Primordial Non-Gaussianity in Multi-Scalar Inflation

0.29 MB·English

by  Shuichiro Yokoyama

KUNS-2110
Primordial Non-Gaussianity in Multi-Scalar Inflation
Shuichiro Yokoyama,1,∗ Teruaki Suyama,2,† and Takahiro Tanaka1,‡
1Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa 277-8582, Japan
(Dated: April21, 2008)
We givea concise formula for thenon-Gaussianity of theprimordial curvatureperturbation gen-
eratedonsuper-horizonscalesinmulti-scalarinflationmodelwithoutassumingslow-rollconditions.
This is an extension of our previous work. Using this formula, we study the generation of non-
Gaussianityforthedoubleinflationmodelsinwhichtheslow-rollconditionsaretemporarilyviolated
afterhorizonexit,andweshowthatthenon-linearparameterfNL forsuchmodelsissuppressedby
theslow-roll parameters evaluated at thetime of horizon exit.
8
0 PACSnumbers: 98.80.Bp,98.80.Cq,98.80.Es
0
2
I. INTRODUCTION II. FORMULATION
r
p
A
Non-Gaussianityoftheprimordialcurvatureperturba- In this section, we derive a formula which is an exten-
1 tion is a potentially useful discriminatorof the many ex- sion of that given in our previous paper [19] to the non
2 istinginflationmodels[1,2]. Planck[3]isexpectedtode- slow-rollcase. We use the unit M2 =(8πG)−1 =1.
pl
tect the primordial non-Gaussianity if the so-callednon-

] linear parameter, f , which parameterizes the magni-
h NL
tudeofthebispectrum,islargerthan3 5[1,4]. Higher
p ∼ A. Background equations
- order correlation functions such as trispectrum would
o also be a useful probe of the primordialnon-Gaussianity
r We consider a -component scalar field whose action
t [5, 6, 7]. Hence it is important to theoretically under- N
s is given by
stand the generation of non-Gaussianity.
a
[ Standard single slow-roll inflation model predicts
2 rather small level of the non-linear parameter, fNL, S =− d4x√−g 12hIJgµν∂µφI∂νφJ +V(φ) , (1)
v suppressed by the slow-roll parameters [8]. In multi- Z (cid:20) (cid:21)
scalarfield inflationmodels with separablepotential, ex- (I,J =1,2, , ) ,
0 ··· N
2 plicit calculations in the slow-roll approximation show
9 that fNL is also of the order of the slow-roll parame- where gµν is the spacetime metric and hIJ is the metric
2 ters[9,10,11,12,13,14,15,16]. Thisfeatureislikelyto on the scalar field space. In the main text, we restrict
1. holdformoregeneralpotentialthatsatisfiestheslow-roll our discussion to the flat field space metric hIJ =δIJ to
1 conditions[17,18,19]. Ontheotherhand,thegeneration avoid inessential complexities due to non-flat field space
7 of non-Gaussianity due to the violation of the slow-roll metric. Extension to the general field space metric is
0 conditions remains an open problem. given in appendix B.
v: In this paper, based on δN formalism [20, 21, 22, 23, We define ϕIi(i=1,2) as
i 24],wegiveausefulformulaforcalculatingthenon-linear
X
parameter in the multi-scalar inflation models includ- ϕI φI , ϕI d φI , (2)
r ing the models in which the slow-roll approximation is 1 ≡ 2 ≡ dN
a
(temporarily) violated after the cosmological scales exit
wheredN =HdtwithH andtbeingtheHubbleparame-
the horizon scale during inflation. Current observations
ter andcosmologicaltime, respectively. Namely, we take
do not exclude such models. We also study the possi-
e-folding number, N, as a time coordinate. For brevity,
bility of the large non-Gaussianity for double inflation
hereinafter,weuseLatinindicesatthebeginningofLatin
model [25, 26], as an example of such a model.
alphabet, a, b or c, instead of the double indices, I, i,
In section II, we derive the formula for the non-linear
i.e., Xa =XI.
parameter f , which is an extension of our previous i
NL Then, the backgroundequation of motion for ϕa is
paper[19]. InsectionIII,weanalyzetheprimordialnon-
Gaussianity in the double inflation model using our for-
d
mula. We give a summary in section IV. ϕa =Fa(ϕ) , (3)
dN
where Fa(=FI) is given by
i
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] FI =ϕI , FI = V ϕI + VI , (4)
‡Electronicaddress: [email protected] 1 2 2 −H2 2 V
(cid:18) (cid:19)
2
where VI = δIJ(∂V/∂φI), and the Friedmann equation Evolution equation for δ(ϕ2)a is given by
is
H2 = 6 2ϕVIϕ , (5) ddNδ(ϕ2)a(N) = Pab(N)δ(ϕ2)b(N)
− 2 2I +Qa (N)δ(ϕ1)b(N)δ(ϕ1)c(N) , (12)
bc
with ϕ =δ ϕJ.
