Inflation and Conformal Invariance: The Perspective from Radial Quantization

TL;DR

This paper explores how conformal invariance constrains inflationary correlators in the dS/CFT correspondence, linking boundary CFT properties to bulk de Sitter physics and analyzing asymptotic symmetries.

Contribution

It provides a novel perspective on inflationary correlators through radially quantized CFT and elucidates the role of asymptotic symmetries in de Sitter space.

Findings

Higuchi bound arises from reflection positivity in CFT

Partial massless states correspond to boundary states with highest weight

Asymptotic symmetries have a quantum mechanics interpretation

Abstract

According to the dS/CFT correspondence, correlators of fields generated during a primordial de Sitter phase are constrained by three-dimensional conformal invariance. Using the properties of radially quantized conformal field theories and the operator-state correspondence, we glean information on some points. The Higuchi bound on the masses of spin-s states in de Sitter is a direct consequence of reflection positivity in radially quantized CFT$_3$ and the fact that scaling dimensions of operators are energies of states. The partial massless states appearing in de Sitter correspond from the boundary CFT$_3$ perspective to boundary states with highest weight for the conformal group. We discuss inflationary consistency relations and the role of asymptotic symmetries which transform asymptotic vacua to new physically inequivalent vacua by generating long perturbation modes. We show that on the CFT$_3$ side, asymptotic symmetries have a nice quantum mechanics interpretation. For instance, acting with the asymptotic dilation symmetry corresponds to evolving states forward (or backward) in "time" and the charge generating the asymptotic symmetry transformation is the Hamiltonian itself. Finally, we investigate the symmetries of anisotropic inflation and show that correlators of four-dimensional free scalar fields can be reproduced in the dual picture by considering an isotropic three-dimensional boundary enjoying dilation symmetry, but with a nonvanishing vacuum expectation value of the boundary stress-energy momentum tensor.

Equations

$${\eta }_{AB}{X}^A{X}^B=-X_0^2+{\vec{X}}^2+X_5^2=\frac{1}{H^2}A,B=(0,i,,5)\mathrm{with}i=1,2,3,$$

$$X^0=\frac{1}{2H}\left(H\eta -\frac{1}{H\eta }\right)-\frac{1}{2}\frac{{\vec{x}}^2}{\eta },$$

$$X^i=\frac{x^i}{H\eta },$$

$$X^5=-\frac{1}{2H}\left(H\eta +\frac{1}{H\eta }\right)+\frac{1}{2}\frac{{\vec{x}}^2}{\eta },$$

$$\mathrm{d}{s}_5^2={\eta }_{AB}\mathrm{d}{X}^A\mathrm{d}{X}^B.$$

$$\mathrm{d}{s}^2=g_{\mu \nu }\mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }=\frac{1}{H^2{\eta }^2}\left(-\mathrm{d}{\eta }^2+\mathrm{d}{\vec{x}}^2\right).$$

$$J_{AB}=X_A\frac{\partial }{\partial X^B}-X_B\frac{\partial }{\partial X^A}.$$

$$J_{ij},P_0=J_{05},{\Pi }_i^{+}=J_{i5}+J_{0i},{\Pi }_i^{-}=J_{i5}-J_{0i}.$$

$$J_{ij}=x_i\frac{\partial }{\partial x_j}-x_j\frac{\partial }{\partial x_i},$$

$$P_0=\eta \frac{\partial }{\partial \eta }+x_i\frac{\partial }{\partial x_i},$$

$${\Pi }_i^{-}=-2H\eta x_i\frac{\partial }{\partial \eta }+H\left({\vec{x}}^2{\delta }_{ij}-2x_i{x}_j\right)\frac{\partial }{\partial x_j}-H{\eta }^2\frac{\partial }{\partial x_i},$$

$${\Pi }_i^{+}=\frac{1}{H}\frac{\partial }{\partial x_i},$$

$$[J_{ij},J_{kl}]={\delta }_{il}{J}_{jk}-{\delta }_{ik}{J}_{jl}+{\delta }_{jk}{J}_{il}-{\delta }_{jl}{J}_{ik},$$

$$[J_{ij},{\Pi }_k^{\pm }]={\delta }_{ik}{\Pi }_j^{\pm }-{\delta }_{jk}{\Pi }_i^{\pm },$$

$$[{\Pi }_k^{\pm },P_0]=\mp {\Pi }_k^{\pm },$$

$$[{\Pi }_i^{-},{\Pi }_j^{+}]=2J_{ij}+2{\delta }_{ij}{P}_0.$$

$$L_{ij}=i{J}_{ij},D=-i{P}_0,P_i=-i{\Pi }_i^{+},K_i=i{\Pi }_i^{-},$$

$$P_i=-\frac{i}{H}{\partial }_i,$$

$$D=-i\left(\eta \frac{\partial }{\partial \eta }+x^i{\partial }_i\right),$$

$$K_i=-2iH{x}_i\left(\eta \frac{\partial }{\partial \eta }+x^j{\partial }_j\right)-iH(-{\eta }^2+{\vec{x}}^2){\partial }_i,$$

$$L_{ij}=i\left(x_i\frac{\partial }{\partial x_j}-x_j\frac{\partial }{\partial x_i}\right),$$

$$[D,P_i]=i{P}_i,$$

$$X^0=-\frac{1}{2H^2\eta }-\frac{1}{2}\frac{{\vec{x}}^2}{\eta },$$

$$X^i=\frac{x^i}{H\eta },$$

$$X^5=-\frac{1}{2H^2\eta }+\frac{1}{2}\frac{{\vec{x}}^2}{\eta },$$

$$-X_0^2+X_i^2+X_5^2=0.$$

$$x_i^{\prime }=a_i+M_i^j{x}_j,$$

$$x_i^{\prime }=\lambda x_i,$$

$$x_i^{\prime }=\frac{x_i+b_i{\vec{x}}^2}{1+2b_i{x}_i+b^2{\vec{x}}^2}.$$

$$x_i\to x_i^{\prime }=\frac{x_i}{{\vec{x}}^2},$$

$$\mathrm{special}\mathrm{conformal}\mathrm{transformations}\sim \mathcal{I}\circ (\mathrm{translation})\circ \mathcal{I}.$$

$$[L_{ij},{\varphi }_s(0)]={\Sigma }_{ij}^s{\varphi }_s(0),$$

$$[D,{\varphi }_s(0)]=-i\Delta {\varphi }_s(0),$$

$$[K_i,{\varphi }_s(0)]=0,$$

$$[J,\varphi (\vec{x})]=e^{i\vec{P}\cdot \vec{x}}[\hat{J},\varphi (0)]e^{-i\vec{P}\cdot \vec{x}},$$

$$\hat{J}=e^{-i\vec{P}\cdot \vec{x}}J{e}^{i\vec{P}\cdot \vec{x}}=\sum _n\frac{(-i{)}^n}{n!}{x}^{i_1}{x}^{i_2}\cdots x^{i_n}[P_{i_1}[P_{i_2}\cdots [P_{i_n},J],\cdots ]],$$

$${\hat{L}}_{ij}=L_{ij}+x_i{P}_j-x_j{P}_i,$$

$$\hat{D}=D+x^i{P}_i,$$

$${\hat{K}}_i=K_i+2(x_{i}D-x^j{L}_{ij})+2x_i{x}^j{P}_j-{\vec{x}}^2{P}_i.$$

$$i[L_{ij},\varphi (\vec{x})]=\left(x_i{\partial }_j-x_j{\partial }_i\right)\varphi (\vec{x}),$$

Full text

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**Inflation and Conformal Invariance:

The Perspective from Radial Quantization

**

Alex Kehagiasa,b and Antonio Riottoc

aPhysics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece

bTheoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland

c*Department of Theoretical Physics and Center for Astroparticle Physics (CAP)

24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland*

**Abstract

According to the dS/CFT correspondence, correlators of fields generated during a primordial de Sitter phase are constrained by three-dimensional conformal invariance. Using the properties of radially quantized conformal field theories and the operator-state correspondence, we glean information on some points. The Higuchi bound on the masses of spin-$s$ states in de Sitter is a direct consequence of reflection positivity in radially quantized CFT3 and the fact that scaling dimensions of operators are energies of states. The partial massless states appearing in de Sitter correspond from the boundary CFT3 perspective to boundary states with highest weight for the conformal group. We discuss inflationary consistency relations and the role of asymptotic symmetries which transform asymptotic vacua to new physically inequivalent vacua by generating long perturbation modes. We show that on the CFT3 side, asymptotic symmetries have a nice quantum mechanics interpretation. For instance, acting with the asymptotic dilation symmetry corresponds to evolving states forward (or backward) in “time” and the charge generating the asymptotic symmetry transformation is the Hamiltonian itself. Finally, we investigate the symmetries of anisotropic inflation and show that correlators of four-dimensional free scalar fields can be reproduced in the dual picture by considering an isotropic three-dimensional boundary enjoying dilation symmetry, but with a nonvanishing vacuum expectation value of the boundary stress-energy momentum tensor.**

1 Introduction

Over the last decade we have become more and more convinced that the cosmic microwave background anisotropies and the large-scale structure of the universe have been originated from some seeds generated in the very early universe. The leading paradigm to explain these primordial fluctuations is inflation [1]. During a stage of accelerated expansion, dubbed de Sitter (dS) period, quantum fluctuations are stretched to cosmological scales and, upon horizon re-entry and thanks to the phenomenon of gravitational instability, they provide the seeds for the structures of the universe.

