Quartic Horndeski-Cartan theories in a FLRW universe

We consider the Quartic Horndeski theory with torsion on a FLRW background in the second order formalism. We show that there is a one parameter family of Quartic Horndeski Cartan Lagrangians and all such theories only modify the dispersion relations of the graviton and the scalar perturbation that are usually found in the standard Horndeski theory on a torsionless spacetime. In other words, for the theories in this class torsion does not induce new degrees of freedom but it only modifies the propagation. This holds for first order perturbations in spite of a kinetic mixing between the Horndeski scalar with the torsion field in the action. We also show that for most Lagrangians within the family of Quartic Horndeski Cartan theories the dispersion relation of the scalar mode is radically modified. We find only one theory within the family whose scalar mode has a regular wave-like dispersion relation.

In this work we start the analysis of Horndeski theories with torsion in the second order (metric) formalism and examine if the degrees of freedom are modified with respect to Horndeski theories on a torsionless spacetime.In the second order formalism we consider a connection that is a priori metric compatible and written in terms of the (metric-dependent) Levi-Civita connection and the torsion tensor.One compelling feature in this formalism is that in contrast to Einstein-Cartan theory, torsion may be dynamical because it couples to second derivatives of the scalar in the Lagrangian and simultaneously, the absence of the Ostrogradsky ghost is guaranteed by the Horndeski construction.The latter may not follow for Horndeski theories in other formalisms different than the second order [24,25].
We first show that there is a one parameter family of Quartic Horndeski theories with torsion which reduces to the usual Horndeski theory on a torsionless spacetime.We find in a perturbative expansion at linear order that contrary to the expectation, these Quartic Horndeski Cartan theories on a FLRW background do not introduce additional degrees of freedom and that the torsionful connection only modifies the usual tensor and scalar degrees of freedom that are found in the standard Quartic Horndeski theory without torsion.
We compute the speed of sound for the graviton, which is the same for all the theories with torsion, and find that the subluminality, no ghost and stability conditions are similar to the standard torsionless Horndeski theory.We also show that in most of the Quartic Horndeski Cartan theories the dispersion relation of the scalar mode is radically modified and has no counterpart with the usual scalar mode in the torsionless Horndeski theory.We consider a particular example for the latter and observe that the unusual dispersion relation does not necessarily imply an instability.Furthermore, we show that there are Horndeski Cartan theories within the family for which both the graviton and the scalar mode are simultaneously ghost-free in the high momentum limit, conditional to the assumption that the graviton is also stable and subluminal.Finally, we find that there is only one theory within the family in which the scalar field perturbation propagates with a regular wave-like dispersion relation.
We proceed as follows: In section II we describe the theories to be analyzed in this work.In section II A we explicitly introduce torsion in the second order formalism and in section II B we write the decomposition of the torsion perturbations into irreducible components under rotation group.
The main results are presented in section III.In particular, in section III B we show a classification of the scalar mode according to the parameter of the theory.In section IV we consider a particular example in order to examine the unusual dispersion relation in a subclass of the theories considered.
We give a summary in section V.In section V A we make an observation on the possibility of strong coupling for cosmological applications of these novel theories and we discuss further avenues of research to assess this question.

