§ INTRODUCTION

The interplay between semiclassical gravity and the quantum effects of Quantum Field Theory (QFT) is a topic of research that has been in the spotlight for several decades.
The most important area of applicability motivating these issues arise from cosmology. In cosmology,  we treat gravity as semiclassical[Exceptions exist where gravity  is treated in perturbation theory, [34] or where the curvature is very strong and a more fundamental theory (an example is string theory) can take over.]
and we couple it with QFTs. The class of semiclassical metrics relevant in this case is cosmological metrics, with the most prominent example being the maximally symmetric cosmological metric, ie, de Sitter.

Other contexts also are relevant, like semiclassical effective actions of string theory with asymptotically AdS or flat asymptotics.  In this context, quantum effects in gravity and matter seem to go hand in hand as they are controlled by the same underlying parameter, the string coupling constant.  However, we understand that there are two types of quantum effects associated with string loops or the $\a'$ expansion, although their separation is “duality-frame" dependent.
There are however limits, (known as double scaling limits) in which gravitational quantum effects can be made subleading to “matter" quantum effects.
In the context of holography, relevant for AdS or asymptotically AdS spaces typically (bulk) gravity and matter are treated semiclassically, but subleading corrections in N
involve quantum effects in both sectors.

The case of QFTs on de Sitter space is an especially hot issue, as we believe that our universe has been near de Sitter at least twice during its history.
Defining a QFT on de Sitter space and, in particular,
answering questions about QFT backreaction on the geometry is a
subtle issue. Perturbative field quantization (and
renormalization) on fixed classical curved space-times is text-book material [1].
However, answering concrete questions about the observable effects of quantum
fields backreacting on classical geometry is not
straightforward.  This concerns particularly theories which are gapless
in the infrared, due to the presence of infrared divergences (and in most cases strong IR dynamics).

It is well known that in classical GR, both de Sitter space and Flat Minkowski space are non-linearly stable, [2]-[6].
However, in the presence of quantum effects from matter, instabilities can appear. For the case of flat space this was established in [7]-[17].

In (quasi) de Sitter space, there are several other issues arising when one considers QFTs.
In  [18]-[21] a divergence of scalar correlators was observed at large times.
This was addressed in [22, 23, 24] using a stochastic approach. The case of interacting massive scalar fields has been treated thoroughly more recently in [25, 26].  A systematic approach to compute corrections in the massless case is lacking and the problem remains still open.

The accumulation of long wavelength fluctuations in an expanding universe
is another issue that has been studied,  starting with references [27, 28]. Their analysis was extended further in [29].
Another issue concerned the fact that the two-point function of a massless scalar in de Sitter space had to break de Sitter symmetry, due to the presence of a zero mode, [30, 31]. This issue is of a different nature and is more similar to the fact that in two dimensions, a massless scalar is IR singular.
The resolution of this issue may be therefore similar: massless scalars are not good acceptable fields on de Sitter space[We shall later conclude in this paper that in all the theories we examine and which are all gapless, there is no breaking of de Sitter invariance, and the de Sitter invariant vacuum is chosen.] as argued in [32].

The expectation that QFTs in de Sitter space render the
manifold unstable at the non-linear level has been entertained for a long time [33]-[37]. In particular, destabilizing effects were most important from massless particles, and a gravity 2-loop computation in [34] suggested such an instability. Similar calculations with massless scalars implied similar effects, [27, 28], see however [38].

The topic of quantum effects has been revived after cosmological (CMB) data became precise, and in [39, 40, 41] it was argued that large time-dependent logs from quantum effects of quantum  fields, could give large corrections to inflationary observables. A different approach in [42],  provided different results.
Therefore, the question of  the consequences  of the
secular terms (growing with time) which arise in perturbation theory
of a massless scalar field in the cosmological patch of de Sitter remains controversial. Do
these contributions  indicate an instability of de
Sitter space against quantum perturbations? Or is this conclusion an artefact of
finite orders in perturbation theory, which is expected to disappear  once an
appropriate resummation is performed (as is the case for infra-red
effects in thermal perturbation theory)?
A review of these developments and additional references can be found in [43]. Furthermore, the persistent difficulty of constructing de Sitter vacua in string theory, see [44] for a review, has led to the conjecture that de Sitter space cannot be attained in a weakly curved/coupled  quantum theory of gravity [45].

There is another issue where de Sitter instabilities induced by quantum dynamics can be important: they trigger an exit from the inflationary regime that is an important ingredient of any inflationary model. Indeed, scalar instabilities triggered by coupling gravity to a CFT$_4$, provided an exit from  inflation, [46, 47, 48]
induced by the conformal anomaly, [49].
However, this exit was not good enough and the model was modified to what is now called the Starobinsky model, where inflation is triggered by a (rather large) $R^2$ term. This is a very successful model when compared to current data [50].  In that model also the exit from inflation is triggered by the unstable scalar mode, that in this context is the scalaron of $R^2$ gravity.

So far the quantum effects studied and the backreaction  on de Sitter and Minkowski spaces used weakly-coupled QFTs.
The holographic gauge/gravity duality provides a way to tackle non-perturbative (large-$N$) four-dimensional quantum field theories by mapping them to higher-dimensional semiclassical General Relativity. Moreover, one controls the dynamics of holographic QFTs even when the manifold they are defined upon is curved.
Holography has been applied to cosmological issues already in several works, [51, 52, 48], [53]-[67].
One important arena where this technique can provide important information is when   four-dimensional gravity (described by Einstein General Relativity or its higher-derivative extensions) is coupled to a strongly coupled QFT. Among the issues that may arise are for example the stability under small (metric and matter) perturbations as well as the non-perturbative stability of cosmological backgrounds.

The holographic approach allows for recasting these questions in terms of a classical higher-dimensional gravity theory, which via holography captures the effects of the QFT, coupled to a classical four-dimensional gravitational theory. To be more  specific, the holographic setup consists of two coupled sectors:

* The holographic sector describes the strongly coupled (holographic)  CFT, whose dual is living in a higher-dimensional space-time (the bulk) with metric $\mathcal{G}_{ab}$.

* The four-dimensional gravity sector is defined in terms of a metric $g^{(0)}_{\omega\s}$. This metric plays the role of a boundary condition for  $\mathcal{G}_{ab}$, and it has no bulk dynamics. From the CFT point of view, it corresponds to the source of the field theory stress tensor.

The action describing classical 4d gravity coupled to the holographic QFT has the form:

S = S_grav[g^(0)_ω]  + S_bulk[𝒢_ab,…]
The first term can be taken to be the usual Einstein-Hilbert action plus eventually higher curvature terms, and it has the effect of making $g^{(0)}_{\omega\s}$  dynamical. The second term describes  the  higher-dimensional holographic dual of the QFT, and   the dots represent  bulk fields other than the metric. Both terms are treated classically, but the bulk action encodes holographically the full quantum dynamics of the dual field theory. The two sectors are coupled by the requirement that  on the conformal boundary $\mathcal{G}_{\omega\s}$ asymptotes  to  $g^{(0)}_{\omega\s}$.  “Integrating out” the CFT consists in evaluating   $S_{bulk}[\mathcal{G}_{ab},\ldots] $ on shell. This  results in an effective gravitational action for $g^{(0)}_{\omega\s}$ alone,

S_eff[g^(0)_ω]  = S_grav[g^(0)_ω]  +  S_bulk^on-shell[g^(0)_ω,…]
The second term in (<ref>) is now a functional of the boundary value $g^{(0)}_{\omega\s}$, i.e. the four-dimensional metric.  Varying the effective action results in the semi-classical Einstein equation :

E_ω  =   T_ω _CFT
where $E_{\omega\s}$ is the variation of the first term in  (<ref>) (i.e. the Einstein tensor if  $S_\text{grav}$ is purely GR) and $ \< T_{\omega\s} \>_{CFT}$ is obtained by varying $ S_{bulk}^{on-shell}$ with respect to $g^{(0)}_{\omega\s}$.

In a previous work [66], some of the authors have used the  setup described above to address the issue of non-perturbative existence of 4d de Sitter space coupled to  a (gap-less) holographic field theory.  For this,  the attention was limited to boundary metrics $g^{(0)}_{\omega\s}$ of maximal symmetry. In this case, the effective action (<ref>)  takes the form of an effective $f(R)$ theory,

S_eff[g^(0)_ω] = ∫d^4x √(g^(0)) f(R),
where $R$ is the Ricci scalar of $g^{(0)}_{\omega\s}$. It was shown in [66] that  such  a theory generically still admits de Sitter solutions (albeit with a smaller cosmological constant than that of the “bare” 4d gravity) after the full quantum effects of the field theory are taken into account.

The results of [66] suggest that, at least in the holographic context, IR effects from a QFT do not always  destroy de Sitter space-time. However, that analysis could only be applied   to constant curvature 4d space-times.  Therefore, it says nothing about the stability of the solutions under non-homogeneous perturbations. The  form  (<ref>)  of the effective action is only good for obtaining maximally symmetric solutions, and it would be misleading to expand it in perturbations around one such background. Rather, to study small perturbations of a holographic QFT coupled to gravity one has to go back to the original theory (<ref>) and study its perturbation spectrum.
This has been considered already in [52], albeit in a slightly different context (Randall-Sundrum cosmology), and more recently in [68] for the case of a two derivative gravitational action coupled to a holographic CFT around de Sitter space.
In [68], beyond the scalar instabilities that are known for several decades, [47],  a spin-2 instability was found for small enough de Sitter curvatures.

Our goal in this paper is to extend previous results on the stability of maximally symmetric spacetimes due to quantum effects in several ways:

* We consider not only de Sitter but also flat space and Anti-de Sitter space.

* We consider a classical gravitational theory with all couplings necessary for renormalization which are relevant at low energies. This implies that we have a cosmological constant and Einstein terms as well as the two independent $R^2$ terms with (finite) dimensionless renormalized couplings $\a,\b$.

\item In our case, the only non-trivial (i.e. in which the CFT degrees of freedom participate) quadratic action for the fluctuations is that in the spin-2 sector.  We analyze not only the possible tachyonic poles that are responsible for the instabilities but also the presence of negative residues that signal the presence of ghosts

\item Moreover, we investigate when such tachyons or ghosts are below the effective UV cutoff for the classical gravitational theory, which is given by the so-called species cutoff, [69].

\end{itemize}

Concretely, we consider  general perturbations around maximally symmetric four-dimensional space-times  in a holographic setup, given by (\ref{intro1}),  where the matter content is a four-dimensional holographic conformal field theory. The coupling to gravity is entirely described by the exact one-point correlation function of the CFT stress-energy tensor, as in equation (\ref{intro3}). This stress-tensor is obtained on a boundary of $AdS_5$ from a holographic calculation and has the correct conformal Weyl anomaly (see [49] for a review of the conformal quantum anomaly, and [70] for the holographic CFT stress-tensor).

The perturbation analysis is obtained by fluctuating the bulk and boundary metrics and then writing the  linearized version of the effective Einstein equation (\ref{intro3}) for the four-dimensional metric $g^(0)_ω$. We perform this analysis  around maximally symmetric 4d space-times of positive curvature (dS), negative curvature\footnote{The case of negative curvature is special. It corresponds to foliating AdS$_5$ by AdS$_4$ slices, which leads to a geometry with two {connected}  boundaries, which is dual  to two copies of the CFTs with {an interface between them }(see e.g. [71] and the recent discussion in [72]). One has then different options on how to couple dynamical gravity to the system, the most general case being a bi-gravity theory with each metric coupled to one of the CFTs. Here, we discuss the special cases in which only one metric is dynamical.}  (AdS) and vanishing curvature (Minkowski).

In this work, we pursue two main objectives:
\begin{enumerate}
\item   We first  obtain analytic  spectral equations  (in terms of transcendental functions)  for the boundary metric fluctuations  of the 4d gravity+ holographic CFT system, around a general maximally symmetric background;
\item We then perform a full numerical analysis of  the spectrum in momentum space, (defined by the eigenvalue of the Laplacian on the corresponding maximally symmetric space-time) and determine criteria for the presence of  instabilities  of both tachyonic and ghostlike type.
This way, we obtain a detailed  map of  the stable and unstable regions of parameter space.
\end{enumerate}

Our approach is similar in spirit to other works in the context of weakly coupled field theory: a similar analysis was performed with a matter content given by quantum corrections of free massless scalar CFT [47], for {\it homogeneous}  time-dependent metric perturbations around de Sitter.  A similar perturbation analysis around flat space was carried out for a free scalar coupled to higher-derivative gravity in [14].

Here,  we use the holographic setup to perform a full parameter-space analysis of the gravity+CFT system, and we establish stable and unstable regions of parameter space around background solutions with zero, negative and positive constant curvature. Specifically, the parameters of the model are:
\begin{itemize}
\item The  boundary gravity renormalized couplings $\Lambda, G, \alpha, \beta$, corresponding to local covariant functions of $g^{(0)}_{\omega\s}$ of dimension up to four, which we define as follows:
\be \label{intro4.1}
{\Lambda\over 8\pi G} \sqrt{g^{(0)}},  \quad  {1\over 16\pi G} \sqrt{g^{(0)}} R, \quad {\alpha \over 384 \pi}  \sqrt{g^{(0)}} R^2, \quad {\beta \over 64\pi} \sqrt{g^{(0)}} \left[ R_{\omega\s}R^{\omega\s} - {1\over 3}R^2 \right]
\ee
Here $\Lambda$ is the 4d cosmological constant, $G$ is  4d Newton's constant,    $R_{\omega\s}$ is the Ricci tensor of  $g^{(0)}_{\omega\s}$ and $R$ the corresponding Ricci scalar and $\alpha,\beta$ two dimensionless parameters.

\item The parameter $N$, counting the degrees of freedom of the CFT. This can be traded for the central charge of the CFT.

\item  The value  $R$ of the curvature  of the background solution. This is not an independent parameter, as it is determined by the other parameters via the background solution. However, it is convenient to use this as an independent parameter instead of e.g. the 4d cosmological constant.

\item An extra parameter is the {\it renormalization scale} $\mu$ of the CFT, which arises from the conformal anomaly. Since this is universal, its   only effect is  to shift the parameter $\beta$ so that it enters only in the combination:
\be \label{intro4.2}
\beta_\text{eff} = \b - {N^2\over \pi} \log \left(4 \mu^2 G N^2 \right).
\ee
\end{itemize}

Although the  QFT we couple to gravity is a holographic CFT$_4$, it is important to stress that  our result holds for {\it any} generic CFT coupled to gravity: as we shall observe  below, the spectral properties of the system are determined by the stress-tensor 2-point function, which for a CFT  in any conformally flat space-time is completely fixed by the central charge. Therefore, although the method we use to compute the fluctuation spectra is specific to a large-$N$ holographic CFT, to obtain the result for a generic CFT it is enough to trade the parameter $N$ with the appropriate central charge.

In order to have semiclassical 4d gravity, we require that the 4d curvature $R$ is small compared to the cutoff of the theory.
Typically, this is assumed to be the Planck scale or the string scale in the case of string theory. However, as argued in [69] if the matter theory has many degrees of freedom, the  perturbativity condition imposes a lower cutoff (the so-called ``species scale") than the Planck scale. For example if, like in our case, the matter theory has $O(N^2)$ degrees of freedom the species cutoff $M_species $ is
\be
M_\text{species} \equiv {M_\text{Planck} \over N} \sim {1\over \sqrt{G} N}.
\label{sp-a}
\ee
In our case, we require $N≫1$ in order to have semiclassical  gravity in the bulk (dual to the large N CFT). At the same time,  we require that boundary curvatures are below the species cut-off,
\be \label{sp-b}
R\lesssim M_\text{species}^2={M^2_\text{Planck} \over N^2}
\ee
The condition (\ref{sp-b}) is enough to neglect all higher curvature corrections which we have not included in (\ref{intro4.1}):   in general, the gravity effective action may contain higher derivative terms that we can write schematically as
\be
S=M^2_\text{Planck}  R+\a R^2+\sum_{n=1}^{\infty}a_n R^{n+2}
\ee
The coefficients will get renormalized by the CFT and the  estimate is that, in the large-$N$ limit,
\be
\a\sim {\cal O}(N^2)\sp a_{n}\sim {N^2\over M^{2n}}
\ee
At the cutoff (\ref{sp-b}) we can estimate the importance of the various terms:
\be
M^2_\text{Planck}R \sim {M^4_\text{Planck}\over N^2}\sp \a R^2\sim {M^4_\text{Planck}\over N^2}\sp    a_n R^{n+2}\sim {M^4_\text{Planck}\over N^{2n+2}}
\ee
For this reason, we can neglect all powers of the curvature above $R^2$.

The same considerations about the cut-off apply to  the analysis of fluctuations, and in particular  of instabilities.
In effective field theory, any mode with a mass of the order or above the cutoff is outside the reach of the theory.
In particular,  it is only when the ghost or tachyon mass is well below the EFT cut-off that one can unambiguously conclude that the theory truly has stability issues.

Indeed, as it is well known (and as we review in a simple example in Appendix \ref{app:free scalars}), an EFT originating from an otherwise healthy UV theory may display some unstable modes as an artefact of the low-energy expansion. In  this case, the unstable modes will have masses of the order of the EFT cut-off (the scale of the fields that have been integrated out). Turning this around, we can say that  one cannot conclude anything about the actual stability or instability  of an EFT based on  the occurrence of ghosts or tachyons whose mass scale is at or above the EFT cut-off: one would have to know the UV completion to reach a definite conclusion. For the same reason, within EFT one cannot reach any definite conclusion about the stability of spacetimes whose curvature scale ($H$ or $χ$) is at or larger than the cut-off.

Throughout this work, we shall encounter many instabilities (tensor and scalar ghosts and tachyons) and   will  be careful to compare their mass scale with the cut-off (\ref{sp-a}) to determine whether they can be considered physical.

