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

A general action for scalar–tensor gravity is given by the Horndeski action [51, 54]:

$$\begin{equation}S=\int {\mathrm{d}}^{4}x\sqrt{-g}\sum _{n=2}^{5}{L}_{n},\end{equation} \tag{ 1 }$$

where the component Horndeski Lagrangians are given by:

$$\begin{align}\hfill {L}_{2}& ={G}_{2}\left(\phi ,X\right)\hfill \\ \hfill {L}_{3}& =-{G}_{3}\left(\phi ,X\right)\square \phi \hfill \\ \hfill {L}_{4}& ={G}_{4}\left(\phi ,X\right)R+{G}_{4X}\left(\phi ,X\right)\left({\left(\square \phi \right)}^{2}-{\phi }^{\alpha \beta }{\phi }_{\alpha \beta }\right)\hfill \\ \hfill {L}_{5}& ={G}_{5}\left(\phi ,X\right){G}_{\alpha \beta }{\phi }^{\alpha \beta }-\frac{1}{6}{G}_{5X}\left(\phi ,X\right)\left({\left(\square \phi \right)}^{3}-3{\phi }^{\alpha \beta }{\phi }_{\alpha \beta }\square \phi +2{\phi }_{\alpha \beta }{\phi }^{\alpha \sigma }{\phi }_{\sigma }^{\beta }\right),\hfill \end{align} \tag{ 2 }$$

where ϕ is the scalar field with kinetic term X = −ϕ_α ϕ^α /2, ϕ_α = ∇_α ϕ, ϕ_αβ = ∇_α ∇_β ϕ, 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 ϕ and X, with derivatives G_iX with respect to X. GR is given by the choice ${G}_{4}={M}_{\mathrm{P}}^{2}/2$ 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 [56–59].

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 [63–67]. 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 10¹⁵ of that of light, leading to tight constraints on the Horndeski action (as well as other modified gravity theories) [68–73]. Specifically, we require G₄ = G₄(ϕ) and G₅ = 0 to satisfy the gravitational wave speed constraint.

These constraints were, however, made under the assumption that the Horndeski scalar field plays a significant 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 G₄ and G₅ 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 fields to maintain a cosmological significance, see [75].

For a spherically symmetric black hole solution in Horndeski gravity we assume the following form for the metric g and scalar field ϕ in 'Schwarzschild-like' coordinates:

$$\begin{equation}\mathrm{d}{s}^{2}={g}_{\mu \nu }\mathrm{d}{x}^{\mu }\;\mathrm{d}{x}^{\nu }=\;-A\left(r\right)\mathrm{d}{t}^{2}+B{\left(r\right)}^{-1}\mathrm{d}{r}^{2}+C\left(r\right)\mathrm{d}{{\Omega}}^{2}\end{equation} \tag{ 3a }$$

$$\begin{equation}\phi =\phi \left(r\right)\end{equation} \tag{ 3b }$$

where dΩ² 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 = r² and ϕ = ϕ₀ = 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 field profile, leading to a modified 'almost' Schwarzschild black hole. Using as a book keeping parameter to track the order of smallness of the hair, we make the following ansatz to second order in :

$$\begin{equation}A\left(r\right)=B\left(r\right)=1-\frac{2M}{r}+{\epsilon}\delta {A}_{1}\left(r\right)+{{\epsilon}}^{2}\delta {A}_{2}\left(r\right)+\mathcal{O}\left({{\epsilon}}^{3}\right)\end{equation} \tag{ 4a }$$

$$\begin{equation}C\left(r\right)=\left(1+{\epsilon}\delta {C}_{1}\left(r\right)+{{\epsilon}}^{2}\delta {C}_{2}\left(r\right)\right){r}^{2}+\mathcal{O}\left({{\epsilon}}^{3}\right)\end{equation} \tag{ 4b }$$

$$\begin{equation}\phi \left(r\right)={\phi }_{0}+{\epsilon}\delta {\phi }_{1}\left(r\right)+{{\epsilon}}^{2}\delta {\phi }_{2}\left(r\right)+\mathcal{O}\left({{\epsilon}}^{3}\right),\end{equation} \tag{ 4c }$$

where we are remaining agnostic as to the exact form of the modifications, 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 $\mathcal{O}\left({\epsilon}\right)$ terms to decay as r → ∞ so as to recover flat space far away from the black hole. It is also conceivable, however, that modified 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.

