outside the horizon
TRANSCRIPT
Remarks on non-Gaussian fluctuations of the inflaton and constancy of � outside the horizon
Namit Mahajan* and Raghavan Rangarajan†
Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India(Received 20 May 2010; revised manuscript received 21 October 2010; published 11 February 2011)
We point out that the non-Gaussianity arising from cubic self-interactions of the inflaton field is
proportional to �Ne where �� V 000 and Ne is the number of e-foldings from horizon exit till the end of
inflation. For scales of interest Ne ¼ 60, and for models of inflation such as new inflation, natural inflation,
and running mass inflation � is large compared to the slow-roll parameter �� V02. Therefore, thecontribution from self-interactions should not be outrightly ignored while retaining other terms in the non-
Gaussianity parameter fNL. However, the Ne-dependent term seems to imply the growth of non-
Gaussianities outside the horizon. Therefore, we briefly discuss the issue of the constancy of correlations
of the curvature perturbation � outside the horizon. We then calculate the 3-point function of the inflaton
fluctuations using the canonical formalism and further obtain the 3-point function of �k. We find that the
Ne-dependent contribution to fNL from self-interactions of the inflaton field is canceled by contributions
from other terms associated with nonlinearities in cosmological perturbation theory.
DOI: 10.1103/PhysRevD.83.043510 PACS numbers: 98.80.�k, 98.80.Cq
Non-Gaussianities arising from self-interactions of theinflaton field are usually treated as negligible compared tothose arising from nonlinearities in cosmological perturba-tion theory [1,2] because they are proportional to higher-order slow-roll parameters, such as �. In this paper we pointout earlier results indicating that this contribution includes aterm proportional to Ne, the number of e-foldings after thescale of interest has left the horizon, which for our horizonscale is approximately 60 by the end of inflation. Then, forcertain models of inflation with relatively larger higher-order slow-roll parameters, such as new inflation, small-field natural inflation (f < 1:5MPl), and running-massinflation, this contribution can be comparable to other con-tributions from non-Gaussianities in cosmological pertur-bation theory.
However, the dependence on Ne would seem to implythat n-point (n > 2) correlations of the curvature perturba-tion � grow outside the horizon. Hence, we discuss therelated issue of constancy of � outside the horizon. Thederivations regarding the constancy of �k are classical.
They only imply that the 2-point function of � to lowestorder, �j�kj2�3ðk� k0Þ, is constant outside the horizon.(Hatted quantities denote quantum operators.) For higher
point functions of � , there is an additional requirementassociated with convergence of an integral over time, as wediscuss below.
Having clarified this, we then work in a gauge in which��, the fluctuation in the inflaton field, is not zero, and
recalculate the 3-point function of ��k, using the canoni-cal formalism which we believe has not been done before.This agrees with results of earlier calculations using thepath integral approach and field equations.
We then relate ��k to �k. We argue that the quantizationof the relation between � and �� derived from the �Nformalism is not ideal if we wish to study any possible
growth of n-point functions of � outside the horizon. Usinga slightly different relation which explicitly includes de-pendence on the final time, we then obtain the 3-point
function of �k, including the term associated with self-interactions of the inflaton field which is proportional to
Ne. Our calculation of h�3i is straightforward and does notinvolve complicated field redefinitions. Finally we find thatthe time-dependent self-interaction contribution is can-celed by other terms associated with non-Gaussianities incosmological perturbation theory.For an inflaton with a cubic interaction the bispectrum
parameter fNL is
6
5fNL ¼ �
�1
3þ �� Ne þ 3P
ik3i
�ktXi<j
kikj � 4
9k3t
��
þ 3
2�� �þ �P
ik3i
�4
kt
Xi<j
k2i k2j þ
1
2
Xi�j
kik2j
�; (1)
where Ne is the number of e-foldings of inflation from thetime the mode of interest leaves the horizon at tex till thetime t, which can be at any later time during inflation. Ourexpression is similar to that in Eq. (38) of Ref. [3] withtheir t� replaced by t, and we have replaced N� by �Ne ¼�Hðt� texÞ. We shall explain the distinction between ourexpressions later. The slow variation inH can be ignored inNe since fNL is to first order in slow roll. �, �, and � are theslow-roll parameters evaluated at t:
� ’ 1
2
�V 0
V
�2 ’ 1
2
_�2
H2; (2)*[email protected]
PHYSICAL REVIEW D 83, 043510 (2011)
1550-7998=2011=83(4)=043510(8) 043510-1 � 2011 American Physical Society
� � V00
V’ �
€�
H _�þ �; (3)
� � V 0V 000
V2: (4)
We use � rather than �2 as used by some authors andMP ¼MPl=
ffiffiffiffiffiffiffi8�
phas been set to 1. � � 0:577216 is Euler’s
constant.The contribution due to the cubic interaction of the
inflaton is the �-dependent term above and was obtainedin Refs. [3–6]. The argument to ignore this term is that it isproportional to �� V 000 and is hence negligible comparedto terms proportional to �� V 02 and �� V 00. This is validfor models such as chaotic inflation, for which � > �> �.However, in general, the hierarchy is �> �> > . . . ,while � may be larger or smaller than other slow-rollparameters. In particular, for new inflation, small-fieldnatural inflation (f < 1:5MPl), and running-mass inflation,� > �. In general, for small-field models with a concave-downward potential, � & 0:0001 [7]. Therefore the � termshould not be outrightly ignored while retaining otherterms in fNL above.
