G-Curvaton

0.24 MB·English

by  Hua Wang

G-Curvaton
Hua Wang1,∗ Taotao Qiu2,3,4,† and Yun-Song Piao1‡
1 College of Physical Sciences, Graduate University of Chinese Academy of Sciences, Beijing 100049, China
2. Department of Physics and Center for Theoretical Sciences,
National Taiwan University, Taipei 10617, Taiwan
3. Leung Center for Cosmology and Particle Astrophysics National Taiwan University, Taipei 106, Taiwan and
4. TPCSF, Institute of High Energy Physics, CAS, P.O. Box 918-4, Beijing 100049, P.R. China
Abstract
2
1
In this paper, we study a curvaton model where the curvaton is acted by Galileon field. We cal-
0
culatethepowerspectrumoffluctuationofG-curvatonduringinflationanddiscusshowitconverts
2
to the curvature perturbation after the end of inflation. We estimate the bispectrum of curvature
n perturbationinduced,andshowthedependenceofnon-Gaussianity ontheparametersofmodel. It
a is found that our model can have sizable local and equilateral non-Gaussianities to up to O(102),
J
which is illustrated by an explicit example.
0
1
PACSnumbers:

]
h
p
-
p I. INTRODUCTION
e
h
Inflationhasnowbeenconsideredasoneofthemostsuccessfultheorytodescribeouruniverseatearlyepoches[1–3].
[
Makingouruniverseexpandfastenough,itcannaturallysolvemanynotoriousproblemsbroughtoutbyhotBigBang,

2 such as flatness problem, horizon problem, monopole problem and so on. Moreover, during inflation the quantum
v
fluctuations generated at the initial stage can be stretched out of the horizon to form classical perturbations, which
5
can provide seeds for the formation of the structure of our universe. An inflation model also succeeds in producing
9
7 nearly scale-invariant power spectrum of scalar perturbation and tiny gravitational waves, which fits very well with
1 today’s observational data [4]. The non-Gaussian corrections of perturbations during inflation can also be large or
. not, according to various inflation models, which is waiting for constraints from new and more accurate data in the
0
near future [5].
1
1 Usually, the perturbation of inflation is generated by the inflaton field itself, which is the simplest way to have
1 curvature perturbation. However, it is not the only choice. Perturbations generated in such a way depends on the
: potential of the inflaton field, and thus puts very severe constraint on inflation models. In order to relax such a
v
constraint, as Lyth and Wands have pointed out, perturbation can also be generated from another field that has
i
X nothing to do with the inflaton field, namely, the curvaton [6], see also relevant works on curvaton mechanism in
r Refs. [7, 8] and earlier [9, 10]. Curvaton field is usually assumed to be a scalar field with light mass and decoupled
a
from all the other kinds of perturbations, thus the perturbation produced by curvaton can be independent on the
nature of inflation. Moreover, since the curvaton is subdominant during inflation, it can only produce isocurvature
perturbation. This isocurvature perturbation has to be convertedinto curvature perturbationat the end of inflation,
so it depends on what happened after the inflation terminated. Usually, there are two cases in which this conversion
can be available: First, when the inflaton decays into radiation after inflation, the curvaton field becomes dominate;
Second,thecurvatonfielddecaysaswellbeforeitsdomination,andreachesequilibriumwithradiationthatdecaysfrom
inflaton. Accordingtodifferentcase,the amountofthe curvatureperturbationconvertedfromcurvatonperturbation
may be different. Curvaton scenarios have been widely studied in, for instance, [11].
Intheoriginalcurvatonpaper,itwassuggestedthatcurvatonismadeofacanonicalscalarfield. However,otherfield
models can be considered to act as a curvaton. People have considered curvaton of Pseudo-Nambu-Goldstone-Boson
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
2
[12], DBI-type [13] or its curvaton brane implement [14], multi-field [15], lagrangian multiplier field [16], Horava-
Lifshitz scenario [17] and so on. Very recently, a new kind of models has been proposed and studied extensively,
which is called “Galileon” models [18]. The original version of these models is a generalization of an effective field
description of the DGP model [19]. These models includes high derivative operator of the scalar field, however, due
to some “delicated design”, its equation of motion remains second order, 1 thus it can violate the NEC without
incorporating any instability modes. Due to such an interesting property, such kind of model generically can admit
superluminal propagation [21] and perhaps even closed timelike curves (CTCs) [22] (see also [23] for CTCs without
Galileon). Moreover,these models can be applied onto various evolution period of our universe, such as dark energy
[24, 25], inflation [26–28], reheating [29], bouncing [30], the slow expansion scenario [31] of primordial universe [32],
[33], and so on. As an extension, these models can also be generalized to DBI version [34], the K-Mouflage scenario
[35], the supersymmetric Galileon [36], the Kinetic Gravity Braiding models [24], the generic Galileon-like action
[20, 37, 38] and others. See also e.g. Refs. [39–46] for various study of their phenomenologies.
In the present work, we study the scenario where a Galileon field behaves as a curvaton, which we dubbed as
“G-Curvaton” scenario. Due to the higher derivative term, Galileon is expected to have some features on generating
perturbations,suchasgettinglargetensor-scalarratioin“G-inflation”scenario. Ourpaperisorganizedasfollows: in
Sec. II we review the original curvaton scenario, by taking curvaton to be the simplest one, i.e., the canonical scalar
field. In Sec. III we study our “G-Cuvaton” model. We first investigate its perturbation, obtaining a scale-invariant
power spectrum. Then we discuss how it converts to the curvature perturbation at the end of inflation, considering
both cases where curvaton decays before and after it dominates the universe. We also study the non-Gaussianities
generated from our model, both local type and equilateral type. Finally we present an explicit example to show how
the observable quantities could be effected. Sec. IV comes our conclusion and discussions.
