Black hole solutions in shift-symmetric degenerate higher-order scalar-tensor theories

TL;DR

This paper investigates black hole solutions in shift-symmetric degenerate higher-order scalar-tensor theories, finding that cubic theories are disfavoured by gravitational wave constraints, while quadratic theories remain promising, and provides exact solutions within certain classes.

Contribution

It identifies which DHOST theories admit exact black hole solutions and clarifies their compatibility with degeneracy and observational constraints.

Findings

Cubic DHOST theories are strongly disfavoured by gravitational wave speed constraints.

Exact static black hole solutions are found in quadratic and some cubic theories.

Conditions for coupling functions to admit solutions are explicitly derived.

Abstract

We show that most classes of shift-symmetric degenerate higher-order scalar-tensor (DHOST) theories which satisfy certain degeneracy conditions are not compatible with the conditions for the existence of exact black hole solutions with a linearly time-dependent scalar field whose canonical kinetic term takes a constant value. Combined with constraints from the propagation speed of gravitational waves, our results imply that cubic DHOST theories are strongly disfavoured and that pure quadratic theories are likely to be the most viable class of DHOST theories. We find exact static and spherically symmetric (Schwarzschild and Schwarzschild-(anti-)de Sitter) black hole solutions in all shift-symmetric higher-derivative scalar-tensor theories which contain up to cubic order terms of the second-order derivatives of the scalar field, especially the full class of Horndeski and Gleyzes-Langlois-Piazza-Vernizzi theories. After deriving the conditions for the coupling functions in the DHOST Lagrangian that allow the exact solutions, we clarify their compatibility with the degeneracy conditions.

Tables

Table 1. Table 1: Compatibility of degeneracy conditions in the cubic DHOST theories with the conditions in Case 1(- Λ Λ \Lambda )-cubic and Case 2(- Λ Λ \Lambda )-cubic.

	Case 1-cubic	Case 2-cubic	Case 1-$\Lambda$-cubic	Case 2-$\Lambda$-cubic
3M-I	✓	$\times$	$\times$	$\times$
3M-II	$\times$	$\times$	$\times$	$\times$
3M-III	$\times$	$\times$	$\times$	$\times$
3M-IV	✓111The conditions of Case 1-cubic are compatible with Classes-3M-IV, 3M-VI, 3M-VII, and 3N-II only for $B_1=B_2=B_3=0$ because of the different ratios of $B_2$ and $B_3$ to $B_1$.	✓	✓	✓
3M-V	$\times$	$\times$	$\times$	$\times$
3M-VI	✓111The conditions of Case 1-cubic are compatible with Classes-3M-IV, 3M-VI, 3M-VII, and 3N-II only for $B_1=B_2=B_3=0$ because of the different ratios of $B_2$ and $B_3$ to $B_1$.	✓	✓	✓
3M-VII	✓111The conditions of Case 1-cubic are compatible with Classes-3M-IV, 3M-VI, 3M-VII, and 3N-II only for $B_1=B_2=B_3=0$ because of the different ratios of $B_2$ and $B_3$ to $B_1$.	✓	✓	✓
3N-I	$\times$	$\times$	$\times$	$\times$
3N-II	✓111The conditions of Case 1-cubic are compatible with Classes-3M-IV, 3M-VI, 3M-VII, and 3N-II only for $B_1=B_2=B_3=0$ because of the different ratios of $B_2$ and $B_3$ to $B_1$.	✓	✓	✓

Table 2. Table 2: Compatibility of degeneracy conditions in full DHOST theories with the conditions in Case 1(- Λ Λ \Lambda )-full and Case 2(- Λ Λ \Lambda )-full. The combinations denoted by “ − - ” were already excluded in terms of the incompatibility between degeneracy conditions, while the combinations denoted by “ × \times ” are ruled out in terms of the incompatibility of degeneracy conditions with the conditions for the existence of Schwarzschild and Schwarzschild- (anti-) de Sitter BH solutions.

Class	3M-I	3M-II	3M-III	3M-IV	3M-V	3M-VI	3M-VII	3N-I	3N-II
2M-I	Case-1-full only	$\times$	$\times$	$-$	$\times$	$-$	$-$	$-$	$-$
2M-II111Class 2M-II has the additional condition $A_1=A_2=A_3=0$ and 2M-III has the condition $A_1=A_2=0$ for all solutions.	$-$	$-$	$\times$	✓	$-$	$-$	✓	$-$	$-$
2M-III111Class 2M-II has the additional condition $A_1=A_2=A_3=0$ and 2M-III has the condition $A_1=A_2=0$ for all solutions.	$-$	$-$	$-$	$-$	$\times$	✓	✓	$-$	$-$
2N-I	Case 1-full only	$\times$	$\times$	$-$	$\times$	$-$	$-$	$\times$	$-$
2N-II	$-$	$-$	$-$	$-$	$-$	$-$	✓	$-$	✓
2N-III	$\times$	$\times$	$\times$	$-$	$\times$	$-$	$-$	$-$	$-$
2N-IV	$\times$	$\times$	$\times$	$-$	$\times$	$-$	$-$	$-$	$-$

Equations

$$\displaystyle S=\int d^{4}x\sqrt{-g}\bigg{[}F_{0}(\phi,X)+F_{1}(\phi,X)\Box\phi+F_{2}(\phi,X)R+F_{3}(\phi,X)G_{\mu\nu}\phi^{\mu\nu}+\sum_{I=1}^{5}A_{I}(\phi,X)L_{I}^{(2)}+\sum_{J=1}^{10}B_{J}(\phi,X)L_{J}^{(3)}\bigg{]},~{}~{}~{}~{}~{}$$

$$L_1^{(2)}={\varphi }_{\mu \nu }{\varphi }^{\mu \nu }{L}_2^{(2)}={\left(\square \varphi \right)}^2,L_3^{(2)}=\square \varphi {\varphi }^{\mu }{\varphi }_{\mu \nu }{\varphi }^{\nu },L_4^{(2)}={\varphi }^{\mu }{\varphi }_{\mu \rho }{\varphi }^{\rho \nu }{\varphi }_{\nu },L_5^{(2)}={\left({\varphi }_{\mu }{\varphi }^{\mu \nu }{\varphi }_{\nu }\right)}^2,$$

