Novel Black-Hole Solutions in
[2mm] Einstein-Scalar-Gauss-Bonnet Theories
[4mm] with a Cosmological Constant
A. Bakopoulos 111Email: firstname.lastname@example.org, G. Antoniou 222Email: email@example.com and P. Kanti 333Email: firstname.lastname@example.org
Division of Theoretical Physics, Department of Physics,
University of Ioannina, Ioannina GR-45110, Greece
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
We consider the Einstein-scalar-Gauss-Bonnet theory in the presence of a cosmological constant , either positive or negative, and look for novel, regular black-hole solutions with a non-trivial scalar hair. We first perform an analytic study in the near-horizon asymptotic regime, and demonstrate that a regular black-hole horizon with a non-trivial hair may be always formed, for either sign of and for arbitrary choices of the coupling function between the scalar field and the Gauss-Bonnet term. At the far-away regime, the sign of determines the form of the asymptotic gravitational background leading either to a Schwarzschild-Anti-de Sitter-type background () or a regular cosmological horizon (), with a non-trivial scalar field in both cases. We demonstrate that families of novel black-hole solutions with scalar hair emerge for , for every choice of the coupling function between the scalar field and the Gauss-Bonnet term, whereas for , no such solutions may be found. In the former case, we perform a comprehensive study of the physical properties of the solutions found such as the temperature, entropy, horizon area and asymptotic behaviour of the scalar field.
As the ultimate theory of Quantum Gravity, that would robustly describe gravitational interactions at high energies and facilitate their unification with the other forces, is still eluding us, the interest in generalised gravitational theories remains unabated in the scientific literature. These theories include extra fields or higher-curvature terms in their action [1, 2], and they provide the framework in the context of which several solutions of the traditional General Relativity (GR) have been re-examined and, quite often, significantly enriched.
In this spirit, generalised gravitational theories containing scalar fields were among the first to be studied. However, the quest for novel black-hole solutions – beyond the three well-known families of GR – was abruptly stopped when the no-hair theorem was formulated , that forbade the existence of a static solution of this form with a non-trivial scalar field associated with it. Nevertheless, counter-examples appeared in the years that followed and included black holes with Yang-Mills , Skyrme fields  or with a conformal coupling to gravity . A novel formulation of the no-hair theorem was proposed in 1995  but this was, too, evaded within a year with the discovery of the dilatonic black holes found in the context of the Einstein-Dilaton-Gauss-Bonnet theory  (for some earlier studies that paved the way, see [9, 10, 11, 12, 13]). The coloured black holes were found next in the context of the same theory completed by the presence of a Yang-Mills field [14, 15], and higher-dimensional  or rotating versions [17, 18, 19, 20] were also constructed (for a number of interesting reviews on the topic, see [21, 22, 23, 24]).
This second wave of black-hole solutions were derived in the context of theories inspired by superstring theory . During the last decade, though, the construction of generalised gravitational theories was significantly enlarged via the revival of the Horndeski  and Galileon  theories. Accordingly, novel formulations of the no-hair theorems were proposed that covered the case of standard scalar-tensor theories  and Galileon fields . However, these recent forms were also evaded  and concrete black-hole solutions were constructed [31, 32, 33]. More recently, three independent groups [34, 35, 36] almost simultaneously demonstrated that a generalised gravitational theory that contains a scalar field and the quadratic Gauss-Bonnet (GB) term admits novel black-hole solutions with a non-trivial scalar hair. In a general theoretical argument, that we presented in , it was shown that the presence of the GB term was of paramount importance for the evasion of the novel no-hair theorem . In addition, the exact form of the coupling function between the scalar field and the GB term played no significant role for the emergence of the solutions: as long as the first derivative of the scalar field at the horizon obeyed a specific constraint, an asymptotic solution describing a regular black-hole horizon with a non-trivial scalar field could always be constructed. Employing, then, several different forms of the coupling function , a large number of asymptotically-flat black-hole solutions with scalar hair were determined . Additional studies presenting novel black holes or compact objects in generalised gravitational theories have appeared [37, 38, 39, 40, 41, 42, 43, 44, 45] as well as further studies of the properties of these novel solutions [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65].
