---
abstract: 'The odd parity gravitational Quasi-Normal Mode spectrum of black holes with non-trivial scalar hair in Horndeski gravity is investigated. We study ‘almost’ Schwarzschild black holes such that any modifications to the spacetime geometry (including the scalar field profile) are treated perturbatively. A modified Regge-Wheeler style equation for the odd parity gravitational degree of freedom is presented to quadratic order in the scalar hair and spacetime modifications, and a parameterisation of the modified Quasi-Normal Mode spectrum is calculated. In addition, statistical error estimates for the new hairy parameters of the black hole and scalar field are given.'
author:
- 'Oliver J. Tattersall'
bibliography:
- 'RefModifiedGravity.bib'
date: 'Received ; published – 00, 0000'
title: 'Quasi-Normal Modes of Hairy Scalar Tensor Black Holes: Odd Parity'
---

Introduction
============

Gravitational wave (GW) astronomy is now in full swing, thanks to numerous and frequent observations of compact object mergers by advanced LIGO and VIRGO [@LIGOScientific:2018mvr]. With next generation ground and space based GW detectors on the horizon, the prospect of performing black hole spectroscopy (BHS) [@Dreyer:2003bv; @Berti:2005ys; @Gossan:2011ha; @Meidam:2014jpa; @Berti:2015itd; @Berti:2016lat; @Berti:2018vdi; @Baibhav:2018rfk; @2019arXiv190209199B; @Giesler:2019uxc; @Bhagwat:2019dtm; @Bhagwat:2019bwv; @Maselli:2019mjd; @Ota:2019bzl; @Cabero:2019zyt] (the gravitational analog to atomic spectroscopy) is tantalisingly close. With BHS, one aims to discern multiple distinct frequencies of gravitational waves emitted during the ringdown of the highly perturbed remnant black hole of a merger event.

These frequencies, known as Quasi-Normal Modes (QNMs), act as fingerprints for a black hole, being dependent on both the background properties of a black hole (e.g. its mass) and on the laws of gravity [@1975RSPSA.343..289C; @0264-9381-16-12-201; @Kokkotas:1999bd; @Berti:2009kk; @Konoplya:2011qq]. In General Relativity (GR), the QNM spectrum of a Kerr black hole is entirely determined by its mass and angular momentum, and the black hole is said to have no further ‘hairs’ [@Kerr:1963ud; @Israel:1967wq; @Israel:1967za; @Carter:1971zc; @1972CMaPh..25..152H; @PhysRevD.5.2403]. Thus the detection of multiple QNMs in the ringdown portion of a gravitational wave signal allows a consistency check between the inferred values of $M$ and $J$ from each frequency.

In gravity theories other than GR, however, the situation can be markedly different. For example, black holes may not be described by the Kerr solution, and may have properties other than mass or angular momentum that affect its QNM spectrum. Such black holes are said to have ‘hair’ and, despite no-hair theorems existing for various facets of modified gravity, finding and studying hairy black hole solutions is at the forefront of strong gravity research [@Blazquez-Salcedo:2016enn; @Blazquez-Salcedo:2017txk; @Silva:2017uqg; @Antoniou:2017acq; @Antoniou:2017hxj; @Bakopoulos:2018nui; @Silva:2018qhn; @Minamitsuji:2018xde; @Sullivan:2019vyi; @Ripley:2019irj; @Macedo:2019sem; @Konoplya:2001ji; @Dong:2017toi; @Endlich:2017tqa; @Cardoso:2018ptl; @Brito:2018hjh; @Franciolini:2018uyq; @Okounkova:2019zjf]. On the other hand, even if black holes in modified gravity theories are described by the same background solution as in GR (i.e. they have no hair), their perturbations may obey modified equations of motion that alter the emitted gravitational wave signal [@Barausse:2008xv; @Molina:2010fb; @Tattersall:2017erk; @Tattersall:2018nve; @Tattersall:2019pvx].

In this paper we will investigate the first possibility, where modified gravity black holes are altered from their usual description in GR due to their interactions with new gravitational fields. We will, however, assume that black holes are (to first order at least) well described by the GR solutions, and any modifications to the background spacetime are treated perturbatively. As various observations appear to suggest that black holes are well described by the suite of GR solutions [@2016PhRvL.116v1101A; @Isi:2019aib], this approach seems sensible. In this way we can treat the new modified QNM spectrum of these hairy black holes as a small correction to the original GR spectrum, greatly simplifying the analytical and numerical analysis.

We will specifically focus on the Horndeski family of scalar-tensor theories of gravity [@Horndeski:1974wa], where a new gravitational scalar field interacts non-minimally with the metric. Furthermore, for simplicity, we will restrict ourselves to looking only at the odd parity sector of perturbations to spherically symmetric black holes, i.e. we will assume that the black holes studied here are described by a slightly modified Schwarzschild metric. The extension of this work to the even parity sector of spherically symmetric black holes, and to include the effects of rotation, are left as future exercises.