There is another reason why the � term in fNL is notautomatically small compared to other terms. For scales ofthe order of our horizon today, Ne is about 60 if t is at theend of inflation. Therefore, the term �Ne in fNL above canbe large. For a new inflation potential of the form V ¼V0 ��3ð�> 0Þ, � ¼ 0:5�2. Also, � ¼ 0:5ðns � 1Þ þ3� � 0:5ðns � 1Þ [8]. If we take ns ¼ 0:96 [9], then � is�0:02 and �Ne is 0.012. � is much smaller. Thus, we seethat �Ne is comparable to � and � � �Ne.
1 2
The presence of the time-dependent Ne in fNL seems to
contradict the notion that n-point functions of � do notgrow outside the horizon. It may be argued that since thecurvature perturbation � does not grow outside the horizonthe Ne contribution from the non-Gaussianities of theinflaton can not contribute to fNL. Note, however, thatthis term has been obtained independently in Refs. [3–6].Thus, we now consider the literature on the constancy of �in the context of nonlinear cosmological perturbationtheory.
Salopek andBond [11] first introduced the generalizationof the Bardeen-Steinhardt-Turner variable � [12] for thecase when one wishes to consider non-Gaussianities in thecurvature perturbation. The Salopek-Bond �ðxÞ is constantoutside the horizon. In other works, such as Refs. [13–18],
the constancy of � is shown for a �k mode. The aboveworksdeal with classical � , and the constancy of classical �ðxÞ or�k only implies that the quantum 2-point function
h�k1�k2
i ¼ ð2�Þ3j�k1 j2�ðk1 � k2Þ is constant outside the
horizon (at lowest order).3 It is not obvious, however, that
other higher n-point functions of � are constant outside thehorizon.One may argue that after horizon exit the curvature
perturbations are classical and so the constancy of �should imply constancy of n-point function for n > 2also. As argued by Weinberg in Ref. [19], the perturba-tions are classical in the sense that commutators involving
� and its time derivative go to 0 for large t. However, thisimplies that quantum n-point functions of zeta can never-theless grow outside the horizon, but only as powers oflna and not as powers of a. It is the time dependence dueto lna� Ne for the 3-point function that is the focus ofthis article.
The n-point functions of � are given by [1,19]
hOðtÞi ¼ X1N¼0
iNZ t
t0
dtNZ tN
t0
dtN�1 � � �Z t2
t0
dt1
� h½HIðt1Þ; ½HIðt2Þ; � � � ½HIðtNÞ; OIðtÞ � � �i;(5)
where OðtÞ can be any product of � operators, OIðtÞ is OðtÞin the interaction picture generated by the quadratic part ofthe Hamiltonian, and t0 is some early time. Note that theexpectation values are obtained in the in-in formalism and
so the bras and kets refer to inh0j and j0iin respectively. HI
is the interaction Hamiltonian and includes terms that are
third or higher order in � . For the 3-point function at lowestorder this reduces to
h�3ðtÞi ¼ iZ t
t0
dt0h½HIðt0Þ; �3I ðtÞi: (6)
If �k is constant, � � eik:x�kck þ e�ik:x��k cyk is constant.
But for h�3ðtÞi to be constant outside the horizon one mustensure that the contribution to the integral above from t� tot is suppressed. Note that � is related to ^�� and h ^ð��Þnigrows outside the horizon.The convergence of the integral for large t, and certain
other conditions for the constancy of h�ni outside thehorizon have been discussed in general in Ref. [20]. Butin Ref. [20] [see Eq. (29)] only Gaussian fluctuations of theinflaton are considered.4 So one should verify whether or
1Interestingly, Appendix B of Ref. [10] obtains a similar resultbut erroneously concludes that the Ne-dependent result ofRef. [4] and the Ne-independent result of Ref. [1] are the samebecause they are of the same order.
2In Ref. [3] it is argued that evaluating expectation values atthe end of inflation may not be valid for large Ne � 60 becauseof divergences of the form �mþ2Nm
e ðm 1Þ. However, for thepotential we are considering, � is much smaller than 1=60.
3Higher-order corrections from loops can give time-dependentor Ne-dependent contributions, as mentioned in Sec. VI ofRef. [19].
4Note that Ref. [11] also considers only Gaussian fluctuationsof the inflaton, since �� is set equal to H=ð2�Þ in the evaluationof � in Secs. IIIB and IIID.