$$L_1^{(3)}={\left(\square \varphi \right)}^3{L}_2^{(3)}=\square \varphi {\varphi }_{\mu \nu }{\varphi }^{\mu \nu },L_3^{(3)}={\varphi }_{\mu \nu }{\varphi }^{\nu \rho }{\varphi }^{\mu }{}_{\rho },L_4^{(3)}=(\square \varphi {)}^2{\varphi }_{\mu }{\varphi }^{\mu \nu }{\varphi }_{\nu },L_5^{(3)}=\square \varphi {\varphi }_{\mu }{\varphi }^{\mu \nu }{\varphi }_{\nu \rho }{\varphi }^{\rho },$$

$$L_6^{(3)}={\varphi }_{\mu \nu }{\varphi }^{\mu \nu }{\varphi }_{\rho }{\varphi }^{\rho \sigma }{\varphi }_{\sigma }{L}_7^{(3)}={\varphi }_{\mu }{\varphi }^{\mu \nu }{\varphi }_{\nu \rho }{\varphi }^{\rho \sigma }{\varphi }_{\sigma },$$

$$L_8^{(3)}={\varphi }_{\mu }{\varphi }^{\mu \nu }{\varphi }_{\nu \rho }{\varphi }^{\rho }{\varphi }_{\sigma }{\varphi }^{\sigma \lambda }{\varphi }_{\lambda },L_9^{(3)}=\square \varphi ({\varphi }_{\rho }{\varphi }^{\rho \sigma }{\varphi }_{\sigma }{)}^2,L_{10}^{(3)}=({\varphi }_{\rho }{\varphi }^{\rho \sigma }{\varphi }_{\sigma }{)}^3,$$

$$F_i=F_i(X)(i=0,1,2,3),A_I=A_I(X)(I=1,\cdots ,5),B_J=B_J(X)(J=1,\cdots ,10).$$

$$X=X_0=\text{const.}$$

$$d{s}^2=-A(r)d{t}^2+\frac{d{r}^2}{B(r)}+2C(r)dtdr+D(r)r^2(d{\theta }^2+{\cos }^2\theta d{\phi }^2),$$

$$\varphi (t,r)=qt+\psi (r).$$

$$X(r)=-\frac{q^2}{A}+B{\psi }^{\prime 2},$$

$${\psi }^{\prime }=\pm \sqrt{\frac{q^2+X_{0}A}{AB}},$$

$$F_2(X)=\frac{M_p^2}{2},A_I(X)=0(I=1,\cdots ,5).$$

$$A=B=1-\frac{r_g}{r},$$

$$F_0=F_{0X}=F_{1X}=0,$$

$$B_{3X}=\frac{3}{10X_0}\left[-2B_2+B_3-3X_0\left(B_4+B_6+2(B_{1X}+B_{2X})\right)\right],B_3=-18B_1,B_2=9B_1,$$

$$F_0=F_{0X}=F_{1X}=0,$$

$$B_1=B_2=B_3=B_{3X}=B_6=0,2B_{1X}+B_{2X}=0,B_4+B_{2X}=0.$$

$$F_i(X)=\sum _{j=0}^{\infty }{f}_{ij}(X-X_0{)}^j,B_J(X)=\sum _{j=0}^{\infty }{b}_{Jj}(X-X_0{)}^j,$$

$$A=B=1-\frac{r_g}{r}-\frac{\Lambda }{3}{r}^2,$$

$$F_0=-\Lambda M_p^2,F_{0X}=0,F_{1X}-\Lambda \left(-X_0{B}_4+4X_0{B}_{1X}+2X_0{B}_{2X}+F_{3X}\right)=0,$$

$$B_1=B_2=B_3=B_{3X}=0,B_4+B_6+2(B_{1X}+B_{2X})=0,$$

$$F_0=-\Lambda M_p^2,F_{0X}=0,F_{1X}-\Lambda (-2X_0{B}_{1X}+F_{3X})=0,$$

$$B_1=B_2=B_3=B_{3X}=B_6=0,B_{2X}+2B_{1X}=0,B_4-2B_{1X}=0,$$

$$F_0=F_{0X}=F_{1X}=0,$$

$$A_1+A_2=0,A_{1X}+A_{2X}=0,$$

$$B_{3X}=\frac{3}{10X_0}\left[-2B_2+B_3-3X_0\left(B_4+B_6+2(B_{1X}+B_{2X})\right)\right],B_3=-18B_1,B_2=9B_1,$$

$$F_0=F_{0X}=F_{1X}=0,$$

$$A_1=A_2=0,A_{1X}+A_{2X}=0,A_3+2A_{1X}=0,$$

$$B_1=B_2=B_3=B_{3X}=B_6=0,2B_{1X}+B_{2X}=0,B_4+B_{2X}=0.$$

$$F_i(X)=\sum _{j=0}^{\infty }{f}_{ij}(X-X_0{)}^j,A_I(X)=\sum _{j=0}^{\infty }{a}_{Ij}(X-X_0{)}^j,B_J(X)=\sum _{j=0}^{\infty }{b}_{Jj}(X-X_0{)}^j,$$

$$F_0+2\Lambda F_2=2\Lambda X_0{A}_1,F_{0X}+4\Lambda F_{2X}=\frac{\Lambda }{2}(2A_1-3X_0{A}_3-4X_0{A}_{1X}),$$

$$F_{1X}-\Lambda \left(-X_0{B}_4+4X_0{B}_{1X}+2X_0{B}_{2X}+F_{3X}\right)=0,$$

$$A_1+A_2=0,A_{1X}+A_{2X}=0,$$

$$B_1=B_2=B_3=B_{3X}=0,B_4+B_6+2(B_{1X}+B_{2X})=0,$$

$$F_0+2\Lambda F_2=0,F_{0X}+\Lambda \left(4F_{2X}-X_0{A}_{1X}\right)=0,F_{1X}-\Lambda (-2X_0{B}_{1X}+F_{3X})=0,$$

$$A_1=A_2=0,A_{1X}+A_{2X}=0,A_3+2A_{1X}=0,$$

$$B_1=B_2=B_3=B_{3X}=B_6=0,B_{2X}+2B_{1X}=0,B_4-2B_{1X}=0,$$

$$F_0=G_2,F_1=G_3,F_2=G_4,F_3=G_5,$$

Full text

I Introduction

Degenerate Higher-Order Scalar-Tensor (DHOST) theories Langlois and Noui (2016); Ben Achour et al. (2016a, b); Crisostomi and Koyama (2018); Langlois et al. (2018) provide a general framework of single-field scalar-tensor theories which contain higher order terms of second-order derivatives of the scalar field. They allow higher-order Euler-Lagrange equations but possess three degrees of freedom, i.e. two metric and one scalar degrees of freedom, without Ostrogradsky ghosts Woodard (2015), as the system of equations reduces to that with the second order derivatives after a suitable field redefinition. DHOST theories represent the extension of similar analyses in the context of analytical mechanics Motohashi et al. (2016a, 2018a, 2018b) (See Refs. Langlois (2019); Kobayashi (2019) for reviews).