*Summary*: In section \[horndeskisection\] we will introduce the action for Horndeski gravity, the hairy black hole metric and scalar field profile that we are considering, and explore the odd parity gravitational perturbations of this system. In section \[QNMsection\] we will utilise the results of [@Cardoso:2019mqo] to calculate the modified QNM spectrum of the modified black hole, and provide observational error estimates for the new hairy parameters. We will then conclude with a discussion of the results presented here.

Throughout we will use natural units with $G=c=1$, except where otherwise stated. The metric signature will be mostly positive.

Horndeski Gravity {#horndeskisection}
=================

Background
----------

A general action for scalar-tensor gravity with 2$^{nd}$ order-derivative equations of motion is given by the Horndeski action [@Horndeski:1974wa; @Kobayashi:2011nu]: $$\begin{aligned}
S=\int d^4x\sqrt{-g}\sum_{n=2}^5L_n,\label{Shorndeski}\end{aligned}$$ where the component Horndeski Lagrangians are given by: $$\begin{aligned}
L_2&=G_2(\phi,X)\nonumber\\
L_3&=-G_3(\phi,X)\Box \phi\nonumber\\
L_4&=G_4(\phi,X)R+G_{4X}(\phi,X)((\Box\phi)^2-\phi^{\alpha\beta}\phi_{\alpha\beta} )\nonumber\\
L_5&=G_5(\phi,X)G_{\alpha\beta}\phi^{\alpha\beta}-\frac{1}{6}G_{5X}(\phi,X)((\Box\phi)^3 \nonumber\\
& -3\phi^{\alpha\beta}\phi_{\alpha\beta}\Box\phi +2 \phi_{\alpha\beta}\phi^{\alpha\sigma}\phi^{\beta}_{\sigma}),\end{aligned}$$ where $\phi$ is the scalar field with kinetic term $X=-\phi_\alpha\phi^\alpha/2$, $\phi_\alpha=\nabla_\alpha\phi$, $\phi_{\alpha\beta}=\nabla_\alpha\nabla_\beta\phi$, and $G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}R\,g_{\alpha\beta}$ is the Einstein tensor. The $G_i$ are arbitrary functions of $\phi$ and $X$, with derivatives $G_{iX}$ with respect to $X$. GR is given by the choice $G_4=M_{P}^2/2$ with all other $G_i$ vanishing and $M_{P}$ being the reduced Planck mass. Note that eq. (\[Shorndeski\]) is *not* the most general action for scalar-tensor theories, and it has been shown that it can be extended to an arbitrary number of terms [@Zumalacarregui:2013pma; @Gleyzes:2014qga; @Gleyzes:2014dya; @Achour:2016rkg].

For a spherically symmetric black hole solution in Horndeski gravity we assume the following form for the metric $g$ and scalar field $\phi$ in ‘Schwarzschild-like’ coordinates:

$$\begin{aligned}
ds^2=&\;g_{\mu\nu}dx^\mu dx^\nu=\;-A(r)dt^2+B(r)^{-1}dr^2+C(r)d\Omega^2\\
\phi=&\;\phi(r)\end{aligned}$$

where $d\Omega^2$ is the metric on the unit 2-sphere.

Our starting point will be a hairless Schwarzschild solution, as in GR, such that $A=B=1-2M/r$, $C=r^2$ and $\phi=\phi_0=const$, where $M$ is the mass of the black hole. We will now introduce perturbatively small ‘hair’ in both the spacetime geometry and in the scalar field profile, leading to a modified ‘almost’ Schwarzschild black hole. Using $\epsilon$ as a book keeping parameter to track the perturbative order of our expansion, we make the following ansatz to second order in $\epsilon$:

\[ansatz\] $$\begin{aligned}
A(r)=&\;B(r)=1-\frac{2M}{r}+\epsilon \delta A_1(r) + \epsilon^2 \delta A_2(r)+\mathcal{O}(\epsilon^3)\label{gbackground}\\
C(r)=&\;\left(1+\epsilon \delta C_1(r) + \epsilon^2 \delta C_2(r)\right) r^2 + \mathcal{O}(\epsilon^3)\\
\phi(r)=&\;\phi_0+\epsilon \delta \phi_1(r)+ \epsilon^2 \delta\phi_2(r)+\mathcal{O}(\epsilon^3),\label{phibackground}\end{aligned}$$

where we are remaining agnostic as to the exact form of the modifications, merely supposing that such perturbations could exist.

We include second order effects to account for the possibility that a small first order modification to the scalar profile $\delta\phi_1$ may only back-react onto the metric at $\mathcal{O}(\epsilon^2)$ due to the effective energy momentum tensor sourcing the metric being quadratic in $\phi$. Nevertheless we leave open the possibility that the metric is also perturbed at first order in $\epsilon$ through some non-minimal coupling.

