# Gravitational Wave Propagation and Polarizations in the Teleparallel analog
of Horndeski Gravity

Sebastian Bahamonde<EMAIL_ADDRESS>Laboratory of Theoretical Physics,
Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia
Maria Caruana<EMAIL_ADDRESS>Institute of Space Sciences and
Astronomy, University of Malta, Msida, Malta  Department of Physics,
University of Malta, Malta  Konstantinos F. Dialektopoulos<EMAIL_ADDRESS>Center for Gravitation and Cosmology, College of Physical Science and
Technology, Yangzhou University, Yangzhou 225009, China  Viktor Gakis
<EMAIL_ADDRESS>Institute of Space Sciences and Astronomy, University
of Malta, Msida, Malta  Department of Physics, National Technical University
of Athens, Zografou Campus GR 157 73, Athens, Greece  Manuel Hohmann
<EMAIL_ADDRESS>Laboratory of Theoretical Physics, Institute of Physics,
University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia  Jackson Levi Said
<EMAIL_ADDRESS>Institute of Space Sciences and Astronomy, University
of Malta, Msida, Malta  Department of Physics, University of Malta, Msida,
Malta  Emmanuel N. Saridakis<EMAIL_ADDRESS>National Observatory of
Athens, Lofos Nymfon, 11852 Athens, Greece  Department of Astronomy, School of
Physical Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, China  CAS
Key Laboratory for Research in Galaxies and Cosmology,
University of Science and Technology of China, Hefei, Anhui 230026, China
Joseph Sultana<EMAIL_ADDRESS>Department of Mathematics, University
of Malta, Msida, Malta

###### Abstract

Gravitational waves (GWs) have opened a new window on fundamental physics in a
number of important ways. The next generation of GW detectors may reveal more
information about the polarization structure of GWs. Additionally, there is
growing interest in theories of gravity beyond GR. One such theory which
remains viable within the context of recent measurements of the speed of
propagation of GWs is the teleparallel analogue of Horndeski gravity. In this
work, we explore the polarization structure of this newly proposed formulation
of Horndeski theory. In curvature-based gravity, Horndeski theory is almost
synonymous with extensions to GR since it spans a large portion of these
possible extensions. We perform this calculation by taking perturbations about
a Minkowski background and consider which mode propagates. The result is that
the polarization structure depends on the choice of model parameters in the
teleparallel Horndeski Lagrangian with a maximum of seven propagating degrees
of freedom. While the curvature-based Horndeski results follows as a
particular limit within this setup, we find a much richer structure of both
massive and massless cases which produce scalar–vector–tensor propagating
degrees of freedom. We also find that the GW polarization that emerges from
the teleparallel analogue of Horndeski gravity results in analogous massive
and massless modes which take on at most four polarizations in the massless
sector and two scalar ones in the massive sector. In none of the cases do we
find vector polarizations.

##  I Introduction

The observation of gravitational waves (GWs) first by the LIGO collaboration
Abbott:2016blz  and now also by the Virgo collaboration Abbott:2017oio  has
made it possible to test general relativity (GR) in the strong field regime
through template matching Carson:2020rea ; Ghosh:2017gfp . As with GR,
theories beyond GR require a lot of theoretical and numerical work before
competing templates can be produced for even the simplest scenarios such as
binary black hole coalescence events. Due to these limitations, it may mean
that GW polarization may be the most promising way forward in terms of testing
gravitational theories beyond GR. Moreover, this provides a model-independent
way to find the propagation of possible extra degrees of freedom (DoF)
Ezquiaga:2018btd ; Mukhanov:991646 ; Barack:2018yly .

GWs propagate with only two tensor polarizations in GR Will:2005va  whereas a
much richer polarization structure exists in many other theories of gravity.
In general, theories of gravity that depend on Riemann manifolds can host a
maximum of six polarizations which are related to the metric tensor form,
namely two tensor polarization modes (plus and cross), two vector modes
(vector $x$ and vector $y$), and two scalar modes (breathing and longitudinal)
Will:2005va . For instance, $f(\accentset{\circ}{R})$ gravity (over-circles
refers to the quantities calculated with the regular Levi-Civita connection)
propagates one extra scalar mode in certain scenarios in addition to the
regular tensor polarizations DeFelice:2010aj ; Liang:2017ahj ; Gong:2017bru ,
while Brans-Dicke theory just has an extra scalar breathing mode
Eardley:1974nw , whereas in bimetric gravity there is the possibility of
having up to the maximum full six polarization modes Visser:1997hd . On the
other hand, theories exist which have identical GW polarization
characteristics to GR such as $f(T)$ Bamba:2013ooa ; Farrugia:2018gyz  and
$f(Q)$ Soudi:2018dhv  gravity (where $T$ and $Q$ are measures of torsion and
nonmetricity, respectively) which are both non-Riemannian theories of gravity.

Preliminary work on the search for extra polarization modes has already
started in earnest through the LIGO-Virgo collaboration since three detectors
offer the possibility of cross-matching arrival time observations
Abbott:2018utx . This has been done in the context of black hole binary
mergers in GW170814 Abbott:2017oio  and binary neutron star mergers in
GW170817 TheLIGOScientific:2017qsa . However, due to the theoretical
limitations of template matching, the waveforms are assumed to be identical to
GR and the vector and scalar modes searches have a strong element of model-
dependence which constrains the applicability of present studies of this
nature. These issues have been tackled in other studies Takeda:2018uai  but
the matter may ultimately be resolved when more novel detectors come online.

GW polarizations offer the possibility of a clear deviation from GR that may
be detectable by the next generation of detectors such as LISA
Belgacem:2019pkk  and the Einstein Telescope Maggiore:2019uih , among many
others Miller:2019vfc . In this work, we explore the GW polarization structure
of the recently proposed teleparallel analogue of Horndeski gravity
Bahamonde:2019shr . Standard Horndeski gravity Horndeski:1974wa  is the most
general second-order theory of gravity (in terms of metric derivatives) that
contains only one scalar field, in the context of the Levi-Civita connection.
However, other connections can also be used to describe different formulations
of gravity. The Levi-Civita connection is characterized by curvature-based
constructions of gravity. More broadly, a trinity of possible deformations to
Riemann manifolds exists where not only curvature but also torsion and
nonmetricity contribute BeltranJimenez:2019tjy . In addition to the curvature-
based Levi-Civita connection, Riemann manifolds are also compatible with a
teleparallel connection, which is torsion-based Aldrovandi:2013wha , and a
disformation connection, which is based on nonvanishing nonmetricity
BeltranJimenez:2017tkd .

The recent measurements of the speed of propagation of GWs has constrained
their speed to the speed of light to within deviations of at most one part in
$10^{15}$ which has rendered severe limitations on the rich structure of
standard Horndeski gravity Ezquiaga:2017ekz . This has consequently
drastically limited the number of viable paths to constructing physically
realistic models of cosmology within standard Horndeski gravity. In Ref.
Bahamonde:2019shr , Bahamonde-Dialektopoulos-Levi Said (BDLS) propose a
teleparallel analogue of Horndeski gravity. Due to the naturally lower-order
nature of teleparallel, or torsional-based gravity, the framework that ensues
in BDLS theory is much richer than in standard gravity (i.e curvature-based
gravity theories). One direct result of this feature is that the GW
propagation speed in BDLS theory contains more terms related to the
gravitational Lagrangian, as shown in Ref. Bahamonde:2019ipm . Crucially, this
means that previously disqualified models of Horndeski gravity can be revived
in a teleparallel gravity form through carefully chosen additional Lagrangian
contributions.

The gravitational wave propagation equation gives the speed of propagation of
GWs together with amplitude modulations which is addressed in Ref.
Ezquiaga:2018btd . The next question one may ask is whether BDLS theory
propagates further polarizations in addition to the standard Horndeski
picture. In Ref. Hou:2017bqj , the polarization modes for different scenarios
is explored in the context of standard Horndeski gravity. In this work, our
central focus is targeted at the determination of the polarization modes in
BDLS theory. In Sec. II we briefly review the teleparallel gravity basis of
BDLS theory and introduce the exact formulation of the teleparallel analogue
of Horndeski gravity while in Sec. III the perturbation regime for this form
of gravity s prescribed. In Sec.IV the perturbations are applied to BDLS
theory and the propagating degrees of freedom (PDoF) are explored for various
scenarios of the theory with a focus on the contributions from the
decomposition in a scalar–vector–tensor (SVT) regime. Finally, the GW
polarization classification is presented in Sec. V which is followed by a
discussion of the main results in Sec. VI. Throughout the work, geometric
units and the $(+1,-1,-1,-1)$ signature are used.

##  II The Teleparallel Analog of Horndeski Theory

The curvature associated with GR is built on the Levi-Civita connection
$\accentset{\circ}{\Gamma}^{\sigma}_{\phantom{\sigma}\mu\nu}$ which is
torsionless and satisfies the metric compatibility condition. Teleparallel
gravity (TG) is based on torsion through the teleparallel connection,
$\Gamma^{\sigma}{}_{\mu\nu}$, which continues to satisfy metricity but that is
curvatureless. As is well known, in curvature based theories of gravity, the
gravitational field is measured through the Riemann tensor and its
contractions, which have led to a large number of extended theories of gravity
beyond GR Clifton:2011jh ; Capozziello:2011et . This is a fundamental measure
of curvature misner1973gravitation . However, when the Levi-Civita connection
is replaced with the teleparallel connection, the Riemann curvature tensor
identically vanishes which points to the necessity of new measures of
geometric deformation within the TG context.

The metric tensor, $g_{\mu\nu}$, is the fundamental object of GR and is
defined as an inner product. In TG the metric tensor emerges from tetrad
fields, $e^{A}_{\phantom{A}\mu}$, which act as soldering agents between the
general manifold (Greek indices) and the local Minkowski space (capital Latin
indices). This means that they can be used to raise Minkowski space indices to
the general manifold and vice versa through RevModPhys.48.393

$\displaystyle g_{\mu\nu}$
$\displaystyle=e^{A}_{\phantom{A}\mu}e^{B}_{\phantom{B}\nu}\eta_{AB}\,,$ (1)
$\displaystyle\eta_{AB}$
$\displaystyle=E_{A}^{\phantom{A}\mu}E_{B}^{\phantom{B}\nu}g_{\mu\nu}\,,$ (2)

which also define the inverse tetrads, $E_{A}^{\phantom{A}\mu}$, which satisfy
the dual basis conditions

$\displaystyle e^{A}_{\phantom{A}\mu}E_{B}^{\phantom{B}\mu}$
$\displaystyle=\delta_{B}^{A}\,,$ (3) $\displaystyle
e^{A}_{\phantom{A}\mu}E_{A}^{\phantom{A}\nu}$
$\displaystyle=\delta_{\mu}^{\nu}\,,$ (4)

where $\eta_{AB}$ is the Minkowski metric. Since tetrads contain 6 extra DoFs
compared to the metric, different tetrads can reproduce the same metric. This
is manifested due to the local Lorentz transformations (LLTs),
$\Lambda^{A}_{\phantom{A}B}$, on the local space.

In this context, TG is routed on the replacement of the Levi-Civita connection
with the teleparallel connection which implies a corresponding exchange of the
metric tensor with the tetrad as the fundamental dynamical object of the
theory. Note that also GR can be formulated using tetrads, even though this is
less common; in TG, however, tetrads are more commonly used than the
alternative, but equivalent formulation in terms of a metric and an
independent teleparallel connection Hohmann:2021fpr . The latter is the linear
affine connection that satisfies both the curvatureless and metricity
conditions Weitzenbock1923 , and denoted by the teleparallel connection. In
the tetrad-based formulation, this connection can be expressed as Cai:2015emx
; Krssak:2018ywd

$\Gamma^{\lambda}{}_{\nu\mu}=E_{A}^{\phantom{A}\lambda}\partial_{\mu}e^{A}_{\phantom{A}\nu}+E_{A}^{\phantom{A}\lambda}\omega^{A}_{\phantom{A}B\mu}e^{B}_{\phantom{B}\nu}\,,$
(5)

where $\omega^{A}_{\phantom{A}B\mu}$ represents the components of a flat spin
connection, which must satisfy

$\partial_{[\mu}\omega^{A}_{\phantom{A}|B|\nu]}+\omega^{A}_{\phantom{A}C[\mu}\omega^{C}_{\phantom{C}|B|\nu]}\equiv
0\,.$ (6)

Given that this is a flat connection, the DoFs associated with the LLTs play
an active role in the field equations of the theory such that the spin
connection will counter balance these inertial effects so that the theory
retains covariance. In any of these setups there will exist a frame in which
the spin connection naturally vanishes, which is called the Weitzenböck gauge
Krssak:2018ywd .

TG is based on the torsion associated with the teleparallel connection. In
this context, a torsion tensor must be defined to replace the the role that
the Riemann tensor plays111It is important to emphasize that, in general, the
Riemann tensor as calculated with Levi-Civita connection does not vanish
$\accentset{\circ}{R}^{\alpha}_{\phantom{\alpha}\beta\mu\nu}\neq 0$., whose
components identically vanish. By definition, the torsion tensor is defined as
the antisymmetric operator acted on the connection Aldrovandi:2013wha ;
ortin2004gravity

$T^{A}_{\phantom{A}\mu\nu}:=2\Gamma^{A}_{\phantom{A}[\nu\mu]}\,,$ (7)

which acts as a measure of the field strength of gravitation in TG, and where
the square brackets denote antisymmetric operator. The torsion tensor
transforms covariantly under LLTs and is diffeomorphism invariant
Krssak:2015oua . An interesting feature of the torsion tensor is that it can
be decomposed into three irreducible pieces, namely axial, vector and purely
tensorial parts which are given by PhysRevD.19.3524 ; Bahamonde:2017wwk

$\displaystyle a_{\mu}$
$\displaystyle:=\frac{1}{6}\epsilon_{\mu\nu\lambda\rho}T^{\nu\lambda\rho}\,,$
(8) $\displaystyle v_{\mu}$
$\displaystyle:=T^{\lambda}_{\phantom{\lambda}\lambda\mu}\,,$ (9)
$\displaystyle t_{\lambda\mu\nu}$
$\displaystyle:=\frac{1}{2}\left(T_{\lambda\mu\nu}+T_{\mu\lambda\nu}\right)+\frac{1}{6}\left(g_{\nu\lambda}v_{\mu}+g_{\nu\mu}v_{\lambda}\right)-\frac{1}{3}g_{\lambda\mu}v_{\nu}\,,$
(10)

where $\epsilon_{\mu\nu\lambda\rho}$ is the totally antisymmetric Levi-Civita
tensor in four dimensions. The irreducible parts of the torsion tensor all
vanish when contracted with each other due to the symmetries of the torsion
tensor itself. The decomposition of the torsion tensor can be used to form the
three scalar invariants

$\displaystyle T_{\text{ax}}$
$\displaystyle:=a_{\mu}a^{\mu}=-\frac{1}{18}\left(T_{\lambda\mu\nu}T^{\lambda\mu\nu}-2T_{\lambda\mu\nu}T^{\mu\lambda\nu}\right)\,,$
(11) $\displaystyle T_{\text{vec}}$
$\displaystyle:=v_{\mu}v^{\mu}=T^{\lambda}_{\phantom{\lambda}\lambda\mu}T_{\rho}^{\phantom{\rho}\rho\mu}\,,$
(12) $\displaystyle T{{}_{\text{ten}}}$
$\displaystyle:=t_{\lambda\mu\nu}t^{\lambda\mu\nu}=\frac{1}{2}\left(T_{\lambda\mu\nu}T^{\lambda\mu\nu}+T_{\lambda\mu\nu}T^{\mu\lambda\nu}\right)-\frac{1}{2}T^{\lambda}_{\phantom{\lambda}\lambda\mu}T_{\rho}^{\phantom{\rho}\rho\mu}\,,$
(13)

that can be summed to form the torsion scalar, and which are parity preserving
Bahamonde:2015zma . These three scalars can be combined to form the most
general purely gravitational Lagrangian containing only parity preserving
scalars, $f(T_{\text{ax}},T_{\text{vec}},T{{}_{\text{ten}}})$, that is
quadratic in contractions of the torsion tensor Bahamonde:2017wwk , and where
the resulting field equations are second order in derivatives of the tetrads.
The torsion scalar can then be written as

$T:=\frac{3}{2}T_{\text{ax}}+\frac{2}{3}T_{\text{ten}}-\frac{2}{3}T{{}_{\text{vec}}}=\frac{1}{2}\left(E_{A}{}^{\lambda}g^{\rho\mu}E_{B}{}^{\nu}+2E_{B}{}^{\rho}g^{\lambda\mu}E_{A}{}^{\nu}+\frac{1}{2}\eta_{AB}g^{\mu\rho}g^{\nu\lambda}\right)T^{A}{}_{\mu\nu}T^{B}{}_{\rho\lambda}\,,$
(14)

which is equivalent to the Ricci scalar, $\accentset{\circ}{R}$ (calculated
using the Levi-Civita connection), less a total divergence term represented by
Bahamonde:2015zma

$R=\accentset{\circ}{R}+T-\frac{2}{e}\partial_{\mu}\left(eT^{\lambda\phantom{\lambda}\mu}_{\phantom{\lambda}\lambda\phantom{\mu}}\right)=0\,,$
(15)

where $R$ is the Ricci scalar as calculated using the teleparallel connection,
which organically vanishes due to the symmetry properties of the flat
teleparallel connection and
$e=\text{det}\left(e^{A}_{\phantom{A}\mu}\right)=\sqrt{-g}$ is the tetrad
determinant. This means that we can define a boundary term, $B$, such that

$\accentset{\circ}{R}=-T+\frac{2}{e}\partial_{\mu}\left(eT^{\lambda\phantom{\lambda}\mu}_{\phantom{\lambda}\lambda\phantom{\mu}}\right):=-T+B\,.$
(16)

Now, given the boundary term nature of $B$, the action produced by a linear
contribution of $T$ alone is guaranteed to produce a teleparallel equivalent
of general relativity (TEGR) Hehl:1994ue ; Aldrovandi:2013wha . An interesting
feature of TEGR is that it can be formulated as native gauge theory of
translations Aldrovandi:2013wha ; Hayashi:1967se .

