# Perturbative unitarity and NEC violation in genesis cosmology

Yong Cai,11footnotetext: Corresponding author<EMAIL_ADDRESS>Ji
Xu,22footnotetext: Corresponding author<EMAIL_ADDRESS>Shuai
Zhao,33footnotetext: Corresponding author<EMAIL_ADDRESS>Siyi
Zhou44footnotetext: Corresponding author<EMAIL_ADDRESS>
###### Abstract

Explorations of the violation of null energy condition (NEC) in cosmology
could enrich our understanding of the very early universe and the related
gravity theories. Although a fully stable NEC violation can be realized in the
“beyond Horndeski” theory, it remains an open question whether a violation of
the NEC is allowed by some fundamental properties of UV-complete theories or
the consistency requirements of effective field theory (EFT). We investigate
the tree-level perturbative unitarity for stable NEC violations in the
contexts of both Galileon and “beyond Horndeski” genesis cosmology, in which
the universe is asymptotically Minkowskian in the past. We find that the
constraints of perturbative unitarity imply that we may need some unknown new
physics below the cut-off scale of the EFT other than that represented by the
“beyond Horndeski” operators.

##  1 Introduction

Inflation [1, 2, 3, 4] has achieved great successes in simultaneously
explaining several puzzles of the Big Bang cosmology. More significantly,
inflation predicted a nearly scale-invariant power spectrum of the primordial
scalar perturbations, which has been confirmed by observations of the cosmic
microwave background temperature anisotropy [5, 6]. However, an inflationary
universe is geodesically incomplete in the past [7, 8]. Furthermore, the
swampland conjecture [9] and the trans-Planckian censorship conjecture [10]
may also reinforce the inference that inflation is not the final story of the
early universe [11, 12].

Alternatives to or completions of the inflationary scenario generally involve
violating the null energy condition (NEC),555In modified theories of gravity,
the NEC may need to be replaced by a more general condition, i.e., the null
convergence condition [13]. which is quite robust and is crucial to the proof
of the Penrose’s singularity theorem, see [14] for a review. Is it possible to
realize a completely healthy NEC violation? What is the underlying physics
required for a healthy NEC violation? Whether violations of the NEC did took
place in the very early universe? Considerable progress have been made in
looking for answers to these questions in gravity and cosmology, especially in
the study of nonsingular cosmology, including cosmological bounce [15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] and genesis [33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] (see [46, 47] for studies on
slow expansion), see also [48, 49, 50, 51, 52, 53]. Additionally, NEC
violations could also occur during inflation and induce enhanced power
spectrum of primordial gravitational waves (GWs) [54, 55].

Nonetheless, challenges remain in constructing a consistent effective field
theory (EFT) to violate the NEC. The “no-go” theorem proved in [56, 57]
indicates that pathological instabilities of perturbations appear in either
the NEC-violating period or sooner or later in spatial flat nonsingular
cosmology constructed by Horndeski theory [58, 59, 60], see also [61, 62, 63,
64, 65, 66, 67, 68, 69]. It is then demonstrated explicitly with the EFT
method [70, 71, 72, 73, 74] that fully stable NEC-violating nonsingular
cosmological models can be constructed in “beyond Horndeski” theories [75,
76], see [77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90] for later
developments. Notably, the physics represented by higher derivative “beyond
Horndeski” operators plays an essential role in realizing fully stable NEC-
violating nonsingular cosmology. So far, it remains an open question whether
some fundamental properties of UV-complete theories or the consistency
requirements of EFT allow a violation of the NEC.

Studies of perturbative unitarity in cosmology may throw some light upon the
unknown new physics related to the very early universe, see e.g., [50, 91, 92,
93, 94, 95, 96, 97, 98, 99]. It would be interesting to see what the
constraints of perturbative unitarity can tell about those higher derivative
operators which are essential in fully stable NEC-violating nonsingular
cosmology, see also [93] for the case of $P(X)$ theory in the context of
bouncing cosmology. However, applying the results of quantum field theory to
cosmological background requires great care. Additionally, the calculation of
amplitudes (even at tree-level) could be a formidable (though not impossible)
task for NEC violations constructed by “beyond Horndeski” theories, since
there are too many perturbative interacting terms.

In this paper, we investigate the tree-level perturbative unitarity of a
stable NEC violation in the context of Galilean and “beyond Horndeski”
genesis. The calculation of scattering amplitudes is carried out at sufficient
past time so that the spacetime can be treated as asymptotical Minkowski.
Consequently, the calculation can be greatly simplified due to the asymptotic
behavior of the genesis solution. Throughout this paper we adopt natural units
$c=\hbar=1$ and have a metric signature $(-,+,+,+)$.

##  2 Perturbative unitarity and NEC violation in Galileon genesis

In this section and the next, we investigate perturbative unitarity for a
stable NEC violation in the context of genesis cosmology. For simplicity, we
start by considering a genesis model constructed by the cubic Galileon theory,
which is only able to guarantee the stability of perturbations during the
genesis phase. A genesis model which is fully stable throughout the entire
history can be constructed with the “beyond Horndeski” higher derivative
operators, which will be carried out in Sec. 3. In the following, we will
focus our discussion on the physics of the NEC-violating phase.

###  2.1 Setup

A stable cosmological genesis can be realized with the action (see e.g., [33,
37, 50])

$\displaystyle S=\int d^{4}x\sqrt{-g}\left({M_{\rm P}^{2}\over
2}R+\lambda_{1}e^{2\phi/M}X+{\lambda_{2}\over M^{4}}X^{2}+{\lambda_{3}\over
M^{3}}X\Box\phi\right)\,,$ (2.1)

where $\phi$ is a scalar field with a dimension of mass,
$X=\nabla_{\mu}\phi\nabla^{\mu}\phi$, $\Box\phi=\nabla_{\mu}\nabla^{\mu}\phi$;
$\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ are dimensionless constants,
$M_{\rm P}$ is the reduced Planck mass, $M<M_{\rm P}$ is some energy scale. We
will work with the flat Friedmann-Robertson-Walker metric, i.e.,

$ds^{2}=-dt^{2}+a^{2}(t)d\vec{x}^{2}\,.$ (2.2)

The background equations can be given as

$\displaystyle 3H^{2}M_{\rm P}^{2}$ $\displaystyle=$
$\displaystyle-\lambda_{1}e^{2\phi/M}\dot{\phi}^{2}+{3\lambda_{2}\over
M^{4}}\dot{\phi}^{4}+{6\lambda_{3}\over M^{3}}H\dot{\phi}^{3}\,,$ (2.3)
$\displaystyle\dot{H}M_{\rm P}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{1}e^{2\phi/M}\dot{\phi}^{2}-{2\lambda_{2}\over
M^{4}}\dot{\phi}^{4}-{3\lambda_{3}\over
M^{3}}H\dot{\phi}^{3}+{\lambda_{3}\over M^{3}}\dot{\phi}^{2}\ddot{\phi}\,,$
(2.4) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\lambda_{1}e^{2\phi/M}\left(\ddot{\phi}+3H\dot{\phi}+\frac{1}{M}\dot{\phi}^{2}\right)-{6\lambda_{2}\over
M^{4}}\left(H\dot{\phi}^{3}+\dot{\phi}^{2}\ddot{\phi}\right)$ (2.5)
$\displaystyle-{3\lambda_{3}\over
M^{3}}\left(3H^{2}\dot{\phi}^{2}+\dot{H}\dot{\phi}^{2}+2H\dot{\phi}\ddot{\phi}\right)\,,$

where only two of Eqs. (2.3) to (2.5) are independent.

In the unitary gauge, the action (2.1) can be mapped to the EFT action (A.1)
(see Appendix. A), where those non-zero functions are

$\displaystyle f=1\,,$ (2.6) $\displaystyle\Lambda(t)={\lambda_{2}\over
M^{4}}\dot{\phi}^{4}+{\lambda_{3}\over
M^{3}}\dot{\phi}^{2}(\ddot{\phi}+3H\dot{\phi})\,,$ (2.7) $\displaystyle
c(t)=-\lambda_{1}e^{2\phi/M}\dot{\phi}^{2}+{2\lambda_{2}\over
M^{4}}\dot{\phi}^{4}-{\lambda_{3}\over
M^{3}}\dot{\phi}^{2}(\ddot{\phi}-3H\dot{\phi})\,,$ (2.8) $\displaystyle
M_{2}^{4}(t)={2\lambda_{2}\over M^{4}}\dot{\phi}^{4}+{\lambda_{3}\over
2M^{3}}\dot{\phi}^{2}(\ddot{\phi}+3H\dot{\phi})\,,$ (2.9) $\displaystyle
m_{3}^{3}(t)={2\lambda_{3}\over M^{3}}\dot{\phi}^{3}\,,$ (2.10)

up to quadratic order.

We will set
$h_{ij}=a^{2}\mathrm{e}^{2\zeta}\left(\mathrm{e}^{\gamma}\right)_{ij}$ and
$\gamma_{ii}=0=\partial_{i}\gamma_{ij}$ in the unitary gauge. The quadratic
action of tensor perturbation is

$S_{\gamma}^{(2)}=\frac{M_{\rm P}^{2}}{8}\int
d^{3}xdt\,a^{3}\left[\dot{\gamma}_{ij}^{2}-\frac{\left(\partial_{k}\gamma_{ij}\right)^{2}}{a^{2}}\right]\,,$
(2.11)

which is same as that in general relativity. The quadratic action of curvature
perturbation in the unitary gauge can be written as

$S^{(2)}_{\zeta}=\int
d^{4}xa^{3}Q_{s}\left[\dot{\zeta}^{2}-c_{s}^{2}{(\partial\zeta)^{2}\over
a^{2}}\right]\,,$ (2.12)

where

$\displaystyle Q_{s}$ $\displaystyle=$
$\displaystyle{1\over\gamma^{2}}\left[-\dot{H}M_{\rm
P}^{2}+\frac{4\lambda_{2}\dot{\phi}^{4}}{M^{4}}+\frac{3\lambda_{3}^{2}\dot{\phi}^{6}}{M^{6}M_{\rm
P}^{2}}+\frac{\lambda_{3}\dot{\phi}^{2}}{M^{3}}\left(\ddot{\phi}+3H\dot{\phi}\right)\right]\,,$
(2.13) $\displaystyle c_{s}^{2}$ $\displaystyle=$ $\displaystyle
1+{4\lambda_{2}M^{-1}\dot{\phi}^{4}+2\lambda_{3}\dot{\phi}^{2}(H\dot{\phi}+2\lambda_{3}M_{\rm
P}^{-2}M^{-3}\dot{\phi}^{4}-\ddot{\phi})\over M_{\rm
P}^{2}M^{3}\dot{H}-4\lambda_{2}M^{-1}\dot{\phi}^{4}-\lambda_{3}\dot{\phi}^{2}(3H\dot{\phi}+3\lambda_{3}M_{\rm
P}^{-2}M^{-3}\dot{\phi}^{4}+\ddot{\phi})}\,,$ (2.14)

and $\gamma=H-\frac{\lambda_{3}\dot{\phi}^{3}}{M^{3}M_{\rm P}^{2}}$, see e.g.,
[70] for details. In order to avoid the ghost and gradient instabilities of
the scalar perturbations, we should have $Q_{s}>0$ and $c_{s}^{2}>0$.

Since $\gamma\neq H$ for a nonzero $\lambda_{3}$ in general, the region where
pathological instabilities appear does not necessarily overlap with the region
of NEC-violation [61]. Therefore, it is possible to obtain a stable NEC-
violation with action (2.1) by removing the instabilities of perturbations to
the later NEC-preserving phase. These instabilities can be eliminated by
“beyond Horndeski” higher derivative operators. However, we will not go into
the details of curing these instabilities thoroughly in this section for
simplicity, since we focus only on the physics of the NEC-violating phase
(i.e., the genesis phase).

In order to apply the bounds of perturbative unitarity, it is more convenient
to work with the spatial flat gauge, in which $\zeta=0$, $\phi(t,{\bf
x})=\phi_{0}(t)+\delta\phi(t,{\bf x})$. For convenience, we define
$\sigma(t,{\bf x})\equiv\delta\phi(t,{\bf x})$. The quadratic, cubic and
quartic actions of $\sigma$ can be given as

$\displaystyle S^{(2)}_{\sigma}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\Big{[}\lambda_{1}e^{\frac{2\phi_{0}}{M}}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma+{4\over
M}\partial_{\mu}\phi_{0}\partial^{\mu}\sigma\sigma+{2\over
M^{2}}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{0}\sigma^{2}\right)$ (2.15)
$\displaystyle\qquad\qquad\,\,\,+\frac{2\lambda_{2}}{M^{4}}\left(\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{0}\partial_{\nu}\sigma\partial^{\nu}\sigma+2\partial_{\mu}\phi_{0}\partial_{\nu}\phi_{0}\partial^{\mu}\sigma\partial^{\nu}\sigma\right)$
$\displaystyle\qquad\qquad\,\,\,+\frac{2\lambda_{3}}{M^{3}}\left(\Box\phi_{0}\partial_{\mu}\sigma\partial^{\mu}\sigma-\nabla_{\mu}\nabla_{\nu}\phi_{0}\partial^{\mu}\sigma\partial^{\nu}\sigma\right)\Big{]}\,,$
$\displaystyle S^{(3)}_{\sigma}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\Big{[}\frac{2\lambda_{1}}{M}e^{\frac{2\phi_{0}}{M}}\left(\sigma\partial_{\mu}\sigma\partial^{\mu}\sigma+\frac{2}{M}\partial_{\mu}\phi_{0}\partial^{\mu}\sigma\sigma^{2}+\frac{2}{3M^{2}}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{0}\sigma^{3}\right)$
(2.16)
$\displaystyle\qquad\qquad\,\,\,+\frac{4\lambda_{2}}{M^{4}}\partial_{\mu}{\phi}_{0}\partial^{\mu}{\sigma}\partial_{\nu}\sigma\partial^{\nu}\sigma+\frac{\lambda_{3}}{M^{3}}\partial_{\mu}\sigma\partial^{\mu}\sigma\Box\sigma\Big{]}\,,$
$\displaystyle S^{(4)}_{\sigma}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\Big{[}\frac{2\lambda_{1}}{M^{2}}e^{\frac{2\phi_{0}}{M}}\left(\sigma^{2}\partial_{\mu}\sigma\partial^{\mu}\sigma+\frac{4}{3M}\partial_{\mu}\phi_{0}\partial^{\mu}{\sigma}\sigma^{3}+\frac{1}{3M^{2}}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{0}\sigma^{4}\right)$
(2.17)
$\displaystyle\qquad\qquad\,\,\,+\frac{\lambda_{2}}{M^{4}}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma\right)^{2}\Big{]}\,.$

###  2.2 A solution of Galileon genesis

For a genesis background (i.e., $H\simeq 0$), Eq. (2.3) suggests

$e^{2\phi/M}={3\lambda_{2}\over\lambda_{1}}{\dot{\phi}^{2}\over M^{4}}\,.$
(2.18)

The solution is

$\dot{\phi}={M\over(-t)}\,,\quad t<0\,.$ (2.19)

We have $\phi(t)=-M\ln(-Mt)+{M\over 2}\ln{3\lambda_{2}\over\lambda_{1}}$ so
that Eq. (2.18) is satisfied, which is valid in the regime $|t|\gg 1/M$. With
Eq. (2.4), we find $\dot{H}={\lambda_{2}+\lambda_{3}\over M_{\rm
P}^{2}(-t)^{4}}$. Genesis requires the violation of the NEC, which indicates
$\dot{H}>0$. Therefore, the condition

$\lambda_{2}+\lambda_{3}>0$ (2.20)

should be satisfied.

