[1]organization=Departamento de Física Teórica and IPARCOS, Universidad
Complutense de Madrid, addressline=Pl. de las Ciencias 1, postcode=ES 28040,
city=Madrid, country=Spain.

[2]organization=IKERBASQUE, Basque Foundation for Science, postcode=ES 48011,
city=Bilbao, country=Spain.

[3]organization=Department of Physics and EHU Quantum Center, University of
the Basque Country, UPV/EHU, addressline=P.O. Box 644, postcode=ES 48080,
city=Bilbao, country=Spain.

# Dark energy with a shift-symmetric scalar field: obstacles, loophole hunting
and dead ends

Teodor Borislavov Vasilev<EMAIL_ADDRESS>Mariam Bouhmadi-López
<EMAIL_ADDRESS>Prado Martín-Moruno<EMAIL_ADDRESS>
###### Abstract

We discuss the possibility of a scalar field being the fundamental description
of dark energy without the need of self-tuning any parameter. Thereby, we
focus on shift-symmetric scalar-tensor theories satisfying also that the
propagation speed of gravitational waves is equal to the speed of light.
Analysing the stability of scalar linear perturbations, we discuss the
conditions that seems to be necessary to describe (super) accelerated cosmic
expansion without introducing instabilities. It has been established, however,
that this stability can be ruined when taking into account the interaction
with tensor perturbations (essentially gravitational waves). Indeed, although
we shall point out that the standard proof of absence of dark energy stable
braiding models due to this interaction has a possible way-out, we find
general arguments suggesting that there are no dark energy stable solutions
that can exploit this loophole. Thus, we discuss future research directions
for finding viable fundamental descriptions of dark energy. We also provide a
dictionary between the covariant version of the scalar field theory and that
of the Effective Field Theory approach, explicitly computing the parameters in
the latter formalism in terms of the functions appearing in the covariant
version, and its derivatives. To the best of our knowledge, this is the first
time these expressions are explicitly obtained up-to arbitrary order in
perturbation theory.

###### keywords:

dark energy, kinetic gravity braiding, perturbation theory, gravitational
waves

##  1 Introduction

Scalar field theories could be crucial in providing an underlying theoretical
framework for addressing the mysteries of dark energy (DE). These theories
introduce a scalar field that can be used to describe dynamical DE, in
contrast to the cosmological constant case (see, for instance, references [1,
2]). To select the appropriate scalar field theory for cosmology, it is
reasonable to demand that the corresponding field equations to be of second
order. Otherwise, the new degrees of freedom (DOF) may destabilise the
dynamics of the theory by means of the Ostrogradski ghost [3]. In this
context, a natural framework is that provided by Horndeski theory [4] (see
also [5] for a review), although some theories with higher-order derivatives
in the action can also avoid the Ostrogradski instability through degeneracy
conditions in the Lagrangian (see, for instance, [6, 7]). Rather recently, the
speed of propagation of gravitational waves (GWs) has also become an important
consideration when formulating theories beyond general relativity. After the
detection of the event GW170817 [8] and its electromagnetic counterpart [9],
it has been concluded that the propagation speed of GWs is very close to that
of light. Matching this observation with the theory reduces the original
Horndeski family of scalar-tensor theories to essentially the so-called
Kinetic Gravity Braiding (KGB) models [10], (which include the well-known
$k$-essence theory as a special case [11, 12, 13, 14, 15]) plus a possible
non-minimal coupling to gravity via the scalar field (see, for instances,
reference [16]).

Scalar fields minimally coupled to gravity and with a canonical kinetic term
have been already used to describe the current accelerated expansion of the
Universe. However, in that simple case one has to rely on the particular form
of a potential term, re-introducing the fine-tuning problem usually associated
with the cosmological constant. This issue can be avoided by focusing on
models that are invariant under constant shifts in the scalar field, for which
potential-like terms do not appear in the Lagrangian. This symmetry-based
argument is an elegant way to make the evolution of the system to depend only
on the rate of change of the scalar field but not on the scalar field itself
(see discussion in reference [10]). However, the simplest minimally coupled
scalar field with a canonical kinetic term in the action is only able to
describe a stiff fluid when imposing shift-symmetry (see, for example,
references [2, 17]); leading to a non-interesting phenomenology for the (late-
time) DE sector. On the other hand, in the Horndeski subfamily previously
mentioned this argument prevents from having a non-minimal coupling to gravity
via the scalar field itself, although a derivative coupling to matter is still
allowed in principle. In either case, evading fifth-force constraints coming
from astrophysical scales is naturally easy under this symmetry consideration
[10]. The Horndeski models that satisfy this symmetry condition and predict a
propagation speed for GWs equal to the speed of light are known as shift-
symmetric KGB theories [10].

Shift-symmetric KGB theories are well-known for producing self-tuning de-
Sitter future solutions [10] (see also references [18, 19, 20, 21, 22, 23,
24]) and they can also describe different DE phenomenology [25, 26].
Remarkably, these theories can describe a phantom regime that is stable in the
sense of no ghost- or gradient-like instabilities for small scalar
perturbation [10]. Moreover, KGB theories can produce a smooth phantom
crossing in the DE sector [10], contrary to the single scalar field
$k$-essence scenario [27, 28]. To this end, it should be highlighted that the
violation of the Null Energy Condition (NEC) in the DE sector is not only
still allowed by the observational data (see discussion in reference [29]),
but it has been indeed proven that a phantom regime (with phantom crossing) is
a necessary prerequisite to ease both the $H_{0}$ and $S_{8}$ cosmological
tensions simultaneously by taking into account new physics that is relevant
only at late cosmic times [30, 31]. Therefore, the motivation, simplicity and
apparent stability properties of the shift-symmetric KGB scalar field models
makes this framework an interesting proposal for modelling DE. Nevertheless,
the absence of instabilities in linear scalar perturbations is a necessary but
not a sufficient condition for the stability of the classical theory. Indeed,
it has been shown that the interaction mediated by the braiding term between
tensor perturbations (essentially GWs) and DE fluctuations may induce a ghost-
like and/or gradient-like instability in the scalar sector [32]. (Se also
references [33, 34] for a discussion on the decay of GWs into scalar
fluctuations when Lorentz invariance is spontaneously broken.) Consequently,
it was concluded that the braiding term in the KGB theory should not produce
any sizeable effect in order to trivially escape from the GWs-induced
instabilities [32]. According to these results, the available parameter-space
of the shift-symmetric KGB models seems to shrink to that of the shift-
symmetric $k$-essence theory only [32].

In the present work, we first re-analyse the inter-relation of a phantom
behaviour with classical and semi-classical instabilities of the scalar field
in the framework of shift-symmetric KGB models. By implementing a novel
characterization of the already-known stability conditions, we shed some light
on previous results in the literature for $k$-essence models [35] (see also,
for instance, references [11, 12, 36, 37, 38]) and also obtain new compact
results for the braiding term. In the second place, we reflect about the
instability induced by the interaction of the braiding term with GWs [32]. We
propose a possible way to circumvent this obstacle when departing from the
Galileon terms for the KGB functions. Unfortunately, we shall argue that the
apparent loophole seems to lack practical applicability for constructing
viable dark energy models with a non-vanishing braiding term. In order to
perform this study, we have derived the detailed form of the mass parameters
of the KGB theory in the Effective Field Theory (EFT) approach. Although the
expressions for these parameters were already computed at leading order, to
the best of our knowledge, it is the first time that the complete family of
parameters up-to arbitrary order is presented.

This work is organised as follows: we review the general properties of shift-
symmetric KGB theories in the first part of section 2. The stability of linear
scalar perturbations in the well-known $k$-essence subclass of the more
general KGB framework is reviewed in section 2.1. In sections 2.2 and 2.3 we
discuss the viability of linearised scalar perturbation for the remaining part
of the KGB theory. Section 3 is devoted to the interaction of the braiding
term with GWs. We review the arguments presented in reference [32] in section
3.1. We identify a potential mechanism to ease the constraints found in
reference [32] in section 3.2. In section 3.3 we discuss that the possibility
for a viable KGB model to implement such a mechanism is an elusive ambition in
practice. Therefore, in section 4, we reflect about the open paths that can
still be further explored to find fundamental viable descriptions for dark
energy. Finally, auxiliary results for discussing the stability of linear
perturbations are summarised in A. A self-contained guide to the EFT approach
to DE and useful calculations are presented in B.

##  2 Stability of linearised scalar perturbations

The KGB theory is given by the action [10] (see also [39, 40])

$\displaystyle S_{g}=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}R+K(\phi,X)-G(\phi,X)\Box\phi\right],$ (1)

where we have adopted the geometric unit system $8\pi G=c=1$, $K(\phi,X)$ and
$G(\phi,X)$ are arbitrary functions of the scalar field $\phi$ and its
canonical kinetic term
$X\coloneqq-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$, the box
represents the covariant d’Alembertian operator
$\Box\phi=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi$ and we have used the
signature $(-,+,+,+)$.

In this work we will focus on the shift-symmetric sector of the KGB theory,
since it has been proven to be extremely fruitful for cosmological
applications [10] (see also [18, 19, 20, 21, 22, 23, 24, 25, 26]). This is the
case when the above action is invariant under constant shifts in the scalar
field given by

$\displaystyle\phi\to\phi+c,$ (2)

being $c$ some constant. In practise, this implies that the functions $K$ and
$G$ do not depend on the scalar field but only on its kinetic term. That is,
we will focus on the particular case of $K=K(X)$ and $G=G(X)$. (Nevertheless,
this assumption will be relaxed when discussing the EFT approach in B.) In
addition, we will also consider the spatially flat cosmological background
described by the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker
(FLRW) space-time. That corresponds to the line element

$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)\,dx_{3}^{2},$ (3)

where $a(t)$ stands for the scale factor and $dx_{3}^{2}$ are the spatial
three-dimensional flat sections. Taking matter and radiation as external
sources to the action (1), the field equations of the theory for this
background read [10]

$\displaystyle 3H^{2}=$ $\displaystyle\,\rho_{m}+\rho_{r}-K+\dot{\phi}J,$ (4)
$\displaystyle\dot{H}=$
$\displaystyle-\frac{1}{2}\left(\rho_{m}+\frac{4}{3}\rho_{r}\right)+XG_{X}\ddot{\phi}-\frac{1}{2}\dot{\phi}J,$
(5) $\displaystyle\dot{\rho}_{m}=$ $\displaystyle-3H\rho_{m},$ (6)
$\displaystyle\dot{\rho}_{r}=$ $\displaystyle-4H\rho_{r},$ (7)

where a dot represents derivation w.r.t. the cosmic time $t$ and $G_{X}$
stands for ${\rm d}G/{\rm d}X$. In addition,

$\displaystyle J(\dot{\phi},H)\coloneqq\dot{\phi}K_{X}+6HXG_{X},$ (8)

is precisely the shift-current related to the global shift-symmetry of the
theory [10]. In view of these equations, the energy density and pressure of
the dark fluid can be defined as

$\rho_{\phi}(\dot{\phi},H)\coloneqq\dot{\phi}J-K,$ (9)

and

$p_{\phi}(\dot{\phi},\ddot{\phi})\coloneqq K-2XG_{X}\ddot{\phi},$ (10)

respectively. Then, the equation of state parameter and the partial energy
density for the scalar field take the form $w_{\phi}\coloneqq
p_{\phi}/\rho_{\phi}$ and $\Omega_{\phi}\coloneqq\rho_{\phi}/(3H^{2})$,
respectively. It should be noted that the energy density, $\rho_{\phi}$,
depends on the Hubble parameter throught $J$ while the pressure, $p_{\phi}$,
contains $\ddot{\phi}$ [10]. Moreover, the pressure will also depend on the
Hubble function when the dependence on $\ddot{\phi}$ is removed using the
field equations.

On the other hand, the field equation of the scalar field can be obtained
varying the action (1) w.r.t. the scalar field. This yields

$\displaystyle A(\dot{\phi},H)\ddot{\phi}+6XG_{X}\dot{H}+3HJ=0,$ (11)

where

$\displaystyle A(\dot{\phi},H)\coloneqq
K_{X}+2XK_{XX}+6H\dot{\phi}\left(G_{X}+XG_{XX}\right),$ (12)

has been defined for the compactness of the notation. This equation is
completely equivalent to the covariant conservation of the $\phi$-fluid in the
hydrodynamic approach, with the scalar field energy density and pressure as
defined in expressions (9) and (10), respectively. However, it should be noted
that the above scalar field equation of motion is not independent from the
system of equations given in expressions (4) to (7).

Alternatively to the scalar field equation (11), one can obtain the same
information from the conservation of the shift-current; that is As noted in
the reference [10], this conservation trivially leads to a first integral of
motion for the system given by

$\displaystyle J=\frac{Q_{0}}{a^{3}},$ (13)

where $Q_{0}$ is a constant. This property of the shift-symmetric KGB theories
can be used to significantly simplify the discussion on the evolution of these
cosmological models [10]. This is because $J=0$ defines a surface in the
configuration space towards which all trajectories evolve if the scale factor
is growing large. Therefore, the locus $J=0$ can be used to simplify the field
equations (4) and (5), at least asymptotically. However, this simplification
should not always be taken without further considerations since it may
drastically reduce the possible future phenomenology of the model; see
discussion in references [25, 26]. In general, the future phenomenology of
this shift-symmetric scalar field theory has been shown to encompass a great
variety of options. From self-tuning de-Sitter-like future solutions [10] (see
also references [18, 19, 20, 21, 22, 23, 24]) to DE-driven cosmological
singularities [25, 26].