II. REVIEW OF THE SIMPLEST CURVATON MODEL
In this section, we would like to review how the mechanism works for the simplest curvaton model, which is made
of a canonical scalar field. The Lagrangianfor the curvaton field is:
1
= ∂ σ∂µσ V(σ) . (1)
σ µ
L −2 −
Notethathereweareusingthe metricwithnotationds2 = dt2+a2(t)δ dxidxj. Asforcurvaton,the effectivemass
ij
of σ needs to be very light, which put the constraint V − H2 to the potential V(σ), where V ∂2V/∂σ2 and
,σσ ,σσ
| |≪ ≡
H denotes the Hubble parameterofthe universe. Moreover,duringinflationH isalmostaconstant. Thus ingeneral,
one may define the parameters
H˙ V
,σσ
ǫ , ξ = , (2)
≡−H2 3H2
which are required to be far smaller than 1 and slowly varying. Moreover, since curvaton is subdominant part of
the universe during inflation, the energy scale of the potential should also be lower than that of inflation, namely
V(σ) 3M2H2.
≪ pl
Suppose the field generates fluctuation during inflation, namely σ(x) σ (t)+δσ(t,x), then from Eq. (1) we can
0
→
easily get the equation of motion for the field fluctuation δσ as:
δ¨σk+3Hδ˙σk+((k/a)2+V,σσ)δσk =0 , (3)
whereδσk istheFourierpresentationofδσ withmomentummodek,andaisthescalefactor. Notethatherewehave
already neglected the coupling of δσk to the metric perturbation, as has been done in [6]. Thus the power spectrum
of the field fluctuation δσk at the horizon crossing is given by
k3 H 2
Pσ ≡ 2π2|δσk|2 = 2π∗ , (4)
(cid:18) (cid:19)
where the star denotes the time of horizon exit, k =a H . The spectral index of the spectrum can also be given by:
∗ ∗
dln
σ
n 1 P = 2ǫ+2ξ 1 , (5)
σ
− ≡ dlnk − ≪
1 ThisideaisactuallypioneeredbyHorndeskithirtyyearsago,in[20].
3
which shows the nearly scale-invariance of the spectrum .
σ
P
However, since the curvaton is the subdominant part of the universe during inflation, the perturbation generated
by it can only be of isocurvature type. This type of perturbation can only be converted into curvature perturbation
when curvaton dominates, or become equilibrium with other part of the universe. Either these cases will not happen
during inflation, however after inflation, when inflaton field decays into radiation whose energy density may decrease
more rapidly than curvaton field, both the two cases will happen, depending on when the curvaton field will decay.
In the case when the curvaton decays late, it will exceed over the radiation decayed from inflaton and dominate the
universe, however in the case when the curvaton decays early, it may become equilibrium with the radiation. The
curvature perturbations for a given matter with energy density ρ in spatial-flat slicing is given by [47]:
δρ
ζ = H , (6)
− ρ˙
and the separately conserved curvature perturbations for radiation and curvaton field therefore read:
δρ 1δρ
r r
ζ = H = , (7)
r
− ρ˙ 4 ρ
r r
δρ δρ
σ σ
ζ = H = , (8)
σ
− ρ˙ 3(ρ +p )
σ σ σ
respectively, where p is the pressure of the curvaton field. Using these results, and assuming that the isocurvature
σ
perturbationconverttocurvatureperturbationinstantly,thenthecurvatureperturbationgeneratedbysuchconversion
reads:
4ρ ζ +3(ρ +p )ζ 3(ρ +p )ζ
r r σ σ σ σ σ σ
ζ = , (9)
4ρ +3(ρ +p ) ≃ 4ρ +3(ρ +p )
r σ σ r σ σ
where in the last step we neglected the curvature perturbation of radiation, ζ . Define the energy density ratio of σ
r
over radiation, r ρ /ρ , then in the first case where the curvaton dominates, we have r 1, Eq. (9) becomes:
σ r
≡ ≫
ζ ζ , (10)
σ
≃
while in the second case where σ become equilibrium with radiation, we have ρ ρ , Eq. (9) becomes:
σ r
≪
3
ζ r(1+w )ζ , (11)
σ σ
≃ 4
where w p /ρ is the equation of state of the curvaton field.
