Accelerating Universes from Compactification on a Warped Conifold

0.15 MB·English

by  Ishwaree P. Neupane

Accelerating Universes from Compactification on a Warped Conifold
Ishwaree P. Neupane
Department of Physics and Astronomy, University of Canterbury,
Private Bag 4800, Christchurch 8020, New Zealand and
Theory Division, CERN, CH-1211 Geneva 23, Switzerland
E-mail: [email protected]
Wefindacosmologicalsolutioncorrespondingtocompactificationof10Dsupergravityonawarped
conifoldthateasilycircumvents“no-go”theoremgivenforawarpedorfluxcompactification,provid-
ingnewperspectivesforthestudyofsupergravityorsuperstringtheoryincosmologicalbackgrounds.
With fixed volume moduli of the internal space, the model can explain a physical Universe under-
going an accelerated expansion in the 4D Einstein frame, for a sufficiently long time. The solution
foundinthelimitthatthewarpfactordependentontheradialcoordinatey isextremized(givinga
constant warping) is smooth and it supports a flat four-dimensional Friedmann-Robertson-Walker
7 cosmology undergoing a period of accelerated expansion with slowly rolling or stabilized moduli.
0
0 PACSnumbers: 98.80.Cq,11.25.Mj,11.25.Yb. arXiv: hep-th/0609086 CERN-PH-TH/2006-171
2

n
a Introduction.– Recent astronomical data, notably the string effects are available at much lower energy scale,
J observations of high redshift type Ia supernovae [1] and such as ρ 10−3 eV or H 10−60M . There is an-
vac Pl
∼ ∼
0 measurements of the cosmic microwave background [2], other particular difficulty in this program in stabilizing
3 not only provide emerging evidence for the ongoing ac- the common modulus associated with the overall shape

celeratedexpansionoftheUniversebutalsoprovidesup- and size of the internal Calabi-Yau spaces. Freezing the
3
port for the concept of inflation, or a rapid exponential volume moduli using non-perturbative dynamics seems
v
6 expansionoflargemagnitudeinamuchearliercosmolog- beyondanythingvisibleinsupergravity. Time andspace
8 ical epoch. Although it is not difficult to construct cos- are not independent, so any idea that the geometry of
0 mological models that exhibit these features, one would spacetimeisfixedandnon-dynamicalisprobablywrong.
9
prefer any such model to be derivable from a fundamen- The advent of string or M theory in time-dependent
0
tal,andmathematicallyconsistentmicroscopictheoryof backgrounds is an important and promising subject. It
6
0 (de Sitter) quantum gravity, such as string theory. Su- has the potential to offer a resolution to the dilemma
/ perstring theory lives in 10 dimensions but we live in a posed by the observed cosmic acceleration within a nat-
h
four-dimensional Universe. Clearly, any attempt to de- ural theoretical framework. In [4] (and generalizations
t
- rive a viable cosmology from string or M theory (com- thereof [5]) the no–go theorem was circumventedjust by
p
e pactification) must produce a four-dimensional de Sitter the choice of negatively curved internal spaces, once the
h Universesimilar to oursandthe size ofextra dimensions fluxes are turned off. The restriction on the curvature
v: should remain much smaller than the physical three di- seems a severe one, especially given the view that flat
i mensions. spacecompactificationsonCalabi-Yauspacesareamong
X
the most natural in string theory.