In GR, the equivalence principle offers a procedure by which to raise local
Lorentz frames to general ones by the exchange of the Minkowski metric with it
general metric tensor while also raising partial derivatives to covariant ones
(with respect to the Levi-Civita connection) misner1973gravitation ;
nakahara2003geometry . TG is distinct in that this coupling prescription is
largely preserved for additional fields while the gravitational connection is
changed to the teleparallel connection Aldrovandi:2013wha . Thus, the coupling
prescription for this context is described by the raising of Minkowski tetrads
to their arbitrary analogs, while the tangent space partial derivatives are
raised to the regular Levi-Civita connection covariant derivatives, namely

$\partial_{\mu}\rightarrow\mathring{\nabla}_{\mu}\,,$ (17)

which emphasises the close relationship both theories have with each other
BeltranJimenez:2019tjy .

Now that both the purely gravitational and scalar field sectors have been
adequately developed, we can consider the recently proposed teleparallel
analog of Horndeski gravity Bahamonde:2019shr ; Bahamonde:2019ipm ;
Bahamonde:2020cfv . This new formulation of Horndeski gravity was built in TG
using just three limiting conditions, namely that (i) the field equations must
be at most second order in their derivatives of the tetrads; (ii) the scalar
invariants will not be parity violating; and (iii) the number of contractions
with the torsion tensor is limited to being at most quadratic. These
conditions are necessary because TG is organically lower order in derivatives
of the tetrad (since the torsion tensor depends only on first derivatives of
the tetrad), and thus can produce a potentially infinite number of second
order theories of gravity. This automatically implies a much weaker form of
the Lovelock theorem Lovelock:1971yv ; Gonzalez:2015sha ; Gonzalez:2019tky .
Condition (iii) limits the number of such terms to a finite number, where such
higher order contractions may play a role in other phenomenology.

The TG Horndeski conditions have been shown to lead directly to a finite
number of contributing scalar invariants that explore possible nonminimal
coupling terms between the scalar field and the torsion tensor, which for the
linear case gives Bahamonde:2019shr

$I_{2}=v^{\mu}\phi_{;\mu}\,,$ (18)

while for the quadratic scenario gives

$\displaystyle J_{1}$ $\displaystyle=a^{\mu}a^{\nu}\phi_{;\mu}\phi_{;\nu}\,,$
(19) $\displaystyle J_{3}$
$\displaystyle=v_{\sigma}t^{\sigma\mu\nu}\phi_{;\mu}\phi_{;\nu}\,,$ (20)
$\displaystyle J_{5}$
$\displaystyle=t^{\sigma\mu\nu}t_{\sigma\phantom{\alpha}\nu}^{\phantom{\sigma}\alpha}\phi_{;\mu}\phi_{;\alpha}\,,$
(21) $\displaystyle J_{6}$
$\displaystyle=t^{\sigma\mu\nu}t_{\sigma}^{\phantom{\sigma}\alpha\beta}\phi_{;\mu}\phi_{;\nu}\phi_{;\alpha}\phi_{;\beta}\,,$
(22) $\displaystyle J_{8}$
$\displaystyle=t^{\sigma\mu\nu}t_{\sigma\mu}^{\phantom{\sigma\mu}\alpha}\phi_{;\nu}\phi_{;\alpha}\,,$
(23) $\displaystyle J_{10}$
$\displaystyle=\epsilon^{\mu}_{\phantom{\mu}\nu\sigma\rho}a^{\nu}t^{\alpha\rho\sigma}\phi_{;\mu}\phi_{;\alpha}\,,$
(24)

where semicolons represent covariant derivatives with respect to the Levi-
Civita connection.

The new scalar invariants can be arbitrarily combined to produce a new
Lagrangian contribution to the regular Horndeski terms. The standard gravity
Horndeski terms naturally emerge since TG is lower order, meaning that we at
least find the regular contributions. The additional Lagrangian term appears
as

$\mathcal{L}_{\text{Tele}}:=G_{\text{Tele}}\left(\phi,X,T,T_{\text{ax}},T_{\text{vec}},I_{2},J_{1},J_{3},J_{5},J_{6},J_{8},J_{10}\right)\,,$
(25)

where the kinetic term is defined as
$X:=-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi$. An aesthetic difference
occurs for the standard terms in that they are now described by the tetrad
rather than the metric but their values remain identical. Hence, the TG analog
of Horndeski gravity is described by Bahamonde:2019shr ; Horndeski:1974wa ;
Saridakis:2021lqd

$\mathcal{S}_{\text{BDLS}}=\frac{1}{2\kappa^{2}}\int
d^{4}x\,e\mathcal{L}_{\text{Tele}}+\frac{1}{2\kappa^{2}}\sum_{i=2}^{5}\int
d^{4}x\,e\mathcal{L}_{i}+\int d^{4}x\,e\mathcal{L}_{\rm m}\,,$ (26)

where

$\displaystyle\mathcal{L}_{2}$ $\displaystyle:=G_{2}(\phi,X)\,,$ (27)
$\displaystyle\mathcal{L}_{3}$
$\displaystyle:=G_{3}(\phi,X)\mathring{\Box}\phi\,,$ (28)
$\displaystyle\mathcal{L}_{4}$
$\displaystyle:=G_{4}(\phi,X)\left(-T+B\right)+G_{4,X}(\phi,X)\left[\left(\mathring{\Box}\phi\right)^{2}-\phi_{;\mu\nu}\phi^{;\mu\nu}\right]\,,$
(29) $\displaystyle\mathcal{L}_{5}$
$\displaystyle:=G_{5}(\phi,X)\mathring{G}_{\mu\nu}\phi^{;\mu\nu}-\frac{1}{6}G_{5,X}(\phi,X)\left[\left(\mathring{\Box}\phi\right)^{3}+2\phi_{;\mu}^{\phantom{;\mu}\nu}\phi_{;\nu}^{\phantom{;\nu}\alpha}\phi_{;\alpha}^{\phantom{;\alpha}\mu}-3\phi_{;\mu\nu}\phi^{;\mu\nu}\,\mathring{\Box}\phi\right]\,,$
(30)

where $\mathcal{L}_{\rm m}$ is the matter Lagrangian in the Jordan conformal
frame, $\kappa^{2}:=8\pi G$, $\accentset{\circ}{G}_{\mu\nu}$ is the standard
Einstein tensor, and comma represents regular partial derivatives. Standard
Horndeski theory is clearly recovered when $G_{\text{Tele}}=0$ is selected.
Due to the invariance under LLTs and general covariance that stems from TG,
the TG analog of Horndeski gravity organically adheres to these critical
properties.

##  III Perturbations in Teleparallel Gravity

In this section we introduce our prescription for taking perturbations about a
Minkowski background in the context of tetrads together with a scalar field.
To do this, we expand the tetrad about the Minkowski background through

$\displaystyle\tilde{e}^{A}_{\phantom{A}\mu}$
$\displaystyle=\delta_{\mu}^{A}+\epsilon\,\delta
e^{A}_{\phantom{A}\mu}=\delta_{\mu}^{A}+\epsilon\,\delta_{B}^{\nu}\eta^{AB}\tau_{\mu\nu}\,,$
(31) $\displaystyle\phi$ $\displaystyle=\phi_{0}+\epsilon\,\delta\phi\,,$ (32)

where

$\displaystyle\tau_{\mu\nu}$ $\displaystyle:=\eta{}_{AB}e^{B}{}_{\mu}\delta
e{}^{A}{}_{\nu}\,,$ (33)

and where $\epsilon$ represents the order of the perturbation, $e^{B}{}_{\mu}$
represents the background value of the tetrad which is just
$\delta^{B}{}_{\mu}$ for a Minkowski background, and $\phi_{0}$ is a constant
(in general, the background value of $\phi_{0}$ can be taken to be time
dependent as $\phi=\phi(t)$). Thus, the metric perturbation becomes (1)

$\tilde{g}_{\mu\nu}=\eta_{\mu\nu}+\epsilon\,h_{\mu\nu}+\frac{1}{2}\epsilon^{2}\,\delta^{2}g_{\mu\nu}=\eta_{\mu\nu}+2\epsilon\,\tau_{(\mu\nu)}+\epsilon^{2}\,\tau_{\alpha\mu}\tau^{\alpha}_{\phantom{\alpha}\nu}\,.$
(34)

To obtain the BDLS perturbation equations, we consider the perturbations of
the action in Eq. (26) which then results in the linearized equations of
motion by taking an appropriate variation (see Appendix A). In the case of
BDLS theory, Taylor expansion is performed about the point $\phi=\phi_{0}$
while the other arguments do not contribute at the background (since the
background is the Minkowski space). Hence, we can write

$\displaystyle
G_{\text{Tele}}\left(\phi,X,T,T_{\text{ax}},T_{\text{vec}},I_{2},J_{1},J_{3},J_{5},J_{6},J_{8},J_{10}\right)=G_{\text{Tele}}+\epsilon\,G_{\text{Tele},\phi}\delta\phi$
$\displaystyle\quad+\epsilon^{2}\Big{[}\frac{1}{2}G_{\text{Tele},\phi\phi}\delta\phi^{2}+G_{\text{Tele},X}X+G_{\text{Tele},T}T+G_{\text{Tele},T_{\text{ax}}}T_{\text{ax}}+G_{\text{Tele},T_{\text{vec}}}T_{\text{vec}}+G_{\text{Tele},I_{2}}I_{2}\Big{]}+\mathcal{O}(\epsilon^{3})\,,$
(35) $\displaystyle
G_{j}(\phi,X)=G_{j}+\epsilon\,G_{j,\phi}\,\delta\phi+\epsilon^{2}\,\Big{[}\frac{1}{2}G_{j,\phi\phi}\delta\phi^{2}+G_{j,X}X\Big{]}+\mathcal{O}(\epsilon^{3})\,,$
(36)

where $G_{\text{Tele},i}=G_{\text{Tele},i}(\phi_{0},0,0,0,0,0,0,0,0,0,0,0)$
for $i=\\{\phi,\phi\phi,X,T,T_{\text{ax}},T_{\text{vec}},I_{2}\\}$ such that
$\phi\phi$ subscript stands for the second order derivative with respect to
$\phi$ and $G_{j,k}=G_{j,k}(\phi_{0},0)$ for $j=\\{2,3,4,5\\}$ and
$k=\\{\phi,\phi\phi,X\\}$ are constants. In these expansions, we also note
that the $J_{i}$ contributions do not appear due to their generically higher
order nature. Therefore, the linearized field equations $W_{\mu\nu}$,
$\widehat{W}$ are obtained by varying the action with respect to
$\tau_{\mu\nu}$ and $\delta\phi$ accordingly.

$\displaystyle W_{\mu\nu}$
$\displaystyle=\left(G_{\text{Tele}}+G_{2}\right)\eta_{\mu\nu}$
$\displaystyle\quad+\epsilon\,\Big{[}\left(G_{\text{Tele}}+G_{2}\right)\left(\tau\,\eta_{\mu\nu}-\tau_{\mu\nu}\right)-2G_{\text{Tele},T_{\text{vec}}}\left(\partial^{\lambda}\partial_{\mu}\tau_{\lambda\nu}-\partial_{\mu}\partial_{\nu}\tau-\partial_{\lambda}\partial_{\beta}\tau^{\lambda\beta}\eta_{\mu\nu}+\Box\tau\,\eta_{\mu\nu}\right)$
$\displaystyle\quad+\left(-2G_{\text{Tele},T}+2G_{4}\right)\left(\Box\tau_{(\mu\nu)}-\partial^{\lambda}\partial_{\mu}\tau_{(\nu\lambda)}-\partial^{\lambda}\partial_{\nu}\tau_{(\mu\lambda)}+\partial_{\mu}\partial_{\nu}\tau+\partial_{\lambda}\partial_{\beta}\tau^{\lambda\beta}\eta_{\mu\nu}-\Box\tau\,\eta_{\mu\nu}\right)$
$\displaystyle\quad+\frac{4}{9}G_{\text{Tele},T_{\text{ax}}}\left(\Box\tau_{[\mu\nu]}-\partial^{\lambda}\partial_{\nu}\tau_{[\mu\lambda]}+\partial^{\lambda}\partial_{\mu}\tau_{[\nu\lambda]}\right)+\left(-G_{\text{Tele},I_{2}}+2G_{4,\phi}\right)\left(\partial_{\mu}\partial_{\nu}\delta\phi-\eta_{\mu\nu}\Box\delta\phi\right)\Big{]}\,,$
(37) $\displaystyle\widehat{W}$
$\displaystyle=G_{\text{Tele},\phi}+G_{2,\phi}+\epsilon\,\Big{[}\left(G_{\text{Tele},\phi}+G_{2,\phi}\right)\,\tau+\left(G_{\text{Tele},I_{2}}-2G_{4,\phi}\right)\left(\Box\tau-\partial_{\lambda}\partial_{\beta}\tau^{\lambda\beta}\right)$
$\displaystyle\quad+\left(G_{\text{Tele},\phi\phi}+G_{2,\phi\phi}\right)\,\delta\phi+\left(G_{\text{Tele},X}+G_{2,X}-2G_{3,\phi}\right)\Box\delta\phi\Big{]}\,,$
(38)

where $\tau:=\eta^{\mu\nu}\tau_{\mu\nu}$ and
$\Box\coloneqq\partial_{\lambda}\partial^{\lambda}$ is the d’Alembert operator
of Minkowski space. In order to obtain the on-shell perturbed field equations,
we impose the background field equations

$\displaystyle 0$ $\displaystyle=$ $\displaystyle G_{\text{Tele}}+G_{2}\,,$
(39) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
G_{\text{Tele},\phi}+G_{2,\phi}\,.$ (40)