Understanding inflation, its dynamics and observational predictions is therefore of extreme importance. From this point of view, symmetries may be of much help. It is well known that they play a crucial role in high energy physics and they have proved to be extremely useful in cosmology too. For instance, the consistency relations for single-field models of inflation [2, 3], which allow to express the squeezed limit of the $(n+1)$-point correlators in terms of $n$-point functions and whose violation would rule out single-field models of inflation, are derived from non-linearly realized symmetries of inflation corresponding to conformal transformations of spatial ${\mathbb{R}}^{3}$ slices. Similarly, late-time universe consistency relations for the fluctuations in the dark matter density or in the number density of galaxies have also been recently obtained [4, 5, 6, 7, 8] based on symmetry arguments.

A crucial step to deepen our knowledge about the properties of the inflationary fluctuations has been taken by Strominger who has formulated the so-called dS/CFT correspondence (CFT standing for conformal field theory) [9]. It is based on the fact that the de Sitter isometry SO(1,4) group acts as conformal group on $\mathbb{R}^{3}$ when the fluctuations are on super-Hubble scales. Bulk fields $\phi(\vec{x},\eta)$ evolving in four-dimensional de Sitter spacetime and behaving near the boundary $\eta=0$ ($\eta$ being the conformal time) as $\phi(\vec{x},\eta)\sim(-\eta)^{\Delta}\phi(\vec{x})$ correspond on the three-dimensional space, where the symmetry is CFT3, to a field of conformal weight $h=\Delta-3$. Correlators of fields excited during a de Sitter phase are therefore expected to be constrained by conformal invariance and the literature on this topic is rich and diverse [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. In particular, much attention has been devoted to the relation among the consistency relations, soft theorems and the asymptotic symmetries of de Sitter which transform asymptotic vacua to new physically inequivalent vacua by generating long perturbation modes [35, 34].

The isometries of de Sitter impose also stringent lower bounds on the masses of particles with higher spin, the so-called Higuchi bound found in Ref. [36] for massive graviton. An extensive study of the role of spinning states in de Sitter has recently been carried out in Refs. [28, 32] where it was shown that peculiar signatures are present in the squeezed limit of the correlation functions of the primordial fluctuations with their angular dependence providing information about the spin. In particular, the link between the Higuchi bound and the properties of CFT3, its relation to the (violation of the) tensor consistency relation and the absence of curly hair in de Sitter have been nicely discussed in Ref. [31].

The goal of this paper is to reinterpret and discuss the Higuchi bound, the partial massless states in de Sitter as well as the asymptotic symmetries and inflationary consistency relations through a new perspective offered by radially quantized CFT3 and the fact that conformal operators are in one-to-one correspondence with the states of the CFT3. We will also discuss the role of symmetries in determining the properties of the correlators in anisotropic inflation.

We will show that the Higuchi bound is a simple consequence of reflection positivity in radially quantized CFT3 assuring that the ground state has finite energy. Indeed, energies must be bounded from below and in radial quantization energies of states are the scaling dimensions of operators.

We will also elucidate the phenomenon of partial masslessness on the boundary CFT3 side. Helicities of states of spinning states may become massless in de Sitter for some particular values of masses. In the dS/CFT3 correspondence, such bulk fields correspond to boundary fields and through the operator-state correspondence of CFT3, they correspond on the boundary to rank-$s$ symmetric tensors which are partially conserved. This in turn means that they correspond to states which must be highest weight tensors for the conformal group: the descendants of the corresponding symmetric traceless tensors have vanishing norm.

We will use the techniques of radial quantization to understand in more detail the role of asymptotic symmetries and consistency relations during inflation. For instance, asymptotic dilation symmetries transform asymptotic vacua to new physically inequivalent vacua by generating a long mode for the curvature perturbation. We will show that on the CFT3 side, this corresponds to evolve states forward (or backward) in “time” along a cylinder obtained by relating the metric on the Euclidean flat space ${\mathbb{R}}^{3}$ to the metric of a cylinder S${}^{2}\otimes{\mathbb{R}}$ by a Weyl tranformation. The charge generating the symmetry transformation will be the Hamiltonian obtained in the radial quantization on the cylinder of the CFT3, thus allowing a nice interpretation of the asymptotic symmetries in terms of a quantum mechanics analogy. More generically, the asymptotic symmetries are generated by the topological charges of the CFT3.

Finally, we will investigate the case of anisotropic de Sitter expansion which is relevant for the case of anisotropic inflation. While anisotropic de Sitter space times are not maximally symmetric, they nevertheless maintain the dilation isometry. This residual symmetry allows to fix the correlators of free four-dimensional scalar fields together with the anisotropy in the power spectrum. By conjecturing that anisotropic de Sitter is dual to a three-dimensional isotropic three-dimensional boundary where the stress energy momentum tensor acquires a nonvanishing vacuum expectation value, we are able to reproduce the four-dimensional results.

The paper is organized as follows. In section 2 we provide a summary of the symmetries of de Sitter. Section 3 contains a brief introduction to the notion of radial quantization of CFT3. The Higuchi bound is obtained in section 4, while the case of partially massless states is presented in section 5. Consistency relations and asymptotic symmetries are discussed in section 6. Section 7 contains our findings regarding anisotropic inflation. Conclusions are presented in section 8. The paper contains also three appendices.

2 Symmetries of the de Sitter geometry

Let us summarize here the more salient features about the symmetries of four-dimensional de Sitter. The expert reader can skip this part.

The four-dimensional de Sitter spacetime of radius $H^{-1}$, where $H$ is the Hubble rate can be described by the hyperboloid [38]

[TABLE]

embedded in five-dimensional Minkowski spacetime $\mathbb{M}^{1,4}$ with coordinates $X^{A}$ and flat metric $\eta_{AB}=\rm{diag}(-1,1,1,1,1)$. A particular parametrization of the de Sitter hyperboloid is then provided by (being $\eta$ the conformal time)

[TABLE]

which satisfies Eq. (2.1). The de Sitter metric is the induced metric on the hyperboloid from the five-dimensional ambient Minkowski spacetime

[TABLE]

For the particular parametrization (2.2) we find the standard de Sitter metric

[TABLE]

The group SO(1,4) acts linearly on $\mathbb{M}^{1,4}$. The generators are

[TABLE]

We may split them as

[TABLE]

In this way, they act on the de Sitter hyperboloid as

[TABLE]

with the corresponding commutator relations

[TABLE]

The key point is that this reproduces the conformal algebra. Indeed,

[TABLE]

one has

[TABLE]

are the Killing vectors of de Sitter spacetime corresponding to symmetries under space translations $P_{i}$, dilitations $D$, special conformal transformations $K_{i}$ and space rotations $L_{ij}$. The corresponding conformal algebra has the following commutation rules

[TABLE]

Let us now take the super-Hubble limit case $H\eta\ll 1$. The parametrization (2.2) reduces to

[TABLE]

where the hyperboloid degenerates to the hypercone

[TABLE]

By identifying the points $X^{A}\equiv\lambda X^{A}$ on the cone, the conformal group acts linearly and induces the conformal transformations $x_{i}\to x_{i}^{\prime}$ with

[TABLE]

on Euclidean $\mathbb{R}^{3}$ with coordinates $x_{i}$. These transformations are the translations and rotations (generated by $P_{i}$ and $L_{ij}$ respectively), dilations (generated by $D$) and special conformal transformations (generated by $K_{i}$). They act on the constant time hypersurfaces of de Sitter spacetime. Finally we note the special conformal transformations can be written in terms of inversion ${\cal I}$

[TABLE]

as

[TABLE]

This property will be useful in the following.

