a a institutetext:  Departamento de Física, Centro de Matemática e Aplicaçoes
(CMA-UBI), Universidade da Beira Interior, 6200 Covilhã, Portugal b b
institutetext: Department of Physics, Tokyo Institute of Technology, 2-12-1
Ookayama, Meguro-ku, Tokyo 152-8551, Japanc c institutetext:  L. D. Landau
Institute for Theoretical Physics RAS, Chernogolovka, Moscow region 142432,
Russian Federationd d institutetext: Kazan Federal University, Kazan 420008,
Republic of Tatarstan, Russian Federation

# Generalized non-local $R^{2}$-like inflation

Alexey S. Koshelev b K. Sravan Kumar c,d Alexei A. Starobinsky<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The $R^{2}$ inflation which is an extension of general relativity (GR) by
quadratic scalar curvature introduces a quasi-de Sitter expansion of the early
Universe governed by Ricci scalar being an eigenmode of d’Alembertian
operator. In this paper, we derive a most general theory of gravity admitting
$R^{2}$ inflationary solution which turned out to be higher curvature non-
local extension of GR. We study in detail inflationary perturbations in this
theory and analyse the structure of form factors that leads to a massive
scalar (scalaron) and massless tensor degrees of freedom. We argue that the
theory contains only finite number of free parameters which can be fixed by
cosmological observations. We derive predictions of our generalized non-local
$R^{2}$-like inflation and obtain the scalar spectral index $n_{s}\approx
1-\frac{2}{N}$ and any value of the tensor-to-scalar ratio $r<0.036$. In this
theory, tensor spectral index can be either positive or negative
$n_{t}\lessgtr 0$ and the well-known consistency relation $r=-8n_{t}$ is
violated in a non-trivial way. We also compute running of the tensor spectral
index and discuss observational implications to distinguish this model from
several classes of scalar field models of inflation. These predictions allow
us to probe the nature of quantum gravity in the scope of future CMB and
gravitational wave observations. Finally we comment on how the features of
generalized non-local $R^{2}$-like inflation cannot be captured by established
notions of the so-called effective field theory of single field inflation and
how we must redefine the way we pursue inflationary cosmology.

###### Keywords:

Models of Quantum Gravity, Cosmology of Theories beyond the SM

††arxiv: arXiv:2209.XXXXX

##  1 Introduction

Ever since the first release of the Planck CMB data $R^{2}$ inflation or
Starobinsky inflation based on modification of general relativity (GR) by
quadratic scalar curvature (which we call shortly as $R^{2}$ gravity) has
became a cue for understanding the physics of early Universe
Starobinsky:1980te ; Starobinsky:1981vz ; Starobinsky:1983zz ; Vilenkin:1985md
; Mijic:1986iv ; Maeda:1987xf ; Akrami:2018odb  because of its success with
the predictions of a quasi-flat (scale invariant) scalar power spectrum of
primordial scalar perturbations $\mathcal{P}_{\mathcal{R}}\sim 2.1\times
10^{-9}$ with the small tilt $n_{s}-1=-\frac{2}{N}$ and primordial
gravitational waves given by the tensor-scalar ratio
$r=\frac{12}{N^{2}}=3(1-n_{s})^{2}$ (where $N=50-60$ e-folds before the end of
inflation, depending on the duration of a post-inflationary stage of creation
and thermalization of known matter), obtained by unambiguously fixing the mass
of the scalaron as $M\approx 1.3\times 10^{-5}\,M_{p}$ in the units of reduced
Planck mass ($M_{p}$) from the measured amplitude of
$\mathcal{P}_{\mathcal{R}}$. The emergence of $R^{2}$ inflation from the semi-
classical modification of gravity due to matter quantum fields clearly points
out the importance of fundamental physics and the first principles in the
construction of successful cosmological models. Along with the observational
success, $R^{2}$ gravity is also an important step beyond general relativity
(GR) towards quantum gravity Stelle:1976gc . In this spirit, $R^{2}$-inflation
has been successfully embedded in a geometric (á la Jordan frame) construction
of supergravity Ketov:2012jt  and also recently in an ultraviolet (UV)
complete framework of analytic non-local gravity Koshelev:2016xqb ;
Koshelev:2017tvv ; Koshelev:2020foq ; Koshelev:2020xby . Despite these
developments especially the later one that we explore further in this paper,
several realizations of Starobinsky-like inflatonary scenarios have been
emerged in recent years in the context of UV-complete frameworks such as in
string theory and SUGRA Ellis:2013nxa ; Kehagias:2013mya ; Linde:2014nna . The
important aspect of all these several generalizations is that they modify the
scalar power spectrum which results in modification of the key inflationary
observable - the tensor-to-scalar ratio which is bounded above from the latest
BICEP/Keck Array CMB data $r<0.036$ BICEP:2021xfz , while the prediction of
scalar tilt $n_{s}\approx 1-\frac{2}{N}$ being almost unchanged in agreement
with the Planck result Akrami:2018odb .

Apart from the scalar field models of inflation which predominantly attempts
to modify the scalar power spectrum, there is another possibility of changing
the key parameter tensor-to-scalar ratio $r$ by modifying the gravitational
power spectrum. Such a possibility has been recently found in geometrical
modifications of GR, in particular in an ultraviolet (UV) non-local
generalization of $R^{2}$-inflation. Such a non-local gravity theory has
emerged with several motivations of UV completing $R^{2}$ gravity (see
Craps:2014wga ; Koshelev:2016xqb ; Koshelev:2017tvv ; Koshelev:2020foq ;
Koshelev:2020xby  and the references therein). Recently found non-local
extension of $R^{2}$ gravity leads to testable predictions especially with
respect to the single field consistency relation $r=-8n_{t}$ which gets
violated solely due to the presence of non-locality. Here in this paper, we
establish $R^{2}$-like inflation in an analytic non-local gravity construction
including all possible higher curvatures which goes beyond the previously
studied model Koshelev:2017tvv ; Koshelev:2020foq . Non-local higher curvature
terms are motivated from quantum gravity point of view which can be witnessed
from the studies of trace-anomaly driven effective theory of quantum gravity
Barvinsky:1990up ; Barvinsky:1994cg ; Barvinsky:1994hw  and from asymptotic
safety (AS) approach to quantum gravity Bonanno:2017pkg ; Eichhorn:2018yfc ;
Bosma:2019aiu . According to the AS approach, a generic (effective) quantum
gravity action must contain all the higher curvatures and infinite
derivatives. In this paper, we study cosmological implications of non-local
higher curvature terms in the view of $R^{2}$-like inflationary paradigm. In a
nutshell, our primary goal is to expose all aspects of higher curvature
gravity embodied with non-locality in the form of analytic infinite
derivatives and higher curvature terms. We present features of this theory
with respect to the almost scale invariant scalar power spectrum and the
tensor power spectrum that can have non-trivial scale dependence.

The paper is organized as follows. In Sec. 2 we provide a concise summary of
results obtained in this paper which will be discussed in detail in the later
sections of the paper. We highlight all the new features of our geometric
framework of non-local $R^{2}$-like inflation in the scope of upcoming ground
based and space based cosmological and gravitational wave observations. In
Sec. 3 we present a most general action for $R^{2}$-like inflation and explain
how the action turns out to be non-local in nature. We demonstrate how the
theory, despite containing infinite derivative and infinite curvature terms
(written as the so-called form factors), can be formulated with finite
parameter space using physically motivated conditions for the theory to be
ghost-free in the Minkowski and quasi-de Sitter (dS) limits. In Sec. 3.4 we
derive the scalar and tensor power spectrum predictions of our general higher
curvature non-local gravity exploiting the results derived in Koshelev:2016xqb
; Koshelev:2017tvv ; Koshelev:2020foq . We discuss the structure of form
factors and their relation to the degrees of freedom in the theory. We derive
predictions for $\left(n_{s},\,r,\,\frac{dn_{t}}{d\ln
k},\,\frac{d^{2}n_{t}}{d\ln k^{2}}\right)$ in this model and compare them to
the latest observational bounds. In Sec. 4 we provide an extensive discussion
of distinguishing the features of generalized non-local $R^{2}$-like inflation
against the effective field theory (EFT) of single field inflation. We present
how generalized non-local $R^{2}$-like inflation reshapes our understanding of
early Universe cosmology. In Sec. 5 we provide a brief outlook to go beyond
what has been studied in this paper. Also we will comment on implications of
our study to the context of quantum gravity. In Appendix A we discuss several
models of $f(R)$ gravity and the prominence of $R^{2}$ inflation in explaining
the inflationary observables. In Appendix B we derive the equations of motion
of the most general gravity action for $R^{2}$-like inflation we constructed
in Sec. 3. In Appendix C we study cosmological perturbations of theory
obtained in Sec. 3 and analyze the degrees of freedom and the ghost-free
conditions for different form factors.

Notations: we use the metric signature $\left(-,\,+,\,+,\,+\right)$, overdot
and ′ denotes derivative with respect to cosmic time (t) and conformal time
($\tau$) respectively, †, ††, ††† to denote first, second and third derivative
with respect to the argument. (n) denotes n-th order perturbation, overbar
denotes background values for flat Friedmann-Lemaître-Robertson-Walker (FLRW)
spacetime, 4-dimensional indices are labelled by small Greek letters and three
dimensional quantities are denoted by $i,j=1,2,3$. We also set $\hbar=c=1$ and
$M_{p}=1/\sqrt{8\pi G}$ is the reduced Planck mass. Throughout this paper we
use $\approx$ to denote dS approximation, subscripts ${}_{(s)},\,_{(h)}$ to
indicate scalar and tensor parts of the perturbations and subscript dS refers
to the quantities evaluated in the quasi-dS (slow-roll) approximation.
Everywhere we perform computations in the leading order of the slow-roll
approximation. Whenever we discuss scalar field EFT models of inflation, we
use the notation for slow-roll parameters, Hubble parameter with subscript E
to distinguish them from the quantities defined in the Jordan frame. We use
the letter O to denote order of magnitude and $\mathcal{O}$ to denote an
arbitrary operator.

##  2 Summary of our results

Apart from discovering a new era in the remote past of our Universe, one of
the main goals of inflationary cosmology is to look for signatures of high
energy physics including gravity through CMB, astrophysical and gravitational
wave observations. Plethora of inflationary mechanisms have been developed in
the last decades in order to derive predictions of physics at UV energy and
space-time curvature in terms of the following set of observables related to
two-point correlations Martin:2013tda

$\Bigg{\\{}n_{s},\,r,\,n_{t}\Bigg{\\}}$ (1)

The latest Planck data with their result on the scalar spectral index
$n_{s}\approx 1-\frac{2}{N}$ and constraint on its running $\frac{dn_{s}}{d\ln
k}$ strongly favour $R^{2}$ inflation where inflationary stage is a phase of
quasi-dS evolution with the Ricci scalar R being the eigenmode of the
d’Alembertian operator (3) which in the Einstein frame is understood as
inflaton rolling down the plateau-like potential (or the Starobinsky
potential) Starobinsky:1980te ; Mukhanov:1990me . $R^{2}$ inflation
interpolates the regime of higher curvature dominance when $M^{2}\ll R\ll
M_{p}^{2}$ (where $M\approx 1.3\times 10^{-5}M_{p}$ is the restmass of the
propagating scalar in $f(R)$ gravity dubbed scalalon) to the matter dominated
phase $|R|\ll M^{2}$ producing graceful exit from inflation, and when to the
radiation dominated stage after the decay of scalaron into other elementary
particles and antiparticles and their final thermalization. The success of
$R^{2}$ inflation puts stringent constraints on coefficients of additional
higher order terms in the action like $R^{n}$.

Before we attempt to understand what quantum gravity is, it is vital to
understand how one can constructively write an action beyond GR. It is trivial
to think that a generic action for quantum gravity (that is metric compatible)
contains all possible curvature invariants that one can write using the Ricci
scalar, Ricci tensor, Weyl tensor or Riemann tensor and an infinite number of
their covariant derivatives. But it is non-trivial to frame out the rules to
obtain an action that is apt for quantum gravity and cosmological application.
Inspired from the attempts of formulating a generic action of quantum gravity
with all possible curvature invariant terms Barvinsky:1990up ;
Barvinsky:1994cg ; Barvinsky:1994hw  and several other approaches of quantum
gravity Koshelev:2020xby , here in this paper we took a phenomenological
approach and formulated a most general action (4) that one can write to have
$R^{2}$-like inflation dictated by the eigenvalue equation (3) as its
particular solution. We found that our generalized action (4) contains
infinite order curvature terms and infinite order of their derivatives, thus
becoming a non-local theory of gravity. Furthermore, we realized that the
earlier studied $R^{2}$-like inflation in quadratic curvature non-local theory
Koshelev:2016xqb ; Koshelev:2017tvv ; Koshelev:2020foq ; Koshelev:2020xby  is
an effective description of our more fundamental action (4). The crux of the
paper is about inflationary predictions of our gravity action (4) for the
following inflationary observables related to two-point scalar and tensor
correlations, scalar and tensor spectral tilts and their running

$\Bigg{\\{}n_{s},\,\frac{dn_{s}}{d\ln k},\,r,\,n_{t},\,\frac{dn_{t}}{d\ln
k},\,\frac{d^{2}n_{t}}{d\ln k^{2}}\Bigg{\\}}$ (2)

Figure 1: In this plot we confront the predictions of generalized non-local
$R^{2}$-like inflation against the latest BICEP/Keck array analysis (depicting
the ruled out monomial models of inflation and the natural inflation which is
in purple colour) BICEP:2021xfz  which has constrained the tensor-to-scalar
ratio as $r<0.036$ and the spectral index as $n_{s}=0.9649\pm 0.0042$. The
black colour vertical line from the left represents
$\left(n_{s},\,r\right)=\left(0.964,\,<0.036\right)$ for $N=55$ whereas the
one on the right represents
$\left(n_{s},\,r\right)=\left(0.967,\,<0.036\right)$ for $N=60$.

We have shown that the power spectrum predictions of our generalized action
remain the same as in the earlier studied model of $R^{2}$-like inflation in
non-local gravity Koshelev:2020foq  with the theoretically consistent choice
of form factors which do not explicitly depend on the background curvature.
Especially, we have established in detail the structure of form factors and
degrees of freedom of the theory (See Sec. 3.4 along with Appendix C for
details). The predictions of $\left(n_{s},\,r\right)$ are depicted in Fig. 1
against the latest bounds on the tensor-to-scalar ratio from BICEP2/Keck Array
BICEP:2021xfz . In this theory we modify the tensor power spectrum as in (25),
whereas the scalar power spectrum remains nearly scale invariant, and also the
scalar spectral index $n_{s}$ and its running $\frac{dn_{s}}{d\ln k}$ remain
the same as in the local $R^{2}$ theory.

Here are the key consequences of the modification of the tensor power spectrum
which is due to terms of infinite order in powers of Ricci scalar, Weyl tensor
squared and their covariant derivatives in (4).

  * •

Sound speed of tensor fluctuations remains Unity that is compatible with the
latest bounds on the sound speed of gravitational waves from GW170817
baker2022measuring . Also the sound speed of scalar fluctuations in our model
remains Unity.

  * •

Single field tensor consistency relation is modified, $r\neq-8n_{t}$, due to
modification of the tensor tilt by non-local higher curvature effects.

  * •

Scale invariant nature of tensor power spectrum111In GR, scale invariance of
the power spectrum of tensor perturbations generated during a metastable de
Sitter (inflationary) stage in the early Universe was first derived in
Starobinsky:1979ty  without introducing a concrete model of inflation. can be
broken and the predictions of $\left(n_{t},\,\frac{dn_{t}}{d\ln k}\right)$ can
be an order of magnitude larger than in conventional EFT models of inflation
(See Sec. 3.4 and the discussion in Sec. 4).