II. QUARTIC HORNDESKI CARTAN LAGRANGIANS
Lifting the assumption of a Christoffel connection to define covariant differentiation on the spacetime introduces new fields in the theory.In this work we assume from the beginning, when we formulate the theory, that the connection can be expressed in terms of the metric and a torsion tensor.Furthermore, we assume a vanishing nonmetricity.Namely, we only consider metric compatible covariant derivatives.
This approach is usually known as second order formalism, in contrast to previous works where one could, for instance, start from a connection initially assumed to be independent of the metric [21,[23][24][25].
Within the second order formalism the natural approach to Horndeski theory on a spacetime with torsion is to promote torsionless to torsionful covariant derivatives in the Horndeski Lagrangian.However, this prescription "Torsionless to Torsionful" does not lead to an unique choice of Lagrangian function.More precisely, there is at least one parameter family of Lagrangian functions which reduces to the usual Quartic Horndeski Lagrangian when we assume a Christoffel connection.
To write this precisely let us start with the Quartic Horndeski Lagrangian in the generalized Galileon notation on a spacetime without torsion where G 4 is an arbitrary function of ϕ and X = − 1 2 g µν ∂ µ ϕ∂ ν ϕ, g is the metric with mostly + signature, G 4,X = ∂G 4 /∂X, and the metric compatible covariant derivative on a vector V on a spacetime without torsion is written as where such that As such, it is clear that two torsionless covariant derivatives commute on the scalar and therefore there is no ambiguity in the contraction of Lorentz indices in the rightmost term G 4,X (∇ µ ∇ ν ϕ) 2 in the Quartic Horndeski Lagrangian without torsion (1).Now we consider the case on a spacetime with torsion such that the connection on the spacetime is not symmetric (we use tilde notation for torsionful quantities) Writing the torsionful metric compatible covariant derivative as (we stick to the convention to sum over the second index of the non symmetric connection) we can write a one parameter (c) family of Quartic Horndeski Cartan Lagrangians by considering all possible contractions with the metric of the terms of the form ( ∇µ ∇ν ϕ) 2 in the G 4,X "counterterm", where Torsion only appears implicitly in the torsionful covariant derivatives with c + s = −1, such that the terms (8) reduce to the standard counterterm proportional to G 4,X in the Horndeski theory (1), when we assume a Christoffel connection (namely, by the prescription Γ → Γ).Thus, these Horndeski Cartan theories take the form where R is the Ricci scalar with torsion and c is a real constant.Below we will show that the choice of parameter c is important to determine the dynamics.Furthermore, let us stress that the term with coupling c does not introduce higher derivatives because the antisymmetric two tensor c ∇µ , ∇ν ϕ clearly vanishes the symmetric second order derivatives on the scalar ∂ µ ∂ ν ϕ.
Indeed, let us note from the start that there are no higher than second order derivatives of any of the fields in the equations of motion derived from ( 9) for all values of c (See the Appendix VI A).Hence, as expected from Horndeski theories in the second order formalism, all the Lagrangians in ( 9) are free of the Ostrogradsky ghost.
A. Torsion in the Quartic Horndeski Lagrangian in the second order formalism As opposed to the usual Horndeski G 4 Lagrangian where the metric and the scalar are the only fields, now there is an additional Torsion field that is necessary to specify the geometry of the spacetime [36].
To write explicitly the torsion in the Quartic Horndeski Cartan action (9) in the second order formalism (See for instance [21,36]), we proceed as follows: provided the assumption (6) and the fact that every difference of connections is a tensor, we define the torsion tensor as and for latter convenience the contortion tensor as where we notice the antisymmetry Now, a direct computation shows that the torsionful connection and the Christoffel connection are related by the contortion tensor as With these definitions for torsion, contortion and previous relations to the Christoffel connection, we can rewrite the torsionful covariant derivative explictly in terms of contortion and the covariant derivative (∇) associated with the Christoffel symbol as and we can write the commutator of torsionful covariant derivatives on a scalar as With this commutator we can rewrite the action (9) in a form more reminiscent of the usual Quartic Horndeski, but with torsionful covariant derivatives plus a lower derivative c term.This last term parameterizes different choices of Lorentz index contractions in the Quartic Horndeski Cartan theories.In terms of contortion, (9) takes the form Finally, let us write R in terms of the Ricci scalar without torsion (R) and contortion, as, All in all, with the previous definitions we can rewrite the Quartic Horndeski theories (9) in terms of three explicit tensor fields: namely, with the metric, the scalar and the contortion tensor K as the three fundamental fields1 B. Linearization: Decomposition of contortion perturbations into irreducible components We will perform a perturbative expansion at linear order about a spatially flat FLRW background.It is convenient to decompose the perturbations into irreducible components under small rotation group as follows: we consider the perturbed metric where is a spatially flat FLRW background metric, η is conformal time, and we denote spatial indices with latin letters such as i = 1, 2, 3 and space-time indices with greek letters, such as µ = 0, 1, 2, 3.The metric perturbation is written as with α, B, ψ, E scalar perturbations, S i , F i transverse vector perturbations, and h ij , a symmetric, traceless and transverse tensor perturbation.
For the contortion perturbation, which satisfies the symmetry (14), there are 24 independent components that can be written in terms of irreducible components under small rotation group as: eight scalars denoted as C (n) with n = 1, . . ., 8, six (two-component) transverse vectors denoted as V (m)   i with m = 1, . . ., 6 and two (two-component) traceless, symmetric, transverse tensors T (1)  ij , T (2)  ij .Explicitly, the decomposition of contortion perturbation reads, for the scalar sector for the vector sector and for the tensor sector where we have not written explicitly the vanishing components and those related to (24)(25)(26) by the symmetry (14).All in all, the components of contortion perturbation are On the other hand, the non-vanishing components of the background contortion tensor on a homogeneous and isotropic background spacetime are such that we write at linearized level the contortion tensor with all indices down as Finally, let us write the scalar field as where Π is a spacetime dependent scalar field perturbation and φ is the background scalar field.All in all, there are 4 background quantities for the scalar, metric and contortion: φ, a, x, y which satisfy 5 equations of motion, of which only 4 are independent.Namely, first using the equation for 0 K ijk which fixes y = 0, we have where repeated spatial indices are not summed in expression (35) and the remaining background equations are where we denote derivative with respect to conformal time η with dot as, for instance, ȧ = ∂a/∂η.

