Ghost-free theories with arbitrary higher-order time derivatives

We investigate static and spherically symmetric black hole (BH) solutions in shift-symmetric quadratic-order degenerate higher-order scalar-tensor (DHOST) theories. We allow a nonconstant kinetic term X = g µν ∂µφ∂ν φ for the scalar field φ and assume that φ is, like the spacetime, a pure function of the radial coordinate r, namely φ = φ(r). First, we find analytic static and spherically symmetric vacuum solutions in the so-called Class Ia DHOST theories, which include the quartic Horndeski theories as a subclass. We consider several explicit models in this class and apply our scheme to find the exact vacuum BH solutions. BH solutions obtained in our analysis are neither Schwarzschild or Schwarzschild (anti-) de Sitter. We show that a part of the BH solutions obtained in our analysis are free of ghost and Laplacian instabilities and are also mode stable against the oddparity perturbations. Finally, we argue the case that the scalar field has a linear time dependence φ = qt + ψ(r) and show several simple examples of nontrivial BH solutions with a nonconstant kinetic term obtained analytically and numerically. I. INTRODUCTIONScalar-tensor theories have provided the unified mathematical description of modified gravity theories [1]. In classic scalar-tensor theories, whose Lagrangian density depends on the metric g µν , the scalar field φ, and its first-order derivative φ µ := ∇ µ φ, L = L(g µν , φ, φ µ ), the Euler-Lagrange (EL) equations are given by the second-order differential equations. However, the Lagrangian density of modern scalar-tensor theories may also contain the second-order derivatives of the scalar field φ µν := ∇ µ ∇ ν φ. Although generically the EL equations in such theories contain higher derivative terms, the appearance of Ostrogradsky ghosts [2] can be avoided using certain degeneracy conditions [3]. Degenerate higher-order scalar-tensor (DHOST) theories [4-9] (see also [10,11]) provide the most general framework of scalar-tensor theories which are free from Ostrogradsky instabilities [2], and hence the system contains only three degrees of freedom (DOFs), namely, two tensorial and one scalar polarizations (See § II for details). Applications of DHOST theories to cosmological and astrophysical problems have been investigated in Refs. [9,[12][13][14][15].The application of modern scalar-tensor theories to BH physics has attracted great interest. Besides the Schwarzschild or Kerr solutions in General Relativity (GR) with or without a constant scalar field [16,17], i.e. GR BH solutions, they also allow BH solutions which are absent in GR. A typical nontrivial BH solution is the stealth Schwarzschild solution [18,19] obtained in shift-symmetric Horndeski theories with the assumptions of a linearly time-dependent scalar field φ = qt + ψ(r) and a constant kinetic term X = const, where t and r are the time and radial coordinates of the static and spherically symmetric spacetime and X := g µν φ µ φ ν represents the canonical kinetic term of the scalar field. In stealth solutions, the spacetime geometry ...

Section: The Dhost Theoriesmentioning

confidence: 99%

“…Since γ 1,± − 2β 1 = ±4n β 1 (β 1 − 1)/3, only the plus branch is consistent with Eq. (37). For γ 1 = γ 1,+ , choosing C 1 = −2M and C 2 = 1 with M being a constant,…”

Section: Model Amentioning

confidence: 99%

Black holes with a nonconstant kinetic term in degenerate higher-order scalar tensor theories

Minamitsuji

Edholm

2020

“…The first condition required for the single scalar propagating degree of freedom is degeneracy in the temporal structure of (26). Of the three possibilities, we focus on the case 1 type where the condition c 3 = c 2 /c 2 1 is satisfied and the combinationζ…”

Section: B Background and Quadratic Lagrangianmentioning

confidence: 99%

Effective field theory of degenerate higher-order inflation

Motohashi

2020

Self Cite

We extend the effective field theory of inflation to a general Lagrangian constructed from Arnowitt-Deser-Misner variables that encompasses the most general interactions with up to second derivatives of the scalar field whose background breaks temporal diffeomorphism invariance. Degeneracy conditions, corresponding to 8 distinct types -only one of which corresponds to known degenerate higher-order scalar-tensor models -provide necessary conditions for eliminating the Ostrogradsky ghost in a covariant theory at the level of the quadratic action in unitary gauge. Novel implications of the degenerate higher-order system for the Cauchy problem are illustrated with the phase space portrait of an explicit inflationary example: not all field configurations lead to physical solutions for the metric even for positive potentials; solutions are unique for a given configuration only up to a branch choice; solutions on one branch can apparently end at nonsingular points of the metric and their continuation on alternate branches lead to nonsingular bouncing solutions; unitary gauge perturbations can go unstable even when degenerate terms in the Lagrangian are infinitesimal. The attractor solution leads to an inflationary scenario where slow-roll parameters vary and running of the tilt can be large even with no explicit features in the potential far from the end of inflation, requiring the optimized slow-roll approach for predicting observables.

“…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.…”

Section: A Dhost Theoriesmentioning

confidence: 99%

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

Minamitsuji

Edholm

2019

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.