Partially massless fields during inflation
HJE
massless elds during in ation
Daniel Baumann 0 1 3 6
Garrett Goon 0 1 3 4 6
Hayden Lee 0 1 2 3 5
Guilherme L. Pimentel 0 1 3 6
Clear Water Bay 0 1 3
Kowloon 0 1 3
Hong Kong 0 1 3
and W Symmetry, Higher Spin Gravity
0 17 Oxford Street, Cambridge, MA 02138 , U.S.A
1 Princetonplein 5 , Utrecht, 3584 CC , The Netherlands
2 Institute for Advanced Study, Hong Kong University of Science and Technology
3 Science Park 904 , Amsterdam, 1098 XH , The Netherlands
4 Institute for Theoretical Physics, Utrecht University
5 Department of Physics, Harvard University
6 Institute of Theoretical Physics, University of Amsterdam
The representation theory of de Sitter space allows for a category of partially massless particles which have no at space analog, but could have existed during in ation. We study the couplings of these exotic particles to in ationary perturbations and determine the resulting signatures in cosmological correlators. When in ationary perturbations interact through the exchange of these elds, their correlation functions inherit scalings that cannot be mimicked by extra massive elds. We discuss in detail the squeezed limit of the tensorscalarscalar bispectrum, and show that certain partially massless elds can violate the tensor consistency relation of single eld in ation. We also consider the collapsed limit of the scalar trispectrum, and nd that the exchange of partially massless elds enhances its magnitude, while giving no contribution to the scalar bispectrum. These characteristic signatures provide clean detection channels for partially massless elds during in ation.
Cosmology of Theories beyond the SM; Higher Spin Symmetry; Conformal

Partially
1 Introduction
2 Higherspin elds during in ation
De Sitter representations Free theory of partially massless elds 2.3 Couplings to in ationary uctuations
3 Imprints on cosmological correlators
2.1
2.2
3.1
3.2
3.3 Trispectrum: h
Consistency relations
{ 1 {
Introduction
Observations suggest that our Universe started as de Sitter space [1] and will end as de
Sitter space [2, 3]. Understanding the physics of de Sitter (dS) is therefore of particular
relevance [4{6]. There exist a number of interesting features of this spacetime which do not have
counterparts in at space. For example, while particles in Minkowski are either massive or
massless [7, 8], the representation theory of de Sitter space allows for an extra category of
partially massless (PM) particles [9{11]. At special discrete values of the masstoHubble
ratio, m=H, the theory gains an additional gauge symmetry and some of the lowest helicity
modes of the wouldbe massive particles become pure gauge modes. In this paper, we revisit
the theoretical and observational status of partially massless particles during in ation.
Partially massless particles have a number of intriguing features that motivate us to
study their e ects during in ation, despite their somewhat exotic nature and their uncertain
status as interacting quantum
eld theories.1 First of all, if PM particles existed during
in ation, they would lead to rather distinct imprints in cosmological correlation functions.
Moreover, the chances of detecting these signals may even be bigger than for massive
particles, since the rate at which PM particles would be produced during in ation is larger
than that for massive particles. Furthermore, while massive elds decay on superhorizon
scales, the amplitude of certain PM
elds can remain constant or even grow. They can
therefore survive until the end of in ation and their contributions to the soft limits of
in ationary correlators are unsuppressed.
Another interesting feature of PM particles
is that their masses are protected against radiative corrections by the gauge symmetry.
Finally, since the existence of PM particles is tied to the nonzero (and nearly constant)
Hubble parameter during in ation, their detection would provide further evidence for an
in ationary, de Sitterlike period of expansion in the early universe.
To describe the e ects of PM particles on cosmological correlators, we will construct
gaugeinvariant couplings between higherspin particles and the in ationary scalar and
tensor
uctuations. Using these couplings, we will show that partially massless
higherspin particles leave unique imprints in the soft limits of in ationary correlation functions.
These soft limits are a wellknown detection channel for extra elds during in ation since
a symmetry
xes their form in single eld in ation [19, 20]. Most of the focus, so far, has
been on the soft limit of the scalar bispectrum h
i, where
is the primordial curvature
perturbation, which would receive characteristic nonanalytic contributions from massive
particles [21{27]. A strict violation of the consistency relation  that is, a modi cation of
the leading term in the squeezed limit  would require extra massless scalars.
An even more robust consistency relation exists for the soft limit of h
i, where
is a tensor uctuation. In that case, the leading term in the tensor squeezed limit cannot
be altered by the addition of light scalars. Moreover, contributions from massive spins
1In order to write a local theory with the right number of degrees of freedom, one imposes a gauge
symmetry that must be obeyed by the higherspin Lagrangian. It turns out to be hard to
nd selfconsistent
interacting theories of these degrees of freedom. It might be that a single light degree of freedom in the
spectrum implies the existence of many other higherspin degrees of freedom. An extreme example of this is
the occurrence of in nitely many elds in the Vasiliev theory of massless higherspin particles in (A)dS [12,
13]. Similarly, interacting theories of an in nite tower of partially massless elds were studied in [14{18].
{ 2 {
elds are constrained by the HiguchiDeserWaldron (HDW) bound in dS space, m2
1)H2 [28{30]. Achieving a strict violation of the tensor consistency relation has so far
been restricted to models where a subset of the de Sitter isometries are fully broken. This
includes models with large corrections to the kinetic terms of the spinning elds2 [31, 32] and
models with broken spatial isometries due to positiondependent background elds [33, 34].
In this paper, we will show that the presence of PM
elds during in ation can lead to
a strict violation of the consistency relation for h
i with a characteristic angular
dependence. In addition, the exchange of PM particles creates an enhanced scalar trispectrum
i, without producing a scalar bispectrum h
i
. These features of the cosmological
correlators are rather unique and provide a clean detection channel for PM
elds during
in ation.
h
h
Outline.
The paper is organized as follows. In section 2, we review the higherspin
representations in de Sitter space, and derive the interaction vertices with the in ationary scalar
and tensor uctuations. In section 3, we determine the imprints of partially massless elds
in in ationary correlators, with particular emphasis on the tensorscalarscalar bispectrum
i and the scalar trispectrum h
i. In section 4, we present our conclusions. The
appendices contain technical details of the computations presented in the main text. In
appendix A, we expand on the free theory of higherspin
elds in de Sitter space. In
appendix B, we derive the interaction vertices between a spin4
eld and the in ationary
uctuations, and discuss generalizations to higher spin. In appendix C, we present the
calculation of h
i and h
i in the inin formalism. Finally, in appendix D, we derive
the scaling behavior of the collapsed trispectrum and the squeezed bispectrum using the
operator product expansion and the wavefunction of the universe.
Notation and conventions.
Throughout the paper, we use natural units, c = ~ = 1,
with reduced Planck mass Mp2l = 1=8 G. The metric signature is ( + + +), spacetime
indices are denoted by Greek letters, ; ;
= 0; 1; 2; 3, and spatial indices by Latin
letters, i; j;
= 1; 2; 3. Conformal time is , and a prime (overdot) on a eld refers to a
derivative with respect to conformal (physical) time. Threedimensional vectors are written
in boldface, k, and unit vectors are hatted, k^. Expectation values with a prime, hfk1 fk2 i0,
indicate that the overall momentumconserving delta function has been dropped. The spin
and depth of a eld are labelled by s and t, respectively. Partial and covariant derivatives
r 1
r s
.
2It is possible to evade the HDW bound by writing quadratic actions that are noncovariant, but respect
the preferred slicing of the spacetime during in ation. If the noncovariant terms are comparable in
magnitude to the covariant terms, then the signs of the kinetic terms for the di erent degrees of freedom of the
higherspin
eld can be tuned separately. In this case, one can protect lowerhelicity modes from becoming
ghostlike, while allowing for mass terms that evade the unitarity bound. In other words, by assuming a
large breaking of de Sitter symmetry, there is enough freedom to write ghostfree quadratic actions that
violate the HDW bound. We thank Andrei Khmelnitsky for discussions on this point [31].
{ 3 {
Higherspin elds during in ation
In this section, we introduce the free theory of higherspin elds during in ation and their
couplings to the in ationary perturbations. We begin, in section 2.1, by reviewing the
classi cation of particles in de Sitter space. We highlight the existence of a class of partially
elds. The free theory of these
elds will be presented in section 2.2. Finally,
in section 2.3, we discuss the allowed couplings between these elds and the in ationary
scalar and tensor uctuations.
2.1
De Sitter representations
The relativistic equations of motion for particles of arbitrary spin in at space were derived
by Fierz and Pauli in [35], based on the requirement of positive energy. This is equivalent to
the condition that oneparticle states transform under unitary irreducible representations
of the Poincare group [7, 8]. These representations are characterized by the eigenvalues of
the two Casimirs of the group:
HJEP04(218)
C1
C2
P P
W W
= m2 ;
=
s(s + 1) m2 ;
where P is the fourmomentum and W is the PauliLubanski pseudovector. The mass m of
a particle is a nonnegative real number, whereas its spin s is a nonnegative half integer. We
distinguish between massive and massless particles. In four spacetime dimensions, massive
particles carry 2s + 1 degrees of freedom (transverse and longitudinal polarizations). For
the massless case, the theory gains a gauge symmetry which eliminates the longitudinal
degrees of freedom, and only the two transverse polarizations remain.
Similarly, particles in de Sitter space are classi ed as unitary irreducible representations
of the isometry group SO(
1,4
). The Casimirs of the de Sitter group have eigenvalues [9, 36]
C1
C2
1
2
MABM AB = m2
2(s
1)(s + 1)H2 ;
WAW A =
s(s + 1) m2
s2 + s
H2 ;
1
2
(2.1)
(2.2)
(2.3)
(2.4)
where MAB are the generators of SO(
1,4
) with A; B 2 f0;
; 4g and WA is the
vedimensional PauliLubanski pseudovector, constructed out of two Lorentz generators.
There is no globally timelike Killing vector in de Sitter space. This is clear when
describing de Sitter space in embedding coordinates, where all isometry generators correspond to
rotations or Lorentz boosts. This implies that the positivity constraints are imposed on
certain combinations of mass and spin of the representations, rather than mere positivity
of the mass. The (nonscalar, bosonic) representations of the de Sitter group fall into three
distinct categories [10, 11]:
principal series
complementary series
discrete series
m2
H2
s
correspond to masses in the discrete series.
