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Operator Product Expansion of Inflationary Correlators
and Conformal Symmetry of de Sitter
A. Kehagiasaand A. Riottob
aPhysics Division, National Technical University of Athens,
15780 Zografou Campus, Athens, Greece
bDepartment of Theoretical Physics and Center for Astroparticle Physics (CAP)
24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland
Abstract
We study the multifield inflationary models where the cosmological perturbation is sourced by light scalar
fields other than the inflaton. The corresponding perturbations are both scale invariant and special con-
formally invariant. We exploit the operator product expansion technique of conformal field theories to
study the inflationary correlators enjoying the symmetries present during the de Sitter epoch. The op-
erator product expansion is particularly powerful in characterizing inflationary correlation functions in
two observationally interesting limits, the squeezed limit of the three-point correlator and the collapsed
limit of the four-point correlator. Despite the fact that the shape of the four-point correlators is not
fixed by the symmetries of de Sitter, its exact shape can be found in the collapsed limit making use of
the operator product expansion. By employing the fact that conformal invariance imposes the two-point
cross-correlations of the light fields to vanish unless the fields have the same conformal weights, we are
able to show that the Suyama-Yamaguchi inequality relating the coefficients fNLof the bispectrum in
the squeezed limit and τNLof the trispectrum in the collapsed limit also holds when the light fields are
intrinsically non-Gaussian. In fact, we show that the inequality is valid irrespectively of the conformal
symmetry, being just a consequence of fundamental physical principles, such as the short-distance ex-
pansion of operator products. The observation of a strong violation of the inequality will then have
profound implications for inflationary models as it will imply either that multifield inflation cannot be
responsible for generating the observed fluctuations independently of the details of the model or that
some new non-trivial degrees of freedom play a role during inflation.
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arXiv:1205.1523v1 [hep-th] 7 May 2012
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Contents
1Introduction2
2Symmetries of the de Sitter geometry7
2.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
3 Symmetry Constraints13
3.1Scale Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Conformal Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
4 The operator product expansion and the NG correlators19
4.1The three-point function from the OPE and its squeezed limit . . . . . . . . . . . . . . . .23
4.2The four-point function from the OPE and its collapsed limit . . . . . . . . . . . . . . . . 26
5 On the Suyama-Yamaguchi inequality31
5.1A further generalization of the Suyama-Yamaguchi inequality . . . . . . . . . . . . . . . . 36
6 Logarithmic Conformal Field Theories38
7 Some considerations and conclusions43
1Introduction
One of the basic ideas of modern cosmology is that there was an epoch early in the history of the universe
when potential, or vacuum, energy associated to a scalar field, the inflaton, dominated other forms of
energy density such as matter or radiation. During such a vacuum-dominated era the scale factor grew
exponentially (or nearly exponentially) in time. During this phase, dubbed inflation [1, 2], a small,
smooth spatial region of size less than the Hubble radius could grow so large as to easily encompass the
comoving volume of the entire presently observable universe. If the universe underwent such a period of
rapid expansion, one can understand why the observed universe is so homogeneous and isotropic to high
accuracy.
Inflation has also become the dominant paradigm for understanding the initial conditions for structure
formation and for Cosmic Microwave Background (CMB) anisotropy. In the inflationary picture, primor-
dial density and gravity-wave fluctuations are created from quantum fluctuations “redshifted” out of the
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horizon during an early period of superluminal expansion of the universe, where they are “frozen” [3–7].
Perturbations at the surface of last scattering are observable as temperature anisotropy in the CMB. The
last and most impressive confirmation of the inflationary paradigm has been recently provided by the
data of the Wilkinson Microwave Anistropy Probe (WMAP) mission which has marked the beginning
of the precision era of the CMB measurements in space [8]. The WMAP collaboration has produced
a full-sky map of the angular variations of the CMB, with unprecedented accuracy. WMAP data con-
firm the inflationary mechanism as responsible for the generation of curvature (adiabatic) superhorizon
fluctuations.