Little gauge transformations
The action (20) at quadratic order in perturbations is invariant under the following gauge transformations of metric perturbations the transformation of the scalar field perturbation and the gauge transformations of torsion perturbations (in momentum space) 5)  i → V (5)   i V (6)  i → V (6)   i (39) where we have decomposed the gauge 4−vector into two scalar gauge parameters ξ 0 (η, ⃗ p), ξ(η, ⃗ p) and the transverse vector ξ i (η, ⃗ p) (with ∂ i ξ i = 0).For convenience, we have occasionally written the transverse vector ξ i in terms of ω i as

III. LINEARIZED DYNAMICS IN QUARTIC HORNDESKI CARTAN THEORIES
In contrast to the case of Einstein-Cartan [36], we do not generally expect torsion to decouple as constraint equations in the Quartic Horndeski Cartan theory because there are terms of the form G 4 ∇K and G 4,X ∇ϕ ∇ ∇ϕ K in the action (20).These terms generate second derivatives of contortion (K) in the Euler Lagrange equation for the scalar (ϕ) and second derivatives of the scalar in the Euler Lagrange equation for contortion.More precisely, the Euler Lagrange equations for ϕ, g µν and K µ νσ computed from the action ( 20) are where we have shown the dependances of E ϕ , E gµν , E K µ νσ on the highest derivatives of the fields.They take the form where we have explicitly written down all of the second derivatives of contortion in E ϕ , which come from the terms G 4 ∇K and G 4,X ∇ϕ ∇ ∇ϕ K in the action (20), where we stress on the dependance of E K µ νσ on second order terms ∇2 ϕ.And E gµν and F are shown in the Appendix VI A. All in all, there are three field equations (41) with up to second order derivatives of all of the three fields.Therefore we could expect new degrees of freedom besides the usual tensor modes and the single scalar mode that are usually found in the Quartic Horndeski theory without torsion 2 .
The main objective of this work, in the second order formalism, is to precisely determine the degrees of freedom for the theory (20).We explore this question at linearized order in a perturbative expansion.