1 is called the \depth" of the eld. Masses that do not
belong in one of the above categories correspond to nonunitary representations and are
not allowed in the spectrum (see gure 1). For spinning
elds, the complementary series
consists of a narrow range of mass values. For the representations in the discrete series, the
system has gauge symmetries which remove wouldbe ghost degrees of freedom from the
spectrum. The discrete series can be further divided into the following two subcategories:
Partially massless elds. Representations of the discrete series with t 6= s 1
correspond to partially massless elds [37, 38]. These elds share some features of massive
and massless elds in at space. On the one hand, they carry more than two degrees
of freedom, akin to massive elds. On the other hand, their correlation functions have
powerlaw behavior, which is similar to that of massless elds. Unlike massive elds,
PM
elds may survive until the end of in ation and therefore be directly observable.
To derive this e ect, however, one needs to extend the de nition of PM
elds beyond
the de Sitter limit to general FRW cosmologies.3 This remains an open problem (but
see appendix A and refs. [39{42]). In this paper, we instead consider the imprints
of PM
elds through their conversion to massless scalar and tensor perturbations
during in ation. This doesn't require us to follow their evolution after in ation.
Massless
t = s
elds.
Representations of the discrete series with maximum depth,
1, correspond to massless elds. Interacting theories of massless higherspin
particles are notoriously di cult to construct in at space, restricted by various
powerful nogo theorems [43{45] (see [46, 47] for reviews). In order for particles to remain
massless, their interactions need to be protected by a gauge symmetry. Typically, this
condition is su cient to uniquely determine the structure of the nonlinear theory.4
3We emphasize that the irreducible representations of the dS group carried by PM elds have no
counterpart in Minkowski space. This can also be understood from the perspective of eld theory; only when m
takes values that are particular multiples of H 6= 0, the action develops new gauge invariance that removes
certain lower helicity components. As a result, there is a discontinuity in the (m; H)plane at (0; 0) for
s
2, since the number of degrees of freedom depends on the way we take the limit H ! 0.
4For massless spin1 particles, the couplings must satisfy a sum rule which corresponds to charge
conservation, as in electromagnetism. For a massless spin2 particle, the couplings to other matter elds must
be universal, thus implying the equivalence principle, as in general relativity.
{ 5 {
The constraints on the couplings of massless higherspin elds to charged matter
imply that these elds cannot induce any longrange forces. The argument relies heavily
on the atspace Smatrix, and one might wonder whether the same conclusion holds
in (A)dS. It does not: a nonzero cosmological constant allows for the existence of
interacting theories of massless higherspin elds (for reviews, see [48, 49]). Having said
that, all known examples require an in nite tower of massless higherspin elds, and
their interactions are highly constrained. It is not known if these examples exhaust
the list of possibilities for theories of massless higherspin elds in curved spacetimes.
The in ationary phenomenology of massive particles in the principal and complementary
series was studied in [22{24, 27, 32, 50{52], while the observational prospects were
considered in [53{57]. In this paper, we will study novel e ects arising from the presence of
particles in the discrete series. In particular, we will study the cubic couplings of partially
massless elds to the in ationary perturbations and the resulting signatures in cosmological
correlators. These give the leading contributions to the cosmological correlators that we
consider in this paper, while selfinteractions of PM
elds do not contribute at tree level.
The cubic couplings can be made gauge invariant at the desired order, and we will show
an explicit construction of this coupling for PM spin4 particles.
2.2
Free theory of partially massless elds
Let us brie y review the free theory of PM
elds. Further details are presented in
appendix A. The onshell equations of motion for a spins, deptht eld are a generalization
of the Fronsdal equations for massless particles in de Sitter space [58]:
h
s + 2
t(t + 1) H2i
1 s = 0 ;
r
2 s = 0 ;
2 s = 0 :
(2.5)
These equations have a gauge symmetry that reduces the number of degrees of freedom from
the naive estimate. For massless elds, this gauge invariance allows us to set all 0 2 s
components to zero. In the case of PM
elds, only a subset of the timelike components of
1 s can be set to zero. The remaining components are labeled by a \spatial spin" n.
Moreover, each component has a helicity label, denoted by . We can therefore write the
components of a PM
eld of spin s and with n spatial indices as
i1 in
=
X
n;s "i1 in ;
with s
n
j j and j j = t + 1;
; s. The conventions for the polarization tensors "i1 in ,
as well as explicit expressions for the mode functions
n;s, can be found in appendix A.
A few features of the power spectra of elds in the discrete series deserve to be
highlighted. First of all, the mode function
n;s takes the following schematic form:
1
1
n;s = p2k ( H )n 1 a0 +
at
+
( k )t e ik ;
with constant coe cients an. Notice that this function is elementary rather than
transcendental (as in the generic massive case), being of the form e ik
multiplied by a rational
{ 6 {
(2.6)
(2.7)
function in k . The fact that these mode functions are very similar to at space mode
functions begs for an explanation. In appendix A, we provide a heuristic explanation for this
feature of PM
elds [59{61]. The argument shows, in a sense, why PM
elds exist and also
suggests a way to extend the de nition of PM
elds away from the perfect de Sitter limit.
Second, the twopoint function of elds in the discrete series scales at late times as
1 s
i
!0
!
2 ;
(2.8)
where
= 1
t is the conformal dimension of a eld with spin s and depth t. We see that
the twopoint function freezes at late times for t = 1 and diverges for t > 1. It is unclear
whether this carries any physical signi cance, since this twopoint function is, of course,
gauge dependent. On the one hand, the gaugeinvariant curvature tensors built from these
higherspin elds have additional derivatives and their twopoint functions therefore vanish
at late times. On the other hand, when PM
elds are minimally coupled to matter, these
latetime divergences can become physical.5 We will return to this issue in section 3.3.2.
2.3
Couplings to in ationary uctuations
When the in ationary uctuations are coupled to PM
elds, one faces the additional
challenge that the allowed interactions must respect the PM gauge symmetry responsible for
protecting the masses of these
elds. We nd that it is possible to write consistent
couplings which generate an interaction of the form
, as well as quadratic mixing terms
on the in ationary background. The consistency of these couplings should be viewed as an
e ective eld theory (EFT) statement. The actions written below will likely require
additional degrees of freedom to remain selfconsistent at higher orders in interactions, i.e. in
order to maintain the gauge symmetry at each order in the
uctuations. Nonetheless, if
such selfconsistent theories exist, and are weakly coupled, then the main contribution to
in ationary correlations should come from the leading, minimal vertices presented below.
We write a minimal coupling of the form6
ge
Z
d x
4 p
g
1 s
J 1 s ( ) ;
(2.9)
where J 1 s ( ) is a spins current, which depends quadratically on the in aton
eld
, and ge is an e ective coupling strength.7 This is analogous to the minimal coupling
between the photon and charged matter in quantum electrodynamics (QED), A J . As
in QED, the gauge transformation of
1 s forces the elds contained in J 1 s to be
5The latetime growth of the power spectrum of certain higherspin
elds is worrisome, as it could be
the sign of an instability. In order to determine whether the backreaction on the dS geometry is large,
it would be most natural to compute the contribution of the higherspin
elds to the background energy
density. However, because the form of the higherspin action on a generic curved background is unknown,
the stress tensor is ambiguous. To address the issue, we will therefore consider other physical observables,
namely correlation functions of
and
. Since the couplings to
and
are gauge invariant, we expect
these expectation values to carry physical information about the backreaction of the higherspin
elds.
6We will only consider couplings to evenspin
elds in this paper. For odd spins, the conserved currents
need to involve at least two di erent elds, which can be achieved e.g. with a complex scalar.
7We will sometimes set H = 1, so that ge becomes a dimensionless quantity.
{ 7 {
charged under the PM gauge symmetry, i.e. they must transform nontrivially if the action
including (2.9) is to be gauge invariant o shell. If we were to extend the theory to
higher orders in the interactions, then the combined transformations would dictate the
form of additional couplings needed for consistency, as well as possible deformations of the
gauge transformations. This is analogous to starting with a at space coupling between a
massless spin2 eld and a scalar, h
, demanding the linear gauge symmetry for h
and, due to consistency, being led directly to the fully nonlinear, di eomorphisminvariant
theory of a minimallycoupled scalar in general relativity (see e.g. [62, 63]). Finding such
a nonlinear completion of the PM theory is challenging, and we will content ourselves with
studying the leading coupling presented above. For previous literature on the construction
1 s
. For concreteness, we will study the example of a spin4 eld, but generalizations
to higher spin are, in principle, straightforward. We will consider two special cases: a
massless spin4 eld and a partially massless spin4 eld with depth t = 1.
We rst determine the conserved currents that can couple to these spin4 elds. Under
the gauge symmetry, the in aton must transform in a nontrivial fashion.8 Leaving the
details to appendix B, we quote here the form of the current for a partially massless spin4 eld
J
( ) = r(
r ) +
;
where the ellipses represent terms that vanish onshell when contracted with
the purpose of computing quantum expectation values, we expand the in aton into the
timedependent background value and its perturbations,
(t; x) =
(t) +
spatially
at gauge, (2.10) and (2.9) imply a cubic coupling between the eld
. In terms of the gaugeinvariant curvature perturbation,
= (H= _)
, this is
L
/ ge
a8
where we have only kept the spatial components of the spin4 eld. While this is the unique
nonvanishing term for a massless eld, not all of the zero components vanish in the partially
massless case. Although these nonspatial components can be included in our analysis,
because they have the kinematical structure of lowerspin elds, we will opt for the full spatial
components of the spinning eld to show the characteristic spins e ect. Similar couplings
exist for general spins, deptht elds, see appendix B. The mixing between the graviton
and a partially massless eld can naturally be obtained by evaluating the coupling (2.9)
on the in aton background and perturbing the metric. The resulting coupling is
(2.10)
1 4 . For
(t; x). In
1 4 and
(2.11)
L
/ ge
_2 ij00 _ ij :
a2
(2.12)
The size of this mixing term is correlated with that of the cubic coupling.
8Since the in aton is charged under the gauge symmetry, it is possible for the in aton background (t) to
a ect the quadratic structure of the higherspin action via higherorder couplings. To analyse this requires
knowing the precise form of these couplings at quadratic order in the higherspin eld. In the present work,
we assume that any such modi cations are small. We thank Paolo Creminelli and Andrei Khmelnitsky for
discussions on this point.
