Complete power spectrum for an induced gravity open inflation model

12 Pages·0.25 MB·English

by  Juan Garcia-Bellido

Complete power spectrum for an induced gravity open inflation model
Juan Garc´ıa-Bellido
Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
and Theory Division, CERN, CH-1211 Geneva 23, Switzerland
Andrew R. Liddle
Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
(February 1, 2008)
We study the phenomenological constraints on a recently proposed model of open inflation in the
contextofinducedgravity. Themaininterestofthismodelistherelativelysmallnumberofparam-
eters, which may be constrained by many different types of observation. Weevaluate the complete
spectrum of density perturbations, which contains continuum sub-curvature modes, a discrete su-
7 per curvature mode, and a mode associated with fluctuations in the bubble wall. ¿From these, we
9 computetheangularpowerspectrumoftemperaturefluctuationsinthemicrowavebackground,and
9 derive bounds on the parameters of the model so that the predicted spectrum is compatible with
1 the observed anisotropy of the microwave background and with large-scale structure observations.
n We analyze the matter era and the approach of the model to general relativity. The model passes
a all existing constraints.
J
PACS numbers: 98.80.Cq Preprint CERN-TH/96-302, SUSSEX-AST96/10-2, astro-ph/9610183
2
2

2
v I. INTRODUCTION ingalow-densityuniverse,inanattempttomaketheage
3 compatible with globular cluster ages. Most frequently
8 Theinflationaryparadigm[1]notonlyprovidesasolu- a cosmological constant is introduced to restore spatial
1
tiontotheclassicalproblemsofthehotbigbangcosmol- flatness, but open inflation models (see Ref. [10] for an
0
1 ogy,butalsopredictsanalmostscaleinvariantspectrum introduction) offer the alternative of a genuinely open
6 of metric perturbations which could be responsible for universe. Suchmodelsgenericallycontainafieldtrapped
9 the observed anisotropy of the cosmic microwave back- in a false vacuum which tunnels to its true vacuum via
h/ ground (CMB), as well as the origin of the large-scale nucleation of a single bubble, inside which a second pe-
p structure. Presentmicrowavebackgroundanisotropyex- riod of inflation drives the universe to almost flatness.
- periments offer only weak constraints; for example the This way one solves the homogeneity problem indepen-
o COBE satellite [2] gives a very accurate determination dently from the flatness problem, allowing for an open
r
t of the amplitude of large-angle anisotropies (which can homogeneous universe inside the bubble.
as beusedtonormalizetheories)butonlyweaklyconstrains In an open universe, the analysis of density pertur-
: the shape of the spectrum. Information is beginning to bations andmicrowaveanisotropiesis considerablymore
v
comeinondegreescales,andfromcombiningmicrowave complicatedthanintheusualflatspacecase. Earlystud-
i
X anisotropy constraints with those from large-scale struc- ies by Lyth and Stewart [11] and by Ratra and Peebles
r ture, but the present situation still offers considerable [12] evaluated the spectrum for slow-roll models leading
a freedom. However, we can expect this to change dra- to an open Universe, using the conformal vacuum as an
matically in the near future, especially with the launch initial condition. In the single-bubble models a differ-
of new-generation microwave anisotropy satellites MAP ent vacuum choice is appropriate, leading to a slightly
[3]andCOBRAS/SAMBA [4]whichpromise tomeasure different spectrum [13,7]. It was later realised however
both cosmological parameters such as Ω , H and Ω , thatextraperturbations,withdiscretewavenumbers,can
0 0 B
and parameters associated with the primordial spectra also be generated. In all, three different types of pertur-
to great accuracy [5]. It is therefore desirable to provide bation have been identified: a continuous spectrum of
avarietyofinflationarymodelswithdefinite predictions, modes with wavenumber k greater than the curvature
which could be used to test and exclude them. scale, known as sub-curvature modes; a super-curvature
Until recently, inflation was always associated with a mode associatedwith the open de Sitter vacuum [13,14],
flat universe, due to its ability to drive the spatial cur- and a mode associated with perturbations in the bubble
vature so effectively to zero. However, it is now un- wallattunneling[15–17]. Theobservedlarge-scalestruc-
derstood that inflation comprises a wider class of mod- turesareduetothesub-curvaturemodes,butlarge-angle
els, some of which may give rise to an open universe at microwave anisotropies are generated by all three types,
present [6–8]. Observations suggesting a high value of with observations seeing the combined total anisotropy.
the Hubble parameter, such as those using the Hubble Thefirstcomputationofallthreetypesofmodetogether
SpaceTelescope[9],havemotivatedthe ideaofconsider- foraparticularmodelwasmadeinRef.[15]forarbitrary
1
Ω in the context of the two-field models of Linde and T ;ν =0, (5)
0 µν
Mezhlumian [8]. Later on, a thorough calculation of all
three contributions from the point of view of quantum even in the presence of a potential for the scalar field.