Obviously, the Hubble parameter

$H={\lambda_{2}+\lambda_{3}\over 3M_{\rm P}^{2}}{1\over(-t)^{3}}+C\,,$ (2.21)

where the constant $C$ should be set as $0$ for genesis. A non-zero $C$ can be
used for realizing NEC-violating inflation, see e.g., [54, 55]. From Eq.
(2.21), we have

$a(t)=e^{\int Hdt}=\exp\left[{\lambda_{2}+\lambda_{3}\over 6M_{\rm
P}^{2}}{1\over(-t)^{2}}\right]\simeq 1+{\lambda_{2}+\lambda_{3}\over 6M_{\rm
P}^{2}}{1\over(-t)^{2}}$ (2.22)

for $|t|\gg 1/M$ while we have set $a(-\infty)=1$. As can be seen from Eq.
(2.2), the universe asymptotically tends to the Minkowski space in infinite
past.

With Eqs. (2.18) to (2.21), we find in the unitary gauge that

$\displaystyle Q_{s}$ $\displaystyle\approx$
$\displaystyle\frac{27\lambda_{2}M_{\rm
P}^{4}}{\left(2\lambda_{3}-\lambda_{2}\right)^{2}}(-t)^{2}\,,$ (2.23)
$\displaystyle c_{s}^{2}$ $\displaystyle\approx$
$\displaystyle\frac{2\lambda_{3}-\lambda_{2}}{3\lambda_{2}}\,,$ (2.24)

where we have kept only the leading order terms in both $Q_{s}$ and
$c_{s}^{2}$. Therefore, we should have

$0<\lambda_{2}<2\lambda_{3}$ (2.25)

so that $Q_{s}>0$ and $c_{s}^{2}>0$ when $|t|\gg 1/M$. It should be pointed
out that $\gamma={\lambda_{2}-2\lambda_{3}\over 3M_{\rm P}^{2}(-t)^{3}}<0$
during the genesis phase under the condition (2.25). If we assume that the
genesis phase eventually enters the standard hot Big Bang expansion, the
$\gamma-$crossing problem would be inevitable. Therefore, instabilities of the
scalar perturbations cannot be eliminated from the entire history of the
universe for the action (2.1). These instabilities are assumed to be cured by
physics (e.g., the higher order “beyond Horndeski” operators [70, 71, 72])
outside the genesis phase. However, we will focus only on a stable NEC-
violating genesis phase in the following for our purpose. Additionally, in
order to avoid superluminal propagation of the scalar perturbations, we should
have $c_{s}^{2}\leq 1$, i.e.,

$\lambda_{3}\leq 2\lambda_{2}\,.$ (2.26)

In the spatial flat gauge, by using Eqs. (2.18) to (2.22), we find

$\displaystyle S^{(2)}_{\sigma}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\Big{[}{\cal A}(t)\dot{\sigma}^{2}-{\cal
B}(t)(\partial_{i}\sigma)^{2}\Big{]}\,,$ (2.27)

where

${\cal A}={3\lambda_{2}\over M^{2}(-t)^{2}}+{\cal O}\left({1\over
M^{3}(-t)^{3}}\right)\,,\quad{\cal B}={2\lambda_{3}-\lambda_{2}\over
M^{2}(-t)^{2}}+{\cal O}\left({1\over M^{3}(-t)^{3}}\right)$ (2.28)

for $|t|\gg M^{-1}$. The mass term appears in the fist line of Eq. (2.15) can
be safely disregarded since it implies $m^{2}\sim M^{-2}|t|^{-4}$. Apparently,
the sound speed squared in the spatial flat gauge can be given as
$c_{s}^{2}={\cal B}/{\cal A}\approx(2\lambda_{3}-\lambda_{2})/(3\lambda_{2})$,
which is consistent with Eq. (2.24). Since $\zeta\simeq
H\delta\rho/\dot{\rho}\sim\sigma(-t)^{-2}$, Eq. (2.27) is also consistent with
Eq. (2.12).

Similarly, in the regime $|t|\gg M^{-1}$, we can obtain

$\displaystyle S^{(3)}_{\sigma}=\int
d^{4}x\left[\frac{\lambda_{3}}{M^{3}}\partial_{\rho}\sigma\partial^{\rho}\sigma
g^{\mu\nu}\partial_{\mu}\partial_{\nu}\sigma+{\cal O}\left({\sigma^{3}\over
M^{3}(-t)}\right)\right]\,,$ (2.29) $\displaystyle S^{(4)}_{\sigma}=\int
d^{4}x\left[\frac{\lambda_{2}}{M^{4}}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma\right)^{2}+{\cal
O}\left({\sigma^{4}\over M^{4}(-t)^{2}}\right)\right]\,.$ (2.30)

In the following, we will consider only the leading order terms in Eq. (2.29)
and (2.30) for simplicity. Consequently, the asymptotic behavior of genesis
solution is able to greatly simplify the calculation of scattering amplitudes.

According to the “no-go” theorem [56, 57], requiring the absence of
instabilities in the entire history of a nonsingular Universe constructed by
Horndeski theory generally indicates the strong coupling issue. It seems that
the strong coupling issue appears in the limit $t\rightarrow-\infty$, as can
be seen in Eqs. (2.27) and (2.28). However, this issue is not physical since
it disappears in the unitary gauge as $Q_{s}\sim(-t)^{2}$ in Eq. (2.12). This
is because we require the absence of instabilities only in the genesis phase.
Whether such an issue is problematic even in the spatial flat gauge may
require further investigation, see [67, 68, 69] for recent studies.
Additionally, the consistency of the perturbative expansion implies
constraints on $\sigma$ or its derivatives, such as
$|\partial\sigma/M|\ll(-t)^{-1}$ and $|\partial^{2}\sigma/M|\ll(-t)^{-2}$,
since the coefficients ${\cal A}$ and ${\cal B}$ of the quadratic action
(2.27) are suppressed by $|t|^{2}$. As is well-known, the genesis solution is
an attractor, namely, $\sigma$ decays as time goes on, see e.g., [33].
Therefore, we will assume that we are working in a regime which is free from
the above strong coupling issue and the inconsistency of perturbation theory.

Additionally, for simplicity, we set ${\cal A}\equiv{\cal B}$ so that
$c_{s}^{2}\equiv 1$, which indicates $\lambda_{3}=2\lambda_{2}$. We define

$\tilde{\sigma}={\cal A}^{1/2}\sigma\,,\qquad\tilde{M}=M{\cal
A}^{1/2}\approx\sqrt{3\lambda_{2}}/(-t)\,,$ (2.31)

such that

$\displaystyle S^{(2)}_{\tilde{\sigma}}=\int
d^{4}x\left[\dot{\tilde{\sigma}}^{2}-(\partial_{i}\tilde{\sigma})^{2}\right]\,,$
(2.32) $\displaystyle S^{(3)}_{\tilde{\sigma}}=\int
d^{4}x\left[\frac{\lambda_{3}}{\tilde{M}^{3}}\partial_{\rho}{\tilde{\sigma}}\partial^{\rho}{\tilde{\sigma}}g^{\mu\nu}\partial_{\mu}\partial_{\nu}{\tilde{\sigma}}\right]\,,$
(2.33) $\displaystyle S^{(4)}_{\tilde{\sigma}}=\int
d^{4}x\left[\frac{\lambda_{2}}{\tilde{M}^{4}}\left(\partial_{\mu}{\tilde{\sigma}}\partial^{\mu}{\tilde{\sigma}}\right)^{2}\right]\,,$
(2.34)

where we have neglected higher order terms. From Eq. (2.32), we find that the
dispersion relation is simply $F=\omega^{2}-k^{2}=0$. Apparently, the
interactions of $\tilde{\sigma}$ depend on $\tilde{M}$, or equivalently, on
time $t$. In the regime $|t|\gg 1/M$, the time scale of the scattering
processes we consider will be $\Delta t\ll|t|$. Therefore, we can
approximately treat $\tilde{M}$ as a constant in the calculation of the
scattering amplitudes. It should be pointed that $\Delta t\ll|t|$ indicates
the energy scale of the scattering process satisfies $M>E\sim 1/\Delta t\gg
E_{\rm IR}\simeq|t|^{-1}$, where $M$ is the UV cut-off scale of the EFT (2.1)
and $E_{\rm IR}$ can be treated as an IR cut-off scale of these scattering
processes we consider. The earlier era we go into the genesis phase, the
smaller $E_{\rm IR}$ we get.

###  2.3 Perturbative unitarity

Some bounds on scattering amplitudes can be established in terms of optical
theorem, which is a straightforward consequence of the unitarity of the ${\cal
S}$-matrix, i.e., ${\cal S}{\cal S}^{\dagger}=\mathbbm{1}$. Inserting ${\cal
S}=\mathbbm{1}+i{\cal T}$, we have

$\displaystyle-i({\cal T}-{\cal T}^{\dagger})={\cal T}{\cal T}^{\dagger}\,.$
(2.35)

Let us take the matrix element of this equation between initial states
$|A\rangle$ and final states $|B\rangle$. Then express the ${\cal T}$-matrix
elements as invariant matrix elements $\mathcal{M}$ times 4-momentum-
conserving delta functions, where

$\displaystyle\langle
B|\mathcal{T}|A\rangle=(2\pi)^{4}\delta^{4}(p_{A}-p_{B}){\cal M}(A\rightarrow
B)\,.$ (2.36)

Eq. (2.35) becomes

$\displaystyle-i\left[\mathcal{M}\left(A\rightarrow
B\right)-\mathcal{M}^{*}\left(B\rightarrow A\right)\right]$ $\displaystyle=$
$\displaystyle\sum_{m}\left(\prod_{i=1}^{m}\int\frac{d^{3}q_{i}}{(2\pi)^{3}}\frac{1}{2E_{i}}\right)\mathcal{M}^{*}\left(A\rightarrow\left\\{q_{i}\right\\}\right)\mathcal{M}\left(B\rightarrow\left\\{q_{i}\right\\}\right)\times(2\pi)^{4}\delta^{(4)}(P_{B}-\sum_{i}q_{i})\,,$

where $\\{q_{i}\\}$ is the inserted complete set of intermediate states. We
are interested in the magnitude of scattering amplitude
$\mathcal{M}\left(A\rightarrow\left\\{q_{i}\right\\}\right)$ with a typical
high energy scale $E$, thus we take $A=B$,

$\displaystyle 2\textrm{Im}\mathcal{M}\left(A\rightarrow
A\right)=\sum_{m}\left(\prod_{i=1}^{m}\int\frac{d^{3}q_{i}}{(2\pi)^{3}}\frac{1}{2E_{i}}\right)(2\pi)^{4}\delta^{(4)}(P_{A}-\sum_{i}q_{i})\left|\mathcal{M}\left(A\rightarrow\left\\{q_{i}\right\\}\right)\right|^{2}\,.$
(2.38)

The mass dimensions of an $n$-particle scattering amplitude is

$\displaystyle[\mathcal{M}]=4-n\,,$ (2.39)

where $n$ is the total number of particles involved in the process, i.e., the
in+out particles. According to Eq.(2.39), a natural bound can be established
on $\textrm{Im}\mathcal{M}\left(A\rightarrow A\right)$,

$\displaystyle\textrm{Im}\mathcal{M}\left(A\rightarrow
A\right)\leq\left|\mathcal{M}\left(A\rightarrow A\right)\right|\leq
E^{4-2n_{A}}\,.$ (2.40)

Substituting this bound back into Eq.(2.38), we have

$\displaystyle\left|\mathcal{M}\left(A\rightarrow\left\\{q_{i}\right\\}\right)\right|\leq
E^{4-(n_{A}+n_{q_{i}})}\,.$ (2.41)

If weakly coupled UV completion was respected, this bound condition have to be
satisfied by tree-level amplitudes. However, the requirement of UV completion
may be too stringent for the EFT of a stable NEC-violating nonsingular
cosmology. In the regime of EFT, the constraint of perturbative unitarity can
be given as [93]

$\left|\mathcal{M}_{4,\ell}\right|\leq 8\pi^{2}\,.$ (2.42)

For four point scattering, where $n_{A}=n_{q_{i}}=2$, the constraint can be
translated into the following partial wave representation

$\displaystyle\mathcal{M}_{4,\ell}(\mathsf{s})=\int_{-1}^{1}\mathrm{~{}d}\cos\theta
P_{\ell}(\cos\theta)\mathcal{M}_{4}(\mathsf{s},\theta)\,,$ (2.43)

where $\mathsf{s}$ is the Mandelstam variable, the subscript $4$ denotes the
number of external legs. As for the $2N$-point functions, we will have the
constraint $\left|\mathcal{M}_{2N}\right|\lesssim\mathsf{s}^{2-N}$.
Specifically, we will consider the five-point scattering, for which the
constraint is $\left|\mathcal{M}_{5}\right|\lesssim\mathsf{s}^{-1/2}$.