NAMIT MAHAJAN AND RAGHAVAN RANGARAJAN PHYSICAL REVIEW D 83, 043510 (2011)
043510-2
not h�ni is indeed constant outside the horizon when oneincludes non-Gaussian fluctuations of the inflaton. We now
check this explicitly for the 3-point function of � whileincluding a cubic interaction of the inflaton field. The
3-point function of � has been obtained by other authors.Largely, the self-interactions of the inflaton are ignored.Moreover, assuming that there is no contribution outsidethe horizon, the integral in Eq. (6) is cut off at t�. In cases
where one first relates � to �� using the �N formalism andthen calculates the 3-point function, obtaining any contri-bution from evolution outside the horizon is precluded bythe adopted formalism.
We work in a gauge in which �� � 0. We first calculate
h ^ð��Þ3i using the equivalent of Eq. (6) for the 3-point
function of ^��. Our results agree with those obtained inRef. [3] using field equations. We then relate � to �� and
use this to obtain h�3i and fNL. The Ne-dependent termmentioned above associated with the cubic self-interactionof the inflaton does appear in fNL. The 3-point function
h�3i has also been obtained in a gauge in which �� is 0[1,21]. Since these calculations do not include self-interactions of the inflaton their results do not include theNe term.
The relevant part of the action S for �� can be expressedas the sum of terms quadratic and cubic in ��. For nota-tional convenience we hereafter replace ��withQ, and let� represent the background homogeneous field. Then
S ¼ S2 þ S3; (7)
where [22,23]
S2 ¼Z
dtd3xa3�1
2ð _QÞ2 � 1
2a2ð@iQÞ2
� 1
2
�V 00ð�Þ � 1
a3d
dt
�a3
H_�2
��Q2
�; (8)
and S3 is given in Eq. (3.6) of Ref. [1]. Retaining terms toleading order in slow-roll parameters we obtain
S3 ¼Z
dtd3xa3��
_�
4HQ _Q2 � 1
a2
_�
4HQð@iQÞ2
� 1
a2@iQ@ic _Q� 1
6V 000ð�ÞQ3
�; (9)
where
@2c ¼ � a2
2H_� _Q : (10)
In Ref. [1] the last term in S3 above, which is proportionalto �, was ignored. As we have argued earlier the contribu-tion of this term could actually be larger than that of otherterms above for certain models of inflation. Therefore, itought not be ignored at this juncture.
The shift function Ni ¼ @ic , and c differs from � ofRef. [1] by a factor of a2. Using Eq. (10), S3 can berewritten as5
S3 ¼Z
dtd3xa3��
_�
4HQ _Q2 � 1
a2
_�
4HQð@iQÞ2
þ_�
2H@iQð@�1
i_QÞ _Q� 1
6V000ð�ÞQ3
�: (11)
One can obtain H, and thus HI, from the Lagrangianin S2 þ S3. We follow Ref. [24] for dealing with the_Q-dependent interaction terms. After obtainingHIðQ;�QÞ we replace Q and �Q by Qin and �in, respec-
tively, and then set �in ¼ a3 _Qin. Keeping terms up to first
order in _�=H, as in the action, we then get HIðQ; _QÞ ¼�Lint. Hereafter, we drop the subscript in.
The contribution to hQð ~k1; tÞQð ~k2; tÞQð ~k3; tÞi from eachterm in HI is given below. We use the conformal time �,defined by dt ¼ ad�, instead of t and let the initial timecorrespond to � ¼ �1. The final time corresponds to thereheat time.1. The Q _Q2 term:
I1¼ð�iÞQk1ð�ÞQk2ð�ÞQk3ð�Þð2�Þ3�3
�Xi
~ki
�
�Z �
�1d�0a2
_�
4H
�Q�
k1ð�0ÞdQ�
k2ð�0Þd�0
dQ�k3ð�0Þd�0
þperm
�
þc:c:: (12)
Qk is given by [25]
Qk ¼ iH
kffiffiffiffiffi2k
p�1� i
k
aH
�exp
�ik
aH
�
¼ iH
kffiffiffiffiffi2k
p ð1þ k�Þ expð�ik�Þ; (13)
which reduces to iH=ðk ffiffiffiffiffi2k
p Þ for jk�j � 1.6 There are 6permutations of k1; k2; k3 for the expression within theintegral. Then
I1 ¼ � i
4
H3ð�ÞQið2k3i Þ
ð2�Þ3�3
�Xi
~ki
�
�Z �
�1d�0 _�ð�0Þ½k22k23ð1� ik1�
0Þeikt�0 þ permþ c:c:; (14)
5The action S3 provided in Ref. [22] may contain typographi-cal errors. The prefactor for the last term of S3 in Eq. (53) ofRef. [22] should be a�2 rather than a�4, and the definition of cand Ni in Eq. (43) is not in agreement with Eq. (2.24) of Ref. [1].However, Eq. (54) for c is correct.