field theory in open de Sitter space was carried out by Substituting R into the equation of motion of the scalar
Yamamoto et al. [17] (see also Ref. [18]). Here we shall field, we obtain
carry out a similar calculation for a different two-field
1
model, for which we shall also discuss some of the impli- (1+6ξ) 2ϕ2 =4V(ϕ) ϕV′(ϕ)+Tλ . (6)
cations of large-scale structure observations. 2 ∇ − λ
In addition to scalar metric perturbations, we expect
We will consider a potential of the type [21,20]
openinflationtoleadtotheproductionofagravitational
wave spectrum, as in conventional inflationary models. λ
V(ϕ)= (ϕ2 ν2)2. (7)
Unfortunately, no-one has yet formulated a method of
8 −
calculating this spectrum evenapproximately,and so we
shall not be able to consider them here. In chaotic in- DuringaradiationdominatederaTλ =0andthescalar
λ
flation models, gravitational waves are negligible in the field will sit at its minimum; matching the present-day
slow-rolllimit (see e.g. Ref. [19]); one can hope that this Planck mass demands
is also true in the open inflation case, but that remains
to be confirmed. 8πξν2 =m2 . (8)
Pl
Theparticularopeninflationmodelweshallstudy,in-
troduced in Ref. [20], is based on the induced gravity During a matter era,the scalarfield oscillatesaroundits
Lagrangian [21]. The interest of this model is the rel- minimum with a large frequency and negligible ampli-
atively small number of parameters, which can be con- tude, passing all the tests associated with an oscillating
strainedbyseveraldifferenttypesofobservation. Thein- gravitationalcoupling[25]aswewillshowinSectionVII.
flatonis a dilatonfield, whosevacuum expectation value
at the end of inflation determines the present value of
III. INDUCED GRAVITY OPEN INFLATION
the gravitational constant. We will constrain the model
fromCMBandlarge-scalestructureobservations,aswell
as ensuring that post-Newtonianand oscillating gravita- In the induced gravity open inflation model [20], the
tional coupling bounds are satisfied. initial period of inflation is driven by the false vacuum
energy of a second scalar field σ. This energy density is
abletoholdthedilatonatafixedlocationdisplacedfrom
II. THE MODEL the minimum of its potential. After the false vacuum
decays, the rolling of the dilaton to its minimum drives
We consider the model of Ref. [20], with an induced the secondperiodofinflationnecessaryto giveΩ0 inthe
gravity Lagrangian[21] desired range.
1 1
= ξϕ2R ∂ ϕ∂µϕ+V(ϕ)+ . (1)
µ mat
L 2 − 2 L A. False vacuum inflation
The dilatonfield ϕdetermines the effective gravitational
coupling, which is positive for ξ > 0. In the absence of Initially the scalar field σ is in its false vacuum. The
a potential, this action corresponds to the usual Brans– details of its potential are not particularly important;
Dicke action [22], where we will parametrize them later. In the false vacuum,
the universe expands driving the spatial curvature and
1
Φ=8πξϕ2 , ω = . (2) any previous inhomogeneities to zero. Later on, the σ
4ξ
field tunnels to its true minimum at V(σ) = 0, via the
The Einstein and scalar field equations are [23,24] production of a bubble.
Theequationsofmotionofthedilatonfieldbeforeand
ξϕ2Gµν =gµνV(ϕ)+ξ( µ ν gµν 2)ϕ2 after the tunneling can be written as [20]
− ∇ ∇ − ∇
1
+ ∂µϕ∂νϕ− 2gµν(∂ϕ)2 +Tµν, (3) H2+2Hϕ˙ + K = 1 1ϕ˙2+V(ϕ)+V(σ) , (9)
(cid:16) (cid:17) ϕ a2 3ξϕ2 2
and (cid:20) (cid:21)
ϕ˙2 4V(ϕ) ϕV′(ϕ)+4V(σ)
2ϕ= V′(ϕ)+ξϕR, (4) ϕ¨+3Hϕ˙ + = − . (10)
∇ − ϕ (1+6ξ)ϕ
where G is the Einstein tensor. Using the identities
µν
G ;ν = 0 and R νΦ = ( 2Φ) 2( Φ), to- Here V(σ)=V0 in the false vacuum and vanishes in the
µν µν∇ ∇µ ∇ − ∇ ∇µ true vacuum, while the curvature K is effectively zero
gether with the ϕ equation of motion, we find that the
before tunnelling and is negative afterwards. The basis
energy-momentum tensor is conserved,
2
for the open inflation model is the existence of a stable Theminimumofthepotentialisnowatϕ=ν <ϕ ,and
st
static solution in the false vacuum [20], with the field slow-rolls from ϕ driving a second stage of in-
st
flation. The dynamics ofthis situation wereinvestigated
ϕ2 =ν2 1+ 8V0 ν2(1+α), (11) longagoinRef. [27]. The rateofexpansionandeffective
st λν4 ≡ massinthetruevacuumimmediatelyaftertunnelingare
(cid:16) (cid:17)
8πV
Hs2t = 3m2P0l . (12) HT2 = 83πmU2T − aK2 = λ24νξ2 1+αα 2− aK2 , (23)
Pl (cid:18) (cid:19)
ItsstabilityisbestseenintheEinsteinframe[20]. Under λν2 1 α
the transformation m2T ≡UT′′(ϕst)= 1+6ξ (1+−α)2 . (24)
ν ν
dt= dt˜, a(t)= a˜(t˜), (13)
ϕ ϕ Thecurvaturetermquicklybecomesnegligibleasthesec-
ond phase of inflation progresses.
