Linear stability of black holes with static scalar hair in full Horndeski theories: generic instabilities and surviving models

Abstract

In full Horndeski theories, we show that the static and spherically symmetric black hole (BH) solutions with a static scalar field φ whose kinetic term X is nonvanishing on the BH horizon are generically prone to ghost/Laplacian instabilities. We then search for asymptotically flat hairy BH solutions with a vanishing X on the horizon free from ghost/Laplacian instabilities. We show that models with regular coupling functions of φ and X result in no-hair Schwarzschild BHs in general. On the other hand, the presence of a coupling between the scalar field and the Gauss-Bonnet (GB) term R 2 GB , even with the coexistence of other regular coupling functions, leads to the realization of asymptotically flat hairy BH solutions without ghost/Laplacian instabilities. Finally, we find that hairy BH solutions in power-law F (R 2 GB ) gravity are plagued by ghost instabilities. These results imply that the GB coupling of the form ξ(φ)R 2 GB plays a prominent role for the existence of asymptotically flat hairy BH solutions free from ghost/Laplacian instabilities. I. INTRODUCTIONGeneral Relativity (GR) has been tested by numerous experiments in the Solar System. While gravity can be well described by GR on the weak gravitational background in our local Universe [1], the dawn of gravitational-wave (GW) astronomy [2] and black hole (BH) shadow measurements [3] have started to allow us to probe the physics of extremely compact objects like BHs and neutron stars [4][5][6][7]. On the other hand, we also know that the Universe recently entered a phase of accelerated expansion [8,9]. While the cosmological constant is the simplest candidate for the source of this acceleration, i.e., dark energy, the theoretical value of vacuum energy mimicking the cosmological constant is enormously larger than the observed dark energy scale [10]. The cosmological constant has also been plagued by tensions of today's Hubble constant H 0 constrained from high-and low-redshift measurements [11,12]. These facts led to the question for the validity of GR on large distances relevant to today's cosmic acceleration. A simple and robust alternative to GR is provided by scalar-tensor theories possessing a scalar degree of freedom coupled to gravity [13].The most general class of scalar-tensor theories with second-order Euler-Lagrange equations of motion is called Horndeski theories [14-17]. 1 There have been numerous attempts for the theoretical construction of dark energy models compatible with observations [25][26][27][28][29][30]. Although such a new scalar degree of freedom potentially manifests itself in the Solar System, fifth forces mediated by the scalar field can be screened [31][32][33][34] by Vainshtein [35] or chameleon [36] mechanisms around a compact body on the weak gravitational background. In the vicinity of a BH, on the other hand, a nonvanishing charge of the scalar field, i.e., scalar hair, gives rise to a nontrivial field profile affecting the background geometry. This offers an interesting possibility for probing the possible deviation f...

Instability of hairy black holes in regularized 4-dimensional Einstein-Gauss-Bonnet gravity

“…This shows that either F, K, or B 2 is negative, so the BH solution (2.19) with the field derivative (2.16) is inevitably plagued by instabilities around the outer horizon. The same property holds for hairy BH solutions where X is an analytic function of r with a non-vanishing constant X s on the horizon [61,62]. Here, we have shown that the BH instability also persists for the non-analytic function (3.2) of X in the vicinity of the outer horizon.…”

“…Although X is finite on the horizon, it is not an analytic function of r. In Refs. [61,62] the analytic property of X was assumed to study the BH stability around the horizon, so we need to handle the present case separately from those discussed in Refs. [61,62].…”

“…The recent general analysis of BH and NS perturbations in Horndeski theories [60] shows that the angular propagation of even-parity modes with large multipoles ℓ ≫ 1 plays an important role to exclude some hairy BH solutions by the instability in the vicinity of an event horizon [61]. This instability can arise for BHs with static scalar hair whose kinetic term X is a non-vanishing constant X s on the horizon [62]. For example, the derivative coupling with an Einstein tensor gives rise to exact hairy BH solutions [63][64][65] with X s = 0, but they are prone to either ghost or Laplacian instabilities.…”

Stability of Hairy Black Holes in Shift-Symmetric Scalar-Tensor Theories via the Effective Field Theory Approach

“…These black holes exist in a wide class of scalar-tensor theories satisfying a shift symmetry φ → φ + c. They are akin to the time-dependent scalar profile that drives cosmological expansion in ghost condensation [39,40]. These time-dependent hairy black hole solutions require careful study, since it has been shown in certain circumstances that they may suffer from a strong coupling problem [33,[41][42][43] and/or a gradient instability [38,[44][45][46].…”

On the effective metric of axial black hole perturbations in DHOST gravity

“…For this reason, axial perturbations are easier to study and they have already been investigated in several works in the context of DHOST theories, or subfamilies: from generic BH backgrounds in [6][7][8][9][10] to "stealth" solutions (solutions with a nontrivial scalar hair but whose metric coincides with the GR Schwarzschild metric) in [11]. Other works also include polar perturbations [12][13][14][15][16][17]. Many of these studies rely on the computation of the quadratic Lagrangian that describes the dynamics of the perturbations.…”