A. Modification of Horndeski degrees of freedom on a spacetime with torsion
Contrary to the expectation from the expression (42) there are in fact no explicit kinetic terms of contortion perturbations about a spatially flat FLRW background.This can be explicitly seen in the quadratic action for the tensor and scalar sectors, respectively (45), (46).On the other hand, the vector sector which is shown in the Appendix VI B is trivial in the sense that all vector perturbations are non-dynamical.The nontrivial part of the quadratic action is written as where the part relevant to the three tensor perturbations where the coefficients v i with i = 1 . . .7, shown in the Appendix VI B, depend only on background quantities.In particular, they are independent of the parameter of the theory c.Hence the tensor modes are the same within the familiy of Quartic Horndeki Cartan theories.On the other hand, let us also point out in advance that after integrating out the torsion perturbations T (1) ij , T (2)  ij and after using the equations of motion for the background fields the mass term for the graviton vanishes, similar as in the torsionless Horndeski theory.
The part of the action relevant to the 13 scalar perturbations C (n) , α, B, ψ, E, Π with n = 1, . . ., 8, without fixing gauge is: where the coefficients f i with i = 1 . . .59 depend only on background quantities and the parameter of the theory c.They are presented in the Appendix VI B.
With respect to the difference in dynamics within the family of Quartic Horndeski Cartan Lagrangians, it is important to notice that in the scalar sector the theory with c = 0 has three less terms, as shown in the first line in (46).In particular, only for the c = 0 theory, the metric perturbation B and the first torsion scalar C (1) are Lagrange multipliers.As we show below, these constraints are critical in the sense that they lead to different dynamics of the scalar field perturbation in the c = 0 theory compared to the Quartic Horndeski Cartan theories with non zero parameter c.
Even though there are no explicit kinetic terms for the contortion tensor and scalar perturbations in (45) and ( 46), there could still arise new kinetic terms after using some of the equations of motion.As we show below this is not the case for the Quartic Horndeski Cartan theories (20).Namely, for the scalar and tensor sectors on the spatially flat FLRW background (45), (46) it is possible to use all constraint equations to integrate out all torsion perturbations C (n) (n = 1, . . .8), T (1)  ij , T (2)  ij as well as the metric perturbations α, B before fixing the gauge.To obtain a final result we assume that the background scalar field φ(η) is not constant ( φ ̸ = 0), that the torsion background x is non vanishing and that coefficients in the action (45) and (46) and after using constraint equations do not vanish 3 .All in all, the second order action in the unitary gauge, where Π = 0 and E = 0, divides only into tensor modes h ij and a single scalar mode ψ, thus showing that generally, a torsionful connection on the spacetime does not introduce additional degrees of freedom but it only modifies the usual tensor and scalar degrees of freedom that are found in the standard, torsionless Quartic Horndeski Lagrangian.The final form of the action for general c parameter reads where where the background functions m i with i = 1, 2, 3 are not very illuminating and are shown in the Appendix VI C.
The crucial aspect evident in the action (47) is the parameter of the theory c in the coefficient of the term ψ∂ i ∂ i ψ.This contribution radically modifies the dispersion relation of the scalar in the case it does not vanish.Hence, this result singles out the Quartic Horndeski Cartan theory (9) with c = 0 as the only one with a scalar degree of freedom with a regular wave-like behavior.In such a case the scalar sector is not necessarily the same in the torsionless and the torsionful Quartic Horndeski theory, because the speed of sound squared is different in both cases (See for instance section 3.2.1 in [20]).
For the theories S 4c with c ̸ = 0 there are some crucial aspects different in comparison to the theory with c = 0. Namely, when the parameter of the theory c is non vanishing: 1) C (1) is not a Lagrange multiplier.2) C (1) and B are coupled.3) The scalar B is not a Lagrange multiplier, as opposed to the standard Quartic Horndeski theory on a torsionless spacetime.All these three aspects are evident from the first line in expression (46).Let us clarify how these features finally amount to the disparity in the propagation of the scalar mode in theories with c = 0 and c ̸ = 0: the essential aspects that we discuss in the scalar sector in expression (47) can be written in a toy model with a Lagrangian of the form for a field x 1 in momentum space.This toy model has the property that if c ̸ = 0 the field x 1 has an unusual dispersion relation of the form discussed before.However, there is an equivalent toy model to (54) which off-shell seems to be the Lagrangian for a degree of freedom x 1 with a standard wave-like dispersion relation, but coupled to an additional auxiliary field x 2 .Indeed, plugging back in the Lagrangian (55) the equation of motion for x 2 (x 2 = − ẋ1 ) we recover the toy model (54).
For the theories S 4c there is a simplified analogy with the toy model (55), which follows identifying ψ and C (6) with x 1 and x 2 , respectively.Indeed, on one hand, the terms of the toy model (55), namely ψ2 , p 2 ψ 2 , p 2 (C (6) ) 2 are present in the initial action (46).On the other hand, the remaining key term to complete the analogy to the toy model (55), namely p 2 C (6) ψ, is generated by means of the term p 2 f 23 C (6) α in the initial action (46) because on-shell (up to background dependent coefficients) Now, it is clear that in order to obtain the unusual dispersion relation in the toy model (55) x 2 must not vanish.Analogously, the auxiliary field C (6) must not vanish.This is only the case when c ̸ = 0. Indeed, the equation of motion for the first torsion scalar C (1) computed from the action (46) in the unitary gauge for a theory with general parameter c is Only if c ̸ = 0 the first torsion scalar C (1) is not a Lagrange multiplier (let us here recall the first critical difference stated above between theories with zero and non zero c parameter) and therefore C (6) does not vanish.As a consequence, in analogy with the equivalence between toy models (55) and (54), when c ̸ = 0 we obtain a non regular wave-like dispersion relation as we found in the result (47).This scalar degree of freedom for theories with c ̸ = 0 has no counterpart with the Quartic Horndeski theory on a torsionless spacetime.We explore further this unusual dispersion relation for the theories with c ̸ = 0 with a concrete example in section IV.
Finally, for the tensor modes the sound speed squared is modified from the standard (torsionless) Quartic Horndeski case, from [20] to The difference between c H τ and c HC τ boils down to the torsion perturbation T (1)  ij , which couples to h ij .On the other hand, T (2)  ij does not couple to the graviton.Let us note that this result on the tensor sector is independent of the torsion background x(η).

