On the Existence of Solutions with a Horizon in Pure Scalar-Gauss-Bonnet Theories

We consider the Einstein-scalar-Gauss-Bonnet theory and assume that, at regimes of large curvature, the Ricci scalar may be ignored compared to the quadratic Gauss-Bonnet term. We then look for static, spherically-symmetric, regular black-hole solutions with a non-trivial scalar field. Despite the use of a general form of the spacetime line-element, no black-hole solutions are found. In contrast, solutions that resemble irregular particle-like solutions or completely regular gravitational solutions with a finite energy-momentum tensor do emerge. In addition, in the presence of a cosmological constant, solutions with a horizon also emerge, however, the latter corresponds to a cosmological rather than to a black-hole horizon. It is found that, whereas the Ricci term works towards the formation of the positively-curved topology of a black-hole horizon, the Gauss-Bonnet term exerts a repulsive force that hinders the formation of the black hole. Therefore, a pure scalar-Gauss-Bonnet theory cannot sustain any black-hole solutions. However, it could give rise to interesting cosmological or particlelike solutions where the Ricci scalar plays a less fundamental role. Email: abakop@cc.uoi.gr Email: pkanti@uoi.gr Email: npappas@uoi.gr ar X iv :1 91 0. 14 63 7v 1 [ he pth ] 3 1 O ct 2 01 9

1 Introduction a scalar hair.
The Einstein-scalar-Gauss-Bonnet theory has also interesting cosmological implications, and these were in fact the first ones to be studied in the literature. In the context of the effective heterotic superstring theory, where the scalar field was identified with one of the moduli fields of the theory, it was shown that this theory leads to singularity-free cosmological solutions [142]. Later, it was demonstrated that similar type of solutions emerge for a variety of coupling functions, with a number of common features, between the scalar field and the Gauss-Bonnet term [143,144]. More recently, the same theory was studied from a novel perspective [145]: as we go backwards in time, the curvature of spacetime considerably increases and the quadratic Gauss-Bonnet term becomes eventually as important as, or even larger than, the linear Ricci term. Assuming that such a time period exists, the Ricci term was altogether ignored from the theory and the coupled system of the scalar field and the Gauss-Bonnet was studied on its own. It was found [145] that this simplified theory supported singularity-free solutions with the same characteristics as the ones emerging in the context of the complete theory. In addition, a family of attractive inflationary solutions with a natural exit mechanism was found, all in an analytical way due to the simplification of the set of field equations.
In the context of the present work, we will keep the same perspective but focus on the emergence of solutions with a horizon. Our main priority will be to investigate whether the simplified, pure scalar-Gauss-Bonnet theory leads to regular black-hole solutions with a non-trivial scalar field. We will therefore assume that there exists a part of spacetime where the Gauss-Bonnet term may dominate over the linear Ricci term. We will seek for static, spherically-symmetric, regular black holes, and attempt to solve the simplified set of field equations by using both analytical and numerical methods. We will also use alternative forms of line-elements in an effort to increase the flexibility of our ansatz. The presence of a cosmological constant will also be employed, and its role to the formation of a horizon will be investigated. As we will demonstrate, the pure scalar-Gauss-Bonnet theory cannot support by itself black-hole spacetimes -this will be due to the conflicting roles of the Gauss-Bonnet and Ricci terms in the formation of a regular black-hole horizon. Nevertheless, our quest for black-hole solutions will lead us instead to a number of alternative solutions -all parts of the phase-space of solutions of the pure scalar-Gauss-Bonnet theory -with a number of interesting characteristics.
The outline of the present work is as follows: in Section 2, we present our theoretical framework and field equations. In Section 3, we look, in an analytical way, for solutions with a horizon in the case where the cosmological constant vanishes -we also perform an exact numerical study of a family of regular solutions we derive. In Section 4, we employ a generalised form of the spacetime line-element, and repeat our previous analysis. We reinstate the cosmological constant in Section 5, and interpret the solutions we obtain. In Section 6, we use exact numerical results for black-hole solutions found in the context of the complete Einstein-scalar-Gauss-Bonnet theory, and investigate the role of the Gauss-Bonnet and Ricci terms in the formation of a black-hole horizon. We finish with our conclusions in Section 7.