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 field). 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 field 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 field 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 fields' h₀(r) and h₁(r) in the following way:

$$\begin{equation}{h}_{\mu \nu ,\ell m}^{\text{odd}}=\begin{pmatrix}0& 0& -{h}_{0}\left(r\right)\frac{1}{\mathrm{sin} \theta }\frac{\partial }{\partial \phi }& {h}_{0}\left(r\right)\mathrm{sin} \theta \frac{\partial }{\partial \theta }\\ 0& 0& -{h}_{1}\left(r\right)\frac{1}{\mathrm{sin} \theta }\frac{\partial }{\partial \phi }& {h}_{1}\left(r\right)\mathrm{sin} \theta \frac{\partial }{\partial \theta }\\ -{h}_{0}\left(r\right)\frac{1}{\mathrm{sin} \theta }\frac{\partial }{\partial \phi }& -{h}_{1}\left(r\right)\frac{1}{\mathrm{sin} \theta }\frac{\partial }{\partial \phi }& 0& 0\\ {h}_{0}\left(r\right)\mathrm{sin} \theta \frac{\partial }{\partial \theta }& {h}_{1}\left(r\right)\mathrm{sin} \theta \frac{\partial }{\partial \theta }& 0& 0\end{pmatrix}{Y}^{\ell m}{\mathrm{e}}^{-\mathrm{i}\omega t}\end{equation} \tag{ 5 }$$

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 field h₂(r) (which would have populated the bottom right hand corner of ${h}_{\mu \nu }^{\text{odd}}$) to zero.

After expanding the action given by equation (1) to second order in the perturbation fields h_i , and integrating over θ and ϕ, it was shown in [79] that the following action is obtained:

$$\begin{equation}{S}^{\left(2\right)}=\int \;\mathrm{d}t\mathrm{d}r\left[{a}_{1}{h}_{0}^{2}+{a}_{2}{h}_{1}^{2}+{a}_{3}\left({\dot {h}}_{1}^{2}{h}_{0}^{\prime 2}-2{\dot {h}}_{1}{h}_{0}^{\prime }+2\frac{{C}^{\prime }}{C}{\dot {h}}_{1}{h}_{0}\right)\right]\end{equation} \tag{ 6 }$$

where a dot represents a time derivative and a prime a radial derivative. The coefficients a_i are given in appendix A.

We can see in equation (6) that h₀ is an auxiliary field (i.e. without a time derivative). Through further manipulation of equation (6), a redefined field Q(h₁) is shown in [79] to obey the following equation of motion:

$$\begin{equation}\left[\frac{{\mathrm{d}}^{2}}{\mathrm{d}{r}_{{\ast}}^{2}}+\frac{\mathcal{F}}{\mathcal{G}}{\omega }^{2}-\mathcal{V}\right]Q=0\end{equation} \tag{ 7 }$$

where r_* is the tortoise coordinate defined by $\mathrm{d}r=\sqrt{AB}\mathrm{d}{r}_{{\ast}}$, and the potential $\mathcal{V}$ is given by:

$$\begin{equation}\mathcal{V}=\;\ell \left(\ell +1\right)\frac{A}{C}\frac{\mathcal{F}}{\mathcal{H}}-\frac{{C}^{2}}{4{C}^{\prime }}{\left(\frac{AB{C}^{\prime 2}}{{C}^{3}}\right)}^{\prime }-\frac{{C}^{2}{\mathcal{F}}^{2}}{4{\mathcal{F}}^{\prime }}{\left(\frac{AB{\mathcal{F}}^{\prime 2}}{{C}^{2}{\mathcal{F}}^{3}}\right)}^{\prime }-\frac{2A\mathcal{F}}{C\mathcal{H}}.\end{equation} \tag{ 8 }$$