Black Hole Perturbations {#perturbsec}
------------------------

We now consider odd parity perturbations to the ‘almost Schwarzschild’ black hole described by eq. (\[gbackground\]) - (\[phibackground\]). For simplicity we will only be considering odd parity perturbations, and as such we do not need to consider the coupling of the metric perturbations to the scalar degree of freedom, but rather only the odd parity metric degree of freedom. An analysis of the even parity sector for perturbatively hairy black holes in Horndeski gravity is left as a future extension to this work; the stability of generic spherically symmetric black holes in Horndeski gravity was studied in [@Kobayashi:2012kh; @Kobayashi:2014wsa], whilst [@Franciolini:2018uyq] builds an effective field theory for QNMs in scalar-tensor gravity in the unitary gauge.

In the Regge-Wheeler gauge [@Regge:1957td], odd parity perturbations $h_{\mu\nu}$ to the metric $g_{\mu\nu}$ can be written in the following way: $$\begin{aligned}
 h_{\mu\nu,\ell m}^{odd}=&
 \begin{pmatrix}
 0&0&h_0(r)B^{\ell m}_\theta&h_0(r)B^{\ell m}_\phi\\
 0&0&h_1(r)B^{\ell m}_\theta&h_1(r)B^{\ell m}_\phi\\
 sym&sym&0&0\\
 sym&sym&0&0
 \end{pmatrix}e^{-i\omega t}\end{aligned}$$ where $sym$ indicates a symmetric entry, $B^{\ell m}_\mu$ is the odd parity vector spherical harmonic and $Y^{\ell m}$ is the standard scalar spherical harmonic: $$\begin{aligned}
B^{\ell m}_\theta = -\frac{1}{\sin\theta}\frac{\partial}{\partial\phi}Y^{\ell m},\quad B^{\ell m}_\phi = \sin\theta\frac{\partial}{\partial\theta}Y^{\ell m}.\end{aligned}$$

Through manipulation of the perturbed Horndeski equations of motion one can show that $h_0$ becomes an auxiliary field, whilst a redefined field $Q(h_1)$ is shown in [@Ganguly:2017ort] to obey the following equation of motion: $$\begin{aligned}
\left[\frac{d^2}{dr_\ast^2}+\frac{\mathcal{F}}{\mathcal{G}}\omega^2-\mathscr{V}\right]Q=0\label{RWgen}\end{aligned}$$ where $r_\ast$ is the tortoise coordinate defined by $dr=\sqrt{AB}dr_\ast$, and the potential $\mathscr{V}$ is given by:

$$\begin{aligned}
\mathscr{V}=&\;\ell(\ell+1)\frac{A}{C}\frac{\mathcal{F}}{\mathcal{H}}-\frac{C^2}{4C^\prime}\left(\frac{ABC^{\prime 2}}{C^3}\right)^\prime-\frac{C^2\mathcal{F}^2}{4\mathcal{F}^\prime}\left(\frac{AB\mathcal{F}^{\prime 2}}{C^2\mathcal{F}^3}\right)^\prime -\frac{2A\mathcal{F}}{C\mathcal{H}}.\label{Vgen}\end{aligned}$$

In the above a prime denotes a derivative with respect to $r$ and the functions $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$ are combinations of the Horndeski $G_i$ functions evaluated at the level of the background:

$$\begin{aligned}
\mathcal{F}=&\;2\left(G_4+\frac{1}{2}B\phi^\prime X^\prime G_{5X}-X G_{5\phi}\right)\\
\mathcal{G}=&\;2\left[G_4-2XG_{4X}+X\left(\frac{A^\prime}{2A}B\phi^\prime G_{5X}+G_{5\phi}\right)\right]\\
\mathcal{H}=&\;2\left[G_4-2XG_{4X}+X\left(\frac{C^\prime}{2C}B\phi^\prime G_{5X}+G_{5\phi}\right)\right].\end{aligned}$$

Furthermore note that we have suppressed spherical harmonic indices for compactness, but eq. (\[RWgen\]) is assumed to hold for each $\ell$.

Eq. (\[RWgen\]) is the analog of the Regge-Wheeler equation [@Regge:1957td] for a generic spherically symmetric black hole in Horndeski gravity. Imposing the boundary conditions that gravitational radiation should be purely ‘ingoing’ at the black hole horizon, and purely ‘outgoing’ at spatial infinity, one can find find the discrete spectrum of QNM frequencies $\omega$ that satisfies eq. (\[RWgen\]).

