On graviton non-gaussianities during inflation

# On graviton non-gaussianities during inflation

Juan M. Maldacena School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540    Guilherme L. Pimentel Joseph Henry Laboratories, Princeton University, Princeton, NJ, 08544.
###### Abstract

We consider the most general three point function for gravitational waves produced during a period of exactly de Sitter expansion. The de Sitter isometries constrain the possible shapes to only three: two preserving parity and one violating parity. These isometries imply that these correlation functions should be conformal invariant. One of the shapes is produced by the ordinary gravity action. The other shape is produced by a higher derivative correction and could be as large as the gravity contribution. The parity violating shape does not contribute to the bispectrum Soda:2011am (); Shiraishi:2011st (), even though it is present in the wavefunction. We also introduce a spinor helicity formalism to describe de Sitter gravitational waves with circular polarization.

These results also apply to correlation functions in Anti-de Sitter space. They also describe the general form of stress tensor correlation functions, in momentum space, in a three dimensional conformal field theory. Here all three shapes can arise, including the parity violating one.

preprint: PUPT-2371

## I Introduction

Recently there has been some effort in understanding the non-gaussian corrections to primordial fluctuations generated during inflation. The simplest correction is a contribution to the three point functions of scalar and tensor fluctuations. For scalar fluctuations there is a classification of the possible shapes for the three point function that appear to the leading orders in the derivative expansion for the scalar field Cheung:2007st (); Cheung:2007sv (); Chen:2006nt (); Weinberg:2008hq ().

In this paper we consider tensor fluctuations. We work in the de Sitter approximation and we argue that there are only three possible shapes for the three point function to all orders in the derivative expansion. Thus the de Sitter approximation allows us to consider arbitrarily high order corrections in the derivative expansion. The idea is simply that the three point function is constrained by the de Sitter isometries. At late times, the interesting part of the wavefunction becomes time independent and the de Sitter isometries act as the conformal group on the spatial boundary. We are familiar with the exact scale invariance, but, in addition, we also have conformal invariance. The conformal invariance fixes the three point functions almost uniquely. By “almost”, we simply mean that there are three possible shapes allowed, two that preserve parity and one that violates parity. We compute explicitly these shapes and we show that they are the only ones consistent with the conformal symmetry. In particular, we analyze in detail the constraints from conformal invariance. In order to compute these three shapes it is enough to compute them for a simple Lagrangian that is general enough to produce them. The Einstein gravity Lagrangian produces one of these three shapes Arutyunov:1999nw (); Maldacena:2002vr (). The other parity conserving shape can be obtained by adding a term to the action, where is the Weyl tensor. Finally, the parity violating shape can be obtained by adding , where is the Weyl tensor with two indices contracted with an tensor. The fact that the gravitational wave expectation value is determined by the symmetries is intimately connected with the following fact: in four dimensional flat space there are also three possible three point graviton scattering amplitudes Benincasa:2007xk () 111 In flat space, one has to complexify the momenta to have nonvanishing three point amplitudes. In de Sitter, they are the natural observables.. Though the parity violating shape is contained in the wavefunction of the universe (or in related AdS partition functions), it does not arise for expectation values Soda:2011am (); Shiraishi:2011st ().222 The previous version of this paper incorrectly stated that the parity violating shape contributed to de Sitter expectation values. The fact that there is no parity violation in the gravity wave bispectrum was shown in Soda:2011am (); Shiraishi:2011st (). . Thus for gravitational wave correlators in we only have two possible shapes, both parity conserving.

We show that, under general principles, the higher derivative corrections can be as large as the term that comes from the Einstein term, though still very small compared to the two point function. In fact, we expect that the ratio of is of order one for the ordinary gravity case, and can be as big as one for the other shape. When it becomes one for the other shape it means that the derivative expansion is breaking down. This happens when the scale controlling the higher derivative corrections becomes close to the Hubble scale. For example, the string scale can get close to the Hubble scale. Even though ordinary Einstein gravity is breaking down, we can still compute this three point function from symmetry considerations, indicating the power of the symmetry based approach for the three point function. This gravity three point function appears to be outside the reach of the experiments occurring in the near future. We find it interesting that by measuring the gravitational wave three point function we can directly assess the size of the higher derivative corrections in the gravity sector of the theory. Of course, there are models of inflation where higher derivatives are important in the scalar sector Alishahiha:2004eh (); ArmendarizPicon:1999rj () and in that case too, the non-gaussian corrections are a direct way to test those models Garriga:1999vw (); Silverstein:2003hf (); Chen:2006nt (). The simplicity of the results we find here is no longer present when we go from de Sitter to an inflationary background. However, the results we find are still the leading approximation in the slow roll expansion. Once we are away from the de Sitter approximation, one can still study the higher derivative corrections in a systematic fashion as explained in Cheung:2007st (); Weinberg:2008hq (); Senatore:2010wk ().