We devote the rest of this introduction to an extended summary of these techniques as well as a discussion of  the results obtained in this work. Although they stem from a holographic calculation, we insist that they are valid for a generic CFT, and  describe the results mostly in the language  of the field theory side. We leave the details of the holographic approach to the rest of the paper.

\subsection{Summary and results} \label{sec:intro-sum}

The setup studied  here consists of higher curvature gravity coupled to a Conformal Field Theory (CFT). For the gravitational part, we  consider the Einstein-Hilbert action with a cosmological constant plus quadratic curvature term:
\be \label{res-0}
S_\text{grav} = S_{EH} + S_2
\ee
\be \label{res-1}
S_{EH} =- \frac{1}{16 \pi G}\int d^4x\sqrt{-g^{(0)}}(R - 2\Lambda)
\ee
\begin{equation} \label{res-2}
S_{2} = \frac{\a}{384 \pi}\int d^4x\sqrt{-g^{(0)}}R^2 + \frac{\b}{64\pi} \int d^4x\sqrt{-g^{(0)}} \left( R_{\omega\s}R^{\omega\s}-\frac{1}{3}R^2 \right)
\end{equation}
Here, $g^(0)_μν$ is the 4d  space-time metric, $g^(0)$ its determinant and $R$ its curvature. The parameters $G,Λ,α,β$ are the (finite) renormalized parameters which already contain the contributions of the CFT\footnote{These  parameters are defined so that they are finite in an appropriate scaling limit after removing the UV cut-off. The detailed procedure is described in section \ref{section:action}.  }

Coupling to a large-$N$ CFT is implemented via holography: we identify the 4d space-time with the conformal boundary of AdS$_5$ bulk manifold, and $g^(0)_μν$ with the leading term in the Fefferman-Graham  expansion of the 5d  bulk metric:
\be \label{res-3}
ds^2_{bulk} = L^2 {d\rho^2 \over 4\rho^2} + {1\over \rho}\left[g_{\mu\nu}^{(0)} + O(\rho) \right] dx^\mu dx^\nu \qquad \rho \to 0
\ee
where $ρ→0$ corresponds to the AdS boundary and $L$ is the AdS length.
This way, one can obtain 4d maximally symmetric metrics $g^0$ whose Ricci curvature $R̅$ satisfies the relation:
\be \label{res-5}
\Lambda=\frac{1}{4} \left( \bar R-\frac{GN^2 \bar R^2}{48\pi} \right).
\ee
The Ricci curvature $R̅$  of the  maximally symmetric background space-time can be positive, negative or vanishing. It is convenient to parametrize it in the various cases as follows:
\bar R= \left\{ \begin{array}{lll}
\displaystyle 12H^2,&\phantom{aa} &{\rm de~~ Sitter},\\ \\
\displaystyle 0 ,&\phantom{aa}&{\rm Minkowski} \\ \\
\displaystyle -12\chi^2 ,&\phantom{aa}& {\rm Anti~~ de~~ Sitter}
\end{array}\right.
\label{bc}\ee}
The parameter $N$, characterizing the number of degrees of freedom of the CFT,   is related to the bulk Planck scale $M$ and bulk AdS length $L$ by  $N^2 ∝(M L)^3 $.  The first term in equation (\ref{res-5})  is the contribution from the vacuum Einstein equation, and the second term is the CFT contribution.

To conclude, the {\it independent} parameters of the theory are the curvature of the background space\footnote{Even if  $\bar{R}$ is not a parameter in the action, we can trade  $\Lambda$  for $\bar{R}$ using (\ref{res-5}). Although the relation between $\Lambda$ and $\bar{R}$ is not one-to-one, by scanning over all values of $\Lambda$ we can obtain any value of $\bar{R}$.} $R̅$, the four-dimensional Newton constant $G$, the two $R^2$ couplings $$ and $_̱eff$, where $β_eff$ is defined in (\ref{intro4.2})  and the number of colors\footnote{In fact, one should take the central charge as a parameter for the CFT.  We shall use $N$ as a proxy for the central charge} $N$ of the holographic CFT$_4$. It will be convenient to  express  quantities in terms of the following  ``reduced'' parameters:
\be\label{tildes}
\tilde{\alpha} = {\pi \alpha \over N^2}, \qquad \tilde{\beta}_\text{eff} = {\pi \beta_\text{eff} \over N^2}.
\ee

Our goal is to determine, as a function of the parameters of the model,  the spectrum of gravitational  fluctuations of the boundary metric  around any maximally symmetric  4d boundary metric $ζ̅_ν$. We  use this information to determine the perturbative stability of the system.

The perturbed boundary metric is taken to be:
\be
g^{(0)}_{\mu\nu} = \bar{\zeta}_{\m \nu} + \delta \zeta^b_{\mu\nu}
\ee

In an appropriate gauge, the boundary perturbation  can be written as
\begin{equation} \label{res-6}
\d \z^b_{\omega\s}=\psi\bar{\z}_{\omega\s}+h_{\omega\s}^{(0)}
\end{equation}
where $ψ$ is a scalar degree of freedom, and $h_ω^(0)$ is a tensor perturbation which is transverse and traceless with  respect to the boundary metric. The scalar is a pure boundary mode\footnote{ {The scalar mode couples to the trace of the stress-energy tensor of the CFT$_4$. Since this theory is conformally invariant, the two-point function of the trace vanishes. Therefore the non-trivial action for the scalar mode is generated by the boundary $R^2$ terms as well as the conformal anomaly of the CFT$_4$,  \protect{[47]}. If the theory is instead a QFT, extra contributions are expected for the dynamics of the scalar mode.}}, whereas the four-dimensional gravity tensor modes couple to tensor perturbations in the bulk.

The metric perturbations are coupled to the CFT via the  bulk dynamics: the boundary field $h_ω^(0)(x)$ is the leading term in a near-boundary expansion of the perturbation of the {\it bulk} metric.
\\

\noindent {\bf Spectral functions}\\
The spectral analysis around the holographic background is tightly connected to the holographic two-point function of the  boundary stress tensor.   When working at linear order in fluctuations  both in the bulk and on the boundary, all one needs is the structure of the effective action (\ref{intro2})  at quadratic order as a function of the boundary metric perturbation $δζ^b$:
\be \label{intro5}
S_\text{eff}^{(2)} =
\int d^4x  {1\over 2}\delta \zeta^b_{\mu\nu} O_\text{grav}^{\mu\nu,\rho\sigma} \delta \zeta^b_{\rho\sigma}   - {1\over 2} \int d^4x \int d^4y  \, \delta \zeta^b(x)_{\mu\nu}  \, \< T^{\mu\nu}  (x) T^{\rho\sigma}   (y) \>_{CFT}\,  \delta \zeta^b_{\rho\sigma}    (y)
\ee
These two terms correspond to the quadratic order approximation of each of the two terms in (\ref{intro2}):  $O_grav $ is the local  kinetic operator of the  quadratic term in the 4d gravity action  $S_grav$ in (\ref{res-0});   $ T^μν T^ρσ  _CFT $ is the holographic two-point function of the stress tensor, which is by definition:
\be \label{intro6}
\< T^{\mu\nu}(x) T^{\rho\sigma}(y) \>_{CFT} = -  {\delta  \over \delta \zeta^b_{\mu\nu}(x) } {\delta  \over \delta \zeta^b_{\rho\sigma}(y) } S^{on-shell}_{bulk}
\ee

The stress tensor two-point function contains both local and non-local contributions. The local contributions simply renormalize the coefficients of local terms which are already present in $O_grav $. The non-local contributions are genuine new effects of the CFT which one cannot find in a local gravity theory.

Equation (\ref{intro5})  shows that by computing the holographic two-point function we have access to
the full   propagator, which we denote by $F^-1$,  of the boundary metric fluctuations: the inverse propagator  is
\be \label{intro7}
{\cal F}^{\mu\nu\rho\sigma} \equiv  O_\text{grav}^{\mu\nu,\rho\sigma} - \< T^{\mu\nu} T^{\rho\sigma} \>_{CFT}
\ee
and the spectrum  of the system are the solutions of the integrodifferential equation
\be \label{intro8}
{\cal F}^{\mu\nu\rho\sigma}  \delta \zeta^b_{\rho\sigma} = 0.
\ee

The linear equation (\ref{intro8})  can be recast into two separate {\it scalar} spectral equations for the scalar and tensor modes defined in (\ref{res-6}),  by going to the appropriate ``momentum space'' of the boundary coordinate. This is done by  decomposing  the modes in eigenfunctions of the d'Alembert operator $∇^2$ of the background boundary metric $_ω$: in the positive, zero and negative curvature case we take the  fluctuation to satisfy
\be \label{res-8}
\left(\nabla^2 - r {\bar{R}\over 12}\right)\delta\f(x) = \left\{ \begin{array}{ll} -H^2\left(\nu^2 -{9\over4}\right)  \delta \f(x) & \quad dS \\ & \\ -  k^2    \delta  \f & \quad \text{Minkowski}  \\ & \\
\chi^2\left(\nu^2 -{9\over4}\right)  \delta \f(x) & \quad AdS \\\end{array}  \right.
\ee
where $δ(x)$ stands for either $ψ$ or $h^(0)_μν$, $r$ is the spin of the perturbation ($r=0$ for $ψ$ and $r=2$ for $h^(0)_ω$), $H$ is the  Hubble parameter in the case of positive curvature boundary (de Sitter) and  $χ$ is  the inverse AdS length in the case of negative curvature boundary, as in (\ref{bc}).
For flat space, this is the usual Fourier decomposition where $k^2 = k^μk_μ$.  In both curved cases,  $ν$ is a dimensionless eigenvalue measuring the invariant ``momentum'' in units of the background curvature.

The values of  $ν^2$ (or $k^2$ in the flat case) are determined by the spectral equation (\ref{intro8}), which in momentum space becomes a transcendental equation for $ν^2$   (or $k^2$) of the form:
\be \label{res-9}
{\cal F}(\nu) = 0
\ee
where the precise form of the function $F$ depends both on the nature of the mode (scalar or tensor) and the background curvature and the parameters in the action. The expressions obtained from  a holographic calculation can be found below: \\

\noindent{\bf Scalar mode}
In this case, the inverse propagator is a polynomial in $ν^2$ (or $k^2$), because it results from a quadratic action which is local on the boundary. The expression of the inverse propagator is given by
\begin{itemize}
\item {\bf Minkowski}
\be \label{res-10}
{\cal F}_{scalar}(k) =  -{3\over 16 \pi G}\left( k^2 -  {4\over \alpha G } \right)
\ee
\item {\bf de Sitter and Anti-de Sitter }
\be \label{res-11}
{\cal F}_{scalar}(\nu) =  -{1\over 64 \pi G}\left[\a G \bar{R} - 12 + {GN^2\bar{R}\over 2\pi}\right]\left\{ {4\over G\a} - {N^2 \bar{R} \over 6\pi\a} - {\bar{R}\over 12}\left(\nu^2 -{9\over 4}\right) \right\}.
\ee
This is the "physical" scalar inverse propagator. For the details, see section \ref{bdy pert}.
\end{itemize}

\noindent{\bf Tensor modes} For tensor modes, the non-local contribution from the CFT stress-tensor correlator in (\ref{intro7}) gives rise to non-polynomial expressions for the inverse propagators:
\begin{itemize}
\item {\bf Minkowski}
\be \label{res-12}
{\cal F}_\text{tensor,Mink}(k) = {N^2 \over 64\pi^2} k^2 \left\{-{2\pi \over GN^2} + {k^2\over 2}\left[{1\over 2} - 2\g_E - \log\left(GN^2k^2\right)- {{\pi\b_\text{eff}\over N^2}}\right] \right\}
\ee
\item {\bf de Sitter}
\bea \label{res-13}
{\cal F}_\text{tensor,dS}(\n) = && {N^2 H^2 \over 64\pi^2} \left(\nu^2 - {9\over 4}\right) \left\{1 - {2\pi\over GN^2H^2} + {2\pi \alpha\over N^2} - {1\over 2}\left(\n^2-{1\over 4}\right)\bigg[ \right.  \nn \\
&& \left.\left. 2\log\left(GN^2H^2\right) -{1\over 2} +  2\mathcal{H}\left(\n-\frac{1}{2}\right)  + {\pi\beta_\text{eff}\over N^2}\right]\right\}.
\eea
where $\mathcal{H}$ is the harmonic number function defined in (\ref{hn}).
The expression (\ref{res-13})  with  $\a=0$  was already obtained in [68]\footnote{In [68]  $\b$ was fixed but the renormalization scale $\mu$ (called E in that paper) was allowed  to vary.}. In this work,  we rederive it in our setup and generalise it to negative and zero curvature and arbitrary values of the $\alpha$ parameter.
\item {\bf Anti-de Sitter}\\
In this case, there are two connected  boundaries, corresponding to two - a priori independent - copies of the CFT. Therefore, one has freedom in how to couple 4d gravity to the system.
Here, we discuss two concrete cases: \\
\textbf{a) Dynamical gravity on one side:} In this case only one of the two CFTs is coupled to dynamical gravity, and the metric on the second boundary is frozen.
\bea\label{res-13a}
{\cal F}_\text{tensor,AdS}^{-}(\n) =&& {N^2 \chi^2 \over 64\pi^2}\left(\n^2-\frac{9}{4} \right) \left\{ 1 + {2\pi\over N^2}\left({1\over G \chi^2} + \a \right) +\right.  \nonumber \\
&& \left. - {1\over 2} (\n^2- 1/4)\left[ {\pi\beta_\text{eff}\over N^2} +  \log\left(GN^2\chi^2\right) - {1\over 2} +  \nn  \right. \right. \\
&&\left. \left. +  \mathcal{H}\left(-{1\over 2}-\n \right) + \mathcal{H} \left(-{1\over 2} + \n \right) \right] \right\}.
\eea
\textbf{b) Symmetric boundary conditions:} In this case, there is effectively a single boundary (see [72] for a recent discussion),  and there is again a  single dynamical gravity theory coupled to a  single 4d  CFT on AdS. This leads to the following spectral density:
\bea \label{res-14a}
{\cal F}_\text{tensor,AdS}^{sym}(\n) =&& {N^2 \chi^2 \over 64\pi^2}\left(\n^2-\frac{9}{4} \right)\left\{   1 + {2\pi\over N^2}\left({1\over G \chi^2} + \a \right)  + \right.  \nonumber \\
&& \left.  - {1\over 2} (\n^2- 1/4)\left[ {\pi\beta_\text{eff}\over N^2} + \log\left(GN^2\chi^2\right) - {1\over 2} \right. \right. \nn \\
&& \left. \left. + \mathcal{H}\left(\n-\frac{1}{2}\right) + \mathcal{H}\left(-\n-\frac{1}{2}\right) - {\pi\over \cos\pi\n}\right] \right\}.
\eea
\end{itemize}

\noindent {\bf Stability}\\
Instabilities of the system are encoded in the properties of the zeros of ${\cal F}$. We perform a full analysis of all parameter space, which we summarize below.   As a byproduct, by setting $N=0$  we  obtain the pure gravity spectral functions and study the corresponding zeros, which give indications about the stability of quadratic gravity around any constant curvature background.

When discussing the gravity + CFT system, we  always compare the results with those of pure gravity theories with the appropriate renormalized parameters. This allows us to identify the new effects (if any) which arise specifically from the coupling to the CFT.
For a CFT with parameter $N$, the comparison should be made by choosing the pure gravity parameters $\alpha$ and $\beta$  such that $\alpha = \tilde{\alpha}/\pi$ and $\beta = \tilde{\beta}_\text{eff}/\pi$, in terms of the quantities  defined in (\ref{tildes}): these are the quantities which are expected to be of order unity after renormalization of the local terms by  the CFT is taken into account.

Depending on  the curvature, there are different criteria for  instabilities. On any background,  instabilities can be of  two types:
\begin{itemize}
\item {\bf Tachyonic instabilities} correspond to modes which grow  exponentially in time and are related  to the {\it position} of the root $\nu$ in the complex plane. Specifically, a root $\nu$ of ${\cal F}(\nu)$ is {\it tachyon-stable} in the  following cases:
\be \label{res-14}
\left\{ \begin{array}{ll} \left| Re (\nu)\right| \leq {3\over 2}   & \quad dS \\ & \\ k^2 \leq 0  & \quad \text{Minkowski}  \\ & \\
Re(\nu) \neq 0 & \quad AdS \\\end{array}  \right\} \quad \Rightarrow \quad \text{tachyon-stable}
\ee
In all other cases, the mode is tachyonic.
\item {\bf Ghost instabilities} correspond to a  mode  with eigenvalue $\nu_0^2$ (or $k^2_0$) developing a ``wrong sign'' kinetic term, and are related to the value of the residue of ${\cal F}$ at the pole:
\be \label{res-15}
\left\{ \begin{array}{ll} Res{\cal F}^{-1}(\nu_0^2) < 0   & \quad dS \\ & \\ Res{\cal F}^{-1}(k^2_0) < 0 & \quad \text{Minkowski}  \\ & \\
Res{\cal F}^{-1}(\nu_0^2) > 0 & \quad AdS \\\end{array}  \right\} \quad \Rightarrow \quad  \text{ghost-stable}
\ee
The sign conventions are discussed in  section \ref{decoupling}.

A heavy ghost can be tolerated if its mass is above the cut-off of the theory because in this case it cannot be described in the context of effective theory (and it may become healthy in the UV-completion).

