Table Of ContentNovember 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla
ModernPhysicsLetters A
(cid:13)c WorldScientificPublishingCompany
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THE SUYAMA-YAMAGUCHI CONSISTENCY RELATION IN THE
o
N PRESENCE OF VECTOR FIELDS
2
2
JUANP.BELTRA´NALMEIDA
] Centro de Investigaciones, Universidad Antonio Narin˜o,
O
Cra 3 Este # 47A-15, Bogot´a D.C. 110231, Colombia
C [email protected]
.
h
YEINZONRODR´IGUEZ
p
- Centro de Investigaciones, Universidad Antonio Narin˜o,
o Cra 3 Este # 47A-15, Bogot´a D.C. 110231, Colombia,
r and
t
s Yukawa Institute for Theoretical Physics, KyotoUniversity,
a Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto606-8502, Japan,
[ and
Escuela de F´ısica, Universidad Industrial de Santander,
2
Ciudad Universitaria, Bucaramanga 680002, Colombia
v
[email protected]
9
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1 CE´SARA.VALENZUELA-TOLEDO
6 Departamento de F´ısica, Universidad del Valle,
. Ciudad Universitaria Mel´endez, Santiago de Cali 760032, Colombia
2
[email protected]
1
1
1
: Received22Nov2012
v Revised(DayMonthYear)
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X
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a Weconsiderinflationarymodelsinwhichvectorfieldsareresponsibleforpartoreventu-
allyalloftheprimordialcurvatureperturbationζ.Suchmodelsarephenomenologically
interesting since they naturally introduce anisotropies in the probability distribution
functionoftheprimordialfluctuationsthatcanleaveameasurableimprintinthecosmic
microwave background. Assuming that non-Gaussianity is generated due to the super-
horizon evolution, we use the δN formalism to do a complete tree level calculation of
thenon-GaussianityparametersfNL andτNL inthepresenceofvectorfields.Weisolate
theisotropicpiecesofthenon-Gaussianityparameters,whichanywayhavecontributions
fromthevector fields,andshowthattheyobeytheSuyama-Yamaguchi consistencyre-
lation τNisLo > (56fNisLo)2. Other ways of defining the non-Gaussianity parameters, which
couldbeobservationallyrelevant,arestatedandtherespectiveSuyama-Yamaguchi-like
consistency relationsareobtained.
Keywords: Non-gaussianity;statisticalanisotropy;vector fieldmodels.
PACSNo.:98.80.Cq
1
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2 J. P. Beltr´an Almeida, Y. Rodr´ıguez, and C. A. Valenzuela-Toledo
1. Introduction
The study of non-Gaussianities in the primordial curvature perturbation ζ is a
subject of major interest in modern cosmology because the evaluation of the non-
Gaussianity(NG)parametersprovidecriteriatodiscriminateamongthemanymod-
1
elsproposedtoexplaintheoriginofthelarge-scalestructurethatweobservetoday .
Particular attention has been paid to the study of a consistency relation between
the f and τ parameters2,3, the Suyama-Yamaguchi (SY) consistency relation
NL NL
(τ > (6f )2 at least at tree level), because its violation would rule out many
NL 5 NL
of the popular models of inflation. So far, it has been shown that this consistency
2
relation works, in principle, on models that only include scalar fields under just a
fewassumptions,butthereisnoanyconclusiveresultformodelsthatincludeother
type of fields, for instance, vector fields (VF)a. Vector field models are particularly
interestingbecausetheyaresuitablecandidatestoexplaintheapparentviolationof
4
statistical isotropy observed in recent analysis of data from the WMAP satellite ,
giventhat VF define inherently a preferreddirectionfor the expansion,for the dis-
tribution of primordial fluctuations, or for both5,6,7. Because of this reason, it is
pertinent to study consistency relations in models that include scalar fields as well
as vector fields as these not only include NG parameters but also the amount of
8
primordial statistical anisotropy g .