We now Taylor expand all of the terms in eq. (\[RWgen\]) to $O(\epsilon^2)$ using eq. (\[gbackground\]) - (\[phibackground\]) to take into account the effects of the perturbative black hole hair that we introduced in eq. (\[ansatz\]), resulting in the following:

$$\begin{aligned}
\left[\frac{d^2}{dr_\ast^2}+\omega^2\left(1+\epsilon^2\alpha_{T}(r)\right)-A(r)\left(\frac{\ell(\ell+1)}{r^2}-\frac{6M}{r^3}+\epsilon\delta V_1+\epsilon^2\delta V_2 \right)\right]Q=0\label{RWhairy}\end{aligned}$$

where $\alpha_T$ is the speed excess of gravitational waves [@DeFelice:2011bh; @Bellini:2014fua] given by, to $O(\epsilon^2)$: $$\begin{aligned}
\alpha_T(r)=&\;-\left(1-\frac{2M}{r}\right)\frac{G_{4X}-G_{5\phi}}{G_4}\delta\phi_1^{\prime 2},\label{alphaT}\end{aligned}$$ whilst the potential perturbations are given by:

\[deltaVs\] $$\begin{aligned}
\delta V_1=&\;\frac{1}{2r^2}\left[4\delta A_1-2r\delta A_1^\prime-2(\ell+2)(\ell-1)\delta C_1+2(r-3M)\delta C_1^\prime-r(r-2M)\delta C_1^{\prime\prime}-\frac{G_{4\phi}}{G_4}\left(r\left(r-2M\right)\delta\phi_1^{\prime\prime}-2(r-3M)\delta\phi_1^\prime\right)\right]\label{deltaV1}\\
\delta V_2=&\;\frac{1}{4r^2}\left[8\delta A_2-4r\delta A_2^\prime+4(\ell+2)(\ell-1)\left(\delta C_1^2-\delta C_2\right)+3r(r-2M)\delta C_1^{\prime2}+4(r-3M)\delta C_2^\prime-2r(r-2M)\delta C_2^{\prime\prime}+4r\delta A_1\delta C_1^\prime\right.\nonumber\\
&\left.-2r^2\left(\delta A_1^\prime \delta C_1^\prime+\delta A_1\delta C_1^{\prime\prime}\right)-4(r-3M)\delta C_1\delta C_1^{\prime\prime}+2r(r-2M)\delta C_1 \delta C_1^{\prime\prime}\right]\nonumber\\
&-\frac{1}{2r^2}\frac{G_{4\phi}}{G_4}\left[-2(r-3M)\delta\phi_2^\prime + r \left( r\delta A_1^\prime\delta\phi_1^\prime -\delta A_1\left(2\delta \phi_1^\prime - r \delta\phi_1^{\prime\prime}\right)+(r-2M)\left(\delta\phi_2^{\prime\prime}-\delta C_1^\prime \delta\phi_1^\prime\right)\right)\right]\nonumber\\
& + \frac{1}{4r^2}\left(\frac{G_{4\phi}}{G_4}\right)^2\left[3r(r-2M)\delta\phi_1^{\prime 2}+2\delta\phi_1\left(r(r-2M)\delta\phi_1^{\prime\prime}-2(r-3M)\delta\phi_1^\prime\right)\right]\nonumber\\
& - \frac{1}{2r^2}\frac{G_{4\phi\phi}}{G_4}\left[r(r-2M)\delta\phi_1^{\prime 2}+\delta\phi_1\left(r(r-2M)\delta\phi_1^{\prime\prime}-2(r-3M)\delta\phi_1^\prime\right)\right]\nonumber\\
& - \frac{\alpha_T(r)}{2r^3}\left[-5M+Mr(r-2M)^{-1}-2r(\ell+2)(\ell-1)+r^2(r-2M)\left(\frac{\delta\phi_1^{\prime\prime}}{\delta\phi_1^\prime}\right)^2 + r\left(r(r-2M)\frac{\delta\phi_1^{\prime\prime\prime}}{\delta\phi_1^\prime}-2(r-5M)\frac{\delta\phi_1^{\prime\prime}}{\delta\phi_1^\prime}\right)\right].\label{deltaV2}\end{aligned}$$

We emphasise that in the above expressions all of the $G_i$ Horndeski functions are evaluated at $\phi=\phi_0$ and $X=0$ (i.e. to zeroth order in the book-keeping parameter $\epsilon$), and as such are *constants*. Note that this approach assumes that the $G_i$ are amenable to an expansion around $\phi=\phi_0$ and $X=0$; this is not the case for Einstein-scalar-Gauss-Bonnet gravity, for example, where the $G_i$ include $\log |X|$ terms [@Kobayashi:2011nu].

As expected, to $\mathcal{O}(\epsilon^0)$ eq. (\[RWhairy\]) is simply the well known Regge Wheeler equation describing odd parity gravitational perturbations to a Schwarzschild black hole [@Regge:1957td]. At $\mathcal{O}(\epsilon^0)$ the effective potential of the Regge-Wheeler equation is modified by $\delta V_1$, which is linear in the first order modifications to the spacetime geometry and scalar profile (and their derivatives). Our expression for $\delta V_1$ with $\delta\phi_1=0$ matches that of eq. (5.9) in [@Franciolini:2018uyq], which concerns perturbations of hairy black holes in the unitary gauge (i.e. with $\delta\phi_1=0$).