In this work we compare ghost masses with two cut-offs: the 4d Planck scale $G^{-1/2}$ (the ultimate cut-off in the semiclassical approach) and the {\it species cutoff scale}
$\Lambda_{\rm species}$
\be
\Lambda_{\rm species}\equiv (GN^2)^{-1/2}\;,
\label{spec}\ee
which can be argued to be the true cut-off of a gravity theory coupled to $N^2$ degrees of freedom [69]. In  fact, it seems that  the latter  is the natural scale in which to measure boundary  curvature $R$  in the present set-up: it  always appears in the combination
$$G N^2 R\sim {R\over M_P^2 N^2}\sim {R\over \Lambda_{\rm species}^2}\;.$$
\end{itemize}

An unstable mode can be a ghost, a tachyon, or both.  In what follows we summarize our results in the scalar and tensor sector and for zero, positive and negative background curvature.
One important point to which we have to pay attention is whether the unstable mode is within the limits of effective field theory, i.e. whether it is light in Planck units (in the case of pure gravity) or light in units of the species scale (\ref{sp-a})  (in the case of gravity coupled to the CFT). \\

\noindent {\bf Stability in the scalar sector}\\
For the scalar mode, it is straightforward to read off the conditions (\ref{res-14}-\ref{res-15}) from equations (\ref{res-10}-\ref{res-11}): this leads to the following conclusions:
\begin{itemize}
\item In Minkowski space, the scalar spectral function (\ref{res-10})  is the same in pure gravity and in the presence of the CFT and does not depend on $N$ as the conformal anomaly of the CFT is not relevant. The scalar mode is never a ghost, and it is  tachyonic if $\alpha > 0$. This agrees with previous analysis (e.g. [14]). The tachyonic mode is within the bounds of the theory if its mass is below the cut-off, which in terms of the ``reduced'' $\tilde\alpha$ parameter defined in (\ref{tildes}) requires  $\tilde{\alpha} \gg 1$ (the same condition as in pure gravity, since $\alpha = \tilde{\alpha}$).

\item In de Sitter space,  {scalar} tachyon-stability requires:
\begin{equation} \label{res-17}
\frac{1}{\a}\left(1-\frac{GN^2 H^2}{2\pi} \right)\leq 0.
\end{equation}
For consistency, $GN^2 H^2\ll 1$ and therefore the second factor is always positive in effective field theory.
Therefore, tachyon stability implies $\a<0$.

Moreover, the scalar mode is a ghost if:
\be\label{res-19}
\left({\pi \alpha \over N^2} + {1\over 2} \right) {G N^2 H^2 \over 12 \pi} > 1.
\ee
{ie. when $\a\gg 1$}.
For pure gravity, (\ref{res-17}) with $N=0$ is the same condition ($\alpha<0$) as for Minkowski space. In pure  de Sitter gravity, the scalar can also be a ghost, if $\alpha$ is very large (at least of order $1/(GH^2) \gg 1$). This mode is light in Planck units if $|\alpha| \gg 1$.

In the presence of the CFT, the tachyon stability condition is modified by the second term proportional to $N^2$ in  (\ref{res-17}). However, note that this term is small if we insist the curvature is below the species cutoff, which requires $GN^2H^2 \ll 1$. If this is the case,  the tachyon-stability condition is not affected much by the CFT in the context of low-energy EFT\footnote{A notable case in  which this condition is violated is the Starobinsky realization of de Sitter (or more generally, inflation), in which the cosmological constant term is absent and the de Sitter curvature is fixed to $GN^2H^2 = 4\pi$ [46, 47]. In this case, the tachyon stability condition is reversed to $\alpha >0$. We comment on this case in Appendix \ref{staro}.}.
The scalar mode is below the species cut-off if $|\tilde{\alpha}| \gg 1$ (the same condition as for pure gravity). Finally, both with and without the CFT   the  time-scale $\tau$ of the tachyonic instability is roughly the inverse tachyon mass, $\tau \sim \sqrt{G|\alpha|}$. In effective field theory ($GH^2 \ll 1 $) this is much faster than the de Sitter Hubble rate $H^{-1}$ (i.e. the tachyon instability is very strong), unless $|\alpha| \gg (GH^2)^{-1}$. Therefore, for $\tilde{\alpha}$ in the interval
\be
1 \ll \tilde{\alpha} \ll {1\over GH^2 N^2}
\label{strtacdS}
\ee
we have a strong instability (faster than one Hubble time) within effective field theory. This condition also applies to pure gravity, if we set $N=1$ and $\tilde{\alpha} = \alpha$.

\item The discussion is similar for Anti-de Sitter. Scalar tachyon-stability requires:
\be \label{res-18}
{9\over 4} - {4\over \a G\chi^2}\left(1 + {GN^2\chi^2\over 2\pi}\right) \geq 0.
\ee
and the scalar mode is a ghost if
\be\label{res-19-ii}
-\left({\pi \alpha \over N^2} + {1\over 2} \right) {G N^2 \chi^2 \over 12 \pi} > 1.
\ee
Like before, to be in the effective field theory we must require that
$G N^2 \chi^2\ll 1$.
The condition for the scalar modes (whether a tachyon or a ghost) to be within the bounds of effective field theory is $|\tilde{\alpha}| \gg 1$.
\end{itemize}

\noindent {\bf Stability in the tensor sector}\\
Unlike the case of the scalar, exploring the roots of the tensor spectral function can only be done numerically, except in some corners where analytic approximations for the transcendental functions can be used (in particular the large eigenvalue limit $\nu\to \infty$). Below we give the broad features of the stability results  in the three cases (zero, positive and negative curvature). More details can be found in the main body of the paper.

In each case, we emphasize what happens for two  special parameter values:

(a)  $N=0$ which corresponds to pure gravity with higher curvature terms;

(b) $\alpha=\beta_\text{eff}=0$, which  corresponds to setting the (renormalized) local quadratic curvature terms to zero. This   gives a measure  of the truly non-local contributions from the CFT.

\begin{itemize}
\item {\bf Minkowski}\\

\begin{itemize}

\item In the special case of  pure gravity ($N=0$), for $\beta \neq 0$ (and independently of $\alpha$)  the quadratic Ricci tensor term always generates a ghost, whose mass is,
\be
m^2_{ghost} = {4\over \beta G}.
\ee
For $\beta<0$ this is also a tachyon. This mode is light compared to the cut-off $G^{-1/2}$  when $|\beta| \gg 1$.
Therefore, the gravity theory is a good and stable effective theory only if $\beta$ is positive and $\beta\lesssim 1$.

\item In the presence of the CFT, the spectral function is  (\ref{res-12}) and its  non-trivial roots   are the solutions of a transcendental  equation of the type $X \log X = a$, where $X$ is proportional to $k^2$ and $a$ is a real constant. The analysis can be done semi-analytically and it leads to the conclusion that for any value of $\tilde{\alpha}$ and $\tilde{\beta}_{\text{eff}}$ {\it Minkowski space always contains two tachyonic tensor modes}. The theory becomes eventually tachyon-stable only in the extreme limit $\tilde{\beta}_{\text{eff}} \to +\infty$.
In this limit,  one always finds also a light  ghost  (light compared to the ``species'' scale $(G N^2)^{-1/2}$), as in the pure gravity case. All in all, the masses of the unstable tensor modes are above the species cut-off for $O(1)$ values of $\beta_{\text{eff}}$ (this includes  the special case $\a=\beta_{\text{eff}}=0$), while Minkowski space is unstable within EFT iff $|\tilde{\beta}_{\text{eff}}|\gg 1$ and independently of $\tilde{\alpha}$.

\end{itemize}

\item {\bf de Sitter}

\begin{itemize}
\item  In the special case of pure higher curvature gravity ($N=0$), there are always two tensor modes, one of which is the massless graviton, and the other is massive. The massive mode is  tachyonic if the following condition is violated:
\be \label{ds tach intro}
{2\over \beta} \left(\alpha - {1\over GH^2}\right) < 1.
\ee
Because $GH^2 \ll 1$, the condition (\ref{ds tach intro}) is violated if $\beta< 0$ for $\alpha$ and $\beta$ of order unity (this matches the Minkowski result).

Whether or not (\ref{ds tach intro}) holds, either mode is necessarily a ghost. If $\beta - 2\alpha < 2 (GH^2)^{-1}$, the ghost is the massless spin-2 mode, otherwise it is the massive mode. For $O(1)$ values of $\alpha$ and $\beta$, the ghost is the massive mode, and its mass is of  order  ${\cal O}(M_p)$. One can have a light ghost only if $\alpha \gg  (GH^2)^{-1} \gg 1$ (in which case the ghost is the massless graviton)  or if $|\beta| \gg 1$ (in this case which one is the ghost depends on the sign of $\beta$). { All in all,   in pure gravity the theory is stable and ghost-free within EFT (i.e. below the cut-off $M_p$) if $\alpha$ and $\beta$ are both $O(1)$. }

\item We now turn to the case of gravity coupled to the CFT. In de Sitter,
the presence of  tachyonic  tensor modes depends on the curvature, on $N$  and the parameters $\tilde{\alpha}$ and $\tilde{\beta}_{\text{eff}}$. The dS curvature $H$  always enters in the combination $GN^2H^2$, i.e. the natural scale to which the curvature is compared is the ``species'' scale (\ref{sp-a}).

For a de Sitter background, the presence or absence of tachyonic instabilities is illustrated in figure \ref{fig:dS}. For a fixed  value of $GN^2H^2$,  tachyon-stability corresponds to values of $\tilde{\beta}_{\text{eff}}$ larger  than a certain critical value, which is typically of order unity. For fixed  $\tilde{\beta}_{\text{eff}}$, there are two regimes, depending on the value of $\tilde{\alpha}$: for small  $\tilde{\alpha}$,  and into negative values, the theory is tachyon-stable for
$GN^2H^2$  larger than  a certain critical value (generically of order unity); for large and positive $\tilde{\alpha}$ there are also intermediate regions of stability: the theory goes from unstable at small $GN^2H^2$,  to stable as  $GN^2H^2$ increases, to again unstable, and finally to stable at large $GN^2H^2$. In the specific case   $\tilde{\a} = \tilde{\beta}_{\text{eff}} = 0$ there is a critical value for $GN^2H^2$ below  which de Sitter space is  tachyon-unstable, as it was also shown in [68]. The critical value corresponds to $GN^2H^2 \approx 0.32$.  For small curvatures, and for $\tilde{\alpha}$ of order unity, the tachyon pole is generically located around the cut-off scale, unless one takes $|\tilde{\beta}_{\text{eff}}| \gg1$.

For any values of the parameters, there are tensor ghosts (tachyonic or not). However, generically, these ghosts are heavy (in units of the ``species'' cut-off $(GN^2)^{-1/2}$) or they occur for curvatures of the order of the cut-off.

\item In the special case $\tilde{\a}=\tilde{\beta}_\text{eff}=0$, like in the generic case above,  for any curvature (including zero-curvature flat spacetime), the mass of the ghost is always larger but comparable to the species scale.

\end{itemize}

\item{\bf Anti-de Sitter.}
\begin{itemize}
\item In the special case of  pure gravity ($N=0$), the situation is similar to the one in de Sitter. There are two tensor modes,   one of them  massless and the other  massive.  For generic $O(1)$ parameters $\alpha$ and $\beta$, the massive mode  is a tachyon for  $\beta < 0$ (up to small corrections).  One of the two tensor modes is always a ghost, and it is light only when $\alpha$ and/or $\beta$ are very large. Therefore, as in de Sitter, for $O(1)$ values of the parameters, the theory does not have instabilities within EFT. This is what happens  in top-down string theory [74]. On the other hand, this analysis means that one has to be careful in taking $\alpha$ and $\beta$ too large. This is standard practice in order to  obtain  a  qualitatively different  behaviour from Einstein AdS gravity.
This is common in phenomenological holographic models and some examples with commentary are [75, 76].

\item In the presence of the CFT,  as in de Sitter,  tensor modes can be tachyonic or not depending on the parameters $\tilde{\alpha}$, $\tilde{\beta}_{\text{eff}}$.
The situation is represented in figure \ref{fig:AdS}.
For fixed  AdS  curvature, there are tachyonic modes  for large and negative values of $\tilde{\b}_{\text{eff}}$ up to a certain critical value (which depends on $\tilde{\alpha}$ for large curvatures but is independent of $\tilde{\alpha}$ for small curvatures) above which the theory is tachyon-stable. The critical value is generically $\mathcal{O}(1)$. For a fixed $\tilde{\beta}_{\text{eff}}$ there are different possibilities: the theory may be  tachyon-stable ($\tilde{\beta}_\text{eff}$ large and positive, $\alpha \gtrsim 0$ ), or be tachyon-stable only above a certain curvature ($\tilde{\beta}_{\text{eff}}$ large and negative) or cross from tachyon-stability to instability to stability again for $\tilde{\beta}_{\text{eff}}\sim O(1)$ and $\alpha <0$. Unless  $|\tilde{\beta}_{\text{eff}}| \gg1$, the tachyonic modes are above the species cut-off.

\item The special case  $\tilde{a} = \tilde{\b}_{\text{eff}}=0$. Here,  AdS space-time is tachyon-stable for any curvature below the species cut-off.
\end{itemize}

\end{itemize}

Finally, we note that until now it was the scalar instability in de Sitter (or near de Sitter) that was employed as a mechanism for exiting inflation.
However, our results show that, depending on the parameters, the ``fastest'' instability may be in  the scalar or the tensor sector.
It should be stressed  though that if the fastest instability is the spin-2 one, this is a disaster for cosmology. The reason is that this instability generates large transverse
variations of the background metric destroying fast its homogeneity and therefore the main principle of cosmology.
Consequently, for cosmology, spin-2 instabilities must be avoided.

Up to specific details which may vary depending on the parameters, the general features of the spectra  discussed above can be  summarised as follows: \\

{\bf Pure gravity:}
\begin{itemize}
\item {\bf Minkowski:}
\begin{itemize}
\item scalar tachyon if $\alpha>0$
\item  if $\beta\neq 0$, two tensor modes: one massless graviton and one  massive ghost (tachyonic or not).
\end{itemize}
\item {\bf dS and AdS}
\begin{itemize}
\item scalar tachyon if $\alpha>0$ if $GN^2H^2\ll 1$.
\item  scalar (light) ghost if $\alpha \gg 1/(GH^2)$.
\item If $\beta\neq 0$,  two tensor modes, one massless  and one  massive. One of them is necessarily a ghost.
\end{itemize}
\end{itemize}
In all these cases, these ghosts/tachyons are below the cutoff  $M_p$ {\it only} if $|\alpha| \gg 1$ and/or  $|\beta| \gg 1$.\\

{\bf Gravity coupled to the CFT:}
\begin{itemize}
\item[-]
The bounds on the ghost/tachyon regions  vary, and there may be more massive tensor modes in the spectrum  (in particular in AdS).
\item[-] The cut-off is now lowered to the species scale, $M_p/\sqrt{N}$
\item[-] The presence of {\it light} ghosts/tachyons still requires  the effective coefficients of the $R^2$ and $R_{\mu\nu}R^{\mu\nu}$ to be large $|\tilde{\alpha}| \gg 1$ and/or  $|\tilde{\beta}_\text{eff}| \gg 1$.
\end{itemize}

\subsection{Discussion}

Our findings show that there are whole regions of parameter space where  the  holographic matter + gravity theory suffers from both scalar and tensor instabilities, for all signs  of the curvature. In particular, the unstable region contains the whole of flat space except eventually in the limit where we decouple the CFT.

Even though ghosts and tachyons seem ubiquitous, as we argued earlier, only when  the unstable modes are lighter than our EFT cut-off (\ref{sp-a})  do they signal an unequivocal instability. From our analysis, it emerges that, at small curvatures (compared to the cut-off),  if the renormalized coefficients of the local  quadratic curvature terms,   (\ref{tildes}),   are $O(1)$,  unstable modes generically  have masses above the UV cut-off $M_p/N$.  On the other hand, the presence of light ghosts or tachyons  requires very large values of the parameters  (\ref{tildes}).

It turns out that, for  large values of the higher curvature parameters (\ref{tildes}), one {\it also} finds light ghosts in pure quadratic-curvature gravity (in dS, AdS or Minkowski) without the CFT. Based on our analysis of parameter space, we can make the following statement: \\

\noindent
{\it Within the validity of EFT, for parameter values for which pure gravity shows no pathologies, neither does the gravity+CFT system.}\\

In other words, for  background curvatures  below the cut-off, light unstable modes   in the gravity + CFT  system are due  essentially to (effective) large {\it local} higher curvature terms which would result in the same instabilities in the pure gravity with the same  parameters. It should be remembered though, that in pure gravity the cutoff is taken to be the Planck scale while in the gravity+CFT system, the cutoff is taken to be the (renormalized) species cutoff in (\ref{spec}).

{ Note that if we insist instead on taking the EFT cut-off to be the Planck scale (rather than the species scale) even in the presence of the CFT, this conclusion changes, and we are led to the fact that coupling a healthy CFT to gravity {\it does} introduce instabilities within EFT. This  is another indication that the correct  cut-off is indeed $M_p/{\sqrt{N}}$. }

From the holographic standpoint, in the case of a CFT coupled to gravity, the scalar modes are the simplest since they do not propagate in the bulk, the only dynamical scalars are boundary degrees of freedom whose dynamics is determined by the $R^2$ terms and the conformal anomaly  [52].
Therefore, any  scalar instabilities can be simply traced purely to a local boundary gravity action.