2I IJ 2
where Qa is defined by
bc
B. Perturbations
∂2Fa ∂Pa
Qa = b . (13)
bc ≡ ∂ϕb∂ϕc (0) ∂ϕc (0)
In the δN formalism [22, 23, 24], the evolution of the (cid:12)ϕ=ϕ(N) (cid:12)ϕ=ϕ(N)
(cid:12) (cid:12)
differencebetweentwoadjacentbackgroundsolutionsde- (cid:12) (cid:12)
termines that of the primordial curvature perturbation The explicit form of Q(cid:12)a (=QIjk ) is s(cid:12)hown in appendix
bc iJK
on super-horizon scales. In this paper, we use the word A. Let us choose the integral constants λa as the initial
”perturbation” to denote the difference between two ad- values of ϕa at N =N , namely, λa =ϕa(N ). Then we
∗ ∗
jacent background solutions. In this subsection, we an- haveδϕa(N )=δλa. Henceδ(ϕ2)a(N)vanishesatN . Un-
∗ ∗
alyze the time evolution of the perturbation and relate
der this initial condition, the formal solution of Eq. (12)
the result to the curvature perturbation.
is given by
Thesolutionofthebackgroundequation(3)islabelled
by 2 integral constants λa. Let us define δϕa as the
pertuNrbation, δ(ϕ2)a(N) = NdN′Λa(N,N′)Qb (N′)
b cd
ZN∗
δϕa(N) ϕa(λ+δλ;N) ϕa(λ;N) , (6)
≡ − δ(ϕ1)c(N′)δ(ϕ1)d(N′) . (14)
×
whereλisabbreviationofλa andδλa isasmallquantity
of (δ). δϕa(N) defined by Eq. (6) represents a pertur- According to the δN formalism [22, 24], the curva-
O
bationofthescalarfieldontheN =constantgauge[23]. ture perturbation on large scales evaluated at a final
For the purpose of calculating the leading bispectrum time, N = N , is given by the perturbation of the e-
c
of the curvature perturbation, it is enough to know the folding number between an initial flat hypersurface at
evolution of δϕa(N) up to second order in δ. For later N =N and a final uniformenergy density hypersurface
∗
convenience, we decompose δϕa as atN =N . LetustakeN tobeacertaintimesoonafter
c ∗
the relevant length scale crossed the horizon scale, H−1,
δϕa =δ(ϕ1)a+ 1δ(ϕ2)a , (7) duringthe scalardominantphase andN to be a certain
2 c
time after the complete convergence of the background
where δ(ϕ1)a and δ(ϕ2)a are first and second order quantities trajectories has occurred. At N > Nc the dynamics of
the universe is characterized by a single parameter and
in δ, respectively.
only the adiabatic perturbations remain1. Then, the e-
Evolution equation for δ(ϕ1)a is given by folding number between N and N can be regarded as
∗ c
the function of the final time N and ϕa(N ), which we
ddNδ(ϕ1)a(N)=Pab(N)δ(ϕ1)b(N) , (8) denote N(Nc,ϕ(N∗)). c ∗
Based on δN formalism, the curvature perturbation
where Pa(=PIj) is defined by on the uniform energy density hypersurfaceevaluated at
b iJ
N =N is given by
c
∂Fa
Pa . (9)
b ≡ ∂ϕb(cid:12)ϕ=(ϕ0)(N) ζ(Nc) ≃ δN(Nc,ϕ(N∗))
(cid:12) 1
(cid:12) = N δϕa+ N δϕaδϕb + , (15)
Here (ϕ0)(N) represents the u(cid:12)nperturbed trajectory. The a∗ ∗ 2 ab∗ ∗ ∗ ···
explicit form of Pa is shown in appendix A. Formally,
b where δϕa = δϕa(N ) represents the field perturbations
solution of this equation can be written as ∗ ∗
and their time derivative on the initial flat hypersurface
δ(ϕ1)a(N)=Λa(N,N )δ(ϕ1)b(N ) , (10) at N = N∗. The left hand side in Eq. (15) is obvi-
b ∗ ∗ ously independent of the initial time N , and hence so
∗
where Λa is a solution of is δN(Nc,ϕ(N∗)). Here we also defined Na∗ = Na(N∗)
b
d
Λa(N,N′)=Pa(N)Λc (N,N′) , (11)
dN b c b
1 Weconsiderthecaseinwhichisocurvatureperturbationsdonot
with the condition Λa(N,N)=ΛIj(N,N)=δI δ j. persistuntillater.
b iJ J i
3
and N =N (N ) by In fact, sub-horizon perturbations of fields give differ-
ab∗ ab ∗
ent k-dependent form of the bispectrum [17]. However,
∂N(Nc,ϕ) the sub-horizon contribution to the bispectrum is sup-
N (N) , (16)
a ≡ ∂ϕa (0) pressedbytheslow-rollparametersevaluatedatthetime
(cid:12)ϕ=ϕ(N)
(cid:12) ofhorizonexit2. Incontrast,thesuper-horizonevolution
N (N) ∂2N(Nc,ϕ)(cid:12)(cid:12) , (17) alwaysgivesthebispectruminthe formofEq.(22)inde-
ab ≡ ∂ϕa∂ϕb (cid:12)ϕ=(ϕ0)(N) pendent of the number of fields (see below). If fNL &1,
(cid:12)
(cid:12) which is aninteresting case fromthe observationalpoint
evaluated at N =N∗. (cid:12) of view, then the contribution due to super-horizon evo-
It is well known that the curvature perturbations on
lution dominates the total bispectrum.