r The past few yearshave witnessed significantprogress
a
in building of inflation models within string theory via An interesting observation in [10] is that a time-
flux compactifications of the ten- or eleven-dimensional dependent compactification of classical supergravities
spacetime of superstring or M theory with the desire to with Ricci-flat extra spaces, involving certain twists in
find models for late-time cosmology [3, 4, 5] supporting thegeometry,cangiverisetoapositivepotentialinlower
a small positive cosmological constant. If one wishes to dimensions and hence a period of accelerated expansion
stay within the realm of low energy supergravity models in the 4D Einstein frame. It has been learned that the
derivedfromsuperstrings,cosmicinflationisruledoutfor time-varying volume moduli with no initial fine tuning
warped flux compactifications of classical supergravities among the scalars lead only to a transient period of cos-
onthebasisofa“no-go”theorem[6,7],whichforbidsac- mic acceleration,except in the case that we live in a hy-
celeratingsolutionsfor warped(and static)extradimen- perbolicUniverse[5]. Inordertofullyaccountforthena-
sions. For a way out, one may possibly include higher ture of an effective four-dimensional cosmology, it is im-
curvaturecorrections[8]to the leadingorderLagrangian portantto gaina properdescriptionofspacetime depen-
′
inα expansionorextendedsources(branes,anti-branes) dent compactifications (of higher dimensional gravity),
that are present in string theory [9] or even invoke cer- rather than the time-dependent or the space-dependent
tain non-perturbative effects (such as, gaugino conden- (warped) compactification alone. This is because upon
sateandEuclidianD3branes)[3]. Theseallachievesome dimensionalreductionaninternalspaceofpositivecurva-
limited success in overcoming the no-go theorem. How- ture gives a negative potential in time-dependent back-
ever, there is no good reason to suppose that all these grounds, while it gives a positive potential in standard
2
warped backgrounds. In view of this observation, all the metric g is related to g˜ via
E,µν µν
studiesonfluxandtime-dependentcompactifications(of
g =e2φg˜ . (4)
string or M theory) to date are either incomplete, or are E,µν µν
at best part of a more complete story.
Onemustchoosethescalarstosatisfyα+4β+σ 2φ,so
InthisLetterweconsideraparticularmodelwherethe ≡
that the 4D Newton constant is then time-independent;
internalspaceshavegeometriesspecifiedby morescalars
φ is a 4D scalar rather than the 10D dilaton.
than just the volume modulus. The type IIB supergrav-
The background solution.– The metric considered by
ity theory in ten dimensions, with a warped 6D conifold
KT [11] corresponds to the choice f = 1,α = 0,β =
geometry provides an example of this kind, as originally
ln√6,σ = ln3,g = 1 and k = y2. By turning on N
studiedbyKlebanovandTseytlin[11],andKlebanovand − −
units of the NS 5-form flux on X and M units of the
Strassler[12]. Inthismodel,theinternalspaceisaRicci- 5
RR 3-form flux through the S3 of T1,1, one finds [11]
flat6DconeY whosebaseisa5DEinstein-K¨ahlerspace,
6
X5 T1,1. Theintroductionofbranesmaybeimportant L4 3g M2 y
for ≡constructing gauge field theories (of the elementary h(y)=h + 1+ s 1+4ln (5)
0 y4 (cid:18) 8πN (cid:18) y (cid:19)(cid:19)
0
particles) at the tip of a warped conifold, given a view
that both gravitational and non-gravitational forces can with L4 27πg Nα′2/4, which satisfies the standard
be localized on D3-branes. We show that the “no-go” quantizat≡ion consditions: (4π2α′)−2 F = N and
theoremdoesnotapplytoatime-dependentbackground (4π2α′)−1 F = M. The singulaRrTit1y,1at5y = 0 may
even if the extra dimensions are warped. S3 3
be resolvedR by deforming the conifold [12, 13] or by al-
The model.– We shall assume that ten- or eleven- lowing time-dependence to the internal space.
