Compact objects in scalar-tensor theories after GW170817
CCompact objects in scalar-tensor theories after GW170817
Javier Chagoya ∗ and Gianmassimo Tasinato † Department of Physics, Swansea University, Swansea, SA2 8PP, U.K.
The recent observations of neutron star mergers have changed our perspective on scalar-tensor theories of gravity, favouring models where gravitational waves travel at the speed oflight. In this work we consider a scalar-tensor set-up with such a property, belonging to abeyond Horndeski system, and we numerically investigate the physics of locally asymptoti-cally flat black holes and relativistic stars. We first determine regular black hole solutionsequipped with horizons: they are characterized by a deficit angle at infinity, and by largecontributions of the scalar to the geometry in the near horizon region. We then study con-figurations of incompressible relativistic stars. We show that their compactness can be muchhigher than stars with the same energy density in General Relativity, and the scalar fieldprofile imposes stringent constraints on the star properties. These results can suggest newways to probe the efficiency of screening mechanisms in strong gravity regimes, and canhelp to build specific observational tests for scalar-tensor gravity models with unit speed forgravitational waves.
I. INTRODUCTION
Scalar-tensor theories with non-minimal couplings between scalar fields and gravity find in-teresting applications to cosmology (dark energy and dark matter problems, see e.g. the review[1]) and quantum gravity (including Lorentz violating systems as Horava-Lisfshitz gravity, see [2]for a recent review). Moreover, they are able to screen fifth forces by means of the Vainshteinmechanism (see for example [3–6]). Over the years, many advances have been made in developingconsistent scalar-tensor theories, going from Brans-Dicke systems [7], to Galileons and Horndeskitheories [8, 9], to beyond Horndenski and DHOST/EST scenarios [10–15]. The study of black holesand compact relativistic stars in these richer scalar-tensor theories is relevant for phenomenologicalinvestigations of screening mechanisms inside compact sources [17–20], and for our theoretical un-derstanding of no-hair and singularity theorems in Einstein General Relativity (GR) non-minimallycoupled with scalar fields (see e.g. the discussion in [21]). The purpose of this work is to investigateasymptotically flat black holes and relativistic stars in a class of scalar-tensor theories compatiblewith the stringent constrains recently obtained from the observation of neutron star mergers.Asymptotically flat black hole solutions with non-trivial scalar profiles have been found inHorndeski gravity (see [22] for a review), and some are known for beyond Horndeski theories [23].A non-vanishing scalar field profile may or may not affect the properties of the geometry. Evenif black hole solutions in these theories exhibit only small deviations from their GR counterparts,it is possible that scalar field effects become more relevant in presence of matter, thus leadingto sizeable consequences that can be constrained with observational data. This phenomenon wasfirst pointed out in a theory of Brans-Dicke gravity applied to neutron star objects in [24–26] anddubbed spontaneous scalarisation ; it has also been analysed recently in more general scalar-tensortheories [27–31]. Investigations of explicit solutions for compact relativistic objects are necessaryfor acquiring a better understanding of how these systems can be distinguished from GR. Studiesalong these lines typically focus on neutron stars, since the strong gravitational field around these ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ g r- q c ] M a r objects provides a good laboratory to test modified gravity theories. These investigations haveshown that configurations compatible – within the error bars – with the measured masses andradii of neutron stars are common in scalar-tensor gravity [32–34]. The analysis of these systems inmodified gravity is at an early stage in comparison with the theoretical advances made in GR overthe past decades, although new developments concerning equation-of-state independent relationsbetween properties of relativistic compact objects indicate promising tools to constrain modifiedgravity theories [35–37].Recently, gravitational and electromagnetic radiation emitted by NS mergers was detected al-most simultaneously by LIGO, VIRGO, and an array of observatories on earth and in space [38],placing a strong constraint on the difference between the propagation speed of gravitational waves( c GW ) and the speed of light, − × − ≤ c GW − ≤ × − [39], where the speed of light isnormalized to unity. Besides the quadratic and cubic Horndeski Lagrangians, scalar-tensor theoriesof gravity generically predict gravitational waves that do not travel at the speed of light. There are,on the other hand, particular combinations of Horndeski and beyond Horndeski Lagrangians whichpredict c GW = 1 [40–43]. Observational consequences of these Lagrangians have been recentlyanalysed, and it has been shown that – in absence of a canonical kinetic term for a scalar field– the screening mechanism allows to recover exact GR solutions in vacuum, although screeningeffects are broken in presence of matter [30, 31].On the other hand, on general physical grounds we expect that the standard scalar kinetic termshould be present in the scalar action, being the leading dimension four operator that governs thescalar dynamics, at least around nearly flat backgrounds. In this paper we focus on a specificscalar-tensor theory that includes, besides the kinetic term of the scalar field, a combination ofquartic Horndeski and beyond Horndeski contributions satisfying the condition c GW = 1. Thepresence of the standard kinetic terms affects considerably the geometry, and we find several newphenomena associated with the non-linearity of our system of equations. In Section II we presentthe theory under consideration. In Section III we determine the conditions to satisfy for obtaining(locally) asymptotically flat black hole solutions supporting a non-trivial scalar field profile. Wenumerically analyse how the scalar affects the size of the horizons, and we find conditions to avoidnaked singularities. The corresponding solutions are characterized by a deficit angle induced bythe scalar field kinetic terms. In Section IV we proceed to study relativistic compact objects inthis theory and we find that, in contrast to other scenarios of beyond Horndeski systems, theangular deficit does not produce a singularity at the centre of these objects. The non-linearitiesof the equations lead to new phenomenological consequences, as for example specific relationsbetween radius and energy content of the objects we investigate. By matching interior and exteriorsolutions we find situations where the scalar field contributions to the geometry are dominantinside a compact object, but negligible in the exterior, pointing towards a sizeable breaking of aVainshtein screening mechanism. Finally, we discuss how the compactness of this scalar-tensorconfigurations, and the deficit angle itself, can be used to constrain this theory. II. THEORIES OF HORNDESKI AND BEYOND AFTER GW170817