Despite the simplicity of the inflationary paradigm, the mechanism by which cosmological adiabatic
perturbations are generated is not yet fully established. In the standard picture, the observed density
perturbations are due to fluctuations of the inflaton field itself. When inflation ends, the inflaton oscillates
about the minimum of its potential and decays, thereby reheating the universe. As a result of the
fluctuations each region of the universe goes through the same history but at slightly different times.
The final temperature anisotropies are caused by the fact that inflation lasts different amounts of time in
different regions of the universe leading to adiabatic perturbations. Under this hypothesis, the WMAP
dataset already allows to extract the parameters relevant for distinguishing among single-field inflation
models.
An alternative to the standard scenario is represented by the curvaton mechanism [9–11] where the
final curvature perturbations are produced from an initial isocurvature perturbation associated to the
quantum fluctuations of a light scalar field (other than the inflaton), the curvaton, whose energy density is
negligible during inflation. The curvaton isocurvature perturbations are transformed into adiabatic ones
when the curvaton decays into radiation much after the end of inflation. Alternatives to the curvaton
model are those models characterized by the curvature perturbation being generated by an inhomogeneity
in the decay rate [12] or the mass [13] of the particles responsible for the reheating after inflation. Other
opportunities for generating the curvature perturbation occur at the end of inflation [14] and during
preheating [15].
All these alternative models to generate the cosmological perturbations have in common that the
comoving curvature perturbation in generated on superhorizon scale when the isocurvature perturbation,
which is associated to the fluctuations of these light scalar fields different from the inflaton, is converted
into curvature perturbation after (or at the end) of inflation. The very simple fact that during inflation
the fluctuation associated to these light fields is of the isocurvature type, that is the energy density
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stored in these fields is small compared to the vacuum energy responsible for inflation, implies that
the de Sitter isometries are not broken by the presence of these light fields. Therefore their statistical
correlators should enjoy all the symmetries present during the de Sitter epoch and therefore be not
only scale invariant, but also conformal invariant. Building up on the results of Ref. [16] (where the
most general three-point function for gravitational waves produced during a period of exactly de Sitter
expansion was studied) and of Ref. [17], in Ref. [18] the consequences of scale invariance and special
conformal symmetry of scalar perturbations were discussed. Further extensions appeared in Ref. [19]
where conformal consistency relations for single-field inflation have been investigated and in Ref. [20]
where the existence of non-linearly realized conformal symmetries for scalar adiabatic perturbations in
cosmology has been pointed out.
In this paper we are concerned with the large class of multifield models where the non-Gaussianity
(NG) of the curvature perturbation is sourced by light fields other than the inflaton. By the δN formalism
[21], the comoving curvature perturbation ζ on a uniform energy density hypersurface at time tfis, on
sufficiently large scales, equal to the perturbation in the time integral of the local expansion from an initial
flat hypersurface (t = t∗) to the final uniform energy density hypersurface. On sufficiently large scales,
the local expansion can be approximated quite well by the expansion of the unperturbed Friedmann
universe. Hence the curvature perturbation at time tf can be expressed in terms of the values of the
relevant scalar fields σI(t∗,? x) at t∗
ζ(tf,? x) = NIσI+1
2NIJσIσJ+ ··· ,(1.1)
where NIand NIJare the first and second derivative, respectively, of the number of e-folds
N(tf,t∗,? x) =
?tf
t∗
dtH(t,? x).(1.2)
with respect to the field σI. From the expansion (1.1) one can read off the n-point correlators. For
instance, the three- and four-point correlators of the comoving curvature perturbation, the so-called
bispectrum and trispectrum respectively, is given by
Bζ(?k1,?k2,?k3) = NINJNKBIJK
?k1?k2?k3+ NINJKNL
?
PIK
?k1PJL
?k2+ 2 permutations
?
(1.3)
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and
Tζ(?k1,?k2,?k3,?k4)=NINJNKNLTIJKL
?k1?k2?k3?k4
?
+NIJNKNLNM
PIK
?
PIL
?k1BJLM
?k12?k3?k4+ 11 permutations
?
+NIJNKLNMNN
PJL
?k12PIM
?k1PKN
?k3
+ 11 permutations
?
,
+NIJKNLNMNN
?