σ σ σ
≡
Furthermore, we investigate the non-Gaussianities of the perturbation generated by the curvaton field. The local
type non-Gaussianities of curvature perturbation are given by:
3
ζ =ζ + flocalζ2 , (12)
g 5 NL g
wherethe subscript“g”denotesthe Gaussianpartofζ while f is the so-callednonlinearestimator. Forlocaltype,
NL
flocal can be estimated by using the so-called δN formalism, where δN is the variation of the number of efolds N of
NL
inflation [48]:
1
ζ =δN =N δσ+ N δσ2+... , (13)
,σ ,σσ
2
where N,σ...σ ∂nN/∂σn and the same notations hereafter. Comparing Eqs. (12) and (13) one can easily find that
≡
n
flocal can be presented using N and N , namely,
NL ,σ ,σσ
|{z}
5N
flocal = ,σσ . (14)
NL ζ 6 N,2σ
(cid:12)
(cid:12)
For nonlocal type, however, things will become a(cid:12)little bit more complicated, since non-Gaussianities also exists
in the field fluctuation δσ itself, and δN formalism will not be valid any longer. In this case, we could express the
3-point correlation function of ζ as:
ζ(k )ζ(k )ζ(k ) =(2π)3δ3( k ) (k ,k ,k ) (15)
1 2 3 i 1 2 3
h| |i B
i
X
4
where (k ,k ,k ) is the shape of the non-Gaussianties and
1 2 3
B
t
ζ(k )ζ(k )ζ(k ) = i dt′ [ζ(t,k )ζ(t,k )ζ(t,k ), p (t′)] , (16)
h| 1 2 3 |i − T h| 1 2 3 Hint |i
Zt0
with p being interaction Hamiltonian of curvaton in momentum space. The nonlinear estimator is defined as:
Hint
10 3 k3 (k ,k ,k )
fnonlocal i=1 i B 1 2 3 . (17)
NL (cid:12)ζ ≡ 3 (2πQ)4 3i=1ki3[Pζ(k1)Pζ(k2)+2perm.]
(cid:12)
(cid:12) P
III. THE G-CURVATON MODEL
In this section, we study our model of which the curvaton field is of Galileon type. For simplicity but without
losing generality, we consider the Lagrangianof curvatonhas the generalthird order Galileonfield, which is givenby
[18, 24]:
= d4x√ g[K(σ,X) G(σ,X)✷σ] , (18)
GC
S − −
Z
where K and G are generic functions of σ and X ∂ σ∂µσ/2 is the kinetic term of the field σ. The Lagrangian
µ
≡ −
used is inspired by the original Galileon construction but does not respect the Galileon symmetry, which was called
‘Generalised Galileons’ by Deffayet et al.[24]. Note that more generalized Galileon model containing higher order
operators of ✷σ or the couplings of σ to the gravitational part was constructed in Ref. [37]. From action (18), the
energy-momentum tensor T has the form of:
µν
T =K ∂ σ∂ σ+Kg ∂ G∂ σ ∂ G∂ σ+g ∂ G∂λσ G ✷σ∂ σ∂ σ . (19)
µν ,X µ ν µν µ ν µ ν µν λ ,X µ ν
− − −
Taking the homogeneous and isotropic background where T has the form of diag ρ ,a2(t)p ,a2(t)p ,a2(t)p , we
µν σ σ σ σ
{ }
can furtherly obtain the energy density and pressure of σ as:
ρ = 2K X K+3HG σ˙3 2G X , (20)
σ ,X ,X ,σ
− −
p = K 2(G +G σ¨)X . (21)
σ ,σ ,X
−
Moreover,by varying action (18) with respect to σ, we obtain the equation of motion for σ:
K ✷σ K (∂ ∂ σ)(∂µσ∂νσ) 2K X+K 2(G G X)✷σ
,X ,XX µ ν ,Xσ ,σ ,σ ,Xσ
− − − −
+G (∂ ∂ σ)(∂µ∂νσ) (✷σ)2+R ∂µσ∂νσ +2G (∂ ∂ σ)(∂µσ∂νσ)
,X µ ν µν ,Xσ µ ν
−
+2G,σ(cid:2)σX G,XX ∂µ∂λσ gµλ✷σ (∂µ∂νσ)∂ν(cid:3)σ∂λσ =0 . (22)
− −
which can be simplified in FRW universe(cid:0)where the backg(cid:1)round are homogeneous and isotropic:
K (σ¨+3Hσ˙)+2K Xσ¨+2K X K 2(G G X)(σ¨+3Hσ˙)
,X ,XX ,Xσ ,σ ,σ ,Xσ
− − −
+6G (HX)˙+3H2X 4G Xσ¨ 2G X +6HG XX˙ =0 . (23)
,X ,Xσ ,σσ ,XX
− −
(cid:2) (cid:3)
A. Scale-Invariant Power Spectrum for Field Fluctuation
To study the perturbations of the curvaton field, we can split the scalar field σ into σ(x) σ (t) + δσ(t,x),
0
→
where σ represents the spatially homogeneous background field, and the δσ stands for the linear fluctuation which
0
correspondstotheisocurvatureperturbationduringinflation. Takingthespatial-flatgaugeandusingtheassumption
thatthereisnocouplingbetweenδσ andotherperturbations,onecanhavetheequationofmotionforfieldfluctuation
in momentum space δσk as:
˙ c2k2
δ¨σk+ 3+ HD Hδ˙σk+ sa2 δσk+M2effδσk =0 , (24)
(cid:18) D(cid:19)
5
where we have defined c2 / , with
s ≡C D
= K +2G (σ¨ +2Hσ˙ )+2G Xσ¨ 2(G G X) , (25)
,X ,X 0 0 ,XX 0 ,σ ,Xσ
C − −
= K +2K X+6HG σ˙ 2(G +G X)+6HG Xσ˙ . (26)
,X ,XX ,X 0 ,σ ,Xσ ,XX 0
D −
Note that in order to avoid ghost or gradiant instabilities in our model, one must require 0, >0, which leads
to the non-negativity of the sound speed squared: c2 0. The effective mass squared is: C ≥ D
s ≥
1 d
2 = a3(K σ˙ +3HG σ˙2 G σ˙ ) K +G ✷σ . (27)
Meff a3dt ,Xσ 0 ,Xσ 0 − ,σσ 0 − ,σσ ,σσ 0
(cid:2) (cid:3)
For later convenience, we also introduce the following “slow variation” parameters:
H˙ c˙ ˙ 2
ǫ = , s = s , δ = D , ξ = Meff, (28)
G −H2 G Hc G H G 3H2
s
D
whichareassumedto besmallbutnotneglected. Whenreducedtothe simplestcurvatonmodelgivenby(1), ǫ =ǫ,
G
ξ =ξ and s =δ =0.