The most general scalar-tensor theory leading to second order equations of motion is a combi-nation of the Horndenski Lagrangians [9]. Calling φ the scalar field, such Lagrangian densities are[44, 45] L = G , L = G [Φ] , L = G R + G ,X (cid:8) [Φ] − [Φ ] (cid:9) , L = G G µν Φ µν − G ,X (cid:8) [Φ] − ] + 2[Φ ] (cid:9) , (1)where Φ is a matrix with components ∇ µ ∇ ν φ , and X = − ∂ µ φ∂ µ φ , [Φ n ] = tr (Φ n ) , (cid:104) Φ (cid:105) = ∂ µ φ∂ µ ∂ ν φ∂ ν φ . (2) G i are arbitrary functions of φ and X , or only of X if we impose a shift symmetry φ → φ +const.The equations of motion associated with Lagrangians (1) are second order ensuring that the systemis free of Ostrogradsky instabilities. On the the other hand, it is possible to have healthy scalar-tensor theories also with higher order equations of motion, provided that constraint conditionsforbid the propagation of would-be Ostrogradsky ghosts. Explicit examples are the theories ofbeyond Horndeski [10], and their generalizations dubbed DHOST/EST theories [11, 12]. Thetheory of beyond Horndeski is constructed with the Lagrangian densities L bH = − F (cid:15) µνρσ (cid:15) µ (cid:48) ν (cid:48) ρ (cid:48) σ ∂ µ φ∂ µ (cid:48) φ Φ µµ (cid:48) Φ νν (cid:48) Φ ρρ (cid:48) , L bH = F (cid:15) µνρσ (cid:15) µ (cid:48) ν (cid:48) ρ (cid:48) σ (cid:48) ∂ µ φ∂ (cid:48) µ φ Φ νν (cid:48) Φ ρρ (cid:48) Φ σσ (cid:48) , (3)with F , arbitrary functions of φ , X . Theories described by a combination of the previous La-grangians – apart from systems only containing L and L – generally lead to a modification ofthe speed of propagation of gravity waves, hence they are disfavoured by the recent observationof gravitational waves from a neutron star merger GW170817 and its associated electromagneticcounterpart GRB 170817A. On the other hand, there are specific combinations of the Horndeskiand beyond Horndeski Langrangians which do not change the speed of gravitational waves [46]. Aparticular example is the combination L c = X + L + L bH , with F = G ,X /X , = X + G R + G ,X X (cid:0) (cid:104) Φ (cid:105) − (cid:104) Φ (cid:105) [Φ] (cid:1) , (4)which we consider in this work. This Lagrangian density includes the standard scalar kineticterm, accompanied by derivative self-interactions and non-minimal couplings with the metric whichbecome important in strong gravity regimes such as in proximity of black holes or in dense objects.For simplicity, and definiteness, we study this theory with the function G chosen as G = M P l + βM P l
X , (5)where β is a dimensionless constant. Black hole configurations for similar systems have been studiedin the past both in beyond Horndeski and vector-tensor theories [47–49]. In particular, a stealthSchwarzschild solution was fist discovered for vector-tensor systems with the same choice of G and a special value of β [50]. When the time component A of the vector is constant, this solutionis equivalent to a scalar-tensor stealth configuration for a scalar field of the form φ = q t + φ ( r ),with a constant q [51]. Further generalisations based on this solution can be found in [52, 53],where neutron stars and asymptotically flat black holes are constructed for arbitrary values of β and vector-tensor generalisations of (5).In this work we focus on the scalar-tensor theory (4), studying new black hole solutions invacuum with novel features (Section III), and the physics of gravitationally bound compact objectsmade of incompressible matter (Section IV). III. BLACK HOLES
The study of black hole solutions in vacuum for scalar-tensor theories with non-minimal cou-plings to gravity is interesting at least for two reasons. First, it allows to probe a strong gravityregime for the theory one considers, where non-perturbative contributions to screening mechanismscan make manifest sizeable deviations from GR results (see, e.g. [54, 55]). Second, it allows to testno-hair and singularity theorems in new settings, possibly revealing new geometries or topologiescharacterized by additional scalar charges (see, e.g. [21, 56]). In this section we aim to investigatewhether there exist asymptotically flat black hole configurations for the beyond Horndeski theoryof Lagrangian (4), answering almost affirmatively – in the sense that we find locally asymptoticallyflat black hole solutions, for which the curvature invariants vanish for large r , but that are char-acterized by a constant angular deficit at infinity. The existence of an angular deficit in beyondHorndeski theories was first identified in [57] as a potential source of singularities at the centre ofconfigurations of matter. As we show below, both in vacuum and inside compact objects the angu-lar deficit of the model we are considering does not affect the regularity of the solutions, providedthat some conditions are satisfied.The covariant form of the equations of motion (EOMs) for the the scalar φ and the metric g µν is given in Appendix A. Since we are interested on static, spherically symmetric space-times, westart imposing the following Ansatz for the metric ds = − f ( r ) dr + h ( r ) − dr + r dθ + r sin θdϕ , (6)while we allow for a linear time dependence in the scalar configuration φ = M P l φ t + φ ( r ) , (7)where φ is a dimensionless constant . This Ansatz for the scalar field is compatible with a staticspacetime (recall that the equations of motion always contain derivatives of the scalar) and havebeen extensively studied in the recent literature on scalar-tensor black hole solutions, since thetime dependence explicitly breaks the assumptions of no-hair theorems in Horndeski theory [22],thus opening up the possibility of finding asymptotically flat black holes dressed with a scalar field(see, e.g. [22, 23, 51], and the review [20]).Using these Ansatz for metric and scalar, we find that the ( t, r ) component of the metric EOMs(the ξ tr component in eq (A3)) reduces to an algebraic condition for the derivative of the radialscalar field profile φ , which reads βφ (cid:48) (cid:2) f h ( h + 1) φ (cid:48) + f (cid:0) h rφ (cid:48) f (cid:48) + φ ( rh (cid:48) −
