pith. sign in

arxiv: 2503.05651 · v3 · submitted 2025-03-07 · 🌀 gr-qc · astro-ph.CO· astro-ph.HE· hep-th

Inverting no-hair theorems: How requiring General Relativity solutions restricts scalar-tensor theories

Hajime Kobayashi , Shinji Mukohyama , Johannes Noller , Sergi Sirera , Kazufumi Takahashi , Vicharit Yingcharoenrat This is my paper

Pith reviewed 2026-05-23 00:41 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COastro-ph.HEhep-th
keywords scalar-tensor theoriesstealth solutionsno-hair theoremsblack hole perturbationsodd-parity modesgeneral relativityhigher-order scalar-tensor
0
0 comments X

The pith

Requiring all stealth solutions in scalar-tensor theories eliminates deviations from General Relativity in the odd-parity sector.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that demanding the existence of stealth black hole solutions—exact General Relativity metrics paired with time-dependent scalar fields of constant kinetic term—strongly restricts the space of quadratic and cubic scalar-tensor theories. When this demand applies to every possible stealth solution, the linear odd-parity perturbation equations around static spherical black holes are forced to coincide with those of General Relativity. A weaker demand, limited to specific solutions such as Schwarzschild or de Sitter, still permits controlled deviations while requiring stability of the odd modes and allowing altered gravitational-wave speeds. The analysis proceeds by deriving covariant conditions on the theory functions so that the Euler-Lagrange equations are satisfied identically on General Relativity backgrounds.

Core claim

In general quadratic/cubic higher-order scalar-tensor theories, requiring that the Euler-Lagrange equations admit all (or selected) exact General Relativity solutions as stealth configurations with constant scalar kinetic term X forces the odd-parity sector of linear perturbations around static spherically symmetric black holes to be identical to General Relativity when every stealth solution is imposed; weaker requirements on specific solutions such as Schwarzschild-de Sitter allow deviations whose stability conditions and gravitational-wave propagation speeds are derived explicitly.

What carries the argument

Covariant conditions on the Lagrangian functions that make the Euler-Lagrange equations satisfied identically for General Relativity metrics with constant scalar kinetic term X.

If this is right

  • Requiring every stealth solution prevents any deviation from General Relativity in the odd-parity perturbation equations.
  • Requiring only Schwarzschild or de Sitter stealth solutions permits deviations from General Relativity while still satisfying the existence condition.
  • Stability criteria for the odd modes can be obtained explicitly in the less restrictive cases.
  • A non-trivial propagation speed for gravitational waves appears as a possible deviation when only specific stealth solutions are required.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Models seeking observable scalar hair must therefore drop the requirement that every stealth solution exists.
  • The same logic can be applied to even-parity perturbations or rotating backgrounds to obtain further restrictions.
  • Ringdown observations of black-hole mergers could directly test whether a candidate theory satisfies the all-stealth condition.

Load-bearing premise

Stealth solutions are defined to have a constant scalar kinetic term X in static spherically symmetric spacetimes, allowing exact solution of the Euler-Lagrange equations on General Relativity metrics.

What would settle it

An observed odd-parity gravitational-wave signal from a black-hole merger whose speed, damping rate, or waveform deviates from General Relativity predictions in a scalar-tensor theory asserted to admit every stealth solution.

Figures

Figures reproduced from arXiv: 2503.05651 by Hajime Kobayashi, Johannes Noller, Kazufumi Takahashi, Sergi Sirera, Shinji Mukohyama, Vicharit Yingcharoenrat.

Figure 1
Figure 1. Figure 1: Here we display the theory space of cubic HOST theories which allow the existence of [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Coefficients in the quadratic Lagrangian in covariant form ( [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Contribution from all cubic HOST functions to the different [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Speed of GWs in the radial direction as a function of distance for different values of the [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
read the original abstract