  * •

The tensor tilt can be positive as well as negative. Especially, it is
possible to achieve blue-tilt222Alternative theories of inflation such as
string gas cosmology can have blue tilt, too Wang:2014kqa , but such
frameworks predict undetectably small non-Gaussianities. In our case we can
have blue tilt, but there can be relatively large non-Gaussianities KKS ; KKS2
that is a distinguishable feature from string gas cosmology $n_{t}>0$ for any
value of $r<0.036$.

  * •

In our framework, the Hubble parameter during inflation remains the same for
any value of tensor-to-scalar ratio $r<0.036$. Therefore, energy scale of
inflation remains unchanged with changing the value of $r$.

  * •

We have shown in detail that our non-local gravity inflation cannot fall into
any category of EFT of single field inflation Cheung:2007st ; Cabass:2022avo
which is established as the unified framework of all single scalar field
models of inflation.

All together we successfully establish in this paper that geometric
modifications of $R^{2}$ inflation can lead to definite predictions for
inflation which are in-line but still distinguishable from several classes of
scalar field inflationary models. Thus, we open a new door for understanding
and development of cosmic inflation. Our study will be continued in the
companion papers where we study primordial scalar and tensor non-Gaussianities
KKS ; KKS2 . All of the above predictions make the generalized non-local
$R^{2}$-like inflation a viable target for the future experiments such as
CMBS4 CMB-S4:2020lpa , LiteBIRD LiteBIRD:2022cnt , e-LISA Ricciardone:2016ddg
and CORE CORE:2016ymi  which are aimed for detection of primordial
gravitational waves.

##  3 Generalized $R^{2}$-like inflation in analytic non-local gravity

###  3.1 Action of the model

Given the success of $R^{2}$ inflation, any extension of it must preserve the
values of the scalar power spectrum and its tilt which are accurately measured
by the Planck data Akrami:2018odb . Several finite derivative higher curvature
extensions of $R^{2}$ gravity have been studied in the literature (See
Appendix A for a detailed discussion) which all conclude that
$R^{2}$-inflation is the consistent setup to describe current bounds on
spectral index and its running for $N\sim 50-60$ e-foldings. In this Section,
we present the most general gravity action mostly up to cubic in curvature
terms which can be written such that the original inflationary background
solution remains the exact one. The latter is known to form the eigenvalue
problem333This is the trace equation of $R^{2}$ theory with the action $S^{\rm
local}_{R+R^{2}}=\frac{1}{2}\int d^{4}x\sqrt{-g}\,(M_{p}^{2}R+f_{0}R^{2})\,$
with $f_{0}=\frac{M_{p}^{2}}{6M^{2}}$.

$\square R=M^{2}R\,$ (3)

A tedious but straightforward calculation leads to the following gravity
action possessing this property that contains infinite order of derivatives
and in some places infinite order curvature terms

$\displaystyle S_{H}^{\rm Non-local}=$ $\displaystyle\,\frac{1}{2}\int
d^{4}x\sqrt{-g}\,\Bigg{(}M_{p}^{2}R+\Bigg{[}R\mathcal{F}_{R}\left(\square_{s}\right)R+\left(\frac{M_{p}^{2}}{2\mathcal{M}_{s}^{2}}+f_{0}R_{s}\right)W_{\mu\nu\rho\sigma}{\mathcal{F}}_{W}\left(\square_{s},\,R_{s}\right)W^{\mu\nu\rho\sigma}$
(4)
$\displaystyle+\frac{f_{0}\lambda_{c}}{\mathcal{M}_{s}^{2}}\mathcal{L}_{1}\left(\square_{s}\right)R\,\mathcal{L}_{2}\left(\square_{s}\right)R\,\mathcal{L}_{3}\left(\square_{s}\right)R$
$\displaystyle+\frac{f_{0}\lambda_{R}}{\mathcal{M}_{s}^{2}}\mathcal{D}_{1}\left(\square_{s}\right)R\mathcal{D}_{2}\left(\square_{s}\right)W^{\mu\nu\gamma\lambda}\mathcal{D}_{3}\left(\square_{s}\right)W_{\mu\nu\gamma\lambda}$
$\displaystyle+\frac{f_{0}\lambda_{W}}{\mathcal{M}_{s}^{2}}\mathcal{C}_{1}\left(\square_{s}\right)W_{\mu\nu\rho\sigma}\mathcal{C}_{2}\left(\square_{s}\right)W^{\mu\nu\gamma\lambda}\mathcal{C}_{3}\left(\square_{s}\right)W_{\gamma\lambda}^{\quad\rho\sigma}\Bigg{]}+\cdots\Bigg{)}\,,$

where $\cdots$ represent higher order curvature terms which do not contribute
neither to two-point nor to 3-point correlations of inflationary fluctuations.
In the companion papers KKS ; KKS2  we further disclose details about these
terms. Here $R,\,W_{\mu\nu\rho\sigma}$ are Ricci scalar and Weyl tensor
respectively and $\square_{s}=\frac{\square}{\mathcal{M}_{s}^{2}}$ with
$\square$ being the d’Alembertian operator. $\mathcal{M}_{s}$ is the scale of
higher derivative corrections, $R_{s}=\frac{R}{\mathcal{M}_{s}^{2}}$, and

$\Bigg{\\{}\mathcal{F}_{R}\left(\square_{s}\right),\,\mathcal{F}_{W}\left(\square_{s},\,R_{s}\right),\,\mathcal{L}_{i}\left(\square_{s}\right),\,\mathcal{C}_{i}\left(\square_{s}\right),\,\mathcal{D}_{i}\left(\square_{s}\right)\Bigg{\\}}$
(5)

are analytic infinite derivative (AID) operators often called form-factors.
Analyticity at zero curvature is essential to restore the Einstein gravity in
IR.

Let us elaborate on how action (4) emerges. In general, any higher curvature
extension of GR is constructed through adding all possible curvature
invariants involving the following three curvature tensors

$\mathbb{R}=\Bigg{\\{}R\quad R_{\mu\nu}\quad
W_{\mu\nu\rho\sigma}\Bigg{\\}}\,,$ (6)

where $\mathbb{R}$ denotes either Ricci scalar, or Ricci tensor, or the Weyl
tensor. Since Riemann tensor can be expressed in terms of the basics (6), we
can drop it from our construction of the action. Weyl tensor is a convenient
choice for FLRW backgrounds compared to Riemann tensor. In the context of
inflationary backgrounds, since
$\bar{R}_{\mu\nu}\approx\frac{\bar{R}}{4}\,g_{\mu\nu}$, it is wise to
eliminate terms in the action containing Ricci tensor to simplify our
analysis. Moreover, our task here is to find a pure gravitational action
compatible with (3). Therefore, we can resort for the moment to just terms
containing the scalar curvature $R$ leaving the detailed analysis of the
impact of the terms containing Ricci tensor for future studies.

Extending Einstein’s gravity to a 4-derivative action, we end up with the
Stelle gravity

$S_{4}=\frac{1}{2}\int
d^{4}x\sqrt{-g}\,(M_{p}^{2}R+f_{0}R^{2}+f_{W0}W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma})\,,$
(7)

and the trace equation here is identical to (3). Stelle gravity was found to
be renormalizable Stelle:1977ry  but non-unitary because of the presence of
the Weyl ghost444However at quantum level, the unitarity problem resolutions
for Stelle gravity were elaborated by changing the Feynman prescription via
the so-called Fakeon prescription or PT-symmetric quantization or by
Renormalization Group approach by including matter couplings Anselmi:2017ygm ;
Mannheim:2021oat ; Bender:2007wu ; Salvio:2014soa ; Salvio:2017qkx . Motivated
by these approaches, there have been studies of $R^{2}$-inflation and their
respective UV-completions Anselmi:2020lpp ; Salvio:2019agg ; Salvio:2022mld ..
Even though non-unitary, Stelle gravity tells us that the higher curvature
modification is the right approach towards quantum gravity due to manifest
renormalizability of the model. Assuming the ghost mass to be heavy, i.e.
$f_{0}\gg f_{W0}$, Stelle gravity successfully leads to Starobinsky inflation
Starobinsky:1980te . Let us remind here that $f_{0}$ is very large indeed in
the local $R^{2}$ inflation,
$f_{0}=\frac{N^{2}}{144\pi^{2}\mathcal{P}_{\mathcal{R}}}\approx 10^{9}$, that
follows from the smallness of the primordial scalar perturbation spectrum
$\mathcal{P}_{\mathcal{R}}$ only ($\mathcal{P}_{\mathcal{R}}(N)\propto
N^{2},N\gg 1$ for Starobinsky inflation, so $f_{0}$ is $N$-independent).

Later on it has become clear that by adding higher (infinite) derivative
terms, an infinite derivative extension of Stelle gravity Krasnikov:1987yj ;
Koshelev:2020xby  can lead to a ghost-free theory around Minkowski555The
action (8) is found to have good UV complete properties such as super-
renormalizability by power counting and unitarity Tomboulis:1997gg ;
Tomboulis:2015esa ; Tomboulis:2015gfa ; Modesto:2011kw ; Biswas:2011ar ;
Koshelev:2016xqb ; Biswas:2016egy ; Koshelev:2017ebj ; Calcagni:2014vxa .
Still several open questions exist about unitarity of the theory at the non-
perturbative level, behavior of scattering amplitudes and the uniqueness of
higher derivative operators.

$S^{\rm Non-local}_{q}=\frac{1}{2}\int
d^{4}x\sqrt{-g}\,(M_{p}^{2}R+R\mathcal{F}_{R}\left(\square_{s}\right)R+W_{\mu\nu\rho\sigma}\mathcal{F}_{C}\left(\square_{s}\right)W^{\mu\nu\rho\sigma})$
(8)

with the form-factors (see Biswas:2016egy ; Koshelev:2016xqb  for more
details)

$\displaystyle\mathcal{F}\left(\square_{s}\right)$ $\displaystyle=\sum_{n\geq
0}f_{n}\square_{s}^{n}=f_{0}M^{2}\,\frac{1-\left(1-\frac{\square}{M^{2}}\right)e^{\gamma_{S}\left(\square_{s}\right)}}{\square}$
(9) $\displaystyle\mathcal{F}_{C}\left(\square_{s}\right)$
$\displaystyle=\sum_{n\geq
0}f_{Wn}\square_{s}^{n}=\frac{M_{p}^{2}}{2}\,\frac{e^{\gamma_{T}\left(\square_{s}\right)}-1}{\square}\,.$

where
$\gamma_{S}\left(\square_{s}\right),\,\gamma_{T}\left(\square_{s}\right)$ are
entire functions, quite arbitrary in principle. The fact that these functions
are entire and come in the exponent guarantees that extra degrees of freedom
do not emerge. Note that finite number of derivatives will not kill ghosts in
the spectrum, we need infinite number of derivatives! From the point of view
of convergence of loop integrals, one want that form-factors $\mathcal{F}$ at
least do not decay for large momenta (meaning that the propagator behaves at
least like in Stelle gravity).

Though looking promising, this construction suffers from the inability to be
healthy around any background but Minkowski. Indeed, it was shown explicitly
in Biswas:2016etb  how the form-factors would look like around dS background
(assuming explicit cosmological constant term in the action), and the crucial
point is that Taylor coefficients of form-factors should implicitly depend on
the background curvature. Even though one can do adjust the coefficient in the
right way, it becomes possible for one given background (one given dS
curvature) only. On the other hand, the very structure of the above action (8)
does not assume an inclusion of extra curvature factors anywhere meaning that
this model is yet incomplete.

In the present paper we want to break free out of the above problem by
considering extra dependence on curvature. To start with and give a feeling of
what we are looking for, we notice that since Weyl tensor is zero for any FLRW
background, in principle we can include any curvature dependence inside the
form-factor $\mathcal{F}_{C}\left(\square_{s}\right)$ without changing the
propagator around Minkowski space-time (or more generically, as long as we are
interested in FLRW background). In this regard, we can generalize the non-
local Weyl square term in (8) as

$S_{W^{2}}^{\rm Non-local}=\frac{1}{2}\int
d^{4}x\sqrt{-g}\Bigg{[}\left(\frac{M_{p}^{2}}{2\mathcal{M}_{s}^{2}}+f_{0}\frac{R}{\mathcal{M}_{s}^{2}}\right)W_{\mu\nu\rho\sigma}\mathcal{F}_{W}\left(\square_{s},\,R_{s}\right)W^{\mu\nu\rho\sigma}\Bigg{]}$
(10)

where $\mathcal{F}_{W}\left(\square_{s},\,R_{s}\right)$ is a general function
of d’Alembertian and Ricci scalar being analytic at zero values of its
arguments. We can also include any arbitrary function of a generic Riemann
tensor analytic at zero argument, but for the moment we restrict ourselves to
the scalar curvature $R$ only.

In the construction of (10), we exploit the following identities of Weyl
tensor Barvinsky:1994cg ; Barvinsky:1994hw

$\displaystyle R^{\sigma}_{\lambda}W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\lambda}$
$\displaystyle=\frac{R}{4}W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}$ (11)
$\displaystyle\implies
R^{\sigma}_{\lambda}\mathcal{O}\left(\square_{s}\right)W_{\mu\nu\rho\sigma}\mathcal{O}\left(\square_{s}\right)W^{\mu\nu\rho\lambda}$
$\displaystyle=\frac{R}{4}\mathcal{O}\left(\square_{s}\right)W_{\mu\nu\rho\sigma}\mathcal{O}\left(\square_{s}\right)W^{\mu\nu\rho\lambda}\,$

Following the papers Barvinsky:1994cg ; Barvinsky:1994hw , one can realize
that at the cubic order in curvatures all possible terms can be recasted to
have only d’Alembertians and not single covariant derivatives as
follows666Indeed we can eliminate all terms with single covariant derivatives
acting on curvature tensor by adding arbitrary number of total derivatives.
For example, suppose we have a term of the form
$\mathbb{R}\nabla_{\mu}\mathbb{R}\nabla^{\mu}\mathbb{R}$ in the action. Then
we can eliminate this term by adding a total derivative of the form
$\square\left(\mathbb{R}^{3}\right)$. We can continue this procedure for any
arbitrary number of derivatives until we arrive to (12).

$\mathcal{O}_{1}(\square_{s})\mathbb{R}\mathcal{O}_{2}(\square_{s})\mathbb{R}\mathcal{O}_{3}(\square_{s})\mathbb{R}$
(12)

Then recalling that we use only terms involving Ricci scalar and Weyl tensor
which do not spoil the Minkowski propagator, we arrive at (4). It is useful to
recall here the following identities of Weyl tensor which confirms that the
cubic in Weyl tensor term that we have in (4) is indeed the only possible
combination

$\displaystyle
W_{[\mu\nu}^{\quad\gamma\lambda}\mathcal{O}\left(\square_{s}\right)W_{\gamma\lambda}^{\quad\alpha\beta}\mathcal{O}\left(\square_{s}\right)W_{\alpha]\beta}^{\quad\mu\nu}=0\,,$
(13) $\displaystyle
W^{\mu\nu\rho\sigma}\mathcal{O}\left(\square_{s}\right)W_{\mu\nu\rho\lambda}=\frac{1}{4}\delta^{\sigma}_{\lambda}W^{\mu\nu\rho\alpha}\mathcal{O}\left(\square_{s}\right)W_{\mu\nu\rho\alpha}\,,$

where in the first line we have complete anti-symmetrization over the five
indices. Moreover, we can make the form factors
$\mathcal{C}\left(\square_{s}\right)$ to further depend on $R_{s}$ but we drop
this generalization for brevity.