Thus, the linearised field equations are reduced to

$\displaystyle W_{\mu\nu}$
$\displaystyle=-2G_{\text{Tele},T_{\text{vec}}}\left(\partial^{\lambda}\partial_{\mu}\tau_{\lambda\nu}-\partial_{\mu}\partial_{\nu}\tau-\partial_{\lambda}\partial_{\sigma}\tau^{\lambda\sigma}\eta_{\mu\nu}+\Box\tau\,\eta_{\mu\nu}\right)$
$\displaystyle\quad+\left(-2G_{\text{Tele},T}+2G_{4}\right)\left(\Box\tau_{(\mu\nu)}-\partial^{\lambda}\partial_{\mu}\tau_{(\nu\lambda)}-\partial^{\lambda}\partial_{\nu}\tau_{(\mu\lambda)}+\partial_{\mu}\partial_{\nu}\tau+\partial_{\lambda}\partial_{\sigma}\tau^{\lambda\sigma}\eta_{\mu\nu}-\Box\tau\,\eta_{\mu\nu}\right)$
$\displaystyle\quad+\frac{4}{9}G_{\text{Tele},T_{\text{ax}}}\left(\Box\tau_{[\mu\nu]}-\partial^{\lambda}\partial_{\nu}\tau_{[\mu\lambda]}+\partial^{\lambda}\partial_{\mu}\tau_{[\nu\lambda]}\right)+\left(-G_{\text{Tele},I_{2}}+2G_{4,\phi}\right)\left(\partial_{\mu}\partial_{\nu}\delta\phi-\eta_{\mu\nu}\Box\delta\phi\right)\,,$
(41) $\displaystyle\widehat{W}$
$\displaystyle=\left(G_{\text{Tele},I_{2}}-2G_{4,\phi}\right)\left(\Box\tau-\partial_{\lambda}\partial_{\sigma}\tau^{\lambda\sigma}\right)+\left(G_{\text{Tele},\phi\phi}+G_{2,\phi\phi}\right)\,\delta\phi+\left(G_{\text{Tele},X}+G_{2,X}-2G_{3,\phi}\right)\Box\delta\phi\,.$
(42)

Here, Eq. (41) can be decomposed into symmetric and antisymmetric parts as
follows

$\displaystyle W_{(\mu\nu)}$
$\displaystyle=-2G_{\text{Tele},T_{\text{vec}}}\left(\partial^{\lambda}\partial_{(\mu}\tau_{|\lambda|\nu)}-\partial_{\mu}\partial_{\nu}\tau-\partial_{\lambda}\partial_{\beta}\tau^{\lambda\beta}\eta_{\mu\nu}+\Box\tau\eta_{\mu\nu}\right)+\left(-G_{\text{Tele},I_{2}}+2G_{4,\phi}\right)\left(\partial_{\mu}\partial_{\nu}\delta\phi-\eta_{\mu\nu}\Box\delta\phi\right)$
$\displaystyle\quad+2\left(-G_{\text{Tele},T}+G_{4}\right)\left(\Box\tau_{(\mu\nu)}-\partial^{\lambda}\partial_{\mu}\tau_{(\nu\lambda)}-\partial^{\lambda}\partial_{\nu}\tau_{(\mu\lambda)}+\partial_{\mu}\partial_{\nu}\tau+\partial_{\lambda}\partial_{\beta}\tau^{\lambda\beta}\eta_{\mu\nu}-\Box\tau\eta_{\mu\nu}\right)\,,$
(43) $\displaystyle W_{[\mu\nu]}$
$\displaystyle=-2G_{\text{Tele},T_{\text{vec}}}\partial^{\lambda}\partial_{[\mu}\tau_{|\lambda|\nu]}+\frac{4}{9}G_{\text{Tele},T_{\text{ax}}}\left[\Box\tau_{[\mu\nu]}-\partial^{\lambda}\partial_{\nu}\tau_{[\mu\lambda]}+\partial^{\lambda}\partial_{\mu}\tau_{[\nu\lambda]}\right]\,,$
(44)

where the antisymmetric part coincides with the field equations one would
obtain by varying with respect to a non-trivial inertial spin connection.
Using the symmetry operators gives a more workable formulation to the
perturbation equations which will then be studied in the sections that follow.

##  IV Scalar–Vector–Tensor Decomposition and Propagating Degrees of Freedom
Analysis

Following the notation of Ref. Bahamonde:2020lsm , the tetrad perturbation
$\delta e^{A}{}_{\mu}$, in Minkowski space, can be split as follows

$\delta
e^{A}{}_{\mu}:=\left[\begin{array}[]{cc}-\varphi&-\left(\partial_{i}\beta+\beta_{i}\right)\\\
\delta^{I}{}_{i}\left(\partial^{i}b+b^{i}\right)&\delta^{Ii}\left(-\psi\delta_{ij}+\partial_{i}\partial_{j}h+2\partial_{(i}h_{j)}+\frac{1}{2}h_{ij}+\epsilon_{ijk}\left(\partial^{k}\sigma+\sigma^{k}\right)\right)\end{array}\right],$
(45)

which will further generate $\tau_{\mu\nu}$, using Eq. (33), to give

$\tau_{\mu\nu}=\left[\begin{array}[]{cc}-\varphi&-\left(\partial_{i}\beta+\beta_{i}\right)\\\
(\partial_{i}b+b_{i})&\left(-\psi\delta_{ij}+\partial_{i}\partial_{j}h+2\partial_{(i}h_{j)}+\frac{1}{2}h_{ij}+\epsilon_{ijk}\left(\partial^{k}\sigma+\sigma^{k}\right)\right)\end{array}\right],$
(46)

and one can also produce the perturbation of the metric perturbation since
$\delta g_{\mu\nu}=2\tau_{(\mu\nu)}$, and so

$\delta
g_{\mu\nu}=\left[\begin{array}[]{cc}-2\,\varphi&\left(\partial_{i}\mathcal{B}+\mathcal{B}_{i}\right)\\\
\left(\partial_{i}\mathcal{B}+\mathcal{B}_{i}\right)&2\left(-\psi\delta_{ij}+\partial_{i}\partial_{j}h+2\partial_{(i}h_{j)}+\frac{1}{2}h_{ij}\right)\end{array}\right],$
(47)

where $\left\\{\varphi,\beta,b,\psi,h\right\\}$ are scalars with $\sigma$
being a pseudo scalar, $\left\\{\beta_{i},b_{i},h_{i}\right\\}$ are vectors
with $\sigma^{k}$ being a pseudo vector and $h_{ij}$ the tensor modes, and
which represent the regular perturbation of the metric tensor about Minkowski
space. All indexed quantities are divergence-less, meaning that
$\partial^{i}X_{ijk..l}=0$, $h_{ij}=h_{(ij)}$ and $\delta^{ij}h_{ij}=0$. We
also have denoted $\mathcal{B}:=b-\beta$ and
$\mathcal{B}_{i}:=b_{i}-\beta_{i}$ to stress the fact that in the perturbation
of the metric Eq. (47) the off diagonal elements are a single scalar and
vector. While Latin indixes are used for inertial coordinates, we reserve the
middle range Latin indices $I,J,K,\dots$ and $i,j,k,\dots$ for purely spacial
coordinates in the spacial inner bundle and the spacial spacetime manifold
respectively.

The gauge transformation of $\delta e^{A}{}_{\nu}$, under a linear change of
coordinates described by $\widetilde{x}^{\mu}\rightarrow x^{\mu}+\xi^{\mu}$,
can be expressed as

$\displaystyle\widetilde{\delta e}^{A}{}_{\mu}$
$\displaystyle\rightarrow\delta
e{}^{A}{}_{\mu}+\mathcal{L}_{\xi}e{}^{A}{}_{\mu}.$ (48)

where $\mathcal{L}_{\xi}$ is the Lie derivative along the flow of $\xi^{\mu}$.
We further split $\xi^{\mu}$ as
$\xi^{\mu}=\left\\{\xi^{0},\omega(\xi^{i}+\delta{}^{ij}\partial{}_{j}\xi)\right\\}$
where $\partial_{i}\xi^{i}=0$. One can then unpack Eq. (48) to all its
individual components as

$\displaystyle\widetilde{\varphi}$ $\displaystyle=$
$\displaystyle\varphi-\dot{\xi}{}^{0}\,,\quad\widetilde{\psi}=\psi\,,\quad\widetilde{\beta}=\beta-\xi{}^{0}\,,\quad\widetilde{\beta_{i}}=\beta_{i}\,,$
(49a) $\displaystyle\widetilde{b}$ $\displaystyle=$ $\displaystyle
b-\dot{\xi}\,,\quad\widetilde{b}_{i}=b_{i}+\dot{\xi_{i}}\,,\quad\widetilde{\sigma}=\sigma\,,\quad\widetilde{\sigma}_{i}=\sigma^{i}-\frac{1}{2}\epsilon^{i}{}_{jk}\partial^{j}\xi^{k}\,,$
(49b) $\displaystyle\widetilde{h}$ $\displaystyle=$ $\displaystyle
h-\xi\,,\quad\widetilde{h}_{i}=h_{i}+\frac{1}{2}\xi_{i}\,,\quad\widetilde{h}_{ij}=h_{ij}\,.$
(49c)

As it is evident $\psi,\sigma$, $\beta_{i}$ and $h_{ij}$ are already gauge
invariant in the Minkowski background. In addition, $\delta\phi$ is also gauge
invariant in this setting since its background value is constant
$\phi(t)\equiv\phi_{0}$. In general one can built a few other gauge invariant
quantities from Eq. (49) according to their similarities in the gauge
transformations as

$\displaystyle\chi$ $\displaystyle:=b-\dot{h},$ (50) $\displaystyle\Phi$
$\displaystyle:=\varphi-\dot{\beta},$ (51) $\displaystyle\Sigma_{j}$
$\displaystyle:=h_{j}k{}^{2}+i\varepsilon{}_{jlp}k{}^{l}\sigma{}^{p},$ (52)
$\displaystyle\Xi_{j}$
$\displaystyle:=ik^{2}b{}_{j}-2\varepsilon{}_{jlp}k{}^{l}\dot{\sigma}{}^{p},$
(53) $\displaystyle\Lambda_{j}$ $\displaystyle:=-b_{j}+2\dot{h}{}_{j}.$ (54)

We raise and lower the indices as $X_{0}=X^{0}$, $X_{j}=-X^{j}$ and use the
conventions
$\Box:=\partial_{\mu}\partial^{\mu}=\partial_{0}^{2}-\partial^{2}$, where the
spacial Laplacian is defined as
$\partial^{2}:=-\eta^{ij}\partial_{j}\partial_{i}=\delta^{ij}\partial_{i}\partial_{j}$.
Also we decompose the perturbations in Fourier space as $f(x^{\mu})\rightarrow
f(k^{\mu})e^{-i\omega t+ik_{j}x^{j}}$ and we define the norm of the wave
covector as $k^{2}:=-\eta^{ij}k_{i}k_{j}=\delta^{ij}k_{i}k_{j}$.

We split the field equations in Eqs. (41–42) in a SVT manner by inserting Eq.
(46) and the we switch to purely gauge invariant variables. This is realised
by using the gauge invariant variables
$\left(\Phi,\chi,\Sigma{}_{i},\Lambda{}_{i}\right)$ introduced in Eqs.
(50-54). The scalar sector consists of 5 linearly independent equations
(considering no possible soft-properties extension Saridakis:2021qxb ) for the
corresponding gauge invariant scalars
$\left(\delta\phi,\psi,\Phi,\chi,\sigma\right)$ which are given as

$\displaystyle W_{00}$
$\displaystyle=k^{2}\Big{(}(G_{\text{Tele},I_{2}}-2G_{4,\phi})\delta\phi+2G_{\text{Tele},T_{\text{vec}}}\Phi-4(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}})\psi\Big{)}\,,$
(55) $\displaystyle k^{j}W{}_{j0}$
$\displaystyle=ik^{2}\bigl{(}(G_{\text{Tele},I_{2}}-2G_{4,\phi})\delta\dot{\phi}+2G_{\text{Tele},T_{\text{vec}}}\chi
k{}^{2}-2(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}})\dot{\psi}\bigr{)}\,,$
(56) $\displaystyle\eta_{jl}W{}^{jl}$
$\displaystyle=3(G_{\text{Tele},I_{2}}-2G_{4,\phi})\delta\ddot{\phi}+2k{}^{2}\Big{(}(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}})\dot{\chi}+(G_{\text{Tele},I_{2}}-2G_{4,\phi})\delta\phi$
$\displaystyle\,\,\,\,\,+2(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}})\Phi-2(G_{4}-G_{\text{Tele},T}+2G_{\text{Tele},T_{\text{vec}}})\psi\Big{)}$
$\displaystyle\,\,\,\,\,-6\Big{(}2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}}\Big{)}\ddot{\psi}\,,$
(57) $\displaystyle k_{l}\epsilon^{ljk}W_{jk}$
$\displaystyle=-\frac{8}{3}iG_{\text{Tele},T_{\text{ax}}}k{}^{2}\bigl{(}\ddot{\sigma}+k{}^{2}\sigma\bigr{)}\,,$
(58) $\displaystyle\widehat{W}$
$\displaystyle=(G_{\text{Tele},X}+G_{2,X}-2G_{3,\phi})\delta\ddot{\phi}+\delta\phi\Big{(}G_{\text{Tele},\phi\phi}+G_{2,\phi\phi}+(G_{\text{Tele},X}+G_{2,X}-2G_{3,\phi})k{}^{2}\Big{)}$
$\displaystyle\,\,\,\,\,+(G_{\text{Tele},I_{2}}-2G_{4,\phi})\bigl{(}-k{}^{2}(\dot{\chi}+\Phi-2\psi)+3\ddot{\psi}\bigr{)}\,.$
(59)

On the other hand, the vector part consists of 3 linearly independent
equations for the gauge invariant variables
$\left(\beta{}_{i},\Sigma{}_{i},\Lambda{}_{i}\right)$

$\displaystyle W_{0j}$
$\displaystyle=\frac{1}{9}\Big{\\{}18G_{\text{Tele},T_{\text{vec}}}\ddot{\beta}{}_{j}+k{}^{2}\Big{[}\Big{(}2G_{\text{Tele},T_{\text{ax}}}-9(G_{4}-G_{\text{Tele},T})\Big{)}\Lambda{}_{j}-\Big{(}9(G_{4}-G_{\text{Tele},T})+2G_{\text{Tele},T_{\text{ax}}}\Big{)}\beta{}_{j}\Big{]}$
$\displaystyle\,\,\,\,\,-2\Big{(}9G_{\text{Tele},T_{\text{vec}}}+2G_{\text{Tele},T_{\text{ax}}}\Big{)}\dot{\Sigma}{}_{j}\Big{\\}}\,,$
(60) $\displaystyle W_{j0}$
$\displaystyle=\frac{1}{9}\Big{\\{}k{}^{2}\Big{[}\Big{(}2G_{\text{Tele},T_{\text{ax}}}-9(G_{4}-G_{\text{Tele},T})\Big{)}\beta{}_{j}-\Big{(}9(G_{4}-G_{\text{Tele},T})+2G_{\text{Tele},T_{\text{ax}}}\Big{)}\Lambda{}_{j}\Big{]}+4G_{\text{Tele},T_{\text{ax}}}\dot{\Sigma}{}_{j}\Big{\\}}\,,$
(61) $\displaystyle k^{l}W{}_{lj}$
$\displaystyle=-\frac{1}{9}i\Big{\\{}k{}^{2}\Big{[}\Big{(}9(G_{4}-G_{\text{Tele},T})+2(9G_{\text{Tele},T_{\text{vec}}}+G_{\text{Tele},T_{\text{ax}}})\Big{)}\dot{\beta}{}_{j}+\Big{(}9(G_{4}-G_{\text{Tele},T})-2G_{\text{Tele},T_{\text{ax}}}\Big{)}\dot{\Lambda}{}_{j}$
$\displaystyle\,\,\,\,\,-18G_{\text{Tele},T_{\text{vec}}}\Sigma{}_{j}\Big{]}+4G_{\text{Tele},T_{\text{ax}}}\ddot{\Sigma}{}_{j}\Big{\\}}\,,$
(62)

while the tensor field is described by one equation

$\displaystyle W_{ij}$
$\displaystyle=(G_{4}-G_{\text{Tele},T})(\ddot{h}_{ij}+k{}^{2}h_{ij})\,.$ (63)