The Theoretical Framework
The starting point of our analysis will be the following action, describing a generalised theory of gravity The theory contains the Ricci scalar curvature R, a scalar field φ and the higher-curvature, quadratic Gauss-Bonnet (GB) term defined as in terms of the Riemann tensor R µνρσ , Ricci tensor R µν and Ricci scalar R. The GB term, being a topological invariant in four dimensions, must be coupled to the scalar field φ. This is realised via the arbitrary coupling function f (φ); choosing different forms for the coupling function, one may study the emergence of solutions within a whole class of theories. The theory includes also a cosmological constant Λ. Throughout our work, we will use units in which G = c = 1. Taking the variation of the action with respect to the metric g µν and the scalar field φ, we end up with the gravitational field equations and the equation of motion for the scalar field. These have the following forms respectively. In the above, G µν is the Einstein tensor and T µν is the total energy-momentum tensor of the theory that receives contributions from the kinetic term of the scalar field, the GB term and the cosmological constant. The dot over the coupling function denotes its derivative with respect to the scalar field (i.e.ḟ = df /dφ). We have also used the definitioñ The emergence of regular black-hole solutions with a non-trivial scalar field, in the context of the theory (1) and for a variety of coupling functions f (φ), was demonstrated in [38,130] with either Minkowski or Anti-de Sitter asymptotic behaviour. Here, we will investigate whether regular, black-hole solutions with scalar hair emerge in the context of the pure scalar-GB theory, i.e. in the absence of the Ricci scalar from the theory. In that case, the derived solutions would rely solely on the synergy between the scalar field and the GB term. From the field equations (3)-(4), we may see that such a synergy is in principle possible: a non-trivial scalar field ensures the presence of the GB term in the theory whereas the GB term provides in its turn a non-trivial potential for the scalar field. It is this same synergy that leads to singularity-free or inflationary cosmological solutions, even in the absence of the Ricci scalar from the theory, as was analytically demonstrated in [145].
The assumption that the linear gravitational Ricci term may be ignored from the theory when compared to the quadratic GB term may be justified only in regimes of spacetime where the curvature is particularly large. This may be realised only near the black-hole horizon whereas in the asymptotic regime the Ricci scalar must be necessarily re-instated. Therefore, the question we would like to pose, and investigate in what follows, is the following: does the curvature of spacetime ever become so strong that a black-hole horizon may be formed only due to the effect of the GB term (supplemented by that of the scalar field)?
To this end, we will assume a static, spherically-symmetric ansatz for the spacetime line-element of the form: In accordance to the above discussion, we will focus on the "near" regime of spacetime, i.e. on the small-r regime, and assume that, there, all terms associated with the Ricci term may be ignored from the field equations. That amounts to ignoring altogether the components of the Einstein tensor G µν from the gravitational field equations (3). Then, employing the ansatz (1), the explicit form of the components of Einstein's equations becomes We observe that, upon ignoring the components of the Einstein tensor from the field equations, the total effective energy-momentum tensor vanishes. However, this is due not to the triviality of the matter distribution in our theory but to the cancellation of the positive contribution of the kinetic term of the scalar field and the negative contribution of the GB term to the effective energy-density and pressure components of the system. We therefore look for non-trivial configurations of the scalar field and metric functions that satisfy the above equations and thus correspond to locally zero-energy and zero-pressure solutions. The scalar field equation (4) on the other hand remains unaltered by the elimination of the Ricci scalar in the small-r regime, and assumes the following explicit form: where we have assumed that the scalar field shares the symmetry of spacetime and thus is also static and spherically-symmetric, φ = φ(r). In all the above equations, the prime denotes differentiation with respect to the radial coordinate r.