{ 8 {
Notice that this coupling vanishes when evaluated on the background metric. However, as
shown in appendix B, it still leads to a nontrivial
vertex of the form
K
( ) = r
+
:
L
/ he
_ ij00 _ ij :
a2
Due to the number of zero components, these mixing terms are only nontrivial if the
higherspin eld has depth t
1. The presence of _ in the prefactor indicates that mixing
arises only when the conformal symmetry of the background is broken, consistent with the
fact that h
i vanishes identically when the conformal symmetry is exact.
In section 3, we will study the cosmological imprints of the interactions in (2.11), (2.12)
and (2.15). To estimate the allowed sizes of the couplings ge and he , we write the
interaction Lagrangian in terms of canonically normalized
elds
Lint
ge
a8

ge
1 + he
{z
~
he
}
p
r
ij00 _ icj :
a2
To get an enhanced coupling, it is desirable to write down an independent mixing operator. For this purpose, we consider the following alternative coupling
he
Z
d x
and he is a coupling constant. Again,
we can nd the form of this coupling by demanding gauge invariance. The main di erence
between (2.9) and (2.13) is that, while we expect the former to be present in the standard
minimal coupling scheme, the latter is an additional allowed coupling that we can introduce
in our e ective theory. For the case of spin 4, the current takes the form
ge . p :
(2.17)
and icj
where
Here we are setting H = 1, so that ge and he are dimensionless and that the elds c
Mpl ij have order one
uctuations. We have used _
1 and Mpl
and
are the scalar and tensor amplitudes, respectively, and introduced the
tensortoscalar ratio r
. This rewriting of the Lagrangian makes manifest that
the strength of the cubic interaction is determined by ge and the quadratic mixing by the
The regime where (2.16) remains perturbative depends on whether or not the
interaccombination h~e
pr.
tion (2.13) is included in the action:
If (2.13) is not included, then h~e = ge
1 and the quadratic and cubic couplings
are both determined by ge . In order for the quadratic mixing to remain weakly
coupled, we then have to impose
2, this implies ge < 1, ensuring that the cubic interaction is also
perturIf (2.13) is included in the action, then the quadratic mixing depends on a
combination of ge and he , which we have denoted by h~e in (2.16), while the cubic coupling
is determined by ge
alone. In principle, this allows a large coupling to the tensor
mode without a correspondingly large coupling to the scalars.9 The requirements of
weak mixing and perturbative interactions now place the following bounds on the
couplings:10
~
he . p ;
1
r
ge . 1 :
(2.18)
2, the couplings can only saturate (2.18) if the two contributions to h~e
cancel to a high degree. The net e ect of including (2.13) in the action is to boost the
possible size of both the tensorscalarscalar bispectrum and the scalar trispectrum;
HJEP04(218)
see section 3.2 and section 3.3.
interactions for massive spinning
elds (i.e. those belonging to the principal
series) were considered in the context of the EFT of in ation [66, 67]. In that case, it
was found that the quadratic mixing
is always tied to the mixing
, so that the weak
coupling constraint of the latter induces a factor of p
implying h~e . 1. In contrast, the fact that PM
r on the size of the graviton coupling,
elds lack a longitudinal mode means that
the mixing
vanishes onshell. As a result, we expect the
vertex to remain weakly
coupled under a less stringent condition as in (2.18), not constrained by the size of the
scalar coupling.
3
Imprints on cosmological correlators
We will now study the imprints of partially massless elds in cosmological correlators.
In section 3.1, we brie y review the scalar and tensor consistency relations in single eld
in ation. We then compute, in section 3.2 and section 3.3, the impact of an intermediate
higherspin eld on both the tensorscalarscalar threepoint function and the scalar
fourpoint function (see
gure 2). We show that these
elds can lead to a strict violation of
the tensor consistency relation and a characteristic scaling and angular dependence in the
fourpoint function, while giving no contribution to the scalar threepoint function.
3.1
Consistency relations
Symmetries play a crucial role in constraining in ationary correlation functions. In
singleeld in ation, acts as the Goldstone boson of spontaneously broken conformal symmetries,
nonlinearly realizing dilatations and special conformal transformations. The corresponding
Ward identities imply consistency relations between correlation functions of di erent orders.
9It would also be interesting to consider the case of strong mixing, h~e
> 1, as a way to boost the tensor
power spectrum without a corresponding e ect on the scalar power spectrum. In this paper, we restrict
ourselves to the more conservative case of weak mixing.
10These bounds should be viewed as orderofmagnitude estimates, since various numerical factors appear
in actual computations of correlation functions. The correct perturbativity conditions are expressed as
bounds on the sizes of correlation functions, see appendix C.
i (left) and h
For example, the Taylor expansion of the threepoint function around the squeezed
limit must take the form
k1!0
lim h k1 k2 k3 i0 = P (k1)P (k3) X an
1
n=0
k1
k3
n
;
with a0 = 1
ns, where ns = 0:968
0:006 is the scalar tilt [68]. The leading term
of the squeezed limit of the bispectrum is therefore
xed by the scale dependence of the
power spectrum [19]. The consistency relation also constrains a1, and the model
dependence in single eld in ation only enters at quadratic order [20, 69{72]. This means that
observing any nonanalytic scaling behavior of the threepoint function between n = 0
and n = 2 would be a clear signature of additional massive elds during in ation. Extra
massless scalars would allow for a strict violation of the single eld consistency relation,
i.e. a modi cation of the coe cient a0.
Similar consistency relations exist in the tensor sector. For example, the
tensorscalarscalar correlator can also be Taylor expanded around the squeezed limit
kl1i!m0h k1 k2 k3 i0 = P (k1)P (k3) E2 (k^1 k^3) X bn
1
n=0
k1
k3
n
;
where Es (k^1 k^3)
spins,
helicityk^i1
3
k^3is "
i1 is (k^1) denotes the contraction between momenta and a
polarization tensor. The two leading coe cients in this expansion are
fully determined by symmetry; for example, b0 = (4
ns)=2. This assumes that the only
longlived quadrupolar anisotropies are those sourced by the graviton. Again, there can be
nonanalytic contributions from extra massive particles as in the scalar case, but this time
only with spin greater than or equal to two [27]. However, achieving a strict violation of
the consistency relation for tensors turns out to be much more di cult. In particular, it
was pointed out in [73] that spin2 particles cannot a ect the leading term in the tensor
consistency relation when the de Sitter symmetries are (approximately) respected, since
unitarity forbids the existence of light spin2 particles with mass 0 < m2 < 2H2. On the
other hand, the particle spectrum allowed by dS representations is much richer and exists
beyond spin two, as was reviewed in section 2. It is then natural to ask whether higherspin
elds can a ect this conclusion.
Another interesting kinematical limit of cosmological correlators is the collapsed limit
of the fourpoint function. This limit probes light states that are being exchanged in the
(3.1)
(3.2)
HJEP04(218)
fourpoint function, and it essentially factorizes into a product of threepoint functions [25,
74{77]. This is analogous to the operator product expansion (OPE) limit of conformal
correlation functions. In schematic form, we can write
where kI = k1 + k2 is an internal momentum. As in the OPE, the expression (3.3) involves
a sum over all intermediate states, and the sum is dominated by the uctuations that decay
the slowest outside of the horizon. See appendix D for more on this OPE perspective.
The SuyamaYamaguchi (SY) relation [78] bounds the size of the collapsed trispectrum
in terms of the size of the squeezed bispectrum [74, 79]:
(3.3)
(3.4)
(3.5)
(3.6)
^NL
lim
lim
56 f^NL
2
;
f^NL
^NL
k1!0 12 P (k1)P (k3)
5 h k1 k2 k3 i0 ;
1
h k1 k2 k3 k4 i0
kI !0 4 P (k1)P (k3)P (kI )
;
where we have introduced the following nonlinearity parameters
and assumed that they have the same momentum scaling. The SY bound (3.4) is
saturated when a single source is responsible for generating the curvature perturbations.11 As
we will show below, higherspin
elds provide an interesting example which nontrivially
satis es this bound. In particular, we will show that PM
elds do not generate any scalar
bispectrum, while sourcing a nontrivial trispectrum.
3.2
We now study the e ects of higherspin particles on the h
i correlator12 and demonstrate
that they violate the consistency condition (3.2). As was discussed in section 2.3, there
exist two types of graviton couplings: one generated from the
coupling (2.11) and
another from the
coupling (2.15). The former has the property that its signature in the
tensor bispectrum is correlated with its e ect on the scalar sector, while the latter allows
the coe cients to be independently tuned, which can lead to an enhanced signal.
11Strictly speaking, the SY bound is only saturated if the single source goes through a signi cant
nonlinear classical evolution on superhorizon scales.
This is because, in single eld slowroll in ation, the
trispectrum from graviton exchange leads to a contribution ^NL = O("), which is in fact parametrically
larger than f^N2L = O("2) [80]. These slowrollsuppressed e ects are typically much smaller than the e ects
that we are interested in.
12Partially massless elds do not contribute to h
i at tree level, as a spinning eld without a longitudinal
degree of freedom is kinematically forbidden to oscillate into a single scalar eld. Hence, h
nonGaussian correlator in which partially massless higherspin
elds can leave an imprint.
i is the simplest
We will use the following measure of the tensorscalarscalar bispectrum
amplitude
hNL
6
17
X
and implies hNL = pr=16 for single eld slowroll in ation [19].
where the bispectrum is evaluated in the equilateral con guration, k1 = k2 = k3
k, with
vectors maximally aligned with the polarization tensor. Our normalization agrees with [81]
The size of the tensor bispectrum due to a higherspin exchange can be estimated as
hNL
h
i
where the top and bottom cases correspond to excluding and including the term (2.13) in
the action, respectively. The perturbativity requirements (2.17) and (2.18) imply,
respectively,
hNL .
8
The tensor bispectrum can be constrained using the hBT T i, hBT Ei, hBEEi correlators
of the CMB anisotropies [81, 82]. The forecasted constraints from the CMB Stage IV
0:1 for the localtype nonGaussianity. The coupling he
is currently unconstrained, whereas the nondetection of the trispectrum puts an upper
bound on ge (see section 3.3) and consequently the size of the tensor bispectrum.
Shape.
