorelikelydepends on competing exponential pressures we describe to result in
whether the suppression of tunneling rates to |B∗i bub- a spatial curvature which is observable but also consis-
bles is compensated by less stringent conditions on the tent with current bounds!)
inflaton potential. For example, it seems plausible that Onemightobjectthat,ifthereisgoingtobeexponen-
the probability distribution for inflaton potentials (ex- tial population growth in our future, then we are quite
tended flat regions that allow slow roll de Sitter epochs) atypical of beings in our universe. However, this objec-
could be exponential in the number of e-foldings [11]. tion requires a very strong use of the typicality assump-
Then, even if B∗ tunneling is very rare relative to CDL tion which we do not find plausible. Used in this way,
events,theremaybe exponentiallymorevacuaforwhich it would imply a dim future for the human race: if we
a B∗ bubble could leadto a universe like ours than for a are typical (i.e., not among the earliest humans to have
CDL bubble. lived),thenthetotalfuturepopulationofhumansisquite
In rough approximation, the ratio of the conditional limited!
4
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ins and M. B. Wise, Phys. Lett. B 600, 15 (2004)
cosmologicalcurvaturecannotbe deducedbyanysimple
[arXiv:hep-th/0407174].
arguments. Regrettably, it does not provide a clean test
[4] D. M. Eardley and S. B. Giddings, Phys. Rev. D 66,
of string theory.
044011(2002)[arXiv:gr-qc/0201034];S.D.H.Hsu,Phys.
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[5] B. Freivogel, M. Kleban, M. Rodriguez Mar-
ACKNOWLEDGEMENTS tinez and L. Susskind, JHEP 0603, 039 (2006)
[arXiv:hep-th/0505232].
[6] M. Tegmark et al., [arXiv:astro-ph/0608632].
S. H. thanks F. Denef, M. Douglas and L. Susskind [7] S.R.Coleman andF.DeLuccia, Phys.Rev.D21, 3305
for discussions concerning the string landscape during (1980).
the 2006 Ettore Majorana school of subnuclear physics. [8] S. R. Coleman, V. Glaser and A. Martin, Commun.
A. Z. thanks F. Wilczek for helpful discussions regard- Math. Phys.58, 211 (1978).
ing the Coleman-De Luccia work. He is also grateful [9] S. K. Blau, E. I. Guendelman and A. H. Guth, Phys.
Rev. D 35, 1747 (1987).
to the Radcliffe Institute for Advanced Study, Harvard
[10] A.D.LindeandA.Mezhlumian, Phys.Rev.D52, 6789
University, where this work was done, for its warm hos-
(1995)[arXiv:astro-ph/9506017];A.D.Linde,Phys.Rev.
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phasizing the problems of eternal inflation, flat poten- [11] Actually, a further assumption is required: that an-
tials and anthropic limits on curvature. S. H. and R. B. thropic probabilities not be weighted by the volume of
are supported by the Department of Energy under DE- spacecreatedbyinflation(i.e.,thenumberofe-foldings).
FG02-96ER40969,whileA.Z.issupportedinpartbythe If that were the case, the flattest potentials might be
the most likely, in which case k is likely to be almost
National Science Foundation under grant number PHY
exactly zero and we will never measure its sign! We
04-56556.
explicitly assume that this is not the case – for exam-
ple, one might weight by probability of life per unit vol-
ume – but it reveals the perils of anthropic reasoning.
For further discussion, see B. Feldstein, L. J. Hall and
T. Watari, Phys. Rev. D 72, 123506 (2005); J. Garriga
∗ Electronic address: [email protected] and A. Vilenkin, Prog. Theor. Phys. Suppl. 163, 245
† Electronic address: [email protected]
(2006) [arXiv:hep-th/0508005].
‡ Electronic address: [email protected]
[1] For a recent review, see M. Douglas and S. Kachru,
[arXiv:hep-th/0610102].