Novel Black-Hole Solutions in Einstein-Scalar-Gauss-Bonnet Theories with a Cosmological Constant

TL;DR

This paper discovers new regular black-hole solutions with scalar hair in Einstein-scalar-Gauss-Bonnet theories with a cosmological constant, showing solutions exist for negative but not positive b3, and analyzing their physical properties.

Contribution

It provides the first comprehensive analysis of black-hole solutions with scalar hair in Einstein-scalar-Gauss-Bonnet theories including a cosmological constant, highlighting the existence for b3<0 and absence for b3>0.

Findings

Solutions with scalar hair exist for b3<0

No such solutions are found for b3>0

Physical properties like temperature and entropy are characterized

Abstract

We consider the Einstein-scalar-Gauss-Bonnet theory in the presence of a cosmological constant $\Lambda$, 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 $\Lambda$ 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 $\Lambda$ determines the form of the asymptotic gravitational background leading either to a Schwarzschild-Anti-de Sitter-type background ($\Lambda<0$) or a regular cosmological horizon ($\Lambda>0$), with a non-trivial scalar field in both cases. We demonstrate that families of novel black-hole solutions with scalar hair emerge for $\Lambda<0$, for every choice of the coupling function between the scalar field and the Gauss-Bonnet term, whereas for $\Lambda>0$, 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.

Equations

$$S=\frac{1}{16\pi }\int d^{4}x\sqrt{-g}\left[R-\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi +f(\varphi )R_{GB}^2-2\Lambda \right].$$

$$R_{GB}^2=R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }-4R_{\mu \nu }{R}^{\mu \nu }+R^2,$$

$$G_{\mu \nu }=T_{\mu \nu },$$

$${\nabla }^2\varphi +\dot{f}(\varphi )R_{GB}^2=0,$$

$$T_{\mu \nu }=-\frac{1}{4}{g}_{\mu \nu }{\partial }_{\rho }\varphi {\partial }^{\rho }\varphi +\frac{1}{2}{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -\frac{1}{2}\left(g_{\rho \mu }{g}_{\lambda \nu }+g_{\lambda \mu }{g}_{\rho \nu }\right){\eta }^{\kappa \lambda \alpha \beta }{\tilde{R}}_{\alpha \beta }^{\rho \gamma }{\nabla }_{\gamma }{\partial }_{\kappa }f(\varphi )-\Lambda g_{\mu \nu }.$$

$${\tilde{R}}_{\alpha \beta }^{\rho \gamma }={\eta }^{\rho \gamma \sigma \tau }{R}_{\sigma \tau \alpha \beta }=\frac{{ϵ}^{\rho \gamma \sigma \tau }}{\sqrt{-g}}{R}_{\sigma \tau \alpha \beta }.$$

$${ds}^2=-e^{A(r)}{dt}^2+e^{B(r)}{dr}^2+r^2({d\theta }^2+{\sin }^2\theta {d\phi }^2).$$

$$G_t^t$$

$$G_r^r$$

$$G_{\theta }^{\theta }$$

$$T_t^t=$$

$$T_r^r=$$

$$T_{\theta }^{\theta }=$$

$$2r{\varphi }^{\prime \prime }+(4+r{A}^{\prime }-r{B}^{\prime }){\varphi }^{\prime }+\frac{4\dot{f}{e}^{-B}}{r}\left[(e^B-3)A^{\prime }{B}^{\prime }-(e^B-1)(2A^{\prime \prime }+A^{\prime 2})\right]=0.$$

$$e^B=\frac{-\beta \pm \sqrt{{\beta }^2-4\alpha \gamma }}{2\alpha },$$

$$\alpha =1-\Lambda r^2,\beta =\frac{r^2{{\varphi }^{\prime }}^2}{4}-(2\dot{f}{\varphi }^{\prime }+r)A^{\prime }-1,\gamma =6\dot{f}{\varphi }^{\prime }{A}^{\prime }.$$

$$B^{\prime }=-\frac{{\gamma }^{\prime }+{\beta }^{\prime }{e}^B+{\alpha }^{\prime }{e}^{2B}}{2\alpha e^{2B}+\beta e^B}.$$

$$A^{\prime \prime }=$$

$${\varphi }^{\prime \prime }=$$

$$e^B=\frac{(2\dot{f}{\varphi }^{\prime }+r)}{1-\Lambda r^2}{A}^{\prime }-\frac{2\dot{f}{\varphi }^{\prime }\left(r^2{\varphi }^{\prime 2}-12\Lambda r^2+8\right)+r\left(r^2{\varphi }^{\prime 2}-4\right)}{4(1-\Lambda r^2)(2\dot{f}{\varphi }^{\prime }+r)}+\mathcal{O}\left(\frac{1}{A^{\prime }}\right).$$

$$A^{\prime \prime }=$$

$${\varphi }^{\prime \prime }=$$

$$W_1$$

$$W_2$$

$$W_3=\left(1-\Lambda r^2\right)\left[r^4+2r^3\dot{f}{\varphi }^{\prime }-16{\dot{f}}^2\left(3-2\Lambda r^2\right)-32\Lambda r{\dot{f}}^3{\varphi }^{\prime }\right].$$

$$W_2{|}_{r=r_h}=0.$$

$$\displaystyle\phi^{\prime}_{h}=-\frac{r_{h}^{3}(1-\Lambda r_{h}^{2})+16\Lambda r_{h}\dot{f}^{2}_{h}(3-\Lambda r_{h}^{2})\pm(1-\Lambda r_{h}^{2})\sqrt{C}}{4\dot{f}\,\Bigl{[}r_{h}^{2}-\Lambda(r_{h}^{4}-16\dot{f}^{2}_{h})\Bigr{]}},$$

$$C=256\Lambda {\dot{f}}_h^4\left(\Lambda r_h^2-6\right)+32r_h^2{\dot{f}}_h^2\left(2\Lambda r_h^2-3\right)+r_h^6\ge 0,$$

