Disformally self-tuning gravity

We extend a previous self-tuning analysis of the most general scalar-tensor theory of gravity in four dimensions with second order field equations by considering a generalized coupling to the matter sector. Through allowing a disformal coupling to matter we are able to extend the Fab Four model and construct a new class of theories that are able to tune away the cosmological constant on Friedmann-Lemaitre-Robertson-Walker backgrounds.


Introduction
Over the past few years there has been a lot of interest in attempts to explain some of the current major issues of modern cosmology, such as dark energy and dark matter, through constructing modified theories of gravity. One particular obstacle that presents itself even before one considers such issues, producing severe complications for proposed explanations of dark energy, is the so-called "cosmological constant problem" [1][2][3][4][5].
The cosmological constant problem is one that arises through combining our knowledge from the two pillars of 20 th century physics, quantum field theory (QFT) and general relativity (GR). Indeed, QFT predicts that the vacuum must have non-trivial structure (in particular, a non-zero energy), the source of which derives from the vacuum fluctuations of each quantum field. If we neglect gravitational effects (as in "ordinary" QFT) then this is not necessarily a problem per se, since physically one cannot measure energy in an absolute sense, only relatively. However, taking gravity into account introduces a dilemma as gravity is sensitive to absolute energies, and as such, upon applying a regularisation procedure to calculate the energy contributions from vacuum loop diagrams, it is found that this leads to a vacuum-energy contribution of order M 4 particle for each particle species. This is problematic as it renders the vacuum energy divergent. Naively, it is possible to fix this issue through introducing a bare term Λ 0 , which is itself divergent, into Einstein's equation such that it counteracts the contribution from the particle content. In doing so one renormalises the cosmological constant such that what actually gravitates is the finite net cosmological constant, (1.1) Current observational data requires that Λ ∼ (meV ) 4 [6] and so we see that a significant amount of fine-tuning is required in order for theory to match experiment. This in itself is a problem, however, it is not catastrophic and it is not the real issue as to why the cosmological constant is so troubling. Indeed, this issue is far more acute than a fine-tuning problem, and arises from the fact that the renormalisation of the cosmological constant is not stable to radiative corrections. Higher order loop corrections, or changes in the matter sector, lead to large changes in Λ at the scale of the QFT cut-off, meaning that Λ 0 is sensitive to high-energy physics, right up to the Planck scale. (For a detailed discussion of the cosmological constant problem we refer you to [7]) 2 Self-tuning Horndeski theory & a disformal coupling to matter Given that current experimental evidence is consistent with GR and Standard Model (SM) QFT, then any attempt to resolve this problem (without introducing any beyond the SM physics) must involve some form of modified theory of gravity. A particular approach to constructing a modified theory of gravity that has been employed often is one involving scalar-tensor combinations. Indeed this has proven to be a useful approach in a wide range of models, from Brans-Dicke gravity [8], to more recent models [9][10][11][12][13][14][15][16][17] inspired by Galileon theory [18]. The starting point for us is Horndeski's scalar-tensor theory of gravity [19], first discovered by G.W. Horndeski in 1974, and independently re-discovered more recently by C. Deffayet et al. [20]. This is the most general scalar-tensor theory that produces secondorder field equations, which is an essential requirement in order to avoid any Ostrogradski instability [21] in the theory. Indeed, Horndeski theory has received a renewed interest in research into modified theories of gravity in recent years [22][23][24][25]. We focus our attention on this particular approach, inspired by its effectiveness at providing a compelling proposal to rectify the cosmological constant problem in recent research. Indeed, such a solution was realised through the derivation of a class of self-tuning theories, the so-called Fab-Four [26,27], the argument here being that instead of concerning oneself over how to treat the vacuum energy contributions and radiative instability of the cosmological constant head-on, one can instead mitigate the effects of the cosmological constant on the geometry seen by matter, i.e. it simply does not gravitate. Preliminary analysis following the derivation of the Fab-Four also suggests that its radiative corrections can be kept under control [26,27], an essential requirement given the source of the cosmological constant problem in the first place. This "screening" of the net cosmological constant from the gravitational sector (such that it is not a source of curvature) is achieved by constructing a theory of gravity, using the Horndeski action, that describes an interaction between gravity and some scalar field φ.
The basic idea of such a screening mechanism, as described by Weinberg [1], is that the cosmological constant is absorbed by the scalar field's dynamics rather than the dynamics of gravity. Importantly, the Fab-Four avoids Weinberg's no-go theorem [1] for self-tuning solutions by forgoing the requirement of Poincaré invariance of the self-tuning scalar field φ. That is to say, φ is spatially homogeneous (satisfying the cosmological principle), but is allowed to evolve in time (i.e. φ = φ(t)) to enable it to "self-tune" relative to the value of the vacuum energy at any given instant in time. In accordance with Einstein's equivalence principle, matter is then introduced into the theory by minimally coupling it to gravity via the metric g µν , however, it is not coupled to the scalar field -this interacts purely with gravity, acting as a screening mechanism to cloak the vacuum energy density ρ m vac (contributed from the matter sector) from the gravitational sector such that ρ m vac has no impact on the spacetime curvature. In doing so, the theory is described by the following action where S H is the Horndeski action, S m is the effective action for matter, and ψ i are the matter fields, which are minimally coupled to g µν . Using this approach, the Fab-Four was shown to be the most general self-tuning theory of gravity in which matter is minimallycoupled to gravity.