ζ
ThemainpurposeofthisletteristofindouthowthewellknownSYconsistency
relation, valid for models involving only scalar fields under just a few assumptions,
is modifiedwhen including VF. Tothis end,we do a complete treelevelcalculation
7
of the f and τ parameters based on the δN formalism , and concentrate on
NL NL
the isotropic pieces of these parameters since these are the ones that are currently
constrained by observations. Under pretty the same few assumptions made in the
scalar fields case, and although the VF do contribute to the isotropic pieces of
the NG parameters, we prove that the SY consistency relation is still obeyed, at
least at tree level. Finally, the modified SY consistency relations for other ways of
defining the NG parameters, which could be observationally relevant when taking
into account the level of statistical anisotropy, are obtained.
2. The curvature perturbation
In the presence of VF, part or even all of the primordial curvature perturbation
ζ can be generated by vector field perturbations. In such scenarios, the curvature
perturbation can be calculated through the δN formalism as an expansion in the
7
perturbations of the fields a few e-folds after horizon exit :
1 1
ζ ≡δN =N δA + N δA δA + N δA δA δA +··· , (1)
a a ab a b abc a b c
2 3!
aVFnaturallygeneratescale-dependenceand,therefore,theproofsinRef.3donotapplytothose
scenarioswhereVFcontributetothegenerationoftheprimordialcurvatureperturbation.
November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla
The Suyama-Yamaguchi consistency relation inthe presence of vectorfields 3
where N is the number of e-folds during inflation and N are the derivatives of
ab···c
N with respect to A . In the previous expression, and in order to avoid notational
a
clustering,we haveassumedthatthereis only onescalarandonevectorfield;thus,
δA ≡δφ for a=0, and δA ≡δA , the latter being the spatial components of the
a a i
6
vector field, for a=i=1,2,3 , where these field perturbations are calculated in a
flat slicing (the threading must be the comoving one). The extension of the results
found in this letter to the case of severalscalarand vectorfields is straightforward.
3. Calculation of the correlators
3.1. The Power Spectrum
The power spectrum (PS) of the primordial curvature perturbation ζ is defined as:
hζ(~k )ζ(~k )i= (2π)3δ(~k )P (~k ) (2)
1 2 12 ζ 1
2π2
= (2π)3δ(~k ) P (~k ), (3)
12 k3 ζ 1
whereP (~k)isthedimensionless power spectrum,and~k =~k +~k .Thestatistical
ζ 12 1 2
anisotropyinζ,duetotheexistenceofapreferreddirectionnˆ,canbeparametrized
9
by a function g as follows :
ζ
P (~k)=Piso(k) 1+g (kˆ·nˆ)2 , (4)
ζ ζ ζ
(cid:16) (cid:17)
where Piso(k) denotes the isotropic part of the PS and the vectors kˆ and nˆ are of
ζ
unit norm.
The first assumption we will make is that the NG in the field perturbations is
negligible, i.e., the NG is generated due to the superhorizon evolution. The second
assumption is to consider that the field perturbationsb are statistically isotropic,
which is equivalent to the fact that there is no correlation between scalar and
6
vectorfieldsandthattheexpansionisisotropic .Thismaybereasonableasagood
7
approximationinseveralcases ,althoughitmustbeprovenineachparticularcase,
and is essential in the development of this letter because, otherwise, there would
not exist any isotropic contributions to the NG parameters which are what really
is constrained by observations. We shall adopt the notation
hδA (~k )δA (~k )i=(2π)3δ(~k )Π (~k ), (5)
a 1 b 2 12 ab 1
with the components
hδA (~k )δA (~k )i =(2π)3δ(~k )P (k ), (6)
0 1 0 2 12 δφ 1
hδA (~k )δA (~k )i =(2π)3δ(~k )Π (~k ), (7)
i 1 j 2 12 ij 1
and zero otherwise. In the latter expression:
Π (~k)=Πeven(~k)P (k)+Πodd(~k)P (k)+Πlong(~k)P (k), (8)
ij ij + ij − ij long
bWerefer,inthecaseofVF,tothescalarperturbationsthatmultiplytherespectivepolarization
vectors inapolarizationmodeexpansion.