$${\dot{f}}_{\pm }^2=\frac{r_h^2\left[3-2\Lambda r_h^2\pm \sqrt{3}\sqrt{3-2\Lambda r_h^2+{\Lambda }^2{r}_h^4}\right]}{16\Lambda (-6+\Lambda r_h^2)}.$$

$$C=({\dot{f}}_h^2-{\dot{f}}_{-}^2)({\dot{f}}_h^2-{\dot{f}}_{+}^2)\ge 0.$$

$${\dot{f}}_h^2\le \frac{r_h^4}{96}\left(1+\frac{\Lambda r_h^2}{6}+...\right),\mathrm{or},{\dot{f}}_h^2\ge \frac{r_h^4}{48}\left(1-\frac{3}{\Lambda r_h^2}+...\right),$$

$$A^{\prime \prime }=-A^{\prime 2}+\mathcal{O}\left(A^{\prime }\right).$$

$$e^A=a_1(r-r_h)+...,$$

Full text

**Novel Black-Hole Solutions in

Einstein-Scalar-Gauss-Bonnet Theories

with a Cosmological Constant**

A. Bakopoulos${}^{\,(a)}$ 111Email: abakop@cc.uoi.gr, G. Antoniou${}^{\,(b)}$ 222Email: anton296@umn.edu and P. Kanti${}^{\,(a)}$ 333Email: pkanti@cc.uoi.gr

*(a)**Division of Theoretical Physics, Department of Physics,

University of Ioannina, Ioannina GR-45110, Greece*

*(b)*School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA

Abstract

We consider the Einstein-scalar-Gauss-Bonnet theory in the presence of a cosmological constant $\Lambda$, 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 $\Lambda$ 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 $\Lambda$ determines the form of the asymptotic gravitational background leading either to a Schwarzschild-Anti-de Sitter-type background ($\Lambda<0$) or a regular cosmological horizon ($\Lambda>0$), with a non-trivial scalar field in both cases. We demonstrate that families of novel black-hole solutions with scalar hair emerge for $\Lambda<0$, for every choice of the coupling function between the scalar field and the Gauss-Bonnet term, whereas for $\Lambda>0$, 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.

1 Introduction

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 [3], 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 [4], Skyrme fields [5] or with a conformal coupling to gravity [6]. A novel formulation of the no-hair theorem was proposed in 1995 [7] 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 [8] (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 [16] 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 [25]. During the last decade, though, the construction of generalised gravitational theories was significantly enlarged via the revival of the Horndeski [26] and Galileon [27] theories. Accordingly, novel formulations of the no-hair theorems were proposed that covered the case of standard scalar-tensor theories [28] and Galileon fields [29]. However, these recent forms were also evaded [30] 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 [34], it was shown that the presence of the GB term was of paramount importance for the evasion of the novel no-hair theorem [7]. In addition, the exact form of the coupling function $f(\phi)$ 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 $\phi_{h}^{\prime}$ 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 $f(\phi)$, a large number of asymptotically-flat black-hole solutions with scalar hair were determined [34]. 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 [34], that aimed at deriving asymptotically-flat black-hole solutions, by introducing in our theory a cosmological constant $\Lambda$, 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 $\Lambda$. 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 [71] was shown later to be unstable [72]. 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 [85], 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 $f(\phi)$, 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 $\Lambda$ 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-$r$ regimes to examine how the presence of $\Lambda$ 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 $\Lambda$ 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 $\Lambda$: whereas, for $\Lambda<0$, solutions emerge with the same easiness as their asymptotically-flat analogues, for $\Lambda>0$, no such solutions were found.

In the former case, i.e. for $\Lambda<0$, 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 $f(\phi)$: 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 $\Lambda<0$ and $\Lambda>0$. 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:

[TABLE]

In this, the quadratic Gauss-Bonnet (GB) term $R^{2}_{GB}$, defined as

[TABLE]

supplements the Einstein-Hilbert term, given by the Ricci scalar curvature $R$, and the kinetic term for a scalar field $\phi$. A coupling term of the scalar field to the GB term, through a general coupling function $f(\phi)$, is necessary in order for the GB term – a total derivative in four dimensions – to contribute to the field equations. A cosmological constant $\Lambda$, 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 $g_{\mu\nu}$ and the scalar field $\phi$, we derive the gravitational field equations and the equation for the scalar field, respectively. These are found to have the form:

[TABLE]

[TABLE]

where $G_{\mu\nu}$ is the Einstein tensor and $T_{\mu\nu}$ is the energy-momentum tensor, with the latter having the form

[TABLE]

In the above, the dot over the coupling function denotes its derivative with respect to the scalar field (i.e. $\dot{f}=df/d\phi$). We have also employed units in which $G=c=1$, and used the definition

[TABLE]

Compared to the theory studied in [34], where $\Lambda$ was zero, the changes in Eqs. (3)-(4) look minimal: the scalar-field equation remains unaffected while the energy-momentum tensor $T^{\mu}_{\;\,\nu}$ receives a constant contribution $-\Lambda\delta^{\mu}_{\;\,\nu}$. 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

[TABLE]

The scalar field will also be assumed to be static and spherically-symmetric, $\phi=\phi(r)$. The coupling function $f(\phi)$ 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 $G^{\mu}_{\;\,\nu}$ may be easily found by employing the line-element (7), and they read

[TABLE]

Throughout our analysis, the prime denotes differentiation with respect to the radial coordinate $r$. Using Eq. (5), the components of the energy-momentum tensor $T^{\mu}_{\;\,\nu}$ take in turn the form

[TABLE]

Matching the corresponding components of $G^{\mu}_{\;\,\nu}$ and $T^{\mu}_{\;\,\nu}$, 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

[TABLE]