?k1PJM
?k2
PKN
?k3
+ 3 permutations
?
(1.4)
where
?σI
?k1σJ
?k2σJ
?k1σJ
?k2σK
?k3σL
?k2?
?k3?
?k4?
= (2π)3δ(?k1+?k2)PIJ
?k1,
?σI
?k1σJ
= (2π)3δ(?k1+?k2+?k3)BIJK
?k1?k2?k3,
?σI
= (2π)3δ(?k1+?k2+?k3?k4)TIJKL
?k1?k2?k3?k4,(1.5)
and?kij= (?ki+?kj). We see that the three-point correlator (and similarly for the four-point one) of the
comoving curvature perturbation is the sum of two pieces. One, proportional to the three-point correlator
of the σIfields, is model-dependent and present when the fields σIare intrinsically NG. The second one
is universal and is generated when the modes of the fluctuations are superhorizon and is present even
if the σIfields are gaussian. One should keep in mind that the relative magnitude between the two
contributions is model-dependent and that the constraints imposed by the symmetries present during
the de Sitter stage apply separately to both the first and the second contribution1. Even though the
intrinsically NG contributions to the n-point correlators are model-dependent, their forms are dictated
by the conformal symmetry of the de Sitter stage (although their amplitudes remain model-dependent).
This is the subject of the present paper.
After a brief summary in of the symmetries of the de Sitter geometry in section 2, we will discuss
in section 3 the constraints imposed by scale invariance and conformal symmetry on the two- and three-
point correlators. In particular, we will demonstrate that the two-point cross-correlations of the light
fields vanish unless their conformal weights are equal. This is a in fact a standard result of field theories
enjoying conformal symmetry.
1The reason is that although the scalar fields σImay have specific scaling dimension and may transform
irreducibly under the conformal group, the comoving curvature perturbations ζ does not have specific scaling
dimension as it is the sum of operators with different dimensions. In other words, ζ is a reducible representation of
the conformal group. However, its n-point functions may be specified by the conformal properties of its irreducible
components of the conformal group.
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We then turn out attention to the operator product expansion technique of conformal field theories
to investigate which kind of informations we can gather on inflationary correlations for fields considered
at coincidence points. The operator product expansion is very powerful to analyze the squeezed limit
of the bispectrum and the collapsed limit of the trispectrum. These limits are particularly interesting
from the observationally point of view because they are associated to the local model of NG (for a review
see [22]) which leads to pronounced effects of NG on the clustering of dark matter halos and to strongly
scale-dependent bias [23].
We use the techniques developed by Ferrara, Gatto and Grillo in the early 70’s to find the model-
independent shape of the three- and point-correlators in the squeezed and collapsed limit, respectively.
While conformal symmetry does not fix uniquely the shape of the four-point correlator, we show that
its shape can be indeed computed in the collapsed limit by using the so-called conformal blocks. This
allows us to prove that the contribution to the three- and four-point correlators of the curvature per-
turbation from the connected three- and four-point correlators of the σIfields (originated from the fact
that these fields can be intrinsically NG) have the same shapes of the universal and model-independent
contribution generated when the modes of the fluctuations are superhorizon and present even if the σI
fields are gaussian. This is done in section 4. This result allows us to extend in section 5 the so-called
Suyama-Yamaguchi inequality [24] which relates the coefficient of the trispectrum τNLof the curvature
perturbation in the collapsed limit to the coefficient fNL of the squeezed limit of its bispectrum and
was proved under the condition that the fluctuations of the scalar fields σIat the horizon crossing are
scale invariant and gaussian. A generalization of this inequality was provided in Refs. [25] to the case
of NG σIfields. However there the crucial assumption was made that the coefficients fNLand τNLwere
momentum-independent, which is not automatically guaranteed if the fields are NG. Based on our results
stemming from scale invariance and special conformal symmetry, we can show that indeed fNLand τNL
are momentum-independent in the squeezed and collapsed limits respectively and therefore we are able
to show that the Suyama-Yamaguchi inequality is valid when the light fields σIare NG. In fact, we
will take a further step and, based on the operators’ short-distance expansion, we will prove that that
the Suyama-Yamaguchi inequality holds on general grounds. It is consequence of fundamental physical
principles (like positivity of the two-point function) and its hard violation would required some new
non-trivial physics to be involved. The observation of a strong violation of the inequality will then have
profound implications for inflationary models as it will imply either that multifield inflation cannot be
responsible for generating the observed fluctuations independently of the details of the model or that
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some new non-trivial degrees of freedom play a role during inflation.