The stability of scalar perturbations around a FLRW background have already
been addressed at linear order for this scalar field theories [10] (see also
references [23, 40, 41]). The absence of ghost and gradient instabilities in
the scalar sector implies [40]

$\displaystyle Q_{\text{S}}$
$\displaystyle\coloneqq\frac{2D}{(2-\alpha_{\textrm{B}})^{2}}>0\,,\qquad
D\coloneqq\alpha_{\textrm{K}}+\frac{3}{2}\alpha_{\textrm{B}}^{2}\,,$ (14)
$\displaystyle c_{\text{s}}^{2}$
$\displaystyle\coloneqq-\frac{\left(2-\alpha_{\textrm{B}}\right)\left[\dot{H}-\frac{1}{2}H^{2}\alpha_{\textrm{B}}\right]-H\dot{\alpha}_{\textrm{B}}+\rho_{\textrm{m}}+\frac{4}{3}\rho_{\textrm{r}}}{H^{2}D}\geq
0,$ (15)

respectively, where the dimensionless functions $\alpha_{K}$ and
$\alpha_{\textrm{B}}$ were first introduced in reference [40]. They are
defined as

$\displaystyle H^{2}\alpha_{\textrm{K}}\coloneqq$
$\displaystyle\,2X\left(K_{X}+2XK_{XX}\right)+12\dot{\phi}XH\left(G_{X}+XG_{XX}\right),$
(16) $\displaystyle H\alpha_{\textrm{B}}\coloneqq$
$\displaystyle\,2\dot{\phi}XG_{X}.$ (17)

Violation of condition (14), on the one hand, leads to scalar perturbations
whose associated kinetic energy takes negative values [42]; i.e. perturbations
are ghost. The presence of a ghost DOF may be detrimental (at the classical
level) if the ghost interacts with a positive-energy mode, as this may lead to
runaway solutions where the total energy is conserved but the relative
energies associated to the ghost and no-ghost sectors diverge (see, for
instance, reference [43]). At the quantum level, the corresponding
$\phi$-particles would have negative energies. In this case the vacuum becomes
unstable due to the spontaneous production of ghost-particles together with
normal-particles with arbitrarily high energies and momenta [43] (see also
reference [42]). Consequently, backgrounds with ghosts are generally
considered pathological at both classical and quantum levels. On the other
hand, violation of condition (15) introduces the so-called gradient
instability. This is when the leading order spatial derivatives have the wrong
sign w.r.t. the time derivatives in the perturbed scalar field equation [42]
(see also references [44]). In Fourier space, the frequencies of the
oscillations become imaginary at high momenta, resulting in perturbations that
grow exponentially fast. Thus, it precludes a stable classical model (at
least, as long as a perturbative treatment is still valid).

It is interesting to note that

$\displaystyle H^{2}D=2X\left(A+6X^{2}G_{X}^{2}\right),$ (18)

and, therefore, the ghost-free condition $D>0$ (see equation (14)) implies

$\displaystyle A+6X^{2}G_{X}^{2}>0,$ (19)

in our notation. Moreover, replacing the time derivative of the Hubble rate
that appears in the scalar field equation (11) by means of the Raychaudhuri
equation (5) leads to

$\displaystyle
H^{2}D\ddot{\phi}+6XJ\left(H-\dot{\phi}XG_{X}\right)-2X^{2}G_{X}\left(4\rho_{r}+3\rho_{m}\right)=0.$
(20)

Hence, the ghost parameter $D$ is just the factor ahead of the scalar field
acceleration.

In the following we will analyse the stability conditions described in
equations (14) and (15) for three different scenarios: $(i)$ the $k$-essence
model ($G\equiv 0$), $(ii)$ when $K\equiv 0$, and $(iii)$ the most general KGB
scenario. These are sections 2.1, 2.2 and 2.3, respectively. In doing so we
will review some well-known facts and also face new discussions about the
viability of these shift-symmetric theories from the point of view of linear
cosmological perturbations.

###  2.1 K-essence

Let us first re-analyze the stability for the well-known $k$-essence case [11,
12, 13, 14, 15], i.e. when the braiding function $G$ is set to zero in the
action (1). (Note that a non-zero constant $G$ would also reduce to the
$k$-essence scenario; see the action (1).) Unlike the scalar field with a
canonical kinetic term, for a $k$-essence field a phantom behaviour is not
necessarily due to the presence of a ghost [38] (see also [35]), since
equation (14) reduces to

$Q_{\text{S}}=\frac{D}{2}=\frac{2X}{H^{2}}\diff{\rho_{\phi}}{X},$ (21)

whereas the violation of the NEC by the scalar field depends on the slope of
the $k$-essence function $K$,

$1+w_{\phi}=\frac{2XK_{X}}{\rho_{\phi}},$ (22)

being

$\displaystyle\rho_{\phi}=$ $\displaystyle\,2XK_{X}-K,$ (23) $\displaystyle
p_{\phi}=$ $\displaystyle\,K,$ (24)

cf. with expressions (9) and (10) when $G_{X}$ is trivial. The slope of the
function $K$ should be negative for a phantom field, but this do not
necessarily force the energy density $\rho_{\phi}$ to be a decreasing function
of the kinetic term. So, the sign on the r.h.s. of equation (21) is in
principle independent of that of equation (22); see reference [38]. However,
the relation between instabilities and phantom behaviour appears when
considering the squared sound speed of perturbations, i.e. $c_{s}^{2}$. Taking
into account the Friedmann equation (5), we can recast the gradient-free
condition (15) in term of the background quantities as

$\displaystyle c_{s}^{2}=\frac{3(1+w_{\phi})\,\Omega_{\phi}}{D}\geq 0,$ (25)

where $\Omega_{\phi}\coloneqq\rho_{\phi}/3H^{2}$ is the partial energy density
for the scalar field and we have also used the fact that
$\alpha_{\textrm{B}}\equiv 0$ for $k$-essence. This new reformulation of the
expression for $c_{s}^{2}$ is extremely useful since we can address the
stability of linear perturbations in terms of the background quantities for
which we have a strong physical intuition. Since we are focusing on the case
$\Omega_{\phi}>0$, this inequality shows an inter-relation between the ghost-
and gradient-free conditions with the possible phantom character of the
$\phi$-fluid. In particular, if the NEC is satisfied for the scalar field
(i.e. if $1+w_{\phi}>0$), there is not a gradient instability if and only if
the ghost-free condition is also satisfied. As a result, there are models with
a completely stable scalar spectrum. On the other hand, the gradient-free and
ghost-free conditions are anti-correlated when the NEC is violated for the
$\phi$-fluid. That is to say, $c_{s}^{2}$ is positive for a phantom fluid only
if a ghost-like instability is present and vice-versa, scalar perturbations
are ghost-free only if a gradient instability is triggered. Please note that
the same conclusion was reached in reference [35], albeit through different
considerations (see also reference [38]).

As already expected, it is not possible to describe an effective phantom fluid
via a scalar field à la $k$-essence in a way that both ghost and gradient
instabilities are absent [35, 38, 42]. However, let us reflect about the
nature of these instabilities (see, for example, reference [42] for a review
on the topic). On the one hand, the presence of gradient instabilities is
related to a wrong sign for the spatial derivatives in the scalar field
perturbed equation. This makes the frequencies of the fluctuations imaginary
at high momenta, resulting in perturbations that grow arbitrary fast. Thus, it
precludes a stable classical model (at least, as long as a perturbative
treatment is still valid). For a ghost mode, on the other hand, the associated
kinetic energy is negative. If the ghost DOF interacts with a positive energy
mode (at the classical level) there may be runaway solutions, where the total
energy is conserved while individual energies diverge; we referred the
interested reader to the discussion in, for example, reference [43]. Thereby,
the common lore states that ghost instabilities are catastrophic (already) at
classical level. Still, the classical background is stable against high
momenta perturbations [42]. Nevertheless, there are examples in the literature
in which the presence of a ghost DOF interacting with a positive energy DOF do
not lead to runaway solutions and, therefore, to the destabilization of the
classical motion of the system [45]. (See also “islands of stability” [46, 47]
and meta-stability of ghosts [48].) Upon canonical quantisation of theories
with ghosts, the energy conservation does not forbid pair creation from vacuum
of ghosts-particles together with normal-particles: the vacuum becomes quantum
mechanically unstable (see, for instance, references [42, 43]). So, the
presence of a ghost is also considered as pathological at the semi-classical
level. However, the possibility of safely living with ghosts at quantum level
has already been explored with positive conclusions [49, 50, 51]. Therefore,
even if it is clear that the presence of any of these instabilities (ghost
and/or gradient) is a red light that has to be seriously taken into account,
it would be interesting to consider the possibility that the scalar field
(understood as being of gravitational nature) should be quantized in a
different way as matter fields (for example, with the space-time itself [52,
53]) or not been quantized at all, when dealing with the ghost instability.
For a review on the topic see, for example, reference [54].

In this context, a gradient instability is hardly tamable in the classical
theory, whereas a ghost-like instability could be further discussed at both
classical and quantum levels. So, if one considers that the $H_{0}$ tension
has to be solved by means of a phantom fluid, then, our best hope in the
$k$-essence case would be selecting the function $K$ in such a way that only
the ghost-free condition is violated. Otherwise, a negative $c_{s}^{2}$ will
inevitably jeopardise the perturbative approximation due to fast growing
solutions. On the other hand, it is also worth to mention that the scalar
field in (shift-symmetric or not) $k$-essence theory cannot cross the phantom
divide [27] (see also [28]). The equation of state parameter for the scalar
field, namely $w_{\phi}$, is always greater or less than $-1$ for a single
scalar field à la $k$-essence. Nevertheless, a phantom-crossing in the DE
sector was analytically shown to be a prerequisite for solving both the
$H_{0}$ and $S_{8}$ tensions simultaneously [30, 31]. Therefore, if both
tensions were considered on the same footing111Note that the disagreement
between the value of $H_{0}$ inferred from Planck-Cosmic Microwave Background
[55] and that coming from the direct local distance ladder measurements
provided by the SH0ES team [56, 57] has reached a statistical significance of
more than 5$\sigma$. Whereas the $S_{8}$ tension is at much lower statistical
significance, reaching between 1.7 to 3 standard deviations; see for instance
[58] and references therein. We refer the interested reader to the review [59]
for a self-contained discussion about the observational tensions in the
standard cosmological model., a scalar field theory beyond $k$-essence should
be explored in order to address both tensions.

###  2.2 Braiding model

Similarly, we now address the case when only the braiding function $G$ is
present. This corresponds to the case of $K=0$ in action (1). Consequently,
the energy density and pressure for the scalar field (9) and (10) simplify to

$\displaystyle\rho_{\phi}$ $\displaystyle=6\dot{\phi}XG_{X}H,$ (26)
$\displaystyle p_{\phi}$ $\displaystyle=-2XG_{X}\ddot{\phi},$ (27)

where the expression for $J$ in equation (8) has been also considered. Then,
the equation of state parameter for the scalar field reads

$\displaystyle w_{\phi}=-\frac{\ddot{\phi}}{3\dot{\phi}H}.$ (28)

The scalar field acceleration, $\ddot{\phi}$, can be removed from the previous
expression by means of the Raychaudhuri and scalar field equations, these are
expressions (5) and (11), respectively. Then, after simple manipulations we
can re-write the scalar field equation of state as

$\displaystyle w_{\phi}$
$\displaystyle=\frac{\Omega_{\phi}(3-\Omega_{r})}{2D},$ (29)

where $\Omega_{r}\coloneqq\rho_{r}/3H^{2}$ is the partial energy density for
radiation and the contribution of matter has been removed through the
constraint $\Omega_{m}=1-\Omega_{r}-\Omega_{\phi}$ coming form the Friedmann
equation (4). Please note that the above expression allow us to re-express the
ghost-free condition (14) in terms of background quantities as

$\displaystyle D=\frac{\Omega_{\phi}(3-\Omega_{r})}{2w_{\phi}}>0.$ (30)

As emphasised in the previous section, this new approach to the evaluation of
the stability conditions is extremely useful since we have a strong physical
intuition for the values of background quantities. The functional form of the
equation of state for the scalar field, that is to say
$w_{\phi}=w_{\phi}(a,\dot{a},\dot{\phi},\ddot{\phi})$, will depend on the
choice for the braiding function $G$. However, if we assume a positive
$\Omega_{\phi}$, it follows from the above inequality that a ghost instability
is always present if $w_{\phi}$ is negative. A significant implication of this
result is that any model that aims to describe a DE component (not necessary
phantom-like) will inevitably have this instability, since $w_{\phi}$ should
be less than $-1/3$.

Figure 1:  Stability of scalar perturbations in the
$(\Omega_{\phi},\,w_{\textup{eff}})$-plane for $K\equiv 0$. The perturbations
are stable in the $w_{\textup{eff}}>1/3$ zone, since $w_{\phi}$ is clearly
positive there in virtue of the relation
$w_{\textup{eff}}=\Omega_{r}/3+w_{\phi}\Omega_{\phi}$. In the dotted region,
which corresponds the horizontal band $0\leq w_{\textup{eff}}\leq 1/3$,
perturbations may be healthy if and only if $w_{\phi}$ is non-negative,
otherwise they will be both ghost- and gradient-unstable. Nevertheless, the
sign of $w_{\phi}$ in that region of the
$(\Omega_{\phi},\,w_{\textup{eff}})$-plane cannot be inferred without solving
the whole evolution of the model and, therefore, the stability conditions
there cannot be further addressed without specifying the function $G$. Next,
scalar perturbations are both ghost- and gradient-unstable in the central
vertical-lined region. Finally, in the lower oblique-lined region, only the
ghost instability is present.