November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla
4 J. P. Beltr´an Almeida, Y. Rodr´ıguez, and C. A. Valenzuela-Toledo
whereΠeven(~k)=δ −kˆ kˆ , Πodd(~k)=iǫ kˆ ,andΠlong(~k)=kˆ kˆ ,P (k)being
ij ij i j ij ijk k ij i j long
the longitudinal powerspectrum andP (k) and P (k) being the parity conserving
+ −
and violating power spectra respectively6,7. The fact that none of P (k), P (k),
δφ +
P (k), and P (k) depend on the direction of the wavevector is a consequence of
− long
the second assumption.
At tree level, the PS is6,7:
P (~k)=N N Π (~k). (9)
ζ a b ab
For concreteness, we shall restrict to the case of vector field perturbations which
preserve parity, then, we just keep the P (k) and P (k) parts of the vector field
+ long
power spectra. We also suppose that all the δA perturbations evolve in the same
i
wayafter horizoncrossingsothat they havethe same spectralindex in their power
spectra.Insuchconditions,thelongitudinalandtheparityconservingpowerspectra
arerelatedbyP (k)=rP (k)whererisascaleindependentnumber.Withthese
long +
considerations in mind, the vector power spectra are:
Π (~k)= δ +(r−1)kˆ kˆ P (k). (10)
ij ij i j +
h i
Our next steps are to evaluate the PS by employing the δN expansion and to
separate the isotropic and anisotropic pieces obtaining in this way:
2
P (~k)=N N P (k)+(r−1) N kˆ P (k)=Piso(k) 1+g (nˆ kˆ )2 , (11)
ζ a b ab i i + ζ ζ i i
(cid:16) (cid:17) (cid:16) (cid:17)
where
Piso(k)= N N P (k), (12)
ζ a b ab
2
Paniso(k)= (r−1) N kˆ P (k), (13)
ζ i i +
(cid:16) (cid:17)
nˆ =N /(N N )1/2 and the relation between the (r−1) factor and the anisotropy
i i j j
parameter g in the power spectrum is given by:
ζ
N N P
i i +
g =(r−1) . (14)
ζ
N N P
a b ab
In the latter expression, the dimensionless power spectra P assume the values
ab
P =P and P =δ P .
00 δφ ij ij +
3.2. The Bispectrum.
The bispectrum (BS) of the primordial curvature perturbation is defined as:
3
h ζ(~k )i =(2π)3δ(~k )B (~k ,~k ,~k ) (15)
i 123 ζ 1 2 3
iY=1
4π4
=(2π)3δ(~k ) B (~k ,~k ,~k ). (16)
123 k3k3 ζ 1 2 3
1 2
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The Suyama-Yamaguchi consistency relation inthe presence of vectorfields 5
Using the δN expansion we get the tree level BS:
B =N N N Π (~k )Π (~k )+2perm. . (17)
ζ a b cd ac 1 bd 2
h i
Expanding the above by employing the expression in (10) and separating the tree
level BS into the isotropic and the anisotropic pieces we get
Biso(k , k , k )=N N N P P (k k )−3, (18)
ζ 1 2 3 a b cd ac bd l m
lX<m
Baniso(~k ,~k ,~k )=(r−1)N N N P P (k k )−3(kˆ kˆ +kˆ kˆ )
ζ 1 2 3 a i bj ab + l m (l)i (l)j (m)i (m)j
lX<m
+(r−1)2N N N P2 (kk )−3kˆ kˆ kˆ kˆ . (19)
i k jn + l m (l)i (l)j (m)k (m)n
lX<m
3.3. The Trispectrum
Finally,weevaluatethe trispectrum(TS)ofthe primordialcurvatureperturbation:
4
h ζ(~k )i=(2π)3δ(~k )T (~k ,~k ,~k ,~k ) (20)
i 1234 ζ 1 2 3 4
iY=1
8π6
=(2π)3δ(~k ) T (~k ,~k ,~k ,~k ).
