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# Cosmological Perturbations in the Teleparallel analog of Horndeski gravity

Bobomurat Ahmedov<EMAIL_ADDRESS>Institute of Fundamental and Applied
Research, National Research University TIIAME, Kori Niyoziy 39, Tashkent
100000, Uzbekistan  National University of Uzbekistan, Tashkent 100174,
Uzbekistan  Konstantinos F. Dialektopoulos<EMAIL_ADDRESS>Department of
Mathematics and Computer Science, Transilvania University of Brasov, 500091,
Brasov, Romania  Laboratory of Physics, Faculty of Engineering, Aristotle
University of Thessaloniki, 54124 Thessaloniki, Greece  Department of Physics,
Nazarbayev University, 53 Kabanbay Batyr avenue, 010000 Astana, Kazakhstan
Jackson Levi Said<EMAIL_ADDRESS>Institute of Space Sciences and
Astronomy, University of Malta, Malta, MSD 2080  Department of Physics,
University of Malta, Malta  Abdurakhmon Nosirov
<EMAIL_ADDRESS>New Uzbekistan University, Mustaqillik ave.
54, 100007 Tashkent, Uzbekistan  Ulugh Beg Astronomical Institute, Astronomy
St. 33, Tashkent 100052, Uzbekistan  Zinovia Oikonomopoulou
<EMAIL_ADDRESS>Institute of Space Sciences and Astronomy,
University of Malta, Malta, MSD 2080  Odil Yunusov<EMAIL_ADDRESS>New Uzbekistan University, Mustaqillik ave. 54, 100007 Tashkent, Uzbekistan
Ulugh Beg Astronomical Institute, Astronomy St. 33, Tashkent 100052,
Uzbekistan

###### Abstract

In this work we study the cosmological perturbations in Bahamonde-
Dialektopoulos-Levi Said (BDLS) theory, i.e. the teleparallel analog of
Horndeski gravity. In order to understand the evolution of structure in a
cosmological model, it is necessary to study its cosmology not only in the
background but also perturbatively. Both Horndeski and its teleparallel analog
have been analyzed a lot in the literature, but in order to study them
quantitatively, we need to know their cosmological perturbations. That is why,
we study here the scalar-vector-tensor decomposition of the theory and we also
express the so-called alpha parameters in terms of the arbitrary functions of
the theory, that designate the deviation from the $\Lambda$CDM model. We have
explored tensor, vector and scalar perturbation of the action up to second
order, which drastically opens up new possibilities on searches in the
parameter space of scalar-tensor theories in the context of observations.

##  I Introduction

The ever increasing precision in the measurement of the expansion of the
Universe SupernovaSearchTeam:1998fmf ; SupernovaCosmologyProject:1998vns  has
led to the possibility that the Universe may be expanding faster than
predicted by the $\Lambda$CDM concordance model DiValentino:2021izs . While
the $\Lambda$CDM model has had many theoretical Peebles:2002gy ;
Copeland:2006wr ; Weinberg:1988cp  and observational open questions
Gaitskell:2004gd , this would constitute a larger disagreement. Indeed, over
the last few years the problem of cosmic tensions has been expressed in
several cosmological parameters DiValentino:2020vhf ; DiValentino:2020zio ;
DiValentino:2020vvd  with the value of the Hubble constant being the parameter
most in contention. These cosmic tensions have primarily emerged as
divergences between reported cosmological parameter values based either on
direct measurements from the late Universe such as Type Ia supernovae, the tip
of the red giant branch measurements, strong lensing as well as many other
approaches Riess:2021jrx ; Wong:2019kwg ; Anderson:2023aga ; Freedman:2020dne
, as compared with indirect measurements from the cosmic microwave background
(CMB) radiation and big bang nucleosynthesis in addition to other approaches
to this regime of the Universe Aghanim:2018eyx ; DES:2021wwk ; eBOSS:2020yzd ;
Zhang:2021yna ; Cooke:2017cwo . The growing tension between cosmological
parameters measured either directly or indirectly has prompted a re-evaluation
of modifications of the concordance models that have been developed in the
literature over the last few decades.

There have been a wide variety of different approaches to modifying the
concordance model given the appearance of cosmic tensions in recent years.
This has taken a variety of forms such as the reconsideration of the
cosmological principle Krishnan:2021jmh ; Krishnan:2021dyb , early Universe
dark energy models Poulin:2023lkg , extra degrees of freedom in the form of
additional neutrino species in the early Universe DiValentino:2021imh ;
DiValentino:2021rjj , modified gravity CANTATA:2021ktz  and many others
Addazi:2021xuf . Modified gravity is particularly interesting since it gives a
clear avenue in which to modify the cosmic evolution both at the background
and perturbative levels at any regimes in the cosmic history of the Universe.
However, there are many directions in which general relativity (GR) can be
modified as the gravitational component of the concordance model. One
interesting approach that has gained momentum in recent years is metric-affine
gravity where the underlying connection on which GR is based is exchanged with
other geometries BeltranJimenez:2019esp ; Hehl:1994ue . This may be a more
natural way of modifying GR since it does not require ad hoc conditions on the
action. Teleparallel gravity (TG) Aldrovandi:2013wha  is one such approach in
which the curvature associated with the Levi-Civita connection
$\accentset{\circ}{\Gamma}^{\sigma}{}_{\mu\nu}$ (over-circles denotes
quantities based on the curvature of the Levi-Civita connection in this work)
is substituted with a torsional teleparallel connection
$\Gamma^{\sigma}{}_{\mu\nu}$.

The teleparallel connection is curvature-free and satisfies metricity
Bahamonde:2021gfp ; Krssak:2018ywd ; Cai:2015emx . The result is an altogether
novel reformulation of gravitational models which can, following a particular
prescription of teleparallel objects produce a teleparallel equivalent of
general relativity (TEGR) Maluf:2013gaa ; aldrovandi1995introduction , which
is dynamically equivalent to GR but constructed in a totally different way.
Thus, GR and TEGR agree on all classical phenomenology but may be different in
the IR limit which may provide more possible directions for quantum theories
of gravity. As in curvature based theories of gravity, TEGR can be modified in
various different directions. The first modification to TEGR and the most
well-known is $f(T)$ gravity Ferraro:2006jd ; Ferraro:2008ey ;
Bengochea:2008gz ; Linder:2010py ; Chen:2010va ; Bahamonde:2019zea ;
Paliathanasis:2017htk ; Farrugia:2020fcu ; Bahamonde:2021srr ;
Bahamonde:2020bbc  where the TEGR Lagrangian $T$ (the torsion scalar) is
generalized to an arbitrary function thereof. This produces a second order
theory in terms of metric components and agrees with a growing number of
observational phenomena. Another direction in which to modify the TEGR action
is to consider it as part of a larger scalar-tensor framework akin to
Horndeski gravity Horndeski:1974wa .

For curvature based settings, Horndeski gravity is the largest class of second
order theories in which a single scalar field is added to the Einstein-Hilbert
action. The observation of the gravitational wave event GW170817
TheLIGOScientific:2017qsa  and its electromagnetic counterpart GRB170817A
Goldstein:2017mmi  has placed severe constraints on this model limiting the
most exotic branches of the theory Ezquiaga:2017ekz . Within this context, a
teleparallel analogue of Horndeski gravity was proposed in Ref.
Bahamonde:2019shr . Here, some further conditions had to be placed on the
ensuing class of theories since TG tends to observe a wider range of Lovelock
terms as compared with curvature based theories Gonzalez:2015sha . This
framework of theories has since been further investigated in various
scenarios. In Ref. Bahamonde:2019ipm  it was found that a much larger class of
models are admitted that tolerate a speed of light propagation speed for
gravitational waves, while in Ref. Bahamonde:2021dqn  the spectrum of
polarization modes was analyzed in the context of the various branches of the
theory. The post-Newtonian parametrization was studied in Ref.
Bahamonde:2020cfv  where it was found that most of the exotic ingredients of
the theory pass the standard tests in this regime. More recently, the class of
models has been explored through the prism of Noether symmetries in Ref.
Dialektopoulos:2021ryi  where the full classification was developed, while in
Refs. Bernardo:2021bsg ; Bernardo:2021izq  the well-tempering approach
developed in Ref. Appleby:2018yci  for regular Horndeski gravity was applied
in this scenario and where the mechanics of well-tempering was better tuned to
the TG setting. There has also been initial work on the stability of the
theory with a particular focus on theoretical conditions that can be derived
from Minkowski space as described Ref. Capozziello:2023foy . It is thus
critical to analyse in a concrete way the full cosmology of this new
teleparallel analogue of Horndeski gravity. The background Friedmann and
Klein-Gordon equations can be found in Ref. Bahamonde:2019shr . We are hence
motivated to determine the full cosmological perturbation equations around a
cosmological background. We do this here together with an initial analysis of
these perturbations through some examples. The work is divided as follows, TG
and its Horndeski analogue are discussed in Sec. II while in Sec. III the
cosmological perturbations are presented. Some example applications of these
perturbation equations are shown in Sec. IV. Specifically, we calulcate the
tensor and scalar primordial power spectra and we express the tensor-to-scalar
ratio in terms of the arbitrary functions of our theory. In addition, we
formulate the alpha parameters that could be used to distinguish between
$\Lambda$CDM and modified descirptions of cosmology. The main results are then
summarized in Sec. V where we describe the impact of this work within the
broader literature on the topic.

##  II Teledeski Gravity: A Teleparallel Analogue to Horndeski Theory

Let us first provide a brief introduction to teleparallel gravity and its
background cosmological dynamics.

###  II.1 Teleparallel Gravity

Curvature-based theories of gravity, such as GR, depend on a geometric
framework in which the Levi-Civita connection
$\accentset{\circ}{\Gamma}^{\sigma}_{\phantom{\sigma}\mu\nu}$ (over-circles
refer to any quantities based on the Levi-Civita connection) is the basis of
the geometric objects of the theory, such as the Riemann tensor
misner1973gravitation . Thus, the Levi-Civita connection is used throughout
curvature-based models of gravity such as in the construction of the Einstein-
Hilbert action through the Ricci scalar. TG offers a different avenue in which
to construct theories of gravity in which the curvature-based connection is
replaced with the torsional teleparallel connection
$\Gamma^{\sigma}_{\phantom{\sigma}\mu\nu}$ Aldrovandi:2013wha ;
Bahamonde:2021gfp ; Cai:2015emx ; Krssak:2018ywd .

On a more practical level, curvature and torsional theories of gravity differ
in that the former is based on the metric tensor $g_{\mu\nu}$ and its
derivatives, whereas TG is built using the tetrad $e^{A}_{\phantom{A}\mu}$,
which accounts for the gravitational variables of the system, and a flat spin
connection $\omega^{B}_{\phantom{B}C\nu}$. Here, Greek indices refer to
coordinates on the general manifold while Latin ones refer to the local
Minkowski spacetime. These objects also appear in GR but they are convoluted
in that setting making them largely impractical, while in TG the spin
connection is an inertial object. The tetrad is directly linked to the metric
tensor through

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

where $E_{A}^{\phantom{A}\mu}$ is the inverse tetrad. Here one can observe
that there are an infinite number of possible choices for the tetrad
components, and so its the spin connection that acts to retain the
diffeomorphism invariance of the system for these different choices.

The tetrad-spin connection pair represent the possible components for a
particular spacetime, and so the teleparallel connection can be written in
these terms through Cai:2015emx ; Krssak:2018ywd

$\Gamma^{\lambda}{}_{\nu\mu}=E_{A}^{\phantom{A}\lambda}\partial_{\mu}e^{A}_{\phantom{A}\nu}+E_{A}^{\phantom{A}\lambda}\omega^{A}_{\phantom{A}B\mu}e^{B}_{\phantom{B}\nu}\,,$
(2)

where the spin connection is guaranteed to be flat through the condition
Bahamonde:2021gfp

$\partial_{[\mu}\omega^{A}_{\phantom{A}|B|\nu]}+\omega^{A}_{\phantom{A}C[\mu}\omega^{C}_{\phantom{C}|B|\nu]}\equiv
0\,.$ (3)

There also exist unique frames for any spacetime in which the spin connection
terms all vanish for particular choices of the tetrad components. This is
called the Weitzenböck gauge Weitzenbock1923 , and is consistently applied
when the spin connection field equations identically vanish for these choices.