A similar derivation can also be applied to the expression for the speed of
sound squared in equation (15). Straightforward differentiation over the
definition for $\alpha_{B}$ in equation (17) leads to

$\displaystyle\frac{\dot{\alpha}_{\textrm{B}}}{H}=3\left(w_{\textup{eff}}-w_{\phi}\right)\Omega_{\phi},$
(31)

where $w_{\textup{eff}}\coloneqq p_{tot}/\rho_{tot}$ represents the effective
equation of state parameter of the total fluid on the r.h.s. of equations (4)
and (5). Taking this result into account, and also substituting $\dot{H}$ by
means of equation (5), the gradient-free condition (15) can be re-formulated
in terms of background densities as

$\displaystyle
c_{s}^{2}=\frac{\Omega_{\phi}(5-\Omega_{\phi}+3w_{\textup{eff}})}{2D}\geq 0.$
(32)

The numerator in the above expression can be positive or negative depending on
the value of $\Omega_{\phi}$ and $w_{\textup{eff}}$. In Figure 1, we represent
the fulfilment of the stability conditions in the
$(\Omega_{\phi},\,w_{\textup{eff}})$-plane. As there can be checked, it is
possible to have ghost-free and gradient-free scalar perturbations during
radiation and matter dominated epoch; that would correspond to
$w_{\textup{eff}}\approx 1/3$ and $w_{\textup{eff}}\approx 0$ for
$\Omega_{\phi}\ll 1$, respectively. A necessary and sufficient condition for
this to happen is that of $w_{\phi}$ being positive; see inequality (30). On
the contrary, perturbations will always become unstable at some point if
$w_{\phi}$ is negative; for example, if the braiding scalar field acts as DE.
Hence, an accelerating universe cannot be safely modelled in this approach.
(Recall that acceleration demands the violation of the Strong Energy Condition
and, therefore, $w_{\textup{eff}}<-1/3$ at some moment in the recent past.
According to Figure 1, this would inevitably produce ghost and gradient
unstable scalar perturbations.) Interestingly, $c_{s}^{2}$ could be positive
for a super-accelerated regime ($w_{\textup{eff}}<-1$), although this would
only be possible at the price of having a ghost mode. This feature may be
interesting for modeling phantom DE since, as we have discussed in the
previous section, the possitivity of the speed of sound squared can be argued
to be more fundamental from a classical point of view than the absence of a
ghost. However, from Figure 1 it is clearly not possible to have a braiding
model that connects this gradient-stable supper-accelerated (late-time) regime
with a realistic early-universe description in a way that $c_{s}^{2}$ remains
always positive. Therefore, modelling DE with only the braiding part of the
KGB theories seems rather unwisely.

###  2.3 The complete KGB theory

In the previous sections we followed a novel approach to the evaluation of the
stability of linear scalar perturbations by rewriting the ghost-free (14) and
gradient-free (15) conditions in terms of background quantities, like the
partial energy densities $\Omega_{i}$ and the equation of state parameters
$w_{\phi}$ and $w_{\textup{eff}}$. Our main purpose with this approach is to
have a better physical intuition about the requirements for a given model to
produce stable scalar perturbations. In this section, we will apply the very
same strategy now for the full KGB theory.

When both $K$ and $G$ functions are non-trivial, the ghost-free condition (14)
cannot be re-expressed solely in terms of the previous background quantities
but

$\displaystyle
D=\frac{3\Omega_{J}^{B}\left[3\Omega_{J}(\Omega_{J}^{B}-2)+\left(3+\Omega_{r}-3\Omega_{\phi}\right)\Omega_{J}^{B}\right]}{6\left[\Omega_{J}-(1+w_{\phi})\Omega_{\phi}\right]}>0,$
(33)

where

$\displaystyle\Omega_{J}\coloneqq$
$\displaystyle\,\frac{\dot{\phi}J}{3H^{2}},$ (34)
$\displaystyle\Omega_{J}^{B}\coloneqq$
$\displaystyle\,\frac{2\dot{\phi}XG_{X}}{H},$ (35)

are the partial energy densities for the contributions of the shift-current,
$J$, and that of the braiding function, $G$, to the different terms in the
expression for scalar field energy density $\rho_{\phi}$ in equation (9),
respectively. Please note that $\Omega_{J}^{B}$ coincide with the definition
of $\alpha_{\rm B}$ [40], see equation (17). It is also important to highlight
that the physical constraint of $\Omega_{\phi}\in[0,1]$ does not imply, in
general, any bound on the value of these auxiliary partial energy densities.
Their value can be arbitrary large and/or negative.

Similarly, the gradient-free condition (15) can be translated to

$\displaystyle
c_{s}^{2}=\frac{1}{D}\left\\{\frac{\Omega_{J}^{B}(2-\Omega_{J}^{B})}{2}+3\Omega_{J}+3(2+\eta)\left[\Omega_{J}-(1+w_{\phi})\Omega_{\phi}\right]\right\\}\geq
0,$ (36)

being

$\displaystyle\eta\coloneqq\frac{2XG_{XX}}{G_{X}},$ (37)

a relevant quantity for the discussion in the next section. Please note that
when $G$ or $K$ are trivial, the above expressions reduce to those obtained in
sections 2.1 and 2.2, respectively.

Contrary to the cases analysed in the previous sections, conditions (33) and
(36) can be satisfied simultaneously for a wide variety of models including
those with phantom behaviour. For instance, considering $\eta$ to be positive
(since this will be crucial in the incoming sections) we show all the possible
cases where both inequalities are satisfied in A. Nevertheless, since
$\Omega_{J}^{B}$ and $\Omega_{J}$ are not physically meaningful quantities as
$\Omega_{\phi}$ is, it is difficult to extract the most general conditions the
functions $K$ and $G$ should meet in order to produce a stable model. Anyway,
there are examples of such models in the literature. For example, in reference
[10] the authors analyse the case of $K$ and $G$ being proportional to $X$.
They also show there that the resulting cosmological model has completely
stable linear scalar perturbations even though the scalar field is violating
the NEC. (Note, however, that their model does not feature super-acceleration
since $1+w_{\textup{eff}}>0$ always.) More examples of stable KGB models can
be found in references [23, 60, 61, 62]. Hence, it is possible to construct
reliable cosmological models with the scalar field coming from shift-symmetric
KGB theories when both $k$-essence and braiding functions are non-trivial.

###  2.4 Summary

Could we describe a stable DE component with a scalar field coming from shift-
symmetric KGB theories? There is no problem to describe it with a $k$-essence
term if it satisfies the NEC. However, it is well-known that $k$-essence
suffers from instabilities if the NEC is violated by the scalar field [35].
Moreover, the scalar field energy density should be always phantom or non-
phantom since $k$-essence cannot produce a phantom-crossing [27] (see also
[28]). Hence, if we interpret current data as favouring the presence of a
slight violation of the NEC in the DE sector at present times, then, the
scalar field should have always violated the NEC. In this scenario, a
$k$-essence proposal will be plagued with a gradient or a ghost instability.
With an appropriate selection for the function $K$, however, the gradient-free
condition can be satisfied for (even) NEC-violating models. Nevertheless, the
resulting scalar field will feature a ghost DOF; yet this may not always be
catastrophic. As we have discussed, it would be interesting to explore whether
it is possible to have a ghost mode in cosmology without destabilising the
classical nor the semi-classical regimes of the theory. While this may be
feasible, addressing both the $H_{0}$ and $S_{8}$ tensions (in case the latter
is confirmed) by means of late-time physics only continues to pose an
insurmountable obstacle in the $k$-essence scenario, since no phantom-crossing
is possible (see discussion in references [30, 31]).

The situation is significantly worsen when only the braiding function $G$ is
present in the scalar field’s action (1). Scalar perturbations are always
ghostly if the scalar field acts as a DE component, i.e. if $w_{\phi}<-1/3$
(see condition (30)) not only for phantom models. This is aggravated even
further by the fact that a gradient instability is also present in any
realistic model that interpolates between an initial radiation dominated epoch
and a late-time accelerated expansion regime; see Figure 1. Hence, the term
$G\Box\phi$ alone in the action (1) is not a viable option for modelling DE.

Finally, when considering the sum of $k$-essence and the braiding term the
situation is different. Following the same strategy we have used to analyse
the preceding marginal examples, we have recovered the well-known result that
the conditions for the avoidance of ghost and gradient instabilities can be
simultaneously fulfilled even when the $\phi$-fluid violates the NEC (see A).
Although the lack of a physical intuition for the auxiliary quantities
$\Omega_{J}$ and $\Omega_{J}^{B}$ did not allow us to restrict the possible
phenomenology of the theory as stringently as for the previous cases, note
that linearly stable KGB models have been already reported in the literature
(see for instance [23, 60, 61, 62] and references therein). This fact points
to the viability (from the point of view of the stability of linear scalar
perturbations) of modelling a realistic DE component with the scalar field
from the KGB theory extra space, at least in principle.

##  3 Beyond scalar perturbations

In section 2 we have discussed the absence of ghost and gradient instabilities
in the linearised scalar perturbations around a flat FLRW background. In the
approach followed, interactions between scalar, vector and tensor
perturbations were not present, since they decoupled at the linear level.
However, these interactions can appear when going beyond the linear regime and
may be important. In fact, the interaction between tensor perturbations
(essentially GWs) and DE fluctuations may induce ghost and/or gradient
instabilities in the scalar sector, as it was pointed out in reference [32].
In that case, the stability conditions discussed in the previous sections do
not guarantee the viability of the theory222See also references [33, 34] for a
discussion on the decay of GWs into scalar fluctuations when Lorentz
invariance is spontaneously broken., since the braiding term in the action can
destabilize the theory through a GWs-DE interaction. As a concrete example,
for cubic Galileon (which corresponds to $G(X)\propto X$) this effect is
detrimental when $|\alpha_{B}|\geq 10^{-2}$; that is to say, if the effects of
the braiding are sizeable [32].

In general, the result of reference [32] raises serious doubts on the
viability of any KGB theory (different from $k$-essence) aiming to describe
the currently observed abundance of DE in the Universe. Nevertheless, as the
authors of that reference commented (see also reference [63]), it is also
important to analyse the fate of the instabilities once they are originated.
The fate of the theory after the instability is reached is, however, uncertain
since it depends on an unknown UV-completion.

In the following, we review the arguments and notation presented in reference
[32]; that is section 3.1. By arguing that the absence of gradient
instabilities is more fundamental from the point of view of the evolution of
classical perturbations, we present in section 3.2 a necessary condition for
the interaction with GWs to induce only a ghost-like instability in the scalar
sector. In section 3.3 we explore whether this mechanism could be implemented
by some reasonable KGB model.

###  3.1 Interaction with gravitational waves: a review

In reference [32] the authors use the EFT approach to discuss the interaction
between DE perturbations and GWs. (See B for a review on the EFT approach to
DE.) For this effect to be sizeable one needs a cubic coupling $\pi\pi\gamma$
in the perturbed action, where $\gamma$ stands for tensor perturbations
(essentially GWs) and $\pi$ denotes scalar field fluctuations [32] (not to be
confused with the scalar field $\phi$ itself). In the covariant version of the
theory, the interaction $\pi\pi\gamma$ is related to the Cubic Horndeski
operator; i.e. the braiding term $G\Box\phi$ in action (1). Nevertheless, this
interaction vertex is also present in beyond Horndeski theories [32, 33, 34].

The interaction between DE perturbations and GWs is conveniently addressed in
the Newtonian gauge [32]

$\displaystyle g_{00}=-(1+2\Phi)\ \ \ \textup{and}\ \ \
g_{ij}=a^{2}(1-2\Psi)(e^{\gamma})_{ij},$ (38)

where $\Phi$ and $\Psi$ are the Newtonian potentials, and $\gamma_{ij}$ is
transverse and traceless. At leading order, the Lagrangian density describing
this interaction reads333This Lagrangian density follows from restoring time
diffeomorphism in the EFT action (B) for the KGB theory using the Stückelberg
procedure. That is, by performing in the action (B) the transformation $t\to
t+\pi(t,x^{i})$, where $\pi$ denotes the scalar field fluctuations. The
relevant transformations in this case are [64] (see also [32])
$\displaystyle\delta g^{00}\to$ $\displaystyle\,\delta
g^{00}+2g^{0\mu}\partial_{\mu}\pi+g^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi,$
(39) $\displaystyle\delta K\to$ $\displaystyle\,\delta
K-(1-\dot{\pi})g^{ij}\partial_{i}\partial_{j}\pi+\frac{2}{a^{2}}\partial_{i}\pi\partial^{i}\dot{\pi}+\dots\,,$
(40) where the former transformation is an exact relation to all order in
$\pi$, whereas the latter has been expanded up-to second order in
perturbations. After restoring full gauge invariance, the Newtonian gauge (38)
can be implemented. The Lagrangian density (3.1) is finally obtained after
solving (at linear order) for the Newtonian potentials in terms of $\pi$, and
canonically normalizing the scalar and tensor fluctuations as [32]
$\displaystyle\pi\to\,\sqrt{D}H\pi\ \ \ \ \rm{and}\ \ \ \
\gamma_{ij}\to\,\frac{1}{\sqrt{2}}\gamma_{ij},$ (41) respectively, being $D$
the ghost function (14).  [32]