The direct detection of gravitational waves from binary black hole (BH) and neutron star (NS) mergers Abbott et al. (2017a, b, c) means that the application of higher-derivative scalar-tensor theories to BH solutions is of great interest. Depending on the profile of the scalar field, these solutions can be classified into several classes. Besides BH solutions in general relativity (GR) with a constant scalar field, higher-derivative scalar-tensor theories also allow BH solutions which are absent in GR. Stealth Schwarzschild solutions were found in Babichev and Charmousis (2014); Kobayashi and Tanahashi (2014) for the shift-symmetric (i.e. symmetric under $\phi\to\phi+c$ where $c$ is a constant) subclasses of the Horndeski theory Horndeski (1974), which are also subclasses of the DHOST theories, for the linearly time-dependent scalar field profile $\phi(t,r)=q\hskip 1.13809ptt+\psi(r)$, with the assumption of a constant kinetic term $X:=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$. Here, “stealth” means that the metric is independent of the scalar field sector of the Lagrangian and the existence of the scalar field is hidden in the spacetime geometry. Stealth BH solutions have also been studied in several different contexts Ayon-Beato et al. (2006); Mukohyama (2005); Bravo Gaete and Hassaïne (2013); Chagoya et al. (2016).

BH solutions with metrics different from those in GR were also found in the shift-symmetric theories with the scalar profile with nonconstant $X$. An asymptotically flat solution with the static scalar field $\phi=\phi(r)$ was also constructed in the Einstein-scalar-Gauss-Bonnet theory with linear coupling, which is a subclass of the shift-symmetric Horndeski theories Sotiriou and Zhou (2014a, b). The key for the existence of this solution is the violation of one of the assumptions in the no-hair theorem for the shift-symmetric theories Hui and Nicolis (2013), that the coupling functions and their derivatives are nonsingular as $X\to 0$. Ref. Babichev et al. (2017) presented the extension of the no-hair theorem to the shift-symmetric Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories (also known as beyond-Horndeski theories) Gleyzes et al. (2015) and asymptotically flat hairy BH solutions that circumvent the assumptions of the theorem. In contrast to Babichev and Charmousis (2014); Kobayashi and Tanahashi (2014), hairy BH solutions obtained in Refs. Sotiriou and Zhou (2014a, b); Babichev et al. (2017) possess the profiles of the scalar field with nonconstant $X$.

Another interesting solution is the self-tuned Schwarzschild-de Sitter solution, in which the metric contains an effective cosmological constant independent of the bare cosmological constant in the action, and hence the spacetime metric may not be affected by a sudden change of the bare cosmological constant via a phase transition Charmousis et al. (2012a, b); Martin-Moruno et al. (2015). The self-tuned Schwarzschild-de Sitter solutions were found in the shift-symmetric Horndeski Babichev and Charmousis (2014); Kobayashi and Tanahashi (2014) and GLPV Babichev and Esposito-Farese (2017); Babichev et al. (2018) theories.

In the context of quadratic DHOST theories, the conditions for the existence of the Schwarzschild and Schwarzschild (anti)-de Sitter solutions under the assumptions of a linearly time-dependent profile of the scalar field $\phi(t,r)=q\hskip 1.42262ptt+\psi(r)$ and a constant value of $X=X_{0}$ were obtained in Refs. Motohashi and Minamitsuji (2019); Ben Achour and Liu (2019). Odd- and even-mode perturbations about the static and spherically BH solutions in pure quadratic DHOST theories were analyzed in Refs. Takahashi et al. (2019); de Rham and Zhang (2019). The analysis of static BH solutions in quadratic DHOST theories was recently extended to Kerr and Kerr- (anti-) de Sitter solutions Charmousis et al. (2019). In this paper, we clarify the conditions for the existence of the Schwarzschild and Schwarzschild-(anti-) de Sitter BH solutions in more general classes of the DHOST theories including cubic DHOST theories. Our analysis follows that for the shift-symmetric quadratic DHOST theories in Ref. Motohashi and Minamitsuji (2019); Ben Achour and Liu (2019); Takahashi et al. (2019). We also present the detailed derivation process of these conditions, and give explicit models and solutions.

As shown in Ref. Langlois et al. (2018) cubic DHOST theories do not appear to satisfy the recent constraint on the propagation speed of gravitational waves $c_{g}$, obtained from the coincident measurements of gravitational waves emitted from a binary NS merger and its associated short gamma-ray burst Abbott et al. (2017c); Creminelli and Vernizzi (2017); Ezquiaga and Zumalac«¡rregui (2017); Baker et al. (2017); Sakstein and Jain (2017), indicating that the difference between $c_{g}$ and the speed of light is at most ${\cal O}(10^{-15})$. However, there are two reasons why the above constraints do not directly apply to our problem and therefore it is worthwhile to investigate exact BH solutions in cubic DHOST theories. The first is that the constraint is based on the assumption that the scalar field has cosmological origins and therefore is distributed over cosmological scales. However, the scalar field may be localized only in the vicinity of BHs or NSs, or may not have a cosmological origin. The second argument is that the frequency band of LIGO $\sim 100{\rm Hz}$ is very close to the cut-off scale of the effective field theory of dark energy de Rham and Melville (2018), and hence the constraints from LIGO may not be directly applicable to the Horndeski or DHOST theories if they provide the effective field theory description for dark energy.

The paper is constructed as follows: In Sec. II, we review the DHOST theories. In Sec. III and Sec. IV, we derive the conditions for the existence of the Schwarzschild and Schwarzschild-(anti-)de Sitter solutions in the pure cubic and full DHOST theories. We check the compatibility of the above conditions with the degeneracy conditions. In Sec. V, we discuss the limits to special subclasses known as Horndeski theory Horndeski (1974) and GLPV theory Gleyzes et al. (2015).