{  The unstable scalar is a pure-gauge mode in Einstein gravity, but it becomes dynamical thanks to the higher curvature terms and the conformal anomaly, and depending on the coefficients,  it may become tachyonic and/or ghost-like.
In this context, scalar  instabilities were studied in 4-dimensional higher-curvature gravity  around flat space in [7]. Around de Sitter, scalar instabilities were investigated in [47] in the (original) Starobinsky model [46], and here we recover the results obtained in the linearized version of Vilenkin's analysis. It is worth mentioning that this model falls outside of the EFT description: indeed, in [46, 47] the 4d cosmological constant is set to zero, which fixes the dS curvature to satisfy $GN^2H^2 = 4\pi$. The value of $H$  is above the species  cut-off $1/(\sqrt{G}N)$  (although for large $N$ it may still be sub-Planckian). In this case, the no-tachyon condition  in the scalar sector is $\alpha >0$, (see equation (\ref{res-17}). However for phenomenological reasons, it is rather desirable to have a  scalar tachyon in order to leave the de Sitter solution in the early universe, and one should choose $\alpha <0$.}

Similar considerations can be made if we want to make a comparison  with  what goes nowadays under the name of the Starobinsky model for inflation\footnote{$\hat \alpha$ in equation (\ref{R+R2}) is related to our $\alpha$ as $\hat \alpha = {\alpha\over 384 \pi}$.} ,
\be \label{R+R2}
S = -\int\sqrt{-g}\left(  {1\over 16\pi G} R - \hat\alpha R^2 \right).
\ee
This may be thought of as a simplified version  of the   anomaly-driven realization of de Sitter in [46, 47] in which one neglects the non-local contribution from the conformal anomaly $\sim R^2\log R$. For this model, one does not need a CFT, but pure higher curvature gravity is enough.  The model (\ref{R+R2}) does not admit de Sitter solutions since the absence of the logarithmic term  pushes this solution to infinite curvature. However,  it admits quasi-de Sitter slowly-rolling FRW solutions (in an appropriately defined Einstein frame)  for $\hat \alpha <0$,  i.e. precisely where one expects a scalar tachyonic instability (see equation  (\ref{res-17}) for large curvature): it is this tachyon that eventually pushes the solution  away from the near-de Sitter geometry, thus ending inflation. This is the choice made in phenomenological models of inflation,  where the parameter
$$\alpha=384\pi\hat \alpha \sim - 5.95\times 10^{11}$$
to reproduce the amplitude of the primordial perturbation spectrum.  For $\hat \alpha<0$, the scalar mode is not a ghost in pure $R+R^2$ gravity, as can be seen by setting $N=0$ in (\ref{res-19}).
More details about how our results compare to  these models can be found in Appendix \ref{staro}.

A general discussion of higher-derivative gravity including the tensor modes can be found in [73], where it was pointed out that the $R_{\mu\nu}R^{\mu\nu}$ term gives rise to a tensor ghost around flat space. Here, we also extend this discussion to  de Sitter  and  AdS.

In the holographic context of gravity coupled to a large-$N$ CFT,  instabilities were already found in certain corners of parameter space by [68], and we agree with their results. A general analysis of the gravitational spectrum around de Sitter space was also performed in [52]  for a specific value of the dS Hubble parameter (namely $GN^2H^2 = 4\pi$)  for which the (renormalized) cosmological term is zero (this case however, is outside of the range of EFT since $R\sim H^2 > M_{species}$).    As we mentioned above, here we find that   these tensor instabilities are either outside of the EFT validity, or they require large effective values of the higher curvature coefficients which would make the pure gravity theory pathological as well. A more detailed comparison with [68] and [52] appears in appendix \ref{compa}.

When there are light tensor tachyons or ghosts, it is  interesting to ask  which direction in solution space the instability leads to. The scalar instability contains a homogeneous mode which can be understood as an instability of the de Sitter solution towards a more general FRW. These are the instabilities of the type considered in  [47]. However, non-homogeneous scalar instabilities and tensor instabilities break FRW.

A related  question is whether  this analysis around maximally symmetric space-times persists in more realistic cosmological solutions such as FRW. The same holographic setup used here can in principle be applied to FRW boundary metrics, by generalizing the bulk solution  along the lines of [54, 55, 57, 77].

This paper is organized as follows.
The setup of our work is presented in section \ref{section:action}, where we start from a theory of gravity in $AdS_5$ and obtain the boundary action with higher curvature terms induced on a regularized boundary.

Metric perturbations are set up in section \ref{bulk pert} for the bulk perspective, and in section \ref{bdy pert} for the boundary theory. Section \ref{bdy pert} also studies the dynamics of the pure boundary scalar perturbation.
The five remaining degrees of freedom for metric perturbations are contained in a transverse-traceless tensor studied in section \ref{linearized eq}, where its equation of motion is obtained.
In section \ref{decoupling} we discuss  tensor instabilities in pure gravity with quadratic curvature terms.
Tensor instability in the general case of the CFT  coupled to gravity is studied in section \ref{sec:flat correlator} for flat space-time, in section \ref{dS poles} for positive curvatures and in section \ref{AdS poles} for negative curvatures.

The appendix contains some of the technical details of this paper. We briefly review here the different sections.
In appendix \ref{app:free scalars}, we provide an explicit example of effective field theory which develops instabilities (ghosts and tachyons) due to the IR expansion. We also find that the mass of these unstable modes is always above the EFT cut-off.
Appendix \ref{renorm} reviews the computation of the counterterms for the bulk renormalization procedure [70].
In appendix \ref{staro}, we relate our setup to Starobinsky's inflation [46].
The geometry of AdS-slicing coordinates is reviewed in appendix \ref{app:AdS-slicing} where we map them to the more usual \textit{global coordinates} of AdS. We also remark that AdS-slicing coordinates are \textit{global} in the sense that they cover the whole AdS manifold.
Appendix \ref{bulk schrodinger} relates the bulk radial equation of spin-2 perturbation to a Schrodinger problem, which allows us to study the normalizability of its solutions.
In appendix \ref{app:tacscale}, we compute the decay rate of Minkowski spacetime in terms of the mass of a tachyonic pole.
Appendices \ref{dS criterion} and \ref{AdS criterion} derive the criteria (\ref{res-14}) for the spin-2 modes in dS and AdS respectively.
appendix \ref{app:legendre} studies the asymptotic behaviour of associated Legendre functions which enter into the solution for spin-2 perturbations in AdS-slicing coordinates of AdS.
Appendix \ref{quadratic action} computes the quadratic terms of the boundary action, which enter into the definition of the two-point functions
for metric perturbations.
Appendix \ref{gauge scalar} proves that one can discard an unphysical scalar mode appearing in the quadratic action, by showing that this mode is a pure gauge.
Appendix \ref{compa} compares our results to previous papers which have used a similar setup. We find the values of our parameters which reduce our setup to their case.
Finally, appendix \ref{moresnapshots} provides supplementary material concerning the poles of the spin-2 propagators in curved space.

The arXiv webpage of this paper contains supplementary material, including 5 animated gifs showing the poles of the spin-2 propagator for different choices of parameters ($GN^2\bar{R}$, $\tilde{\a}$ and $\tbe$). These gifs and their associated parameters are presented in the ``animated\_gifs.pdf" ancillary file.

\section{The theory}
\label{section:action}

We use the following notation for the various metrics:
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
symbol & name & relation/definition\\
\hline
$\mathcal{G}_{ab}$ & 5d bulk metric & \makecell{Fefferman-Graham \\coordinates (\ref{Graham-Fefferman})} \\
\hline
$ g_{\omega\sigma} $ & Fefferman-Graham slice metric & defined in (\ref{Graham-Fefferman}) \\
\hline
$g^{(0)}_{\omega\sigma} $ & 4d space-time metric & leading term of (\ref{hr10})\\
\hline
$\gamma_{\omega\sigma} $ & \makecell{induced metric on \\ the regulated boundary} & related to $g^{(0)}_{\omega\s}$ as (\ref{im})\\
\hline
$ \bar{\zeta}_{\omega\sigma} $ & \makecell{background 4d \\space-time metric} & \makecell{maximally symmmetric\\ background of $g^{(0)}_{\omega\s}$ (\ref{scalar0})} \\
\hline
$ \delta \zeta_{\omega\sigma} $ &  slice metric perturbation & defined in (\ref{b6a})\\
\hline
$ \delta \zeta^b_{\omega\sigma} $ & \makecell{$\delta \zeta_{\omega\s}$ evaluated \\ on the boundary} & \makecell{defined in (\ref{scalar0}),\\ gauge fixed in (\ref{scalar2})} \\ \hline
\end{tabular}
\end{center}

\subsection{Setup}
We consider a semi-classical theory of gravity in four dimensions,  described by a 4d metric $g^{(0)}_{\omega\sigma}$,   including quadratic curvature terms  and coupled to a 4d Conformal Field Theory (CFT). The total action is
\be
S = S_\text{grav} + S_\text{CFT}.
\label{s0}
\ee
The first term, $S_\text{grav}$, is the gravity action:
\be
S_\text{grav} = S_\text{EH} + S_\a + S_\b,
\label{s01}
\ee
which includes the Einstein-Hilbert plus the cosmological constant term\footnote{In our notation, the curvature tensors are understood to be those built from the metric $g^{(0)}_{\omega\sigma}$, unless otherwise specified explicitly.},
\be
S_\text{EH} = -{1 \over 16\pi G} \int d^4x \sqrt{g^{(0)}}(R - 2\Lambda),
\label{s02}
\ee
as well as  two quadratic curvature terms:
\be
S_\a = { \alpha\over 384\pi}\int d^4x \sqrt{g^{(0)}}R^2,
\label{s03}
\ee
\be
S_\b =  {\beta \over 64\pi} \int d^4x \sqrt{g^{(0)}}\left(R^{\omega\s}R_{\omega\s}-{1\over 3}R^2\right).
\label{s1}
\ee
Here, $G$ is the Newton constant,  $\Lambda$  the cosmological constant,  $\alpha$ and $\beta$ are the dimensionless $R$-squared couplings.

The second term $S_\text{CFT}$ in  (\ref{s0}) is the quantum effective action of a  CFT in  a background metric $g^{(0)}_{\omega\s}$,
and it is a functional of the background  metric $g^{(0)}_{\omega\s}$.

The action (\ref{s0}) is meant to be the {\em renormalized} action, in which all the divergences have been renormalized. The parameters $G,\Lambda,\alpha,\beta$ are therefore to be interpreted as  finite, physical  parameters left after the renormalization procedure (which will be described in detail in  subsection \ref{sec:renorm-action} and reference [66]).

Variation of the action with respect to the boundary metric yields the following Einstein equation:
\be
R_{\omega\s}-\frac{1}{2}R g^{(0)}_{\omega\s}+\Lambda g^{(0)}_{\omega\s} +8 \pi G \left( ^{(\a)}H_{\omega\s}+^{(\b)}H_{\omega\s} \right) =8\pi G   \langle T_{\omega\s} \rangle.
\label{EE}
\ee
\be
^{(\a)}H_{\omega\s}= \frac{\a}{96\pi}\left\{ \nabla_\omega \nabla_\sigma R-R R_{\omega\s}-\left(\Box R-\frac{1}{4}R^2 \right)g^{(0)}_ {\omega\s} \right\},
\label{Halpha}
\ee
\be
^{(\b)}H_{\omega\s}=\frac{\b}{32\pi}\left\{ \frac{1}{2} \left(R_{\kappa\lambda}R^{\kappa\lambda}-\frac{1}{3}R^2+\frac{1}{3} \Box R \right)g^{(0)}_{\omega\s}-2 R_{\omega\kappa\s\lambda}R^{\kappa\lambda}-\Box R_{\omega\s}  \right. \nn
\ee
\be
\qquad\qquad\qquad \left. +\frac{1}{3} \left( 2 R R_{\omega\s}+\nabla_\omega \nabla_\s R \right) \right\}.
\label{Hbeta}
\ee
In Eq. \eqref{EE}, the right-hand side is the renormalized CFT stress-energy tensor expectation value,
\be
\label{tcft} \langle T_{\omega\s} \rangle=\frac{2}{\sqrt{g_{(0)}}}\frac{\d S_{CFT} }{\d g^{(0)\omega\s} }.
\ee

Before we present the computation of the CFT stress tensor, we comment on the terms which are explicit on the left-hand side of (\ref{EE}). It is  important to remark that $ ^{(\b)}H_{\omega\s}$ is traceless. Furthermore, $^{(\a)}H_{\omega\s}$ is also traceless if the boundary has a constant curvature.

For convenience, in the following, we define the tensor
\begin{equation}
E_{\omega\s}\equiv -{16\pi G \over \sqrt{g^{(0)}}}{\delta S\over \delta g^{(0)\omega\s}}.
\end{equation}
We then write Einstein's equations as:
\be
\end{equation}
\par

From now on, we shall assume the CFT is a large-$N$ theory which has a holographic description in terms of a (semiclassical)  five-dimensional gravity dual.  We  shall  review how the renormalized stress tensor (\ref{tcft}) is computed in this context [70], and how the renormalized parameters of the effective gravity theory arise.
However, as we shall argue, our results do not depend on this assumption.
Below we present the main results  and give more details in appendix \ref{renorm}.

\subsection{Constructing the renormalized action} \label{sec:renorm-action}

To arrive  at (\ref{s0})  we replace the CFT contribution with its dual description, namely an  Einstein-Hilbert theory on a  5-dimensional manifold $\mathcal{M}$ (the bulk, on which the metric will be denoted by ${\mathcal G}$) together with covariant boundary terms on the boundary,  $\pa\mathcal{M}$. However, this is divergent and to regulate the divergences we move the boundary $ \pa\mathcal{M}$ to the regulated boundary $\pa\mathcal{M}_{\e}$ which is inside $\mathcal{M}$ and this also defines the regulated bulk space $\mathcal{M}_{\e}$. $\gamma$ is the induced metric on the regulated boundary.

The bare regularized gravity dual  action is:
\be
S_\text{reg} = S_\text{bulk} + S_\text{grav}^0.
\label{hr}
\ee
The first term is the usual Einstein-Hilbert action with a boundary $\partial \mathcal{M}_{\e}$,
\be
S_\text{bulk} = -M^3\left[\int_{\mathcal{M}_{\e} } d^5x \sqrt{\mathcal{G}}(R[\mathcal{G}] - 2\Lambda_{5}) -2 \int_{\partial\mathcal{ M}_{\e} } d^4x \sqrt{\gamma}K\right]. \label{hr0}
\ee
where $M$ is the $5$-dimensional Planck mass,  $\gamma$ is the determinant of the induced metric on $\partial \mathcal{M}_{\e} $ and $K$ is the corresponding extrinsic curvature
\footnote{Geometrical tensors follow the same conventions as Wald's book \textit{General Relativity}}.
The second term in (\ref{hr}) is a boundary term which depends only on intrinsic tensors on $\de {\mathcal M}_{\e}$,

\be
S_\text{grav}^0  = S_\text{EH}^0 + S_\a^0 + S_\b^0,  \label{hr0a}
\ee
\be
S_\text{EH}^0 = -{1 \over 16\pi G^0}\int d^4x \sqrt{\gamma}(R[\gamma] - 2\Lambda^0),
\label{hr0b}
\ee
a $R^2$ term
\be
S_\a^0 = {\alpha^0 \over 384\pi}\int d^4x \sqrt{\gamma}\left(R[\g]\right)^2,
\label{hr0c}
\ee
and an additional term proportional to the 4-dimensional Weyl anomaly\footnote{In our notation, the Latin letters will denote the bulk coordinates, and the Greek indices such as $\omega,\s$ denote $4$-dimensional slice coordinates.} [49]
\be
S_\b^0 = {\beta^0\over 64\pi} \int d^4x  \sqrt{\gamma}\left(R[\gamma]^{\kappa\lambda}R[\gamma]_{\kappa\lambda}-{1\over 3}\left(R[\g]\right)^2\right).
\label{hr0d}
\ee
This action looks similar to (\ref{s1}), and depends on a set of bare parameters $\alpha^0$, $\beta^0$, $\Lambda^0$ and $G^0$.  Below we shall relate these bare parameters to the physical ones ($\alpha$, $\beta$, $\Lambda$ and $G$) in the renormalized action (\ref{s0}).