$\displaystyle\mathcal{L}_{\pi}=$
$\displaystyle\frac{1}{2}\left[\dot{\pi}^{2}-{c_{s}^{2}}\,\partial_{i}\pi\partial^{i}\pi\right]-\frac{1}{\Lambda_{\rm
B}^{3}}\square\pi\partial_{\mu}\pi\partial^{\mu}\pi+\frac{\eta}{\Lambda_{\rm
B}^{3}}\ddot{\pi}\partial_{i}\pi\partial^{i}\pi$
$\displaystyle+\Gamma^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi-\frac{\Lambda_{\rm
B}^{3}}{2}\pi\Gamma_{\mu\nu}\Gamma^{\mu\nu},$ (42)

where the expansion of the Universe has been neglected and indices are
raised/lowered with the Minkowski metric. In addition, $c_{s}^{2}$ is given by
expression (15) and

$\displaystyle\Gamma_{\mu\nu}$
$\displaystyle\coloneqq\frac{\dot{\gamma}_{\mu\nu}}{\Lambda^{2}},$ (43)
$\displaystyle\Lambda^{2}$
$\displaystyle\coloneqq\frac{\sqrt{2}H^{2}D}{\bar{m}_{1}^{3}},$ (44)
$\displaystyle\Lambda_{\rm B}^{3}$
$\displaystyle\coloneqq-\frac{2H^{3}D^{\frac{3}{2}}}{\bar{m}_{1}^{3}},$ (45)

being $D$ the ghost factor (14). Moreover, the parameter $\eta$ is defined
through the relation [32]

$\displaystyle 4\bar{m}_{2}^{3}=-(1+\eta)\bar{m}_{1}^{3},$ (46)

where $\bar{m}_{1}^{3}$ and $\bar{m}_{2}^{3}$ are functions on the background
time, $t$, that appear in the EFT action (B). This parameter is useful for
measuring deviations with respect to cubic Galileon (i.e. $G(X)\propto X$) for
which $\eta$ is zero and, therefore, $4\bar{m}_{2}^{3}=-\bar{m}_{1}^{3}$.
(Further details can be found in B.) The covariant version of this parameter
can be obtained from the expression for the mass parameters $\bar{m}_{1}^{3}$
and $\bar{m}_{2}^{3}$ in terms of the function $G$ and its derivatives (see
B). Taking the expressions (116) and (117) into account, it is straightforward
to obtain that the covariant version of $\eta$ is nothing but the quantity we
already have defined in equation (37). That is $\eta=2XG_{XX}/G_{X}$.

The field equation for the scalar field perturbation, $\pi$, follows from
varying the action corresponding to the Lagrangian density (3.1). This reads
[32]

$\displaystyle\ddot{\pi}-c_{s}^{2}\nabla^{2}\pi+\frac{2}{\Lambda^{3}_{\rm
B}}\left[(\partial_{\mu}\partial_{\nu}\pi)(\partial^{\,\mu}\partial^{\nu}\pi)-\left(\Box\pi\right)^{2}\right]+\frac{1}{2}\Lambda_{\rm
B}^{3}\Gamma_{\mu\nu}\Gamma^{\,\mu\nu}$
$\displaystyle+\frac{2\eta}{\Lambda_{\rm
B}^{3}}\left[(\partial_{i}\dot{\pi})(\partial^{i}\dot{\pi})-\ddot{\pi}\nabla^{2}\pi\right]+2\Gamma^{\,\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi=0,$
(47)

where $\nabla^{2}\equiv\eta^{ij}\partial_{i}\partial_{j}$. Here we will
restrict our discussion to the case where $\pi$ propagates subluminally, that
is when $c_{s}^{2}<1$ [32]. (The luminal and superluminal cases are discussed
in reference [32].) In this scenario, the lightcone for $\pi$ is narrower than
the one for the GWs, since $c^{2}_{\rm GWs}=1$ [32, 34]. Consequently, the DE
fluctuations are not sensitive to the source of the GWs, provided we are far
enough from the source of emission [32, 34]. To address the DE-GWs
interaction, therefore, we can consider the following classical GW solution
with linear polarization “+” travelling in the $z$ direction [32, 34]

$\displaystyle\gamma_{ij}=h_{0}^{+}\sin\left[\omega(t-z)\right]\epsilon_{ij}^{+},$
(48)

being $\epsilon_{ij}^{+}=\text{diag}(1,-1,0)$ and $h_{0}^{+}$ the
dimensionless strain amplitude. Then, it follows from the definition (43) that

$\displaystyle\Gamma_{00}=\Gamma_{0i}=0\;,\qquad\Gamma_{ij}=\frac{\beta
c_{s}^{2}}{2}\cos\left[\omega(t-z)\right]\epsilon_{ij}^{+},$ (49)

being

$\displaystyle\beta\coloneqq\frac{2\omega
h_{0}^{+}}{c_{s}^{2}|\Lambda^{2}|}=\frac{\sqrt{2}\omega
h_{0}^{+}}{H^{2}c_{s}^{2}}\left|\frac{\bar{m}_{1}^{3}}{D}\right|,$ (50)

a parameter.

The field equation (3.1) simplifies significantly when written in the null
coordinates [32]

$\displaystyle u\coloneqq t-z\ \ \ \textup{and}\ \ \ v\coloneqq t+z.$ (51)

Since there is no intersection between the region where the source of emission
of the GWs is active and the past lightcone of $\pi$, there is a translational
invariance along the latter null coordinate [32]. This suggests the änstaz
$\pi=\pi(u)$ for the solutions of the field equation (3.1). By defining

$\displaystyle\varphi\coloneqq\frac{\pi}{\Lambda_{\rm B}^{3}},$ (52)

equation (3.1) reduces to [32]

$\displaystyle\varphi^{\prime\prime}(u)=-\frac{\Gamma_{\mu\nu}\Gamma^{\,\mu\nu}}{2(1-c_{s}^{2})}=-\frac{\beta^{2}c_{s}^{4}}{4(1-c_{s}^{2})}\cos^{2}(\omega
u),$ (53)

where prime denotes derivation w.r.t. the null coordinate $u$. In addition,
the relations (49) have been used in the second equality. From this result it
follows that $\varphi$ is always decelerating along the null direction $u$.

The stability of a general solution to the equation (53), namely $\hat{\pi}$,
can be addressed by analysing the tensor structure of the kinetic part related
to the quadratic action for small fluctuations around that solution. That is,
by considering $\pi=\hat{\pi}+\delta\pi$ and expanding the Lagrangian density
(3.1) around the background solution $\hat{\pi}$ coming from equation (53).
This leads to [32] (see also reference [65])

$\displaystyle\mathcal{L}^{(2)}_{\rm
kinetic}=Z^{\mu\nu}(\hat{\pi})\,\partial_{\mu}\delta\pi\,\partial_{\nu}\delta\pi,$
(54)

where the values of the kinetic matrix $Z^{\mu\nu}$ depend on the space-time
coordinates through the background solution $\hat{\pi}$. These are [32]

$\displaystyle Z^{00}$
$\displaystyle=\frac{1}{2}+(2+\eta)\varphi^{\prime\prime}(u)$
$\displaystyle=\frac{1}{2}\left[1-\frac{c_{s}^{4}\beta^{2}}{2(1-c_{s}^{2})}(2+\eta)\cos^{2}\left(\omega
u\right)\right],$ (55a) $\displaystyle Z^{11}$
$\displaystyle=-\frac{c_{s}^{2}}{2}+\Gamma^{11}+\eta\varphi^{\prime\prime}(u)$
$\displaystyle=-\frac{c_{s}^{2}}{2}\left[1-\beta\cos\left(\omega
u\right)+\frac{c_{s}^{2}\beta^{2}}{2(1-c_{s}^{2})}\eta\cos^{2}\left(\omega
u\right)\right],$ (55b) $\displaystyle Z^{22}$
$\displaystyle=-\frac{c_{s}^{2}}{2}+\Gamma^{22}+\eta\varphi^{\prime\prime}(u)$
$\displaystyle=-\frac{c_{s}^{2}}{2}\left[1+\beta\cos\left(\omega
u\right)+\frac{c_{s}^{2}\beta^{2}}{2(1-c_{s}^{2})}\eta\cos^{2}\left(\omega
u\right)\right],$ (55c) $\displaystyle Z^{33}$
$\displaystyle=-\frac{c_{s}^{2}}{2}+(2+\eta)\varphi^{\prime\prime}(u)$
$\displaystyle=-\frac{c_{s}^{2}}{2}\left[1+\frac{c_{s}^{2}\beta^{4}}{2(1-c_{s}^{2})}(2+\eta)\cos^{2}\left(\omega
u\right)\right],$ (55d) $\displaystyle Z^{03}$
$\displaystyle=Z^{30}=(2+\eta)\varphi^{\prime\prime}(u)$
$\displaystyle=-\frac{c_{s}^{4}\beta^{2}}{4(1-c_{s}^{2})}(2+\eta)\cos^{2}\left(\omega
u\right),$ (55e)

where $\varphi^{\prime\prime}$ is given by equation (53). (Recall that we are
considering here only the case of subluminal propagation for $\hat{\pi}$, i.e.
$c_{s}^{2}<1$. See reference [32] for a discussion when $c_{s}^{2}\geq 1$.)

It should be highlighted that for a general time-dependent kinetic term like
(54) there is no clear definition of stability [65]. However, in the limit
where the time and length scales considered are much shorter than the rate of
variation of $\hat{\pi}$, it is perfectly acceptable to analyse the stability
of the system as if the kinetic matrix $Z^{\mu\nu}$ were constant [65] (see
also reference [32]). Within this _local_ approximation to the stability of
the theory, the absence of a gradient instability will be determined by the
correct relative signs between the terms involving time and spatial
derivatives in the corresponding action (54). It is convenient, however, to
move to Fourier space and directly analyse the dispersion relation for a mode
with four-momentum $k^{\mu}=(\omega,k^{i})$ since the kinetic matrix
$Z^{\mu\nu}$ is not in diagonal form. For the Lagrangian density (54), the
dispersion relation reads [65]

$\displaystyle Z^{\mu\nu}k_{\mu}k_{\nu}=0.$ (56)

Moreover, multiplying this relation by $Z^{00}$, it is possible to re-
expressed it as [65]

$\displaystyle\left(Z^{0i}k_{i}-Z^{00}\omega\right)^{2}=\left(Z^{0i}Z^{0j}-Z^{00}Z^{ij}\right)k_{i}k_{j},$
(57)

when $Z^{00}$ is not trivial. We recall that indices have been raised/lowered
with the Minkowski metric. The absence of a gradient instability demands all
frequencies, $\omega$, to be real for any wave vector $k^{i}$. Otherwise,
exponentially growing modes will be present. It is straightforward to notice
from the above expression that this stability is guaranteed if the spatial
matrix $Z^{0i}Z^{0j}-Z^{00}Z^{ij}$ is positive defined [65]. Since this matrix
is diagonal for the $Z^{\mu\nu}$ found in (55), in our case of interest the
positive-definedness condition reduces to

$\displaystyle Z^{0i}Z^{0j}-Z^{00}Z^{ij}\geq 0,$ (58)

for all spatial directions.

In addition to the previous condition, the ghost-like instability is avoided
if [65]

$\displaystyle Z^{00}>0.$ (59)

It is interesting to note that a theory with $Z^{00}<0$ can be also made
stable provided it features superluminal excitations and that one can boost to
a frame with $Z^{00}>0$ [66]. However, we will not discuss this case here.
Also note that the conditions (58) and (59) reduce to the standard ones when
the matrix $Z^{\mu\nu}$ is in diagonal form [42].

When $\eta$ is always trivial (that is when considering the case of cubic
Galileon only, i.e. $G\propto X$), the ghost-free condition (59) leads to

$\displaystyle|\,\beta\,|<\frac{1}{c_{s}^{2}}\sqrt{1-c_{s}^{2}}\,,$ (60)

which follows from demanding the factor ahead of the trigonometric term in
equation (55a) to be less than one. Moreover, the expression
$(Z^{03})^{2}-Z^{00}Z^{33}$ is always positive when $\eta\equiv 0$. Therefore,
there is not a gradient instability in the $z$ direction. The background
solution $\hat{\pi}$, on the other hand, does not affect the entries $Z^{11}$
and $Z^{22}$. For $|\,\beta\,|>1$, these entries oscillate between positive
and negative values in a way that condition (58) could never be fulfilled.
Avoiding this oscillations in the $(x,y)$-plane demands

$\displaystyle|\,\beta\,|<1.$ (61)

For a non-relativistic speed of sound, $c_{s}$, condition (61) is clearly more
restrictive than the ghost-free condition (60). Therefore, by demanding
$|\,\beta\,|<1$ the stability conditions (58) and (59) are simultaneously
fulfilled in the non-relativistic regime.

The complete stability region (including relativistic $c_{s}$) is shown in
green in figure 2. As there can be seen, $|\,\beta\,|<1$ is a necessary
condition for the absence of ghost and gradient instabilities when $\eta$ is
trivial444Note, however, that this conditions is not sufficient to guarantee
the stability for values of $c_{s}$ close to the speed of light.. Owing to the
definition of $\beta$ in equation (50), this condition results in a tight
constraint over $\bar{m}_{1}^{3}$ [32]. Please note that there is a direct
relation between $\bar{m}_{1}^{3}$ and $\alpha_{\textrm{B}}$, which
essentially measures the amount of braiding in the theory. This is

$\displaystyle\bar{m}_{1}^{3}=\alpha_{\rm B},$ (62)

as follows from comparing equations (17) and (116). The observational bounds
for $\alpha_{\rm B}$ coming from the stability condition of $|\,\beta\,|<1$
are discussed in reference [32]. To that aim, the authors there have
considered a GWs-background as that generated by the event GW150914 [67] with
a chirp mass of $M_{c}=28M_{\odot}$ and $f=30$ Hz at a distance of 1 Mpc. They
found that the condition of $|\,\beta\,|<1$ directly translates to
$|\alpha_{\textrm{B}}|\leq 10^{-2}$ for that event [32], see definition of
$\beta$ in equation (50). Thus, they conclude that the braiding part should
not have any sizeable effect in order to avoid the destabilising interaction
with GWs when $\eta$ is zero [32]. Moreover, similar conclusions apply to the
case of $c_{s}^{2}=1$ [32]. It was also argued that the same results should
hold for the superluminal case, $c_{s}^{2}>1$, but no explicit calculations
were provided due to technical difficulties [32].