Our results have also a “dual” use. The computation of the three point function for gravitational waves is mathematically equivalent to the computation of the three point function of the stress tensor in a three dimensional conformal field theory. This is most clear when we consider the wavefunction of the universe as a function of the metric, expanded around de Sitter space at late times Maldacena:2002vr (). This is a simple consequence of the symmetries, we are not invoking any duality here, but making a simple statement 333Of course, this statement is consistent with the idea that such a wavefunction can be computed in terms of a dual field theory. Here we are not making any assumption about the existence of a dual field theory. Discussions of a possible dual theory in the de Sitter context can be found in Strominger:2001pn (); Witten:2001kn (). . From this point of view it is clear why conformal symmetry restricts the answer. If one were dealing with scalar operators, there would be only one possible three point function. For the stress tensor, we have three possibilities, two parity conserving and one parity violating. The parity conserving three point functions were computed in Osborn:1993cr (). Here we present these three point functions in momentum space. Momentum space is convenient to take into account the conservation laws, since one can easily focus on the transverse components of the stress tensor. However, the constraints from special conformal symmetry are a little cumbersome, but manageable. We derived the explicit form of the special conformal generators in momentum space and we checked that the correlators we computed are the only solutions. In fact, we found it convenient to introduce a spinor helicity formalism, which is similar to the one used in flat four dimensional space. This formalism simplifies the algebra involving the spin indices and it is a convenient way to describe gravitational waves in de Sitter, or stress tensor correlators in a three dimensional conformal field theory. In Fourier space the stress tensor has a three momentum , whose square is non-zero. The longitudinal components are determined by the Ward identities. So the non-trivial information is in the transverse, traceless components. The transverse space is two dimensional and we can classify the transverse indices in terms of their helicity. Thus we have two operators with definite helicity, . In terms of gravitational waves, we are considering gravitational waves that have circular polarization. These can be described in a convenient way by defining two spinors and , such that . In other words, we form a null four vector, and we proceed as in the four dimensional case. We only have symmetry, rather than , which allows us to mix dotted and undotted indices. We can then write the polarization vectors as (no bar), etc. This leads to simpler expressions for the three point correlation functions of the stress tensor in momentum space. We have expressed the special conformal generator in terms of these variables. One interesting aspect is that this formalism makes the three point function completely algebraic (up to the delta function for momentum conservation). As such, it might be a useful starting point for computing higher point functions in a recursive fashion, both in dS and AdS. This Fourier representation might also help in the construction of conformal blocks. The connection between bulk symmetries and the conformal symmetry on the boundary was discussed in the inflationary context in Larsen:2002et (); Larsen:2003pf (); Larsen:2004kf (); McFadden:2009fg (); McFadden:2010vh ().

The idea of using conformal symmetry to constrain cosmological correlators was also discussed in Antoniadis:2011ib (), which appeared while this paper was in preparation. Though the point of view is similar, some of the details differ. In that paper, scalar fluctuations were considered. However, scalar fluctuations, and their three point function, crucially depend on departures from conformal symmetry. It is likely that a systematic treatment of such a breaking could lead also to constraints, specially at leading order in slow roll. On the other hand, the gravitational wave case, which is discussed here, directly gives us the leading term in the slow roll expansion.

The paper is organized as follows. In section II we perform the computation of the most general three point function from a bulk perspective. We also discuss the possible size of the higher derivative corrections. In section III we review the spinor helicity formalism in 4D flat spacetime, and propose a similar formalism that is useful for describing correlators of CFTs and expectation values in dS and AdS. We then write the previously computed three point functions using these variables. In section IV we review the idea of viewing the wavefunction of the universe in terms of objects that have the same symmetries as correlators of stress tensors in CFT. We also emphasize how conformal symmetry constrains the possible shapes of the three point function. In section V we explicitly compute the three point function for the stress tensor for free field theories in 3D, and show that, up to contact terms, they have the same shapes as the ones that do not violate parity, computed from the bulk perspective. The appendices contain various technical points and side comments.

Note added: We have revised this paper correcting statements regarding the parity violating terms in the bispectrum, in light of Soda:2011am (); Shiraishi:2011st ().

## Ii Direct computation of general three point functions

In this section we compute the three point function for gravitational waves in de Sitter space. We do the computation in a fairly straightforward fashion. In the next section we will discuss in more detail the symmetries of the problem and the constraints on the three point function.