We also define a length $L$ associated with the bulk cosmological constant,
\be
\Lambda_{5} = -{6\over L^2}. \label{hr5a}
\ee
We consider asymptotically AdS solutions, for which the ansatz for the full metric is written using Fefferman-Graham coordinates [78], given by
\be
ds^2 = \mathcal{G}_{ab}dX^adX^b= L^2\frac{d\rho^2}{4\rho^2} + \frac{1}{\rho}g_{\omega\s}(x,\rho)dx^{\omega} dx^\s,
\label{Graham-Fefferman}
\ee
where $L$ and $x^\s$ have the dimension of a length and $\rho$ is dimensionless. This coordinate system is the one of an asymptotically AdS space with a conformal boundary located at $\rho \rightarrow 0$. We define the  regulated boundary $\pa{\mathcal M}_{\e}$ as the hypersurface $\rho=\epsilon$, on which the induced metric  is:
\be
\gamma_{\omega\s}(\epsilon, x) = {1\over \epsilon}g_{\omega\s}(\epsilon,x) .
\label{im}
\ee
The  metric $g_{\omega\s}$ is determined by solving the bulk Einstein equation order by order in $\rho$ as $\rho \to 0$, starting with  an arbitrary metric $g^{(0)}_{\omega\s}$ to lowest order [70]:
\be
g_{\omega\s}(x,\rho) = g^{(0)}_{\omega\s} +  \rho g^{(2)}_{\omega\s} + \rho^{2}g^{(4)}_{\omega\s} + \hat{g}_{\omega\s}\rho^{2}\log{\rho} + \mathcal{O}(\rho^{3}).
\label{hr10}
\ee
The leading term in this expansion, $g^{(0)}_{\omega\s}$, is identified with the  metric of the dual field theory side.  The terms  $g^{(2)}_{\omega\s}$ and $\hat{g}_{\omega\s}$ are given by\footnote{Recall that all the geometrical tensors are built from the metric $g^{(0)}_{\omega\s}$ unless otherwise stated}:
\be
g^{(2)}_{\omega\s} = -{L^2\over 2}\left(R_{\omega\s} - {R\over 6}g^{(0)}_{\omega\s}\right),
\label{fg161}
\ee
\be
\hat{g}_{\omega\s} = {L^4\over 16}\left\{ 2R_{\omega \kappa \s \lambda}R^{\kappa\lambda} - {1\over 3} \nabla_\omega\nabla_\s R + \nabla^2 R_{\omega\s} - {2\over 3}RR_{\omega\s} + \left({1\over 6}R^2 -,\right.\right.
\label{fg261}
\ee
\left.\left.- {1\over 6}\nabla^2 R - {1\over 2}R_{\kappa\lambda}R^{\kappa\lambda}\right)g^{(0)}_{\omega\s} \right\}
These expressions are found by solving the bulk Einstein equation in a near-boundary expansion  [70].  Note that, comparing Eqs. \eqref{Hbeta} and \eqref{fg261}, we can write:
\begin{equation} \label{trghat}
\tensor[^{(\beta)}]{H}{_\omega_\s}  = -{\beta\over 2\pi }\hat{g}_{\omega\s}.
\end{equation}
Unlike $g^{(2)}_{\omega\s}$ and $\hat{g}_{\omega\s}$, $g^{(4)}_{\omega\s}$ is not fully determined from $g^{(0)}_{\omega\s}$, except for its trace, which is given by\footnote{As in [70], when matrix components are not written, it means that both matrix multiplication and trace operations are done using the metric $g^{(0)}$.} [70]:
\be
\tr\left[ g^{(4)}\right] = {1\over 4}\tr\left[ \left( g^{(2)}\right)^2\right].
\label{trg4}
\ee

Divergences of $S_\text{bulk}$, which arise when we remove the regulator and let $\epsilon \to 0$,  are made explicit when  $S_\text{bulk}$  is written in terms of $g^{(0)}_{\omega\s}$.
The method to obtain these divergences is briefly reviewed in appendix \ref{renorm}, resulting in:

\begin{equation}
S_\text{bulk} =  \frac{M^3}{L} \int d^4 x\sqrt{g^{(0)}}\left\{-{6\over \epsilon^{2}}  + {1\over 8} \log\epsilon \left(R^{\kappa\l}R_{\kappa\l} - {1\over 3}R^2\right) \right\}
+ \mathcal{O}(\epsilon^0), \label{Sdiv1}
\end{equation}
The first term in curly brackets contains all the divergent terms of $S_\text{bulk}$. These can also be written covariantly in a series expansion involving curvature tensors of the induced metric on the boundary.
\begin{equation}
S_\text{bulk} = \frac{M^3}{L} \int d^4x \sqrt{\gamma}\left\{ - 6  - {L^2\over 2} R[\g] + \right. \nn
\ee
\be
+ \left. {L^4\over 8}\left( {1\over 2} + \log \epsilon\right)\left(R^{\kappa\lambda}[\g]R_{\kappa\lambda}[\g] - {1\over 3}\left(R[\g]\right)^2\right)\right\} + ... \label{hr30}
\ee
where $...$ indicates higher curvature invariants.
The explicit $\epsilon$ dependence in (\ref{hr30}) reflects the conformal anomaly.

The  quadratic curvature term in (\ref{Sdiv1}) can be shifted by a finite amount by redefining the cut-off.   This scheme dependence is made explicit by introducing  as an extra parameter, a  scale $\m$, and defining the divergent part of the action $S_\text{div}$ as follows:
\be
S_\text{div} \equiv \frac{M^3}{L} \int d^4 x\sqrt{g^{(0)}}\left\{-{6\over \epsilon^{2}}  + {1\over 4} \log(\sqrt{\epsilon} \m L) \left(R^{\omega\s}R_{\omega\s} - {1\over 3}R^2\right) \right\}.
\label{ig1e}
\ee

We now turn to the bare ``boundary''  gravitational action (\ref{hr0a}). It is also divergent in the limit $\epsilon \to 0$. This  can be made manifest by expressing it  in terms of curvature tensors of $g^{(0)}_{\omega\s}$ using the expansion (\ref{hr10}) and the expressions (\ref{fg161}-\ref{fg261}). The result for the Einstein-Hilbert part is
\be
S_\text{EH}^0 = -{1\over 16\pi G^0} \int d^4x \sqrt{\gamma}(R[\gamma] - 2\Lambda^0),
\ee
\be
=  -{1\over 16\pi G^0} \int d^4x \sqrt{g^{(0)}} \left\{-{2\Lambda_0 \over \epsilon^2 } + {1\over \epsilon}\left(1 + {\Lambda_0L^2\over 6}\right)R + \right.
\nonumber
\ee
\be
\left.  + {L^2\over 4}\left(2 + {\Lambda_0L^2\over 4}\right)\left(R^{\omega\s}R_{\omega\s} - {1\over 3}R^2\right) + \mathcal{O}(\epsilon)\right\},
\label{ig1}
\ee
while $S_{\a}^{0}$ and $S_\b^0$ are finite. Note that  additional finite quadratic terms appear when we expand $S_{EH}^0$ in powers of $\epsilon$.

Therefore, the full boundary action  $S_\text{grav}^0$  written in terms of $g^{(0)}_{\omega\s}$, including   all divergent and finite terms,
has the form:
\be
S_\text{grav}^0 = -{1\over 16\pi G^0} \int d^4x \sqrt{g^{(0)}} \left\{-{2\Lambda_0 \over \epsilon^2 } + {1\over \epsilon}\left(1 + {\Lambda_0L^2\over 6}\right)R \right\} +
\nonumber
\ee
\be
+ \left[\beta^0 - {1\over 16\pi G^0}{L^2\over 4}\left(2 + {\Lambda_0L^2\over 4}\right)\right]\int d^4x \sqrt{g^{(0)}} \left(R^{\omega\s}R_{\omega\s} - {1\over 3}R^2\right)
\nonumber
\ee
\be
+ \alpha^0\int d^4x \sqrt{g^{(0)}} R^2 + \mathcal{O}(\epsilon).
\label{ig1bis}
\ee

The renormalisation procedure we adopt consists in taking the limit $\epsilon \to 0$  by choosing appropriately the bare parameters  ($\Lambda^0$, $G^0$, $\a^0$ and $\b^0$) as a function of the cut-off, such that the quantity
\be\label{ig1b}
S_\text{grav} \equiv S^0_\text{grav} + S_{div}
\ee
remains finite\footnote{This is different from  the standard holographic renormalization procedure [70], in which the bare parameters ($\Lambda^0$, $G^0$, $\a^0$ and $\b^0$) are independent on the cut-off and  a counterterm action (whose coefficients are completely fixed) is introduced to cancel all divergences coming from $S_\text{bulk}$ (\ref{hr30}). This  leaves only finite quadratic curvature terms in the renormalized action and no Einstein-Hilbert term.} in the limit $\epsilon\to 0$.

We write the resulting finite action in terms of new \textit{physical} parameters ($\Lambda$, $G$, $\alpha$ and $\beta$), each  corresponding  to the one  boundary term:

\be
\Lambda = {1\over \epsilon}\left(1 + {\Lambda^0L^2\over 6}\right)^{-1} \left[\Lambda^0 - {48 \pi G^0 M^3 \over  L} \right],
\label{ig7}
\ee
\be
G = {\epsilon G^0\over 1+{\Lambda^0L^2\over 6}},
\label{ig8}
\ee
\be
\alpha = \alpha^0,
\label{ig9}
\ee
\be
\beta = \beta^0 + 16\pi M^3 L^3 \log(\sqrt{\epsilon} L\m) - {2 L^2 \over G^0}\left(1 + {\Lambda^0 L^2\over 8}\right).
\label{ig10}
\ee
and we take $\epsilon \to 0$ together with appropriate limits of ($\Lambda_0$, $G_0$, $\alpha_0$ and $\beta_0$) so that the left hand sides are finite.

Finally, combining (\ref{hr}) and (\ref{ig1b}) we write the renormalized action as
\be
S = \underset{\epsilon \rightarrow 0}{\lim}\left[S_\text{bulk} - S_\text{div} \right] + S_\text{grav},
\label{ig11}
\ee
\be
\equiv S_\text{CFT} + S_\text{grav},
\label{ig11a}
\ee
i.e. equation (\ref{s0}). The bulk contribution inside the square brackets is interpreted as the renormalized effective action of the CFT.

\subsection{The induced stress tensor}
The renormalized stress tensor is defined by
\be
\left<T_{\omega\s}\right> = \underset{\epsilon\rightarrow 0}{\lim}  \left[{1\over \epsilon}{2\over \sqrt{\g}}{\delta S_{CFT}\over \delta \g^{\omega\s}} \right]  =  {2\over \sqrt{g^{(0)}}}{ \delta S_\text{CFT}\over \delta g^{(0)\omega\s}}.
\label{hr42}
\ee
It can be shown that this definition leads to
\be
{2\over \sqrt{\g}}{\delta S_{bulk}\over \delta \g^{\omega\s}} = 2 M^3 (K_{\omega\s} - K\gamma_{\omega\s}).
\label{hr38}
\ee
As shown in [70] the divergent pieces of (\ref{hr38}) cancel the ones of the  $S_{\text{div}}$. We are then left with the renormalized stress tensor given by \footnote{For a notational simplification, there is no difference in subscript and superscript in Fefferman-Graham metric expansion. }
\be
\left<T_{\omega\s}\right>  = - {2 M^3\over  L} \left\{2\left[ 2\log(\m L) - 1\right] \hat{g} - 2g^{(4)} + \left( g^{(2)}\right)^2 - {1\over 4}g^{(0)}\tr\left[ \left( g^{(2)}\right)^2\right] +  \right. \nn
\ee
\be
\left. + {1\over 4}g^{(0)}\left( \tr\left[ g^{(2)}\right]\right)^2 -{1\over 2}g^{(2)}\tr\left[ g^{(2)}\right]\right\}_{\omega\s},
\label{stress}
\ee
where $g^{(2)}_{\omega\s}$, $g^{(4)}_{\omega\s}$ and $\hat{g}_{\omega\s}$ are the terms of the Fefferman-Graham expansion (\ref{hr10}), the expressions for $g^{(2)}_{\omega\s}$ and $\hat{g}_{\omega\s}$ in terms of $g^{(0)}_{\omega\s}$ are given in Eqs. (\ref{fg161},\ref{fg261}). $g^{(4)}_{\omega\s}$ must be obtained from solving the bulk dynamics. The stress-tensor expectation value in (\ref{stress}) is to be inserted into the right-hand side of the Einstein equation (\ref{EE}).

Even if the stress-tensor is not fully constrained by the boundary data, its trace is known using (\ref{trg4}). It gives
\be
g^{(0)\omega\s } \left<T_{\omega\s}\right> = {(M L)^3\over 4  }\left(R^{\omega\s}R_{\omega\s} - {1\over 3}R^2\right).
\label{hr43aa}
\ee

In a generic  CFT with a 5d gravity dual, the parameter $M^3L^3$ is large, and  proportional to the central charge\footnote{Recall that in holographic  CFTs, the two    central charges $a$ and $c$ are equal, up to $1/N^2$ corrections} $a$:
\be
(M L)^3 = {a \over 2\pi^2}.
\label{hr43ab-ii}
\ee

When the CFT is a large-$N$ gauge theory in 4d, then  $a\propto N^2$
For example, in ${\cal N}=4$ SYM we have, in the large-$N$ limit:
\be
a =  {N^2\over 4}.
\label{hr43ab-iii}
\ee

In what follows  we  assume, for definiteness, the  ${\cal N}=4$ relation (\ref{hr43ab-iii}), and set:
\be
M^3 L^3 = {N^2\over 8 \pi^2}.
\label{hr43ab}
\ee
This will allow us to replace  $M^3 L^3$ with $N^2$ and write all equations which pertain to the field theory side purely in terms of 4d parameters.  Readers can keep in mind that, for any other CFT (even  for those that are not large-$N$) they can substitute
$$N^2 \to  4 a.$$

\subsection{Background solutions}
\label{slicings}

In this section, we  discuss the background (i.e. homogeneous) solutions of the equations of motion for the 5d theory (\ref{hr}).
We take these solutions to be the  $AdS_5$ metric with three different maximally symmetric slicings,
\be
ds_5^2 = L^2du^2 + a^2(u) \bar{\zeta}_{\omega\s} dx^\omega dx^\s,
\label{b0}
\ee
where $a(u)$ is a dimensionless scale factor and  the slice metric $\bar{\zeta}_{\omega\s}$ is a $u$-independent  maximally symmetric 4d metric.   This results in three possible coordinate systems for AdS$_5$,  that correspond to the dual CFT on three distinct four-dimensional maximally symmetric metrics: AdS$_4$, dS$_4$ and M$_4$.

\begin{itemize}
\item $\bar \zeta$ being the Minkowski metric. In this case
\be
a(u) = e^{u},
\label{b0flat}
\ee
where $u >0$ and the AdS boundary is located at $u\rightarrow +\infty$.

\item $\bar \zeta$ being the de Sitter metric  with Hubble curvature $H$, in which case
\be
a(u) = LH \sinh u,
\label{b0dS}
\ee
where $u\in \mathbb{R}$. $u=0$ is a horizon. From now on, we take $u$ positive. Therefore, $u \rightarrow + \infty$ is the $AdS_5$ boundary. The curvature of dS is given by
\begin{equation}
\bar{R} =12 H^2.
\label{b0H}
\end{equation}

\item $\bar \zeta$ being the Anti-de Sitter metric  with radius $\chi^{-1}$, in which case
\be
a(u) = L\chi \cosh u.
\label{b0AdS}
\ee
In this case, there are two  asymptotic boundaries, located at $u=\pm \infty$. These two boundaries are connected. More details for the geometry of AdS-slicing coordinates are given in appendix \ref{app:AdS-slicing}.
In the field theory interpretation, these correspond to two independent copies of the CFT on AdS$_4$ that interact via their common AdS$_4$ boundary\footnote{A conformal rescaling of such a setup corresponds to an interface between two copies of the same CFT in flat space, see the extended discussion in [64].}.  It is a matter of choice whether only one or both are coupled to dynamical metric perturbations, as we shall discuss in section \ref{AdS prop}. Since there is no horizon at $u=0$, we shall observe  that both sides are reachable by the bulk metric perturbations. The boundary curvature is related to $\chi$ by:
\begin{equation}
\bar{R} =-12\chi^2.
\label{b0chi}
\end{equation}
\end{itemize}

The bulk metric (\ref{b0})  can also be written in Fefferman-Graham coordinates as
\be
ds_5^2 = L^2 {d\rho^2\over 4\rho^2} +{1\over \rho} f(\rho) ds_4^2.
\label{b1}
\ee
where $ds_4^2 = \bar{\zeta}_{\omega\s}dx^\omega dx^\s $ and the function $f$ for each different slicing is then given by
\be
\label{tabular slicings}
\begin{tabular}{|c|C|C|C|}
\hline
space-time & ds_4^2 & \rho & f(\rho)\\
\hline
M$_{4}$ & ds^2_\text{flat} & \rho = e^{-2u} & 1 \\
\hline
dS$_4$ & ds^2_{dS} & \rho = \left({2\over LH}\right)^2 e^{-2u} &  f_{dS}(\rho) = 1 - {(LH)^2\over 2}\rho + \left({LH\over 2}\right)^4\rho^2 \\
\hline
AdS$_4$ & ds^2_{AdS} & \rho = \left({2\over L\chi}\right)^2 e^{-2\text{sign}(u)u} & f_{AdS}(\rho) = 1 + {(L\chi)^2\over 2}\rho + \left({L\chi\over 2}\right)^4\rho^2 \\
\hline
\end{tabular}
\ee

In these coordinates, $\rho >0$ and the $AdS_5$ boundary is located at $\rho = 0$. In de Sitter slicing,  we are free to choose a sign of $u$ (here we took positive $u$) because of the horizon $u=0$ which separates the two sides of the bulk.
However, in $AdS$ slicing where there is no such horizon, the bulk $AdS_5$ needs two different Fefferman-Graham patches such that $\rho \rightarrow 0$ is the $AdS_5$ boundary $u\rightarrow \pm \infty$. Hence the $\text{sign}(u)$ in the expression for $\rho$ in AdS-slicing.\footnote{The global embedding of dS and AdS slices in AdS$_5$ is discussed in detail in [79] for dS, and in [72] for AdS.}

The background solutions (\ref{b1}) are then related to the general Fefferman-Graham expansion (\ref{Graham-Fefferman}) by
\be
\left. g_{\omega\s}(x,\rho)\right|_\text{background} = f(\rho)\bar{\zeta}_{\omega\s},
\label{b2}
\ee
from which every term of the expansion (\ref{hr10}) are fixed. In particular,   we can read off  the corresponding boundary theory metric $g^{(0)}_{\omega\s}$ as the leading term as $\rho \to 0$:
\be \label{bc-iii}
g^{(0)}_{\omega\s} = \bar{\zeta}_{\omega\s}
\ee
and it  is either the  Minkowski metric or the de Sitter metric with Hubble scale $H$, or the anti-de Sitter metric with AdS length  $\chi^{-1}$. We denote   the background curvature  $R[\bar{\zeta}] \equiv \bar{R}$,
\bar R= \left\{ \begin{array}{lll}
\displaystyle 12H^2,&\phantom{aa} &{\rm de~~ Sitter},\\ \\
\displaystyle 0 ,&\phantom{aa}&{\rm Minkowski} \\ \\
\displaystyle -12\chi^2 ,&\phantom{aa}& {\rm Anti~~ de~~ Sitter}
\end{array}\right.
\label{bc-ii}\ee}
For a maximally symmetric background, the trace of the Einstein equation (\ref{EE}) gives
\be
\Lambda = {1\over 4}\left(\bar{R} - {GN^2\bar{R}^2\over 48\pi}\right),
\label{scalar7}
\ee
where $N$ was defined in (\ref{hr43aa}). Note that, for each value of the boundary parameters $\Lambda$ and $G$, there are either two  values of the curvature $\bar{R}$ satisfying equation (\ref{scalar7}), or there are none. On the other hand, by scanning all values of $\Lambda$, we can obtain any value of $\bar{R}$. Therefore, it is convenient to  trade $\Lambda$ for $\bar{R}$: in what follows we shall replace  $\Lambda$ in terms of  $\bar{R}$ using  (\ref{scalar7}) in all equations.
This leaves $GN^2\bar{R}$ as the only dimensionless background curvature parameter.