####  2.3.1 Constraints from four-point scattering

We will calculate the amplitude of four-point scattering
$\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma}$ to find out what
the constraints of perturbative unitarity, i.e., (2.41) and (2.42), can tell
for a stable NEC-violating Galileon genesis. From the actions (2.32), (2.33)
and (2.34), we can deduce the Feynman rules for $3\tilde{\sigma}$ and
$4\tilde{\sigma}$ vertexes, which are collected in Fig. 2.

Figure 1: Feynman rules for $3\tilde{\sigma}$ and $4\tilde{\sigma}$ vertexes.

Having Feynman rules in hand, the amplitudes for every diagram which is shown
in Fig. 2 can be deduced,

$\displaystyle
i\mathcal{M}^{(a)}(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma})$
$\displaystyle=$ $\displaystyle
i\frac{8\lambda_{2}}{\tilde{M}^{4}}\left(p_{1}\cdot p_{2}~{}p_{3}\cdot
p_{4}+p_{1}\cdot p_{3}~{}p_{2}\cdot p_{4}+p_{1}\cdot p_{4}~{}p_{2}~{}\cdot
p_{3}\right)\,,$ $\displaystyle
i\mathcal{M}^{(b)}(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma})$
$\displaystyle=$
$\displaystyle\frac{i2\lambda_{3}}{\tilde{M}^{3}}\left(-p_{1}^{2}~{}p_{2}\cdot(p_{1}+p_{2})-p_{2}^{2}~{}p_{1}\cdot(p_{1}+p_{2})+(p_{1}+p_{2})^{2}p_{1}\cdot
p_{2}\right)\frac{i}{-(p_{1}+p_{2})^{2}}$
$\displaystyle\times\frac{i2\lambda_{3}}{\tilde{M}^{3}}\left((p_{1}+p_{2})^{2}~{}p_{3}\cdot
p_{4}-p_{3}^{2}~{}(p_{1}+p_{2})\cdot p_{4}-p_{4}^{2}~{}(p_{1}+p_{2})\cdot
p_{3}\right)\,,$ $\displaystyle
i\mathcal{M}^{(c)}(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma})$
$\displaystyle=$
$\displaystyle\frac{i2\lambda_{3}}{\tilde{M}^{3}}\left(p_{1}^{2}~{}p_{3}\cdot(p_{1}-p_{3})-p_{3}^{2}~{}p_{1}\cdot(p_{1}-p_{3})-(p_{1}-p_{3})^{2}p_{1}\cdot
p_{3}\right)\frac{i}{-(p_{1}-p_{3})^{2}}$
$\displaystyle\times\frac{i2\lambda_{3}}{\tilde{M}^{3}}\left(p_{2}^{2}~{}(p_{3}-p_{1})\cdot
p_{4}-(p_{3}-p_{1})^{2}p_{2}\cdot
p_{4}-p_{4}^{2}~{}p_{2}\cdot(p_{3}-p_{1})\right)\,,$ $\displaystyle
i\mathcal{M}^{(d)}(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma})$
$\displaystyle=$
$\displaystyle\frac{i2\lambda_{3}}{\tilde{M}^{3}}\left(p_{1}^{2}~{}p_{4}\cdot(p_{1}-p_{4})-p_{4}^{2}~{}p_{1}\cdot(p_{1}-p_{4})-(p_{1}-p_{4})^{2}p_{1}\cdot
p_{4}\right)\frac{i}{-(p_{1}-p_{4})^{2}}$ (2.44)
$\displaystyle\times\frac{i2\lambda_{3}}{\tilde{M}^{3}}\left(p_{2}^{2}~{}(p_{4}-p_{1})\cdot
p_{3}-(p_{4}-p_{1})^{2}p_{2}\cdot
p_{3}-p_{3}^{2}~{}p_{2}\cdot(p_{4}-p_{1})\right)\,.$

Figure 2: Feynman diagrams for the four-point scattering
$\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma}$.

As a result, we have

$\displaystyle\mathcal{M}(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma})$
$\displaystyle=$
$\displaystyle\frac{2\lambda_{2}}{\tilde{M}^{4}}\Big{(}\mathsf{s}^{2}+\mathsf{t}^{2}+\mathsf{u}^{2}\Big{)}-\frac{\lambda_{3}^{2}}{\tilde{M}^{6}}\Big{(}\mathsf{s}^{3}+\mathsf{t}^{3}+\mathsf{u}^{3}\Big{)}\,,$
(2.45)

where the Mandelstam variables

$\displaystyle\mathsf{s}=-(p_{1}+p_{2})^{2}=-(p_{3}+p_{4})^{2}\,,$
$\displaystyle\mathsf{t}=-(p_{3}-p_{1})^{2}=-(p_{2}-p_{4})^{2}\,,$
$\displaystyle\mathsf{u}=-(p_{4}-p_{1})^{2}=-(p_{2}-p_{3})^{2}\,.$ (2.46)

Recalling Eq. (2.31) and $\lambda_{3}=2\lambda_{2}$, we find without surprise
that the requirement of UV completion, i.e., (2.41), cannot be satisfied by
the stable NEC violation in the context of cubic Galileon genesis (2.1). In
the EFT regime, the unitarity bound (2.42) implies

$\mathsf{s}\lesssim{\lambda_{2}^{1/2}\over|t|^{2}}\sim\lambda_{2}^{1/2}(M_{\rm
P}^{2}H)^{2/3}\ll M^{2}\,,$ (2.47)

where we have used Eq. (2.21) and $|t|\gg M^{-1}$, and neglected the constant
coefficients. Namely, the perturbative unitarity is violated at a scale
$\sqrt{\mathsf{s}}\sim|t|^{-1}\sim\tilde{M}\ll M$. However, the existences of
the IR cut-off scale of the scattering processes we calculated and the UV cut-
off scale of the EFT have already required $M>\sqrt{\mathsf{s}}\gg E_{\rm
IR}\simeq|t|^{-1}$, which implies that the bound of perturbative unitarity
(2.47) cannot be satisfied.

Therefore, although a stable NEC violation can be realized with cubic Galileon
theory in the context of genesis cosmology, the bound of tree-level
perturbative unitarity indicates that new physics should enter the EFT even
blow the cut-off scale $M$. Furthermore, (2.47) implies that the earlier era
we go into the genesis phase, the more urgent we may need new physics at a
lower scale, though the cosmological background is asymptotically Minkowskian.

Notably, Ref. [50] has carried out a similar analysis of the strong coupling
scale indicated by unitarity for the low-energy forward $2\rightarrow 2$
scattering of the background field in the Galileon theory. The second line of
the lagrangian (3.2) in [50] seems equivalent to Eq. (2.1) provided
$c_{2}=f^{2}\simeq\lambda_{1}M^{2}$ and $c_{3}={f^{2}\over
3H_{0}^{2}}\simeq\lambda_{2}$. However, two scales (i.e., $f$ and $H_{0}$)
rather than one (i.e., $M$) were involved in [50]. As a result, the cut-off
scales of the EFT actions are different, which are read as $H_{0}$ in [50] and
$M$ in our case, unless $f=H_{0}=M$.666Note that the background field $\pi$ is
dimensionless in [50]. As for the derivation of the strong coupling scales,
our results in Eqs. (2.45) and (2.47) could be consistent with that of Ref.
[50] provided we set $f=H_{0}=\tilde{M}$.777See Eqs. (3.3) to (3.5) of [50].
The comparison of interpretations of these results is a little tricky, since
the parameter $c_{3}$ in the second line of Eq. (3.2) is not exactly equal to
the $c_{3}$ in the third line of Eq. (3.2) in [50], where the latter one
should depend on $\tilde{\pi}^{-4}$ actually. However, $c_{3}\simeq
f^{2}/H_{0}^{2}$ has been used in the derivations of Eqs. (3.3) to (3.5) in
[50]. Consequently, the third line of Eq. (3.2) in [50] will be equivalent to
our Eqs. (2.32) to (2.34) if $f=H_{0}=\tilde{M}$, which indicates that the
strong coupling scales $\sim\tilde{M}$ in both cases.  Therefore, the
situations and results in the two papers are not completely equivalent, but
they can confirm each other to some extent. Furthermore, we will consider
explicitly also the five-point scattering in Sec. 2.3.2.

####  2.3.2 Constraints from five-point scattering

For the five-point scattering
$\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma}\tilde{\sigma}$,
Eq. (2.41) becomes

$\displaystyle\left|\mathcal{M}\left(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma}\tilde{\sigma}\right)\right|\leq
E^{-1}\,.$ (2.48)

Similarly, the scattering amplitude for this five-point process can be written
down. Here we define a set of Mandelstam variable basis

$\displaystyle\mathsf{s}_{12}$ $\displaystyle=$
$\displaystyle-(p_{1}+p_{2})^{2}\,,\qquad\mathsf{s}_{13}=-(p_{1}-p_{3})^{2}\,,\qquad\mathsf{s}_{14}=-(p_{1}-p_{4})^{2}\,,$
$\displaystyle\mathsf{s}_{23}$ $\displaystyle=$
$\displaystyle-(p_{2}-p_{3})^{2}\,,\qquad\mathsf{s}_{24}=-(p_{2}-p_{4})^{2}\,.$
(2.49)

This amplitude can be written in terms of Mandelstam variables

$\displaystyle\mathcal{M}\left(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma}\tilde{\sigma}\right)=\frac{\lambda_{3}}{\tilde{M}^{9}}\Big{[}4\lambda_{2}\tilde{M}^{2}G_{1}(\mathsf{s}_{ij})+\lambda_{3}^{2}G_{2}(\mathsf{s}_{ij})\Big{]}\,,$
(2.50)

where

$\displaystyle G_{1}(\mathsf{s}_{ij})$ $\displaystyle=$
$\displaystyle-\mathsf{s}_{14}\Big{(}\mathsf{s}_{23}(2\mathsf{s}_{13}+\mathsf{s}_{14}+\mathsf{s}_{23})+\mathsf{s}_{12}(2\mathsf{s}_{23}+\mathsf{s}_{13})\Big{)}$
$\displaystyle-\mathsf{s}_{24}\Big{(}\mathsf{s}_{13}(2\mathsf{s}_{12}+2\mathsf{s}_{14}+\mathsf{s}_{13})+\mathsf{s}_{23}(2\mathsf{s}_{13}+2\mathsf{s}_{14}+\mathsf{s}_{12})\Big{)}-\mathsf{s}_{13}\mathsf{s}_{24}^{2}\,,$
$\displaystyle G_{2}(\mathsf{s}_{ij})$ $\displaystyle=$
$\displaystyle\mathsf{s}_{23}\Big{(}2\mathsf{s}_{13}^{3}+\mathsf{s}_{14}(\mathsf{s}_{14}+\mathsf{s}_{23})^{2}+\mathsf{s}_{13}^{2}(3\mathsf{s}_{14}+4\mathsf{s}_{23})+\mathsf{s}_{13}(3\mathsf{s}_{14}^{2}+4\mathsf{s}_{14}\mathsf{s}_{23}+2\mathsf{s}_{23}^{2})\Big{)}$
$\displaystyle+\mathsf{s}_{24}\Big{(}\mathsf{s}_{13}^{3}+3\mathsf{s}_{13}^{2}\mathsf{s}_{14}+3\mathsf{s}_{13}\mathsf{s}_{14}^{2}+2\mathsf{s}_{14}^{3}+4\mathsf{s}_{23}(\mathsf{s}_{13}+\mathsf{s}_{14})^{2}+3\mathsf{s}_{23}^{2}(\mathsf{s}_{13}+\mathsf{s}_{14})\Big{)}$
$\displaystyle+\mathsf{s}_{24}^{2}\Big{(}2\mathsf{s}_{13}^{2}+4\mathsf{s}_{13}\mathsf{s}_{14}+4\mathsf{s}_{14}^{2}+3\mathsf{s}_{23}(\mathsf{s}_{13}+\mathsf{s}_{14})\Big{)}$
$\displaystyle+\mathsf{s}_{24}^{2}(\mathsf{s}_{13}+2\mathsf{s}_{14})+\mathsf{s}_{12}^{3}(\mathsf{s}_{13}+\mathsf{s}_{14}+\mathsf{s}_{23}+\mathsf{s}_{24})$
$\displaystyle+2\mathsf{s}_{12}^{2}\Big{(}\mathsf{s}_{13}^{2}+\mathsf{s}_{14}^{2}+\mathsf{s}_{23}^{2}+\mathsf{s}_{13}(\mathsf{s}_{14}+3\mathsf{s}_{23})+\mathsf{s}_{24}(\mathsf{s}_{23}+3\mathsf{s}_{14})+\mathsf{s}_{24}^{2}\Big{)}$
$\displaystyle+\mathsf{s}_{13}\Big{(}\mathsf{s}_{12}^{3}+\mathsf{s}_{14}^{3}+(\mathsf{s}_{23}+\mathsf{s}_{24})^{3}+\mathsf{s}_{13}^{2}(3\mathsf{s}_{14}+7\mathsf{s}_{23}+2\mathsf{s}_{24})+\mathsf{s}_{14}^{2}(2\mathsf{s}_{23}+7\mathsf{s}_{24})$
$\displaystyle+\mathsf{s}_{14}(2\mathsf{s}_{23}^{2}+4\mathsf{s}_{23}\mathsf{s}_{24}+7\mathsf{s}_{24}^{2})+\mathsf{s}_{13}(3\mathsf{s}_{14}^{2}+7\mathsf{s}_{23}^{2}+4\mathsf{s}_{23}\mathsf{s}_{24}+2\mathsf{s}_{24}^{2}+4\mathsf{s}_{14}\mathsf{s}_{23}+4\mathsf{s}_{14}\mathsf{s}_{24})\Big{)}\,.$

One can resort to Fig. 3 for relevant Feynman diagrams. The unitarity bound
Eq. (2.48) again implies that the cubic Galileon genesis (2.1) cannot be UV
complete. In the EFT regime, the bound of perturbative unitarity
$\left|\mathcal{M}_{5}\right|\lesssim\mathsf{s}^{-1/2}$ indicates that
$\sqrt{\mathsf{s}}\lesssim|t|^{-1}\sim\tilde{M}\ll M$, which is consistent
with (2.47), where we have assumed that $\mathsf{s}_{ij}\sim\mathsf{s}$ in our
estimation. Therefore, the conclusion of Sec. 2.3.1 remains valid.