3 Solutions in the "near" regime for Λ = 0 The issue of the emergence of regular, black-hole solutions from the system of Eqs. (8)- (11), with Λ = 0, was briefly discussed in [38]. Here, we will first review and expand on the mathematical arguments involved in that analysis and, second, study and characterise the families of solutions we obtain. We will again assume that the cosmological constant is zero and postpone the study of its role for a later section. As our priority is to find solutions with a horizon, we will demand that, for some value of the radial coordinate r = r h , the following conditions should hold The above conditions amount to assuming that A → ∞ and B → ∞. In fact, one may consider any of the two conditions as the starting point of the analysis -in the case of the emergence of a spherically-symmetric black hole, these two conditions are equivalent; however, in the absence of such a solution, each condition allows us to explore different parts of the phase space of the solutions of the theory. We investigate these two different lines of thinking in the following two subsections.

Expanding Around A → ∞
Before applying any limit, we may observe that Eq. (9) can be solved to yield an expression for the metric component e B , that is from which we may easily deduce an expression for the first derivative of B, namely Employing Eqs. (13) and (14), we may eliminate the metric function B(r) and its derivative from the remaining field equations (8), (10) and (11). The latter reduce to a set of two independent second-order, coupled, ordinary differential equations that may be brought to the following form where: In the above expressions, we have eliminated B but, for simplicity of notation, we kept e B -the latter should be considered as a dependent function of the remaining independent variables according to Eq. (13).
We will now assume that, as r → r h , A diverges. The regularity of the horizon, if existent, demands that the scalar field φ as well as its first and second derivative remain there finite. Under these assumptions, Eq. (13) readily gives that By replacing the above in Eqs. (15)- (18), we find that, near the point of interest, the following relations hold The above equations may be easily integrated with respect to the radial coordinate to give: The first of the above equations consistently gives that, near r h , A → ∞ as assumed above.
Integrating once more, we find the complete form of the asymptotic solution for the metric functions A, B and the scalar field φ near r h : In the above, b 1 , a 2 and φ i are arbitrary integration constants. As demanded, the scalar field and its derivatives are regular at r h while the metric function e A vanishes thus exhibiting the expected behaviour near a black-hole horizon. However, the behaviour of the second metric function e B , as given by Eq. (23), does not describe a black hole as it remains regular near r h .
In order to gain more information about the form of spacetime around r h , we calculate the scalar gravitational quantities, the exact expressions of which may be found in the Appendix A. We then find the following results: According to the above, all curvature invariant quantities R, R µν R µν and R µνρσ R µνρσ are diverging near r h while the Gauss-Bonnet combination exhibits a softer divergence than expected as the dominant terms of order (r − r h ) −2 exactly cancel. Under the change of coordinate l = r − r h , the expansions (24) resemble the asymptotic form of a particle-like solution near the origin. As discussed above, our solution is characterised by a spacetime singularity that, at first, may be considered as unphysical. However, particle-like solutions plagued by singularities, either in spacetime or in the profile of the scalar field, are quite common in the literature in the context of scalar-tensor theories of gravity (see, for instance [136,138]). These solutions are physical whenever they are characterised by a finite total energy-momentum tensor as is the case also for the solutions derived here.

Expanding Around B → ∞
Alternatively, we may employ the fact that near a black-hole horizon it holds that e B → ∞. As we will shortly confirm, this amounts to assuming that B → ∞. The metric function A will now be considered as a dependent variable, and Eq. (9) may be readily solved to give: Computing also the second derivative of A from the above expression, we may eliminate the metric function A and its derivatives from the remaining field equations. Then, we form a system of two coupled, ordinary differential equations, one first-order and one second-order, for the metric function B and the scalar field φ, respectively. These have the form: where Again, for notational simplicity we have kept A and A in the expressions above, however, these quantities are now dependent functions of the independent variables B and φ.