II Set-up

II.1 DHOST theories

We consider the most general theories which are composed of terms up to cubic order of the second order covariant derivatives of the scalar field

[TABLE]

where $g_{\mu\nu}$ is the metric; $g:={\rm det}(g_{\mu\nu})$; $R$ is the Ricci scalar associated with $g_{\mu\nu}$; $\phi$ is the scalar field; $\phi_{\mu}:=\nabla_{\mu}\phi$; $X:=\phi_{\mu}\phi^{\mu}$, the quadratic terms are

[TABLE]

and the cubic terms are

[TABLE]

with $\phi_{\mu\nu}:=\nabla_{\nu}\nabla_{\mu}\phi$; $\Box\phi:=g^{\mu\nu}\phi_{\mu\nu}$; $F_{i}(\phi,X)$ ($i=0,1,2,3$), $A_{I}(\phi,X)$ ($I=1,\cdots,5$), and $B_{J}(\phi,X)$ ($J=1,\cdots,10$) are functions of $\phi$ and $X$.

In order to eliminate the Ostrogradsky ghosts, degeneracy conditions are imposed, which depend on the class of DHOST  Langlois and Noui (2016); Ben Achour et al. (2016a, b); Crisostomi and Koyama (2018); Langlois et al. (2018). Following Ref. Ben Achour et al. (2016b), we call degeneracy conditions for the pure quadratic DHOST theories ($F_{3}=B_{1}=B_{2}=\cdots=B_{10}=0$) “Class-2M” and “Class-2N” for the cases of $F_{2}=F_{2}(\phi)$ and $F_{2}=F_{2}(\phi,X)$, respectively111Here we take “minimally coupled” to mean $F_{2}(X,\phi)=F_{2}(\phi)$, because this coupling is free from Ostrogradsky ghosts and does not contribute to the degeneracy conditions, while in Ref. Ben Achour et al. (2016b) the authors define “minimally coupled” as $F_{2}=0$.. Similarly, we call degeneracy conditions for the pure cubic DHOST theories ($F_{2}=F_{2}(\phi)$ and $A_{1}=A_{2}=\cdots=A_{5}=0$) “Class-3M” and “Class-3N” for the cases of $F_{3}=0$ and $F_{3}\neq 0$, respectively. Degeneracy conditions for the pure quadratic, pure cubic, and full DHOST theories are reviewed in Appendices A-C. We note that the terms $F_{0}$ and $F_{1}$ are not relevant for the degeneracy conditions.

We also note that the pure quartic GLPV ($G_{4}=G_{4}(\phi,X)$ and $f_{4}=f_{4}(\phi,X)$) and the pure quartic beyond-Horndeski ($G_{4}=G_{4}(\phi)$ and $f_{4}=f_{4}(\phi,X)$) theories belong to Class-2N-I and Class-2M-I, and the pure quintic GLPV ($G_{5}=G_{5}(\phi,X)$ and $f_{5}=f_{5}(\phi,X)$) and the pure quintic beyond-Horndeski ($G_{5}=0$ and $f_{5}=f_{5}(\phi,X)$) theories belong to Class-3N-I and Class-3M-I, respectively, where we will use Eqs. (2.1)-(2.3) and Eqs. (2.6)-(2.7) in Ref. Langlois (2019) for the conventions of the Horndeski and beyond-Horndeski theories 222 Here, “pure beyond-Horndeski” means that within the quartic or quintic order GLPV theory, only the function $f_{4}$ or $f_{5}$ is a nontrivial function of $X$, while $G_{4}=G_{4}(\phi)$ or $G_{5}=0$ respectively. . In full DHOST theories in which all the terms in Eq. (1) are nontrivial, not all the combinations of the pure quadratic and cubic DHOST theories are degenerate, and even if they can be degenerate some additional conditions may have to be imposed Ben Achour et al. (2016b).

In the rest of the paper, we study the shift-symmetric subclass of DHOST, where

[TABLE]

The shift symmetry simplifies the equations of motion dramatically. With the shift symmetry, the value of the scalar field itself has no physical meaning, and one has to integrate the scalar field equations of motion only once. We also impose the ansatz that the canonical kinetic term is constant

[TABLE]

II.2 Static and spherically symmetric solutions

As in Refs. Babichev and Charmousis (2014); Kobayashi et al. (2014); Motohashi and Minamitsuji (2019); Takahashi et al. (2019), we focus on a static spherically symmetric spacetime

[TABLE]

and a linearly time-dependent scalar field

[TABLE]

Our choice of linear time-dependence is motivated by the fact that this gives equations of motion which are time-independent (whereas time-dependent equations of motion would be at odds with the static BHs we are investigating). On the other hand, choosing time-independence (i.e. setting $q=0$) would mean that only a trivial solution ($\phi=$ constant) is allowed because of the no-hair theorem Hui and Nicolis (2013); Babichev et al. (2017).

According to the gauge fixing theorem in Ref. Motohashi et al. (2016b), we can set $D(r)=1$ at the level of the action without losing any independent equation of motion, while choosing $C(r)=0$ at the level of the action loses an independent equation for $C$. So that we maintain an independent equation, the action must be varied before imposing $C(r)=0$. 333This is different to the static scalar field in a static spherically symmetric spacetime, where it is possible to choose both $C(r)=0$ and $D(r)=1$ at the level of the action Motohashi et al. (2016b). We derive the equations of motion by varying the action, and afterwards impose the gauge conditions $C(r)=0$ and $D(r)=1$. Once we have fixed the gauge, the kinetic term reads

[TABLE]

where a prime denotes a derivative with respect to $r$. Using Eq. (8), one can solve Eq. (5) for $\psi^{\prime}$ as

[TABLE]

and with this relation higher derivatives of $\psi$ can be erased in terms of $q$, $X_{0}$, $A(r)$, $B(r)$, and derivatives of $A(r)$, $B(r)$. By taking the positive branch of Eq. (9) and using ingoing Eddington$-$Finkelstein coordinates $(v,r)$ defined by $dv=dt+dr/\sqrt{AB}$, we obtain $\phi\simeq q\hskip 0.85358ptv$ in the vicinity of the future event horizon. Thus, the scalar field is regular at the future event horizon Babichev and Charmousis (2014); Kobayashi and Tanahashi (2014). If we take the negative branch of Eq. (9), we obtain the scalar field solution regular at the past event horizon. From now on, we focus on the BH solutions and take the positive branch of Eq. (9). We note that when choosing shift symmetry using Eq. (4) and a constant $X$ using Eq. (5), the couplings and their derivatives in the equations of motion evaluated at $X=X_{0}$ reduce to numbers rather than functions, which makes solving the equations of motion significantly easier.