φ ϕ
=(1+6ξ)1/2ln , (14) Forlateruse,wealsodefinethe usualslow-rollparam-
ν ν eters [19] soon after tunneling
the effective potential in the false vacuum becomes
m2 U′(φ) 2 8ξ 1
ǫ Pl T = , (25)
λν4 ν2 ν4 ≡ 16π U (φ) 1+6ξ α2
U (φ)= 1 2 +(1+α) , (15) (cid:18) T (cid:19)
F 8 (cid:20) − ϕ2 ϕ4(cid:21) m2 U′′(φ) 8ξ 1 α
η Pl T = − . (26)
λν3 ν2 ν2 ≡ 8π U (φ) 1+6ξ α2
U′(φ)= 1 (1+α) , (16) T
F 2(1+6ξ)1/2 ϕ2 − ϕ2
(cid:20) (cid:21) Thescalarfieldφslow-rollsdowntheeffectivepotential
λν2 ν2 ν2 given by Eq. (20) until it starts oscillating around its
U′′(φ)= 2(1+α) 1 , (17)
F 1+6ξ ϕ2 ϕ2 − minimumandinflationends. Thevalueofφattheendof
(cid:20) (cid:21) inflationcanbecomputedfromthecondition H˙ H2
− T ≃ T
where primes denote derivatives with respect to the (or equivalently φ˙2 U (φ)), giving
T
Einstein-framescalarfield φ. It is clearthatU′(φ )=0 ≃
F st
atthestaticvalue,whiletheeffectivesquaremassispos- 8ξ
ϕ2 =ν2 1+ ν2(1+β). (27)
itive ensuring stability. At the static point, we have end 1+6ξ ≡
(cid:18) (cid:19)
H2 = 8πUF = λν2 α , (18) Thenumberofe-foldsduringthesecondstageofinflation
F 3m2 24ξ 1+α from φ to φ canbe computed in the Einstein frame,
Pl st end
λν2 1
m2F ≡UF′′(ϕst)= 1+6ξ 1+α, (19) N = 1 φst dφUT(φ)
ξν2 U′(φ)
Zφend T
where H is the rate of expansion of the universe in the
F 1 1+α
Einstein frame and m is the mass of the φ field at the = α β ln . (28)
F β − − 1+β
static point. (cid:20) (cid:18) (cid:19)(cid:21)
In order to produce an open universe, the number of e-
folds after tunneling has to be around N = 60, the pre-
B. True vacuum inflation
cisenumberdependingonthereheatingtemperatureand
otherdetailsofthepost-inflationaryevolution. Weadopt
Eventually the σ field tunnels to its true vacuum by the number 60 for definiteness. This gives a relation be-
nucleatinga bubble, inside whichthe universeinflates to tween the two dimensionless parameters α and β.
almostflatness. Asufficiently low tunneling rateensures
that the bubble stays isolated [26,20]. After tunneling,
the effective potential in the true vacuum, again in the IV. METRIC PERTURBATIONS AND
Einstein frame, becomes TEMPERATURE ANISOTROPIES
λν4 ν2 2 Quantum fluctuations of the inflaton field φ produce
U (φ)= 1 , (20)
T 8 − ϕ2 long-wavelength curvature perturbations; we will use
(cid:18) (cid:19) R
todenotethecurvatureperturbationoncomovinghyper-
λν3 ν2 ν2
U′(φ)= 1 , (21) surfaces (in the Einstein frame).
T 2(1+6ξ)1/2 ϕ2 − ϕ2
(cid:18) (cid:19) Openinflationgeneratesthreedifferenttypesofmodes:
λν2 ν2 ν2 those that cross outside during the second stage of infla-
UT′′(φ)= 1+6ξ ϕ2 2ϕ2 −1 . (22) tionandconstituteacontinuumofsub-curvaturemodes;
(cid:18) (cid:19)
3
adiscretesuper-curvaturemodeassociatedwiththeopen mass during tunneling [17]. The coth(πq) factor can be
deSittervacuum,andamodeassociatedwiththebubble interpretedasdue tothe initialtransientbehaviorasthe
wall fluctuations at tunneling. The mode functions are curvature term dies away.
eigenvalues of the Laplacian, with eigenvalue k2 where Notice that Eq. (34) has only been derived in the case
−
k isthewavenumber. Definingq2 =k2 1,thenthesub- of perfect de Sitter expansion after tunneling [17]. It
curvature modes have positive q2 and−the other modes seems very plausible that it also holds when there are
have negative q2. We label the former mode functions deviationsfromde Sitter,where the righthandside isto
Π (r) and the latter Π¯ (r). In Appendix A we give be evaluated when k = aH. This is the only simple for-
ql |q|l
explicit forms for these; see also Ref. [28]. mulawhichreducestothecorrectresultbothfordeSitter
Thespectrum (q)ofthecurvatureperturbationcan expansion and in the flat-space limit (see e.g. Ref. [31])
R
P
be defined from the mode expansion of by [14] whichmustbeattainedafterthecurvaturetermhasdied
R
away sufficiently.
hRqlmRq′l′m′i= q(q22π+2 1)PR(q)δ(q−q′)δll′δmm′. (29) theFoslploecwtirnugmnocarmusaeldpbrayctthicee,tiwmeedveasrciraitbieonthoefvHaraiantdioǫnbiny
apower-law. Noticethatthispower-lawissuperimposed
In order to compare with observations, we must com-
onthe cothbehavior,sothe complete spectrumdoesnot
pute the effect that such a perturbation has on the tem-
have a power-law form on very large scales. The power-
peratureoftheCMB,expandedasusualinsphericalhar-
lawindex of (q)/coth(πq)canbe derivedinthe usual
R
monics P
way from the slow-roll parameters as [19]
∆T
(θ,φ)= a Yl(θ,φ). (30) 8ξ 2(2+α)
T lm m n−1=−6ǫ+2η =−1+6ξ α2 , (35)
lm
X
andgivesthestandardresultintheflat-spacelimitwhere
The main contribution on large scales comes from the
the coth term in Eq. (34) equals unity. This expression
Sachs–Wolfe effect [29]. The complete angular power
is valid provided both ǫ and η are much less than one.