Figure 3: Feynman diagrams for the five-point scattering
$\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma}\tilde{\sigma}$.
The topological inequivalent diagrams are presented and the momentum labels
are omitted.

##  3 Perturbative unitarity and NEC violation in “beyond Horndeski” genesis

The results of Secs. 2.3.1 and 2.3.2 motivate us to explore whether the new
physics implied by the bounds of perturbative unitarity can be represented by
“beyond Horndeski” operators, which are required by the absence of
instabilities in the entire history of an NEC-violating nonsingular cosmology.
In this section, we apply the constraints of perturbative unitarity to a
stable NEC violation in the context of “beyond Horndeski” genesis.

###  3.1 Setup

It is discovered in Refs. [70, 71, 72] that the EFT operator $\delta
g^{00}R^{(3)}$ plays a crucial role in thoroughly eliminating the pathological
instabilities of perturbations induced by a violation of the NEC while
$m_{4}^{2}\neq{\tilde{m}}_{4}^{2}$ (i.e., $\alpha_{\mathrm{H}}\neq 0$), see
Appendix. A. The covariant action

$S=\int d^{4}x\sqrt{-g}\left[\frac{M_{\rm
P}^{2}}{2}R+P(\phi,X)\right]+S_{\delta g^{00}R^{(3)}}$ (3.1)

can be used to construct nonsingular cosmology which is fully stable in the
entire history, where $S_{\delta g^{00}R^{(3)}}=\int d^{4}x\sqrt{-g}L_{\delta
g^{00}R^{(3)}}$,

$\displaystyle L_{\delta g^{00}R^{(3)}}$ $\displaystyle=$
$\displaystyle\frac{f_{1}(\phi)}{2}\delta g^{00}R^{(3)}$ $\displaystyle=$
$\displaystyle\frac{f}{2}R-\frac{X}{2}\int f_{\phi\phi}d\ln
X-\left(f_{\phi}+\int\frac{f_{\phi}}{2}d\ln X\right)\square\phi$
$\displaystyle+\frac{f}{2X}\left[\phi_{\mu\nu}\phi^{\mu\nu}-(\square\phi)^{2}\right]-\frac{f-2Xf_{X}}{X^{2}}\left[\phi^{\mu}\phi_{\mu\rho}\phi^{\rho\nu}\phi_{\nu}-(\square\phi)\phi^{\mu}\phi_{\mu\nu}\phi^{\nu}\right]\,,$
$\displaystyle f(\phi,X)$ $\displaystyle=$ $\displaystyle
f_{1}\left(1+{X/f_{2}}\right)\,,\qquad\delta
g^{00}={X/\dot{\phi}^{2}(t)}+1={X/f_{2}(t(\phi))}+1\,,$ (3.3)

$\phi_{\mu}=\nabla_{\mu}\phi$, $\phi^{\mu}=\nabla^{\mu}\phi$, $f_{X}=df/dX$,
$f_{\phi}=df/d\phi$ and $f_{\phi\phi}=d^{2}f/d\phi^{2}$. It is demonstrated in
Refs. [73, 74] that the action (3.1) belongs to the “beyond Horndeski” (GLPV)
theory [75, 76], which corresponds to $m_{4}^{2}\neq{\tilde{m}}_{4}^{2}$ in
the EFT action (A.1).

The integral terms in Eq. (3.1) and the constraints (3.3) guarantee that the
background evolution and the quadratic action of tensor perturbation are
unaffected by $S_{\delta g^{00}R^{(3)}}$. However, we can relax such
requirements so that we can get rid of the integral terms and the constraints
(3.3). Inspired by (3.1), we can construct a fully stable NEC-violating
genesis model with the extended action

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\Big{\\{}\frac{1}{2}[M_{\rm
P}^{2}+F(\phi,X)]R+P(\phi,X)+G(\phi,X)\Box\phi$ (3.4)
$\displaystyle+\frac{F}{2X}\left[\phi_{\mu\nu}\phi^{\mu\nu}-(\square\phi)^{2}\right]-\frac{F-2XF_{X}}{X^{2}}\left[\phi^{\mu}\phi_{\mu\rho}\phi^{\rho\nu}\phi_{\nu}-(\square\phi)\phi^{\mu}\phi_{\mu\nu}\phi^{\nu}\right]\Big{\\}}\,,$

see [86, 87, 88, 89] for recent studies. It can be checked that the action
(3.4) still belongs to the “beyond Horndeski” (GLPV) theory [75, 76],888See
e.g. Eq. (4) of [100] or Eq. (2.17) of [101], see also the footnote on page 2
of [89].  which is free from the Ostrogradsky instabilities. In the special
case of $F=2XF_{X}$, the action (3.4) reduces to a subclass of Horndeski
theory.

Expanding around a cosmological background, the quadratic order of action
(3.4) in perturbations can be written as (A.2), in which

$\displaystyle{\cal M}^{2}=M_{\rm P}^{2},\quad\alpha_{\rm T}=F/M_{\rm
P}^{2}\,,$ (3.5) $\displaystyle\alpha_{\rm
K}=(4\dot{\phi}^{4}P_{XX}-2\dot{\phi}^{2}P_{X}+12H\dot{\phi}^{3}G_{X}-12H\dot{\phi}^{5}G_{XX}+2\dot{\phi}^{2}G_{\phi}-2\dot{\phi}^{4}G_{\phi
X})/(H^{2}M_{\rm P}^{2}),$ $\displaystyle\alpha_{\rm
B}=\frac{-2\dot{\phi}^{3}G_{X}-2\dot{\phi}\ddot{\phi}F_{X}+\dot{\phi}F_{\phi}}{2HM_{\rm
P}^{2}},\quad\alpha_{\rm H}={F-2XF_{X}\over M_{\rm P}^{2}}$
$\displaystyle\alpha_{\rm L}=0,\quad\beta_{1}=\beta_{2}=\beta_{3}=0\,.$

Since $\alpha_{\rm L}=\beta_{i}=0$, the action (3.4) does not involve those
characteristic degenerate higher-order scalar-tensor (DHOST) operators in
(3.4). The parameter $\alpha_{\rm H}$ vanishes (i.e.,
$m_{4}^{2}={\tilde{m}}_{4}^{2}$) for Horndeski theory and becomes non-zero
(i.e., $m_{4}^{2}\neq{\tilde{m}}_{4}^{2}$) for “beyond Horndeski” (GLPV)
theory, as we have mentioned.

The background equations of (3.4) can be obtained as

$\displaystyle 3M_{\rm P}^{2}H^{2}$ $\displaystyle=$
$\displaystyle-P+6H\dot{\phi}^{3}G_{X}+\dot{\phi}^{2}G_{\phi}-2\dot{\phi}^{2}P_{X}-3H\dot{\phi}F_{\phi}+6H\dot{\phi}^{3}F_{\phi
X}\,,$ (3.6) $\displaystyle M_{\rm P}^{2}\dot{H}$ $\displaystyle=$
$\displaystyle\dot{\phi}^{2}P_{X}-\dot{\phi}^{2}G_{\phi}-3H\dot{\phi}^{3}G_{X}+\dot{\phi}^{2}\ddot{\phi}G_{X}$
(3.7)
$\displaystyle+\frac{3}{2}H\dot{\phi}F_{\phi}-\frac{1}{2}F_{\phi}\ddot{\phi}-\frac{1}{2}\dot{\phi}^{2}F_{\phi\phi}+\dot{\phi}^{2}\ddot{\phi}F_{\phi
X}-3H\dot{\phi}^{3}F_{\phi X}\,.$

Obviously, $F(\phi,X)$ will not appear in the background equations when $F$ is
independent on $\phi$.

If we write the action (3.4) in terms of the operators in (A.1), we do not
need to include the operators $\delta K\dot{\delta g^{00}}$, $(\dot{\delta
g^{00}})^{2}$ or $(\partial_{i}{\delta g^{00}})^{2}$. The coefficients $c(t)$
and $\Lambda(t)$ can be determined by $c(t)={M_{\rm P}^{2}\over
2}(\dot{f}H-2f\dot{H}-\ddot{f})$ and $\Lambda(t)={M_{\rm P}^{2}\over
2}(5\dot{f}H+6fH^{2}+2f\dot{H}+\ddot{f})$, where $f=1+F/M_{\rm P}^{2}$, $H$ is
solved by Eqs. (3.6) and (3.7). The explicit expressions of $c(t)$,
$\Lambda(t)$ and $M_{2}^{4}$ are clumsy, while
$m_{3}^{3}=2\dot{\phi}^{3}G_{X}$, $m_{4}^{2}=-F/2$ and
$\tilde{m}_{4}^{2}=-XF_{X}$.

The quadratic action of tensor perturbation is

$S_{\gamma}^{(2)}=\frac{M_{\rm P}^{2}}{8}\int
d^{3}xdt\,a^{3}Q_{T}\left[\dot{\gamma}_{ij}^{2}-c_{T}^{2}\frac{\left(\partial_{k}\gamma_{ij}\right)^{2}}{a^{2}}\right]\,,$
(3.8)

where

$Q_{T}=1,\quad c_{T}^{2}=1+\alpha_{\rm T}\,.$ (3.9)

Apparently, for a non-vanishing $F(\phi,X)$, the propagating speed of
primordial gravitational waves will be modified during the genesis phase,
which could be able to generate interesting features in the power spectrum of
primordial gravitational waves ([102, 103, 104, 55]). We require $-M_{\rm
P}^{2}<F\leq 0$ so that $0<c_{T}^{2}\leq 1$.

The quadratic action of curvature perturbation in the unitary gauge can be
written as

$S^{(2)}_{\zeta}=\int
d^{3}xdt\,a^{3}Q_{s}\left[\dot{\zeta}^{2}-c_{s}^{2}{(\partial\zeta)^{2}\over
a^{2}}\right]\,,$ (3.10)

where

$\displaystyle Q_{s}$ $\displaystyle=$ $\displaystyle{M_{\rm P}^{2}\over
2}{\alpha_{\rm K}+6\alpha_{\rm B}^{2}\over(1+\alpha_{\rm B})^{2}}\,,$ (3.11)
$\displaystyle c_{s}^{2}$ $\displaystyle=$ $\displaystyle{M_{\rm P}^{2}\over
Q_{s}}\left\\{{1\over a}{{\rm d}\over{\rm d}t}\left[{a(1+\alpha_{\rm H})\over
H(1+\alpha_{\rm B})}\right]-(1+\alpha_{\rm T})\right\\}\,,$ (3.12)

see e.g., [105]. In order to avoid the ghost and gradient instabilities of
$\zeta$, $Q_{s}>0$ and $c_{s}^{2}>0$ are required, respectively. We also
require $c_{s}^{2}\leq 1$ so that there is no superluminal propagation of the
scalar perturbation modes. It is convenient to define $\gamma\equiv
H(1+\alpha_{\rm B})$ and $Q_{\tilde{m}_{4}}\equiv 1+\alpha_{H}$, which are
usually used in the analysis of these instability problems. The ways to
overcome the “no-go” theorem have been discussed explicitly in [72] with these
quantities $\gamma$ and $Q_{\tilde{m}_{4}}$.

###  3.2 A solution of “beyond Horndeski” genesis

Based on the previous section, a model of fully stable genesis can be
constructed. In this section, we will set the free functions $F(\phi,X)$,
$P(\phi,X)$ and $G(\phi,X)$ in (3.4) as

$\displaystyle F(\phi,X)$ $\displaystyle=$ $\displaystyle M_{\rm
P}^{2}\left({\kappa_{1}\over M^{4}}X+{\kappa_{2}\over M^{8}}X^{2}\right)\,,$
(3.13) $\displaystyle P(\phi,X)$ $\displaystyle=$
$\displaystyle\lambda_{1}e^{2\phi/M}X+{\lambda_{2}\over M^{4}}X^{2}\,,$ (3.14)
$\displaystyle G(\phi,X)$ $\displaystyle=$ $\displaystyle{\lambda_{3}\over
M^{3}}X\,,$ (3.15)

where $\kappa_{1}$, $\kappa_{2}$, $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$
are dimensionless constants. Apparently, the action (3.4) can be viewed as the
addition of higher order “beyond Horndeski” operators to the action (2.1) in
Sec. 2. The background evolution of the universe will not be affected by $F$.
As a result, the solution of background evolution of genesis will be exactly
the same as that introduced in Sec. 2.2.

