Multibrane Inflation and Dynamical Flattening of the Inflaton Potential

0.52 MB·English

by  James M. Cline

Multibrane Inflation and Dynamical Flattening of the Inflaton Potential
James M. Cline, Horace Stoica
Physics Department, McGill University, 3600 University Street, Montr´eal, Qu´ebec, Canada H3A 2T8
e-mail: [email protected], [email protected]
(Dated: August,2005)
Weinvestigate theproblemoffinetuningofthepotentialin theIKLIMTwarped fluxcompactifi-
cationscenarioforbrane-antibraneinflationinTypeIIBstringtheory. Wearguefortheimportance
of an additional parameter ψ0 (approximated as zero by IKLIMT), namely the position of the an-
tibrane, relative to the equilibrium position of the brane in theabsence of theantibrane. We show
that for a range of values of a particular combination of the K¨ahler modulus, warp factor, and ψ0,
the inflaton potential can be sufficiently flat. We point out a novel mechanism for dynamically
achieving flatness within this part of parameter space: the presence of multiple mobile branes can
lead to a potential which initially has a metastable local minimum, but gradually becomes flat as
someofthebranestunnelout. Eventuallythelocalminimumdisappears andtheremainingbranes
6
slowly roll together, with assisted inflation further enhancing the effective flatness of thepotential.
0
WiththeadditionofK¨ahlerandsuperpotentialcorrections, thismechanismcancompletelyremove
0
thefinetuningproblem of braneinflation, within large regions of parameter space. Themodel can
2
befalsified if futurecosmic microwave backgroundobservations confirm thehint of a large running

n spectral index.
a
J PACSnumbers: 11.25.Wx, 98.80.Cq

9

3 I. INTRODUCTION servation also applies to other models of inflation.
v Inthispaperwereexaminethefine-tuningissue,start-
9 ing within the context of the original IKLIMT potential,
2 The past few yearshave witnessed significantprogress
withoutintroducingadditionalcorrections. Aclosercon-
0 in the building of realistic inflation models within string
sideration of the potential between the brane and an-
8 theory [1]. The most popular approach has used the in-
tibranerevealsthe importanceofa parameter(which we
0 teractionpotentialbetweenbranesandantibranesasthe
5 source of energy driving inflation; hence the interbrane callψ0) whichwas set to a certain value in IKLIMT. It is
0 the distancewhichthe branewilltravelbefore annihilat-
separation plays the role of the inflaton. Early attempts
/ ing with the antibrane in the warped throat, assuming
h suffered from the assumption that moduli could be fixed
that the brane starts somewhere closer to the the inte-
t by fiat, without specifying the stabilization mechanism.
- rior of the Calabi-Yau at an initial position ψ = 0. The
p However,explicitexampleshaveshownthatproperlyfix- ∼
other important parameters governing the shape of the
e ingthemodulicaninterferewiththeflatnesswhichisre-
h inflatonpotentialarethewarpfactorofthethroat,which
quiredfor the inflatonpotential[2](IKLIMT). Flux com-
: we denote as ǫ1/4, and the K¨ahler modulus at its stabi-
v pactifications in type IIB string theory have provided a
lized value, σ. The warp factor is tunable through the
Xi robust way of stabilizing the dilaton and complex struc- ratio of NS-NS and RR fluxes K,M: ǫ1/4 =e−2πK/3gsM
ture moduli of the Calabi-Yau manifold [3], with the
r [3, 4]. The VEV of the K¨ahler modulus σ is determined
a added benefit of providing warped throats [4] which can by the nonperturbative superpotential W = Ae−aσ in-
be useful for tuning the scale of inflation or of the stan-
voked by ref. [5], where a = 2π/N in the case of gluino
dardmodel. Nonperturbativeeffectshavebeenplausibly
condensationinanSU(N)gaugetheorysectorwhichcan
invoked to stabilize the remaining K¨ahler moduli [5, 6].
naturally arise in these compactifications.
This last step, unfortunately, generically spoils what- If the parameter ψ is set to zero as in IKLIMT, then
0
ever flatness of the inflaton potential which was accom- indeed it is necessary to introduce additional sources of
plishedbywarping,andthusrequiresadditionalfinetun- dependence on the inflaton into the potential, in order
ing of parameters. One can introduce extra parameters tocancelthe unwantedcontributionto the inflatonmass
associated with corrections to the superpotential, which coming fromthe K¨ahlermodulus. However,in this work
must be tuned to a part in 100 in order to obtain the wepoint outthatfora rangeofvalues 0.259<b<0.286
minimum amount of 60 e-foldings of inflation [2]. Alter- ofthe combinationb2 2σǫψ−6, it is possible to achieve
natively,itispossibletotuneparametersalreadypresent 60 or more e-foldings o≡f inflat0ion without any additional
within the model but at the level of a part in 1000 [7]. corrections to the potential. Fortunately this restriction
Thefine-tuningproblemoftheIKLIMTmodelhasbeen onb applies onlyto the simplest IKLIMT model. We will
considered by several authors [8, 9]. Shift symmetry of show that the restriction disappears when one considers
the Lagrangianhas been proposed as a possible solution moregeneralexamplesinwhichcorrectionstotheK¨ahler
[10]. It has also been pointed out that if Lyman α con- and superpotentials are taken into account.