The functions $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$ are combinations of the Horndeski G_i functions evaluated at the level of the background:

$$\begin{equation}\mathcal{F}=2\left({G}_{4}+\frac{1}{2}B{\phi }^{\prime }{X}^{\prime }{G}_{5X}-X{G}_{5\phi }\right)\end{equation} \tag{ 9a }$$

$$\begin{equation}\mathcal{G}=2\left[{G}_{4}-2X{G}_{4X}+X\left(\frac{{A}^{\prime }}{2A}B{\phi }^{\prime }{G}_{5X}+{G}_{5\phi }\right)\right]\end{equation} \tag{ 9b }$$

$$\begin{equation}\mathcal{H}=2\left[{G}_{4}-2X{G}_{4X}+X\left(\frac{{C}^{\prime }}{2C}B{\phi }^{\prime }{G}_{5X}+{G}_{5\phi }\right)\right].\end{equation} \tag{ 9c }$$

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 infinity, one can find the discrete spectrum of QNM frequencies ω that satisfies equation (7).

We now Taylor expand all of the terms in equation (7) to O(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:

$$\begin{align}\hfill \left[\frac{{d}^{2}}{\mathrm{d}{r}_{{\ast}}^{2}}+{\omega }^{2}\left(1+{{\epsilon}}^{2}{\alpha }_{T}\left(r\right)\right)-A\left(r\right)\left(\frac{\ell \left(\ell +1\right)}{{r}^{2}}-\frac{6M}{{r}^{3}}+{\epsilon}\delta {V}_{1}+{{\epsilon}}^{2}\delta {V}_{2}\right)\right]Q=0\end{align} \tag{ 10 }$$

where α_T is the speed excess of gravitational waves [80, 81] given by, to O(²):

$$\begin{equation}{\alpha }_{T}\left(r\right)=\;-\left(1-\frac{2M}{r}\right)\frac{{G}_{4X}-{G}_{5\phi }}{{G}_{4}}\delta {\phi }_{1}^{\prime 2},\end{equation} \tag{ 11 }$$

whilst the potential perturbations are given by:

$$\begin{align}\delta {V}_{1}& =\frac{1}{2{r}^{2}}\left[4\delta {A}_{1}-2r\delta {A}_{1}^{\prime }-2\left(\ell +2\right)\left(\ell -1\right)\delta {C}_{1}+2\left(r-3M\right)\delta {C}_{1}^{\prime }\right.\\ & \quad \left.-r\left(r-2M\right)\delta {C}_{1}^{\prime \prime }-\frac{{G}_{4\phi }}{{G}_{4}}\left(r\left(r-2M\right)\delta {\phi }_{1}^{\prime \prime }-2\left(r-3M\right)\delta {\phi }_{1}^{\prime }\right)\right]\end{align} \tag{ 12a }$$