6There are higher-order corrections to the late-time modefunctions, as discussed in Ref. [21].
REMARKS ON NON-GAUSSIAN FLUCTUATIONS OF THE . . . PHYSICAL REVIEW D 83, 043510 (2011)
043510-3
where kt ¼ k1 þ k2 þ k3. Replacing the lower limit, �1,by �1ð1� i�Þ and setting � to 0 after taking the limiteliminates the contribution of the lower limit. Integratingby parts, taking the limit ki� � 1, and using the complexconjugate to avoid listing some terms, we get
I1 ¼ � i
4
H3ð�ÞQið2k3i Þ
ð2�Þ3�3
�Xi
~ki
��k22k
23
� _�ð�Þikt
þ ik1 _�ð�ÞðiktÞ2
þ ik1� _�0ð�ÞðiktÞ2
þ_�00ð�ÞðiktÞ3
þ 3ik1 _�00ð�ÞðiktÞ4
þ ik1� _�000ð�ÞðiktÞ4
þ_�0000ð�ÞðiktÞ5
�Z �
�1d�0½�5ik1 _�0000ð�0Þ
þ ð1� ik1�0Þ _�00000ð�0Þ e
ikt�0
ðiktÞ5�þ perm þ c:c:; (15)
where _�0 ¼ d _�=d� and so on.In the Appendix we assess the higher derivative terms.
We find that the�:::term is proportional to�2e2Ne . Similarly,
the �::::
term is proportional to �4e4Ne . These higher-orderterms in slow-roll parameters are (increasingly) larger than
the terms proportional to _� (for � ¼ 0:02 and Ne ¼ 60).
However, it is not consistent to consider them here, as wehave ignored higher-order terms in slow-roll parameters inthe action. But this is an indication that there may beconvergence issues at higher orders in the slow-roll parame-ters, and these will have to be handled with care.7 Similarbehavior may be expected while working in the �� ¼ 0
gauge as, for example, in Ref. [21] where integrals for h�3iinclude powers of � ’ 0:5 _�2=H2 in the integrand.Having noted our concern above, we hereafter do not
include terms of higher order in slow-roll parameters.Then,
I1 ¼ �2� 1
4
H3ð�ÞQið2k3i Þ
ð2�Þ3�3
�Xi
~ki
�_�ð�Þ
��k22k
23
ktþ k1k
22k
23
k2tþ perm
�: (16)
The prefactor of 2 comes from the complex conjugate.There are a total of 6 permutations of the variablesðk1; k2; k3Þ. The interchange of k2 and k3 gives the sameexpression as above.2. The Qð@iQÞ2 term:
I2 ¼ ð�iÞQk1ð�ÞQk2ð�ÞQk3ð�Þð2�Þ3�3
�Xi
~ki
�Z �
�1d�0a2
_�
4H½ð� ~k2 � ~k3ÞQ�
k1ð�0ÞQ�k2ð�0ÞQ�
k3ð�0Þ þ perm þ c:c:
¼ � i
4
H3ð�ÞQið2k3i Þ
ð2�Þ3�3
�Xi
~ki
�Z �
�1d�0 _�ð�0Þ
��� ~k2 � ~k3�02
�ð1� ik1�
0Þð1� ik2�0Þð1� ik3�
0Þeikt�0 þ perm
�þ c:c:
¼ �21
4
H3ð�ÞQið2k3i Þ
ð2�Þ3�3
�Xi
~ki
�_�ð�Þð ~k2 � ~k3Þ
��kt þ
Xi�j
kikjkt
þ k1k2k3k2t
þ perm
�: (17)
Clearly there is a symmetry in the ð ~k2; ~k3Þ interchange. ~k2 � ~k3 can be replaced by ðk21 � k22 � k23Þ=2 usingP ~ki ¼ 0. Note
that the first term above is obtained from the real part of i exp½ikt�=� in the limit kt� � 1.3. The @iQð@�1
i_QÞ _Q term:
I3 ¼ ðþiÞQk1ð�ÞQk2ð�ÞQk3ð�Þð2�Þ3�3
�Xi
~ki
��
Z �
�1d�0a2
_�
2H
~k1 � ~k2k22
�Q�
k1ð�0ÞdQ�
k2ð�0Þd�0
dQ�k3ð�0Þd�0
þ perm þ c:c:
¼ 21
2
H3ð�ÞQið2k3i Þ
ð2�Þ3�3
�Xi
~ki
�_�ð�Þ
~k1 � ~k2k22
�k22k
23
ktþ k1k
22k
23
k2tþ perm
�
¼ 21
2
H3ð�ÞQið2k3i Þ
ð2�Þ3�3
�Xi
~ki
�_�ð�Þð ~k1 � ~k2Þ
�k23kt
þ k1k23
k2tþ perm
�: (18)
The 3-point function of Q is I1 þ I2 þ I3 plus the contribution from the cubic self-interaction. For the self-interactioncontribution we use the expression given in Ref. [3] which agrees with Refs. [4,6]. Using MATHEMATICA to include all thepermutations and then simplify their sum gives
7A concern regarding using perturbation theory in slow-roll parameters may also be found in Ref. [3], as mentioned earlier.
