Quasi-normal modes of hairy scalar tensor black holes: odd parity

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 as small quantities. 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.


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 [1]. With next generation ground and space based GW detectors on the horizon, the prospect of performing black hole spectroscopy (BHS) [2][3][4][5][6][7][8][9][10][11][12][13][14] (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 ngerprints for a black hole, being dependent on both the background properties of a black hole (e.g. its mass) and Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. on the laws of gravity [15][16][17][18][19]. 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' [20][21][22][23][24][25]. 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 [26] the least damped QNM (assumed to be the ℓ = m = 2 fundamental overtone) of the rst gravitational wave detection GW150914 is observed. The values of the oscillation frequency and damping time of this mode are consistent with the predictions of GR, assuming a Kerr black hole with the inferred mass and spin of the GW150914 remnant. As mentioned, however, multiple modes must be observed to test the consistency of the GR spectrum.
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 modi ed gravity, nding and studying hairy black hole solutions is at the forefront of strong gravity research [27][28][29][30][31][32][33][34][35][36][37][38][39][40]. On the other hand, even if black holes in modi ed gravity theories are described by the same background solution as in GR (i.e. they have no hair), their perturbations may obey modi ed equations of motion that alter the emitted gravitational wave signal [41][42][43][44][45].
In this paper we will investigate the rst possibility, where modi ed gravity black holes are altered from their usual description in GR due to their interactions with new gravitational elds. We will, however, assume that black holes are (to rst order at least) well described by the GR solutions, and any modi cations to the background spacetime are treated as small quantities. As various observations appear to suggest that black holes are well described by the suite of GR solutions [26,46], this approach seems sensible. In this way we can treat the new modi ed QNM spectrum of these hairy black holes as a small correction to the original GR spectrum, greatly simplifying the analytical and numerical analysis. This is analogous to the study of slowly rotating black holes, where the Kerr background solution is treated as a small modi cation to the Schwarzschild metric (with the dimensionless black hole spin considered as an 'expansion' parameter), and gravitational wave perturbations are studied on top of this new background (see, for example, [47][48][49][50]). In this way, even without full knowledge of an exact black hole solution, we can probe how the QNM spectrum will be affected by modi cations to the geometry and scalar pro le.
We will speci cally focus on the Horndeski family of scalar-tensor theories of gravity [51], where a new gravitational scalar eld interacts non-minimally with the metric. The motivation for working with Horndeski gravity is that it encompasses a large family of scalar-tensor theories of gravity (perhaps the simplest extension to GR that one could imagine), ranging from models which describe dark energy to those inspired by string theory [52]. 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 modi ed 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 2 we will introduce the action for Horndeski gravity, the hairy black hole metric and scalar eld pro le that we are considering, and explore the odd parity gravitational perturbations of this system. In section 3 we will utilise the results of [53] to calculate the modi ed QNM spectrum of the modi ed black hole, and provide observational error estimates for the new hairy parameters given a power law ansatz for the hairy modi cations. 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.

Background
A general action for scalar-tensor gravity is given by the Horndeski action [51,54]: where the component Horndeski Lagrangians are given by: where φ is the scalar eld with kinetic term X = −φ α φ α /2, φ α = ∇ α φ, φ αβ = ∇ α ∇ β φ, and G αβ = R αβ − 1 2 R g αβ is the Einstein tensor. The G i are arbitrary functions of φ and X, with derivatives G iX with respect to X. GR is given by the choice G 4 = M 2 P /2 with all other G i vanishing and M P being the reduced Planck mass.
The Horndeski action is formulated in such a way as to ensure 2nd order-derivative equations of motion, free from any Ostrogradski instability related to higher order time derivatives [55]. Note that equation (1) 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 [56][57][58][59].
Horndeski theories can be consistent with Solar System tests of gravity through screening mechanisms [60], and the conditions for Horndeski theories to have appropriate Einstein gravity limits have been studied in [61]. Recently, strong constraints were placed on the Horndeski action by the observation of gravitational waves from a binary neutron star merger GW170817 by LIGO and VIRGO [62], along with its optical counterpart the gamma ray burst GRB 170817A [63][64][65][66][67]. Due to the almost coincident arrival of both gravitational waves and photons at Earth from this event, the speed of gravitational waves was constrained to be within 1 part in 10 15 of that of light, leading to tight constraints on the Horndeski action (as well as other modi ed gravity theories) [68][69][70][71][72][73]. Speci cally, we require G 4 = G 4 (φ) and G 5 = 0 to satisfy the gravitational wave speed constraint.