### ii.1 Setup and review of the computation of the gravitational wave spectrum

The gravitational wave spectrum in single-field slow roll inflation was derived in Starobinsky:1979ty (). Here we will compute the non-gaussian corrections to that result. As we discussed above, we will do all our computations in the de Sitter approximation. Namely, we assume that we have a cosmological constant term so that the background spacetime is de Sitter. There is no inflaton or scalar perturbation in this context. This approximation correctly gives the leading terms in the slow roll expansion. We leave a more complete analysis to the future.

It is convenient to write the metric in the ADM form

 g00=−N2+gijNiNj, g0i=gijNj, gij=e2Htexp(γ)ij (II.1)

Where is Hubble’s constant. and are Lagrange multipliers (their equations of motion will not be dynamical), and parametrizes gravitational degrees of freedom. The action can be expressed as

 S=M2Pl2∫√−g(R−6H2)=M2Pl2∫√g3(NR(3)−6NH2+N−1(EijEij−(Eii)2)) (II.2)

Where and we define . We fix the gauge by imposing that gravity fluctuations are transverse traceless, and . Up to third order in the action, we only need to compute the first order values of the Lagrange multipliers and Maldacena:2002vr (). By our gauge choice, these are and as there cannot be a first order dependence on the gravity fluctuations. Expanding the action up to second order in perturbations we find

 S2=M2Pl8∫(e3Ht˙γij˙γij−eHt∂lγij∂lγij) (II.3)

We can expand the gravitational waves in terms of polarization tensors and a suitable choice of solutions of the classical equations of motion. If we write

 γij(x,t)=∫d3k(2π)3∑s=+,−ϵsijeik.xγcl(t)a†s,→k+h.c. (II.4)

the gauge fixing conditions imply that the polarization tensors are traceless, , and transverse . The helicities can be normalized by . The equations of motion are then given by

 0= γ′′cl(η)−2ηγ′cl(η)+k2γcl(η) ,           η=−e−HtH (II.5)

where we introduced conformal time, . We take the classical solutions to be those that correspond to the Bunch-Davies vacuum Bunch:1978yq (), so . Here we have denoted by the absolute value of the 3-momentum of the wave. We are interested in the late time contribution to the two-point function, so we take the limit where . After Fourier transforming the late-time dependence of the two-point function and contracting it with polarization tensors of same helicities we find:

 ⟨γs1kγs2k′⟩=(2π)3δ3(k+k′)12k3(HMPl)24δs1s2 (II.6)

In the inflationary context (with a scalar field), higher derivative terms could give rise to a parity breaking contribution to the two point function. This arises from terms in the effective action of the form Lue:1998mq (); Alexander:2004wk (). This parity breaking term leads to a different amplitude for positive and negative helicity gravitational waves, leading to a net circular polarization for gravitational waves. If is constant this term is a total derivative and it does not contribute. Thus, in de Sitter there is no contribution from this term. In other words, the parity breaking contribution is proportional to the time derivative of . Some authors have claimed that one can get such parity breaking terms even in pure de Sitter Contaldi:2008yz (). However, such a contribution would break CPT. Naively, we would expect that CPT is spontaneously broken because of the expansion of the universe. However, in de Sitter we can go to the static patch coordinates where the metric is static. For such an observer we expect CPT to be a symmetry. A different value of the left versus right circular polarization for gravitational waves would then violate CPT.

In the AdS case, or in a general CFT, there can be parity violating contact terms in the two point function444A contact term is a contribution proportional to a delta function of the operator positions.. This is discussed in more detail in appendix C.

### ii.2 Three point amplitudes in flat space

In order to motivate the form of the four dimensional action that we will consider, let us discuss some aspects of the scattering of three gravitational waves in flat space. This is relevant for our problem since at short distances the spacetime becomes close to flat space.

In flat space we can consider the on shell scattering amplitude between three gravitational waves. Due to the momentum conservation condition we cannot form any non-zero Mandelstam invariant from the three momenta. Thus, all the possible forms for the amplitude are exhausted by listing all the possible ways of contracting the polarization tensors of the gravitational waves and their momenta, Benincasa:2007xk ()555In flat space, the three point amplitude is non-trivial only after analytically continuing to complex values of the momentum.. There are only two possible ways of doing this, in a parity conserving manner. One corresponds to the amplitude that comes from the Einstein action. The other corresponds to the amplitude we would get from a term in the action that has the form , where is the Weyl tensor. In addition, we can write down a parity violating amplitude that comes from a term of the form , where . These terms involving the Weyl tensor are expected to arise from higher derivative corrections in a generic gravity theory. By using field redefinitions, any other higher derivative interaction can be written in such a way that it does not contribute to the three point amplitude.