2.1 Representations

The representations of the SO(1,4) algebra can be constructed by using the method of induced representations. Let us first investigate the stability subgroup at the origin of the coordinates $x_{i}$, that is is the group generated by $(L_{ij},D,K_{i})$. From the conformal algebra one can see that $P_{i}$ and $K_{i}$ are raising and lowering operators for the dilation operator $D$. Therefore there are states which will be annihilated by $K_{i}$ and each irreducible representation can be specified by an irreducible representation of the rotational group SO(3) (i.e. its spin) and a definite conformal dimension annihilated by $K_{i}$. Representations $\phi_{s}(0)$ of the stability group at the origin with spin $s$ and dimension $\Delta$ are specified by the following relations

[TABLE]

where we have indicated by $\Sigma^{s}_{ij}$ the spin-$s$ representation of SO(3). The representations $\phi_{s}(0)$ that satisfy the relations (2.25) are called primary fields and once the primary fields have been identified, all other fields, called the descendants, are deduced by taking derivatives of the primaries $\partial_{i}\cdots\partial_{j}\phi_{s}(0)$. Furthermore, for all the generators of the stability subgroup, collectively indicated by $J=(L_{ij},D,K_{i})$, since $\phi(\vec{x})=e^{i\vec{P}\cdot\vec{x}}\phi(0)e^{-i\vec{P}\cdot\vec{x}}$, we have that

[TABLE]

with

[TABLE]

and $\phi(0)$ a representation of the stability subgroup. For $J=L_{ij},D$ and $J=K_{i}$ we therefore have

[TABLE]

In particular, for scalar degrees of freedom, the right-hand side of the first equation in (2.25) is zero and

[TABLE]

We recall now that the de Sitter algebra SO(1,4) has two Casimir invariants

[TABLE]

Using Eqs. (2.6) and (2.9), it is easy to show that

[TABLE]

which, using the the explicit representation (2.10), becomes

[TABLE]

Eq. (2.37) then gives a fundamental relation between the mass and the conformal weight. Indeed,

[TABLE]

implies

[TABLE]

for a massive scalar field in de Sitter. Appendix A offers a more intuitive way of getting the same relation.

The method of the induced representations for which we have worked out the case of the scalar can be adopted to include higher-spin fields as well. For the case of a higher-spin field described by a symmetric-traceless tensor $\phi_{i_{1}\cdots i_{s}}$, we obtain

[TABLE]

and the spin operator $\Sigma_{ij}^{s}$ acts as

[TABLE]

It is then easy to verify that

[TABLE]

The equation of motion for the spin $s$ field $\Phi_{\mu_{1}\cdots\mu_{s}}$ is

[TABLE]

where $\nabla_{\mu}$ is the covariant derivative in de Sitter space and $\Phi_{\mu_{1}\cdots\mu_{s}}$ is subject to the conditions

[TABLE]

For $m=0$, Eq. (2.47) enjoys the extra gauge invariance transformation

[TABLE]

In addition, it can be verified that [39]

[TABLE]

and therefore, we find from Eqs. (2.46), (2.47) and (2.50) that the mass of the spin $s$ field $\Phi_{\mu_{1}\cdots\mu_{s}}$ in de Sitter space is

[TABLE]

Note that for $s=0$, Eq. (2.51) does not coincide with Eq. (2.40). The reason is that Eq. (2.51) for $s=0$ gives the mass of a conformally coupled scalar, that is a scalar whose action is

[TABLE]

where $R=12H^{2}$ is the scalar curvature of the de Sitter space.

3 The radial quantization of CFT3

This section contains known results too and more details can be found, for instance, in Refs. [40, 37]. In the radial quantization of the CFT3 one foliates the space ${\mathbb{R}}^{3}$ by spheres S2 centered at the origin, see Fig. 1. The unitary operator that takes points from one sphere to the other is constructed through the dilation operator $D$ as

[TABLE]

From this expression, one already hints that the operator $D$ plays the role of the Hamiltonian. The states living on the various spheres are classified according to their scaling dimension $\Delta$

[TABLE]

and possibly by their spin $s$ under the representation of SO($3$)

[TABLE]

where we have exploited the fact that the angular momentum $L_{ij}$ is the only one commuting with the operator $D$. States are therefore generated by inserting operators inside the sphere. Conformal operators are in one-to-one correspondence with the states of the CFT3

[TABLE]

This is the so-called operator-state correspondence. Thus, by inserting an operator ${\cal O}_{\Delta}$ of scaling dimension $\Delta$ at the origin the state $|\Delta\rangle={\cal O}_{\Delta}(0)|0\rangle$ has also dimension $\Delta$

[TABLE]

Primary operators are those for which $\vec{K}|\Delta\rangle=0$ and a conformal multiplet in radial quantization is given by acting with momentum generators on a primary state $|\Delta\rangle$, $P_{i}|\Delta\rangle$, $P_{i}P_{j}|\Delta\rangle$, $\cdots$. Tthis is equivalent to act with derivatives at the origin

[TABLE]

Since as the dilation operator moves points long the spheres, if we insert an operator not at the origin, the corresponding state $|\Psi\rangle={\cal O}_{\Delta}(\vec{x})|0\rangle$ is not an eigenstate of the dilation operator $D$. We can though decompose the state $|\Psi\rangle$ in terms of states with different energies

[TABLE]

Notice that the operator $\vec{P}$, when applied to the state $|\Delta\rangle$ $n$ times, it raises the energy from $\Delta$ to $(\Delta+n)$. This is the consequence of the commutation relation (2.17). Similarly, the operator $\vec{K}$, when applied to the state $|\Delta\rangle$ $n$ times, it lowers the energy from $\Delta$ to $(\Delta-n)$ as a consequence of the commutation relation (2.17). We will now discuss the radial quantization mapping the theory on the cylinder.

3.1 The mapping to the cylinder

In a conformal field theory we can relate the metric on the Euclidean flat space ${\mathbb{R}}^{3}$ to the metric of a cylinder S${}^{2}\otimes{\mathbb{R}}$ by a Weyl tranformation

[TABLE]

where we have introduced the radial coordinates $r$ and $\vec{n}\in$ S2 on ${\mathbb{R}}^{3}$ and

[TABLE]

For instance, expressed in these new coordinates on the cylinder, see Fig. 2, the local operator correlation functions of a scalar field with dimension $\Delta$ become

[TABLE]

The function $f$ can only depend on differences of the type $(\tau_{i}-\tau_{j})$ and the unit vectors $n_{i}$ as the scaling factors $r_{i}^{-\Delta_{\phi}}$ are already accounted for. What remains is already scale invariant and therefore can only depend on ratios of the distances $r_{i}/r_{j}$ and therefore on the differences $(\tau_{i}-\tau_{j})$.

This logic suggests a definition of the fields on the cylinder of the type

[TABLE]

where $\phi(r,\vec{n})$ are the fields in ${\mathbb{R}}^{3}$. The correlation functions of the fields on the cylinder reduce to

[TABLE]

that is the dynamics on S${}^{2}\otimes{\mathbb{R}}$ is invariant under translations.

On the cylinder we can define the “time” reflection operation which takes $\tau$ into $-\tau$. In the Hamiltonian formulation we have

[TABLE]

so that

[TABLE]

This has several consequences:

Any $2n$-point correlatator with operator inserted symmetrically under reflection is positive in a unitary theory. For instance

[TABLE]

This property is called reflection positivity. 2. 2.

If we remember that the operators $\vec{K}$ and $\vec{P}$ are related by the inversion operator

[TABLE]

then we have

[TABLE] 3. 3.

Since “time” reflection operation on the cylinder corresponds to an inversion ${\cal I}$ in flat space where $x_{i}$ is mapped into $x_{i}/\vec{x}^{2}$, this implies that in radial quantization the conjugate of a state $|\Delta_{\phi}\rangle=\phi(\vec{x})|0\rangle$ is given by

[TABLE]

This means that, while the state $|\Delta_{\phi}\rangle=\lim_{\vec{x}\to 0}\phi(\vec{x})|0\rangle$ is obtained by acting with $\phi(0)$ on the in vacuum the state $\langle\Delta_{\phi}|$ is obtained by acting with $\phi(\infty)$ on the out vacuum:

[TABLE]

and

[TABLE]

The definition of $\phi(\infty)$ as the conjugate of $\phi(0)$ contains as usual a rescaling factor necessary to get a finite limit, that is

[TABLE]

Notice also that, using the expression for the two-point correlator

[TABLE]

one automatically gets

[TABLE] 4. 4.

If we think as $\tau$ as the “time” coordinate, in the Schrödinger picture, states on the cylinder evolve as

[TABLE]

A time translation $\tau\to\tau+\lambda$ on the cylinder generates a rescaling $r\to e^{\lambda}r$. The dilation operator displaces points along the time direction on the cylinder

[TABLE]

In other words, in radial quantization, states live on spheres, and we evolve from one state to another with the dilation operator. Viewed from the point of view of the cylinder, the dilation operator moves states along the “time”-direction.