These constraints were, however, made under the assumption that the Horndeski scalar eld plays a signi cant cosmological role. If one is instead content to allow φ to become unimportant for cosmology (and thus to the propagation of gravitational waves over cosmological distances), we are still free to consider G 4 and G 5 in their entirety 1 This is what we will do in this paper, allowing the full suite of Horndeski functions to be relevant to black hole solutions, whilst knowing that those terms which contribute to the speed excess of gravitational waves must vanish in a cosmological setting. For a discussion of black hole solutions where we do require any non-GR elds to maintain a cosmological signi cance, see [75].
For a spherically symmetric black hole solution in Horndeski gravity we assume the following form for the metric g and scalar eld φ in 'Schwarzschild-like' coordinates: where dΩ 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 φ = φ 0 = const, where M is the mass of the black hole. We will now introduce small deviations as 'hair' in both the spacetime geometry and in the scalar eld pro le, leading to a modi ed 'almost' Schwarzschild black hole. Using ǫ as a book keeping parameter to track the order of smallness of the hair, we make the following ansatz to second order in ǫ: where we are remaining agnostic as to the exact form of the modi cations, merely supposing that such perturbations could exist. Note that the functions δA i will shift the location of the horizon from r = 2M. We will look into this further in a later section. Asymptotically we might expect any O(ǫ) terms to decay as r → ∞ so as to recover at space far away from the black hole. It is also conceivable, however, that modi ed gravity effects might manifest themselves as an apparent cosmological constant term, leading to an asymptotically de Sitter (or anti-de Sitter) form for the metric.

Black hole perturbations
We now consider odd parity perturbations to the 'almost Schwarzschild' black hole described by equations (4a)-(4c). For simplicity we will only be considering odd parity perturbations, and as such we do not need to consider the coupling of the even parity metric perturbation to the scalar degree of freedom (nor, indeed, perturbations to the effective energy momentum tensor of the scalar eld). In this way we can see the effect on the QNM spectrum of the black hole due entirely to the hairy nature of the background, and not to (for example) couplings to scalar eld perturbations. 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 [76,77], whilst [40] builds an effective eld theory for QNMs in scalar-tensor gravity in the unitary gauge.
In the Regge-Wheeler gauge [78], odd parity perturbations h µν to the metric g µν can be decomposed into tensorial spherical harmonics and written in terms of two 'perturbation elds' h 0 (r) and h 1 (r) in the following way: where Y ℓm is the usual scalar spherical harmonic. Note that in the Regge-Wheeler gauge we have been able to set a third perturbation eld h 2 (r) (which would have populated the bottom right hand corner of h odd µν ) to zero. After expanding the action given by equation (1) to second order in the perturbation elds h i , and integrating over θ and φ, it was shown in [79] that the following action is obtained: where a dot represents a time derivative and a prime a radial derivative. The coef cients a i are given in appendix A.
We can see in equation (6) that h 0 is an auxiliary eld (i.e. without a time derivative). Through further manipulation of equation (6), a rede ned eld Q(h 1 ) is shown in [79] to obey the following equation of motion: where r * is the tortoise coordinate de ned by dr = √ ABdr * , and the potential V is given by: The functions F , G, and H are combinations of the Horndeski G i functions evaluated at the level of the background: Furthermore note that we have suppressed spherical harmonic indices for compactness, but equation (7) is assumed to hold for each ℓ. Equation (7) is the analog of the Regge-Wheeler equation [78] 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 in nity, one can nd the discrete spectrum of QNM frequencies ω that satis es equation (7).