1234 k3k3k3 ζ 1 2 3 4
1 2 23
(21)
Once again, we use the δN expansion to get the tree level contribution to the TS,
which results in
T =TτNL +TgNL (22)
ζ ζ ζ
=N N N N Π (~k )Π (~k )Π (~k )+11perm.
a b cd ef ac 1 be 2 df 13
h i
+N N N N Π (~k )Π (~k )Π (~k )+3perm. .
a b c def ad 1 be 2 cf 3
h i
(23)
As we are interested in the τ amplitude, we only need the term TτNL given that
NL ζ
the term TgNL enters in the definition of the g parameter. The decomposition of
ζ NL
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6 J. P. Beltr´an Almeida, Y. Rodr´ıguez, and C. A. Valenzuela-Toledo
TτNL in isotropic and anisotropic pieces is given by
ζ
Tζiso(τNL) = NaNbNcdNefPacPbePdf (klkmkls)−3, (24)
l<m,s6=m,l
X
Tζaniso(τNL) = (r−1)NiNbNjdNefPbePdfP+ (klkmkls)−3(kˆ(l)ikˆ(l)j+kˆ(m)ikˆ(m)j)
l<mX,s6=m,l
+ (r−1)NbNdNeiNfjPbePdfP+ (klkmkls)−3kˆ(ls)ikˆ(ls)j
l<mX,s6=m,l
+ (r−1)2NiNkNjdNlfPdfP+2 (klkmkls)−3kˆ(l)ikˆ(l)jkˆ(m)kkˆ(m)n
l<mX,s6=m,l
+ (r−1)2NiNdNjkNfnPdfP+2 (klkmkls)−3kˆ(ls)kkˆ(ls)n(kˆ(m)ikˆ(m)j+kˆ(l)ikˆ(l)j)
l<mX,s6=m,l
+ (r−1)3NiNkNjmNhnP+3 (klkmkls)−3kˆ(l)ikˆ(l)jkˆ(m)kkˆ(m)hkˆ(ls)mkˆ(ls)n. (25)
l<mX,s6=m,l
4. The NG parameters fNL and τNL
In this work we consider the definition of the non-gaussianity parameters f and
NL
τ as follows
NL
6
B = f (P (~k )P (~k )+2perm.), (26)
ζ NL ζ 1 ζ 2
5
T =τ (P (~k )P (~k )P (~k )+11perm.). (27)
ζ NL ζ 1 ζ 2 ζ 14
Noticethat,inthedefinitions above,wehaveusedthefullPSincludingbothscalar
and vector field perturbations which, accordingly,include isotropic and anisotropic
contributions. Nevertheless, it is usual to find in the literature a definition which
1
considers only the isotropic piece of the PS , so the NG parameters are given by
B = 6f′ (Piso(~k )Piso(~k )+2perm.) and T = τ′ (Piso(~k )Piso(~k )Piso(~k )+
ζ 5 NL ζ 1 ζ 2 ζ NL ζ 1 ζ 2 ζ 14
11perm.). We can relate both definitions using the expression in (11) for the PS:
1 1
f = f′ = f′ , (28)
NL 1+χ1 NL 1+ 2i=1mi(gζ)i NL
1 P 1
τ = τ′ = τ′ , (29)
NL 1+χ2 NL 1+ 3i=1li(gζ)i NL
P
wherethe parametersm andl areobtainedfromthe expansionofthe products of
i i
the PS in (11) which are in the definitions of the NG parameters. They depend on
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The Suyama-Yamaguchi consistency relation inthe presence of vectorfields 7
the derivatives of N and the momenta~k . Explicitly:
i
Pζ(~k1)Pζ(~k2)+2perm.=(1+χ1) Pζiso(k1)Pζiso(k2)+2perm.
= 1+g nˆ nˆ l<m(klkm)−3(kˆ(hl)ikˆ(l)j+kˆ(m)ikˆ(m)j) i
" ζ i j P l<m(klkm)−3
+gζ2nˆinˆjnˆknˆn Pl<m(klkm)Pl<−m3kˆ((kl)likkˆm(l))−jkˆ3(m)kkˆ(m)n#(NaNbPab)2lX<m(klkm)−3,
P (30)
Pζ(~k1)Pζ(~k2)Pζ(~k13)+11perm.=(1+χ2) Pζiso(k1)Pζiso(k2)Pζiso(k13)+11perm.