straints are viewedwith skepticism, the degree of tuning Another crucial parameter which we consider is the
required is considerably relaxed [11]. Of course, this ob- number of mobile mobiles branes, N, generalizing the
2
simplest case of only a single brane playing the role of antibranes such that the K¨ahler modulus remains sta-
theinflaton. Werefertothispossibilityasmultibranein- bilized and heavy. Thus it does not act like a second
flation. Within the aforementionedrange of b values, we inflatonfield,since itchangesverylittle duringtheslow-
observe a very interesting phenomenon, whereby start- rolling of the brane-antibrane separation ψ.
ingwithasufficiently largenumberN dynamicallyleads
Not all models are suitable for constructing such a
to a sufficiently flat inflaton potential. At large N, the
multi-brane setup. We will give here two examples, one
branesstartoutbeingconfinedinametastableminimum
whichworksandonewhichdoesnot. TheoriginalKKLT
of the potential. But as successive branes tunnel out of
[5] setup does not accommodate a large number of mo-
this minimum, it becomes shallower, until finally all the
bile branes; just one additional antibrane is sufficient to
remaining branes can roll together into the throat. This
lift the vacuum from anti-deSitter (AdS) to Minkowski
is illustrated in figure 1. The potential is often very flat
(or deSitter, dS). The barrier that prevents tha K¨ahler
whenthe branesstartrolling,leading to a longperiodof
modulus from rolling has a height roughly equal to the
inflation. This is a novel, dynamical mechanism for flat-
depth ofthe AdS minimum. Addind a secondbrane lifts
tening the inflaton potential, which appears to be quite
the potential by an amount that is roughly equal to the
specific to brane-antibrane inflation.
height of the barrier,which destroys the localminimum.
Infigure2wetaketheparametersofthepotentialandthe
antibrane tension as given by [5]. One sees that adding
N branes
a second brane results in a very shallow minimum and
three branes destabilizes the K¨ahler modulus. The po-
tential of the K¨ahler modulus is given by
N − 1 branes
.. V = aAe−aσ 1σaAe−aσ+W +Ae−aσ + ND (1)
.. 2σ2 3 0 σ3
(cid:18) (cid:19)
N − n branes
with W = 10−4, A=1, a=0.1 D =3 10−9.
0
− ×
6E−15
FIG.1: Sequenceofinflatonpotentialsduetosuccessivetun-
neling of branestoward theright. 4E−15
N = 3
2E−15
II. WHEN ARE MULTIPLE BRANES N = 2
ALLOWED?
N = 1
0
Beforelookingatthedynamicsofinflationwithstacks 120 160 200 240 280
of branesand antibranes,we consider under what condi- σ
N = 0
tions multiple branes are compatible with the stabiliza-
−2E−15
tion of the extra dimensions, i.e., the K¨ahler modulus,
σ, in supergravity language. For each extra brane we
have to introduce a corresponding antibrane such that
the tadpole cancellation condition is statisfied. Adding
FIG. 2: Potential energy of the K¨ahler modulus in the the
antibraneslifts the localminimum ofthe potentialby an
original KKLT scenario for N = 0,1,2 and 3 antibranes,
additional amount, given by the sum of their tensions, showing there can be at most 2 antibranes without desta-
reduced by the warp factor at their location in the bot- bilizing theK¨ahler modulus.
tom of the Klebanov-Strasslerthroat. There is a critical
number of antibranes that can be added such that the
potential still has a local minimum along the σ direc- The situation is dramatically improved if one uses a
tion and therefore the K¨ahler modulus remains stable. racetrack superpotential as in ref. [12]. The minimum
The stronger the warping, the larger the number of an- is already Minkowski and adding antibranes in a highly
tibranes that can be added. In our analysis we will add warpedthroatlifts the vacuumto dS without destabiliz-
3
ing the K¨ahler modulus. The potential is: where r =(2σ ψ2) [13], T is relatedto the tension τ of
the 3-brane or−antibrane by T =2τ/g4 [5], ǫ is the warp
s
e−2(a+b)σ factor, ψ is the 3-brane position, and ψ is the location
V = (bBeaσ+aAebσ) (2) 0
6σ2 of the antibrane. To keep our work self-contained, this
ND expression is derived in Appendix A, in just the same
Beaσ(3+bσ)+ebσ(A(3+aσ)+3eaσW ) +
× 0 σ3 wayasinIKLIMT exceptfortheomissionoftwoapprox-
imations: (1)wedonotassumethatψ ψ ,and(2)we
(cid:2) (cid:3) 0
where we choose the same parameters as in [12], A = do not Taylor-expandthe term (1+ǫ/(≫ψ ψ )4)−1.
0
1, B = 1.03, a=2π/100, b=2π/99, W = 2 10−4 Let us first comment on point (1) abov−e. In IKLIMT
0
and D =−10−9. With these values the critical n−um×ber of and certain other investigations of brane-antibrane in-
branes turns out to be N =28, illustrated in fig. 3 flation, it was assumed that inflation takes place suffi-
ciently far from the antibrane to justify the approxima-
tion ψ ψ =ψ . Ontheotherhand,theψ parameter
| − 0|∼ 0 0
was kept in [7] and [14]. Physically, it may be impor-
N = 29
tant to allow for nonzero ψ because of the possibility
0
that the ψ-dependence coming from the (2σ ψ2)2 fac-
−
tor in the potential will tend to stabilize the brane at
some point other than the bottom of the warped throat,
in the absence of the brane-antibrane interaction. An
example where this is particularly clear is that of two
symmetrically placed warped throats, each with its own
antibrane [14]. In that case the potential (translated to
1E−13
our notation) was taken to be
N = 28 Tǫ ǫ ǫ
V = 2 (4)
r2 − (ψ ψ )4 − (ψ+ψ )4
(cid:18) − 0 0 (cid:19)
Fromsymmetryconsiderationsitisobviousinthismodel
that there exists a metastable or unstable equilibrium
pointexactlymidwaybetweenthetwothroats,atψ =0,
which cannot coincide with the bottom of either throat.