At $\mathcal{O}(\epsilon^2)$, the potential is further modified by $\delta V_2$, which is quadratic in first order ‘hairy’ terms, and linear in the second order modifications. Furthermore, at second order in the perturbative expansion, we see that the frequency term $\omega^2$ is rescaled by a factor of $c_T=1+\epsilon^2\alpha_T$ where $c_T$ is the propagation speed of gravitational waves in Horndeski gravity [@DeFelice:2011bh; @Bellini:2014fua].

Eqs. (\[RWhairy\]) - (\[deltaVs\]) are the main results of this section. In the next section, we will explore how the modifications introduced to eq. (\[RWhairy\]) by our perturbative hair approach affect the spectrum of QNM frequencies $\omega$ of the black hole. A note of interest, however, is that in the $\omega=0$ limit, eq. (\[RWgen\]) could be used to study the tidal deformation of black holes in Horndeski gravity.

Parameterised QNM Spectrum {#QNMsection}
==========================

In [@Cardoso:2019mqo] (henceforth referred to as Cardoso et al) a formalism is developed such that, given a Schr[ö]{}dinger style QNM style equation: $$\begin{aligned}
\left[f(r)\frac{d}{dr}\left(f(r)\frac{d}{dr}\right)+\omega^2-f(r)\tilde{V}\right]\psi=0\label{RWcardoso}\end{aligned}$$ where $f(r)=1-r_H/r$ with $r_H$ the horizon radius, and $\tilde{V}$ is a modified Regge-Wheeler potential in the following form: $$\begin{aligned}
\tilde{V}=\frac{\ell(\ell+1)}{r^2}-\frac{6M}{r^3}+\frac{1}{r_H^2}\sum_{j=0}^{\infty}\alpha_j\left(\frac{r_H}{r}\right)^j,\end{aligned}$$ the spectrum of frequencies $\omega$ can be described in terms of corrections to the standard GR QNM spectrum. The new frequencies are given by: $$\begin{aligned}
\omega = \; \omega_{\,0} + \sum_{j=0}^{\infty}\alpha_j e_j\end{aligned}$$ where $\omega_{\,0}$ is the unperturbed GR frequency and the $e_j$ are a ‘basis set’ of complex numbers which have been calculated using high precision direct integration of the equations of motion (the reader should consult [@Cardoso:2019mqo] for a detailed explanation of this formalism).

We will now use this approach to calculate the modifications to the QNM spectrum induced by the perturbative black hole hair (with an appropriate power law ansatz for the $\delta (A,C,\phi)_i$). For simplicity and compactness we will present results to only first order in the book-keeping parameter $\epsilon$, such that we are seeking the leading order corrections to $\omega$ in the following form: $$\begin{aligned}
\omega=\;\omega_{\,0} + \epsilon \omega_1.\end{aligned}$$ First, however, we must make sure that our eq. (\[RWhairy\]) is transformed into the same form as eq. (\[RWcardoso\]) so that we can correctly read off the $\alpha_j$ coefficients.

Equation manipulation
---------------------

To first order in $\epsilon$, the modified Regge-Wheeler equation is given by $$\begin{aligned}
\left[A(r)\frac{d}{dr}\left(A(r)\frac{d}{dr}\right)+\omega^2-A(r)\overline{V}\right]Q=0\label{RWnat}\end{aligned}$$ where the effective potential $\overline{V}$ is given by: $$\begin{aligned}
\overline{V}=\frac{\ell(\ell+1)}{r^2}-\frac{6M}{r^3}+\epsilon\delta V_1\label{Vbar}\end{aligned}$$ Following the procedure introduced in Cardoso et al, the first step to obtain an equation in the form of eq. (\[RWhairy\]) is to write: $$\begin{aligned}
A(r)=f(r)Z(r)\label{AtoF}\end{aligned}$$ where $f(r)=1-r_H/r$, and find appropriate expressions for $r_H$ and $Z$ to $O(\epsilon)$. The location of the horizon in our modified spacetime will not be exactly at $r=2M$, but will be corrected due to $\delta A_1$. We thus make the following expansion for the horizon radius: $$\begin{aligned}
r_H=2M+\epsilon \delta r_{H,1}.\end{aligned}$$ To find the new position of the horizon, we require $A(r_H)=0$. Solving order by order in $\epsilon$, we find the following for the location of the horizon: $$\begin{aligned}
\delta r_{H,1}=&\;-2M\delta A_1(2M)\label{deltarH}\end{aligned}$$ with $Z$ thus given by: $$\begin{aligned}
Z(r)=&\; 1+ \epsilon \delta Z_1\nonumber\\
=&\; 1+ \epsilon \frac{\delta A_1(r)-\frac{2M}{r}\delta A_1(2M)}{1-2M/r}\end{aligned}$$ in order to make eq. (\[AtoF\]) hold to $O(\epsilon)$.