Given that it has now been shown that a self-tuning theory as a solution to the cosmological constant problem is possible, it is natural to question whether it is feasible to construct a generalisation of this idea. Indeed, a minimal extension of the Fab-Four theory has recently been proposed by E. Babichev et al. in which the starting point involves replacing the potentials appearing in the self-tuning Fab-Four Lagrangian with more general functions that depend upon both the scalar field φ and the corresponding canonical kinetic term X = − 1 2 ∂ µ φ∂ µ φ [28]. Nevertheless, in the quest to construct the most general self-tuning Horndeski theory, we further extend the Fab-Four model to encompass the case in which the scalar field φ is allowed to enter the matter-sector, such that it interacts with matter directly. This can most readily be achieved through a disformal coupling of matter to gravity [29].
The aim, therefore, is to use the general idea of self-tuning, where a scalar field absorbs the effects of Λ, but in a more general context where the self-tuning scalar is allowed to couple directly to matter. In such an approach, one considers two distinct (but related) geometries: one defining the geometry on which matter plays out its dynamics, and one describing gravitation. It was shown by Bekenstein [29] that the most general relation between the physical and gravitational geometries, described by the two metricsḡ µν and g µν respectively, involving a scalar field φ that adheres to the weak equivalence principle and causality, is given by the following disformal transformation The inverse to (2.2) is found to be [30] g µν (x) = 1 which leads to an expression forX of the form, .
In principle, this relation between X andX can be inverted such that X = X(φ,X). With this in mind we can then rearrange (2.2) to obtain an expression for the inverse transformation between the two metrics g µν andḡ µν , and from this we can imply thatĀ andB are related to A and B in the following manner Given recent research into modified theories of gravity involving disformal couplings [31][32][33][34][35], it is natural to consider such an approach in an attempt to generalise the Fab-Four theory. Indeed, we aim to construct a self tuning theory of gravity described by the action (2.7) In this representation of the theory both gravity and matter are directly coupled to the scalar field φ(t) since there is a non-trivial dependence on φ contained inḡ µν . The reason being that we treat g µν , φ and ψ as the dynamical variables, with the physical geometry, g µν , determined via (2.2), i.e.ḡ µν =ḡ µν [g µν , φ, ψ]. We follow the convention of referring to these different representations of the theory as "frames", and shall refer to this particular representation as the Horndeski frame, which is the analogue of the Einstein frame. An alternative representation of the theory can be found by expressing the action in terms ofḡ µν , φ and ψ, and using these as the dynamical variables. In doing so we remove any direct coupling of the scalar field φ to the matter-sector at the level of the action, but gravity remains directly coupled to φ. We refer to this representation as the Jordan frame, and in this frame the action can be expressed as 1 (2.8) Writing the action in the Jordan frame is clearly advantageous as in this particular representation matter follows the geodesics defined by the physical metricḡ µν (as opposed to in the Horndeski frame where the matter geodesics defined by the metric are also influenced by variations in the scalar field φ), such that the associated energy-momentum tensor is covariantly conserved thus corresponding to the physical frame.
As the aim here is to construct a generalisation of the Fab-Four theory, in order to define the notion of self-tuning we shall follow the structure laid out in the derivation of the Fab-Four theory [26,27]. Thus, by a self-tuning theory, we postulate that their exists some scalar field, φ, that evolves dynamically such that it absorbs any energy density contributed by the net cosmological constant, and in doing so screens it from the gravitational sector such that it has no effect on the spacetime curvature that matter sees. In other words, the existence of this scalar field means that (effectively) the net cosmological constant (regardless of its value) does not gravitate. We do not require that such a scalar field is Poincaré invariant, and in doing so avoid Weinberg's no-go theorem for such self-tuning modifications of gravity [1].
In addition to this, the requirements that must be satisfied in order for the theory to be self-tuning, a so-called self-tuning filter [26,27], are as follows: 1. The vacuum solution, for the metric that matter sees (ḡ µν ), to such a theory should always be Minkowski spacetime no matter the value of the net cosmological constant; 2. This should remain so even after any phase transition in which the cosmological constant jumps instantaneously by a finite amount; 3. The theory should permit a non-trivial cosmology (ensuring that Minkowski spacetime is not the only solution, a condition that is certainly required by observation). 1 Note that SJ will not in general be of Horndeski form, and is more likely to be of beyond-Horndeski form [36,38]. This is because the Horndeski Lagrangian changes its form under general disformal transformations and so the gravitational action will no-longer resemble that of Horndeski theory in the Jordan frame.

Towards a self-tuning disformal theory of gravity
In analogy to the derivation of the Fab-Four we first consider the Horndeski Lagrangian, given by Up to total derivative terms that do not contribute to the equations of motion, the different pieces can be written as [30] 3) where G µν is the Einstein tensor, R the scalar curvature, To enable its construction we first study the cosmological set-up of this theory. Indeed, we shall require that the geometry in both frames (Horndeski and Jordon, respectively) is Friedmann-Lemaître-Robertson-Walker (FLRW). To this end, in the Horndeski frame, we treat g µν , φ and ψ i as the dynamical variables, with the Jordan-frame metricḡ µν being determined via (2.2). We then assume that the Horndeski-frame metric g µν abides by the cosmological principle, i.e. we require that at any given instant in time t the geometry defined by g µν is spatially homogeneous and isotropic. This requirement is achieved by foliating spacetime into a set of spacelike hypersurfaces, Σ t , such that the spatial "slice" at a given instant in time is homogeneous and isotropic. Accordingly the geometry defined by g µν is of the form, where N (t) is the lapse function, a(t) is the scale factor, and γ ij (x) is the (maximally symmetric) metric on the plane (k = 0), sphere (k = 1), or hyperboloid (k = −1).