B. Stability and classification of the scalar mode in Horndeski Cartan theories
The stability conditions to avoid ghost and gradient instabilities for the tensor modes are which are independent of the parameter of the theory c.Expressions (60) can be written altogether with the requirement of subluminality of the graviton c HC τ 2 < 1 as and easily compared the analogous conditions for the tensor modes in the torsionless Quartic Horndeski theory [20] On the other hand, for the scalar mode we require a separate analysis for the theories with c = 0 and non zero c.
a. Stability in the scalar mode for the theory S 4c with c = 0 In this theory the scalar mode has the usual wave-like dispersion relation and the ghost-free and stability conditions are as usual provided that we have also required a positive sign for the kinetic term of the graviton (G τ > 0).
b. Ghost-free condition in the scalar mode for the theories S 4c with c ̸ = 0 In this case, the non wave-like dispersion relation only allows to state the ghost-free condition as Also, drawing an analogy with an oscillatory system with a restaurative force and bounded energy from below, we can demand a similar condition to gradient stability To advance further in the analysis let us consider the case of high momentum.Then, only G SII and F S are relevant.
In such a case, from equation (51) the no ghost condition for the scalar reduces to Assuming the stability, subluminality and no ghost condition for the graviton (61), which explicitly restricts to because for the background fields X = 1 2 a 2 φ2 > 0, we can rewrite the no ghost condition for the scalar (66), as Using again the stability of the graviton (61) rewritten as we find from (68) that the scalar degree of freedom for theories S 4c with c ̸ = 0 can be classified in the high momentum approximation as non ghost for theories with c < 0, and as a ghost if 0 < c ≤ 2. Hence, we reach the important conclusion that there are Horndeski Cartan theories (c < 0) for which both the graviton and the scalar mode are simultaneously ghost-free in the high momentum limit.Let us stress that this classification of the scalar mode only holds by simultaneously assuming that the graviton is subluminal, stable and ghost-free (61) and cannot be deduced by stability considerations of the scalar mode on its own.Table I summarizes these results.
The stability of the scalar mode in a particular example with c ̸ = 0 is further examined in section IV.

