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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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