In the Jordan frame, there is no direct interaction between the scalar field φ(t) and matter, insteadḡ µν , φ and ψ i are treated as the dynamical variables, with g µν determined via (2.5). It seems reasonable to consider this as the physical frame since matter follows the geodesics defined purely by metric in this frame (as opposed to in the Horndeski frame where the matter geodesics defined by the metric are also influenced by variations in the scalar field φ), and its corresponding energy-momentum tensorT µν is locally conserved (i.e.∇ µT µν = 0, whereas in the Horndeski frame ∇ µ T µν = 0 due to the direct coupling between φ and matter). Since we require that the matter sector is completely screened from any gravitational effects that would be introduced by the vacuum energy this amounts to stipulating that geometry defined byḡ µν is asymptotically Minkowski in form, An immediate consequence of imposing the geometries (3.6) and (3.7), along with the requirement that the scalar field is time-dependent (i.e. φ = φ(t)), is that the canonical kinetic term X takes the form when evaluated in the Horndeski frame, and when evaluated in the Jordan frame. We can also determine a mapping between the Horndeski and Jordan frames for the lapse function N (t) and the scale factor a(t) by noting that the geometry defined by g µν (3.6) can be expressed in the Jordan frame via the inverse disformal transformation (2.5) (3.10) Now the two expressions (3.6) and (3.10) must be equivalent, thus enabling us to infer the following relations as well as confirming the earlier relation (2.4). Finally, from the disformal relationship between the two metrics g µν ,ḡ µν (2.2) and the form of X (3.8), we can ascertain that the integration measures √ −g and √ −ḡ are related by [30,32] Before continuing, a remark must be made about the cosmological set-up and its effect on the action for the theory. Since we are considering a homogeneous scalar field φ = φ(t) it follows that the Lagrangian L H (x) = L H (t) will also be homogeneous. As such we can effectively neglect spatial dependence in the theory in the following manner: (3.14) where we have used that √ γ, and noted that γ ij (x) is a maximally symmetric spatial metric. Note that we have also factored out the spatial part of the determinant of the metric, as it plays no role. Accordingly, we define an effective metric determinant g →g such that Having set-up the cosmological structure we are now in a position to evaluate the Horndeski Lagrangian (3.1) on a background FLRW cosmology. As the Horndeski Lagrangian provides a description of the dynamics in the gravitational sector, we evaluate the appropriate curvature terms on the metric defined in this sector. We then use the fact that the two metrics are disformally related to transform the pieces of (3.1) to their expressions in terms of the physical metric via the inverse disformal transformation (2.5). After some work we obtain the following expression for the Horndeski Lagrangian (3.1) evaluated on a background FLRW cosmology whereH =ǡ a is the Hubble parameter in the Jordan frame, with d dt ( ) ≡ ( ) • , ( ) ,φ ≡ ∂ ∂φ ( ) and ( ) ,X ≡ ∂ ∂X ( ). Furthmore, we have defined to auxilliary functionsG 3 andG 5 , It is then possible to express (3.15) in the form As matter is decoupled from the scalar field φ (up to gravitational interactions) in the Jordan frame the on-shell equation of motion (EOM) for φ can be determined purely from (3.17). Indeed, from (3.17) and using the generalised Euler-Lagrange equation it follows that And on-shell this satisfies The Lagrangian (3.17) can also be employed to calculate the gravitational Hamiltonian. Indeed, upon calculating the canonical kinetic momenta of each of the dynamical fields, a(t), φ(t) andφ(t), (defined accordingly as pā ≡ respectively), we find that where, for example, Z i,φ ≡ ∂Z i ∂φ and Z i,φ,φ ≡ ∂ 2 Z i ∂φ∂φ (the other derivative terms are defined in the same manner).
Accordingly, the full Hamiltonian H total of the theory can be constructed from the contribution from the gravitational-sector H and a source from the matter-sector in the form of a cosmological fluid of energy density ρ m . Consequently, the full Hamiltonian is given by H total = H + ρ m . Importantly, as a result of the diffeomorphism invariance of the theory, the full Hamiltionian satisfies the constraint H total = 0, such that 4 Applying the self-tuning filter Up to this point we have kept things fairly general, only applying minimal constraints to our theory, however we now wish to pass it through our self-tuning filter (laid out at the end of section 2) and determine the contraints that we must apply in order for the theory to be self-tuning. We shall apply the filter in the Jordan frame in which the spacetime geometry is described byḡ µν . The reasoning being that this is the metric to which matter couples, and as such is required to be screened from the effects of the cosmological constant, i.e. we want the theory to self-tune with respect to this metric. We first consider the implications of applying the self-tuning filter in the situation where our cosmological background is in vacuo. Now, the matter-sector is expected to contribute a constant vacuum energy density that (as in the case of the Fab-Four derivation [26,27]) we identify with the cosmological constant, ρ m = Λ. According to the first filter the vacuum energy should have no effect on the spacetime curvature seen by matter, thus we require a flat spacetime regardless the value of Λ. It also follows from the second filter that this should remain true even in the case where the matter-sector undergoes a phase transition and in doing so alters the the net value of Λ by a constant amount (over an effectively infinitesimal time interval). This translates to the requirement that any abrupt change in the matter-sector is completely absorbed by the scalar field φ, leaving the geometry unaffected. Consequently the scalar field tunes itself to each change in the vacuum energy Λ and this must be permitted independently of the time of transition.
Given these considerations, our initial observation is that in order to be consistent with the first filter we seek cosmological vacuum solutions that are Ricci flat i.e. R = 0. Insisting on this provides us with so-called "on-shell-in-ā" conditions where we have defined s ≡ √ −k such thatH = s 2 a 2 when on-shell-in-ā. To proceed, we shall assume that the scalar φ is a continuous function, but thatφ, φ and ... φ may be discontinuous. With this in mind, we then go on-shell-in-ā at the level of the field equations. This means that we impose the conditions (4.1), but leave φ to be determined dynamically.