65 66 67 σ 68 69 70 If we simply remove one of the throats, the result corre-
sponds to our starting point (3).
The presence or absence of the ψ parameter explains
FIG. 3: Using a racetrack superpotential and a throat with 0
differences in the amount of required fine-tuning of the
strongerwarpingonecanaddalargenumber(28inthisexam-
potential found in different investigations. To elucidate
ple) of antibranes while keepingtheK¨ahler modulus stable.
this,recallthattheψ-dependenceinthevolumemodulus
r =(2σ ψ2) is the origin of the η problem pointed out
In the remaining analysis, we will not be concerned −
inIKLIMT—the massofthe inflatonis oforder√V H
with the particular model of K¨ahler modulus stabiliza- ∼
in Planck units, ruining the slow roll condition. It was
tion, beyond the assumption that it admits the required
noticed by [7] that it is possible to tune parameters of
number of additional branes. It will also be found that
a model with a potential similar to that in (3) so that
a large number is not required, so the racetrack model
theslow-rollconditionsaresatisfied,evenwithoutadding
mentioned here is sufficient to establish the existence of
anyextraψ-dependentcorrectionstothesuperpotential,
a viable model.
by varying only the parameters which already appear.
However, the degree of tuning needed was more severe
than in IKLIMT where corrections to the superpotential
III. INFLATON ACTION
wereinvoked. Onewayofunderstandingthisistorealize
thatifψ =0andonlycorrectionsoforderψ2 areadded,
0
Our starting point for the dynamics of inflation is the then the potential automatically has V′(0) = 0, so that
LagrangianforthemobilebranepositionψintheIKLIMT one of the slow roll parameters V′/V is small without
framework. Tosimplifytheanalysis,wewillassumethat any tuning. Then only the η parameter (V′′/V) needs
all moduli (notably the K¨ahler modulus σ) are heavy tobe tuned. Inourpotentialhowever,bothslow-rollpa-
except for the inflaton, and that the 4D effective cos- rameters must be tuned, for generic parameters in the
mological constant vanishes at the end of inflation. The potential. In [7] it was found that 60 e-foldings of infla-
Lagrangiancan be parametrized as tion could be obtained by tuning the warp factor to one
part in 2000, whereas IKLIMT only needed one part in
6σ Tǫ ǫ −1 100 tuning. Although this sounds contrary to the spirit
= ψ˙2 1+ (3)
L r2 − r2 (ψ ψ )4 of the present paper, it was due to the fact that no sys-
(cid:18) − 0 (cid:19)
4
tematic study of the effect of varying ψ was done in [7]; These equations can be solved analytically, as the so-
0
instead only one fixed value was considered. lutionof a cubic equation, but the expressionis notvery
We note that the behavior of the potential as ψ ψ enlightening. Instead we give an approximate descrip-
0
→
is classically correct,since by derivation it is of the form tion, by first considering b as a free parameter, and de-
1/h(ψ)whereh(ψ)satisfies 2h=Cδ(ψ ψ )inthe6D termining as a function of b the critical values a and
0 c
∇ −
Calabi-Yau manifold. Although the full brane position x which satisfy (9). For consistency, we should also re-
c
must be specified by 3 complex ψ coordinates, we are quire that b2 a; otherwise the potential blows up at
≥
keeping the angular ones fixed and following only the x = b/√a < 1, before the brane reaches the antibrane.
radial coordinate for the brane-antibrane separation. Therangeofvalueswherea >0(obviouslyalsorequired
c
We alsocommentonourchoiceofavoidingthe second for a physically sensible solution) and a < b2 turns out
c
approximation (point (2) above), by not Taylor expand- to be rather narrow,
ing (1 + ǫ/(ψ ψ )4)−1. We choose not to do so be-
0
−
causethereisnoadvantageinmakingthisexpansion—it 0.259<b<0.286, (10)
only leads to an unphysical divergence in the potential
energy as the brane-antibrane distance shrinks. The un- and in this regionac is to a good approximationa linear
expanded form has good behavior as ψ ψ , including function of b:
0
→
the vanishing of the potential at the minimum.
a =3b 0.77 (11)
c ∼ −
A. Multibrane Lagrangian One can also approximate the critical value xc by a
quadratic form, x = 0.146+1.21b+6(b 0.259)(b
c ∼ − − −
0.286).
We will be interested below in the generalization to
We will show later that the narrowness of the inter-
more than one brane rolling. The Lagrangian for N
val (10) is an accident, in the sense that physically well-
branes and N antibranes is
motivatedperturbationstothepotentialcanremovethis
−1
N N restriction. Therefore we will ultimately not regardit as
6σ NTǫ ǫ
= ψ˙2 1+ (5) a fine tuning. However the special value a for a given
L r2 i=1 i − r2 i=1 (ψi−ψ0)4! value of b clearly does represent a fine tunincg of the pa-
X X
where now the volume modulus is rameters. It is this condition which will be addressed
by a dynamical mechanism. Namely, by starting with a
N
large enough number N of branes, one can initially have
r =2σ ψ2 (6)
− i a > ac (but still a < b2). The potential always has a
i=1
X metastable minimum from which branes can tunnel in
Later we will specialize to the situation where all the this case. After each tunneling event, the value of a di-
branes are coincident, ψi = ψ. Then the potential takes minishes by a discrete amount. At some critical number
the form NǫTr−2(1+Nǫ(ψ−ψ0)−4)−1. of branes, Nc, the curvature changes so that the branes
roll instead of being trapped. It becomes a quantitative
question, depending on the values of σ and ǫ, whether
IV. CONDITIONS FOR INFLATION
the potential is sufficiently flat to give enough inflation.