l<m,s6=m,l(klkmkls)−3(kˆ(hl)ikˆ(l)j +kˆ(m)ikˆ(m)j+kˆ(ls)ikˆ(ls)j) i
= 1+g nˆ nˆ
" ζ i j P l<m,s6=m,l(klkmkls)−3
+gζ2nˆinˆjnˆknˆn Pl<m,s6=m,l(klkmkPls)−3hkˆ(l)ikˆ(l)l<jkˆm(m,s6=)kmkˆ(,lm(k)nlk+mkkˆl(sls))−ik3ˆ(ls)j(kˆ(m)kkˆ(m)n+kˆ(l)kkˆ(l)n)i
+gζ3nˆinˆjnˆknˆhnˆtnˆn Pl<m,s6=m,l(klkmlk<lms),−s6=3mkˆP(,ll)(ikkˆl(kl)mjkˆk(lms))−kk3ˆ(m)hkˆ(ls)tkˆ(ls)n#
×(NaNbPab)3 (klkmkls)−3P. (31)
l<mX,s6=m,l
Certainly, both definitions of the NG parameters are approximately equal if the
correctionsarisingfromtheexistenceofsomestatisticalanisotropieswerenegligible.
Otherwise, such corrections would be very important from the observational point
of view.
4.1. The fNL parameter
For the f parameter we can use the results (12), (13), (18), and (19) to sepa-
NL
rate the isotropic and the anisotropic contributions as follows: f =fI +fII =
NL NL NL
Biso/(P P + 2p.)+ Baniso/(P P + 2p.). The labels I and II are related to the
ζ ζ ζ ζ ζ ζ
isotropicandthe anisotropicpieces ofthe bispectrum in (18) and(19) respectively.
Using this decomposition, we can compare the fI and fII contributions by eval-
NL NL
uating its ratio
fII 2
NL ≡ξ ≡ f (g )l
fI 1 l ζ
NL Xl=1
N N N N N P P (k k )−3(kˆ kˆ +kˆ kˆ )
=g a bj i c d ab cd l<m l m (l)i (l)j (m)i (m)j
ζN N N N N P P P (k k )−3
a b cd k k ac bd l<m l m
(N N P )2N N N (k kP)−3kˆ kˆ kˆ kˆ
+g2 a b ab i k jn l<m l m (l)i (l)j (m)k (m)n . (32)
ζ(N N )2N N N P P P (k k )−3
i i a b cd ac bd l<m l m
P
In the latter expression, we have used the expression in (14) to write the series
expansion in terms of the level of statistical anisotropy g . The coefficients f can
ζ l
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8 J. P. Beltr´an Almeida, Y. Rodr´ıguez, and C. A. Valenzuela-Toledo
be read off directly from the latter expression. Using the ratio fII /fI , we can
NL NL
write the total f as: f = (1+ξ )fI = (1+ξ )/(1+χ )fiso, where fiso is
NL NL 1 NL 1 1 NL NL
the piece of f that depends only on the isotropic BS in (18) and the isotropic
NL
PS in (12). A very appealing feature that we can see from the previous analysis
is that the ratio of the terms related to the anisotropic and the isotropic parts of
the NG parameter f depends on the parameters of the model; for instance, it
NL
depends on the value of the ratio r, the level of statistical anisotropy g , and the
ζ
shape of the triangle defined by the ~k momenta. Moreover, interestingly enough,
i
for some values of g and for some configurations of the momenta, the anisotropic
ζ
contribution could be of the same order of magnitude as the isotropic piece and
cannot be neglected.