In doing so we find that such that the on-shell-in-ā field equations are where, to adhere to the second filter, the matter sector contributes Λ to the vacuum energy, where Λ is a piece-wise constant function of time. A subscript/superscript k on a variable will denote that it is on-shell-in-ā.
From (3.20) it can be seen that the gravitational Hamiltonian H is constructed from a set of functions Z i = Z i (ā, φ,φ,φ) (and their derivatives), with terms depending onφ, φ and ... φ . As such, requiring that it satisfy the condition (4.4) imposes retrictions on hoẅ φ, ... φ appear in H (as H k = −Λ there must be some discontinuity in H k to account for the discontinuous nature of Λ). Noting also from (3.18) that the scalar EOM is similarly constructed from the set of functions Z i = Z i (ā, φ,φ,φ) (and their derivatives), we can use these restrictions to impose constraints on the functional form of ε φ k . This leaves us with three possible cases to consider: Indeed, requiring that H k contains a discontinuity, accounting for the discontinuous nature of Λ, imposes the following contraints, which may be seen from (3.20): is linear inφ then we require thatφ ∼ step-function which then implies that ... φ ∼ delta-function. Note also that, from (3.20), this automatically implies that is independent ofφ, thenφ ∼ step-function implying thatφ ∼ delta-function and H k must be independent ofφ and ... φ .
Following on from this analysis, we can then study the implications of these results on the equations of motion for the scalar field ε φ k . Again, working on a case-by-case basis, and using the expression ε φ k (3.18) we find the following: However, we require that ε φ k = 0 and noting that in this case .... φ ∼ delta-function, we must therefore conclude that Z k i,φ,φ = 0 (since there is no support for a delta-function on the left-hand side of the equation); is linear inφ then clearly Z k i,φ,φ = 0 and it follows that ε φ k will be, at most, linear inφ (note that Z k i,φ,φ ∼ α(ā, φ,φ)φ, since Z k i is linear inφ, and hence Z k i,φ,φ,φ ∼ α(ā, φ,φ) ⇒φ 2 Z k i,φ,φ,φ ∼φ 2 α(ā, φ,φ). Accordingly, we see that this term will cancel with the term −φZ k i,φ,φ ∼φ 2 α(ā, φ,φ) in (3.20). As such, any non-linear terms inφ in the Hamiltonian will cancel out); iii) If Z k i (ā, φ,φ,φ) is independent ofφ it is trivially found that ε φ k will be, at most, linear inφ.
Hence, it is seen that in all three cases Z k i can be, at most, linear inφ and consequently the on-shell-in-ā Lagrangian L k will be also.
Following this analysis to its logical conclusion it is found that, in actual fact, in order to satisfy (4.1) the on-shell-in-ā Lagrangian must be equivalent to a total derivative (a detailed discussion on this analysis is provided in appendix A).
We are now in the position to construct a preliminary definition for a self-tuning Lagrangian that satisfies the on-shell-in-ā condition (4.1). To this end, we take into account that two Lagrangians that differ by a total derivative describe the same dynamical theory (i.e. they lead to the same equations of motion). Thus, we are working within an equivalence class of Lagrangians, [L, ≡], where two Lagrangians are considered equivalent if (and only if) they differ by a total derivativẽ We further note that when on-shell-in-ā,H = s a , hence (4.6) From our earlier analysis we also know that the on-shell-in-ā Lagrangian must be equal to a total derivative As such, we can construct the following "Horndeski-like" Lagrangiañ Such a Lagrangian certainly adheres to the self-tuning criteria (c.f. end of section 2); it is, in a sense, sufficient for self-tuning, but to what extent is it necessary? Indeed, a priori, it cannot be taken to be necessary as there could possibly be other equivalent Lagrangians, with Z i =Z i + ∆Z i , that admit the same set of self-tuning solutions. To establish whether this is the case we need to demand that the "tilded" and "untilded" systems each have equations of motion that give the same dynamics when on-shell (generically, i.e. not just when on-shell-in-ā). That is, we require that when on-shell In general we cannot imply from this statement that ε φ ≡ε φ , nor even that ε φ ∝ε φ , as there could well be a non-linear relation between all the relevant equations. Despite this it turns out that, in order for a general "Horndeski-like" self-tuning theory to be viable, we are forced to have In other words our putative self-tuning LagrangianL describes the general case of a selftuning theory, satisfying the self-tuning constraints of section 2. Furthermore, as this result implies that ∆Z i = 0 we find that Z i =Z i and that these functions can at most be dependent onā, φ andφ, i.e. Z i = Z i (ā, φ,φ). (Refer to appendix A for a detailed discussion on this analysis).
Given the functional expression for the Lagrangian (3.15) (when evaluated on an FRW background) we observe that the functions Z i may be expressed in the following form and importantly, by comparison with (3.15), it is clear that It is then possible to derive a set of equations relating the remaining non-trivial functions X i (i = 0, . . . , 3) and Y i (i = 0, 1). Indeed, from the analysis above we know that L ≡L and as such they differ by a total derivative at most. Taking this into account we find the following set of equations for X i and Y i are arbitrary potential terms. (Refer to appendix B for a detailed discussion).
We can also derive a set of equations that relate X i and Y i to the component functions K = K(φ, X), G i = G i (φ, X) (i = 3, 4, 5) of the Horndeski Lagrangian when evaluated on an FLRW background (3.15) by direct comparison of (4.11) with (3.15) Thus through equating the corresponding equations in (4.12) and (4.13) we obtain a set of partial differential equations in X that the component functions K(φ, X) and G i (φ, X) (i = 3, 4, 5) must satisfy in order for the theory to be self-tuning.