We now Taylor expand all of the terms in equation (7) to O(ǫ 2 ) using equations (4a)-(4c) to take into account the effects of the perturbative black hole hair that we introduced in equation (4), resulting in the following: where α T is the speed excess of gravitational waves [80,81] given by, to O(ǫ 2 ): whilst the potential perturbations are given by: We emphasise that in the above expressions all of the G i Horndeski functions are evaluated at φ = φ 0 and X = 0 (i.e. to zeroth order in the book-keeping parameter ǫ), and as such are constants. Note that this approach assumes that the G i are amenable to an expansion around φ = φ 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 [54]. As expected, to O(ǫ 0 ) equation (10) is simply the well known Regge Wheeler equation describing odd parity gravitational perturbations to a Schwarzschild black hole [78].
At O(ǫ 0 ) the effective potential of the Regge-Wheeler equation is modi ed by δV 1 , which is linear in the rst order modi cations to the spacetime geometry and scalar pro le (and their derivatives). Our expression for δV 1 with δφ 1 = 0 matches that of equation (5.9) in [40], which concerns perturbations of hairy black holes in the unitary gauge (i.e. with δφ 1 = 0).
At O(ǫ 2 ), the potential is further modi ed by δV 2 , which is quadratic in rst order 'hairy' terms, and linear in the second order modi cations. Furthermore, at second order in the perturbative expansion, we see that the frequency term ω 2 is rescaled by a factor of c T = 1 + ǫ 2 α T where c T is the propagation speed of gravitational waves in Horndeski gravity [80,81]. As discussed previously, this term would have to vanish in cosmological settings to satisfy the constraints obtained from GW/GRB170817. Interestingly, we see that even in the case of pure Schwarzschild geometry (i.e. with δA i = δC i = 0), if there is non-minimal coupling between the scalar eld and metric such that G 4φ or G 4φφ = 0, the QNM spectrum can be modi ed by the presence of a non-trivial scalar radial pro le.
With regards to the stability of this slightly hairy black hole, in [76,79] it is given that a necessary condition to avoid 'no-ghost' and 'Laplacian' instabilities is that F , G, and H are all positive. Looking at equation (9), we see that for the ansatz given by equation (4) each of the functions is equal to 2G 4 (φ 0 , 0)(1 + O(ǫ)). As G 4 (φ 0 , 0) plays the role of the constant background Planck mass in Horndeski, this is clearly positive. Thus each of F , G, and H must also be positive for ǫ ≪ 1.
Turning now to stability against odd parity perturbations, if the potential given by equation (8) is positive everywhere then we are ensured stability. Using equation (10), we see that the potential has the following form: Clearly V vanishes on the horizon r H where A(r H ) = 0, and is positive for intermediate values of r (the Regge-Wheeler potential is everywhere positive, and the O(ǫ) terms are assumed to be much smaller than the standard GR terms). If each of the functions δA i , δC i , and δφ i also decay at least as quickly as 1/r, then the leading order term of V is ℓ(ℓ + 1)/r 2 as r → ∞ (as in the usual GR case). This ensures that the potential is everywhere positive, including as r → ∞, and can be seen graphically with an example power law ansatz for the hairy functions in gure 1. For other forms of the hairy functions, e.g. those with de Sitter asymptotics, the stability will have to be analysed separately (examples of applying the S-deformation stability analysis to Horndeski theories are given in [79]). Equation (10)- (12) are the main results of this section. In the next section, we will explore how the modi cations introduced to equation (10) by the small amounts of hair affect the spectrum of QNM frequencies ω of the black hole. A note of interest, however, is that in the ω = 0 limit, equation (7) could be used to study the tidal deformation of black holes in Horndeski gravity.

Parameterised QNM spectrum
In [53] (henceforth referred to as Cardoso et al) a formalism is developed such that, given a Schrödinger style QNM style equation: where f(r) = 1 − r H /r with r H the horizon radius, andṼ is a modi ed Regge-Wheeler potential in the following form: the spectrum of frequencies ω can be described in terms of corrections to the standard GR QNM spectrum. The new frequencies are given by: where ω 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 [53] for a detailed explanation of this formalism).