###  3.2 Restrictions on form-factors (5)

The requirement that (3) is an exact background can be met by imposing the
following constraints on the operators (see Appendix B for a detailed
derivation using equations of motion following from the action (4))

$\mathcal{F}_{R}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)=f_{0}=\frac{M_{p}^{2}}{6M^{2}}\,,\quad\mathcal{F}^{\dagger}_{R}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)=0\,,\quad\mathcal{L}_{i}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)=0\,,$
(14)

where † denotes derivative with respect to the argument. Since Weyl tensor
vanishes on FLRW backgrounds, we do not get any constraints on the form
factors
$\mathcal{F}_{W}\left(\square_{s},\,\frac{2R}{3\mathcal{M}_{s}^{2}}\right)$,
$\mathcal{C}_{i}\left(\square_{s}\right)$ and
$\mathcal{D}_{i}\left(\square_{s}\right)$ as long as the background solution
is concerned. Conditions on $\mathcal{F}_{R}$ are naturally the same as they
were in the previous studies Koshelev:2016xqb ; Koshelev:2017tvv  based on the
action (8). The last equality in (14) is a new condition which is sufficient,
but not necessary though, and can be weakened as only 2 out of 3 operators
$\mathcal{L}_{i}$ must vanish at the point $M^{2}/\mathcal{M}_{s}^{2}$ to have
(3) as an exact background.

Another requirement is that the action (4) must be ghost-free containing only
a scalaron and a massless graviton degree of freedom in the Minkowski and in
the quasi-dS limits (i.e., $\bar{R}_{\rm dS}\approx\mathrm{const}$ and
$f_{0}\bar{R}_{\rm dS}\gg M_{p}^{2}$, though still $\bar{R}_{\rm dS}\ll
M_{p}^{2}$) during inflation. This leads to Koshelev:2016xqb ;
Koshelev:2017tvv

$\displaystyle\mathcal{F}_{R}\left(\square_{s}\right)$
$\displaystyle=f_{0}M^{2}\frac{1-\left(1-\frac{\square}{M^{2}}\right)e^{\gamma_{S}\left(\square_{s}\right)}}{\square_{s}}$
(15) $\displaystyle\mathcal{F}_{W}\left(\square_{s},\,R_{s}\right)$
$\displaystyle=\frac{e^{\gamma_{T}\left(\square_{s}-\frac{2}{3}R_{s}\right)}-1}{\square_{s}-\frac{2}{3}R_{s}}\,$
$\displaystyle\mathcal{D}_{1}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)$
$\displaystyle=0\,,$

where $\gamma$-s are entire functions. With the last condition in (15), we can
notice that the contribution to the second order perturbed action from the 3rd
line of (4) vanishes exactly (see Sec. 3.4 for details). Additionally
combining (14) with (15), one yields

$\gamma_{S}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)=0\implies\gamma_{S}\left(\square_{s}\right)=\left(\square_{s}-\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)\tilde{\gamma}_{S}\left(\square_{s}\right)\,.$
(16)

where $\tilde{\gamma}_{S}$ is another entire function.

Indeed, we count the degrees of freedom perturbatively by computing the second
order action around a given background. Around Minkowski space-time, all the
cubic and higher order curvature terms are irrelevant, and with (15) the
graviton propagator as a function of the 4-momentum square $p^{2}$ is given by
Krasnikov:1987yj ; Koshelev:2016xqb

$\Pi\left(p^{2}\right)\sim-\frac{P^{(2)}}{p^{2}e^{\gamma_{T}\left(p^{2}\right)}}+\frac{P^{(0)}}{2p^{2}\left(1+\frac{p^{2}}{M^{2}}\right)e^{\gamma_{S}\left(p^{2}\right)}}\,,$
(17)

where $P^{(2)},\,P^{(0)}$ are spin projection operators.

This implies the theory contains only a scalaron and a massless graviton in
the spectrum Krasnikov:1987yj ; Koshelev:2016xqb . From the UV-completion
point of view, it is expected that the propagator for the high momenta
$p\to\infty$ is suppressed at least as in the Stelle gravity. Similarly, it is
preliminary expected that
$\mathcal{L}_{i}\left(\frac{p^{2}}{\mathcal{M}_{s}^{2}}\right)$,
$\mathcal{C}_{i}\left(\frac{p^{2}}{\mathcal{M}_{s}^{2}}\right)$ and
$\mathcal{D}_{i}\left(\frac{p^{2}}{\mathcal{M}_{s}^{2}}\right)$ at least do
not grow in the high energy limit. Note that the action (4) with the form
factors (15) is a generalization of the earlier studied non-local quadratic
curvature gravity Koshelev:2016xqb ; Koshelev:2017tvv ; Koshelev:2020foq
which can be seen as inclusion of the curvature dependence in the form-factors
compared to the previous studies. This move is motivated by the understanding
that the form-factors which depend on the d’Alembertian only cannot
accommodate ghost-free conditions around Minkowski and (quasi)-dS spaces
simultaneously. We note that the conditions (14) are sufficient but not
necessary. Indeed those conditions guarantee that (3) is an exact background
solution as we demanded from the beginning of our consideration. However,
regarding operators $\mathcal{L}_{i}$ it is enough that only any 2 out of 3
vanish at the point $M^{2}/\mathcal{M}_{s}^{2}$. Then any term in the
equations of motion coming from $R^{3}$ part would vanish anyway. This can be
traced by examining equations of motion given explicitly in Appendix B.
However, imposing
$\mathcal{L}_{i}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)=0$ on all 3
operators will make the contribution of cubic curvature term to the second
order perturbed action vanish (see Sec. (3.4) for details) that will simplify
our analysis. We defer the other possible but complicated choices of
$\mathcal{L}_{i}\left(\square_{s}\right)$ for future investigation, but we
greatly expect that observational aspects of the model remain almost the same
even in the general case.

###  3.3 On the number of model parameters

Examining the construction of our action (4) which involve infinite order
derivatives and infinite order terms in curvature, one might think the theory
has infinite number of parameters. However we can end up with a theory with
only finite parameter space. For example, let us take the entire functions

$\Bigg{\\{}\bar{\gamma}_{S}\left(\square_{s}\right),\,\gamma_{T}\left(\square_{s}\right)\Bigg{\\}}$
(18)

defined in (15) and (16) can be chosen to be either polynomials or functions
of polynomials777For example, we can consider a choice prescribed in the
context of super-renormalizable quadratic curvature non-local theories
Koshelev:2016xqb
$\displaystyle\bar{\gamma}_{S}\left(\square_{s}\right)=\gamma_{T}\left(\square_{s}\right)$
$\displaystyle=\Gamma\Big{(}0,\,H_{T}(\square_{s})^{2}\Big{)}+\gamma_{E}+\ln\Big{(}H_{T}(\square_{s})^{2}\Big{)}.$
(19) where $\Gamma\left(0,\,z\right)=\int_{z}^{\infty}dt\frac{e^{-t}}{t}$ is
the incomplete gamma function with first argument being zero, $\gamma_{E}$ is
the Euler-Maschoroni constant and $H_{T}\left(\square_{s}\right)$ is some
polynomial which determines the high energy behaviour of the graviton
propagator around Minkowski. We can clearly see here that the entire functions
depend on the choice of polynomial function $H_{T}\left(\square_{s}\right)$,
thus leading us to finite parameter space.  which leads to finite parameter
space and it can be observationally probed through inflationary observables
that we shall study in the rest of the paper. In this paper, we study two-
point inflationary correlations which do not receive any contributions from
the last three lines of the action in (4). Thus, the parameter space
associated with
$\mathcal{L}_{i}\left(\square_{s}\right),\,\mathcal{D}_{i}\left(\square_{s}\right),\,\mathcal{C}_{i}\left(\square_{s}\right)$
is irrelevant for this paper, but we address it in the companion works KKS ;
KKS2 .

###  3.4 Power spectrum predictions of generalized non-local $R^{2}$-like
inflation with (4)

In this section, we derive that the power spectrum predictions of $R^{2}$-like
inflation in (4) using the several results obtained in Koshelev:2016xqb ;
Koshelev:2017tvv ; Koshelev:2020foq  as explained below. Calculating the
second order perturbation of the action (4) around the background solution
$\bar{\square}\bar{R}=M^{2}\bar{R}$, we obtain

$\delta^{(2)}S_{H}^{\rm Non-local}=\delta^{(2)}S_{R+R^{2}}^{\rm
local}+\delta^{(2)}S_{R+R^{2}}^{\rm Non-
local}+\delta^{(2)}S_{\mathbb{R}^{3}}^{\rm Non-local}$ (20)

where $S_{R+R^{2}}^{\rm local}$ is the local $R^{2}$ action and

$\displaystyle S_{R+R^{2}}^{\rm Non-local}=$ $\displaystyle\,S_{R^{2}}^{\rm
Non-local}+S_{W^{2}}^{\rm Non-local}$ (21) $\displaystyle=$
$\displaystyle\,\frac{1}{2}\int
d^{4}x\sqrt{-g}\Bigg{\\{}R\Bigg{[}\mathcal{F}_{R}\left(\square_{s}\right)-f_{0}\Bigg{]}R$
$\displaystyle+\left(\frac{M_{p}^{2}}{2\mathcal{M}_{s}^{2}}+f_{0}\frac{R}{\mathcal{M}_{s}^{2}}\right)W_{\mu\nu\rho\sigma}{\mathcal{F}}_{W}\left(\square_{s},\,\frac{R}{\mathcal{M}_{s}^{2}}\right)W^{\mu\nu\rho\sigma}\Bigg{\\}}$

and $S_{\mathbb{R}^{3}}^{\rm Non-local}$ constitute from the last 3-lines in
(4) which are cubic in curvature. Remarkably, applying
$\mathcal{L}_{i}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)=0$ from (14),
we can see that the cubic non-local term has no contribution to the above
second order action around the flat FLRW background satisfying888 It is also
not very surprising that the cubic scalar curvature term in the action (4) do
not affect the second order action around the inflationary solution
$\bar{\square}\bar{R}=M^{2}\bar{R}$. An analogous simple example is the
following action  $S^{\rm Min}_{R^{3}}=\frac{1}{2}\int
d^{4}x\sqrt{-g}\,\left[M_{p}^{2}R+f_{0}R^{2}+f_{0}\Lambda^{2}R^{3}\right]\,$
(22) where $\Lambda$ is some mass scale. The above Lagrangian has the
following property that the cubic term $R^{3}$ does not effect the second
order action (or the linearized equations of motion) around Minknowski,
however, it is obviously not true in other backgrounds. If we study inflation
in any local higher curvature extension of $R^{2}$, we do modify inflationary
solution unless we assume $R^{2}$ is the most relevant term during inflation
and remaining terms are highly negligible. In fact, the general statement is
that slow-roll inflation in local $f(R)$ gravity occurs for the range of $R$
for which the Lagrangian density ${\cal L}$ is close to $R^{2}$, namely,
${\cal L}=A(R)R^{2}$ where $A(R)$ is a slowly changing function of
$R,~{}|\frac{d\ln A}{d\ln R}|\ll 1$ Appleby:2009uf . In our case the cubic
scalar curvature non-local terms (4) are introduced in order not to effect the
second order scalar perturbations around the inflationary phase given by
$\bar{\square}\bar{R}=M^{2}\bar{R}$.  (3).

$\delta^{(2)}S_{\mathbb{R}^{3}}\Bigg{|}_{\bar{\square}\bar{R}=M^{2}\bar{R}}=0\,.$
(23)

Therefore, non-local contribution to the two-point inflationary correlations
comes only from the second variation of (21). The cubic in curvature terms in
(4) will give non-trivial shapes of non-Gaussianities which will be
investigated in a separate paper KKS . We present in detail the computation of
the second order action (20) and analyze the corresponding perturbed degrees
of freedom in the quasi-dS limit in Appendix C. After long computations we
deduce that our second order action in (20) exactly coincides with the result
obtained in (4.5) of Koshelev:2017tvv . As a result, the predictions for
scalar and tensor power spectra become exactly the same as in the theory
studied in Koshelev:2016xqb ; Koshelev:2017tvv ; Koshelev:2020foq . The
crucial difference in our study is that we consider a choice of
$\mathcal{F}_{R}\left(\square_{s}\right),\,\mathcal{F}_{W}\left(\square_{s}\right)$
that does not explicitly depend on the background curvatures during inflation
which was the case of Craps:2014wga ; Koshelev:2017tvv ; Koshelev:2020foq .
Therefore, the scalar power spectrum $\mathcal{P}_{\mathcal{R}}$ and spectral
index $n_{s}$ of generalized non-local $R^{2}$ inflation are

$\mathcal{P}_{\mathcal{R}}(k)\approx\frac{1}{3f_{0}\bar{R}_{\rm
dS}}\frac{H^{2}}{16\pi^{2}\epsilon^{2}}\Bigg{|}_{k=aH}\,,\quad
n_{s}-1\equiv\frac{d\ln\mathcal{P}_{\mathcal{R}}}{d\ln
k}\Bigg{|}_{k=aH}\approx-\frac{2}{N}\,,$ (24)

where $a(t)H(t)$ is estimated at the first Hubble radius crossing during
inflation for a given mode wave number $k$. The above result (24) is almost
independent of $\mathcal{F}_{R}\left(\square_{s}\right)$ as far as
$M^{2}\ll\mathcal{M}_{s}^{2}$, and the dimensionless coefficients of
$\mathcal{F}_{R}\left(\square_{s}\right)$ (15) are of the $O(1)$ (see
Koshelev:2017tvv ; Koshelev:2020foq  for further details, where it was shown
that the non-local corrections to $n_{s}$ are of the order of
$O\left(\frac{M^{4}}{\mathcal{M}_{s}^{4}}\right)$).

The inflationary tensor power spectrum of generalized non-local $R^{2}$ model

$\displaystyle\mathcal{P}_{T}$
$\displaystyle=\frac{1}{12\pi^{2}f_{0}}\left(1-3\epsilon\right)e^{-2\gamma_{T}\left(\frac{\bar{R}_{dS}}{2\mathcal{M}_{s}^{2}}\right)}\Bigg{|}_{k=aH}\,,$
(25)

Computing the tensor-to-scalar ratio from (24) and (25) we obtain

$r=\frac{12}{N^{2}}e^{-2\gamma_{T}\left(-\frac{\bar{R}}{2\mathcal{M}_{s}^{2}}\right)}\Bigg{|}_{k=aH}\,.$
(26)

The tensor spectral index and its running and running of running can be
computed as

$\displaystyle n_{t}\equiv\frac{d\ln\mathcal{P}_{T}}{d\ln k}\Bigg{|}_{k=aH}$
$\displaystyle\approx-\frac{3}{2N^{2}}-\left(\frac{2}{N}+\frac{3}{2N^{2}}\right)\frac{\bar{R}_{\rm
dS}}{2\mathcal{M}_{s}^{2}}\gamma_{T}^{\dagger}\left(-\frac{\bar{R}_{\rm
dS}}{2\mathcal{M}_{s}^{2}}\right)\,$ (27) $\displaystyle\frac{dn_{t}}{d\ln
k}\Bigg{|}_{k=aH}$
$\displaystyle\approx-\frac{3}{N^{3}}-\frac{3}{4N^{3}}\frac{\bar{R}_{\rm
dS}}{\mathcal{M}_{s}^{2}}\gamma_{T}^{\dagger}\left(-\frac{\bar{R}_{\rm
dS}}{2\mathcal{M}_{s}^{2}}\right)-\frac{1}{2N^{2}}\frac{\bar{R}_{\rm
dS}^{2}}{\mathcal{M}_{s}^{4}}\gamma_{T}^{\dagger\dagger}\left(-\frac{\bar{R}_{\rm
dS}}{2\mathcal{M}_{s}^{2}}\right)\,$ $\displaystyle\frac{d^{2}n_{t}}{d\ln
k^{2}}\Bigg{|}_{k=aH}$
$\displaystyle\approx-\frac{9}{N^{4}}-\frac{3}{2N^{4}}\frac{\bar{R}_{\rm
dS}}{\mathcal{M}_{s}^{2}}\gamma_{T}^{\dagger}\left(-\frac{\bar{R}_{\rm
dS}}{2\mathcal{M}_{s}^{2}}\right)+\frac{3}{4N^{3}}\frac{\bar{R}_{\rm
dS}}{\mathcal{M}_{s}^{2}}\gamma_{T}^{\dagger\dagger}\left(-\frac{\bar{R}_{\rm
dS}}{2\mathcal{M}_{s}^{2}}\right)$
$\displaystyle\quad-\frac{1}{4N^{3}}\frac{\bar{R}_{\rm
dS}^{3}}{\mathcal{M}_{s}^{6}}\gamma_{T}^{\dagger\dagger\dagger}\left(-\frac{\bar{R}_{\rm
dS}}{2\mathcal{M}_{s}^{2}}\right)$

where ${}^{\dagger},\,^{\dagger\dagger},\,^{\dagger\dagger\dagger}$ indicate
first, second and third derivative with respect to the argument, we used
$\frac{d}{d\ln k}\approx-\left(1+\frac{1}{2N}\right)\frac{d}{dN}$ and
$\frac{d\bar{R}_{\rm dS}}{dN}\approx 2\bar{R}_{\rm dS}\epsilon$. From (25),
(26) and (27) we can conclude the following

  * •

The tensor power spectrum is modified due to the non-local contributions from
the higher curvature non-local terms involving Ricci scalar and Weyl tensor in
(4) i.e., due to the effect of (10) in the limit $M_{p}^{2}\ll f_{0}R$.