In the present work, we will extend our previous analyses , that aimed at deriving asymptotically-flat black-hole solutions, by introducing in our theory a cosmological constant , either positive or negative. In the context of this theory, we will investigate whether the previous, successful synergy between the Ricci scalar, the scalar field and the Gauss-Bonnet term survives in the presence of . The question of the existence of black-hole solutions in the context of a scalar-tensor theory, with scalar fields minimally-coupled or conformally-coupled to gravity, and a cosmological constant has been debated in the literature for decades [66, 67, 68, 69, 70]. In the case of a positive cosmological constant, the existing studies predominantly excluded the presence of a regular, black-hole solution with an asymptotic de Sitter behaviour - a counterexample of a black hole in the context of a theory with a conformally-coupled scalar field  was shown later to be unstable . On the other hand, in the case of a negative cosmological constant, a substantial number of solutions with an asymptotically AdS behaviour have been found in the literature (for a non-exhaustive list, see [73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84].
Here, we perform a comprehensive study of the existence of black-hole solutions with a non-trivial scalar hair and an asymptotically (Anti)-de Sitter behaviour in the context of a general class of theories containing the higher-derivative, quadratic GB term. To our knowledge, the only similar study is the one performed in the special case of the shift-symmetric Galileon theory , i.e. with a linear coupling function between the scalar field and the GB term. In this work, we consider the most general class of this theory by considering an arbitrary form of the coupling function , and look for regular black-hole solutions with non-trivial scalar hair. Since the uniform distribution of energy associated with the cosmological constant permeates the whole spacetime, we expect to have an effect on both the near-horizon and far-field asymptotic solutions. We will thus repeat our analytical calculations both in the small and large- regimes to examine how the presence of affects the asymptotic solutions both near and far away from the black-hole horizon. As we will see, our set of field equations admits regular solutions near the black-hole horizon with a non-trivial scalar hair for both signs of the cosmological constant. At the far-away regime, the analysis needs to be specialised since a positive or negative sign of leads to either a cosmological horizon or an asymptotic Schwarzschild-Anti-de Sitter-type gravitational background, respectively. Our results show that the emergence of a black-hole solution with a non-trivial scalar hair strongly depends on the type of asymptotic background that is formed at large distances, and thus on the sign of : whereas, for , solutions emerge with the same easiness as their asymptotically-flat analogues, for , no such solutions were found.
In the former case, i.e. for , we present a large number of novel black-hole solutions with a regular black-hole horizon, a non-trivial scalar field and a Schwarzschild-Anti-de Sitter-type asymptotic behaviour at large distances, These solutions correspond to a variety of forms of the coupling function : exponential, polynomial (even or odd), inverse polynomial (even or odd) and logarithmic. Then, we proceed to study their physical properties such as the temperature, entropy, and horizon area. We also investigate features of the asymptotic profile of the scalar field, namely its effective potential and rate of change at large distances since this greatly differs from the asymptotically-flat case.
The outline of the present work is as follows: in Section 2, we present our theoretical framework and perform our analytic study of the near and far-way radial regimes as well as of their thermodynamical properties. In Section 3, we present our numerical results for the two cases of and . We finish with our conclusions in Section 4.
2 The Theoretical Framework
We consider a general class of higher-curvature gravitational theories described by the following action functional:
In this, the quadratic Gauss-Bonnet (GB) term , defined as
supplements the Einstein-Hilbert term, given by the Ricci scalar curvature , and the kinetic term for a scalar field . A coupling term of the scalar field to the GB term, through a general coupling function , is necessary in order for the GB term – a total derivative in four dimensions – to contribute to the field equations. A cosmological constant , that may take either a positive or a negative value, is also present in the theory.