If we now define $\tilde{Q}=\sqrt{Z}Q$, we transform eq. (\[RWnat\]) into $$\begin{aligned}
\left[f(r)\frac{d}{dr}\left(f(r)\frac{d}{dr}\right)+\frac{\omega^2}{Z^2}-f(r)V\right]\tilde{Q}=0\label{RWmod}\end{aligned}$$ where the new potential $V$ is given by: $$\begin{aligned}
V=\frac{\overline{V}}{Z}-\frac{f\left(Z^\prime\right)^2-2Z(fZ^\prime)^\prime}{4Z^2}.\end{aligned}$$ and $\overline{V}$ is still given by eq. (\[Vbar\]).

We can expand the $\omega^2$ term in eq. (\[RWmod\]) to $O(\epsilon)$ and write it in the following way: $$\begin{aligned}
\frac{\omega^2}{Z^2}=&\;\omega^2(1-2\epsilon\delta Z_1(r_H))-2\epsilon\omega^2(\delta Z_1(r)-\delta Z_1(r_H))\label{omegamod}\end{aligned}$$ The first term on the right hand side of eq. (\[omegamod\]) can be seen as a (constant) rescaling of the frequencies, whilst the second term can be absorbed into the perturbed potential $V$ by setting $\omega=\omega_{\,0}$. The final form of the modified Regge Wheeler equation is now in the same form as eq. (\[RWcardoso\]): $$\begin{aligned}
\left[f(r)\frac{d}{dr}\left(f(r)\frac{d}{dr}\right)+\tilde{\omega}^2-f(r)\tilde{V}\right]\tilde{Q}=0\end{aligned}$$ where $$\begin{aligned}
\tilde{\omega}^2=&\;\omega^2(1-2\delta Z_1(r_H))\\
\tilde{V}=&\;V+\frac{2\omega_0^2}{f(r)}(\delta Z_1(r)-\delta Z_1(r_H))\label{Vomega}\end{aligned}$$

The final step before we are able to calculate numerically the modified QNM spectrum of our hairy Horndeski black holes is to assume an appropriate functional form for $\delta A_i$, $\delta C_i$ and $\delta \phi_i$. We will make the following simple power law choices: $$\begin{aligned}
\delta \phi_1 = &\;Q_1\left(\frac{2M}{r}\right),\;\; \delta A_1 = \;a_1\left(\frac{2M}{r}\right)^{2},\;\;\delta C_1 = \; c_{1}\left(\frac{2M}{r}\right)\label{deltaansatz}\end{aligned}$$ so that, in addition to the Horndeski $G_i$ parameters, we have 3 ‘hairs’, $Q_{1}$, $a_{1}$, and $c_{1}$ that can affect our QNM spectrum. Of course the hairy parameters may be related when considering specific solutions, but for now we will assume that they are independent.

With the above ansatz we find the non-zero $\alpha_j$ are given by (absorbing $\epsilon$ into the definitions of $(a,c,Q)_1$):

\[alphas\] $$\begin{aligned}
\alpha_0=&\ 8M^2\omega_0^2a_1\\
\alpha_3=&\;\ell(\ell+1)(a_1-c_1)-a_1-2Q_1\frac{G_{4\phi}}{G_4}\\
\alpha_4=&\;\frac{5}{2}\left(a_1+c_1+Q_1\frac{G_{4\phi}}{G_4}\right)\end{aligned}$$

leading to the following corrections to the QNM frequency spectrum for the $\ell=2,3$ modes (for example):

\[qnmspec\] $$\begin{aligned}
M\omega_1^{\ell=2}=&\;Q_1\frac{G_{4\phi}}{G_4}\left[-0.0126+0.0032i\right]+a_1\left[-0.0267+0.0621i\right]+c_1\left[-0.1296+0.0106i\right]\\
M\omega_1^{\ell=3}=&\;Q_1\frac{G_{4\phi}}{G_4}\left[-0.0075+0.0008i\right]+a_1\left[-0.1326+0.0677i\right]+c_1\left[-0.2040+0.0110i\right]\end{aligned}$$

The choices made in eq. (\[deltaansatz\]) were simply to give a concrete example of a modified QNM in terms of the Horndeski (and new ‘hairy’) parameters; one could of course make a different ansatz of one’s choosing to calculate $\omega_1$ (though it should noted that the numerical results of Cardoso et al only apply for those potentials which can be expressed as a series in inverse integer powers of $r$ - for potentials that do not fit this form alternative methods of calculating the QNM spectrum will have to be deployed [@1985RSPSA.402..285L; @2009CQGra..26v5003D; @Pani:2012zz]).

Figure \[fig1\] shows the effect that each of $a_1$, $c_1$, and $\tilde{Q}_1=Q_1G_{4\phi}/G_4$ has on the form of the (real part of the) effective potential $f(r)\tilde{V}$ for $\ell=2$. We see that a non-zero $a_1$ leads to the most noticeable modification to the effective potential. This is unsurprising due to a non-zero $a_1$ leading to a shift in the position of the horizon through eq. (\[deltarH\]), as well as giving rise to an effective mass-squared term due to $\alpha_0$ in eq. (\[alphas\]).