By analogy, in our de Sitter computation we will consider only the following terms in the gravity action

 Seff=∫d4x[√−g(M2Pl2(−6H2+R)+Λ−2(a WabcdWcdmnWmnab))++ Λ−2(b ϵabefWefcdWcdmnWmnab)] (II.7)

Here is a scale that sets the value of the higher derivative corrections. We will discuss its possible values later. This form of the action is enough for generating the most general gravity three point function that is consistent with de Sitter invariance. This will be shown in more detail in section IV, by using the action of the special conformal generators. For the time being we can accept it in analogy to the flat space result. Instead of the Weyl tensor in (II.7) we could have used the Riemann tensor. The disadvantage would be that the term would not have vanished in a pure de Sitter background and it would also have contributed to the two point function. However, these extra contributions are trivial and can be removed by field redefinitions. So it is convenient to consider just the term.

### ii.3 Three-point function calculations

In this subsection we compute the three point functions that emerge from the action in (II.7). First we compute the three point function coming from the Einstein term, and then the one from the term.

### ii.4 Three point function from the Einstein term

This was done in Arutyunov:1999nw (); Maldacena:2002vr ()666In Arutyunov:1999nw () the case was considered.. For completeness, we review the calculation and give some further details. To cubic order we can set , in (II.1). Then the only cubic contribution from (II.2) comes from the term involving the curvature of the three dimensional slices, . Let us see more explicitly why this is the case. On the three dimensional slices we define , with . All indices will be raised and lowered with . The action has the form

 S(3)R=12∫dtd3x[eHt^R(3)+e−Ht(^Eij^Eij−(^Eii)2)] (II.8)

Now we prove that the second term does not contribute any third order term to the action. To second order in we have . Then we find

 ^Eij^Eij−(^Eii)2=˙γij˙γij+o(γ4) (II.9)

which does not have any third order term. Thus, the third order action is proportional to the curvature of the three-metric. This is then integrated over time, with the appropriate prefactor in (II.8). Of course, if we were doing the computation of the flat space three point amplitude, we could also use a similar argument. The only difference would be the absence of the factor in the action (II.8). Thus, the algebra involving the contraction of the polarization tensors and the momenta is the same as the one we would do in flat space (in a gauge where the polarization tensors are zero in the time direction). Thus the de Sitter answer is proportional to the flat space result, multiplied by a function of only, which comes from the fact that the time dependent part of the wavefunctions is different.

We want to calculate the tree level three-point function that arises from this third-order action. In order to do that, we use the in-in formalism. The general prescription is that any correlator is given by the time evolution from the “in” vacuum up to the operator insertion and then time evolved backwards, to the “in” vacuum again, . We are only interested in the late-time limit of the expectation value. We find

 ⟨γs1(x1,t)γs2(x2,t)γs3(x3,t)⟩t→+∞=−i∫+∞−∞dt′ [Hint(t′), γs1(x1,+∞)γs2(x2,+∞)γs3(x3,+∞)] (II.10)

We write the gravitational waves in terms of oscillators as in (II.4). We calculate correlators for gravitons of specific helicities and 3-momenta. Note that, because there are no time derivatives in the interaction Lagrangian, then it follows that . Once we put in the wavefunctions, the time integral that we need to compute is of the form (in conformal time). Two aspects of the calculation are emphasized here. One is that we need to rotate the contour to damp the exponential factor at early times, which physically corresponds to finding the vacuum of the interacting theory Maldacena:2002vr (), as is done in the analogous flat space computation. Another aspect is that, around zero, the primitive is of the form , where is our late-time cutoff. This divergent contribution is real and it drops out from the imaginary part. We get

 ⟨γs1k1γs2k2γs3k3⟩R=(2π)3δ3(k1+k2+k3)(HMPl)44(2k1k2k3)3×[(k2ik2jϵ1ij)ϵ2klϵ3kl−2ϵ1ij(k3lϵ2li)(k2mϵ3mj)+cyclic]×(k1+k2+k3−k1k2+k1k3+k2k3k1+k2+k3−k1k2k3(k1+k2+k3)2) (II.11)

The second line is the one that is the same as in the flat space amplitude. The third line comes from the details of the time integral. Below we will see how this form for the expectation value is determined by the de Sitter isometries, or the conformal symmetry.