In order to solve the full system of equations for the scalar, vector and
tensor sector we rewrite the corresponding equations as matrices in the
following way:

Scalar sector:

$\left(\begin{array}[]{ccccc}M_{S11}&M_{S12}&M_{S13}&0&0\\\
M_{S21}&M_{S22}&0&M_{S24}&0\\\ M_{S31}&M_{S32}&M_{S33}&M_{S34}&0\\\
0&0&0&0&M_{S45}\\\
M_{S51}&M_{S52}&M_{S53}&M_{S54}&0\end{array}\right)\left(\begin{array}[]{c}\delta\phi\\\
\psi\\\ \Phi\\\ \sigma\\\ \chi\end{array}\right)=:M_{S}Y_{S}=0\,,$ (64)

where the components of the matrix $M_{S}$ are

$\displaystyle M_{S11}$
$\displaystyle=(G_{\text{Tele},I_{2}}-2G_{4,\phi})k{}^{2}\,,\quad
M_{S12}=-4(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}})k{}^{2}\,,\quad
M_{S13}=M_{S12}-\frac{M_{S22}}{\omega}\,,$ (65a) $\displaystyle M_{S21}$
$\displaystyle=\omega M_{S11}\,,\quad
M_{S22}=-2(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}})\omega
k{}^{2}\,,\quad M_{S24}=\frac{ik^{2}}{\omega}(\omega M_{S12}-M_{S22})\,,$
(65b) $\displaystyle M_{S31}$
$\displaystyle=\bigl{(}2-\frac{3\omega^{2}}{k^{2}}\bigr{)}M_{S11}\,,\quad
M_{S32}=-M_{S12}+\frac{M_{S22}}{\omega}\bigl{(}2-\frac{3\omega^{2}}{k^{2}}\bigr{)}\,,\quad
M_{S33}=-M_{S12}\,,$ (65c) $\displaystyle M_{S34}$
$\displaystyle=iM_{S22}\,,\quad
M_{S45}=\frac{8}{3}i\,G_{\text{Tele},T_{\text{ax}}}k^{2}(\omega^{2}-k^{2})\,,$
(65d) $\displaystyle M_{S51}$
$\displaystyle=G_{\text{Tele},\phi\phi}+G_{2,\phi\phi}-(G_{\text{Tele},X}+G_{2,X}-2G_{3,\phi})(\omega^{2}-k^{2})\,,\quad
M_{S52}=\bigl{(}2-\frac{3\omega^{2}}{k^{2}}\bigr{)}M_{S11}\,,$ (65e)
$\displaystyle M_{S53}$ $\displaystyle=-M_{S11}\,,\quad M_{S54}=i\omega
M_{S11}\,.$ (65f)

where we have expressed all the matrix elements wrt to the unique elements
$M_{S11},M_{S12},M_{S22}$ and $M_{S51}$ whenever possible.

Vector sector:

$\left(\begin{array}[]{ccc}M_{V11}&M_{V12}&M_{V13}\\\
M_{V21}&M_{V22}&M_{V23}\\\
M_{V31}&M_{V32}&M_{V33}\end{array}\right)\left(\begin{array}[]{c}\beta{}_{j}\\\
\Sigma{}_{j}\\\ \Lambda{}_{j}\end{array}\right)=:M_{V}Y_{V}=0\,,$ (66)

where the components of the vectorial matrix $M_{V}$ are

$\displaystyle M_{V11}$
$\displaystyle=-2G_{\text{Tele},T_{\text{vec}}}\omega^{2}-\tfrac{1}{9}(9G_{4}-G_{\text{Tele},T}+2G_{\text{Tele},T_{\text{ax}}})k{}^{2}\,,$
(67a) $\displaystyle M_{V12}$ $\displaystyle=\frac{i}{k^{2}}(\omega
M_{V13}-M_{V31})\,,\quad
M_{V13}=\tfrac{1}{9}(-9G_{4}-G_{\text{Tele},T}+2G_{\text{Tele},T_{\text{ax}}})k{}^{2}\,,$
(67b) $\displaystyle M_{V21}$ $\displaystyle=M_{V13}\,,\quad
M_{V22}=\frac{-i\omega}{k^{2}(\omega^{2}-k{}^{2})}\Bigl{(}k{}^{2}M_{V11}+\bigl{(}\omega^{2}-k{}^{2}\bigr{)}M_{V13}-\omega
M_{V31}\Bigr{)}\,,$ (67c) $\displaystyle M_{V23}$
$\displaystyle=\frac{1}{\omega^{2}-k{}^{2}}\bigl{(}-k{}^{2}M_{V11}+\omega
M_{V31}\bigr{)}\,,$ (67d) $\displaystyle M_{V31}$
$\displaystyle=-\tfrac{1}{9}\bigl{(}9G_{4}-G_{\text{Tele},T}+2(9G_{\text{Tele},T_{\text{vec}}}+G_{\text{Tele},T_{\text{ax}}})\bigr{)}\omega
k{}^{2}\,,$ (67e) $\displaystyle M_{V32}$ $\displaystyle=\frac{i}{\omega
k{}^{2}}\Bigl{(}\omega(k{}^{2}M_{V11}+\omega^{2}M_{V13})-\bigl{(}\omega^{2}+k{}^{2}\bigr{)}M_{V31}\Bigr{)}\,,\quad
M_{V33}=\omega M_{V13}\,,$ (67f)

where we have expressed all the matrix elements wrt to the unique elements
$M_{V11},M_{V13}$ and $M_{V31}$ whenever possible.

Tensor sector:

$\displaystyle-(G_{4}-G_{\text{Tele},T})(\omega^{2}-k^{2})=:M_{T}Y_{T}=0\,.$
(68)

where we stress that $G_{4}-G_{\text{Tele},T}\neq 0$. An interesting feature
of this representation is that all three sector equations can be collected
into one master in a block diagonal manner

$\displaystyle\left(\begin{array}[]{ccc}M_{S}&0&0\\\ 0&M_{V}&0\\\
0&0&M_{T}\end{array}\right)\left(\begin{array}[]{c}Y_{S}\\\ Y_{V}\\\
Y_{T}\end{array}\right)$ $\displaystyle=:MY=0\,,$ (75)

where $M_{V}$ and $M_{T}$ are to be understood as $M_{V}\otimes I_{2}$ and
$M_{T}\otimes I_{2}=M_{T}I_{2}$, and $I_{2}$ being the identity matrix of
dimension 2. This is to account for the fact that all indexed fields carry 2
components, i.e, $\beta{}_{j}=\left(\beta{}_{1},\beta{}_{2}\right)$. The
propagation of GWs can analogously be characterized by a principal polynomial
Hohmann:2018jso ; hormander2015analysis ; hormander2015analysisII  of the
perturbation equations which vanishes. Thus, we can calculate the principal
polynomial $P(k):=\det(M)=\det(M_{S})\det(M_{V})\det(M_{T})$ of $M$ as

$\displaystyle P(k)$
$\displaystyle=-2^{14}3^{-5}(G_{4}-G_{\text{Tele},T}){}^{5}G_{\text{Tele},T_{\text{vec}}}{}^{3}G_{\text{Tele},T_{\text{ax}}}{}^{3}k{}^{12}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{8}\Bigl{(}\tilde{c}_{1}+\tilde{c}_{2}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}\Bigr{)}\,,$
(76)

where

$\displaystyle\tilde{c}_{1}$
$\displaystyle:=2(G_{\text{Tele},\phi\phi}+G_{2,\phi\phi})(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}}),$
(77) $\displaystyle\tilde{c}_{2}$
$\displaystyle:=-3(G_{\text{Tele},I_{2}}-2G_{4,\phi})^{2}-2(G_{\text{Tele},X}+G_{2,X}-2G_{3,\phi})(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}})\,.$
(78)

it is also convenient to define the quantities

$\displaystyle\tilde{c}_{3}$
$\displaystyle:=-G_{\text{Tele},\phi\phi}-G_{2,\phi\phi}\,,$ (79)
$\displaystyle\tilde{c}_{4}$
$\displaystyle:=G_{\text{Tele},X}+G_{2,X}-2G_{3,\phi}\,,$ (80)

and

$\displaystyle Z_{1}$
$\displaystyle:=-\frac{(G_{4}-G_{\text{Tele},T})G_{2,\phi\phi}}{-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})(2G_{3,\phi}-G_{2,X})}\,,$
(81) $\displaystyle Z_{2}$
$\displaystyle:=\frac{\bigl{(}3(G_{\text{Tele},I_{2}}-2G_{4,\phi})^{2}+2(2(G_{4}-G_{\text{Tele},T})+3(G_{\text{Tele},T_{\text{vec}}}))(G_{\text{Tele},X}+G_{2,X})\bigr{)}}{4(2(G_{4}-G_{\text{Tele},T})+3(G_{\text{Tele},T_{\text{vec}}}))}\,.$
(82)

Given the form of $P(k)$ in Eq. (76), we expect that in some branches of the
theory there exist massive or massless modes or even both. A detailed analysis
needs to be carried out for all the cases where $P(k)$ is degenerate, i.e
$P(k)=0$ for any $k$. One of these cases can be generated by
$G_{\text{Tele},T_{\text{vec}}}=0$ for example. Thus, in general this is an
iterative process in order to find the reduced and non-degenerate instances of
$P(k)$ and then find the corresponding dispersion relations
$\omega=\omega(k)$. Hence after obtaining the dispersion relations that
correspond to non-degenerate $P(k)$ we obtain the solutions of the system in
Eq. (75), all of which span a null space (the space of all solutions of the
system of Eq. (75)).

Cases | Conditions | Sectors | PDoF
---|---|---|---
Massless$\left(\omega^{2}=k{}^{2}\right)$ | Massive$\left(\omega^{2}-k{}^{2}=m^{2}\right)$
Scalar | Vector | Tensor | Scalar | $m^{2}$
– | $G_{4}-G_{\text{Tele},T}\neq 0$ | - | - | 1 | - | - | 2
0 | $G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}=0,$ |  |  |  |
$G_{\text{Tele},I_{2}}=0,G_{\text{Tele},X}=0,G_{\text{Tele},\phi\phi}=0$ |  |  |  |
0.I |  $G_{2,\phi\phi}\neq 0$ and | - | - | 1 | 1 | $Z_{1}$ | 3
$-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})\left(2G_{3,\phi}-G_{2,X}\right)\neq
0$
0.II |  $G_{2,\phi\phi}=0$ and | - | - | 1 | - | - | 2
$-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})\left(2G_{3,\phi}-G_{2,X}\right)=0$
1 | $G_{\text{Tele},T_{\text{vec}}}\neq 0,\,G_{\text{Tele},T_{\text{ax}}}\neq 0$ | 2 | 1 | 1 | 1 | $-\tilde{c}_{1}/\tilde{c}_{2}$ | 7
$\tilde{c}_{1}\neq 0,\tilde{c}_{2}\neq 0$
2 | $G_{\text{Tele},T_{\text{vec}}}\neq 0,\,G_{\text{Tele},T_{\text{ax}}}\neq 0,\tilde{c}_{1}=0,\tilde{c}_{2}=0$ |  |  |  |
2.I | $G_{\text{Tele},T_{\text{ax}}}\neq 0,\tilde{c}_{3}\neq 0,\tilde{c}_{4}\neq 0$ |
2.I.a | $G_{\text{Tele},T_{\text{ax}}}\neq 0,\tilde{c}_{3}\neq 0,\tilde{c}_{4}\neq 0$ | 1 | 1 | 1 | 1 | $-\tilde{c}_{3}/\tilde{c}_{4}$ | 6
2.I.b | $G_{\text{Tele},T_{\text{ax}}}\neq 0,\tilde{c}_{3}=0,\tilde{c}_{4}=0$ | 1 | 1 | 1 | - | - | 5
2.II | $G_{3,\phi}=Z_{2},\,G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi}$ |  |  |  |
2.II.a | $G_{\text{Tele},I_{2}}-2G_{4,\phi}\neq 0$ | 2 | 1 | 1 | - | - | 6
2.II.b | $G_{\text{Tele},I_{2}}-2G_{4,\phi}=0$ | 1 | 1 | 1 | - | - | 5
3 | $G_{\text{Tele},T_{\text{vec}}}\neq 0,\,G_{\text{Tele},T_{\text{ax}}}=0,\tilde{c}_{1}\neq 0,\tilde{c}_{2}\neq 0$ | 1 | - | 1 | 1 | $-\tilde{c}_{1}/\tilde{c}_{2}$ | 4
4 | $G_{\text{Tele},T_{\text{vec}}}\neq 0,\,G_{\text{Tele},T_{\text{ax}}}=0,\tilde{c}_{1}=0,\tilde{c}_{2}=0$ |  |  |  |
4.I | $G_{\text{Tele},T_{\text{vec}}}=-\tfrac{2}{3}(G_{4}-G_{\text{Tele},T}),\,G_{4,\phi}=\tfrac{1}{2}G_{\text{Tele},I_{2}}$ |
4.I.a | $\tilde{c}_{3}\neq 0,\tilde{c}_{4}\neq 0$ | - | - | 1 | 1 | $-\tilde{c}_{3}/\tilde{c}_{4}$ | 3
4.I.b | $\tilde{c}_{3}=0,\tilde{c}_{4}=0$ | - | - | 1 | - | - | 2
4.II | $G_{3,\phi}=Z_{2},\,G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi}$ |  |  |  |
4.II.a | $G_{\text{Tele},I_{2}}-2G_{4,\phi}\neq 0$ | 1 | - | 1 | - | - | 3
4.II.b | $G_{\text{Tele},I_{2}}-2G_{4,\phi}=0$ | - | - | 1 | - | - | 2
5 | $G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}\neq 0,\tilde{c}_{1}\neq 0,\tilde{c}_{2}\neq 0$ | 1 | - | 1 | 1 | $-\tilde{c}_{1}/\tilde{c}_{2}$ | 4
6 | $G_{\text{Tele},T_{\text{vec}}}=0,G_{\text{Tele},T_{\text{ax}}}\neq 0,\tilde{c}_{1}=0,\tilde{c}_{2}=0$ | 1 | - | 1 | - | - | 3
7 | $G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,\tilde{c}_{1}\neq 0,\tilde{c}_{2}\neq 0$ | - | - | 1 | 1 | $-\tilde{c}_{1}/\tilde{c}_{2}$ | 3
8 | $G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}=0,\tilde{c}_{1}=0,\tilde{c}_{2}=0$ | - | - | 1 | - | - | 2

Table 1: All Branches of the theory are represented with their respective
PDoF. To each of the scalar, vector and tensor components correspond 1,2 and 2
DoF respectively. The quantities $\tilde{c}_{i}$ are defined in (77), (78),
(79) and (80) while $Z_{1}$ and $Z_{2}$ are defined in Eqs. (81)-(82)) and ,
respectively.