On the contrary, if $\eta\neq 0$, then, the oscillations in the spatial
entries $Z^{11}$ and $Z^{22}$ can be avoided even for $|\,\beta\,|>1$ provided
that $\eta$ is sufficiently large and positive (see appendix A of reference
[32]). Hence, this may provide a mechanism for having a stable theory with
$|\,\beta\,|>1$. If that is the case, then, the strong observational bounds
for $\alpha_{\rm B}$ discussed in reference [32] shall be revised. The larger
the value of $\eta$, however, the harder avoiding a ghost instability will be
since this quantity enters linearly in the factor ahead of the trigonometric
term in expression (55a). Therefore, one should carefully explore the region
of viability for the optimal effect when $\eta>0$. We address this discussion
in the next sections.

###  3.2 Loophole hunting

Condition (61) places a strong restriction to the maximum amount of braiding a
theory may have in order to avoid the destabilising interaction with GWs
described in reference [32]. As a result from that constraint, the braiding
part should not have any sizeable effects when data from GWs events are taken
into account [32]. However, this restrictions holds only for the special case
of $\eta\equiv 0$. In a more general situation, it would be possible to ease
the constraint in equation (61) provided that $\eta$ is positive enough to
ensure that all spatial direction in the kinetic matrix $Z^{\mu\nu}$ are
negative. This is obviously not the case for $G\propto X$ for which $\eta$ is
trivial [32], but may be possible for models beyond a linear function $G$.

Inspired by the fact that the ghost-free condition (60) is less restrictive
than the constraint in equation (61) for non-relativistic sound velocities
when $\eta$ is trivial, we explore here the possibility of a ghost-free and
gradient-free theory with $|\,\beta\,|>1$ when $\eta\neq 0$. If that
possibility comes to be true, then, the previously mentioned observational
bound of $|\alpha_{\rm B}|\leq 10^{-2}$ [32] should be revised. (Recall the
definition of $\beta$ in equation (50).) This may indicate that KGB theory
could have sizeable effects at cosmological scales after all.

In general, the ghost-free condition (59) implies

$\displaystyle|\,\beta\,|<\frac{1}{c_{s}^{2}}\sqrt{\frac{2(1-c_{s}^{2})}{2+\eta}},$
(63)

which follows from demanding the factor ahead of the trigonometric term in
equation (55a) to be less than one. The gradient-free condition (58) is
automatically satisfied in the $z$ direction since the combination

$\displaystyle(Z^{03})^{2}-Z^{00}Z^{33}=\frac{c_{s}^{2}}{4}\left[1+\frac{(2+\eta)\beta^{2}c_{s}^{2}}{2}\cos^{2}(\omega
u)\right],$ (64)

is always positive for the values of interest for $\eta$. Whereas in the
$xy$-plane the absence of a gradient instability implies

$\displaystyle Z^{00}\,Z^{11}<0,$ (65) $\displaystyle Z^{00}\,Z^{22}<0,$ (66)

as follows from comparing the gradient-free condition (58) with the
expressions for the $Z^{\mu\nu}$ found in equations (55). Contrary to the case
of $\eta\equiv 0$, the oscillations in the spatial entries $Z^{11}$ and
$Z^{22}$ can be avoided even for $|\,\beta\,|>1$ provided that $\eta$ is
sufficiently large and positive. Taking into account their explicit
expressions in equations (55b) and (55c), respectively, we found that for

$\displaystyle\eta>\frac{1-c_{s}^{2}}{2c_{s}^{2}},$ (67)

these terms are always negative. Therefore, a model satisfying the above
condition will be free from gradient instabilities in the $xy$-plane if and
only if $Z^{00}$ is positive. That is to say, if the ghost-free condition (63)
is also satisfied. In other words, condition (63) and the new constraint found
in equation (67) ensure the total stability of the system.

We should emphasise the fact that the above condition does not depend on the
specific properties of the GWs (local properties) but only on the value of
$c_{s}^{2}$, which depends on the KGB functions. It is also interesting to
note that if scalar perturbations propagate close to the speed of light, then,
condition (67) is easily satisfied if $\eta$ is just slightly positive.
However, this regime would strongly constraint $\beta$ to vanish, see equation
(63). For a model with a non-negligible amount of braiding, this suggests that
avoiding GWs-induced instabilities becomes more difficult if DE fluctuations
propagate with a relativistic speed of sound. Conversely, the ghost-free
condition (63) is compatible with $|\,\beta\,|\gg 1$ if $c_{s}^{2}$ is small.
Nevertheless, $\eta$ should diverge strongly than $1/c_{s}^{2}$ in order to
avoid the gradient instability in this regime. This behaviour is summarised in
figure 2. There we show different slices at constant $\eta$ of the stability
region where conditions (63) and (67) are simultaneously fulfilled. The green
region corresponds to the case of $\eta\equiv 0$ (compare with figure 2 in
reference [32]). As discussed before, the absolute value of $\beta$ is always
confined there to be less than one. Alternatively, this region becomes
significantly bigger for small speed of sound if $\eta$ is positive. In fact,
stability at $|\,\beta\,|\approx 10$ is possible for $\eta$ of the order of
twenty. Please note that this suppose an enlargement of the stability region
by a factor of ten w.r.t. the green zone. Moreover, this region can be
amplified even more at low $c_{s}$ for larger values of $\eta$. In the high
$c_{s}$ regime, however, the stability zone narrows compared to the case of
$\eta$ trivial.

In view of these results, we postulate that the would-be KGB candidate that
could ease (to some extent) the restrictions from the interaction with GWs
described in reference [32] must meet the following necessary (but not
sufficient) conditions. It should produce a non-relativistic speed of sound,
$c_{s}$, and a large value of $\eta$. The definition of both quantities in
terms of the KGB functions and its derivatives can be found in equations (15)
and (37), respectively. However, if for the selected model DE perturbations
propagate at relativistic speeds, then, smaller values of $\eta$ are
preferred. In any case, the contribution of the braiding part in that regime
should be negligible.

Figure 2: Slices at constant $\eta$ of the stability region where conditions
(63) and (67) are simultaneously satisfied. Within these regions, the
interaction between DE fluctuations and GWs described in reference [32] does
not induced a ghost or gradient instability. The green zone represent the
stability at $\eta=0$. This is the corresponding region for cubic Galileon,
i.e. $G\propto X$ (compare with figure 2 in reference [32]). The darker shaded
areas represent the stability at different non-zero values of $\eta$.

It should be also mentioned that, within the spirit of the discussion in past
sections, one may consider the possibility to relax the ghost-free condition
(63) and be concerned only with ensuring a gradient-free spectrum. Of course,
this will drastically reduce the restrictions on the theory. Although the
possibility of a ghost-mode in cosmology sets an interesting discussion, here
there is a technical difficulty in pursuing that direction. Given the
structure of $Z^{00}$ in equation (55a), if the condition (63) is violated
then the sign of $Z^{00}$ is not always negative (ghost mode excited) but
fluctuates. Moreover, this fluctuation will jeopardise the relations (65) and
(66) since the functional dependence of $Z^{00}$, $Z^{11}$ and $Z^{22}$ on
$\cos(\omega u)$ is different. Thus, if the ghost-free condition is violated,
then, gradient instabilities in the $xy$-plane will also (periodically) appear
within each fluctuation of the GWs.

###  3.3 No natural dark energy

A natural KGB candidate to test our hypothesis is that when $\eta$ is a
constant. From its expression in terms of $G$ and its derivatives (37), this
implies that

$\displaystyle G(X)=c_{G}X^{\frac{2+\eta}{2}},$ (68)

being $c_{G}$ a coupling constant. That is a power-law prescription for the
braiding function $G$. Consequently, the case of $\eta$ constant provide not
only a natural choice to test our proposal but a physically relevant model.
For this model, the resulting speed of sound squared should be constrained as

$\displaystyle\frac{1}{1+2\eta}<c_{s}^{2}<1,$ (69)

in order to fulfil condition (67). This results in a lower limit for
$c_{s}^{2}$ that should be satisfied (at least) during the cosmological
evolution where interactions with GWs are expected. Please note that this is
in agreement with the results presented in reference [68], where the
possibility of a vanishing speed of sound was shown to be disfavoured. It
should be also highlighted that, even though having a constant $\eta$ is a
very restrictive consideration, the results from exploring this case should
also hold (at least asymptotically) for any other model for which $\eta$
converges to a constant value. In other words, the physical conclusions from
exploring a model like (68) should also apply asymptotically to any KGB theory
for which the function $G$ converges, at some point, to a power-law function.

Figure 3: Numerical evolution for $\eta=0.4$ and $\alpha=0.6$ in expressions
(68) and (70), respectively. In addition, the coupling constant $c_{K}$ equals
minus unity, whereas $c_{G}$ ranges from $9\cdot 10^{-11}$ to $9.3\cdot
10^{-11}$. Left panel represents the evolution of the partial energy densities
for radiation, matter and the scalar field. Right panels show the evolution of
the Hubble rate at low redshift compared to direct measurements collected from
references [69, 70, 71, 72, 73] (see also Table II in reference [74]).

In order to explore whether this mechanism can be conducted by a reasonable
KGB model, we select a power-law function for the $k$-essence part as well.
That is, we also consider

$\displaystyle K(X)=c_{K}X^{\alpha},$ (70)

where $\alpha$ is a constant and $c_{K}$ equals to $+1$ or $-1$. For the KGB
model given by the functions (68) and (70), we have numerically explored the
parameter-space spanned by $\\{(\eta>0)\times\alpha\times c_{G}\\}$. As our
main purpose, at this stage, is to examine the stability of the KGB theory,
not to confront it with observational data, we have considered the initial
conditions $H_{0}=73.2$ km s-1 Mpc-1, $\Omega_{\phi 0}\approx 0.68$,
$\Omega_{m0}\approx 0.32$ and $\Omega_{r0}\approx 10^{-5}$ at face value for
the numerical integrations (in line with local direct measurements as those
reported in reference [56]).

Our numerical analysis has shown two main tendencies for these models. First,
we found that the ghost-free and gradient-free conditions discussed in section
2 (where scalar perturbations decouple from GWs at linear order) are not
fulfilled for all the parameter-space, but only in some regions. Second, for
the models that satisfy the ghost-free and gradient-free at linear order, we
have found that the condition (67) is always violated at some point during the
(past) evolution of the system. We summarise these findings with the two proxy
KGB candidates shown in figures 3 to 6. In the former, we consider $\eta=0.4$,
$\alpha=0.6$ and $c_{G}$ ranging form $9\cdot 10^{-11}$ to $9.3\cdot
10^{-11}$. As can be appreciate in figure 3, this simple KGB model has a
compelling behaviour on the background level. Depending on the value of the
coupling constant $c_{G}$, the fractional contribution of the scalar field to
the total energy of the universe picks at matter-radiation equality, rendering
this model as a potentially interesting Early Dark Energy (EDE) candidate (see
a review on EDE in, for instance, reference [75]). Moreover, the resulting
evolution of the Hubble rate at low redshift can be, in principle, made
compatible with the current data available from direct measurements555We
recall that we have not fitted the parameters of the model at hands with
observational data. Contrary, the previous statement comes from just comparing
the resulting $H(z)$ with what to be expected taking into account the current
data coming from direct measurements collected from references [69, 70, 71,
72, 73].. At the level of linear scalar perturbations, the ghost-free and
gradient-free conditions are always satisfied, even though the model at hands
shows a slight phantom behaviour at present time; see figure 4. Nevertheless,
as can be seen in figure 5, the condition (67) is never fulfilled for the
domain of the numerical integration (that is from $z_{i}=10^{6}$ to
$z_{f}=-0.999$). Since this bound is always violated, the signs of $Z^{11}$
and $Z^{22}$ will alternate between positive and negative values if
$|\,\beta\,|>1$. In that case, conditions (65) and (66) could not be always
satisfied during the period of a single oscillation of the GWs. As a result,
the interaction between DE fluctuations and GWs will induce (at least) a
gradient instability for certain (periodic) values of the argument
$\omega(t-z)$ if the amount of braiding is non-negligible. Hence, this model
cannot provide a working example for the mechanism we discussed in section
3.2.

A second example is shown in figure 6. This corresponds to $\eta=2$,
$\alpha=2$ and $c_{K}=-1$ in expressions (68) and (70). In this case, the
coupling constant $c_{G}$ takes values from $4.5\cdot 10^{-4}$ to $4.8\cdot
10^{-4}$. The background evolution of this proxy model is still compatible
with low redshift measurements, but shows some tensions with data above
$z\approx 1$ for some values of $c_{G}$; see left panel in figure 6. The right
panel shows the fulfillment of the condition (67). This condition is always
satisfied in the asymptotic past of the model (within the limits of the
integration domain). Unfortunately, it is consistently violated when the
scalar field becomes dominant, if not even before that moment. As a result,
interaction between DE fluctuations and GWs will induce (at least) a gradient
instability if the amount of braiding is non-negligible.