4 The Higuchi bound from the dS/CFT3 correspondence and radial quantization

The goal of this section is to show that one can deduce the Higuchi bound [36], which makes impossible the existence spin-1 fields in de Sitter spacetime with masses $m^{2}<0$ and of spin-2 fields with masses $m^{2}<2H^{2}$, based on the dS/CFT3 correspondence.

The origin of the Higuchi bound from a CFT3 point of view has been already nicely discussed in the literature recently [28, 31]. We offer here a different perspective using the radial quantization of CFT3. Essentially, the Higuchi bound derives from the fact that, even if primary field satisfies the reflection positivity condition on the CFT3 side, a descendant can violate it. This leads to a operator dimension requirement to avoid negative norm of the descendants.

4.1 First derivation

Let us first consider a spin-1 state in de Sitter. Such a state contains a transverse free and traceless helicity-1 state and a helicity-0 state $\pi(\vec{x})$ of scaling dimension $\Delta_{\pi}$. According to the dS/CFT3 correspondence, this bulk field corresponds to a dual boundary field $L_{\pi}(\vec{x})$ of scaling dimension $h_{\pi}=3-\Delta_{\pi}$ and, through the operator-state correspondence, to a state $|h_{\pi}\rangle$. The two-point correlator of the field boundary field $L_{\pi}$ can be written as

[TABLE]

where the last inequality comes from reflection positivity. On the cylinder we can write

[TABLE]

From this expression we deduce the general rules

[TABLE]

They imply

[TABLE]

so that

[TABLE]

With this expression we can now associate to the vector field in the bulk $A^{\mu}(\vec{x},\eta)$ of conformal dimension $\Delta$ a state on the boundary $|L^{i}\rangle$ for which the conformal dimension is $h=3-\Delta$ and

[TABLE]

We consider now the scalar descendent $P_{i}|L^{i}\rangle$ of the primary vector state $|L^{i}\rangle$. Since $P_{i}|L^{i}\rangle=-\vec{P}^{2}|\Delta_{\pi}\rangle$, if we impose

[TABLE]

then we get

[TABLE]

From reflection positivity of the primary helicity-0 state we therefore get the Higuchi bound $h<2$ or $\Delta>1$. Since

[TABLE]

the Higuchi bound for vectors leads to $m^{2}>0$. Making the necessary changes, we can repeat similar steps to arrive at the Higuchi bound for the massive spin-2 state $\Phi_{\mu\nu}$. By defining

[TABLE]

one can show that

[TABLE]

In the dS/CFT3 correspondence, a bulk field $\Phi_{\mu\nu}(\vec{x},\eta)$ that behaves near the boundary with a given scaling dimension $\Delta$ corresponds to a boundary field $S^{\mu\nu}(\vec{x})$ of dimension $h=3-\Delta$ and coupling

[TABLE]

and one can compute

[TABLE]

From this expression, knowing that $P_{i}P_{j}|S^{ij}\rangle=\vec{P}^{4}|h_{\pi}\rangle$, and writing

[TABLE]

after a lenghty, but straightforward calculation we find

[TABLE]

This leads to $h<2$ or $\Delta>1$. Since

[TABLE]

one finally gets

[TABLE]

In the next subsection we derive the Higuchi bound in an alternative, and maybe more physically intuitive, manner.

4.2 Second derivation

Let us again first consider the simplest case of vector field $A_{\mu}$. The condition $\nabla^{\mu}A_{\mu}=0$ leads on super-Hubble scales to the condition

[TABLE]

On the other side, the operator-state correspondence leads to

[TABLE]

If $A_{i}(\vec{x},\eta)$ has scaling dimension $\Delta$, then in the dS/CFT3 correspondence such a bulk field corresponds to the boundary field $L^{i}$ of dimension $h=3-\Delta$. Correspondingly we obtain the following relation

[TABLE]

Using the conformal algebra in Eqs. (2.17)-(2.17), the fact that in radial quantization $P_{i}^{\dagger}=K_{i}$, the transformations of rank-1 tensors $T^{i}$ under rotations

[TABLE]

and the fact that $|L^{i}\rangle$ is a primary

[TABLE]

we can now evaluate the norm of the vector $|L^{\eta}\rangle$

[TABLE]

Imposing that the norm is positive we get again $\Delta>1$ (or $h<2$), which leads to the Higuchi bound.

For the spin-2 state the procedure is similar. The equation of motion for the spin $2$ field $\Phi_{\mu\nu}$ in de Sitter is

[TABLE]

where $\Phi_{\mu\nu}$ is subject to the conditions

[TABLE]

The equation of motion of the helicity-0 part $\Phi_{\eta\eta}$ on super-Hubble scales is

[TABLE]

From this expression we extract that leading time behaviour $\Phi_{\eta\eta}(\vec{x},\eta)\sim(-\eta)^{\Delta}\Phi_{\eta\eta}(\vec{x})$. The conditions (4.25) impose

[TABLE]

from which we deduce

[TABLE]

where $\hat{\Phi}_{ij}$ stands for the traceless and transverse-free part of ${\Phi}_{ij}$

[TABLE]

Conformal operators are in one-to-one correspondence with the states of the conformal field theory,

[TABLE]

Correspondingly we obtain the following relation

[TABLE]

Using again the conformal algebra in Eqs. (2.17)-(2.17) and the transformations of rank-2 tensors $T^{ij}$ under rotations

[TABLE]

and the fact that $|\hat{S}^{ij}\rangle$ is a primary

[TABLE]

we can now evaluate the norm of the vector $|S^{\eta\eta}\rangle$

[TABLE]

where we have recalled that conformal dimension of a spin-2 field is defined with half indices up and half down [41] and therefore raising up indices increases the corresponding scaling dimension by a factor $+2$. Imposing that the norm is positive ($\Delta>0$ is always satisfied) we get again $\Delta>1$ (or $h<2$), which leads to the Higuchi bound.

One can generalize this result to any spin-$s$ state by requiring that the descendants of the corresponding $s^{\rm th}$-rank symmetric traceless tensor have positive norm

[TABLE]

Taking $m=s$, we get again the bound $h<2$ or $\Delta>1$ which, from Eq. (2.51), delivers

[TABLE]

The origin of the Higuchi bound can therefore be interpreted from the state-operator correspondence in radial quantization in the following way. On the CFT3 side of the dS/CFT correspondence the various helicities of the vector correspond to states living on a boundary. If the Higuchi bound is violated a descendant of these states acquires a negative norm. The Higuchi bound assures that the ground state having the smallest energy has finite energy.

5 The limit of enhanced symmetry and partial masslessness

As we have seen in the previous section, the case $\Delta=1$ is special for both spin-1 and spin-2 states. For $m^{2}=0$, extra gauge invariance is acquired for vectors and only the helicity-1 modes are physical. For spin-2 states with squared mass $m^{2}=2H^{2}$, the field becomes partially massless, and the number of propagating degrees of freedom becomes four.

While in Minkowski space massless fields of spin-$s$ state have helicities $\pm s$ and massive ones have all helicities running from $-s$ to $s$, in de Sitter space-time there exist “partially massless” fields [42] with mass

[TABLE]

Helicities range from $-s$ to $s$ with the helicities $-n,-n+1,\dots n$ removed for all $n\leq s-2$. Such fields are symmetric tensors $\Phi_{\mu_{1}\cdots\mu_{n}}$ and the linear action is invariant under the transformation

[TABLE]

where the dots stand for terms obtained both by symmetrizing the indices and adding terms with fewer than $s-n-1$ derivatives. To the best of our knowledge, in the literature there is no proof that the gauge-invariance of the partially massless field can be maintained exactly beyond linear order (see for instance [43, 44] for recent discussions on this matter).

Vectors in de Sitter do not have partial massless states. However, enhanced symmetry is acquired for $m^{2}=0$. Spin-2 states have $n=0$ and there exists one partial massless degree of freedom with mass $m^{2}=2H^{2}$. In both cases the corresponding scaling dimension is $\Delta=1$.

It is interesting to see what the special case $\Delta=1$ corresponds to on the dual theory in terms of states of spin 1 and 2. We discuss first the case of a vector $A_{\mu}$, which is maybe less interesting than the case of the massive spin-2 particle, but is technically less involved. Let us take it massless to start with. The action is

[TABLE]

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. We can work in the gauge $A_{0}$=0 and let us check that the action is conformally invariant for the special value $\Delta=1$ of the conformal weight of the fields $A_{i}$. We should first recall that a vector $A_{i}$ with scaling dimension $\Delta$ in $D$-dimensions transforms under the conformal group as $A_{i}(x)\to A_{i}^{\prime}(x^{\prime})$ where

[TABLE]

Under dilations we have

[TABLE]

For inversions one gets

[TABLE]

where one had made use of $\partial_{i}=\vec{x}^{\prime 2}\,J^{j}_{\,i}\partial^{\prime}_{j}$, with $J^{j}_{\,i}=(\delta^{j}_{\,i}-2x^{i}x_{j}/\vec{x}^{2})$ (recall that special conformal transformations are obtained by a subsequent operations of inversion, translation and inversion).