NAMIT MAHAJAN AND RAGHAVAN RANGARAJAN PHYSICAL REVIEW D 83, 043510 (2011)
043510-4
hQð ~k1; tÞQð ~k2; tÞQð ~k3; tÞi ¼ ð2�Þ3�ð ~k1 þ ~k2 þ ~k3Þ�H2V 000
4Qik3i
�� 4
9k3t þ kt
Xi<j
kikj þ 1
3
�1
3þ �þ lnjkt�j
�Xi
k3i
�
þ H4
8Qik3i
_�
H
1
kt
�1
2
Xi
k4i � 5Xi<j
k2i k2j �
Xi�j�k
k2i kjkk
��(19)
¼ ð2�Þ3�ð ~k1 þ ~k2 þ ~k3Þ�H2V 000
4Qik3i
��� 4
9k3t þ kt
Xi<j
kikj þ 1
3
�1
3þ �þ lnjkt�j
�Xi
k3i
�
þ H4
8Qik3i
_�
H
�1
2
Xi
k3i �4
kt
Xi<j
k2i k2j �
1
2
Xi�j
kik2j
��; (20)
where i; j ¼ 1; 2; 3, kt ¼P
iki, and we take all ~ki to haveapproximately the same magnitude. � ¼ �1=ðaHÞ. Theterm proportional to V000, which is due to the cubic self-interaction of the inflaton, was obtained in Refs. [3–6].lnjkt�j ¼ ln½ðaHÞex=ðaHÞ � ln½aex=a ¼ �Ne, and thusone gets an Ne-dependent term.8 This term is not explicitlycanceled by any other term in the expression above.Moreover, as we show later, this term is also not canceledby the time variation of slow-roll parameters in other terms.
The above uses the action/Lagrangian and the canonical
formalism to calculate the 3-point function of Q. Ref. [22]uses the path integral formalism to obtain the 3-pointfunction. (It does not consider the contribution from thecubic self-interaction term.) Ref. [3] uses the solutions of
the Heisenberg field equations to obtain hQ3i, including theself-interaction contribution, and our results agreewith Eqs. (20) and (29) of Ref. [3]. Ref. [27] explicitly
shows the equivalence of the form of hQ3i obtained inRefs. [3,22].
In the �N formalism, the gauge-invariant quantity �ð ~x; tÞis the difference in the number of e-foldings of evolutionbetween some time t� and t at ~x and the number ofe-foldings between t� and t for an isotropic homogeneousbackground, where t� lies on a spatially flat slice of space-time with field values �ð ~x; t�Þ while t belongs to a space-time slice of uniform energy density. t� is typically chosento be a few e-foldings after the relevant scale has left thehorizon.
�ð ~x; tÞ ¼ N½ ðtÞ; �ð ~x; t�Þ � N½ ðtÞ; �ðt�Þ
¼ @N½ ðtÞ; �ð ~x; t�Þ@�ð ~x; t�Þ
���������ðt�Þ��ð ~x; t�Þ þ 1
2
� @2N½ ðtÞ; �ð ~x; t�Þ@�ð ~x; t�Þ2
���������ðt�Þ��ð ~x; t�Þ2 þ � � � ;
(21)
where �ðt�Þ is the spatial average value of � at t�,��ð ~x; t�Þ ¼ �ð ~x; t�Þ ��ðt�Þ ¼ Qð ~x; t�Þ, and � � � refers tohigher-order terms that have been omitted. (We havetemporarily reintroduced �ð ~x; tÞ). Dependence of N on_�ðt�Þ is ignored in the slow-roll approximation [23].N½ ðtÞ; �ð ~x; t�Þ is given by
N½ ðtÞ; �ð ~x; t�Þ ¼Z t
t�H½�ð ~x; tÞdt
¼Z �ðtÞ
�ð ~x;t�ÞH½�ð ~x; tÞ d�ð ~x; tÞ
_�ð ~x; tÞ : (22)
Then,
@N½ ðtÞ; �ð ~x; t�Þ@�ð ~x; t�Þ
���������ðt�Þ¼ �H½�ðt�Þ
_�ðt�Þ’ V
V0
���������ðt�Þ’ � 1ffiffiffiffiffiffi
2��p
(23)
@2N½ ðtÞ; �ð ~x; t�Þ@�ð ~x; t�Þ2
���������ðt�Þ’ @
@�
�Vð�ÞV0ð�Þ
����������ðt�Þ’ 1� ��
2��:
(24)
The þð�Þ in the first equation is for V 0ð�Þ> 0 (< 0), or,
equivalently, for _�< 0 (> 0). Below, we will consider thesign as for V0ð�Þ< 0 as relevant for new inflation.