1) + hφ (cid:1) − hrφ f (cid:48) (cid:3) f r (cid:0) f hφ (cid:48) − φ (cid:1) + 12 φ (cid:48) = 0 . (8) From now on we set M Pl = 1. The correct dimensions of all expressions are recovered after reinstating theappropriate factors of M Pl . If one chooses β = 0 – corresponding to GR plus a standard kinetic term for the scalar field –the only solution of the previous equation is φ (cid:48) = 0. On the other hand, if β (cid:54) = 0, we have a cubicequation for φ (cid:48) , which additionally admits the following two branches of solutions: φ (cid:48) = ± φ (cid:115) βhrf (cid:48) + f h r + 2 βf h + 2 βf rh (cid:48) − βf h βf + f hr + 2 βf h + 2 βf rf (cid:48) . (9)Notice that such branches are well defined also in the limit β →
0, giving φ (cid:48) = ± φ : hencethese branches are disconnected from the β = 0 branch, even when φ is turned on. The presenceof different branches is common in Horndeski and beyond Horndeski theories where the scalarfield derivative satisfies a non-linear algebraic equation, and the non-trivial scalar field profile isresponsible for providing a screening mechanism that recovers GR solutions in the strong gravityregime [6, 30, 31, 33]. In what follows, we will concentrate on the upper branch of the algebraicsolution (9). The remaining independent equations, that we take as the ( t, t ) and ( r, r ) componentsof the metric equations, are hard to solve exactly for f, h , but we can study the system numerically,or analytically in certain regimes.Flat space, corresponding to the choice f = h = 1 using Ansatz (6), is not a solution of theEOMs. Asymptotically de Sitter solutions can be easily found (very similar to the ones originallyfound in [58]), but here we will focus on black hole solutions that are at least locally asymptoticallyflat. This branch of solutions has been less studied in the literature, and it is important toinvestigate in detail the corresponding phenomenology. In order to take into account local insteadof global flatness, it is compulsory to slightly generalize the metric Ansatz (6) by including a deficitangle, ds = − f ( r ) dt + h ( r ) − dr + s − r d Ω , (10)with s not necessarily equal to one. This modification does not change the branch structure ofthe solutions of the scalar field equation. With such Ansatz, it is possible to analytically determineasymptotic solutions for the functions f, h expanded in inverse powers of the radial distance r ,imposing the condition that f = h = 1 at asymptotic infinity. The corresponding equations ofmotion with this Ansatz are given in (A4-A6). We find, up to second order in an 1 /r expansion, s = 1 − βφ , (11) f ( r ) = 1 − Mr − β φ (cid:0) βφ − (cid:1) r + O (cid:18) r (cid:19) , (12) h ( r ) = 1 − Mr + 4 β φ (cid:0) − βφ (cid:1) r + O (cid:18) r (cid:19) , (13) φ (cid:48) ( r ) = φ + 2 M φ r + 2 φ (cid:2) β φ + (cid:0) M − β (cid:1) − β φ (cid:3) r + O (cid:18) r (cid:19) , (14)with M an integration constant. We notice that, since s (cid:54) = 1, the geometry has a deficit angle:this is a consequence of including the kinetic terms of the scalar in our action. On the other hand,the radial dependence of the functions f , h gives us hope that a would-be conical singularity atthe origin r = 0 can be absent, or covered by horizons. In what follows, we discuss conditions forensuring that this is the case for the system under consideration. Notice that, besides the deficitangle, the standard ‘1 − M/r ’ behaviour (plus subleading corrections) of the metric componentsindicates that the metric is asymptotically flat and approaches GR results at large distances.Conical deficits covered by horizons have a long history in black hole physics, starting from [59],and physical realizations and interpretations – related with strings piercing the black hole horizons rg rr ϕ C FIG. 1:
Numerical BH solutions for M = β = 1 . The left panel shows the metric component g rr in GR(black line) and in the theory we consider, with φ spanning between . and . (blue lines). The size ofthe black hole horizon decreases as φ increases. For φ (cid:38) . we do not find solutions that form an eventhorizon. The dashed line shows a solution for φ = 0 . . The compactness of these black holes is shown inthe right panel. in Abelian-Higgs models [56] – can be subtle [21]. It is interesting that conical deficits appear alsoin the context of a single scalar field coupled with gravity, and we will later attempt to connectthem with no hair theorems for this system. Geometries with similar deficit angles arise whenconsidering gravitational monopoles [60–62].The presence of conical singularities in solutions of beyond Horndeski theories has been firstpointed out in [57, 63]: they focus on systems that are not shift symmetric, finding harmful conicalsingularities at the origin unless the parameters of the theory are appropriately tuned. A set-up more similar to ours has been analysed in a vector-tensor system [64], showing that conicalsingularities can then be avoided. We will make more detailed comparisons with these works inlater Sections. A. Numerical evidence for regular black holes
We now provide numerical evidence that spherically symmetric, locally asymptotically flat so-lutions of the EOMs (A4-A6) are free of naked conical singularities at the origin, when appropriateconditions are satisfied. As we shall see, despite the fact that the solution for the metric compo-nents has the standard 1 /r behaviour at large distances from the origin, there arise large deviationsfrom GR configurations near the black hole horizon.We numerically solve equations (A4-A6), using the asymptotic fields given in eqs. (12) and (13)as boundary conditions, and proceed integrating inwards towards small r until we encounter theposition r h of a horizon, defined by the condition g rr = h ( r h ) = 0. The system of equations (A4-A6)is reduced to two equations for f and h , since we impose s = 1 − βφ and we algebraically solvethe equation ξ tr = 0 for φ (cid:48) . We fix β = 1, so that the size of the angular deficit is controlled onlyby the scalar parameter φ . The boundary conditions are then specified in terms of two quantities:the black hole mass M , and φ . For definiteness we fix M = 0 . M P l = 1) and construct black hole solutions characterised by different values of φ . In orderto ensure that the conical deficit is positive ( s > ≤ φ < / √ β ∼ .
57. Our numerical results are shown in Fig. 1.The left panel of Fig. 1 shows h ( r ) – the inverse of the radial metric component. The black lineshows the quantity h ( r ) for the Schwarzschild metric, and the blue lines correspond to differentvalues of φ between 0 .
02 and 0 .