Black hole solutions in general scalar-tensor theories are known to permit hair, i.e. non-trivial scalar profiles and/or metric solutions different from the ones of General Relativity (GR). Imposing that some such solutions$\unicode{x2013}$e.g. Schwarzschild or de Sitter solutions motivated in the context of black hole physics or cosmology$\unicode{x2013}$should exist, the space of scalar-tensor theories is strongly restricted. Here we investigate precisely what these restrictions are within general quadratic/cubic higher-order scalar-tensor theories for stealth solutions, whose metric is given by that in GR, supporting time-dependent scalar hair with a constant kinetic term. We derive, in a fully covariant approach, the conditions under which the Euler-Lagrange equations admit all (or a specific set of) exact GR solutions, as the first step toward our understanding of a wider class of theories that admit approximately stealth solutions. Focusing on static and spherically symmetric black hole spacetimes, we study the dynamics of linear odd-parity perturbations and discuss possible deviations from GR. Importantly, we find that requiring the existence of all stealth solutions prevents any deviations from GR in the odd-parity sector. In less restrictive scenarios, in particular for theories only requiring the existence of Schwarzschild(-de Sitter) black holes, we identify allowed deviations from GR, derive the stability conditions for the odd modes, and investigate the generic deviation of a non-trivial speed of gravitational waves. All calculations performed in this paper are reproducible via companion $\texttt {Mathematica}$ notebooks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper derives, in a fully covariant manner, the conditions on the coefficients of general quadratic/cubic scalar-tensor Lagrangians such that every static spherically symmetric GR metric is admitted as an exact stealth solution (constant scalar kinetic term X, time-dependent scalar profile). Substituting these conditions into the linearized odd-parity perturbation equations around these backgrounds shows that the requirement of all GR stealth solutions forces the equations to coincide exactly with GR, leaving no free deviation parameters. For less restrictive cases (e.g., only Schwarzschild or Schwarzschild-de Sitter solutions required), allowed deviations are identified, stability conditions for odd modes are derived, and the speed of gravitational waves is shown to deviate from the GR value in general. All steps are supported by companion Mathematica notebooks.

Significance. If the derivation holds, the result inverts the logic of no-hair theorems by showing how demanding the existence of stealth GR solutions carves out a highly restricted subclass of scalar-tensor theories, with the strongest restriction eliminating all odd-parity deviations. The fully covariant approach, explicit stability conditions, and reproducible notebooks are strengths that make the central claim testable and extendable to approximate stealth solutions.

minor comments (2)
  1. [§2] §2, around Eq. (2.7): the notation for the general quadratic/cubic action could include an explicit statement that the coefficients are allowed to be functions of the scalar and X, to avoid any ambiguity with the later specialization to constant X.
  2. [Table 1] Table 1: the column labels for the restricted cases (Schwarzschild only vs. all SSS) would benefit from a footnote clarifying which GR solutions are imposed in each row.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, accurate summary of the results, and recommendation to accept. We are pleased that the fully covariant approach, stability conditions, and reproducibility via notebooks were noted as strengths.

Circularity Check

0 steps flagged

Derivation is self-contained; no circular reductions identified

full rationale

The paper starts from a general quadratic/cubic scalar-tensor action and derives, via direct substitution into the Euler-Lagrange equations, the coefficient conditions required for every static spherically symmetric GR metric to be an exact stealth solution (constant X). These conditions are then inserted into the linearized odd-parity perturbation equations. The resulting system is algebraically forced to match GR because the same coefficient restrictions that enforce the background solutions also eliminate all free deviation parameters in the odd sector. This is a straightforward consequence of the field equations rather than a redefinition, a fit, or a self-citation chain. The constant-X stealth definition is an explicit part of the problem statement, not an unexamined assumption. No load-bearing step reduces to its own input by construction, and the companion notebooks make the algebraic steps externally verifiable.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on restricting the general quadratic/cubic scalar-tensor action to admit exact GR metrics as solutions; the free coefficients in that action are constrained rather than fitted to data.

free parameters (1)
  • coefficients in the general quadratic/cubic scalar-tensor Lagrangian
    The general action contains several free functions or constant coefficients that are then restricted by the requirement that GR solutions satisfy the equations of motion.
axioms (2)
  • domain assumption Theories are restricted to quadratic and cubic higher-order scalar-tensor theories
    Paper explicitly limits scope to this class of theories.
  • domain assumption Stealth solutions have constant kinetic term X
    Definition used throughout the derivation of conditions.

pith-pipeline@v0.9.0 · 5846 in / 1294 out tokens · 91676 ms · 2026-05-23T00:41:59.374501+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Master Equation for Screening in Luminal Horndeski Gravity

    gr-qc 2026-05 unverdicted novelty 7.0

    A master screening equation is derived for luminal Horndeski gravity that recovers Vainshtein and Chameleon mechanisms and introduces Phaedrus screening with screening radius scaling linearly with source mass.