G G G
ForsolvingEq. (24)andcomputingthepowerspectrum,itisconvenienttoturntotheconformalcoordinatewhere
conformal time τ is defined as dτ dt/a. Using a new variable uk zδσk where z a√ , the equation of motion
≡ ≡ ≡ D
can be written in the Fourier space as
z′′
u′′+ c2k2 u =0 , (29)
k s − z k
(cid:18) (cid:19)
and the prime denotes differentiation with respect to τ. Under the slow roll approximation where all the parameters
in (28) are small, we find:
z′′ 2
. (30)
z ≃ τ2
Thus we finally can obtain the power spectrum of δσk at horizon crossing as (using definition (4)):
H2
= ∗ (31)
Pδσk 4π2c3
sD
with the spectral index given by
dln
σ
n 1= P = 2ǫ 3s δ +2ξ 1. (32)
σ G G G G
− dlnk − − − ≪
From Eq.(32) we can see that the isocurvature perturbation generated by our curvaton field can give rise to a very
flat power spectrum which is nearly scale-invariant. Moreover, comparing to usual curvaton case (4), the amplitude
of the power spectrum is suppressed by Galileon-like nonlinear terms such as , however, when G(σ,X) = 0 and
D
K(σ,X) = X V(σ), the result exactly reduces to that of usual curvaton case. When G(σ,X) = 0 and K(σ,X) is
−
DBI-type,the resultisthatofDBI-curvaton[13]. TheresultgivenhereisalsoconsistentwiththecasewhereGalileon
field act as an inflaton field and generate curvature perturbations [26]2, and the similar property has also been found
in other featured inflation models, such as DBI inflation [49].
B. Generating the Curvature Perturbation from Curvaton
From last paragraphwe learned that our G-Curvaton model is able to give rise to perturbations with nearly scale-
invariant power spectrum, however these perturbations are of isocurvature ones. As has been mentioned before,
curvature perturbations can be obtained after inflation, so now we consider the epoch when inflation has ceased and
the inflaton has already decayed to radiation, during which the universe is filled with the curvaton field ρ and the
σ
2 Strictly speaking, as authors of Ref. [26] pointed out themselves, what they calculated is not usual comoving curvature perturbation
butanother variablewhichcoincidewithcomovingcurvatureperturbationonlyinlargescales.
6
radiation ρ . From this moment on, the isocurvature perturbations began to convert into curvature ones, and this
r
conversion will complete in two possible cases, namely, either when the curvaton dominates the universe, or decay
as well, whichever is earlier 3. We assume that the conversion as well as curvaton decay happens instantaneously, so
that we can separately consider the curvature perturbations for each component (radiation and curvaton) and just
use Eq. (9) to calculate the final total curvature perturbation of the universe. From Eqs. (8) and (9) we know that
the final curvature perturbation can be expressed as:
δρ
σ
ζ = , (33)
4ρ +3(ρ +p )
r σ σ
wherethedensityperturbationδρ canfurtherlybeexpandedwithrespecttoδσ. The linearorderofδρ isgivenby:
σ σ
δ(1)ρ ρ δσ
σ σ,σ
≃
= (2K X K +3HG σ˙3 2G X)δσ , (34)
,Xσ − ,σ ,Xσ 0 − ,σσ
while the second order of δρ reads:
σ
1
δ(2)ρ ρ δσ2
σ σ,σσ
≃ 2
1
= (2K X K +3HG σ˙3 2G X)δσ2 (35)
2 ,Xσσ − ,σσ ,Xσσ 0 − ,σσσ
respectively.