Gravitational scalars in TG are built by replacing the Levi-Civita connection
with its teleparallel analogue. The result of this is that the Riemann tensor
vanishes identically
$R^{\alpha}_{\phantom{\alpha}\beta\gamma\epsilon}(\Gamma^{\sigma}_{\phantom{\sigma}\mu\nu})\equiv
0$ (while the regular Riemann tensor remains nonzero
$\accentset{\circ}{R}^{\alpha}_{\phantom{\alpha}\beta\gamma\epsilon}(\accentset{\circ}{\Gamma}^{\sigma}_{\phantom{\sigma}\mu\nu})\neq
0$). Thus, we need to define a torsion tensor that is based solely on the
teleparallel connection, namely Aldrovandi:2013wha ; ortin2004gravity

$T^{A}_{\phantom{A}\mu\nu}:=2\Gamma^{A}_{\phantom{A}[\nu\mu]}\,,$ (4)

where square brackets denote the antisymmetric operator. The torsion tensor
represents the field strength of the theory Bahamonde:2021gfp , and is
invariant under local Lorentz and diffeomorphic transformations Krssak:2015oua
.

The torsion tensor can be decomposed into three irreducible parts
PhysRevD.19.3524 ; Bahamonde:2017wwk

$\displaystyle a_{\mu}$
$\displaystyle:=\frac{1}{6}\epsilon_{\mu\nu\lambda\rho}T^{\nu\lambda\rho}\,,$
(5) $\displaystyle v_{\mu}$
$\displaystyle:=T^{\lambda}_{\phantom{\lambda}\lambda\mu}\,,$ (6)
$\displaystyle t_{\lambda\mu\nu}$
$\displaystyle:=\frac{1}{2}\left(T_{\lambda\mu\nu}+T_{\mu\lambda\nu}\right)+\frac{1}{6}\left(g_{\nu\lambda}v_{\mu}+g_{\nu\mu}v_{\lambda}\right)-\frac{1}{3}g_{\lambda\mu}v_{\nu}\,,$
(7)

which are the axial, vector, and purely tensorial parts, respectively, and
where $\epsilon_{\mu\nu\lambda\rho}$ is the totally antisymmetric Levi-Civita
tensor in four dimensions. Using this decomposition, unique gravitational
scalar invariants can be built Bahamonde:2015zma

$\displaystyle T_{\text{ax}}$
$\displaystyle:=a_{\mu}a^{\mu}=-\frac{1}{18}\left(T_{\lambda\mu\nu}T^{\lambda\mu\nu}-2T_{\lambda\mu\nu}T^{\mu\lambda\nu}\right)\,,$
(8) $\displaystyle T_{\text{vec}}$
$\displaystyle:=v_{\mu}v^{\mu}=T^{\lambda}_{\phantom{\lambda}\lambda\mu}T_{\rho}^{\phantom{\rho}\rho\mu}\,,$
(9) $\displaystyle T{{}_{\text{ten}}}$
$\displaystyle:=t_{\lambda\mu\nu}t^{\lambda\mu\nu}=\frac{1}{2}\left(T_{\lambda\mu\nu}T^{\lambda\mu\nu}+T_{\lambda\mu\nu}T^{\mu\lambda\nu}\right)-\frac{1}{2}T^{\lambda}_{\phantom{\lambda}\lambda\mu}T_{\rho}^{\phantom{\rho}\rho\mu}\,,$
(10)

which form the set of all general scalar invariants that are not parity
violating and involve, at most, quadratic contractions of the torsion tensor.

There is a particular combination of the axial, vector, and purely tensorial
scalar invariants that produce the so-called torsion scalar Bahamonde:2021gfp

$T:=\frac{3}{2}T_{\text{ax}}+\frac{2}{3}T_{\text{ten}}-\frac{2}{3}T{{}_{\text{vec}}}=\frac{1}{2}\left(E_{A}{}^{\lambda}g^{\rho\mu}E_{B}{}^{\nu}+2E_{B}{}^{\rho}g^{\lambda\mu}E_{A}{}^{\nu}+\frac{1}{2}\eta_{AB}g^{\mu\rho}g^{\nu\lambda}\right)T^{A}{}_{\mu\nu}T^{B}{}_{\rho\lambda}\,,$
(11)

which turns out to be an incredibly important scalar since it is equal to the
Ricci scalar (up to a total divergence term) Bahamonde:2015zma

$R=\accentset{\circ}{R}+T-\frac{2}{e}\partial_{\mu}\left(eT^{\lambda\phantom{\lambda}\mu}_{\phantom{\lambda}\lambda\phantom{\mu}}\right)=0\,,$
(12)

where the torsion connection calculated Ricci scalar $R$ vanishes as described
above, and where $e=\text{det}\left(e^{A}_{\phantom{A}\mu}\right)=\sqrt{-g}$
is the tetrad determinant. The regular curvature-based Ricci scalar can be
equivalently expressed as

$\accentset{\circ}{R}=-T+\frac{2}{e}\partial_{\mu}\left(eT^{\lambda\phantom{\lambda}\mu}_{\phantom{\lambda}\lambda\phantom{\mu}}\right):=-T+B\,,$
(13)

where $B$ is this total divergence term.

The action that is based on the linear form of the torsion scalar is
guaranteed to produce a teleparallel equivalent of general relativity (TEGR)
(up to a boundary term), which will be dynamically equivalent Hehl:1994ue ;
Aldrovandi:2013wha . Using the same rationale as in curvature-based theories
of gravity, one can direct generalize the TEGR action to an f(T) gravity
Ferraro:2006jd ; Ferraro:2008ey ; Bengochea:2008gz ; Finch:2018gkh ;
Paliathanasis:2017htk ; Linder:2010py ; Chen:2010va ; Bahamonde:2019zea ;
Cai:2015emx ; Farrugia:2016qqe ; Finch:2018gkh ; Farrugia:2016xcw ;
Iorio:2012cm ; Deng:2018ncg  in which the Lagrangian of the theory is raised
from $T$ to $f(T)$. The f(T) field equations have an advantage over their
curvature-based analogue in that they remain generically second order in terms
of the metric derivatives at the level of the field equations.

In this work we investigate the teleparallel analogue of Horndeski gravity
which is built on the interactions of gravitational objects with a single
scalar field. In TG, the scalar fields couple to matter identically as in GR
(using the minimal coupling prescription), where partial derivatives are
raised to the Levi-Civita covariant derivative, that is Aldrovandi:2013wha ;
BeltranJimenez:2020sih

$\partial_{\mu}\rightarrow\mathring{\nabla}_{\mu}\,,$ (14)

which only applies to the matter sector. With this in hand, we can review the
recently proposed teleparallel analog of Horndeski gravity Bahamonde:2019shr ;
Bahamonde:2019ipm ; Bahamonde:2020cfv , also called Bahamonde-Dialektopoulos-
Levi Said (BDLS) theory. This is based on three base conditions, namely (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. Without these conditions, the ensuing action would not be finite in
nature. Due, in part, to the second order nature of many extensions of TG, it
turns out that the resulting action is an extension of the regular form of
Horndeski gravity. The result of these conditions is that the terms of regular
Horndeski gravity are found, as well as additional terms that are linear in
contractions with the torsion tensor Bahamonde:2019shr

$I_{2}=v^{\mu}\phi_{;\mu}\,,$ (15)

where $\phi$ is the scalar field, as well as terms that are quadratic in this
respect

$\displaystyle J_{1}$ $\displaystyle=a^{\mu}a^{\nu}\phi_{;\mu}\phi_{;\nu}\,,$
(16) $\displaystyle J_{3}$
$\displaystyle=v_{\sigma}t^{\sigma\mu\nu}\phi_{;\mu}\phi_{;\nu}\,,$ (17)
$\displaystyle J_{5}$
$\displaystyle=t^{\sigma\mu\nu}t_{\sigma\phantom{\alpha}\nu}^{\phantom{\sigma}\alpha}\phi_{;\mu}\phi_{;\alpha}\,,$
(18) $\displaystyle J_{6}$
$\displaystyle=t^{\sigma\mu\nu}t_{\sigma}^{\phantom{\sigma}\alpha\beta}\phi_{;\mu}\phi_{;\nu}\phi_{;\alpha}\phi_{;\beta}\,,$
(19) $\displaystyle J_{8}$
$\displaystyle=t^{\sigma\mu\nu}t_{\sigma\mu}^{\phantom{\sigma\mu}\alpha}\phi_{;\nu}\phi_{;\alpha}\,,$
(20) $\displaystyle J_{10}$
$\displaystyle=\epsilon^{\mu}_{\phantom{\mu}\nu\sigma\rho}a^{\nu}t^{\alpha\rho\sigma}\phi_{;\mu}\phi_{;\alpha}\,,$
(21)

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

Therefore, we can write the teleparallel analogue of Horndeski gravity as an
action through

$\mathcal{S}_{\text{BDLS}}=\int
d^{4}x\,e\mathcal{L}_{\text{Tele}}+\sum_{i=2}^{5}\int
d^{4}x\,e\mathcal{L}_{i}+\int d^{4}x\,e\mathcal{L}_{\rm m}\,,$ (22)

where the contributions from regular Horndeski gravity continue to appear as
Horndeski:1974wa

$\displaystyle\mathcal{L}_{2}$ $\displaystyle:=G_{2}(\phi,X)\,,$ (23)
$\displaystyle\mathcal{L}_{3}$
$\displaystyle:=-G_{3}(\phi,X)\mathring{\Box}\phi\,,$ (24)
$\displaystyle\mathcal{L}_{4}$
$\displaystyle:=G_{4}(\phi,X)\left(-T+B\right)+G_{4,X}(\phi,X)\left[\left(\mathring{\Box}\phi\right)^{2}-\phi_{;\mu\nu}\phi^{;\mu\nu}\right]\,,$
(25) $\displaystyle\mathcal{L}_{5}$
$\displaystyle:=G_{5}(\phi,X)\mathring{G}_{\mu\nu}\phi^{;\mu\nu}-\frac{1}{6}G_{5,X}(\phi,X)\left[\left(\mathring{\Box}\phi\right)^{3}+2\phi_{;\mu}^{\phantom{;\mu}\nu}\phi_{;\nu}^{\phantom{;\nu}\alpha}\phi_{;\alpha}^{\phantom{;\alpha}\mu}-3\phi_{;\mu\nu}\phi^{;\mu\nu}\,\mathring{\Box}\phi\right]\,,$
(26)

which turn out to be equivalent to their corresponding regular Horndeski
contributions except that they are calculated using teleparallel objects
rather than the metric, but which continue to produce the same contributions
to equations of motion for particular systems, and where

$\mathcal{L}_{\text{Tele}}:=G_{\text{Tele}}\left(\phi,X,T,T_{\text{ax}},T_{\text{vec}},I_{2},J_{1},J_{3},J_{5},J_{6},J_{8},J_{10}\right)\,,$
(27)

where the kinetic term is defined as
$X:=-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi$, $\mathcal{L}_{\rm m}$
is the matter Lagrangian in the Jordan conformal frame,
$\accentset{\circ}{G}_{\mu\nu}$ is the standard Einstein tensor, and where
commas represent regular partial derivatives. For the limit where
$G_{\text{Tele}}=0$, we recover regular Horndeski gravity.

###  II.2 Background Cosmology

By varying the action with respect to the tetrad, spin connection and scalar
field, we can determine the field equations, as presented in Ref.
Bahamonde:2020cfv . The increased generality of the theory produces vastly
larger field equations. For this reason, we consider only the equations of
motion for a (maximally symmetric) flat FLRW cosmology

$\mathrm{d}s^{2}=-N(t)^{2}\mathrm{d}t^{2}+a(t)^{2}(\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2})\,,$
(28)

where $N(t)$ is the lapse function (set to unity once the equations of motion
are determined), and $a(t)$ is the scale factor. To obtain the modified
equations of motion, we consider the tetrad choice
$e^{a}_{\phantom{a}\mu}=\textrm{diag}(N(t),a(t),a(t),a(t))$ which is
compatible with the Weitzenböck gauge Krssak:2018ywd ; Bahamonde:2021gfp .