an uniform density hypersurface, ζ, remain constant in
Weassumethattheslow-rollconditionsaresatisfiedat
time for N >N . Hence, ζ(N ) gives the final spectrum
c c N =N . Then, to a good approximation, δϕI becomes
of the primordial perturbation. ∗ 1∗
a Gaussian variable [8, 17] with its variance given by
Let us take N to be a certain late time during the
F
scalar dominant phase. Then we have 2
H
δϕI δϕJ =δIJ ∗ , (25)
ζ(N ) δN(N ,ϕ(N )) h 1∗ 1∗i 2π
c c F (cid:18) (cid:19)
≃
1
= N δϕa + N δϕaδϕb + . (18) and ϕI becomes function of ϕI. Differentiating ϕI
aF F 2 abF F F ··· VI, w2e have 1 2 ≃
where δϕa = δϕa(N ), N = N (N ) and N = − V
F F aF a F abF
Ncaanb(uNseFt)h.eDsoulruitnigontshfeorpδeϕriaodgivweinthbyNE∗q<s.(N10)<anNdF(,14w)e. δϕI2∗ = VVIV2J − VVIJ δϕJ1∗+··· . (26)
Using these solutions, we obtain the relations; (cid:18) (cid:19)
The higher order terms are also suppressed by the slow-
Na∗ =NbFΛba(NF,N∗) , (19) roll parameters. Hence, δϕI is Gaussian as is δϕI to
2∗ 1∗
Nab∗ =NcdFΛca(NF,N∗)Λdb(NF,N∗) a good approximation. Then, we can write down the
NF variance of δϕa∗ as
+2 dN′N (N′)Qc (N′)
c de
ZN∗ H 2
×Λda(N′,N∗)Λeb(N′,N∗) , (20) hδϕa∗δϕb∗i≃Aab(cid:18)2π∗(cid:19) . (27)
with
Atthefirstorderbothinthefieldperturbationandslow-
roll limit, the matrix Aab =AIJ can be written as
N (N) N Λb (N ,N). (21) ij
a ≡ bF a F
AIJ =δIJ , AIJ =AIJ =ǫIJ ,
11 12 21
C. Non-linear parameter AIJ =ǫI ǫKJ , (28)
22 K
where
In this subsection, we derive a formula for the non-
linear parameter f by making use of the δN formal-
NL VI(φ)VJ(φ) VIJ(φ)
ism. We first give the definition of f . It is defined as ǫIJ . (29)
NL ≡ V(φ)2 − V(φ) (0)
the magnitude of the bispectrum of the curvature per- (cid:20) (cid:21)φ=φ(N∗)
turbation ζ,
Since ǫIJ = O(ǫ,η), we find that δϕI δϕJ and
6 f δϕI δϕJ are suppressed by the slow-hroll1∗para2m∗ieters.
B (k ,k ,k )= NL P (k )P (k ) h 2∗ 2∗i
ζ 1 2 3 5(2π)3/2 ζ 1 ζ 2 At the same order, O(ǫ,η), it is known that we need to
(cid:20) add the slow-roll correction terms to AIJ in Eq. (28).
11
+P (k )P (k )+P (k )P (k ) , Such corrections to AIJ are given in [28].
ζ 2 ζ 3 ζ 3 ζ 1 11
(cid:21)
(22)
where Pζ is the power spectrum of ζ. The definitions of 2 Here,weconsiderthecasethatinflationisinducedbythecanon-
P and B are, respectively,
ζ ζ icalscalarfields. Insuchacase,currentobservationsofthespec-
trum of curvature perturbation restrict the models so that the
hζk1ζk2i≡δ(k1+k2)Pζ(k1) , (23) slow-rollconditionsaresatisfieduntilthecosmologicallyrelevant
hζk1ζk2ζk3i≡δ(k1+k2+k3)Bζ(k1,k2,k3) .(24) sBcoarlnes-Ienxfietldof(tDhBeIh)oirnizfloantioscnalme.odOetlh,etrhweisseu,bf-ohroerxizaomnpcloen,tirnibDuitriaocn-
to the bispectrum not only has different k-dependent form,but
Equation (22) restricts the form of the bispectrum. The
also can be large enough to be detectable in the future experi-
bispectrum in general does not take that simple form. mentwithoutinconsistencywithcurrentobservations [27].
4
Using these equations, to the leading order, the non- Here,letusevaluateζ(N )ontheuniformenergyden-
F
linear parameter is written as sity hypersurface at N = N , neglecting the later evo-
F
lution of the curvature perturbations during the period
6 N N N AacAbd
a∗ b∗ cd∗ with N > N > N . In this case we can obtain explicit
f c F
5 NL ≃ (Ne∗Nf∗Aef)2 forms of NaF and NabF, which appeared in Eqs. (21)
and (30), written in terms of the background quantities
1
= (Na∗Θa∗)2"NabFΘa(NF)Θb(NF) aenteNrgy=deNnFsi.tyOhnyptehresusrufapceer-ihsoerqizuoivnalsecnatletso,tthheecuonnisftoarnmt
NF Hubble hypersurface. Then, ζ(NF) is evaluated by the
+ dN′N (N′)Qc (N′) time shift δN measured from the H = constatnt hyper-
c ab
ZN∗ surface. Therefore at N =NF we have the equation;
×Θa(N′)Θb(N′)# , (30) H(ϕa(NF +ζ(NF)))=H (ϕ0)a(NF) . (35)
where The Hubble parameter H is given(cid:0) by Eq.(cid:1)(5). Solving
Eq. (35) with respect to ζ(N ), we obtain
F
Θa(N) Λa(N,N )AcbN , (31)
≡ c ∗ b∗ 1
and Θa∗ = Θa(N∗). As we mentioned before, we have ζ(NF)≈NaFδϕaF + 2NabFδϕaFδϕbF, (36)
neglectedthe non-Gaussianityfromthesub-horizoncon-
tributionsinderivingEq.(30). Eq.(30)showsthat,aside with
from N and N whose explicit form will be given in
aF abF H (ϕ)
the next subsection IID, f is completely determined N = a , (37)
by the quantities Na(N) aNndLΘa(N). These quantities aF −Hc(ϕ)Fc(ϕ)(cid:12)ϕ=(ϕ0)(NF)
(cid:12)
obey the following closed differential equations, Uab(ϕ) (cid:12)
N = (cid:12) , (38)
ddNNa(N)=−Nb(N)Pba(N) , (32) abF −Hc(ϕ)Fc(ϕ)(cid:12)(cid:12)(cid:12)ϕ=(ϕ0)(NF)
where (cid:12)
d
Θa(N)=Pa(N)Θb(N) . (33)
dN b U =H 2Ha (H Pc +FcH )
ab ab− H Fd c b cb
First, we solve Eq. (32) backward till N = N under d
∗
2H H
Ethqe. i(n3i3t)ialfocrownadridtiotinllsNNa=(NNF) =unNdearF.theThineintiawlecosonldvie- −(H Ha e)b2 FcHcdFd+FcPcdHd .