III The pure cubic DHOST theories

In this section, we derive the conditions that guarantee the existence of Schwarzschild or Schwarzschild-(anti-)de Sitter solutions in the pure cubic theories, defined by

[TABLE]

where $M_{p}$ is the reduced Planck mass, such that the $F_{2}$ term reduces to the Einstein-Hilbert term. We also provide the general form of the coupling functions which satisfy the conditions.

III.1 Schwarzschild solution

First, let us focus on the Schwarzschild solution

[TABLE]

where $r_{g}$ is a constant which corresponds to the location of the event horizon.

We will clarify the conditions that the metric Eq. (11) and the scalar field Eq. (7) with Eq. (9) are the solutions of the scalar-tensor theories Eq. (1) with Eq. (10). We substitute Eqs. (11) and (7) with Eq. (9) into the equations of motion to produce a Taylor series in $r-r_{g}$. We impose that the coefficient which depends on the coupling functions and their derivatives evaluated at $X=X_{0}$ vanishes at each order of $r-r_{g}$, and repeat the same procedure until the sufficient number of the conditions are obtained Babichev and Esposito-Farese (2017); Ben Achour and Liu (2019); Motohashi and Minamitsuji (2019). As one goes to higher order in $r-r_{g}$, the series expansion covers a larger domain of $r>r_{g}$. After the consistent conditions are obtained, all the higher order terms of $r-r_{g}$ trivially vanish and the entire domain $0<r<\infty$ can be covered. We note that our conditions do not depend on the reference position of the Taylor expansion of the equations of motion, and the same conditions are obtained if we expand the same equations with respect to the other radius, for instance, $r=\infty$. As in the case of the pure quadratic DHOST theories, the conditions that the solutions exist are classified into the following two cases: one is satisfied for $Q:=X_{0}+q^{2}=0$, and the other is for $Q\neq 0$.

For $Q:=X_{0}+q^{2}=0$ (Case 1-cubic), the conditions for the existence of the stealth Schwarzschild solutions are

[TABLE]

which are evaluated at $X=X_{0}=-q^{2}$.

For $Q\neq 0$ (Case 2-cubic):

[TABLE]

which are evaluated at $X=X_{0}$. We note that functions which do not appear in these conditions are not relevant for the existence of the solutions. As we will see in Sec. III.2, the constant $F_{0}(X_{0})$ serves as the effective cosmological constant, and should vanish for an asymptotically flat solution.

The general form of the couplings satisfying Eqs. (III.1) and (III.1) is given by the Taylor series:

[TABLE]

where $i=0,1,3$ ($F_{2}=M_{p}^{2}/2$) and $J=1,2,3,\cdots,10$. The relations between the coefficients are given by Eqs. (D.1) and (74).

III.2 Schwarzschild-(anti-)de Sitter solution

Second, let us focus on the Schwarzschild- (anti-) de Sitter solution

[TABLE]

where $r_{g}$ is a constant which corresponds to the mass of the Schwarzschild- (anti-) de Sitter spacetime and $\Lambda$ represents the effective cosmological constant. We substitute Eqs. (15) and (7) with Eq. (9) into the equations of motion to produce an inverse power series of $r$ with respect to $r=\infty$. We impose that the coefficient which depends on the coupling functions and their derivatives evaluated at $X=X_{0}$ vanishes at each order of $1/r$, and repeat the same procedure until the sufficient number of the conditions are obtained. The conditions can be classified into two cases: One is satisfied for $Q:=X_{0}+q^{2}=0$, and the other is for $Q\neq 0$.

For $Q=0$ (Case 1-$\Lambda$-cubic):

[TABLE]

where the functions are evaluated at $X=X_{0}=-q^{2}$.

For $Q\neq 0$ (Case 2-$\Lambda$-cubic):

[TABLE]

where the functions are evaluated at $X=X_{0}$. We note that the functions which do not appear in these conditions are not relevant for the existence of the solutions. The general form of the couplings satisfying Eqs. (III.2) and (III.2) is given by the Taylor series Eq. (14), with the relations between coefficients given by (D.1) and (D.1).

Finally, we should note that for both the cases of the Schwarzschild and Schwarzschild- (anti-) de Sitter cases, there would also be solutions with solid deficit angles with $B=\alpha\left(1-r_{g}/r-\Lambda r^{2}/3\right)$ ($\alpha\neq 1$) as in the pure quadratic DHOST theories Motohashi and Minamitsuji (2019); Takahashi et al. (2019). These solutions are very similar to the global monopole solutions obtained in Barriola and Vilenkin (1989). Since in contrast to the case of the global monopole, solutions with deficit angles are not the unique solutions in our theory and not physically important in comparison with the exact Schwarzschild or Schwarzschild-(anti-) de Sitter solutions. Thus in this paper we will not consider these solutions.

III.3 Compatibility with the degeneracy conditions

Now, we impose degeneracy conditions of pure cubic DHOST theories shown in Appendix B, and check their compatibility with the conditions (III.1), (III.1), (III.2), and (III.2). In Table 1, we show which class of pure cubic DHOST theories defined in Appendix. B are compatible with the different BH solutions, recalling that Case 1(-$\Lambda$)-cubic and 2(-$\Lambda$)-cubic refer to $X_{0}=-q^{2}$ and $X_{0}\neq-q^{2}$ respectively:

Thus the conditions of the stealth Schwarzschild solutions are compatible with Classes-3M-I, 3M-IV, 3M-VI, 3M-VII, and 3N-II; the conditions of the Schwarzschild- (anti-) de Sitter solutions are compatible with Classes-3M-IV, 3M-VI, 3M-VII, and 3N-II, respectively.