The action for the gauge field expressed for the de Sitter metric in conformal time becomes

[TABLE]

The action can only be invariant under inversions if the dimension of the vector $A_{i}$ is $\Delta=1$. Let us check it. Under inversions, the vector $A_{i}$ transforms as in Eq. (5.4). This is a coordinate transformation, but with the extra term $J^{(1-\Delta)/D}$ where J=\Big{|}\det(\partial{x^{\prime}}^{j}/\partial{x}^{i})\Big{|}. When transforming $f_{ij}=(\partial_{i}A_{j}-\partial_{j}A_{i})$, there will be cross terms of the form $A_{j}\partial_{i}J$ which cannot be canceled unless the $J$ factor is missing and this imposes $\Delta=1$. Then, by using that

[TABLE]

and the orthogonality relation $J^{i}_{\,j}J^{j}_{\,k}=\delta^{i}_{\,k}$, the action is invariant under inversion

[TABLE]

Let us now find the result $\Delta=1$ in a more convoluted manner, but useful for what we wish to obtain later on. We follow Ref. [41] here. We assume that the vector $A_{\mu}$ has a non-vanishing mass $m$ and action

[TABLE]

In the limit of $m=0$ the action acquires an extra gauge invariance

[TABLE]

Through the dS/CFT3 correspondence, we know that a massless field of spin $s$ corresponds on the boundary to a rank-$s$ conserved symmetric tensor. This means that in the limit $m=0$ the massless vector must corresponds on the boundary to a partially conserved current $L^{\mu}$. Indeed, since the coupling

[TABLE]

should be gauge-invariant, the partial conservation must have the form

[TABLE]

Furthermore, on the boundary $\eta=0$, the vector field behaves like

[TABLE]

In the dS/CFT3 correspondence, such a bulk field corresponds to the boundary field $L^{i}$ of dimension $h=3-\Delta$. Via the operator-state correspondence of CFT, the vector $L^{i}$ corresponds to a state $|L^{i}\rangle$ which must be a highest weight vector for the conformal group. Indeed, Eq. (5.13) in flat space reduces to $\partial_{i}L^{i}=0$. This means that the first descendent of $|L^{i}\rangle$, that is $P_{i}|L^{i}\rangle$, must a null vector. This can occur only for a particular conformal dimension of $L^{i}$. Since conformal invariance is obtained for $\Delta=1$, one should then recover $h=2$. Let us check that this is the case. From

[TABLE]

and the fact that $|{L}^{i}\rangle$ is a primary

[TABLE]

we get

[TABLE]

which vanishes for $h=2$ as it should.

Similar to what discussed for the massless vector state, for the spin-2 case, from general arguments we know that a partially massless field must correspond in the boundary theory to a partially conserved tensor $L^{\mu\nu}$. Indeed, since the coupling (4.12) should be invariant under the extra gauge symmetry

[TABLE]

then

[TABLE]

Through the operator-state correspondence the field $L^{\mu\nu}$ corresponds to a state $|S^{\mu\nu}\rangle$ which must be a highest weight vector for the conformal group. The partially conservation of $S^{\mu\nu}$ implies that a certain level descendant of $|\hat{S}^{ij}\rangle$ is a null vector. From Eq. (4.34) we see that this is achieved for $h=2$ or $\Delta=1$ which correctly corresponds to the state $|S^{\eta\eta}\rangle$ being null.

5.1 Generalization to higher spins

What described above can be generalized to any spin-$s$ state by requiring that the descendants of the corresponding $s^{\rm th}$-rank symmetric traceless tensor have vanishing norm, corresponding to the fact that is $\hat{S}^{i_{1}i_{2}\cdots i_{s}}$ is partially conserved [41]

[TABLE]

Imposing that this norm vanishes for $m=r$, but is nonvanishing for $m<r$, so that

[TABLE]

then $h=2+s-r$. Such state corresponds in the four-dimensional de Sitter space to a partially massless field with a range of helicities missing, depending on $r$.

Notice that for $s>2$ there is always a partial massless state for which $\Delta=0$. Indeed, being

[TABLE]

the dominant scaling dimension at $\eta=0$, the scaling dimensions of the partial massless states become

[TABLE]

For

[TABLE]

one gets $\Delta=0$ and since $n\leq s-2$, such state for which $n=1$ always exists. For $s=3$, there are two partially massless states, one for $n=1$ and $\Delta=0$ and the other for $n=0$ and $\Delta=1$. They correspond in Eq. (LABEL:mji) to $m=s-n=2$ and $m=s-n=3$, respectively. For $m=r=2$, one has $h=2+3-2=3$, corresponding to $\Delta=0$; for $m=r=3$, one has $h=2+3-3=2$, corresponding to $\Delta=1$. These values correctly reproduce the conformal weights of the partially massless states.

Partially massless states with $\Delta=0$ might be relevant during inflation because these states will not decay on super-Hubble scales. It remains to be seen if these partial massless states survive beyond linear order.

6 Consistency relations from the dS/CFT3 correspondence, radial quantization and asymptotic symmetries

Inflationary consistency relations have attracted a lot of attention since, if violated, they would rule out single-field models of inflation. There exist both scalar consistency relations relating the squeezed limit of the $(n+1)$-point correlators to the $n$-point correlators of scalar perturbations and tensor consistency relations involving tensor and scalar modes [2] (see also [3]). Recently, it has been shown that the consistency relations are in close connection with the asymptotic symmetries of de Sitter space [34, 35] since soft degrees of freedom produced by the expansion of de Sitter can be interpreted as the Nambu-Goldstone bosons of spontaneously broken asymptotic symmetries of the de Sitter spacetime.

We will see that the radial quantization allows to identify the charges generating these asymptotic symmetries with the topological charges of the CFT3. For instance, in the case of the scalar consistency relations we will see that the corresponding charge is nothing else than the Hamiltonian and that the action of the charge simply evolves the states on the cylinder forward (or backward) in “time”.

Let us see first how the consistency relations arise in the CFT3 using radial quantization.

6.1 Scalar consistency relations

Suppose that inflation has generated a long mode $\zeta_{L}$ (with wavelength larger than the Hubble radius) for the comoving curvature perturbation such that the perturbed metric reads

[TABLE]

Under a dilation symmetry $\eta\to\lambda\eta$ and $\vec{x}\to\vec{x}^{\prime}=\lambda\vec{x}$, the long mode transforms non-linearly as a Nambu-Goldstone mode

[TABLE]

The constant zero mode of the curvature perturbation can be removed (or generated) by simply choosing $\lambda=\zeta_{L}$. Furthermore, one can approximate the effect of such a constant long-wavelength mode on an $n$-point function as a rescaling of the coordinates

[TABLE]

This argument implies that in that case the squeezed limit of the $(n+1)$-point function would be

[TABLE]

where we have indicated by $P_{\zeta}$ the power spectrum and primes indicate we have removed $\pi$’s and Dirac delta functions. For $n=2$, the relation above provides the famous Maldacena’s consistency relation for the three-point correlator of the comoving curvature perturbation in the squeezed limit stating that its size is proportional to the deviation of the two-point function from scale invariance and therefore proportional to the slow-roll parameters.

Similar arguments lead to the so called conformal consistency relation [12] where the long mode of the curvature perturbation in the metric can be removed not only at the level of the constant zero mode, but also at its first gradient. This is achieved simply by a special conformal transformations

[TABLE]

which can be neutralized by transforming the long mode $\zeta_{L}$ as

[TABLE]

and taking $\vec{b}=-1/2\vec{\nabla}\zeta_{L}$. Consequently, the effect a constant long-wavelength gradient mode acts on the $n$-point function as a rescaling of the coordinates (6.5)

[TABLE]

which in momentum space becomes

[TABLE]

6.2 Scalar consistency relations from the dS/CFT3 correspondence and radial quantization

The fact that the de Sitter isometry SO(1,4) group acts as conformal group CFT3 when the fluctuations are on super-Hubble scales allows a simple interpretation of the consistency relations in radial quantization on the cylinder.