We will now use this approach to calculate the modi cations to the QNM spectrum induced by the perturbative black hole hair (with an appropriate power law ansatz for the δ(A, C, φ) i ). For simplicity and compactness we will present results to only rst order in the book-keeping parameter ǫ, such that we are seeking the leading order corrections to ω in the following form: First, however, we must make sure that our equation (10) is transformed into the same form as equation (14) so that we can correctly read off the α j coef cients.

Equation manipulation
To rst order in ǫ, the modi ed Regge-Wheeler equation is given by where the effective potential V is given by: Following the procedure introduced in Cardoso et al, the rst step to obtain an equation in the form of equation (10) is to write: where f(r) = 1 − r H /r, and nd appropriate expressions for r H and Z to O(ǫ). The location of the horizon in our modi ed spacetime will not be exactly at r = 2M, but will be corrected due to δA 1 . We thus make the following expansion for the horizon radius: To nd the new position of the horizon, we require A(r H ) = 0. Solving order by order in ǫ, we nd the following for the location of the horizon: with Z thus given by: in order to make equation (20) hold to O(ǫ).
If we now de neQ = √ ZQ, we transform equation (18) into where the new potential V is given by: and V is still given by equation (19). We can expand the ω 2 term in equation (24) to O(ǫ) and write it in the following way: The rst term on the right hand side of equation (26) can be seen as a (constant) rescaling of the frequencies. As the second term on the right hand side is already O(ǫ), it can be absorbed into the perturbed potential V by setting ω = ω 0 . This is because any frequency corrections in this term would result in terms O(ǫ 2 ) or higher (which we are neglecting in this section). The nal form of the modi ed Regge Wheeler equation is now in the same form as equation (14): The nal step before we are able to calculate numerically the modi ed QNM spectrum of our hairy Horndeski black holes is to assume an appropriate functional form for δA i , δC i and δφ i . We will make the following simple power law choices: 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 speci c solutions, but for now we will assume that they are independent.
With the above ansatz we nd the non-zero α j are given by (absorbing ǫ into the de nitions of (a, c, Q) 1 ): leading to the following corrections to the QNM frequency spectrum for the ℓ = 2, 3 modes (for example) where the unperturbed GR frequencies are given by [18]: Looking at the real and imaginary parts of the frequencies separately, with ω GR = ω R + iω I and ω 1 = δω R + iδω I , we nd the following fractional differences for the ℓ = 2 QNM: We see that with, for example 2 , (a 1 , c 1 ,Q 1 ) all ∼ 10 −1 , we might expect fractional deviations from the GR spectrum of around 5% and 9% for the real and imaginary parts of the ℓ = 2 frequency, and of 6% and 9% for ℓ = 3. In [26] the least damped QNM of the GW150914 remnant black hole is predicted using the posterior distributions on the mass and spin of the remnant; the predicted fractional uncertainties on the real and imaginary parts of the frequency in that case are around 3% and 8% respectively. Given that in [26] the uncertainties on the real and imaginary parts of the frequency obtained from searching for the QNM in the data are clearly greater than those obtained from using the predicted nal mass and spin of the black hole, the example ∼ 5-10% fractional deviations postulated here will likely be dominated by other uncertainties with current detectors. Figure 1 shows the effect that each of a 1 , c 1 , andQ 1 has on the form of the rst order 'hairy' potential A(r)V(r) (with V given by equation (19)) for ℓ = 2. We see that a non-zero a 1 leads to the most noticeable modi cation to the effective potential. This is unsurprising due to a nonzero a 1 leading to a shift in the position of the horizon through equation (22), as well as giving rise to an effective mass-squared term due to α 0 in equation (31). Looking at equations 34 and 35, a 1 has the most signi cant effect on δω I , while δω R is most greatly impacted by c 1 .
The choices made in equation (30) were simply to give a concrete example of a modi ed QNM in terms of the Horndeski (and new 'hairy') parameters; one could of course make a different ansatz of one's choosing to calculate ω 1 (though it should be 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 t this form alternative methods of calculating the QNM spectrum will have to be deployed [48,82,83]).
Moving beyond the toy model considered above, we could consider more generic forms for δA 1 , δC 1 and δφ 1 . For example: where m, n, and p are positive integers. The contributions to the perturbed potential that these terms lead to are given in appendix B. In appendix C we consider the case of a 'stealth' black hole, one that has Schwarzschild geometry while being endowed with a non-trivial scalar pro le (that is, δA i = δC i = 0, δφ i = 0). We consider both power law and general scalar pro les.