### ii.5 Three point amplitude from W3 in flat space

Let us calculate the following term in flat space, to which we will refer as : . We can write the following first order expressions for the components of the Weyl tensor

 W0i0j = 12¨γij Wij0k = 12(˙γki,j−˙γkj,i) W0ijk = 12(˙γik,j−˙γij,k) (II.12) Wijkl = 12(−δik¨γjl+δil¨γjk+δjk¨γil−δjl¨γik)

where we used that is an on shell gravitational wave. i.e. obeys the flat space equations of motion. We also used that , , . We can then write

 WαβγδWγδσρWσραβ=WijklWklmnWmnij+6W0ijkWjklmWlm0i++12W0i0jW0jklWkl0i+8W0i0jW0j0kW0k0i (II.13)

Evaluating these terms leads us to

 S(3)=∫Λ−2[2¨γij¨γjk¨γki+3¨γij˙γkl,i˙γkl,j+3¨γij˙γik,l˙γjl,k−6¨γij˙γik,l˙γkl,j] (II.14)

Plugging where , we get the following expression for the vertex due to the term:

 VW3,flat=6 k1k2k3[k1 ki2kj2ϵ1ijϵ2klϵ3kl+cyclic −(k1+k2+k3)(ϵ1ijkk3ϵ2kikl2ϵ3lj+cyclic)−2 k1k2k3ϵ1ijϵ2jkϵ3ki] (II.15)

By choosing a suitable basis for the polarization tensors, one can show that this agrees with the gauge invariant covariant expression .

### ii.6 Three point function from W3 in dS

The straightforward way of performing the computation would be to insert now the expressions for the wavefunctions in the term in de Sitter space, etc. There is a simple observation that allows us to perform the de Sitter computation. First we observe that the Weyl tensor is designed so that it transforms in a simple way under overall Weyl rescaling of the metric. Thus the Weyl tensor for the metric in conformal time is simply given by . Note also that, for this reason, the Weyl tensor vanishes in the pure de Sitter background. Thus, we only need to evaluate the Weyl tensor at linearized order777Note that the term does not contribute to the two point function.. For on shell wavefunctions we can show that

 Wμνδσ(γ)=−i|→k|ηWflatμνδσ(eikη+i→k.→x) (II.16)

where is the expression for the flat space Weyl tensor that we computed in the previous section, computed at linearized order for a plane wave around flat space. Thus, when we insert these expressions in the action we have

 S=∫W3=∫0−∞dηd3x(k1k2k3η3)(H2η2)(Wflat)3 (II.17)

The whole algebra involving polarization tensors and momenta is exactly the same as in flat space. The only difference is the time integral, which now involves a factor of the form , where we have defined , and we rotated the contour appropriately. Putting all this together, we get the following result for the three-point function due to the term

 ⟨γs1k1γs2k2γs3k3⟩W3=(2π)3δ3(k1+k2+k3)×(HMPl)6(HΛ)2a(−30)(k1+k2+k3)6(k1k2k3)2VW3,flat (II.18)

where was introduced in (II.15). There are also factors of that were included to get this result. The parity violating piece will be discussed after we introduce spinor variables, because they will make the calculation much simpler.

### ii.7 Estimating the size of the corrections

Let us write the effective action in the schematic form

 S=M2Pl2[∫[√gR−6H2√g]+L4∫W3]+⋯ (II.19)

where the dots denote other terms that do not contribute to the three point function. Here is a constant of dimensions of length. We have pulled out an overall power of for convenience. The gravitational wave expectation values coming from this Lagrangian have the following orders of magnitude

 ⟨γγ⟩∼H2M2Pl ,       ⟨γγγ⟩R=H4M4Pl        ⟨γγγ⟩W3=H4M4Pl(LH)4 (II.20)

Thus the ratio between the two types of non-gaussian corrections is

 ⟨γγγ⟩W3⟨γγγ⟩R∼L4H4 (II.21)

We know that is small. This parameter controls the size of the fluctuations. In the context, we know that when the right hand side in (II.21) becomes of order one we have causality problems Hofman:2008ar (); Brigante:2007nu (); Brigante:2008gz (); Hofman:2009ug (). We expect that the same is true in , but we have not computed the precise value of the numerical coefficient where such causality violation would occur. So we expect that

 HL≲1 (II.22)

In a string theory context we expect to be of the order of the string scale, or the Kaluza Klein scale. Thus the four dimensional gravity description is appropriate when . In fact, in string theory we expect important corrections when . In that case, the string length is comparable to the Hubble scale and we expect to have important stringy corrections to the gravity expansion. Note that in the string theory context we can still have being quite small. So we see that there are scenarios where the higher derivative corrections are as important as the Einstein contribution, while we still have a small two point function, or small expansion parameter . In general, in such a situation we would not have any good argument for neglecting higher curvature corrections, beyond the term. However, in the particular case of the three point function, we can just consider these two terms and that is enough, since these two terms (the Einstein term and the term) are enough to parametrize all the possible three point functions consistent with de Sitter invariance. If we define an , then we find that the Einstein gravity contribution of is of order one. This is in contrast to the for scalar fluctuations which, for the simplest models, is suppressed by an extra slow roll factor888 Note that we have divided by the gravity two point function to define . If we had divided by scalar correlators, we would have obtained a factor of . .