spectrum Cl alm 2 has contributions from the | |
≡ h| | i For later comparison, we write the spectrum as
continuum of sub-curvature modes, the super-curvature
mode and the bubble-wall mode, (q)=A2 coth(πq) 1+q2 (n−1)/2 , (36)
PR C
C =C(C)+C(S)+C(W). (31) where (cid:2) (cid:3)
l l l l
8U (φ )
The contribution of each mode to the C is measured A2 = T st (37)
l C 3ǫ(φ )m4
by a window function W , given by [29] st Pl
ql
isameasureoftheamplitudeattheq =0limit. Formally
η0 thespectrumdivergesthere,thoughnotinaharmfulway
5W =Π (η )+6 drF′(η r)Π (r), (32)
ql ql 0 0− ql thanks to the window function given by Eq. (32).
Z0
For our model, Eq. (37) becomes
for the sub-curvature modes; the same expression with
Π¯ gives the window function W¯ for the negative q2 λ 1+6ξ α2 2
|q|l |q|l A2 = . (38)
modes. Here C (16πξ)2 6ξ 1+α
(cid:18) (cid:19)
sinh2η 3ηsinhη+4(coshη 1) Theangularpowerspectrumforthe continuummodes
F(η)=5 − (coshη 1)3 − , (33) can be written as [13]
−
∞ dq
gives the growth rate of perturbations during the mat- C(C) =2π2 (q)W2 . (39)
ter era [30], and η0 = cosh−1(2/Ω0 1) is the distance l Z0 q(1+q2)PR ql
−
to the last scattering surface. The normalization of the We compute the angular power spectrum for different
contribution to the C is given in the expressions below. values of Ω in the low-density range 0.2 Ω 0.6. In
l 0 0
≤ ≤
Fig. 1 we show the first twelve multipoles, adopting the
notation D =l(l+1)C .∗
l l
Sub-curvature modes
A detailed computation of the amplitude of the sub-
curvature modes gives the result [13,7,17] ∗Our calculation only includes the Sachs–Wolfe effect, as
is appropriate for computing the amplitude on the largest
8UT angular scales. We therefore don’t include the rise to the
(q)=coth(πq) , (34)
PR 3ǫm4 acoustic peak, caused by the first oscillation of the photon–
Pl
baryon fluid, which is known to induce an effective extra tilt
where ǫ is the slow-rollparameter defined earlier and we of around 0.15 [32]; see for example Fig. 8 in Ref. [33] (who
havedroppedasmallcorrectiontermfromthe changein only consider sub-curvaturemodes).
4
FIG.1. Thefirst12multipolesoftheangularpowerspec- FIG. 2. As Fig. 1, but for the discrete supercurvature
trumassociatedwiththecontinuummodes,normalizedtothe mode,showing Ω0 =0.6,0.5,0.4,0.3,0.2, reading from top to
tenthmultipole,forΩ0 =0.2,0.3,0.4,0.5,0.6asreadfromtop bottom at in thecentre of the figure.
to bottom at low l.
mode in its spectrum, since its mass at the false vacuum
The normalization to COBE for tilted open models shouldbemuchlargerthantherateofexpansioninorder
has recently been given by Bunn and White [33], under topreventtunnelingviatheHawking-Mossinstanton[8].
the assumption that only the continuum modes are im- The wavenumber associated with this mode is given by
portant. They specify a quantity δ , which measures
H
thenormalizationofthe presentmatterpowerspectrum. 9 m2 1/2 1 2
The preferred value depends on n and Ω ; however our k2 =1 F . (42)
model is nearly scale-invariant and the d0ependence on −"(cid:18)4 − HF2(cid:19) − 2#
Ω is quite weak and can be ignored at the accuracy we
0 The amplitude of this mode is [18]
are working. Therefore we take the value δ =2 10−5
H
×
regardlessofΩ0. InanopenuniverseδH isrelatedtoPR A2 8UF =A2 UF . (43)
as [33] S ≃ 3ǫm4 C U
Pl T
δH = 25PR1/2 g(ΩΩ00), (40) fwohrmeruelathfeornotrhmeaalinzgautiloanr opfowAe2Sr isspdecetfirnuemd tohfroteumghpetrhae-
ture anisotropies induced by this super-curvature mode,
where g(Ω ) is a function measuring the suppression in
0 namely [17,18]
the growth perturbations relative to a critical-density
universe, and R is evaluated at around the 10th multi- D(S) l(l+1)C(S) =4πA2W¯2 . (44)
pole where cotPh(πq) 1. The Ω dependence can give a l ≡ l S 1l
0
≃
factorofupto1.5intheregionofinterest,butwecanig- Fig. 2 shows the first twelve multipoles, showing quite
nore it as we do not require such accuracy. Reproducing a complicated dependence. For example, it is not auto-
the amplitude of temperature anisotropies is the main matic that the quadrupole receives the biggest contribu-
constraint on the parameters of the model, and yields tion [34].