We will show that in fact it is almost always flat enough
The optimal potential for achieving slow roll inflation to give at least 60 e-foldings of inflation.
is one where both the first and secondderivatives vanish To findouthow likelyit is that adequateinflationwill
simultaneously at some critical point ψc. Although this occur, we have explored the parameter space ǫ, σ, b,
requires fine tuning, we will discuss a mechanism where subject to the aforementioned restrictions and the as-
the tuning canautomaticallyoccurthrougha dynamical sumptionthatN startsoutbeing largerthanthe critical
process. To characterize the tuning, we introduce new value, where all the branes start to roll. For each value
parameters of ǫ, σ, b , the quantities ψ and N are determined.
0 c
{ }
The fields start from rest at the value where the poten-
Nǫ 2σǫ ψ
a = , b2 = , x= (7) tialhasalocalminimumwithN =N +1branes,onthe
ψ4 ψ6 ψ c
0 0 0 assumption that the nontunneling branes do not move
such that the potential is proportional to during the tunneling event. The equations of motionare
−1
1 a
Vˆ(x)= 1+ (8)
(b2/a x2)2 (x 1)4 dπi ∂V dNe
− (cid:18) − (cid:19) dt =−3Hπi− ∂ψ , dt =H (12)
i
The conditions for the potential to have a flat point, as
defined above, are for the canonical momentum
12σ
a=(6x 1)(x 1)4, 5x2(x 1)4 =b2 (9) πi = r2 ψ˙i (13)
− − −
5
andnumber ofe-foldingsNe lna(t), wherethe Hubble Ne
≡
parameter is given by H2 = (6σNψ˙2/r2 + V)/(3M2).
p
Since all the fields move together, we need consider only
one equation for the inflatons, which we do numerically.
The result is that, for b satisfying (10), there is a high
probabilityofgettingenoughinflation,forrandomvalues
of ǫ,σ selected from within a reasonable range, 5
− ≤
log ǫ 2.4, 1 log 2σ 3.6. Scanning uniformly
10 ≤ − ≤ 10 ≤
over this region, we find that 85% of tries result in more
than 55 e-foldings of inflation when b = 0.27, and an
evengreaterpercentageoccursforothervaluesofb. The
results are illustrated for b = 0.27 in Figure 4, showing
ε
log ( )
thenumberofe-foldings,N (omittingvalueswhereN >
e e 10
300 for clarity) versus log ǫ and log 2σ. log ( 2 σ )
10 10
Infigure4itcanbeseenthatN isconstantalonglines 10
e
of constant A (2σ)2/3ǫ−1/3. This can be understood
≡
by rewriting the potential in terms of the canonically FIG. 4: Ne versus log10ǫ and log102σ for b = 0.27, in the
normalized field φ=ψi 3N/σ: range Ne <300.
T N 1 p−2 36(N/A)3 −1
V = 1 φ2 1 (14)
A3 − 6 − (φ φ )4
(cid:18) (cid:19) (cid:18) − 0 (cid:19)
5
where φ0 =b−1/3 3N/A. Thus for fixed b, the shape of actual values
the potential, power spectrum, and duration of inflation empirical relation
p
depend only upon combination of parameters in A, and 4
not on the orthogonalcombination B =(2σ)1/3ǫ2/3. We Ne
cantherefore restrictour explorationofparameterspace 10
g
toaone-dimensionalscanalongthedirectionofA. More- o 3
l
over,we canshow that the number of branesN depends
in a very simple way on the value of A: N = ab−4/3A.
2
This follows from eliminating ψ from the definitions of
0
a and b. Thereforethe criticalnumber of branesis given
by
1
0.5 1 1.5 2 2.5 3
(2σ)2/3 log N
Nc =acb−4/3 ǫ1/3 (15) 10 c
andfor a fixedvalue ofb, we cantreatthe parametersA FIG. 5: logNe versus the critical number of branes logNc,
andN interchangeably. Ofcourse(15)can’tbesatisfied b=0.27. The empirical relationship (16) is also plotted.
c
exactly, because N has to be an integer. The acciden-
c
tal discrepancies between the optimal value of N and
c
the closest integer below this value will affect how long The factor √N in (16) can be partly understood in
c
inflation actually lasts. terms of assisted inflation [15]. With N fields rolling si-
In figure 5 we show the correlation between number multaneously,thekinetictermismultipliedbyN relative
of e-foldings and the critical number of branes, scanning to onefield. We shouldrescalethe fields byψ ψ/√N,
uniformly in the interval 1 < log10A < 4 of parameter and the slow-roll parameters by 1/N. One the→refore ex-
space. Although the two quantities are generally corre- pects an enhancement in the length of inflation by some
lated,there areorder-of-magnitudevariationsinNe over power of Nc. It would of course be satisfying to under-
narrow ranges of Nc. Figure 5 demonstrates that both stand why that power is 1/2 in the present model. We
features can be matched fairly well by an empirical rela- havenotyetsucceededinanalyticallyderivingthisresult.