As we described earlier, only higherspin elds with depths t = 0 or 1 contribute
to h
i at tree level. For t = 1, the tensor bispectrum induced by the exchange of a PM
eld takes the following form in the soft limit
k1!0
lim h k1 k2 1k3 i0 = P (k1)P (k3) Y^s ( ; ') ;
(3.7)
(3.8)
(3.9)
(3.10)
where Y^s is a spherical harmonic, is an e ective coupling constant de ned below, and the
angles are de ned by cos
= k^1 k^3 and ei' = " k^3. The polarization tensor for the graviton
"ij is built out of two polarization vectors ", " that span the plane perpendicular to k^1,
which are
xed up to a phase (see appendix A). We note that the bispectrum (3.10) has
the same scaling as the leading term in the tensor consistency relation (3.2). Thus, elds
with t = 1, s > 2 generate strict violations of (3.2) by moving the value of b0 away from its
predicted form. The case t = 1, s = 2 doesn't constitute a violation, as it corresponds to
the massless graviton. We dub this shape \local tensor nonGaussianity", in analogy to the
nonGaussian shape that violates the scalar consistency relation. Due to the kinematics
of the mixing with the graviton, only the helicity
=
2 modes of the higherspin
eld
contribute in the threepoint function. Since these modes have the same amplitude, we
have absorbed all of the numerical factors into
than order unity in the weakly nonGaussian regime. (The precise overall normalization as
ge h~e
pr, which is required to be less
ˆ Y
2 +s 0
ˆ Y
−200
π/2
θ
π
correspond to spins 4, 6, and 8, respectively.
= cos 1(k^1 k^3). The solid, dashed, and dotted lines
a function of the spin of the eld can be found in appendix C.) For t = 0, the bispectrum
is suppressed by a factor of k1=k3 relative to the t = 1 case.
The angular dependence is given by the usual spherical harmonic of degree (spin) s
and order (helicity) , which can be factorized into longitudinal and transverse parts
Y^s ( ; ') = E ( ; ')P^s (cos ) ;
(3.11)
where P^s (x) / (1
x2)
=2Ps (x) is a version of the associated Legendre polynomial with a
suitable normalization. On top of the usual quadrupole moment due to the external tensor
mode, we see that the exchange of a higherspin eld induces an extra longitudinal angular
component. Aligning k1 with the zaxis, we can express the hard momentum in spherical
polar coordinates as k^3 = (sin cos '; sin sin '; cos ). The angular dependence can then
be written as a product of E
only of . For example, some explicit expressions are
= sin2 e i ' and the longitudinal part which is a function
X
= 2
Y^s ( ; ') = <> 36
(35 + 60 cos 2 + 33 cos 4 ) sin2 cos 2'
(210 + 385 cos 2 + 286 cos 4 + 143 cos 6 ) sin2 cos 2'
s = 8
s = 4
s = 6 :
(3.12)
Figure 3 shows the angular dependence as a function of the angle
for ' = 0. We can read
o the spin of the particle by measuring the period of the oscillations. For the purpose
of data analysis, having the full bispectrum shape available would also be helpful. The
expression of the tensor bispectrum for general momentum con gurations can be found in
appendix C.
Treating the soft tensor mode as a classical background, the squeezed tensor bispectrum
would also contribute an anisotropic correction to the power spectrum in the following
schematic way [83]:
h
h k k0 i0 q = P (k) 1 + hNL q Y^s ( ; ')
i
h
P (k) 1 + Qi1 is ki1
^
k^is i ;
(3.13)
generated by higherspin elds.13
for a given realization of the tensor mode, with the angles and ' now de ned in terms of k
and q. The external tensor mode is not directly observable, but its variance Q
is, after averaging over all long momenta. It would be interesting to measure this anisotropic
2
2
hQi1 is i
e ect due to PM
elds in largescale structure observations. See e.g. [73, 84, 85] for forecasts
on the detection limit of Q2 from future experiments and [86] for related work on anisotropy
We next consider the contribution of the interaction (2.11) to the correlator h
i. We will
rst focus on the special case of depth t = 1 elds which freeze on superhorizon scales, and
then study the imprints of t > 1 elds which continue to grow after they exit the horizon.
Fields of depth t = 0 are not addressed in detail, as they decay on superhorizon scales and
are hence phenomenologically less interesting. Details of the relevant inin calculations can
be found in appendix C.
3.3.1
Depth t = 1
elds
For a given spin s
2, there exist s
1 partially massless states. The depth t = 1 case is
rather special, as it is the only state which freezes on superhorizon scales. In the following,
we discuss the signature of such states on the h
i correlator via the second diagram in
We use the standard measure of the size of the trispectrum
where the righthand side is evaluated at the tetrahedral con guration with ki = k and k^i
1=3. The estimated size of nonGaussianity induced by the exchange of a PM eld is
Imposing the weak coupling constraints (2.17) and (2.18), we nd
gure 2.
k^j =
(3.14)
(3.15)
(3.16)
g
2
e
2
:
NL .
8
<r 1
;
:
2
;
where the top and bottom cases correspond to excluding and including the term (2.13)
in the action, respectively. Of course, the shape of the correlator crucially a ects the
13When higherspin elds acquire classical background values, they can also leave statistically anisotropic
imprints in higherorder correlators [87]. We thank G. Franciolini, A. Kehagias, and A. Riotto for sharing
their draft with us.
observability of the signal. As we will describe below, the trispectrum under consideration
has a similar scaling behavior in the soft limit as the \local" trispectrum, which arises
in multi eld in ationary models (see [88] for a review). For comparison, the current
observational bound on the size of the latter is NloLcal = ( 9:0
7:7) 104 [89]. This bound
roughly translates into ge < 10 2. Since there is no bispectrum counterpart for this signal,
our scenario is an example which satis es the SY relation in the most extreme manner.
Shape. Let us describe the collapsed limit of the trispectrum, leaving the details of the
full shape to appendix C. For partially massless elds with depth t = 1, we get
kI !0
g
2
e
lim h k1 k2 k32k4 i0 = P (k1)P (k3)P (kI ) X
s
j j=2
where the amplitude of each helicity mode is
Es =
64 2(2s
1)!!]2s!(s + 1)!(s
)!(
1
)!!(s
2)!(s + )!( + 1)!
2)!
:
We see that the overall scaling behavior for a given depth is independent of the particle's
spin. As advertised before, the scaling is the same as the local trispectrum, being
proportional to P (k1)P (k3)P (kI ). The amplitude of each helicity mode is uniquely determined
by its spin. The angular dependence is again factorized into the transverse part E and the
longitudinal part P^s . The transverse polarization tensors project the momenta k1 and k3
onto the plane perpendicular to kI . The trispectrum is therefore a function of the angles
cos
k1 k^I and cos 0
^
k3 k^I between the vectors, as well as the angles ' and '0 on the
^
projected plane with respect to the polarization tensor. The values of the projection angles
depend on the two chosen polarization directions on the plane, but the di erence
is independent of this choice (see gure 4). We can write the full angular dependence as
Y^s ( ; ')Y^s ( 0; '0) / e
i
Ps (cos )Ps (cos 0) ;
where the factor of (1
cos2 ) =2 = sin
inside the associated Legendre polynomial
contributes to the transverse part. Note that, as a consequence of the addition theorem
for spherical harmonics, we have [24]
Esj j Y^s ( ; ')Y^s ( 0; '0) ;
is the angle between k^1 and k^3. This is not
quite the angular dependence that we observe, since each helicity of the higherspin eld
has a di erent amplitude, and some of the helicities are missing for the PM
eld. For
fs; tg = f4; 1g, we have
4
X
j j=2
E4j j Y^4 ( ; ')Y^4 ( 0; '0) / 14
15
(5 + 7 cos 2 )(5 + 7 cos 2 0) sin2 sin2 0 cos 2
+ 75 cos cos 0 sin3 sin3 0 cos 3
+
sin4 sin4 0 cos 4 : (3.21)
105
4
the exchanged (partially) massless eld is contained in the ( ; 0; ) dependence of the trispectrum.
As follows from (3.19), the helicity component of the eld is responsible for the / cos
term in (3.21).
We see that each helicity mode contributes with a distinct angular
dependence, with all of the amplitudes being roughly of the same size. The factorization
of the angular dependence into the sum over polarizations of the intermediate particle
is analogous to what happens for atspace scattering amplitudes when the intermediate
particle goes onshell, which is a consequence of unitarity. In the case of cosmological
correlators, there is the precise relationship between the amplitudes of di erent helicities,
which is a consequence of conformal symmetry.
3.3.2
Depth t > 1 elds
We now study the impact of depth t > 1 partially massless elds on the scalar
trispectrum, which includes, in particular, the massless case t = s
1. At tree level, this class
of elds generates characteristic divergences in h
i, when evaluated in the collapsed
con guration.
In section 2.2, we encountered the peculiar result that the twopoint functions of depth
t > 1 partially massless elds diverge at late times; cf. eq. (2.8). However, the physical
content of this fact is not clear, since the twopoint function is gauge dependent. The
in uence of the higherspin
eld can instead be tested in a gaugeinvariant manner by
calculating its e ect on h
i via the exchange diagram in gure 2. In the collapsed limit,
this trispectrum is directly sensitive to the higherspin twopoint function14 and its scaling
behavior due to the exchange of a spins, deptht PM particle is given by
kI !0
g
2
e
lim h k1 k2 k32k4 i0 = P (k1)P (k3)P (kI )
k1k3
k2
I
t 1 s
X
j j=2
Esj;tj Y^s ( ; ')Y^s ( 0; '0) ;
(3.22)
14This fact is made manifest in the wavefunction of the universe formalism; see appendix D.
with Es;t de ned in (C.29). Notice that this diverges at a rate that is faster than P (kI )
for t > 1.
There are several e ects which could make the strongly divergent behavior of (3.22)
less extreme. First, loop diagrams with more intermediate higherspin elds would also lead
to a singular behavior that goes as ge2n(k1k3=kI2)n(t 1), where n is the number of loops.15
Although these contributions are higher order in ge , they would be more singular than the
treelevel diagram if t > 1. In that case, it does not make sense to only consider an
individual diagram, but we would instead need to sum all of them. It is conceivable that the result
after the resummation will behave more tamely in the collapsed limit. Second, consistency
of the theory may require the introduction of many new degrees of freedom and interactions
which will also contribute to the scalar trispectrum. If so, the multitude of particles may
soften the collapsed limit of h
i. This is the behavior claimed in [90] which calculates,
via a dual description, the fourpoint function of a conformallycoupled scalar due to the
exchange of the tower of massless higherspin elds in the minimal Vasiliev theory. In that
case, the complete correlator behaves much more softly in the collapsed limit than the
individual exchange diagrams do, which have similarly divergent behavior to (3.22).