The models represented in figures 3 to 6 illustrate the general behaviour we
found for the functions (68) and (70). Even though there exist regions in the
parameter-space for which $D>0$ and $c_{s}^{2}\geq 0$ always, the condition
(67) is, in the best case scenario, fulfilled only for a finite time during
the (past) evolution of the model. Moreover, this condition is systematically
violated (at the latest) when the scalar field becomes dominant. Hence, a GWs-
induced instability, like that described in reference [32] if the amount of
braiding is non-negligible, seems to be inevitable for the models at hands.

However, it is important to emphasize that the numerical screening of the
$(\eta,\alpha)$-plane we have performed is not an analytic proof of the non-
viability of the KGB example given by the functions (68) and (70). Moreover, a
different choice for the $K$ function may also affect the results.
Nevertheless, given the naturalness of power-law functions in physics and the
lack of a positive conclusion in our numerical analysis, we suspect that
finding a KGB model that could conduct the mechanism we discussed in section
3.2 for avoiding a GWs-induced instability [32] would be rather difficult, if
not impossible.

Figure 4: Numerical evolution for $\eta=0.4$ and $\alpha=0.6$ in expressions
(68) and (70), respectively. In addition, the coupling constant $c_{K}$ equals
minus unity, whereas $c_{G}$ ranges from $9\cdot 10^{-11}$ to $9.3\cdot
10^{-11}$. Left panel represents the speed of sound squared, $c_{s}^{2}$, as
defined in equation (15). Right figure shows the evolution of the ghost
parameter $D$ defined in equation (14). As can be seen, linear order
perturbations (where scalar perturbations are decoupled form GWs) are always
stable even though the scalar field displays a phantom behaviour at present
epoch.  Figure 5: Numerical evolution for $\eta=0.4$ and $\alpha=0.6$ in
expressions (68) and (70). Taking $c_{K}=-1$, the curves represent the ratio
$(1-c_{s}^{2})/2c_{s}^{2}$ from equation (67) for different values of the
coupling constant $c_{G}$. The horizontal line depicts the upper-bound for
this ratio found in expression (67). Clearly the model never satisfies this
bound for the domain of the numerical integration and, therefore, the
interaction with GWs always introduce (at least) a gradient instability in the
scalar sector.

Figure 6: Numerical evolution for $\eta=2$ and $\alpha=2$ in expressions (68)
and (70), respectively. In addition, the coupling constant $c_{K}$ equals
minus unity, whereas $c_{G}$ ranges from $4.5\cdot 10^{-11}$ to $4.8\cdot
10^{-11}$. Left panel shows the evolution of the Hubble rate at low redshift
compared to direct measurements collected from references [69, 70, 71, 72, 73]
(see also Table II in reference [74]). Right panel represents the ratio
$(1-c_{s}^{2})/2c_{s}^{2}$ from equation (67). The horizontal line there
depicts the upper-bound this ratio should satisfy to avoid a GWs-induced
gradient instability; see inequality (67).

##  4 What is left?

The braiding term $G\Box\phi$ in the scalar field action (1) has been shown to
interact with GWs in such a way that it produces a ghost and/or a gradient
instability in the scalar perturbations if the amount of braiding present at
cosmological scales is non-negligible [32]. Motivated by the fact that the
stability analysis in reference [32] was mainly performed for the special case
of cubic Galileon, i.e. $G\propto X$, in this paper we have proposed a
theoretical mechanism to avoid these GW-induced instabilities for KGB models
beyond cubic Galileon. This mechanism relies on having a large and positive
$\eta$ (quantity that measure deviations w.r.t. a linear function $G$) for
expanding the stability region to $|\,\beta\,|\gg 1$ at low values of the
speed of sound, as shown in figure 2. It should be mentioned, however, that
the parameter $\beta$ is not only proportional to the amount of braiding in
the theory (through $\bar{m}_{1}^{3}$ or, equivalently, $\alpha_{\rm B}$) but
to the strain amplitude of the GWs as well, see equation (50). If
observational data of GWs events with higher and higher strain amplitudes is
considered, then the resulting $\bar{m}_{1}^{3}$ (or $\alpha_{\rm B}$) will
still be constrained to be negligible, even though our mechanism could extend
the stability region to larger values of $\beta$. In that scenario, the
conclusion in reference [32] about the non-viability of the braiding at
cosmological scales will be only marginally affected by the discussion we have
proposed here. Nevertheless, before finally concluding that the KGB theory
should not have any sizeable effects, the fate of these instabilities, once
originated, should be addressed. This lays beyond the scope of the present
work.

We have also explored whether some reasonable KGB candidate could conducted
the theoretical mechanism proposed here for evading GWs-induced instabilities.
Unfortunately, using numerical simulations with a power-law prescription for
the functions $K$ and $G$ we have not been able to find a working example for
this effect. As we commented in section 3.3, the power-law choice for $G$
provides a physically relevant case for our mechanism to work. Therefore, we
consider the lack of a working example with power-law functions as a solid
hint towards the non-viability of a cosmological braiding like the one present
in the shift-symmetric KGB theories.

Conversely, the $k$-essence part in the KGB action (1) trivially evades the
constraints coming from references [32, 33, 34]. Moreover, this scalar field
theory can feature ghost-free and gradient-free scalar perturbations if the
NEC is satisfied for the DE sector. As we have discussed, however, the ghost-
free and gradient-free conditions for $k$-essence are anti-correlated when the
NEC is violated. Therefore, if we were to describe a DE component that
violates the NEC (a possibility that is observationally viable [29]) with a
single scalar field within the $k$-essence theory, then we will inevitably
face a ghost or a gradient instability. Whether the instability will be ghost-
or gradient-like depends, ultimately, on the function $K$. It is also
important to note that $k$-essence cannot produce a phantom crossing [27, 28].
Hence, the scalar field will always be phantom or non-phantom, with the
stability issues this may entail in the former case. This points towards the
impossibility of solving simultaneously the $H_{0}$ and $S_{8}$ tensions with
late-time modifications to $\Lambda$CDM à la $k$-essence should the $S_{8}$
tension be still present in future observations, as this resolution would
demand a phantom crossing [30, 31]. With this potential limitation in mind, an
interesting extension of the $k$-essence scenario is that when more than one
scalar field is present [76, 77]. In that case, the effective DE fluid can
cross the phantom divide [78] even though each scalar field, namely
$\phi_{i}$, is restricted to the phantom or non-phantom regime in the same way
as described in section 2.1.

It should be noted that the discussion presented in this work will be only
partially affected if we allow the shift-symmetry to be (weakly) broken at
some regimes. This would introduce an explicit dependence on the scalar field
$\phi$ in the functions $K$ and $G$. The destabilising interaction between GWs
and the braiding term will still occur even for $G=G(\phi,X)$ [32]. However,
it would be interesting to explore whether the mechanism proposed in section
3.2 will have a better performance in that case. If not, then, the braiding
term should not have any sizeable effect and the surviving theory will be
effectively that of $k$-essence. For this non-shift-symmetric $k$-essence
theory, no additional terms will appear in the formulas for the scalar field
energy density and pressure, or in the equations for $c_{s}^{2}$ and $D$; see
the corresponding expressions in, for instance, reference [10]. Thus, our
compulsions about the linear stability of the theory shall remain the same as
for the shift-symmetric $k$-essence case. In addition to the previous
considerations, the $k$-essence scalar field cannot produce a phantom crossing
even if $K=K(\phi,X)$; see reference [27, 28]. Therefore, the conclusions
presented in section 2.1 shall remain unchanged.

Along this line of thought, another possible extension of the $k$-essence
theory is that when a non-minimal coupling to gravity is allowed for the
scalar field. Assuming that the external matter field are minimally coupled to
the metric $g_{\mu\nu}$, the gravitational part of the total action would read

$\displaystyle S_{g}=\int d^{4}x\sqrt{-g}\left[f(\phi)R+K(\phi,X)\right],$
(71)

where the non-minimal coupling function $f$ can only depend on the scalar
field, but not on its kinetic term $X$. This is due to strong restrictions on
the speed of propagation of GWs; see reference [16]. Please note that this
scalar field theory would also trivially evade the constraints from [32].
Nevertheless, the non-minimal coupling may introduce a fifth-force at local
scales. Therefore, some mechanism should be invoked to screen this force on
astrophysical scales; see, for instance, references [79, 80, 81] and
references therein.

Furthermore, within the spirit of analysing fundamental symmetries in order to
address DE, we should also briefly mention the possibility of considering a
scalar field with an action that is invariant just under transverse
diffeomorphisms [74, 82, 83, 84]. In this case, one can describe phenomenology
of interest for the dark sector even with a shift-symmetric field with a
canonical kinetic term [74, 84]. On the other hand, one can consider more
general theories invariant only under transverse diffeomorphisms [85, 86],
being the most prominent example that of Unimodular Gravity [87] (see also
[88, 89, 90, 91, 92]). Finally, as it is well-known, more general alternative
theories of gravity goes beyond the consideration of a scalar field to
describe the dark sector [93].

Acknowledgments

The research of T.B.V. and P.M.M. is supported by the project
PID2022-138263NB-I00 funded by MICIN/AEI/10.13039/501100011033 and by FEDER,
UE. T.B.V. also acknowledge financial support from Universidad Complutense de
Madrid and Banco de Santander through Grant No. CT63/19-CT64/19. The work of
M.B.L. is supported by the Basque Foundation of Science Ikerbasque. Her work
has been also financed by the Spanish project PID2020-114035GB-100
(MINECO/AEI/FEDER, UE). She would like to acknowledge the financial support
from the Basque government Grant No. IT1628-22 (Spain).

##  Appendix A Stability of linear scalar perturbation

In this appendix we discuss the possibility of satisfying simultaneously the
ghost and gradient-free conditions for linear scalar perturbations given by
the inequalities (33) and (36), respectively. In order to make our notation
more compact, it will be useful to introduce the two following reference
scales

$\displaystyle\Omega^{\star}$
$\displaystyle\coloneqq\frac{(3+\Omega_{r}-3\Omega_{\phi})\Omega_{J}^{B}}{3(2-\Omega_{J}^{B})},$
(72) $\displaystyle\bar{\Omega}$
$\displaystyle\coloneqq\frac{2+\eta}{3+\eta}(1+w_{\phi})\Omega_{\phi}-\frac{\Omega_{J}^{B}(2-\Omega_{J}^{B})}{6(3+\eta)},$
(73)

where $\Omega^{\star}$ is well-defined only for $\Omega_{J}^{B}\neq 2$. The
case of $\Omega_{J}^{B}=2$ can be directly addressed using the expressions
(33) and (36). We consider here the parameter $\eta$ to be positive since that
is the case proposed in section 3.2 for avoiding a GWs-induced gradient
instability. Also note that the quantity $\Omega_{r}+3-3\Omega_{\phi}$ is
always positive666This is because
$\Omega_{r}+3-3\Omega_{\phi}=4\Omega_{r}+3\Omega_{m}$, which is clearly
positive since the energy densities of radiation and matter are non-negative..
Hence, $\Omega^{\star}$ is positive if $\Omega_{J}^{B}\in(0,2)$, whereas it is
negative for $\Omega_{J}^{B}<0$ and $\Omega_{J}^{B}>2$.

Combining the ghost and the gradient-free conditions lead to the following
scenarios in which both conditions are satisfied for $\eta$ non-negative:

#### Scenario 1

($\Omega_{J}^{B}<0,\,\Omega_{J}>\max\\{(1+w_{\phi})\Omega_{\phi},\,\Omega^{\star},\,\bar{\Omega}\\}$):
Both the numerator and the denominator in equation (33) are positive. The
reference scale $\Omega^{\star}$ is negative. Phantom behaviour is allowed.

#### Scenario 2

($0<\Omega_{J}^{B}<2,\,\max\\{(1+w_{\phi})\Omega_{\phi},\,\bar{\Omega}\\}<\Omega_{J}<\Omega^{\star}$):
The numerator and the denominator in equation (33) are positive. The reference
scale $\Omega^{\star}$ is also positive. Clearly,
$\max\\{(1+w_{\phi})\Omega_{\phi},\,\bar{\Omega}\\}<\Omega^{\star}$ should
hold for consistency. Phantom behaviour is allowed.

#### Scenario 3

($0<\Omega_{J}^{B}<2,\,\max\\{\Omega^{\star},\bar{\Omega}\\}<\Omega_{J}<(1+w_{\phi})\Omega_{\phi}$):
The numerator and the denominator in equation (33) are negative. The reference
scale $\Omega^{\star}$ is positive and, therefore, so it is $\Omega_{J}$.
Hence, from the upper bound for $\Omega_{J}$ it follows that the scalar field
cannot be phantom.

#### Scenario 4

($\Omega_{J}^{B}=2,\Omega_{J}>\max\\{(1+w_{\phi})\Omega_{\phi},\,\bar{\Omega}\\}$):
The numerator and the denominator in equation (33) are positive. Note that
$\bar{\Omega}>(1+w_{\phi})\Omega_{\phi}$ if the scalar field is phantom-like
and, therefore, $\Omega_{J}$ should be greater that $\bar{\Omega}$. In the
non-phantom regime, the lower bound for $\Omega_{J}$ is given by
$(1+w_{\phi})\Omega_{\phi}$.

#### Scenario 5

($\Omega_{J}^{B}>2,\,\Omega_{J}>\max\\{(1+w_{\phi})\Omega_{\phi},\,\Omega^{\star},\,\bar{\Omega}\\}$):
The numerator and the denominator in equation (33) are positive. The reference
scale $\Omega^{\star}$ is negative. Phantom behaviour is allowed.