  * •

In comparison with local $R^{2}$ gravity, the tensor power spectrum contains a
strong scale dependence due to the exponential term
$e^{\gamma_{T}\left(\frac{\bar{R}_{\rm dS}}{\mathcal{M}_{s}^{2}}\right)}$
where $\bar{R}_{\rm dS}(k)$ depends on wave number $k$.

  * •

With tensor tilt and its running and running of running computed in from (27)
we can probe $\gamma_{T}\left(\frac{\bar{R}_{\rm
dS}}{\mathcal{M}_{s}^{2}}\right)$ and reconstruct the form factor (15) from
the future primordial gravitational wave observations CMB-S4:2020lpa ;
Calcagni:2020tvw .

  * •

The single field tensor consistency relation $r=-8n_{t}$ gets violated solely
due to the modification of tensor-power spectrum.

To illustrate the above conclusions we consider a choice of entire function

$\gamma_{T}\left(\square_{s}-\frac{2R}{3\mathcal{M}_{s}^{2}}\right)=\beta_{1}\left(\square_{s}-\frac{2R}{3\mathcal{M}_{s}^{2}}\right)^{2}+\beta_{2}\left(\square_{s}-\frac{2R}{3\mathcal{M}_{s}^{2}}\right)^{3}\,.$
(28)

and present the predictions of the model for the different ranges of parameter
space $\left(\beta_{1},\,\beta_{2}\right)$ in Fig. 2 and Fig. 3.

Figure 2: In the upper left and right panel we plot the tensor-to-scalar ratio
($r$) and and the tensor tilt ($n_{t}$) versus the ratio of scalaron mass
$(M)$ and the scale of non-locality ($\mathcal{M}_{s}$) respectively. In the
lower panel we plot the running $\left(\frac{dn_{t}}{d\ln k}\right)$ and
running of running of the tensor spectral index $\left(\frac{d^{2}n_{t}}{d\ln
k^{2}}\right)$ versus $\frac{M}{\mathcal{M}_{s}}$. The parameters for all
these plots are
$\left(\beta_{1},\,\beta_{2}\right)=\left(0.875,\,0.1575\right)$,
$\left(\beta_{1},\,\beta_{2}\right)=\left(0.625,\,0.1125\right)$,
$\left(\beta_{1},\,\beta_{2}\right)=\left(0.625,\,0.1125\right)$ for full,
dashed and dotted lines in magenta respectively.

Figure 3: In the upper left and right panel we plot the tensor-to-scalar ratio
($r$) and and the tensor tilt ($n_{t}$) versus the ratio of scalaron mass
$(M)$ and the scale of non-locality ($\mathcal{M}_{s}$) respectively. In the
lower panel we plot the running $\left(\frac{dn_{t}}{d\ln k}\right)$ and
running of running of the tensor spectral index $\left(\frac{d^{2}n_{t}}{d\ln
k^{2}}\right)$ versus $\frac{M}{\mathcal{M}_{s}}$. The parameters for all
these plots are
$\left(\beta_{1},\,\beta_{2}\right)=\left(0.2282,\,0.04\right)$,
$\left(\beta_{1},\,\beta_{2}\right)=\left(0.1843,\,0.0326\right)$,
$\left(\beta_{1},\,\beta_{2}\right)=\left(0.1316,\,0.0233\right)$ for full,
dashed and dotted lines in brown respectively.

From (26) and (27) we can derive the following relation between tensor tilt
$n_{t}$, running of tensor to scalar ratio $\frac{d\ln r}{d\ln k}$.

$n_{t}\approx-\left(1-n_{s}\right)+\frac{d\ln r}{d\ln k}$ (29)

Nature of scalaron: From (66) we can extract that the sound speed of curvature
perturbation is unity. Therefore, we have a canonical scalar field propagating
during inflation similar to the local $R^{2}$ theory. Thus, we can also
conclude that there is no super-horizon evolution of the curvature
perturbation $\mathcal{R}$ and it can be assumed to be conserved on
superhorizon scales as its solution generally contain a growing mode and a
decaying mode DeFelice:2010aj

$\mathcal{R}=c_{1}+c_{2}\int\frac{dt}{a^{3}\epsilon^{2}}\,.$ (30)

##  4 Non-local $R^{2}$-like inflation versus effective field theory of scalar
field inflation

The goal of this section is to provide a qualitative and somewhat quantitative
understanding of how the framework of inflation in non-local gravity (4) is
different from the effective field theory of (single-field) inflation (EFTI)
Cheung:2007st ; Weinberg:2008hq . Of course, the obvious difference between
the two is locality, but here we discuss in detail how EFTI cannot capture
everything about our ignorance of UV-completion. We argue that even though
EFTI is a useful tool to probe the unknown UV-physics, the well-motivated
inputs from UV-completion point of gravity may shed new lights on inflationary
cosmology beyond what is promised by EFTI.

###  4.1 Nature of primordial fields in EFTI

Ever since the EFTI was proposed Cheung:2007st ; Weinberg:2008hq ;
Creminelli:2014wna ; Creminelli:2015oda , it is widely understood and
established that EFTI captures all our ignorance about UV-completion and the
approach unifies all our models of inflation through the so-called EFT
parameters. As far as the inflationary two-point correlations are concerned,
EFTI prescribes that everything about inflation can be captured by the
following set of slowly varying time-dependent parameters

$\Biggl{\\{}H_{\rm\inf},\,\epsilon_{E},\,\eta_{E},\,c_{s},\,c_{t}\Biggr{\\}}$
(31)

where $H_{\rm\inf}$ is the nearly constant Hubble parameter during inflation
in an EFT. The corresponding slow-roll parameters are

$\epsilon_{E}=-\frac{\dot{H}_{\rm inf}}{H_{\rm
inf}^{2}},\quad\eta_{E}=\frac{\dot{\epsilon}_{E}}{H_{\inf}\epsilon_{E}}\,.$
(32)

The additional two-parameters (which are also nearly constant)
$\left(c_{s},\,c_{t}\right)$ are the sound speed of scalar and tensor
perturbations defined through their second order actions

$\displaystyle\delta_{s}^{(2)}S_{\rm EFTI}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\Bigg{[}\frac{M_{p}^{2}}{c_{s}^{2}}\epsilon\left(\dot{\zeta}^{2}-c_{s}^{2}\frac{\left(\partial_{i}\zeta\right)^{2}}{a^{2}}\right)\Bigg{]}$
(33) $\displaystyle\delta_{t}^{(2)}S_{\rm EFTI}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\Bigg{[}\frac{M_{p}^{2}}{c_{t}^{2}}\epsilon\left(\dot{h}_{jk}^{2}-c_{t}^{2}\frac{\left(\partial_{i}h_{jk}\right)^{2}}{a^{2}}\right)\Bigg{]}$

where $\zeta=-{\cal R}$ is the curvature perturbation defined in the so-called
Unitary gauge of EFTI and $h_{ij}$ is the tensor standard tensor perturbation.
During inflation the scalar perturbation can be approximated to be massless in
the first approximation. In standard canonical scalar field inflation (for
example Starobinsky or Higgs inflation Kehagias:2013mya ), the sound speeds
$c_{s}=c_{t}=1$ which means that the speed of propagation of scalar and tensor
perturbed modes equals to the velocity of light.

The construction of EFTI conveys that our ignorance of UV-completion beyond
standard canonical scalar field inflation should be encoded in
$\left(c_{s},\,c_{t}\right)$ different from Unity. As a consequence, the key
observables of inflation are modified in the leading order of the slow-roll
approximation as Burgess:2017ytm ; Baumann:2015xxa

$n_{s}=1-2\epsilon_{E}-\eta_{E}-\epsilon_{s},\quad
r=16\frac{c_{s}}{c_{t}}\epsilon_{E},\quad
n_{t}=-2\epsilon_{E}-\epsilon_{t},\quad r\neq-8n_{t}\,.$ (34)

where $\epsilon_{s}=\frac{\dot{c}_{s}}{Hc_{s}}\ll
1,\,\epsilon_{t}=\frac{\dot{c}_{t}}{Hc_{t}}\ll 1$ are the slow-roll parameters
associated with the sound speeds. Therefore, the central observationally
relevant message from EFTI is that any violation of single field tensor
consistency relation $r=-8n_{t}$ must arise from
$0\leq\left(c_{s},\,c_{t}\right)\leq 1$. The generalized non-local
$R^{2}$-like inflation is a counterexample to this which we discuss below.

###  4.2 Differences between non-local $R^{2}$-like inflation and EFTI

Here we make theoretical and observational comparisons between our framework
of geometric inflation with (4) and the EFTI. In the theoretical part we
exploit the differences between the construction of EFTI and (4) with respect
to the nature of perturbed degrees of freedom and the cut-off scale and the
roads to UV-completion. In the observational part we make qualitative and few
quantitative remarks on distinguishing EFTI with our generalized non-local
$R^{2}$-like inflation.

####  4.2.1 With respect to theoretical framework

Here we provide a comparative discussion of EFTI Cheung:2007st ;
Weinberg:2008hq  and our non-local gravity inflation.

  * •

EFTI is motivated from particle physics and identifying carefully the
symmetries based on the adiabatic evolution of the inflationary background.
EFTI construction by Cheung et al Cheung:2007st  captures the nature of
inflationary perturbations and their interactions around a standard slow-roll
inflation by exploiting time dependent spatial diffeomorphisim invariance in
the frame of the broken time diffeomorphisms. This procedure further employs
the so-called Stuckelberg trick to non-linearly recover time-diffeomorphism
invariance and to obtain an effective Lagrangian for a scalar field that is
valid below a cut-off scale $\Lambda\gtrsim H_{\rm inf}$. Our geometric
framework of inflation studied in this paper goes beyond any of the
assumptions of symmetries and perturbative expansion of terms of order
$\frac{H}{\Lambda}$ in the EFTI. We collected all possible curvature invariant
terms one can write around an exact inflationary background (3). We have ended
up with infinite number of curvature invariant terms in the action (4) which
cannot be truncated at any order as far as the
$\mathcal{M}_{s}^{2}\lesssim\bar{R}_{\rm dS}\ll M_{p}^{2}$. Moreover, our
diffeomorphism invariant action (4) captures a significant set of terms which
can be relevant for quantum gravity.

  * •

The construction of EFTI relies on an existence of a cut-off scale
$\Lambda>H_{\inf}$, and it assumes that all effects (of possible existence of
heavy modes that can be integrated out) above this cut-off scale can be
encoded in terms of EFTI parameters. In our construction, we do have a scale
$\mathcal{M}_{s}\gtrsim H$ which is the scale of non-locality, but we have
found that collecting all possible non-local terms we get new effects for
inflationary correlations which cannot be mapped to EFTI parameters. Indeed
comparing (33) with (66) and (68), we can deduce that the sound speeds of
perturbations in the generalized non-local $R^{2}$-like inflation are Unity.
But in our study we do obtain violation of tensor consistency relation (read
from (26) and (27)) due to the non-local effect, more precisely due to the
terms involving Weyl tensor in (4). Therefore, perturbations in generalized
non-local $R^{2}$-like inflation do not belong to any special case of EFTI
which thus proves that the assumptions of EFTI capturing all possible UV-
signatures breaks down at least in our context.

  * •

In EFTI the sound speeds deviation from Unity is signature of non-relativistic
effects that can occur in the low-energy (at the scale of inflation) theory
which may or may not persist in the UV-theory. For example, if the fundamental
theory is Lorentz invariant, the non-relativistic effects are emergent in the
low-energy limit. In the alternative case, Lorentz invariance may not be the
fundamental symmetry in the UV Baumann:2015nta ; Aoki:2021ffc . Our theory (4)
is Lorentz invariant and we do not have any non-relativistic regime at any
energy scale. Moreover, we entirely fix the gravitational degrees of freedom
at all energy scales (see Sec. 3.4 and Appendix C).

  * •

In Weinberg:2008hq  Weinberg has proposed a version of EFTI by constructing a
covariant action which is a bit more general than the one by Cheung et al
Cheung:2007st . Weinberg has introduced new terms that indeed are relevant in
the EFTI, especially the term relevant to our discussion is999In the
Weinberg’s paper Weinberg:2008hq  the scalar field $\phi$ is dimensionless and
the scalar field kinetic term is written as
$-\frac{\Lambda^{2}}{2}\left(\partial\phi\right)^{2}$, but here we do the
rescaling $\phi\to\frac{\phi}{\Lambda}$ that is reflected in (35).

$\int
d^{4}x\sqrt{-g}\Biggl{[}f_{1}\left(\frac{\phi}{\Lambda}\right)W^{\mu\nu\rho\sigma}W_{\mu\nu\rho\sigma}\Biggr{]}$
(35)

where $\phi$ is a canonically normalized inflaton field or the Goldstone boson
in the language of EFTI in Cheung:2007st . The presence of Weyl square terms
in the action does not effect the background equations for FLRW, but they do
effect perturbations. However, for any effective scale $\Lambda<M_{p}$, Weyl
square term (35) may not be the lowest order term and in principle we could
add higher derivative and higher curvature terms which do not effect the FLRW
solutions

$\displaystyle\int d^{4}x\sqrt{-g}\Biggl{[}$ $\displaystyle
f_{1}\left(\frac{\phi}{\Lambda}\right)W^{\mu\nu\rho\sigma}\mathcal{F}_{1}\left(\frac{\square}{\Lambda^{2}},\,\frac{R}{\Lambda^{2}}\right)W_{\mu\nu\rho\sigma}\Biggr{]}$
(36)

where
$\mathcal{F}_{1}\left(\frac{\square}{\Lambda^{2}},\,\frac{R}{\Lambda^{2}}\right)$
is an analytic non-polynomial functions of d’Alembertian and the Ricci scalar.
The Weyl tensor contributions in (36) cannot be truncated to any finite order
especially if $R\gtrsim\Lambda^{2},\,\square\gtrsim\Lambda^{2}$ and if
$\phi\sim O\left(M_{p}\right)$ during inflation101010This is true in the
context of large field inflation, e.g., local $R^{2}$ or Higgs inflation. one
can expect $f_{1}\left(\frac{\phi}{\Lambda}\right)\gg 1$. The scalar field
model of inflation with (36) is not the subject of present paper at all, but
(36) expands the current understanding of EFTI and inevitability of the higher
curvature and higher derivative terms, especially when the square of the cut-
off scale is comparable to Ricci scalar. Analogously, we can think of (35) as
the one similar to the term
$\frac{R}{\mathcal{M}_{s}^{2}}W^{\mu\nu\rho\sigma}W_{\mu\nu\rho\sigma}$ in our
action (4), and it is indeed not the lowest order term of the action.
Therefore, our construction in a way goes beyond the EFTI proposed by Weinberg
Weinberg:2008hq . Moreover, including just the Weyl square terms like (35) can
lead to non-trivial sound speed for tensor modes as well as for a tensor ghost
Baumann:2015xxa , but it need not be the case if we sum up all possible terms
(which is precisely true in our case). Therefore, in this sense EFTI cannot
actually capture fully the UV-complete theory, but on the contrary, EFTI could
lead us to the physics that may not be compatible with UV-theory. This is
certainly because EFTI is built on just writing the lowest order terms with
symmetry assumptions which are subjected to only particular class of UV-
complete theories Burgess:2017ytm .