By varying the action (1) with respect to the metric tensor and the scalar field , we derive the gravitational field equations and the equation for the scalar field, respectively. These are found to have the form:
where is the Einstein tensor and is the energy-momentum tensor, with the latter having the form
In the above, the dot over the coupling function denotes its derivative with respect to the scalar field (i.e. ). We have also employed units in which , and used the definition
Compared to the theory studied in , where was zero, the changes in Eqs. (3)-(4) look minimal: the scalar-field equation remains unaffected while the energy-momentum tensor receives a constant contribution . However, as we will see, the presence of the cosmological constant affects both of the asymptotic solutions, the properties of the derived black holes and even their existence.
In the context of this work, we will investigate the emergence of regular, static, spherically-symmetric but non-asymptotically flat black-hole solutions with a non-trivial scalar field. The line-element of space-time will accordingly take the form
The scalar field will also be assumed to be static and spherically-symmetric, . The coupling function will retain a general form during the first part of our analysis, and will be chosen to have a particular form only at the stage of the numerical derivation of specific solutions.
The non-vanishing components of the Einstein tensor may be easily found by employing the line-element (7), and they read
Throughout our analysis, the prime denotes differentiation with respect to the radial coordinate . Using Eq. (5), the components of the energy-momentum tensor take in turn the form
Matching the corresponding components of and , the explicit form of Einstein’s field equations may be easily derived. These are supplemented by the scalar-field equation (4) whose explicit form reads
Although the system of equations involve three unknown functions, namely , and , only two of them are independent. The metric function may be easily shown to be a dependent variable: the -component of field equations takes in fact the form of a second-order polynomial with respect to , i.e. , which easily leads to the following solution
Employing the above expression for , the quantity may be also found to have the form
Therefore, by using Eqs. (15) and (17), the metric function may be completely eliminated from the field equations. The remaining three equations then form a system of only two independent, ordinary differential equations of second order for the functions and :
The expressions for the quantities , and , in terms of , are given for the interested reader in Appendix A as they are quite complicated.
2.1 Asymptotic Solution at Black-Hole Horizon
As we are interested in deriving novel black-hole solutions, we will first investigate whether an asymptotic solution describing a regular black-hole horizon is admitted by the field equations. As a matter of fact, instead of assuming the usual power-series expression in terms of , where is the horizon radius, we will construct the solution as was done in [8, 34]. To this end, we demand that, near the horizon, the metric function should vanish (and should diverge) whereas the scalar field must remain finite. The first demand is reflected in the assumption that should diverge as – this will be justified a posteriori – while and must be finite in the same limit.
Assuming the aforementioned behaviour near the black-hole horizon, Eq. (15) may be expanded in terms of as follows444Note, that only the (+)-sign in the expression for in Eq. (15) leads to the desired black-hole behaviour.
From Eq. (20), we conclude that the combination near the horizon must be non-zero and positive for the metric function to have the correct behaviour, that is to diverge as while being positive-definite. Then, Eq. (22) dictates that, if we want to be finite, we must necessarily have
The above constraint may be written as a second-order polynomial with respect to , which can then be solved to yield
where all quantities have been evaluated at The quantity under the square root stands for the following combination
and must always be non-negative for to be real. This combination may be written as a second-order polynomial for with roots
Then, the constraint on becomes
Therefore, the allowed regime for the existence of regular, black-hole solutions with scalar hair is given by or , since . To obtain some physical insight on these inequalities, we take the limit of small cosmological constant; then, the allowed ranges are
respectively. In the absence of , Eq. (28) results into the simple constraint , and defines a sole branch of solutions with a minimum allowed value for the horizon radius (and mass) of the black hole . In the presence of a cosmological constant, this constraint is now replaced by , or by the first inequality presented in Eq. (31) in the small- limit. This inequality leads again to a branch of solutions that – for chosen , and – terminates at a black-hole solution with a minimum horizon radius . We observe that, at least for small values of , the presence of a positive cosmological constant relaxes the constraint allowing for smaller black-hole solutions, while a negative cosmological constant pushes the minimum horizon radius towards larger values. The second inequality in Eq. (31) describes a new branch of black-hole solutions that does not exist when ; this was also noted in  in the case of the linear coupling function. This branch of solutions describes a class of very small GB black holes, and terminates instead at a black hole with a maximum horizon radius .