Equation (\ref{scalar7}) does not depend on $\alpha$ or $\beta$ since they multiply tensors that are traceless when evaluated on the background metric $\bar{\zeta}_{\omega\s}$ with constant curvature.

The maximally symmetric backgrounds were discussed in detail (for  holographic CFTs and holographic RG flows on de Sitter) in [66]. We now move to the perturbations around these background solutions. These are described by turning on perturbations in both the bulk and the boundary metric and solving the corresponding Einstein's equation and boundary conditions. This will be the subject of the next section.

\section{Bulk metric perturbations}
\label{bulk pert}

Equation (\ref{b2})  holds for the unperturbed, background metric. In this section, we study perturbations of the bulk metric, by adopting  the same gauge invariant decomposition of metric perturbations as in [52].

In a perturbed geometry, the bulk metric reads
\be
ds_5^2 = (\mathcal{G}_{ab} + \delta \mathcal{G}_{ab})dX^a dX^b.
\label{b6}
\ee
Using (\ref{b0}), one can relate the slice component perturbations to actual perturbations of the slice metric $\delta\zeta_{\omega\s}$ defined as
\be
\delta \mathcal{G}_{\omega\s} = a^2(u)\delta \zeta_{\omega\s},
\label{b6a}
\ee
such that the full metric can now be written as
\be
ds_5^2 = (\mathcal{G}_{uu} + \delta \mathcal{G}_{uu})du^2 + 2(\mathcal{G}_{u\s} +\delta \mathcal{G}_{u \s}) dudX^\s + a^2(u)(\bar{\zeta}_{\omega\s} + \delta \zeta_{\omega\s})dx^\omega dx^\s.
\label{b6b}
\ee

Even if a $5\times5$ symmetric matrix contains 15 independent elements,
only 10 degrees of freedom are invariant under the gauge transformation
\be
\delta \mathcal{G}_{ab} \rightarrow \delta \mathcal{G}_{ab} + 2\nabla^{(\mathcal{G})}_{\left(a\right.}\xi_{\left.b\right)}.
\label{b7}
\ee
One can construct these 10 invariant quantities by decomposing the perturbation  $\mathcal{G}_{ab}$ into transverse and traceless elements for the slice covariant derivative $\hat{\nabla}$ built with $\zeta_{\omega\s}$ as follows [52]:
\be
\delta \mathcal{G}_{uu} = A,
\label{b8}
\ee
\be
\delta \mathcal{G}_{u\s} = B_\s + \hat{\nabla}_\s B,
\label{b9}
\ee
\be
\delta \zeta_{\omega\s} = h_{\omega\s} + 2\hat{\nabla}_{(\omega}\chi_{\s)} + \bar{\zeta}_{\omega\s}\psi + \hat{\nabla}_{(\omega}\partial_{\s)} \phi,
\label{b10}
\ee
where $B_\s$, $\chi_\s$ are transverse and $h_{\omega\s}$ is transverse-traceless:
\be
\hat{\nabla}^\s\chi_\s = 0 = \bar{\zeta}^{\omega\s}h_{\omega\s},
\label{b11}
\ee
\be
\hat{\nabla}^\omega h_{\omega\s} = 0.
\label{b12}
\ee
As it is well known (and rederived in [52] in the present context), the only propagating degree of freedom in the bulk of  $AdS$ with pure gravity is the \textit{tensor} (transverse-traceless) perturbation $h_{\omega\s}$  which contains 5 degrees of freedom. On top of this, there exists a  scalar mode which has purely boundary dynamics, and that will be discussed in the next section. Therefore, here we set to zero all components of the perturbation except for the tensor mode.

The next step is  to obtain the equation of motion for this tensor mode. The bulk Einstein equation is:
\be
R_{ab}[\mathcal{G}] = -{4\over L^2}\mathcal{G}_{ab}.
\label{b13}
\ee

When linearized with respect to $h_{\omega\s}$, the above  equation yields:
\be
(L^2 \nabla^{(\mathcal{G)}2}  + 2 )a^2(u)h_{\omega\s} = 0,
\label{b14}
\ee
where the differential operator into parenthesis is known as the Lichnerowicz operator for AdS. This operator can be decomposed into the $(u,x^\omega)$ slicing coordinates (\ref{b0}). Equation (\ref{b14}) then takes the following form
\be
\left\{\partial_u^2 + 4 {a'\over a} \partial_u + 2\left[1-\left({a'\over a}\right)^2\right]  + L^2a^{-2}\hat{\nabla}^2\right\} h_{\omega\s} = 0.
\label{b14bis}
\ee
This equation will be specialized to different slicings and solved in section \ref{linearized eq}.

Tensor perturbations $h_{\omega\s}$ can be expanded in a similar way as in (\ref{Graham-Fefferman}):
\be
h_{\omega\s} = h^{(0)}_{\omega\s} + \rho h^{(2)}_{\omega\s} + \rho^2 h^{(4)}_{\omega\s} + \rho^2 \log \rho \hat{h}_{\omega\s} + \mathcal{O}(\rho^3).
\label{b15}
\ee

We shall now  linearize the  boundary Einstein field equation (\ref{EE}) and obtain an equation which involves the various terms in the near-boundary expansion (\ref{b15}). To this end, we need to relate perturbations of the metric $g_{\omega\s}$ defined in (\ref{Graham-Fefferman}) to the slice perturbations $h_{\omega\s}$ defined in (\ref{b10}).
We introduce the following notation, for any tensor $A$ of the slice metric:
\be
(\delta_h A)[\bar{\zeta}] \equiv \underset{\varepsilon \rightarrow 0}{\lim} {A[\bar{\zeta} + \varepsilon h^{(0)}] - A[\bar{\zeta}]\over \varepsilon}.
\label{dS14a}
\ee
We identify term by term the expansion (\ref{hr10}) with the expansion of the bulk metric (\ref{b0}) close to the boundary $\rho\rightarrow 0$ where $\rho(u)$ is given in table \ref{tabular slicings}. The result for both AdS and dS is given by
\begin{subequations}
\label{dS13aaa}
\be
\delta_h g^{(0)}_{\omega\s} = h^{(0)}_{\omega\s},
\label{dS13ab}
\ee
\be
\delta_h g^{(2)}_{\omega\s}  =  h^{(2)}_{\omega\s} - {L^2R\over 24}h^{(0)}_{\omega\s},
\label{dS13ac}
\ee
\be
\delta_h g^{(4)}_{\omega\s} =  h^{(4)}_{\omega\s} -  {L^2R\over 24} h^{(2)}_{\omega\s} + \left({L^2R\over 48}\right)^2h^{(0)}_{\omega\s},
\label{dS13ad}
\ee
\be
\delta_h \hat{g}_{\omega\s} = \hat{h}_{\omega\s}.
\label{dS13ae}
\ee
\label{i11b}
\end{subequations}

The quantities  $h^{(2)}$ and $\hat{h}$ can be written in  terms of $h^{(0)}$ and  of boundary curvature tensors: as is summarized in appendix \ref{renorm}, $g^{(2)}_{\omega\s}$ and $\hat{g}_{\omega\s}$ are obtained in terms of $g^{(0)}$ by solving perturbatively the bulk Einstein equation for small $\rho$ [70]. By varying the solution for $g^{(2)}$ and $\hat{g}$ given in (\ref{fg16}, \ref{fg26}) with respect to $h^{(0)}_{\omega\s}$, we obtain using (\ref{dS13aaa}):
\be
h^{(2)}_{\omega\s} = {L^2\over 4}\left(\nabla^2- {R\over 6}\right)h^{(0)}_{\omega\s},
\label{b16}
\ee
\be
\hat{h}_{\omega\s}= -{2\pi L^4\over\b}  \delta_h \tensor[^{(\b)}]{H}{_\omega_\s} = -{L^4\over 32}\left( \nabla^2 - \frac{R}{6}\right)\left( \nabla^2 - \frac{R}{3}\right)h^{(0)}_{\omega\s},
\label{b17}
\ee
where the laplacian operator $\nabla^2$ is constructed with the Fefferman-Graham metric $g^{(0)}_{\omega\s}$.
On the contrary, we need to solve (\ref{b14}) in the whole bulk (with appropriate conditions in the interior) to find $h^{(4)}_{\omega\s}$. We postpone this to section 5.

Using the equations above, all linearized quantities can be expressed purely in terms of  $h^{(0)}_{\omega\s}$ and $h^{(4)}_{\omega\s}$,  which for now are independent.

The variation of the holographic stress-tensor (\ref{stress}) in the presence of a perturbation $\delta h_{\mu\nu}$ is given by:
\be
\delta_h \left<T_{\omega\s}\right> = {N^2\over 2\pi^2L^4}\left[\delta_h g^{(4)}_{\omega\s} - \left({L^2R\over 24}\right)^2\delta_h g^{(0)}_{\omega\s} + (1-2\log (\m L))\delta_h \hat{g}_{\omega\s}\right],
\label{dSTh}
\ee
where $\delta_h g^{(4)}_{\omega\s}$, $\delta_h g^{(0)}_{\omega\s}$ and $\delta_h \hat{g}_{\omega\s}$ have to be written using equations (\ref{dS13ab}-\ref{dS13ae}) and  (\ref{b16}-\ref{b17}).

We now turn to the linearization of the left-hand side of Einstein's equation (\ref{EE}), in which the cosmological constant can be replaced by a function of the background curvature $\bar{R}$ using (\ref{scalar7}).  Note that the CFT stress-tensor also contributes to the value of $\Lambda$ through the trace of the background Einstein equation (\ref{scalar7}). By moving all the CFT contributions (i.e. those proportional to $N$) to the right-hand-side of the linearized Einstein equation, we find
\be
\left(-\nabla^2 + {R\over 6}\right)h^{(0)}_{\omega\s} + 8\pi G\delta_h (\tensor[^{(\alpha)}]{H}{_\omega_\s} + \tensor[^{(\beta)}]{H}{_\omega_\s})  =  8\pi G \delta_h \left<T_{\omega\s}\right>^T,
\label{dSTh1}
\ee
\be
\left<T_{\omega\s}\right>^T \equiv  \left<T_{\omega\s}\right> - {1\over 4}g^{(0)}_{\omega\s}\left<T^\m_\m\right>.
\label{dSTh2}
\ee
The curvature squared terms $\tensor[^{(\a)}]{H}{_\omega_\s}$ and $\tensor[^{(\b)}]{H}{_\omega_\s}$ are then linearized with respect to the tensor perturbation. Then, equation (\ref{dSTh1}) is written as a sum of contributions from the Fefferman-Graham terms $h^{(0)}_{\omega\s}$, $h^{(4)}_{\omega\s}$ and $\hat{h}_{\omega\s}$:
\be
h^{(4)}_{\omega\s} + {L^4R\over 24}\left\{{3\pi \over GN^2R}- {1\over 4} - {\pi\alpha\over 4 N^2} \right\}\left(\nabla^2 -{R\over 6} \right)h^{(0)}_{\omega\s} + \left(1 - 2\log(\m L) + \beta {\pi\over N^2}\right)\hat{h}_{\omega\s} = 0,
\label{dS13b}
\ee
where $\hat{h}_{\omega\s}$ is to expressed in terms of $h^{(0)}_{\omega\s}$ using (\ref{b17}).

Equation (\ref{dS13b}) is a linear equation relating $h^{(4)}_{\omega\s}$ to $h^{(0)}_{\omega\s}$. As usual in holography however,  $h^{(4)}_{\omega\s}$ is determined by  $h^{(0)}_{\omega\s}$ by solving the bulk equation and imposing a regularity condition in the interior. This makes $h^{(4)}_{\omega\s}$ into a (non-local) linear functional of $h^{(0)}_{\omega\s}$.  Therefore, all in all, equation (\ref{dS13b}) takes the form of a dynamical equation for  $h^{(0)}_{\omega\s}$, of the form:
\be \label{bb21}
{\mathcal F}^{\mu\nu\omega\sigma}(\nabla^2, \bar{R})  h^{(0)}_{\omega\s} = 0.
\ee

Determining the explicit form of the functional ${\mathcal F}$ will be the goal of section 5. Here  we conclude by the remark that equation  (\ref{bb21}) can also be obtained by varying the quadratic part of the action (\ref{s0}) evaluated on-shell: indeed, as  shown in appendix  \ref{quadratic action},  once it is evaluated on the solution of the linear  bulk equation,  the quadratic part of the action (\ref{s0}) is equal to the boundary expression:
% %
\be
S^{(2)}[h^{(0)}] =  {N^2\over 2\pi^2}\int d^4 x\,  \sqrt{\bar{\zeta}}h^{(0)\omega\s }\left\{h^{(4)}_{\omega\s} + \left({{\pi\beta\over N^2}} + 1 - 2\log(\m L)\right)\hat{h}_{\omega\s} + \right.
\nonumber
\ee
\be
\left. + {RL^4\over 24}\left(\nabla^2 - {R\over 6} \right) \left({3\pi\over GN^2R}- {1\over 4} -{\pi\a\over 4N^2} \right) h^{(0)}_{\omega\s}  \right\}.
\label{res11}
\ee
Using (\ref{b17}) for $\hat{h}$ and the determination of $h^{(4)}$ in terms of $h^{(0)}$ from the bulk solution, this expression can be written again as a quadratic functional of $h^{(0)}$:
\be
S^{(2)}[h^{(0)}]  = \int d^4x \,\sqrt{g^{(0)}}\int d^4y\,\sqrt{g^{(0)}}  h^{(0)\mu\nu} (x){\mathcal F}_{\mu\nu\omega\sigma}(\nabla^2, \bar{R})[x,y]  h^{(0)\omega\s(y)}
\label{S2_1}
\ee
where ${\mathcal F}$ is the same functional which gives the  equation of motion (\ref{bb21}), as it is clear  by varying (\ref{S2_1}) with respect to $ h^{(0)\mu\nu}$.  The quantity ${\mathcal F}_{\mu\nu\omega\sigma}$  is the  inverse propagator of the induced boundary gravity tensor fluctuations $h^{(0)}_{\omega\s}$:

\be
\mathcal{F}_{\mu\nu\omega\sigma} \equiv {1\over \sqrt{g^{(0)}(x)}} {1\over \sqrt{g^{(0)}(y)}} {\delta^2 S^{(2)}\over \delta h^{(0)\mu\nu}(x)\delta h^{(0)\omega\s}(y)}
\label{b20}
\ee

Using   the definition (\ref{ig11a}) in the quadratic action,
\be
S^{(2)} = S^{(2)}_\text{grav} +  S^{(2)}_{CFT},
\ee
the right-hand side of equation (\ref{b20}) can be seen as the sum of two contributions, one from  $S_\text{grav}$ and one from $S_{CFT}$.

As $S_\text{grav}$ is local (it is  quadratic in the boundary curvature), the   first contribution is a {\it local} 4-derivative differential operator,
\be  \label{b22-i}
{1\over \sqrt{g^{(0)}(x)}} {1\over \sqrt{g^{(0)}(y)}} {\delta^2 S^{(2)}_\text{grav}\over \delta h^{(0)\mu\nu}\delta h^{(0)\omega\s}} =  \delta(x,y) O_{\mu\nu\rho\sigma}(\nabla^2,\bar{R}) .
\ee
The part coming from the CFT is by definition the renormalized stress tensor correlator of the  CFT:
\be\label{b22}
{1\over \sqrt{g^{(0)}(x)}} {1\over \sqrt{g^{(0)}(y)}} {\delta^2 S^{(2)}_{CFT}\over \delta h^{(0)\mu\nu}\delta h^{(0)\omega\s}} = - \< T_{\mu\nu}(x) T_{\omega\s}(y) \>_{CFT}.
\ee
Therefore, the full inverse graviton propagator (\ref{b20}) has the form :
\be \label{b23}
\mathcal{F}_{\mu\nu\omega\sigma} = \delta(x,y) O_{\mu\nu\rho\sigma}(\nabla^2,\bar{R})   - \< T_{\mu\nu}(x) T_{\omega\s}(y) \>_{CFT}.
\ee
The non-local part is fully contained in the term $h^{(4)}$ in equation (\ref{res11}), and to determine it one has to solve the bulk radial equations.

We make a final comment on the appearance  of the bulk AdS radius $L$ in equation (\ref{res11}). As $L$ is {\it not} a parameter of the 4d theory (only $M L$ is, see equation (\ref{hr43ab-ii}), this quantity should not enter the full spectral operator ${\cal F}_{\mu\nu\rho\sigma}$. This is indeed the case: as will become obvious  with the explicit computations in section \ref{linearized eq}, there is a similar logarithmic contribution  to (\ref{res11}) coming  from $h^{(4)}$, which will effectively replace $\log \mu L \to -2\log (2\mu \sqrt{G} N)$.  These terms come from the variation of Weyl anomaly in the CFT, which has the form of the term with coefficient $\beta$ in the gravitational action (see  equation (\ref{s1})).  This implies that effectively  $\beta$ and $\mu$ will always appear in the combination:
\be \label{b24}
\beta_\text{eff} = \beta - {N^2 \over \pi}\log (4\mu^2 G N^2).
\ee

\section{The boundary scalar perturbation}
\label{bdy pert}

In this section, we focus on scalar perturbations\footnote{Here by scalar we mean with respect to the slice isometry group.}. This scalar mode decouples in pure  Einstein's gravity but reappears due to the higher curvature terms in the boundary action\footnote{If the QFT is not conformal, then the coupling of the QFT to gravity will also contribute to the scalar dynamics via the two-point function of the trace of the energy-momentum tensor.}

In the present setup,  where gravity lives on the boundary of  $AdS_5$, the scalar mode exists only on the boundary because only tensor perturbations are dynamical in the bulk (see e.g. [52]) as the trace of the energy-momentum tensor has trivial dynamics in a CFT.