  * •

Since our action (4) contains higher derivative terms involving Weyl tensor,
it is not possible to rewrite this action in the Einstein frame by using a
conformal transformation of the metric. This means that we cannot cast our
action (4) to the form where gravity part is described by the Einstein-Hilbert
term only and all the rest is written as a (non-local) scalar field with a
potential. This can be also understood focusing on the fact that the graviton
propagator (17) in our case gets modified compared to GR exactly due to the
presence of the Weyl tensor terms. This is a significant difference in
comparison to many models of scalar-tensor theories that belong to the class
of EFTI Burgess:2017ytm  where inflation can be rewritten in the Einstein
frame.

####  4.2.2 With respect to inflationary observables

Here we discuss the ways to distinguish EFTI with non-local gravity inflation
in the context of inflationary observables built from two-point correlation
functions. With the 3-point correlations (i.e., Non-Gaussianities) which we
study in the companion paper KKS , we can further distinguish the non-local
gravity with EFTI.

  * •

Inflationary observables (2) in the EFTI depends on the time dependent
parameters $\left(c_{s},\,c_{t}\right)$ which are determined by the background
scalar field evolution whereas the in the case of non-local gravity we have
constant parameters associated with the choice of entire functions (18).

  * •

Since the sound speeds must be bounded $0\leq\left(c_{s},\,c_{t}\right)\leq 1$
to avoid instabilities, they must be slowly-varying or almost constant during
inflation. Let us consider EFTI with $c_{t}=1$ and $c_{s}\neq 1$ i.e., the
case of a non-canonical scalar field inflation in which $r=-8c_{s}n_{t}$,
$n_{t}=-2\epsilon$ and $r<-8n_{t}$. In our non-local gravity inflation tensor
tilt is modified as well as its running (27). We can see this in Fig. 2 and
Fig. 3. Moreover, we can have $r>-8n_{t}$ and also blue tensor tilt $n_{t}$
that is not possible to achieve in the context of inflation with non-canonical
scalar field Chen:2006nt . Inflationary models with $c_{t}\neq 1$ suggested
within EFTI (and also generalized Galileon theories Kobayashi:2011nu  and the
models where inflaton is coupled to Weyl tensor square term like in (35)
Baumann:2015xxa  ) can lead to varied predictions with respect to $n_{t}$ and
its running, which are in common with our inflation in non-local gravity. But
$c_{t}$ is expected to be slowly varying, so we greatly expect
$\frac{dn_{t}}{d\ln k}\ll n_{t}$ in these models. But in our case we can read
from Fig. 2 and Fig. 3 that both tensor tilt and its running can be of
comparable order $\frac{dn_{t}}{d\ln k}\sim O\left(n_{t}\right)$, or running
of $n_{t}$ can dominate over $n_{t}$, that is $\frac{dn_{t}}{d\ln k}\gg n_{t}$
and vice versa. These new features place the non-local gravity inflation as
the unique interesting target in the scope of future observations CORE:2016ymi
.

  * •

Moreover, possible relation between $c_{s}$ and $c_{t}$ via (dis-) conformal
transformation sheds further light on the inflationary observables
Creminelli:2014wna ; Burrage:2016myt  which might help us to distinguish the
EFTI and our framework of non-local gravity where we cannot perform a
transformation to the Einstein frame.

Primordial non-Gaussianities is another powerful window to further probe the
signatures of generalized non-local $R^{2}$-like inflation which we shall be
presenting in the companion paper KKS . In this section we have only discussed
EFTI of single field inflation in the face of non-local gravity inflation, but
there exist two other categories known as multi-field inflation (where more
than one scalar fields of comparable mass drive inflation) and the quasi-
single field inflation(where inflation is driven by a single scalar field with
additional degrees of freedom whose masses are being of the order of Hubble
perarameter square Chen:2015lza ). These scenarios lead to notable
observational signatures in the context of primordial non-Gaussianities
Cabass:2022avo . Therefore, we defer the detailed discussion of these effects
to the paper on primordial non-Gaussianities in non-local gravity inflation
KKS .

##  5 Conclusions and outlook

In this paper, we take a significant step forward on understanding cosmic
inflation beyond the $R^{2}$ model. Our quest for most general theory of
gravity consistent with $R^{2}$-like inflation has lead to the construction of
gravity theory containing terms of infinite order in derivatives and curvature
which goes beyond the previous construction of non-local gravity theories
Koshelev:2020xby . Our theory opens doors for new understanding of quantum
gravity. So far only non-local quadratic curvature theories were expected to
be good candidates for quantum gravity, but here we propose higher order terms
which are inevitable if we wish to explain inflation in non-local gravity.

It would be interesting to see renormalizability of our action (4) beyond the
power-counting methods which were employed earlier Koshelev:2016xqb . We
constrained the form factors in the action (4) with the aim of achieving
ghost-free degrees of freedom in the Minkowski and quasi-dS limits, but it is
important to constrain them from the UV-completion point of view. In this
regard, studying scattering amplitudes and perturbative renormalizability of
(4) is the most important direction to progress further. In Appendix C we
discuss how the form factors can lead to infinitely many complex conjugate
poles in the propagator of perturbed modes in curved space-time. In our study
we assumed zero initial conditions for degrees of freedom associated with
these complex conjugate poles, but it is worth to investigate this issue
futher by formulating the optical theorem in curved space-time rigorously.

In our formulation of gravity action (4), we explicitly dropped terms
involving Ricci tensor, but it is interesting to study further the
implications of including these terms for quantum gravity and inflationary
observables. Also in this study we do not include curvature terms which
violate parity, but in principle one can include them and compute their
implications in the context of inflationary observables. Finally, it is
interesting to explore the frameworks of multi-field and quasi-single field
inflation in the non-local context that might lead us to new directions in the
field.

###### Acknowledgements.

AK is supported by FCT Portugal investigator project IF/01607/2015. This
research work was supported by grants UID/MAT/00212/2019, COST Action CA15117
(CANTATA). KSK acknowledges the support from JSPS and KAKENHI Grant-in-Aid for
Scientific Research No. JP20F20320. AAS was supported by the RSF grant
21-12-00130. We thank L. Buoninfante, A. De Felice, T. Noumi, A. Tokareva and
especially M. Yamaguchi for very useful discussions.

##  Appendix A Lessons from local extensions of $R^{2}$ inflation

Higher curvature modification of $R^{2}$ inflation is motivated both from
quantum gravity and phenomenological point of view Ferrara:2013rsa ;
Asaka:2015vza . Higher curvature extensions of $R^{2}$ model with $R^{n}$
terms such as

$S_{R^{3}}=\frac{1}{2}\int
d^{4}x\sqrt{-g}\left[M_{p}^{2}R+f_{0}\left(R^{2}+\frac{f_{1}}{6M^{2}}R^{3}\right)\right]$
(37) $S_{R^{4}}=\frac{1}{2}\int
d^{4}x\sqrt{-g}\left[M_{p}^{2}R+f_{0}\left(R^{2}+\frac{f_{2}}{4M^{4}}R^{4}\right)\right]$
(38)

were extensively studied in recent years Huang:2013hsb ; Ivanov:2021chn ;
Motohashi:2014tra ; Bamba:2015uma ; Asaka:2015vza  and it was found that the
dimensionless coefficients of $R^{3}$ and $R^{4}$ terms must be
$f_{1}\lesssim\mathcal{O}\left(10^{-4}\right)$ and
$f_{2}\lesssim\mathcal{O}\left(10^{-7}\right)$ (reading from Ivanov:2021chn )
to be compatible with the present constraints on $\left(n_{s},\,r\right)$.
This conveys that any higher curvature correction beyond $R^{2}$ term should
not modify significantly the background evolution during $N\sim(50-60)$
e-folds from the end of inflation, otherwise we severely spoil the approximate
scale invariance of the observed power spectrum of scalar perturbations and
especially the prediction $n_{s}=1-\frac{2}{N}$, so that the model will no
longer be compatible with the CMB data. Furthermore, the effects of these
higher curvature corrections are expected to be even more negligible after the
end of inflation, i.e., at the stage of reheating when the $R$ and $R^{2}$
terms dominate. Similar conclusions can be drawn with other higher curvature
modifications such as $R^{3/2}$ Martins:2020oxv  and those inspired from the
asymptotic safety (AS) approach. In the latter case, suitable renormalization
group (RG) based re-summation of quantum corrections to $R^{2}$ term gives
Demmel:2015oqa

$S_{\rm AS}=\frac{1}{2}\int
d^{4}x\sqrt{-g}\Bigg{[}M_{p}^{2}R+\frac{aR^{2}}{1+b\ln\left(\frac{R}{\mu}\right)}\Bigg{]}\,.$
(39)

From the conclusions of Liu:2018hno  where inflationary observables for this
action were calculated, we learn that if $b\ll 10^{-3}$ the inflationary
predictions are very close to those of $R^{2}$ model, whereas for $b\geq
10^{-3}$ predictions slightly deviate and the tensor-to-scalar ratio can be as
large as $r\sim 10^{-2}$ which can be verified in the future detection of
B-modes CMB-S4:2020lpa . Apart from $\left(n_{s},\,r\right)$, the running of
spectral index is also a crucial parameter that gets affected by deviations
from $R^{2}$ model (see Liu:2018hno  for details). Currently the running of
spectral index is severely constrained:

$\frac{dn_{s}}{d\ln k}\Bigg{|}_{k=k_{\ast}}=-0.0045\pm
0.0067\,\,at\,\,68\%\,{\rm CL}\,.$ (40)

So, future CMB probes and even more precise measurement of $n_{s}$ will tell
us more about the effects of higher curvature terms.

Instead of higher curvature extensions, higher derivative (6th order)
corrections to the $R^{2}$-inflation studied in Castellanos:2018dub  with the
following action

$S^{6}_{R^{2}}=\frac{M_{p}^{2}}{2}\int\Bigg{[}R+\alpha R^{2}+\gamma R\square
R\Bigg{]}$ (41)

where the $6^{\rm th}$ order term $R\square R$ is considered as a small
perturbation with $\beta=\frac{\gamma}{6\alpha^{2}}\ll 1$. The inflationary
predictions $\left(n_{s},\,r\right)$ (41) were estimated within the background
solution $\bar{\square}\bar{R}\approx M^{2}\square{R}$ as

$n_{s}=1-\frac{2}{N}\pm\frac{\beta}{N},\quad
r=\frac{12}{N^{2}}\pm\frac{\beta}{N^{2}}\,,$ (42)

where $|\beta|\ll 1$ and its maximum value is $|\beta|_{\rm max}=0.3$ inferred
from the Planck 2018 data Castellanos:2018dub . Moreover, studies of
reconstructing f(R) based on the Planck data of $\left(n_{s},\,r\right)$
Sebastiani:2015kfa  also revealed that $R^{2}$ inflation is the best fit
paradigm once we assume the number of e-foldings to be $N=55-60$. Therefore,
from all this knowledge of geometrical modifications of $R^{2}$ theory we can
fairly assume that observationally compatible FLRW background must be very
close to the solution of the eigenvalue equation (3).

##  Appendix B Equations of motion of (4)

In this appendix we work out the equations of motion (EOMs) of our model (4).
First, we reiterate the action in a bit more general way as follows. EOMs are
obtained by computing

$\delta S_{H}^{\rm Non-local}=\int d^{4}x\frac{\delta(\sqrt{-g}L)}{\delta
g_{\mu\nu}}h_{\mu\nu}=\int d^{4}x\frac{E^{\mu\nu}}{2}h_{\mu\nu}$ (43)

We start by deriving the part of the EOMs coming from terms quadratic in Weyl
keeping only the contributions where one of the Weyl tensors is varied. These
terms are relevant for the quadratic variation and derivation of the
propagator. Using the explicit expression for the Weyl tensor in 4 dimensions

$W^{\mu\alpha}_{\phantom{\alpha}\nu\beta}=R^{\mu\alpha}_{\phantom{\alpha}\nu\beta}-\frac{1}{2}(\delta^{\mu}_{\nu}R^{\alpha}_{\beta}-\delta^{\mu}_{\beta}R^{\alpha}_{\nu}+R^{\mu}_{\nu}{\delta}^{\alpha}_{\beta}-R^{\mu}_{\beta}{\delta}^{\alpha}_{\nu})+\frac{R}{6}(\delta^{\mu}_{\nu}{\delta}^{\alpha}_{\beta}-\delta^{\mu}_{\beta}{\delta}^{\alpha}_{\nu})$

which uses the position of indexes such that explicit metrics are avoided, one
can compute

$\begin{split}&\delta(\mathcal{O}_{1}R\mathcal{O}_{2}W^{\mu\alpha}_{\phantom{\alpha}\nu\beta}\mathcal{O}_{3}W_{\mu\alpha}^{\phantom{\mu\alpha}\nu\beta})\\\
=&\mathcal{O}_{1}R\left(\mathcal{O}_{2}\delta
R^{\mu\alpha}_{\phantom{\alpha}\nu\beta}\mathcal{O}_{3}W_{\mu\alpha}^{\phantom{\mu\alpha}\nu\beta}+\mathcal{O}_{2}W^{\mu\alpha}_{\phantom{\alpha}\nu\beta}\mathcal{O}_{3}\delta
R_{\mu\alpha}^{\phantom{\mu\alpha}\nu\beta}\right)+O(W^{2})\\\
=&h^{\mu}_{\beta}(R_{\alpha\nu}+2\nabla_{\alpha}\nabla_{\nu})\left[\mathcal{O}_{2}(\mathcal{O}_{1}R\mathcal{O}_{3}W_{\mu}^{\phantom{\mu}\alpha\nu\beta})+\mathcal{O}_{3}(\mathcal{O}_{1}R\mathcal{O}_{2}W_{\mu}^{\phantom{\mu}\alpha\nu\beta})\right]+O(W^{2})\end{split}$

From the above derivation we can deduce

$\delta(W_{\mu\nu\rho\sigma}\mathcal{O}\left(\square_{s},\,R_{s}\right)W^{\mu\nu\rho\sigma})=h^{\mu}_{\beta}2(R_{\alpha\nu}+2\nabla_{\alpha}\nabla_{\nu})\mathcal{O}\left(\square_{s},\,R_{s}\right)W_{\mu}^{\phantom{\mu}\alpha\nu\beta}+O(W^{2})$

The part of EOMs linear in Weyl tensor thus reads (we use twice variation of
the action)

${E^{\beta}_{\mu}}_{W}=(R_{\alpha\nu}+2\nabla_{\alpha}\nabla_{\nu})\left\\{2\mathcal{F}_{W}W_{\mu}^{\phantom{\mu}\alpha\nu\beta}+2f_{0}\frac{\lambda_{R}}{\mathcal{M}_{s}^{2}}\left[\mathcal{O}_{2}(\mathcal{O}_{1}R\mathcal{O}_{3}W_{\mu}^{\phantom{\mu}\alpha\nu\beta})+\mathcal{O}_{3}(\mathcal{O}_{1}R\mathcal{O}_{2}W_{\mu}^{\phantom{\mu}\alpha\nu\beta})\right]\right\\}$
(44)

We notice that terms linear in Weyl in EOMs would vanish in the trace of
Einstein equations due to the total tracelessness of the Weyl tensor.