IV. EXAMPLE: THE SCALAR MODE IN HORNDESKI CARTAN THEORIES S4c WITH NONZERO c
We have shown that only the Quartic Horndeski Cartan theory S 4 c with c = 0 has a scalar mode that propagates with a regular wave-like dispersion relation on the FLRW background.For other theories, namely, when c ̸ = 0, we showed that the dispersion relation is unusual.In this section we consider an example for a theory with c ̸ = 0 and we observe that the unusual dispersion relation does not necessarily imply an instability.
Let us consider a G 4 function that can satisfy the subluminality, ghost-free and stability conditions of the graviton (61) where, in natural units κ is a constant with dimension of length squared.For simplicity we work in Planck units, so, we choose κ = 1.Furthermore, let us consider a theory with negative c parameter, c = − 1 2 , which, as discussed in the last section, has the property that whenever the healthy graviton conditions (61) are met, then the scalar is not a ghost in the high momentum approximation (See also Table I).
Using the final form of the action (47) with the G 4 choice (70) and for the theory c = − 1 2 , the action of the scalar sector is where Let us notice that the scalar is not a ghost whenever the graviton is healthy: namely, the subluminality, stability and ghost-free condition for the graviton (61) is satisfied for time domains where When the graviton is healthy the scalar is not a ghost because the inequality (75) implies Furthermore, in accordance with our assumptions while deriving (47), the torsion background does not vanish where the graviton is healthy (75) and where ϕ, a and φ do not vanish, which were also part of the assumptions: FIG. 1: Evolution of the scalar mode ψ for widely different momenta.In both cases the amplitude of oscillation of the scalar mode is decreasing, approaching zero before a late time when the assumptions are violated and a singularity is developed at η ≈ 3983.The position of this singularity (not shown in the figures) is associated to the time when a → 0 and independent of the momentum.The initial data is such that the conditions for a healthy graviton (75) are met at the initial time and also they are such that φ φ| η=0 > 0, such that the assumptions φ(η) ̸ = 0, φ(η) ̸ = 0 are satisfied for the time domains of numerical evolution (Notice that φ(η) seems monotonic increasing within these time domains).Initial Conditions φ(0) = 0.5, φ(0) = 0.01, a(0) = 0.5, ψ(0, |⃗ p|) = 0, ψ(0, |⃗ p|) = −0.9.
The numerical solutions in Figures 1a, 1b suggest that the amplitude of oscillation of the scalar perturbation ψ is decreasing and oscillates close to zero for late times, for a wide range of momenta.This behavior was observed restricted to initial data that satisfies the conditions for a healthy graviton (75) and for as long as the assumptions hold.In particular there is a singularity not shown in the Figures 1a, 1b that only develops at late times, which is associated with the vanishing of a.
On the other hand, Figure 2 shows a typical case where a singularity quickly develops at early times 4 , because the initial conditions quickly lead to violate the assumption φ ̸ = 0 upon which the validity of the result (71) and figures rely.Approaching this singularity seems to lead to a growing amplitude of the scalar perturbation, even though the gradient stability condition F S > 0 holds in the tested time domain in Figure 2. Thus, we avoid initial conditions of the scalar that quickly lead to φ = 0 and singularities.More precisely, because φ is monotonic at least within the tested time domain, we can easily avoid this singularity by restricting to initial conditions of the scalar that satisfy φ φ| η=0 > 0.
All in all, despite the unusual dispersion relation for the Quartic Horndeski Cartan theories with c ̸ = 0, the results in this section for c = − 1 2 suggest that the scalar mode is not necessarily unstable because at least in some cases and within the tested time domains, which we chose to coincide with domains where the graviton is also healthy, the amplitude of oscillation of the scalar perturbation decreases with time, settling to zero for late times, whenever the initial conditions do not lead to violate the assumptions upon which the computations were derived.

V. CONCLUSION
We considered the Quartic Horndeski theory with torsion in the second order formalism, written as a one parameter (c) family where tilde denotes torsionful quantities and the theories (81) for all c reduce to the standard Horndeski action without torsion by taking R → R, and ∇ → ∇ such that [∇ µ , ∇ ν ] ϕ = 0, where ∇ denotes covariant derivative defined with a Christoffel connection.We showed in a perturbative expansion that these Quartic Horndeski Cartan theories at linear order on a spatially flat FLRW background do not introduce additional degrees of freedom and that the torsionful connection only modifies the usual tensor and scalar degrees of freedom when compared to the standard Quartic Horndeski theory without torsion.Indeed, the quadratic action for the one parameter family of theories (81) can be finally written for the graviton h ij and a single scalar field perturbation ψ in the form where G τ , F τ , G SI , G SII , F S are functions of G 4 and its derivatives.We computed the speed of sound for the graviton, which is the same in all of the Quartic Horndeski Cartan theories, and found that the subluminality, ghost-free and stability conditions are similar to the standard Horndeski theory without torsion, as shown in Table I.We also showed that for the theories (81) with parameter c ̸ = 0 the dispersion relation of the scalar mode is radically modified and it has no counterpart with the usual scalar mode in the torsionless Horndeski theory, as can be seen in the term proportial to c in expression (82).We analyzed the latter in a particular example and observed that the unusual non wave-like dispersion relation when c ̸ = 0 does not necessarily imply an instability.Furthermore, we showed that there are Horndeski Cartan theories (c < 0) for which both the graviton and the scalar mode are simultaneously ghost-free in the high momentum limit, conditional to the assumption that the graviton is also stable and subluminal.
We found that the theory with parameter c = 0 is the only one within the family in which the scalar field perturbation propagates with a regular wave-like dispersion relation and yet, it is not necessarily the same in comparison to the torsionless Quartic Horndeski theory because its speed of sound is different.TABLE I: Summary of tensor and scalar modes classified according to the parameter c of the theory.* We refer to a stable, non ghost and subluminal graviton as healthy.The ghost free conditions in the scalar mode are only written if they are tied to the assumption of a healthy graviton.Otherwise they require further assumptions depending on G 4 .