32. For each of the blue lines the function h ( r ) crosses zero,indicating the position of an horizon, whose size shrinks as φ increases. Solutions for 0 . < φ (cid:46) .
40 can be found as well, but they require a higher numerical precision near the horizon. For φ (cid:38) .
40 we do not find regular solutions equipped with an horizon: an interpretation for thisfact will be provided below.The right panel shows the compactness of such black holes, C BH = Mr h , (15)where r h is the radius of the event horizon, and M = 0 . C Schw = 0 .
5. The compactness increases non-linearly with φ , showing that – thanks to the O (1 /r )corrections to the metric – our solutions are different from the Schwarzschild configuration whenapproaching the horizon.Let us return to specifically discuss the behaviour of the system for φ (cid:38) .
40. For our valuesof M = 0 . β = 1, we could not find solutions with an event horizon for φ (cid:38) .
40. Indeed,when changing from φ = 0 .
40 to φ = 0 .
41 the solution for the radial metric component changesdrastically from profiles like those shown in the blue lines of Fig. 1 to a profile as the one shown inred in the same figure. This limiting value of φ is well lower than the bound one would infer fromrequiring the angular part of the metric to have a positive signature, φ < .
57. The reason forthis behaviour is the following: for any given φ , there exists a corresponding minimum mass M min that the black hole must have, in order for ensuring that φ ( r ) is real everywhere. For φ (cid:38) . M = 0 . M min , the scalar field becomes imaginary at a finiteradius r c > M min ). In this regime, since the action and equations of motionremain real after the replacement φ → iφ , one might accept the possibility that the scalar fieldcan be imaginary, and a solution with real metric can be found for r < r c . The metric components g tt and g rr match continuously to the solution for r > r c : but they and the Ricci scalar diverge at r = 0. In order to avoid such singular geometries, associated with imaginary scalar fields, we mustrequire that the mass parameter M characterizing the metric components f ( r ), h ( r ) is larger than M min . It would be interesting to find a dynamical method to generate such minumum mass forthe system.We can numerically plot the behaviour of the solutions for the set-up we are considering. Theleft panel of Fig. 2 shows the behaviour of the scalar field solution near the minimum mass M min corresponding to φ = 0 .
30. For
M > M min , the spacetime has an event horizon, and φ (cid:48) divergesthere, but the geometry and the trace of the energy-momentum tensor are regular at the horizon.For M = M min the spacetime is regular everywhere and does not have horizons, as shown in theright panel of Fig. 2. This is the only solution for which φ (cid:48) is always real and vanishes preciselyat r = 0. For M < M min , φ (cid:48) vanishes at r c . This solution can be extended to r < r c if the scalarfield is allowed to be imaginary, at the price of introducing a naked singularity in the geometry.In summary, for M > M min we have black hole geometries with an event horizon, for M = M min we have a regular geometry without horizons, and for M < M min we have geometries that do nothave a horizon, but they become singular in a region where the scalar field is imaginary: inorder to have a regular geometry, we need to impose M ≥ M min . A similar situation exists inquartic Horndeski models with a scalar field that depends only on r , where a minimum mass thatseparates black holes from naked singularities is given in terms of the coupling constants of themodel, which also determine the (secondary) asymptotic scalar hair [23]. Another analogy is theReissner-Nordstrom black hole: given a electric charge Q , a minimum black hole mass is requiredto keep the singularity at r = 0 protected by an event horizon. r − . − . . . . . . . . φ φ = 0 . M > M min M = M min M = 0 0 1 2 3 4 5 r . . . . . . { g rr , g tt } φ = 0 . , M = M min g tt g rr FIG. 2:
Vacuum geometries. The left panel shows φ (cid:48) for a black hole geometry (dashed line) with horizonrepresented by the dashed vertical line, a regular geometry (solid line), and a singular geometry. The radius r c where the scalar field becomes imaginary in the singular case is indicated by the dotted vertical line, andthe region r < r c shows the opposite of the norm of φ (cid:48) . The right panel shows the metric fields of the regulargeometry, which exists only for M = M min . By repeating the analysis described above for different values of φ , we numerically found thatthe minimum mass depends quadratically on φ .Despite the fact that the geometry does not correspond to flat space at spatial infinity, thecurvature invariants go to zero, so the space-time is locally asymptotically flat, and the blackhole is isolated and not affected by far away contributions to the energy momentum tensor. Theasymptotic properties of the black hole geometry seem to depend on the scalar field properties,through the deficit angle s , which at first sight enters in the computation of asymptotic charges.On the other hand, some care is needed to compute the gravitational mass through an ADMintegral in theories with deficit angles. This topic has been clarified in [60] for a geometry with thesame asympotics as ours. Their work explains that the ADM energy should be properly normalizedby the total angular volume of the asymptotic geometry, which includes the deficit angle. Followingtheir procedure, we find that the ADM mass for our system is M ADM = M (16)with M the coefficient of the 1 /r terms in the metric components f , h . Since the gravitationalADM mass is the only asymptotic charge for these black hole configurations, and it does notdepend on the scalar parameter φ , we conclude that our black holes do not have scalar hairs .This conclusion is in agreement with the recent paper [65]. IV. RELATIVISTIC COMPACT OBJECTS
In this Section we analyse non-singular, gravitationally bound star-like objects with sphericalsymmetry, studying how the non-minimally coupled scalar field we consider modifies their prop-erties with respect to GR configurations. We find numerical solutions that represent sizeabledeviations from GR solutions when the scalar parameter φ is large (see our scalar Ansatz (7)),and that are nevertheless connected to GR in the limit φ →