Recovering the Fab-Four
Having determined the self-tuning contraints we now wish to begin analysing particular cases. An important first step is checking the consistency of the theory, i.e. that in the special case whereĀ = 1, N = 1 and B = 0 it reduces to the Fab-Four. In this case the set of differential equations given in (4.13) take the form (4.14) Utilising the equations given in (4.12) we can solve these iteratively, starting with X 3 + Y 1 and rewriting it purely in terms ofG 5 . This then gives us a differential equation forG 5 which we can solve and subsequently deduce G 5 (using thatG 5,X = NĀ φ G 5 ). Then, by inserting our solutions forG 5 and G 5 into the next equation in (4.14), X 2 + Y 0 , we can solve for G 4 ; continuing in this fashion we can also determineG 3 , G 3 and finally K.
Accordingly, the following set of solutions are obtained: where X = 1 2φ 2 . (Note that we can identify the constant in the expression for K with the vacuum energy, such that const. = −Λ).
In this case we can follow a similar analysis as in the Fab-Four analysis [26,27] to deduce the covariant form of the self-tuning Lagrangian. Indeed, starting from the Horndeski Lagrangian evaluated on an FLRW background (3.15) we note that each of the arbitrary potential terms V i (i = 0, . . . , 3) and integration functions f 3 , f 4 , f 5 and g 5 are completely de-coupled from one another. As such we can analyse the form of (3.15) on a case-by-case basis and in doing so we find that the functions V 0 , V 2 , f 3 lead to vanishing contributions in (3.15); V 1 , V 3 , f 4 , f 5 and g 5 give non trivial terms, but f 4 and f 5 lead to the same type of expression, meaning that only four of the eight functions yield independent terms in the self-tuning FLRW Lagrangian. Upon lifting this from the FLRW Lagrangian to the full covariant form, we find that the functions that gave vanishing a FRW contribution to the Lagrangian were in fact total derivatives in the covariant form, just as in the original fab-four construction [26,27]. The remaining four non-trivial FLRW contribuitons can be expressed covariantly as where R is the scalar curvature,Ĝ = R 2 − 4R µν R µν + R µνρλ R µνρλ is the Gauss-Bonnet combination, G µν is the Einstein tensor and P µναβ = − 1 4 ε µνλσ R λσγδ ε γδαβ is the double dual of the Riemann tensor.
Note that the curvature terms contained in these component Lagrangians are evaluated in the Jordan frame, however in this particular case the Jordan and Horndeski frames coincide. Comparing these covariant expressions with those found in the Fab-Four [26,27] we see that they are indeed the component Lagrangians that constitute the Fab-Four, as required.

Investigating the conformally coupled case
Another check of our system of equations (4.12), (4.13) is to setĀ =Ā(φ) and B = 0, which is equivalent to a conformal transformation. Given that the Horndeski Lagrangian maintains its form under such transformations then in fact the original calculation of [26,27] actually also includes the case where matter is minimally coupled not to the Horndeski metric, but to a conformally related one, as long as the conformal factor depends on φ, but not X. What this means is that if we setĀ =Ā(φ) and B = 0 we should recover the Fab-Four, but where the curvature terms in the Lagrangian are expressed using the conformally scaled metric. We find that this is indeed the case.

The most general disformal case
Now that we have confirmed the consistency of our disformal generalisation of the Fab-Four we wish to study the disformal properties of the theory, in other words, we would like to study the effects of "switching on" the disformal part of (2.2). Given that a special disformal transformation [37] (Ā =Ā(φ),B =B(φ)) does not change the form of the Horndeski action, in order to provide any generalisation beyond Fab-Four theory we need to analyse the most general case in whichĀ andB (and, implicitly, N ) are in principle dependent on both φ andX =X(φ, X). As we shall see, requiring that the theory adheres to the self-tuning conditions of section 2 leads to important contraints on the general form of the disformal transformation (2.2).
An initial observation is that the functions X i and Y i on the left-hand side (LHS) of the equations given in (4.13) depend on φ andφ, however, the right-hand side (RHS) of each of these equations contains terms proportional toȦ(φ,X), which in this most general case will be dependent on φ,φ andφ. Therefore in order for LHS = RHS we require that the sum of terms proportional to powers inφ must vanish in each case. Concentrating on the X 0 equation (first equation given in (4.13)), this means we require that where α, β and γ are the coefficients ofȦ,Ȧ 2 andȦ 3 in the X 0 equation respectively, and are functions of φ andφ. ExpandingȦ(φ,X) we find that it can be expressed in the forṁ where we have noted from (3.9) thatẊ = 1 2φ 2 • =φφ, and also thatĀ ,φ andĀ ,X will be, at most, functions of φ andφ. Thus, upon inserting (4.21) into (4.20) we arrive at the following equation αλ + βλ 2 + γλ 3 + ασ + 2βλσ + 3γλ 2 σ φ + βσ 2 + 3γλσ 2 φ 2 + γσ 3φ3 = 0, (4.22) Now, as we are assuming thatĀ is (in general) non-trivially dependent on both φ andX, i.e.Ā =Ā(φ,X), it must be thatĀ ,φ = 0 andĀ ,X = 0, which from (4.21), implies that λ = 0 and σ = 0. We require that the coefficient of each power inφ vanishes, in order for (4.22) to hold for all values ofφ. Observe the chain of contraints that this necessitates and from this we see that the final term in (4.22), αλ + βλ 2 + γλ 3 , is trivially zero. Applying these results to our original expression (4.20), it follows that the coefficient for each power inȦ must vanish identically. The same argument can then be applied to the remaining equations given in (4.13) to conclude that this result holds for each equation.