Taking a variation with respect to the dynamical variables of $N(t)$, $a(t)$,
and $\phi(t)$, we obtain the equations of motion of the system for a flat
homogeneous and isotropic background. This results in the Friedmann equation

$\mathcal{E}_{\rm Tele}+\sum_{i=2}^{5}\mathcal{E}_{i}=0\,,$ (29)

where

$\displaystyle\mathcal{E}_{\rm Tele}$
$\displaystyle=6H\dot{\phi}\tilde{G}_{6,I_{2}}+12H^{2}\tilde{G}_{6,T}+2X\tilde{G}_{6,X}-\tilde{G}_{6}\,,$
(30) $\displaystyle\mathcal{E}_{2}$ $\displaystyle=2XG_{2,X}-G_{2}\,,$ (31)
$\displaystyle\mathcal{E}_{3}$
$\displaystyle=6X\dot{\phi}HG_{3,X}-2XG_{3,\phi}\,,$ (32)
$\displaystyle\mathcal{E}_{4}$
$\displaystyle=-6H^{2}G_{4}+24H^{2}X(G_{4,X}+XG_{4,XX})-12HX\dot{\phi}G_{4,\phi
X}-6H\dot{\phi}G_{4,\phi}\,,$ (33) $\displaystyle\mathcal{E}_{5}$
$\displaystyle=2H^{3}X\dot{\phi}\left(5G_{5,X}+2XG_{5,XX}\right)-6H^{2}X\left(3G_{5,\phi}+2XG_{5,\phi
X}\right)\,,$ (34)

and

$\mathcal{L}_{\rm Tele}=\tilde{G}_{6}(\phi,X,T,I_{2})\,,$ (35)

which represents all the nonvanishing scalars for $G_{\rm{Tele}}$, the Hubble
parameter is defined as $H=\dot{a}/a$, and dots denote derivatives with
respect to cosmic time. The torsion scalar takes on the form $T=6H^{2}$, while
$I_{2}=3H\dot{\phi}$ and $X=\frac{1}{2}\dot{\phi}^{2}$, and commas denote
partial derivatives. Taking a variation with respect to the scale factor leads
to the second Friedmann equation

$\mathcal{P}_{\rm Tele}+\sum_{i=2}^{5}\mathcal{P}_{i}=0\,,$ (36)

where

$\displaystyle\mathcal{P}_{\rm Tele}$
$\displaystyle=-3H\dot{\phi}\tilde{G}_{6,I_{2}}-12H^{2}\tilde{G}_{6,T}-\frac{d}{dt}\Big{(}4H\tilde{G}_{6,T}+\dot{\phi}\,\tilde{G}_{6,I_{2}}\Big{)}+\tilde{G}_{6}\,,$
(37) $\displaystyle\mathcal{P}_{2}$ $\displaystyle=G_{2}\,,$ (38)
$\displaystyle\mathcal{P}_{3}$
$\displaystyle=-2X\left(G_{3,\phi}+\ddot{\phi}G_{3,X}\right)\,,$ (39)
$\displaystyle\mathcal{P}_{4}$
$\displaystyle=2\left(3H^{2}+2\dot{H}\right)G_{4}-12H^{2}XG_{4,X}-4H\dot{X}G_{4,X}-8\dot{H}XG_{4,X}$
$\displaystyle\phantom{gggg}-8HX\dot{X}G_{4,XX}+2\left(\ddot{\phi}+2H\dot{\phi}\right)G_{4,\phi}+4XG_{4,\phi\phi}+4X\left(\ddot{\phi}-2H\dot{\phi}\right)G_{4,\phi
X}\,,$ (40) $\displaystyle\mathcal{P}_{5}$
$\displaystyle=-2X\left(2H^{3}\dot{\phi}+2H\dot{H}\dot{\phi}+3H^{2}\ddot{\phi}\right)G_{5,X}-4H^{2}X^{2}\ddot{\phi}G_{5,XX}$
$\displaystyle\phantom{gggg}+4HX\left(\dot{X}-HX\right)G_{5,\phi
X}+2\left[2\frac{d}{dt}\left(HX\right)+3H^{2}X\right]G_{5,\phi}+4HX\dot{\phi}G_{5,\phi\phi}\,.$
(41)

Finally, the modified Klein-Gordon equation can be determined by taking a
variation with respect to the scalar field, giving

$\frac{1}{a^{3}}\frac{\mathrm{d}}{\mathrm{d}t}\Big{[}a^{3}(J+J_{\rm
Tele})\Big{]}=P_{\phi}+P_{\rm Tele}\,,$ (42)

where the standard Horndeski terms appear as $J$ and $P_{\phi}$ which come
from the Lagrangian terms $\mathcal{L}_{i}$, where $i=2,..,5$ and
Kobayashi:2011nu

$\displaystyle J$
$\displaystyle=\dot{\phi}G_{2,X}+6HXG_{3,X}-2\dot{\phi}G_{3,\phi}+6H^{2}\dot{\phi}\left(G_{4,X}+2XG_{4,XX}\right)-12HXG_{4,\phi
X}$
$\displaystyle\phantom{gggggggg}+2H^{3}X\left(3G_{5,X}+2XG_{5,XX}\right)-6H^{2}\dot{\phi}\left(G_{5,\phi}+XG_{5,\phi
X}\right)\,,$ (43) $\displaystyle P_{\phi}$
$\displaystyle=G_{2,\phi}-2X\left(G_{3,\phi\phi}+\ddot{\phi}G_{3,\phi
X}\right)+6\left(2H^{2}+\dot{H}\right)G_{4,\phi}$
$\displaystyle\phantom{gggggii}+6H\left(\dot{X}+2HX\right)G_{4,\phi
X}-6H^{2}XG_{5,\phi\phi}+2H^{3}X\dot{\phi}G_{5,\phi X}\,,$ (44)

while $J_{\rm Tele}$ and $P_{\rm Tele}$ are new terms related to the
teleparallel Horndeski, given by

$\displaystyle J_{\rm Tele}$ $\displaystyle=\dot{\phi}\tilde{G}_{6,X}\,,$ (45)
$\displaystyle P_{\rm Tele}$
$\displaystyle=-9H^{2}\tilde{G}_{6,I_{2}}+\tilde{G}_{6,\phi}-3\frac{\mathrm{d}}{\mathrm{d}t}\left(H\tilde{G}_{6,I_{2}}\right)\,.$
(46)

Interestingly, the arguments of $\tilde{G}_{6}$ do not depend on $T_{\rm ax}$
and $T_{\rm ten}$ since they are zero for the flat FLRW metric. For this
reason, we can simply write the contributions in terms of $T_{\rm vec}$ since
$T=-(2/3)T_{\rm vec}=6H^{2}$ for this case. Naturally, one should not expect
this to also be true at perturbative level Bahamonde:2019ipm . Indeed, in this
work. explore the cosmological perturbations of the teleparallel analogue of
Horndeski gravity. This will open the way for deeper investigations into
specific models of the framework that may lead to a better understanding of
which models are most compatible with observations of the cosmic evolution of
the Universe.

##  III Cosmological Perturbations

The goal of cosmological perturbations in general, is to associate the physics
of the early Universe to CMB anisotropies and large-scale structure and to
provide the initial conditions for numerical simulations of structure
formation. In this scenery, the physical quantities can be decomposed into a
homogeneous background where their dependence is restricted only to the cosmic
time and the perturbative parts which depend on spacetime. Linear
perturbations around a spatially flat FLRW spacetime in Horndeski gravity have
been studied in Kobayashi:2011nu . Following that, our starting point is to
establish the adequate form of the tetrad matrix and then proceed to the
evaluation of the action for tensor, vector and scalar perturbations. In what
follows, we work on the unitary gauge, where the scalar field perturbations
vanish, i.e. $\delta\phi=0$.

The first-order expansion of the metric can be parametrized as

$g_{\mu\nu}=g_{\mu\nu}^{0}+\delta g_{\mu\nu}\,,$ (47)

where $g_{\mu\nu}^{0}$ in our case is the spatially flat FLRW background,
while $\delta g_{\mu\nu}$ can be expressed in an irreducible form as

$\displaystyle\delta
g_{\mu\nu}\rightarrow\left(\begin{matrix}-2\alpha&aB_{i}+a^{2}\partial_{i}\beta\\\
aB_{i}+a^{2}\partial_{i}\beta&2a^{2}\left[\zeta\delta_{ij}+\frac{1}{2}h_{ij}+2\partial_{(i}h_{j)}\right]\end{matrix}\right),$
(48)

where $h_{ij}$ is symmetric, traceless ($h_{ij}\delta^{ij}$) and divergencless
($\partial^{i}h_{ij}$) tensor perturbation, $B_{i}=b_{i}+\beta_{i}$, $h_{i}$
are solinoidal vector perturbations (i.e. $\partial_{i}X^{i}=0$) and $\alpha$,
$\beta$ as well as $\zeta$ are scalar perturbations. According to (1) there
are many tetrads that could reproduce the above metric. In what follows, we
choose to work with

$e^{A}{}_{\mu}=\bar{e}^{A}{}_{\mu}+\delta e^{A}{}_{\mu}\,,$ (49)

where $\bar{e}^{A}{}_{\mu}={\rm diag}(N,a,a,a)$ is the background flat FLRW
tetrad and $\delta e^{A}{}_{\mu}$ is the perturbed tetrad that is given by

$\displaystyle\delta
e^{A}{}_{\mu}\rightarrow\left(\begin{matrix}\alpha&-a\beta_{i}\\\
\delta^{I}_{i}(\partial^{i}\beta+b^{i})&a\delta^{Ii}\left[\zeta\delta_{ij}+\frac{1}{2}h_{ij}+\frac{1}{8}h_{ik}h_{kj}+2\partial_{(i}h_{j)}\right]\end{matrix}\right),$
(50)

where capital Latin indices indicate components at the Lorentz spacetime,
while small ones, the associated 3 dimensional spatial part. For greater
details one can check Bahamonde:2021gfp ; Capozziello:2023foy .

###  III.1 Tensor perturbation

We consider the tensor perturbations of Eq. (50) to be described by two
functions of spacetime, $h_{+}(t,x,y,z)$ and $h_{\times}(t,x,y,z)$ as the two
components of a symmetric, traceless and divergenceless tensor $h_{ij}$. We
assume for the sake of simplicity, that the perturbations lie on the x-y
plane, so that the z-axis is in the direction of the wavevector $\vec{k}$,
meaning $\hat{k}=z$. Setting $a,\beta,\zeta$ and $b_{i},\beta_{i},h_{i}$ to
zero, the tetrad (49) takes the form

$\displaystyle e^{A}_{\phantom{A}\mu}\rightarrow\left(\begin{matrix}1&0&0&0\\\
0&-a-\frac{1}{2}ah_{+}-\frac{1}{8}a\left(h_{+}^{2}+h_{\times}^{2}\right)&-\frac{1}{2}ah_{\times}&0\\\
0&-\frac{1}{2}ah_{\times}&-a+\frac{1}{2}ah_{+}-\frac{1}{8}a\left(h_{+}^{2}+h_{\times}^{2}\right),&0\\\
0&0&0&a\end{matrix}\right),$ (51)

where the $a$ stands for the scale factor. The associated metric from the Eq.
(1) becomes

$g_{\mu\nu}=-N^{2}\mathrm{d}t^{2}+a^{2}(\delta_{ij}+h_{ij})\mathrm{d}x^{i}\mathrm{d}x^{j}\,,$
(52)

where the background value of the lapse function is considered to be unity.
The scalar invariants of the decomposition of the torsion tensor become

$\displaystyle T_{\rm ax}$ $\displaystyle=\,\,\,\,\,0\ ,$ (53) $\displaystyle
T_{\rm vec}$ $\displaystyle=\,-\frac{9\dot{a}^{2}}{a^{2}}\ ,$ (54)
$\displaystyle T_{\rm ten}$
$\displaystyle=\,\,\,\frac{3}{4a^{2}}\,\left[\,(\bm{\nabla}h_{+})^{2}+(\bm{\nabla}h_{\times})^{2}-a^{2}\dot{h}_{+}^{2}-a^{2}\dot{h}_{\times}^{2}\,\right]\,.$
(55)

Taking these into account, the action (22) becomes up to second-order
perturbations,

$S_{\rm
T}^{(2)}=\int\mathrm{d}t\mathrm{d}^{3}x\,\,\frac{a^{3}}{4}\left[\,\mathcal{G}_{T}\dot{h}_{ij}^{2}-\frac{\mathcal{F}_{T}}{a^{2}}(\bm{\nabla}h_{ij})^{2}\,\right],$
(56)

where

$\displaystyle\mathcal{G}_{\rm
T}=2\left(G_{4}-2XG_{4,X}+XG_{5,\phi}-HX\dot{\phi}G_{5,X}+2XG_{{\rm
Tele},J_{8}}+\frac{X}{2}G_{{\rm Tele},J_{5}}-G_{{\rm Tele},T}\right),$ (57)
$\displaystyle\mathcal{F}_{\rm
T}=2\left(G_{4}-XG_{5,\phi}-X\ddot{\phi}G_{5,X}-G_{{\rm Tele},T}\right),$ (58)

where $G_{i,A}$ denotes the derivative of $G_{i}$ with respect to $A$. Notice
that in the limiting case, where $G_{\rm Tele}$ vanishes, Eq. (57) and (58),
the standard Hordeski results are recovered for the perturbed action
Kobayashi:2011nu .