It is instructive to go through the simplest cases of GR and $f(T)$ gravity
for which only the combination $(G_{4}-G_{\text{Tele},T})\neq 0$ in Eq. (75)
to illustrate the core steps of the procedure. It is straightforward to see
that after reducing the system to only linearly dependent equations, we obtain
that the system with $M_{s},M_{V}$ and $M_{T}$ being defined in (64), (66) and
(68), respectively, becomes

$\left(\begin{array}[]{cccccc}M_{11}&0&0&0&0&0\\\ M_{21}&-M_{11}&0&0&0&0\\\
0&0&\frac{M_{11}}{4}&0&0&0\\\ 0&0&0&\frac{M_{11}}{4}&0&0\\\
0&0&0&0&M_{55}&0\\\
0&0&0&0&0&M_{55}\end{array}\right)\left(\begin{array}[]{c}\psi\\\ \Phi\\\
\beta{}_{i}\\\ h_{ij}\end{array}\right)=MY\,,$ (83)

with the components $M_{i}$ being

$\displaystyle M_{11}$
$\displaystyle=-4(G_{4}-G_{\text{Tele},T})k{}^{2}\,,\quad
M_{21}=4(G_{4}-G_{\text{Tele},T})\bigl{(}3\omega^{2}-k{}^{2}\bigr{)}\,,$ (84)
$\displaystyle M_{55}$
$\displaystyle=(G_{4}-G_{\text{Tele},T})\bigl{(}-\omega^{2}+k{}^{2}\bigr{)}\,.$
(85)

from which it is already obvious that only the tensor modes propagate,
nevertheless we will go through the steps of the standard procedure used
throughout the cases. The principal polynomial is
$P(k)=-16(G_{4}-G_{\text{Tele},T}){}^{6}k{}^{8}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}$,
which is not degenerate, and thus our dispersion relation describes the usual
massless speed of light propagation $\omega^{2}-k{}^{2}=0$. Substituting in
the dispersion relation into Eq. (75) and solving the system, we generate the
null space of $M$ i.e. all the space of all linearly independent solutions
that the system can have. We then calculate the general solution
$Y_{\left|k\right|}$ which is a linear combination, that uses as coefficients
the $A_{i}\in\mathbb{C}$, of all the elements in the null space of $M$ as

$\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,0,A_{1},A_{2}\right)^{T}\,.$ (86)

Therefore, in the solution the only non-trivial values are in the last two
slots occupied by the tensor perturbations, as expected. Also, notice that the
maximum value of the subscript index of the $A_{i}$ indicates the number of
PDoF of the theory.

In Appendix B we use this approach to categorically analyse each possible case
that ultimately assumes a non-degenerate principal polynomial. These results
are summarized in Table 1. Here, we present the various classes of sub-cases
through various conditions together with a thorough classification of their
PDoF by means of their SVT decomposition as well as whether each sub-case
assumes a massless or massive sector or both. Case 0 revisits the situation of
standard Horndeski gravity which reproduces the results in Ref. Hou:2017bqj ;
Gong:2018ybk  but we further find an extra sub-case that only entails a
massless sector. In this sector only the tensor perturbations propagate and in
this way we exhaustively complete the standard Horndeski scenario. In case 1,
we develop the general case of the full BDLS scenario where none of the
contributions from the action in Eq. (26) vanish. This case corresponds to 7
PDoF, which can be seen as two extra scalars and a vector in the massless
sector compared to the standard Horndeski gravity. Case 2 then explores the
possibility of the $\tilde{c}_{1}$ and $\tilde{c}_{2}$ vanishing which is a
natural class of models that emerges from the study of principlal polynomial
(76) and leads to a rich structure of sub-classes with various expressions of
PDoF. This situation is then further developed in cases 3, 5, and 7 where the
combinations of vanishing $c_{2}$ and $c_{3}$ parameters are analyzed. In
cases 4, 6, and 8, we take $\tilde{c}_{2}=0=\tilde{c}_{3}$ and a similar
series of sub-classes are developed.

Theory | Case | PDoF | Lagrangian Density $\mathcal{L}_{i}$ $\left(S_{i}=\frac{1}{2\kappa^{2}}\int d^{4}x\,e\,\mathcal{L}_{i}\right)$
---|---|---|---
GR or $f(T)$ | - | 2 |  $\accentset{\circ}{R}$ or $f(T)$
Horndeski | 0.I | 3 | Eqs. (27)-(30)
$G_{{\rm Tele}}$ | 1 | 7 | $G_{\text{Tele}}\left(\phi,X,T,T_{\text{ax}},T_{\text{vec}},I_{2},J_{1},J_{3},J_{5},J_{6},J_{8},J_{10}\right)$
Generalized NGR | 2.I.b | 5 | $f(T,T_{{\rm ax}},T_{{\rm vec}})$
Generalized teleparallel dark energy | 7 | 3 | $-A(\phi)T-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)$
Generalized Teleparallel Scalar Tensor | 7 | 3 | $F(\phi)T+P(\phi,X)-G_{3}(\phi,X)\Box\phi$
Tachyonic teleparallel gravity | 7 | 3 | $f(T,X,\phi)$
Table 2: These are some of the literature models shown against our analysis,
as presented in Table. 1.

In totality, BDLS theory predicts a wide range of branches which exhaustively
include all forms of SVT PDoF regarding the massless sector. These branches
may also include a massive sector which is always comprised of a single
(massive) scalar field, a common feature just like with the standard (scalar-
tensor) Horndeski scenario. The branches of the BDLS theory are either just
the massless sector or can be a combination of both the massless and massive
sectors. The standard Horndeski theory, on top of the new massless branch is
explored in Case 0.II, which also falls in the BDLS branch of Case 7. An
interesting feature of this branching is that there are cases distinct from GR
and standard Horndeski which, although enjoy the same number and type of PDoF,
they are actually represent a super-class of the aforementioned standard
paradigms.

##  V Polarizations of GWs

It is a well known fact that metric theories of gravity allow a maximum of six
GW polarizations Eardley:1973br ; Eardley:1974nw . These six polarizations are
classified according to their helicity as two tensor (helicity $\pm 2$) modes
plus (+) and cross ($\times$), two vector (helicity $\pm 1$) modes called x
and y and two scalar (helicity 0) modes named breathing and longitudinal
modes. These polarizations are further illustrated in Fig. 1. We can probe the
polarization content of GWs by measuring the outputs of their relative
amplitudes in the detectors Will:2014kxa ; Chatziioannou:2012rf ; Isi:2017equ
; Callister:2017ocg ; Isi:2017fbj .

$\Phi_{22}\leftrightarrow$ b mode$x$$y$$\odot$$\Psi_{2}\leftrightarrow$ l
mode$z$$x,y$$\rightarrow$$\Re\Psi_{3}\leftrightarrow$ x
mode$z$$x$$\rightarrow$$\Im\Psi_{3}\leftrightarrow$ y
mode$z$$y$$\rightarrow$$\Re\Psi_{4}\leftrightarrow$ $+$
mode$x$$y$$\odot$$\Im\Psi_{4}\leftrightarrow$ $\times$ mode$x$$y$$\odot$
Figure 1: All possible polarizations of GW travelling in z-direction, starting
with the scalar modes breathing, longitudinal, the vector modes x,y, and the
tensor modes $+,\times$. The GW deforms a sphere of freely falling test
particles.

At the moment there is a three-detector network that can be used to
distinguish among very specific subsets of all possible polarization
combinations. On the other hand, to more accurately reconstruct the
polarization content of a GW we would need a five-detector network also
determine the effect of degeneracies. Thus, future measurements will shed more
light on the status on the types of polarization content. Using a simplified
analysis that strictly relies on GR templates Isi:2017fbj , there have been
reported some constraints on pure combinations such as only tensor against
full vector and full tensor against full scalar in Abbott:2017oio ;
Abbott:2018lct ; LIGOScientific:2019fpa . It turned out that in this analysis
using (GW170814, GW170817, and GW170818) only the tensorial polarizations were
not disfavoured.

Let us point out that this analysis states that a GW cannot only have scalar
or only vector polarizations but is highly more likely to have only tensor
polarizations. Still, it does not exclude theories that can have a combination
of tensor with scalar–vector polarizations. It also does not exclude GWs
produced with only vector or only scalar polarizations directly from their
source. Thus, a viable theory must predict at the very least tensor
polarizations.

The way we measure the polarizations is directly linked to the electric
components of the linearized Riemann tensor $\accentset{\circ}{R}_{i0j0}$.
These components control the response of test particles in the presence of a
GW. In analytical terms this is described through the geodesic deviation
equation Ref. Carroll:2004st

$\ddot{x}_{i}=-\accentset{\circ}{R}_{i0j0}x^{j}\,,$ (87)

where dots represent coordinate time derivatives, $(t,x,y,z)=(0,1,2,3)$,
$i=\\{1,2,3\\}$ and $x^{j}=(x,y,z)$. As a matter of fact there is a very
useful tool in studying the polarizations of massless (and almost massless)
GWs which is based on the Newman-Penrose (NP) formalism called the E(2)
classification. This tool facilitates the categorization of the allowed
polarizations of massless gravitational waves in a given gravitational theory
via the representation of the little group, which is the two-dimensional
Euclidean group $\mathrm{E}(2)$ Eardley:1973br ; Eardley:1974nw ; Will:1993ns
. Thus, within the E(2) framework we can parametrize the six independent
components of $\accentset{\circ}{R}_{i0j0}$, using the NP variables as

$\displaystyle\accentset{\circ}{R}{}_{0i0j}$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{2}(\Re\Psi_{4}+\Phi_{22})&\frac{1}{2}\Im\Psi_{4}&-2\Re\Psi_{3}\\\
\frac{1}{2}\Im\Psi_{4}&-\frac{1}{2}(\Re\Psi_{4}-\Phi_{22})&2\Im\Psi_{3}\\\
-2\Re\Psi_{3}&2\Im\Psi_{3}&-6\Psi_{2}\end{array}\right)\,.$ (91)

where $\Phi_{22},\Psi_{2},\Psi_{3},\Psi_{4}$ are the NP variables, $\Re$
represents the real part and $\Im$ the imaginary one. These NP variables can
also be classified with respect to their helicity states through

$\begin{array}[]{lllllc}\Psi_{2}&:&s=0\,,&\Phi_{22}&:&s=0\,,\\\
\Psi_{3}&:&s=-1\,,&\overline{\Psi}_{3}&:&s=1\,,\\\
\Psi_{4}&:&s=-2\,,&\overline{\Psi}_{4}&:&s=2\,,\end{array}$ (92)

where the overbar denotes complex conjugation. One can directly deduce from
Eq. (92) that $\Phi_{22},\Psi_{2}$ are related to scalar DoF, $\Psi_{3}$ is
related to vectorial DoF and finally $\Psi_{4}$ is related to tensorial DoF.
Finally, we can visualize the parametrization of Eq. (91) as in Fig. 1.

Cases | Conditions | Polarizations
---|---|---
Massless sector | Massive sector
$\omega^{2}=k{}^{2}$ | $\omega^{2}-k{}^{2}=m^{2}$
Scalar | Vector | Tensor | Scalar
b | l | x | y | $+$ | $\times$ | b | l
- | $G_{4}-G_{\text{Tele},T}\neq 0$ | - | - | - | - | $\surd$ | $\surd$ | - | -
0 | $G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,G_{\text{Tele},I_{2}}=0\,,$ |  |  |
$G_{\text{Tele},X}=0,G_{\text{Tele},\phi\phi}=0$ |  |  |
0.I |  $G_{2,\phi\phi}\neq 0$ and | - | - | - | - | $\surd$ | $\surd$ | $\surd$ | $\surd$
$-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})\left(2G_{3,\phi}-G_{2,X}\right)\neq
0$
0.II |  $G_{2,\phi\phi}=0$ and $-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})\left(2G_{3,\phi}-G_{2,X}\right)=0$ | - | - | - | - | $\surd$ | $\surd$ | - | -
1 | $G_{\text{Tele},T_{\text{vec}}}\neq 0,\,G_{\text{Tele},T_{\text{ax}}}\neq 0,\,\tilde{c}_{1}\neq 0,\,\tilde{c}_{2}\neq 0$ | $\surd$ | - | - | - | $\surd$ | $\surd$ | $\surd$ | $\surd$
2 | $G_{\text{Tele},T_{\text{vec}}}\neq 0,\,G_{\text{Tele},T_{\text{ax}}}\neq 0,\,\tilde{c}_{1}=0,\,\tilde{c}_{2}=0$ |  |  |  |  |  |  |  |
2.I | $G_{\text{Tele},T_{\text{ax}}}\neq 0,\tilde{c}_{3}\neq 0,\tilde{c}_{4}\neq 0$ |  |  |  |  |  |  |  |
2.I.a | $G_{\text{Tele},T_{\text{ax}}}\neq 0,\,\tilde{c}_{3}\neq 0,\,\tilde{c}_{4}\neq 0$ | - | - | - | - | $\surd$ | $\surd$ | - | -
2.I.b | $c\neq 0,\,\tilde{c}_{3}=0,\,\tilde{c}_{4}=0$ | - | - | - | - | $\surd$ | $\surd$ | - | -
2.II | $G_{3,\phi}=Z_{2},\,G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi}$ |  |  |  |  |  |  |  |
2.II.a | $G_{\text{Tele},I_{2}}-2G_{4,\phi}\neq 0$ | $\surd$ | - | - | - | $\surd$ | $\surd$ | - | -
2.II.b | $G_{\text{Tele},I_{2}}-2G_{4,\phi}=0$ | - | - | - | - | $\surd$ | $\surd$ | - | -
3 | $G_{\text{Tele},T_{\text{vec}}}\neq 0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,\tilde{c}_{1}\neq 0,\,\tilde{c}_{2}\neq 0$ | $\surd$ | - | - | - | $\surd$ | $\surd$ | $\surd$ | $\surd$
4 | $G_{\text{Tele},T_{\text{vec}}}\neq 0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,\tilde{c}_{1}=0,\,\tilde{c}_{2}=0$ |  |  |  |  |  |  |  |
4.I | $G_{\text{Tele},T_{\text{vec}}}=-\tfrac{2}{3}(G_{4}-G_{\text{Tele},T}),\,G_{4,\phi}=\tfrac{1}{2}G_{\text{Tele},I_{2}}$ |  |  |  |  |  |  |  |
4.I.a | $\tilde{c}_{3}\neq 0,\tilde{c}_{4}\neq 0$ | - | - | - | - | $\surd$ | $\surd$ | - | -
4.I.b | $\tilde{c}_{3}=0,\tilde{c}_{4}=0$ | - | - | - | - | $\surd$ | $\surd$ | - | -
4.II | $G_{3,\phi}=Z_{2},\,G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi}$ |  |  |  |  |  |  |  |
4.II.a | $G_{\text{Tele},I_{2}}-2G_{4,\phi}\neq 0$ | $\surd$ | - | - | - | $\surd$ | $\surd$ | - | -
4.II.b | $G_{\text{Tele},I_{2}}-2G_{4,\phi}=0$ | - | - | - | - | $\surd$ | $\surd$ | - | -
5 | $G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}\neq 0,\,\tilde{c}_{1}\neq 0,\,\tilde{c}_{2}\neq 0$ | - | - | - | - | $\surd$ | $\surd$ | $\surd$ | $\surd$
6 | $G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}\neq 0,\,\tilde{c}_{1}=0,\,\tilde{c}_{2}=0$ | - | - | - | - | $\surd$ | $\surd$ | - | -
7 | $G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,\tilde{c}_{1}\neq 0,\,\tilde{c}_{2}\neq 0$ | - | - | - | - | $\surd$ | $\surd$ | $\surd$ | $\surd$
8 | $G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,\tilde{c}_{1}=0,\,\tilde{c}_{2}=0$ | - | - | - | - | $\surd$ | $\surd$ | - | -
Table 3: All Branches of the BDLS theory with corresponding Polarizations. The
quantity $Z_{2}$ is defined in the appendix (see Eqs. (81)-(82)) and the
quantities $\tilde{c}_{i}$ are defined in (77), (78), (79) and (80),
respectively.

For our analysis, we will use the SVT decomposition of the electric components
of the Riemann tensor Eq. (87) which in terms of gauge invariant variables
(see Eq. (49)) assumes the form

$\displaystyle\accentset{\circ}{R}{}_{0i0j}$
$\displaystyle=\left(\begin{array}[]{ccc}\ddot{\psi}-\tfrac{1}{2}\ddot{h}_{+}&-\tfrac{1}{2}\ddot{h}_{\times}&-\tfrac{1}{2}ik(\dot{\beta}_{1}+\dot{\Lambda}_{1})\\\
-\tfrac{1}{2}\ddot{h}_{\times}&\ddot{\psi}+\tfrac{1}{2}\ddot{h}_{+}&-\tfrac{1}{2}ik(\dot{\beta}_{2}+\dot{\Lambda}_{2})\\\
-\tfrac{1}{2}ik(\dot{\beta}_{1}+\dot{\Lambda}_{1})&-\tfrac{1}{2}ik(\dot{\beta}_{2}+\dot{\Lambda}_{2})&\ddot{\psi}-k^{2}(\dot{\chi}+\Phi)\end{array}\right)\,.$
(96)

In contrast to the NP representation in Eq. (91), the SVT counterpart Eq. (96)
is valid for both a massless and massive GWs. As a matter of fact when the GW
is massless then Eq. (96) and Eq. (91) will coincide.