The implication of this result is a non-trivial one. Indeed, it is found that by assuming thatĀ is a function of both φ and X and applying the ensuing contraints leads to a contradiction. We therefore conclude from this that in order for the theory to be self-tuning, A can be a function of φ at most, i.e.Ā =Ā(φ). This is a powerful result as the form of the Horndeski Lagrangian does not change under φ-dependent conformal transformations, meaning that ifĝ µν = A 2 (φ)g µν , we find where B has been redefined such thatB = AB, andÂ = 1. Thus, given that L H maintains its form under conformal transformations, along with the self-tuning requirement, implies that we can effectively set A = 1 which, due to the relation between the disformal transformation (2.2) and its inverse (2.5), further implies that upon moving to the Jordan frame (as in the previous analysis) we can also effectively setĀ = 1 (c.f. (2.6)).
Indeed, given this freedom to setĀ = 1 it is found that, upon several integrationby-parts, the Horndeski Lagrangian evaluated on an FLRW background L F LRW can be expressed as follows Accordingly, if we can solve the corresponding set of equations (4.12) and (4.13) with then the theory is guaranteed to be self-tuning 2 . Clearly, as these equations involve an unknown function of X, B(φ, X), expressed in the above set as an unkown function N , we cannot integrate these in general, but rather must do so by first specifying the arbitrary function. Another approach would be to specify the various G i , K, and solve for what N has to be. Even having done so, one would then like to be able to write down the covariant form of the Lagrangian, which may not be an easy task.

A particular solution for the most general disformal case
In this section we shall present a simple solution to the set of differential equations (4.26), and in doing so explicitly show that the corresponding Lagrangian can not be put into Fab-Four form. Before proceeding we would like to remind the reader of a few of the equations that have been used earlier in the paper as they shall be employed heavily in this analysis. The first we draw attention to is the set of differential equations (4.26), derived in the previous subsection, from which one can in principle determine the forms of the functions K, G 3 , G 4 and G 5 once the disformal coupling has been given in the form of B(φ, X), or equivalently 3 N (t). Futhermore, we shall be using the kinetic term (3.8) for the scalar field φ, along with the auxilliary functions,G 3 andG 5 , both of which are defined in (3.16) in terms of G 3 and G 5 . We first note from (4.26) that we have a system of four differential equations with five unknown functions, N, K, G 3 , G 4 and G 5 . The first equation in (4.26) is, of course, trivial to solve (for K) leaving us with three remaining equations and four unknown functions. Thus, whichever way we look at it, our system is under-determined, which is what allows us to choose how matter is to be disformally coupled. It is important to note that one cannot choose N to be of the form N = f (φ) (2X) −1/2 , the reason being that we require from (3.8) that N √ 2X =φ, hence if N were of this form one would arrive at inconsistent solutions since φ andφ are independent variables.
Having set-up our preliminary framework we now proceed to solve the differential equation for G 5 (fourth and final equation in (4.26)). To keep matters simple we shall make the 2 Note that solving this system with N = 1 gives the Fab-Four case. 3 From (2.6), (3.8) and (3.11) we see that Given this, and using our relation between G 5 andG 5 (3.16) we are left with the following differential equation whose solution is given byG 5 = f (φ) √ 2X + 1 3 V 2 and thus implying that Using (5.1), this leads to N √ 2XV ′ 3 = 0, which we solve with as we take N to be non-vanishing. We therefore see that in fact (5.1) gives no constraint on the form of N . We now turn our attention to G 4 which can be determined from the third equation in (4.26). Again, in the interest of obtaining an analytic solution, we note that as the integration function f (φ) is arbitrary we restrict to the case f = const, and in doing so we find that at which point we make our choice of N to be Using that the function V 1 (φ) is still free we further simplify restrict to V 1 = 0, giving the following solution to (5.7) This leads us onto to the penultimate differential equation to be solved, the second equation of (4.26), which we simplify by taking g(φ) = g = const. and h(φ) = h = const. We now have that G 4,φ = 0 and so one finds where I(φ) is an arbitrary integration function. (note that we have made use of the relation between G 3 andG 3 (c.f. (3.16)), and taken into account from earlier that we set V 1 = 0). Hence, G 3 (φ, X) = I(φ). In summary then, we find the following solution for the set of functions Upon inserting these into our disformally self-tuning Lagrangian (4.25), we find that for this particular scenario it has the following form This is an important result as it shows explicitly that the system of coupled differential equations (4.26) is solvable and leads to non-trivial (consistent) results, even for a simplified case as was analysed here. The analysis of this particular case has also identified which choices of N are not permissible if one is to obtain a consistent solution set. Furthermore, whilst the Lagrangian (5.13) is clearly of self-tuning form (4.8), it cannot be put into Fab-Four form [27], highlighting the fact that in the most general disformal case our theory extends beyond Fab-Four.