Even though the physical intuition behind each of these scenarios is not so
transparent as for the marginal models studied in sections 2.1 and 2.2, their
very existence is a solid proof that it is possible to have a ghost-free and
gradient-free scalar perturbations at linear level (recall we consider here
only the case of $\eta$ non-negative) in the shift-symmetric KGB theory even
when $w_{\phi}<-1$.

##  Appendix B Quick guide to the EFT of dark energy

The framework of EFT was first applied to DE in reference [35] (see also [94,
95] for applications to inflationary models) and further developed in
references [96, 97, 98, 99] (see also reference [44] for a review). This
approach considers the most general form for the gravitational action
(including cosmological perturbations up-to an arbitrary order) built-up only
on symmetry arguments. Originally, this framework was based on two assumptions
[96]: $i)$ the validity of the weak equivalence principle and the existence of
a Jordan metric $g_{\mu\nu}$ universally coupled to matter fields; $ii)$ the
existence of a unitary gauge compatible with the residual symmetries of
unbroken spatial diffeomorphisms.

The last of these assumptions advocates for the presence of a scalar field
$\phi$ in the DE sector. This is so because the presence of a homogeneous
scalar field $\phi(t)$ in a FLRW background defines a preferred time slicing
of space-time. These are the hypersurfaces with $\phi$ constant and,
therefore, $\delta\phi\equiv 0$ on them. So, for this choice (unitary gauge)
only metric degrees of freedom are explicitly displayed in the action. The
scalar field perturbation can be reintroduced explicitly in the theory via the
Stüeckelberg trick. That is, by performing infinitesimal time diffeomorphism
$t\to t+\pi(t,x)$ being $\pi$ the scalar field fluctuations. However, in this
approach $\pi$ does not represent the original scalar field $\phi$ in the DE
sector, but the perturbations encoding the scalar degree of freedom in the
theory.

In order to construct the most general expression for the action satisfying
the previous ansätze, note that hypersurfaces with constant $\phi$ can be
defined as those orthogonal to the unit four-vector

$\displaystyle n_{\mu}\coloneqq-\epsilon\frac{\partial_{\mu}\phi}{\sqrt{2X}},$
(74)

provided that $\partial_{\mu}\phi$ is time-like, i.e. $X>0$, where
$\epsilon\coloneqq\textup{sgn}(\dot{\phi})=\pm 1$ is to ensure the four-vector
is future-oriented. (Here we are not going to consider oscillating solutions
where the background field velocity, $\dot{\phi}$, crossed zero since this
case may be problematic in standard perturbation theory [40].) For a
homogeneous scalar field, that is $\phi=\phi(t)$, this four-vector reduces to

$\displaystyle n_{\mu}=-\frac{\delta_{\mu}^{0}}{\sqrt{-g^{00}}}.$ (75)

This slicing induces the spatial metric

$\displaystyle h_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu},$ (76)

where $n_{\mu}n^{\mu}=-1$ and $n^{\mu}h_{\mu\nu}=0$.

Note that invariance under time-translations is spontaneously broken (in the
sense discussed in reference [99]) since the scalar field signals out a
preferred time. The terms allowed in the EFT action are, therefore, those
invariant under the residual symmetries of the unbroken spatial
diffeomorphisms, such as the contravariant time-time component of the Jordan
metric $g^{00}$. The extrinsic curvature of constant time hypersurfaces is
also allowed to appear. This is defined as [100]

$\displaystyle\mathcal{K}_{\mu\nu}\coloneqq h_{\mu}^{\
\alpha}\nabla_{\alpha}n_{\nu},$ (77)

and represents the spatial projection of the covariant derivative of
$n_{\nu}$. (Note that the extrinsic curvature is sometimes defined with a
different sign convention, see, for instance, references [101, 102].) In
addition, its trace reads

$\displaystyle\mathcal{K}=\nabla_{\mu}n^{\mu}.$ (78)

It will be useful for latter calculations to note that

$\displaystyle\int\textup{d}^{4}x\sqrt{-g}\,l(t)\mathcal{K}=\int\textup{d}^{4}x\sqrt{-g}\,l(t)\nabla_{\mu}n^{\mu}$
$\displaystyle=-\int\textup{d}^{4}x\sqrt{-g}\,n^{\mu}\partial_{\mu}l=-\int\textup{d}^{4}x\sqrt{-g}\sqrt{-g^{00}}\,\dot{l}\,,$
(79)

up-to boundary terms and for any function $l$ on time.

The Ricci scalar $R$, any curvature invariants, and contractions of tensors
with $g_{\mu\nu}$, $n_{\mu}$ and the covariant derivative $\nabla_{\mu}$ are
also invariant under spatial diffeomorphisms. Therefore, the most general EFT
action can be expressed as [94]

$\displaystyle
S_{g}=\int\textup{d}^{4}x\sqrt{-g}\mathcal{L}\left(R_{\mu\nu\rho\sigma},\mathcal{K}_{\mu\nu},g^{00},t\right),$
(80)

where time is also allowed to appear explicitly. For addressing the
cosmological perturbation, however, it is more convenient to re-write the
above action into the part contributing to the background level and that
enclosing the perturbations around a flat FLRW metric (at any order). This is
[96] (sea also [94, 98])

$\displaystyle
S_{g}=\int\textup{d}^{4}x\sqrt{-g}\left[\frac{M^{2}_{*}}{2}f(t)R-\Lambda(t)-c(t)g^{00}\right]+S^{(2)}_{DE},$
(81)

where $f$, $\Lambda$ and $c$ are functions of the cosmic time $t$. Moreover,
the latter two background quantities read

$\displaystyle c$
$\displaystyle=\frac{M^{2}_{*}}{2}(-\ddot{f}+H\dot{f})+\frac{1}{2}(\rho_{\textup{DE}}+p_{\textup{DE}}),$
(82) $\displaystyle\Lambda$
$\displaystyle=\frac{M^{2}_{*}}{2}(\ddot{f}+5H\dot{f})+\frac{1}{2}(\rho_{\textup{DE}}-p_{\textup{DE}}),$
(83)

being $p_{\textup{DE}}$ and $\rho_{\textup{DE}}$ the pressure and energy
densities of the DE fluid. In addition, $M^{2}_{*}$ is the Planck mass. Hereon
we will take $M^{2}_{*}=1$ since we have adopted the geometric unit system in
this paper. Conversely, $S^{(2)}_{DE}$ contains all terms that start at
quadratic order in perturbations and, therefore, they do not affect the
background. (Note, however, that the background quantities do affect all
perturbation levels.) This part can be expressed as follows [96]

$\displaystyle S^{(2)}_{DE}=\,\frac{1}{2}\int\textup{d}^{4}x$
$\displaystyle\sqrt{-g}\bigg{[}M_{2}^{4}(\delta
g^{00})^{2}-\bar{m}_{1}^{3}\,\delta
g^{00}\delta\mathcal{K}-\bar{M}_{2}^{2}\,\delta\mathcal{K}^{2}$
$\displaystyle-\bar{M}_{3}^{2}\,\delta\mathcal{K}_{\mu}^{\
\nu}\delta\mathcal{K}_{\
\nu}^{\mu}+m_{2}^{2}h^{\mu\nu}\partial_{\mu}g^{00}\partial_{\nu}g^{00}$
$\displaystyle+\lambda_{1}\delta R^{2}+\lambda_{2}\delta R_{\mu\nu}\delta
R^{\mu\nu}+\mu_{1}^{2}\delta g^{00}\delta R$
$\displaystyle+\gamma_{1}C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}+\gamma_{2}\epsilon^{\mu\nu\rho\sigma}C_{\mu\nu}^{\
\ \ \kappa\lambda}C_{\rho\sigma\kappa\lambda}$
$\displaystyle+\frac{M_{3}^{4}}{3}(\delta g^{00})^{3}-\bar{m}_{2}^{3}\,(\delta
g^{00})^{2}\delta\mathcal{K}+\dots\bigg{]},$ (84)

where $M^{4}_{i},\,\bar{M}^{2}_{i},\,m^{2}_{i},\,\bar{m}^{3}_{i}$ and
$\mu_{i}$ are mass parameters whereas $\lambda_{i}$ and $\gamma_{i}$ are
dimensionless quantities. Generally, these parameters are allowed to depend on
the time coordinate $t$. In addition, $\delta g^{00}\coloneqq g^{00}+1$ is the
perturbation of the metric, $\delta\mathcal{K}^{\mu}_{\,\nu}$ is the
perturbation of the extrinsic curvature, and $\delta\mathcal{K}$ the
perturbation of its trace. Likewise, $\delta R_{\mu\nu}$ is the perturbation
of the Ricci tensor and $\delta R$ the perturbation of its trace, whereas
$C_{\mu\nu\rho\sigma}$ represents the Weyl tensor. The action (81) describes
virtually all DE models encompassing a single scalar degree of freedom.

For the case of the KGB theories given by action (1), the background
quantities in (81) read [96]

$\displaystyle f=$ $\displaystyle\,1,$ (85) $\displaystyle c=$
$\displaystyle\,\frac{1}{2}(\rho_{\phi}+p_{\phi}),$ (86)
$\displaystyle\Lambda=$ $\displaystyle\,\frac{1}{2}(\rho_{\phi}-p_{\phi}),$
(87)

with $\rho_{\phi}$ and $p_{\phi}$ given by equations (9) and (10),
respectively. Whereas the action (B) for the perturbations reduces to [96]

$\displaystyle S^{(2)}_{DE}=\,\frac{1}{2}\int$
$\displaystyle\textup{d}^{4}x\sqrt{-g}\bigg{[}M_{2}^{4}(\delta
g^{00})^{2}-\bar{m}_{1}^{3}\,\delta g^{00}\delta\mathcal{K}$
$\displaystyle+\frac{1}{3}M_{3}^{4}(\delta
g^{00})^{3}-\bar{m}_{2}^{3}\,(\delta
g^{00})^{2}\delta\mathcal{K}+\dots\bigg{]}.$ (88)

That is, only the mass parameters $M_{n}^{4}$ and $\bar{m}_{n-1}^{3}$ (for
$n\geq 2$) are present for the KGB theory (see also, for instance, reference
[44]). Recall that these parameter are, in general, functions on the time
coordinate $t$. Moreover, note that $M_{2}^{4}$ is connected with the
kineticity term $\alpha_{K}$, whereas $\bar{m}_{1}^{3}$ is equal to the
braiding term $\alpha_{B}$ up-to sign conventions; see the discussion in B.3
(see also table 2 in reference [40]). It is also important to highlight that
the operator introducing the kinetic mixing with gravity at leading order is
$\bar{m}_{1}^{3}$. This can be understood in an intuitive way by noting that
the perturbation $\delta g^{00}$ introduces a term proportional to $\dot{\pi}$
at linear order after the Stückelberg procedure, see equation (39). The
extrinsic curvature, on the other hand, contains time derivative of the
spatial metric, $h_{ij}$, when time diffeomorphism is fully restored [100]
(see also, for instance, references [96, 103, 104]). Schematically, the
product $\delta g^{00}\delta\mathcal{K}$, therefore, introduces a term
proportional to $\dot{\pi}\dot{h}$ (at second order) into the Lagrangian. In
the Newtonian gauge (38), this contribution is proportional to
$\dot{\pi}\dot{\Psi}$; see reference [96]. For the same reason, the operators
$\bar{m}_{n-1}^{3}$ (for $n\geq 2$) will also be responsible for introducing
the kinetic mixing between scalar perturbations and the GWs at higher orders.

In the following part of the appendix, we compute the expression of these mass
parameters in terms of the KGB functions $K$ and $G$, and their derivatives.
These calculations will be done without invoking a shift-symmetry and,
therefore, the resulting expression are valid for the most general KGB
scenario.

###  B.1 Mass parameters for k-essence

As a warm-up, we consider first the $k$-essence theory. In the most general
scenario, the $k$-essence function depends on both the scalar field and its
kinetic term; i.e. $K=K(\phi,X)$. (We remind the reader that we will not
invoke a shift-symmetry here; thus, the results obtained in this appendix are
valid for the most general case.) When perturbing this function, one may
naively consider that perturbations around the background values of both
$\phi$ and $X$ should be taken into account. However, in the unitary gauge
(defined by $\delta\phi\equiv 0$) only the perturbations of the kinetic term
remains. Let the perturbed kinetic term be defined as

$\displaystyle\mathcal{X}\coloneqq X+\delta X,$ (89)

where $X$ is the background value and $\delta X$ the perturbation. In the
unitary gauge, the perturbed kinetic term $\mathcal{X}$ is related to the
perturbed metric through

$\displaystyle\mathcal{X}=$
$\displaystyle-\frac{1}{2}\dot{\phi}^{2}g^{00}=-Xg^{00},$ (90)

being $g^{00}$ the perturbed time-time component of the metric. Hence,

$\displaystyle\delta X=$ $\displaystyle-X\delta g^{00},$ (91)

in the unitary gauge. Consequently, the perturbed $k$-essence function becomes

$\displaystyle K(\phi,\mathcal{X})\xrightarrow{\text{unitary
gauge}}K(t,\,g^{00}),$ (92)

where we have assumed that $\phi$ and its kinetic term are functions of time
only at the FLRW background. Note that the scalar field does not appear
explicitly in the unitary gauge: it is “eaten” in the metric degrees of
freedom [44, 96]. However, in order to obtain the expressions for the mass
parameters in the EFT language in terms of $K(\phi,X)$ and its derivatives, it
will be useful to maintain the usual covariant notation for the function $K$
and its arguments instead of $K(t,\,g^{00})$ when there is no risk for
confusion. Please note that this is an abuse of notation since neither $\phi$
nor $X$ can appear explicitly in this gauge. They are simply functions on the
time coordinate.