F e
tions Θa(N∗) = AabNb∗. Substituting these solutions (cid:0) (cid:1) (39)
into Eq. (30), we obtain f .
NL
The evolution equation of Θa(N), Eq. (33), is iden- and H ∂H/∂ϕa, H ∂2H/∂ϕa∂ϕb. The explicit
a ab
tical to that of δϕa, Eq. (8), at the linear level. Since forms of≡H and H are≡shown in appendix A. The
a ab
Eqs. (32) and (33) are mutually dual, a variable com- right-handsidesofEqs.(37)and(38)arewritteninterms
posed of N (N ) and Θa(N ); of local quantities, i.e., those depending only on φI and
a ∗ ∗
dφI/dN at N = N . Hence, once we specify a cen-
W(N )=N (N )Θa(N ) , (34) F
∗ a ∗ ∗ tralbackgroundtrajectory(ϕ0)a, we canreadily determine
becomes constant irrespective of N . This constancy of N , N . δN evaluated using the determined N
∗ aF abF aF
W(N ) corresponds to the constancy of δN(N ,ϕ(N )) andN dependsonN unlessN >N ,whereN isa
∗ c ∗ abF F F c c
which was mentioned below Eq. (15). time when the complete convergence of the background
trajectories occurs.
D. Non-linearity generated in scalar dominant
phase III. DOUBLE INFLATION MODEL WITH
LARGE MASS RATIO
In order to evaluate Eq. (30), we need to know N
aF
and N . If one takes into account the evolution of the Substituting Eqs. (37) and (38) to Eq. (30), the non-
abF
curvatureperturbationsafterthescalardominantphase, linear parameter, f , can be also defined as a function
NL
isocurvature perturbations may remain during preheat- ofthefinaltime,N . Asmentionedabove,thecurvature
F
ing/reheating era after inflation. In such cases, in order perturbationsonuniformenergydensityhypersurface,ζ,
tocalculateN , N ,weneedtoinvestigatetheevolu- evolve when N <N , and hence f also evolves.
aF abF F c NL
tionofthebackgrounde-foldingnumberwiththeeffectof In this section, using the formulationgivenin the pre-
short wavelength/radiationcomponent, which is beyond vious section II, we calculate the non-linear parameter
the scope of the present paper. f (N ) for the double inflation models, which violate
NL F
5
theslow-rollconditionsforacertainperiodwhenthecos- set the initial time N to 0. The central background
∗
mologically relevant scales are well outside the horizon. trajectory in the field space is shown in Fig 1 and the
The potential is given by [25, 26], evolution of these two fields as a function of N is shown
in Fig 2. We define slow-roll parameters as
1 1
V(φ,χ)= m2φ2+ m2χ2 . (40)
2 φ 2 χ 1 dH
ǫ , (41)
≡−H dN
Weassumealargemassratio,i.e.,m /m 1. Because
of mχ/mφ ≫1, the energy density oχf χ-fiφel≫d, ρχ, decays ηφφ ≡ mV2φ , ηχχ ≡ mV2χ . (42)
fasterthanthatofφ-field,ρ . Henceifρ ρ atinitial
φ φ χ
≫
time, ρχ never dominates the energy density during the The evolution of the slow-roll parameters is shown in
later evolution of the universe. In such a model, only a Figs. 3, 4 and 5.
single chaotic inflation induced by φ-field occurs. It is
known that in a single scalar slow-roll inflation, f is 12
NL
suppressed by the slow-rollparameters. Here we assume
the opposite, ρ ρ , at the initial time. In this case
χ φ
≫ 10
the slow-roll conditions are badly violated when φ χ
−
equality (ρ ρ ) occurs. We assume that both fields
φ χ
≃
are in the slow-roll phase at the initial time and the in-
8
flationinducedbyφ-fieldalsooccursafterφ χequality.
−
Denoting the initialvalues offields atN =N∗ as φ∗ and ÿ
χ∗, this assumption implies φ∗,χ∗ 1. 6
≫
There are several works in which the primordial non-
Gaussianity in the two-scalar chaotic inflation model
whose potential is given by Eq. (40) was investigated. 4
In Ref. [9, 10], the authors analyzed the primordial non-
Gaussianity generated during -flation [29] model and
N 2
gave a simple analytic formula using the slow-roll ap-
proximation even on super-horizon scales based on δN
formalism. In Ref. [10], using their slow-rollformula the
0
authorsalsocalculatedthe non-linearparameterf for
NL
the two-scalarchaotic inflation model whose potential is
given by Eq. (40) with the mass ratio m /m = 9. In 0 2 4 6 8 10 12
φ χ
suchcase,theslow-rollconditionsarenotviolatedduring þ
scalar dominant phase, so they could use the slow-roll
formula. They compared the result obtained by using
their analytic formula with that obtained numerically,
and they found that the non-linear parameter does not FIG. 1: This figure shows a background trajectory in field
become large in such case. In Ref. [16], the authors also space. Weset initial values of fields to φ∗ =χ∗ =10.