0. Using the results of the previousSection, we match the interior configurations for these compact objects to the exterior solutions we We consider ‘scalar hair’ as any conserved quantity which can be measured asymptotically far from the black hole,and that depends on the scalar parameter φ . previously determined, in order to investigate the efficiency of Vainshtein screening right outsideour configurations describing compact objects.We wish to study static, spherically symmetric configurations of matter minimally coupled togravity, S = (cid:90) d x √− g L c + S m , (17)where L c is defined in eq. (4), and the matter action defines the corresponding matter energy-momentum tensor as T µν = − √− g δS m δg µν . (18)The equations of motion for the metric result in ξ µν = T µν , (19)where ξ µν is the tensor defined in eq (A3), including metric and scalar contributions. We considera perfect fluid, so that the only non-vanishing components of the energy-momentum tensor are T tt = − ρ ( r ) , T ij = p ( r ) δ ij , (20)where the Latin indices denote the spatial components of the energy-momentum tensor, and ρ and p characterise the density and pressure of the perfect fluid.We take the metric Ansatz (10), with s ≡ − βφ . (21)In this way we guarantee that these solutions are in the same coordinate frame as the exteriorsolutions determined in the previous section. The fluid energy density and pressure can then beexpressed in terms of the metric components and scalar field through the relations ρ ( r ) = − ξ ( r ) , p ( r ) = ξ rr ( r ) , (22)with ξ rr the ( r, r ) component of the tensor ξ µν obtained by raising one index in equation (A3).In order to describe the fluid we also need to consider an equation of state. We will considerconfigurations of constant density, ρ ( r ) = ρ . (23)Although it is not fully realistic, this set-up allows us to obtain some analytic results, as well asexact numerical solutions. We are interested in configurations that are everywhere regular: weimpose that f (cid:48) (0) = h (cid:48) (0) = p (cid:48) (0) = 0 to ensure regularity at the origin of the configuration. Theradial size R s of the compact object is defined as the point where the pressure profile for mattervanishes, p ( R s ) = 0.Since the energy-momentum tensor is diagonal and matter is not directly coupled to the scalarfield, the component ξ tr of the metric equations of motion and the scalar field equation remainunchanged with respect to the vacuum case, and can be solved algebraically for φ (cid:48) . In addition tothe Einstein and scalar field equations, we impose the condition that the matter energy-momentumtensor is covariantly conserved, ∇ µ T µν = 0 . (24) This is indeed implied by the Einstein equations through the Bianchi Identities, given that we do not directlycouple the scalar with matter. f ( r ) with solution ( f is a constant) f ( r ) = f ( p ( r ) + ρ ) − . (25)Plugging the algebraic solution for φ (cid:48) and (25) into the equations of motion we reduce the systemto two equations for h ( r ) and p ( r ). Before entering into this topic, it is interesting to consider thesmall r limit of our system, and compute the Ricci scalar. We find R ( r (cid:28)
1) = 2(1 − βφ − h ) r − r ρ h (cid:48) (0) + h (cid:48) (0) p (0) − h p (cid:48) (0) ρ + p (0) + regular terms . (26)The coefficient of 1 /r vanishes since h = s = 1 − βφ , and the coefficient of 1 /r vanishes dueto the regularity conditions at the origin h (cid:48) (0) = 0 , p (cid:48) (0) = 0. This fact distinguishes our systemfrom the beyond Horndeski set-up studied in [57, 63], where it was shown that the angular deficitinduces a singularity at r = 0 when the scalar field depends only on r , due to an 1 /r divergence inthe Ricci scalar. In [64], it was shown that this singularity can be removed in beyond GeneralisedProca theories thanks to the presence of a time component of the vector field. This is heuristicallyrelated to our results, since the linear dependence in t of our scalar field can be seen as the timecomponent of a vector field A µ in the scalar limit A µ = ∇ µ φ .Let us now return to discuss the solutions to our equations. We fix the constant density ρ inthe star interior, and the radius R s of the object: we would like then to determine solutions of ourequations with the appropriate boundary conditions discussed above. In the limit of small β , werecover the standard GR solutions: expanding h ( r ) = h ( r ) + βh ( r ) + . . . , and similarly for p ( r ),we find that the leading terms are the GR ones corresponding to a TOV incompressible solution[66]: h ( r ) = 1 − ρ r , (27) p ( r ) = ρ (cid:112) − ρ R s / − (cid:112) − ρ r / (cid:112) − ρ r / − (cid:112) − ρ R s / . (28)It is interesting that the GR results are recovered for small β , although we are working in a branchof solutions that is formally disconnected from GR, and includes a non-trivial profile for the scalarfield, φ (cid:48) ( r ) = (cid:114) f ρ φ (cid:112) − R s ρ (cid:112) (6 − r ρ ) (6 − R s ρ ) − r ρ . (29)which survives in the small β limit (analogously to the vacuum configurations, as discussed aroundeq (9)).Outside the regime of β small we cannot find analytical solutions, but we can attempt an ap-proximation for low density, or investigate the system numerically. We consider the two possibilitiesin what follows. A. Analytic solutions for low density
We assume that h and p can be expanded as h ( r ) = h + ρ h ( r ) + ρ h ( r ) + . . . and p ( r ) = ρ p ( r ) + ρ p ( r ) + . . . . These expansions are motivated by the GR solutions for the same system1[66]. Solving the equations of motion for h ( r ) and p ( r ) order by order in ρ we find h ( r ) = s − r ρ βρ φ s f r (cid:18) r − rβs r + 4 βs − (cid:112) βs tan − r √ βs (cid:19) , (30) p ( r ) = (cid:0) R s − r (cid:1) ρ s + ρ R s R s − r s + ρ f β / φ s / (cid:18) r tan − r √ βs − R s tan − R s √ βs (cid:19) . (31)We remind the reader that s = 1 − βφ . These radial profiles of the interior configuration arequite different from the GR ones. To obtain the previous solutions we impose appropriate boundaryconditions at the origin, and we demand a fixed radius R s for the star. We set to zero an integrationconstant in h ( r ) by demanding the metric to be regular at the origin, and express the integrationconstant in p ( r ) in terms of the radius R s where p ( r ) vanishes. Notice that by requiring the starradius R s to remain always the same as we go to higher orders in ρ , we allow the central pressureto change due to the perturbative corrections. Up to third order, the central pressure changes to p (0) = R s ρ s + ρ (cid:32) R s s + βf φ − β / φ s / R s f tan − R s √ βs (cid:33) . (32)On the other hand, we have checked that up to third order in ρ , the central value of h ( r ) remainsfixed to h = s due to non-trivial cancellations between the higher order corrections. (This ensuresthat the Ricci scalar R remains regular at the origin, see eq (26)).The limit of empty object, ρ → h and p shown in eqs. (30, 31) has tobe taken with some care. These profiles solve the equations of motion obtained after imposing thecovariant conservation of matter, eq. (25), and are not necessarily continuously connected to thevacuum solutions (12-14). When ρ = 0 the pressure vanishes as well, and the continuity equationsloses its physical interpretation. However, to be consistent with the system of equations that wesolved, we have to require f ∼ ρ , so that f acquires a finite value. On the other hand, the vacuumsolutions do not admit in general a constant profile for f : the only way to make this possible is toimpose that M and φ vanish. Thus, if we want that the limit ρ → φ = 0 when ρ = 0, and the solution reduces to Minkowski spacetime with a constant scalar field. B. Numerical solutions
We now investigate interior configurations using numerical methods, in a regime where β , φ and ρ are not necessarily small. As we shall learn, we find interesting conditions on the parametersinvolved in order to get regular solutions, which can indicate new ways to constrain the scalar-tensor theories under consideration. Our analysis will focus to study the compactness of the stellarobject, a physical quantity that will be helpful to point out differences with GR results.We compute interior solutions for different values of φ and ρ by solving numerically the systemof equations derived from (17)-(20), with the metric Ansatz (10) and s = 1 − βφ . The initialconditions are set at small radius, and are determined by Taylor expanding the equations of motionaround r = 0, imposing that at the origin the fields behave as h (0) = 1 − φ β , h (cid:48) (0) = 0, and p (cid:48) (0) = 0, and solving for h (cid:48)(cid:48) (0) and p (cid:48)(cid:48) (0). We work in units where M P l = 1, and we fix fordefinitiness β = 1; we work imposing a fixed radius R s for the star, R s = 1 . h ( r ), the pressure p ( r ), and their derivatives near the origin are the2 . . . . . . φ . . . . . C s . ρ crit . ρ crit . ρ crit . ρ crit . ρ crit FIG. 3:
Compactness of constant density objects for β = 1 and different values of φ . The dashed line showsthe GR limit for the compactness, which is obtained only for an object with the critical density ρ crit . Thedensity of each solution is indicated by the point color. The gap in the sequence of solutions with ρ = 0 . ρ crit and ρ = 0 . ρ crit is an effect of the scalar field contributions, as explained in the main text. constant density ρ , the value of φ , and the central pressure p ( r = 0) = p – which controlsthe radius R s of the resulting configuration. To explore this parameter space we choose arbitraryvalues of ρ and φ , and we select p by requiring that the resulting configurations have a givenradius (that we choose arbitrarily). In GR, the central pressure that satisfies this requirement canbe computed exactly for stars of radius R s , by evaluating the TOV incompressible solution for thepressure at the origin [66]: p ,GR = ρ (cid:112) − R s ρ / − √ √ − (cid:112) − R s ρ / . (33)For our beyond Horndeski system we do not have an analytic method for determining the centralpressure associated with a configuration with a given radius R s . Thus, we proceed numerically byfixing ρ and φ and shooting p until we find a solution with the desired radius. For any ρ andsmall φ , Eq. (33) serves as seed for p : then the resulting p serves respectively as seed for thecentral pressure of a configuration with a higher φ , and this process is repeated until φ ∼ . R s = 1 .
5, hence it is convenient to parametrise thedensity ρ in terms of the critical density in GR for an object of a given R s . We do so by writing ρ = Aρ crit , where A is a constant in the range (0 ,
1) and ρ crit is the critical density of a compactobject of constant density in GR (see, e.g., [67]), ρ crit = 163 R s . (34)Solutions with ρ ≥ ρ crit do not exist in GR, and we do not find evidence of their existence in thebeyond Horndeski model under consideration.3 (cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159)(cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159)(cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242) (cid:242) (cid:242) Ρ (cid:61) Ρ crit (cid:72) R (cid:42) (cid:61) (cid:76) Φ R (cid:42) FIG. 4: R s - φ parameter space for a fixed ρ , equal to 60 % of the critical density of an object of radius R s = 1 . . Green points correspond to the solutions shown in red in Fig. 3. For R s (cid:38) . there is a regionof the parameter space where we do not find solutions. The points coloured in orange are referred to in thenext Figure. We apply the results of this numerical method to investigate a physically relevant quantity, thestellar compactness, which allows us to find constraints on the parameters involved, and also topoint out differences with GR configurations. The intrinsic stellar compactness, which we plot inFig. 3, is defined as C s = m ( R s ) R s . (35)In the previous expression, the mass of the star m ( R s ) corresponds the value at R s of the the massfunction m ( r ) defined by expressing the metric component h ( r ) in the stellar interior as h ( r ) = 1 − m ( r ) r , (36)that is, including within m ( r ) all the radial dependence of corrections to the Schwarzschild metricdue to matter and scalar field. The compactness defined in this way only includes contributions ofthe interior of the star – this is why we call it intrinsic – and it is in principle different from thecompactness as measured by an asymptotic observer, which we shall discuss in the next subsec-tion. Such difference is important for characterizing the efficiency of the screening mechanism inproximity of the object surface.Each point in Fig. 3 represents a configuration of matter with radius R s = 1 .
5, density indicatedby the point colour, φ by the x -axis, and stellar compactness by the y -axis. We observe thefollowing properties: • High stellar compactness is possible for configurations with a low density of matter: this isdue to the large contributions of the scalar profile for characterizing the internal geometryof the system. • For ρ (cid:38) . ρ c , there exists a range of values of φ where we cannot find configurations withthe desired radius R s . The reasons for this will be explored in some length below. • The intrinsic stellar compactness does not exceed the GR limit C = 4 / ≈ .