Scalar mode
Non wave-like dispersion relation.

Not a ghost (in high momentum)
if the graviton is healthy * .
Wave-like dispersion relation.

Non wave-like dispersion relation.
A ghost (in high momentum) if the graviton is healthy * .
Non wave-like dispersion relation.

Graviton
Is massless.The no ghost, stability and subluminality conditions (G τ > 0, Vector sector Non dynamical.

A. Discussions and outlook
It is important to notice that the result in (82) showing only a graviton and a single scalar perturbation about the FLRW background is contrary to a naive expectation, because there are terms in the action (81) that suggest a kinetic mixing of the Horndeski scalar with Torsion, as we explained at the beginning of section III.For instance, there are terms where torsion couples to second covariant derivatives of the scalar.This opens the question whether there are hidden symmetries in the full theory, or accidental symmetries for linear order perturbations on this specific background, such as in other theories with torsion [37][38][39] and with non-metricity [40].This could mean a strong coupling, evident in a discontinuity in the number of degrees of freedom between perturbative expansions about different backgrounds.A Hamiltonian analysis could shed light on this regard, but given the complexity of these theories other potentially simpler approaches would be to perform higher order perturbations as in [39] or to explore less symmetric backgrounds.However, the latter may need a case by case analysis depending on the background (See for instance [40]).All in all, contrasting the evidence presented in this work to results following the latter approaches could be relevant to understand whether the theories in this work also suffer from strong coupling or not, and to assess the viability of these theories at least for applications in Cosmology where the FLRW background is of primary importance5 .

Equation for the scalar
A generally covariant form of the equation for the scalar was written in equation (42) in the form E ϕ = 0, with (83) and for a function G(ϕ, X) N in expression (84) can be used to write the standard Quartic Horndeski equation for the scalar on a torsionless spacetime as N = 0 by taking ∇ → ∇ and R → R.

Equation of motion for metric
A generally covariant form of the equation for the metric was written in equation (41) in the form E gµν = 0, with where higher covariant derivatives can be re expressed in terms of curvature tensors and lower derivatives using their commutator.
B. Vector sector and coefficients of the quadratic action in its initial form and after using constraints a. Tensor sector: The coefficients v i , i = 1, . . .7 in the action for the tensor sector (45) are (93) b. Vector sector: The quadratic action for the vector perturbations can be written as Let us note that even though this action depends on c, the dynamics is independent of this parameter.Namely, there is no dynamical vector perturbation.The coefficients h i , i = 1, . . . 25 are (95) c. Scalar sector: The coefficients f i , i = 1, . . .59 in the action for the scalar sector (46) are (170) where the equations of motion for the background fields have been used only in some coefficients such as f 51 .Using these equations one can get rid of second derivatives of φ and a, and of x, but this leads in many cases to longer expressions.

C. Coefficients of the quadratic action in its final form
The background functions m i with i = 1, 2, 3 relevant to the final form of the quadratic action ( 47  Using the equations for the background fields one can get rid of second derivatives of φ, first derivatives of a and of x, but this leads in m 2 and m 3 to much longer expressions.