Let us make an infinitesimal conformal transformation of the coordinates on the cylinder $y^{\mu}\to y^{\mu}+\xi^{\mu}(y)$, where we identify $y^{0}=\tau$ and $y^{i}$ with the angular coordinates ($i=1,2$). In particular, we consider a sphere which encloses all the points at which we wish to evaluate the correlators and such that the transformation is conformal within the sphere, and the identity outside it. This gives rise to an (infinitesimal) discontinuity on the surface of the sphere $\Sigma$, and, at least classically, to a modification of the action $S$ according to (after integrating by parts)

[TABLE]

where the integral is on the area of the surface of the sphere. This change is balanced by the explicit change in the correlation function under the conformal transformation

[TABLE]

In particular, if we perform a dilation transformation $r\to e^{\lambda}r$, this corresponds to a shift in the cylinder coordinate $\tau\to\tau+\lambda$, that is $\xi^{0}=\lambda$ and $\xi^{i}=0$. Therefore

[TABLE]

where in the last passage we have introduced the topological surface operator

[TABLE]

for which

[TABLE]

Ward identities guarantee that the correlator of $P^{\mu}(\Sigma)$ with other operators is unchanged as we move the surface, as long as it does not cross any operator insertions. Indeed, from the Ward identity

[TABLE]

integrating over the boundary of a ball containing, say $\vec{x}_{1}$ and no other insertions, one concludes that the results are independent from the surface.

Since $\langle P^{0}(\Sigma)\phi(\vec{x})\cdots\rangle=\partial_{\tau}\langle\phi(\vec{x})\cdots\rangle$, identifying the scalar field with the comoving curvature perturbation $\zeta$ and $\lambda$ with its long mode $\zeta_{L}$, going to Fourier space one finds the expression (6.4). This operation amounts to leaving the decoupling limit in which gravity is not dynamical. Adding gravity, things may be more transparent in the $\zeta$-gauge where the inflaton field driving inflation is unperturbed and the time slicing is fixed. In such a case, and since we wish to obtain for the time being only conformal rescaling of the spatial part of the metric, transformations of the three-dimensional conformal group SO(1,4) for every fixed time can be performed. When gravity is switched off, SO(1,4) is a non-linearly realized symmetry of the action in de Sitter, in the presence of gravity SO(1,4) is the symmetry group of CFT3.

Since the long wavelength states $\zeta_{L}$ populate the future boundary of de Sitter where the CFT3 is living, we can interpret these scalars as the Nambu-Goldstone bosons of spontaneously broken asymptotic symmetries of the de Sitter spacetime. These scalars are physical adiabatic modes [45]. The associated charge [35] has therefore a nice interpretation in the framework of radial quantization: it is nothing else that the Hamiltonian which in radial quantization is associated with the dilation operator.

Acting with this charge on the state creates a new state equivalent to a change in the local coordinates induced by the soft scalar and adding a constant long wavelength mode results in evolving the states forward (or backward) in “time” along the cylinder through the Hamiltonian. This charge operates on states by evolving them to other states

[TABLE]

in a way that correlators feel the evolution only through scaling dimensions. By expanding in powers of small $\zeta_{L}$ one finds the standard result that the dependence of the short modes on the long mode is proportional to the scaling dimension of $\zeta$. This scaling dimension differs from zero only through the slow-roll parameters.

6.3 Another perspective

All considerations above indeed follow from the basic property that correlation functions on conformally flat backgrounds in CFT3 can be calculated by rescaling the flat space correlation functions

[TABLE]

This on the cylinder becomes

[TABLE]

Let us demonstrate the rule (6.16) by computing the two point function on the cylinder starting from a simple dilation $r\to e^{\lambda}r$.

This rescaling corresponds to a time translation $\tau\to\tau+\lambda$ on the cylinder and that the dilation operator displaces points along the “time” direction on the cylinder. Consider the two-point function of the curvature perturbation $\zeta(\vec{x})$ of scaling dimension $\Delta_{\zeta}$

[TABLE]

In the Hamiltonian formulation we can then write

[TABLE]

Since the Hamiltonian is the dilation operator we can also write

[TABLE]

The dilation $r\to r^{\prime}=e^{\lambda}r$ does not have an impact on the cylinder (remember that the dilation operator leaves the vacuum invariant) and one can therefore write

[TABLE]

which is the relation (6.16) with $\Omega(\vec{x})=\lambda$. Here we have also made use of the basic property that a dilation transformation does not change the vectors $\vec{n}_{1}$ and $\vec{n}_{2}$ since the two-point correlator depends only on such an angle, we have had the freedom to replace the vectors $\vec{n}_{i}$ with the vectors $\vec{n}_{i}$. Appendix B offers an alternative way to prove that that correlation functions on conformally flat backgrounds can be calculated by rescaling the flat space correlation functions.

Similar considerations hold for the conformal consistency relation in which one removes the constant gradient of $\zeta_{L}$ for which Eq. (6.16) holds for $\Omega(\vec{x})=-2\vec{b}\cdot\vec{x}=\vec{\nabla}\zeta_{L}\cdot\vec{x}$. Here one has to make again use of the basic property that the special conformal transformation does not change the angle between the vectors $\vec{n}_{1}$ and $\vec{n}_{2}$ and since the two-point correlator depends only on such an angle, there is the freedom to replace the vectors $\vec{n}^{\prime}_{i}$ with the vectors $\vec{n}_{i}$.

6.4 Tensor consistency relation

As for the consistency relations involving tensors, one can generate a long wavelength tensor mode by the transformations of the coordinates

[TABLE]

where $\gamma_{ij}$ is the traceless transverse tensor mode. Considering the long mode as a background if seen on small scales, one can write the relation

[TABLE]

Expanding at linear order in the long tensor mode and going to momentum space one finds

[TABLE]

where we have introduced the polarization vectors by $\gamma^{ij}_{\vec{q}}=\sum_{s=1,2}\epsilon^{s}_{ij}\gamma^{s}_{\vec{q}}$, $P_{\gamma}$ is the power spectrum of the tensor mode and the dots stand for terms sub-leading in $\Delta_{\zeta}$. Notice that the tensor consistency relation is then a consequence of the fact that the partition function in the CFT3 side is invariant under diffeomorphism. The transformation generating the long tensor mode cannot be reproduced by a conformal transformations since the metric $\delta_{ij}{\rm d}x^{i}{\rm d}x^{j}$ on ${\mathbb{R}}^{3}$ transforms into $(\delta_{ij}+\gamma_{ij}){\rm d}x^{i}{\rm d}x^{j}$ and this is not of the form dictated by conformal transformations $\delta_{ij}{\rm d}x^{i}{\rm d}x^{j}\to e^{2\Omega(\vec{x})}\delta_{ij}{\rm d}x^{i}{\rm d}x^{j}$. This is the reason why the tensor consistency relation is not suppressed by deviation from de Sitter.

6.5 Tensor consistency relation from the dS/CFT3 correspondence and radial quantization

Let us reproduce the tensor consistency relation starting from the CFT3 and radial quantization. We follow the procedure of the previous subsection, with the appropriate differences. First, we locate the operators $\zeta(\vec{x})$ close to the origin and consider the long tensor mode constant on a sphere surrounding such operators. As we mentioned already, under a general infinitesimal non-conformal coordinate transformation $x^{i}\to x^{i}+\xi^{i}(\vec{x})$, the action response has the form

[TABLE]

where $T_{ij}$ is the stress-energy momentum tensor. Since the correlation functions involving $\zeta(\vec{x})$ with respect to the original action are equal to those of $\zeta(\vec{x})+\delta\zeta(\vec{x})$ with respect to the modified action, one finds close to the origin and sphere of radius $\epsilon$

[TABLE]

where the integration is over the complement of the sphere of radius $\epsilon$ surrounding the operator at the origin. By integrating by parts and using the conservation of energy one finds

[TABLE]

where $\Sigma_{\epsilon}$ is the surface surrounding the sphere. We find

[TABLE]

where we have used again the topological operator (6.12) and the fact that the result is independent of the surface as long as we do not cross any operator. Since $\langle P_{i}(\Sigma)\zeta(\vec{x})\cdots\rangle=\partial_{i}\langle\zeta(\vec{x})\cdots\rangle$ we see that the operator $\zeta(0)$ creating the state $|\Delta_{\zeta}\rangle$ gets shifted by an amount $1/2\gamma^{ij}_{L}x_{j}\partial_{i}\zeta(\vec{x})|0\rangle$ evaluated at the origin. Going to momentum-space one thus recovers the consistency relation (6.24).

In agreement with Ref. [34], we see that soft gravitons produced by the de Sitter expansion can be viewed as the Nambu-Goldstone bosons of spontaneously broken asymptotic symmetries of the de Sitter spacetime. The corresponding charge is the topological operator

[TABLE]

We conclude that asymptotic symmetries are generated by the topological charges of CFT3.

7 Anisotropic de Sitter

In this section we consider the case of anisotropic inflation, see for instance Refs. [46, 47, 48] and [49] for a review. The spacetime can be approximated by a de Sitter expansion with different Hubble rates along different directions. Our goal is to obtain some general results based on symmetry arguments and to find a dual interpretation of them. The first step is to study the corresponding isometries.