Summary & outlook
We have been able to show that it is indeed possible to generalise the Fab-Four theory and obtain a self-tuning theory of gravity in which the self-tuning scalar field φ is disformally coupled to matter. It has been shown that this generalisation is consistent with known results, reproducing the Fab-Four theory for both minimal coupling to the Horndeski metric and a minimal coupling to a Weyl-rescaled Horndeski metric -as long as the scaling function depends on φ but not X. Furthermore, we have found that the requirement that the scalar field φ is able to self-tune, and thus screen the cosmological constant, places strong constraints on any form of disformal coupling to matter in the theory. Indeed, it was found that, in general, the conformal part of any disformal coupling to matter necessarily must be a function of φ alone (as opposed to being a function of both φ and its canonical kinetic term X). Given this result, the general disformal case can be simplified by effectively setting the conformal function A(φ) to unity (due to the Horndeski Lagrangian maintaining its form under Weyl rescaling by a function of φ), and it is subsequently found that the theory can automatically be expressed in a self-tuning form in general. Thus if one can determine the Horndeski functions K(φ, X), G i (φ, X) (i = 3, 4, 5), then the theory is guaranteed to be self-tuning. The caveat of this result is that the differential equations that must be solved in order to determine the Horndeski functions can not be solved in general, only on a caseby-case basis, owing to the presence of an arbitrary function. We have, however, been able to solve the system for a particular case with non-trivial disformal coupling. This simple analysis also provided information on inadmissible choices of the lapse function N (t), and served to highlight that the resulting Lagrangian cannot be expressed in Fab-Four form.
Throughout this paper all analysis was carried out by evaluating the theory on an FLRW background, as such the theory, in its current form, is not covariant. Ideally, the aim would be to find a covariant form of the theory however at present there does not appear to be an "obvious" approach to take in accomplishing this task (we are unable to utilise the same procedure as in the Fab-Four case due to the additional terms introduced by a disformal coupling). It is possible that future research into this area may uncover an analogue approach to that taken in deriving a covariant form of the Fab-Four. Indeed, further analysis of the results presented in this paper may enable one to formulate a covariant expression using the particular case of the form of the Lagrangian on an FLRW background as a starting point.
Finally, it is worth noting that there has been a recent gain in interest into the possiblity of extending beyond Horndeski theory (see, for example, [38][39][40][41][42][43]), and given that the results obtained for the general disformal case cannot be expressed in Fab-Four form these may well prove useful as a starting point for such an extension.
If we now go (generically) on-shell, then where we have noted that the expression in the brackets on the second line vanishes by virtue of the on-shell requirements (4.9). Hence we find that ∆H should vanish when the on-shell conditions are satisfied, however, we are yet to determine whether ∆H vanishes algebraically or identically. Now, by assumption, ∆H cannot depend on ρ m , as if it did then this would imply that H also depends on ρ m , however, H generates the time evolution of the scalar field, φ, which (again by assumption) does not directly couple to the matter sector, and hence H should be independent of ρ m (as otherwise this would imply that φ interacts directly with matter). Consequently, ∆H cannot vanish by virtue of the equationH = −ρ m . In cases I) and II) (Z i non-linear inφ and linear inφ, respectively, c.f. end of section 4), we see that ε φ (c.f. (3.18)) contains ... a , howeverH (c.f. (3.20)) does not and so we cannot useε φ = 0 to enforce ∆H = 0 (as in both cases there would be non-trivial terms remaining with no corresponding term to cancel with). In case III) (Z i independent ofφ, c.f. end of section 4), we see thatε φ containsä, howeverH does not and so we cannot useε φ = 0 to enforce ∆H = 0 in this case either. Hence, as we cannot use the dynamical equationsH = −ρ m andε φ = 0 to enforce ∆H = 0 we are forced to conclude that ∆H is identically zero. In other words, We now turn our attention to the scalar equation of motion, ε φ (3.18). The analysis in this case is a little more involved than for the Hamiltonian case, and to aid ourselves we first consider the following.
As was the case with the Hamiltonian, in principle, the Lagrangians of our putative self-tuning theory and a general self-tuning theory will differ by a function Given this, we claim that To prove this we refer to our earlier derivation of H , (3.20), from which we can infer that We now know that when we are (generically) on-shell ∆H vanishes identically and this implies that each of the terms in the above equation vanish individually. By equating powers inH we immediately see that and this holds whatever dependence Z i has onφ. Hence, we find that which leaves us with a first-order differential equation for ∆Z i , and as ∆Z i,φ = 0 we can infer that it is a function ofā, φ andφ, at most, i.e. ∆Z i = ∆Z i (ā, φ,φ). Upon integrating (A.7) with respect toφ we find where σ i (ā, φ) is an arbitrary function ofā and φ. Now that we are equipped with this additional information we shall proceed with our analysis of the scalar equation of motion. As was the case for the Hamiltonian, in principle, the scalar equation of motion,ε φ , for our putative self-tuning theory will differ from that of a general self-tuning theory, ε φ , by a function ∆ε φ = ∆ε φ (ā,ȧ,ä, φ,φ,φ) as follows Now, as ε φ describes the motion of the scalar field, φ, which by assumption does not directly couple to the matter sector, it therefore cannot depend on ρ m (for the same reasons as discussed in the Hamiltonian case). Consequently, this implies that ∆ε φ should be independent of ρ m also. Thus, when on-shell, ∆ε φ cannot vanish by