More important than the shape is the amplitude of
ξ3 1/2 1+α theseanisotropiesrelativetothesub-curvatureones. We
√λ=6 10−3 , (41)
× 1+6ξ α2 will compare their contributions in Section V.
(cid:18) (cid:19)
asfoundinRef.[20]. Thisrelationcanreadilybesatisfied
for reasonable values of the parameters [20]. Bubble wall mode
In addition to the sub- and super-curvature modes,
Super-curvature mode there is a contribution from the bubble wall fluctua-
tions. Thesefluctuationscontributeasatransversetrace-
We now consider the contribution to the CMB aniso- less curvature perturbation mode with k2 = 3, see
−
tropies coming from the discrete super-curvature mode Refs. [15–17], which still behaves as a homogeneous ran-
associatedwiththedilatonfieldφ. Thismodeappearsin dom field [35,28].
theopendeSitterspectrumwhenm2 <2H2 inthefalse Unlike the modes we’ve discussed so far, these modes
F F
vacuum [13]. The tunneling field σ does not have this needextraparametersfortheirdescription,becausetheir
5
4U 3ǫ
A2 = T =A2 , (50)
W m4 C 2
Pl
which is much smaller than the amplitude of the contin-
uum modes.
The angular power spectrum associatedwith the bub-
ble wall mode is [15]
4πA2
D(W) l(l+1)C(W) = W W¯2 . (51)
l ≡ l (l+2)(l 1) 2l
−
Fig.3 showsthe first twelve multipoles. The quadrupole
has the largest amplitude for all Ω .
0
We willcomparetheir contributionto the CMB inthe
next Section.
FIG.3. As Fig. 1, but for the discrete bubblewall mode,
for Ω0 =0.6,0.5,0.4,0.3,0.2, from top to bottom at low l.
V. COMPARISON WITH OBSERVATIONS
amplitude depends on the details of the bubble wall,
A. Microwave background anisotropies
whichisdeterminedbythepotentialfortheσ field. This
extra freedom allows the bubble wall fluctuations to be
We can now examine constraints on the shape of the
tuned relative to the others.
combined spectrum. The COBE data alone do not offer
The perturbation amplitude for the bubble wall mode
particularly strong constraints in this respect; for exam-
is given by [15,36,17]
ple, although Yamamoto and Bunn [37] argued that the
4U 1/2 inclusion of super-curvaturemodes could harmthe fit to
A2W = a2bmT4 a2+(1+a2b)2 , (45) COBE based on the two-year data, a recent comprehen-
Pl h i sive analysis of the four-year data by G´orski et al. [38]
where finds no useful constraint. Those papers however dis-
cussed only a particular model for the super-curvature
a2 = 24πUTS12 , b= UF−UT . (46) modes and didn’t include the bubble-wall modes at all.
m2 (U U )2 4U As Sasaki and Tanaka discussed [18], there can be in-
Pl F− T T
teresting constraints if the super-curvature modes have
and S1 gives the bubble wall contribution to the bounce their amplitude enhanced, and we shall also se that the
action, Bwall = 2π2R3S1 (see e.g. Ref. [15]). In order bubblewallmodesaretypicallymoreimportantthanthe
to compute S1, we will consider a symmetry breaking super-curvature ones.
potential of the type Unfortunately, the complicated structure of the per-
turbationspectrainopeninflationmodelsmeansthatfor
γ σ 4
U(σ)=U + σ2(σ σ )2 µU , (47) a full analysis each model would have to be confronted
F 0 0
4 − − σ0 with the COBE data set on a case-by-case basis. Such
(cid:16) (cid:17)
an analysis is outside the scope of this paper. We shall
whereσ =M 2/γ correspondstothetruevacuumand
0 adopt a more simplistic approach, which is to demand
U =M4/16γ is the value of potential at the maximum.
0 thatthe discrete modes do notdominate the quadrupole
p
With µ 1 for the thin-wall approximation to be valid,
whilecontributingnegligiblytothetenthmultipole. This
≪
S can be computed as [15]
1 wouldgiveanunacceptableshapetothespectrumonthe
scales sampled by COBE.
σ0 M3
S = dσ[2(U(σ) U )]1/2 . (48) InFig.4,weshowthecontributionstothequadrupole
1 F
Z0 − ≃ 3γ andtothetenthmultipole,asfunctionsofΩ0,normalized
to the size of the corresponding metric perturbation. It
In the limit a2b 1 of small gravitational effects at
≪ does not require much effort to keep the contribution to
tunneling, we recover the result of Ref. [8], namely
the tenth multipole from the discrete modes low (unless
Ω isverysmall),butwemustensurethatthequadrupole
2U (U U ) 3ǫ 0
A2 = T F− T =A2 , (49) is not dominated by the discrete modes.
W πm2 S2 C 2a2b
Pl 1 Consideringfirstthesuper-curvaturemodes,acrossthe
whole range of interesting Ω we find that the super-
where ǫ is the slow-roll parameter. However, in the op- 0
curvature mode contributes about a factor thirty less
posite limit of strong gravitational effects, a2b 1, we
≫ to the low multipoles than do the continuum modes,
have [15]
6
tributing substantially even to the tenth multipole, and
this will force down the normalization ot COBE and
make it impossible to fit large-scale structure observa-
tions. We can therefore adopt the above constraint even
at low Ω values. This imposes only a very mild con-
0
straint on the parameters of the model, namely
α>0.01, (53)
whichiseasytosatisfyaswewillseeinSectionVI. Since
the value of Ω at present is not known with any rea-
0
sonable accuracy, it would be excessive to give a precise
Ω -dependent constraint. However,in the future, with a
0
narrow range of values for Ω , one would be able to give
0
amorerefinedconstraintontheparametersofthemodel
from a full comparison of the detailed spectrum against
COBE or its successors.