\subsection{Gauge fixing}

To study the dynamics of this  boundary scalar mode, we define  boundary metric perturbations:
\be
ds_4^2 =  g^{(0)}_{\omega\s} dx^\omega dx^\s = (\bar{\zeta}_{\omega\s} + \delta \zeta_{\omega\s}^{b})dx^\omega dx^\s,
\label{scalar0}
\ee
where $\bar{\zeta}_{\omega\s}$ is the background metric (flat, dS or AdS) defined in \ref{slicings} and $\delta \zeta_{\omega\s}^{b}$ is a perturbation which, unlike the general perturbation in equation (\ref{b6b}),  depends only on the slice coordinates $x^\mu$.

The decomposition (\ref{b10}) still applies, and the boundary gauge transformations are:
\be
\delta \zeta_{\omega\s}^b \rightarrow \delta \zeta_{\omega\s}^b + 2\hat{\nabla}_{(\omega}\xi_{\s)}.
\label{scalar1}
\ee
One can do a gauge transformation to eliminate the transverse and longitudinal vector components, by choosing $\xi_b$ in (\ref{b7}) to be:
\be
\xi_\s = - \chi_\s - {1\over 2}\partial_\s \phi.
\label{scalar2}
\ee
Keeping only the scalar mode, one is left with:
\be
\delta \zeta_{\omega\s}^b = \psi\, \bar{\zeta}_{\omega\s}.
\label{scalar2a}
\ee
Equation (\ref{scalar2a})  is the definition of the scalar perturbation, and we study its dynamics in the following subsections.

\subsection{Scalar equation of motion}
\label{scalar}
The classical equations of motion for $\psi$ are obtained by linearizing the Einstein equation (\ref{EE}), which we rewrite here for convenience:
\be
0= E_{\omega\s}[g^{(0)}] \equiv  - {16\pi G\over \sqrt{g^{(0)}}}{\delta S[g^{(0)}] \over \delta g^{(0)\omega\s}}
\label{def E}
\ee
\be
= R_{\omega\s} - {1\over 2}Rg^{(0)}_{\omega\s} + \Lambda g^{(0)}_{\omega\s} + 8\pi G(\tensor[^{(\alpha)}]{H}{_\omega_\s} + \tensor[^{(\beta)}]{H}{_\omega_\s}) - 8\pi G \left<T_{\omega\s}\right>,
\label{scalar4}
\ee
where quadratic curvature terms $\tensor[^{(\alpha)}]{H}{_\omega_\s}$, $\tensor[^{(\beta)}]{H}{_\omega_\s}$ are defined in (\ref{Halpha}) (\ref{Hbeta}) and the CFT stress tensor is given in (\ref{stress}).
The equation of motion for $\psi$ is obtained by linearizing $E_{\omega\s}[g^{(0)}]$ around $E_{\omega\s}[\bar{\zeta}]$. Linear and quadratic curvature terms are linearized using the definitions (\ref{scalar0}-\ref{scalar2a}). However, to obtain the CFT stress tensor, one needs to solve the bulk equations.  Nevertheless, its trace (\ref{hr43aa}) and its divergence (which is zero) are fully constrained by the boundary geometrical tensors. Hence, one can take a shortcut and perturb the trace of Einstein's equation.
It will then be convenient to define the trace of the generalized Einstein tensor (\ref{def E}) as
\be
E[g^{(0)}] \equiv g^{(0)\omega\s} E_{\omega\s}[g^{(0)}].
\label{scalar17a}
\ee
Then, the full, non-linear, traced Einstein equation is given by
\be
0  = E[g^{(0)}] = -R + 4\Lambda - {\alpha G\over 4} \Box R - {GN^2\over 4\pi}\left(R^{\omega\s}R_{\omega\s} - {1\over 3}R^2\right),
\label{psi6}
\ee
where $\Box$ is the Laplacian operator $\nabla^2$ applied to a scalar quantity.
Equation (\ref{psi6}) only contains scalar geometric quantities of the boundary. When evaluated on the background metric $\bar{\z}$, equation (\ref{psi6}) reduces to (\ref{scalar7}).

The linearization of geometrical quantities which appear in (\ref{psi6}) for an arbitrary perturbation (\ref{scalar0}) around $\bar{\z}$ are given by
\be
\delta R = -\left(3\Box + \bar{R} \right)\psi,\qquad \d \left(R^{\omega\s}R_{\omega\s} \right)=\frac{\bar{R}}{2}\d R.
\label{scalar9}
\ee
We observe that these linearized scalar quantities depend on $\psi$ only, due to $h^{(0)}$ being traceless.
This leads to  the linearized version of equation (\ref{psi6}):
\be
\left[1 + {\a G\over 4}\Box - {GN^2 \bar{R}\over 24\pi}\right](3\Box + \bar{R})\psi = 0.
\label{scalar16}
\ee
Equation (\ref{scalar16}) is  a linear {\it local} equation for $\psi(x)$.

We now turn to the computation of the scalar propagator from varying the action as in (\ref{def E}). Because of a residual gauge variance of $\psi$, the action cannot formally be differentiated by $\psi$. In appendix \ref{quadratic action}, we still define the following quantity:
\be
\widetilde{\mathcal{F}} \equiv {1\over \sqrt{\bar{\z}}}\left.{\delta^2 S[g^{(0)}] \over \delta\psi^2 }\right|_{\bar{\z}}.
\label{scalar19}
\ee
If $\widetilde{\mathcal{F}}$ was the scalar propagator, then the result would be
\be
\widetilde{\mathcal{F}} = -{1\over 16\pi G}\left[1 + {\a G\over 4}\Box - {GN^2 \bar{R}\over 24\pi}\right](3\Box + \bar{R}).
\label{2pt Psi a}
\ee
By taking the inverse of (\ref{2pt Psi a}),
the scalar propagator one would obtain is then:
\be
\widetilde{\mathcal{F}}^{-1} = -{64\pi G}\left[\a G R -12 + {GN^2 \bar{R}\over 2\pi}\right]^{-1}\left\{{1\over \Box + {4\over G\a} - {N^2 \bar{R}\over 6\pi \a} } - {1\over  \Box + {R\over 3}}\right\},
\label{2pt Psi b}
\ee
which describes two propagating modes.
However, the second pole in (\ref{2pt Psi b})  is unphysical, as we show in Appendix  \ref{gauge scalar}. The reason it appears in (\ref{2pt Psi b}) is that $\psi$ is not a gauge invariant quantity, and this propagator was derived by considering only the trace of Einstein's equation. To show that the second pole is a pure gauge, one has to look at the non-diagonal components of the linearized Einstein equation. This was discussed in [7] for flat space with quadratic curvature terms only, and in appendix  \ref{gauge scalar} we present the argument to non-zero curvature and with a holographic stress tensor on the right-hand side of (\ref{scalar4}), see also [80, 52] for similar discussions.

Therefore, the only physical pole of the putative propagator  (\ref{2pt Psi b}) is the first term, and the true scalar propagator is:
\be
\displaystyle{\mathcal{F}^{-1} = -{64\pi G}\left[\a G \bar{R} -12 + {GN^2 \bar{R}\over 2\pi}\right]^{-1}\left\{{1\over \Box+ { 4\over G\a} - {N^2 \bar{R}\over 6\pi \a}} \right\}.}
\label{2pt Psi b-ii}
\ee
Both the position of the pole and the sign of the residue depend on the parameters of the model: there are regions in parameter space where the scalar can be a ghost or a tachyon.

The condition for a mode to be tachyonic changes depending on the sign of the background curvature $\bar{R}$, and will be discussed in the next subsection.  Here we  discuss  under which condition the scalar is ghostlike.

In any background, the scalar mode  is a ghost if the residue of the pole has the ``wrong'' sign, the ``right'' sign being that of the pole of the massless spin-2 pole in pure gravity, which  in our conventions is:
\be
{\cal F}_{massless spin2}^{-1} = 32\pi G{1\over \nabla^2 - {\bar{R}\over 6}}.
\ee

Therefore, the scalar mode is a ghost if the residue of (\ref{2pt Psi b-ii})  is positive, i.e:
\be
\left({\pi\a\over N^2} + {1\over 2}\right){GN^2 \bar{R}\over 12\pi} > 1 \quad  \Rightarrow \quad \text{scalar mode is a ghost}.
\label{2pt Psi c}
\ee
This condition is valid both for positive and negative $\bar{R}$. In sections \ref{dS poles} and \ref{AdS poles}, we shall see that this inequality also appears in the context of the tensor sector, but not exactly for the same reasons.

It is useful to compare the  equation of motion (\ref{scalar16}) to other discussions in the literature. If either $\bar{R} = 0$ or $N = 0$, the equation of motion (\ref{scalar16}) agrees with the $R+R^2$ modified gravity analysis of [73].
The first case, $N =0$, corresponds to pure gravity with no CFT; the second case, $\bar{R}=0$, corresponds to flat space, in which the boundary scalar mode decouples even in the presence of the CFT.%
\subsection{Scalar tachyonic instabilities}
Neglecting the unphysical solution in (\ref{scalar16}), we are left with a single scalar mode satisfying  a massive Klein-Gordon equation:
\be
\Box\psi = {4\over G\a}\left[{GN^2R\over 24\pi} - 1\right]\psi.
\label{nd15}
\ee
It is useful to parametrize the eigenvalues in terms of a complex `` total momentum eigenvalue''  $\nu^2$ (or  $k^2$ in flat space)   as follows:
\be
\Box \psi = \left\{\begin{array}{ll} -{\bar{R}\over 12}\left(\n^2-{9\over 4}\right)\psi, & \bar{R}\neq 0, \\ & \\ - k^2 \psi, & \bar{R} = 0. \end{array} \right.
\label{psi9}
\ee
Equation (\ref{nd15}) then translates into
\be
\n^2-{9\over 4} = {48 \over G\alpha \bar{R}}\left(1 - {GN^2 \bar{R}\over 24\pi}  \right), \quad dS \, \text{or} \, AdS,
\label{psi10}
\ee
\be
k^2 = {4\over G\alpha}, \quad  \text{Minkowski.}
\label{psi10-i}
\ee
To determine whether the solution corresponds to a tachyonic mode,  we need to specify the geometry of the boundary.
\begin{itemize}
\item {\bf Minkowski.} The theory is tachyon-stable if the invariant four-momentum $k^2$  is timelike or null, $k^2 \leq 0$. From equation (\ref{psi10-i}) we conclude that:
\be
\alpha >0 \quad \Rightarrow \quad \text{scalar mode is tachyonic.}
\ee
Note that for  $\a = 0$, the scalar mode is decoupled (at quadratic order).

\item {\bf de Sitter.}
We choose to use
Poincar\'e coordinates, which cover the  expanding patch of de Sitter. In these coordinates, the de Sitter metric is given by
\be
ds^2_{dS} = {1\over (H\tau)^2}\eta_{\omega\s}dx^\omega dx^\s, \qquad \bar{R} = 12H^2,
\label{psi1}
\ee
where $ x^0 \equiv \tau = {e^{-H t}\over H}$ is the conformal time.

The solutions of Eq. \eqref{psi9}  are given in terms of Bessel functions:
\be
\psi = (H\tau)^{3\over 2} J_{\pm\n}(\tau p) \underset{\tau \rightarrow 0}{\sim} e^{Ht(\pm \n-3/2)},
\label{psi9a}
\ee
where $p\equiv \sqrt{\delta_{ij}p^ip^j}$ is the norm of the 3-dimensional Fourier momentum in the spatial directions.
Both solutions  (\ref{psi9a})  are bounded as $\tau\to 0$   if [68]:
\be
|\text{Re}(\n)|  \leq 3/2.
\label{psi2}
\ee
Defining a tachyon as a mode which grows exponentially in time,  equation (\ref{psi10}) then translates into  the statement\footnote{In an expanding background such a mode is still ok to have around as long as the growth rate is much smaller than the Hubble time, i.e.  $|Re(\nu)|- 3/2| \ll 1$. In this case one can say that de Sitter space is long-lived.}:
\be
{1\over \a}\left(1- {GN^2H^2\over 2\pi}\right) > 0 \quad \Rightarrow \quad \text{scalar mode is tachyonic.}
\label{psi_tachyonic}
\ee
By inserting  equation  (\ref{psi10}) into the exponent of (\ref{psi9a}), we obtain the ``decay rate'' $\Gamma$ of de Sitter due to the tachyon instability:
\be
\G = H\left[ \sqrt{{9\over 4} + {4\pi \over GN^2H^2 \tilde{\a}}\left(1 - {GN^2H^2\over 2\pi}\right)} -{3\over 2} \right],
\label{Gamma-scalar}
\ee
which is real and positive if we are in the tachyonic regime (\ref{psi_tachyonic}).

We now  discuss a few special cases.
\begin{itemize}
\item As in flat space, if $\alpha=0$, the scalar mode is non-propagating.
\item The special case $GN^2H^2 = 4\pi$ corresponds to a vanishing 4d cosmological constant  $\Lambda= 0$ (by equation (\ref{scalar7})).  This is the case  studied in [52]. We find, in agreement with that work, that scalar tachyonic  instabilities occur for $\alpha<0$. This also includes the homogeneous scalar mode from the original Starobinsky model [46, 47].
\item In the absence of the  CFT, i.e. for  $N^2= 0$,   de Sitter is  tachyon-unstable in the scalar sector for $\alpha> 0$. This is the opposite sign compared to the previous paragraph case.
There is no contradiction here,  since (\ref{scalar7}) shows that de Sitter is not a solution when both $\Lambda$ {\it and} $N^2$ vanish.
\end{itemize}

\item {\bf anti-de Sitter}:
In this case, a massive scalar mode is tachyonic if it violates the BF bound,
[81], which in 4 space-time dimensions means that the mass $m$ satisfies
\be
m^2 \chi^2 < - {9\over 4},
\ee
where $\chi$ is the AdS length.  Comparing  with equation (\ref{psi9}) with $\bar{R}=-12\chi^2$,   violation of the BF bound is equivalent to:
\be
\n^2 < 0.
\label{psi15}
\ee
Using (\ref{psi10}) we then conclude:
\be
{9\over 4} - {4\over \a G\chi^2}\left(1 + {GN^2\chi^2\over 2\pi}\right) < 0 \quad \Rightarrow \quad \text{scalar mode is tachyonic.}
\label{psi16}
\ee
\begin{itemize}
\item For $\alpha = 0$ the scalar mode decouples as in the other cases.
\item For pure gravity, in the absence of the CFT ($N^2=0$), the tachyonic condition (\ref{psi16}) becomes
\be
{16\over  \alpha   G \chi^2}~~ >~~ 9.
\label{psi17}
\ee
In particular, this cannot be satisfied  for $\alpha<0$. This is the opposite compared to the de Sitter case.
\end{itemize}
\end{itemize}

\section{The  spin-two spectral equations}
\label{linearized eq}

We now move to  tensor perturbations $h^{(0)}_{\omega\s}$ defined in (\ref{b20}). The linearized Einstein equation (\ref{dS13b}) contains  $h^{(4)}$, which can only be specified by solving the perturbation equations in the bulk.  This section is devoted to expressing $h^{(4)}$ in terms of the boundary perturbation $h^{(0)}$ by solving the bulk tensor equation (\ref{b14}). This has to be done separately for each slice geometry  (flat, positive and negative curvature). We treat each case in a separate subsection.

\subsection{Flat-slicing} \label{sec:flat prop}
The bulk equation of motion for the tensor mode (\ref{b14}) simplifies significantly in flat slicing coordinates.
First, one can write the bulk metric as a conformally flat space by defining the usual Poincar\'e coordinate $Z$ as
\be
Z \equiv e^{-u} = \sqrt{\rho}.
\label{flat1}
\ee
The perturbed bulk metric (\ref{b6b}) in which we only keep the propagating tensor $h_{\omega\s}$ is then written as
\be
ds_5^2 = {1\over Z^2}[L^2 dZ^2 + (\eta_{\omega\s} + h_{\omega\s}) dx^\omega dx^\s].
\label{flat1a}
\ee
The bulk equation of motion (\ref{b14}) describes the dynamics of a massless graviton in AdS. In flat slicing coordinates (\ref{flat1a}), we can insert $a = e^{u}$ into (\ref{b14bis}), which boils down to the massless scalar equation
\be
\Box_5 h_{\omega\s} = 0,
\label{flat2}
\ee
where $\Box_5$ is the $AdS$ scalar Laplacian in Poincar\'e coordinates (\ref{b0flat}), given by
\be
L^2\Box_5 = Z^2(\partial_Z^2 + L^2\eta^{\k\l}\partial_\k\partial_\l) - 3Z\partial_Z.
\label{flat3}
\ee

Now the strategy is to search for separable solutions, which we write as
\be
h_{\omega\s}(Z,x) = F(Z,k) \tilde{h}^{(0)}_{\omega\s}(x,k),
\label{flat4}
\ee
where $\tilde{h}^{(0)}$ solves the eigenvalue equation parametrized by $k^2$ as
\be
\partial^\s\partial_\s \tilde h^{(0)}_{\omega\k} = -k^2 \tilde h^{(0)}_{\omega\k} \equiv (m_2)^2 \tilde h^{(0)}_{\omega\k},
\label{flat5}
\ee
and $(m_2)^2$ can be a complex number in general.