$$\begin{align}\hfill \delta {V}_{2}& =\frac{1}{4{r}^{2}}\left[8\delta {A}_{2}-4r\delta {A}_{2}^{\prime }+4\left(\ell +2\right)\left(\ell -1\right)\left(\delta {C}_{1}^{2}-\delta {C}_{2}\right)+3r\left(r-2M\right)\delta {C}_{1}^{\prime 2}\right.\hfill \\ \hfill & \quad +4\left(r-3M\right)\delta {C}_{2}^{\prime }-2r\left(r-2M\right)\delta {C}_{2}^{\prime \prime }+4r\delta {A}_{1}\delta {C}_{1}^{\prime }-2{r}^{2}\left(\delta {A}_{1}^{\prime }\delta {C}_{1}^{\prime }+\delta {A}_{1}\delta {C}_{1}^{\prime \prime }\right)\hfill \\ \hfill & \quad \left.-4\left(r-3M\right)\delta {C}_{1}\delta {C}_{1}^{\prime \prime }+2r\left(r-2M\right)\delta {C}_{1}\delta {C}_{1}^{\prime \prime }\right]-\frac{1}{2{r}^{2}}\frac{{G}_{4\phi }}{{G}_{4}}\left[-2\left(r-3M\right)\delta {\phi }_{2}^{\prime }\right.\hfill \\ \hfill & \quad \left.+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)+\left(r-2M\right)\left(\delta {\phi }_{2}^{\prime \prime }-\delta {C}_{1}^{\prime }\delta {\phi }_{1}^{\prime }\right)\right)\right]\hfill \\ \hfill & \quad +\frac{1}{4{r}^{2}}{\left(\frac{{G}_{4\phi }}{{G}_{4}}\right)}^{2}\left[3r\left(r-2M\right)\delta {\phi }_{1}^{\prime 2}+2\delta {\phi }_{1}\left(r\left(r-2M\right)\delta {\phi }_{1}^{\prime \prime }-2\left(r-3M\right)\delta {\phi }_{1}^{\prime }\right)\right]\hfill \\ \hfill & \quad -\frac{1}{2{r}^{2}}\frac{{G}_{4\phi \phi }}{{G}_{4}}\left[r\left(r-2M\right)\delta {\phi }_{1}^{\prime 2}+\delta {\phi }_{1}\left(r\left(r-2M\right)\delta {\phi }_{1}^{\prime \prime }-2\left(r-3M\right)\delta {\phi }_{1}^{\prime }\right)\right]\hfill \\ \hfill & \quad -\frac{{\alpha }_{T}\left(r\right)}{2{r}^{3}}\left[-5M+Mr{\left(r-2M\right)}^{-1}-2r\left(\ell +2\right)\left(\ell -1\right)+{r}^{2}\left(r-2M\right){\left(\frac{\delta {\phi }_{1}^{\prime \prime }}{\delta {\phi }_{1}^{\prime }}\right)}^{2}\right.\hfill \\ \hfill & \quad \left.+r\left(r\left(r-2M\right)\frac{\delta {\phi }_{1}^{\prime \prime \prime }}{\delta {\phi }_{1}^{\prime }}-2\left(r-5M\right)\frac{\delta {\phi }_{1}^{\prime \prime }}{\delta {\phi }_{1}^{\prime }}\right)\right].\hfill \end{align} \tag{ 12b }$$

We emphasise that in the above expressions all of the G_i Horndeski functions are evaluated at ϕ = ϕ₀ and X = 0 (i.e. to zeroth order in the book-keeping parameter ), and as such are constants. Note that this approach assumes that the G_i are amenable to an expansion around ϕ = ϕ₀ 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 $\mathcal{O}\left({{\epsilon}}^{0}\right)$ equation (10) is simply the well known Regge Wheeler equation describing odd parity gravitational perturbations to a Schwarzschild black hole [78].

At $\mathcal{O}\left({{\epsilon}}^{0}\right)$ the effective potential of the Regge–Wheeler equation is modified by δV₁, which is linear in the first order modifications to the spacetime geometry and scalar profile (and their derivatives). Our expression for δV₁ with δϕ₁ = 0 matches that of equation (5.9) in [40], which concerns perturbations of hairy black holes in the unitary gauge (i.e. with δϕ₁ = 0).

At $\mathcal{O}\left({{\epsilon}}^{2}\right)$, the potential is further modified by δV₂, 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 ω² is rescaled by a factor of c_T = 1 + 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 field and metric such that G_4ϕ or G_4ϕϕ ≠ 0, the QNM spectrum can be modified by the presence of a non-trivial scalar radial profile.

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 $\mathcal{F},\mathcal{G}$, and $\mathcal{H}$ are all positive. Looking at equation (9), we see that for the ansatz given by equation (4) each of the functions is equal to $2{G}_{4}\left({\phi }_{0},0\right)\left(1+\mathcal{O}\left({\epsilon}\right)\right)$. As G₄(ϕ₀, 0) plays the role of the constant background Planck mass in Horndeski, this is clearly positive. Thus each of $\mathcal{F},\mathcal{G}$, and $\mathcal{H}$ must also be positive for ≪ 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:

$$\begin{equation}\mathcal{V}=A\left(r\right)\left(\frac{\ell \left(\ell +1\right)}{{r}^{2}}-\frac{6M}{{r}^{3}}+\mathcal{O}\left({\epsilon}\right)\right).\end{equation} \tag{ 13 }$$

Clearly $\mathcal{V}$ 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 $\mathcal{O}\left({\epsilon}\right)$ 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 $\mathcal{V}$ is ℓ(ℓ + 1)/r² 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 figure 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]).