Let us now focus on the regime near r h where e B and B are assumed to diverge. First, we expand there Eq. (29) to obtain: Substituting the above result into Eqs. (30)-(33), we find that near r h the following relations hold: According to Eq. (35), near r h , B indeed diverges as assumed. If we integrate this equation with respect to the radial coordinate, we find that: Indeed, as demanded, the metric function e B exhibits the desired behaviour near a black-hole horizon. In order for this horizon to be also regular, the scalar field and its derivatives should be finite. Then, Eq. (36) dictates that we must necessarily have φ (r h ) = 0. We may in fact solve analytically Eqs. (34) and (36) to obtain the solutions and where a 0 and φ i are again integration constants, and where we have defined the variable Also, F D (x) and Erfi(x) are the Dawson and error function, respectively, defined as: Although the solution for the metric function e B (37) hints to the presence of a horizon, on which the scalar field remains regular according to Eqs. (38), the behaviour of the metric function e A reveals that this solution is not a black hole: in the limit r → r h , or ξ → 0, A remains finite and A adopts an arbitrary constant value. Using the above asymptotic solutions, we may calculate once again the scalar curvature quantities, the expressions of which are listed below: The above asymptotic values reveal that, near r h , the spacetime remains regular and no singularities emerge. All curvature invariants assume constant values apart from the GB combination that is vanishing at exactly r = r h . We have performed a numerical integration of the system (30) to determine the solutions for the metric function B and scalar field φ in the whole radial regime. To this end, we have used Eqs. (37) and (38) as boundary conditions, and finally integrated Eq. (29) with a randomly chosen boundary condition e A(r h ) = 2. In Fig. 1(a,b), we depict the profiles of the two metric components −g tt and g rr , respectively, for a variety of forms of the coupling function f (φ). We observe that the qualitative behaviour of the two metric components remains largely unaffected by the exact form of f (φ), especially at the small-r regime. As  our analytic calculations revealed, the g rr component diverges as r → r h but the g tt remains finite thus failing to adopt a black-hole profile. The regularity of the spacetime is reflected in the finiteness of the GB term depicted in Fig. 2: this scalar gravitational quantity vanishes very close to and far away from r h while having non-trivial values at finite, intermediate distance.
The solution for the scalar field φ is presented in Fig. 3(a). As clearly shown, it adopts finite constant values near r h and at asymptotic infinity which leads to its first derivative having a vanishing value at both these regimes. This profile results into the form of the contribution of the scalar kinetic-term to the effective energy-momentum tensor that is shown in Fig. 3(b): as expected, this contribution has a non-zero value at intermediate values of the radial coordinate where the scalar field has a non-trivial profile. The same is true for the contribution of the GB term to the total energy-momentum tensor (5): this is again non-trivial at intermediate distances, in accordance to Fig. 2, and has exactly the same form as the T t φ t contribution, depicted in Fig. 3(b), but with the exact opposite sign: it is this behaviour that guarantees the vanishing total energy-momentum tensor as our analysis demanded. Let us stress that both families of solutions derived in this section correspond to non-trivial gravitational solutions with finite, zero total energy-momentum tensor whose physical significance will be studied elsewhere.

Alternative Spacetime Line-elements
The main objective of this work is to find an asymptotic solution that describes a regular black-hole horizon. The line-element (1) employed in the previous section has failed to accomplish this task. While retaining the assumptions of staticity and spherical symmetry, we could consider alternative ansatzes for the line-element of the spacetime that could perhaps allow for more general families of solutions. To this end, we consider the following line-element ds 2 = −e A(r) dt 2 + e B(r) dr 2 + H 2 (r)(dθ 2 + sin 2 θ dϕ 2 ).
Note that, by setting H(r) = r, we recover the line-element employed in the previous section. A line-element of this general form was used in the context of the Einstein-scalar-GB theory in order to find traversable wormhole solutions [135] -there, it was demonstrated that the above line-element led to more extended families of solutions even in cases where the lineelement (1) did not allow for any solutions at all. It seems therefore justified that such a line-element could be used in our quest for black-hole solutions, too.