Substituting the genesis solution (i.e., Eqs. (2.19), (2.21) and (2.22)) into
Eqs. (3.9), (3.11) and (3.12), we have

$\displaystyle Q_{T}$ $\displaystyle=$ $\displaystyle 1,\quad
c_{T}^{2}=1-\frac{\kappa_{1}}{M^{2}(-t)^{2}}+\frac{\kappa_{2}}{M^{4}(-t)^{4}}\,.$
(3.16) $\displaystyle Q_{s}$ $\displaystyle\approx$
$\displaystyle\frac{27\lambda_{2}M_{\rm
P}^{4}}{\left(\lambda_{2}-2\lambda_{3}-3\kappa_{1}M_{\rm
P}^{2}M^{-2}\right)^{2}}(-t)^{2}\,,$ (3.17) $\displaystyle c_{s}^{2}$
$\displaystyle\approx$
$\displaystyle-\frac{\lambda_{2}-2\lambda_{3}-3\kappa_{1}M_{\rm
P}^{2}M^{-2}}{3\lambda_{2}}$ (3.18)

for $|t|\gg M^{-1}$. We have kept only the leading order terms in $Q_{s}$ and
$c_{s}^{2}$, where $\kappa_{2}$ does not appear. Apparently, we should require
$\kappa_{1}>0$ so that $c_{T}\leq 1$. In order to guarantee that $Q_{s}>0$ and
$0<c_{s}^{2}\leq 1$, we should have $\lambda_{2}>0$,
$\lambda_{2}-2\lambda_{3}-3\kappa_{1}M_{\rm P}^{2}M^{-2}<0$ and
$4\lambda_{2}-2\lambda_{3}-3\kappa_{1}M_{\rm P}^{2}M^{-2}\geq 0$.
Additionally, we may also consider adding a term $\sim(-X/M^{4})^{n}$ in
$F(X)$, where $n$ is a constant. However, we will find $c_{s}>1$ in the limit
$|t|\gg M^{-1}$ for $n<1$. Therefore, given the genesis solution and the
requirement of $0<c_{s}^{2}\leq 1$, the formulation of $F(X)$ in Eq. (3.13) is
quite general.

The quantities

$\gamma=\frac{\lambda_{2}-2\lambda_{3}-3\kappa_{1}M_{\rm P}^{2}M^{-2}}{3M_{\rm
P}^{2}(-t)^{3}}+\frac{2\kappa_{2}}{M^{4}(-t)^{5}}\,,$ (3.19)
$Q_{\tilde{m}_{4}}=1+\frac{\kappa_{1}}{M^{2}(-t)^{2}}-\frac{3\kappa_{2}}{M^{4}(-t)^{4}}\,.$
(3.20)

Hence, $\gamma<0$ and $Q_{\tilde{m}_{4}}>0$ in the limit $|t|\gg M^{-1}$. If
we assume that the universe eventually enters the standard hot Big Bang
expansion, during which $\gamma=H>0$, we will find $\gamma$ cross $0$ at some
time $t_{\gamma}$ after the end of the genesis phase.999Note that the genesis
solution is no longer applicable at $t_{\gamma}$. In order to avoid the
gradient instability induced by the $\gamma-$crossing around $t_{\gamma}$, we
should carefully design the behavior of $Q_{\tilde{m}_{4}}$, see e.g., the
condition (13) of [72], see also [89] for an example of the numerical
simulation. In this paper, instead of handling the explicit constructions of
$\gamma$ and $Q_{\tilde{m}_{4}}$, we will focus our discussion on the behavior
of genesis in the limit $|t|\gg M^{-1}$ for our purpose.

In the spatial flat gauge, we have

$\displaystyle S^{(2)}_{\sigma}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\Big{[}{\cal A}(t)\dot{\sigma}^{2}-{\cal
B}(t)(\partial_{i}\sigma)^{2}\Big{]}\,,$ (3.21)

where

${\cal A}={3\lambda_{2}\over M^{2}(-t)^{2}}+{\cal O}\left({1\over
M^{3}(-t)^{3}}\right)\,,\quad{\cal
B}={2\lambda_{3}-\lambda_{2}+3\kappa_{1}M_{\rm P}^{2}M^{-2}\over
M^{2}(-t)^{2}}+{\cal O}\left({1\over M^{3}(-t)^{3}}\right)$ (3.22)

for $|t|\gg M^{-1}$. The sound speed squared in the spatial flat gauge can be
given as $c_{s}^{2}={\cal B}/{\cal
A}\approx(2\lambda_{3}-\lambda_{2}+3\kappa_{1}M_{\rm
P}^{2}M^{-2})/(3\lambda_{2})$, which is consistent with Eq. (3.18). The
arguments given in Sec. 2.2 remain applicable.

In the limit $|t|\gg M^{-1}$, the leading order terms in the cubic and quartic
actions of $\sigma$ can be found as

$\displaystyle S^{(3)}_{\sigma}=\int d^{4}x{2M_{\rm P}^{2}\kappa_{1}(-t)\over
M^{5}}\left[g^{\mu\nu}g^{\alpha\beta}\partial_{\nu}\sigma\partial_{\alpha}\dot{\sigma}\partial_{\mu}\partial_{\beta}\sigma-\partial_{i}\sigma\partial_{i}\dot{\sigma}g^{\mu\nu}\partial_{\mu}\partial_{\nu}\sigma+g^{\mu\nu}\partial_{\mu}\dot{\sigma}\partial_{\nu}\dot{\sigma}\dot{\sigma}+{\cal
O}(-t)^{-1}\right]\,,$ (3.23) $\displaystyle S^{(4)}_{\sigma}=\int
d^{4}x\frac{M_{\rm
P}^{2}\kappa_{1}(-t)^{2}}{M^{6}}\Big{[}g^{\mu\alpha}g^{\nu\beta}\partial_{\mu}\sigma\partial_{\nu}\sigma\partial_{\alpha}\partial_{\beta}\sigma
g^{\rho\gamma}\partial_{\rho}\partial_{\gamma}\sigma-g^{\mu\alpha}g^{\rho\beta}g^{\nu\gamma}\partial_{\alpha}\sigma\partial_{\nu}\sigma\partial_{\rho}\partial_{\mu}\sigma\partial_{\gamma}\partial_{\beta}\sigma$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad-g^{\alpha\beta}\partial_{\alpha}\sigma\partial_{\beta}\sigma
g^{\mu\nu}\partial_{\mu}\dot{\sigma}\partial_{\nu}\dot{\sigma}-4g^{\alpha\beta}g^{\mu\nu}\partial_{\nu}\sigma\partial_{\alpha}\dot{\sigma}\partial_{\mu}\partial_{\beta}\sigma\dot{\sigma}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+4\dot{\sigma}\partial_{i}\sigma\partial_{i}\dot{\sigma}g^{\mu\nu}\partial_{\mu}\partial_{\nu}\sigma-4\dot{\sigma}^{2}g^{\mu\nu}\partial_{\mu}\dot{\sigma}\partial_{\nu}\dot{\sigma}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+g^{\alpha\beta}\partial_{\alpha}\sigma\partial_{\beta}\sigma
g^{\mu\nu}\partial_{\mu}\partial_{\nu}\sigma\ddot{\sigma}+{\cal
O}(-t)^{-1}\Big{]}\,.$

Apparently, the interactions are distinctive from that of the cubic Galileon
genesis.

Similar to Sec. 2.2, we set ${\cal A}\equiv{\cal B}$ so that $c_{s}^{2}\equiv
1$ for simplicity, which indicates
$\lambda_{3}=2\lambda_{2}-{3\tilde{\kappa_{1}}}/2$, where
$\tilde{\kappa}_{1}\equiv\kappa_{1}M_{\rm P}^{2}M^{-2}$. We define

$\tilde{\sigma}={\cal A}^{1/2}\sigma\,,\qquad\tilde{M}=M{\cal
A}^{1/2}\approx\sqrt{3\lambda_{2}}/(-t)\,,$ (3.24)

such that

$\displaystyle S^{(2)}_{\tilde{\sigma}}=\int
d^{4}x\left[\dot{\tilde{\sigma}}^{2}-(\partial_{i}\tilde{\sigma})^{2}\right]\,,$
(3.25) $\displaystyle S^{(3)}_{\tilde{\sigma}}=\int
d^{4}x{2\tilde{\kappa}_{1}(-t)\over\tilde{M}^{3}}\left(g^{\mu\nu}g^{\alpha\beta}\partial_{\nu}\tilde{\sigma}\partial_{\alpha}\dot{\tilde{\sigma}}\partial_{\mu}\partial_{\beta}\tilde{\sigma}-\partial_{i}\tilde{\sigma}\partial_{i}\dot{\tilde{\sigma}}g^{\mu\nu}\partial_{\mu}\partial_{\nu}\tilde{\sigma}+g^{\mu\nu}\partial_{\mu}\dot{\tilde{\sigma}}\partial_{\nu}\dot{\tilde{\sigma}}\dot{\tilde{\sigma}}\right)\,,$
$\displaystyle S^{(4)}_{\tilde{\sigma}}=\int
d^{4}x{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\Big{(}g^{\mu\alpha}g^{\nu\beta}\partial_{\mu}\tilde{\sigma}\partial_{\nu}\tilde{\sigma}\partial_{\alpha}\partial_{\beta}\tilde{\sigma}g^{\rho\gamma}\partial_{\rho}\partial_{\gamma}\tilde{\sigma}-g^{\mu\alpha}g^{\rho\beta}g^{\nu\gamma}\partial_{\alpha}\tilde{\sigma}\partial_{\nu}\tilde{\sigma}\partial_{\rho}\partial_{\mu}\tilde{\sigma}\partial_{\gamma}\partial_{\beta}\tilde{\sigma}$
$\displaystyle\qquad\qquad\qquad\qquad\quad-g^{\alpha\beta}\partial_{\alpha}\tilde{\sigma}\partial_{\beta}\tilde{\sigma}g^{\mu\nu}\partial_{\mu}\dot{\tilde{\sigma}}\partial_{\nu}\dot{\tilde{\sigma}}-4g^{\alpha\beta}g^{\mu\nu}\partial_{\nu}\tilde{\sigma}\partial_{\alpha}\dot{\tilde{\sigma}}\partial_{\mu}\partial_{\beta}\tilde{\sigma}\dot{\tilde{\sigma}}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+4\dot{\tilde{\sigma}}\partial_{i}\tilde{\sigma}\partial_{i}\dot{\tilde{\sigma}}g^{\mu\nu}\partial_{\mu}\partial_{\nu}\tilde{\sigma}-4\dot{\tilde{\sigma}}^{2}g^{\mu\nu}\partial_{\mu}\dot{\tilde{\sigma}}\partial_{\nu}\dot{\tilde{\sigma}}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+g^{\alpha\beta}\partial_{\alpha}\tilde{\sigma}\partial_{\beta}\tilde{\sigma}g^{\mu\nu}\partial_{\mu}\partial_{\nu}\tilde{\sigma}\ddot{\tilde{\sigma}}\Big{)}\,,$
(3.27)

where we have neglected higher order terms. From Eq. (3.25), we can see that
the dispersion relation is simply $F=\omega^{2}-k^{2}=0$. As explained in Sec.
2.2, the time scale of the scattering processes we consider will be $\Delta
t\ll|t|$ in the regime $|t|\gg 1/M$. Therefore, we can approximately treat the
coefficients ${2\tilde{\kappa}_{1}(-t)\over\tilde{M}^{3}}$ and
${\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}$ in (3.2) and (3.27) as
constants in the calculation of scattering amplitudes. Again, the energy scale
of the scattering processes we consider should satisfy $M>E\sim 1/\Delta t\gg
E_{\rm IR}\simeq|t|^{-1}$.

###  3.3 Perturbative unitarity

In this subsection, by using the constraints of perturbative unitarity
introduced in Sec. 2.3, we revisit the specific “beyond Horndeski” action
(3.4) in the context of genesis cosmology. Due to the complexity of the
interactions given by Eqs. (3.2) and (3.27), we will calculate only the
amplitude for the four-point scattering processes in the following.

####  3.3.1 Constraints from four-point scattering

The Feynman diagrams for
$\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma}$ can be
categorized into four kinds which are presented in Fig. 4 and Fig. 5
separately.

Figure 4: First kind of Feynman diagrams for
$\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma}$ which only
utilize one four-point vertex. The presented four types (B, C, D, F) whose
contributions are nonzero correspond to the second, third, fourth and sixth
terms in $S^{(4)}_{\tilde{\sigma}}$ in Eq. (3.23) respectively.

Figure 5: Three kinds of Feynman diagrams for
$\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma}$ which utilize
two three-point vertices.