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Equation (10)–(12) are the main results of this section. In the next section, we will explore how the modifications 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.

To first order in , the modified Regge–Wheeler equation is given by

$$\begin{equation}\left[A\left(r\right)\frac{\mathrm{d}}{\mathrm{d}r}\left(A\left(r\right)\frac{\mathrm{d}}{\mathrm{d}r}\right)+{\omega }^{2}-A\left(r\right)\overline{V}\right]Q=0\end{equation} \tag{ 18 }$$

where the effective potential $\overline{V}$ is given by:

$$\begin{equation}\overline{V}=\frac{\ell \left(\ell +1\right)}{{r}^{2}}-\frac{6M}{{r}^{3}}+{\epsilon}\delta {V}_{1}\end{equation} \tag{ 19 }$$

Following the procedure introduced in Cardoso et al, the first step to obtain an equation in the form of equation (10) is to write:

$$\begin{equation}A\left(r\right)=f\left(r\right)Z\left(r\right)\end{equation} \tag{ 20 }$$

where f(r) = 1 − r_H/r, and find appropriate expressions for r_H and Z to O(). The location of the horizon in our modified spacetime will not be exactly at r = 2M, but will be corrected due to δA₁. We thus make the following expansion for the horizon radius:

$$\begin{equation}{r}_{\mathrm{H}}=2M+{\epsilon}\delta {r}_{\mathrm{H},1}.\end{equation} \tag{ 21 }$$

To find the new position of the horizon, we require A(r_H) = 0. Solving order by order in , we find the following for the location of the horizon:

$$\begin{equation}\delta {r}_{\mathrm{H},1}=\;-2M\delta {A}_{1}\left(2M\right)\end{equation} \tag{ 22 }$$

with Z thus given by:

$$\begin{equation}Z\left(r\right)=\;1+{\epsilon}\delta {Z}_{1}=\;1+{\epsilon}\frac{\delta {A}_{1}\left(r\right)-\frac{2M}{r}\delta {A}_{1}\left(2M\right)}{1-2M/r}\end{equation} \tag{ 23 }$$

in order to make equation (20) hold to O().

If we now define $\tilde {Q}=\sqrt{Z}Q$, we transform equation (18) into

$$\begin{equation}\left[f\left(r\right)\frac{\mathrm{d}}{\mathrm{d}r}\left(f\left(r\right)\frac{\mathrm{d}}{\mathrm{d}r}\right)+\frac{{\omega }^{2}}{{Z}^{2}}-f\left(r\right)V\right]\tilde {Q}=0\end{equation} \tag{ 24 }$$

where the new potential V is given by:

$$\begin{equation}V=\frac{\overline{V}}{Z}-\frac{f{\left({Z}^{\prime }\right)}^{2}-2Z{\left(f{Z}^{\prime }\right)}^{\prime }}{4{Z}^{2}}.\end{equation} \tag{ 25 }$$

and $\overline{V}$ is still given by equation (19).

We can expand the ω² term in equation (24) to O() and write it in the following way:

$$\begin{equation}\frac{{\omega }^{2}}{{Z}^{2}}=\;{\omega }^{2}\left(1-2{\epsilon}\delta {Z}_{1}\left({r}_{\mathrm{H}}\right)\right)-2{\epsilon}{\omega }^{2}\left(\delta {Z}_{1}\left(r\right)-\delta {Z}_{1}\left({r}_{\mathrm{H}}\right)\right)\end{equation} \tag{ 26 }$$

The first 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 $\mathcal{O}\left({\epsilon}\right)$, it can be absorbed into the perturbed potential V by setting ω = ω₀. This is because any frequency corrections in this term would result in terms $\mathcal{O}\left({{\epsilon}}^{2}\right)$ or higher (which we are neglecting in this section). The final form of the modified Regge Wheeler equation is now in the same form as equation (14):

$$\begin{equation}\left[f\left(r\right)\frac{\mathrm{d}}{\mathrm{d}r}\left(f\left(r\right)\frac{\mathrm{d}}{\mathrm{d}r}\right)+{\tilde {\omega }}^{2}-f\left(r\right)\tilde {V}\right]\tilde {Q}=0\end{equation} \tag{ 27 }$$