4
Conclusions
In this paper, we have studied the eld theory of partially massless elds during in ation
and discussed their imprints on cosmological correlators. Our main conclusions are:
Partially massless elds can have a consistent linearized coupling to a scalar eld
with arbitrary mass. We have constructed the corresponding conserved currents, and
derived the relevant couplings between these higherspin elds and the in ationary
scalar and tensor perturbations.
Partially massless elds lead to a vanishing scalar bispectrum, but a nonzero
trispectrum. The trispectrum has an unsuppressed behavior in the collapsed limit and a
distinct angular dependence.
Partially massless elds can lead to a strict violation of the tensor consistency relation
while respecting de Sitter symmetry, providing a loophole to the theorem of [73]. This
local tensor nonGaussianity is analogous to the sensitivity of the scalar bispectrum
to extra light scalar species.
Partially massless elds can mix quadratically with tensor modes, but not scalar
modes.
The reason is purely kinematical  there is no longitudinal mode that
mixes with the single scalar leg. This means that we can potentially enhance the
tensor power spectrum, while not altering the scalar power spectrum. To realize this
intriguing possibility would require understanding the regime of strong mixing of PM
elds with the in ationary tensor modes.
15This scaling behavior follows from a simple estimate of the contribution from ladder diagrams involving
the cubic vertex at n loops. For the contribution due to vertices with more legs, a more detailed analysis
is necessary.
(top) [89, 91{93] and tensorscalarscalar bispectrum (bottom) [81, 82, 94, 95] for local
nonGaussianity. The red and green regions correspond to the sensitivity levels of Planck [89] and
forthcoming experiments, respectively. The \gravitational oor" refers to the guaranteed level of
nonGaussianity sourced by gravitational nonlinearities during in ation, while \nonperturbative"
denotes the strongly nonGaussian regime.
Partially massless elds do not decay outside the horizon, making them, in principle,
directly observable after in ation. Predicting the nal spectrum of PM
elds is
challenging, because it requires coupling these elds to the matter uctuations in the late
universe, something we did not address in this paper. It also requires understanding
the meaning of PM
elds away from the de Sitter limit.
Figure 5 is a schematic illustration of current and future constraints on the scalar
trispectrum and the tensorscalarscalar bispectrum. We see that future surveys will probe deeper
into the allowed parameter space of primordial nonGaussianities. We also see that there
are still many orders of magnitude of parameter space left to be explored before we would
hit the gravitational oor. Perhaps, within this unexplored territory, there will be new
surprises and a rich cosmological fossil record waiting to be discovered.
Acknowledgments
We thank Nima ArkaniHamed, Matteo Biagetti, Cora Dvorkin, Kurt Hinterbichler, Austin
Joyce, Andrei Khmelnitsky, Juan Maldacena, SooJong Rey, David Vegh, and Matthew
Walters for helpful discussions, and Dionysios Anninos, Paolo Creminelli and Antonio
Riotto for comments on a draft. D.B., G.G. and G.P. acknowledge support from a Starting
Grant of the European Research Council (ERC STG Grant 279617). G.G. is also supported
by the DeltaITP consortium, a program of the Netherlands organization for scienti c
research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science
(OCW). H.L. thanks the Institute for Advanced Study for hospitality while this work was
completed. G.P. acknowledges funding from the European Union's Horizon 2020 research
and innovation programme under the MarieSklodowska Curie grant agreement number
751778. We acknowledge the use of the Mathematica package xAct [96].
By increasing the depth, we obtain lighter elds, whose number of degrees of freedom is
smaller than those of their heavier cousins. In more detail, the depth interpolates between
the number of degrees of freedom of a massive eld of spin s in at space; i.e., 2s + 1, and
2, the number of degrees of freedom of a massless eld.20
Finally, we should mention an interesting proposal to select the massless graviton
action, using conformal gravity as a starting point [60] (see also [99]). By imposing Dirichlet
boundary conditions on the latetime amplitudes for the uctuations,21 Maldacena showed
that the wavefunction of the universe for Einstein gravity can be computed using conformal
gravity. It would be interesting to study whether a similar procedure would select a partially
massless eld, using (A.29) as a starting point. The procedure outlined in [60] does not
work for the case of the spin2
eld with m2 = 2H2 [100], as ghost degrees of freedom
appear in the interaction vertices beyond cubic order. Nonetheless, it was shown in [100]
that the spin2 PM
eld can propagate in an Einstein spacetime (see also [39]). For our
purposes, having a quadratic action for PM
elds in a cosmological FRW background
would su ce to determine whether the power spectra of higherspin elds leave interesting
imprints in the late universe.
B
Spin4 couplings
In this appendix, we derive the form of the couplings of a scalar eld and the graviton to a
(partially) massless spin4 eld that were used in the main text. We also comment on the
generalization of our couplings to partially massless elds of arbitrary spin.
B.1
Conserved current
We will construct the linearized coupling between (partially) massless spin4 elds and
a scalar
eld via the standard Noether procedure.22
For concreteness, we will present
expressions for dS4, but these results can easily be generalized to AdS or any number of
dimensions. We consider a coupling of the form
ge
Z
d x
4 p
g
1 s
J 1 s ( ) :
First, we write the most general totally symmetric ranks tensor up to s derivatives as
s=2
X
k=0
J 1 s =
kr( 1 k r k+1 s) +
;
s = 2; 4;
;
BunchDavies vacuum.
in AdS.
20In reality, the counting should begin with 2s + 2 and descend in pairs from the highest weight
representation; the extra degree of freedom is ghostlike and decouples in the free theory [98]. In practice, the
highest depth partially massless eld has 2s degrees of freedom, and the reduction of its number of degrees
of freedom by unity is attributed to Weyl symmetry.
21These are the natural boundary conditions one must impose in order to compute the wavefunction of
the universe. From the wavefunction of the universe one can extract in ationary correlation functions by
using j j2 as a probability distribution.
When we compute in ationary expectation values directly, we
do not impose late time boundary conditions on the eld variables, but only that the initial state is the
22See [101{103] for similar Noether constructions of higherspin currents for complex and conformal scalars
(B.1)
(B.2)
where the ellipses denote terms that have contractions among the derivatives with
appropriate factors of the metric. To begin with, let us construct a current for a massless spin4
eld. This eld changes under a gauge transformation as
= r(
) :
r J
four derivatives:
Invariance of the coupling under this transformation implies that the current is conserved,
= 0. We begin by writing down the most general form of a spin4 current up to
J =
1 r
4
+ 2r r
3
+ 3r
2
r
2
+ 4gr2
r
2
+ 5gr r
3
+ 6H2gr r
r
2
+ 9H2g2r r
+ 10H4g2 2
sym
;
where we have suppressed the indices and g stands for the metric. Under the gauge
transformation (B.3), the coupling
these terms, we also introduce a transformation rule for the scalar eld
J will generate terms such as r r
4 . To cancel
= ( 1r
+ 2r
r
+ 3r
r
+ 4r
)
:
We demand that the coupling is invariant under gauge transformations for both the
spinning and scalar elds. We nd that a massless spin4 eld can couple to scalar elds with
arbitrary mass, at least at the linearized order.23 After many integrations by parts and
dropping boundary terms, the o shell gauge invariance xes the coe cients to be
(B.3)
(B.4)
(B.5)
3 =
7 =
3 1 +
26 1 ;
3
where we have given the result for a massless scalar eld. These conditions leave four
parameters unconstrained, while we expect them to be fully
xed if the full theory were
known. Nevertheless, this ambiguity does not have any observable consequences and goes
away when we evaluate the coupling onshell. In the transverse and traceless gauge, all
terms in the current that are proportional to the metric drop out in the coupling.
Essentially, the form of the coupling is uniquely xed onshell (up to terms that are related by
integrations by parts). The current can be put in the form
J
= r(
r ) :
(B.7)
This leaves only one free parameter for the onshell coupling, namely the coupling constant
ge . Moreover, all the zero components of the spin4 eld vanish onshell. The covariant
derivatives will lead to terms that involve ij / 0 ij , but these do not contribute since
23Further imposing the tracelessness condition of the current forces the scalar to be conformally coupled
with mass m2 = 2H2.
the polarization tensor of
is traceless, "iijk = 0. This means that we can simply replace
the covariant derivatives in the current with partial derivatives. The onshell current thus
takes a particularly simple form
One can perform a similar procedure of constructing an o shell gaugeinvariant linearized
coupling between massless spinning
elds and a scalar eld for general spin. However, it
is easy to see that the onshell coupling will, again, take a unique form. Up to integration
by parts, this is
a8
J =
a2s
We will compute the correlation functions that arise from this coupling in appendix C.
Next, let us consider the partially massless case. The onshell gauge transformation of
a spin4, depth1 partially massless eld is [cf. (A.10)]
= r( r r
) + H2g
( r
) ;
with the gauge parameter subject to the conditions (A.11).24 The coupling (B.1) is
invariant under this transformation if the current satis es the condition
r
J
+ H2r J
= 0 :
Again, we will start with the most general ansatz for the current (B.4). Since the gauge
transformation (B.10) contains more derivatives than that for massless elds, this time we
require a scalar transformation involving
ve derivatives. It turns out that we can recast
the most general totally symmetric scalar transformation in the form
= (~1H4
r + ~2H2
r
+ ~3r
r + ~4 r
r + ~5
2
r
+ ~6r
via the use of the onshell conditions for the gauge parameter
. The coupling will be
invariant under the transformations (A.10) and (B.12) if
(B.8)
(B.9)
(B.10)
(B.11)
r
) ;
(B.12)
3 =
~1 =
1 + 2
~6 ;
5 = 4
1
~
2~4
for a massless scalar eld. Again, the form of the coupling simpli es greatly onshell, taking
the form (B.7). Evaluating the coupling in components, we can determine the coe cients of
the couplings for both the spatial and nonspatial components of the PM
eld. Generalizing
to higher spins and focusing on spatial components, the onshell coupling, again, uniquely
takes the form (B.9) for elds of any depth.
24To ensure full o shell gauge invariance, one needs to introduce a number of auxiliary lowerspin elds.
Here we will restrict ourselves to the physical degrees of freedom of the onshell PM
eld.