Although the system of equations involve three unknown functions, namely $A(r)$, $B(r)$ and $\phi(r)$, only two of them are independent. The metric function $B(r)$ may be easily shown to be a dependent variable: the $(rr)$-component of field equations takes in fact the form of a second-order polynomial with respect to $e^{B}$, i.e. $\alpha e^{2B}+\beta e^{B}+\gamma=0$, which easily leads to the following solution

[TABLE]

where

[TABLE]

Employing the above expression for $e^{B}$, the quantity $B^{\prime}$ may be also found to have the form

[TABLE]

Therefore, by using Eqs. (15) and (17), the metric function $B(r)$ 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 $A(r)$ and $\phi(r)$:

[TABLE]

The expressions for the quantities $P$, $Q$ and $S$, in terms of $(r,\phi^{\prime},A^{\prime},\dot{f},\ddot{f})$, 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 $(r-r_{h})$, where $r_{h}$ 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 $e^{A(r)}$ should vanish (and $e^{B(r)}$ should diverge) whereas the scalar field must remain finite. The first demand is reflected in the assumption that $A^{\prime}(r)$ should diverge as $r\rightarrow r_{h}$ – this will be justified a posteriori – while $\phi^{\prime}(r)$ and $\phi^{\prime\prime}(r)$ must be finite in the same limit.

Assuming the aforementioned behaviour near the black-hole horizon, Eq. (15) may be expanded in terms of $A^{\prime}(r)$ as follows444Note, that only the (+)-sign in the expression for $e^{B}$ in Eq. (15) leads to the desired black-hole behaviour.

[TABLE]

Then, substituting the above into the system (18)-(19), we obtain

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

From Eq. (20), we conclude that the combination $(2\dot{f}\phi^{\prime}+r)$ near the horizon must be non-zero and positive for the metric function $e^{B}$ to have the correct behaviour, that is to diverge as $r\rightarrow r_{h}$ while being positive-definite. Then, Eq. (22) dictates that, if we want $\phi^{\prime\prime}$ to be finite, we must necessarily have

[TABLE]

The above constraint may be written as a second-order polynomial with respect to $\phi^{\prime}$, which can then be solved to yield

[TABLE]

where all quantities have been evaluated at $r=r_{h}$ The quantity $C$ under the square root stands for the following combination

[TABLE]

and must always be non-negative for $\phi^{\prime}_{h}$ to be real. This combination may be written as a second-order polynomial for $\dot{f}^{2}_{h}$ with roots

[TABLE]

Then, the constraint on $C$ becomes

[TABLE]

Therefore, the allowed regime for the existence of regular, black-hole solutions with scalar hair is given by $\dot{f}^{2}_{h}\leq\dot{f}^{2}_{-}$ or $\dot{f}^{2}_{h}\geq\dot{f}^{2}_{+}$, since $\dot{f}^{2}_{+}>\dot{f}^{2}_{-}$. To obtain some physical insight on these inequalities, we take the limit of small cosmological constant; then, the allowed ranges are

[TABLE]

respectively. In the absence of $\Lambda$, Eq. (28) results into the simple constraint $\dot{f}^{2}_{h}\leq r_{h}^{4}/96$, and defines a sole branch of solutions with a minimum allowed value for the horizon radius (and mass) of the black hole [34]. In the presence of a cosmological constant, this constraint is now replaced by $\dot{f}^{2}_{h}\leq\dot{f}^{2}_{-}$, or by the first inequality presented in Eq. (31) in the small-$\Lambda$ limit. This inequality leads again to a branch of solutions that – for chosen $f(\phi)$, $\phi_{h}$ and $\Lambda$ – terminates at a black-hole solution with a minimum horizon radius $r_{h}^{min}$. We observe that, at least for small values of $\Lambda$, 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 $\Lambda=0$; this was also noted in [85] 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 $r_{h}^{max}$.

Returning now to Eq. (18) and employing the constraint (27), the former takes the form

[TABLE]

Integrating the above, we find that $A^{\prime}(r)\sim 1/(r-r_{h})$, a result that justifies the diverging behaviour of this quantity near the horizon that we assumed earlier. A second integration yields $A(r)\sim\ln(r-r_{h})$, which then uniquely determines the expression of the metric function $e^{A}$ in the near-horizon regime. Employing Eq. (20), the metric function $B$ 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 $r\rightarrow r_{h}$, is given by the following expressions

[TABLE]

where $a_{1},b_{1}$ and $\phi_{h}$ 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 [34] for the case of vanishing $\Lambda$. The presence of the cosmological constant modifies though the exact expressions of the basic constraint (27) for $\phi^{\prime}_{h}$ and of the quantity $C$ given in (28), the validity of which ensures the existence of a regular black-hole horizon. As in [34], the exact form of the coupling function $f(\phi)$ 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

[TABLE]

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 $T^{\mu}_{\;\,\nu}$ for $\Lambda<0$, while it subtracts a positive contribution for $\Lambda>0$. 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

[TABLE]

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 $\Lambda$.