Now we can consider the two cases separately. First, if the curvaton dominates the energy density before decays,
we can use Eqs. (8), (10) as well as (34) to have the final curvature perturbation as:
δ(1)ρ ρ
ζ(I) σ σ,σ δσ
≃ 3(ρ +p ) ≃ 3(ρ +p )
σ σ σ σ
1 K σ˙2 K +3HG σ˙3 G σ˙2
= ,Xσ 0 − ,σ ,Xσ 0 − ,σσ 0 δσ , (36)
3 K σ˙2+3HG σ˙3 2G σ˙2 G σ¨ σ˙2
(cid:18) ,X 0 ,X 0 − ,σ 0 − ,X 0 0(cid:19)
and the power spectrum of curvature perturbation is:
k3
(I) ζ 2
Pζ ≡ 2π2| |
1 H2 K σ˙2 K +3HG σ˙3 G σ˙2 2
= ∗ ,Xσ 0 − ,σ ,Xσ 0 − ,σσ 0 , (37)
9 4π2c3 K σ˙2+3HG σ˙3 2G σ˙2 G σ¨ σ˙2
(cid:18) sD(cid:19)(cid:18) ,X 0 ,X 0 − ,σ 0 − ,X 0 0(cid:19)
where we also made use of our previous results (31). Second, if the curvatondecays before its dominance, it will only
contribute part of the energy density of the universe with some fraction r as defined before. Using Eq. (11), in this
case the curvature perturbation is given by
rδ(1)ρ rρ
ζ(II) σ σ,σδσ
≃ 4 ρ ≃ 4 ρ
σ σ
r K σ˙2 K +3HG σ˙3 G σ˙2
= ,Xσ 0 − ,σ ,Xσ 0 − ,σσ 0 δσ , (38)
4 K σ˙2 K+3HG σ˙3 G σ˙2
(cid:18) ,X 0 − ,X 0 − ,σ 0 (cid:19)
and the curvature perturbation power spectrum is:
r2 K σ˙2 K +3HG σ˙3 G σ˙2 2 k3
(II) = ,Xσ 0 − ,σ ,Xσ 0 − ,σσ 0 δσ 2
Pζ 16 K σ˙2 K+3HG σ˙3 G σ˙2 2π2| |
(cid:18) ,X 0 − ,X 0 − ,σ 0 (cid:19)
r2 H2 K σ˙2 K +3HG σ˙3 G σ˙2 2
= ∗ ,Xσ 0 − ,σ ,Xσ 0 − ,σσ 0 . (39)
16 4π2c3 K σ˙2 K+3HG σ˙3 G σ˙2
(cid:18) sD(cid:19)(cid:18) ,X 0 − ,X 0 − ,σ 0 (cid:19)
3 Actually,besidesthestandardmodelradiation,thecurvatoncanalsodecayintootherproductssuchasdarkradiationordarkmatter.
Inthat case, theremayhave residual isocurvatureperturbations andthe non-Gaussianities mayalsobe different[50] (see also[51] for
areview). Inthispaperweassumethatcurvatondecays tostandardmodelradiationforsimplicity. WethankA.Mazumdarforpoint
outthisforusviaprivatecommunication.
7
Here the supscripts (I) and (II) refer to the two cases respectively.
FromEqs. (37)and(39)onecanseethatcomparingtousualcurvatoncase,theamplitudeofcurvatureperturbation
will also be modified with a different prefactor, which contains contribution from the high derivative term G(σ,X).
So one can naturally expect G-Curvaton model has some feature that could be distinguished from usual K-essence
curvaton models.
C. Non-Gaussianities generated by G-Curvaton
In this paragraph, we extend our study to non-linear perturbations of our model, i.e., non-Gaussianities. First
of all, we consider the local type non-Gaussianities generated by the curvaton model. For local type, the nonlinear
parameter flocal has been defined in (12). From Eqs. (33-35) and using the δN formalism (13), we can also express
NL
flocal in terms of energy density as:
NL
5 ρ
flocal = [4ρ +3(ρ +p )] σ,σσ . (40)
NL ζ 6 r σ σ ρ2σ,σ
(cid:12)
(cid:12)
Now we consider the two cases separately(cid:12). For the first case where curvaton dominates the energy density before
decays, one gets:
(I) 5(ρ +p )ρ
flocal σ σ σ,σσ
NL ζ ≃ 2 ρ2σ,σ
(cid:12)
(cid:12) 5 (K σ˙2+3HG σ˙3 2G σ˙2 G σ¨ σ˙2)(K σ˙2 K +3HG σ˙3 G σ˙2)
(cid:12) = ,X 0 ,X 0 − ,σ 0 − ,X 0 0 ,Xσσ 0 − ,σσ ,Xσσ 0 − ,σσσ 0 , (41)
2 (K σ˙2 K +3HG σ˙3 G σ˙2)2
(cid:20) ,Xσ 0 − ,σ ,Xσ 0 − ,σσ 0 (cid:21)
and for the second case where the curvaton decays and never dominates the energy density, we have
(II) 10ρ ρ
flocal σ σ,σσ
NL ζ ≃ 3r ρ2σ,σ
(cid:12)
(cid:12) 10 (K σ˙2 K+3HG σ˙3 G σ˙2)(K σ˙2 K +3HG σ˙3 G σ˙2)
(cid:12) = ,X 0 − ,X 0 − ,σ 0 ,Xσσ 0 − ,σσ ,Xσσ 0 − ,σσσ 0 , (42)
3r (K σ˙2 K +3HG σ˙3 G σ˙2)2
(cid:20) ,Xσ 0 − ,σ ,Xσ 0 − ,σσ 0 (cid:21)
respectively.