  2. Testing Dark Energy with Black Hole Ringdown

    gr-qc 2026-03 unverdicted novelty 7.0

    Dynamical dark energy imprints O(1) shifts on black hole quasi-normal modes via cosmological hair, enabling constraints at 10^{-2} (LVK) to 10^{-4} (LISA) precision using the cubic Galileon as example.

  3. Stable black hole solutions with cosmological hair

    gr-qc 2026-03 unverdicted novelty 7.0

    Stable black hole solutions with cosmological scalar hair are explicitly derived in the cubic Galileon theory, recovering cosmological behavior at large distances and regular short-range dynamics.

Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages · cited by 3 Pith papers · 37 internal anchors

  1. [1]

    ringdown-calculations

    S. Sirera, “ringdown-calculations.” https://github.com/sergisl/ringdown-calculations

    work page

  2. [2]

    Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability

    D. Langlois and K. Noui, “Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability,” JCAP 02 (2016) 034, 1510.06930

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  3. [3]

    Extended Scalar-Tensor Theories of Gravity

    M. Crisostomi, K. Koyama, and G. Tasinato, “Extended Scalar-Tensor Theories of Gravity,” JCAP 1604 (2016), no. 04 044, 1602.03119

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  4. [4]

    Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order

    J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui, and G. Tasinato, “Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order,” JHEP 12 (2016) 100, 1608.08135

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  5. [5]

    Second-order scalar-tensor field equations in a four-dimensional space,

    G. W. Horndeski, “Second-order scalar-tensor field equations in a four-dimensional space,” Int. J. Theor. Phys. 10 (1974) 363–384

    work page 1974

  6. [6]

    From k-essence to generalised Galileons

    C. Deffayet, X. Gao, D. A. Steer, and G. Zahariade, “From k-essence to generalised Galileons,” Phys. Rev. D 84 (2011) 064039, 1103.3260

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  7. [7]

    Generalized G-inflation: Inflation with the most general second-order field equations

    T. Kobayashi, M. Yamaguchi, and J. Yokoyama, “Generalized G-inflation: Inflation with the most general second-order field equations,” Prog. Theor. Phys. 126 (2011) 511–529, 1105.5723. 35

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  8. [8]

    "Shadowy" modes in Higher-Order Scalar-Tensor theories

    A. De Felice, D. Langlois, S. Mukohyama, K. Noui, and A. Wang, “Generalized instantaneous modes in higher-order scalar-tensor theories,” Phys. Rev. D 98 (2018), no. 8 084024, 1803.06241

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  9. [9]

    Nonlinear definition of the shadowy mode in higher-order scalar-tensor theories,

    A. De Felice, S. Mukohyama, and K. Takahashi, “Nonlinear definition of the shadowy mode in higher-order scalar-tensor theories,” JCAP 12 (2021), no. 12 020, 2110.03194

    work page arXiv 2021

  10. [10]

    Avoidance of Strong Coupling in General Relativity Solutions with a Timelike Scalar Profile in a Class of Ghost-Free Scalar-Tensor Theories,

    A. De Felice, S. Mukohyama, and K. Takahashi, “Avoidance of Strong Coupling in General Relativity Solutions with a Timelike Scalar Profile in a Class of Ghost-Free Scalar-Tensor Theories,” Phys. Rev. Lett. 129 (2022), no. 3 031103, 2204.02032

    work page arXiv 2022

  11. [11]

    Invertible disformal transformations with higher derivatives,

    K. Takahashi, H. Motohashi, and M. Minamitsuji, “Invertible disformal transformations with higher derivatives,” Phys. Rev. D 105 (2022), no. 2 024015, 2111.11634

    work page arXiv 2022

  12. [12]

    Takahashi, M

    K. Takahashi, M. Minamitsuji, and H. Motohashi, “Generalized disformal Horndeski theories: Cosmological perturbations and consistent matter coupling,” PTEP 2023 (2023), no. 1 013E01, 2209.02176

    work page arXiv 2023

  13. [13]

    Effective description of generalized disformal theories,

    K. Takahashi, M. Minamitsuji, and H. Motohashi, “Effective description of generalized disformal theories,” JCAP 07 (2023) 009, 2304.08624

    work page arXiv 2023

  14. [14]