Integrating the above, we find that , a result that justifies the diverging behaviour of this quantity near the horizon that we assumed earlier. A second integration yields , which then uniquely determines the expression of the metric function in the near-horizon regime. Employing Eq. (20), the metric function is also determined in the same regime. Therefore, the asymptotic solution of Eqs. (15), (18) and (19), that describes a regular, black-hole horizon in the limit , is given by the following expressions
where and are integration constants. We observe that the above asymptotic solution constructed for the case of a non-zero cosmological constant has exactly the same functional form as the one constructed in  for the case of vanishing . The presence of the cosmological constant modifies though the exact expressions of the basic constraint (27) for and of the quantity given in (28), the validity of which ensures the existence of a regular black-hole horizon. As in , the exact form of the coupling function does not affect the existence of the asymptotic solution, therefore regular black-hole solutions may emerge for a wide class of theories of the form (1).
The regularity of the asymptotic black-hole solution is also reflected in the non-diverging behaviour of the components of the energy-momentum tensor and of the scale-invariant Gauss-Bonnet term. The components of the former quantity in this regime assume the form
Employing the asymptotic expansions (33)-(35), one may see that all components remain indeed finite in the vicinity of the black-hole horizon. For future use, we note that the cosmological constant adds a positive contribution to all components of the energy-momentum tensor for , while it subtracts a positive contribution for . Also, all scalar curvature quantities, the explicit form of which may be found in Appendix B, independently exhibit a regular behaviour near the black-hole horizon – when these are combined, the GB term, in the same regime, takes the form
exhibiting, too, a regular behaviour as expected.
2.2 Asymptotic Solutions at Large Distances
The form of the asymptotic solution of the field equations at large distances from the black-hole horizon depends strongly on the sign of the cosmological constant. Therefore, in what follows, we study separately the cases of positive and negative .
2.2.1 Positive Cosmological Constant
In the presence of a positive cosmological constant, a second horizon, the cosmological one, is expected to emerge at a radial distance . We demand that this horizon is also regular, that is that the scalar field and its derivatives remain finite in its vicinity. We may in fact follow a method identical to the one followed in section 2.1 near the black-hole horizon: we again demand that, at the cosmological horizon, while ; then, using that diverges there, the regularity of from Eq. (19) eventually leads to the constraint
with now being given by the non-negative expression
Employing Eq. (40) in Eq. (18), the solution for the metric function may be again constructed. Overall, the asymptotic solution of the field equations near a regular, cosmological horizon will have the form
where care has been taken for the fact that . One may see again that the above asymptotic expressions lead to finite values for the components of the energy-momentum tensor and scalar invariant quantities. Once again, the explicit form of the coupling function is of minor importance for the existence of a regular, cosmological horizon.
2.2.2 Negative Cosmological Constant
For a negative cosmological constant, and at large distances from the black-hole horizon, we expect the spacetime to assume a form close to that of the Schwarzschild-Anti-de Sitter solution. Thus, we assume the following approximate forms for the metric functions
where , , and are, at the moment, arbitrary constants. Substituting the above expressions into the scalar field equation (14), we obtain at first order the constraint
The gravitational equations, under the same assumptions, lead to two additional constraints, namely
Contrary to what happens close to the horizons (either black-hole or cosmological ones), the form of the coupling function now affects the asymptotic form of the scalar field at large distances. The easiest case is that of a linear coupling function, - that case was first studied in , however, we review it again in the context of our analysis as it will prove to play a more general role. The scalar field, at large distances, may be shown to have the approximate form
The third first-order constraint, Eq. (49), is then trivially satisfied. In order to determine the values of the remaining coefficients, one needs to derive higher-order constraints. For example, the coefficients , and are found at third-order approximation to have the forms
while for or one needs to go even higher. In contrast, the coefficient remains arbitrary and may be interpreted as the gravitational mass of the solution.