This action represents how tensor perturbations propagate on a flat FLRW
background cosmology. By taking a variation of Eq. (56), we find the
propagation equation of gravitational waves, which confirms the result in Ref.
Bahamonde:2019ipm . We can similarly use this result in the standard
$\alpha_{i}$ parametrization of cosmological perturbations, as we do later on.
Specifically, the squared speed of the gravitational waves is given by

$c_{\rm T}^{2}=\frac{\mathcal{F}_{\rm T}}{\mathcal{G}_{\rm T}}\,,$ (59)

which is not equal to unity for arbitrary $G_{i}$’s. In addition, one can see
from the action (56) that the following conditions should hold

$\mathcal{F}_{\rm T}>0\quad\text{and}\quad\mathcal{G}_{\rm T}>0\,,$ (60)

in order to avoid ghost and gradient instabilities.

###  III.2 Vector perturbation

Another important component of the cosmological perturbations decomposition is
vector perturbations, which are divergence-free in nature. Usually, vector
perturbations decay in an expanding background cosmology, unless they are
driven by anisotropic stress. We do not expect here to happen something
different, but to be sure we have to study them in detail. Fixing $\alpha$,
$\beta$, $\zeta$ and $h_{ij}$ to vanish in (50) the tetrad becomes

$\displaystyle
e^{A}\text{}_{\mu}\rightarrow\left(\begin{matrix}1&-a\beta_{i}\\\
\delta^{I}_{i}b^{i}&a\delta^{Ii}\left[\delta_{ij}+2\partial_{(i}h_{j)}\right]\end{matrix}\right).$
(61)

The scalar invariants of the theory (50) take the form

$\displaystyle T_{\rm ax}=$
$\displaystyle\frac{1}{9a^{2}}\left[\bm{\nabla}\times(\bm{\beta}-\bm{b})\right]^{2},$
(62) $\displaystyle T_{\rm vec}=$
$\displaystyle-\frac{9\dot{a}^{2}}{a^{2}}+\frac{1}{a^{2}}\Bigg{\\{}9\dot{a}^{2}\left(2\bm{b}\bm{\beta}+\bm{\beta}^{2}\right)-6\dot{a}\left[a\left[\bm{b}\dot{\bm{\beta}}-2(\bm{\nabla}\times\bm{h})(\bm{\nabla}\times\dot{\bm{h}})\right]+(\bm{\nabla}\times\bm{b})(\bm{\nabla}\times\bm{h})\right]-$
(63)
$\displaystyle-6\dot{a}(\bm{b}+\bm{\beta})\left(\bm{\nabla}^{2}\bm{h}-a\dot{\bm{\beta}}\right)+\left(\bm{\nabla}^{2}\bm{h}-a\dot{\bm{\beta}}\right)^{2}\Bigg{\\}},$
$\displaystyle T_{\rm ten}=$
$\displaystyle-\frac{1}{a^{2}}\left[a^{2}\dot{\bm{\beta}}^{2}-(\bm{\nabla}\times\bm{b})^{2}-(\bm{\nabla}\times\bm{b})(\bm{\nabla}\times\bm{\beta})-(\bm{\nabla}\times\bm{\beta})^{2}+3a(\bm{\nabla}\times\dot{\bm{h}})(\bm{b}-a\bm{h})+a\dot{\bm{\beta}}\bm{\nabla}^{2}\bm{h}+(\nabla^{2}\bm{h})^{2}\right]\
.$ (64)

We expand the action keeping up to quadratic order terms in the perturbations
and we get

$\displaystyle S^{(2)}_{\rm V}=$
$\displaystyle\int\mathrm{d}t\mathrm{d}^{3}xa^{3}\Bigg{[}\frac{A_{1}}{a^{2}}(\bm{\nabla}^{2}\bm{h})^{2}+A_{2}\dot{\bm{\beta}}^{2}+A_{3}(\bm{\nabla}\times{\dot{\bm{h}}})^{2}-\frac{A_{3}}{a}(\bm{\nabla}\times{\dot{\bm{h}}})(\bm{\nabla}\times{\bm{b}})+\frac{A_{4}}{a^{2}}(\bm{\nabla}\times{\bm{b}})^{2}+\frac{A_{5}}{a^{2}}(\bm{\nabla}\times{\bm{\beta}})^{2}$
(65)
$\displaystyle+\frac{A_{6}}{a}(\bm{\nabla}\times{\bm{h}})(\bm{\nabla}\times{\bm{\beta}})+\frac{A_{7}}{a}(\bm{\nabla}\times{\bm{h}})(\bm{\nabla}\times{\dot{\bm{\beta}}})+\frac{A_{8}}{a^{2}}(\bm{\nabla}\times{\bm{\beta}})(\bm{\nabla}\times{\bm{b}})\Bigg{]}\
,$

where

$\displaystyle A_{1}:=$ $\displaystyle\frac{X}{18}\left(2XG_{{\rm
Tele},J_{6}}-6G_{{\rm Tele},J_{3}}-5G_{{\rm Tele},J_{5}}-2G_{{\rm
Tele},J_{8}}\right)+G_{{\rm Tele},\rm T_{\rm vec}},$ (66) $\displaystyle
A_{2}:=$ $\displaystyle\frac{2X}{9}\left(2XG_{{\rm Tele},J_{6}}+3G_{{\rm
Tele},J_{3}}-5G_{{\rm Tele},J_{5}}-2G_{{\rm Tele},J_{8}}\right)+G_{{\rm
Tele},T_{\rm vec}},$ (67) $\displaystyle A_{3}:=$ $\displaystyle
X\left(-4G_{4,X}-2G_{5,X}H\dot{\phi}+2G_{5,\phi}+G_{{\rm Tele},J_{5}}+4G_{{\rm
Tele},J_{8}}\right)+2\left(G_{4}-G_{{\rm Tele},T}\right),$ (68) $\displaystyle
A_{4}:=$
$\displaystyle\frac{a}{18}\left[3X\left(-6G_{4X}-3G_{5X}H\dot{\phi}+3G_{5\phi}+3G_{\rm
Tele,J5}+6G_{\rm Tele,J8}+2G_{\rm Tele,J10}\right)+9G_{4}-9G_{\rm
Tele,T}+2G_{\rm Tele,T_{ax}}\right]\ ,$ (69) $\displaystyle A_{5}:=$
$\displaystyle\frac{X}{2}\left(-2G_{4,X}-G_{5,X}H\dot{\phi}+G_{5,\phi}+2G_{{\rm
Tele},J_{5}}\right)+\frac{1}{2}G_{4}-\frac{1}{2}G_{{\rm
Tele},T}+\frac{1}{9}G_{{\rm Tele},T_{\rm ax}}-\frac{2X}{3}G_{{\rm
Tele},J_{10}},$ (70) $\displaystyle A_{6}:=$
$\displaystyle-2\frac{d}{dt}(2XG_{4X}-XG_{5\phi}-G_{4})-2\dot{\phi}XG_{5,X}\dot{H}-\dot{\phi}G_{{\rm
Tele},I_{2}}-4XG_{5.X}H^{2}\dot{\phi}$ (71)
$\displaystyle-H\Bigg{\\{}2X\left[4G_{4,X}+\left(2XG_{5,XX}+3G_{5,X}\right)\ddot{\phi}+2XG_{5,\phi
X}-2G_{5,\phi}\right]-4G_{4}+4G_{{\rm Tele},T}-6G_{{\rm Tele},T_{\rm
vec}}\Bigg{\\}},$ $\displaystyle A_{7}:=$
$\displaystyle\frac{X}{9}\left(-36G_{4,X}-18G_{5,X}H\dot{\phi}+18G_{5,\phi}-4XG_{{\rm
Tele},J_{6}}+3G_{{\rm Tele},J_{3}}+10G_{{\rm Tele},J_{5}}+4G_{{\rm
Tele},J_{8}}\right)+$ (72) $\displaystyle+2\left(G_{4}-G_{{\rm
Tele},T}+G_{{\rm Tele},T_{\rm vec}}\right),$ $\displaystyle A_{8}:=$
$\displaystyle X\left(-2G_{4,X}-G_{5,X}H\dot{\phi}+G_{5,\phi}+G_{{\rm
Tele},J_{5}}\right)+G_{4}-G_{{\rm Tele},T}-\frac{2}{9}G_{{\rm Tele},T_{\rm
ax}}+\frac{X}{3}G_{{\rm Tele},J_{10}}.$ (73)

The coefficients of $\bm{\beta}^{2},\bm{\beta}\bm{b}$ and
$\bm{\nabla}\times\bm{h}$ vanish thanks to background equations. If we choose
the condition with $A_{1}=0$ and $A_{3}=0$ we have one dynamic variable
$\bm{\beta}$ and two auxiliaries fields $\bm{h}$ and $\bm{b}$. By varying with
respect to these two auxiliaries fields, we have the following constraint
equations:

$\displaystyle
2\frac{A_{4}}{a^{2}}\left(\bm{\nabla}\times\bm{b}\right)+\frac{A_{8}}{a^{2}}\left(\bm{\nabla}\times\bm{\beta}\right)=0,$
(74)
$\displaystyle\frac{A_{6}}{a}\left(\bm{\nabla}\times\bm{\beta}\right)+\frac{A_{7}}{a}\left(\bm{\nabla}\times\dot{\bm{\beta}}\right)=0.$
(75)

Using Eq. (75) we rewrite the equation for the action as given below:

$\displaystyle S_{\mathrm{V}}^{(2)}$
$\displaystyle=\int\mathrm{d}t\mathrm{d}^{3}xa^{3}\Bigg{[}\mathcal{G}_{V}\dot{\bm{\beta}}^{2}-\frac{\mathcal{F}_{V}}{a^{2}}(\bm{\nabla}\times\bm{\beta})^{2}\Bigg{]},$
(76)

where

$\displaystyle\mathcal{G}_{\rm V}=A_{2}\quad\rm{and}\quad\mathcal{F}_{\rm
V}=\frac{A_{8}^{2}}{4A_{4}}-A_{5}.$ (77)

From Eq. (76) we can see that in order to avoid ghost and gradient
instabilities, following inequalities should be satisfied

$\displaystyle\mathcal{F}_{\rm V}>0,\quad{\rm}\quad\mathcal{G}_{\rm V}>0.$
(78)

As in the curvature-based Horndeski cosmology case, these perturbations are
found to contribute little to the resulting cosmology, and by and large, do
not impact the evolution of the Universe after inflation.