    Invertible disformal transformations with arbitrary higher-order derivatives,

    K. Takahashi, “Invertible disformal transformations with arbitrary higher-order derivatives,” Phys. Rev. D 108 (2023), no. 8 084031, 2307.08814

    work page arXiv 2023

  15. [15]

    Derivative-dependent metric transformation and physical degrees of freedom

    G. Dom` enech, S. Mukohyama, R. Namba, A. Naruko, R. Saitou, and Y. Watanabe, “Derivative-dependent metric transformation and physical degrees of freedom,” Phys. Rev. D 92 (2015), no. 8 084027, 1507.05390

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  16. [16]

    General invertible transformation and physical degrees of freedom

    K. Takahashi, H. Motohashi, T. Suyama, and T. Kobayashi, “General invertible transformation and physical degrees of freedom,” Phys. Rev. D 95 (2017), no. 8 084053, 1702.01849

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  17. [17]

    Novel “no-scalar-hair

    J. D. Bekenstein, “Novel “no-scalar-hair” theorem for black holes,” Phys. Rev. D 51 (1995), no. 12 R6608

    work page 1995

  18. [18]

    Black Uniqueness Theorems

    P. O. Mazur, “Black hole uniqueness theorems,” hep-th/0101012

    work page internal anchor Pith review Pith/arXiv arXiv

  19. [19]

    A no-hair theorem for the galileon

    L. Hui and A. Nicolis, “No-Hair Theorem for the Galileon,” Phys. Rev. Lett. 110 (2013) 241104, 1202.1296

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  20. [20]

    Black hole hairs in scalar-tensor gravity and the lack thereof,

    L. Capuano, L. Santoni, and E. Barausse, “Black hole hairs in scalar-tensor gravity and the lack thereof,” Phys. Rev. D 108 (2023), no. 6 064058, 2304.12750

    work page arXiv 2023

  21. [21]

    Black Holes in the Ghost Condensate

    S. Mukohyama, “Black holes in the ghost condensate,” Phys. Rev. D 71 (2005) 104019, hep-th/0502189

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  22. [22]

    Dressing a black hole with a time-dependent Galileon

    E. Babichev and C. Charmousis, “Dressing a black hole with a time-dependent Galileon,” JHEP 08 (2014) 106, 1312.3204. 36

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  23. [23]

    Exact black hole solutions in shift symmetric scalar-tensor theories

    T. Kobayashi and N. Tanahashi, “Exact black hole solutions in shift symmetric scalar–tensor theories,” PTEP 2014 (2014) 073E02, 1403.4364

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  24. [24]

    Hairy Schwarzschild-(A)dS black hole solutions in DHOST theories beyond shift symmetry

    J. Ben Achour and H. Liu, “Hairy Schwarzschild-(A)dS black hole solutions in degenerate higher order scalar-tensor theories beyond shift symmetry,” Phys. Rev. D 99 (2019), no. 6 064042, 1811.05369

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  25. [25]

    Exact black hole solutions in shift-symmetric quadratic degenerate higher-order scalar-tensor theories

    H. Motohashi and M. Minamitsuji, “Exact black hole solutions in shift-symmetric quadratic degenerate higher-order scalar-tensor theories,” Phys. Rev. D 99 (2019), no. 6 064040, 1901.04658

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  26. [26]

    Linear stability analysis of hairy black holes in quadratic degenerate higher-order scalar-tensor theories: Odd-parity perturbations

    K. Takahashi, H. Motohashi, and M. Minamitsuji, “Linear stability analysis of hairy black holes in quadratic degenerate higher-order scalar-tensor theories: Odd-parity perturbations,” Phys. Rev. D 100 (2019), no. 2 024041, 1904.03554

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  27. [27]

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

    M. Minamitsuji and J. Edholm, “Black hole solutions in shift-symmetric degenerate higher-order scalar-tensor theories,” Phys. Rev. D 100 (2019), no. 4 044053, 1907.02072

    work page arXiv 2019

  28. [28]

    General Relativity solutions with stealth scalar hair in quadratic higher-order scalar-tensor theories,

    K. Takahashi and H. Motohashi, “General Relativity solutions with stealth scalar hair in quadratic higher-order scalar-tensor theories,” JCAP 06 (2020) 034, 2004.03883

    work page arXiv 2020

  29. [29]