In the perturbative limit (i.e. for small values of the coupling constant of the GB term), one may show that the above asymptotic solution is valid for all forms of the coupling function . Indeed, if we write
and define , then, at first order, . Therefore, independently of the form of , at first order in the perturbative limit, is a constant, as in the case of a linear coupling function. Then, a solution of the form of Eqs. (45)-(46) and (50) is easily derived 555In the perturbative limit, at first order, one finds , , , , and . For more details on the perturbative analysis of the black-hole solutions that arise in the context of the general class of theories (1) and are either asymptotically-flat or (Anti)-de Sitter, see . with in Eqs. (51) and (52) being now replaced by .
For arbitrary values of the coupling constant , though, or for a non-linear coupling function , the approximate solution described by Eqs. (45), (46) and (50) will not, in principle, be valid any more. Unfortunately, no analytic form of the solution at large distances may be derived in these cases. However, as we will see in section 3, numerical solutions do emerge with a non-trivial scalar field and an asymptotic Anti-de Sitter-type behaviour at large distances. These solutions are also characterised by a finite GB term and finite, constant components of the energy-momentum tensor at the far asymptotic regime.
2.3 Thermodynamical Analysis
In this subsection, we calculate the thermodynamical properties of the sought-for black-hole solutions, namely their temperature and entropy. The first quantity may be easily derived by using the following definition [87, 88]
that relates the black-hole temperature to its surface gravity . The above formula is valid for spherically-symmetric black holes in theories that may contain also higher-derivative terms such as the GB term. The final expression of the temperature in Eq. (54) is derived by employing the near-horizon asymptotic forms (33)-(34) of the metric functions.
The entropy of the black hole may be calculated by using the Euclidean approach in which the entropy is given by the relation 
where is the Helmholtz free-energy of the system given in terms of the Euclidean version of the action , and . The above formula has been used in the literature to determine the entropy of the asymptotically-flat coloured GB black holes  and of the family of novel black-hole solutions found in  for different forms of the GB coupling function. However, in the case of a non-asymptotically-flat behaviour, the above method needs to be modified: in the case of a de-Sitter-type asymptotic solution, the Euclidean action needs to be integrated only over the causal spacetime whereas, for an Anti-de Sitter-type asymptotic solution, the Euclidean action needs to be regularised [90, 92], by subtracting the diverging, ‘pure’ AdS-spacetime contribution.
Alternatively, one may employ the Noether current approach developed in  to calculate the entropy of a black hole. In this, the Noether current of the theory under diffeomorphisms is determined, with the Noether charge on the horizon being identified with the entropy of the black hole. In , the following formula was finally derived for the entropy
where is the Lagrangian of the theory, the binormal to the horizon surface , and the 2-dimensional projected metric on . The equivalence of the two approaches has been demonstrated in , in particular in the context of theories that contain higher-derivative terms such as the GB term. Here, we will use the Noether current approach to calculate the entropy of the black holes as it leads faster to the desired result.
To this end, we need to calculate the derivatives of the scalar gravitational quantities, appearing in the Lagrangian of our theory (1), with respect to the Riemann tensor. In Appendix C, we present a simple way to derive those derivatives. Then, substituting in Eq. (56), we obtain
The first term inside the curly brackets of the above expression comes from the variation of the Einstein-Hilbert term and leads to:
We recall that is antisymmetric, and, in addition, satisfies . Therefore, we easily obtain the result
where is the horizon surface. The remaining terms in Eq. (57) are all proportional to the coupling function and follow from the variation of the GB term. To facilitate the calculation, we notice that, on the horizon surface, the binormal vector is written as: . This means that we may alternatively write:
Therefore, the terms proportional to may be written as
To evaluate the above integral, we will employ the near-horizon asymptotic solution (33)-(35) for the metric functions and scalar field. The asymptotic values of all quantities appearing inside the square brackets above are given in Appendix C. Substituting in Eq. (61), we straightforwardly find
The above describes the entropy of a GB black hole arising in the context of the theory (1), with a general coupling function between the scalar field and the GB term, and a cosmological constant term. We observe that the above expression matches the one derived in  in the context of the theory (1) but in the absence of the cosmological constant. This was, in fact, expected on the basis of the more transparent Noether approach used here: the term does not change the overall topology of the black-hole horizon and it does not depend on the Riemann tensor; therefore, no modifications are introduced to the functional form of the entropy of the black hole due to the cosmological constant. However, the presence of modifies in a quantitative way the properties of the black hole and therefore the value of the entropy, and temperature, of the found solutions.