(However, the various contributions to h�3i are ultimately
dependent on only even powers of _�, and so the final
expression for h�3i, or fNL, is independent of the sign ofV0ð�Þ.) Then � is related to Q by
�ð ~k; tÞ ¼ � 1ffiffiffiffiffiffiffiffi2��
p Qð ~k; t�Þ þ 1
2
�1� ��
2��
�
�Z d3q
ð2�Þ3 Qð ~k1 � ~q; t�ÞQð ~q; t�Þ þ � � � : (25)
Theþð�Þ in front of the first term is for V 0ð�Þ> 0 (< 0),
or equivalently, for _�< 0 (> 0).In the �N formalism, � is independent of t, and there is
no t dependence on the r.h.s. of Eq. (25). If one directlyquantizes the above relation, as in Refs. [3,22], thenEq. (25) implies
8The result in Ref. [5] differs by a sign and a term [3]. Inaddition, lnjkt�j in Ref. [5] should be set to �Ne and not þNe.Ref. [26] also obtains a V000 term with a lna-dependence.
REMARKS ON NON-GAUSSIAN FLUCTUATIONS OF THE . . . PHYSICAL REVIEW D 83, 043510 (2011)
043510-5
h�ð ~k1; tÞ�ð ~k2; tÞ�ð ~k3; tÞi¼ � 1
ð2��Þ3=2hQð ~k1; t�ÞQð ~k2; t�ÞQð ~k3; t�Þi
þ 1
2��1
2
�1� ��
2��
�Qð ~k1; t�ÞQð ~k2; t�Þ
�Z d3q
2�3Qð ~k3 � ~q; t�ÞQð ~q; t�Þ þ perm
(26)
The r.h.s. above does not depend on t, but that is because of
the way �ðtÞ was defined as in terms of Qðt�Þ.One may instead argue that the quantum field �ðtÞ
should be expressed as a function of quantum operatorsat t. Furthermore, t and t� are defined on different hyper-surfaces and so correspond to different ‘‘definitions’’ oftime. We need a relation between the quantum operators
�ðtÞ and QðtÞwhich is valid at all times t, including prior toand after horizon exit, and both operators should be func-tions of the same t. For this one may use Eq. (A8) ofRef. [1] and
�ð ~k; tÞ ¼ � 1ffiffiffiffiffiffi2�
p Qð ~k; tÞ þ 1
2
�1� �
2�
�
�Z d3q
ð2�Þ3 Qð ~k1 � ~q; tÞQð ~q; tÞ þ � � � : (27)
‘‘� � �’’ includes terms quadratic in Q but which will notcontribute to the 3-point function because their coefficientsare suppressed for t > tex, i.e., outside the horizon. (Asdiscussed in Sec. 3 of Ref. [1], � and �� are defined indifferent gauges, i.e, they are functions of differenttime variables t (uniform density gauge) and ~t (spatially flator uniform curvature gauge), respectively. But ~t ¼tþ Tðt; ~xÞ, and so Q on the r.h.s. of the equation abovecan be expressed in terms of the same time variable as onthe l.h.s. Then,
h�ð ~k1; t; Þ�ð ~k2; tÞ�ð ~k3; tÞi¼ � 1
ð2�Þ3=2 hQð ~k1; tÞQð ~k2; tÞQð ~k3; tÞi
þ 1
2�
1
2
�1� �
2�
�Qð ~k1; tÞQð ~k2; tÞ
�Z d3q
2�3Qð ~k3 � ~q; tÞQð ~q; tÞ þ perm
: (28)
The first term can be evaluated using Eq. (20). The expec-tation value in the second term above is
I4 ¼ ð2�Þ3�3
�Xi
~ki
�½2Qk1ðtÞQ�
k1ðtÞQk2ðtÞQ�k2ðtÞ þ perm
(29)
¼ 8�2ð2�Þ3�3
�Xi
~ki
�½P� ðk1ÞP� ðk2Þ þ perm: (30)
The factor of 2 on the first line above comes from different
ways of contracting the Q’s, and there are 3 permutationsinvolving k1; k2; k3. P� is defined as
P� ðkÞ ¼ 1
2�
H2
2k3: (31)
The terms dropped in Eq. (27) would have contributedsimilarly to the second term in Eq. (28), i.e., without anytime integral as for the first term, and hence would beevaluated only at a time t > tex, when their contributionwill be suppressed as mentioned above. Defining fNL as inRef. [3]:
h�ð ~k1; tÞ�ð ~k2; tÞ�ð ~k3; tÞi� ð2�Þ3�ð ~k1 þ ~k2 þ ~k3Þ 65 fNL
Xi<j
P� ðkiÞP� ðkjÞ; (32)
one gets fNL as in Eq. (1). (fNL as defined in Eq. (32) is thenegative of fNL in Refs. [1,22].) The key difference betweenour expression for fNL and that in Ref. [3] and other worksis that our fNL is a function of t rather than t� because weexpressed �ðtÞ as a function of QðtÞ instead of Qðt�Þ.Our calculation of h�3i above has been rather straight-
forward. We neither use extensive integration by parts torewrite the action, nor do we invoke any field redefinition,as in Refs. [1,21,22]. Furthermore, if t corresponds to theend of inflation then Ne � 60 for modes entering thehorizon today. Then, as mentioned earlier, the �Ne termin fNL is comparable to the � term and a priori should notbe ignored.Now, we consider the constancy of the 3-point function