4.2. The τNL parameter.
For the τ parameter, we apply the same decomposition related to the isotropic
NL
andanisotropicpiecesthatwedidbeforewiththef parameter:τ =τI +τII =
NL NL NL NL
Tiso/(P P P +11p.)+Taniso/(P P P +11p.). We follow the same steps that we
ζ ζ ζ ζ ζ ζ ζ ζ
did in the case of the f parameter and write the ratio of both contributions as a
NL
series in powers of the g parameter:
ζ
τII 3
τNIL ≡ξ2≡ τl(gζ)l
NL Xl=1
=g (NaNbPab)NiNcNjdNefPcePdf l<m,s6=m,l(klkmkls)−3(kˆ(l)ikˆ(l)j+kˆ(m)ikˆ(m)j)
ζ(NiNi)NaNbNcdNefPacPbePdf P l<m,s6=m,l(klkmkls)−3
+g (NaNbPab)NbNdNeiNfjPbePdf l<m,s6=m,l(Pklkmkls)−3kˆ(ls)ikˆ(ls)j
ζ(NiNi)NaNbNcdNefPacPbePdf P l<m,s6=m,l(klkmkls)−3
+g2 (NaNbPab)2NiNkNjdNnfPdf l<Pm,s6=m,l(klkmkls)−3kˆ(l)ikˆ(l)jkˆ(m)kkˆ(m)n
ζ(NiNi)2NaNbNcdNefPacPbePdf P l<m,s6=m,l(klkmkls)−3
+g2 (NaNbPab)2NiNdNjkNfnPdf l<m,s6=mP,l(klkmkls)−3kˆ(ls)kkˆ(ls)n(kˆ(m)ikˆ(m)j+kˆ(l)ikˆ(l)j)
ζ(NiNi)2NaNbNcdNefPacPbePdf P l<m,s6=m,l(klkmkls)−3
+g3 (NaNbPab)3NiNkNjmNhn l<m,s6=m,l(klkmPkls)−3kˆ(l)ikˆ(l)jkˆ(m)kkˆ(m)hkˆ(ls)mkˆ(ls)n .
ζ(NiNi)3NaNbNcdNefPacPbePdf P l<m,s6=m,l(klkmkls)−3
P (33)
The coefficients τ can be read off directly from the expression in (33). Then, the
l
τ parameter canbe written as: τ =(1+ξ )τI =(1+ξ )/(1+χ )τiso, where
NL NL 2 NL 2 2 NL
τiso isthepartofτ thatdependsonlyontheisotropicTSin(24)andtheisotropic
NL NL
PS in (12). The ratio again depends on the g parameter and on the configuration
ζ
described by the~k momenta.
i
5. A consistency relation for fNL and τNL
Searching for relations between the NG parameters, we start exploring their
isotropic pieces. Here we give an explicit proof that the isotropic pieces of the
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The Suyama-Yamaguchi consistency relation inthe presence of vectorfields 9
NG parameters, in presence of scalar and vector field perturbations, obey the SY
2
consistency relation . Using the expressions in (12), (18), and (24) we find that:
6 N N N P P
f(iso) = a b cd ac bd , (34)
5 NL (N N P )2
a b ab
N N N N P P P
(iso) a b cd ef ac be df
τ = . (35)
NL (N N P )3
a b ab
Now, we can write the scale invariant power spectra P as:
ab
1 0
P =D P = P , (36)
ab ab δφ (cid:18)0sδ (cid:19) δφ
ij ab
wheres=P /P isthevectortoscalarPSratio.DefiningthevectorsK =D N
+ δφ a ab b
and T =N K , we evaluate the ratio A =τiso/(6fiso)2:
a ab b NL NL 5 NL
(N K )(D T T ) (D N N )(D T T )
A = a a ab a b = ab a b ab a b >1,
NL (K T )2 (D N T )2
a a ab a b
(37)
which implies:
6
τiso >( fiso)2. (38)
NL 5 NL
The latter expression is a direct consequence of the Cauchy-Schwartz inequality
for the vectors N and T with the positive definite metric D , and reflects the
a a ab
fact that the SY consistency relation, at least at tree level, applies for the isotropic
pieces of the NG parameters even in the presence of vector fields.
Now,wecantrytoestablisharelationforthecompleteNGparametersf and
NL
τ .ThefactthattheisotropicNGparametersobeytheSYinequalityimpliesthat
NL
the total NG parametersobey a modified consistency relation.The modificationto
this relation will depend on several aspects, for instance, on the level of statistical
anisotropy, the momenta configuration, etc. Employing the SY inequality for the
isotropic parameters, we deduce that the total NG parameters obey the relation:
(1+ξ )(1+χ )2 6 2
τ > 2 1 f . (39)
NL (1+χ )(1+ξ )2 (cid:18)5 NL(cid:19)
2 1
On the other hand, the relation obeyed by the NG parameters f′ and τ′ is the
NL NL
following:
2
(1+ξ ) 6
τ′ > 2 f′ . (40)
NL (1+ξ )2 (cid:18)5 NL(cid:19)
1
All of these relations represent a constraint on the NG and statistical anisotropy
parameters for models of inflation that involve vector fields.