In section 6 we study, even though briefly, the possible implications of another class of conformal
theories, namely the logarithmic conformal field theories, which can be of interest from the cosmological
point of view. These are theories characterized by the appearance of logarithms in correlation functions
due to logarithmic short-distance singularities in the operator product expansion. As a consequence, the
spectral index of the curvature perturbation power spectrum gets a new contribution due to logarithmic
short-distance singularities in the OPE. This contribution is present even if the fields light involved are
massless. Finally, section 7 present our conclusions.
2 Symmetries of the de Sitter geometry
Conformal invariance in three-dimensional space R3is connected to the symmetry under the group
SO(1,4) in the same way conformal invariance in a four-dimensional Minkowski spacetime is connected
to the SO(2,4) group. As SO(1,4) is the isometry group of de Sitter spacetime, a conformal phase during
which fluctuations were generated could be a de Sitter stage. In such a case, the kinematics is specified
by the embedding of R3as flat sections in de Sitter spacetime. The de Sitter isometry group acts as
conformal group on R3when the fluctuations are superhorizon. It is in this regime that the SO(1,4)
isometry of the de Sitter background is realized as conformal symmetry of the flat R3sections [17,18].
Correlators are expected to be constrained by conformal invariance. All these reasonings apply in the
case in which the cosmological perturbations are generated by light scalar fields other than the inflaton.
Indeed, it is only in such a case that correlators inherit all the isometries of de Sitter.
It is also important to stress that the two-point correlator cannot capture the full conformal invariance
and is only sensitive to the scale invariance symmetry. To reveal the full conformal symmetry one needs
to consider higher-order correlators. This is what we will do in the following. Before though, and for
the sake of self-completeness, we would like to remind the reader of some basic geometrical and algebraic
properties of de Sitter spacetime and group [26]. The expert reader may skip the following two sections
many details of which are contained already in, for instance, Ref. [17].
The four-dimensional de Sitter spacetime of radius H−1is described by the hyperboloid
ηABXAXB= −X2
0+ X2
i+ X2
5=
1
H2
(i = 1,2,3),(2.1)
embedded in five-dimensional Minkowski spacetime M1,4with coordinates XAand flat metric ηAB =
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diag(−1,1,1,1,1). A particular parametrization of the de Sitter hyperboloid is provided by
1
2H
xi
Hη,
X5= −1
2H
X0=
?
Hη −
1
Hη
?
−1
2
x2
η,
Xi=
?
Hη +
1
Hη
?
+1
2
x2
η,
(2.2)
which may easily be checked that satisfies Eq. (2.1). The de Sitter metric is the induced metric on the
hyperboloid from the five-dimensional ambient Minkowski spacetime
ds2
5= ηABdXAdXB.(2.3)
For the particular parametrization (2.2), for example, we find
ds2=
1
H2η2
?−dη2+ d? x2?. (2.4)
The group SO(1,4) acts linearly on M1,4. Its generators are
JAB= XA
∂
∂XB− XB
∂
∂XA
A,B = (0,1,2,3,5) (2.5)
and satisfy the SO(1,4) algebra
[JAB,JCD] = ηADJBC− ηACJBD+ ηBCJAD− ηBDJAC.(2.6)
We may split these generators as
Jij, P0= J05, Π+
i= Ji5+ J0i, Π−
i= Ji5− J0i, (2.7)
which act on the de Sitter hyperboloid as
Jij= xi
∂
∂xj
∂η+ xi∂
− xj
∂
∂xi,
P0= η∂
∂xi,
∂η+ H?x2δij− 2xixj
Π−
i= −2Hηxi∂
1
H∂xi
? ∂
∂xj
− Hη2∂
∂xi,
Π+
i=
∂
(2.8)
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Let us now consider the three-point functions. Here we want to calculate the correlator
Gabc(? x1,? x2,? x3) = ?σa(? x1)σb(? x2)σc(? x3)?. (6.18)
Again we will use Ward identities for dilations and special conformal transformations. From dilation we
get
∆i
aGibc+ ∆i
bGaic+ ∆i
cGabi+
?