We are also interested in getting a
interaction vertex, which would contribute to the
correlator h
i. This coupling can naturally be generated by evaluating the scalar elds
in the coupling (B.1) on the background and perturbing the metric.25 In principle, this
procedure could produce terms with and without derivatives acting on ij . Naively, the
latter would violate the tensor consistency relation even when
is massive, in contrast to
the result of [73]. However, we will show that such terms are indeed absent.
We take _ to be constant, which implies that r0
r
= ijklrij rkl = ijkl 0 0 _2
ij kl
= 0. Starting from (B.7), we have
where the last expression follows from perturbing the metric in spatially at gauge. We
see that this also produces a tadpole term
iijj . Note that the spatial trace itself gives
gij gkl ijkl = a 4( iijj
2 ijkk ij ) :
Demanding the absence of the tadpole exactly cancels the ij term. Consequently, the
vertex only involves _ ij . Using the traceless condition, we will denote this coupling for the
general spin case by
ge
_2 ij0 0 _ ij :
a2s 2
he
Z
d x
4 p
g
vertex is proportional to _2 and is correlated with the cubic vertex
An alternative coupling between and is of the form
where K is linearly dependent on .26 The most general form of K up to four derivatives is
K
= ~1r(
) + ~2g( r ) + ~3g( r )
+ ~4g( g )
+ ~5g( g )
The last three terms vanish identically upon using the equation of motion of . Imposing
the onshell conservation condition r
K
+ H2r K
= 0, we obtain
~1 =
16H2 ~2 :
imposed, g
r
graviton. For example, we have
While the remaining terms also vanish when the background onshell gauge conditions are
= 0, g
= 0, they can still induce nontrivial couplings to the
r
= ijklrijkl = ijkl 0 p 0 _
ij 0k pl
= a 4H2 H iijj
2H ijkk ij +
3
2 ijkk _ ij
_
:
25The same procedure would also yield
and
interactions, which can contribute to the correlators
h 26Ni oatnedthhat this coupling induces terms of order O( ;
i, respectively.
. The o shell consistency of keeping both
J and
K types of couplings then requires introducing an
extra scalar transformation rule of the form
= O( ; ). This can always be done at linear order in .
) under the gauge transformations of
and
(B.15)
(B.16)
(B.17)
Again, only the _ ij term remains after cancelling the tadpole. The other term gives
r
= g
ij r
ij
= g gimgjn
= a 4H ~ii
ij 0mn _
ij ijkl
ij ~ij +
1
2 ~ij _ ij
_
;
(B.21)
= g
denotes the trace. On the other hand, the tadpole gives
g gij
ij = g gij
ij + g gij
ij + g
gij
ij
= ~ii
ij ijkk
ij ~ij :
(B.22)
(B.23)
(C.1)
(C.2)
(C.3)
The same logic prevents couplings involving ij without a time derivative.27 We see that
both terms will then lead to the generic form of the coupling
_ ij0 0 _ ij ;
a2
which, in contrast to (B.16), is proportional to a single factor of _ and can be independent
from the cubic vertex
In this appendix, we compute cosmological correlators involving an exchange of a (partially)
eld. We provide details of the computation of the scalar trispectrum and the
tensor bispectrum in section C.1 and section C.2, respectively. In appendix D, we will
analyze the soft limits of these correlators by applying the operator product expansion to
both the wavefunction of the universe and to the nal inin correlator.
Preliminaries. The expectation value of an operator Q at time 0 is computed by
h jQ( 0)j i = h0j hTei R 01 d HI ( )i
Q( 0) hTe i R 01 d HI ( )i j0i ;
where j i (j0i) is the vacuum of the interacting (free) theory, T (T) denotes (anti)time
ordering, and HI is the interaction Hamiltonian. We use the i prescription and replace
(1 + i ) in the time integrals to evaluate the expectation value in the interacting vacuum.
To compute quantum expectation values, we follow the usual procedure of quantization.
We promote the elds
and
to operators and expand in Fourier space
( ; k) = k( )ay(k) + h:c: ;
i1 in
(k) =
i1 in (k^) s;s(k; )by(k; ) + h:c: ;
where the creation and annihilation operators obey the canonical commutation relations
[a(k); ay(k0)] = (2 )3 (k
k0) ; [b(k; ); by(k0; 0)] = (2 )
0 (k
k0) :
3
27The tadpole cancellation is more manifest in the language of the EFT of in ation, where graviton
couplings are generated through the extrinsic curvature Kij
12 _ij in comoving gauge.
X
j j=t+1
The mode functions of and are given by
k =
H
_ p
2k3
(1 + ik )e ik ;
k =
H
p
Mpl 2k3
(1 + ik )e ik :
(C.4)
The mode functions of elds in the discrete series can be derived using the formulas given
in appendix A. The treelevel diagrams that we will compute have two interaction vertices.
Expanding the inin master formula (C.1) to quadratic order gives
where it is understood that each side is evaluated in the appropriate vacuum state.
The result for the treelevel exchange of a spins eld is given by
d~ hHI ( )Q( 0)HI (~)i
2Re
d~ hQ( 0)HI ( )HI (~)i ;
Z
1
d
1
(C.5)
HJEP04(218)
T1
2Re[T2] + 23 perms ;
1 2 3 4E12I E34I
1 2 3 4E12I E34I
Z
1 a2s 4 3 4 I ;
1 a2s 4 3 4 I ;
hQ( 0)i =
1
d
1
C.1
s
X
j j=t+1
(C.6)
(C.7)
(C.8)
(C.9)
(C.10)
(C.11)
(C.12)
(C.13)
(C.14)
where we have suppressed the time arguments (e.g. the mode functions outside of the
integrals are evaluated at
= 0) and de ned
Eabl
s;s(ki) ;
k^i1
a
k^ais=2 k^is=2+1
b
k^bis "i1 is (k^l) :
We have also set H = 1, which can be trivially restored. These integral formulas are even
applicable for massive elds in the complementary and principle series upon setting t =
1
in (C.6).
Spin4 PM particles. Before presenting a formula for arbitrary spin, let us consider
the example of a fs; tg = f4; 1g
eld. Using the mode functions (A.18){(A.20), the
nontimeordered integral T1 can be expressed in the following compact form
T1 =
I1 = (
4
X I1 ;
j j=2
E12I E34I
)4Aj j k1k2k3k4kI3 Kj j(k1; k2; kI )Kj j(k3; k4; kI ) ;
where we have expressed the amplitude in terms of the dimensionless power spectrum and
de ned A2 = 1=32, A3 = 1=1960, A4 = 1=686000, and
K2(ki; kj ; kl) = J0(ki; kj ; kl) + J1(ki; kj ; kl)
K3(ki; kj ; kl) = 5J0(ki; kj ; kl) + 5J1(ki; kj ; kl) + J2(ki; kj ; kl)
K4(ki; kj ; kl) = 140J0(ki; kj ; kl) + 140J1(ki; kj ; kl) + 55J2(ki; kj ; kl) + 13J3(ki; kj ; kl) ; (C.15)
with ki1 is
ki1 +
+ kis . The momentum scaling in (C.12) is correct, since Kn (or
Jn) scales as 1=k. For the timeordered integral T2, the inner layer instead consists of an
inde nite integral
Z
kln ki2jl (1 + n; ikijl ) + kij kijl (2 + n; ikijl ) + kikj (3 + n; ikijl )
an integer
where
The function T2 can then be written as
1
Jn(ki; kj ; kl)
d (1 + iki )(1 + iki )(ikl )ne ikijl
=
kln n! ki2jl + (n + 1)! kij kijl + (n + 2)! kikj ;
ki3j+l n
ki3j+l n
(n; x) = (n
n 1
1)! e x X xm
:
m=0
T2 =
4
X I2 ;
j j=2
in terms of the function
(C.16)
(C.17)
(C.18)
(C.19)
(C.20)
(C.21)
(C.22)
(C.23)
(C.24)
where (n; x) is the incomplete gamma function, which takes the following form when n is
I2 4 = L00 + Lf01g + L11 ;
I2 3 = 25 L00 + 25 Lf01g + 5 Lf02g + +25 L11 + 5 Lf12g + L22 ;
I2 2 = 19600 L00 + 19600 Lf01g + 7700 Lf02g + 1820 Lf03g
+ 19600 L11 + 7700 Lf12g + 1820 Lf13g + 3025 L22 + 715 Lf23g + 169 L33 ;
with Lfmng
Lmn = (
Lmn + Lmn and
4 E12I E34I
k1k2k3k4kI3
1
not very illuminating.
Although it is possible to obtain a closedform expression for Lmn, the result is lengthy and
In the collapsed limit, kI
k1
k2, the expressions simplify dramatically. This is
because the spin4 mode function in the longwavelength limit simpli es to
d (1 + ik1 )(1 + ik2 )(ikI )m Ln(k3; k4; kI ) ei(kI k1 k2) :
4;4(k ! 0) / H3k3=2 4
:
e ik
In this case, the kI dependence can be pulled out of the integrals, after which it is easy to
see that the trispectrum scales as kI 3 in the limit kI ! 0. In this limit, we nd
I1 + I2 = 25(
)4Cj j E1(1kI1Ek333kIIjs)=34 ;
with coe cients C2 = 1=32, C3 = 5=392 and C4 = 1=35. Di erent helicity modes contribute
with di erent but roughly similar amplitudes. The angular dependence now becomes
E11I = Es (k^1 k^I ) = E (k^1 k^I )Ps (k^1 k^I ) ;
factorizing into the transverse and longitudinal parts. The nal expression for the
trispec
P (k1)P (k3)P (kI ) X Cj j Y^s ( ; ')Y^s ( 0; '0) ; (C.27)
Arbitrary spin. The general formula for the exchange of a spins, deptht eld in the
lim h k1 k2 k32k4i0 = P (k1)P (k3)P (kI )
X Ft Esj;tj Y^s ( ; ')Y^s ( 0; '0) ;
k1k3
k2
I
trum in the collapsed limit is then
with angles de ned in section 3.2.
collapsed limit is
where
g
2
e
;s( ! 0) = Zs
21=2 t(2t)!
e ik
i
p t! ( k ) +t 1
n+1;s( ! 0) =
1 + t + n
k
n;s( ! 0) :
times as
relation
4
j j=2
t 1 s
j j=2
(2
Es;t
s!(s + t)!(s
t 1)!
2t+2 t!