###  III.3 Scalar perturbation

Like in tensor and vector perturbation, by making $b_{i}$, $\beta_{i}$,
$h_{i}$ vectors and $h_{ij}$ tensor zero, tedrad in (50) takes the form

$\displaystyle e^{A}\text{}_{\mu}\rightarrow\left(\begin{matrix}1+\alpha&0\\\
a\delta^{I}_{i}\partial^{i}\beta&a\delta^{I}_{i}(1+\zeta)\end{matrix}\right).$
(79)

As we will see later on, we can use the constraint equations to remove
$\alpha$ and $\beta$ and thus get the quadratic action only in terms of a
single variable, $\zeta$. In the case of scalar perturbations, axial,
vectorial and tensorial parts of the torsion tensor are

$\displaystyle T_{\rm ax}=$ $\displaystyle\,0,$ (80) $\displaystyle T_{\rm
vec}=$
$\displaystyle-\frac{9\dot{a}^{2}}{a^{2}}+\frac{6\dot{a}}{a^{2}}\left[3\dot{a}\alpha+a(\bm{\nabla}^{2}\beta-3\dot{\zeta})\right]+\frac{1}{a^{2}}\Big{\\{}-27\alpha^{2}\dot{a}^{2}+6\,a\dot{a}\left[3\bm{\nabla}\beta\,\bm{\nabla}\zeta-2\alpha(\bm{\nabla}^{2}\beta-3\,\dot{\zeta})\,\right]\,+$
(81)
$\displaystyle+\left[\,(2\bm{\nabla}\zeta+\bm{\nabla}\alpha)^{2}-a^{2}(\bm{\nabla}^{2}\beta-3\dot{\zeta})^{2}\,\right]\Big{\\}},$
$\displaystyle T_{\rm ten}=$
$\displaystyle\left[\frac{(\bm{\nabla}\zeta-\bm{\nabla}\alpha)^{2}}{a^{2}}-(\bm{\nabla}^{2}\beta)^{2}\right].$
(82)

After plugging perturbed tetrad Eq. (79) into the action and expanding it to
the second order, we get

$S_{\rm
S}^{(2)}=\int\mathrm{d}t\mathrm{d}^{3}xa^{3}\Bigg{[}-3\mathcal{A}\dot{\zeta}^{2}+\frac{\mathcal{B}}{a^{2}}(\bm{\nabla}\zeta)^{2}+\Sigma\alpha^{2}-2\Theta\alpha\bm{\nabla}^{2}\beta+2\mathcal{A}\dot{\zeta}\bm{\nabla}^{2}\beta+6\Theta\alpha\dot{\zeta}-2\mathcal{C}\alpha\frac{\bm{\nabla}^{2}}{a^{2}}\zeta\Bigg{]},$
(83)

where the new coefficients

$\displaystyle\mathcal{A}:=$ $\displaystyle
2\Big{[}G_{4}-2XG_{4,X}+XG_{5,\phi}-G_{\rm Tele,T}+\frac{3}{2}(G_{{\rm
Tele},T_{\rm vec}}-XG_{{\rm Tele},I_{2}I_{2}})-3H^{2}(4G_{{\rm
Tele},TT}-12G_{{\rm Tele},TT_{\rm vec}}$ $\displaystyle+9G_{{\rm Tele},T_{\rm
vec}T_{\rm vec}})-H(XG_{5,X}+6G_{{\rm Tele},TI_{2}}-9G_{{\rm Tele},T_{\rm
vec}I_{2}})\dot{\phi}\Big{]},$ (84) $\displaystyle\mathcal{B}:=$
$\displaystyle\frac{2}{9}\Big{(}9G_{4}-9XG_{5,\phi}-6XG_{{\rm
Tele},J_{3}}-5XG_{{\rm Tele},J_{5}}+2X^{2}G_{{\rm Tele},J_{6}}-2XG_{{\rm
Tele},J_{8}}-9G_{{\rm Tele},T}+18G_{{\rm Tele},T_{\rm vec}}$
$\displaystyle-9XG_{5,X}\ddot{\phi}\Big{)},$ (85) $\displaystyle\mathcal{C}:=$
$\displaystyle 2\big{(}G_{4}-2XG_{4,X}+XG_{5,\phi}-HXG_{5,X}\dot{\phi}-G_{\rm
Tele,T}+G_{\rm Tele,T_{\rm vec}}\big{)}$ $\displaystyle+\frac{X}{9}(3G_{\rm
Tele,J_{3}}+10G_{\rm Tele,J_{5}}-4XG_{\rm Tele,J_{6}}+4G_{\rm Tele,J_{8}})\,,$
(86) $\displaystyle\Sigma:=$ $\displaystyle X\left(G_{{\rm Tele},X}+2XG_{{\rm
Tele},XX}+2XG_{2,XX}+G_{2,X}-2XG_{3,\phi
X}-2G_{3,\phi}\right)+3H\dot{\phi}\big{(}4XG_{{\rm Tele},XI_{2}}+G_{{\rm
Tele},I_{2}}$ $\displaystyle+2X^{2}G_{3,XX}+4XG_{3,X}-4X^{2}G_{4,\phi
XX}-10XG_{4,\phi X}-2G_{4,\phi}\big{)}+3H^{2}\big{(}12XG_{{\rm
Tele},I_{2}I_{2}}+8XG_{{\rm Tele},XT}$ $\displaystyle+2G_{{\rm
Tele},T}-2G_{4}-3G_{{\rm Tele},T_{\rm
vec}}+8X^{3}G_{4,XXX}+32X^{2}G_{4,XX}+14XG_{4,X}-12XG_{{\rm Tele},XT_{\rm
vec}}-4X^{3}G_{5,\phi XX}$ $\displaystyle-18X^{2}G_{5,\phi
X}-12XG_{5,\phi}\big{)}+2H^{3}\dot{\phi}\left(36G_{{\rm
Tele},TI_{2}}-54G_{{\rm Tele},T_{\rm
vec}I_{2}}+2X^{3}G_{5,XXX}+13X^{2}G_{5,XX}+15XG_{5,X}\right)$
$\displaystyle+18H^{4}\left(-12G_{{\rm Tele},TT_{\rm vec}}+4G_{{\rm
Tele},TT}+9G_{{\rm Tele},T_{\rm vec}T_{\rm vec}}\right),$ (87)
$\displaystyle\Theta:=$ $\displaystyle-6H^{3}(4G_{{\rm Tele},TT}-12G_{{\rm
Tele},TT_{\rm vec}}+9G_{{\rm Tele},T_{\rm vec}T_{\rm
vec}})+H(2G_{4}-8XG_{4,X}-8X^{2}G_{4,XX}+6XG_{5,\phi}$
$\displaystyle+4X^{2}G_{5,\phi X}-2G_{{\rm Tele},T}+3G_{{\rm Tele},T_{\rm
vec}}-6XG_{{\rm Tele},I_{2}I_{2}}-4XG_{{\rm Tele},XT}+6XG_{{\rm Tele},XT_{\rm
vec}})$ $\displaystyle-H^{2}(5XG_{5,X}+2X^{2}G_{5,XX}+18G_{{\rm
Tele},TI_{2}}-27G_{{\rm Tele},T_{\rm vec}I_{2}})\dot{\phi}$
$\displaystyle-(XG_{3,X}-G_{4,\phi}-2XG_{4,\phi X}+\frac{1}{2}G_{{\rm
Tele},I_{2}}+XG_{{\rm Tele},XI_{2}})\dot{\phi}\,,$ (88)

have been introduced. The coefficients of $\zeta^{2}$ and $\alpha\zeta$ vanish
due to the background equations. If we check this result to the limiting
cases, it is easy to see that if $G_{\rm Tele}\xrightarrow[]{}0$, we can
recover the same results for the action and its coefficient as in Horndeski
case Ref. Kobayashi:2011nu . To be more precise, if we consider the vanishing
of the teleparallel terms, we get $\mathcal{A}=\mathcal{C}=\mathcal{G}_{\rm
T}$ and $\mathcal{B}=\mathcal{F}_{\rm T}$.

By varying the action (83) with respect to $\alpha$ and $\beta$, one obtains a
set of equations

$\displaystyle\Sigma\alpha-\Theta\bm{\nabla}^{2}\beta+3\Theta\dot{\zeta}-\mathcal{C}\frac{\bm{\nabla}^{2}}{a^{2}}\zeta=0\,,$
(89) $\displaystyle\Theta\alpha-\mathcal{A}\dot{\zeta}=0\,.$ (90)

Using these equations, we can eliminate $\alpha$ and $\beta$ from the action
(83) and rewrite it as

$\displaystyle S_{\rm
S}^{(2)}=\int\mathrm{d}t\mathrm{d}^{3}xa^{3}\Bigg{[}\mathcal{G_{S}}\dot{\zeta}^{2}-\frac{\mathcal{F_{S}}}{a^{2}}(\bm{\nabla}\zeta)^{2}\Bigg{]},$
(91)

where the new coefficients are

$\displaystyle\mathcal{G}_{\rm
S}=3\mathcal{A}+\frac{\Sigma\mathcal{A}^{2}}{\Theta^{2}}\,,$ (92)
$\displaystyle\mathcal{F}_{\rm
S}=\frac{1}{a}\frac{d}{dt}\Bigg{(}\frac{a\mathcal{A}\mathcal{C}}{\Theta}\Bigg{)}-\mathcal{B}\,.$
(93)

We can also express $\mathcal{A}$ , $\mathcal{B}$ and $\mathcal{C}$
coefficients with $\mathcal{F}_{\rm T}$, $\mathcal{G}_{\rm T}$ and $G_{\rm
Tele}$ like

$\displaystyle\mathcal{A}$ $\displaystyle=\mathcal{G}_{\rm
T}+f_{1}\left(G_{\rm Tele}\right),$ (94) $\displaystyle\mathcal{B}$
$\displaystyle=\mathcal{F}_{\rm T}+f_{2}\left(G_{\rm Tele}\right),$ (95)
$\displaystyle\mathcal{C}$ $\displaystyle=\mathcal{G}_{\rm
T}+f_{3}\left(G_{\rm Tele}\right).$ (96)

where

$\displaystyle f_{1}\left(G_{\rm Tele}\right)$ $\displaystyle=3G_{\rm
Tele,T_{\rm vec}}-X\left(G_{\rm Tele,J_{5}}+G_{\rm Tele,J_{8}}+3G_{\rm
Tele,I_{2}I_{2}}\right)+6H\dot{\phi}\left(3G_{\rm Tele,T_{\rm
vec}I_{2}}-2G_{\rm Tele,TI_{2}}\right)+$ (97)
$\displaystyle+6H^{2}\left(12G_{\rm Tele,TT_{\rm vec}}-4G_{\rm
Tele,TT}-9G_{\rm Tele,T_{\rm vec}T_{\rm vec}}\right),$ $\displaystyle
f_{2}\left(G_{\rm Tele}\right)$ $\displaystyle=\frac{2}{9}\left(18G_{\rm
Tele,T_{\rm vec}}-6XG_{\rm Tele,J_{3}}-5XG_{\rm Tele,J_{5}}-2XG_{\rm
Tele,J_{8}}+2X^{2}G_{\rm Tele,J_{6}}\right),$ (98) $\displaystyle
f_{3}\left(G_{\rm Tele}\right)$ $\displaystyle=\frac{1}{9}\left(18G_{\rm
Tele,T_{\rm vec}}+3XG_{\rm Tele,J_{3}}+XG_{\rm Tele,J_{5}}-32XG_{\rm
Tele,J_{8}}-4X^{2}G_{\rm Tele,J_{6}}\right).$ (99)

We can write $\mathcal{G}_{S}$ and $\mathcal{F}_{S}$ in terms of
$\mathcal{G}_{T}$ and $\mathcal{F}_{T}$ as follows

$\displaystyle\mathcal{G}_{\rm S}$
$\displaystyle=3\mathcal{G}_{T}+\frac{\Sigma}{\Theta^{2}}\left[\mathcal{G}_{\rm
T}+f_{1}\left(G_{\rm Tele}\right)\right]^{2}+3f_{1}(G_{\rm Tele})\,,$ (100)
$\displaystyle\mathcal{F}_{\rm S}$
$\displaystyle=\frac{1}{a}\frac{d}{dt}\Bigg{\\{}\frac{a\left[\mathcal{G}_{\rm
T}+f_{1}\left(G_{\rm Tele}\right)\right]\left[\mathcal{G}_{\rm
T}+f_{3}\left(G_{\rm
Tele}\right)\right]}{\Theta}\Bigg{\\}}-\mathcal{F}_{T}-f_{2}(G_{\rm Tele})\,.$
(101)

The square sound speed is given by $c_{\rm S}^{2}=\mathcal{F}_{\rm
S}/\mathcal{G}_{\rm S}$ and ghost and gradient stability are obtained

$\displaystyle\mathcal{F}_{\rm S}>0,\quad{\rm}\quad\mathcal{G}_{\rm S}>0.$
(102)

These considerations are essential for understanding the speed at which scalar
perturbations propagate from the early Universe which can impact the predicted
size of baryonic acoustic oscillations.