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10 J. P. Beltr´an Almeida, Y. Rodr´ıguez, and C. A. Valenzuela-Toledo
6. Conclusions
In this paper,we have studied under which conditions new kinds ofthe well known
Sayama-Yamaguchiconsistencyrelationareallowedwhenvectorfieldsareincluded
in the inflationary dynamics. In particular we have shown that the isotropic pieces
ofthenon-gaussianityparametersobeytheSuyama-Yamaguchiconsistencyrelation
τiso >(6fiso)2 when its calculation is performed at tree level.
NL 5 NL
We have also derived a modified consistency relation for the complete non-
gaussianity parameters f and τ and for a set of new parameters f′ and τ′
NL NL NL NL
that could be observationally relevant; these relations are given by the Eqs. (39)
and (40). Since the latter depend on the level of statistical anisotropy and the
configuration of the wavevectors, the naive relations τ > (6f )2 and τ′ >
NL 5 NL NL
(6f′ )2 could eventually be violated in some particular inflationary model that
5 NL
includes vector fields. On this basis, we expect that in a near future the derived
relationscouldprovideacriteriatodiscriminateamongthedifferentmodelsforthe
generationoftheprimordialcurvatureperturbationwhenvectorfieldsareinvolved.
Acknowledgments
J.P.B.A. and Y.R. are supported by Fundaci´on para la Promoci´on de la Investi-
gaci´onylaTecnolog´ıadelBancodelaRepu´blica(COLOMBIA)grantnumber3025
CT-2012-02. J.P.B.A. is supported by VCTI (UAN) grant number 2010251. Y.R.
was a JSPS postdoctoral fellow (P11323) and, in addition, is supported by DIEF
de Ciencias (UIS) grant number 5177. C.A.V.-T. is supported by Vicerrector´ıa de
Investigaciones (UNIVALLE) grant number 7858.
References
1. E. Komatsu et. al., arXiv:0902.4759 [astro-ph.CO]; E. Komatsu, Class. Quantum
Grav. 27, 124010 (2010); A. P. S. Yadav and B. D. Wandelt, Adv. Astron. 2010,
565248 (2010).
2. T.SuyamaandM.Yamaguchi,Phys. Rev.D77,023505 (2008);T.Suyama,T.Taka-
hashi, M. Yamaguchi, and S. Yokoyama, JCAP 1012, 030 (2010); N. S. Sugiyama,
E.Komatsu,andT.Futamase,Phys. Rev. Lett. 106,251301 (2011); N.S.Sugiyama,
JCAP 1205, 032 (2012); A.Kehagias and A. Riotto, Nucl. Phys. B864, 492 (2012).
3. K. M. Smith, M. LoVerde, and M. Zaldarriaga, Phys. Rev. Lett. 107, 191301 (2011);
V.Assassi, D.Baumann, and D.Green, arXiv:1204.4207 [hep-th].
4. N.E.Groeneboom,L.Ackerman,I.K.Wehus,andH.K.Eriksen,Astrophys. J.722,
452 (2010); N.E. Groeneboom and H. K.Eriksen, Astrophys. J. 690, 1807 (2009).
5. S.YokoyamaandJ.Soda,JCAP0808,005(2008); K.Dimopoulos, Phys. Rev. D74,
083502(2006);K.Dimopoulos,Int.J.Mod.Phys.D21,1250023(2012)[Erratum-ibid.
D21, 1292003 (2012)]; T. R. Dulaney and M. I. Gresham, Phys. Rev. D81, 103532
(2010);A.E.Gumrukcuoglu,B.Himmetoglu,andM.Peloso,Phys.Rev.D81,063528
(2010);M.-a.Watanabe,S.Kanno,andJ.Soda,Prog.Theor.Phys.123,1041(2010).
6. C. A. Valenzuela-Toledo, Y. Rodr´ıguez, and J. P. Beltr´an Almeida, JCAP1110, 020
(2011).
7. K. Dimopoulos, M. Karˇciauskas, D. H. Lyth, and Y. Rodr´ıguez, JCAP 0905, 013
(2009).