IV The full DHOST theories

In this section, we derive the conditions that guarantee the existence of Schwarzschild or Schwarzschild- (anti-)de Sitter solutions in the full DHOST theories, namely, in the presence of both quadratic and cubic DHOST couplings $F_{i}(X)\neq 0$, $A_{I}(X)\neq 0$, and $B_{J}(X)\neq 0$, where $i=0,1,2,3;I=1,\cdots,5;\,J=1,\cdots,10$, and provide the general form of the coupling functions which satisfy these conditions. We find that there is no mixing between quadratic and cubic terms, i.e. there are no conditions containing both $A_{I}$s and $B_{J}$s.

IV.1 Schwarzschild solution

First, let us focus on the Schwarzschild solution given by the metric Eq. (11). The procedure for obtaining the solution is the same as that in Sec. III.1, i.e. we substitute the metric Eq. (11) and the scalar field Eq. (7) with Eq. (9) into the equations of motion and find the conditions such that the coefficients of the Taylor series of the equations of motion vanish. Here, we only focus on the result.

The conditions that the solutions exist are as follows:

For $Q=X_{0}+q^{2}=0$ (Case 1-full):

[TABLE]

which are evaluated at $X=X_{0}=-q^{2}$. We note that the limit to pure quadratic DHOST theories ($F_{3}=0$ and $B_{1}=B_{2}=\cdots=B_{10}=0$) agrees with Case 1 obtained in Ref. Motohashi and Minamitsuji (2019).

For $Q\neq 0$ (Case 2-full), the conditions for the existence of the stealth Schwarzschild solutions are given by

[TABLE]

which are evaluated at $X=X_{0}$. We note that the limit to pure quadratic DHOST theories ($F_{3}=0$ and $B_{1}=B_{2}=\cdots=B_{10}=0$) agrees with Case 2 obtained in Ref. Motohashi and Minamitsuji (2019). Thus, for the existence of the Schwarzschild solution, the conditions for the pure quadratic and pure cubic DHOST theories can be independently satisfied, and no extra conditions with the mixture of $A_{I}$s and $B_{J}$s, and their derivatives, are imposed.

The general form of the couplings satisfying Eqs. (IV.1) and (IV.1) is given by the Taylor series

[TABLE]

where $i=0,1,2,3$, $I=1,2,\cdots,5$, and $J=1,2,3,\cdots,10$, with the relations between coefficients given in (D.2) and (D.2), respectively.

IV.2 Schwarzschild- (anti-) de Sitter solution

Second, let us focus on the Schwarzschild-(anti-) de Sitter solution given by the metric Eq. (15), i.e. we substitute the metric and the scalar field Eq. (7) with Eq. (9) into the equations of motion and find the conditions such that the coefficients of the Taylor series of the equations of motion vanish. The procedure for obtaining the solution is the same as that in Sec. III.2. Here, we only focus on the result.

The conditions that we get the solutions are as follows:

For $Q:=X_{0}+q^{2}=0$ (Case 1-$\Lambda$-full):

[TABLE]

where the functions are evaluated at $X=X_{0}=-q^{2}$. We note that the limit to pure quadratic DHOST theories ($F_{3}=0$ and $B_{1}=B_{2}=\cdots=B_{10}=0$) agrees with Case 1-$\Lambda$ obtained in Ref. Motohashi and Minamitsuji (2019).

For $Q\neq 0$ (Case 2-$\Lambda$-full):

[TABLE]

where the functions are evaluated at $X=X_{0}$. We note that the limit to pure quadratic DHOST theories ($F_{3}=0$ and $B_{1}=B_{2}=\cdots=B_{10}=0$) agrees with Case 2-$\Lambda$ obtained in Ref. Motohashi and Minamitsuji (2019).

Thus, for the existence of the Schwarzschild- (anti-) de Sitter solution, the conditions for the pure quadratic and pure cubic DHOST theories have to be independently satisfied, and no extra conditions with the mixture of $A_{I}$s and $B_{J}$s, and their derivatives, are imposed. The general form of the couplings satisfying Eqs. (IV.2) and (IV.2) is given by the Taylor series Eq. (20), with the relations between coefficients given by (D.2) and (D.2).

As we noted in the case of the pure cubic DHOST theories, for both the cases of the Schwarzschild and Schwarzschild-(anti-) de Sitter cases, there would also be solutions with solid deficit angles. Again, these solutions are not physically important in comparison with the exact Schwarzschild or Schwarzschild-(anti-) de Sitter solutions, and therefore we focus on these solutions.

IV.3 Compatibility with the degeneracy conditions

We then impose degeneracy conditions of the full DHOST theories shown in Appendix C, and check the compatibility of the conditions (IV.1), (IV.1), (IV.2), and (IV.2) with them. Combining all conditions, we find that only a few combinations of quadratic and cubic theories allow BH solutions. Our results are summarized in Table 2, which shows the combinations of theories which allow both Schwarzschild and Schwarzschild-(anti-) de Sitter solutions. 444The combination of Class-3M-I with Class-2M-I and that of Class-3M-I with Class-2N-I permit Schwarzschild solutions as long as $X_{0}=-q^{2}$, but not de Sitter-Schwarzschild solutions. In other words they only permit Case-I-full solutions. We note that the additional condition $A_{1}=A_{2}=0$ is required for both of these combinations when looking for Case 2-full and Case 2-$\Lambda$-full theories.

Since the combination of the pure quartic order Horndeski theory ($G_{4}\neq 0$ while $f_{4}=0$) among Class-2N DHOST and the pure quintic order beyond-Horndeski theory ($f_{5}\neq 0$ while $G_{5}=0$) among Class-3M DHOST cannot be degenerate, Case 1-full solutions do not exist in this limit (See Sec. V).

V GLPV and Horndeski theories

Particular choices of the quadratic and cubic DHOST theories correspond to the Horndeski and GLPV theories.