44 (see, e.g.,[67]). This is in contrast to what happens in vector-tensor theories [53], but similar findings4 Ρ (cid:61) Ρ crit (cid:45) (cid:45) Φ (cid:68) p Ρ (cid:61) Ρ crit Φ (cid:68) p (cid:68) p FIG. 5:
Difference between the pressure in GR and in the beyond Horndeski model under consideration forconfigurations with of the critical density in GR for an object of radius R s = 1 . . The left panel shows ∆ p for solutions before the gap along the sequence of green points in Fig. 4, while the right panel correspondsto solutions in the gap (orange curves correspond to orange points in Fig. 4) and along the sequence of greenpoints after the gap. have been reported for a subset of Horndeski gravity [32]. In the next section we show thatthis is true even when the effects from the exterior solution are taken into account.The fact that we find a gap in the range of allowed stellar densities is interesting, and deservessome more words since it can suggest ways to test and constrain the parameter space of relativisticcompact objects in scalar-tensor theories. We investigate in more detail what happens in the regionwhere we cannot find solutions with R s = 1 .
5. We fix the density to be 60% of the critical density:the green points in Fig. 4 correspond to the configurations shown in red in Fig. 3. From φ ≈ . φ ≈ .
10 we do not find solutions with R s = 1 .
5. Indeed, around φ ≈ .
02 there is a drasticchange in the maximum radius, which falls to about R s = 1 .
3, as shown by the orange points inFig. 4. The blue points in the same figure show configurations with the same density as the greenand orange points, but for different radius: these are drawn in order to outline the region wheresolutions do not exist.The origin of an interval in the parameter space where solutions do not exist, and in particularof the drastic change of R s near the lower end of this interval in φ , can be understood with thehelp of Fig. 5, where we plot a quantity defined as∆ p ( r ) = p ( r ) − p GR ( r ) , = ξ rr − G rr , (37)where ξ rr is the ( r, r ) component of the left-hand-side of the equations of motion for the metric(see eqs (19) and (22)), while G rr the ( r, r ) component of the Einstein tensor calculated on theconfiguration we examine, with no contributions from the scalar field (recall we work in units M P l = 1). The quantity ∆ p ( r ) describes the specific contributions to the total pressure whichcan be associated with the scalar field. The left panel shows ∆ p for solutions with φ < .
02 and R s = 1 .
5; the solid line corresponds to the last configuration along the sequence of green pointsbefore the gap in Fig. 4. We see that ∆ p has a minimum at some radius significantly smaller than R s . Based on this, we speculate that for φ (cid:38) .
02, ∆ p acquires large negative values, whose sizeis sufficient to drive the total pressure p ( r ) to zero at a radius smaller than the value R s = 1 . Φ C FIG. 6:
Compactness of constant density configurations for β = 1 . The case ρ = 0 . ρ crit is shown in blue,and ρ = 0 . ρ crit in red. The dashed lines show the stellar compactness, and the points show the compactnessmeasured asymptotically. The scalar field can have a relevant impact in the asymptotic compactness, but notenough as to get values of C higher than the GR limit C ≈ . . scalar-tensor theory under consideration where – due to large contributions associated with thescalar field – there do not exist compact configurations for certain radii and energy densities.As mentioned above, we can overcome the problem and find solutions by changing some of theconditions, for example by reducing the stellar size R s . The physical requirement is that the totalpressure vanishes at R s . In order to find the correct value of R s where this happens maintainingthe same energy density ρ , we thus need to change the central pressure to a smaller value suchthat both the GR and scalar field contributions to the pressure vanish at the same point. Theconfigurations with maximum radius that we find are shown with orange markers in Fig. 4, andthe profiles of ∆ p associated to them are shown with orange lines in the right panel Fig. 5. Thecurves shown in blue in the same plot instead correspond to configurations along the sequence ofgreen points, to the right of the gap.These results show that the scalar-tensor theory under consideration imposes more stringentconstraints on the stellar properties with respect to GR, since we identified forbidden regions onthe the energy density-radius plane, which depend on the value of φ , and where regular starconfigurations do not exist. In a more refined version of our analysis, considering a polytropicequation of state, this fact can suggest observable tests for the parameter space of these scalar-tensor theories, which would be excluded in case compact objects are found within the forbiddenregions. C. Matching of interior and exterior solutions
In section III we learned that static, spherically symmetric vacuum solutions to the equationsof motion derived from (4) do not correspond exactly to GR configurations in vacuum, since theydiffer from the Schwarzschild solution by an amount controlled by φ and β . Therefore, the extrinsiccompactness measured by an observer far away from a compact object can be different from theintrinsic quantity we studied in Section IV B, – eq (35) and below – due to contributions fromthe exterior part of the geometry. To investigate how large these contributions are, we take thevery same values of the metric and scalar field at a position R s from the solutions shown foundin Section IV B, and we use these values as initial conditions to integrate numerically the vacuumequations from R s outwards. At large r , we compute the gravitational mass using the asymptotic6solutions (12)-(13). The matching of the interior and exterior solutions at R s is straightforward,and we match φ (cid:48) , g rr , g (cid:48) rr and g tt at that point.In Fig. 6 we reproduce the intrinsic stellar compactness of configurations with ρ = 0 . ρ crit and ρ = 0 . ρ crit (dashed lines), and we show the asymptotic, extrinsic compactness for some of theseconfigurations (points). Interestingly, even when the effects from the exterior solution are taken intoaccount, the compactness does not exceed the GR limit C = 4 /