The same arguments can be applied to the bubble-
wall modes; again we must prevent the domination of
the quadrupole by these modes. For the interesting Ω
0
values(i.e.thosenottooclosetoone),weseefromFig.4
that this simply requires
A2 <A2 , (54)
W C
∼
again independent of Ω in the interesting range. This
0
againimposes onlya verymild constraintonthe param-
eters of the model. For a2b 1, we have 3ǫ < 2a2b
≪
giving
1+6ξ M3
FIG.4. Themultipolesassociatedwitheachofthemodes, γ <9 104 α5/2, (55)
normalizedtothecorrespondingmetricperturbation(A2C,A2S × 6ξ m3Pl
or A2W as appropriate), as a function of Ω0. The top panel
shows the quadrupole, while the lower shows the tenth mul- which is easy to satisfy for sufficiently large M. For
tipole. a2b 1, the amplitude of the metric perturbation
≫
Eq. (50) is completely negligible since ǫ 1, see Sec-
≪
tion VI.
for the same size of metric perturbation.† Unless Ω is
0
very small, then if the super-curvature mode contribu-
tion is comparable to that of the continuum modes for B. Implications for large-scale structure
the quadrupole, then it is negligible at the tenth multi-
pole. Conservatively then, we shall require
We can now combine the COBE normalization with
observations of large-scale structure. This has the ad-
A2 <100A2 , (52)
S C vantage of being relatively insensitive to the inclusion
∼
of super-curvature or bubble-wall modes, because with
independently of Ω , to prevent the super-curvature
0
the constraints above these modes affect only the low-
modes from dominating the low multipoles. In principle
est multipoles, and while these can influence whether
this limit could become inappropriate at small enough
or not the spectral shape is a good fit to observations,
Ω , because then the shape of the super-curvature spec-
0
they are not very significant at the higher multipoles
trum is not so steep as to be ruled out by observations.
(around the tenth to fifteenth) which are most impor-
However, by then the super-curvature modes are con-
tant for determining the normalization. The drawback
thoughisthatlookingtolarge-scalestructureintroduces
a dependence on all the other cosmological parameters,
namely the Hubble parameter h, the baryon density Ω
†AlthoughthereisadiparoundΩ0 ≃0.4causedbyan‘acci- andthenatureofthedarkmatter. Despitethis,interestB-
dental’cancellation between intrinsicand line-ofsight terms,
ing constraints can still be found, and two analyses have
thedipisnotatthesamelocationforthel=3,4...multipoles
appeared which discuss tilted open models — Liddle et
so the dip doesn’t allow one to weaken the constraint in its
al. [39] who used the two-year COBE data and, more
neighborhood [34].
7
recently, White and Silk [40] who used an accurate nor- λ 2.5 10−5 6ξ3 4 10−14. (58)
malization to the four-year COBE data. Both of these ≃ × 1+6ξ ≃ ×
considered only cold dark matter; other choices tend to
strengthen the constraints so we shall do likewise. See Ref. [20]for a range of values, under the assumption
In the model we are considering, the spectral index is ξ 1.
≪
always tilted to n less than one, as seen from Eq. (35). WenowconsiderthedeSittervacuumsuper-curvature
Whether or not this is allowed depends quite sensitively mode. Thismodeexists,sincem2F <2HF2 forξ ≪1from
on both Ω and h. If Ω is too low, below around 0.30, Eqs. (18) and (19). The amplitude of curvature pertur-
0 0
thentheopencolddarkmattermodelfaresbadlyagainst bations that give rise to temperature anisotropies in the
observations; this conclusion is consistent with a similar CMB is constrainedby Eq.(52), which imposes the con-
constraint from velocity flows [41] which is independent dition Eq. (53). This is satisfied as long as α > 0.01.
ofthepowerspectrum. Forexample,WhiteandSilk[40] Since we are considering values of α 1, the contri-
≃
findthatthis valueisallowedonlyifthe powerspectrum bution from the super-curvature mode will be negligible
is‘blue’,withnatleast1.10. Thisconclusionisenforced comparedto that ofthe sub-curvaturemodes, regardless
both by the cluster abundance and by the shape of the of Ω0.‡
galaxy correlationfunction. Our model will therefore be Considernowthebubblewallmodecontributiontothe
ruled out if it turns out that the universe is indeed as CMBanisotropies. Therearetwopossibilities,depending
open as this. on the relative strength of the gravitational effects at
However, one does not have to increase Ω by very tunneling [15]. For a2b 1 we are in the weak gravity
0 ≪
much to radically change this conclusion. For Ω = 0.5, regime of Ref. [8], and the condition on the parameters
0
for example, White and Silk find viable models for n becomes, from Eq. (55),
as low as 0.85, with the preferred value depending on
1+6ξ M3
the Hubble parameter h. Our model can therefore be γ <105 . (59)
comfortablycompatiblewiththedataforthisΩ0,andat 6ξ m3Pl
least marginally compatible for Ω as low as 0.4.