? x1·?∇1+ ? x2·?∇2+ ? x3·?∇3
?
Gabc= 0, (6.19)
whereas from special conformal transformations we have (?b being the parameter vector of the special
conformal transformation, see Eq. (2.28))
δ?b ·
?
2? x1∆i
?
aGibc+ 2? x2∆i
bGaic+ 2? x3∆i
cGabi+
(? x1+ ? x2)x12
∂
∂x12
+ (? x1+ ? x3)x13
∂
∂x13
+ (? x2+ ? x3)x23
∂
∂x23
??
Gabc= 0.(6.20)
Combining the two equations above we get
∆i
aGibc= ∆i
bGaic= ∆i
cGabi,(6.21)
which leads us to
wG122+ xij
∂
∂xijG122= 0,
∀ i < j(6.22)
wG222+ G122+ xij
∂
∂xijG122= 0,
∀ i < j(6.23)
G111= G112= 0. (6.24)
The solution to Eqs.(6.22) and (6.23) is given by
G122= cx−w
12x−w
23x−w
13,(6.25)
G222= cx−w
12x−w
23x−w
13(2ln(x12x23x13) + a).(6.26)
As a result, the three-point functions in the theory are given by
?σ1(? x1)σ2(? x2)σ2(? x3)? =
c
23xw
c
23xw
xw
12xw
13
,(6.27)
?σ2(? x1)σ2(? x2)σ2(? x3)? =
?σ1(? x1)σ1(? x2)σ2(? x3) = 0,
?σ1(? x1)σ1(? x2)σ1(? x3) = 0.
xw
12xw
13
?
− ln?x12x23x13
?+ a
?
, (6.28)
(6.29)
(6.30)
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The correlators in momentum space are easily evaluated. For example we find
?σ1(?k1)σ2(?k2)??=C0(w)
k3−2w
1
,
?σ2(?k1)σ2(?k2)??= a?σ1(?k1)σ2(?k2)??+
?σ1(?k1)σ1(?k2)? = 0,
∂
∂w?σ1(?k1)σ2(?k2)??=C0(w)
k3−2w
1
?
− 2lnk1+ a + C0,w
?
,
(6.31)
where C0,w denotes derivative of C0 with respect to w. Similar expression holds for the three-point
functions. For example we have that
?σ2(?k1)σ2(?k2)σ2(?k3)??= a?σ1(?k1)σ2(?k2)σ2(?k3)??+
∂
∂w?σ1(?k1)σ2(?k2)σ2(?k3)??.(6.32)
In particular, in the squeezed k1? k2,k3we get
?σ1(?k1)σ2(?k2)σ2(?k3)??∼
C1(w)
k3−w
1
C1(w)
k3−w
1
k3−2w
2
, (6.33)
?σ2(?k1)σ2(?k2)σ2(?k3)??∼
?σ1(?k1)σ1(?k2)σ1(?k3)??= ?σ1(?k1)σ2(?k2)σ2(?k3)??= 0.
k3−2w
2
?
ln(k1k2
2) + a + C1,w
?
,(6.34)
(6.35)
The corresponding correlators of the comoving curvature perturbations can be easily calculated. For
example, by using the expression (1.1) we get for the spectrum
?ζ?k1ζ?k2??∼
A
k3−2w
1
(1 + 2γ lnk1) ∼
A
k3−2w−2γ
1
,(6.36)
where
A = 2N1N2+ aN2
2C0+ C0,w,γ =C0
N2
2
A, (6.37)
and in the last step in (6.36) we have assumed that γ ? 1. We see that the spectral index of the curvature
perturbation power spectrum, nζ−1 = dlnk3Pζ/dlnk, gets a new contribution equal to 2γ from the due
to logarithmic short-distance singularities in the OPE, even if the fields involved are massless. Further
considerations will be presented elsewhere.