dimensional supergravity is the relevant starting point. Let us momentarily set h const (or take y L),
≡ ≫
The model below corresponds to the dimensional reduc- g 1 and k y2, which is relevant to finding a pure
≡ ≡
tion to 4 dimensions of type IIB supergravity,where the time-dependent solution. Einstein’s equations admit the
spacetimeisawarpedproductofasix-dimensionalspace following explicit solution (in the gauge δ =3)
Y and M ( R1,3). In particular, the 10D metric is
6 4
ds≡2 =h−1/2g˜ dxµdxν +h1/2ds2. (1) a = a0e±2c1u, α=c1u+α0,
10 µν 6 1 3f2
β = c u+α ln√6=σ+ ln . (6)
The metric of the large three dimensions (plus time) is 1 0− 2 2
For the branch c u < 0, the size of the internal space
dx dxµ = a2δdu2+a2(dx2+dx2+dx2), (2) 1
µ − 1 2 3 shrinks with time, while the size of the physical three
where a a(u) and δ is a constant, the choice of which spaces can grow if we choose the negative exponent.
fixes the≡nature of the time coordinate u. In the gauge Thisresultisremarkableasitwasimpossibleforinternal
δ = 0, u becomes the proper time t. The metric on the spaces with a single (common) volume modulus.
transverse 6D space is Afewremarksmayberelevantbeforeweproceed. We
have chosen a factorizable geometry: the dependence of
2 the warp factor h on time t (or u) has been absorbed
ds2 = e2αgdy2+e2βk dθ2+sin2θ dφ2
6 i i i into the metric g (or the scalars α,β,σ) as we would
µν
Xi=1(cid:0) (cid:1) like to write the metric in 4D Einstein conformal frame;
2 a time dependenece in h would render it difficult for such
+e2σm(dψ+f cosθ dφ )2. (3)
i i an interpretation. Time-dependent solutions of our sort
Xi=1
were studied in the past, for example, by Kodama and
The ranges of the angular coordinates are 0 θ < π, Uzawa [14]. However, it was assumed there rather im-
i
≤
0 φ <2πand0 ψ <4π. Weassumethatthemoduli plicitly that α=β =σ and also g =k =m. In the work
i
≤ ≤
parameters other than the volume scalars are stabilized of Buchel, for example [15], the metric was not written
(frozen); the scalars α, β and σ are functions of u (or in the 4D Einstein frame, and also no time-dependence
thepropertime t),whileg,k,mandharefunctionsonly was allowed for internal spaces. These or other similar
of the radial coordinate y. We will also consider some assumptions exhaust some (or all) of the interesting cos-
examples where β and σ are functions of the radial co- mological solutions that we have found in this Letter.
ordinate y. As discussed in [12], due to the twist along Kachru et al. [3] proposed to fix the volume mod-
the normal S1 bundle, the model preserves only 1/4 of uli using some non-perturbative dynamics, such as a
the = 4 supersymmetries and gives a mass to the gaugino condensate. This is an interesting proposal but
N
scalar fields. Even though we put in different functions such a construction is model or scheme dependent. For
of y for the two 2-spheres and the twisted S1, Einstein’s the warped(conifold) geometryunder consideration,the
equationssimplifyalotwhenthesefunctionsarepropor- gaugino condensate is related to the deformation of the
tional, so henceforth k(y) m(y). The Einstein-frame conifold,so it is alreadyvisible in the classicalgeometry,
∝
3
and one does not need instantons to see the condensate. write down the result only for β. For the solution above
However, we show that the volume moduli can be sta- only the region y4 >√2c2 is physical; the singularity at
bilised spontaneously due to a natural expansion of the y = 0 is due to the choice k y2, not because of any
≡
Universe,evenleadingtoatransientperiodofcosmicac- specific ansatz for form fields. This is clear also from
celeration at late times. In general, the volume factors the deformed conifold solutions of [12]. To quantify this,
are dependent on both time and space; in order to write suppose that h h , without specifying k(y). We find
≃ 0
an effective action in four dimensions, it is necessary to
integrate out the y-coordinate. This can be done only if 3(k′/k)′+k−1e−2β =0. (10)
the solutions for both space- and time-dependent parts
of the volume factors are known, simultaneously. Clearly,ifk k y2,thenwegetβ = ln 6k ,while,if
≡ 0 − 0
Cosmological solution.– We shall consider the case k k sech(y), then β = ln 3k +3lnpcosh(y), which
≡ 0 − 0 2
α = 0 and g = 1, so as to maintain the inter- is regular everywhere. p
pretation of y as the holographic energy scale. In The presence of external fluxes would modify the
the zero flux limit of the 10D Einstein equations, and solution for warp and volume factors as in (9) for
with the choice k y2, the symmetries of the met- a a , or in a more complicated way for a a(t).