tionship. Let ∆Nc be the discrepancy between the ideal Howeveritis notsurprisingthat Ne fails to be enhanced
valueofNcandtheclosestintegerbelowthisvalue. Then by a full factor of Nc as would be the case for ordinary
we find that assisted inflation, because the potential contains Nψ2 in
√N the K¨ahler modulus. Only if the potential was indepen-
c
Ne ∼=10∆Nc (16) dtreanntsfoofrNminwgoutlodtwheebceansounreictahllaytnNoerm∼aNlizce.dInfietlhdatwcoausled,
The factor 1/∆N , is understandable since if ∆N = 0, cause V(ψ) V(φ/√N) and the slow roll parameters
c c
→ ′′
thepotentialistunedtobeperfectlyflatatsomeposition would go like η V /V 1/N. The surprise is that
∼ ∼
ψ =ψ . thereisanyassistedinflationeffectatall. Thereasonfor
c
6
4
b = 0.28
-3 b = 0.27
3.5 b = 0.26
)
p
3 M -3.5
Ne 4/
10 1/V
g 2.5 (
o 0
l g1 -4
o
2 l
b = 0.26
1.5 b = 0.27 -4.5
b = 0.28
0 1 2 3 0.5 1 1.5 2
log N log N
10 c 10 c
FIG. 6: logNe versuslogNc for b=0.26, 0.27, and 0.28. FIG. 7: Log of the inflationary scale as a function of the
numberof branes.
itisthatatthecriticalvalueofN,thepotentialisflatter
0.15
than it would generically be for an arbitrary function of b = 0.28 0.02
b = 0.27
Nψ2. b = 0.26 0
Theaboveresultswereforb=0.27. Allowingbtovary, 0.1
onesfindsthatsmallerbtendstogivemoreinflationwith -0.02
a smaller number of branes. This dependence is shown
1 0.05 -0.04
in figure 6. The main observation, though, is that the -
s
great majority of all cases yield more than 60 e-foldings n -0.06
of inflation.
0 20 40 60 80 100
V. SCALAR POWER SPECTRUM -0.05
Inadditiontohavingalongenoughperiodofinflation, 0 200 400 600
N
we must obtain a spectrum of density perturbations in C
agreement with observational constraints from the Cos-
micMicrowaveBackground. Tocomputethepowerspec- FIG. 8: Deviation of scalar spectral index from unity versus
trum, we use the following expression [16] which applies Nc
to a set of noncanonically normalized fields with La-
grangian = 1G ψ˙iψ˙j V:
L 2 ij − ǫ which determine N .) From the figure one sees that
c
P(k)= V ∂Ne∂NeGij = NcVH2 r (17) rthoueginhfllyatcioonnsaisrtyesnctawleitrhanoguersabsesutwmepetnio1n0−of3an5d51e0-−fo4l.d5iMngps,
75π2Mp2 ∂ψi ∂ψj 75π2Mp2 6ψ˙2 ofinflation. Thestringscale,whichdeterm∼inesthebrane
tension,mustbe inaboutthe samerange. Ifthenumber
These quantities are evaluated at horizoncrossingof the
of branes is small enough, the inflationary scale can be
relevant wave number, when k = aH. To normalize the
sufficientlyhighforthetensorcontributiontothecosmic
spectrum, we evaluate P(k) for modes whichcrossedthe
microwave background to be observable in future exper-
horizon 55 e-foldings before inflation ended, correspond-
ing to a scale of inflation around V1/4 4 1014 GeV. iments.
The consistency of this assumption will∼be v×erified. The The scalar spectral index is given by ns = 1 +
COBEnormalizationimpliesthatP =4 10−10forthese dlnP/dlnk. We have numerically evaluated it over the
modes. This allows us to normalize∼the×scale of the po- relevantrangeofparameters. Liketheotherquantities,it
tential by fixing the value of the parameter T which is is most conveniently displayed as a function of the num-
proportionalto the unwarpedbranetension. The results ber of branes. The deviation ns 1 at 55 e-foldings be-
−
are expressed in terms of the scale of inflation relative fore the end of inflation is shown as a function of Nc
to the Planck scale. In figure 7 we plot log(V1/4/M ) in figure 8. There is a wide range of possible values,
p
as a function of the number of branes. (In the previous 0.07<ns 1<0.15.
− −
sectionitwasexplainedthat, justlikeN , spectralprop- It is also interesting to consider the running of the
e
erties depend only on the same combination of σ and spectral index, dn/dlnk = d2lnP/dlnk2, for which
7
0 0(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
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(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
-0.002 -0.002(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)a(cid:0)(cid:1)l(cid:0)(cid:1)lo(cid:0)(cid:1)(cid:0)(cid:1)wed region
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
)-0.004 )-0.004(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
k k (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
dn / d ln(--00..000068 bbb === 000...222868 dn / d ln(--00..000068(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)1(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)σ(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)2(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)σ(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
-0.01 -0.01(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)bbb(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) ===(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)000(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)...(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)222(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)868(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)
-0.012 -0.012(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
0.95 1 1.05 1.1 1.15 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)0.(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)9(cid:0)(cid:1)5(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)1(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)1(cid:0)(cid:1).0(cid:0)(cid:1)5(cid:0)(cid:1)(cid:0)(cid:1) 1.1 1.15
n n
s s
FIG. 9: dn/dlnk versus ns. FIG.10: dn/dlnkwithn−1,withWMAP+SDSSconstraints
shown.