1)!!(2t)! 2
:
denotes the amplitude for each helicity and Ft(k1; k3; kI ; 0) is a function that can depend
logarithmically on some of its arguments, with an IR cuto
0. The cases without any log
dependences are F0 = O(kI3), F1 = 25 and F3 = 36. Since this function is just constant for
t = 1; 3, the scaling in kI for these cases is given by (C.28). On the other hand, we see that
there is an extra suppression for t = 0 compared to the naive scaling that is suggested by
the latetime behavior of the intermediate eld. For general depths, the functional form of
Ft becomes more complicated as t increases. However, the overall scaling behavior in kI
is always xed by t as in (C.28) (with a few exceptions including t = 0). The trispectrum
therefore becomes singular in the collapsed limit for t > 1, in the sense that it diverges at
a rate faster than P (kI ) does. The derivation of (C.28) can be found in the insert below.
Derivation of (C.28). By taking the collapsed limit, we probe the latetime behavior of
the intermediate eld. The spins,
helicitymode with spatial spin n =
behaves at late
The latetime behavior of the n =
+ 1 mode can then be obtained using the recursion
(C.25)
(C.26)
(C.28)
(C.29)
(C.30)
(C.31)
where the numerical prefactor given by (C.29) xes the amplitude of the
helicitymode.
In the small kI limit, the kI dependence drops out of the integrals, giving the overall scaling
behavior as in (C.28). The form of Ft can be determined by computing the integral
ik1 0)2(1 + ik3 0)2e2i(k1 k3) 0
Z 0 d
1
1 t (1 + ik1 )2e 2ik1 Z 0 d~
1 ~1 t (1
ik3 ~)2e2ik3 ~ ;
(C.34)
This means that the n = s mode will behave as
It can then be shown that the twopoint function in the latetime limit becomes
e ik
p t!
( + t)! ( k )s+t 1
:
s;s s;s(
and then taking 0 ! 0 limit and multiplying by an appropriate symmetry factor.
C.2
The tensorscalarscalar correlator with a general spins eld exchange is given by
X
0= 2
h k1 k2 k3 i0 2ge he
Re[B1
B2
B3] + 5 perms ;
B1 = _3(k1k2)s=2
E1203["ij (k^3)"ij0 (k^3)] 1 2 3
E1203["ij (k^3)"ij0 (k^3)] 1 2 3
E1203["ij (k^3)"ij0 (k^3)] 1 2 3
Z 0
Z 0
Z 0
1
1
d
d
d
a2s 4 1 2 3
a2s 4 1 2 3
0Z 0
0Z
Z
1
1
as 3 ( 3 )0 ~3 0 ; (C.36)
as 3 ( 3 )0 ~3 0 ; (C.37)
1 as 3 ( 3 )0 ~3 0
1 a2s 4 1 2 3 0 ; (C.38)
2;s(ki) denotes the mode function with n = 2 spatial components. These
integral formulas are valid for elds with t =
1; 0; 1.28
When summing over the PM
eld helicities, note that some of the integrals that involve taking the contraction of the
transverse polarization tensors of the same helicity will vanish. This is because transverse
polarization tensors are built out of two polarization vectors "i , which are null, "i "i = 0,
and satisfy ("i ) = "i . This means that the only combination
= 0 to B3. Our normalization for polarization tensors "is1 is "i1 is = 2s
s
gives a factor of 4 from the contraction.
Spin4 PM particles.
function for n =
= 2 is given by
Again, let us rst specialize to the case fs; tg = f4; 1g. The mode
(C.32)
(C.33)
(C.35)
(C.39)
~k=2 =
i p
e ik k2(1 + ik )
2k3 10p70 2H3
28Recall that t > 1 lacks a helicity2 degree of freedom and that t = 1 indicates that the eld belongs
to either the complementary or principal series.
B1 =
B2 =
B3 =
3 2
3 2
Z
0
0
0
3 3 2
E123
where we have suppressed the arguments (k1; k2; k3) of the functions Kn, Jn, and Pn, the
latter of which is de ned by
Pn(ki; kj; kl)
i (1 + ikl )Ln(ki; kj; kl) = (n + 1)!
n(n5ki + kjl)(2n3kj2 + n4kl2 + n42kjkl) + ki2 2n3ki + 2(n42
1)kj + 3(n52
Not surprisingly, this has exactly the same structure as the graviton mode function. The
only di erence is the spindependent normalization constant and extra powers of the scale
factor due to the (conformal) time components of the PM
With the mode functions being simple algebraic functions, it is lengthy but
straightforward to compute the integrals (C.36){(C.38) as in the previous section. Skipping the
details of the computation, we nd that the integrals can be expressed in the following
compact form
(C.40)
(C.41)
(C.42)
(C.43)
3)kl o ;
(C.44)
(C.45)
When the external tensor mode becomes soft, k1
k2
k3, a bunch of terms becomes
unimportant and the bispectrum takes a considerably simpler form. First, we note that
Pn scales as k1=k3 relative to Kn and Jn in the squeezed limit, which implies that we can
neglect B3. Moreover, only J0 survives in the squeezed limit and Jn>0 is subleading in k1.
Taking the permutations for which the PM eld carries the soft momentum, the nal result
is given by
k1!0
~
lim h k1 k2 k3i0t=1 = P (k1)P (k3) Y^4 ( ; ') ;
1
denotes an e ective coupling strength, with angles de ned
in section 3.2. Imposing the perturbativity bounds of section 2.3, this parameter naively
needs to be much smaller than unity. However, the smallness of the overall numerical factor
is due to the normalization of the
00ij mode function in the
vertex, which should be
taken into account when setting a bound on h~e . The correct perturbativity condition of
the above correlator is then ~ . 1.
Arbitrary spin. The tensor squeezedlimit bispectrum for an arbitrary spins eld with
depth t 2 f0; 1g is
k1!0
1
lim h k1 k2 k3i0 = P (k1)P (k3)
k1
k3
1 t
Y^s ( ; ') ;
where we de ned
Ns;t ge h~e
pr, with
405 s!(s + t)!
Ns;t =
4t+2 (2s
1)!!(s + 1)!(s + 2)!(t + 2)!
t!
(2t)! 2
:
(C.46)
Unlike in the case of the trispectrum, only the helicities
=
2 contribute. As a result,
only t
1 elds can contribute, and the t = 0 eld leads to an extra suppression in the
squeezed limit relative to the t = 1 case. The derivation of the spindependent amplitude
can be found in the insert below. This amplitude would be di erent for di erent types
of
vertices. Typically, the higher the spatial spin of , the larger the amplitude. In
other words, Ns;t would be larger for interactions with more number of spatial derivatives,
e.g. ijk @k _ ij . Nonetheless, the weak coupling constraints imply
. 1 for all types of
interaction vertices.
Derivation of (C.45). Taking
= 2 in (C.32), we get
HJEP04(218)
! 0) = Z =2 21=2 t(2t)! (s + t)!
s
i
p t!
(2 + t)! ( k )s+t 1
e ik
:
2;s = Z =2 21=2 t(2t)! e ik
2 s
i
p t!
( k )t+1
s;s 22;s(
2
4 N~s;t
k2t s+3H2(s 1) s+2t ;
while (C.30) gives
In the squeezed limit, the product of these mode functions becomes relevant
(C.47)
(C.48)
(C.49)
(C.50)
where the N~s;t is related to (C.46) by some numerical factors. In the squeezed limit, B1 B2
becomes proportional to
K4(k3; k3; k1 ! 0) + 140 J0(k3; k3; k1 ! 0) = 280 J0(k3; k3; k1 ! 0) =
350
k1k3
:
Also, the integration involving the
vertex gives a factor of 34 . Combining with other
numerical and momentumdependent factors in (C.36) and (C.37), we arrive at (C.45).
D
Operator product expansions
In this appendix, we use the operator product expansion (OPE) to argue for the form of
cosmological correlators in various limits. In particular, we will show how these arguments
x the form of the scalar trispectrum in the collapsed limit and the tensorscalarscalar
bispectrum in the squeezed limit.
The OPE is used in two di erent ways in the derivation of the collapsed trispectrum.
First, it is applied directly to the collapsed limit of h i
4 . Second, we assume that the
coe cients in the wavefunction of the universe have a good OPE and construct the collapsed
limit of h
4i given the restricted forms of these coe cients. Both methods yield the same
result, but only when su cient care is taken in deriving the OPE in momentum space.
We start, in section D.1, with a review of the wavefunction of the universe, and its
connection to correlation functions in conformal
eld theory (CFT). The behaviour of
cosmological correlators in the soft limits can be recast in terms of the OPE in momentum
space, which we consider in section D.2. We will address certain subtleties in performing
the Fourier transform of more standard OPE expressions in position space. In section D.3,
we calculate the collapsed limit of h
4i using the two methods mentioned above. Finally,
the soft limit of h
i is analyzed in section D.4, using similar techniques.
D.1
Wavefunction of the universe
Consider a theory of a scalar
and a spins eld29
on dS4. The wavefunction for
this system is determined semiclassically as
exp(iScl), where Scl is the action
evaluated on the classical solutions which interpolate from the BunchDavies vacuum at
early times to the indicated value at late times: k( )
k and
k( )
classical action is then expanded as a function of the latetime eld values
!0
[ k; k]
exp
d3k1d3k2 hOk1 Ok2 i k1 k2 + h k1 k2 i k1 k2
d3k1d3k2d3k3 hOk1 Ok2 k3 i k1 k2 k3
d3k1d3k2d3k3d3k4 hOk1 Ok2 Ok3 Ok4 i k1 k2 k3 k4 +
; (D.1)
and kij
ki + kj .
The wavefunction of the universe calculation is similar to AdS/CFT, a fact which we
will return to below. As in standard holographic computations, two conformal weights are
associated to each bulk eld:
=
3
2
r 9
4
m2
H2
s
3
2
s
s
1 2
2
m2
H2
;
29Indices on , and related quantities will often be suppressed in this and following sections.
where the quantities in angled brackets are simply functions of the indicated momenta.