![image](plots.pdf){width="90.00000%"}\

Parameter Estimation {#paramsec}
--------------------

We will now follow the Fisher matrix approach of [@Berti:2005ys] (to which the reader should refer to for an in-depth treatment of statistical errors and ringdown observations) for performing a parameter estimation analysis on the modified QNM spectrum calculated above.

The ringdown signal observed at a gravitational wave detector from a black hole can be modelled as $h=h_+ F_+ + h_\times F_\times$, where $h$ is the total strain, $h_{+,\times}$ is the strain in each of the $+$ and $\times$ polarisations, and $F_{+,\times}$ are pattern functions which depend on the orientation of the detector with respect to the source, and on polarisation angle. In frequency space, the strain in each polarisation is given by: $$\begin{aligned}
h_+=&\;\frac{A^+_{\ell m}}{\sqrt{2}}\left[e^{i\phi_+} S_{\ell m} b_+(f) + e^{-i\phi_+}S^\ast_{\ell m} b_-(f)\right]\\
h_\times=&\;\frac{A^+_{\ell m}N_\times}{\sqrt{2}}\left[e^{i\phi_\times} S_{\ell m} b_+(f) + e^{-i\phi_\times}S^\ast_{\ell m} b_-(f)\right]\end{aligned}$$ where the amplitude $A_+$, amplitude ratio $N_\times$, and phases $\phi_{+,\times}$ are real. The $S$ are complex spin weight 2 spheroidal harmonics, and $b_{\pm}$ are given by: $$\begin{aligned}
b_{\pm}=&\;\frac{1/\tau_{\ell m}}{(1/\tau_{\ell m})^2+4\pi^2(f\pm f_{\ell m})}\end{aligned}$$ where for a given $(\ell, m)$, $\omega_{\ell m}=2\pi f_{\ell m} - i/\tau_{\ell m}$.

We are interested in calculating the statistical errors in determining the ‘hairy’ parameters that affect the QNM spectrum, and as such we assume that the mass $M$ of the black hole, and thus the unperturbed QNM frequency $\omega_{\,0}$, is known. Furthermore, we will assume that $N_\times=1$ and $\phi_+=\phi_\times=0$, and that $A_+$ is known (effectively resulting in us fixing a specific signal-noise-ratio $\rho$). In [@Berti:2005ys] it is shown that the results for statistical errors are not strongly affected by the values of $N_\times$ or of the phases.

For simplicity we will assume that $c_1=0$, thus we are effectively considering a Reissner-Nordstrom-like black hole with a $1/r$ scalar profile. Defining $\tilde{Q}_1=Q_1G_{4\phi}/G_4$ and setting our book-keeping parameter $\epsilon=1$, we can write the oscillation frequency and damping time of the perturbed $\ell=2$ QNM (for example) as follows:

$$\begin{aligned}
2\pi f=&\;2\pi f_0-0.0126\tilde{Q}_1/M-0.0267a_1/M\label{f2}\\
\tau^{-1}=&\;\tau_0^{-1}-0.0032\tilde{Q}_1/M-0.0621a_1/M.\label{tau2}\end{aligned}$$

Using the Fisher matrix formalism laid out in [@Berti:2005ys], and remembering that we are assuming $M$ to be known exactly, we calculate the following errors for $\tilde{Q}_1$ and $a_1$:

$$\begin{aligned}
\sigma^2_{\tilde{Q}_1}=&\;\frac{1}{2\rho^2q^2}\frac{f^{\prime 2}q^2(1+4q^2)-2fqf^\prime q^\prime + f^2q^{\prime 2}}{\left(\dot{f}q^\prime-f^\prime \dot{q}\right)^2}\\
\sigma^2_{a_1}=&\;\frac{1}{2\rho^2q^2}\frac{\dot{f}^2q^2(1+4q^2)-2fq\dot{f}\dot{q} + f^2\dot{q}^2}{\left(\dot{f}q^\prime-f^\prime \dot{q}\right)^2}\end{aligned}$$

where $q=\pi f \tau$ is the ‘quality factor’ of a given oscillation mode, and we now use the notation $F^\prime\equiv\frac{\partial F}{\partial a_1}$ and $\dot{F}\equiv\frac{\partial F}{\partial \tilde{Q}_1}$ for a quantity $F$.

Assuming a detection of the $\ell=2$ mode, we can use the expressions given in eq. (\[f2\]) and (\[tau2\]) to calculate the errors numerically. Additionally setting $\tilde{Q}_1=a_1=0$, we interpret the following errors as ‘detectability’ limits on the parameters: $$\begin{aligned}
\rho\sigma_{\tilde{Q}_1}\approx&\;12,\quad\rho\sigma_{a_1}\approx\;2.\end{aligned}$$ With an SNR of $\rho\sim10^2$, which could be typical of LISA events, we thus have that $\sigma_{\tilde{Q}_1}\approx0.1$ whilst $\sigma_{a_1}\approx0.02$ (assuming that the mass of the black hole is known with absolute precision).