    Rotating Black Holes in Higher Order Gravity,

    C. Charmousis, M. Crisostomi, R. Gregory, and N. Stergioulas, “Rotating Black Holes in Higher Order Gravity,” Phys. Rev. D 100 (2019), no. 8 084020, 1903.05519

    work page arXiv 2019

  30. [30]

    Perturbations of stealth black holes in degenerate higher-order scalar-tensor theories,

    C. de Rham and J. Zhang, “Perturbations of stealth black holes in degenerate higher-order scalar-tensor theories,” Phys. Rev. D 100 (2019), no. 12 124023, 1907.00699

    work page arXiv 2019

  31. [31]

    Ghost Condensation and a Consistent Infrared Modification of Gravity

    N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, “Ghost condensation and a consistent infrared modification of gravity,” JHEP 05 (2004) 074, hep-th/0312099

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  32. [32]

    Weakly-coupled stealth solution in scordatura degenerate theory,

    H. Motohashi and S. Mukohyama, “Weakly-coupled stealth solution in scordatura degenerate theory,” JCAP 01 (2020) 030, 1912.00378

    work page arXiv 2020

  33. [33]

    Spontaneous Lorentz Breaking at High Energies

    H.-C. Cheng, M. A. Luty, S. Mukohyama, and J. Thaler, “Spontaneous Lorentz breaking at high energies,” JHEP 05 (2006) 076, hep-th/0603010

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  34. [34]

    Approximately stealth black hole in higher-order scalar-tensor theories,

    A. De Felice, S. Mukohyama, and K. Takahashi, “Approximately stealth black hole in higher-order scalar-tensor theories,” JCAP 03 (2023) 050, 2212.13031

    work page arXiv 2023

  35. [35]

    Black Hole Ringdown as a Probe for Dark Energy,

    J. Noller, L. Santoni, E. Trincherini, and L. G. Trombetta, “Black Hole Ringdown as a Probe for Dark Energy,” Phys. Rev. D 101 (2020) 084049, 1911.11671

    work page arXiv 2020

  36. [36]

    Bridging dark energy and black holes with EFT: frame transformation and gravitational wave speed,

    S. Mukohyama, E. Seraille, K. Takahashi, and V. Yingcharoenrat, “Bridging dark energy and black holes with EFT: frame transformation and gravitational wave speed,” JCAP 01 (2025) 085, 2407.15123. 37

    work page arXiv 2025

  37. [37]

    Perturbations and quasinormal modes of black holes with time-dependent scalar hair in shift-symmetric scalar-tensor theories,

    K. Tomikawa and T. Kobayashi, “Perturbations and quasinormal modes of black holes with time-dependent scalar hair in shift-symmetric scalar-tensor theories,” Phys. Rev. D 103 (2021), no. 8 084041, 2101.03790

    work page arXiv 2021

  38. [38]

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

    D. Langlois, K. Noui, and H. Roussille, “On the effective metric of axial black hole perturbations in DHOST gravity,” JCAP 08 (2022), no. 08 040, 2205.07746

    work page arXiv 2022

  39. [39]

    Black hole perturbations in DHOST theories: master variables, gradient instability, and strong coupling,

    K. Takahashi and H. Motohashi, “Black hole perturbations in DHOST theories: master variables, gradient instability, and strong coupling,” JCAP 08 (2021) 013, 2106.07128

    work page arXiv 2021

  40. [40]

    C. M. Will, Theory and Experiment in Gravitational Physics. Cambridge University Press, 2 ed., 2018

    work page 2018

  41. [41]

    Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian

    M. Zumalac´ arregui and J. Garc´ ıa-Bellido, “Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian,” Phys. Rev. D 89 (2014) 064046, 1308.4685

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  42. [42]

    Healthy theories beyond Horndeski

    J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, “Healthy theories beyond Horndeski,” Phys. Rev. Lett. 114 (2015), no. 21 211101, 1404.6495

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  43. [43]

    Exploring gravitational theories beyond Horndeski

    J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, “Exploring gravitational theories beyond Horndeski,” JCAP 02 (2015) 018, 1408.1952

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  44. [44]

    Effective Description of Higher-Order Scalar-Tensor Theories

    D. Langlois, M. Mancarella, K. Noui, and F. Vernizzi, “Effective Description of Higher-Order Scalar-Tensor Theories,” JCAP 05 (2017) 033, 1703.03797

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  45. [45]