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 $r=r_{c}>r_{h}$. We demand that this horizon is also regular, that is that the scalar field $\phi$ 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, $g_{tt}\rightarrow 0$ while $g_{rr}\rightarrow\infty$; then, using that $A^{\prime}$ diverges there, the regularity of $\phi^{\prime\prime}$ from Eq. (19) eventually leads to the constraint

[TABLE]

with $\tilde{C}$ now being given by the non-negative expression

[TABLE]

Employing Eq. (40) in Eq. (18), the solution for the metric function $A$ may be again constructed. Overall, the asymptotic solution of the field equations near a regular, cosmological horizon will have the form

[TABLE]

where care has been taken for the fact that $r\leq r_{c}$. 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 $f(\phi)$ 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

[TABLE]

where $k$, $M$, $\Lambda_{eff}$ and $q_{1,2}$ are, at the moment, arbitrary constants. Substituting the above expressions into the scalar field equation (14), we obtain at first order the constraint

[TABLE]

The gravitational equations, under the same assumptions, lead to two additional constraints, namely

[TABLE]

[TABLE]

Contrary to what happens close to the horizons (either black-hole or cosmological ones), the form of the coupling function $f(\phi)$ now affects the asymptotic form of the scalar field at large distances. The easiest case is that of a linear coupling function, $f(\phi)=\alpha\phi$ - that case was first studied in [85], 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

[TABLE]

where again $(\phi_{\infty},d_{1},d_{2},d_{3})$ are arbitrary constant coefficients. The coefficients $d_{1}$ and $\Lambda_{eff}$ may be determined through the first-order constraints (47) and (48), respectively, and are given by

[TABLE]

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 $k$, $q_{1}$ and $d_{2}$ are found at third-order approximation to have the forms

[TABLE]

[TABLE]

while for $q_{2}$ or $d_{3}$ one needs to go even higher. In contrast, the coefficient $M$ 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 $\alpha$ of the GB term), one may show that the above asymptotic solution is valid for all forms of the coupling function $f(\phi)$. Indeed, if we write

[TABLE]

and define $f(\phi)=\alpha\tilde{f}(\phi)$, then, at first order, $\dot{f}\simeq\alpha\,\dot{\tilde{f}}(\phi_{0})$. Therefore, independently of the form of $f(\phi)$, at first order in the perturbative limit, $\dot{f}$ 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 $d_{1}=8\Lambda\dot{f}(\phi_{0})/3$, $\Lambda_{eff}=\Lambda$, $k=1$, $q_{1}=0$, and $d_{2}=-4\dot{f}(\phi_{0})$. 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 [89]. with $\alpha$ in Eqs. (51) and (52) being now replaced by $\dot{f}(\phi_{0})$.

For arbitrary values of the coupling constant $\alpha$, though, or for a non-linear coupling function $f(\phi)$, 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]

[TABLE]

that relates the black-hole temperature $T$ to its surface gravity $k_{h}$. 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 [86]

[TABLE]

where $F=I_{E}/\beta$ is the Helmholtz free-energy of the system given in terms of the Euclidean version of the action $I_{E}$, and $\beta=1/(k_{B}T)$. The above formula has been used in the literature to determine the entropy of the asymptotically-flat coloured GB black holes [15] and of the family of novel black-hole solutions found in [34] 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 $r_{h}\leq r\leq r_{c}$ 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 [91] 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 [93], the following formula was finally derived for the entropy

[TABLE]

where $\mathcal{L}$ is the Lagrangian of the theory, $\hat{\epsilon}_{ab}$ the binormal to the horizon surface $\mathcal{H}$, and $h_{(2)}$ the 2-dimensional projected metric on $\mathcal{H}$. The equivalence of the two approaches has been demonstrated in [92], 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

[TABLE]

The first term inside the curly brackets of the above expression comes from the variation of the Einstein-Hilbert term and leads to:

[TABLE]

We recall that $\hat{\epsilon}_{ab}$ is antisymmetric, and, in addition, satisfies $\hat{\epsilon}_{ab}\,\hat{\epsilon}^{\,ab}=-2$. Therefore, we easily obtain the result

[TABLE]

where $A_{\mathcal{H}}=4\pi r_{h}^{2}$ is the horizon surface. The remaining terms in Eq. (57) are all proportional to the coupling function $f(\phi)$ 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: \hat{\epsilon}_{ab}=\sqrt{-g_{00}\,g_{11}}\big{|}_{\mathcal{H}}\left(\delta^{0}_{a}\delta^{1}_{b}-\delta^{1}_{a}\delta^{0}_{b}\right). This means that we may alternatively write:

[TABLE]

Therefore, the terms proportional to $f(\phi)$ may be written as

[TABLE]

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

[TABLE]

Combining the expressions (59) and (62), we finally derive the result

[TABLE]

The above describes the entropy of a GB black hole arising in the context of the theory (1), with a general coupling function $f(\phi)$ between the scalar field and the GB term, and a cosmological constant term. We observe that the above expression matches the one derived in [34] 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 $\Lambda$ 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 $\Lambda$ 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 $r\approx r_{h}+\mathcal{O}(10^{-5})$ (for simplicity, we set $r_{h}=1$). The metric function $A$ and scalar field $\phi$ in that regime are described by the asymptotic solutions (33) and (35). The input parameter $\phi^{\prime}_{h}$ is uniquely determined through Eq. (27) once the coupling function $f(\phi)=\alpha\tilde{f}(\phi)$ is selected and the values of the remaining parameters of the model near the horizon are chosen. These parameters appear to be $\alpha$, $\phi_{h}$ and $\Lambda$. However, the first two are not independent: since it is their combination $\alpha\tilde{f}(\phi_{h})$ 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 $\alpha$ and vary only $\phi_{h}$. The values of $\phi_{h}$ and $\Lambda$ also cannot be totally uncorrelated as they both appear in the expression of $C$, 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 $\dot{f}_{h}^{2}\leq\dot{f}_{-}^{2}$ and $\dot{f}_{h}^{2}\geq\dot{f}_{+}^{2}$ 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 $\dot{f}_{h}$. However, in what follows we will assume that $\dot{f}_{h}>0$, and thus study the two regimes $\dot{f}_{h}\leq\dot{f}_{-}$ and $\dot{f}_{h}\geq\dot{f}_{+}$; similar results emerge if one assumes instead that $\dot{f}_{h}<0$.