It should be noted that logarithmic field theories have OPE which contains short distance logarithmic
singularities. For example the OPE in Eq. (4.13) is modified as [51,59]
?
σI(? x1)σJ(? x2)? x1→? x2
∼
1
x12
?wI+wJ?
CIJ
0 + DIJ
0ln|x12| +CIJK
xwK
12
σK(? x2) + ···
?
.(6.38)
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Repeating the analysis of the previous section, we can calculate the four-point function at the collapsed
limit by using the above OPE in the (12) and (34) channels. The only difference is that thye matrix
CIJin Eq. (5.36) is not positive definite. In fact, a simple inspection of (6.31) reveals that, in the
simplest case of two fields (I,J = 1,2), CIJhas a positive and a negative eigenvalues. Therefore, the
Cauchy-Schwarz inequality (5.39) gets inverted and leads to
τNL≤
?6
5fNL
?2
.(6.39)
Thus, logarithmic conformal field theories provide an example, consistent with the de Sitter symmetries,
which leads to violation of the SY inequality. Such theories violate unitarity but there is no obvious
reason for a CFT to be unitary [29], and logaritmic CFT’s is an example. It remains to be seen if
logarithmic conformal field theories play a real role in cosmology or not.
7Some considerations and conclusions
In this paper we have studied the implications of the symmetries present during a de Sitter phase for the
statistical correlations of the light fields present during a multifield inflationary dynamics. In particular,
we have assumed that the NG is generated by light fields other than the inflaton field. The cosmological
perturbations are both scale invariant and conformally invariant.
We have first shown that, as a consequence of the conformal symmetries, the two-point cross-
correlation of the light fields vanish if their conformal weights are different. Therefore, no assumption is
needed on such a cross-correlation, it is simply dictated by the conformal symmetry.
Secondly, we have pointed out that the OPE technique is very suitable to analyze two interesting
limits: the squeezed limit of the three-point correlator and the collapsed limit of the four-point correlator.
Despite the fact that the conformal symmetry does not fix the shape of the four-point correlators of the
light NG fields, we have been able to compute it in the collapsed limit. Both the resulting shapes of the
squeezed limit of the bispectrum and the collapsed limit of the trispectrum of the NG light fields turn
out to be of the same form of the shapes of the corresponding bispectrum and trispectrum universally
generated on superhorizon scales of the comoving curvature perturbation. Thanks to this result, we
have succeeded in showing that the SY inequality relating the two NG observables fNLand τNLis valid
independently of the NG nature of the light scalar fields at horizon crossing. In fact, we have been able to
show that the SY inequality is valid irrespectively of the conformal symmetry, being just a consequence
of the short-distance expansion of the two-operator product expansion.
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In most of this paper the working assumption was that the cosmological perturbations enjoy both the
scale invariant and the conformal symmetry of pure de Sitter. The inflaton background spontaneously
breaks this symmetry, so that the variation of a correlation function of the curvature pertubation under
the de Sitter isometry group should always be connected with the soft emission of one or many soft
inflaton perturbations [19,60–63]. It would be interesting to understand how our results will change
under the assumption of a slight breaking of the de Sitter isometries. Under the assumption that the
NG is generated by scalar fields other than the inflaton, we expect that our results in the squeezed and
collapsed limits of the bispectrum and trispectrum respectively are still valid up to small corrections of
the order of the slow-roll parameters.
Finally, while it is clear that a detection of a non-conformal correlation function, for example an
equilateral three-point function, would imply that the source of perturbations is not decoupled from the
inflaton [18], it would be interesting to understand if it possible to find other cosmological observables
which can robustly test the conformality of the primordial cosmological perturbations. This might be a
non trivial task as post inflationary nonlinear evolution of the correlators contaminate such a primordial
input.
Acknowledgments
We thank C. Byrnes, P. Creminelli, A. Petkou and M. Sloth for useful conversations. A.R. is supported by
the Swiss National Science Foundation (SNSF), project ‘The non-Gaussian Universe” (project number:
200021140236).
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