0 For M 10−3m this gives γ < 0.03, which is a rea-
Pl
≃
sonable bound on the coupling γ. On the other hand,
for a2b 1, condition Eq. (54) is easily satisfied for
VI. TWO POSSIBLE SCENARIOS
≫
ǫ 1/200,see Eq. (50).
≃
In this section we will explore two different scenarios.
B. ξ 1
≫
A. ξ 1
≪ This case was considered in Ref. [42]. The dilaton ex-
pectationvalueνismuchsmallerthanthepresentPlanck
This case was considered in Ref. [20]. Here the dila-
mass. For ξ 1, we have β 4/3 and the value of α
ton expectation value ν is much larger than the present ≫ ≃
is now determined from the required number of e-folds
Planckmass,seeEq.(8). Fordefiniteness,wewillchoose
Eq. (28),
a particular value, 8ξ = 1/200. From the required num-
ber of e-folds, Eq. (28), this determines α to be of order 4N 4 1+α
one. α ln , (60)
3 ≃ − 3 − 1+4/3
Let us study now the contribution of the different (cid:16) (cid:17)
modes to the CMB anisotropies. We first consider the whichgivesα 85. This is aregimequite different from
≃
continuum spectrum of sub-curvature modes. The slow- the previous case.
roll parameters, given by Eqs. (25) and (26), become We first consider the continuum spectrum of sub-
curvature modes. The slow-rollparameters are
1
ǫ , η 0, (56)
≃ 200 ≃ ǫ 1.85 10−4, η 0.016, (61)
≃ × ≃−
which determine the tilt of the primordial spectrum of which determine the tilt of the primordial spectrum of
density perturbations via Eq. (35) as density perturbations via Eq. (35) as
n 1 6ǫ 0.03, (57) n 1 2η 0.032. (62)
− ≃− ≃− − ≃ ≃−
which is compatible with large- and small-scale observa-
tionsforΩ >0.4,accordingtoRef.[40]. The constraint
0
on the ampl∼itude of the angular power spectrum deter-
‡Inthelimitofsmallαandβ,thee-foldingsrelationEq.(28)
mines the value of λ via Eq. (41) as
gives α 10√β, so the super-curvatureconstraint will even-
tually b≃ecome important once ξ<10−7.
∼
8
Thisisverysimilartothepreviouscaseandthuscompat- λν4, the fractional shift in the gravitational constant at
iblewithlarge-andsmall-scaleobservationsforΩ >0.4. this static point is tiny, and so too is the energy density
0
However,the constrainton the amplitude of the an∼gular associated with the potential, which contributes only a
power spectrum now determines the value of the combi- minute fraction(perhaps 10−100!) of the criticaldensity.
nation λ/ξ2 rather than λ alone, as We have analyzed the detailed behavior, described in
Appendix B. When matter domination starts, the field
λ 9 10−10. (63) risesfromitsminimumtooscillateaboutthestaticpoint,
ξ2 ≃ × which it does on a very rapid timescale. As ρ de-
creases,thestaticpointmovestowardsthetrueminimum
If we choose λ 1, we have ξ 3 104, which gives a (with the oscillation amplitude also decreasing though
∼ ∼ ×
very reasonable expectation value for the dilaton, from rather more slowly). At all times, the oscillations are
Eq. (8), of of such small amplitude that general relativity holds to
extremely high accuracy.
ν 10−3m 1016GeV. (64)
Pl
≃ ≃
WenowconsiderthedeSittervacuumsuper-curvature
VIII. CONCLUSIONS
mode. Thismodealsoexistsinthiscase,sincem2 <2H2
F F
for α 1 from Eqs. (18) and (19). The amplitude of
curvat≫ureperturbationsisconstrainedbyEq.(52),which Inthispaperwehaveanalyzedavarietyofphenomeno-
imposes the condition Eq. (53). Since we have α 1, logicalconstraints on a recently proposed model of open
the contribution from this super-curvature mode wi≫ll be inflation in the context of induced gravity theories [20].
negligible. The most stringent constraints come from observations
of the temperature anisotropies in the microwave back-
Concerning the bubble wall mode contribution, again
there are two possibilities. In the weak gravity regime ground. The model predicts a matter power spectrum
a2b 1, the condition on the parameters becomes tiltedto n<1,whichwill be incompatible withobserva-
≪ tions if the universe turns out to have Ω < 0.4. Other-
0
M3 wise,itispossibletochoosetheparameters∼ofthemodel
γ <105 α5/2. (65)
m3 so that it is in agreement with observations.
Pl
During the matter era, the large dilaton mass and the
which gives a trivial constraint of γ < 7 for M extremely small amplitude of oscillationsaroundits vac-
10−3m . Fora2b 1,theconditioninE∼q.(54)iseasil≃y uumexpectationvalueensurethatthetheoryapproaches
Pl
satisfied for ǫ 10≫−4. general relativity very efficiently, passing all the post-
≃ Newtonian and oscillating gravitationalcoupling tests.