7.1 The isometries of anisotropic de Sitter

Let us consider a metric of the form (we use cosmic time)

[TABLE]

parametrizing an anisotropic de Sitter expansion with unequal expansion rates $H_{i}$ along the three cartesian axes. The isometries are transformations of the form

[TABLE]

which leave the metric invariant. The infinitesimal functions $\xi^{\mu}(x)$ are the solutions of the Killing equation

[TABLE]

being $\nabla^{\mu}$ the covariant derivative. Recasting Eq. (7.3) under the form

[TABLE]

we find the following set of equations

[TABLE]

As one can imagine, the isometries are much less rich than in the isotropic de Sitter case. We loose of course the three-dimensional isotropy and the special conformal symmetry, but we keep

three translations with Killing vectors

[TABLE] 2. 2.

dilations $\mathscr{D}$ with Killing vectors

[TABLE]

These dilational Killing vectors are the infinitesimal form of the finite dilational symmetry,

[TABLE]

where we have defined the average expansion rate $H$ and the corresponding rate deviation $\epsilon_{i}$ from isotropy

[TABLE]

Dilations will play a crucial role in what follows. For convenience, we write the main equations using the conformal time

[TABLE]

defined with respect to the isotropic Hubble rate $H$. The metric (7.1) becomes

[TABLE]

which is invariant under the dilation transformations

[TABLE]

7.2 Correlators of free fields in anisotropic de Sitter

We investigate here the simplest case of a free massive spectator field $\sigma(\vec{x},\eta)$. The action

[TABLE]

is manifestly invariant under the transformations (7.12) if the scalar field transforms as a scalar. Let us reparametrize these transformations as

[TABLE]

and let us see what the invariance implies. We write down the $n$-point correlator of the field $\sigma$. The variation of the latter is

[TABLE]

Going to momentum space we obtain

[TABLE]

Integrating by parts and imposing that the variation is vanishing we get

[TABLE]

where we have used the fact that $C_{ij}k_{ai}\partial_{k_{aj}}\delta^{(3)}(\vec{k}_{1}+\cdots+\vec{k}_{n})=0$, the last of Eq. (7.9) and we have introduced the scaling dimension of the scalar field. Finally, Eq. (LABEL:long) becomes

[TABLE]

At this point we should stress that the momenta involved in this relation are those with indices down, which do not depend on time in anisotropic backgrounds. For the case of the two-point correlator, one can easily check that the solution is

[TABLE]

where $\hat{u}_{i}$ are the unit vectors of the three different axes. We conclude that the invariance under dilations imply the presence of anisotropy in the power spectrum (as well as in the higher correlation functions).

To make contact with the more standard way of expressing the anisotropic contribution to the power spectrum, we assume that the anisotropy direction is along a generic unit vector $\hat{n}$, which we can arbitrarily set to be the $x_{3}$-direction, and that in the orthogonal directions the expansion rate is the same. The metric (7.11) becomes

[TABLE]

The average expansion rate $H$ and deviation from isotropy $\epsilon_{H}$ become

[TABLE]

With these definitions we have

[TABLE]

and from Eq. (7.2) we get

[TABLE]

It is important to stress at this point that the momenta in these expressions are referring to the Fourier transform with respect to coordinates where rotational invariance holds. This means that at the end of inflation at time $\eta_{*}$, all the momenta have to be properly rescaled as the coordinates in the metric do not exhibit rotational invariance. In other words, we have to rescale the coordinates in the following way

[TABLE]

Performing this necessary rescaling and restoring the arbitrary anisotropy direction along $\hat{n}$, we finally get, denoting by $q=(-H\eta_{*})k$ the physical wavelength of the mode of interest at the end of inflation)

[TABLE]

Let us notice that we can cast the expression (7.2) in an alternative way by expanding the power spectrum to first-order in $\epsilon_{H}$ (we rename the physical momentum now as $k$)

[TABLE]

where $P_{2}$ is the $\ell=2$ Legendre polynomials. In Appendix C we check this result with a direct computation.

In this simple set-up the anisotropy is solely due to the anisotropic expansion of the universe. This was the case considered first in the so-called ACW model [46] where the anisotropic de Sitter expansion was due to a vector field $B_{\mu}$ which, by a Lagrange constraint, satisfies the relation $B_{\mu}B^{\mu}=m^{2}$ and has a vacuum expectation value along the $x_{3}$-direction. On the top there is an isotropic vacuum energy. The ACW model suffers of instabilities related to the negative energy of the longitudinal mode of the vector field, but we have considered it just for the sake of comparison.

We will start from expression (7.2) to elaborate our considerations about the correspondence between anisotropic four-dimensional de Sitter spacetimes and a three-dimensional boundary theory.

7.3 Anisotropic de Sitter and its three-dimensional dual perspective

When the de Sitter spacetime is anisotropic, the isometry of the metric is not any longer SO(1,4). Nevertheless, dilations are still isometries. As a consequence, the power spectrum and the higher correlator functions acquire an anisotropic dependence. We would like to show in this subsection that this result has a dual interpretation: the four-dimensional anisotropic de Sitter spacetime is in correspondence with an isotropic three-dimensional boundary enjoying dilation symmetry. In this picture, the angular anisotropic dependence of the spectrum can be attributed to a non-zero expectation value of the stress tensor $\langle T_{ij}\rangle$ of the three-dimensional dual theory. Our procedure follows the one nicely introduced by Cardy for anisotropic corrections to correlation functions in conformal systems [50].

Let us consider a three-dimensional theory enjoying dilation symmetry and recall that the stress tensor is determined by the response of the action under a general coordinate transformation. Indeed, under the transformation $x^{i}\to x^{i}+\alpha^{i}(\vec{x})$, the action $S$ changes as

[TABLE]

For dilations we have $\alpha^{i}=\lambda x^{i}$, and hence invariance of the action simply means that the stress tensor is traceless $T^{i}_{\,\,\,i}=0$. In fact, the stress tensor may be viewed as the generator of dilation transformations in the following sense. Consider a coordinate transformation $x^{i}\to x^{i}+\alpha^{i}(\vec{x})$. We assume that, with $\epsilon_{\rm in}<\epsilon_{\rm out}$, inside the ball $|\vec{x}|<\epsilon_{\rm in}$ this transformation is a dilation, i.e. $\alpha^{i}=\lambda x^{i}$, outside the ball $|\vec{x}|>\epsilon_{\rm out}$, $\alpha^{i}=0$, and in between $\epsilon_{\rm in}<|\vec{x}|<\epsilon_{\rm in}$, $\alpha^{i}$ is a general differentiable function.

This transformation induces a corresponding transformation to the action $S\to S+\delta S$ where, using Eq. (7.27) and the conservation of the stress tensor $\partial^{i}T_{ij}=0$,

[TABLE]

where ${\rm d}\Omega_{2}$ is the integral measure on the unit two-dimensional sphere. The surface integral at $|\vec{x}|=\epsilon_{\rm out}$ vanishes due to the fact that $\alpha^{i}=0$ by continuity there, and so we are left with

[TABLE]

We are now interested in the response to the transformation $x^{i}\to x^{i}+\alpha^{i}(\vec{x})$ of the correlator of a field $\pi(0)$ of scaling dimension $h$ with operators ${\cal O}_{1}(\vec{x}_{1}){\cal O}_{2}(\vec{x}_{2})\cdots{\cal O}(\vec{x}_{n})$ of arbitrary spin and dimension at $x_{a}^{i}$ ($a=1,\cdots,n$) with $|\vec{x}_{a}|>\epsilon_{\rm out}$. The boundary field $\pi(\vec{x})$ of scaling dimension $h=3-\Delta$ is supposed to be the dual of the free scalar bulk scalar field $\sigma$ with dimension $\Delta$.