Turning to the part containing only $R$, we start by noting that

$\int d^{4}x\sqrt{-g}(\delta(A\mathcal{O}(\Box_{s})B))=\int
d^{4}x\sqrt{-g}\left(\delta
A\mathcal{O}\left(\square_{s}\right)B+A\mathcal{O}\left(\square_{s}\right)\delta
B+\sum_{n\geq 1}f_{n}\sum_{l=0}^{n-1}A^{(l)}\delta\Box_{s}B^{(n-1-l)}\right)$

as long as $\mathcal{F}=\sum\limits_{n\geq 0}f_{n}\Box_{s}^{n}$ and total
derivatives are omitted. Also $A^{(n)}\equiv\Box_{s}^{n}A$. d’Alembertian
acting on scalars is simply given $g^{\mu\nu}\nabla_{\mu}\partial_{\nu}$ and
its variation reads

$\delta(\Box_{s})B=\frac{1}{\mathcal{M}_{s}^{2}}(-h^{\mu\nu}\nabla_{\mu}\partial_{\nu}-g^{\mu\nu}\gamma^{\rho}_{\mu\nu}\partial_{\rho})B$

The double summation term upon moving derivatives transforms into

$\int
d^{4}x\sqrt{-g}\left(\frac{h^{\mu}_{\beta}}{\mathcal{M}_{s}^{2}}\sum_{n\geq
1}f_{n}\sum_{l=0}^{n-1}\left[\partial^{\beta}A^{(l)}\partial_{\mu}B^{(n-l-1)}-\frac{1}{2}\delta^{\beta}_{\mu}\partial^{\nu}A^{(l)}\partial_{\nu}B^{(n-l-1)}-\frac{1}{2}\delta^{\beta}_{\mu}A^{(l)}\Box
B^{(n-l-1)}\right]\right)$

When $A=B=R$ and $\mathcal{O}=\mathcal{F}_{R}\left(\square_{s}\right)$, we
have

${E^{\beta}_{\mu}}_{R^{2}}=2(-R^{\beta}_{\mu}+\mathcal{D}^{\beta}\mathcal{D}_{\mu}-\delta^{\beta}_{\mu}\Box)\mathcal{F}_{R}R+\mathcal{K}^{\beta}_{\mu}-\frac{1}{2}\delta^{\beta}_{\mu}\mathcal{K}^{\nu}_{\nu}-\frac{1}{2}\delta^{\beta}_{\mu}\tilde{\mathcal{K}}$
(45)

where we have defined

$\mathcal{K}_{\mu}^{\beta}=\frac{1}{\mathcal{M}_{s}^{2}}\sum_{n=1}^{\infty}f_{Rn}\sum_{l=0}^{n-1}\partial^{\beta}R^{(l)}\partial_{\mu}R^{(n-l-1)}\,,\quad\tilde{\mathcal{K}}=\sum_{n=1}^{\infty}f_{Rn}\sum_{l=0}^{n-1}R^{(l)}R^{(n-l)}$

Dealing with the cubic in curvature term when
$A=\mathcal{L}_{1}\left(\square_{s}\right)R\mathcal{L}_{2}\left(\square_{s}\right)R$,
$B=R$ and $\mathcal{O}=\mathcal{L}_{3}\left(\square_{s}\right)$, we have

$\delta(\mathcal{L}_{1}R\mathcal{L}_{2}R)\mathcal{L}_{3}R+(-R^{\beta}_{\mu}+\mathcal{D}^{\beta}\mathcal{D}_{\mu}-\delta^{\beta}_{\mu}\Box)\mathcal{L}_{3}(\mathcal{L}_{1}R\mathcal{L}_{2}R)+{\mathcal{K}_{3}}_{\mu}^{\beta}-\frac{1}{2}\delta^{\beta}_{\mu}{\mathcal{K}_{3}}_{\nu}^{\nu}-\frac{1}{2}\delta^{\beta}_{\mu}{\tilde{\mathcal{K}}_{3}\mu}^{\beta}$
(46)

where we have defined

${\mathcal{K}_{3}}_{\mu}^{\beta}=\frac{1}{\mathcal{M}_{s}^{2}}\sum_{n=1}^{\infty}l_{3n}\sum_{l=0}^{n-1}\partial^{\beta}(\mathcal{L}_{1}R\mathcal{L}_{2}R)^{(l)}\partial_{\mu}R^{(n-l-1)}\,,\quad\tilde{\mathcal{K}}_{3}=\sum_{n=1}^{\infty}l_{3n}\sum_{l=0}^{n-1}(\mathcal{L}_{1}R\mathcal{L}_{2}R)^{(l)}R^{(n-l)}$

and the first term in (46) results in a cycling rotation of indices. The
result with all normalizations and the factor $2$ reads

${E^{\beta}_{\mu}}_{R^{3}}=2\frac{\lambda_{3}}{\mathcal{M}_{s}^{2}}\left[(-R^{\beta}_{\mu}+\nabla^{\beta}\nabla_{\mu}-\delta^{\beta}_{\mu}\Box)\mathcal{R}_{3}+\sum_{i=1,2,3}\left({\mathcal{K}_{i}}_{\mu}^{\beta}-\frac{1}{2}\delta^{\beta}_{\mu}{\mathcal{K}_{i}}_{\nu}^{\nu}-\frac{1}{2}\delta^{\beta}_{\mu}{\tilde{\mathcal{K}}_{i}}\right)\right]$
(47)

where

$\mathcal{R}_{3}=\mathcal{L}_{1}(\mathcal{L}_{2}R\mathcal{L}_{3}R)+\mathcal{L}_{2}(\mathcal{L}_{1}R\mathcal{L}_{3}R)+\mathcal{L}_{3}(\mathcal{L}_{1}R\mathcal{L}_{2}R)$

The remaining is the Einstein term and the square root of $-g$ variation
canonically leading to

${E_{\mu}^{\beta}}_{R}=\delta^{\beta}_{\mu}L_{H}^{\rm Non-
local}-M_{P}^{2}R^{\beta}_{\mu}$ (48)

Performing the routine one has

$\displaystyle E^{\beta}_{\mu}=$ $\displaystyle\delta^{\beta}_{\mu}L_{H}^{\rm
Non-
local}-M_{P}^{2}R^{\beta}_{\mu}+2(-R^{\beta}_{\mu}+\mathcal{D}^{\beta}\mathcal{D}_{\mu}-\delta^{\beta}_{\mu}\Box)\mathcal{F}_{R}R+\mathcal{K}^{\beta}_{\mu}-\frac{1}{2}\delta^{\beta}_{\mu}\mathcal{K}^{\nu}_{\nu}-\frac{1}{2}\delta^{\beta}_{\mu}\tilde{\mathcal{K}}$
(49) $\displaystyle+$
$\displaystyle\frac{2}{3}\frac{\lambda_{c}}{\mathcal{M}_{s}^{2}}\left[(-R^{\beta}_{\mu}+\mathcal{D}^{\beta}\mathcal{D}_{\mu}-\delta^{\beta}_{\mu}\Box)\mathcal{R}_{3}+\sum_{i=1,2,3}\left({\mathcal{K}_{i}}_{\mu}^{\beta}-\frac{1}{2}\delta^{\beta}_{\mu}{\mathcal{K}_{i}}_{\nu}^{\nu}-\frac{1}{2}\delta^{\beta}_{\mu}{\tilde{\mathcal{K}}_{i}}\right)\right]$
$\displaystyle+$
$\displaystyle(R_{\alpha\nu}+2\nabla_{\alpha}\nabla_{\nu})\left\\{2\mathcal{F}_{W}W_{\mu}^{\phantom{\mu}\alpha\nu\beta}+\frac{2}{3}\frac{\lambda_{R}}{\mathcal{M}_{s}^{2}}\left[\mathcal{D}_{2}(\mathcal{D}_{1}R\mathcal{D}_{3}W_{\mu}^{\phantom{\mu}\alpha\nu\beta})+\mathcal{D}_{3}(\mathcal{D}_{1}R\mathcal{D}_{2}W_{\mu}^{\phantom{\mu}\alpha\nu\beta})\right]\right\\}+\cdots$

where auxiliary definitions from the above are used and $\cdots$ contain terms
irrelevant for both the FLRW background and linear perturbations because all
these additional terms $\cdots$ are quadratic in Weyl tensor. Computing the
trace one gets

$\begin{split}E=&{M_{p}^{2}}R+\frac{4}{3}\frac{\lambda_{3}}{\mathcal{M}_{s}^{2}}\mathcal{L}_{1}\left(\square_{s}\right)R\,\mathcal{L}_{2}\left(\square_{s}\right)R\,\mathcal{L}_{3}\left(\square_{s}\right)R-6\Box\mathcal{F}_{R}R-(\mathcal{K}_{\nu}^{\nu}+2\tilde{\mathcal{K}})\\\
-&2\frac{\lambda_{c}}{\mathcal{M}_{s}^{2}}\left[(R+3\Box)\mathcal{R}_{3}+\sum_{i=1,2,3}\left({\mathcal{K}_{i}}_{\nu}^{\nu}+2{\tilde{\mathcal{K}}_{i}}\right)\right]+\dots\end{split}$
(50)

where we see that linear in Weyl tensor terms went away identically due to the
tracelessness of the Weyl tensor and $\cdots$ are the terms irrelevant even
for linear variations. We remind that in the absence of matter, solving the
trace equation on a spatially-flat FLRW background is enough to solve all the
system modulo perhaps some radiation source. As such we start solving the EOMs
with the latter equation as it seems to be the simplest one, even though
highly non-trivial. Now we try to see whether Starobinsky solution (3) can be
a solution here. This would preserve a lot of the famous local $R^{2}$
inflation model. Substituting this into the EOMs simplifies them considerably.
First we compute

$\mathcal{K}_{\mu}^{\beta}=\frac{1}{\mathcal{M}_{s}^{2}}\mathcal{F}_{R}^{\dagger}\left(r_{s}\right)\partial^{\beta}R\partial_{\mu}R\,,\quad\tilde{\mathcal{K}}=r_{s}\mathcal{F}_{R}^{\dagger}\left(r_{s}\right)R^{2}$
(51) $\displaystyle{\mathcal{K}_{3}}_{\mu}^{\beta}$
$\displaystyle=\frac{1}{\mathcal{M}_{s}^{2}}\mathcal{L}_{1}(r_{s})\mathcal{L}_{2}(r_{s})\partial^{\beta}\left(\frac{\mathcal{L}_{3}(\Box_{s})-\mathcal{L}_{3}(r_{s})}{\Box_{s}-r_{s}}R^{2}\right)\partial_{\mu}R$
$\displaystyle\tilde{\mathcal{K}}_{3}$
$\displaystyle=r_{s}\mathcal{L}_{1}(r_{s})\mathcal{L}_{2}(r_{s})\left(\frac{\mathcal{L}_{3}(\Box_{s})-\mathcal{L}_{3}(r_{s})}{\Box_{s}-r_{s}}R^{2}\right)R$

where $r_{s}=M^{2}/\mathcal{M}_{s}^{2}$. Now we, can notice that (3) passes
through all equations proving it is a solution as long as FLRW backgrounds are
considered (meaning that Weyl tensor vanishes) and conditions (14) are met.

##  Appendix C Degrees of freedom and two point correlations of (4) in quasi-
dS limit

In this section we compute the inflationary observables related to two point
correlations of fluctuations. To study quantum fluctuations generated during
inflation, we first define metric perturbations by the following line element
in terms of the gauge invariant Bardeen potentials $\left(\Phi,\,\Psi\right)$
and the transverse and traceless tensor fluctuations $h_{ij}$

$\displaystyle ds^{2}=a^{2}\left(\tau\right)\Bigg{[}-$
$\displaystyle\left(1+2\Phi\right)d\tau^{2}+\left(\left(1-2\Psi\right)\delta_{ij}+2h_{ij}\right)dx^{i}dx^{j}\Bigg{]}\,.$
(52)

The above perturbed metric remains the same when we choose the Newtonian
gauge.

By studying the linearized perturbed equations (49) in the dS approximation,
i.e. assuming $\bar{R}\approx{\rm constant}$, we obtain

$\Bigg{[}f_{0}\bar{R}_{\rm dS}+f_{0}\bar{R}_{\rm dS}\left(\bar{\square}_{\rm
dS}-\frac{\bar{R}_{\rm
dS}}{6}\right)\mathcal{F}_{R}\left(\frac{\bar{\square}_{\rm
dS}}{\mathcal{M}_{s}^{2}}+\frac{\bar{R}_{\rm
dS}}{2\mathcal{M}_{s}^{2}}\right)\Bigg{]}\frac{\Phi+\Psi}{a^{2}}=0.$ (53)

Substituting the form factor (15) in (53), we obtain

$e^{\gamma_{T}\left(\frac{\bar{\square}_{\rm
dS}}{\mathcal{M}_{s}^{2}}-\frac{\bar{R}_{\rm
dS}}{6\mathcal{M}_{s}^{2}}\right)}\left(\Phi+\Psi\right)=0\,,$ (54)

whose general solution is $\Phi+\Psi=0$ since the exponent of an entire
function of d’Alembertian operator do not generate any new solutions that is
extensively explored in Koshelev:2017tvv . At the second order level around
dS, the part of the action (4) containing Weyl tensor terms becomes equivalent
to the one studied in Koshelev:2016xqb ; Koshelev:2017tvv ; Koshelev:2020foq .
Therefore, our result (54) matches with Eq. 5.33 of Koshelev:2016xqb . We,
however, show this explicitly in Appendix C.

Imposing (53) it is straight forward to see from (55) that the scalar part of
the Weyl tensor perturbation vanishes Koshelev:2016xqb

$\delta_{(s)}W^{\alpha\beta}_{\quad\mu\nu}=\frac{\Phi+\Psi}{a^{2}}K^{\alpha\beta}_{\quad\mu\nu}\,,$
(55)

where

$\displaystyle K^{0i}_{\quad 0j}$
$\displaystyle=-\frac{1}{6}\delta^{i}_{j}k^{2}+\frac{1}{2}k^{i}k_{j}$ (56)
$\displaystyle K^{il}_{\quad jm}$
$\displaystyle=\frac{1}{3}k^{2}\left(\delta^{l}_{m}\delta^{i}_{j}-\delta^{l}_{j}\delta^{i}_{m}\right)-\frac{1}{2}\delta^{l}_{m}k^{i}k_{j}-\frac{1}{2}\delta^{i}_{j}k^{l}k_{m}+\frac{1}{2}\delta^{l}_{j}k^{i}k_{m}+\frac{1}{2}\delta^{i}_{m}k^{l}k_{j}\,.$

Thus, the second order action for the scalar perturbations for (4) become

$\delta_{(s)}^{(2)}S=\frac{1}{2f_{0}\bar{R}_{\rm dS}}\int
d^{4}x\sqrt{-\bar{g}}\Upsilon\frac{\mathcal{W}\left(\bar{\square}_{s}\right)}{\mathcal{F}_{R}\left(\bar{\square}_{s}\right)}\left(\bar{\square}_{\rm
dS}-M^{2}\right)\Upsilon\,.$ (57)

where $\Upsilon=2f_{0}\bar{R}_{\rm dS}\Psi$ is the canonical variable, and it
is related to the curvature perturbation as
$\Upsilon\approx-2f_{0}\bar{R}_{\rm dS}\mathcal{R}$. The operator
$\mathcal{W}\left(\square_{s}\right)$ is related to the form factor
$\mathcal{F}_{R}\left(\square_{s}\right)$ as

$\mathcal{W}\left(\square_{s}\right)=3\mathcal{F}_{R}\left(\square_{s}\right)+\left(\bar{R}_{\rm
dS}+3M^{2}\right)\frac{\mathcal{F}_{R}\left(\square_{s}\right)-f_{0}}{\square-M^{2}}\,,$
(58)

