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Theoretical filters for shift-symmetric Horndeski gravities

Athanasios Bakopoulos, Christos Charmousis, Nikos Chatzifotis, Theodoros Nakas

TL;DR

This work establishes a vacuum-based filter for shift-symmetric Horndeski gravity by deriving consistency conditions that permit nontrivial Minkowski and de Sitter vacua, including stealth configurations where the scalar is active without altering the metric. It then develops a disformal-transformation framework to generate exact beyond-Horndeski solutions from Horndeski seeds, yielding solitons and black holes with primary scalar hair. The results identify a narrow subset of models that sustain canonical vacua and provide explicit methods to explore rich compact-object solutions in extended theories, with clear criteria for the transformation’s validity and physical viability. Collectively, the study offers both a principled model-selection tool and a constructive path to probe beyond-Horndeski phenomenology, with implications for gravitational-wave and black-hole physics.

Abstract

We investigate the structure of nontrivial maximally symmetric vacua and compact-object solutions in shift-symmetric scalar-tensor theories. Focusing on Horndeski gravity, we derive consistency conditions directly from the field equations to identify the subclasses that admit Minkowski and de Sitter vacua with a nontrivial scalar field. In doing so, we obtain a filtering mechanism that operates independently of observational data. In this context, we introduce the notion of stealth vacua, where the scalar field remains active without altering the vacuum. Following this, we examine the theoretical framework of Horndeski theories that admit homogeneous geometries and we extract the implicit form of the solution pertaining to the entire family of theories. Building upon these frameworks, we construct exact solutions in beyond-Horndeski gravity by applying a linear disformal transformation to the regularized Einstein-Gauss-Bonnet black hole. This procedure yields solitonic spacetimes with scalar hair as well as black holes carrying primary scalar hair, demonstrating how disformal maps can qualitatively modify solution properties. We delineate the parameter space in which the transformation is well-defined and analyze the solutions. Our results provide both a principled criterion for selecting viable Horndeski models and a framework for exploring rich solution spaces in beyond-Horndeski gravity.

Theoretical filters for shift-symmetric Horndeski gravities

TL;DR

This work establishes a vacuum-based filter for shift-symmetric Horndeski gravity by deriving consistency conditions that permit nontrivial Minkowski and de Sitter vacua, including stealth configurations where the scalar is active without altering the metric. It then develops a disformal-transformation framework to generate exact beyond-Horndeski solutions from Horndeski seeds, yielding solitons and black holes with primary scalar hair. The results identify a narrow subset of models that sustain canonical vacua and provide explicit methods to explore rich compact-object solutions in extended theories, with clear criteria for the transformation’s validity and physical viability. Collectively, the study offers both a principled model-selection tool and a constructive path to probe beyond-Horndeski phenomenology, with implications for gravitational-wave and black-hole physics.

Abstract

We investigate the structure of nontrivial maximally symmetric vacua and compact-object solutions in shift-symmetric scalar-tensor theories. Focusing on Horndeski gravity, we derive consistency conditions directly from the field equations to identify the subclasses that admit Minkowski and de Sitter vacua with a nontrivial scalar field. In doing so, we obtain a filtering mechanism that operates independently of observational data. In this context, we introduce the notion of stealth vacua, where the scalar field remains active without altering the vacuum. Following this, we examine the theoretical framework of Horndeski theories that admit homogeneous geometries and we extract the implicit form of the solution pertaining to the entire family of theories. Building upon these frameworks, we construct exact solutions in beyond-Horndeski gravity by applying a linear disformal transformation to the regularized Einstein-Gauss-Bonnet black hole. This procedure yields solitonic spacetimes with scalar hair as well as black holes carrying primary scalar hair, demonstrating how disformal maps can qualitatively modify solution properties. We delineate the parameter space in which the transformation is well-defined and analyze the solutions. Our results provide both a principled criterion for selecting viable Horndeski models and a framework for exploring rich solution spaces in beyond-Horndeski gravity.

Paper Structure

This paper contains 11 sections, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. Minkowski Vacuum
  4. Extension of the regularized EGB black hole
  5. Non-homogeneous black-hole solution
  6. Special cases
  7. Field-theoretic constraints for homogeneous solutions
  8. De Sitter Vacuum
  9. Solutions with constant kinetic term
  10. From vacuum to compact-object solutions through disformal transformation
  11. Concluding remarks

Figures (3)

  • Figure 1: (a) The metric function $h$, and (b) the inhomogeneity factor $K$ for $q M=0.5$, $\eta/M=1$, $\alpha/M^2=0.1$. The horizontal axis is logarithmic in both graphs.
  • Figure 2: Parameter space spanned by the dimensionless parameters $M/\sqrt{\alpha}$ and $q\sqrt{\alpha}$. The upper blue region indicates the permissible range of parameter values.
  • Figure 3: The disformed metric functions $h(r)$ and $f(r)$ for two different sets of the dimensionaless parameters $\{\xi/\alpha^2,M/\sqrt{\alpha},q\sqrt{\alpha} \}$. (a) Single-horizon black hole, (b) black hole with multiple horizons.