The Horndeski theories corresponds to the choice

[TABLE]

Similarly, the GLPV theories correspond to the choice

[TABLE]

The case of (V) with $f_{4}=0$ and $f_{5}=0$ corresponds to Horndeski theories. The pure quartic order GLPV theory ($G_{4}\neq 0$ and $f_{4}\neq 0$ while $G_{5}=f_{5}=0$) belongs to Class-2N-I DHOST, and the pure quintic order GLPV theory ($G_{5}\neq 0$ and $f_{5}\neq 0$ while $G_{4}=M_{p}^{2}/2$ and $f_{4}=0$) belongs to Class- 3N-I DHOST Ben Achour et al. (2016b). We note that the combination of the quartic order Horndeski term ($G_{4}$) and the pure quintic order beyond-Horndeski term ($f_{5}$), and that of the pure quintic order beyond-Horndeski term ($f_{5}$) and the quartic order Horndeski term ($G_{4}$), are both nondegenerate Langlois and Noui (2016); Crisostomi et al. (2016a). On the other hand, the combination of the quartic order Horndeski ($G_{4}$) and pure beyond-Horndeski ($f_{4}$) terms, and that of the quintic order Horndeski term ($G_{5}$) and pure beyond-Horndeski ($f_{5}$) terms are degenerate Langlois and Noui (2016); Crisostomi et al. (2016a).

V.1 The pure cubic DHOST theories

V.1.1 GLPV theories

The GLPV limit of the pure cubic DHOST theories corresponds to the choice of $G_{4}(X)=M_{p}^{2}/2$ and $f_{4}(X)=0$. In the GLPV limit, the conditions in Case 1-cubic reduce to

[TABLE]

evaluated at $X=X_{0}=-q^{2}$. The conditions in Case 2-cubic reduce to

[TABLE]

evaluated at $X=X_{0}$.

The conditions in Case 1-$\Lambda$-cubic reduce to

[TABLE]

evaluated at $X=X_{0}=-q^{2}$. The conditions in Case 2-$\Lambda$-cubic reduce to

[TABLE]

evaluated at $X=X_{0}$.

V.1.2 Horndeski limit

In the case of the Horndeski theories, $f_{5}(X)=0$, the conditions for Case 1-cubic and Case 2-cubic further reduce to

[TABLE]

evaluated at $X=X_{0}=-q^{2}$, and $X=X_{0}\neq-q^{2}$, respectively. In the Horndeski limit, the conditions for Case 1$-\Lambda$-cubic and Case 2$-\Lambda$-cubic further reduce to

[TABLE]

evaluated at $X=X_{0}=-q^{2}$ and $X=X_{0}\neq-q^{2}$, respectively.

V.2 The full DHOST theories

V.2.1 GLPV theories

In the GLPV limit, the conditions in Case 1-full reduce to

[TABLE]

evaluated at $X=X_{0}=-q^{2}$. The conditions in Case 2-full reduce to

[TABLE]

evaluated at $X=X_{0}$.

The conditions in Case 1-$\Lambda$-full reduce to

[TABLE]

evaluated at $X=X_{0}=-q^{2}$. The conditions in Case 2-$\Lambda$-full reduce to

[TABLE]

evaluated at $X=X_{0}$. These results agree with those in Ref. Motohashi and Minamitsuji (2019) which takes $G_{5}(X)=f_{5}(X)=0$, and the particular subclass of the quartic order GLPV theories which provides the self-tuning mechanism of the cosmological constant Babichev and Esposito-Farese (2017); Babichev et al. (2018).

In the case that both the quartic- and quintic- order beyond-Horndeski interactions ($f_{4}(X)\neq 0$ and $f_{5}(X)\neq 0$) are present, the theory can be degenerate only if $G_{4}(X)=G_{5}(X)=0$ Langlois and Noui (2016); Crisostomi et al. (2016a). Then, the conditions in Case 1-full and Case 2-full further reduce to

[TABLE]

evaluated at $X=X_{0}=-q^{2}$, and

[TABLE]

evaluated at $X=X_{0}\neq-q^{2}$, respectively. The conditions in Case 1-$\Lambda$-full and Case 2-$\Lambda$-full further reduce to

[TABLE]

evaluated at $X=X_{0}=-q^{2}$, and

[TABLE]

evaluated at $X=X_{0}\neq-q^{2}$, respectively. All these conditions are not compatible with the assumption that $f_{4}(X)\neq 0$ and $f_{5}(X)\neq 0$. Thus, the existence of the Schwarzschild and Schwarzschild-(anti-) de Sitter solutions cannot be compatible with degeneracy of GLPV theories with both the quartic and quintic beyond-Horndeski interactions.

V.2.2 Horndeski limit

In the case of the Horndeski theories, $f_{4}(X)=f_{5}(X)=0$, the conditions for Case 1-full and Case 2-full further reduce to

[TABLE]

evaluated at $X=X_{0}=-q^{2}$, and

[TABLE]

evaluated at $X=X_{0}$, respectively. In the Horndeski limit, the conditions for Case 1$-\Lambda$-full and Case 2$-\Lambda$-full further reduce to

[TABLE]

evaluated at $X=X_{0}=-q^{2}$, and

[TABLE]

evaluated at $X=X_{0}$, respectively.

These conditions coincide with those obtained in Refs. Babichev and Charmousis (2014); Kobayashi and Tanahashi (2014) which takes $G_{3}(X)=G_{5}(X)=0$.

VI Conclusions

We obtained conditions for the existence of the Schwarzschild and Schwarzschild-(anti-) de Sitter black hole solutions in the shift-symmetric DHOST theories by assuming that the scalar field has a nontrivial profile with a linear time dependence and that the ordinary kinetic term of the scalar field is constant. Our analysis extended the work of Ref. Motohashi and Minamitsuji (2019), exhausted all the known classes of DHOST theories, and included the shift-symmetric subclasses of Horndeski and GLPV theories as special cases.

We have also investigated the compatibility between the degeneracy conditions and the conditions to allow the exact BH solutions obtained in this paper. We found that in most classes of DHOST theories with both quadratic and cubic interactions, these conditions are not compatible, and cubic DHOST theories are not the scalar-tensor theories giving rise to viable BH solutions with a non-trivial scalar field which are absent in GR. Combined with the fact that none of the cubic DHOST theories satisfy the latest bound on the propagation speed of gravitational waves Langlois et al. (2018), our analysis means that cubic DHOST theories are disfavoured in all the regimes of our Universe. In other words, the only DHOST theories which are likely to be viable alternatives to GR are quadratic DHOST theories. Ref. de Rham and Zhang (2019) recently argued that even-mode perturbations about stealth Schwarzschild solutions in pure quadratic DHOST theories suffer strong coupling problems, indicating that predictions at the level of linear perturbations cannot be trusted. If this is true, the same issue may also be present for the case of the cubic DHOST theories. All these facts would strongly disfavor DHOST theories in terms of the properties of BHs, in comparison with those in GR.