providedanotherformulationforthenon-linearitygener-
atedonsuper-horizonscaleswithoutslow-rollconditions. FromFig.2,weseethatinitiallyχ-fielddecaysrapidly
Theyalsocalculatedthenon-linearparameterusingtheir while φ-field remains almost at its initial value due to
formulation for the potential given by Eq. (40) with the largeHubble friction. At aroundN =26,χ-field reaches
mass ratio mφ/mχ =12. They also found that the non- zero and starts damped oscillation. As shown in Fig. 3,
linear parameter evaluated at the end of inflation does aroundthis time, the slow-rollparameter ǫ exceeds 1 for
not become large in such model. a moment and the inflation ends once. From Figs. 4 and
Here, using the formulation given in the previous sec- 5,alsoη exceeds1afterχ-fieldsettlestotheminimum,
χχ
tion,we numericallycalculatedfNL forthe massparam- while, φ-field is slow-rolling during this phase.
etersmφ =0.05, mχ =1.0andcomparedtheresultwith Incalculatingthenon-linearparameterfNL,weregard
the previous work [16]. f as a function of N in the same sense as ζ(N ) in
NL F F
Eq.(36). Weshowtheevolutionofthenon-linearparam-
eter f (N ) in Fig. 6. From this figure, we find that
NL F
A. Numerical calculation for mχ/mφ =20 thenon-linearparameterfNL(NF)temporarilyoscillates
with the maximum amplitude reaching 8. As soon as
∼
Using the formulation given in the previous section, χ-fieldsettlestoitsminimum,theuniversebecomesdom-
we numerically calculated f for the mass parameters inatedbyφ-fieldduetothedecayofρ ,andtheuniverse
NL χ
m =0.05, m =1.0. Hencethemassratioism /m = starts the second inflation driven by φ-field. After the
φ χ χ φ
20. Initial value of fields are set to φ = χ = 10. We secondinflationstarts,iso-curvatureperturbationdecays
∗ ∗
choose the e-folding number as a time coordinate and rapidlyduetothedecayofχ. HencethetimeN isbefore
c
6
þ;ÿ ñ
ÿÿ
10 100
þ
ÿ
8 80
6 60
4 40
2 20
N N
10 20 30 40 50 10 20 30 40 50
FIG.2: (coloronline)Thisfigureshowsevolutionofφ(dashed FIG. 5: This figure shows the evolution of the slow-roll pa-
red line) and χ (solid blue line). We choose the e-folding rameter ηχχ definedby Eq. (42).
number,N,asatimecoordinateandsettheinitialtime,N∗,
to 0.
ï f
NL
0.03
1.2
0.02
1
0.8 0.01
0.6 0
0.4
-0.01
0.2
-0.02
N
10 20 30 40 50
N
0 10 20 30 40 50 F
FIG. 3: This figure shows the evolution of the slow-roll pa- f
NL 8
rameter ǫ defined byEq. (41).
6
4
ñ
þþ
2
2
0
1.5 -2
-4
1 N
F
27 28 29 30 31
0.5
FIG. 6: This figureshows theevolution of non-linear param-
10 20 30 40 50 N eter, fNL(NF). After the second inflation, NF > 30, the
background trajectories in phase space have completely con-
verged. Thus,thecurvatureperturbationonaconstantHub-
ble hypersurface, ζ(NF), remains constant on large scales at
FIG. 4: This figure shows the evolution of the slow-roll pa- NF >30 and thefNL(NF) also remains constant in thisera.
rameter η defined by Eq.(42).
φφ
7
the end of the second inflation in this model. Then, the Here we recall that we have set N = 0. After the time
∗
sumoff (N >30)andthe sub-horizoncontributions when χ 1, χ-field is no longer in the slow-roll phase
NL F
∼
givestheprimordialnon-Gaussianity,independentofthe and ρ evolves as;
χ
reheating process. The final value of f (N > 30) is
NL F
0.01004. Hence the generation of large non-Gaussianity ρχ(N) ρχ(χ(Nos))exp[ 3(N Nos)] , (49)
≃ − −
due to the violation of the slow-roll conditions in this
where N represents the time when χ-field starts oscil-
model does not occur. os
lation. From Eq. (48), we have
In Ref. [16], the authors numerically calculated the
non-linear parameter for the model whose potential is
χ2 χ(N )2
given by Eq. (40) with mass ratio mφ/mχ = 12. They N ∗ os . (50)
os
used their own formalism, which is different from δN ≃ 4 − 4
formalism used in this paper. The behavior of the non-
Then, Eq. (49) can be rewritten as
linearparameterf obtainedbytheircalculationseems
NL
similar to our result shown in Fig. 6.