Employing then the line-element (46), we find the following explicit expressions for the components of the gravitational field equations and the equation for the scalar field We may, as before, work first with the limit e A → 0, or A → ∞, as r → r h . Then, Eq. (48) may be solved to yield an expression for the dependent function e B : From the above, we may again find an expression for B that, together with the one for e B , could be used to eliminate the metric function B from the remaining field equations. These, then, reduce to a set of three, coupled, second-order, ordinary differential equations for the metric functions A and H and the scalar field φ, that may be written as 4 In order to reduce the technicalities of our analysis, we present the expression for B and the explicit forms of the functionsS,P ,Q andK in Appendix B.
Let us instead focus on the physical implications of this system of equations. In the limit A → ∞ while H, φ and φ remain finite, we obtain By replacing the above in Eqs. (52), we get near the point of interest, r r h , the results Again, in order to keep φ finite we demand that φ (r h ) = 0. Integrating twice the last two equations with respect to the radial coordinate, we obtain the asymptotic solutions where a 1 and h 1 are integration constants. We observe that the metric function e A vanishes indeed at r = r h , as expected near a black-hole horizon. However, the circumferential radius H 2 (r) also vanishes at the same point thus signalling the presence of an additional pathology in the coordinate system. If we define a new radial coordinate as l ≡ h 1 (r − r h ) 4/3 , then the line-element (46) becomes ds 2 = −eÃ dt 2 + eB dl 2 + l 2 (dθ 2 + sin 2 θ dϕ 2 ) , with eÃ ∼ a 2 l 2 + ... and eB ∼ b 0 = const.. We may thus conclude that the obtained asymptotic solution is again not a black hole but more likely a particle-like solution with similar characteristics as the ones derived in subsection 3.1. Let us now use the alternative condition that, as r → r h , we have e B → ∞, or equivalently B → ∞. As in subsection 3.2, we may then solve Eq. (48) with respect to A , obtaining the result For a regular black-hole spacetime, we will demand that both the scalar field φ and the circumferential radius H remain finite near the horizon. Then, there are two distinct cases: • If H remains finite near r h , then, in the limit e B → ∞, Eq. (59) leads to If, as assumed, H 2 , φ and φ adopt constant, finite values at the black-hole horizon, then A a 0 + a 1 (r − r h ) + ... and the metric component e A adopts a constant, instead of a vanishing, value at r r h .
• If H is allowed to diverge, then a more careful analysis should be performed. As an indicative example, we may consider the "conformal" case H = re B/2 , that was used in [135]. Then, Eq. (59) becomes For the conventional black-hole dependence where e B b 1 (r − r h ) + ..., the aforementioned equation gives A → 0, thus leading to a constant value for the g tt component near r h . For an alternative dependence, such as the logarithmic given in Eq. (37), we obtain instead that A → const., which however leads again to a constant g tt at r r h .
Therefore, even in the context of the more general line-element (46), the condition e B → ∞ leads to a solution with similar characteristics as the ones that emerged in subsection 3.2. Overall, the system of field equations fails to admit an asymptotic black-hole solution with the anticipated behaviour (12) for both the g tt and g rr metric components.

Solutions with Horizons when Λ = 0
In this section, we reinstate the cosmological constant Λ in order to investigate its role in the emergence or not of solutions with a horizon in the context of the pure scalar-Gauss-Bonnet theory. For Λ = 0, Eq. (9) takes the form of a polynomial with respect to the metric function e B , i.e. αe 2B + βe B + γ = 0. This may be solved to yield where: Taking the derivative of the above expression with respect to the radial coordinate, we may eliminate B from the field equations replacing it by The field equations then reduce to a system of two coupled, second-order, ordinary differential equations for A and φ. This system now has the form where We will assume that, near the black-hole horizon r h , e A → ∞, or equivalently A → ∞. Then, Eq. (62) takes the expanded form: If we replace the above expansion, too, into the system (65), we obtain the following equations in the limit A → ∞: The finiteness of φ at the horizon dictates that the following relation should hold The above condition on the first derivative of the scalar field at the horizon is analogous to the one derived in the context of the Einstein-scalar-Gauss-Bonnet theory -there, that condition also related φ h to the parameters of the theory and ensured the regularity of the black-hole horizon. Here, we observe that the condition (72) relates φ with the value of the cosmological constant. Let us proceed to derive the complete form of the asymptotic solution near r h . By using the above condition on φ h , the system (70)-(71) takes the simplified form: φ =O(1).