Following the analysis presented in Sec. IV and also using the electric
components of the Riemann tensor as formulated in Eq. (96), we exhaustively
calculated all possible polarizations contained in the BDLS theory. The
calculation process, at this stage, is straightforward since we only had to
substitute the solutions $Y_{\left|k\right|}$ and $Y_{\left|m\right|}$, as
already illustrated in Appendix B, into Eq. (96). Then, we gathered all the
results into Table 3 which closely follows the structure of the PDoF Table 1.

We first considered the most trivial cases of GR and $f(T)$ gravity theories
where we properly reproduced the well known and only tensor polarizations.
Following up with the standard Horndeski Case 0, for the first branch Case
0.I, which assumes 3 PDoF, we found tensor polarizations for the massless
sector and a mix of breathing and longitudinal modes for the massive sector.
Regarding the new branch labeled Case 0.II, we found only tensor
polarizations, as expected, since only tensorial PDoF were found in Sec. IV.

Delving into the TG realm, in Case 1 (the full BDLS theory), we found the
breathing mode along the usual tensor modes in the massless section and the
breathing along with the longitudinal for the massive sector. In terms of the
polarizations content the full BDLS theory has an extra breathing mode for the
massless sector compared to the standard full Horndeski (Case 0.I). This
unique polarization imprint is only shared between Case 1 and Case 3 inside
the BDLS class although Case 1 includes one additional massless scalar and one
massless vectorial PDoF.

Another interesting type of polarization imprint was produced in Cases 2.I.a,
2.I.b, 2.II.b, 5 and 6 where although there exists a massless scalar PDoF it
does not leave a polarization imprint. This also happens for Cases 2.I.a and
4.I.a for a massive scalar DoF. The elusiveness of DoF also happens for the
vectorial DoFs found in Cases 1, 2.I.a, 2.I.b, 2.II.a and 2.II.b, which are
the only cases predicting vectorial DoF. This, phenomenon of a DoF not leaving
a polarization imprint is directly linked to the fact that it is not coupled
to the corresponding metrical DoF to which the Riemann tensor is sensitive to.

We also have a few trivial subclasses described by Cases 2.I.b, 2.II.b, 4.I.b,
4.II.b, 6 and 8 where only tensor polarizations were found. This, is also not
so straightforward to expect since except for the Cases 4.I.b, 4.II.b and 8
which only enjoy tensorial PDoF, the cases 2.I.b, 2.II.b and 6 predict more
than 2 DoF as as explained in Sec. IV.

On the whole, in the BDLS theory although scalar, vectorial and tensorial
propagating DoF are predicted as illustrated in Sec. IV we can have only
combinations of scalar, tensor polarizations for the massless sectors and only
scalar ones for the massive sectors. Thus, although vectorial propagating DoF
are predicted they leave no polarization imprint.

##  VI Conclusion

This work explores the number and nature of the PDoF associated with BDLS
theory in a Minkowski background. In this context, we also expose the
expressions of GW polarization through the various subclasses of models of the
model. BDLS theory is a TG analogous formulation of Horndeski gravity in which
the connection is changed from Levi-Civita, which is curvature-based, to the
teleparallel connection, which is torsional. This simple change drastically
impacts the breath of the theory by adding a term in the Lagrangian (26) which
starkly increases the potential models that ensue. We cover this background in
Sec. II where we also introduce the gravitational tools in which BDLS theory
is constructed.

In Sec. III, we then describe the transition to tetrad based perturbations.
The use of metric tensor perturbations is well known Mukhanov:991646 .
However, given that metrics do not define unique tetrads, we must be careful
to define tetrads that not only produce the metric but also that continue to
satisfy the field equations (up to perturbative order). In Eqs. (31,32) we
prescribe the tetrad together with its scalar field perturbation which
reproduces the correct perturbed metric in Eq. (34). The equations of motion
are then determined (41,42) and simplified into an appropriate form using
symmetry operators (43,44).

The perturbed field equations are explored in Sec. IV where the tetrad is
decomposed into its SVT constituents (45) which can be correlated with their
the usual SVT decomposed metric (47). Through this prism, we explore the
number of PDoF which is an important property to determine to accurately
understand how this freedom is then expressed. In Table 1 we display the
nature and number of DoF and determine whether they are massive or massless
for the various subclasses of models within BDLS theory (which expands on the
Horndeski branches). The details of these tedious calculations are shown in
Appendix B where each branch is explained.

It is important to highlight that we identify a number of cases in which extra
PDoF are identified where we find a maximum of seven PDoF for one of these
cases. In general, we find both massive and massless cases and a distribution
over the scalar–vector–tensor modes for these DoF. In the massive instance, we
naturally find that this is associated with the scalar field. To better
represent the impact of these results, we list how this will effect a number
of well-known TG theories that are second order in Table 2. While some of
these have already been determined elsewhere, we are in agreement with
literature results. The prism of the BDLS derivation presented here is that it
can also collect these results together in one formalism.

PDoF can be expressed as GW polarization which we explore in Sec. V. Here, we
describe the possible SVT polarizations of GWs as well as the way in which
they can be determined though the Levi-Civita Riemann tensor quantity in the
geodesic deviation equation (87). The results are presented in Table 1 where
we describe the polarization characteristics for each of the subclasses
defined in Table 3. Of particular interest, we expand on the work done in
standard Horndeski gravity in which the massive branch was explored in Ref.
Hou:2017bqj  but where the massless case is omitted, which we expand upon
here.

We also find that a maximum of four polarization modes can be produced by BDLS
theory and that none of them express a vector mode. Despite this, we continue
to have massless and massive scalar mode cases. Beyond the standard Horndeski
gravity, we find a wide variety on polarization states in these subclasses
which may be interesting for future polarization detection analyses in the
coming years.

GW polarization is a key test of gravity that is yet to be exposed to the same
level of details as other features of gravity such as the speed of propagation
of GW radiation. In general, TG has not been exhaustively studied but has
shown promise in a number of interesting areas of astrophyscs Bamba:2012cp ;
Linder:2010py ; Escamilla-Rivera:2019ulu ; Finch:2018gkh ; Briffa:2020qli ;
LeviSaid:2020mbb ; Farrugia:2016xcw ; Farrugia:2016qqe ; Bahamonde:2020bbc .
As expressed in this work, BDLS theory offers a wider landscape on which to
produce second-order theories of gravity where only one scalar field is
incorporated. It would be interesting to further study the features of this
theory.

### Acknowledgements

J.L.S. would also like to acknowledge funding support from Cosmology@MALTA
which is supported by the University of Malta. S.B. and M.H. were supported by
the Estonian Research Council grants PRG356 “Gauge Gravity” and by the
European Regional Development Fund through the Center of Excellence TK133 “The
Dark Side of the Universe”. The authors would like to acknowledge networking
support by the COST Action CA18108. The research in this work is partially
funded by the Tertiary Education Scholarship Scheme (TESS, Malta).

##  Appendix A Expansion of Multivariable Function

The perturbative expansion of the field equations is obtained by first
expanding the functions in the Lagrangian. This is done by performing a Taylor
expansion about background values. Consider a function of scalars, say
$G(\alpha,\beta)$, such that the parameters of the function are expanded as
follows

$\displaystyle\alpha$
$\displaystyle=\alpha^{(0)}+\epsilon\,\alpha^{(1)}+\epsilon^{2}\alpha^{(2)}\,,$
(97) $\displaystyle\beta$
$\displaystyle=\beta^{(0)}+\epsilon\,\beta^{(1)}+\epsilon^{2}\beta^{(2)}+\epsilon^{3}\beta^{(3)}\,,$
(98)

where $\alpha^{(0)}$ and $\beta^{(0)}$ are constants at the zeroth order which
represent the background parameters. Taylor expanding function $G$ about the
zeroth order results in the following equation:

$\displaystyle G(\alpha,\beta)$
$\displaystyle=G(0)+G_{,\alpha}(0)\left[\alpha-\alpha^{(0)}\right]+G_{,\beta}(0)\left[\beta-\beta^{(0)}\right]+G_{,\alpha\beta}(0)\left[\alpha-\alpha^{(0)}\right]\left[\beta-\beta^{(0)}\right]$
(99)
$\displaystyle\quad+\frac{1}{2}G_{,\alpha\alpha}(0)\left[\alpha-\alpha^{(0)}\right]^{2}+\frac{1}{2}G_{,\beta\beta}(0)\left[\beta-\beta^{(0)}\right]^{2}+...$
$\displaystyle=G(0)+\epsilon\left[G_{,\alpha}(0)\alpha^{(1)}+G_{,\beta}(0)\beta^{(1)}\right]$
$\displaystyle\quad+\epsilon^{2}\left[G_{,\alpha}(0)\alpha^{(2)}+G_{,\beta}(0)\beta^{(2)}+G_{,\alpha\beta}(0)\alpha^{(1)}\beta^{(1)}+\frac{1}{2}\left(G_{,\alpha\alpha}(0){\alpha^{(1)}}^{2}+G_{,\beta\beta}(0){\beta^{(1)}}^{2}\right)\right]+\mathcal{O}(\epsilon^{3})\,,$

where $G(0)=G\left(\alpha^{(0)},\beta^{(0)}\right)$ stands for the background
of the function $G$. This equation shows that scalars with contributions to
perturbation orders higher two, such as $\beta^{(3)}$, will not appear in the
second order expansion of the function. In the case of BDLS theory, the scalar
invariants $J_{1}$, $J_{3}$, $J_{5}$, $J_{6}$, $J_{8}$ and $J_{10}$ do not
appear in the expansion as seen in Eq. (35) since they do not have
contributions that are smaller than the perturbative order of 3. Moreover, in
the case of $X$, there is only a second order contribution which leads to no
second order derivative terms in the function expansion.

##  Appendix B Solutions and Branching

The general solution of the system in Eq. (75) will be a linear combination of
the elements of the null space of matrix $M$. We will denote the coefficients
of the linear combinations as $A_{i}\in\mathbb{C}$ for the massless dispersion
relation $\omega^{2}-k^{2}=0$ and $B_{i}\in\mathbb{C}$ for the massive
dispersion relation $\omega^{2}-k^{2}=m^{2}$. These coefficients also act as
labels for the number of PDoF. Hence each $A_{i}$ or $B_{i}$ denotes one PDoF.
If there are both massless and massive branches we have a solution space for
each sector and thus the sum of their dimensions is the total number PDoF. We
will denote the solutions with respect to to the massless dispersion relation
$\omega^{2}-k^{2}=0$ as $Y_{\left|k\right|}$ and the ones corresponding to a
massive dispersion relation $\omega^{2}-k^{2}=m^{2}$ as $Y_{\left|m\right|}$.
Finally these solutions will be equal and compared with the column vector $Y$
defined in Eq. (75).

After each solution $Y_{\left|k\right|}$ or $Y_{\left|m\right|}$ we will
calculate the electric components of the Riemann tensor that correspond to
each of them as $\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$ or
$\accentset{\circ}{R}_{0i0j}(Y_{\left|m\right|})$ accordingly. Finally, we
will comment only on the first few interesting cases since, although the list
is exhaustive, the rest of the cases have similar or identical interpretation.

Let us now explore all the possible cases.

Case 0: Horndeski
$\left(G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,G_{\text{Tele},I_{2}}=0,\,G_{\text{Tele},X}=0,\,G_{\text{Tele},\phi\phi}=0\right)$

The Horndeski case has the following determinant

$\displaystyle P(k)$
$\displaystyle=-16(G_{4}-G_{\text{Tele},T}){}^{5}k{}^{8}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\Bigl{(}(G_{4}-G_{\text{Tele},T})G_{2,\phi\phi}+(-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})(2G_{3,\phi}-G_{2,X}))\bigl{(}\omega^{2}-k{}^{2}\bigr{)}\Bigr{)}\,,$
(100)

from which we further have two sub-cases:

Case 0.I $(G_{2,\phi\phi}\neq 0$ and
$-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})\left(2G_{3,\phi}-G_{2,X}\right)\neq
0)$.

In this case the determinant reads

$\displaystyle P(k)$
$\displaystyle=-16(G_{4}-G_{\text{Tele},T}){}^{5}k{}^{8}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\Bigl{(}(G_{4}-G_{\text{Tele},T})G_{2,\phi\phi}+(-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})(2G_{3,\phi}-G_{2,X}))\bigl{(}\omega^{2}-k{}^{2}\bigr{)}\Bigr{)}\,,$
(101)

from which it is evident that it is non-degenerate and there are two
dispersion relations $\omega^{2}-k{}^{2}=0$ massless speed of light
propagation and

$(G_{4}-G_{\text{Tele},T})G_{2,\phi\phi}+(-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})(2G_{3,\phi}-G_{2,X}))\bigl{(}\omega^{2}-k{}^{2}\bigr{)}=0$

which defines an effective mass as

$m^{2}=-\frac{(G_{4}-G_{\text{Tele},T})G_{2,\phi\phi}}{-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})(2G_{3,\phi}-G_{2,X}))}>0\,,$

and then

$\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\Phi,\beta{}_{i},h{}_{ij}\right)\,,$
(102) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,0,0,\frac{2A_{1}}{k^{2}},\frac{2A_{2}}{k^{2}}\right)^{T}\,,$
(103) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}&A_{2}&0\\\ A_{2}&-A_{1}&0\\\
0&0&0\end{array}\right)\,,$ (107) $\displaystyle Y_{\left|m\right|}$
$\displaystyle=\left(2(G_{4}-G_{\text{Tele},T})B_{1},-G_{4,\phi}B_{1},G_{4,\phi}B_{1},0,0,0,0\right)^{T},$
(108) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|m\right|})$
$\displaystyle=G_{4,\phi}\left(\begin{array}[]{ccc}\bigl{(}m^{2}+k{}^{2}\bigr{)}B_{1}&0&0\\\
0&\bigl{(}m^{2}+k{}^{2}\bigr{)}B_{1}&0\\\ 0&0&m^{2}B_{1}\end{array}\right)\,.$
(112)

The full Horndeski theory assumes 3 PDoF, 2 of which are the tensor modes
described by the parameters $\left(A_{1},A_{2}\right)$ in the massless sector
Eq. (103) and the remaining DoF is a scalar described by the parameter $B_{1}$
in the massive sector Eq. (108). It is evident that by setting $G_{4,\phi}=0$
one can completely hide the massive scalar from the polarization detectors Eq.
(112) but it will still propagate as can be seen from Eq. (108).

Case 0.II ($G_{2,\phi\phi}=0$ and
$-3G_{4,\phi}{}^{2}+(G_{4}-G_{\text{Tele},T})\left(2G_{3,\phi}-G_{2,X}\right)=0$).

A solution of this system that covers the whole solution manifold is

$G_{2,X}=-\frac{3G_{4,\phi}{}^{2}}{(G_{4}-G_{\text{Tele},T})}+2G_{3,\phi}\,,$

and then the determinant becomes

$\displaystyle P(k)$
$\displaystyle=-8(G_{4}-G_{\text{Tele},T}){}^{5}G_{4,\phi}k{}^{8}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\,,$
(113)

and for $G_{4,\phi}\neq 0$ the solution is

$\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\beta{}_{i},h{}_{ij}\right)\,,$ (114)
$\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,0,\frac{2A_{1}}{k^{2}},\frac{2A_{2}}{k^{2}}\right)^{T}\,,$
(115)

whereas for $G_{4,\phi}=0$

$\displaystyle Y_{\omega}$
$\displaystyle=\left(\psi,\beta{}_{i},h{}_{ij}\right)\,,$ (116) $\displaystyle
Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,\frac{2A_{1}}{k^{2}},\frac{2A_{2}}{k^{2}}\right)^{T},$
(117)

and thus for both cases $G_{4,\phi}=0$ and $G_{4,\phi}\neq 0$

$\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}&A_{2}&0\\\ A_{2}&-A_{1}&0\\\
0&0&0\end{array}\right),$ (121)

Although the algebraic solutions are different as can be seen from Eq. (115)
and Eq. (117) the final physical solution is the same, i.e only tensor modes.
This is also reflected in the polarizations described by Eq. (121) since their
value does not depend at all by the choice of $G_{4,\phi}$.