Before starting our quest for black holes with an (Anti)-de Sitter asymptotic behaviour at large distances, we first consider the case with $\Lambda=0$ where upon we successfully reproduce the families of asymptotically-flat back holes derived in [34]. Then, we select non-vanishing values of $\Lambda$ and look for novel black-hole solutions. We will start with the case of a negative cosmological constant ($\Lambda<0$) in the next subsection and consider the case of a positive cosmological constant ($\Lambda>0$) 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 $\Lambda<0$, the asymptotic solution (45)-(46) describing an Anti-de Sitter-type background. The arbitrary coefficient $a_{1}$, that does not appear in the field equations, may be fixed by demanding that, at very large distances, the metric functions satisfy the constraint $e^{A}\simeq e^{-B}$. We have considered a large number of forms for the coupling function $f(\phi)$, 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 $f(\phi)$.

We will first discuss the case of an exponential coupling function, $f(\phi)=\alpha e^{-\phi}$. The solutions for the metric functions $e^{A(r)}$ and $e^{B(r)}$ are depicted in the left plot of Fig. 1. We may easily see that the near-horizon behaviour, with $e^{A(r)}$ vanishing and $e^{B(r)}$ 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 $\Lambda=-1$ (in units of $r_{h}^{-2}$), $\alpha=0.1$ and $\phi_{h}=1$, 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 $\dot{f}_{h}\geq\dot{f}_{+}$ 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 [85]. $\dot{f}_{h}\leq\dot{f}_{-}$, 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 $\alpha=0.01$, $\phi_{h}=1$ 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 $f(\phi)$, 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 $T^{\mu}_{\;\,\nu}$ over the whole radial regime, for the indicative solution $\Lambda=-1$, $\alpha=0.1$ and $\phi_{h}=1$. Far away from the black-hole horizon, all components reduce to $-\Lambda$, 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 $T^{r}_{\;\,r}\approx T^{t}_{\;\,t}$, since, at $r\simeq r_{h}$, $A^{\prime}\simeq-B^{\prime}$; also, the $T^{\theta}_{\;\,\theta}$ component always has the opposite sign to that of $T^{r}_{\;\,r}$ since $A^{\prime\prime}\simeq-A^{\prime 2}$. This qualitative behaviour of $T^{\mu}_{\;\,\nu}$ 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 $T^{r}_{\;\,r}\approx T^{t}_{\;\,t}>0$. Comparing this behaviour with the asymptotic forms (36)-(38), we deduce that, close to the black-hole horizon where $A^{\prime}>0$, we must have $(\phi^{\prime}\dot{f})_{h}<0$. 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 [7] and the emergence of novel, asymptotically-flat black-hole solutions [34]. We observe that also in the context of the present analysis with $\Lambda\neq 0$, 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 $\alpha>0$, the constraint $(\phi^{\prime}\dot{f})_{h}<0$ means that $\phi^{\prime}_{h}>0$ independently of the value of $\phi_{h}$. 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 $\alpha=0.1$, $\phi_{h}=0.5$ and for different values of the cosmological constant. The scalar field satisfies indeed the constraint $\phi^{\prime}_{h}>0$ and increases away from the black-hole horizon777A complementary family of solutions arises if we choose $\alpha<0$, with the scalar profile now satisfying the constraint $\phi^{\prime}_{h}<0$ and decreasing away from the black-hole horizon.. At large distances, we observe that, for small values of the cosmological constant, $\phi(r)$ assumes a constant value; this is the behaviour found for asymptotically-flat solutions [34] that the solutions with small $\Lambda$ are bound to match. For increasingly larger values of $\Lambda$ though, the profile of the scalar field deviates significantly from the series expansion in powers of $(1/r)$ thus allowing for a $r$-dependent $\phi$ even at infinity – in the perturbative limit, as we showed in the previous section, this dependence is given by the form $\phi(r)\simeq d_{1}\ln r$.

We will now consider the case of an even polynomial coupling function of the form $f(\phi)=\alpha\phi^{2n}$ with $n\geq 1$. 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 $\alpha$ or the cosmological constant $\Lambda$, 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 $T^{r}_{\,\,r}$ near the black-hole horizon implies again that, there, we should have $(\dot{f}\phi^{\prime})_{h}<0$, or equivalently $\phi_{h}\phi^{\prime}_{h}<0$, for $\alpha>0$. Indeed, two classes of solutions arise in this case: for positive values of $\phi_{h}$, we obtain solutions for the scalar field that decrease away from the black-hole horizon, while for $\phi_{h}<0$, 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. $n=1$), for $\phi_{h}=-1$ and $\alpha=0.01$, arising for different values of $\Lambda$ – since $\phi_{h}<0$, the scalar field exhibits an increasing behavior as expected.

Let us examine next the case of an odd polynomial coupling function, $f(\phi)=\alpha\phi^{2n+1}$ with $n\geq 0$. 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 $\alpha(\phi^{2n}\phi^{\prime})_{h}<0$ or simply $\phi_{h}^{\prime}<0$, when $\alpha>0$. As this holds independently of the value of $\phi_{h}$, 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. $n=1$) is presented for $\alpha=0.1$, $\phi_{h}=0.1$ and various values of $\Lambda$.

The case of an inverse polynomial coupling function, $f(\phi)=\alpha\phi^{-k}$, with $k$ either an even or odd positive integer, was also considered. For odd $k$, i.e. $k=2n+1$, the positivity of $T^{r}_{\,\,r}$ near the black-hole horizon demands again that $(\dot{f}\phi^{\prime})_{h}<0$, or that $-\alpha/\phi^{2n+2}\phi^{\prime}<0$. For $\alpha>0$, the solution for the scalar field should thus always satisfy $\phi^{\prime}_{h}>0$, regardless of our choices for $\phi_{h}$ or $\Lambda$. As an indicative example, in the left plot of Fig. 4, we present the case of $f(\phi)=\alpha/\phi$ with a family of solutions arising for $\alpha=0.1$ and $\phi_{h}=2$. 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 $k$, i.e. $k=2n$, the aforementioned constraint now demands that $\phi_{h}\,\phi^{\prime}_{h}<0$. As in the case of the odd polynomial coupling function, two subclasses of solutions arise: for $\phi_{h}>0$, solutions emerge with $\phi^{\prime}_{h}<0$ whereas, for $\phi_{h}<0$, we find solutions with $\phi^{\prime}_{h}>0$. The profiles of the solutions in this case are similar to the ones found before, with $\phi$ approaching, at large distances, an almost constant value for small $\Lambda$ 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, $f(\phi)=\alpha\ln\phi$. Here, the condition near the horizon of the black hole gives $\alpha\phi^{\prime}/\phi<0$, therefore, for $\alpha>0$, we must have $\phi^{\prime}_{h}\phi_{h}<0$; for $\phi_{h}>0$, 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 $\alpha=0.01$ and $\phi_{h}=1$, while varying the cosmological constant $\Lambda$. 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 $-\Lambda$.