We find that

[TABLE]

Since, only $\pi(0)$ transforms whereas all other fields ${\cal O}_{n}(\vec{x}_{n})$ do not change ($\alpha^{i}=0$ at $x_{n}^{i}$, and thus $\delta{\cal O}_{n}=0$), we get

[TABLE]

This form of the transformation and the fact that the stress tensor is symmetric, traceless and has scaling dimension three are enough to determine part of the operator product expansion (OPE) of $T_{ij}(x)$ with $\pi(0)$. The latter will contain among others the term

[TABLE]

where

[TABLE]

Since, the dimension of $\pi$ is $h$, we have for a dilation $\alpha^{i}=\lambda x^{i}$,

[TABLE]

On the other hand, using the OPE of Eq. (7.32) in Eq. (7.31), we find

[TABLE]

from which it follows

[TABLE]

In order to find the induced angular dependence of the spectrum due to a non-zero vacuum expectation value of the stress tensor, we should consider the contribution of the latter to the OPE of the field $\pi$ with itself. Since, the dimensions of $\pi$ and $T_{ij}$ are $h$ and three, respectively, the OPE of a scalar field $\pi(\vec{x})$ with itself will have the form

[TABLE]

where

[TABLE]

The parameter $c_{\sigma}$ is not arbitrary, but it is specified by $c_{1}$ and the central charge $c_{\rm T}$ defined in the two-function of the energy-momentum tensor

[TABLE]

where

[TABLE]

In addition, the three-function $\left\langle T_{ij}\pi\pi\right\rangle$ is given by

[TABLE]

where $x_{ij}=|\vec{x}_{i}-\vec{x}_{j}|$. For $\vec{x}_{3}=0$, $\vec{x}_{1}=\vec{x}$ and $\vec{x}_{2}=\vec{y}$ with $|\vec{y}|\to 0$, the leading behaviour of Eq. (7.33) turns out to be

[TABLE]

which should be equal, after using Eq. (7.37), to

[TABLE]

By comparing, Eqs. (7.43), (7.42) and (7.39) one finally finds [50]

[TABLE]

The OPE is may be written in momentum space as

[TABLE]

and therefore

[TABLE]

For an anisotropy along a direction specified by the unit vector $\vec{n}$, the only possibility for the expectation value of the energy-momentum tensor is

[TABLE]

where $A$ is a constant. Since

[TABLE]

we may write Eq. (7.46) as

[TABLE]

with

[TABLE]

Eq. (7.49) reproduces correctly the angular dependence of the spectrum found in Eq. (7.2). In addition, the function $\tilde{C}(k)$ can be found by solving the Ward identity corresponding to the dilation symmetry

[TABLE]

As a consequence $\widetilde{C}(k)$ satisfies the equation

[TABLE]

which has the solution

[TABLE]

In the dual picture the field $\pi$ is a dual operator in the putative three-dimensional dual theory of the field $\sigma$. In the wavefunction of the universe approach [51] and in the Gaussian approximation this means that such a wavefunction is

[TABLE]

This means that the two-point correlator of the bulk field $\sigma$ is given by

[TABLE]

where we have used the relation $h=3-\Delta$ and expanded for small anisotropies. This result exactly matches the result in Eq. (7.2).

We can also examine the case in which the anisotropy is generated by the vacuum expectation value of a generic spin $\ell$ operator ${\cal{O}}_{i_{1}i_{2}\cdots i_{\ell}}$. Indeed, let us recall that the OPE of a scalar field takes the form [52]

[TABLE]

where $\Phi(a;b;z)$ is the confluent hypergeometric function and ${\cal{O}}_{i_{1}i_{2}\cdots i_{\ell}}$ is a spin $\ell$ operator (i.e., symmetric and traceless). By using the expansion

[TABLE]

we find that the leading contribution of the spin $\ell$ operators to the two-function in momentum space will take the general form

[TABLE]

In an anisotropic background specified by a unit vector $n^{i}$, the expectation value of a spin $\ell$ operators can take the form

[TABLE]

where $A_{\ell}$ is a constant and \big{[}n_{i_{1}}\cdots n_{i_{n}}\big{]} denotes the traceless symmetric part of the polynomial $n_{i_{1}}\cdots n_{i_{n}}$. For example,

[TABLE]

and so on. Using the relation

[TABLE]

where $P_{\ell}$ are the Legendre polynomials, we get that the two-function can be written as

[TABLE]

Using again the Ward identity we find that $\widetilde{C}_{\ell}(k)$ satisfies the equation

[TABLE]

A solution to this equation exists only for $\ell=2$, which corresponds to a non-zero vacuum expectation value of the stress tensor as discussed above. In other words, the only vacuum expectation value that it is consistent with scale invariance in the anisotropic de Sitter spacetime is the one of a spin $\ell=2$ operator.

8 Conclusions

In this paper we have discussed several topics related to inflation and de Sitter spacetimes from a three-dimensional perspective by making use of the properties of radially quantized CFT3. Despite the fact that some of the results, e.g. on the Higuchi bound, are not totally new, we have offered new interpretations and tools which might turn out to be useful in addressing other topics.

One interesting question we are investigating is if, under some assumptions, the convergence of the operator product expansion and conformal block decomposition in unitary CFT3 [53] may deliver an upper bound on four-point correlators. This will be quite exciting because it will have a strong impact on the next future experimental efforts of detecting primordial four-point functions.

Another issue worth understanding is the following. As already pointed out by in Ref. [9], in the dS/CFT correspondence the dual CFT3 may be non-unitary. This happens when there are sufficiently massive stable scalars which correspond to complex conformal weights. While in inflation we are mainly interested in very light degrees of freedom and this issue might not represent a real problem, it represents nevertheless a strong motivation to study the dS/CFT3 in more detail.

Unitarity, or reflection positivity, of the CFT3 is the property we made use of to derive the Higuchi bound. Since the bound corresponds to real scaling dimensions the derivation is consistent. In AdS/CFT there Lorentzian theories on the boundary and in the bulk. There is time evolution and this implies that if one theory is unitary, the same must be true for the other. In dS/CFT3 the equivalence is between an Euclidean theory (future boundary) and a Lorentzian theory (bulk). No “real” time evolution in the Euclidean theory is present and so there is no right to require unitary a priori. Nevertheless, if $\mathscr{H}_{\rm cyl}$ has arbitrarily large and negative eigenvalues, then the Hamiltonian is unbounded and one is obliged to consider states that have no overlap with arbitrarily high-energy eigenstates.

Finally, it will be interesting to investigate more in details the role played by symmetries in anisotropic inflation. We hope to come back to these issues soon.

Acknowledgments

We thank M. Sloth and J. Sonner for comments. A.R. is supported by the Swiss National Science Foundation (SNSF), project Investigating the Nature of Dark Matter, project number: 200020-159223.

Appendix A An alternative derivation of the relation between mass and scaling dimension for scalar fields

Another, maybe more intuitive way to obtain the relation (2.40) is to consider a massive scalar field with action

[TABLE]

where $a(\eta)=-1/H\eta$. By integrating by parts one obtains

[TABLE]

By posing $\phi(\eta,\vec{x})=\eta^{\Delta}\varphi(\vec{x})$, the action becomes

[TABLE]

If the coefficient of the first piece vanishes one is left with a free massless scalar field and therefore with a conformal field theory on ${\mathbb{R}}^{3}$. This corresponds precisely to the condition (2.40).

Appendix B Calculation of CFT correlation functions through the state-operator correspondence

We present here an alternative proof that correlation functions on conformally flat backgrounds can be calculated by rescaling the flat space correlation functions. It makes use of the state-operator correspondence. One has

[TABLE]

and

[TABLE]

The two-point correlator of a scalar field with scaling dimension $\Delta_{\zeta}$ becomes

[TABLE]

where

[TABLE]

and we have taken into account that all the cross-terms with different powers of $\vec{K}$ and $\vec{P}$ give vanishing matrix elements as states with different energies are orthogonal. For $m=1$ for instance, we have

[TABLE]

and similarly for the higher terms. The important point though is that a rescaling $r\to e^{\lambda}r$ does not alter neither the scalar products nor the ratio $(r_{1}/r_{2})^{\Delta_{\zeta}+m}$. It appears only on the overall terms $(r_{1}r_{2})^{-\Delta_{\zeta}}$.

Appendix C Anisotropy in the power spectrum: direct computation

We reobtain the generic result (7.2) with a more explicit computation. Consider a free massless scalar field $\sigma$ (therefore from now $\Delta=0$) in the background metric

[TABLE]

After redefining the scalar field $\sigma$ as

[TABLE]

we find that the equation of motion for the Fourier modes $\chi_{\vec{k}}(\eta)$ turns out to be (primes indicate derivatives with respect to the conformal time)

[TABLE]

where $k_{\perp}^{2}=k_{1}^{2}+k_{2}^{2}$. Expanding in $\epsilon_{H}$, we find that to first order we have

[TABLE]

where $k^{2}=k_{1}^{2}+k_{2}^{2}+k_{3}^{2}$. Looking for solutions of the form

[TABLE]

we find that $\chi_{\vec{k}}^{(0)}(\eta)$ satisfies the standard equation in de Sitter

[TABLE]

and the appropriately normalized solution is

[TABLE]

The field $\chi_{k}^{(1)}(\tau)$ satisfies

[TABLE]

whose solution is

[TABLE]

where $E_{i}(x)=-\int_{-x}^{\infty}e^{-t}{\rm d}t/t$ is exponential integral. In the limit $-k\eta\ll 1$, the solution is given in the leading $\ln(k/H)$ order

[TABLE]

The final power spectrum is therefore

[TABLE]

which coincides with the result (7.2) obtained by symmetry arguments.