Final note: We commented in the introduction that
VII. MATTER ERA no method had been formulated to compute the gravi-
tational wave spectrum, which we therefore did not con-
sider. Aswewererevisingforthefinalversionofthispa-
One of the remaining issues is to make sure that af-
per, preprints appeared [44] making significant progress
ter inflation the scalar field ϕ remains close to the mini-
inthisdirection. Itwillbeinterestingtoapplythesenew
mum of its potential. Deviations from this would result
resultstospecificopeninflationmodelsincludingtheone
in time variations of the gravitational constant, which
discussed in this paper.
are strongly constrained [43]. During the radiation era
the scalar field will remain at the minimum due to the
vanishing trace of the energy momentum tensor, as seen
ACKNOWLEDGMENTS
fromEq.(6). However,duringthemattererathedilaton
couples to the matter fluid and thus will be subject to a
force which shifts the field from its minimum. J.G.B.wassupportedinpartbyPPARCandA.R.L.by
However,itiseasytoshowthatthis effectistiny. The the Royal Society. We thank Anne Green, Andrei Linde
relevant equation, from Eq. (6), is and Martin White for useful discussions. J.G.B. thanks
StanfordUniversityforits hospitalityduring partofthis
ϕ˙2 4V(ϕ) ϕV′(ϕ)+ρ work, with the visit funded by NATO Collaborative Re-
ϕ¨+3Hϕ˙ + = − . (66)
ϕ (1+6ξ)ϕ search Grant Ref. CRG.950760.
For a given ρ, there is a static solution at
APPENDIX A: OPEN UNIVERSE MODE
ϕ2 =ν2 1+ 2ρ . (67) FUNCTIONS
st λν4
(cid:18) (cid:19)
The open universe mode functions are discussed in
Since the matter-era energy density is tiny in compari-
Refs. [14,28]
son to the inflationary energy density which determines
9
Sub-curvature modes APPENDIX B: MATTER ERA OSCILLATIONS
OF THE GRAVITATIONAL COUPLING
The sub-curvature modes can be written as [45,14]
Here we carry out a detailed analysis of the evolution
Π (r)=N Π˜ (r), (A1)
ql ql ql ofthedilatonduringtheradiationandmattereras. Here
weshallassumethattheoscillationsaredampedonlyby
with
the Hubble expansion and not by any particle decays –
l
2 2 if such decays were present the general relativistic limit
N = (n2+q2)−1/2, N = , (A2)
ql π q0 π would be even more quickly approached.
r nY=1 r The energy-momentum tensor conservation Eq. (5) in
where the unnormalized modes Π˜ (r) can be generated theJordanframeensuresρa3(1+w) =constantduringthe
ql
from the first two radiation (w = 1/3) and matter (w = 0) eras. In order
tostudythecosmologicalevolutionduringtheseeras,let
sinqr
Π˜ (r)= , (A3) us redefine our variables as
q0
sinhr
cothr sinqr qcosqr ϕ2
Π˜ (r)= − , (A4) u= 1, z =mt, (B1)
q1 sinhr ν2 −
through the recurrence relation where m is given by
Π˜ql(r)=(2l 1) cothrΠ˜q,l−1(r) λν2
− m2 = . (B2)
[(l 1)2+q2]Π˜ (r). (A5) 1+6ξ
q,l−2
− −
TheϕequationofmotionEq.(6)andtheFriedmanequa-
tion become
Super-curvature modes
a′ 2(ρ 3p)
u′′+3 u′+u= − , (B3)
a λν4
The first (l 1) multipoles are [34]
≥ a′ u′ 2 1+6ξ
1 r 2 + =
Π¯ (r)= cothr , (A6) a 1+u 6ξ ×
11 2 − sinh2r (cid:20) (cid:21)
h i u′ 2 u2 8ρ
Π¯12(r)= 21 1+ 3(1−sinrhc2orthr) . (A7) "(cid:18)1+u(cid:19) + 1+u + λν4(1+u)# , (B4)
h i
The rest can be obtained with the recurrence relation where primes denote derivatives w.r.t. z. During the
2l 1 radiation era, the right hand side of Eq. (B3) vanishes
Π¯1l(r)= − cothrΠ¯1,l−1(r) and u = u′ = 0 is a stable fixed point. Very soon one
l 1
− can neglect the u-terms in the Friedman equation, and
l
Π¯ (r). (A8) we find the radiation era attractor, a′/a = 1/2z. The
1,l−2
−l 1 scalarfield equationofmotion, u′′+3u′/2z+u=0, has
−
an exact solution,
Bubble wall modes z1/4u(z)=c1J1(z)+c2Y1(z), (B5)
4 4
where J,Y are Bessel functions. Its amplitude decays
The first (l 2) multipoles are [15]
≥ asympt{otica}lly as u(z) z−3/4, so we expect the matter
Π¯ (r)= sinh4r−8sinh2r+12r, (A9) era to start with initial∝conditions at u=u′ =0.
22 4sinh3r During the matter era,u=u′ =0 is a spiral attractor
Π¯ (r)= (A10) and we can always neglect the u-terms in the Friedman
23
equation,
sinh5r 15sinh3r 80sinhr+120rcoshr
− − .
8sinh4r a′ 2 1+6ξ A 4
= , (B6)
The rest can be obtained from the recurrence relation a ≃ 6ξ a3 9z2
(cid:18) (cid:19) (cid:16) (cid:17)
2l 1
Π¯ (r)= − cothrΠ¯ (r) where A = 2ρa3/λν4 is a constant of order 10−120 in
2l 2,l−1
l 2
− Planck units. The equation of motion for u becomes
l+1
Π¯ (r). (A11)
2,l−2 2 A K
−l 2 u′′+ u′+u= = , (B7)
− z a3 z2
10