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. $V_{\phi}\equiv\dot{f}(\phi)\,R^{2}_{GB}$. 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 $f(\phi)$. As expected, the potential $V_{\phi}$ 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, $V_{\phi}$ 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 $V_{\infty}$ in the place of $V_{\phi}$ in the scalar-field equation (14), we arrive at the intermediate result

[TABLE]

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 $\alpha$.

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 $V_{\phi}$, combined graphs for different forms of the coupling function $f(\phi)$. 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 ${\cal O}(1/r)$ 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 $d_{1}$, 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 $\Lambda=-0.1$ of the cosmological constant. We see that, for small values of the mass $M$, 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 [34] 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 $\Lambda=-0.001$ and $\Lambda=-0.1$ 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 $f(\phi)$ - this is in complete agreement with the profile found in the asymptotically-flat case [34]. This behaviour persists for even larger values of the negative cosmological constant for all classes of solutions apart from the one emerging for the logarithmic function whose horizon area is significantly increased in the small-mass regime, as may be seen from the right plot of Fig. 6. These plots verify also the termination of all branches of solutions at the point of a minimum horizon, or minimum mass, that all our GB solutions exhibit as a consequence of the inequality (28). We also observe that, as hinted by the small-$\Lambda$ approximation given in Eq. (30), an increase in the value of the negative cosmological constant pushes upwards the lowest allowed value of the horizon radius of our solutions.

We now move to the thermodynamical quantities of our black-hole solutions. We start with their temperature $T$ given by Eq. (54) in terms of the near-horizon coefficients $(a_{1},b_{1})$. In the left plot of Fig. 7, we display its dependence in terms of the cosmological constant $\Lambda$, for several forms of the coupling function. We observe that $T$ increases, too, with $|\Lambda|$; we thus conclude that the more negatively-curved the spacetime is, the hotter the black hole, that is formed, is. Note that the form of the coupling function plays almost no role in this relation with the latter thus acquiring a universal character for all GB black-hole solutions. The dependence of the temperature of the black hole on its mass, as displayed in the right plot of Fig. 7, exhibits a decreasing profile, with the obtained solution being colder the larger its mass is. For small black-hole solutions, the exact dependence of $T$ on $M$ depends on the particular form of the coupling function but for solutions with a large mass its role becomes unimportant as a common ‘Schwarzschild-AdS regime’ is again approached.

Let us finally study the entropy of the derived black-hole solutions. In Fig. 8, we display the ratio of the entropy of our GB solutions over the entropy of the corresponding Schwarzschild-Anti-de Sitter solution with the same mass, for the same indicative values of the negative cosmological constant as for the horizon area. i.e. for $\Lambda=-0.001$ (left plot) and $\Lambda=-0.1$ (right plot). We observe that the profile of this quantity depends strongly on the choice of the coupling function $f(\phi)$, for solutions with small masses, whereas in the limit of large mass, where our solutions reduce to the SAdS ones, this ratio approaches unity as expected. For small values of $\Lambda$, the left plot of Fig. 8 depicts a behaviour similar to the one found in the asymptotically-flat case [34]: solutions emerging for the linear and the quadratic coupling functions exhibit smaller entropy compared to the SAdS one, while solutions for the exponential, logarithmic and inverse-linear coupling functions lead to GB black holes with a larger entropy over the whole mass range or for particular mass regimes. As we increase the value of the cosmological constant (see right plot of Fig. 8), the entropy ratio is suppressed for all families of GB black holes apart from the one emerging for the logarithmic coupling function, which exhibits a substantial increase in this quantity over the whole mass regime. Together with the solutions for the exponential and inverse-linear coupling functions, they have an entropy ratio larger than unity while this ratio is now significantly lower than unity for all the other polynomial coupling functions. Although the question of the stability of the derived solutions is an important one and must be independently studied for each family of solutions found, the entropy profiles presented above may provide some hints regarding the thermodynamical stability of our solutions compared to the Schwarzschild-Anti-de Sitter ones.

3.2 de Sitter Gauss-Bonnet Black Holes

We now address the case of a positive cosmological constant, $\Lambda>0$. We start our integration process at a distance close to the black-hole horizon, using the asymptotic solutions (33)-(35) and choosing $\phi_{h}$ to satisfy again the regularity constraint (27). The coupling function $f(\phi)$ is assumed to take on a variety of forms – namely exponential, even and odd polynomial, inverse even and odd polynomial, and logarithmic forms – as in the case of the negative cosmological constant. The numerical integration then proceeds outwards to meet the corresponding asymptotic solution (42)-(44) near the cosmological horizon.

Unfortunately, and despite our persistent efforts, no complete black-hole solution interpolating between the asymptotic solutions (33)-(35) and (42)-(44) was found. The same negative result concerning the existence of a black hole solution with an asymptotically de Sitter behaviour was obtained in [85], where the case of a linear coupling function between the GB term and the scalar field was considered. It is, however, worth noting that the two asymptotic solutions near the black-hole and cosmological horizons do independently emerge – it is the effort to match them in a smooth way via an intermediate solution that fails.