where

$$\begin{equation}{\tilde {\omega }}^{2}={\omega }^{2}\left(1-2{\epsilon}\delta {Z}_{1}\left({r}_{\mathrm{H}}\right)\right)\end{equation} \tag{ 28 }$$

$$\begin{equation}\tilde {V}=V+\frac{2{\epsilon}{\omega }_{0}^{2}}{f\left(r\right)}\left(\delta {Z}_{1}\left(r\right)-\delta {Z}_{1}\left({r}_{\mathrm{H}}\right)\right)\end{equation} \tag{ 29 }$$

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 δA_i , δC_i and δϕ_i . We will make the following simple power law choices:

$$\begin{align}\hfill \delta {\phi }_{1}=\;{Q}_{1}\left(\frac{2M}{r}\right),\quad \delta {A}_{1}=\;{a}_{1}{\left(\frac{2M}{r}\right)}^{2},\quad \delta {C}_{1}=\;{c}_{1}\left(\frac{2M}{r}\right)\end{align} \tag{ 30 }$$

so that, in addition to the Horndeski G_i parameters, we have 3 'hairs', Q₁, a₁, and c₁ 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 α_j are given by (absorbing into the definitions of (a, c, Q)₁):

$$\begin{equation}{\alpha }_{0}=\;8{M}^{2}{\omega }_{0}^{2}{a}_{1}\end{equation} \tag{ 31a }$$

$$\begin{equation}{\alpha }_{3}=\ell \left(\ell +1\right)\left({a}_{1}-{c}_{1}\right)-{a}_{1}-2{Q}_{1}\frac{{G}_{4\phi }}{{G}_{4}}\end{equation} \tag{ 31b }$$

$$\begin{equation}{\alpha }_{4}=\frac{5}{2}\left({a}_{1}+{c}_{1}+{Q}_{1}\frac{{G}_{4\phi }}{{G}_{4}}\right)\end{equation} \tag{ 31c }$$

leading to the following corrections to the QNM frequency spectrum for the ℓ = 2, 3 modes (for example)

$$\begin{align}\hfill M{\omega }_{1}^{\ell =2}& ={Q}_{1}\frac{{G}_{4\phi }}{{G}_{4}}\left[-0.0126+0.0032\mathrm{i}\right]+{a}_{1}\left[-0.0267+0.0621\mathrm{i}\right]\hfill \\ \hfill & \quad +{c}_{1}\left[-0.1296+0.0106\mathrm{i}\right]\hfill \end{align} \tag{ 32a }$$

$$\begin{align}\hfill M{\omega }_{1}^{\ell =3}& ={Q}_{1}\frac{{G}_{4\phi }}{{G}_{4}}\left[-0.0075+0.0008\mathrm{i}\right]+{a}_{1}\left[-0.1326+0.0677\mathrm{i}\right]\hfill \\ \hfill & \quad +{c}_{1}\left[-0.2040+0.0110\mathrm{i}\right]\hfill \end{align} \tag{ 32b }$$

where the unperturbed GR frequencies are given by [18]:

$$\begin{align}\hfill M{\omega }_{\mathrm{G}\mathrm{R}}^{\ell =2}=0.3737-0.0890\mathrm{i},\quad M{\omega }_{\mathrm{G}\mathrm{R}}^{\ell =3}=0.5994-0.09270\mathrm{i}.\end{align} \tag{ 33 }$$

Looking at the real and imaginary parts of the frequencies separately, with ω_GR = ω_R + iω_I and ω₁ = δω_R + iδω_I, we find the following fractional differences for the ℓ = 2 QNM:

$$\begin{equation}\frac{\delta {\omega }_{\mathrm{R}}^{\ell =2}}{{\omega }_{\mathrm{R}}^{\ell =2}}=-\left(3.37\;{\tilde {Q}}_{1}+7.14\;{a}_{1}+34.68\;{c}_{1}\right)\%\end{equation} \tag{ 34a }$$