The amplitudes corresponding to Fig. 4 are

$\displaystyle
i\mathcal{M}_{B}(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma})$
$\displaystyle=$
$\displaystyle-2i{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\Bigg{\\{}-p_{12}\big{(}p_{14}p_{23}+p_{13}p_{24}\big{)}+p_{13}\big{(}p_{14}(p_{23}+p_{24})+p_{24}(p_{23}-p_{34})\big{)}$
$\displaystyle+p_{14}p_{23}\big{(}p_{24}-p_{34}\big{)}+p_{12}p_{34}\big{(}p_{13}+p_{14}+p_{23}+p_{24}\big{)}\Bigg{\\}}\,,$
$\displaystyle
i\mathcal{M}_{C}(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma})$
$\displaystyle=$
$\displaystyle-4i{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\Bigg{\\{}p_{14}p_{23}(p_{1t4t}+p_{2t3t})+p_{13}p_{24}(p_{1t3t}+p_{2t4t})-p_{12}p_{34}(p_{1t2t}+p_{3t4t})\Bigg{\\}}\,,$
$\displaystyle
i\mathcal{M}_{D}(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma})$
$\displaystyle=$
$\displaystyle-4i{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\Bigg{\\{}-p_{12}\Big{(}p_{13}p_{2t4t}+p_{14}p_{2t3t}+p_{24}p_{1t3t}+p_{23}p_{1t4t}\Big{)}$
$\displaystyle+p_{13}\Big{(}p_{14}(p_{2t3t}+p_{2t4t})+p_{1t2t}p_{34}+p_{23}(p_{1t4t}+p_{2t4t})-p_{2t4t}p_{34}\Big{)}$
$\displaystyle+p_{34}\Big{(}p_{14}(p_{1t2t}-p_{2t3t})+p_{1t2t}(p_{23}+p_{24})-p_{1t3t}p_{24}-p_{1t4t}p_{23}\Big{)}$
$\displaystyle+p_{24}\Big{(}p_{14}(p_{1t3t}+p_{2t3t})+p_{23}(p_{1t3t}+p_{1t4t})\Big{)}+p_{12}p_{3t4t}\Big{(}p_{13}+p_{14}+p_{23}+p_{24}\Big{)}\Bigg{\\}}\,,$
$\displaystyle
i\mathcal{M}_{F}(\tilde{\sigma}\tilde{\sigma}\to\tilde{\sigma}\tilde{\sigma})$
$\displaystyle=$
$\displaystyle-16i{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\Big{(}p_{1t2t}p_{3t4t}\Big{)}\left(-p_{12}+p_{13}+p_{14}-p_{34}+p_{23}+p_{24}\right)\,,$
(3.28)

where $p_{ij}\equiv p_{i}\cdot p_{j}$. This expression can be written in
compact form

$\displaystyle\mathcal{M}_{4}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{B}+\mathcal{M}_{C}+\mathcal{M}_{D}+\mathcal{M}_{F}$
(3.29) $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\frac{1}{2}\left(\mathsf{s}^{3}+\mathsf{t}^{3}+\mathsf{u}^{3}-2\mathsf{s}^{2}\mathsf{s}_{\mathsf{t}}-2\mathsf{t}^{2}\mathsf{t}_{\mathsf{t}}-2\mathsf{u}^{2}\mathsf{u}_{\mathsf{t}}\right)\,,$

where the Mandelstam variables

$\displaystyle\mathsf{s}=-(p_{1}+p_{2})^{2}=-(p_{3}+p_{4})^{2}\,,$ (3.30)
$\displaystyle\mathsf{t}=-(p_{3}-p_{1})^{2}=-(p_{2}-p_{4})^{2}\,,$
$\displaystyle\mathsf{u}=-(p_{4}-p_{1})^{2}=-(p_{2}-p_{3})^{2}\,,$

and

$\displaystyle\mathsf{s}_{\mathsf{t}}=2p_{1t2t}=2p_{3t4t}\,,\quad\quad\,\,\,\mathsf{s}_{\mathsf{s}}=-2\vec{p}_{1}\cdot\vec{p}_{2}=-2\vec{p}_{3}\cdot\vec{p}_{4}\,,$
(3.31)
$\displaystyle\mathsf{t}_{\mathsf{t}}=-2p_{1t3t}=-2p_{2t4t}\,,\quad\mathsf{t}_{\mathsf{s}}=2\vec{p}_{1}\cdot\vec{p}_{3}=2\vec{p}_{2}\cdot\vec{p}_{4}\,,$
$\displaystyle\mathsf{u}_{\mathsf{t}}=-2p_{1t4t}=-2p_{2t3t}\,,\quad\mathsf{u}_{\mathsf{s}}=2\vec{p}_{1}\cdot\vec{p}_{4}=2\vec{p}_{2}\cdot\vec{p}_{3}\,,$

where $p_{itjt}\equiv p_{it}p_{jt}$.

The amplitudes for $\mathsf{s}$-channel in Fig. 5 are

$\displaystyle\mathcal{M}_{\mathsf{s}I\to I}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}p_{12}(p_{1t}+p_{2t})\bigg{(}p_{34}(p_{14}+p_{24})(p_{1t}+p_{2t}-p_{3t})+p_{34}(p_{13}+p_{23})(p_{1t}+p_{2t}-p_{4t})$
$\displaystyle+(p_{13}+p_{23})(p_{14}+p_{24})(p_{3t}+p_{4t})\bigg{)}\,,$
$\displaystyle\mathcal{M}_{\mathsf{s}I\to II}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\bigg{(}-2p_{12}^{2}(p_{1t}+p_{2t})(p_{3t}+p_{4t})\vec{p}_{3}\cdot\vec{p}_{4}\bigg{)}\,,$
$\displaystyle\mathcal{M}_{\mathsf{s}I\to III}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\bigg{(}2p_{12}p_{3t4t}(p_{1t}+p_{2t})^{2}(p_{13}+p_{14}+p_{23}+p_{24}-p_{34})\bigg{)}\,,$
$\displaystyle\mathcal{M}_{\mathsf{s}II\to I}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\vec{p}_{1}\cdot\vec{p}_{2}(-p_{1t}-p_{2t})\bigg{(}p_{34}(p_{14}+p_{24})(p_{1t}+p_{2t}-p_{3t})+p_{34}(p_{13}+p_{23})(p_{1t}+p_{2t}-p_{4t})$
$\displaystyle+(p_{13}+p_{23})(p_{14}+p_{24})(p_{3t}+p_{4t})\bigg{)}\,,$
$\displaystyle\mathcal{M}_{\mathsf{s}II\to II}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\bigg{(}2p_{12}(\vec{p}_{1}\cdot\vec{p}_{2})(\vec{p}_{3}\cdot\vec{p}_{4})(p_{1t}+p_{2t})(p_{3t}+p_{4t})\bigg{)}\,,$
$\displaystyle\mathcal{M}_{\mathsf{s}II\to III}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\bigg{(}-2(p_{1t}+p_{2t})^{2}p_{3t4t}\vec{p}_{1}\cdot\vec{p}_{2}(p_{13}+p_{14}+p_{23}+p_{24}-p_{34})\bigg{)}\,,$
$\displaystyle\mathcal{M}_{\mathsf{s}III\to I}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}p_{1t2t}(p_{1t}+p_{2t})\bigg{(}p_{34}(p_{14}+p_{24})(p_{1t}+p_{2t}-p_{3t})+p_{34}(p_{13}+p_{23})(p_{1t}+p_{2t}-p_{4t})$
$\displaystyle+(p_{13}+p_{23})(p_{14}+p_{24})(p_{3t}+p_{4t})\bigg{)}\,,$
$\displaystyle\mathcal{M}_{\mathsf{s}III\to II}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\bigg{(}-2p_{12}p_{1t2t}\vec{p}_{3}\cdot\vec{p}_{4}(p_{1t}+p_{2t})(p_{3t}+p_{4t})\bigg{)}\,,$
$\displaystyle\mathcal{M}_{\mathsf{s}III\to III}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\bigg{(}2p_{1t2t}p_{3t4t}(p_{1t}+p_{2t})^{2}(p_{13}+p_{14}+p_{23}+p_{24}-p_{34})\bigg{)}\,.$
(3.32)

This expression can be written in compact form in terms of Eq. (3.30) and Eq.
(3.31),

$\displaystyle\mathcal{M}_{\mathsf{s}}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{\mathsf{s}I\to I}+\mathcal{M}_{\mathsf{s}I\to
II}+\mathcal{M}_{\mathsf{s}I\to III}+\mathcal{M}_{\mathsf{s}II\to
I}+\mathcal{M}_{\mathsf{s}II\to II}+\mathcal{M}_{\mathsf{s}II\to III}$ (3.33)
$\displaystyle+\mathcal{M}_{\mathsf{s}III\to I}+\mathcal{M}_{\mathsf{s}III\to
II}+\mathcal{M}_{\mathsf{s}III\to III}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\frac{1}{4}{\mathsf{s}(\mathsf{s}-\mathsf{s})}\Bigg{\\{}\mathsf{s}(p_{1t}^{2}+p_{2t}^{2})-\mathsf{s}_{\mathsf{t}}(p_{1t}^{2}+p_{2t}^{2}-\mathsf{s}+\mathsf{s}_{\mathsf{t}})+\mathsf{s}_{\mathsf{s}}(\mathsf{t}_{\mathsf{t}}+\mathsf{u}_{\mathsf{t}})\Bigg{\\}}\,.$

Similarly, the amplitudes for $\mathsf{u}$-channel and $\mathsf{t}$-channel in
Fig. 5 are

$\displaystyle\mathcal{M}_{\mathsf{t}}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{\mathsf{t}I\to I}+\mathcal{M}_{\mathsf{t}I\to
II}+\mathcal{M}_{\mathsf{t}I\to III}+\mathcal{M}_{\mathsf{t}II\to
I}+\mathcal{M}_{\mathsf{t}II\to II}+\mathcal{M}_{\mathsf{t}II\to III}$ (3.34)
$\displaystyle+\mathcal{M}_{\mathsf{t}III\to I}+\mathcal{M}_{\mathsf{t}III\to
II}+\mathcal{M}_{\mathsf{t}III\to III}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\frac{1}{4}{\mathsf{t}(\mathsf{t}-\mathsf{t})}\Bigg{\\{}\mathsf{t}(p_{1t}^{2}+p_{3t}^{2})-\mathsf{t}_{\mathsf{t}}(p_{1t}^{2}+p_{3t}^{2}-\mathsf{t})-\mathsf{s}_{\mathsf{t}}^{2}+\mathsf{t}_{\mathsf{s}}(\mathsf{s}_{\mathsf{t}}+\mathsf{u}_{\mathsf{t}})\Bigg{\\}}\,,$

and

$\displaystyle\mathcal{M}_{\mathsf{u}}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{\mathsf{u}I\to I}+\mathcal{M}_{\mathsf{u}I\to
II}+\mathcal{M}_{\mathsf{u}I\to III}+\mathcal{M}_{\mathsf{u}II\to
I}+\mathcal{M}_{\mathsf{u}II\to II}+\mathcal{M}_{\mathsf{u}II\to III}$ (3.35)
$\displaystyle+\mathcal{M}_{\mathsf{u}III\to I}+\mathcal{M}_{\mathsf{u}III\to
II}+\mathcal{M}_{\mathsf{u}III\to III}$ $\displaystyle=$
$\displaystyle{\tilde{\kappa}_{1}(-t)^{2}\over\tilde{M}^{4}}\frac{1}{4}{\mathsf{u}(\mathsf{u}-\mathsf{u})}\Bigg{\\{}\mathsf{u}(p_{1t}^{2}+p_{4t}^{2})-\mathsf{u}_{\mathsf{t}}(p_{1t}^{2}+p_{4t}^{2}-\mathsf{u})-\mathsf{s}_{\mathsf{t}}^{2}+\mathsf{u}_{\mathsf{s}}(\mathsf{s}_{\mathsf{t}}+\mathsf{t}_{\mathsf{t}})\Bigg{\\}}\,.$

The total amplitude is

$\displaystyle\mathcal{M}_{\rm
total}=\mathcal{M}_{4}+\mathcal{M}_{\mathsf{s}}+\mathcal{M}_{\mathsf{t}}+\mathcal{M}_{\mathsf{u}}\,.$
(3.36)

Therefore, the stable NEC violation in the context of “beyond Horndeski”
genesis, which is constructed by the specific action (3.4) with $F$ given by
Eq. (3.13), cannot satisfy the requirement (2.41) of UV completion. In the EFT
regime, by using the constraint (2.42) of perturbative unitarity, we obtain
approximately

$\sqrt{\mathsf{s}}\lesssim(M/M_{\rm P})^{1/3}|t|^{-1}\sim(M/M_{\rm
P})^{1/3}\tilde{M}\ll M\,,$ (3.37)

where we have disregarded the coefficient. Namely, the perturbative unitarity
is violated at a scale $\sqrt{s}\sim(M/M_{\rm P})^{1/3}|t|^{-1}\sim(M/M_{\rm
P})^{1/3}\tilde{M}$ $\ll M$. Given the IR cut-off scale $E_{\rm IR}$ of the
scattering processes we considered, the tree-level perturbative unitarity is
already violated. This result is actually consistent with the results of Secs.
2.3.1 and 2.3.2, despite the differences in coefficients.

Consequently, although we are able to realize fully stable NEC violation with
the “beyond Horndeski” theory, the constraints of tree-level perturbative
unitarity imply that we may need some unknown new physics below the cut-off
scale $M$ in the EFT other than that represented by the “beyond Horndeski”
operators in Eq. (3.1) or (3.4).

##  4 Summary and outlook

The NEC is a crucial and quite robust condition in gravity and cosmology. The
explorations of a fully stable NEC violation have made some significant
progress. However, it is still an open question whether some fundamental
properties of UV-complete theories or the consistency requirements of EFT
forbid a violation of the NEC. The constraints of perturbative unitarity could
provide us with some novel insights into the EFT of a fully stable NEC
violation.

In this paper, we investigated the tree-level perturbative unitarity for
stable NEC violations in the contexts of both Galileon genesis and “beyond
Horndeski” genesis cosmology, in which the universe is asymptotically
Minkowskian in the infinite past. The calculations of scattering amplitudes
could be simplified by the genesis solution provided we consider only the
leading order interacting terms. It is found that the tree-level perturbative
unitarity gets broken at an energy scale $\sqrt{\mathsf{s}}\sim|t|^{-1}\ll M$
in both Galileon genesis and “beyond Horndeski” genesis, where $M$ is the cut-
off scale of the EFT action, $t$ is the cosmological time. Therefore, the
constraints of perturbative unitarity imply that the earlier era (i.e., the
larger of $|t|$) we go into the genesis phase, the more urgent we may need the
unknown new physics at a lower scale other than that represented by the
“beyond Horndeski” operators.

In the calculations of the scattering amplitudes, we have assumed that the
sound speed squared $c_{s}^{2}=1$ during the NEC-violating phase for
simplicity. Additionally, the models of genesis cosmology we considered are
constructed by some specific Galileon and “beyond Horndeski” theories, which
are representative to some extent. Whether our conclusion remains unchanged
for a general $c_{s}^{2}$ and other more complicated “beyond Horndeski”
theories (or a different construction of the stable genesis model as discussed
in [72]) requires further investigations. It would also be interesting to see
whether the required unknown new physics indicated by perturbative unitarity
can be represented by those higher order DHOST operators or some modified
dispersion relations. Our study might be extended to the context of bouncing
cosmology as well. Furthermore, taking into account the constraints from
cosmological observations will place tighter constraints on the EFT of a fully
stable NEC violation.