Parameter estimation
We will now follow the Fisher matrix approach of [3] (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 modi ed QNM spectrum calculated above. In GR, the ringdown signal observed at a gravitational wave detector from a black hole can be modelled as h = h + F + + h × F × , where h is the total strain, h +,× is the strain in each of the + and × polarisations, and F +,× 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: where the amplitude A + , amplitude ratio N × , and phases φ +,× are real. The S are complex spin weight 2 spheroidal harmonics, and b ± are given by: where for a given (ℓ, m), ω ℓm = 2πf ℓm − i/τ ℓm . In general, Horndeski gravity admits another polarisation of gravitational waves in addition to the usual + and × polarisations present in GR: that of the 'breathing' mode (for a massless scalar eld) or mixed breathing-longitudinal mode (for a massive eld) [84,85]. This mode is associated with the oscillations of the additional scalar degree of freedom. As we are only interested in considering the effect of black hole hair on the frequency spectrum associated with the odd parity gravitational degree of freedom in this paper, and not of the scalar degree of freedom, we will neglect the additional polarisation. This is equivalent to setting the scalar eld perturbation to zero, and focussing solely on the gravitational perturbations. Thus we will model the strain h as the usual sum of the GR polarisations in order to gain an estimate of how well one could constrain deviations from the GR QNM spectrum. In [86] it is shown that the odd parity metric degree of freedom contributes to both + and × polarisations.
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 ω 0 , is known. Furthermore, we will assume that N × = 1 and φ + = φ × = 0, and that A + is known (effectively resulting in us xing a speci c signalnoise-ratio ρ). In [3] it is shown that the results for statistical errors are not strongly affected by the values of N × or of the phases.
Furthermore we will assume that c 1 = 0, thus we are effectively considering a Reissner-Nordstrom-like black hole with a 1/r scalar pro le. Again withQ 1 = Q 1 G 4φ /G 4 , and absorbing the book-keeping parameter ǫ into the de nition of our hairy parameters (such that ǫa 1 → a 1 etc), we can write the oscillation frequency and damping time of the perturbed ℓ = 2 mode (for example) as follows: Using the Fisher matrix formalism laid out in [3], and remembering that we are assuming M to be known exactly, we calculate the following errors forQ 1 and a 1 : where q = πfτ is the 'quality factor' of a given oscillation mode, and we now use the notation for a quantity F. Assuming a detection of the ℓ = 2 mode, we can use the expressions given in equations (40a) and (40b) to calculate the errors. Additionally settingQ 1 = a 1 = 0, we interpret the following errors as 'detectability' limits on the parameters: With an SNR of ρ ∼ 10 2 , which could be typical of LISA events, we thus have that σQ 1 ≈ 0.1 whilst σ a 1 ≈ 0.02 (assuming that the mass of the black hole is known with absolute precision). As discussed previously with regards to GW150914, however, this is clearly a highly optimistic scenario for current detectors.
Equation (34) shows that the metric Reissner-Nordstrom-like hair a 1 has a more signi cant effect on the frequency and damping time than the scalar hairQ 1 , so it is unsurprising that we nd 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 modi cations to the spacetime geometry than to the scalar pro le, 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 eld are well described to rst order by the hairless Schwarzschild solution, we can treat the effect of any black hole 'hair' as small modi cations.
Making use of the results for generic spherically symmetric black holes derived in [79], we present a modi ed Regge-Wheeler style equation, equation (10), describing odd parity gravitational perturbations. Equation (10) takes into account effects induced by generic modications to both the spacetime geometry and to the background radial pro le of the Horndeski scalar eld. Labelling the background modi cations by a book-keeping parameter ǫ to keep track of the order of 'smallness', we present results to O(ǫ 2 ). We show that the odd parity perturbations are not only affected by changes to the background spacetime, but also by the scalar eld pro le, with the 'nonminimal' and 'derivative' couplings to curvature G 4 and G 5 in the Horndeski action playing a role in equation (10).