From (57) we can first deduce that the kinetic term of $\Upsilon$ has one real
zero corresponding to $\bar{\square}_{\rm dS}=M^{2}$ which indicates there is
one propagating degree of freedom which we call ‘scalaron’. If there are any
other degrees of freedom, they must arise from zeros of the operator
$\mathcal{W}\left(\square_{s}\right)$ (58).111111If
$\mathcal{W}\left(\square_{s}\right)=3f_{0}e^{\gamma_{0}\left(\square_{s}+\frac{\bar{R}_{\rm
dS}}{3\mathcal{M}_{s}^{2}}\right)}$, where $\gamma_{0}$ is an entire function
of $\square_{s}+\frac{\bar{R}_{\rm dS}}{3\mathcal{M}_{s}^{2}}$ operator, then
it will have no zeros in the entire complex plane. But as a consequence the
form factor $\mathcal{F}_{R}\left(\square_{s}\right)$ must depend on the
background $\bar{R}_{\rm dS}$ as Craps:2014wga ; Koshelev:2020foq
$\mathcal{F}_{R}\left(\square_{s}\right)\equiv\mathcal{F}_{R}\left(\square_{s},\,\bar{R}_{\rm
dS}\right)=f_{0}\frac{e^{\gamma_{0}\left(\square_{s}+\frac{\bar{R}_{\rm
dS}}{3\mathcal{M}_{s}^{2}}\right)}\left(\square_{s}-\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)+\left(\bar{R}_{\rm
dS}+3M^{2}\right)}{3\square+\bar{R}_{\rm dS}}$ (59) This choice of form factor
was considered in Craps:2014wga ; Koshelev:2020foq , but it leads to
background dependence in the action (4) through the factor $\bar{R}_{\rm dS}$
in (59). In this paper we rather consider the choice of form factor
$\mathcal{F}_{R}\left(\square_{s}\right)$ corresponding to (9), so that we can
take a smooth low-energy limit and avoid background dependence.  Computing it
for the form factor (9), we obtain

$\mathcal{W}\left(\square_{s}\right)=-f_{0}\bar{R}_{\rm
dS}\left(\frac{1-e^{\gamma_{S}\left(\square_{s}\right)}}{\square}\right)+3f_{0}e^{\gamma_{S}\left(\square_{s}\right)}$
(60)

To obtain zeros of $\mathcal{W}\left(\square_{s}\right)$, we need to solve the
following characteristic equation

$\mathcal{W}\left(\frac{Z}{\mathcal{M}_{s}^{2}}\right)=0\implies\bar{R}_{\rm
dS}\left(\frac{1-e^{\gamma_{S}\left(\frac{Z}{\mathcal{M}_{s}^{2}}\right)}}{Z}\right)=3e^{\gamma_{S}\left(\frac{Z}{\mathcal{M}_{s}^{2}}\right)}\,.$
(61)

To solve (61), first we must specify the choice of entire function
$\gamma_{S}\left(\square_{s}\right)$. For simplicity, let us take a compatible
choice of entire function (62) corresponding to
$p_{i}\left(\square_{s}\right)=\alpha_{1}\square_{s}$ that gives

$\gamma_{S}\left(\square_{s}\right)=\alpha_{1}\square_{s}\left(\square_{s}-\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)\,.$
(62)

Substituting (62) into the characteristic equation (61), we obtain no real
solutions but instead we get infinite tower of complex conjugate solutions
given by

$\displaystyle Z=$
$\displaystyle\,\frac{M^{2}}{2}+\frac{\mathcal{M}_{s}^{2}}{2}\left[\left(2\pi+4q\pi\right)^{2}+\left(\frac{M}{\mathcal{M}_{s}}\right)^{8}\right]^{1/4}\Bigg{\\{}\cos\left[\frac{1}{2}{\rm
Arg}\left[4\pi
i\left(q+\frac{1}{2}\right)+\frac{M^{4}}{\mathcal{M}_{s}^{4}}\right]\right]$
(63) $\displaystyle+i\sin\left[\frac{1}{2}{\rm Arg}\left[4\pi
i\left(q+\frac{1}{2}\right)+\frac{M^{4}}{\mathcal{M}_{s}^{4}}\right]\right]\Bigg{\\}}$
$\displaystyle\approx$
$\displaystyle\,\Bigg{|}_{M^{2}\ll\mathcal{M}_{s}^{2}}\pm\mathcal{M}_{s}^{2}\sqrt{q+\frac{1}{2}}\left(1\pm
i\right)$

where $q\geq 1$ is a positive integer. In the context of string field theory,
it was known that complex conjugate poles give (classical) degrees of freedom
for a coupled system of scalar fields (both with positive and negative kinetic
terms in equal numbers) Koshelev:2007fi ; Koshelev:2009ty ; Koshelev:2010bf ;
Arefeva:2008zru . The primary question here is whether degrees of freedom
corresponding to the infinite tower of complex conjugate poles are physical
and contribute to the inflationary correlations. In this context, we can take
inspiration from the studies in different context Buoninfante:2018lnh ;
Buoninfante:2020ctr  where non-local scalar field theories with infinitely
many complex conjugate poles are considered with the following type of
Lagrangians

$\mathcal{L}_{\phi}=\frac{1}{2}\phi\left(e^{\gamma_{\phi}\left(\square_{s}\right)}-1\right)\phi-V(\phi)\,,$
(64)

where $\gamma_{\phi}$ being an arbitrary entire function. It was shown that
the Lagrangians of this type satisfy the optical theorem both at the tree-
level Buoninfante:2018lnh ; Buoninfante:2020ctr  and one-loop level Luca  due
to the exact cancellation of contributions from complex conjugate poles.
Unitarity due to the presence of complex conjugate poles has been studied in
the context of Lee-Wick theories, and it was proposed that one can project
away the states corresponding to complex conjugate poles using new quantum
field theory prescriptions (such as fakeon). Thus, one can disregard these
states as unphysical121212One can alternatively chose a suitable entire
function such that the imaginary part of the pole is small enough to avoid any
classical instabilities Koshelev:2021orf . But however, such a choice of
entire function necessarily should depend on the background value of
$\bar{R}_{\rm dS}$. In the present construction we aim to avoid any background
dependence of form factors. Modesto:2015ozb ; Modesto:2016ofr ;
Anselmi:2017lia ; Anselmi:2017ygm ; Anselmi:2018kgz ; Anselmi:2020lpp ;
Anselmi:2021hab ; Anselmi:2022toe ; Liu:2022gun ; Frasca:2022gdz .

All the above mentioned studies are about the non-local scalar field theories
in Minkowski space-time. Our case is in the context of gravity and our
gravitational action action (4) has no degrees of freedom (classically) with
complex conjugate poles around Minkowski space-time. Indeed, we can observe
that in the Minkowski limit we have $\mathcal{W}\left(\square_{s}\right)\to
3f_{0}e^{\gamma_{S}\left(\square_{s}\right)}$ (see (60)), and therefore we
have only one real pole for the kinetic operator of $\Upsilon$ at
$\bar{\square}=M^{2}$. This is expected as our form factor (9) is fixed for
the ghost-free condition around Minkowski background Biswas:2016egy . It is
well known that $R^{2}$ inflation is an intermediate phase of evolution which
is unstable to the past as well as to the future Starobinsky:1980te ;
Muller:1987hp ; Muller:2017nxg . Moreover, when we quantize the inflationary
modes, we take them to be deep inside the Hubble radius $k\gg aH$ and impose
adiabatic vacuum initial conditions Mukhanov:1990me ; Koshelev:2020foq . The
Fourier modes deep inside the Hubble radius $k\gg aH$ are the ones close to
the locally Minkowski space-time since the effects of dS curvature can be
neglected Mukhanov:1990me . Therefore, physically for the purposes of
inflationary quantum fluctuations, we ignore all the complex conjugate modes
(63) by setting initial conditions of them to be zero deferring the detailed
study of non-local (quantum) theories in curved space-time for future
investigations.

Thus, in addition to massless graviton, we have only one propagating scalar
(scalaron) with mass $M^{2}$ in Minkowski space-time. Thus, we can deduce from
(57) that solutions of $\Upsilon$ are governed by

$\left(\bar{\square}_{\rm dS}-M^{2}\right)\Upsilon=0\,,$ (65)

Following the previous studies Craps:2014wga ; Koshelev:2017tvv , we redefine
the field $\Upsilon$ after imposing the on-shell condition (65) as
$\Upsilon\to\sqrt{\frac{\mathcal{W}\left(\frac{M^{2}}{\mathcal{M}_{s}^{@}}\right)}{f_{0}\bar{R}_{\rm
dS}\mathcal{F}_{R}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)}}\Upsilon$.
As a result, we obtain the second order action for the curvature perturbation
$\mathcal{R}=\Psi+H\frac{\delta
R}{\dot{\bar{R}}}\approx-\frac{\Psi}{\epsilon}$ identical to that in the local
$R^{2}$-theory:

$\delta^{(2)}_{(s)}S_{H}^{Non-local}=\frac{M_{p}^{2}}{2}\epsilon\int
d^{4}x\sqrt{-\bar{g}}\,\mathcal{R}\left(\bar{\square}_{\rm
dS}-M^{2}\right)\mathcal{R}\,,$ (66)

where $\bar{\square}_{\rm dS}$ is the d’Alembertian operator in the dS
approximation.

###  C.1 Tensor fluctuations around $\bar{\square}\bar{R}=M^{2}\bar{R}$ and dS
approximation

Computing the second order action of (4) for transverse and traceless tensor
perturbations $h_{ij}$ (52) in the dS approximation, i.e., $\bar{R}_{\rm
dS}\approx\mathrm{const}$, we get

$\displaystyle\delta^{(2)}_{(h)}S_{H}^{\rm Non-local}=$ $\displaystyle\int
d^{4}x\sqrt{-\bar{g}}\Bigg{[}\left(\frac{M_{p}^{2}}{2}+f_{0}\bar{R}_{\rm
dS}\right)\left(\delta^{(2)}R+\delta^{(2)}_{g}\right)-\frac{f_{0}}{2}\delta^{(2)}_{g}f_{0}\bar{R}_{\rm
dS}$ (67) $\displaystyle+f_{0}\bar{R}_{\rm dS}\,\delta
W_{\mu\nu\rho\sigma}\mathcal{F}_{W}\left(\frac{\bar{\square}_{\rm
dS}}{\mathcal{M}_{s}^{2}},\,\frac{\bar{R}_{\rm
dS}}{\mathcal{M}_{s}^{2}}\right)\delta W^{\mu\nu\rho\sigma}$
$\displaystyle+\frac{f_{0}\lambda_{R}}{\mathcal{M}_{s}^{2}}\mathcal{D}_{1}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)\mathcal{D}_{2}\left(\frac{\bar{\square}_{\rm
dS}}{\mathcal{M}_{s}^{2}}\right)\delta
W_{\mu\nu\rho\sigma}\mathcal{D}_{3}\left(\frac{\bar{\square}_{\rm
dS}}{\mathcal{M}_{s}^{2}}\right)\delta W^{\mu\nu\rho\sigma}\Bigg{]}\,.$

Imposing $\mathcal{D}_{1}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)=0$ and
using $\mathcal{F}_{W}\left(\square_{s},\,R_{s}\right)$ from (15) with the
approximation $M_{p}^{2}\ll f_{0}\bar{R}_{\rm dS}$, the second order action
for tensor fluctuations becomes

$\delta_{(t)}^{(2)}S_{H}^{\rm Non-local}=\int
d^{4}x\sqrt{-g}\Bigg{[}h_{ij}\,e^{\gamma_{T}\left(\frac{\bar{\square}_{\rm
dS}}{\mathcal{M}_{s}^{2}}-\frac{2\bar{R}_{\rm
dS}}{3\mathcal{M}_{s}^{2}}\right)}\left(\bar{\square}_{\rm
dS}-\frac{\bar{R}_{\rm dS}}{6}\right)h^{ij}\Bigg{]}$ (68)

that coincides with the results of Koshelev:2016xqb ; Koshelev:2020foq .