Expectation values are then calculated by integrating the desired elds against j [ k; k]j2,
as in quantum mechanics. For instance, this yields the latetime twopoint functions
h k
ki0 =
1
2 RehOkO ki0
h k
ki0 =
1
2 Reh k
ki0
;
and the fourpoint function
with
h k1 k2 k3 k4 i0 =
hO4i0A + hO4i0B
Qj4=1 2 RehOkj O kj i0
;
4 0
hO iA
4 0
hO iB
Reh k12
k12 i0
k12 i0 Reh k12 Ok3 Ok4 i0 + 2 perms ;
2 RehOk1 Ok2 Ok3 Ok4 i0 ;
2
(D.2)
(D.3)
(D.4)
(D.5)
(D.6)
for the scalar and spins eld, respectively. In the limiting case where
massless, spins, deptht eld, the weights
s reduce to
+ t :
The weights
+ and
the bulk
elds themselves are assigned the remaining weights,
and
s for
and ,
respectively. More explicitly, the quantities in (D.7) have the following scalings:
Reh k
+ +
RehOkO ki0 / k
2 + 3
ki0 / k2 s+ 3
h k
h k
2
ki0 / k
ki0 / k2 s
3
;
3
;
where the relation
= 3 was used, justifying the weight assignments.
D.2
OPE in momentum space
In order to analyze the in ationary correlators in the kinematical regimes of interest, we
must review some features of the OPE in momentum space. We begin with a discussion
of the momentum space OPE between two scalar operators. This is a subtle object,30 as
the process of Fourier transforming, in general, does not commute with taking the OPE
limit [104, 105].
As an illustrative example, consider the O
for three scalar operators with respective weights
(a) (b)
O
! O
a, b and
1
(c) position space OPE channel
c. It is of the form
where xab = xa
xb. Fourier transforming both sides of (D.10) gives
lim
xa!xb O
(a)(xa)O(b)(xb) / xaba+ b
c O
(c)(xb) ;
(a) (b)
qli!m0 Ok q=2O k q=2 / k a+ b
c d Oq(c) ;
in ddimensions. The result (D.11) is suspicious as it involves integrating an expression
which only holds at close (but separated) points (D.10) over all possible separations, and
indeed (D.11) is not in general correct.
The proper momentumspace OPE can instead be derived by directly Fourier
transforming the explicit position space expression for the correlator hO
The Fourier transform which isolates the desired momentum con guration is
(a)(xa)O(b)(xb)O(c)(xc)i.
Z
(a) (b)
hOk q=2O k q=2Oq(c)i0 =
ddx ddy eik x+iq y
hO
(a)(x=2)O(b)( x=2)O(c)(y)i ;
and careful treatment [104, 105] reveals distinct results depending on the weight
correct form of this OPE channel is
(a) (b)
qli!m0 Ok q=2O k q=2 / Oq(c) k a+ b
c d
8
>1
>
<
ln q
>>:(q=k)d 2 c
c < d=2
c = d=2 :
c > d=2
30We would like to thank Matthew Walters for discussions on this point.
(D.7)
(D.8)
(D.9)
(D.10)
(D.11)
(D.12)
c. The
(D.13)
Hence, the naive result (D.11) is only correct in the regime of small weights,
c
We will require the same computation for the OPE channel from two scalars to a
traceless, symmetric, spins operator of weight
s: O
O
! Oi1 is . The end result is
analogous to (D.13). The calculation requires the knowledge of the corresponding position
space correlator, which is uniquely xed by conformal invariance [106]:
hO
(a)(x1)O(b)(x2)Os(x3)i /
x(12 a+ b
z x13 x223
s+s)=2x(13 a+ s
z x23 x213 s
b+s)=2x(23 b+ s
a+s)=2 : (D.14)
Above, we have contracted all loose indices of the spins operator with an auxiliary null
vector zi to turn it into the indexfree operator Os. In order remove the auxiliary vectors,
one repeatedly acts on the above with a particular derivative operator whose detailed
form we will not need.
We then Fourier transform (D.14) in the con guration (D.12)
following [104]. The process is straightforward and results in31
>1
>
<
ln q
>>:(q=k)d 2 s
(a) (b)
qli!m0 Ok q=2O k q=2 / Oi1 is (q) k^i1
k^is k a+ b
s d
In deriving (D.15), we have made use of the momentum space twopoint function for
spinning operators:
hOi1 is (q)Oj1 js ( q)i0 / q2 s d i1 isj1 js (q^) ;
where i1 isj1 js (q^) is a symmetric, traceless tensor structure.32 The result for
is the result which one would obtain from naively Fourier transforming the position space
OPE of two scalar operators,33 while the cases of larger dimension are di erent, completely
analogously to the scalar result (D.13). The di erent behavior for large and small operator
weights will be crucial for nding agreement between results derived via wavefunction of
s < d=2
s = d=2 :
s > d=2
(D.15)
(D.16)
s < d=2
the universe and the inin computations.
D.3
Collapsed trispectrum
We now use the OPE to derive the form of the collapsed limit of h
discussed above. We specialize to the case of massless
and take
4i using the two methods
to be a PM
eld of
spin s and depth t.
First, we use the OPE to calculate the collapsed limit of h
4i directly. The calculation
is relatively straightforward. The
and
elds have weight assignments
= 0 and
31Where the weights are such that the Fourier transform does not converge, these results are de ned by
33See [75], for instance, for the form of this OPE.
analytic continuation.
32The form of
P jZs j2"
i1 is "j1
js .
(q^) can be found using the arguments in appendix A of [24]. It is also related to the
polarization vectors and normalization factors in appendix A via a completeness relation i1:::isj1:::js /
t, respectively, and hence the
s < d=2 branch of the OPE applies, as d = 3
here
operator, we nd
qli!m0 k+q=2 k+q=2 =
X Cs;t i1 is (q)k^i1
k^is kt 4
Performing the contractions and only keeping the contribution of a single spins, deptht
qli!m0h k1 q=2 k1 q=2 k2+q=2 k2+q=2i0 /
1
kt 4kt 4
2
q2t+1
^s
(q^) k^s2 ;
where (q^) is the tensor structure de ned in (D.16). This reproduces the scaling behavior
HJEP04(218)
in (3.17).
The angular dependence can also be matched when the tensor structure is
expanded into the helicity basis.
Next, we turn to the wavefunction of the universe. Wavefunction coe cients arise
as analytic continuations of AdS/CFT calculations, simply because in both cases one is
computing the onshell action as a function of boundary data [19, 73, 107]. As the AdS
quantities are CFT correlators which have a good OPE, it is feasible that the OPE can
also be applied to wavefunction coe cients. We now apply this logic to derive OPE limits
of coe cients which in turn determine the collapsed limit of h
4i. The following arguments
are intended to be more heuristic than rigorous and indeed we will nd some technical
disagreements between parts of the OPE prediction and concrete wavefunction calculations,
though the discrepancies don't a ect the predicted overall scaling of h i
The fourpoint function is determined by hO4i0A and hO4i0B, de ned in (D.4). In
the collapsed con guration (D.18), the hO4i0A term is dominated by just one of its three
4 .
4 0
hO iA
lim
qOk2+q=2Ok2+q=2i0 ;
Reh q
qi0
due to the strong scaling of the denominator with q, via the Reh q
simply the PM power spectrum h q
qi0, which is proportional to q 1 2t. The terms in
the numerator are evaluated using the OPE (D.15) with
s > d=2, as opposed to (D.17):
qi0 1 factor. This is
qli!m0 Ok+q=2O k+q=2 =
X Cs;t Oi1 is (q)k^i1
s;t
k^is kt+2q 2t 1
(D.18)
(D.19)
(D.20)
(D.21)
(D.22)
yielding
Hence, in the collapsed limit, we nd
q!0
lim RehOk q=2O k q=2 qi0 / kt+2 :
hO4i0A /
1
kt+2kt+2
2
q2t+1
^s
1
(q^) k^s2 ;
where we restored the tensor structure. The case of graviton exchange is calculated in [108]
and the result is consistent with the scaling in (D.22). We have also calculated the result
for massless spin4 exchange and we again nd agreement with (D.22).
The OPE (D.20) can also be used to analyze the collapsed limit of hO4i0B, and here
we nd tension with concrete calculations. Performing the contractions and keeping only
the contribution from spins, deptht operators, one nds the same scaling as in (D.22),
hO4i0B /
1
kt+2kt+2
2
q2t+1
^s
(q^) k^s2 :
(D.23)
However, unlike (D.22), the result (D.23) is not in agreement with the explicit calculations
for massless spin234 or spin4 particles, where this contribution is instead found to be
subleading as q is taken soft. While this
nding doesn't a ect the
it does imply that the OPE cannot be naively applied35 to wavefunction coe cients and
nal scaling for h i
4 ,
that this second OPE application is only a rough argument. In the end, after using (D.3),
wavefunction heuristics give the same scaling result for the collapsed trispectrum as directly
applying the OPE to (D.18).
D.4
Soft tensor bispectrum
We can apply similar methods to analyze the contribute of
to the following correlator:
qli!m0h q k q=2 k q=2i0 :
However, in order for the answer to be nontrivial, we need to assume that the conformal
symmetry is broken. Applying the OPE (D.17) directly and keeping only the
channel contribution, one nds
qli!m0h q k q=2 k q=2i0 / kt 4Y^s ( ; ') h q
qi0 ;
where we have expanded i1 is into the helicity basis and contracted the momenta, with
the angles de ned by cos
= k^ q^ and cos ' = " k^. The mixed twopoint function is only
nontrivial if the
eld has a spin2 component, corresponding to the restriction t
the conformal symmetry is preserved, then h q
qi0 = 0. However, if we assume that scale
symmetry holds, but special conformal symmetries are softly broken, then it is possible to
have a nonzero twopoint function:
h q
qi0 =
qt+2 ;
where
is a small parameter characterizing the breaking of special conformal symmetry.
Inserting (D.26) into (D.25) and assuming that the OPE (D.17) holds up to O( )
corrections, we nd the leading order scaling:
qli!m0h q k q=2 k q=2i0 / q3k3
q 1 t
Y^s ( ; ') ; t
1 ;
consistent with (C.44). Similar results follow from a wavefunction analysis.
34See [108] for more on the subdominance of hO4i0B to hO4i0A in the spin2 case.
35We can make an interesting speculation for the origin of the mismatch: in the putative dual theory,
there may be more than one operator associated to . In particular, if there is not only the operator
of
weight
that the OO !
for example.
s+, but also the associated shadow operator ~ of weight
s in the spectrum, then it's possible
of the shadow operator ~ leaves hO4i0A una ected. The cancellation can only happen for ~ exchange, as
follows from the (D.15). Shadow operators have appeared previously in the dS/CFT literature; see [90, 109],
and OO ! ~ OPE channel contributions to hO4i0B can cancel out, while the existence
(D.24)
(D.25)
(D.26)
(D.27)
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