In an inflationary situation we know that the fact that the fluctuations are small is an indication that the theory was weakly coupled when the fluctuations were generated. However, it could also be that the stringy corrections, or higher derivative corrections were sizable. In that case, we see that the gravitational wave three point function (or bispectrum) gives a direct measure of the size of higher derivative corrections. Other ways of trying to see these corrections, discussed in Kaloper:2002uj (), involves a full reconstruction of the potential, etc. In an inflationary context terms involving the scalar field and its time variation could give rise to new shapes for the three point function since conformal symmetry would then be broken. However, one expects such terms to be suppressed by slow roll factors relative to the ones we have considered here. However a model specific analysis is necessary to see whether terms that contain slow roll factors, but less powers of dominate over the ones we discussed. For example, a term of the form is generically present in the effective actionWeinberg:2008hq (). Such a term could give a correction of the order . Here is a small quantity of the order of a slow roll parameter, involving the time derivatives of . Whether this dominates or not relative to (II.21) depends on the details of the inflationary scenario. In most cases, one indeed expects it to dominate. It would be very interesting if (II.21) dominates because it is a direct signature of higher derivative corrections in the gravitational sector during inflation.

Notice that the upper bound (II.22) is actually smaller than the naive expectation from the point of view of the validity of the effective theory. From that point of view we would simply demand that the correction due to at the de Sitter scale should be smaller than one. This requires the weaker bound . This condition is certainly too lax in the context, where one can argue for the more restrictive condition (II.22).

In summary, we can make the higher derivative contribution to the gravity three point function of the same order as the Einstein Gravity contribution. Any of these two terms are, of course, fairly small to begin with.

## Iii Spinor helicity variables for de Sitter computations

In this section we introduce a technical tool that simplifies the description of gravitons in de Sitter. The same technique works for anti-de Sitter and it can also be applied for conformal field theories, as we will explain later.

The spinor helicity formalism is a convenient way to describe scattering amplitudes of massless particles with spin in four dimensions. We review the basic ideas here. For a more detailed description, see Witten:2003nn (); Benincasa:2007xk (); ArkaniHamed:2008gz (); Cheung:2009dc (). In four dimensions the Lorentz group is . A vector such as can be viewed as having two indices, . The new indices run over two values. A 4-momentum that obeys the mass shell condition, can be represented as a product of two (bosonic) spinors . Note that if we rescale and we get the same four vector. We shall call this the “helicity” transformation. Similarly, the polarization vector of a spin one particle with negative helicity can be represented as

 ξ−a˙b=λa¯μ˙b⟨¯λ,¯μ⟩ (III.1)

where we used the invariant contraction of indices , where is the SL(2) invariant epsilon tensor. We have a similar tensor to contract the dotted indices. We cannot contract an undotted index with a dotted index. Note that this polarization vector (III.1) is not invariant under the helicity transformation. In fact, we can assign it a definite helicity weight, which we call minus one. This polarization tensor (III.1) is independent of the choice of . More precisely, different choices of correspond to gauge transformations on the external particles. For negative helicity we exchange in (III.1). For the graviton we can write the polarization tensor as a “square” of that of the vector

 ξ+ab˙a˙b=μaμb¯λ˙a¯λ˙b⟨μ,λ⟩2, ξ−ab˙a˙b=λaλb¯μ˙a¯μ˙b⟨¯λ,¯μ⟩2 (III.2)

The product of two four vectors can be written as .

Now let us turn to our problem. We are interested in computing properties of gravitational waves at late time. We still have the three momentum . This is not null. However, we can just define a null four momentum . This is just a definition. We can now introduce and as we have done above for the flat space case. In other words, given a three momentum we define , via

 (|→k|,→k)a˙b=(|k|σ0a˙b+→k.→σa˙b)=λa¯λ˙b (III.3)

In the de Sitter problem we do not have full symmetry. We only have one symmetry which corresponds to the rotation group in three dimensions. This group is diagonally embedded into the group we discussed above. In other words, as we perform a spatial rotation we change both the and indices in the same way. This means that we now have one more invariant tensor, which allows us to contract the dotted with the undotted indices. For example, out of and we can construct by contracting with . This is proportional to . Thus, this contraction is equivalent to picking out the zero component of the null vector. When we construct the polarization tensors of gravitational waves, or of vectors, it is convenient to choose them so that their zero component vanishes. But, we have already seen that extracting the zero component involves contracting dotted and undotted indices. We can now then choose a special in (III.1) which makes sure that the zero component vanishes. Namely, we choose . This would not be allowed under the four dimensional rules, but it is perfectly fine in our context. In other words, we choose polarization vectors of the form

 ξ+a˙b=¯λa¯λ˙b⟨¯λ,λ⟩ ,          ξ−a˙b=λaλ˙b⟨λ,¯λ⟩ (III.4)

Notice that the denominator is just what we were calling . Note also that the zero component of is zero, since this involves contracting the and indices. This gives a vanishing result due to the antisymmetry of the inner product. In our case we have a delta function for momentum conservation due to translation invariance, but we do not have one for energy conservation. The delta function for momentum conservation can be written by contracting with in order to get the spatial momentum. Alternatively we can say that . This is just saying that the fourvector has only a time component.