3 Numerical Solutions
In order to construct the complete black-hole solutions in the context of the theory (1), i.e. in the presence of both the GB and the cosmological constant terms, we need to numerically integrate the system of Eqs. (18)-(19). The integration starts at a distance very close to the horizon of the black hole, i.e. at (for simplicity, we set ). The metric function and scalar field in that regime are described by the asymptotic solutions (33) and (35). The input parameter is uniquely determined through Eq. (27) once the coupling function is selected and the values of the remaining parameters of the model near the horizon are chosen. These parameters appear to be , and . However, the first two are not independent: since it is their combination that determines the strength of the coupling between the GB term and the scalar field, a change in the value of one of them may be absorbed in a corresponding change to the value of the other; as a result, we may fix and vary only . The values of and also cannot be totally uncorrelated as they both appear in the expression of , Eq. (28), that must always be positive; therefore, once the value of the first is chosen, there is an allowed range of values for the second one for which black-hole solutions arise. This range of values are determined by the inequalities and according to Eq. (30), and lead in principle to two distinct branches of solutions. In fact, removing the square, four branches emerge depending on the sign of . However, in what follows we will assume that , and thus study the two regimes and ; similar results emerge if one assumes instead that .
Before starting our quest for black holes with an (Anti)-de Sitter asymptotic behaviour at large distances, we first consider the case with where upon we successfully reproduce the families of asymptotically-flat back holes derived in . Then, we select non-vanishing values of and look for novel black-hole solutions. We will start with the case of a negative cosmological constant () in the next subsection and consider the case of a positive cosmological constant () in the following one.
3.1 Anti-de Sitter Gauss-Bonnet Black Holes
As mentioned above, the integration starts from the near-horizon regime with the asymptotic solutions (33) and (35), and it proceeds towards large values of the radial coordinate until the form of the derived solution for the metric resembles, for , the asymptotic solution (45)-(46) describing an Anti-de Sitter-type background. The arbitrary coefficient , that does not appear in the field equations, may be fixed by demanding that, at very large distances, the metric functions satisfy the constraint . We have considered a large number of forms for the coupling function , and, as we will now demonstrate, we have managed to produce a family of regular black-hole solutions with an Anti-de Sitter asymptotic behaviour, for every choice of .
We will first discuss the case of an exponential coupling function, . The solutions for the metric functions and are depicted in the left plot of Fig. 1. We may easily see that the near-horizon behaviour, with vanishing and diverging, is eventually replaced by an Anti-de Sitter regime with the exactly opposite behaviour of the metric functions at large distances. The solution presented corresponds to the particular values (in units of ), and , however, we obtain the same qualitative behaviour for every other set of parameters satisfying the constraint 666Here, we do not present black-hole solutions that satisfy the alternative choice since this leads to solutions plagued by numerical instabilities, that prevent us from deducing their physical properties in a robust way. The same ill-defined behaviour of this second branch of solutions with very small horizon radii was also found in . , that follows from Eq. (28). The spacetime is regular in the whole radial regime, and this is reflected in the form of the scalar-invariant Gauss-Bonnet term: this is presented in the right plot of Fig. 1, for , and for a variety of values of the cosmological constant. We observe that the GB term acquires its maximum value near the horizon regime, where the curvature of spacetime is larger, and reduces to a smaller, constant asymptotic value in the far-field regime. This asymptotic value is, as expected, proportional to the cosmological constant as this quantity determines the curvature of spacetime at large distances.