WMAP has given a hint of a larger-than-expected neg-
of mobile branes exceeded the critical number N . But
ative value, dn/dlnk = 0.02 [17]. Although the er- c
∼ − since b looks fine-tuned, this is not a complete solution
ror bars are large enough to be consistent with no run-
to the problem of naturalness for the inflaton potential.
ning, the suggestion of large running has attracted at-
Could this be an accident of the particular form of the
tention. Generically, within the slow-roll formalism, one
simplest IKLIMT potential? To answer this, we explore
expects dn/dlnk (n 1)2, which is usually smaller
∼ s − the effect of perturbing the model by corrections which
than the measured central value. Our numerical survey
are expected to be present, but which we have so far
shows that it is possible to have rare instances of run-
ignored for the sake of simplicity.
ning as negative as 0.01, even though inflation lasts
− AtleasttwokindsofmodificationstothebasicIKLIMT
long enough. However, as shown in figure 9, it is still
modelaregenericallyexpected: correctionstothe super-
true that dn/dlnk (n 1)2. The rare cases of large
∼ s − potentialand to the K¨ahlerpotential. A simple example
running correspond to occasional extreme values of the
of the former was considered in [2], where the superpo-
spectral index.
tential for the K¨ahler modulus T = σ+iθ and inflaton
We have checked that the spectral index and running
was taken to be
are in good agreement with the predictions of slow roll
inflation,n 1=2η 6ε,dn/dlnk = 2ξ+16ǫη 24ε2, W =W +g(T)(1+δψ2) (18)
where ε =−1(V′/V)−2, η = V′′/V an−d ξ = V′V−′′′/V2, 0
2
in units where Mp = 1. The derivatives must be taken leading to a contribution to the potential of the form
with respect to the canonically normalized fields, φ =
i
(r/√12σ)ψ . In all cases, ε 10−6 10−9, whereas α+βψ2+γψ4
η 0.03iand ξ 10−3. T∼he runni−ng is completely δV = (2σ ψ2)2 (19)
do∼min−ated by ξ. ∼ −
It is interestingto comparethe predictedvalues ofthe with σ-dependent coefficients α, β, γ. This should be
running and the spectral index with experimental con- added to the brane-antibrane potential which we have
straints from combined WMAP and Sloan Digital Sky been using so far. In the case of N branes,we take W =
Survey Lyman α data [18] The allowed region is super- W0 + g(T)(1+ δ iψi2). To simplify the analysis, we
imposed on the predictions in figure 10. Although most consider examples where α, β, γ are chosen in such a
P
of the predicted points are allowed,we note that the few way that the full potential continues to have a global
examples with large running have too large a deviation minimum at ψ =ψ0, where V(ψ0) is assumed to vanish:
of n from unity. Thus if large running should be ex-
s
perimentally confirmed in the future, it will rule out the
(ψ2 ψ2)2
models under consideration. δV Nδ − 0 (20)
∼ (2σ Nψ2)2
−
We have included the factors of N appropriate to the
VI. GENERALIZED MODELS
multibrane case, when all branes are coincident.
The second kind of correction involves the K¨ahler
The previous analysis shows that within a narrow potential, which for N branes was taken to be K =
range of the parameter b = √2σǫ/ψ3, a long period of 3ln(T+T¯ ψ 2)[13]. Thiswasalwaysunderstood
inflation will always ensue, as long as0the initial number t−o be an ap−proxii|mai|te expression, good when ψ2 σ,
P i ≪
8
−6
a = 9.1 x 10 4
a = 33.75
a = 9.1 x 10−6 a = 31.25 3.5
V
3
e
N
02.5
1
b = 0.01 b = 5 og
l
2
x x
1.5 b= 0.01
b = 5
FIG. 11: Rescaled brane-antibrane potential (22) including
K¨ahler and superpotential corrections. 1
1 2 3
log N
10 c
but subject to higher order corrections in ψ , for exam-
i
ple ψ 4. The denominator in V + δV would then be FIG.12: Logofnumberofe-foldingsversuslogofthecritical
i
repla|ce|d by numberof branes, for theexamples illustrated in figure 9.
(2σ Nψ2)2 (2σ N(ψ2+dψ4))2 (21)
− → − 0
As shown in Appendix B, the K¨ahler corrections also
-0.005
givecontributionstothenumeratoroftheF-termpoten-
tial, whose Taylor series starts with a term of order ψ4.
-0.01
For small ψ, we can absorb this leading term into the γ k)
bcOou(etdffiψtch2ie)esncetoraorrfeec(tn1ieo9gn)l.sig,TiwbhhleeicwhkihnceaennteicvbeetreirntmhfeerKrbeard¯baφnf˙raeoφs¯˙mbaraeeqls.sol(oBgwe6lt)ys, dn / d ln(s-0-0.0.0125 bb == 50.01
rolling.