9. Also, notice that for low densitythe screening of the exterior solution is highly efficient: even though the scalar field introduces largemodifications to the stellar compactness, the scalar contributions in the exterior are negligible, andextrinsic and intrinsic values of the compactness almost coincide. On the other hand, for highervalues of the stellar energy density, the values of the extrinsic and intrinsic compactness differ forlarge values of the parameter φ . This implies that the effect of the scalar field in this regime isrelevant also outside the object, and not only on its interior. V. DISCUSSION
The recent observation of gravitational waves from a neutron star merger GW170817 and itsassociated electromagnetic counterpart GRB170817A has changed our perspective on scalar-tensortheories. One possibility is to focus only on the simplest theories where the graviton speed c GW isautomatically equal to one; the other is to consider richer systems where this condition is obtainedat the price of tuning some parameters. In this work we considered the second possibility, studyingthe physics of compact objects in a theory of beyond Horndeski with c GW = 1 that includes thescalar kinetic term.We focussed on black hole and relativistic star configurations which are locally asymptoticallyflat, that can be continuously connected to GR configurations, and that have been less exploredin the literature. Depending on a parameter controlling the scalar field, φ , our solutions can bevery similar to GR when φ is small, while they can provide sizeable corrections to it when φ islarger. This shows that a Vainshtein screening mechanism, which is very effective to reproduce GRpredictions in a weak gravity limit, can be less so in strong gravity regimes.For what respect black hole configurations, we shown that our geometries are characterizedasymptotically by an angular deficit, due to presence of the scalar kinetic term, and are equippedwith regular horizons provided that the black hole mass is larger than a value depending on thescalar parameter φ . Our geometries have not scalar hairs, despite the fact that the scalar has aprofile that extends asymptotically far from the black hole. The black hole solutions can be morecompact than the Schwarzschild black hole, thanks to the effect of the scalar field. The angulardeficit could be detected by its effect on geodesics and light propagation [61, 62].We also studied regular relativistic compact objects corresponding to incompressible stars withconstant energy density. The scalar field modifies properties of the star as its compactness, allowingfor stars that are twice as compact as neutron stars with the same matter density. These deviationsfrom GR can be accessed observationally, for example through quantities that depend on the tidaldeformability of a star, which is directly affected by the compactness [68, 69]. We also found thatthere are forbidden regions in parameter space where regular star configurations of given radiusand energy density cannot be found, depending of the scalar field profile. In a more refined versionof our analysis, considering a polytropic equation of state, this fact can suggest observable testsfor the parameter space of these scalar-tensor theories, which would be excluded in case objectsare found in the forbidden regions.By analysing the difference between our interior and exterior solutions and their GR coun-terparts, we numerically investigated the efficiency of the screening of the scalar field inside andoutside the relativistic star. We found that including the standard kinetic term of the scalar field7breaks the perfect screening of vacuum solutions, not only because of the angular deficit but alsobecause the time and radial components of the metric acquire corrections that distinguish themfrom the Schwarzschild solution in the exterior of the object. Nevertheless, there are situationswhere such deviations from a Schwarzschild solution are small in the exterior, while the correc-tions to the interior metric are large with respect to GR. We cannot find the opposite situation –corrections that are large in the exterior but small in the interior. This indicates that the breakingof screening is more severe in the interior solutions.Much work is left for the future. It is interesting to continue to investigate the physics ofcompact objects in other scalar-tensor theories with c GW = 1, for realistic equations of state forthe star interior. Acknowledgments
We are partially supported by the STFC grant ST/P00055X/1.
Appendix A: Equations of motion
The covariant equations of motion derived from action (4) with G = M P l + M − P l β X are0 = M − P l β (cid:20) [Φ] R + ∇ α R ∇ α φ + R αβ (cid:104) Φ (cid:105) − [Φ] (cid:104) Φ (cid:105) + (cid:104) Φ (cid:105) [Φ ] + 2[Φ] (cid:104) Φ (cid:105) − (cid:104) Φ (cid:105) X − [Φ] − R αβ φ α φ β − ] − φ α φ β φ σ ∇ σ R αβ + 2[Φ ] + 2 R ασβδ φ α φ β Φ δσ X (cid:21) + [Φ] , (A1)0 = (cid:18) M P l + βXM P l (cid:19) G µν − g µν X − φ µ φ ν − M − P l β X (cid:0) g µν (cid:104) Φ (cid:105) − φ α (cid:104) Φ (cid:105) Φ ( µ | α φ | ν ) + 2 (cid:104) Φ (cid:105) φ µ φ ν (cid:1) − βM P l (cid:20) g µν ( ∇ α [Φ] φ α + R αβ φ α φ β + [Φ ]) − ∇ α Φ µν φ α − Φ να Φ αµ − R µανβ φ α φ β + Rφ µ φ ν (cid:21) − βM − P l X (cid:104) g µν (3 (cid:104) Φ (cid:105) + φ α φ β ∇ σ Φ αβ φ σ ) − φ α φ β (Φ µα Φ νβ + ∇ β Φ ( µ | α φ | ν ) ) + 2[Φ] φ α Φ ( µ | α φ | ν ) − βα φ α Φ β ( µ φ ν ) + φ µ φ ν (2 R αβ φ α φ β + ∇ α [Φ] φ α + 2[Φ ] − [Φ] ) (cid:105) (A2) ≡ ξ µν . (A3)Under the spherically symmetric ansatz (10), the ( r, r ), ( t, t ) and ( t, r ) components of the equationsof motion become0 = (cid:0) f hφ (cid:48) − φ (cid:1) (cid:0) − f s + r φ + f (cid:0) h + 4 rh (cid:48) + hr φ (cid:48) (cid:1)(cid:1) + 2 β (cid:8) φ (cid:0) h − s + rh (cid:48) (cid:1) +2 f hφ φ (cid:48) (cid:2)(cid:0) h + 3 rh (cid:48) (cid:1) φ (cid:48) + 4 hrφ (cid:48)(cid:48) (cid:3) − f h φ (cid:48) (cid:2)(cid:0) h − s + 3 rh (cid:48) (cid:1) φ (cid:48) + 4 hrφ (cid:48)(cid:48) (cid:3)(cid:9) , (A4)0 = f (cid:0) φ − f hφ (cid:48) (cid:1) (cid:0) f s + r φ + h (cid:0) − f − rf (cid:48) + f r φ (cid:48) (cid:1)(cid:1) + 2 β (cid:8) φ (cid:0) f s − f h + hrf (cid:48) (cid:1) − f h (cid:2) f s + 3 h (cid:0) f + rf (cid:48) (cid:1)(cid:3) φ (cid:48) + 2 f h rφ φ (cid:48) (cid:0) f (cid:48) φ (cid:48) + 2 f φ (cid:48)(cid:48) (cid:1)(cid:9) , (A5)0 = φ φ (cid:48) (cid:8) f r (cid:0) f hφ (cid:48) − φ (cid:1) + 2 β (cid:2) φ (cid:0) f h − f s − hrf (cid:48) + f rh (cid:48) (cid:1) + f h (cid:0) f s + hf + rhf (cid:48) (cid:1) φ (cid:48) (cid:3)(cid:9) . 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