≡ ≡ 0 ≡
ρri2c/ya2n,s(aitiz) 4imβp+lyσt=hacton(is)t 4βµ+, σh(≡y) =ϕ(t),λ+h(ρy4)/y=4. Since Ryy = (h4k)−1/2ddy h9/4k3ddy h−1/4k−5/2 +
Upondimensionalreduction≡thefirstbranchpaboveyields 29 ddy(lnh)ddy(lnk) and Rty h= −52β˙ddy[(cid:0)ln(hk)], we(cid:1)ifind
I = vol(X5) d4x dy g R + , where vol(X ) solutions only in the (large volume) limit where
8πG10 − (4)(cid:16) (4) L(cid:17) 5 hk = const (see below), or when the volume mod-
contains onRly theRspacpe-dependent part,
uli are fixed. Moreover, + , where
gr flux
L ≡ K−V =12β˙2+ 32σ˙2+4β˙σ˙ − 12hh′32 e−σ−4β 2Lcfl2uex2σ−∝4βeF−′22hφ−−22σk(−c221)e−w4βithh−K2Lk−3→+c L21+e−2c8βFhL−(y3)k.−5UKn2de−r
1 ≡ 0 1
1 f2eσ−8β 4e−σ−6β +20e−σ−4β , (7) our metric ansatz, equations Rˆpq = 0 (with n = 5) are
−y2h − automatically satisfied. To see the effect of fluxes on the
(cid:0) (cid:1)
spatial sections of the cosmology, one can take k y2
where ′ d/dy. The corresponding scalar potential al- ≡
≡ (and hence h= λ+ρ4/y4). The explicit solution is
lows only an anti-de Sitter minimum or it at best de-
p
scribes only a short period of accelerated expansion due
2 f ρ8e−5β
to a relatively large slope of the potential along the β- a(t)=a eHt, H2 | | , (11)
0 ≡r3(λy4+ρ4)5/2
direction. One can easily modify the form of the poten-
tialbyintroducingabulkcosmologicaltermoradditional
withanarbitrary(constant)β. InsuchacaseR (X )is
source terms (fluxes, branes) or even by invoking some pq 5
supported by five-form (and self-dual three-form) fluxes
particularnon-perturbativedynamics, soas to uplift the
on T1,1. The above solution is stable as long as the vol-
AdSminimumandmakeitametastabledeSitterground
umemoduliarefixed. Forslowlyrollingmoduli,anysuch
state. However, we do not consider this last possibility
de Sitter phase would be only metastable. This example
here, as it hinders our ability to find analytic solutions.
demonstrates that it is indeed possible to maintain fixed
Instead we consider the second branch, (ii). The type
volume modulus of the internal space while the spatial
IIB supergravity equations may be solved by making
sections of the cosmologyundergo a de Sitter expansion.