virtue of the equatioñ H = −ρ m . Noting from the set-up to this analysis, that ∆L =ā 3 3 i=0 ∆Z iH i , we have However, we know from previously, that regardless of the dependence of Z i onφ, ∆Z i,φ = 0 and so Now, when (generically) on-shell we require that the conditions (4.9) are satisfied, and this implies that, on-shell, ∆ε φ = 0. We know that this condition can not be satisfied by virtue of the equationH = −ρ m , and so, at best, it vanishes by virtue of the equationε φ = 0. For cases I) and II) (Z i non-linear inφ and linear inφ, respectively, c.f. end of section 4),ε φ contains ... a and so we cannot use it to substitute in forä in ∆ε φ (as there will be no corresponding term in ∆ε φ to cancel out the ... a term introduced in such a substitution). As such, in these cases, ∆ε φ must vanish identically (i.e. each term in (A.10) must vanish individually). Accordingly, through equating powers inH this implies that and using (A.8), For i = 0, 1 we see that the left-hand side vanishes due to the term i (1 − i) and so permits a non-trivial form for σ i . However, for i = 2, 3, we see that i (1 − i) = 0 and so we are forced to conclude that σ 2 = 0 (A.14) For the case in which Z i is independent ofφ (case III, c.f. end of section 4),ε φ does not contain ... a , but it does still containä and so we must be more careful in our analysis (as, in principle, this could be substituted in to ∆ε φ such that the terms cancel algebraically such that ∆ε φ is not identically zero, but ∆ε φ = 0 is satisfied). To proceed, we note that for Z i,φ = 0, the scalar equation of motion has the form 16) and similarly forε φ (with "tilded" functionsÃ,B,C replacing the functions A, B and C), as in both cases the coupling to matter is the same. It follows then, thaẗ which leads us to the expression where ∆A = A −Ã and similarly for ∆B and ∆C. Now, ∆ε φ ought to vanish by virtue of the equationε φ = 0 when on-shell, and so we can immediately infer that This is because the second term (in the third line) on the right-hand side of (A.18) contains φ whereas the third term does not, and hence they cannot cancel one another out (even in principle). Therefore it must also be the case that Hence, Now, we require thatÃ∆B =B∆A in order for ∆ε φ to vanish by virtue ofε φ = 0, and so utilising the above relations.Ã Assuming that ∆A = 0 (i.e. ∆ε φ vanishes by virtue ofε φ = 0 and not identically) this gives where α i (ā, φ) is an arbitrary 'constant' of integration (with respect toφ). It is evident from this expression that (where, as u i and v i are arbitrary functions we have absorbed any additional terms, introduced through integrating, into them). Accordingly,Z i has the following form(s) for each value of i = 0, 1, 2, 3: Notice, however, from (4.8) thatL vanishes when on-shell-in-ā which leads to the following as, in general, s a = 0. In patricular, this implies that u i = 0 ⇒Z i,φ = 0. Now, this scenario is highly undesirable as it leads to a highly constrained trivial theory in which the only solution permitted is a Minkowski spacetime (in direct violation of the self-tuning filter c.f. section 2. We are therefore forced to conclude that for a non-trivial theory, in actual fact, ∆A = 0 and hence ∆ε φ vanishes identically (as in cases I and II). This result implies that and we see that for i = 0, 1 this condition is satisfied by i (1 − i), however, for i = 2, 3 we see that i (1 − i) = 0 and, as such, we conclude that Therefore, in all three cases, σ 2 = 0 = σ 3 and ∆ε φ vanishes identically, i.e. ε φ =ε φ . B Derivation of system of differential equations for K(φ, X), G i (φ, X) (i = 3, 4, 5) Given the expression found for ∆Z i (A.4) in appendix A and the requirement that ∆ε φ = 0 identically we now know that σ 2 = 0 = σ 3 in all three cases (c.f. end of section 4) and so we aim to determine a more explicit form for the remaining non-trivial functions σ 0 and σ 1 . To this end, note that ∆Z i,φ = 0 in all three cases, and further that ∆Z i takes the form given in (A.8). As such, We now know that ∆ε φ must vanish on-shell and this immediately implies that i∆Z i,φǟ a = 0 (B.2) Furthermore, we now know that ∆ε φ must vanish identically and so the remaining terms must also vanisḣ Thus, through equating powers inH, we can infer from this that σ 1,φ = 3σ 0 +āσ 0,ā = 1 a 2 3ā 2 σ 0 +ā 3 σ 0,ā = 1 a 2 ā 3 σ 0 ,ā ⇒ā 2 σ 1,φ = ā 3 σ 0 ,ā (B.5) and hence, by defining a function µ = µ(ā, φ), we can unambiguously express σ 0 and σ 1 in the following formsā 2 σ 1 (ā, φ) = µ ,ā ,ā 3 σ 0 (ā, φ) = µ ,φ (B.6) Given the analysis thus far we now claim that our self-tuning ansatz,L and the general self-tuning Lagrangian, L, differ by a total derivative, i.e.
To prove this claim we note that ∆Z i has the form (A.8) and that σ 2 = 0 = σ 3 , and as such ∆Z 0 = σ 0φ , ∆Z 1 = σ 1 , ∆Z 2 = 0 = ∆Z 3 (B.8) From this we can deduce that and upon noting the forms of σ 0 and σ 1 , (B.6), we can re-express this as ∆L =ā 3 σ 0φ +ā 3 σ 1H =φµ ,φ +ȧµ ,ā = d dt µ(ā, φ) (B.10) as required. Given the functional expression for the Lagrangian, (3.15), we observe that the functions Z i can be expressed in the following form where s ≡ √ −k (=ȧ when on-shell-in-ā), and in particular, we note that Y 2 = 0 = Y 3 . Our aim now is to determine how X i and Y i are related and their functional forms. We start from our knowledge that the Lagrangian of the general self-tuning theory and our ansatz differ by a total derivative, i.e. L =L + d dt µ(ā, φ) , and hence, on the right-hand side we have that If we now equate powers inH we obtain the following set of equations Upon substituting the coefficients ofH,H 2 andH 3 into the coefficient ofH 0 we find that To analyse this further we shall first expandā −3 µ(ā, φ) as a power series, in terms if s a , around s = 0 in the following manner where V i (φ) are (as of yet) arbitrary functions of the scalar field. We therefore have thaṫ i (φ) ≡ dV i dφ and similarly for higher order derivatives).
Hence, equating powers in s a leads to the following relations X 0 = const. +φV ′ 0 (φ) (B.18) Using (B.11) we can now compare with the general Horndeski Lagrangian (whose form we calculated earlier), (3.15), to determine the forms of X i and Y i and/or their relations to one another.