Again, this resembles the behaviour obtained around a black-hole horizon. Integrating Eq. (73) with respect to r, we obtain the usual behaviour while a second integration leads to the complete asymptotic solution for the metric functions and the scalar field near r h The above describes indeed a solution with a horizon and a regular scalar field. There is, however, a caveat: if we replace φ h from Eq. (72) into Eq. (69), we find: The robustness of the metric is then ensured only if Eqs. (76)- (78) are re-written as Note that a 1 is a completely arbitrary constant and thus may be appropriately chosen. The above reveal that, when Λ > 0, the asymptotic solution we have found corresponds to a cosmological horizon r c with a non-trivial scalar field. Such solutions have indeed been derived in the context of the pure scalar-Gauss-Bonnet theory but from the perspective of a homogeneous, isotropic universe [145]. Here, we have in fact re-derived those solutions describing a de Sitter universe using isotropic, spherically-symmetric coordinates. In the case of an Anti-de Sitter spacetime with Λ < 0, no cosmological horizon exists and no robust black-hole solution emerges either in the context of our theory. Therefore, even in the presence of a non-vanishing cosmological constant, the pure scalar-Gauss-Bonnet theory fails to support a regular black-hole solution with scalar hair. We close this section by noticing that the asymptotic solution (80)- (82) emerges also in the case where we work with the assumption that near the horizon it is e B , or B , that diverges; this is expected for a physical, spherically-symmetric horizon. Also, similar solutions which possess again a cosmological rather than a black-hole horizon emerge if we employ the more general line-element (46) -as their characteristics are similar to the ones derived above, we refrain from presenting the corresponding analysis.

Synergy between the Ricci and GB terms
In order to shed more light on the fact that the pure scalar-Gauss-Bonnet theory does not admit black-hole solutions but allows for cosmological solutions with a horizon to emerge, we turn to the complete Einstein-scalar-GB theory. In the context of the latter, we will investigate the contribution of each gravitational term to the formation of a black hole. To this end, we choose the case of the exponential coupling function f (φ) = αe −φ , where α is a positive coupling constant. This choice leads to the dilatonic black holes that were found in [13] and studied again in [38]. They are a one-parameter family of black-hole solutions, with the independent parameter being any one of the three parameters of the theory, namely (α, r h , φ h ), due to the rescaling symmetries of the theory. Here, we fix the horizon radius r h to unity while the value of the scalar field at the horizon φ h is determined by the boundary condition φ ∞ = 1 at infinity. These leave the coupling constant α as the independent parameter of the theory.
A theoretical argument for the regularity of the formed black-hole horizon, analogous to the one that led to Eq. (72), leads to the following condition on the first derivative of the scalar field at the horizon [38] The reality of the value of φ h imposes the following additional constraint on the coupling functionḟ For the choice f (φ) = αe −φ , the above reduces to a constraint on the maximum allowed value of the coupling parameter, that is Black-hole solutions then arise in the context of the Einstein-scalar-GB theory for the range of values [0, 0.123] of the coupling constant α. When α → 0, the GB term decouples from the theory and the only black-hole solution is the Schwarzschild solution characterized by a trivial scalar field. As α adopts a non-vanishing value, a GB black hole is formed that possess scalar hair and a horizon radius smaller than the one of the Schwarzschild solution [38]. In Table 1, we display a number of indicative values of α in the range [0, 0.123] and the values of the Ricci scalar and the combination f (φ) R 2 GB near the horizon for each of the corresponding black-hole solution. We observe that for small values of the coupling constant, the Ricci term adopts a very small value -this is due to the fact that, for small α, the obtained solution is still very close to the Schwarzschild one that is a vacuum solution with R = 0. The GB term, in contrast, adopts a much larger value from the beginning due to the contribution of the Riemann tensor that is not zero even for the Schwarzschild solution. As α moves towards its maximum value, the curvature of spacetime increases and this enhances the magnitude of both gravitational terms. Above the maximum value a max = 0.123, no black-hole solutions emerge.