## Acknowledgments

We would like to thank Yun-Song Piao, Shuang-Yong Zhou, Toshifumi Noumi, Gen
Ye, Yunlong Zheng, Mian Zhu and Chao Chen for helpful discussions. The work of
Cai is supported in part by the National Natural Science Foundation of China
(Grant No. 11905224), the China Postdoctoral Science Foundation (Grant No.
2021M692942) and Zhengzhou University (Grant No. 32340282). The work of Xu is
supported in part by National Natural Science Foundation of China under Grant
Nos. 12105247 and 12047545, the China Postdoctoral Science Foundation under
Grant No. 2021M702957. The work of Zhao is supported by Jefferson Science
Associates, LLC under U.S. DOE Contract # DE-AC05-06OR23177 and by U.S. DOE
Grant # DE-FG02-97ER41028. The work of Zhou is supported in part by the
Swedish Research Council under grants number 2015-05333 and 2018-03803. We
acknowledge the use of the computing server Arena317@ZZU.

##  Appendix A The EFT of nonsingular cosmology

The approach of the effective field theory (EFT) is powerful in investigating
inflation [106], dark energy [107, 108, 109] as well as nonsingular cosmology
[70, 71, 72, 73]. We work with the $3+1$ decomposed metric
$ds^{2}=-N^{2}dt^{2}+h_{ij}\left(dx^{i}+N^{i}dt\right)\left(dx^{j}+N^{j}dt\right)$,
where $h_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu}$ is the induced metric, $n^{\mu}$
is the unit normal vector of the constant time hypersurfaces, $N$ and $N^{i}$
are the lapse function and the shift vector, respectively. In the unitary
gauge, the EFT action that is able to realize a fully stable NEC violation can
be written as

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\Big{[}{M_{\rm P}^{2}\over 2}f(t)R-\Lambda(t)-c(t)g^{00}$ (A.1)
$\displaystyle\qquad\qquad\quad+{M_{2}^{4}(t)\over 2}(\delta
g^{00})^{2}-{m_{3}^{3}(t)\over 2}\delta K\delta
g^{00}-m_{4}^{2}(t)\left(\delta K^{2}-\delta K_{\mu\nu}\delta
K^{\mu\nu}\right)$ $\displaystyle\qquad\qquad\quad+{\tilde{m}_{4}^{2}(t)\over
2}R^{(3)}\delta g^{00}\Big{]}$

up to quadratic order [70, 71, 72, 73] (see also [106, 107, 108, 109]), where
$R^{(3)}$ is the induced 3-dimensional Ricci scalar, $K_{\mu\nu}$ is the
extrinsic curvature, $\delta g^{00}=g^{00}+1$, $\delta
K_{\mu\nu}=K_{\mu\nu}-h_{\mu\nu}H$, $H$ is the Hubble parameter. We have
disregarded those higher-order spatial derivatives and the action of matter
sector in (A.1).

In the cosmological context where there is an evolving scalar field $\phi$,
the constant time hypersurfaces can be set as the uniform scalar field
hypersurfaces. As a result, we have $n_{\mu}=-\phi_{\mu}/\sqrt{-X}$, where we
have defined $X=\phi_{\mu}\phi^{\mu}$, $\phi_{\mu}=\nabla_{\mu}\phi$ and
$\phi^{\mu}=\nabla^{\mu}\phi$ for convenience. Using
$K_{\mu\nu}=h_{\mu}^{\sigma}\nabla_{\sigma}n_{\nu}$ and the Gauss-Codazzi
relation, we can obtain the corresponding covariant action of (A.1) in
principle, see e.g. [108, 110]. In fact, the EFT action (A.1) is able to
specify a variety of theories of gravity by different choices of the time-
dependent coefficiens $f$, $\Lambda$, $c$, $M_{2}^{4}$, $m_{3}^{3}$,
$m_{4}^{2}$ and $\tilde{m}_{4}^{2}$. For example, the Horndeski theory [58,
59, 60] and the “beyond Horndeski” (GLPV) theory [75, 76] correspond to
$m_{4}^{2}={\tilde{m}}_{4}^{2}$ and $m_{4}^{2}\neq{\tilde{m}}_{4}^{2}$,
respectively.

However, in order to cover more general degenerate higher-order scalar-tensor
(DHOST) theories [111] (see [101] for a review), the EFT action (A.1) has to
incorporate additional operators $\delta K\dot{\delta g^{00}}$, $(\dot{\delta
g^{00}})^{2}$ and $(\partial_{i}{\delta g^{00}})^{2}$ [110, 105, 112], where
there is still no Ostrogradski instability. Following the convention of [105,
112, 101], the action of all quadratic and cubic DHOST theories can be written
as

$\displaystyle S^{\text{quad}}=\int d^{3}xdta^{3}\frac{{\cal
M}^{2}}{2}\left\\{\delta K_{ij}\delta
K^{ij}-\left(1+\frac{2}{3}\alpha_{\mathrm{L}}\right)\delta
K^{2}+\left(1+\alpha_{\mathrm{T}}\right)\left(\delta_{1}R^{(3)}\frac{\delta\sqrt{h}}{a^{3}}+\delta_{2}R^{(3)}\right)\right.$
(A.2)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+H^{2}\alpha_{\mathrm{K}}\delta
N^{2}+4H\alpha_{\mathrm{B}}\delta K\delta
N+\left(1+\alpha_{\mathrm{H}}\right)\delta_{1}R^{(3)}\delta N$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\left.+4\beta_{1}\delta
K\delta\dot{N}+\beta_{2}\delta\dot{N}^{2}+\frac{\beta_{3}}{a^{2}}\left(\partial_{i}\delta
N\right)^{2}\right\\}$

up to quadratic order in the unitary gauge, where $\delta N\equiv 1-N=\delta
g^{00}/2$. Note that the contribution from the first line of (A.1) at
quadratic order is also included in (A.2). The relations between ${\cal
M}^{2}$, $\alpha_{i}$ and the coefficients in (A.2) can be find in [110]. The
covariant scalar-tensor theories can be mapped to (A.2) by using the relations
provided in [105]. Particularly, for the Horndeski theory and the “beyond
Horndeski” (GLPV) theory, which belong to the subclass Ia of the DHOST theory,
we will find $\beta_{i}=0$ in (A.2).