Acknowledgements.

M.M. was supported by the research grant under “Norma Transitória do DL 57/2016”. J.E. acknowledges the support by CENTRA.

Appendix A The pure quadratic DHOST theories

In the Appendices A-C, we summarize the degeneracy conditions for the pure quadratic, pure cubic, and full DHOST theories. In the case of the pure quadratic DHOST theories, $F_{3}=B_{J}=0$ (where $J=1,2,\cdots,10$).

A.1 The case $F_{2}=F_{2}(\phi)$

In the case of $F_{2}=F_{2}(\phi)$, degeneracy conditions for minimally coupled quadratic theories555Here we take “minimally coupled” to mean $F_{2}(X,\phi)=F_{2}(\phi)$, because this coupling is free from Ostrogradsky ghosts and does not contribute to the degeneracy conditions. are classified into the following three classes Crisostomi et al. (2016b):

The Class-2M-I DHOST theory is given by

[TABLE]

The pure quartic beyond-Horndeski theories ($f_{4}\neq 0$ while $G_{4}=G_{4}(\phi)$ and $G_{5}=f_{5}=0$) satisfy the degeneracy conditions of the Class-2M-I DHOST Langlois and Noui (2016); Ben Achour et al. (2016b).

The Class-2M-II DHOST theory is given by

[TABLE]

The Class-2M-III DHOST theory is given by

[TABLE]

Class-2M theories are also called “Class III” theories Ben Achour et al. (2016a); Langlois (2019).

A.2 The case $F_{2}=F_{2}(\phi,X)$

In the case of $F_{2}=F_{2}(\phi,X)$, degeneracy conditions are classified into the following four classes Ben Achour et al. (2016b):

The Class-2N-I DHOST theory with $F_{2}\neq XA_{1}$ is given by

[TABLE]

The pure quartic order GLPV theories ($G_{4}\neq 0$ and $f_{4}\neq 0$ while $G_{5}=f_{5}=0$) satisfy the degeneracy conditions of the Class-2N-I DHOST Langlois and Noui (2016); Ben Achour et al. (2016b).

The Class-2N-II DHOST theory with $F_{2}\neq XA_{1}$ is given by is given by

[TABLE]

Class-2N-I and Class-2N-II theories which satisfy $A_{1}+A_{2}=0$ are also called “Class I” theories Ben Achour et al. (2016a); Langlois (2019).

The Class-2N-III DHOST theory with $A_{1}+A_{2}\neq 0$ and $F_{2}\neq XA_{1}$ is given by

[TABLE]

The Class-2N-IV DHOST theory with $A_{1}+A_{2}\neq 0$ is given by

[TABLE]

Class-2N-III and Class-2N-IV theories which do not satisfy $A_{1}+A_{2}=0$ are also called “Class II” theories Ben Achour et al. (2016a); Langlois (2019).

Appendix B The pure cubic DHOST theories

In the case of pure cubic DHOST theories, $F_{2}=A_{I}=0$ (where $I=1,2,\cdots,5$).

B.1 The minimally coupled case $F_{3}=0$

In the minimally coupled case $F_{3}=0$, degeneracy conditions are classified into seven classes Ben Achour et al. (2016b):

The Class-3M-I DHOST theory ($9B_{1}+2B_{2}\neq 0$) is given by

[TABLE]

The pure quartic order beyond-Horndeski interactions ($f_{4}\neq 0$ while $G_{4}=G_{4}(\phi)$ and $G_{5}=f_{5}=0$) satisfy the degeneracy conditions of Class-3M-I DHOST Langlois and Noui (2016); Ben Achour et al. (2016b).

The Class-3M-II DHOST theory ($9B_{1}-2B_{2}\neq 0$) is given by

[TABLE]

The Class-3M-III DHOST theory is given by

[TABLE]

The Class-3M-IV DHOST theory is given by

[TABLE]

The Class-3M-V DHOST theory is given by

[TABLE]

The Class-3M-VI DHOST theory is given by

[TABLE]

The Class-3M-VII DHOST theory is given by

[TABLE]

B.2 The nonminimally coupled case $F_{3}\neq 0$

In the nonminimally coupled case $F_{3}\neq 0$, degeneracy conditions are classified into two classes Ben Achour et al. (2016b):

The Class-3N-I DHOST theory ($B_{1}\neq 0$) is given by

[TABLE]

The pure quintic order GLPV theories ($G_{5}\neq 0$ and $f_{5}\neq 0$ while $G_{4}=f_{4}=0$) satisfy the degeneracy conditions of the Class-3N-I DHOST Ben Achour et al. (2016b).

The Class-3N-II DHOST theory ($B_{1}=0$) is given by

[TABLE]

Appendix C The full DHOST theories

If both quadratic and cubic terms exist, $A_{I}\neq 0$ ($I=1,2,\cdots,5$) and $B_{J}\neq 0$ ($J=1,2,\cdots,10$), they

can be compatible only for particular combinations of Class-2N/M and Class-3N/M theories.

The compatibility of cubic DHOST theories with Class-2M DHOST theories is summarized as follows Ben Achour et al. (2016b):

[TABLE]

where we have defined

[TABLE]

The compatibility of cubic DHOST theories with Class-2N DHOST theories is summarized as follows Ben Achour et al. (2016b):

[TABLE]

where we have defined

[TABLE]

Appendix D The coefficients in general couplings

D.1 The pure cubic DHOST theories

In the case of the pure cubic DHOST theories, the coefficients in the Taylor series form (14) satisfy the following relations:

Case-1-cubic satisfies

[TABLE]

with $X_{0}=-q^{2}$.

Case-2-cubic satisfies

[TABLE]

Case-1-$\Lambda$-cubic satisfies

[TABLE]

with $X_{0}=-q^{2}$.

Case-2-$\Lambda$-cubic satisfies

[TABLE]

D.2 The full DHOST theories

In the case of the pure cubic DHOST theories, the coefficients in the Taylor series form (20) satisfy the following relations:

Case-1-full satisfies

[TABLE]

with $X_{0}=-q^{2}$.

Case-2-full satisfies

[TABLE]

Case-1-$\Lambda$-full satisfies

[TABLE]

with $X_{0}=-q^{2}$.

Case-2-$\Lambda$-full satisfies

[TABLE]