3
ρ (N) ρ˜ exp 3N + χ2 , (51)
χ ≃ χ − 4 ∗
(cid:18) (cid:19)
B. Approximate analytical expression for fNL where ρ˜ is independent of N, χ and φ . Since ρ de-
χ ∗ ∗ χ
cays in time for N >N , ρ dominates the totalenergy
os φ
In the previous subsection (IIIA), by numerical cal- density of the universe at some time and φ-field starts
culation, we found that the final value of fNL is much slow-rolling. Let us denote the e-folding number at the
lessthan1,althoughthecurvatureperturbationbecomes time when ρ ρ as N . For N > N , the evolution
φ χ eq eq
highly non-Gaussian (fNL 8) for a moment. We can of φ-field is giv≃en by
≃
derivetheanalyticalexpressionforthefinalvalueoff
NL
inthe limitoflargemassratiom /m 1,whichisthe
subject of this subsection. χ φ ≫ φ(N)≃ φ2∗−4(N −Neq) , (52)
q
The backgroundequations are given by
during the slow-roll phase. N is obtained from the
eq
equation,
d dφ dφ
H H +3H2 +m2φ=0 , (43)
dN dN dN φ 1
(cid:18) (cid:19) ρ (N ) m2φ(N )2 ρ (N ) . (53)
d dχ dχ φ eq ≃ 2 φ eq ≃ χ eq
H H +3H2 +m2χ=0 , (44)
dN dN dN χ
(cid:18) (cid:19) Combining Eq. (51) with Eq. (53), we have
m2φ2+m2χ2
H2 = φ χ . (45)
6+ dφ 2+ dχ 2 Neq χ2∗ 1logm2φφ2∗ . (54)
dN dN ≃ 4 − 3 2ρ˜χ
(cid:18) (cid:19) (cid:18) (cid:19)
Within the slow-roll approximation, these equations re- ThetotalenergydensityafterN =Neq isapproximately
duce to ρφ. Hence, using Eqs. (52) and (54), we have
d φ m2φφ , d χ m2χχ , ρ(N) ≃ ρφ(N)
dN ≃−3H2 dN ≃−3H2 m2
1 φφ2(N)
H2 m2φ2+m2χ2 . (46) ≃ 2
≃ 6 φ χ
m2 1 m2φ2
(cid:0) (cid:1) φ φ2+χ2 4 N + log φ ∗ .
At the initial time, N = N∗, both fields are in the slow- ≃ 2 " ∗ ∗− 3 2ρ˜χ !#
roll phase and φ χ 1. Then the ratio of the time
∗ ∗
≃ ≫ (55)
derivatives of the scalar fields is
dφ/dN m2φ φ After N = Neq, the universe is dominated by the sin-
1 , (47) gle slow-roll component φ-field, and the curvature per-
dχ/dN ≃ m2 χ ≪
χ turbation on the uniform density hypersurface becomes
constantin time onsuper-horizonscales. FromEq.(55),
andalsoρ ρ inthe limitoflargemassratio. Hence,
χ φ
≫ we obtain an expression for the e-folding number at the
to a good approximation, φ(N) φ until the energy
∗
≃ final time in terms of φ and χ as
density of φ-field dominates the total energy density of ∗ ∗
the universe, i.e., ρ ρ . On the other hand, solv-
ing Eq. (46), the evoφlu≃tionχof χ-field during the slow-roll N(N ,φ ,χ ) 1 φ2+χ2 2logφ
F ∗ ∗ ≃ 4 ∗ ∗ − 3 ∗
phase can be obtained as
(cid:0)ρ 1(cid:1) m2
F φ
log , (56)
χ(N) χ2 4N . (48) −2m2 − 3 2ρ˜
≃ ∗− φ χ
p
8
where ρ =ρ(N ). This gives density of the universeuntil φN 1 3, then the inflation
F F
is almost like the single field infla∼tion by φN-field. After
6 N N NIJ φN-fielddecays,thenφN−1-fielddecaysandsoon. Hence
I J
f (N )
− 5 NL F ≃ (N NK)2 the leading e-folding number would be given by
K
2
= φ2+χ2 1+O(φ−∗2, χ−∗2) ,(57) N 1 N φI2. (60)
∗ ∗ ≃ 4 ∗
(cid:0) (cid:1) I=1
X
where N = ∂N/∂φI, N = ∂2N/∂φI∂φJ ( Here, φI =
(φ,χ)). HI ence f ∗is I(Jφ−2, χ−2), a∗nd∗it is much less Then the corresponding fNL is
NL O ∗ ∗
than1. f estimatedusingthisexpressionis 0.01for
mφ = 0.0N5L, mχ = 1.0 and φ∗ = χ∗ = 10, wh≈ich agrees 6f 2 N φI2 −1 1 . (61)
wellwith the numericalresultin the previous subsection − 5 NL ∼ ∗ ! ∼ 2N
(see IIIA). XI=1
The reason why we get small fNL in the double infla- Roughly speaking, fNL is suppressed by the inverse of
tionmodelcanbeunderstoodasfollows. Inthecaseofa the e-folding number. This resultis quite similarto that
large mass ratio,evolution of the universe can be clearly inRef. [9], wherethe authorsstudiedunder the slow-roll
divided into three stages. First one is the inflation in- approximation.
duced almost by a single field, i.e., χ-field. The second
one is the non-inflationary phase where the energy den-
sityoftheoscillatingχ-fieldisstilllargerthanthatofthe IV. SUMMARY
φ-field. The thirdoneisthe inflationinducedbyasingle
field,i.e.,φ-field. Thenthetotale-foldingnumberduring
Based on the δN formalism, we have derived a useful
all these stages mainly comes from the two inflationary
formula for calculating the primordial non-Gaussianity
stages, i.e., the first and the third stages. The perturba-
due to the super-horizon evolution of the curvature per-
tionofthee-foldingnumbergeneratedduringthesecond
turbationinmulti-scalarinflationwithoutimposingslow-
stage is negligible. Hence as a crude approximation, the
roll conditions. This formula can apply for the infla-
dynamics can be regarded as just the sum of two single
tion models with general field space metric, h , as far
IJ
fieldinflationaryphases. Thispicturegivesthefirstterm
as super-horizoncontributions are concerned. Generally,
of Eq. (56). The correctionto this picture is represented
whenonecalculatesthenon-Gaussianityofthecurvature
by the second term, which is indeed minor for N 60.
perturbations, one has to solve the second order pertur-
≃
The derivation of Eq. (57) can be straightforwardly bationequations. Indoingsoforamulti-scalarinflation,
extended to more general double inflation models with
there appear tensorial quantities with respect to the in-
dices of the field components. Our formula reduces the
V(φ,χ)=c1φ2p+c2χ2p, (58) problem of calculating the non-linear parameter fNL to
solving only first order perturbation equations for two
where p is anarbitrary positive integer. In this case, the vector quantities. This reduces ( 2) calculations to
final value of fNL is more suppressedthan Eq. (57) by a ( ) ones where is the numOberNof the scalar field
factor 1/p. OcomNponents. HenceNourformalismhasagreatadvantage
forthenumericalevaluationoff intheinflationmodel
NL
composed of a large number of fields.