Eqs. (\[f2\]) and (\[tau2\]) show that the metric Reissner-Nordstrom-like hair $a_1$ has a more significant effect on the frequency and damping time than the scalar hair $\tilde{Q}_1$, so it is unsurprising that we find it possible to constrain $a_1$ to a greater degree than the scalar hair. In fact, in general it perhaps makes intuitive sense that the odd parity QNMs are more affected by modifications to the spacetime geometry than to the scalar profile, given that the scalar perturbations only couple to the even parity sector of the gravitational perturbations.

Discussion
==========

In this paper we have studied the QNMs associated with odd parity gravitational perturbations of spherically symmetric black holes in Horndeski gravity. By assuming that the background solutions for the spacetime geometry and Horndeski scalar field are well described to first order by the hairless Schwarzschild solution, we can treat the effect of any black hole ‘hair’ perturbatively.

Making use of the results for generic spherically symmetric black holes derived in [@Ganguly:2017ort], we present a modified Regge-Wheeler style equation, eq. (\[RWhairy\]), describing odd parity gravitational perturbations. Eq. (\[RWhairy\]) takes into account effects induced by *generic* modifications to both the spacetime geometry and to the background radial profile of the Horndeski scalar field. Labelling the background modifications by a book-keeping parameter $\epsilon$ to keep track of the order of ‘smallness’, we present results to $\mathcal{O}(\epsilon^2)$. We show that the odd parity perturbations are not only affected by changes to the background spacetime, but also by the scalar field profile, with the ‘nonminimal’ and ‘derivative’ couplings to curvature $G_4$ and $G_5$ in the Horndeski action playing a role in eq. (\[RWhairy\]).

Through the formalism of [@Cardoso:2019mqo] the odd parity QNM spectrum of such perturbatively hairy black holes can be calculated (assuming an inverse power law ansatz for the modifications to both the spacetime and scalar profile). In eq. (\[qnmspec\]) the first order modifications to the QNM spectrum are presented for the $\ell=2,3$ modes. It is straightforward to calculate the modifications for other $\ell$ using the results of this paper combined with the numerical data provided in [@Cardoso:2019mqo].

We have thus presented a straightforward way to associate deviations from the expected GR QNM spectrum of black holes to not only modifications to the background spacetime, but also to fundamental parameters of a modified gravity theory (in this case, the $G_i$ of Horndeski gravity). In section \[paramsec\] we perform a simple parameter estimation exercise based on the hypothetical observation of the $\ell=2$ QNM of a black hole whose mass we are assuming to know. With SNRs typical of LISA detections we show that the ‘hairy’ black hole parameters introduced could potentially be well constrained.

There are of course numerous ways to develop the work presented here. As mentioned briefly in section \[perturbsec\], eq. (\[RWgen\]) with $\omega=0$ could be used to study the tidal deformations of black holes in Horndeski gravity. Furthermore, one could attempt to find exact solutions for the $\delta(A,C,\phi)_i$ in different realisations of Horndeski gravity (through finding ‘order-by-order’ solutions of otherwise).

The most natural extension to this work is of course to study the even parity sector of perturbations in Horndeski gravity. In general the even parity sector of gravitational perturbations is more complex than the odd parity sector, and in Horndeski gravity this is only further complicated through the coupling of scalar perturbations to the gravitational modes. The formalism of [@Cardoso:2019mqo] has, usefully, been expanded to apply to coupled QNM equations in [@McManus:2019ulj], thus calculating the modified even parity QNM spectrum should be relatively straightforward once the relevant equations have been derived. Such an analysis will then provide a complete description of ‘almost’ Schwarzschild QNMs in Horndeski theory.

Perhaps the most important extension to this line of research is to include black hole spin, given that the black holes currently observed through merger events appear to possess non-negligible angular momentum [@LIGOScientific:2018mvr]. As a first step, one could consider studying slowly rotating ‘almost Kerr’ black holes in Horndeski gravity by introducing another ‘hairy’ function in the $g_{t\phi}$ component of the slowly rotating Kerr metric. More ambitiously, perhaps an ‘almost’ Teukoslky like equation could be found by introducing perturbations to the full Kerr solution. Perturbations of a stealth Kerr black hole (i.e. a Kerr geometry endowed with a non-trivial scalar profile) in Degenerate Higher Order Scalar Tensor theories were been studied in [@Charmousis:2019fre].

Acknowledgments {#acknowledgments .unnumbered}
===============

OJT would like to thank Pedro Ferreira, Vitor Cardoso, and Adrien Kuntz for useful conversations whilst preparing this work. OJT acknowledges support from the European Research Council.