By the above reasoning, the two kinds of corrections -0.025
enter into the rescaled potential (8) as
-0.03
1 a −1
Vˆ c(x2 1)2+ 1+ 0.8 0.9 1 1.1 1.2
→ ba2 −x2−dx4 2 − (cid:18) (x−1)4(cid:19) ! ns
(22)
(cid:0) (cid:1) FIG. 13: dn/dlnk versus ns for the models with K¨ahler and
By experimenting with this generalized potential, one
superpotential corrections.
finds that the new coefficient c allows for the removal
of the lower bound on b in (10). Moreover,negative val-
ues of d can remove the upper bound. It is easy to find
cases with N < 20, it is possible to get large deviations
values of c and d for which the potential has the desired
from a flat sp∼ectrum with n > 1. These are accompa-
behavior as a function of N, as in figure 1, regardless of s
nied by sizable running, but as in figure 10, such points
the value of b. For example c= 0.1, d= 0.5 yields the
− occur outside of the experimentally allowed region. The
potentialsshowninfigure11,atb=0.01andb=5. Itis
overall conclusion is that the generalized models have
importantthat the new parametersc and d areindepen-
similar qualitative behavior to the simplest class which
dentofthenumberofbranes. Thusacontinuestobethe
was studied in sections III-IV.
only parameter which changes due to a brane tunneling
event.
Asintheprevioussection,wescannedoverparameters
VII. CAVEATS AND CONCLUSIONS
for which A varies, hence the number of branes, while
keeping b, c, d fixed, for the two exampleswith b=0.01
and b = 5. Figure 12 shows the number of e-foldings In this work we have presented a very general means
versus number of branes. The general growth of N whereby a long period of inflation, with a suitably flat
e
∼
√N is evident, as it was for the simple model of the spectrum of density perturbations, can be automatically
c
previous section. obtainedaftertunnelingofbranes,fromametastablepo-
Figure 13 shows the correlation of dn/dlnk with n . sition in a Calabi-Yau manifold, into a warped throat
s
Itshowsthatthesmall-bmodeltendstohavearedspec- containing antibranes. It depends only on the fact
trum, with n < 1, and negligible running. The large-b thatthe brane-antibranepotentialgenericallyhassucha
s
model also has n <1 for most realizations, but for rare metastable minimum when the number of branes is suf-
s
9
ficiently large, and the possibility of starting at a place (referred to as b,c in section V) where multibrane in-
{ }
in the string theory landscape where the number of mo- flation works will be large, given that it was easy to
bile branes exceeds the thresholdfor havinga minimum. findasuccessfulexample,butamorecompleteinvestiga-
Once a certain critical number of branes N is reached, tion could be made. Moreover, such corrections should
c
the minimum disappears and the branes start rolling in principle be computed from a fully string theoretic
slowly to the true minimum. They automatically start startingpoint,whichmightrestricttheallowedrangesof
from the same position, which is guaranteed to be close b,c .
{ }
to the flattened region of the potential. Another issue which deserves further study is the pre-
In this scenario, an arbitrarily long time will be spent cise relationship between the radial coordinate r of the
in a metastable de Sitter phase resembling old inflation, brane within the warped throat, and the field ψ which
beforetheslowrollingperiod(similarlytoref.[19]). Each is used to describe the inflaton position in the super-
tunneling event results in a small bubble of open uni- gravity description. The exact correspondence between
verse, within which another period of quasi-de Sitter ex- these fields was not needed in IKLIMT because the main
pansion will occur. The process repeats itself until the dependence on ψ was taken to be through the K¨ahler
critical number of branes is reached. For us, these long modulus factor (and superpotential corrections), rather
periods of old inflation are irrelevant—we have focused than a combination of these with the brane-antibrane
entirely on the final phase of conventionalinflation asso- potential. We have simply identified the two fields in
ciated with the smooth motion of the branes until they our analysis,but the actual relation could be more com-
annihilate with antibranes in the throat, hopefully lead- plicated,possiblyintroducingadditionalparametersinto
ingtothereheatingofthe universe[20],andperhapsthe the Lagrangian.
production of cosmic superstrings [21].
We numerically scanned over large ranges of the loga-
Acknowledgments
rithmof relevantcombinationsofparameterswhichgov-
ern the shape of the inflaton potential, and found that
large fractions ( 10 90%) of the trials resulted in in- WethankC.BurgessandF.Quevedoforcollaboration
flation with no n∼eed f−or tuning, whenever the flattening during the early stages of this work. We also thank H.
mechanismcouldoccur. Thisseemstobeaquitepromis- Firouzjahi, S. Kachru,R. Kallosh, L. Kofman, A. Linde,
ing approach for ameliorating the need for fine-tuning I. Neupane, E. Silverstein, L. Susskind, and H. Tye for
withintheframeworkofthestring-theorylandscape[23]. helpful comments and discussions. JC and HS are sup-
ported by grants from NSERC of Canada and FQRNT
For convenience of numerical integration, we chose a
of Qu´ebec.
form of the potential which has good behavior as the
branesreachthe bottomof the throat. This form,which
differs from the commonly used Taylor-expanded form,
APPENDIX A: BRANE-ANTIBRANE
is in fact the result of a direct perturbative compu-
POTENTIAL IN WARPED THROAT
tation, and does not require any special assumptions.