C. Comment on fNL in N-flation model We have also studied the primordial non-Gaussianity
in double inflation model as an example that violates
slow-roll conditions by using our formalism. We found
Discussion of the previous subsection about the small
that, although f defined for the curvature perturba-
f in the double inflation model gives us some insight NL
NL tion on a constant Hubble hypersurface exceeds 1 for a
into the generation of the non-Gaussianity in the so-
moment around the time when the slow-roll conditions
called -flation model [29], in which the potential is
given bNy are violated, the final value of fNL is suppressed by the
slow-rollparametersevaluatedatthetimeofhorizonexit.
Wehaveshownthatthis canbeunderstoodevenanalyt-
V(φ)= N m2IφI2. (59) ically in the δN formalism. This result is straightfor-
2
I=1
X
This is a generalization of the double inflation model to
an arbitrary number of fields. We consider the case in 3 If ρN becomes subdominant before φN ∼ 1, then φN is slowly
which all fields have large initial amplitude, i.e., φI 1. rolling until H decreases to mN. Such a case was studied in
We also assume m < m < < mN. In this c≫ase, [9, 10] under the slow-roll and horizon crossing approximation.
φN field decays firsφt. If ρχN do·m··inates the total energy rTohllepfianraalmveatleures.offNL wasfoundtobesuppressedbytheslow-
9
wardly extended to more general double inflation model and Hij by
IJ
and -flation model.
N
H V V V
H11 = IJ + I J ,
IJ 2 V V2
(cid:18) (cid:19)
Acknowledgments H3 H3
H12 = V ϕ , H21 = V ϕ ,
IJ 4V2 I 2J IJ 4V2 J 2I
SY thanks Takashi Nakamura for stimulating com- H3 3H2
H22 = δ + ϕ ϕ . (A4)
ments. We would like to thank F. Vernizzi and G. IJ 2V IJ 2V 2I 2J
(cid:18) (cid:19)
I. Rigopoulos for useful comments. TT is supported
byMonbukagakushoGrant-in-AidforScientificResearch
Nos. 17340075 and 19540285. This work is also sup- APPENDIX B: EXTENSION TO THE GENERAL
ported in part by the 21st Century COE “Center for FIELD SPACE METRIC
Diversity and Universality in Physics” at Kyoto univer-
sity, from the Ministry of Education, Culture, Sports, Inthis appendix, we willextend the formulationgiven
Science and Technology of Japan. The authors thank in sec. II for the flat field space metric h = δ to the
IJ IJ
the Yukawa Institute for Theoretical Physics at Kyoto generalcase. Forgeneralcase,the backgroundequations
University. Discussions during YITP workshop YITP- corresponding to Eq. (3) for flat field space are given by
W-07-10 on ”KIAS-YITP Joint Workshop, String Phe-
nomenologyandCosmology”wereusefultocompletethis dϕI DϕI
1 =FI , 2 =FI , (B1)
work. dN 1 dN 2
where DAI dAI + ΓI AJdϕK, ΓI is Christoffel
≡ JK 1 JK
symbol with respect to the field space metric h , and
IJ
APPENDIX A: SPECIFIC EXPRESSION
V V
FI =ϕI , FI = ϕI +hIJ J . (B2)
In this appendix, we show the explicit forms of Pa(= 1 2 2 −H2 2 V
b (cid:18) (cid:19)
PIj) in Eq. (9), Qa (=QIjk ) in Eq. (13), H (=Hi) in
iJ bc iJK a I In the same way as Eq. (6), we can define the perturba-
Eq. (36) and H (=Hij) in Eq. (39). PIj is given by
ab IJ iJ tion as
P1IJ1 =0 , P1IJ2 =δIJ , δϕa(N)=ϕa(λ+δλ;N) ϕa(λ;N) , (B3)
−
V VI VIV
PI1 = J J , and we decompose δϕI as
2J −H2 V − V2 i
(cid:18) (cid:19)
PI2 =ϕIϕ + VIϕ V δI . (A1) δϕa =δ(ϕ1)a+ 1δ(ϕ2)a . (B4)
2J 2 2J V 2J − H2 J 2
QIjk by The curvature perturbationcanbe expandedinterms of
iJK δϕI as
i
QI11 =QI21 =QI12 =QI22 =0 , 1
1JK 1JK 1JK 1JK ζ N δϕa+ N δϕaδϕb+ . (B5)
a ab