Although in Section 2.2.2, we could not find the analytic form of the scalar field at large distances from the black-hole horizon for different forms of the coupling function , our numerical results ensure that its behaviour is such that the effect of the scalar field at the far-field regime is negligible, and it is only the cosmological term that determines the components of the energy-momentum tensor. In the left plot of Fig. 2, we display all three components of over the whole radial regime, for the indicative solution , and . Far away from the black-hole horizon, all components reduce to , in accordance with Eqs. (11)-(13), with the effect of both the scalar field and the GB term being there negligible. Near the horizon, and according to the asymptotic behaviour given by Eqs. (36)-(38), we always have , since, at , ; also, the component always has the opposite sign to that of since . This qualitative behaviour of remains the same for all forms of the coupling function we have studied and for all solutions found, therefore we refrain from giving additional plots of this quantity for the other classes of solutions found.
From the results depicted in the left plot of Fig. 2, we see that, near the black-hole horizon, we always have . Comparing this behaviour with the asymptotic forms (36)-(38), we deduce that, close to the black-hole horizon where , we must have . In the case of vanishing cosmological constant, the negative value of this quantity was of paramount importance for the evasion of the no-hair theorem  and the emergence of novel, asymptotically-flat black-hole solutions . We observe that also in the context of the present analysis with , this quantity turns out to be again negative, and to lead once again to novel black-hole solutions. Coming back to our assumption of a decreasing exponential coupling function and upon choosing to consider , the constraint means that independently of the value of . In the right plot of Fig. 2, we display the solution for the scalar field in terms of the radial coordinate, for the indicative values of , and for different values of the cosmological constant. The scalar field satisfies indeed the constraint and increases away from the black-hole horizon777A complementary family of solutions arises if we choose , with the scalar profile now satisfying the constraint and decreasing away from the black-hole horizon.. At large distances, we observe that, for small values of the cosmological constant, assumes a constant value; this is the behaviour found for asymptotically-flat solutions  that the solutions with small are bound to match. For increasingly larger values of though, the profile of the scalar field deviates significantly from the series expansion in powers of thus allowing for a -dependent even at infinity – in the perturbative limit, as we showed in the previous section, this dependence is given by the form .
We will now consider the case of an even polynomial coupling function of the form with . The behaviour of the solution for the metric functions matches the one depicted 888Let us mention at this point that, for extremely large values of either the coupling constant or the cosmological constant , that are nevertheless allowed by the constraint (28), solutions that have their metric behaviour deviating from the AdS-type form (45)-(46) were found; according to the obtained behaviour, both metric functions seem to depend logarithmically on the radial coordinate instead of polynomially. As the physical interpretation of these solutions is not yet clear, we omit these solutions from the remaining of our analysis. in the left plot of Fig. 1. The same is true for the behaviour of the GB term and the energy-momentum tensor, whose profiles are similar to the ones displayed in Figs. 1 (right plot) and 2 (left plot), respectively. The positive-definite value of near the black-hole horizon implies again that, there, we should have , or equivalently , for . Indeed, two classes of solutions arise in this case: for positive values of , we obtain solutions for the scalar field that decrease away from the black-hole horizon, while for , solutions that increase with the radial coordinate are found. In Fig. 3 (left plot), we present a family of solutions for the case of the quadratic coupling function (i.e. ), for and , arising for different values of – since , the scalar field exhibits an increasing behavior as expected.
Let us examine next the case of an odd polynomial coupling function, with . The behaviour of the metric functions, GB term and energy-momentum tensor have the expected behaviour for an asymptotically AdS background, as in the previous cases. The solutions for the scalar field near the black-hole horizon are found to satisfy the constraint or simply , when . As this holds independently of the value of , all solutions for the scalar field are expected to decrease away from the black-hole horizon. Indeed, this is the profile depicted in the right plot of Fig. 3 where a family of solutions for the indicative case of a qubic coupling function (i.e. ) is presented for , and various values of .