We also observe that the sign of R remains always positive, which reflects the positive curvature of spacetime around the formed horizon. In the Einstein-Hilbert action, the presence of the Ricci term leads to the attractive force of gravity. The presence of the GB term, on the other hand, produces the opposite effect as it leads to a repulsive force in the theory. Indeed, if we examine the components of the effective energy-momentum tensor near the horizon, we obtain the following expressions for the effective energy-density and radial pressure of the system: Note that the contribution of the scalar-field kinetic term vanishes near the horizon and the dominant contribution to the energy-momentum tensor components comes from its effective potential, i.e. by its coupling to the GB term. The combination φ ḟ is always negative at the horizon due to the regularity constraint (83). Also, near the horizon radius, it holds that A −B 1/(r − r h ) + ..., since e A increases near the horizon, as r increases, while e B decreases. These, then, lead to an effective, local equation of state for the GB contribution of the form Clearly, the GB term saturates the dominant energy condition and violates the weak and strong energy conditions by creating a negative effective energy-density and an equal in magnitude, but positive, radial pressure component at the horizon. It is this violation of the energy conditions that causes the evasion of the novel no-hair theorem [9] and allows for the emergence of black holes with non-trivial scalar hair [13,38]. The positive, outward radial pressure signifies in addition the repulsive role of the GB term in the system.
We may, therefore, interpret now the non-emergence of a black-hole horizon in the context of the pure scalar-GB theory. The presence of the Ricci term is necessary in order to provide the attractive force that will create the positively-curved topology around the formed black hole. Even in vacuum, the Ricci term causes the formation of a black hole in the form of the Schwarzschild solution. When the GB term is turned on, the Schwarzschild solution gets naturally scalarised 5 as it is automatically donned with a non-trivial, regular scalar field. Due to the repulsive effect of the GB term, the black-hole horizon is now formed at a smaller horizon value than in the Schwarzschild case -indeed, in [38], it was demonstrated that, for a fixed coupling constant, a GB black hole has always a smaller horizon radius compared to the one of the Schwarzschild solution with the same mass. Apparently, gravity dominates over a smaller regime of spacetime, creating a black-hole topology, in the presence of the GB term. In other words, the same amount of mass needs to be "squeezed" more to create a black hole when the repulsive GB term is present in the theory.
Therefore, the GB term makes the formation of a black hole more difficult. What it does facilitate is the dressing of the black-hole solution with a non-trivial scalar field, a feature that would have been forbidden in its absence. As the value of the coupling constant increases, the weight of the GB term in the theory gradually increases, too. The same holds for its repulsive effect. Beyond the maximum value (85) of the GB coupling parameter, no black-hole horizon can be formed -or sustained -any more. We may easily then justify the fact that a pure scalar-GB theory can not, in the absence of the Ricci scalar, create by itself a black-hole solution.
In the case of cosmological solutions, the presence of the GB term in the theory leads to the emergence of singularity-free [142,144,145] and inflationary solutions [145]. In a cosmological context, traditional gravity, in the form of the Ricci term, leads to the formation of the initial singularity, a feature that is not desirable in the theory. The addition of the GB term with its repulsive effect manages to provide the necessary outward pressure to the system so that the initial singularity is avoided, and a smooth transition between a collapsing and an expanding phase of the Universe is realised [142,144,145]. In the emergence of de Sitter, inflationary solutions [145], the GB term is again providing the outward pressure component that accelerates the expansion of the Universe even in the absence of any potential for the scalar field.