of � . To determine how fNL changes during inflation after amode crosses the horizon, we take the derivative of fNLwith respect to t. Now, using d=dt ¼ _�d=d� we get
d�
dt’ ½4�2 � 2��H d�
dt’ ½2��� �H
d�
dt’ ½4��� ��� H; (33)
where ¼ V 02V0000=V3. Then, dfNL=dt � ð5=6Þd½��Ne��=dt ¼ 0. Here we have kept terms to first order in slow-roll parameters as in fNL. Thus, for the terms consideredabove, dfNL=dt is zero. indicating that fNLðtÞ ¼ fNLðtexÞ,i.e., the Ne factor in �Ne, which corresponds to growthoutside the horizon, is canceled by changes in other termsand does not contribute to the bispectrum.9 However, notethat hQ3i in Eq. (20) is a function of terms proportional to �and �, and so theNe-dependent term in hQ3i is not canceledby the time variation of other terms.The argument above for the constancy of � closely
follows that in Ref. [3]. In Ref. [3] it is argued that fNL
9Note that these arguments are for adiabatic fluctuations of theinflaton and do not apply to any nonadiabatic fluctuations orfluctuations of other scalar fields considered in Refs. [4–6].
NAMIT MAHAJAN AND RAGHAVAN RANGARAJAN PHYSICAL REVIEW D 83, 043510 (2011)
043510-6
is constant outside the horizon by taking the derivative offNLðt�Þ effectively with respect to t�. One may argue that to
study evolution of h�3ðtÞi outside the horizon one shouldtake derivatives with respect to t. But then, dfNL=dtwill be
0 by construct, as in the �N formalism �ðk; tÞ is expressedin terms of ��ðk; t�Þ and fNL is independent of t.
In conclusion, we have calculated the 3-point function of
the curvature perturbation � using the in-in formalism in agauge in which �� � 0. The calculation of the 3-point
function of � has generally been done ignoring inflatonself-interactions, and using the �N formalism which byconstruct does not allow one to check for evolution outsidethe horizon. We have included the contribution associatedwith non-Gaussian fluctuations of the inflaton due to acubic self-interaction, and this is proportional to �Ne andgrows outside the horizon. If we take Ne to be the numberof e-foldings of inflation after the mode of interest has leftthe horizon till the end of inflation, then for our currenthorizon scale Ne is 60. In new inflation, small-field naturalinflation and running-mass models of inflation, � < � < �,and �Ne is then comparable to other contributions to thenon-Gaussianity parameter fNL and should not be out-rightly ignored. The Ne-dependent term corresponds toevolution outside the horizon. However, on including thetime dependence of other contributions to fNL, this time-dependent growth cancels. Our results also indicate thatthere may be issues related to the convergence at higherorders in perturbation theory. Higher-order contributionscan render fNL to have stronger time dependence.
R. R. would like to acknowledge J. Maldacena and D.Seery for very useful clarifications of their work. R. R.would also like to thank the organizers of the XthWorkshop on High Energy Physics Phenomenology(WHEPP-X) at the Institute for Mathematical Sciences,Chennai, India for discussions on issues related to thiswork.
APPENDIX
In this appendix we compare the _�ð�Þ, _�ð�Þ0, _�ð�Þ00,_�ð�Þ000, and _�ð�Þ0000 terms in Eq. (15). We will ignoreterms proportional to � and derivatives of � which aresmall for inflation models that are of our interest. We will
also use Eqs. (2), (3), and (33), so � � � €�=ðH _�Þ, _H ¼��H2 � 0, and _� � ��H. We take ki � kt.
The _� terms within curly brackets in Eq. (15) are
� _�=kt.
The _�0 term is �k1�
k2t_�ð�Þ0 ¼ k1�
k2ta €� ¼ � k1
k2t
€�
H� 1
kt_��: (A1)
Thus, this is smaller than the _� terms.
The _�00 terms are � _�00=k3t ¼ a2�:::=k3t . Now, €� �
�H� _�. Therefore, �::: � � _�H2 þ �2 _�H2. The _�00 terms
are �a2
k3tð�H2 þ �2H2Þ _� (A2)
¼ a2
a2exH2ex
ð�H2þ�2H2Þ_�
kt¼ expð2NeÞð�þ�2Þ
_�
kt; (A3)
where we use kt ¼ aexHex and ignore the variation in H.With � ¼ 0:5�2, � ¼ �0:02, and Ne ¼ 60, the above is
approximately 1049 times larger than the _� term.