The case of an inverse polynomial coupling function, , with either an even or odd positive integer, was also considered. For odd , i.e. , the positivity of near the black-hole horizon demands again that , or that . For , the solution for the scalar field should thus always satisfy , regardless of our choices for or . As an indicative example, in the left plot of Fig. 4, we present the case of with a family of solutions arising for and . The solutions for the scalar field clearly satisfy the expected behaviour by decreasing away from the black-hole horizon. On the other hand, for even , i.e. , the aforementioned constraint now demands that . As in the case of the odd polynomial coupling function, two subclasses of solutions arise: for , solutions emerge with whereas, for , we find solutions with . The profiles of the solutions in this case are similar to the ones found before, with approaching, at large distances, an almost constant value for small but adopting a more dynamical behaviour as the cosmological constant gradually takes on larger values.
As a final example of another form of the coupling function between the scalar field and the GB term, let us consider the case of a logarithmic coupling function, . Here, the condition near the horizon of the black hole gives , therefore, for , we must have ; for , this translates to a decreasing profile for the scalar field near the black-hole horizon. In the right plot of Fig. 4, we present a family of solutions arising for a logarithmic coupling function for fixed and , while varying the cosmological constant . The profiles of the scalar field agree once again with the one dictated by the near-horizon constraint, and they all decrease in that regime. As in the previous cases, the metric functions approach asymptotically an Anti-de Sitter background, the scalar-invariant GB term remains everywhere regular, and the same is true for all components of the energy-momentum tensor that asymptotically approach the value .
It is of particular interest to study also the behaviour of the effective potential of the scalar field, a role that in our theory is played by the GB term together with the coupling function, i.e. . In the left plot of Fig. 5, we present a combined graph that displays its profile in terms of the radial coordinate, for a variety of forms of the coupling function . As expected, the potential takes on its maximum value always near the horizon of the black hole, where the GB term is also maximized and thus sources the non-trivial form of the scalar field. On the other hand, as we move towards larger distances, reduces to an asymptotic constant value. Although this asymptotic value clearly depends on the choice of the coupling function, its common behaviour allows us to comment on the asymptotic behaviour of the scalar field at large distances. Substituting a constant value in the place of in the scalar-field equation (14), we arrive at the intermediate result
Then, employing the asymptotic forms of the metric functions at large distances (45)-(46), the above may be easily integrated with respect to the radial coordinate to yield a form for the scalar field identical to the one given in Eq. (50). We may thus conclude that the logarithmic form of the scalar field may adequately describe its far-field behaviour even beyond the perturbative limit of very small .
We now proceed to discuss the physical characteristics of the derived solutions. Due to the large number of solutions found, we will present, as for , combined graphs for different forms of the coupling function . Starting with the scalar field, we notice that no conserved quantity, such as a scalar charge, may be associated with the solution at large distances in the case of asymptotically Anti-de Sitter black holes: the absence of an term in the far-field expression (50) of the scalar field, that would signify the existence of a long-range interaction term, excludes the emergence of such a quantity, even of secondary nature. One could attempt instead to plot the dependence of the coefficient , as a quantity that predominantly determines the rate of change of the scalar field at the far field, in terms of the mass of the black hole. This is displayed in the right plot of Fig. 5 for the indicative value of the cosmological constant. We see that, for small values of the mass , this coefficient takes in general a non-zero value, which amounts to having a non-constant value of the scalar field at the far-field regime. As the mass of the black hole increases though, this coefficient asymptotically approaches a zero value. Therefore, the rate of change of the scalar field at infinity for massive GB black holes becomes negligible and the scalar field tends to a constant. This is the ‘Schwarzschild-AdS regime’, where the GB term decouples from the theory and the scalar-hair disappears - the same behaviour was observed also in the case of asymptotically-flat GB black holes  where, in the limit of large mass, all of our solutions merged with the Schwarzschild ones.
We present next the ratio of the horizon area of our solutions compared to the horizon area of the SAdS one with the same mass, for the indicative values of the negative cosmological constant and in the two plots of Fig. 6. These plots provide further evidence for the merging of our GB black-hole solutions with the SAdS solution in the limit of large mass. The left plot of Fig. 6 reveals that, for small cosmological constant, all our GB solutions remain smaller than the scalar-hair-free SAdS solution independently of the choice for the coupling function