##  IV Applications

The results we found in the above analysis can be used in several applications
in cosmology, especially in the early Universe. In order to understand the
formation and evolution of large-scale structure one has to understand the
nature of cosmological perturbations. In this section, we will discuss the
power spectrum of the perturbations above, and we will provide the form of the
alpha parameters that can be used to distinguish between the concordance model
and other alternatives.

###  IV.1 Primordial power spectrum

The seeds of all structure in the Universe are considered to be the primordial
fluctuations, whose origin is most probably related to inflation. Before the
freezing of the horizon at the very early Universe, as the scale factor grew
exponentially, it caused quantum fluctuations of the inflaton field, which at
later stages re-entered the horizon and set the initial conditions for the
formation of large scale structure. These fluctuations are usually described
by their power spectrum that has both scalar and tensor modes. That is what we
discuss in this section.

#### Tensor perturbations

The quadratic action (56) in tensor perturbations can be re-written using the
canonical variables

$\mathrm{d}y_{\rm T}:=\frac{\mathcal{F}^{1/2}_{\rm T}}{a\mathcal{G}_{\rm
T}}\mathrm{d}t\,,\,\,\,z_{\rm T}:=\frac{a}{2}(\mathcal{F}_{\rm
T}\mathcal{G}_{\rm T})^{1/4}\,,\,\,\,v_{ij}(y_{\rm T},\bm{x}):=z_{\rm
T}h_{ij}\,,$ (103)

as

$\mathcal{S}_{\rm T}^{(2)}=\int\mathrm{d}y_{\rm
T}\mathrm{d}^{3}x\left[(v^{\prime}_{ij})^{2}-(\bm{\nabla}v_{ij})^{2}+\frac{z_{\rm
T}^{\prime\prime}}{z_{\rm T}}v_{ij}^{2}\right]\,,$ (104)

where prime denotes differentiation with respect to $y_{\rm T}$. Varying this
action with respect to $v_{ij}$ and solving its equation, we see that at
superhorizon scales we get

$v_{ij}\propto z_{\rm T}\,,\quad v_{ij}\propto z_{\rm
T}\int\frac{\mathrm{d}y_{\rm T}}{z^{2}_{\rm T}}\,,$ (105)

or in terms of the non-canonical variables

$h_{ij}={\rm const}\,,\quad
h_{ij}=\int^{t}\frac{\mathrm{d}t^{\prime}}{a^{3}\mathcal{G}_{\rm T}}\,.$ (106)

To evaluate the power spectral density, we assume that

$\epsilon:=-\frac{\dot{H}}{H^{2}}\simeq{\rm const}\,,\,\,f_{\rm
T}:=\frac{\dot{\mathcal{F}}_{\rm T}}{H\mathcal{F}_{\rm T}}\simeq{\rm
const}\,,\,\,g_{\rm T}:=\frac{\dot{\mathcal{G}}_{\rm T}}{H\mathcal{G}_{\rm
T}}\simeq{\rm const}\,.$ (107)

In order for the canonical time coordinate to run from $-\infty$ to $0$ as the
Universe expands, we have to impose the condition

$\epsilon+\frac{f_{\rm T}-g_{\rm T}}{2}<1\,,$ (108)

while in order for the second solution in Eq. (105)-(106) to decay, we have to
assume that

$\epsilon-g_{\rm T}<3\,.$ (109)

Expanding the tensor perturbations in terms of the eigenfunctions
$e^{i\bm{k}\cdot\bm{x}}$ of the Laplacian and the polarization tensor
$\mathrm{e}_{ij}$ in the Fourier space, we can solve the mode functions
equation to get

$v_{ij}=\frac{\sqrt{\pi}}{2}\sqrt{-y_{\rm T}}H_{\nu_{\rm T}}^{(1)}(-ky_{\rm
T})\mathrm{e}_{ij}\,,$ (110)

with $H_{\nu_{\rm T}}^{(1)}$ being the Hankel function of first kind (plus
sign) and $\nu_{\rm T}$ being a positive scalar defined as

$\nu_{\rm T}:=\frac{3-\epsilon+g_{\rm T}}{2-2\epsilon-f_{\rm T}+g_{\rm T}}\,.$
(111)

Thus, the power spectrum of the primordial tensor fluctuations becomes

$\mathcal{P}_{\rm T}=8\gamma_{\rm T}\frac{\mathcal{G}_{\rm
T}^{1/2}}{\mathcal{F}_{\rm
T}^{3/2}}\frac{H^{2}}{4\pi^{2}}\Big{\rvert}_{-ky_{\rm T}=1}\,,$ (112)

where

$\gamma_{\rm T}=2^{2\nu_{\rm
T}-3}\left|\frac{\Gamma(\nu_{T})}{\Gamma(3/2)}\right|^{2}(1-\epsilon-\frac{f_{\rm
T}}{2}+\frac{g_{\rm T}}{2})\,.$

We evaluate the power spectrum at the sound horizon exit, i.e. $-ky_{\rm
T}=1$, since $c_{\rm T}\neq c$ in general. The tensor spectral index is given
by

$n_{\rm T}=3-2\nu_{\rm T}\,,$ (113)

and the scale-invariant limit for tensor perturbations would be for $\nu_{\rm
T}=3/2$. From Eq. (113) we see that, the gravitational wave spectrum could
have a blue tilt if

$n_{\rm T}>0\Rightarrow 4\epsilon+3f_{\rm T}-g_{\rm T}<0\,.$ (114)

Notice that the conditions (108) and (109) are not affected by this and even
if B-mode polarization were to be detected on CMB, the theory could still be a
good candidate. However, we should mention that the assumptions (107) may not
always hold, as long as the above conditions are satisfied.

#### Scalar perturbations

In order to canonically normalize the action (91) we define the following
variables

$\mathrm{d}y_{\rm S}:=\frac{\mathcal{F}_{\rm S}^{1/2}}{a\mathcal{G}_{\rm
S}^{1/2}}\mathrm{d}t\,,\,\,\,z_{\rm S}:=\sqrt{2}a(\mathcal{F}_{\rm
S}\mathcal{G}_{\rm S})^{1/4}\,,\,\,\,u(y_{\rm S},\bm{x}):=z_{\rm S}\zeta\,.$
(115)

Plugging them in the quadratic action we get

$\mathcal{S}_{\rm S}^{(2)}=\frac{1}{2}\int\mathrm{d}y_{\rm
S}\mathrm{d}^{3}x\left[(u^{\prime})^{2}-(\bm{\nabla}u)^{2}+\frac{z^{\prime\prime}_{\rm
S}}{z_{\rm S}}u^{2}\right]\,,$ (116)

where, as previously, prime denotes differentiation with respect to the
canonical time variable, $y_{\rm S}$.

Following the same procedure as with the tensor perturbations to evaluate the
power spectrum in this case, we assume

$\epsilon:=-\frac{\dot{H}}{H^{2}}\simeq{\rm const}\,,\,\,f_{\rm
S}:=\frac{\dot{\mathcal{F}}_{\rm S}}{H\mathcal{F}_{\rm S}}\simeq{\rm
const}\,,\,\,g_{\rm S}:=\frac{\dot{\mathcal{G}}_{\rm S}}{H\mathcal{G}_{\rm
S}}\simeq{\rm const}\,.$ (117)

Then, the power spectrum will be given by

$\mathcal{P}_{\rm S}=\frac{\gamma_{\rm S}}{2}\frac{\mathcal{G}_{\rm
S}^{1/2}}{\mathcal{F}_{\rm
S}^{3/2}}\frac{H^{2}}{4\pi^{2}}\Big{\rvert}_{-ky_{\rm S}=1}\,,$ (118)

where

$\nu_{\rm S}:=\frac{3-\epsilon+g_{\rm S}}{2-2\epsilon-f_{\rm S}+g_{\rm S}}$

and

$\gamma_{\rm S}=2^{2\nu_{\rm S}-3}\left|\frac{\Gamma(\nu_{\rm
S})}{\Gamma(3/2)}\right|^{2}(1-\epsilon-\frac{f_{\rm S}}{2}+\frac{g_{\rm
S}}{2})\,.$

The scalar spectral index is given by

$n_{\rm S}=4-2\nu_{\rm S}\,,$ (119)

and thus a spectrum with equal amplitudes at horizon crossing should obey

$\epsilon+\frac{3f_{\rm S}}{4}-\frac{g_{\rm S}}{4}=0\,.$ (120)

If we consider the limit $\epsilon,f_{\rm T},g_{\rm T},f_{\rm S},g_{\rm S}\ll
1$, we get $\nu_{\rm T},\nu_{\rm S}\rightarrow 3/2$ and thus $\gamma_{\rm
T},\gamma_{\rm S}\rightarrow 1$. The tensor-to-scalar ration is then given by

$r=16\frac{\mathcal{F}_{\rm S}}{\mathcal{F}_{\rm T}}\frac{c_{\rm S}}{c_{\rm
T}}\,.$ (121)

Given that, one can consider any model of inflation that is a sub-class of the
BDLS theory and find the form $r$ in terms of the slow-roll parameters.

###  IV.2 Alpha parameters

It has been proposed in Bellini:2014fua  that perturbation dynamics can be
solely described by four functions, $\alpha_{i}$. Similarly to effective field
theory methods, one can use observations to constrain the value of these four
parameters, without specifying any particular model or initial conditions and
thus test possible deviations from $\Lambda$CDM.

Given a specific background evolution, which can also be obtained merely by
the observations Dialektopoulos:2023dhb ; Dialektopoulos:2023jam ;
Briffa:2023ern ; Dialektopoulos:2021wde ; LeviSaid:2021yat ; Hwang:2022hla ,
and the value of the matter density today Planck:2018vyg , one can fully
determine the evolution of large-scale structure in the Universe. The physical
meaning of these time-dependent functions is as follows:

  * •

Kineticity: $\alpha_{\rm K}$. Indicates the kinetic energy of curvature
perturbations. It is essentially unconstrained by observations since it has no
actual impact on any of the observables and large values tend to suppress the
sound speed of the perturbations. The contributions for $\alpha_{\rm K}$ are
from all the $G_{2},G_{3},G_{4},G_{5}$ and $G_{\rm Tele}$.

  * •

Braiding: $\alpha_{\rm B}$. Its presence shows evidence of the mixing between
the kinetic term of the scalar field and the metric. As $\alpha_{\rm K}$, it
only affects the curvature perturbations, giving rise to a quintessence-like
force. It is the reason for dark energy clustering and has contributions from
$G_{\rm Tele}$ and all $G_{i}$ functions except the potential-like term
$G_{2}$.

  * •

Planck mass run rate: $\alpha_{\rm M}$. Merely a redefinition of the Planck
mass that does not affect physics. It contributes to both curvature and tensor
perturbations and also creates anisotropic stress in the Jordan frame. It
receives contributions from $G_{4},G_{5}$ and $G_{\rm Tele}$ terms.

  * •

Tensor speed excess: $\alpha_{\rm T}$. Describes the propagation speed of
gravitational waves through the tensor perturbations and specifically its
deviation from the speed of light. This is done through the relation $c_{\rm
T}^{2}=1+\alpha_{\rm T}$ their values could be constrained by cosmological
observations and experiments. Its contributions are from $G_{4},G_{5}$ and
$G_{\rm Tele}$.