To demonstrate this, in Fig. 9 we display the result of our numerical integration for the indicative case of $\alpha=0.01$, $\phi_{h}=-1$ and $\Lambda=0.01$. The coupling function has been chosen to be $f(\phi)=\alpha e^{-\phi}$, however, the same qualitative behaviour was found for every choice of $f(\phi)$ we have considered. From the metric functions and the scalar-field profiles displayed in the two plots, we clearly see that an asymptotic solution describing a regular black-hole horizon is indeed formed. In this, the metric component $|g_{tt}|$ vanishes while the $g_{rr}$ one diverges, as expected. The scalar field near the black-hole horizon assumes a finite, constant value while it decreases away from the horizon, in perfect agreement with the scalar-field profile found in the case of a negative cosmological constant. The integration proceeds uninhibited but stops abruptly close to the regime where the cosmological horizon should form. In fact, from the left plot of Fig. 9, we may clearly see the expected behaviour of the metric components near the cosmological horizon (i.e. the vanishing of $|g_{tt}|$ and divergence of $g_{rr}$) just to emerge.

The emergence of asymptotic solutions and the failure to smoothly match them strongly reminds us of the analysis involved in the no-hair theorems [3, 7], where a similar situation holds. It is, however, difficult to generalise that analysis, or equivalently the argument for their evasion as developed in [34], in the present case of a non-vanishing cosmological constant 999A theoretical analysis is currently under way but has, so far, not given any conclusive results.. One could, nevertheless, gain some understanding of the situation by examining the form of the near-horizon value of the $T^{r}_{\,\,r}$ component of the energy-momentum tensor given in Eq. (37) – the profile of this component is of paramount importance for the evasion of the novel no-hair theorem [7] and the emergence of novel solutions. For the evasion to be realised, this component must be positive and decreasing close to the black-hole horizon [8, 34]. From Eq. (37), it becomes clear that the presence of a negative cosmological constant ($\Lambda<0$) in the theory always gives a positive contribution to $T^{r}_{\,\,r}$, and enhances the probability of obtaining regular black holes. This justifies the easiness in which novel black-hole solutions with an asymptotically Anti-de Sitter behaviour have emerged in the context of our analysis. On the other hand, the contribution of a positive cosmological constant ($\Lambda>0$) to $T^{r}_{\,\,r}$ is always negative, and this makes the evasion of the no-hair theorem less likely. It would be indeed interesting to re-address the arguments presented in [34] as well as the ones employed in the versions of the no-hair theorems for non-asymptotically-flat black holes [66, 67, 68, 69, 70] to cover also the case where the GB term and the cosmological constant appear simultaneously in the theory.

Nevertheless, even if the evasion of the no-hair theorems may be realised for $\Lambda>0$ in the presence of the GB term – for small values of $\Lambda$ this seems quite likely – this merely opens the way to look for novel solutions, it does not guarantee their existence. The emergence of a complete solution interpolating between the two horizons still demands the smooth matching of the two asymptotic solutions. It is quite likely that the system does not have enough freedom to simultaneously satisfy the requirements for the existence of a regular solution, namely Eqs. (27)-(28) and (40)-(41) - this was also noted in [85]. Or, that a very careful selection of parameters may be necessary for such a solution to emerge. In any case, further investigation is necessary, and we hope to return to this topic soon.

4 Conclusions

In this work, we have extended our previous analyses [34], on the emergence of novel, regular black-hole solutions in the context of the Einstein-scalar-GB theory, to include the presence of a positive or negative cosmological constant. Since the uniform distribution of energy associated with the cosmological constant permeates the whole spacetime, we expected $\Lambda$ to have an effect on both the near-horizon and far-field asymptotic solutions. Indeed, our analytical calculations in the small-$r$ regime revealed that the cosmological constant modifies the constraint that determines the value of $\phi^{\prime}_{h}$ for which a regular, black-hole horizon forms. In addition, it was demonstrated that such a horizon is indeed formed, for either positive or negative $\Lambda$ and for all choices of the coupling function $f(\phi)$.

In contrast, the behaviour of the solution in the far-field regime depended strongly on the sign of the cosmological constant. For $\Lambda>0$, a second horizon, the cosmological one, was expected to form at a distance $r_{c}>r_{h}$, whereas for $\Lambda<0$, an Anti-de Sitter type of solution was sought for at asymptotic infinity. Both types of solutions were analytically shown to be admitted by the set of our field equations at the limit of large distances, thus opening the way for the construction of complete black-hole solutions with an (Anti)-de Sitter asymptotic behaviour.

The complexity of the field equations prevented us from constructing such a solution analytically, therefore we turned to numerical analysis. Using our near-horizon analytic solution as a starting point, we integrated the set of field equations from the black-hole horizon and outwards. For a negative cosmological constant ($\Lambda<0$), we demonstrated that regular black-hole solutions with an Anti-de Sitter-type asymptotic behaviour arise with the same easiness that their asymptotically-flat counterparts emerge. We have produced solutions for an exponential, polynomial (even or odd), inverse polynomial (even or odd) and logarithmic coupling function between the scalar field and the GB term. In each and every case, once $f(\phi)$ was chosen, selecting the input parameter $\phi_{h}^{\prime}$ to satisfy the regularity constraint (27) and the second input parameter $\phi_{h}$ to satisfy the inequality (28) a regular black hole solution always emerged. The metric components exhibited the expected behaviour near the black-hole and asymptotic infinity with the scalar invariant GB term being everywhere regular. All solutions possessed non-trivial scalar hair, with the scalar field having a non-trivial profile both close to and far away from the black-hole horizon. For small negative values of $\Lambda$, we recovered the power-law fall-off of the scalar field at infinity, found in the asymptotically-flat case [34] whereas for large negative values of $\Lambda$ the profile of $\phi$ was dominated by a logarithmic dependence on the radial coordinate. This behaviour was analytically shown to emerge both in the linear coupling-function case and in the perturbative limit, in terms of the coupling parameter $\alpha$, but it was numerically found to accurately describe all of our solutions at large distances.