In this notation, the $k$-essence function can be expanded around its
background value as

$\displaystyle
K(\phi,\mathcal{X})=K(\phi,X)+\sum_{n=1}^{\infty}\frac{1}{n!}\left.\diffp[n]{K(\phi,\mathcal{X})}{\mathcal{X}}\right|_{\delta
X=0}(\delta X)^{n},$ (93)

where subscript “$\delta X=0$” denotes evaluation of the corresponding
quantity on the background level. For the sake of the compactness of the
notation, we will avoid writing explicitly the arguments of the function $K$
and its derivatives appearing in the r.h.s. of the expansion, where it should
be kept in mind that these expressions are always evaluated on the background.
We will also omit the subscript “$\delta X=0$” in the following and express
the coefficient in the expansion simply as $\partial^{n}K/\partial X^{n}$
where there is no risk for confusion. Accordingly, the above expansion can be
rewritten as

$\displaystyle
K(\phi,\mathcal{X})=K-XK_{X}-XK_{X}g^{00}+\sum_{n=2}^{\infty}\frac{(-X)^{n}}{n!}\diffp[n]{K}{X}(\delta
g^{00})^{n},$ (94)

where we have substituted $\delta X=\mathcal{X}-X$ in the first order in the
expansion, and used equations (90) and (91) to explicitly introduce $g^{00}$
and $\delta g^{00}$, respectively. Comparing the above expression with
equations (81) and (B), it is straightforward to readout the contribution of
the $k$-essence function to the EFT parameters. These are777Please, note that
the authors in reference [96] used a different convention for the definition
of the kinetic term. This is $X\coloneqq
g^{\mu\nu}\nabla_{\mu}\phi\,\nabla_{\nu}\phi$ [96], in contrast with our
definition for
$X\coloneqq-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\,\nabla_{\nu}\phi$, where
the convention used for the signature of the metric tensor is in both cases
the same. Also note that it is always possible to redefine the scalar field in
a way that $\phi(t)=\sqrt{2}t$ and, therefore, the kinetic term would simply
read $X=1$ when evaluated on the background [96]. However, we will not
consider such redefinition here in order to explicitly keep the kinetic term,
$X$, in the final expressions for the EFT parameters. [96]

$\displaystyle c=$ $\displaystyle\,XK_{X},$ (95) $\displaystyle\Lambda=$
$\displaystyle\,XK_{X}-K,$ (96) $\displaystyle M_{n}^{4}=$
$\displaystyle\,(-X)^{n}\diffp[n]{K}{X},$ (97)

for $n\geq 2$. Therefore, the $k$-essence function $K$ contributes only to the
mass parameters $M^{4}_{n}$ at perturbation level. So, there is no kinetic
mixing between scalar and tensor perturbations.

###  B.2 Mass parameters for $G\Box\phi$

From the definition of the d’Alembertian operator and the slicing vector
$n_{\mu}$ (see equation (74)), it is straightforward to check that

$\displaystyle\Box\phi=-\epsilon\left(\sqrt{2X}\mathcal{K}+\frac{n^{\mu}}{\sqrt{2X}}\partial_{\mu}X\right).$
(98)

To compute the EFT parameters for the braiding part of the KGB action (1), it
will be useful to consider the following expansion in polynomials of $\delta
X$ [96]

$\displaystyle
G(\phi,\mathcal{X})=\epsilon\sqrt{2\mathcal{X}}\sum_{m=0}^{\infty}l_{m}\left(\delta
X\right)^{m},$ (99)

where $\mathcal{X}=X+\delta X$ is the perturbed kinetic term and

$\displaystyle
l_{m}=\frac{1}{m!}\diffp[m]{}{\mathcal{X}}\left.\left(\frac{G(\phi,\mathcal{X})}{\epsilon\sqrt{2\mathcal{X}}}\right)\right|_{\delta
X=0},$ (100)

are the coefficient in the expansion evaluated on the background. Recall that
we are not imposing a shift-symmetry in this section. Consequently, the above
coefficients depend on the background values of both the scalar field and its
kinetic term. That is $l_{m}=l_{m}(\phi,X)$. Let us emphasise again that this
is an abuse of notation, since neither $\phi$ nor its kinetic term can
explicitly appear in the unitary gauge. As mentioned in the previous section,
background quantities as $\phi$ and $X$ are functions of time only, whereas
scalar perturbations are encoded in time-time component of the metric. Hence,
the perturbed braiding function, $G$, and the expansion coefficients, $l_{m}$,
should read $G(t,g^{00})$ and $l_{m}(t)$ in this gauge choice, respectively.
Nevertheless, this abuse of notation will be convenient, where there is no
risk of confusion, for obtaining the expressions of the EFT parameters in
terms of $G(\phi,X)$ and its derivatives.

Perturbing equation (98), and taking into account equations (B) and (99), we
obtain after few integrations by parts that

$\displaystyle\int\textup{d}^{4}x\sqrt{-g}\,\left(-G\Box\phi\right)=\int\textup{d}^{4}x\sqrt{-g}\Bigg{\\{}-\left(\dot{X}l_{0}+2X\dot{l}_{0}\right)\sqrt{-g^{00}}$
$\displaystyle+\sum_{n=1}^{\infty}\left(-X\right)^{n}\Bigg{[}\left(2Xl_{n}+\frac{2n-1}{n}l_{n-1}\right)\mathcal{K}$
$\displaystyle+\left(\dot{X}l_{n}-\frac{\dot{l}_{n-1}}{n}\right)\sqrt{-g^{00}}\Bigg{]}\left(\delta
g^{00}\right)^{n}\Bigg{\\}},$ (101)

up-to boundary terms; cf. with equation (86) in reference7 [96]. The time
derivative of the coefficients $l_{n}$ can be simplified by noting that

$\displaystyle\dot{l}_{n}=\dot{\phi}\diffp{l_{n}}{\phi}+(n+1)\dot{X}l_{n+1},$
(102)

where it should be kept in mind that these coefficients are evaluated on the
background. Since $\mathcal{K}=3H+\delta\mathcal{K}$, and expanding
$\sqrt{-g^{00}}$ in powers of $\delta g^{00}$ as

$\displaystyle\sqrt{-g^{00}}=\sqrt{1-\delta
g^{00}}=1-\sum_{n=1}^{\infty}\lambda_{n}(\delta g^{00})^{n},$ (103)

where

$\displaystyle\lambda_{n}\coloneqq\frac{(2n)!}{4^{n}(n!)^{2}(2n-1)},$ (104)

we can readout the contribution of $-G\Box\phi$ to all terms in the effective
action (B). (Note that $G\Box\phi$ enters with a minus in the action (1), in
contrast with the convention used in reference [96].) These are

$\displaystyle c=$
$\displaystyle\,XG_{X}\left(3H\dot{\phi}-\ddot{\phi}\right)-2XG_{\phi},$ (105)
$\displaystyle\Lambda=$
$\displaystyle\,XG_{X}\left(3H\dot{\phi}+\ddot{\phi}\right),$ (106)
$\displaystyle M_{n}^{4}=$
$\displaystyle\,n!\Bigg{[}a_{n}+3Hb_{n}+2\lambda_{n}X\left(G_{\phi}+G_{X}\ddot{\phi}\right)$
$\displaystyle\left.-\sum_{m=1}^{n-1}\lambda_{m}a_{n-m}\right],$ (107)
$\displaystyle\bar{m}_{n-1}^{3}=$ $\displaystyle-2b_{n-1},$ (108)

for $n\geq 2$, and where we have defined

$\displaystyle a_{n}\coloneqq$
$\displaystyle\,\frac{(-1)^{n+1}\dot{\phi}X^{n}}{n}\diffp{l_{n-1}}{\phi},$
(109) $\displaystyle b_{n}\coloneqq$
$\displaystyle\,(-X)^{n}\left(2Xl_{n}+\frac{2n-1}{n}l_{n-1}\right),$ (110)

with the coefficient $l_{n}$ from in equation (100). Please note that the
background parameters $c$ and $\Lambda$, and the first order mass parameters
$M^{4}_{2}$ and $\bar{m}_{1}^{3}$ were already computed, for instance, in
references [96, 98] (note the different conventions used there). (See also
reference [64] for a discussion on the cubic and quartic order parameters in
(B).) Nevertheless, to the best of our knowledge, this is the first time these
mass parameters are explicitly obtained in terms of the function $G$ and its
derivatives for an arbitrary order.

###  B.3 Mass parameters for the KGB theory

The background quantities and mass parameters for the complete KGB theory (1)
are given by the sum of the results presented in the two previous sections.
These read

$\displaystyle c=$
$\displaystyle\,XK_{X}+XG_{X}\left(3H\dot{\phi}-\ddot{\phi}\right)-2XG_{\phi},$
(111) $\displaystyle\Lambda=$
$\displaystyle\,XK_{X}-K+XG_{X}\left(3H\dot{\phi}+\ddot{\phi}\right),$ (112)
$\displaystyle M_{n}^{4}=$
$\displaystyle\,(-X)^{n}\diff[n]{K}{X}+n!\Bigg{[}a_{n}+3Hb_{n}+2\lambda_{n}X\left(G_{\phi}+G_{X}\ddot{\phi}\right)$
$\displaystyle\left.-\sum_{m=1}^{n-1}\lambda_{m}a_{n-m}\right],$ (113)
$\displaystyle\bar{m}_{n-1}^{3}=$ $\displaystyle-2b_{n-1},$ (114)

for $n\geq 2$. As to be expected, the combinations $c+\Lambda$ and $c-\Lambda$
(see definitions (86) and (87)) coincide with the expressions for
$\rho_{\phi}$ and $p_{\phi}$ for the non shift-symmetric KGB theory [10] (also
compare with the shift-symmetric version introduced in equations (9) and (10),
respectively). In addition, the leading order ($n=2$) mass parameters read

$\displaystyle M_{2}^{4}=$
$\displaystyle\,X^{2}K_{XX}+\frac{1}{2}XG_{X}\left(3H\dot{\phi}+\ddot{\phi}\right)+3H\dot{\phi}X^{2}G_{XX}$
$\displaystyle-X^{2}G_{\phi X},$ (115) $\displaystyle\bar{m}_{1}^{3}=$
$\displaystyle\,2\dot{\phi}XG_{X}.$ (116)

These results coincide with those presented in the literature modulo sign
conventions and numerical factors in the definition of $X$; cf. with, for
instance, references [96, 98]. Moreover, in accordance with the results
presented in Table 2 of reference [40], note that $M_{2}^{4}$ is connected
with the kineticity term $\alpha_{\textrm{K}}$ defined in equation (16)
through $4M_{2}^{4}=H^{2}\alpha_{\textrm{K}}-2c$, whereas
$\alpha_{\textrm{B}}$ is equal to $\bar{m}_{1}^{3}$ (modulo sign conventions);
see definition in equation (17).

For the discussion in section 3, it is also important to compute the next-to-
leading order braiding operator $\bar{m}_{2}^{3}$. This is

$\displaystyle\bar{m}_{2}^{3}=$
$\displaystyle-\frac{1}{2}\dot{\phi}X\left(G_{X}+2XG_{XX}\right),$ (117)

as can be seen from equation (114). Comparing this expression with that of
$\bar{m}_{1}^{3}$ in equation (116), it follows that

$\displaystyle
4\bar{m}_{2}^{3}=-2\dot{\phi}XG_{X}\left(1+\frac{2XG_{XX}}{G_{X}}\right)=-\bar{m}_{1}^{3}\left(1+\frac{2XG_{XX}}{G_{X}}\right).$
(118)

Consequently, the parameter $\eta$ used to measure deviations w.r.t. cubic
Galileon (i.e. $G(X)\propto X$) in expression (46) reads

$\displaystyle\eta=\frac{2XG_{XX}}{G_{X}},$ (119)

which is precisely the quantity first introduced in equation (37).

Finally, for the shift-symmetric case of the KGB theory the functions $K$ and
$G$ depend on the kinetic term $X$ but not on the scalar field itself. In this
scenario, the expressions (111) and (113) simplify as $G_{\phi}$ vanishes. In
addition, the coefficients $a_{n}$ also become trivial since there is no
dependence on $\phi$ in the shift-symmetric case; see definition (109).
Nevertheless, the formulas for the background quantity $\Lambda$ and the
braiding-related parameters $\bar{m}_{n-1}^{3}$ remain absolutely the same.
Thus, the EFT parameters for the shift-symmetric KGB theory simply read

$\displaystyle c=$
$\displaystyle\,XK_{X}+XG_{X}\left(3H\dot{\phi}-\ddot{\phi}\right),$ (120)
$\displaystyle\Lambda=$
$\displaystyle\,XK_{X}-K+XG_{X}\left(3H\dot{\phi}+\ddot{\phi}\right),$ (121)
$\displaystyle M_{n}^{4}=$
$\displaystyle\,(-X)^{n}\diff[n]{K}{X}+n!\left(3Hb_{n}+2\lambda_{n}XG_{X}\ddot{\phi}\right),$
(122) $\displaystyle\bar{m}_{n-1}^{3}=$ $\displaystyle-2b_{n-1},$ (123)

for $n\geq 2$, where $\lambda_{n}$ and $b_{n}$ are defined in equations (104)
and (110), respectively.

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