However, the perturbative derivation does assume that
Nǫ/(ψ ψ )4 1inordertobevalid. Onemightthere- We consider the backgroundof AdS5 X5 space with
0 ×
− ≪ the AdS metric:
fore wonder how sensitive our results are to the behav- 5
ior of the potential when this approximation is break- r2 R2
ing down. We have investigated what happens if the ds25 = R2 −dt2+dx2 + r2 dr2 (A1)
Taylor expansion 1 Nǫ/(ψ ψ )4 is used in place of
(1+Nǫ/(ψ ψ )4)−1−,andwe−also0checkeddifferentways where R4 = 4πag Nα(cid:0)′2 is the rad(cid:1)ius of the AdS space
0 s 5
ofregulatin−gthesingularity,like1 Nǫ/(Nǫ+(ψ ψ )4). createdasthenear-horizongeometryofastackofN 1
0
− − ≫
Forallthesealternatives,wefoundthatalthoughthespe- D3branes. ThereisalsoanRRbackgroundfieldcreated
cific parameter values where a flat potential occurs may by the stack of branes:
be somewhat sensitive to the behavior of the potential
1 r 4
ntheealresψs0w,otrhkesimnetchheansaismme owfaymausltwiberahnaeveinpflraestieonntende.vWere- Ax0,x1,x2,x3 =rli→m01+ R 4 ≃ R (A2)
r (cid:16) (cid:17)
thereforebelieveittobearobustresult,whichisqualita-
If one now writes the action fo(cid:0)r a(cid:1)probe brane is this
tivelyquitedifferentfromfieldtheorymodelsofinflation.
background,one finds:
Although the dynamical flattening of the potential re-
quires a rather special range of parameters in the sim-
S = T d4ξ det ∂ xM∂ xNg
plest IKLIMT model, we showed that this restriction can D3 − 3 − ξa ξb NM
be lifted once corrections to the superpotential and the Z q (cid:0) (cid:1)
K¨ahler potential are taken into account. We exhibited + T3 d4ξAx0,x1,x2,x3 (A3)
an example of such corrections that works, without sys- Z
tematically exploring the parameter space of such cor- We will use the convention that ξ0 = t, but the rest
rections. Subjectively,itseemsthatthe parameterspace ofthecoordinateswilldescribethelocationofthebrane,
10
includingfluctuations(localbendings),xi =xi ξ0,ξi = that only its radial location varies.
xi t,ξi . We make here a further assumption, namely
(cid:0) (cid:1)
(cid:0) (cid:1)
∂ xM∂ xNg =∂ x0∂ x0g +∂ xi∂ xig +∂ r∂ rg (A4)
ξa ξb NM ξa ξb 00 ξa ξb ii ξa ξb rr
The metric will look like:
2
g +g ∂r g ∂r ∂r g ∂r ∂r g ∂r ∂r
00 rr ∂ξ0 rr∂ξ0∂ξ1 rr∂ξ0∂ξ2 rr∂ξ0∂ξ3
 grr∂∂ξr1(cid:16)∂∂ξr0 (cid:17) gxx+grr ∂∂ξr1 2 grr∂∂ξr1∂∂ξr2 grr∂∂ξr1∂∂ξr3 = g000 g0xx 00 00 + g ∂r ∂r
 grr∂∂ξr2∂∂ξr0 grr∂∂ξr2(cid:16)∂∂ξr1 (cid:17) gyy+grr ∂∂ξr2 2 grr∂∂ξr2∂∂ξr3   00 00 g0yy g0  (cid:18) rr∂ξµ∂ξν(cid:19)
 grr∂∂ξr3∂∂ξr0 grr∂∂ξr3∂∂ξr1 grr∂∂ξr3(cid:16)∂∂ξr2 (cid:17) gzz+grr ∂∂ξr3 2   zz 
 (cid:16) (cid:17)  (A5)
We can apply the usual formula:
det(A+ǫB)=detAdet 1+ǫA−1B detA 1+ǫTr A−1B (A6)
≃
Writing the metric at the location of the brane as:(cid:0)g = r2f(cid:1) we obta(cid:0)in: (cid:0) (cid:1)(cid:1)
µν R2 µν
r 8 R 4 ∂r ∂r
det ∂ xM∂ xNg = 1 fµν (A7)
(cid:0) ξa ξb NM(cid:1) −(cid:16)R(cid:17) −(cid:18)r(cid:19) ∂ξµ∂ξν!
The factor r4 in the bracket comes from the product the gravitational + RR field will make it favorable for
R4
the brane to move towards r =0, that is, the bottom of
4
R theAdS throat. Forthesetupinfig.(14)wehavetoadd
gµνg = fµν (A8) 5
rr r the piece corresponding to the graviton + RR exchange
(cid:18) (cid:19)
between the D3 and the D3. However in this setup we
Using these results in the DBI +CS action we obtain:
r 4 R 4 ∂r ∂r
S = T d4ξ f 1 fµν
D3 − 3 − R s − r ∂ξµ∂ξν
Z p (cid:16) (cid:17) (cid:18) (cid:19)
∂x r 4
± T3 d4ξdet ∂ξ R ǫtξ1ξ2ξ3 (A9)
Z (cid:18) (cid:19)(cid:16) (cid:17)
ExpandingthesquareroottermweobtainforaD3brane
(sign +):
1 ∂r ∂r
S =T d4ξ f fµν (A10)
D3 3 − 2 ∂ξµ∂ξν Stack anti−D3 D3
Z
p
while for a D3 (sign ): r = 0 r0 r1
−
1 ∂r ∂r
S = T d4ξ f fµν
D3 3 − 2 ∂ξµ∂ξν
Z
p r 4
2T d4ξ f (A11)
− 3 − R FIG. 14: IKLIMTsetup
Z p (cid:16) (cid:17)
The extra piece in the case of the D3 comes from the
non-BPS nature of the antibrane, since the graviton ex- havetoincludetheeffectoftheD3inadditiontothatof
change and the RR exchange forces now add instead of the stack. The derivation of the background perturbed
canceling (for 3-branes the dilaton is constant and con- bythepresenceoftheD3placedawayfromthestackcan
tributes no force). The potential energy for the D3 in be found in Appendix B of IKLIMT. It starts by writing