appropriate ansatz for the form fields [11, 14]; in the
A constant warping.– In the large volume limit, when
case where the volume moduli (β,σ) are fixed (or time-
the backreaction of the fluxes on Einstein’s equations
independent)thesupergravityequationsmaybereduced
can be ignored (since their contribution to the stress
to the form [14]
tensor is volume suppressed), the warp factor is mini-
R = 0, R =0, mized. This particular case may be related to the large
µν µp
1 τ limit of the resolved conifold metrics in [13], for which
Rˆpq ≡ Rpq− nR(Xn)gpq(Xn)=0. (8) h → h(τ) ≡ h0 + h1e−4τ/3; k(τ) and m(τ) also take
their extremized values. The field equations reduce to
Here (µ,ν) run from 1 to (10 n). In particular, in the equations of motion, and a constraint, for the variables
−
static case, a a , we define β β(y), σ σ(y) (in the (α,β,σ) thatarethe Euler-Lagrangeequationsofthe ef-
≡ 0 ≡ ≡
metric (3)) and take n=6. With k y2, we find fectiveLagrangian = 3α˙2+12β˙2+3σ˙2+4α˙β˙+4β˙σ˙+
≡ L 2 2
1α˙σ˙ e−2φ−4β f2e2σ 4e2β . The explicit solution is
h=h exp[c/y4], β = 6(1 2c2/y8). (9) 2 − h0 −
0 − − (cid:0) (cid:1)
p
In the large y limit, h(y → ∞) ≡ h0 +L4/y4 [11]. All a = eζu(coshχu)−5/8, α=c1u+c0,
our solutions, both for fixed and time-dependent volume 1 1 3f2
moduli, satisfy the relation σ = β − 12ln(3f2/2), so we σ = −4lncoshχu+c2u+c3 =β− 2ln 2 (12)
4
(in the gauge δ =3), where ζ (c +7c )/6 and or without external fluxes. Allowing time dependence
≡ 1 2
in the warped conifold solutions is an excellent route for
h = 64 e−c0−7c3, χ2 16 c2+2c c +7c2 . (13) studying aspects of de Sitter Universe via string com-
0 81 f6χ2 ≡ 15(cid:16) 1 1 2 2(cid:17) pactifications. We considered explicit cosmologies that
ariseinmodelsofgravitywhichcorrespondtothedimen-
The four-dimensional cosmic time t is defined by dt =
sional reduction to 4 dimensions of 10-d supergravity. It
a3du. It follows that this solution exhibits a period of
± is remarkable that a model with so many attractive fea-
accelerated expansion (a˙ >0, a¨ >0) in the 4D Einstein
tures can arise from a simple compactification of type
frame,providedthat2 3√2<c /c <2+3√2. Froma
− 1 2 IIB (as well as type IIA) supergravityon a warped coni-
purelymetricpointofview,allthesolutionswith f >0
| | fold. Further generalizations of the solutions discovered
are non-singular. The constants c and c may be set to
0 3 in this Letter are also possible, including the case where
zerousingashift-symmetryinu,oralternatively,canbe
g =g(y). Finally we note that for slowly rolling moduli,
absorbed into g and k so that each becomes unity even
the effective potential, V(φ), can vary slowly with time,
if they are assigned different values initially. The scalars
while for fixed volume moduli, it acts purely as a cosmo-
β and σ can be stabilized by requiring that
logical term (cf. (11)); thus the model could satisfy the
c /c =2 or c /c = 4. solar system test and other constraints from cosmology.
1 2 1 2 −
Thescalefactorthenevolvesasa e−c2u+e4c2u oras
a e−2c2u+e3c2u . In the first∼ca(cid:0)se the Univers(cid:1)e still I wish to thank Igor Klebanov for constructive re-
∼ marks and recommendations on the draft, and Chiang-
expe(cid:0)riencesashortp(cid:1)eriodofacceleratedaccelerationbe-
Mei Chen, Chris Herzog, Pei-Ming Ho, David Wiltshire
forethevolumescalarsβandσattainnearlyfixedvalues,
forhelpfuldiscussions. Thisresearchissupportedbythe
in which limit V e−α. One takes c u>0, so that the
∝ 2 Foundation for Research, Science and Technology (NZ)
physical three spaces expand faster, in the conventional
under Research Grant No. E5229.
manner. The radialmodulus associatedwithR1 expands
in the first case,providinga “4+1+compactspace”type
background, while it shrinks in the second case, provid-
ing a “3+1+compactspace” type background. Thus the
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