    Cuscuton: A Causal Field Theory with an Infinite Speed of Sound

    N. Afshordi, D. J. H. Chung, and G. Geshnizjani, “Cuscuton: A Causal Field Theory with an Infinite Speed of Sound,” Phys. Rev. D 75 (2007) 083513, hep-th/0609150

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  46. [46]

    Extended Cuscuton: Formulation

    A. Iyonaga, K. Takahashi, and T. Kobayashi, “Extended Cuscuton: Formulation,” JCAP 12 (2018), no. 12 002, 1809.10935

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  47. [47]

    Essentials of k-essence

    C. Armendariz-Picon, V. F. Mukhanov, and P. J. Steinhardt, “Essentials of k essence,” Phys. Rev. D 63 (2001) 103510, astro-ph/0006373

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  48. [48]

    Black holes with primary scalar hair,

    A. Bakopoulos, C. Charmousis, P. Kanti, N. Lecoeur, and T. Nakas, “Black holes with primary scalar hair,” Phys. Rev. D 109 (2024), no. 2 024032, 2310.11919

    work page arXiv 2024

  49. [49]

    Stability and quasinormal modes for black holes with time-dependent scalar hair,

    S. Sirera and J. Noller, “Stability and quasinormal modes for black holes with time-dependent scalar hair,” Phys. Rev. D 111 (2025), no. 4 044067, 2408.01720

    work page arXiv 2025

  50. [50]

    Fundamental theorem on gauge fixing at the action level

    H. Motohashi, T. Suyama, and K. Takahashi, “Fundamental theorem on gauge fixing at the action level,” Phys. Rev. D 94 (2016), no. 12 124021, 1608.00071

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  51. [51]

    Black holes in a cubic Galileon universe

    E. Babichev, C. Charmousis, A. Leh´ ebel, and T. Moskalets, “Black holes in a cubic Galileon universe,” JCAP 09 (2016) 011, 1605.07438. 38

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  52. [52]

    Black hole perturbations in modified gravity,

    D. Langlois, K. Noui, and H. Roussille, “Black hole perturbations in modified gravity,” Phys. Rev. D 104 (2021), no. 12 124044, 2103.14750

    work page arXiv 2021

  53. [53]

    Testing the speed of gravity with black hole ringdowns,

    S. Sirera and J. Noller, “Testing the speed of gravity with black hole ringdowns,” Phys. Rev. D 107 (2023), no. 12 124054, 2301.10272

    work page arXiv 2023

  54. [54]

    Generalized Regge-Wheeler equation from Effective Field Theory of black hole perturbations with a timelike scalar profile,

    S. Mukohyama, K. Takahashi, and V. Yingcharoenrat, “Generalized Regge-Wheeler equation from Effective Field Theory of black hole perturbations with a timelike scalar profile,” JCAP 10 (2022) 050, 2208.02943

    work page arXiv 2022

  55. [55]

    Quasinormal modes from EFT of black hole perturbations with timelike scalar profile,

    S. Mukohyama, K. Takahashi, K. Tomikawa, and V. Yingcharoenrat, “Quasinormal modes from EFT of black hole perturbations with timelike scalar profile,” JCAP 07 (2023) 050, 2304.14304

    work page arXiv 2023

  56. [56]

    Black hole ringdown from physically sensible initial value problem in higher-order scalar-tensor theories,

    K. Nakashi, M. Kimura, H. Motohashi, and K. Takahashi, “Black hole ringdown from physically sensible initial value problem in higher-order scalar-tensor theories,” Phys. Rev. D 109 (2024), no. 2 024034, 2310.09839

    work page arXiv 2024

  57. [57]

    Time-dependent spherically symmetric covariant Galileons

    E. Babichev and G. Esposito-Far` ese, “Time-Dependent Spherically Symmetric Covariant Galileons,” Phys. Rev. D 87 (2013) 044032, 1212.1394

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  58. [58]

    Cosmological self-tuning and local solutions in generalized Horndeski theories

    E. Babichev and G. Esposito-Farese, “Cosmological self-tuning and local solutions in generalized Horndeski theories,” Phys. Rev. D 95 (2017), no. 2 024020, 1609.09798

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  59. [59]

    Asymptotically flat black holes in Horndeski theory and beyond

    E. Babichev, C. Charmousis, and A. Leh´ ebel, “Asymptotically flat black holes in Horndeski theory and beyond,” JCAP 04 (2017) 027, 1702.01938

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  60. [60]