Thenweturntothenonlocaltypeofnon-Gaussianities. Thenonlocaltypeofnon-Gaussianitiesismorecomplicated,
as described in Eqs. (15-17). The interaction hamiltonian is p = , where is the 3-rd order perturbed
Hint −L3 L3
Lagrangian. Startingfromaction(18)andafterstraightforwardbutrathertedious calculation,wegetthe interaction
hamiltonian at the leading order with respect to slow roll parameters:
d3k d3k d3k
Hp 1 2 3(2π)3δ3(k +k +k ) +aL (k k )δ˙σ(t,k )δ˙σ(t,k )δσ(t,k )+a3L δ˙σ(t,k )δ˙σ(t,k )δ˙σ(t,k )
int ⊃ (2π)9 1 2 3 { 1 2· 3 1 2 3 2 1 2 3
Z
+aL (k k )δ˙σ(t,k )δσ(t,k )δσ(t,k )+a−1L (k k )k2δσ(t,k )δσ(t,k )δσ(t,k ) , (43)
3 2· 3 1 2 3 4 1· 2 3 1 2 3 }
where we have defined
L = G σ˙2 , (44)
1 ,XX0 0
1 1
L = (5HG σ˙2+2HG + K σ˙3+HG σ˙4+K σ˙ ) , (45)
2 2 ,XX0 0 ,X0 3 ,XXX0 0 ,XXX0 0 ,XX0 0
1
L = (K σ˙ +2G σ˙ σ¨ +3HG σ˙2+4HG ) , (46)
3 −2 ,XX0 0 ,XX0 0 0 ,XX0 0 ,X0
1
L = G . (47)
4 ,X0
−2
Moreover, from the solution of the canonical variable u we can obtain the mode solution of the perturbation
k
variable (in conformal time) δσ(τ,k) as:
iH
δσ(τ,k)=q(τ,k)ak+q∗(τ,k)a†−k , q(τ,k)= 2 c3k3(1+icskτ)e−icskτ , (48)
D s
p
8
where ak anda†−k areproductionandannihilationoperatorsrespectively. By substituting Eq. (43-48) into (16) , one
can get the result of 3-point correlation function δσ(k )δσ(k )δσ(k ) . Note that when carrying out integrations
1 2 3
h| |i
withrespecttoconformaltimeτ, weassumethattheL ’s(i=1,2,3,4)areallslowvaryingandcanberoughlytaken
i
outofthe integrations,whichgreatlysimplifies ourcalculation. Similar assumptionshasbeenmade whencalculating
more general single field inflation with nonminimal coupling to Gravity as well as Gauss-Bonnet terms [43] though
morerigidcalculationwithnonminimalcouplingonlywasperformedin[52]. Thefinalshapeof δσ(k )δσ(k )δσ(k )
1 2 3
h| |i
than reads:
3L H6 3L H5 L H5
(k ,k ,k ) = 1 2 + 3 (6k k k k3+2 k2k4+3 k k5
Bδσ 1 2 3 3c8k k k K3 − 3c6k k k K3 4 3c8k3k3k3K3 1 2 3 i i j i j
D s 1 2 3 D s 1 2 3 D s 1 2 3 i i6=j i6=j
X X X
L H6
+ k6)+ 4 (2 k2k5 7 k4k3+4 k k6 18k2k2k2 k
i 2 3c10k3k3k3K4 i j − i j i j − 1 2 3 i
i D s 1 2 3 i6=j i6=j i6=j i
X X X X X
4k k k k3k 24k k k k2k2+12k k k k4+ k7) , (49)
− 1 2 3 i j − 1 2 3 i j 1 2 3 i i
i6=j i>j i i
X X X X
and the corresponding non-linear estimator f can be calculated according to Eq. (17). As an example, here we
NL
consider the equilateral case where k =k =k , in which the estimator then reduces to:
1 2 3
10 L H2 L H 17L H 13L H2
fequil = ( 1 2 + 3 4 ) , (50)
NL (cid:12)δσ 27 9Dc2s − 9D 36Dc2s − 18Dc4s
(cid:12)
while using δN formalism (13), the n(cid:12)onlinear estimator for curvature perturbation reads:
fequil =N fequil . (51)
NL ,σ NL
ζ δσ
(cid:12) (cid:12)
(cid:12) (cid:12)
From Eqs. (33) and (34), we can easily get N,σ =(cid:12) ρσ,σ/[4ρr+3(cid:12)(ρσ+pσ)], which gives
(I) ρ
fequil = σ,σ fequil
NL ζ 3(ρσ+pσ) NL δσ
(cid:12) (cid:12)
(cid:12) 10ρ L H(cid:12)2 L H 17L H 13L H2
(cid:12) = σ,σ ( 1 (cid:12) 2 + 3 4 ) (52)
81(ρ +p ) 9 c2 − 9 36 c2 − 18 c4
σ σ D s D D s D s
for the case of which curvaton dominates before decays, and
(II) rρ
fequil = σ,σfequil
NL ζ 4ρσ NL δσ
(cid:12) (cid:12)
(cid:12)(cid:12) = 5rρσ,σ(L1H(cid:12)(cid:12)2 L2H + 17L3H 13L4H2) (53)
54ρ 9 c2 − 9 36 c2 − 18 c4
σ D s D D s D s
for the case of which curvaton decays and never dominates.
D. A concrete G-Curvaton model
Inpreviouspartsofthissection,wehavepresentedthewholeprocessofhowourG-Curvatonmodelworks,including
its background evolution, scale-invariant isocurvature perturbation generation, curvature perturbation conversion as
well as different types of non-Gaussianities. However, due to the involvement of our model, especially containing
high derivative terms, the above generalanalysis can only be rather qualitative. Moreover,unlike the usual curvaton
model,therearemanyuncertaintiesinourmodelwithgeneralform(18)anditcanhavevariousdecayingmechanisms,
each of which may have different results for curvature perturbations and non-Gaussianities. In order to make things
more specific, it is necessary to focus on some explicit models to see how our models can be different from the usual
curvaton models.