The second equation is an ordinary differential  equation for $F(Z,k)$,
\be \label{flat5a}
\left(Z^2\de_Z^2 - 3 Z \de_Z -  k^2 L^2 Z^2\right) F(Z,k) = 0.
\ee
Note that, ultimately,  we want to write an equation for the boundary tensor perturbation of the form (\ref{bb21}) in Fourier space,  with $\nabla^2$ replaced by $-k^2$. The solution will be the physical mass$^2$ of a propagating 4d mode. As the solutions of this equation  may be complex, we must allow for complex values of $k^2$ beyond the usual choices of timelike ($k^2<0$) and spacelike ($k^2>0$) momentum one obtains for real wavenumbers.
With this caveat, it is now convenient to write (\ref{flat5a}) as:
\be
\left[y^2{d^2\over dy^2} - y^2 - 3 y{d\over dy}\right]F(y) = 0, \qquad  y\equiv L k Z.
\label{flat6a}
\ee
where  we define $k$, for any complex  $k^2$ outside of the negative real axis, as  the complex root of $k^2$ with {\it positive} real part\footnote{This prescription is enough to identify tachyonic modes, for which $Re(k) \neq 0$.  Instead, real negative $k^2$ corresponds to non-tachyonic propagating particles, and as usual, their propagator needs a further  prescription. We  use the analytic continuation of the results to purely imaginary values of $k$, which corresponds to taking the retarded stress tensor 2-point function .} and  with a slight abuse of notation, we have replaced $F(Z, k)$ by $F(y)$.

We now solve the equations for $\tilde{h}^{(0)}$ (\ref{flat5}) and for $F$ (\ref{flat6a}). First, the solution of (\ref{flat5}) are the Fourier modes :
\be
\tilde h^{(0)}_{\omega\k}(k^\s) = e^{\pm ik^\s x_\s} = e^{\pm i (-\omega t + \bm{k}.\bm{x})},\;   \bm{k}\in \mathbb{R}^3, \; \omega \equiv \sqrt{-k^2 + \bm{k}^2}.
\label{flat6}
\ee

For a non-negative eigenvalue $k^2$, modes with $|\bm{k}| < |k|$  will necessarily feature an imaginary part in $\omega$ and one of the two solutions in (\ref{flat6}) will diverge with time. This is the usual tachyon instability for flat space, which occurs for massive Klein-Gordon equations with negative mass square, and more generally it persists also for a complex mass.
Therefore, the condition that a mode characterised by $k^2$ is non-tachyonic is:
\be
\text{Re}(k) = 0.
\label{flat7a}
\ee
where, as above, we have defined $k$ as the complex root of $k^2$ with positive real part\footnote{This is simply a convention since the equation has a symmetry in $k\to -k$.}.

Equation (\ref{flat6a})  is solved by modified Bessel functions,
\be
F(y) = y^2 (\lambda_1 K_2(y) +  \lambda_2 I_2(y)).
\label{flat7}
\ee

We must impose that the solution (\ref{flat7}) is  regular at the horizon $Z \rightarrow +\infty$. This   requires $\lambda_2 = 0$ because in this limit $I_2(y) \sim exp[k L Z]$ and   by definition  $\text{Re}(k)>0$. The remaining solution $K_2(y)$ is a vanishing exponential at $Z\to +\infty$.

We fix the remaining parameter $\lambda_1$ by choosing  the normalization at the AdS$_5$ boundary $Z=0$ so that:
\be
F(Z=0) = 1.
\label{flat11}
\ee
This way, the solution (\ref{flat4}) for the bulk tensor perturbation $h_{\omega\s}(x,Z)$ coincides at $Z=0$ with the boundary tensor mode $h^{(0)}_{\omega\s}$ defined in the FG expansion  (\ref{b15}). For this reason, the leading term $\tilde h^{(0)}_{\omega\s}$ of $h_{\omega\s}$ in (\ref{flat4}) is identified as the Fourier mode of the leading term in the Fefferman-Graham expansion (\ref{b15}) which was defined as $h^{(0)}_{\omega\s}$.
We drop the tilde from now on.

For small $y$, the Bessel function $K_2$ behaves as:
\be
K_2(2y) \underset{y\rightarrow 0}{=} {1\over 2}\left\{y^{-2} - 1 + {3\over 4}y^2 - y^2\left[ \gamma_E + \log(y)\right] \right\}+ \mathcal{O}(y^4),
\label{flat10}
\ee
where $\g_E$ is the Euler-Mascheroni constant.
Then, equation (\ref{flat11})  fixes  $\lambda_1 = 1/2$ in (\ref{flat7}).

Having completely fixed $F(Z,k)$, we can read-off   $h^{(4)}_{\omega\s}$ and $\hat{h}_{\omega\s}$ from its near-boundary expansion,  (\ref{flat10}) and (\ref{flat4}) and compare with the corresponding  terms in equation (\ref{b15}), recalling that $Z=\sqrt{\rho}$ . We find:
\begin{subequations}
\label{n21a}
\be
h^{(2)}_{\omega\k} = - \left({kL\over 2}\right)^2 h^{(0)}_{\omega\k},
\label{n21}
\ee
\be
h^{(4)}_{\omega\k} =\left({kL\over 2}\right)^4\left[ {3\over 4} - \gamma_E - \log\left({kL\over 2}\right)\right] h^{(0)}_{\omega\k},
\label{n26}
\ee
\be
\hat{h}_{\omega\k} = -{1\over 2}\left({kL\over 2}\right)^4 h^{(0)}_{\omega\k}.
\label{n26a}
\ee
\end{subequations}
The terms $h^{(2)}_{\omega\k}$ and $\hat{h}_{\omega\k}$ agree with the perturbative solutions of the bulk Einstein equation (\ref{hr20}) that are given in appendix \ref{renorm} by (\ref{fg16}) and (\ref{fg26}).  To perform this comparison and check that they agree, it is enough to linearize $g^{(2)}_{\omega\k}$ (\ref{fg16}) and $\hat{g}_{\omega\k}$ (\ref{fg26}) with respect to the transverse traceless perturbation $h^{(0)}_{\omega\s}$.

The linearization of the stress tensor (\ref{stress}) around a flat background for the tensor perturbation is given by\footnote{The term $h^{(2)}$ does not contribute in (\ref{n26b}) because it always appears in the CFT stress tensor (\ref{stress}) multiplied by $g^{(2)}[\bar{\zeta}]$, which vanishes on a flat background.}:
\be
\delta_h \left<T_{\omega\k}\right> = {N^2\over 2\pi^2L^4}\left[h^{(4)}_{\omega\k} + \left(1- 2\log \mu L\right)\hat{h}_{\omega\k}\right].
\label{n26b}
\ee

We can use the bulk solutions (\ref{n21a}) into (\ref{n26b}), to obtain the perturbed stress-tensor in terms of $h^{0}$ alone:
\be
\delta_h \left<T_{\omega\k}\right> = {N^2\over 2\pi^2}  \left( {k\over 2}\right)^4\left[ {1\over 4} - \gamma_E - \log\left({k\over 2\m}\right)  \right] h^{(0)}_{\omega\k}.
\label{n27}
\ee

As a final step, inserting the expressions  (\ref{n21a})  in   (\ref{dS13b}) we obtain  the  linearized Einstein equation specialized to a flat background, in the form of an equation for $h^{(0)}$ alone:

\be
{N^2\over 64\pi^2} k^2 Q_\text{flat}(k)h^{(0)}_{\omega\s} = 0,
\label{n28ab}
\ee
\be
Q_\text{flat}(k) \equiv \left\{-{2\pi \over GN^2} + k^2\left[{1\over 4} - \g_E - \log\left({k\over 2\m}\right)- {1\over 2}{{\pi\beta\over N^2}}\right]\right\}.
\label{n28b}
\ee
From (\ref{n28b}), as anticipated in Section \ref{bulk pert}, we  can observe that the contributions from the renormalization scale $\m$ coming from the CFT and the quadratic curvature term proportional to $\b$ combine into the parameter $\beta_\text{eff}$ given in equation (\ref{b24}).
For convenience, we also define
\be
\tilde{\b}_\text{eff} \equiv {\pi\b_\text{eff}\over N^2}.
\label{n28d}
\ee
The spectral equation (\ref{n28ab}) is then written as
\be
Q_\text{flat}(k) \equiv \left\{-{2\pi \over GN^2} + {k^2\over 2}\left[{1\over 2} - 2\g_E - \log\left({GN^2k^2}\right)- \tilde{\b}_\text{eff}\right]\right\}.
\label{n28e}
\ee

The quantity multiplying $h^{(0)}$ in (\ref{n28ab}) is the inverse propagator (\ref{b20}) for a flat spacetime. Its expression is given by\footnote{For the overall coefficient in this expression, see  Appendix \ref{quadratic action}.}:

\be
\mathcal{F}_\text{flat}(k) =  {N^2\over 64\pi^2} k^2 Q_\text{flat}(k).
\label{n28a}
\ee
Non-trivial solutions ($h^{(0)}_{\omega\s}\neq 0$) to the equation of motion (\ref{n28ab})  correspond to the propagating momentum modes $k$ of the boundary perturbation $h^{(0)}$ and are found by solving the  spectral equation
\be
k^2 Q_\text{flat}(k) = 0.
\label{n29}
\ee
Solutions of this equation are the poles of the propagator $\mathcal{F}_\text{flat}^{-1}$.
An obvious solution to that equation is the massless mode $k^2=0$ which is  present in pure Einstein-Hilbert gravity. ``Exotic'' Modes with $k^2\neq 0$ satisfy:
\be
1 = {GN^2 k^2 \over 4\pi}\left({1\over 2} - 2\gamma_E - \log\left(GN^2k^2\right) - \tilde{\b}_\text{eff} \right).
\label{n31}
\ee
The only solutions which are non-tachyonic are those for which $k^2<0$.
The absence of tachyon-instabilities of flat space is then equivalent to the absence of solutions $k$ to (\ref{n31}) with a non-zero real part. We study the existence of such unstable solutions in section \ref{sec:flat correlator}.

\subsection{dS-slicing}
\label{dS prop}
We now consider the CFT on de Sitter and we turn to equation (\ref{b14bis}) applied to dS slicing coordinates (\ref{b0dS}). As a result, we obtain
\be
\left\{\partial_u^2 + 4\coth{u}\partial_u + \frac{H^{-2}\hat{\nabla}^2 - 2}{\sinh^2u}\right\}h_{\omega\s} = 0.
\label{dS3}
\ee
The operator inside curly brackets is similar to the expression of the Laplace operator of $AdS_5$ acting on scalars, in which case the numerator of the last term would be replaced by the 4-dimensional slice scalar Laplacian.

Similarly to the flat slicing case (\ref{flat4}), we search for separable solutions of the form:
\be
h_{\omega\s}(x ,u) = F(u,\n)\tilde{h}^{(0)}_{\omega\s}(x,\n).
\label{dS4}
\ee
This results in two equations: the first one is an eigenvalue problem on the slice, which we write as:
\be
(\hat{\nabla}^2 - 2H^2)\tilde{h}^{(0)}_{\omega\s} =  -H^2\left(\n^2 - {9\over4}\right) \tilde{h}^{(0)}_{\omega\s} \equiv (m_2)^2\tilde{h}^{(0)}_{\omega\s}.
\label{dS5}
\ee
The second equation is an ODE in the radial direction:
\be
\left\{{d^2\over du^2} + 4 \coth u{d\over du} - {\n^2 - {9\over 4} \over \sinh^2 u}\right\}F(u,\n) = 0.
\label{dS7}
\ee

The information about tachyonic instabilities is contained in the value of $\n$.
As shown in Appendix \ref{dS criterion}, and as it is pointed out in [52, 68], modes with
\be
|\text{Re}(\n)| >3/2
\label{dS6}
\ee
are tachyonic because (\ref{dS5}) contains a solution which diverges with time (see appendix \ref{dS criterion} for the details).

In pure 4d gravity, the only propagating mode would be the transverse-traceless graviton, which is a zero eigenvalue for the  Lichnerowicz operator of de Sitter (the left-hand side of equation (\ref{dS5})
and corresponds to  $\n = \pm 3/2$.

Turning on the CFT matter content and the quadratic curvature terms will allow for modes with different values of $\nu$. These will be determined by solving the  boundary spectral equation, which we derive below.

As in the flat case, to obtain the boundary spectral equation we have to solve the radial equation (\ref{dS7}).  The most general solution of (\ref{dS7}) is a linear combination of two hypergeometric functions given by
\be
F(u,\n) = C_+ \tanh u^{\n-\frac{3}{2}}\tensor[_2]{F}{_1}\left[\frac{1}{2}\left(\nu-\frac{3}{2}\right),\frac{1}{2}\left(\nu-\frac{1}{2}\right);1+\nu;\tanh^2u\right] +\nonumber
\ee
\be
+ C_- \tanh u^{-\nu-\frac{3}{2}}\tensor[_2]{F}{_1}\left[\frac{1}{2}\left(-\nu-\frac{3}{2}\right),\frac{1}{2}\left(-\nu-\frac{1}{2}\right);1-\nu;\tanh^2u\right],
\label{dS8}
\ee
where $C_\pm$ are integration constants.
This solution may have a   singularity at the horizon $u = 0$, depending on the real part of $\n$. As we show in appendix \ref{bulk schrodinger},
requiring the solution (\ref{dS8}) to be normalizable at $u=0$ gives the following constraints :
\begin{itemize}
\item If $\text{Re}(\n) > 0$, we need to set $C_-=0$ for normalizability at $u=0$.
\item If $\text{Re}(\n) < 0$, we need to set $C_+=0$ for normalizability at $u=0$.
\item If $\text{Re}(\n) = 0$, both solutions oscillate at the horizon $u=0$.
\end{itemize}
Since the problem (\ref{dS7}) is symmetric in $\n\leftrightarrow -\n$, we can choose $\text{Re}(\n) \geq 0$ without loss of generality. In this case, the most general regular solution at $u=0$ is the one with $C_- = 0$. The case where $\n$ is imaginary lies in the stable region (\ref{dS6}) and needs a further prescription (e.g. infalling boundary conditions). We shall define the spectral function by analytic continuation to purely imaginary $\nu$.

We fix  the normalization of $F$ by imposing  that in the UV, at $u\rightarrow +\infty$:
\be
F(u,\nu) \underset{u \rightarrow +\infty}{\longrightarrow} 1.
\label{dS10}
\ee
This condition ensures that
$\tilde{h}^{(0)}_{\omega\s}$ defined in  (\ref{dS4}) identifies with the leading term $h^{(0)}_{\omega\s}$ of the Feffeman-Graham expansion (\ref{b15}). From now on, we drop the tilde on $h^{(0)}$ although it only represents a single mode $\n$ (\ref{dS5}).

Using Gauss' hypergeometric theorem
\be
\tensor[_2]{F}{_1}\left[a,b;c;1\right] = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)},
\label{dS11}
\ee
valid for $\text{Re}(c)>\text{Re}(a+b)$, the boundary condition (\ref{dS10}) fixes the value of the integration constant to
\be
C_+ = \frac{\Gamma\left(\frac{5}{2}+\nu\right) \sqrt{\pi}2^{-\nu-\frac{3}{2}}}{\Gamma(1+\nu)}.
\label{dS12}
\ee

The near-boundary expansion of $F(u,\n)$ can be obtained using a hypergeometric transformation (page 49 of [82]). It allows us to transform $F(u,\n)$ into a power series of $e^{-u}$ instead of $\tanh^2u$ (hypergeometric functions are defined as a power series of their last argument). The first few terms of the result are given by
\be
F(u,\nu) = 1 - e^{-2u}\left(\n^2 - {9\over 4}\right) - e^{-4u}\left(\n^2 - {9\over 4}\right)\bigg\{1 + \nonumber
\ee
\be
+ \left(\n^2 - {1\over 4}\right)\left[-u - {3\over 4} + \mathcal{H}\left(\n-{1\over 2}\right)\right]\bigg\} + \mathcal{O}(e^{-6u}),
\label{dS13}
\ee
where $\mathcal{H}$ is the harmonic number function defined in terms of the Euler Gamma function, $\G$ as
\be
\mathcal{H}(z) = {\G'(z+1)\over \G (z+1)} + \g_E.
\label{hn}
\ee
The terms in (\ref{dS13})  are enough to read all of the Fefferman-Graham expansion (\ref{b15}) using the relation between positive $u$ and $\rho$ in tabular \ref{tabular slicings}:
\begin{subequations}
\label{dS13aa}
\be
h^{(2)}_{\omega\s} = -h^{(0)}_{\omega\s}\left({LH\over 2}\right)^2\left(\nu^2-{9\over 4}\right)
\label{i60d1}
\ee
\be
h^{(4)}_{\omega\s} = - h^{(0)}_{\omega\s} \left({LH\over 2}\right)^4 \left(\nu^2-{9\over 4}\right)\left\{ 1 + \left(\nu^2-{1\over 4}\right)\left[ \log\left({LH\over 2}\right)- {3\over 4}
+ \mathcal{H} \left( \n - {1\over 2}  \right)\right]\right\}
\label{i611}
\ee
\be
\hat{h}_{\omega\s} =  - {h^{(0)}_{\omega\s}\over 2} \left({LH\over 2}\right)^4 \left(\nu^2-{9\over 4}\right) \left(\nu^2-{1\over 4}\right).
\label{i61a1}
\ee
\end{subequations}

As we already discussed for the flat slicing, $h^{(2)}$ and $\hat{h}$ can be found using an independent method discussed in appendix \ref{renorm}. This method consists in solving perturbatively the bulk Einstein equation at small values of the Fefferman-Graham coordinate $\rho$ for an arbitrary boundary metric $g^{(0)}$. The solution for $g^{(2)}$ and $\hat{g}$, given in (\ref{fg16}) and (\ref{fg26}), can be linearized with respect to $h^{(0)}$ to obtain the same result as (\ref{i60d1}) and (\ref{i61a1}) using the formulae (\ref{dS13aaa}). However,   this alternative method does not determine $h^{(4)}$ in terms of $h^{(0)}$, but only its trace and divergence.