    Stealth Schwarzschild solution in shift symmetry breaking theories

    M. Minamitsuji and H. Motohashi, “Stealth Schwarzschild solution in shift symmetry breaking theories,” Phys. Rev. D 98 (2018), no. 8 084027, 1809.06611

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  61. [61]

    Existence and instability of hairy black holes in shift-symmetric Horndeski theories,

    J. Khoury, M. Trodden, and S. S. C. Wong, “Existence and instability of hairy black holes in shift-symmetric Horndeski theories,” JCAP 11 (2020) 044, 2007.01320

    work page arXiv 2020

  62. [62]

    Spherical black hole perturbations in EFT of scalar-tensor gravity with timelike scalar profile,

    S. Mukohyama, K. Takahashi, K. Tomikawa, and V. Yingcharoenrat, “Spherical black hole perturbations in EFT of scalar-tensor gravity with timelike scalar profile,” 2503.00520

    work page arXiv

  63. [63]

    An Ordinary Short Gamma-Ray Burst with Extraordinary Implications: Fermi-GBM Detection of GRB 170817A

    A. Goldstein et al., “An Ordinary Short Gamma-Ray Burst with Extraordinary Implications: Fermi-GBM Detection of GRB 170817A,” Astrophys. J. Lett. 848 (2017), no. 2 L14, 1710.05446

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  64. [64]

    INTEGRAL Detection of the First Prompt Gamma-Ray Signal Coincident with the Gravitational Wave Event GW170817

    V. Savchenko et al., “INTEGRAL Detection of the First Prompt Gamma-Ray Signal Coincident with the Gravitational-wave Event GW170817,” Astrophys. J. Lett. 848 (2017), no. 2 L15, 1710.05449

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  65. [65]

    GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral

    Virgo, LIGO Scientific Collaboration, B. P. Abbott et al., “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett. 119 (2017), no. 16 161101, 1710.05832. 39

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  66. [66]

    Multi-messenger Observations of a Binary Neutron Star Merger

    B. P. Abbott et al., “Multi-messenger Observations of a Binary Neutron Star Merger,” Astrophys. J. Lett. 848 (2017), no. 2 L12, 1710.05833

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  67. [67]

    Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A

    LIGO Scientific, Virgo, Fermi-GBM, INTEGRAL Collaboration, B. P. Abbott et al., “Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A,” Astrophys. J. Lett. 848 (2017), no. 2 L13, 1710.05834

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  68. [68]

    Gravitational Rainbows: LIGO and Dark Energy at its Cutoff

    C. de Rham and S. Melville, “Gravitational Rainbows: LIGO and Dark Energy at its Cutoff,” Phys. Rev. Lett. 121 (2018), no. 22 221101, 1806.09417

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  69. [69]

    Probing the speed of gravity with LVK, LISA, and joint observations,

    I. Harry and J. Noller, “Probing the speed of gravity with LVK, LISA, and joint observations,” Gen. Rel. Grav. 54 (2022), no. 10 133, 2207.10096

    work page arXiv 2022

  70. [70]

    Measuring the propagation speed of gravitational waves with LISA,

    LISA Cosmology Working Group Collaboration, T. Baker et al., “Measuring the propagation speed of gravitational waves with LISA,” JCAP 08 (2022), no. 08 031, 2203.00566

    work page arXiv 2022

  71. [71]

    Testing gravitational wave propagation with multiband detections,

    T. Baker, E. Barausse, A. Chen, C. de Rham, M. Pieroni, and G. Tasinato, “Testing gravitational wave propagation with multiband detections,” JCAP 03 (2023) 044, 2209.14398

    work page arXiv 2023

  72. [72]

    Novel probe of graviton dispersion relations at nanohertz frequencies,

    B. Atkins, A. Malhotra, and G. Tasinato, “Novel probe of graviton dispersion relations at nanohertz frequencies,” Phys. Rev. D 110 (2024), no. 12 124018, 2408.10122

    work page arXiv 2024

  73. [73]

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

    M. Minamitsuji and J. Edholm, “Black holes with a nonconstant kinetic term in degenerate higher-order scalar tensor theories,” Phys. Rev. D 101 (2020), no. 4 044034, 1912.01744

    work page arXiv 2020

  74. [74]

    J. M. Mart´ ın-Garc´ ıa, “xAct.”http://www.xact.es/. 40

    work page