Conclusions
After the derivation, in an analytical way, of cosmological singularity-free and inflationary solutions in the context of the pure scalar-Gauss-Bonnet theory [145], here, we have investigated whether black-hole solutions can be supported in the context of the same theory. We have therefore ignored the presence of all terms associated with the Ricci term in the field equations and used both analytical and numerical means to integrate them. Any solutions that could emerge would rely solely on the synergy between the scalar field and the GB term in the theory and would correspond to zero-energy and zero-pressure gravitational solutions.
We initially focused on the derivation of static, spherically-symmetric solutions that would describe a regular black hole with a non-trivial scalar hair. In section 3, we solved analytically the field equations near the sought-for black-hole horizon as it is only there that ignoring the Ricci scalar, compared to the quadratic GB term, may be justified. Working with the assumption that there the g tt metric component vanishes, we derived a family of gravitational solutions with a finite g rr and a regular scalar field. The spacetime possesses a true singularity but a finite energy-momentum tensor, a profile that resembles the one of particle-like solutions in quadratic gravitational theories [136,138]. Alternatively, demanding that the g rr diverges at a specific value of the radial coordinate, we determined, first analytically and then numerically, a second family of gravitational solutions with no spacetime singularity and a finite, again, energy-momentum tensor. We have postponed the study of these two families of solutions and their physical interpretation for a future study.
Coming back to our quest for black-hole solutions, in Section 4, we considered a more general line-element for the spacetime, that preserved the assumptions of staticity and sphericalsymmetry. Despite the increased flexibility of the line-element, that resulted in the addition of a third unknown metric function, no solutions with a black-hole horizon were found. The asymptotic solutions derived analytically in the small-r regime shared the same characteristics as the ones found in Section 3.
Re-instating the cosmological constant, that was ignored in the first part of our analysis, we looked again for solutions with a horizon. The analysis in this case resembled the one that led to the derivation of regular black-hole solutions with scalar hair [38]. Indeed, solutions with a horizon and a regular scalar field were successfully found in Section 5 in the case of a positive cosmological constant, however, these were shown to correspond to a cosmological rather than to a black-hole horizon.
In Section 6, we looked more carefully at the synergy between the Ricci and GB terms. The values of the components of the energy-momentum tensor near the black-hole horizon reveal the repulsive effect of the GB term as opposed to the attractive effect of the Ricci term. As the latter is necessary in order to create the positive curvature around a black hole, no such solution emerges in the presence of only the GB term, that in fact works towards pushing outwards any distribution of matter. Even in the context of the complete Einsteinscalar-GB theory, black holes emerge only up to a maximum value of the GB coupling parameter -it is only over this restricted parameter regime that the Ricci term manages to form a black-hole horizon despite the presence of the GB term. Nevertheless, the GB term justifies its presence by supporting a non-trivial, regular scalar field, a feature forbidden by General Relativity.
In the context of the effective-field-theory point of view, one could imagine of adding even higher-derivative terms in the quadratic action (1). How are then the previouslyderived solutions [38,130] modified by the presence of these gravitational corrections? In the light of the analysis of Section 6, we conclude that this depends on the role of these terms when it comes to their contributions to the effective energy-momentum tensor. If such a higher-derivative term had a positive contribution to the radial pressure as the GB term, then it would have a destabilising effect and black-hole solutions could emerge for a more restricted range of values for the coupling constant α. If, on the other hand, the produced contribution to the radial pressure of the system was negative, the added term would have a role similar to that of the Ricci scalar and the emergence of black-hole solutions would be facilitated.
Closing this work, let us return to the question we posed in Section 2 to which we may now give a simple answer: we may indeed find spacetime regimes where the GB term is overwhelmingly dominant over the Ricci scalar, however, the gravitational solution that would form will not be a black hole. Nevertheless, solutions the existence of which does not rely on the attractive nature of the Ricci term, and thus on its presence in the theory, do exist and these include a plethora of interesting solutions such as particle-like solutions, cosmological solutions and even wormholes. As that part of the phase-space of solutions of the pure scalar-GB theory has not adequately explored so far, we plan to return to such a careful study soon.