Before constructing models, let’s investigate how many constraints we have to consider for our model. First
of all, as was mentioned in previous section, a curvaton model must have light mass, that is, 2 H2. In
Meff ≪
slow roll approximation, we have σ˙ M2, so the first term of Eq. (27) can then be neglected, which makes
≪ pl
2 K +3HG σ˙. Therefore,ifwechoose K and G tobesmallenough,itwillbe safeforcurvaton.
Meff ≃− ,σσ ,σσ | ,σσ| | ,σσ|
9
AnotherwayisbothK and3Hσ˙G maybelargeinamplitude,butareofsimilarvalueandoppositesign. Inthis
,σσ ,σσ
case,theycanbecanceledtohavearelativelysmallvalue,whichmayneedsomefinetuninginthemodel. Thesecond
constraint comes from the observations. The current observation data gives very tight constraint on the amplitude
of the curvature perturbations, for example, the WMAP-7 measurement of the CMB quadrupole anisotropy requires
2.4 10−9 [53]. In orderfor the amplitude of the curvature perturbationin (37) and (39) to meet the data, one
ζ
P ∼ ×
canfurtherlyconstraintheformofLagrangianofourmodel,namelyK(σ,X)andG(σ,X)andtheirfielddependence.
Taking account to both constraints from above, we can consider that our model may have the Lagrangian with
form of:
K(σ,X)=X V(σ) , G(σ,X)= g(σ)X (54)
− −
as an example. Here we require both V and g be small enough to give rise to small effective mass needed for
,σσ ,σσ
curvaton. For background evolution, from eqs. (20), (21) and (23), we have the following equations:
ρ =X(1 6gHσ˙ +g σ˙2)+V , (55)
σ − 0 ,σ 0
ρ +p =2X(1+gσ¨ 3gHσ˙ +g σ˙2) , (56)
σ σ 0− 0 ,σ 0
σ¨ +3Hσ˙ +2g σ˙2σ¨ + 1g σ˙4 3gH˙σ˙2 6gHσ˙ σ¨ 9gH2σ˙2+V =0 . (57)
0 0 ,σ 0 0 2 ,σσ 0 − 0 − 0 0− 0 ,σ
For perturbations, from Eqs. (25) and (26) we can get:
=1 2gσ¨ 4gHσ˙ , =1+4g X 6gHσ˙ (58)
0 0 ,σ 0
C − − D −
for our model, which can give the sound speed squared c2 = / . One can also get the power spectrum and non-
s C D
Gaussianities of curvature perturbations by making use of the explicit form (54) in the corresponding formulae that
has been derived in the above sections.
For later convenience, let us first introduce some more “slow-variation”parameters, namely
σ¨ g˙ g˙ g˙
0 ,σ ,σσ
η , α , β , γ . (59)
≡ Hσ˙ ≡ gH ≡ g H ≡ g H
0 ,σ ,σσ
We can always choose the form of g(σ) such that the last three parameters α, β , γ 1 all the time. However,
| | | | | | ≪
η can only be small when the field remains slow-rolling. This is important because in curvaton scenario, curvature
| |
perturbations are produced after the end of inflation, and in that case, the slow-rolling of the curvaton field is not
always satisfied.
Now let’s turnonto the two casesofgeneratingcurvatureperturbations oneby one. Inthe firstcase the curvature
perturbationisgeneratedwhencurvatondominatestheuniverse. Thatrequiresthedecayingofthecurvatonisslower
than that of radiation which is transferred from inflaton. For instance, it is reasonable to assume that the curvaton
field is still slow-rolling,and exit in a few number of efolds, so it will not lead to another period of rapid acceleration
[26]. Substituting (54) into Eqs. (37), (41) and (52) and letting all the slow-variation parameters defined in (59) be
small, we can get:
H2
(I) ( ∗ )(H/σ˙ )2 , (60)
Pζ ≃ 4π2c3 0
sD
(I) 5 (3αgHσ˙ 6ǫgHσ˙ +ǫ)
flocal [ 0− 0 ] , (61)
NL ζ ≃ 6 (1 3gHσ˙0)
(cid:12) −
fequil(cid:12)(cid:12)(I) 10gHσ˙0((3−ǫ)gHσ˙0−1)(1+ 17 13 )(H/σ˙ )2 , (62)
NL (cid:12)ζ ≃ 243D 1−3gHσ˙0 2c2s − 4c4s 0
(cid:12)
Similarly, in the second cas(cid:12)e the curvature perturbation is generated when the curvaton decays and becomes
equilibriumwithradiation,andthusthe energydensity ofcurvatonhasthesamescalingasthatofradiationwiththe
ratio r =ρ /ρ . Again using (54) with Eqs. (39), (42) and (53), letting α, β and γ be small but retaining η for the
σ r
reason given above, we can get:
(1+r)2 H2 1
(II) ( ∗ )[(3 ǫ+2η)gHσ˙ (η+3)]2(H/σ˙ )−2 , (63)
Pζ ≃ 16 4π2c3 − 0− 3 0
sD
(II) 20ǫ ( 6(3 ǫ+2η)gHσ˙ +(η+3))
flocal [ − − 0 ](H/σ˙ )2 , (64)
NL ζ ≃ (r+1) (3(3 ǫ+2η)gHσ˙0 (η+3))2 0
(cid:12) − −
fequil(cid:12)(cid:12)(II) 5(r+1)gHσ˙0[(3 ǫ+2η)gHσ˙ 1(η+3)](1+ 17 13) . (65)
NL (cid:12)ζ ≃ 486D − 0− 3 2c2s − 4c4s
(cid:12)
(cid:12)
10
FromabovewecanseethatanothervariablethatappearsrepeatedlyisgHσ˙ . Defining x:=gHσ˙ ,andinorderto