As already seen in Saltas:2014dha ; Riazuelo:2000fc  the most general
parametrization of the gravitational wave perturbation equation on a flat
cosmological background in modified gravity takes the form

$\ddot{h}_{ij}+(3+\alpha_{\rm M})H\dot{h}_{ij}-(1+\alpha_{\rm
T})\frac{k}{a^{2}}h_{ij}=0\,,$ (122)

where dots are derivatives with respect to cosmic time and $k$ the wavenumber
of the perturbation on the Fourier space.

In BDLS theory the excess tensor speed parameter is given by Bahamonde:2019ipm

$M_{\ast}^{2}a_{\rm T}\equiv
2X\left(2G_{4,X}-2G_{5,\phi}-(\ddot{\phi}-H\dot{\phi})G_{5,X}-2G_{{\rm
Tele},J_{8}}-\frac{1}{2}G_{{\rm Tele},J_{5}}\right)\,,$ (123)

and the effective Planck mass by

$M_{\ast}^{2}\equiv
2\left(G_{4}-2XG_{4,X}+XG_{5,\phi}-HX\dot{\phi}G_{5,X}+2XG_{{\rm
Tele},J_{8}}+\frac{1}{2}XG_{{\rm Tele},J_{5}}-G_{{\rm Tele},T}\right)\,.$
(124)

In addition, the Planck mass run rate parameter can be determined by

$HM_{\ast}^{2}\alpha_{\rm
M}\equiv\frac{\mathrm{d}M_{\ast}^{2}}{\mathrm{d}t}\,.$ (125)

The remaining two alpha-parameters, i.e. braiding and kineticity, are
expressed respectively as

$\displaystyle\mathcal{A}H\alpha_{B}=$ $\displaystyle
2\dot{\phi}(XG_{3X}-G_{4\phi}-2XG_{4\phi
X})+2XH(4G_{4X}+8XG_{4XX}-4G_{5\phi}-4XG_{5\phi X}+$ (126)
$\displaystyle+3G_{{\rm Tele},I_{2}I_{2}}+4G_{{\rm Tele},XT}-6G_{{\rm
Tele},XT_{\rm vec}})+2\dot{\phi}XH^{2}(3G_{5X}+2XG_{5XX})+$
$\displaystyle+\dot{\phi}(G_{\rm Tele,I_{2}}+12H^{2}G_{{\rm
Tele},TI_{2}}-18H^{2}G_{{\rm Tele},T_{\rm vec}I_{2}}+2XG_{{\rm
Tele},XI_{2}}),$ $\displaystyle\mathcal{A}H^{2}\alpha_{K}=$ $\displaystyle
2X(G_{2X}+2XG_{2XX}-2G_{3\phi}-2XG_{3\phi
X})+12\dot{\phi}XH(G_{3X}+XG_{3XX}-3G_{4\phi X}-2XG_{4\phi XX})+$ (127)
$\displaystyle+12XH^{2}(G_{4X}+8XG_{4XX}+4X^{2}G_{4XXX}-G_{5\phi}-5XG_{5\phi
X}-2X^{2}G_{5\phi XX})+$
$\displaystyle+4\dot{\phi}XH^{3}(3G_{5X}+7XG_{5XX}+2X^{2}G_{5XXX})+2X(9H^{2}G_{{\rm
Tele},I_{2}I_{2}}+2XG_{{\rm Tele},XX}+6\dot{\phi}HG_{{\rm
Tele},XI_{2}}+G_{{\rm Tele},X}).$

Notice that in the $G_{\rm Tele}\rightarrow 0$ limit, $\mathcal{A}\rightarrow
M_{\ast}$ and one can derive the respective expressions in the standard
Horndeski cosmology Bellini . The presence of the teleparallel contribution
here, in terms of $G_{\rm Tele}$ might seemingly complicate things, but it
also adds interesting phenomenology in the structure of the theory. This can
become evident from the fact that functions that were severely constrained in
the standard Horndeski formulation, such as the $G_{4,X}$ and $G_{5}$, can
survive in this setup.

Furthermore, one can rewrite the expressions in the action (91) in terms of
the alpha parameters and obtain

$\displaystyle\Theta=\frac{\mathcal{A}H}{2}(2-\alpha_{B})\,\,{\rm
and}\,\,\Sigma=-\frac{\mathcal{A}H^{2}}{2}(6-\alpha_{K}-6\alpha_{B}),$ (128)
$\displaystyle\mathcal{G_{S}}=\frac{2\mathcal{A}D}{(2-\alpha_{B})^{2}},\,\,\rm
where\,\,D=\alpha_{K}+\frac{3}{2}\alpha_{B}^{2}.$

If we consider $\mathcal{C}H\alpha_{X}=d\mathcal{C}/dt$ equation holds for
$\alpha_{X}$ parameter, squared sound speed can be found with the following
formula

$\displaystyle c_{s}^{2}$
$\displaystyle=\frac{\mathcal{C}}{\mathcal{A}}\,\,\frac{(2-\alpha_{B})(H^{2}(1+\alpha_{X})-\dot{H})+\dot{\alpha_{B}}H}{DH^{2}}-\frac{\mathcal{B}}{\mathcal{A}}\,\,\frac{(2-\alpha_{B})^{2}}{2D}.$
(129)

Here we introduced a new parameter $\alpha_{X}$ by using the similar approach
like $\alpha_{M}$ in Horndeski gravity, in order to give more compact form to
the formula of squared sound speed. Also, as it is mentioned earlier, if we
make teleparallel terms zero ($G_{\rm Tele}\xrightarrow[]{}0$), then
$\mathcal{A}=\mathcal{C}=\mathcal{G}_{T}$ and $\mathcal{B}=\mathcal{F}_{T}$
which means that we get the same expression for squared sound speed like in
Horndeski case.

##  V Conclusions

A homogeneous and isotropic FLRW Universe is the pedestal of modern cosmology,
even if it has been argued otherwise Krishnan:2021jmh . It is on top of this
background that the cosmological perturbations propagate and become source for
many of our observations.

In this paper, we study in detail the cosmological perturbations around a flat
FLRW spacetime in the teleparallel analog of Horndeski gravity. It has been
shown before Bahamonde:2019shr , that BDLS theory presents much richer
phenomenology compared to the standard/curvature Horndeski, because of the
presence of $G_{\rm Tele}$ function, which depends on all those teleparallel
quantities that in pure Riemannian geometry do not exist. Specifically, it was
found that Bahamonde:2019ipm  in BDLS theory, terms like $G_{4,X}$ and
$G_{5}(\phi,X)$ can survive, in contrast with the standard Horndeski
formulation in Riemannian geometry, where they are severely constrained (if
not eliminated) because they predict a different speed for the propagation of
gravitational waves.

Here we present the scalar, vector and tensor perturbations in this theory and
we show how the propagation speed of both tensor and scalar perturbations is
affected by the presence of the extra teleparallel Lagrangian. In addition,
after normalizing the quadratic scalar and tensor perturbations action, we
switch to the Fourier space and calculate the power spectra of primordial
fluctuations. We also express the tensor-to-scalar ratio in terms of the
perturbations coefficients in the action. In this way, one could assume a
specific teleparallel Horndeski model, fix the $G_{i}$ and $G_{\rm Tele}$
functions and calculate immediately the tensor-to-scalar ratio.

Last but not least, we present the perturbations in terms of the so-called
alpha parameters, that is the four parameters that one can constrain solely
from observations without the need to specify any physical model. In this way,
we could possibly discriminate between the concordance cosmological model and
alternative descriptions.

As far as future projects are concerned, it would be very interesting to see
how this analysis is affected in a cosmological spacetime with non-trivial
spatial curvature. Furthermore, we plan to investigate possible evasions of
the no-go theorem that manifests itself in the Riemannian Horndeski gravity
regarding bouncing solutions.

###### Acknowledgements.

The work was supported by the PNRR-III-C9-2022–I9 call, with project number
760016/27.01.2023, by the Nazarbayev University Faculty Development
Competitive Research Grant No. 11022021FD2926 and by the Hellenic Foundation
for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I.
Research Projects to support Faculty members and Researchers and the
procurement of high-cost research equipment grant” (Project Number: 2251).
This article is also based upon work from COST Action CA21136 Addressing
observational tensions in cosmology with systematics and fundamental physics
(CosmoVerse) supported by COST (European Cooperation in Science and
Technology).

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##  Appendix A Teleparallel scalars in the cosmological setting

Here, all independent scalars in BDLS theory are presented explicitly for
tensor, vector and scalar perturbations separately. The ones that are not
mentioned are vanishing.

The non-trivial scalars of tensor perturbation take the following form

$\displaystyle I_{2}=\frac{3\dot{a}\dot{\phi}}{a},$ (130) $\displaystyle
J_{5}=\frac{\dot{\phi}^{2}}{8}(\,\dot{h}_{+}^{2}+\dot{h}_{\times}^{2}\,),$
(131) $\displaystyle
J_{8}=\frac{\dot{\phi}^{2}}{2}(\,\dot{h}_{+}^{2}+\dot{h}_{\times}^{2}\,).$
(132)

The respective ones in vector perturbations are

$\displaystyle
I_{2}=\frac{3\dot{a}\dot{\phi}}{a}+\frac{1}{a}\Big{\\{}\dot{\phi}\left[-3\dot{a}\left(2\bm{b}\bm{\beta}+\bm{\beta}^{2}\right)+a\big{(}\bm{b}\dot{\bm{\beta}}-2(\bm{\nabla}\times\bm{h})(\bm{\nabla}\times\dot{\bm{h}})\big{)}+(\bm{\nabla}\times\bm{b})(\bm{\nabla}\times\bm{h})\right]+\dot{\phi}(\bm{b}+\bm{\beta})\left(\bm{\nabla}^{2}\bm{h}-a\dot{\bm{\beta}}\right)\Big{\\}},$
(133) $\displaystyle
J_{3}=\frac{1}{6a^{2}}\Big{\\{}\dot{\phi}^{2}\left[2a^{2}\dot{\bm{\beta}}^{2}-(\bm{\nabla}^{2}\bm{h})^{2}-a\dot{\bm{\beta}}\bm{\nabla}^{2}\bm{h}\right]\Big{\\}},$
(134) $\displaystyle
J_{5}=\frac{1}{4a^{2}}\Bigg{\\{}\dot{\phi}^{2}\left[\bm{\nabla}^{2}\times(-a\dot{\bm{h}}+\bm{b}+\bm{\beta})\right]^{2}+\dot{\phi}^{2}\left(a\bm{\nabla}\times\dot{\bm{h}}+\bm{\nabla}\times\bm{\beta}\right)^{2}-\frac{5\dot{\phi}^{2}\left(2a\dot{\bm{\beta}}+\Delta\bm{h}\right)^{2}}{9}\Bigg{\\}},$
(135) $\displaystyle
J_{6}=\frac{1}{36a^{2}}\Bigg{[}\dot{\phi}^{4}\left(2a\dot{\bm{\beta}}+\bm{\nabla}^{2}\bm{h}\right)^{2}\Bigg{]},$
(136) $\displaystyle
J_{8}=\frac{1}{2a^{2}}\Bigg{[}\dot{\phi}^{2}\left(2a\bm{\nabla}\times\dot{\bm{h}}-\bm{\nabla}\times\bm{b}\right)^{2}-\frac{\dot{\phi}^{2}\left(2a\dot{\bm{\beta}}+\Delta\bm{h}\right)^{2}}{9}\Bigg{]},$
(137) $\displaystyle
J_{10}=\frac{\dot{\phi}^{2}}{6a^{2}}\bigg{[}(\bm{\nabla}\times\bm{b})(\bm{\nabla}\times\bm{\beta})+(\bm{\nabla}\times\bm{b})^{2}-2(\bm{\nabla}\times\bm{\beta})^{2}\bigg{]},$
(138)

and the ones related to scalar perturbations are

$\displaystyle
I_{2}=\frac{3\dot{a}\dot{\phi}}{a}-\frac{\dot{\phi}}{a}\,\left[6\alpha\dot{a}+a(\bm{\nabla}^{2}{\beta}-3\dot{\xi})-9\alpha^{2}\dot{a}-2a\alpha(\bm{\nabla}^{2}{\beta}-3\dot{\xi})+3a\bm{\nabla}{\beta}\bm{\nabla}{\xi}\right],$
(139) $\displaystyle
J_{3}=\frac{\dot{\phi}^{2}}{3a^{2}}\,\left[\,(\bm{\nabla}\alpha)^{2}+\bm{\nabla}\zeta\,\bm{\nabla}\alpha-2\,(\bm{\nabla}\zeta)^{2}\,\right],$
(140) $\displaystyle
J_{5}=\frac{\dot{\phi}^{2}}{18a^{2}}\,\left[\,3a^{2}\,(\bm{\nabla}^{2}\beta)^{2}-10\,(\bm{\nabla}\zeta-\bm{\nabla}\alpha)^{2}\,\right],$
(141) $\displaystyle
J_{6}=\frac{\dot{\phi}^{4}}{9a^{2}}\,\left(\bm{\nabla}\alpha-\bm{\nabla}\zeta\,\right)^{2},$
(142) $\displaystyle
J_{8}=\frac{2\dot{\phi}^{2}}{9a^{2}}\,\left[\,3a^{2}\,(\bm{\nabla}^{2}\beta)^{2}-(\bm{\nabla}\zeta-\bm{\nabla}\alpha)^{2}\,\right]\,.$
(143)


# Enhanced power graphs of groups are weakly perfect

Peter J. Cameron111School of Mathematics and Statistics, University of St
Andrews, UK; email<EMAIL_ADDRESS>and Veronica Phan22237 Street 2,
Ward 6, District 8, Ho Chi Minh City, Vietnam; email<EMAIL_ADDRESS>
###### Abstract

A graph is weakly perfect if its clique number and chromatic number are equal.
We show that the enhanced power graph of a finite group $G$ is weakly perfect:
its clique number and chromatic number are equal to the maximum order of an
element of $G$. The proof requires a combinatorial lemma. We give some remarks
about related graphs.