V VI VI V ≃ 2 ···
QI11 = JK J K
2JK −H2 V − V2 However, δϕI defined in this way does not transform as
i
a vector under the ”coordinate transformation”, φI
VIKVJ VIVJK +2VIVJVK , φ¯I(φ) because Eq. (B3) takes the difference of variabl→es
− V2 − V2 V3 ! at different points in field space. Hence the evolution
QI2J12K = VVIJ − VVIV2J ϕ2K , ethqiusadtiroanwfboarcδkϕ(Iithlaocukgshthneotmeasnseifnetsitalc)o,vwareiainntcreo.dTuoceavnoeiwd
(cid:18) (cid:19) (1) (2)
QI21 = VIK VIVK ϕ =QI12 , perturbation variables δϕ˜a =δϕ˜a+ 21δϕ˜a defined by
2JK V − V2 2J 2KJ
(cid:18) (cid:19) (1) dϕI (2) D dϕI
QI22 =δI ϕ +δI ϕ +δ ϕI + VI . (A2) δϕ˜I1 ≡ dλ1δλ , δϕ˜I1 ≡ dλ dλ1δλ2 , (B6)
2JK K 2J J 2K JK(cid:18) 2 V (cid:19) δ(ϕ˜1)I DϕI2δλ , δ(ϕ˜2)I D2ϕI2δλ2 , (B7)
2 ≡ dλ 2 ≡ dλ2
Hi is given by
I
which transform as vectors. Differentiating the back-
H V H3 ground equations (B1) with respect to λ, we obtain the
HI1 = 2 VI , HI2 = 2V ϕ2I , (A3) evolution equations for the perturbations of the scalar
10
fields in curved field space metric. To the first order, we of the first and second order equations are given by
have
(1) (1)
D δ(ϕ˜1)a(N)=(Pa(N)+∆Pa(N))δ(ϕ˜1)b(N) , (B8) δϕ˜a(N)=Λab(N,N∗)δϕ˜b(N∗),
dN b b N
(2)
δϕ˜a(N)= dN′Λa(N,N′)
where the Pab is given by ZN∗ b
(1) (1)
Pa = DFa , (B9) × Qbcd(N′)+∆Qbcd(N′) δϕ˜c(N′)δϕ˜d(N′) .
b ∂ϕb (cid:12)ϕ=(ϕ0)(N) (cid:2) (cid:3) (B15)
(cid:12) Theexpressionforthecurvatureperturbationinterms
and ∆Pa, which vanishes i(cid:12)(cid:12)n the flat field space metric ofδϕa,Eq.(B5),canbealsorewrittenasthatintermsof
b
representingtermsduetothecurvatureofthefieldspace, δϕ˜a. To make the expressionmore concise, we introduce
is given by new variables N˜a and N˜ab as expansion coefficients of ζ
in terms of δϕ˜a so that we have
∆PI1 =∆PI2 =∆PI2 =0 ,
1J 1J 2J
∆P2IJ1 =−RILJKϕL2ϕK2 . (B10) ζ ≃ N˜aFδϕ˜aF + 21N˜abFδϕ˜aFδϕ˜bF +··· . (B16)
Here, RI represents Riemann tensor associated with
LJK
(2) Comparing the two expressions (B5) and (B16), we find
the fieldspace metric h . Evolutionequationfor δϕ˜a is
IJ
also given by
N˜1 =N1 N2ΓK ϕJ , N˜2 =N2 ,
I I − K JI 2 I I
dDNδ(ϕ˜2)a(N)=(Pab(N)+∆Pab(N))δ(ϕ˜2)b(N) N˜I1J1+=NNI1IJ11K2++NNK22I2IK1ΓKMΓKLLΓJLϕNL2JϕM2 ϕN2
+(Qabc(N)+∆Qabc(N))δ(ϕ˜1)b(N)δ(ϕ˜1)c(N) , N˜12−=N(cid:0)NK21∇2 IΓNKLJ2ϕΓL2K(cid:1)−NK1NΓ2K2IJΓK+NϕK2LΓ,KILΓLNJϕN2 ,
(B11) N˜I2J1 =NI2J1−NK2ΓKIJ −NK22JΓKILϕL2 ,
IJ IJ − K IJ − KI JL 2
N˜22 =N22 . (B17)
where IJ IJ
Qabc = ∂Dϕ2b∂Fϕac (0) = D∂Pϕcab (0) ,(B12) From the constancy of N˜aδ(ϕ˜1)a, we find
(cid:12)ϕ=ϕ(N) (cid:12)ϕ=ϕ(N)
and the explicit form of(cid:12)(cid:12)(cid:12)∆Qa is given b(cid:12)(cid:12)(cid:12)y N˜a(N)=N˜bFΛba(NF,N) . (B18)
bc
∆QI11 = RI ϕL , We also redefine ΘIi in Eq. (31) as
1JK − JKL 2
∆QI12 =∆QI21 =∆QI22 =0 ,
1JK 1JK 1JK Θa(N)=Λa(N,N )AbcN˜ , (B19)
∆QI11 = RI ϕMϕL , b ∗ c∗
2JK −∇K MJL 2 2
∆QI2J12K =−2RIKJLϕL2 , ∆QI2J21K =−RILKJϕL2 , where Aab is defined by
∆QI22 =0 , (B13)
2JK
H 2
where is covariant derivative with respect to h . δϕ˜aδϕ˜b =Aab ∗ . (B20)
∇K IJ h ∗ ∗i 2π
By introducing the matrix Λ as the solution of (cid:18) (cid:19)
D Aab for generalcase also depends onthe field space met-
Λa(N,N )=(Pa(N)+∆Pa(N))Λc (N,N ),
dN b ∗ c c b ∗ ric, hIJ.
(B14) Then, the expression for the non-linear parameter in
general cases is also given by Eq. (30) with N (N′) and
c
with the condition ΛIj(N,N) = δIδ i, formal solutions Qc (N′)replacedbyN˜ (N′)andQc (N′)+∆Qc (N′).
iJ J j ab c ab ab
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