Through the formalism of [53] the odd parity QNM spectrum of such perturbatively hairy black holes can be calculated (assuming an inverse power law ansatz for the modi cations to both the spacetime and scalar pro le). In equation (32) the rst order modi cations to the QNM spectrum are presented for the ℓ = 2, 3 modes for a speci c power law model of the black hole hair. It is straightforward to calculate the modi cations for other ℓ using the results of this paper combined with the numerical data provided in [53]. Results for more generic models of the black hole hair and scalar pro le are provided in appendices B and C.
We have thus presented a straightforward way to associate deviations from the expected GR QNM spectrum of black holes to not only modi cations to the background spacetime, but also to fundamental parameters of a modi ed gravity theory (in this case, the G i of Horndeski gravity). In section 3.2 we perform a simple parameter estimation exercise based on the hypothetical observation of the ℓ = 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. The predicted fractional deviations from the GR frequency spectrum are likely to be dominated by other sources of uncertainty when considering current detectors, however.
There are of course numerous ways to develop the work presented here. As mentioned brie y in section 2.2, equation (7) with ω = 0 could be used to study the tidal deformations of black holes in Horndeski gravity. Furthermore, one could attempt to nd exact solutions for the δ(A, C, φ) i in different realisations of Horndeski gravity (through nding '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 [53] has, usefully, been expanded to apply to coupled QNM equations in [87], thus calculating the modi ed 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 nonnegligible angular momentum [1]. As a rst step, one could consider studying slowly rotating 'almost Kerr' black holes in Horndeski gravity by introducing another 'hairy' function in the g t,φ 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 pro le) in degenerate higher order scalar tensor theories have been studied in [88,89], and of a Kerr black hole in f(R) gravity in [90].
where F , G, H are given by equation (9).

Appendix B. General power law model
For the hairy black hole model given by equation (36), we nd the following rst order contributions to the effective potentialṼ: r (2ℓ(ℓ + 1) − n(2n + 5)) + (−2ℓ(ℓ + 1) + n(n + 3) + 4)+ × 2M r 2 (n − 1)(n + 3) + 4r 2 ω 2 0 + 4r 2 ω 2 0 n 1 − 2M r − 1 (B1) whereQ 1 = Q 1 G 4φ /G 4 , andṼ 1 denotes the linear in ǫ component ofṼ. We see that for arbitrary n it is non-trivial to nd the α j that contribute to the modi cations to the QNM spectrum because of the (1 − 2M/r) 2 factor in the denominator of the a 1 term. For 0 < n < 6, however, the factor in the denominator cancels with the numerator, leading to a nite number of terms in powers of 1/r. In cases where the factor does not cancel, one would have to Taylor expand (1 − 2M/r) −2 to an appropriate number of terms until the desired 'accuracy' of the modi cation to the QNM frequency is reached (i.e. until adding additional terms from the Taylor expansion changes ω 1 less than some predetermined threshold).

Appendix C. Stealth black holes
We will now study the special case of a 'stealth' black hole-i.e. a black hole endowed with non trivial scalar hair whilst maintaining Schwarzschild geometry (and therefore with δA i = δC i = 0). Working to linear order in the hair terms, which in this case is just δφ 1 , we nd the following corrections to the ℓ = 2 QNM using the 1/(ℓ + 1/2) expansion method of [83] when the scalar pro le is of the form given in equation (36)  As δV 1 is linear in the scalar pro le δφ 1 , and because we are considering only linear corrections to the QNM frequencies, one can use equation (C1) to construct the corrected ℓ = 2 mode for a scalar pro le of the following form: where each of the Q (p i ) 1 ≪ 1 but of similar magnitude to each other. One would simply use equation (C1) to calculate the δω R,I for each value of p i required and then sum the corrections to nd the nal frequency.
For a generic scalar pro le (i.e. not limited to integer power laws), we have calculated the QNM frequencies as a function of ℓ and δφ 1 using the expansion method of [83]. These include very lengthy expressions involving up to 12 derivatives of δφ 1 so we do not include them here, but they are presented in a mathematica notebook available online [91]. Indeed, the expressions given in equation (C1) are in fact valid for any p.