Case 1: Full BDLS theory $(G_{\text{Tele},T_{\text{vec}}}\neq
0,\,G_{\text{Tele},T_{\text{ax}}}\neq 0,\,\tilde{c}_{1}\neq
0,\,\tilde{c}_{2}\neq 0)$

In this case the determinant reads

$\displaystyle P(k)$
$\displaystyle=-\frac{16384}{243}(G_{4}-G_{\text{Tele},T}){}^{5}G_{\text{Tele},T_{\text{vec}}}{}^{3}G_{\text{Tele},T_{\text{ax}}}{}^{3}k{}^{12}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{8}\Bigl{(}\tilde{c}_{1}+\tilde{c}_{2}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}\Bigr{)},$
(122)

where $\tilde{c}_{1}$ and $\tilde{c}_{2}$ are defined in Eq. (77)-Eq. (78). It
is evident that it is non-degenerate and there are two dispersion relations
$\omega^{2}-k{}^{2}=0$ which describes massless speed 1 propagation and
$\tilde{c}_{1}+\tilde{c}_{2}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}=0$ which
defines an effective mass as $m^{2}=-\frac{\tilde{c}_{1}}{\tilde{c}_{2}}>0$.
The solutions for this case are

$\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\Phi,\chi,\sigma,\beta{}_{j},\Sigma{}_{j},\Lambda{}_{j},h{}_{ij}\right)\,,$
(123) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\Big{(}0,-\frac{A_{1}}{k^{2}},-\frac{2A_{1}}{G_{\text{Tele},T_{\text{vec}}}k{}^{2}}(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}}),$
$\displaystyle\,\,\,\frac{iA_{1}}{G_{\text{Tele},T_{\text{vec}}}\left|k\right|{}^{3}}(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}}),A_{2},-A_{3},-A_{4},i\left|k\right|A_{3},i\left|k\right|A_{4},A_{3},A_{4},\frac{2A_{5}}{k^{2}},\frac{2A_{6}}{k^{2}}\Big{)}^{T}\,,$
(124) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}+A_{5}&A_{6}&0\\\
A_{6}&A_{1}-A_{5}&0\\\ 0&0&0\end{array}\right)\,,$ (128) $\displaystyle
Y_{\left|m\right|}$
$\displaystyle=\Big{(}-2(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}})B_{1},-(G_{\text{Tele},I_{2}}-2G_{4,\phi})B_{1},(G_{\text{Tele},I_{2}}-2G_{4,\phi})B_{1},$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,0,0,0,0,0,0,0,0,0,0\Big{)}^{T}\,,$
(129) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|m\right|})$
$\displaystyle=\left(G_{\text{Tele},I_{2}}-2G_{4,\phi}\right)\left(\begin{array}[]{ccc}\bigl{(}m^{2}+k{}^{2}\bigr{)}B_{1}&0&0\\\
0&\bigl{(}m^{2}+k{}^{2}\bigr{)}B_{1}&0\\\ 0&0&m^{2}B_{1}\end{array}\right)\,.$
(133)

The system in total assumes 7 PDoF which are divided as 6 in the massless
sector described by Eq. (124) and 1 in the massive sector described by Eq.
(129). The massless sector is parametrized by $\left(A_{1},..,A_{6}\right)$
which are packed as 2 scalars $\left(A_{1},A_{2}\right)$, one vector
$\left(A_{3},A_{4}\right)$ and the tensor modes $\left(A_{5},A_{6}\right)$.
Regarding the polarization content, the massless sector enjoys the usual
tensor $(A_{5},A_{6})$ polarizations along with the breathing
$\left(A_{1}\right)$ mode. The massive sector contains one massive scalar
described by $B_{1}$. One can see that by setting
$\left(G_{\text{Tele},I_{2}}-2G_{4,\phi}\right)\rightarrow 0$ in Eq. (133) the
massive scalar mode becomes undetectable to the polarization detectors
although it will still propagating as can be seen from Eq. (129).

Case 2 $(G_{\text{Tele},T_{\text{vec}}}\neq
0,\,G_{\text{Tele},T_{\text{ax}}}\neq 0,\,\tilde{c}_{1}=0,\,\tilde{c}_{2}=0)$

A pair of solutions of the system $\tilde{c}_{1}=0,\tilde{c}_{2}=0$ that
covers the whole solution manifold are

$G_{\text{Tele},T_{\text{vec}}}=-\tfrac{2}{3}(G_{4}-G_{\text{Tele},T}),G_{4,\phi}=\tfrac{1}{2}G_{\text{Tele},I_{2}}$

and

$G_{3,\phi}=\frac{\bigl{(}3(G_{\text{Tele},I_{2}}-2G_{4,\phi})^{2}+2(2(G_{4}-G_{\text{Tele},T})+3(G_{\text{Tele},T_{\text{vec}}}))(G_{\text{Tele},X}+G_{2,X})\bigr{)}}{4(2(G_{4}-G_{\text{Tele},T})+3(G_{\text{Tele},T_{\text{vec}}}))},\,G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi}\,.$

Case 2.I
$(G_{\text{Tele},T_{\text{vec}}}=-\tfrac{2}{3}(G_{4}-G_{\text{Tele},T}),\,G_{4,\phi}=\tfrac{1}{2}G_{\text{Tele},I_{2}})$

The determinant of the system reads as

$\displaystyle P(k)$
$\displaystyle=2^{17}3^{-9}(G_{4}-G_{\text{Tele},T}){}^{8}(G_{\text{Tele},T_{\text{ax}}}){}^{3}k{}^{12}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{7}\Bigl{(}\tilde{c}_{3}+\tilde{c}_{4}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}\Bigr{)}\,,$
(134)

which in turn means that we have more subcases.

Case 2.I.a $(G_{\text{Tele},T_{\text{ax}}}\neq 0,\,\tilde{c}_{3}\neq
0,\tilde{c}_{4}\neq 0)$

The corresponding important quantities in this subcase are:

$\displaystyle P(k)$
$\displaystyle=\tfrac{131072}{19683}(G_{4}-G_{\text{Tele},T}){}^{8}G_{\text{Tele},T_{\text{ax}}}{}^{3}k{}^{12}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{7}\Bigl{(}\tilde{c}_{3}+\tilde{c}_{4}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}\Bigr{)}\,,$
(135) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\Phi,\chi,\sigma,\beta{}_{i},\Sigma{}_{i},\Lambda{}_{i},h{}_{ij}\right)\,,$
(136) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,A_{1},-A_{2},-A_{3},i\left|k\right|A_{2},i\left|k\right|A_{3},A_{2},A_{3},\frac{2A_{4}}{k^{2}},\frac{2A_{5}}{k^{2}}\right)^{T}\,,$
(137) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{4}&A_{5}&0\\\ A_{5}&-A_{4}&0\\\
0&0&0\end{array}\right)\,,$ (141) $\displaystyle Y_{\left|m\right|}$
$\displaystyle=\left(B_{1},0,0,0,0,0,0,0,0,0,0,0\right)^{T}\,,$ (142)
$\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|m\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)\,.$ (146)

This is a peculiar case where the only scalar PDoF that are scalars $\sigma$
which is massless and $\delta\phi$ which is the massive are not at all
involved in the polarizations. This is due to the fact that they are not
coupled to any metric$\left(\Phi,\chi\right)$ scalar DoF and hence they cannot
have a polarization imprint in both the massless and massive sectors.

Case 2.I.b $(G_{\text{Tele},T_{\text{ax}}}\neq
0,\,\tilde{c}_{3}=0,\,\tilde{c}_{4}=0)$

The solution of $\tilde{c}_{3}=0,\tilde{c}_{4}=0$ is

$G_{3,\phi}=\frac{(G_{\text{Tele},X}+G_{2,X})}{2}\,,\quad
G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi}\,,$

and then the important quantities become:

$\displaystyle P(k)$
$\displaystyle=\tfrac{131072}{19683}(G_{4}-G_{\text{Tele},T}){}^{8}G_{\text{Tele},T_{\text{ax}}}{}^{3}k{}^{12}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{7}\,,$
(147) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\Phi,\chi,\sigma,\beta{}_{i},\Sigma{}_{i},\Lambda{}_{i},h{}_{ij}\right)\,,$
(148) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,A_{1},-A_{2},-A_{3},i\left|k\right|A_{2},i\left|k\right|A_{3},A_{2},A_{3},,\frac{2A_{4}}{k^{2}},\frac{2A_{5}}{k^{2}}\right)^{T}\,,$
(149) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{4}&A_{5}&0\\\ A_{5}&-A_{4}&0\\\
0&0&0\end{array}\right)\,.$ (153)

Case 2.II $(G_{3,\phi}=Z_{2},\,G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi})$

In this case $Z_{2}$ is defined in (82). Further we have
$2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}}\neq 0$ and then we
have:

$\displaystyle P(k)$
$\displaystyle=\tfrac{16384}{243}(G_{4}-G_{\text{Tele},T}){}^{5}G_{\text{Tele},T_{\text{vec}}}{}^{3}G_{\text{Tele},T_{\text{ax}}}{}^{3}(G_{\text{Tele},I_{2}}-2G_{4,\phi})k{}^{12}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{8}\,.$
(154)

There are further subsubcases:

Case 2.II.a ($G_{\text{Tele},I_{2}}-2G_{4,\phi}\neq 0.$)

Here, we have the following terms:

$\displaystyle P(k)$
$\displaystyle=\tfrac{16384}{243}(G_{4}-G_{\text{Tele},T}){}^{5}(G_{\text{Tele},T_{\text{vec}}}){}^{3}(G_{\text{Tele},T_{\text{ax}}}){}^{3}(G_{\text{Tele},I_{2}}-2G_{4,\phi})k{}^{12}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{8}\,,$
(155) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\chi,\sigma,\beta{}_{i},\Sigma{}_{i},\Lambda{}_{i},h{}_{ij}\right)\,,$
(156) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(-\frac{4(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}})}{(G_{\text{Tele},I_{2}}-2G_{4,\phi})k{}^{2}}A_{1},-\frac{A_{1}}{k^{2}},\frac{iA_{1}}{k^{3}},A_{2},-A_{3},-A_{4},ikA_{3},ikA_{4},A_{3},A_{4},\frac{2A_{5}}{k^{2}},\frac{2A_{6}}{k^{2}}\right)^{T}\,,$
(157) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}+A_{5}&A_{6}&0\\\
A_{6}&A_{1}-A_{5}&0\\\ 0&0&0\end{array}\right)\,.$ (161)

Case 2.II.b ($G_{\text{Tele},I_{2}}-2G_{4,\phi}=0.$)

The determinant of the system in this case is

$\displaystyle P(k)$
$\displaystyle=-\tfrac{16384}{243}(G_{4}-G_{\text{Tele},T}){}^{4}G_{\text{Tele},T_{\text{vec}}}{}^{3}(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}})G_{\text{Tele},T_{\text{ax}}}{}^{3}k{}^{12}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{7}\,.$
(162)

Case 2.II.b.1 $(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}}\neq 0)$

$\displaystyle P(k)$
$\displaystyle=-\tfrac{16384}{243}(G_{4}-G_{\text{Tele},T}){}^{4}G_{\text{Tele},T_{\text{vec}}}{}^{3}(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}})G_{\text{Tele},T_{\text{ax}}}{}^{3}k{}^{12}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{7}\,,$
(163) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\Phi,\chi,\sigma,\beta{}_{i},\Sigma{}_{i},\Lambda{}_{i},h{}_{ij}\right)\,,$
(164) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,A_{1},-A_{2},-A_{3},i\left|k\right|A_{2},i\left|k\right|A_{3},A_{2},A_{3},\frac{2A_{4}}{k^{2}},\frac{2A_{5}}{k^{2}}\right)^{T}\,,$
(165) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{4}&A_{5}&0\\\ A_{5}&-A_{4}&0\\\
0&0&0\end{array}\right)\,.$ (169)

Case 2.II.b.2 $(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}}=0)$

$\displaystyle P(k)$
$\displaystyle=-\tfrac{4096}{243}i(G_{4}-G_{\text{Tele},T}){}^{7}(G_{\text{Tele},T_{\text{ax}}}){}^{3}\omega
k{}^{8}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{7}\,,$ (170) $\displaystyle
Y_{\omega}$
$\displaystyle=\left(\Phi,\sigma,\beta{}_{i},\Sigma{}_{i},\Lambda{}_{i},h{}_{ij}\right)\,,$
(171) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,A_{1},-A_{2},-A_{3},i\left|k\right|A_{2},i\left|k\right|A_{3},A_{2},A_{3},\frac{2A_{4}}{k^{2}},\frac{2A_{5}}{k^{2}}\right)^{T}\,,$
(172) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{4}&A_{5}&0\\\ A_{5}&-A_{4}&0\\\
0&0&0\end{array}\right)\,.$ (176)

Case 3 $(G_{\text{Tele},T_{\text{vec}}}\neq
0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,\tilde{c}_{1}\neq 0,\,\tilde{c}_{2}\neq
0)$

$\displaystyle P(k)$
$\displaystyle=-32i(G_{4}-G_{\text{Tele},T}){}^{5}G_{\text{Tele},T_{\text{vec}}}{}^{3}\omega^{2}k{}^{10}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{3}\Bigl{(}\tilde{c}_{1}+\tilde{c}_{2}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}\Bigr{)}\,,$
(177) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\Phi,\chi,\beta{}_{i},\Sigma{}_{i},h{}_{ij}\right)\,,$
(178) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\Big{(}0,-\frac{A_{1}}{k^{2}},-\frac{2(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}})}{(G_{\text{Tele},T_{\text{vec}}})k{}^{2}}A_{1},\frac{i(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}})}{(G_{\text{Tele},T_{\text{vec}}})\left|k\right|{}^{3}}A_{1},0$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,,0,0,0,\frac{2A_{2}}{k^{2}},\frac{2A_{3}}{k^{2}}\Big{)}^{T}\,,$
(179) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}+A_{2}&A_{3}&0\\\
A_{3}&A_{1}-A_{2}&0\\\ 0&0&0\end{array}\right)\,,$ (183) $\displaystyle
Y_{\left|m\right|}$
$\displaystyle=\left(-2(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}})B_{1},-(G_{\text{Tele},I_{2}}-2G_{4,\phi})B_{1},(G_{\text{Tele},I_{2}}-2G_{4,\phi})B_{1},0,0,0,0,0,0,0\right)^{T}\,,$
(184) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|m\right|})$
$\displaystyle=(G_{\text{Tele},I_{2}}-2G_{4,\phi})\left(\begin{array}[]{ccc}\bigl{(}m^{2}+k{}^{2}\bigr{)}B_{1}&0&0\\\
0&\bigl{(}m^{2}+k{}^{2}\bigr{)}B_{1}&0\\\ 0&0&m^{2}B_{1}\end{array}\right)\,,$
(188)

where we can hide the massive scalar from the polarizations by setting
$(G_{\text{Tele},I_{2}}-2G_{4,\phi})=0$.