## References

  * [1] A. H. Guth, _The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems_ , _Phys. Rev. D_ 23 (1981) 347–356.
  * [2] A. A. Starobinsky, _A New Type of Isotropic Cosmological Models Without Singularity_ , _Phys. Lett. B_ 91 (1980) 99–102.
  * [3] A. D. Linde, _A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems_ , _Phys. Lett. B_ 108 (1982) 389–393.
  * [4] A. Albrecht and P. J. Steinhardt, _Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking_ , _Phys. Rev. Lett._ 48 (1982) 1220–1223.
  * [5] Planck collaboration, N. Aghanim et al., _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6, [1807.06209].
  * [6] Planck collaboration, Y. Akrami et al., _Planck 2018 results. X. Constraints on inflation_ , _Astron. Astrophys._ 641 (2020) A10, [1807.06211].
  * [7] A. Borde and A. Vilenkin, _Eternal inflation and the initial singularity_ , _Phys. Rev. Lett._ 72 (1994) 3305–3309, [gr-qc/9312022].
  * [8] A. Borde, A. H. Guth and A. Vilenkin, _Inflationary space-times are incompletein past directions_ , _Phys. Rev. Lett._ 90 (2003) 151301, [gr-qc/0110012].
  * [9] P. Agrawal, G. Obied, P. J. Steinhardt and C. Vafa, _On the Cosmological Implications of the String Swampland_ , _Phys. Lett. B_ 784 (2018) 271–276, [1806.09718].
  * [10] A. Bedroya, R. Brandenberger, M. Loverde and C. Vafa, _Trans-Planckian Censorship and Inflationary Cosmology_ , _Phys. Rev. D_ 101 (2020) 103502, [1909.11106].
  * [11] Y. Cai and Y.-S. Piao, _Pre-inflation and trans-Planckian censorship_ , _Sci. China Phys. Mech. Astron._ 63 (2020) 110411, [1909.12719].
  * [12] H.-H. Li, G. Ye, Y. Cai and Y.-S. Piao, _Trans-Planckian censorship of multistage inflation and dark energy_ , _Phys. Rev. D_ 101 (2020) 063527, [1911.06148].
  * [13] F. J. Tipler, _Energy conditions and spacetime singularities_ , _Phys. Rev. D_ 17 (1978) 2521–2528.
  * [14] V. A. Rubakov, _The Null Energy Condition and its violation_ , _Phys. Usp._ 57 (2014) 128–142, [1401.4024].
  * [15] Y.-S. Piao, B. Feng and X.-m. Zhang, _Suppressing CMB quadrupole with a bounce from contracting phase to inflation_ , _Phys. Rev. D_ 69 (2004) 103520, [hep-th/0310206].
  * [16] Y.-S. Piao, _Can the universe experience many cycles with different vacua?_ , _Phys. Rev. D_ 70 (2004) 101302, [hep-th/0407258].
  * [17] Y.-S. Piao, _A Possible explanation to low CMB quadrupole_ , _Phys. Rev. D_ 71 (2005) 087301, [astro-ph/0502343].
  * [18] Y.-F. Cai, T. Qiu, Y.-S. Piao, M. Li and X. Zhang, _Bouncing universe with quintom matter_ , _JHEP_ 10 (2007) 071, [0704.1090].
  * [19] Y.-F. Cai, T.-t. Qiu, R. Brandenberger and X.-m. Zhang, _A Nonsingular Cosmology with a Scale-Invariant Spectrum of Cosmological Perturbations from Lee-Wick Theory_ , _Phys. Rev. D_ 80 (2009) 023511, [0810.4677].
  * [20] T. Qiu, J. Evslin, Y.-F. Cai, M. Li and X. Zhang, _Bouncing Galileon Cosmologies_ , _JCAP_ 10 (2011) 036, [1108.0593].
  * [21] Y.-F. Cai, D. A. Easson and R. Brandenberger, _Towards a Nonsingular Bouncing Cosmology_ , _JCAP_ 08 (2012) 020, [1206.2382].
  * [22] Z.-G. Liu, Z.-K. Guo and Y.-S. Piao, _Obtaining the CMB anomalies with a bounce from the contracting phase to inflation_ , _Phys. Rev. D_ 88 (2013) 063539, [1304.6527].
  * [23] T. Qiu, X. Gao and E. N. Saridakis, _Towards anisotropy-free and nonsingular bounce cosmology with scale-invariant perturbations_ , _Phys. Rev. D_ 88 (2013) 043525, [1303.2372].
  * [24] Y.-F. Cai, E. McDonough, F. Duplessis and R. H. Brandenberger, _Two Field Matter Bounce Cosmology_ , _JCAP_ 10 (2013) 024, [1305.5259].
  * [25] M. Koehn, J.-L. Lehners and B. A. Ovrut, _Cosmological super-bounce_ , _Phys. Rev. D_ 90 (2014) 025005, [1310.7577].
  * [26] L. Battarra, M. Koehn, J.-L. Lehners and B. A. Ovrut, _Cosmological Perturbations Through a Non-Singular Ghost-Condensate/Galileon Bounce_ , _JCAP_ 07 (2014) 007, [1404.5067].
  * [27] Y. Wan, T. Qiu, F. P. Huang, Y.-F. Cai, H. Li and X. Zhang, _Bounce Inflation Cosmology with Standard Model Higgs Boson_ , _JCAP_ 12 (2015) 019, [1509.08772].
  * [28] M. Koehn, J.-L. Lehners and B. Ovrut, _Nonsingular bouncing cosmology: Consistency of the effective description_ , _Phys. Rev. D_ 93 (2016) 103501, [1512.03807].
  * [29] T. Qiu and Y.-T. Wang, _G-Bounce Inflation: Towards Nonsingular Inflation Cosmology with Galileon Field_ , _JHEP_ 04 (2015) 130, [1501.03568].
  * [30] S. Nojiri, S. D. Odintsov and V. K. Oikonomou, _Bounce universe history from unimodular $F(R)$ gravity_, _Phys. Rev. D_ 93 (2016) 084050, [1601.04112].
  * [31] S. Banerjee and E. N. Saridakis, _Bounce and cyclic cosmology in weakly broken galileon theories_ , _Phys. Rev. D_ 95 (2017) 063523, [1604.06932].
  * [32] D. Nandi and L. Sriramkumar, _Can a nonminimal coupling restore the consistency condition in bouncing universes?_ , _Phys. Rev. D_ 101 (2020) 043506, [1904.13254].
  * [33] P. Creminelli, A. Nicolis and E. Trincherini, _Galilean Genesis: An Alternative to inflation_ , _JCAP_ 11 (2010) 021, [1007.0027].
  * [34] Z.-G. Liu, J. Zhang and Y.-S. Piao, _A Galileon Design of Slow Expansion_ , _Phys. Rev. D_ 84 (2011) 063508, [1105.5713].
  * [35] Y. Wang and R. Brandenberger, _Scale-Invariant Fluctuations from Galilean Genesis_ , _JCAP_ 10 (2012) 021, [1206.4309].
  * [36] Z.-G. Liu and Y.-S. Piao, _A Galileon Design of Slow Expansion: Emergent universe_ , _Phys. Lett. B_ 718 (2013) 734–739, [1207.2568].
  * [37] P. Creminelli, K. Hinterbichler, J. Khoury, A. Nicolis and E. Trincherini, _Subluminal Galilean Genesis_ , _JHEP_ 02 (2013) 006, [1209.3768].
  * [38] K. Hinterbichler, A. Joyce, J. Khoury and G. E. Miller, _DBI Realizations of the Pseudo-Conformal Universe and Galilean Genesis Scenarios_ , _JCAP_ 12 (2012) 030, [1209.5742].
  * [39] K. Hinterbichler, A. Joyce, J. Khoury and G. E. Miller, _Dirac-Born-Infeld Genesis: An Improved Violation of the Null Energy Condition_ , _Phys. Rev. Lett._ 110 (2013) 241303, [1212.3607].
  * [40] Z.-G. Liu, H. Li and Y.-S. Piao, _Preinflationary genesis with CMB B-mode polarization_ , _Phys. Rev. D_ 90 (2014) 083521, [1405.1188].
  * [41] D. Pirtskhalava, L. Santoni, E. Trincherini and P. Uttayarat, _Inflation from Minkowski Space_ , _JHEP_ 12 (2014) 151, [1410.0882].
  * [42] S. Nishi and T. Kobayashi, _Generalized Galilean Genesis_ , _JCAP_ 03 (2015) 057, [1501.02553].
  * [43] T. Kobayashi, M. Yamaguchi and J. Yokoyama, _Galilean Creation of the Inflationary Universe_ , _JCAP_ 07 (2015) 017, [1504.05710].
  * [44] Y. Cai and Y.-S. Piao, _The slow expansion with nonminimal derivative coupling and its conformal dual_ , _JHEP_ 03 (2016) 134, [1601.07031].
  * [45] S. Nishi and T. Kobayashi, _Scale-invariant perturbations from null-energy-condition violation: A new variant of Galilean genesis_ , _Phys. Rev. D_ 95 (2017) 064001, [1611.01906].
  * [46] Y.-S. Piao and E. Zhou, _Nearly scale invariant spectrum of adiabatic fluctuations may be from a very slowly expanding phase of the universe_ , _Phys. Rev. D_ 68 (2003) 083515, [hep-th/0308080].
  * [47] Y.-S. Piao, _Primordial perturbations during a slow expansion_ , _Phys. Rev. D_ 76 (2007) 083505, [0706.0981].
  * [48] S. Dubovsky, T. Gregoire, A. Nicolis and R. Rattazzi, _Null energy condition and superluminal propagation_ , _JHEP_ 03 (2006) 025, [hep-th/0512260].
  * [49] P. Creminelli, M. A. Luty, A. Nicolis and L. Senatore, _Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies_ , _JHEP_ 12 (2006) 080, [hep-th/0606090].
  * [50] A. Nicolis, R. Rattazzi and E. Trincherini, _Energy’s and amplitudes’ positivity_ , _JHEP_ 05 (2010) 095, [0912.4258].
  * [51] V. A. Rubakov, _Consistent NEC-violation: towards creating a universe in the laboratory_ , _Phys. Rev. D_ 88 (2013) 044015, [1305.2614].
  * [52] B. Elder, A. Joyce and J. Khoury, _From Satisfying to Violating the Null Energy Condition_ , _Phys. Rev. D_ 89 (2014) 044027, [1311.5889].
  * [53] A. Ijjas, J. Ripley and P. J. Steinhardt, _NEC violation in mimetic cosmology revisited_ , _Phys. Lett. B_ 760 (2016) 132–138, [1604.08586].
  * [54] Y. Cai and Y.-S. Piao, _Intermittent null energy condition violations during inflation and primordial gravitational waves_ , _Phys. Rev. D_ 103 (2021) 083521, [2012.11304].
  * [55] Y. Cai and Y.-S. Piao, _Generating enhanced primordial GWs during inflation with intermittent violation of NEC and diminishment of GW propagating speed_ , _JHEP_ 06 (2022) 067, [2201.04552].
  * [56] M. Libanov, S. Mironov and V. Rubakov, _Generalized Galileons: instabilities of bouncing and Genesis cosmologies and modified Genesis_ , _JCAP_ 08 (2016) 037, [1605.05992].
  * [57] T. Kobayashi, _Generic instabilities of nonsingular cosmologies in Horndeski theory: A no-go theorem_ , _Phys. Rev. D_ 94 (2016) 043511, [1606.05831].
  * [58] G. W. Horndeski, _Second-order scalar-tensor field equations in a four-dimensional space_ , _Int. J. Theor. Phys._ 10 (1974) 363–384.
  * [59] C. Deffayet, X. Gao, D. A. Steer and G. Zahariade, _From k-essence to generalised Galileons_ , _Phys. Rev. D_ 84 (2011) 064039, [1103.3260].
  * [60] 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, [1105.5723].
  * [61] D. A. Easson, I. Sawicki and A. Vikman, _G-Bounce_ , _JCAP_ 11 (2011) 021, [1109.1047].
  * [62] A. Ijjas and P. J. Steinhardt, _Classically stable nonsingular cosmological bounces_ , _Phys. Rev. Lett._ 117 (2016) 121304, [1606.08880].
  * [63] A. Ijjas and P. J. Steinhardt, _Fully stable cosmological solutions with a non-singular classical bounce_ , _Phys. Lett. B_ 764 (2017) 289–294, [1609.01253].
  * [64] R. Kolevatov and S. Mironov, _Cosmological bounces and Lorentzian wormholes in Galileon theories with an extra scalar field_ , _Phys. Rev. D_ 94 (2016) 123516, [1607.04099].
  * [65] S. Akama and T. Kobayashi, _Generalized multi-Galileons, covariantized new terms, and the no-go theorem for nonsingular cosmologies_ , _Phys. Rev. D_ 95 (2017) 064011, [1701.02926].
  * [66] D. A. Dobre, A. V. Frolov, J. T. Gálvez Ghersi, S. Ramazanov and A. Vikman, _Unbraiding the Bounce: Superluminality around the Corner_ , _JCAP_ 03 (2018) 020, [1712.10272].
  * [67] Y. A. Ageeva, O. A. Evseev, O. I. Melichev and V. A. Rubakov, _Horndeski Genesis: strong coupling and absence thereof_ , _EPJ Web Conf._ 191 (2018) 07010, [1810.00465].
  * [68] Y. Ageeva, O. Evseev, O. Melichev and V. Rubakov, _Toward evading the strong coupling problem in Horndeski genesis_ , _Phys. Rev. D_ 102 (2020) 023519, [2003.01202].
  * [69] Y. Ageeva, P. Petrov and V. Rubakov, _Horndeski genesis: consistency of classical theory_ , _JHEP_ 12 (2020) 107, [2009.05071].
  * [70] Y. Cai, Y. Wan, H.-G. Li, T. Qiu and Y.-S. Piao, _The Effective Field Theory of nonsingular cosmology_ , _JHEP_ 01 (2017) 090, [1610.03400].
  * [71] P. Creminelli, D. Pirtskhalava, L. Santoni and E. Trincherini, _Stability of Geodesically Complete Cosmologies_ , _JCAP_ 11 (2016) 047, [1610.04207].
  * [72] Y. Cai, H.-G. Li, T. Qiu and Y.-S. Piao, _The Effective Field Theory of nonsingular cosmology: II_ , _Eur. Phys. J. C_ 77 (2017) 369, [1701.04330].
  * [73] Y. Cai and Y.-S. Piao, _A covariant Lagrangian for stable nonsingular bounce_ , _JHEP_ 09 (2017) 027, [1705.03401].
  * [74] R. Kolevatov, S. Mironov, N. Sukhov and V. Volkova, _Cosmological bounce and Genesis beyond Horndeski_ , _JCAP_ 08 (2017) 038, [1705.06626].
  * [75] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, _Healthy theories beyond Horndeski_ , _Phys. Rev. Lett._ 114 (2015) 211101, [1404.6495].
  * [76] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, _Exploring gravitational theories beyond Horndeski_ , _JCAP_ 02 (2015) 018, [1408.1952].
  * [77] Y. Cai and Y.-S. Piao, _Higher order derivative coupling to gravity and its cosmological implications_ , _Phys. Rev. D_ 96 (2017) 124028, [1707.01017].
  * [78] Y. Cai, Y.-T. Wang, J.-Y. Zhao and Y.-S. Piao, _Primordial perturbations with pre-inflationary bounce_ , _Phys. Rev. D_ 97 (2018) 103535, [1709.07464].
  * [79] S. Mironov, V. Rubakov and V. Volkova, _Bounce beyond Horndeski with GR asymptotics and $\gamma$-crossing_, _JCAP_ 10 (2018) 050, [1807.08361].
  * [80] T. Qiu, K. Tian and S. Bu, _Perturbations of bounce inflation scenario from $f(T)$ modified gravity revisited_, _Eur. Phys. J. C_ 79 (2019) 261, [1810.04436].
  * [81] G. Ye and Y.-S. Piao, _Implication of GW170817 for cosmological bounces_ , _Commun. Theor. Phys._ 71 (2019) 427, [1901.02202].
  * [82] G. Ye and Y.-S. Piao, _Bounce in general relativity and higher-order derivative operators_ , _Phys. Rev. D_ 99 (2019) 084019, [1901.08283].
  * [83] S. Mironov, V. Rubakov and V. Volkova, _Genesis with general relativity asymptotics in beyond Horndeski theory_ , _Phys. Rev. D_ 100 (2019) 083521, [1905.06249].
  * [84] S. Akama, S. Hirano and T. Kobayashi, _Primordial non-Gaussianities of scalar and tensor perturbations in general bounce cosmology: Evading the no-go theorem_ , _Phys. Rev. D_ 101 (2020) 043529, [1908.10663].
  * [85] S. Mironov, V. Rubakov and V. Volkova, _Subluminal cosmological bounce beyond Horndeski_ , _JCAP_ 05 (2020) 024, [1910.07019].
  * [86] A. Ilyas, M. Zhu, Y. Zheng, Y.-F. Cai and E. N. Saridakis, _DHOST Bounce_ , _JCAP_ 09 (2020) 002, [2002.08269].
  * [87] A. Ilyas, M. Zhu, Y. Zheng and Y.-F. Cai, _Emergent Universe and Genesis from the DHOST Cosmology_ , 2009.10351.
  * [88] M. Zhu, A. Ilyas, Y. Zheng, Y.-F. Cai and E. N. Saridakis, _Scalar and tensor perturbations in DHOST bounce cosmology_ , _JCAP_ 11 (2021) 045, [2108.01339].
  * [89] M. Zhu and Y. Zheng, _Improved DHOST Genesis_ , _JHEP_ 11 (2021) 163, [2109.05277].
  * [90] S. Mironov and V. Volkova, _Stable nonsingular cosmologies in beyond Horndeski theory and disformal transformations_ , _Int. J. Mod. Phys. A_ 37 (2022) 2250088, [2204.05889].
  * [91] D. Cannone, N. Bartolo and S. Matarrese, _Perturbative Unitarity of Inflationary Models with Features_ , _Phys. Rev. D_ 89 (2014) 127301, [1402.2258].
  * [92] C. de Rham, S. Melville, A. J. Tolley and S.-Y. Zhou, _Positivity bounds for scalar field theories_ , _Phys. Rev. D_ 96 (2017) 081702, [1702.06134].
  * [93] C. de Rham and S. Melville, _Unitary null energy condition violation in P(X) cosmologies_ , _Phys. Rev. D_ 95 (2017) 123523, [1703.00025].
  * [94] T. Grall and S. Melville, _Inflation in motion: unitarity constraints in effective field theories with (spontaneously) broken Lorentz symmetry_ , _JCAP_ 09 (2020) 017, [2005.02366].
  * [95] H. Goodhew, S. Jazayeri and E. Pajer, _The Cosmological Optical Theorem_ , _JCAP_ 04 (2021) 021, [2009.02898].
  * [96] S. Céspedes, A.-C. Davis and S. Melville, _On the time evolution of cosmological correlators_ , _JHEP_ 02 (2021) 012, [2009.07874].
  * [97] S. Kim, T. Noumi, K. Takeuchi and S. Zhou, _Perturbative unitarity in quasi-single field inflation_ , _JHEP_ 07 (2021) 018, [2102.04101].
  * [98] S. Melville and E. Pajer, _Cosmological Cutting Rules_ , _JHEP_ 05 (2021) 249, [2103.09832].
  * [99] R. Brandenberger and V. Kamali, _Unitarity Problems for an Effective Field Theory Description of Early Universe Cosmology_ , 2203.11548.
  * [100] D. Langlois, R. Saito, D. Yamauchi and K. Noui, _Scalar-tensor theories and modified gravity in the wake of GW170817_ , _Phys. Rev. D_ 97 (2018) 061501, [1711.07403].
  * [101] D. Langlois, _Dark energy and modified gravity in degenerate higher-order scalar–tensor (DHOST) theories: A review_ , _Int. J. Mod. Phys. D_ 28 (2019) 1942006, [1811.06271].
  * [102] Y. Cai, Y.-T. Wang and Y.-S. Piao, _Is there an effect of a nontrivial $c_{T}$ during inflation?_, _Phys. Rev. D_ 93 (2016) 063005, [1510.08716].
  * [103] Y. Cai, Y.-T. Wang and Y.-S. Piao, _Propagating speed of primordial gravitational waves and inflation_ , _Phys. Rev. D_ 94 (2016) 043002, [1602.05431].
  * [104] Y.-T. Wang, Y. Cai, Z.-G. Liu and Y.-S. Piao, _Probing the primordial universe with gravitational waves detectors_ , _JCAP_ 01 (2017) 010, [1612.05088].
  * [105] D. Langlois, M. Mancarella, K. Noui and F. Vernizzi, _Effective Description of Higher-Order Scalar-Tensor Theories_ , _JCAP_ 05 (2017) 033, [1703.03797].
  * [106] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, _The Effective Field Theory of Inflation_ , _JHEP_ 03 (2008) 014, [0709.0293].
  * [107] G. Gubitosi, F. Piazza and F. Vernizzi, _The Effective Field Theory of Dark Energy_ , _JCAP_ 02 (2013) 032, [1210.0201].
  * [108] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, _Essential Building Blocks of Dark Energy_ , _JCAP_ 08 (2013) 025, [1304.4840].
  * [109] F. Piazza and F. Vernizzi, _Effective Field Theory of Cosmological Perturbations_ , _Class. Quant. Grav._ 30 (2013) 214007, [1307.4350].
  * [110] J. Gleyzes, D. Langlois and F. Vernizzi, _A unifying description of dark energy_ , _Int. J. Mod. Phys. D_ 23 (2015) 1443010, [1411.3712].
  * [111] D. Langlois and K. Noui, _Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability_ , _JCAP_ 02 (2016) 034, [1510.06930].
  * [112] D. Langlois, _Degenerate Higher-Order Scalar-Tensor (DHOST) theories_ , in _52nd Rencontres de Moriond on Gravitation_ , pp. 221–228, 2017, 1707.03625.