The _�000 term is �k1�
k4t_�000 ¼ k1�
k4ta3�
::::¼ � k1a2
Hk4t�::::: (A4)
Now, �::: � ð�þ �2ÞH2 _�. Therefore, �
::::� �ð4�þ�2Þ�H3 _�. Then, the _�000 term is �
a2
a2exH2ex
ð4�þ �2Þ�H2_�
kt¼ e2Neð4�þ �2Þ�
_�
kt; (A5)
which is a factor of � less than the _�00 term.
The _�0000 terms are� _�0000=k5t . Using�::::from above, d�
::::
dt �ð11��2 þ 4�2 þ �4ÞH4 _�. Then, the _�0000 terms are �
a4
k5t
d�::::
dt� 10
a4�4H4
a4exH4ex
_�
kt¼ 10e4Ne�4
_�
kt: (A6)
For � ¼ �0:02 and Ne ¼ 60, the above term is a factor of
1098 larger than the _� term. It is also larger than the _�00term by a factor of 10�2 expð2NeÞ.Thus, terms higher in slow-roll parameters in evaluating
the integral in I1 are larger than the lowest order terms.This also holds for I2 and I3.
[1] J.M. Maldacena, J. High Energy Phys. 05 (2003) 013 (see
Sec. 3.1).[2] P. Creminelli, J. Cosmol. Astropart. Phys. 10 (2003) 003.[3] D. Seery, K. A. Malik, and D.H. Lyth, arXiv:0802.0588v4.
This differs in the expression for fNL from the (earlier)
published version: J. Cosmol. Astropart. Phys. 03 (2008)
014.[4] T. Falk, R. Rangarajan, and M. Srednicki, Astrophys. J.
403, L1 (1993).[5] M. Zaldarriaga, Phys. Rev. D 69, 043508 (2004).
REMARKS ON NON-GAUSSIAN FLUCTUATIONS OF THE . . . PHYSICAL REVIEW D 83, 043510 (2011)
043510-7
[6] F. Bernardeau, T. Brunier, and J. P. Uzan, Phys. Rev. D 69,063520 (2004).
[7] L. Boubekeur and D.H. Lyth, J. Cosmol. Astropart. Phys.07 (2005) 010.
[8] L. Alabidi and D.H. Lyth, J. Cosmol. Astropart. Phys. 08(2006) 013.
[9] E. Komatsu et al. (WMAP Collaboration), Astrophys. J.Suppl. Ser. 180, 330 (2009).
[10] N. Barnaby and J.M. Cline, J. Cosmol. Astropart. Phys. 07(2007) 017.
[11] D. S. Salopek and J. R. Bond, Phys. Rev. D 42, 3936(1990).
[12] J.M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys.Rev. D 28, 679 (1983).
[13] D. H. Lyth and D. Wands, Phys. Rev. D 68, 103515 (2003).[14] G. I. Rigopoulos and E. P. S. Shellard, Phys. Rev. D 68,
123518 (2003).[15] K. A. Malik and D. Wands, Classical Quantum Gravity 21,
L65 (2004).[16] D. H. Lyth, K.A. Malik, and M. Sasaki, J. Cosmol.
Astropart. Phys. 05 (2005) 004.[17] D. Langlois and F. Vernizzi, Phys. Rev. Lett. 95, 091303
(2005).
[18] D. Langlois and F. Vernizzi, Phys. Rev. D 72, 103501(2005).
[19] S. Weinberg, Phys. Rev. D 74, 023508 (2006).[20] S. Weinberg, Phys. Rev. D 78, 123521 (2008).[21] X. Chen, M.X. Huang, S. Kachru, and G. Shiu, J. Cosmol.
Astropart. Phys. 01 (2007) 002.[22] D. Seery and J. E. Lidsey, J. Cosmol. Astropart. Phys. 09
(2005) 011.[23] M. Sasaki and E.D. Stewart, Prog. Theor. Phys. 95, 71
(1996).[24] C. Itzykson and J. B. Zuber, Quantum Field Theory,
International Series In Pure and Applied Physics(McGraw-Hill, New York, USA, 1980), Sec. 6–1–4.
[25] A. D. Linde, Particle Physics and InflationaryCosmology, Contemporary Concepts in Physics(Harwood Academic, Chur, Switzerland, 1990), Vol. 5;The same article also appeared in Contemp. ConceptsPhys. 5, 2005, 1.
[26] A. Gangui, F. Lucchin, S. Matarrese, and S. Mollerach,Astrophys. J. 430, 447 (1994) [see the last term ofEq. (26)].
[27] F. Vernizzi and D. Wands, J. Cosmol. Astropart. Phys. 05(2006) 019.
NAMIT MAHAJAN AND RAGHAVAN RANGARAJAN PHYSICAL REVIEW D 83, 043510 (2011)
043510-8