## References

  * (1) A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity,” Phys. Lett. B 91 (1980) 99–102.
  * (2) A. A. Starobinsky, “Nonsingular Model of the Universe with the Quantum Gravitational De Sitter Stage and its Observational Consequences,” in Proceedings of the Second Seminar “Quantum Theory of Gravity”, Moscow Oct. 1981, INR Press, Moscow (1982), pp. 58-72.
  * (3) A. A. Starobinsky, “The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy,” Sov. Astron. Lett. 9 (1983) 302–304.
  * (4) A. Vilenkin, “Classical and Quantum Cosmology of the Starobinsky Inflationary Model,” Phys. Rev. D 32 (1985) 2511.
  * (5) M. B. Mijic, M. S. Morris, and W.-M. Suen, “The R**2 Cosmology: Inflation Without a Phase Transition,” Phys. Rev. D 34 (1986) 2934.
  * (6) K.-i. Maeda, “Inflation as a Transient Attractor in R**2 Cosmology,” Phys. Rev. D 37 (1988) 858.
  * (7) Planck Collaboration, Y. Akrami et al., “Planck 2018 results. X. Constraints on inflation,” arXiv:1807.06211 [astro-ph.CO].
  * (8) K. S. Stelle, “Renormalization of Higher Derivative Quantum Gravity,” Phys. Rev. D16 (1977) 953–969.
  * (9) S. V. Ketov and A. A. Starobinsky, “Inflation and non-minimal scalar-curvature coupling in gravity and supergravity,” JCAP 08 (2012) 022, arXiv:1203.0805 [hep-th].
  * (10) A. S. Koshelev, L. Modesto, L. Rachwal, and A. A. Starobinsky, “Occurrence of exact $R^{2}$ inflation in non-local UV-complete gravity,” JHEP 11 (2016) 067, arXiv:1604.03127 [hep-th].
  * (11) A. S. Koshelev, K. Sravan Kumar, and A. A. Starobinsky, “$R^{2}$ inflation to probe non-perturbative quantum gravity,” JHEP 03 (2018) 071, arXiv:1711.08864 [hep-th].
  * (12) A. S. Koshelev, K. Sravan Kumar, A. Mazumdar, and A. A. Starobinsky, “Non-Gaussianities and tensor-to-scalar ratio in non-local R2-like inflation,” JHEP 06 (2020) 152, arXiv:2003.00629 [hep-th].
  * (13) A. S. Koshelev, K. S. Kumar, and A. A. Starobinsky, “Analytic infinite derivative gravity, $R^{2}$-like inflation, quantum gravity and CMB,” Int. J. Mod. Phys. D 29 (2020) 2043018, arXiv:2005.09550 [hep-th].
  * (14) J. Ellis, D. V. Nanopoulos, and K. A. Olive, “Starobinsky-like Inflationary Models as Avatars of No-Scale Supergravity,” JCAP 1310 (2013) 009, arXiv:1307.3537 [hep-th].
  * (15) A. Kehagias, A. Moradinezhad Dizgah, and A. Riotto, “Remarks on the Starobinsky model of inflation and its descendants,” Phys. Rev. D89 no. 4, (2014) 043527, arXiv:1312.1155 [hep-th].
  * (16) A. Linde, “Inflationary Cosmology after Planck 2013,” in Proceedings, 100th Les Houches Summer School: Post-Planck Cosmology: Les Houches, France, July 8 - August 2, 2013, pp. 231–316.  2015\.  arXiv:1402.0526 [hep-th].  https://inspirehep.net/record/1280019/files/arXiv:1402.0526.pdf.
  * (17) BICEP, Keck Collaboration, P. A. R. Ade et al., “Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season,” Phys. Rev. Lett. 127 no. 15, (2021) 151301, arXiv:2110.00483 [astro-ph.CO].
  * (18) B. Craps, T. De Jonckheere, and A. S. Koshelev, “Cosmological perturbations in non-local higher-derivative gravity,” JCAP 1411 no. 11, (2014) 022, arXiv:1407.4982 [hep-th].
  * (19) A. O. Barvinsky and G. A. Vilkovisky, “Covariant perturbation theory. 2: Second order in the curvature. General algorithms,” Nucl. Phys. B333 (1990) 471–511.
  * (20) A. O. Barvinsky, Y. V. Gusev, G. A. Vilkovisky, and V. V. Zhytnikov, “The One loop effective action and trace anomaly in four-dimensions,” Nucl. Phys. B 439 (1995) 561–582, arXiv:hep-th/9404187.
  * (21) A. O. Barvinsky, Y. V. Gusev, G. A. Vilkovisky, and V. V. Zhytnikov, “The Basis of nonlocal curvature invariants in quantum gravity theory. (Third order.),” J. Math. Phys. 35 (1994) 3525–3542, arXiv:gr-qc/9404061.
  * (22) A. Bonanno and F. Saueressig, “Asymptotically safe cosmology – A status report,” Comptes Rendus Physique 18 (2017) 254–264, arXiv:1702.04137 [hep-th].
  * (23) A. Eichhorn, “An asymptotically safe guide to quantum gravity and matter,” Front. Astron. Space Sci. 5 (2019) 47, arXiv:1810.07615 [hep-th].
  * (24) L. Bosma, B. Knorr, and F. Saueressig, “Resolving Spacetime Singularities within Asymptotic Safety,” Phys. Rev. Lett. 123 no. 10, (2019) 101301, arXiv:1904.04845 [hep-th].
  * (25) J. Martin, C. Ringeval, and V. Vennin, “Encyclopædia Inflationaris,” Phys. Dark Univ. 5-6 (2014) 75–235, arXiv:1303.3787 [astro-ph.CO].
  * (26) V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, “Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions,” Phys. Rept. 215 (1992) 203–333.
  * (27) T. Baker, G. Calcagni, A. Chen, M. Fasiello, L. Lombriser, K. Martinovic, M. Pieroni, M. Sakellariadou, G. Tasinato, D. Bertacca, and I. D. Saltas, “Measuring the propagation speed of gravitational waves with lisa,” 2022.
  * (28) A. A. Starobinsky, “Spectrum of relict gravitational radiation and the early state of the universe,” JETP Lett. 30 (1979) 682–685. [Pisma Zh. Eksp. Teor. Fiz. 30 (1979) 719-723].
  * (29) Y. Wang and W. Xue, “Inflation and Alternatives with Blue Tensor Spectra,” JCAP 10 (2014) 075, arXiv:1403.5817 [astro-ph.CO].
  * (30) A. S. Koshelev, K. Sravan Kumar, and A. A. Starobinsky, “Generalized non-local $R^{2}$-like inflation and scalar non-Gaussianities,” arXiv:2209.XXXX [hep-th].
  * (31) A. S. Koshelev, K. Sravan Kumar, and A. A. Starobinsky, “Generalized non-local $R^{2}$-like inflation and tensor non-Gaussianities,” arXiv:2209.XXXX [hep-th].
  * (32) C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan, and L. Senatore, “The Effective Field Theory of Inflation,” JHEP 03 (2008) 014, arXiv:0709.0293 [hep-th].
  * (33) G. Cabass, M. M. Ivanov, M. Lewandowski, M. Mirbabayi, and M. Simonović, “Snowmass White Paper: Effective Field Theories in Cosmology,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.08232 [astro-ph.CO].
  * (34) CMB-S4 Collaboration, K. Abazajian et al., “CMB-S4: Forecasting Constraints on Primordial Gravitational Waves,” arXiv:2008.12619 [astro-ph.CO].
  * (35) LiteBIRD Collaboration, E. Allys et al., “Probing Cosmic Inflation with the LiteBIRD Cosmic Microwave Background Polarization Survey,” arXiv:2202.02773 [astro-ph.IM].
  * (36) A. Ricciardone, “Primordial Gravitational Waves with LISA,” J. Phys. Conf. Ser. 840 no. 1, (2017) 012030, arXiv:1612.06799 [astro-ph.CO].
  * (37) CORE Collaboration, F. Finelli et al., “Exploring cosmic origins with CORE: Inflation,” JCAP 04 (2018) 016, arXiv:1612.08270 [astro-ph.CO].
  * (38) K. S. Stelle, “Classical Gravity with Higher Derivatives,” Gen. Rel. Grav. 9 (1978) 353–371.
  * (39) D. Anselmi, “On the quantum field theory of the gravitational interactions,” JHEP 06 (2017) 086, arXiv:1704.07728 [hep-th].
  * (40) P. D. Mannheim, “Solution to the ghost problem in higher-derivative gravity,” in Quantum Gravity, Higher Derivatives & Nonlocality.  9, 2021.  arXiv:2109.12743 [hep-th].
  * (41) C. M. Bender and P. D. Mannheim, “No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model,” Phys. Rev. Lett. 100 (2008) 110402, arXiv:0706.0207 [hep-th].
  * (42) A. Salvio and A. Strumia, “Agravity,” JHEP 06 (2014) 080, arXiv:1403.4226 [hep-ph].
  * (43) A. Salvio and A. Strumia, “Agravity up to infinite energy,” Eur. Phys. J. C 78 no. 2, (2018) 124, arXiv:1705.03896 [hep-th].
  * (44) D. Anselmi, E. Bianchi, and M. Piva, “Predictions of quantum gravity in inflationary cosmology: effects of the Weyl-squared term,” JHEP 07 (2020) 211, arXiv:2005.10293 [hep-th].
  * (45) A. Salvio and F. Sannino, eds., From the Fermi Scale to Cosmology.  Frontiers, 2019.
  * (46) A. Salvio, “BICEP/Keck data and Quadratic Gravity,” arXiv:2202.00684 [astro-ph.CO].
  * (47) N. V. Krasnikov, “Nonlocal gauge theories,” Theor. Math. Phys. 73 (1987) 1184–1190. [Teor. Mat. Fiz.73,235(1987)].
  * (48) E. T. Tomboulis, “Superrenormalizable gauge and gravitational theories,” arXiv:hep-th/9702146 [hep-th].
  * (49) E. T. Tomboulis, “Renormalization and unitarity in higher derivative and nonlocal gravity theories,” Mod. Phys. Lett. A30 no. 03n04, (2015) 1540005.
  * (50) E. T. Tomboulis, “Nonlocal and quasilocal field theories,” Phys. Rev. D92 no. 12, (2015) 125037, arXiv:1507.00981 [hep-th].
  * (51) L. Modesto, “Super-renormalizable Quantum Gravity,” Phys. Rev. D86 (2012) 044005, arXiv:1107.2403 [hep-th].
  * (52) T. Biswas, E. Gerwick, T. Koivisto, and A. Mazumdar, “Towards singularity and ghost free theories of gravity,” Phys. Rev. Lett. 108 (2012) 031101, arXiv:1110.5249 [gr-qc].
  * (53) T. Biswas, A. S. Koshelev, and A. Mazumdar, “Consistent higher derivative gravitational theories with stable de Sitter and anti-de Sitter backgrounds,” Phys. Rev. D95 no. 4, (2017) 043533, arXiv:1606.01250 [gr-qc].
  * (54) A. S. Koshelev, K. Sravan Kumar, L. Modesto, and L. Rachwal, “Finite quantum gravity in ds and ads spacetimes,” Phys. Rev. D98 no. 4, (2018) 046007, arXiv:1710.07759 [hep-th].
  * (55) G. Calcagni and L. Modesto, “Nonlocal quantum gravity and M-theory,” Phys. Rev. D91 no. 12, (2015) 124059, arXiv:1404.2137 [hep-th].
  * (56) T. Biswas, A. S. Koshelev, and A. Mazumdar, “Gravitational theories with stable (anti-)de Sitter backgrounds,” Fundam. Theor. Phys. 183 (2016) 97–114, arXiv:1602.08475 [hep-th].
  * (57) S. A. Appleby, R. A. Battye, and A. A. Starobinsky, “Curing singularities in cosmological evolution of F(R) gravity,” JCAP 06 (2010) 005, arXiv:0909.1737 [astro-ph.CO].
  * (58) G. Calcagni and S. Kuroyanagi, “Stochastic gravitational-wave background in quantum gravity,” JCAP 03 (2021) 019, arXiv:2012.00170 [gr-qc].
  * (59) A. De Felice and S. Tsujikawa, “f(R) theories,” Living Rev. Rel. 13 (2010) 3, arXiv:1002.4928 [gr-qc].
  * (60) S. Weinberg, “Effective Field Theory for Inflation,” Phys. Rev. D 77 (2008) 123541, arXiv:0804.4291 [hep-th].
  * (61) P. Creminelli, J. Gleyzes, J. Noreña, and F. Vernizzi, “Resilience of the standard predictions for primordial tensor modes,” Phys. Rev. Lett. 113 no. 23, (2014) 231301, arXiv:1407.8439 [astro-ph.CO].
  * (62) P. Creminelli, D. L. López Nacir, M. Simonović, G. Trevisan, and M. Zaldarriaga, “Detecting Primordial $B$-Modes after Planck,” JCAP 1511 no. 11, (2015) 031, arXiv:1502.01983 [astro-ph.CO].
  * (63) C. P. Burgess, “Intro to Effective Field Theories and Inflation,” arXiv:1711.10592 [hep-th].
  * (64) D. Baumann, H. Lee, and G. L. Pimentel, “High-Scale Inflation and the Tensor Tilt,” JHEP 01 (2016) 101, arXiv:1507.07250 [hep-th].
  * (65) D. Baumann, D. Green, H. Lee, and R. A. Porto, “Signs of Analyticity in Single-Field Inflation,” Phys. Rev. D 93 no. 2, (2016) 023523, arXiv:1502.07304 [hep-th].
  * (66) K. Aoki, S. Mukohyama, and R. Namba, “Positivity vs. Lorentz-violation: an explicit example,” JCAP 10 (2021) 079, arXiv:2107.01755 [hep-th].
  * (67) X. Chen, M.-x. Huang, S. Kachru, and G. Shiu, “Observational signatures and non-Gaussianities of general single field inflation,” JCAP 0701 (2007) 002, arXiv:hep-th/0605045 [hep-th].
  * (68) T. Kobayashi, M. Yamaguchi, and J. Yokoyama, “Generalized G-inflation: Inflation with the most general second-order field equations,” Prog. Theor. Phys. 126 (2011) 511–529, arXiv:1105.5723 [hep-th].
  * (69) C. Burrage, S. Cespedes, and A.-C. Davis, “Disformal transformations on the CMB,” JCAP 08 (2016) 024, arXiv:1604.08038 [gr-qc].
  * (70) X. Chen, M. H. Namjoo, and Y. Wang, “Quantum Primordial Standard Clocks,” JCAP 02 (2016) 013, arXiv:1509.03930 [astro-ph.CO].
  * (71) S. Ferrara, R. Kallosh, A. Linde, and M. Porrati, “Minimal Supergravity Models of Inflation,” Phys. Rev. D 88 no. 8, (2013) 085038, arXiv:1307.7696 [hep-th].
  * (72) T. Asaka, S. Iso, H. Kawai, K. Kohri, T. Noumi, and T. Terada, “Reinterpretation of the Starobinsky model,” PTEP 2016 no. 12, (2016) 123E01, arXiv:1507.04344 [hep-th].
  * (73) Q.-G. Huang, “A polynomial f(R) inflation model,” JCAP 02 (2014) 035, arXiv:1309.3514 [hep-th].
  * (74) V. R. Ivanov, S. V. Ketov, E. O. Pozdeeva, and S. Y. Vernov, “Analytic extensions of Starobinsky model of inflation,” JCAP 03 no. 03, (2022) 058, arXiv:2111.09058 [gr-qc].
  * (75) H. Motohashi, “Consistency relation for $R^{p}$ inflation,” Phys. Rev. D 91 (2015) 064016, arXiv:1411.2972 [astro-ph.CO].
  * (76) K. Bamba and S. D. Odintsov, “Inflationary cosmology in modified gravity theories,” Symmetry 7 no. 1, (2015) 220–240, arXiv:1503.00442 [hep-th].
  * (77) J. S. Martins, O. F. Piattella, I. L. Shapiro, and A. A. Starobinsky, “Inflation with sterile scalar coupled to massive fermions and to gravity,” arXiv:2010.14639 [hep-th].
  * (78) M. Demmel, F. Saueressig, and O. Zanusso, “A proper fixed functional for four-dimensional Quantum Einstein Gravity,” JHEP 08 (2015) 113, arXiv:1504.07656 [hep-th].
  * (79) L.-H. Liu, T. Prokopec, and A. A. Starobinsky, “Inflation in an effective gravitational model and asymptotic safety,” Phys. Rev. D 98 no. 4, (2018) 043505, arXiv:1806.05407 [gr-qc].
  * (80) A. R. R. Castellanos, F. Sobreira, I. L. Shapiro, and A. A. Starobinsky, “On higher derivative corrections to the $R+R^{2}$ inflationary model,” JCAP 12 (2018) 007, arXiv:1810.07787 [gr-qc].
  * (81) L. Sebastiani and R. Myrzakulov, “F(R) gravity and inflation,” Int. J. Geom. Meth. Mod. Phys. 12 no. 9, (2015) 1530003, arXiv:1506.05330 [gr-qc].
  * (82) A. S. Koshelev, “Non-local SFT Tachyon and Cosmology,” JHEP 04 (2007) 029, arXiv:hep-th/0701103.
  * (83) A. S. Koshelev and S. Y. Vernov, “Cosmological perturbations in SFT inspired non-local scalar field models,” Eur. Phys. J. C 72 (2012) 2198, arXiv:0903.5176 [hep-th].
  * (84) A. S. Koshelev and S. Y. Vernov, “Analysis of scalar perturbations in cosmological models with a non-local scalar field,” Class. Quant. Grav. 28 (2011) 085019, arXiv:1009.0746 [hep-th].
  * (85) I. Y. Aref’eva and A. S. Koshelev, “Cosmological Signature of Tachyon Condensation,” JHEP 09 (2008) 068, arXiv:0804.3570 [hep-th].
  * (86) L. Buoninfante, G. Lambiase, and M. Yamaguchi, “Nonlocal generalization of Galilean theories and gravity,” Phys. Rev. D 100 no. 2, (2019) 026019, arXiv:1812.10105 [hep-th].
  * (87) L. Buoninfante, G. Lambiase, Y. Miyashita, W. Takebe, and M. Yamaguchi, “Generalized ghost-free propagators in nonlocal field theories,” Phys. Rev. D 101 no. 8, (2020) 084019, arXiv:2001.07830 [hep-th].
  * (88) L. Buoninfante, “private communication,”.
  * (89) A. S. Koshelev and A. Tokareva, “Unitarity of Minkowski nonlocal theories made explicit,” Phys. Rev. D 104 no. 2, (2021) 025016, arXiv:2103.01945 [hep-th].
  * (90) L. Modesto and I. L. Shapiro, “Superrenormalizable quantum gravity with complex ghosts,” Phys. Lett. B 755 (2016) 279–284, arXiv:1512.07600 [hep-th].
  * (91) L. Modesto, “Super-renormalizable or finite Lee–Wick quantum gravity,” Nucl. Phys. B 909 (2016) 584–606, arXiv:1602.02421 [hep-th].
  * (92) D. Anselmi and M. Piva, “Perturbative unitarity of Lee-Wick quantum field theory,” Phys. Rev. D 96 no. 4, (2017) 045009, arXiv:1703.05563 [hep-th].
  * (93) D. Anselmi, “Fakeons And Lee-Wick Models,” JHEP 02 (2018) 141, arXiv:1801.00915 [hep-th].
  * (94) D. Anselmi, “Diagrammar of physical and fake particles and spectral optical theorem,” JHEP 11 (2021) 030, arXiv:2109.06889 [hep-th].
  * (95) D. Anselmi, “Fakeons versus Lee-Wick ghosts: physical Pauli-Villars fields, finite QED and quantum gravity,” arXiv:2202.10483 [hep-th].
  * (96) J. Liu, L. Modesto, and G. Calcagni, “Quantum gravity with ghost pairs,” arXiv:2208.13536 [hep-th].
  * (97) M. Frasca, A. Ghoshal, and A. S. Koshelev, “Confining the complex ghosts out,” arXiv:2207.06394 [hep-th].
  * (98) V. Müller, H.-J. Schmidt, and A. A. Starobinsky, “The Stability of the De Sitter Space-time in Fourth Order Gravity,” Phys. Lett. B 202 (1988) 198–200.
  * (99) D. Müller, A. Ricciardone, A. A. Starobinsky, and A. Toporensky, “Anisotropic cosmological solutions in $R+R^{2}$ gravity,” Eur. Phys. J. C 78 (2018) 311, arXiv:1710.08753 [gr-qc].