##  1 Introduction

The _directed power graph_ of a finite group $G$, defined in [8], has the
elements of $G$ as vertices, with an arc from $x$ to $y$ if $y=x^{n}$ for some
integer $n$. This relation is reflexive and transitive, hence is a _partial
preorder_. The (undirected) _power graph_ , defined in [5], is obtained by
ignoring directions: that is, $x$ and $y$ are joined if one is a power of the
other. This graph is thus the comparability graph of a partial preorder; a
small extension of Dilworth’s theorem shows that it is _perfect_ , that is,
every induced subgraph has clique number equal to chromatic number.

Both these graphs were first defined for semigroups, but most work on them has
concerned groups.

According to the _strong perfect graph theorem_ [6], a graph is perfect if and
only if it has no induced subgraph which is a cycle of odd length greater than
$3$ or the complement of one.

The _enhanced power graph_ of $G$, defined in [1], again has vertex set $G$,
with $x$ and $y$ joined if there is an element $z$ such that both $x$ and $y$
are powers of $z$. (Equivalently, $x$ and $y$ are joined if and only if the
group they generate is cyclic.) It is shown in [4] that enhanced power graphs
of finite groups are _universal_ , that is, every finite graph occurs as an
induced subgraph of such a graph. Thus, these graphs are not in general
perfect.

Our purpose here is to show that enhanced power graphs are _weakly perfect_ ,
that is, they have chromatic number equal to clique number. Indeed our result
is not restricted to finite groups, but applies to groups in which all
elements have finite and bounded order.

###### Theorem 1

Let $G$ be a finite group, or a torsion group of bounded exponent. Then the
clique number and the chromatic number of $G$ are both equal to the maximal
order of an element of $G$.

The result for clique number is known, and the proof is straightforward; the
result for chromatic number requires the following purely combinatorial
result. We note that the proof is constructive, so gives an easy algorithm for
colouring the enhanced power graph.

###### Theorem 2

For every natural number $n$, there exist subsets $A_{1},A_{2},\ldots,A_{n}$
of $\\{1,2,\ldots,n\\}$ with the properties

  * •

$|A_{q}|=\phi(q)$ for $q\in\\{1,\ldots,n\\}$, where $\phi$ is Euler’s totient;

  * •

if $\mathop{\mathrm{lcm}}(q,q^{\prime})\leq n$, then $A_{q}\cap
A_{q^{\prime}}=\emptyset$, where $\mathop{\mathrm{lcm}}$ denotes the least
common multiple.

These theorems will be proved in the next two sections. In the final section
we give some concluding remarks.

Many further properties of power graphs and enhanced power graphs can be found
in [2] and [12].

##  2 Proof of Theorem 2

Let $D$ be the set of fractions $p/q$ (in their lowest terms) in $(0,1]$, for
$1\leq q\leq n$. We define a function$f:D\to\\{1,2,\ldots,n\\}$ by the rule

$f(p/q)=\lceil np/q\rceil.$

The key observation is the following:

> If $p/q\neq p^{\prime}/q^{\prime}$ and $f(p/q)=f(p^{\prime}/q^{\prime})$,
> then $\mathop{\mathrm{lcm}}(q,q^{\prime})>n$.

For, if $f(p/q)=f(^{\prime}p/q^{\prime})$, then there exists $m$ such that

$m-1<np/q,np^{\prime}/q^{\prime}\leq m.$

Thus $|p/q-p^{\prime}/q^{\prime}|<1/n$. On the other hand,
$|p/q-p^{\prime}/q^{\prime}|$ is a rational number whose numerator is at least
$1$ (since $p/q\neq p^{\prime}/q^{\prime}$), and the denominator is
$\mathop{\mathrm{lcm}}(q,q^{\prime})$. So we have

$\frac{1}{n}>\left|\frac{p}{q}-\frac{p^{\prime}}{q^{\prime}}\right|\geq\frac{1}{\mathop{\mathrm{lcm}}(q,q^{\prime})},$

and so $\mathop{\mathrm{lcm}}(q,q^{\prime})>n$, as required.

Now we let $D_{q}$ be the set of fractions in $D$ with denominator $q$, so
that $|D_{q}|=\phi(q)$, and let $A_{q}=f(D_{q})\subseteq\\{1,\ldots,n\\}$. By
our key observation we see that

  * •

the restriction of $f$ to $D_{q}$ is injective, so $|A_{q}|=\phi(q)$;

  * •

if $q\neq q^{\prime}$ and $\mathop{\mathrm{lcm}}(q,q^{\prime})\leq n$, then
$A_{q}\cap A_{q^{\prime}}=\emptyset$.

So the theorem is proved.

For example, here are the sets generated for $n=12$ by the above procedure.

$\begin{array}[]{l}A_{1}=\\{12\\},A_{2}=\\{6\\},A_{3}=\\{4,8\\},A_{4}=\\{3,9\\},\\\
A_{5}=\\{3,5,8,10\\},A_{6}=\\{2,10\\},A_{7}=\\{2,4,6,7,9,11\\},\\\
A_{8}=\\{2,5,8,11\\},A_{9}=\\{2,3,6,7,10,11\\},A_{10}=\\{2,4,9,11\\},\\\
A_{11}=\\{2,3,4,5,6,7,8,9,10,11\\},A_{12}=\\{1,5,7,11\\}.\end{array}$

##  3 Proof of Theorem 1

We begin with the observation that if a finite set of elements in a group has
the property that any two of its elements generate a cyclic group, then the
whole set generates a cyclic group. A proof can be found in [1, Lemma 32]. It
follows that a maximal clique in the enhanced power graph is a maximal cyclic
subgroup of $G$, and the clique number is equal to the order of the largest
cyclic subgroup, say $n$.

In order to find a colouring with $n$ colours, we take $\\{1,2,\ldots,n\\}$ to
be the set of colours, with the subsets $A_{q}$ given by Theorem 2. We will
use the set $A_{q}$ to colour elements of order $q$. If two elements of order
$q$ are joined, they lie in the same cyclic subgroup of order $q$; this
subgroup has $\phi(q)$ generators, so we have enough colours to give them all
different colours. Other elements of order $q$ are not joined to these ones,
so we may re-use the same set of colours for them. Now, if two elements of
different orders $q$ and $q^{\prime}$ are joined, they generate a cyclic group
of order $\mathop{\mathrm{lcm}}(q,q^{\prime})$, which is at most $n$; so the
sets of colours assigned to them are disjoint. Thus, we obtain a proper
colouring.

##  4 Further remarks

Our combinatorial lemma can deal with any set of element orders, as long as
the largest order $n$ is given. Now there are groups in which the set of
element orders is $\\{1,\ldots,n\\}$ for some $n$. (For example, the orders of
elements in the alternating group $A_{7}$ are $1,2,3,4,5,6,7$.) But, as we
show below, this can only occur for finitely many values of $n$. So, at first
glance, it seems we may be able to simplify the argument for most groups by
using the fact that not all orders occur. We have not attempted to do so, and
indeed it seems unlikely that any simplification can be obtained.

###### Proposition 3

There are only finitely many values of $n$ for which there exists a finite
group in which the set of element orders is $\\{1,\ldots,n\\}$.

###### Proof

We use the _Gruenberg–Kegel graph_ of a group $G$ (sometimes called the _prime
graph_): the vertices are the prime divisors of $|G|$, with vertices $p$ and
$q$ joined if $G$ contains an element of order $pq$. Gruenberg and Kegel
described this graph in an unpublished manuscript on the decomposition of the
augmentation ideal of the group ring; their main theorem, a description of the
groups whose Gruenberg–Kegel graph is disconnected, was published by
Gruenberg’s student Williams [11] and refined by later authors, notably
Kondrat’ev [9].

We will use the fact that the number of connected components of this graph is
at most $6$, for any finite group.

Now suppose that $G$ is a group in which the element orders are
$\\{1,2,\ldots,n\\}$. If $p$ is a prime in the interval $(n/2,n]$, then $p$ is
an isolated vertex in the Gruenberg–Kegel graph of $G$; so there can be at
most five such primes. But, in a strengthening of Bertrand’s postulate, Erdős
[7] showed that the number of primes in this interval tends to $\infty$ with
$n$. The result is proved.

The weak perfect graph theorem asserts that a graph is perfect if and only if
its complement is perfect. This does not hold for weakly perfect graphs.
However, we note that Jitender Kumar Parveen has recently posted on the arXiv
a paper showing (among other things) that the complement of the enhanced power
graph of a finite group is weakly perfect [10].

A related graph is the difference of the power graph and enhanced power graph
of the group $G$, which we will denote by $\Delta(G)$: $x$ and $y$ are joined
in this graph if they are joined in the enhanced power graph but not in the
power graph.

For a group $G$, let $\Omega(G)$ denote the set of orders of elements of $G$.
For a positive integer $n$, let $\alpha(n)$ denote the size of the largest
antichain in the lattice of divisors of $n$. De Bruijn _et al._ [3] showed
that, if $n$ has $m$ prime factors (counted with multiplicity), then a
maximum-size antichain consists of all divisors with $m/2$ prime factors if
$m$ is even, and either all divisors with $\lfloor m/2\rfloor$ prime factors
or all with $\lceil m/2\rceil$ prime factors if $m$ is odd. (This is a
generalisation of Sperner’s lemma.)

###### Proposition 4

For a finite group $G$, the clique number of $\Delta(G)$ is equal to
$\max\\{\alpha(n):n\in\Omega(G)\\}$.

###### Proof

A clique $S$ in $\Delta(G)$ is a clique in the enhanced power graph, and so is
contained in a cyclic group $C$. Now a cyclic group has the property that if
$x$ and $y$ are two elements for which the order of $x$ divides the order of
$y$, then $x$ is a power of $y$. It follows that the elements of $S$ all have
different orders, and these form an antichain in the lattice of divisors of
$|C|$.

###### Proposition 5

Let $G$ be the symmetric group $S_{8}$ on $8$ letters. Then $\Delta(G)$ is not
weakly perfect.

###### Proof

We have $\Omega(G)=\\{1,2,3,4,5,6,7,8,10,12,15\\}$; so the clique number of
$\Delta(G)$ is equal to $2$. But $\Delta(G)$ is not bipartite, since

$\\{(1,2),(3,4,5),(6,7),(1,2,3),(4,5,6,7,8)\\}$

induces a $5$-cycle.

It is an interesting problem to describe the groups $G$ for which $\Delta(G)$
is weakly perfect, but we shall not discuss this here.

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