For the graviton, we likewise take and in (III.2). With these choices we make sure that the polarization vector has zero time components and that it is transverse to the momentum.

Everything we said here also applies for correlation function of the stress tensor in three dimensional field theories. If we have the stress tensor operator in Fourier space, we can then contract it with a polarization vector transverse to constructed from and . In other words, we construct operators of the form with as in (III.4). This formalism applies for any case where we have a four dimensional bulk and a three dimensional boundary, de Sitter, Anti-de Sitter, Hyperbolic space, Euclidean boundary, Lorentzian boundary, etc. The only difference between various cases are the reality conditions. For example, in the de Sitter case that we are discussing now, the reality condition is .

In summary, we can use the spinor helicity formalism tyo describe gravitational waves in de Sitter, or any inflationary background. It is a convenient way to take into account the rotational symmetry of the problem. One can rewrite the expressions we had above in terms of these variables.

### iii.1 Gravitational wave correlators in the spinor helicity variables

Let us first note the form of the two point function. The only non-vanishing two point functions are the and two point functions. This is dictated simply by angular momentum conservation along the direction of the momentum. Since the momenta of the two insertions are opposite to each other, their spins are also opposite and sum to zero as they should. The two point functions are then

 ⟨γ+γ+⟩=δ3(k+k′)⟨λ,λ′⟩2⟨λ,¯λ⟩5=δ3(k+k′)1⟨λ,¯λ⟩3 (III.5)

where in the last formula we have used a particular expression for in terms of . More precisely, if the momentum of one wave if , with its associated and , then for we can choose and . Here we have used that the matrices are symmetric. In the first expression we can clearly see the helicity weights of the expression. For the one we get a similar expression.

We can now consider the three point functions. The simplest to describe are the ones coming from the interaction. In fact, these contribute only to the and correlators, but not to the correlators. This is a feature which is also present in the flat space case. These non vanishing correlators can be rewritten as

 ⟨γ+k1γ+k2γ+k3⟩W3 =M(−28×32×5)(k1+k2+k3)6(k1k2k3)2[⟨¯1,¯2⟩⟨¯2,¯3⟩⟨¯3,¯1⟩]2 ⟨γ−k1γ−k2γ−k3⟩W3 =M(−28×32×5)(k1+k2+k3)6(k1k2k3)2[⟨1,2⟩⟨2,3⟩⟨3,1⟩]2 (III.6) M=(2π)3δ3(k1+k2+k3)(HMPl)6(HΛ)2

where the in the denominators can also be written in terms of brackets such as , if so desired. Note that, when rewritten in terms of the and , the above expressions are just rational functions of the spinor helicity variables (up to the overall momentum conservation delta function). One can check that indeed the and vertices vanish for the term, which is straightforward by using the expressions in appendix B. Note that this is not trivial because we do not have four-momentum conservation, only the three-momenta are conserved. The parity violating interaction does not contribute to the de Sitter expectation values Soda:2011am (); Shiraishi:2011st ().

The Einstein term contributes to all polarization components

 ⟨γ+k1γ+k2γ+k3⟩R =(2π)3δ3(∑iki)(HMPl)42(k1k2k3)5[(k1+k2+k3)3− −(k1k2+k1k3+k2k3)(k1+k2+k3)−k1k2k3][⟨¯1,¯2⟩⟨¯2,¯3⟩⟨¯3,¯1⟩]2 (III.7) ⟨γ+k1γ+k2γ−k3⟩R =(2π)3δ3(∑iki)(HMPl)418(k1k2k3)5[(k1+k2−k3)(k1−k2+k3)(k2+k3−k1)]2 (k1+k2+k3−k1k2+k1k3+k2k3k1+k2+k3−k1k2k3(k1+k2+k3)2)[⟨¯1,¯2⟩3⟨¯1,¯3⟩⟨¯3,¯2⟩]2 (III.8)

and similar expressions for and . Note that the Einstein gravity contribution to or is non-vanishing. This is in contradistinction to what happens in flat space, where it does not contribute to the or cases. This might seem surprising, given that we had said before that the polarization tensor contribution to the time integrand is the same as the flat space one. After doing the time integral, in flat space we get energy conservation, which we do not have here. This explains why we got a non-vanishing answer. In fact, the flat space amplitude is recovered from the above expressions by focusing on the coefficients of the double poles in . The fact that (III.1) does not have a double pole ensures that the flat space answer is zero for those polarizations999 In comparing to the flat space result, there are also factors of that come from the normalization of the wavefunction. . Similarly, the flat space answers for are obtained by looking at the coefficient of the order pole in in (III.1).