Case 4 $(G_{\text{Tele},T_{\text{vec}}}\neq
0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,\tilde{c}_{1}=0,\,\tilde{c}_{2}=0)$

A pair of solutions of the system $\tilde{c}_{1}=0,\tilde{c}_{2}=0$ that cover
the whole solution manifold are

$G_{\text{Tele},T_{\text{vec}}}=-\tfrac{2}{3}(G_{4}-G_{\text{Tele},T})\,,\quad
G_{4,\phi}=\tfrac{1}{2}G_{\text{Tele},I_{2}}\,,$

and

$G_{3,\phi}=\frac{\bigl{(}3(G_{\text{Tele},I_{2}}-2G_{4,\phi})^{2}+2(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}})(G_{\text{Tele},X}+G_{2,X})\bigr{)}}{4(2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}})}\,,\quad
G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi}\,.$

Case 4.I
$(G_{\text{Tele},T_{\text{vec}}}=-\tfrac{2}{3}(G_{4}-G_{\text{Tele},T}),\,G_{4,\phi}=\tfrac{1}{2}G_{\text{Tele},I_{2}})$

By defining

$\displaystyle\tilde{c}_{3}$
$\displaystyle=-G_{\text{Tele},\phi\phi}-G_{2,\phi\phi}\,,$ (189)
$\displaystyle\tilde{c}_{4}$
$\displaystyle=G_{\text{Tele},X}+G_{2,X}-2G_{3,\phi}\,,$ (190)

the determinant of the system reads as

$\displaystyle P(k)$
$\displaystyle=\tfrac{256}{81}i(G_{4}-G_{\text{Tele},T}){}^{8}\omega^{2}k^{10}\bigl{(}\omega^{2}-k^{2}\bigr{)}^{2}\Bigl{(}\tilde{c}_{3}+\tilde{c}_{4}\bigl{(}\omega^{2}-k^{2}\bigr{)}\Bigr{)}\,,$
(191)

which in turn means that we have more subcases

Case 4.I.a $(\tilde{c}_{3}\neq 0,\,\tilde{c}_{4}\neq 0)$

$\displaystyle P(k)$
$\displaystyle=\tfrac{256}{81}i(G_{4}-G_{\text{Tele},T}){}^{8}\omega^{2}k{}^{10}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\Bigl{(}\tilde{c}_{3}+\tilde{c}_{4}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}\Bigr{)}\,,$
(192) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\chi,\beta{}_{i},\Sigma{}_{i},h{}_{ij}\right)\,,$
(193) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,0,0,0,0,\frac{2}{k^{2}}A_{1},\frac{2}{k^{2}}A_{2}\right)^{T}\,,$
(194) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}&A_{2}&0\\\ A_{2}&-A_{1}&0\\\
0&0&0\end{array}\right)\,,$ (198) $\displaystyle Y_{\left|m\right|}$
$\displaystyle=\left(B_{1},0,0,0,0,0,0,0,0\right)^{T}\,,$ (199)
$\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|m\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)\,.$ (203)

Case 4.I.b $(\tilde{c}_{3}=0,\,\tilde{c}_{4}=0)$

The solution of $\tilde{c}_{3}=0,\,\tilde{c}_{4}=0$ is

$G_{3,\phi}=\tfrac{1}{2}(G_{\text{Tele},X}+G_{2,X})\,,\quad
G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi}\,,$

and then we have:

$\displaystyle P(k)$
$\displaystyle=\tfrac{256}{81}i(G_{4}-G_{\text{Tele},T}){}^{8}\omega^{2}k{}^{10}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\,,$
(204) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\psi,\chi,\beta{}_{i},\Sigma{}_{i},h{}_{ij}\right)\,,$
(205) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,0,0,0,\frac{2}{k^{2}}A_{1},\frac{2}{k^{2}}A_{2}\right)^{T}\,,$
(206) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}&A_{2}&0\\\ A_{2}&-A_{1}&0\\\
0&0&0\end{array}\right)\,.$ (210)

Case 4.II
$(G_{3,\phi}=Z_{2},\,G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi},\,2(G_{4}-G_{\text{Tele},T})+3G_{\text{Tele},T_{\text{vec}}}\neq
0)$

In this case with $Z_{2}$ defined in (82) we have

$\displaystyle P(k)$
$\displaystyle=-32i(G_{4}-G_{\text{Tele},T}){}^{5}(G_{\text{Tele},T_{\text{vec}}}){}^{3}(G_{\text{Tele},I_{2}}-2G_{4,\phi})\omega^{2}k{}^{10}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{3}\,,$
(211)

and then more subcases appear.

Case 4.II.a $(G_{\text{Tele},I_{2}}-2G_{4,\phi}\neq 0)$

$\displaystyle P(k)$
$\displaystyle=-32i(G_{4}-G_{\text{Tele},T}){}^{5}G_{\text{Tele},T_{\text{vec}}}{}^{3}(G_{\text{Tele},I_{2}}-2G_{4,\phi})\omega^{2}k{}^{10}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{3}\,,$
(212) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\chi,\beta{}_{i},\Sigma{}_{i},h{}_{ij}\right)\,,$
(213) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(-\frac{4(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}})}{(G_{\text{Tele},I_{2}}-2G_{4,\phi})k{}^{2}}A_{1},-\frac{A_{1}}{k^{2}},\frac{iA_{1}}{\left|k\right|{}^{3}},0,0,0,0,\frac{2A_{2}}{k^{2}},\frac{2A_{3}}{k^{2}}\right)^{T}\,,$
(214) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}+A_{2}&A_{3}&0\\\
A_{3}&A_{1}-A_{2}&0\\\ 0&0&0\end{array}\right)\,.$ (218)

Case 4.II.b $(G_{\text{Tele},I_{2}}-2G_{4,\phi}=0)$

The determinant of the system in this case is

$\displaystyle P(k)$
$\displaystyle=32i(G_{4}-G_{\text{Tele},T}){}^{4}G_{\text{Tele},T_{\text{vec}}}{}^{3}(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}})\omega^{2}k{}^{10}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\,.$
(219)

Case 4.II.b.1 $(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}}\neq 0)$

$\displaystyle P(k)$
$\displaystyle=32i(G_{4}-G_{\text{Tele},T}){}^{4}G_{\text{Tele},T_{\text{vec}}}{}^{3}(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}})\omega^{2}k{}^{10}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\,,$
(220) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\psi,\chi,\beta{}_{i},\Sigma{}_{i},h{}_{ij}\right)\,,$
(221) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,0,0,0,\frac{2A_{1}}{k^{2}},\frac{2A_{2}}{k^{2}}\right)^{T}\,,$
(222) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}&A_{2}&0\\\ A_{2}&-A_{1}&0\\\
0&0&0\end{array}\right)\,.$ (226)

Case 4.II.b.2 $(G_{4}-G_{\text{Tele},T}+G_{\text{Tele},T_{\text{vec}}}=0)$

$\displaystyle P(k)$
$\displaystyle=-8(G_{4}-G_{\text{Tele},T}){}^{7}\omega^{3}k{}^{6}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\,,$
(227) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\psi,\beta{}_{i},\Sigma{}_{i},h{}_{ij}\right)\,,$ (228)
$\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,0,0,\frac{2A_{1}}{k^{2}},\frac{2A_{2}}{k^{2}}\right)^{T}\,,$
(229) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}&A_{2}&0\\\ A_{2}&-A_{1}&0\\\
0&0&0\end{array}\right)\,.$ (233)

Case 5 $(G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}\neq
0,\,\tilde{c}_{1}\neq 0,\,\tilde{c}_{2}\neq 0)$

In this case we find the following quantities:

$\displaystyle P(k)$
$\displaystyle=-\tfrac{2048}{243}i(G_{4}-G_{\text{Tele},T}){}^{5}G_{\text{Tele},T_{\text{ax}}}{}^{3}\omega^{2}k{}^{10}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{3}\Bigl{(}\tilde{c}_{1}+\tilde{c}_{2}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}\Bigr{)}\,,$
(234) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\Phi,\sigma,\beta{}_{i},\Sigma{}_{i},h{}_{ij}\right)\,,$
(235) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,A_{1},0,0,0,0,\frac{2A_{2}}{k^{2}},\frac{2A_{3}}{k^{2}}\right)^{T}\,,$
(236) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{2}&A_{3}&0\\\ A_{3}&-A_{2}&0\\\
0&0&0\end{array}\right)\,,$ (240) $\displaystyle Y_{\left|m\right|}$
$\displaystyle=\left(-4(G_{4}-G_{\text{Tele},T})B_{1},-(G_{\text{Tele},I_{2}}-2G_{4,\phi})B_{1},(G_{\text{Tele},I_{2}}-2G_{4,\phi})B_{1},0,0,0,0,0,0,0\right)^{T}\,,$
(241) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|m\right|})$
$\displaystyle=\left(G_{\text{Tele},I_{2}}-2G_{4,\phi}\right)\left(\begin{array}[]{ccc}\bigl{(}m^{2}+k{}^{2}\bigr{)}B_{1}&0&0\\\
0&\bigl{(}m^{2}+k{}^{2}\bigr{)}B_{1}&0\\\ 0&0&m^{2}B_{1}\end{array}\right)\,.$
(245)

Case 6: $(G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}\neq
0,\,\tilde{c}_{1}=0,\,\tilde{c}_{2}=0)$

A solution set of the system that covers the whole solution manifold is

$G_{3,\phi}=\frac{3(G_{\text{Tele},I_{2}}-2G_{4,\phi})^{2}}{8(G_{4}-G_{\text{Tele},T})}+\tfrac{1}{2}(G_{\text{Tele},X}+G_{2,X}),G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi}\,,$
(246)

which leads us to the determinant

$\displaystyle P(k)$
$\displaystyle=-\tfrac{2048}{243}i(G_{4}-G_{\text{Tele},T}){}^{5}G_{\text{Tele},T_{\text{ax}}}{}^{3}(G_{\text{Tele},I_{2}}-2G_{4,\phi})\omega^{2}k{}^{10}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{3}\,,$
(247)

which leads several subcases:

Case 6.I $(G_{\text{Tele},I_{2}}-2G_{4,\phi}\neq 0)$

$\displaystyle P(k)$
$\displaystyle=-\tfrac{2048}{243}i(G_{4}-G_{\text{Tele},T}){}^{5}G_{\text{Tele},T_{\text{ax}}}{}^{3}(G_{\text{Tele},I_{2}}-2G_{4,\phi})\omega^{2}k{}^{10}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{3}\,,$
(248) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\sigma,\beta{}_{i},\Sigma{}_{i},h{}_{ij}\right)\,,$
(249) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,A_{1},0,0,0,0,\frac{2A_{2}}{k^{2}},\frac{2A_{3}}{k^{2}}\right)^{T}\,,$
(250) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{2}&A_{3}&0\\\ A_{3}&-A_{2}&0\\\
0&0&0\end{array}\right)\,.$ (254)

Case 6.II $(G_{\text{Tele},I_{2}}-2G_{4,\phi}=0)$

$\displaystyle P(k)$
$\displaystyle=\tfrac{2048}{243}i(G_{4}-G_{\text{Tele},T}){}^{5}G_{\text{Tele},T_{\text{ax}}}{}^{3}\omega^{2}k{}^{8}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{3}\,,$
(255) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\psi,\sigma,\beta{}_{i},\Sigma{}_{i},h{}_{ij}\right)\,,$
(256) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,A_{1},0,0,0,0,\frac{2A_{2}}{k^{2}},\frac{2A_{3}}{k^{2}}\right)^{T}\,,$
(257) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{2}&A_{3}&0\\\ A_{3}&-A_{2}&0\\\
0&0&0\end{array}\right)\,.$ (261)

Case 7:
$(G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,\tilde{c}_{1}\neq
0,\,\tilde{c}_{2}\neq 0)$

In this case we obtain:

$\displaystyle P(k)$
$\displaystyle=-4(G_{4}-G_{\text{Tele},T}){}^{5}(\omega^{2}-k{}^{2})^{2}k{}^{8}\Bigl{(}\tilde{c}_{1}+\tilde{c}_{2}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}\Bigr{)}\,,$
(262) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\Phi,\beta{}_{i},h{}_{ij}\right)\,,$
(263) $\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,0,0,\frac{2A_{1}}{k^{2}},\frac{2A_{2}}{k^{2}}\right)^{T}\,,$
(264) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}&A_{2}&0\\\ A_{2}&-A_{1}&0\\\
0&0&0\end{array}\right)\,,$ (268) $\displaystyle Y_{\left|m\right|}$
$\displaystyle=\left(-4(G_{4}-G_{\text{Tele},T})B_{1},-(G_{\text{Tele},I_{2}}-2G_{4,\phi})B_{1},(G_{\text{Tele},I_{2}}-2G_{4,\phi})B_{1},0,0,0,0\right)^{T}\,,$
(269) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|m\right|})$
$\displaystyle=(G_{\text{Tele},I_{2}}-2G_{4,\phi})\left(\begin{array}[]{ccc}\bigl{(}m^{2}+k{}^{2}\bigr{)}B_{1}&0&0\\\
0&\bigl{(}m^{2}+k{}^{2}\bigr{)}B_{1}&0\\\ 0&0&m^{2}B_{1}\end{array}\right)\,.$
(273)

This case gives us exactly the full Horndeski Case 0i. in the limit
$G_{\text{Tele},I_{2}}\rightarrow 0$. Thus the full Horndeski theory is a sub-
branch of Case 7 and just like we discussed in B with
$\left(G_{\text{Tele},I_{2}}-2G_{4,\phi}\right)\rightarrow 0$ where it hides
the massive scalar PDoF from the polarization detectors.

Case 8:
$(G_{\text{Tele},T_{\text{vec}}}=0,\,G_{\text{Tele},T_{\text{ax}}}=0,\,\tilde{c}_{1}=0,\,\tilde{c}_{2}=0)$

A solution set of this system that covers the whole solution manifold is

$G_{3,\phi}=\frac{3(G_{\text{Tele},I_{2}}-2G_{4,\phi})^{2}}{8(G_{4}-G_{\text{Tele},T})}+\tfrac{1}{2}(G_{\text{Tele},X}+G_{2,X})\,,\quad
G_{2,\phi\phi}=-G_{\text{Tele},\phi\phi}\,,$ (274)

which leads us to the determinant

$\displaystyle P(k)$
$\displaystyle=4(G_{4}-G_{\text{Tele},T}){}^{5}(G_{\text{Tele},I_{2}}-2G_{4,\phi})k{}^{8}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\,.$
(275)

The Horndeski Case 0ii. is a subcase for $G_{\text{Tele},I_{2}}\rightarrow 0$.

Case 8.I $(G_{\text{Tele},I_{2}}-2G_{4,\phi}\neq 0)$

$\displaystyle P(k)$
$\displaystyle=4(G_{4}-G_{\text{Tele},T}){}^{5}(G_{\text{Tele},I_{2}}-2G_{4,\phi})k{}^{8}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\,,$
(276) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\delta\phi,\psi,\beta{}_{i},h{}_{ij}\right)\,,$ (277)
$\displaystyle Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,0,\frac{2A_{1}}{k^{2}},\frac{2A_{2}}{k^{2}}\right)^{T}\,,$
(278) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}&A_{2}&0\\\ A_{2}&-A_{1}&0\\\
0&0&0\end{array}\right)\,.$ (282)

Case 8.II $(G_{\text{Tele},I_{2}}-2G_{4,\phi}=0)$

$\displaystyle P(k)$
$\displaystyle=-4(G_{4}-G_{\text{Tele},T}){}^{5}k{}^{6}\bigl{(}\omega^{2}-k{}^{2}\bigr{)}^{2}\,,$
(283) $\displaystyle Y_{\omega}$
$\displaystyle=\left(\psi,\beta{}_{i},h{}_{ij}\right)\,,$ (284) $\displaystyle
Y_{\left|k\right|}$
$\displaystyle=\left(0,0,0,\frac{2A_{1}}{k^{2}},\frac{2A_{2}}{k^{2}}\right)^{T}\,,$
(285) $\displaystyle\accentset{\circ}{R}_{0i0j}(Y_{\left|k\right|})$
$\displaystyle=\left(\begin{array}[]{ccc}A_{1}&A_{2}&0\\\ A_{2}&-A_{1}&0\\\
0&0&0\end{array}\right)\,.$ (289)

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