The absence of a $(1/r)$-term in the expression of the scalar field at large distances excludes the presence of a scalar charge, even a secondary one. The coefficient $d_{1}$ in front of the logarithmic term in the expression of $\phi$ can give us information on how much the large-distance behaviour of the scalar field deviates from the power-law one valid in the asymptotically-flat case. We have found that this deviation is stronger for GB black holes with a small mass whereas the more massive ones have a $d_{1}$ coefficient that tends to zero. The temperature of the black holes was found to increase with the cosmological constant independently of the form of the coupling function. The latter plays a more important role in the relation of $T$ with the black-hole mass: while the temperature decreases with $M$ for all classes of solutions found, the lighter ones exhibit a stronger dependence on $f(\phi)$. The same dependence on the form of the coupling function is observed in the entropy and horizon area of our solutions. For small masses, the entropy of each class of solutions has a different behaviour, with the ones for the exponential, inverse-linear polynomial and logarithmic coupling functions exhibiting a ratio $S_{GB}/S_{SAdS}$ (over the entropy of the Schwarzschild-Anti-de Sitter black hole with the same mass) larger than unity for the entire mass range, for large values of $\Lambda$. This feature hints towards the enhanced thermodynamical stability of our solutions compared to their General Relativity (GR) analogues. In the limit of large mass, the entropy of all classes of our solutions tend to the one of the Schwarzschild-Anti-de Sitter black hole with the same mass. The same holds for the horizon area: while for small masses, each class has its own pattern with $M$, will all solutions being smaller in size than the corresponding SAdS one apart from the logarithmic case, for large masses all black-hole solutions match the horizon area of the SAdS solution.

Based on the above, we conclude that our GB black-hole solutions with a negative cosmological constant smoothly merge with the SAdS ones, in the large mass limit. As in the asymptotically-flat case, it is the small-mass range that provides the characteristic features for the GB solutions. These solutions have a modified dependence of both their temperature and horizon area on their mass compared to the SAdS solution. Another characteristic is also the minimum horizon, or minimum mass, that all our GB solutions possess due to the inequality (28).

Turning to GB solutions with a positive cosmological constant, our quest has failed to find any such solutions. Although the presence of a positive $\Lambda$ does not obstruct the formation of a regular black-hole or cosmological horizon, our numerical integration did not manage to produce a complete solution that would interpolate between the two asymptotic regimes. This result holds independently of the choice of the GB coupling function $f(\phi)$ or the value of $\Lambda$.

In conclusion, we have demonstrated that the general classes of theories that contain the GB term and lead to novel black-hole solutions, continue to do so even in the presence of a negative cosmological constant in the theory. In contrast, the presence of a positive cosmological constant presents a severe obstacle for the formation of these solutions. A further investigation is clearly necessary in both cases: the relevance of the GB solutions with an Anti-de Sitter-type asymptotic solution in the context of the AdS-CFT correspondence should be inquired, and the deeper reason for the absence of solutions with a positive cosmological constant should be investigated further. We hope to return soon with results on both issues.

Acknowledgments G.A. would like to thank Onassis Foundation for the financial support provided through its scholarship program. This research is co-financed by Greece and the European Union (European Social Fund- ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning” in the context of the project “Strengthening Human Resources Research Potential via Doctorate Research” (MIS-5000432), implemented by the State Scholarships Foundation (IKY).

Appendix A Set of Differential Equations

Here, we display the explicit expressions of the coefficients $P$, $S$ and $Q$ that appear in the system of differential equations (18)-(19) whose solution determines the metric function $A$ and the scalar field $\phi$. Note, that in these expressions we have eliminated, via Eq. (17), $B^{\prime}$, that involves $A^{\prime\prime}$ and $\phi^{\prime\prime}$, but retained $e^{B}$ for notational simplicity. They are:

[TABLE]

[TABLE]

and

[TABLE]

Appendix B Scalar Quantities

By employing the metric components of the line-element (7), one may compute the following scalar-invariant gravitational quantities:

[TABLE]

Appendix C Variation with respect to the Riemann tensor

Here, we derive the derivatives of the Lagrangian of the theory (1) with respect to the Riemann tensor. A simple way to do this is to take the derivatives ignoring the symmetries, that the final expression should possess, and restore them afterwards. For example, if $A_{abcd}$ is a 4-rank tensor and $A$ the corresponding scalar quantity, we may write:

[TABLE]

Now, if $A_{abcd}=R_{abcd}$, it should satisfy the following relations:

[TABLE]

Restoring the symmetries, we arrive at:

[TABLE]

Alternatively, we could have explicitly written:

[TABLE]

which clearly furnishes the same result.

We now proceed to the higher derivative terms. Let us start with the Kretchmann scalar for which we find

[TABLE]

The above result does not need any correction as it is already proportional to $R_{abcd}$, and satisfies all the desired identities. We now move to the $R_{\mu\nu}R^{\mu\nu}$ term, and employ again the simple method used above. Then:

[TABLE]

If $A_{abcd}=R_{abcd}$ and $A_{\mu\nu}=R_{\mu\nu}$, the above result will have all the right properties if it is rewritten as

[TABLE]

which is indeed the correct result. Finally, we easily derive that

[TABLE]

In order to compute the integral appearing in Eq. (61), we use the near-horizon solution (33)-(35) for the metric functions and scalar field. Then recalling that, near the horizon, the relations $A^{\prime\prime}\approx-A^{\prime 2}$ and $B^{\prime}\approx-A^{\prime}$ also hold, we find the results

[TABLE]

Substituting these into Eq. (61), we readily obtain the result (62).