0 0
avoid ghost and gradient instability, x should be smaller than unity. We consider two asymoptotic cases: i) x could
be positive or negative, with x 1 and ii) x is negative, with x 1, which means that the Galileon term plays a
| |≪ | |≫
unimportant/importantroleattheendofinflationrespectively. Thespectrumandnon-Gaussianitiesofthecurvature
perturbation can be reduced as:
i) x 1: c2 1
| |≪ s ≃
H2 (I) 5 (I) 125x
(I) ( ∗ )(H/σ˙ )2 , flocal ǫ , fequil (H/σ˙ )2 , (66)
Pζ ≃ 4π2 0 NL ζ ≃ 6 NL ζ ≃− 486 0
(cid:12) (cid:12)
(II) (1+r)2(H∗2)(η+3)(cid:12)(cid:12)2 , flocal (II) (cid:12)(cid:12) 20ǫ(H/σ˙0)2 , fequil (II) 125(r+1)x(η+3) , (67)
Pζ ≃ 144 4π2 (H/σ˙0)2 NL ζ ≃ (r+1)(η+3) NL ζ ≃− 5832
(cid:12) (cid:12)
(cid:12) (cid:12)
ii) x 1: c2 η+2 (η is small for Case I) (cid:12) (cid:12)
| |≫ s ≃ 3
1 3 H2 (I) 5 (I) 35(3 ǫ)
(I) ( ∗ )(H/σ˙ )2 , flocal (2ǫ α) , fequil − (H/σ˙ )2 , (68)
Pζ ≃ 4|x|r2 4π2 0 NL (cid:12)ζ ≃ 6 − NL (cid:12)ζ ≃− 17496 0
(II) (1+r)2|x| 3 (H∗2)(3(cid:12)(cid:12)−ǫ+2η)2 , flocal (II)(cid:12)(cid:12) 40ǫ(H/σ˙0)2 ,
Pζ ≃ 32 s(η+2)3 4π2 (H/σ˙0)2 NL ζ ≃ 3(r+1)(3 ǫ+2η)x
(cid:12) − | |
(II) 5(r+1)(3 ǫ+2η)(103+118η+4η2) (cid:12)
fequil − , (cid:12) (69)
NL ζ ≃ 11664x(2+η)2
(cid:12) | |
(cid:12)
respectively. (cid:12)
From the above results we can have a couple of comments on the perturbations of our G-Curvaton model. Besides
the slow variation parameters which are roughly of order 1, now we have three more free parameters, namely the
value of x, H and σ˙ at the end of inflation, to determine and f . Considering the constraints from CMB that
0 ζ NL
the amplitude of power spectrum to be nearly 10−9, we stPill have large parameter space to have considerable large
non-Gaussianities. For instance, for Case I the curvaton field is still slow-rolling at the end of inflation, where σ˙
0
is small compared to H. If it satisfies σ˙ 10−3H, then from the power spectrum we have H 10−8, which| is|
0
| | ∼ ∼
slightly lower than that of chaotic inflation. For subcase where x 1, we can have fequil 102 just by requiring
| | ≪ NL ∼
g 10−13. For subcase where x 1, however, we can have fequil 102 by requiring g 1018. Moreover, in
b| o|t∼hcasesflocal remainsoforder|u|ni≫ty. For CaseII,the energydeNnsLity∼ofσ is comparablet|o|H∼,whichroughlygives
NL
σ˙2[1+ (1)x] H2. In subcase x 1, we have σ˙ H, which gives H 10−5 in order to meet the constrainton
0 O ∼ | |≪ | 0|∼ ∼
power spectrum. This in turn gives small fequil with flocal roughly of (10). In subcase x 1, however,we obtain
gσ˙3 H. If furtherly it satisfies, i.e., σ˙ NL 10−1H,NitLwill lead to HO 10−2 from con|st|ra≫int on power spectrum,
| 0| ∼ | 0| ∼ ∼
whichinturngivesσ˙ 10−3 aswellas g 107. Thenthe nonlinearestimatorflocal isroughlyof (10),andfequil
becomes oforderunit0y∼4. Moreover,due t|o|t∼he non-lineareffects of the GalileonteNrmL G(X,σ)✷σ inOour model, itNcLan
be expected that large tensor-to-scalar-ratio can be generated as well as large non-Gaussianities [26], which can be
distinguishable from the standard curvaton model, of which large non-Gaussianities are also accompanied with lower
energy scale of inflation, which leads to small tensor-to-scalarratio (cf. the second paper in [11]).
IV. CONCLUSION AND DISCUSSIONS
In this paper, we investigated G-Curvaton scenario, where the curvaton field is acted by Galileon action. This
opens a new access of curvaton scenarios that the fluctuations could be affected by nonlinear terms. After reviewing
the standard curvaton mechanism, we started from the action of Galileon field and calculated the spectrum of the
field perturbation. The power spectrum is suppressed by given in (26), rather than K in the normal single field
,X
D
case. We studied the generation of curvature perturbations in both two possible cases, and apart from curvature
perturbation, we obtained non-Gaussianities of both local and non-local types.
Wehaveshown,inthiswork,thatthereislargepossibilitiesforG-Curvatontohaveconsistentpowerspectrumand
large non-Gaussianities. We presented a concrete model of G-Curvaton as an illustration. With proper choice of the
4 InouranalysiswealsomadetheapproximationofH∗∼H,whereH∗ isthevalueofHubbleparameterwhenthefluctuationsofσexits
thehorizonduringinflation.