The expressions (III.1) can also be written in a form that shows explicitly the effect of changing the helicity of one particle:

 ⟨γ+k1γ+k2γ+k3⟩R=N(k1+k2+k3)2(⟨¯1,¯2⟩⟨¯2,¯3⟩⟨¯3,¯1⟩)2 (III.9) ⟨γ+k1γ+k2γ−k3⟩R=N(k1+k2−k3)2(⟨¯1,¯2⟩⟨¯2,3⟩⟨3,¯1⟩)2 (III.10) N =(2π)3δ3(∑iki)(HMPl)42(k1k2k3)5× ×(k1+k2+k3−k1k2+k1k3+k2k3k1+k2+k3−k1k2k3(k1+k2+k3)2) (III.11)

In the next section we will show that the forms of these results follow from demanding conformal symmetry.

## Iv Gravitational wave correlation function and conformal symmetry

In this section we will show how the three point functions we discussed above are constrained by conformal symmetry.

### iv.1 Wavefunction of the universe point of view

In order to express the constraints of conformal symmetry it is convenient to take the following point of view on the computation of the gravity expectation values. Instead of computing expectation values for the gravitational waves, we can compute the probability to observe a certain gravitational wave, or almost equivalently, the wavefunction . The expectation values are given by simply taking and integrating over . This point of view is totally equivalent to the usual one, where one computes expectation values of . It is useful because it makes the connection to very transparent101010In fact, the perturbative de Sitter computation is simply an analytic continuation of the perturbative Anti-de Sitter computation Maldacena:2002vr ().. It also makes the action of the symmetries more similar to the action of the symmetries in a conformal field theory. This is explained in more detail in Maldacena:2002vr () (see also deBoer:1999xf ()).

One writes the wavefunction in the form:

 Ψ=exp(12∫d3xd3y⟨Ts(x)Ts′(y)⟩γs(x)γs′(y)++16∫d3xd3yd3z⟨Ts(x)Ts′(y)Ts′′(z)⟩γs(x)γs′(y)γs′′(z)+ ⋯) (IV.1)

The first term expresses the simple fact that the wavefunction is gaussian. From this point of view, the quantity is just setting the variance of the gaussian. Namely, this is just a convenient name that we give to this variance. Similarly for the cubic term, which is responsible for the first non-gaussian correction, etc. Here we have ignored local terms that are purely imaginary and which drop out when we take the absolute value of the wavefunction. From this expression for the wavefunction one can derive the following forms for the two and three point functions Maldacena:2002vr (), to leading order in the loop expansion,

 ⟨γs1k1γs2k2⟩ =−12⟨Ts1k1Ts2k2⟩ (IV.2) ⟨γs1k1γs2k2γs3k3⟩ =−⟨Ts1k1Ts2k2Ts3k3⟩+⟨Ts1−k1Ts2−k2Ts3−k3⟩∗Πi(2⟨TsikiTsi−ki⟩) (IV.3)

So we see that it is easy to go from the description in terms of a wavefunction to the description in terms of expectation values of the metric. The complex conjugate arises from doing and we used that . However, if the wavefunction contains terms that are pure phases, we can loose this information when we consider expectation values of the metric. Precisely this happens when we have the parity violating interaction . It contributes to a term that is a pure phase.

Here is the usual Wheeler de Witt wavefunction of the universe, evaluated in perturbation theory. It is expressed in a particular gauge, because we have imposed the , conditions. The usual reparametrization constraints and Hamiltonian constraints boil down to some identities on the functions appearing in (IV.1). These identities are precisely the Ward identities obeyed by the stress tensor in a three dimensional conformal field theory111111Though the Ward identities are the same, some of the positivity constraints of ordinary CFT’s are not obeyed. For example, the two point function is negative. Thus, if there is a dual CFT, it should have this unusual property. . In the case, this is of course familiar from the point of view. In the de Sitter case, it is also true since this wavefunction is a simple analytic continuation of the one. It is an analytic continuation where the radius is changed by times the radius. In any case, one can just derive directly these Ward identities from the constraints of General Relativity. These identities express the fact that the wavefunction is reparametrization invariant. For the case that we have scalar operators (and corresponding scalar fields in ) we get an identity of the form