$$\begin{equation}\frac{\delta {\omega }_{\mathrm{I}}^{\ell =2}}{{\omega }_{\mathrm{I}}^{\ell =2}}=-\left(3.60\;{\tilde {Q}}_{1}+69.77\;{a}_{1}+11.91\;{c}_{1}\right)\%,\end{equation} \tag{ 34b }$$

and for the ℓ = 3 mode:

$$\begin{equation}\frac{\delta {\omega }_{\mathrm{R}}^{\ell =3}}{{\omega }_{\mathrm{R}}^{\ell =3}}=-\left(1.25\;{\tilde {Q}}_{1}+22.12\;{a}_{1}+34.03\;{c}_{1}\right)\%\end{equation} \tag{ 35a }$$

$$\begin{equation}\frac{\delta {\omega }_{\mathrm{I}}^{\ell =3}}{{\omega }_{\mathrm{I}}^{\ell =3}}=-\left(0.86\;{\tilde {Q}}_{1}+73.03\;{a}_{1}+11.87\;{c}_{1}\right)\%.\end{equation} \tag{ 35b }$$

where ${\tilde {Q}}_{1}={Q}_{1}{G}_{4\phi }/{G}_{4}$.

We see that with, for example ² , $\left({a}_{1},{c}_{1},{\tilde {Q}}_{1}\right)$ 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 final 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 a₁, c₁, and ${\tilde {Q}}_{1}$ has on the form of the first order 'hairy' potential $A\left(r\right)\overline{V}\left(r\right)$ (with $\overline{V}$ given by equation (19)) for ℓ = 2. We see that a non-zero a₁ leads to the most noticeable modification to the effective potential. This is unsurprising due to a non-zero a₁ 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 α₀ in equation (31). Looking at equations 34 and 35, a₁ has the most significant effect on δω_I, while δω_R is most greatly impacted by c₁.

The choices made in equation (30) 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 ω₁ (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 fit 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 δA₁, δC₁ and δϕ₁. For example:

$$\begin{align}\hfill \delta {C}_{1}& ={c}_{1}{\left(\frac{2M}{r}\right)}^{m},\;\;\delta {A}_{1}=\;{a}_{1}{\left(\frac{2M}{r}\right)}^{n},\hfill \\ \hfill \delta {\phi }_{1}& ={Q}_{1}{\left(\frac{2M}{r}\right)}^{p},\hfill \end{align} \tag{ 36 }$$

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 profile (that is, δA_i = δC_i = 0, δϕ_i ≠ 0). We consider both power law and general scalar profiles.

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 modified 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:

$$\begin{equation}{h}_{+}=\frac{{A}_{\ell m}^{+}}{\sqrt{2}}\left[{\mathrm{e}}^{\mathrm{i}{\phi }_{+}}{S}_{\ell m}{b}_{+}\left(f\right)+{\mathrm{e}}^{-\mathrm{i}{\phi }_{+}}{S}_{\ell m}^{{\ast}}{b}_{-}\left(f\right)\right]\end{equation} \tag{ 37 }$$

$$\begin{equation}{h}_{{\times}}=\frac{{A}_{\ell m}^{+}{N}_{{\times}}}{\sqrt{2}}\left[{\mathrm{e}}^{\mathrm{i}{\phi }_{{\times}}}{S}_{\ell m}{b}_{+}\left(f\right)+{\mathrm{e}}^{-\mathrm{i}{\phi }_{{\times}}}{S}_{\ell m}^{{\ast}}{b}_{-}\left(f\right)\right]\end{equation} \tag{ 38 }$$

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:

$$\begin{align}\hfill {b}_{{\pm}}=\;\frac{1/{\tau }_{\ell m}}{{\left(1/{\tau }_{\ell m}\right)}^{2}+4{\pi }^{2}\left(f{\pm}{f}_{\ell m}\right)}\end{align} \tag{ 39 }$$

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 field) or mixed breathing-longitudinal mode (for a massive field) [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 field 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 ω₀, is known. Furthermore, we will assume that N_× = 1 and ϕ₊ = ϕ_× = 0, and that A₊ is known (effectively resulting in us fixing a specific signal-noise-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 c₁ = 0, thus we are effectively considering a Reissner–Nordstrom-like black hole with a 1/r scalar profile. Again with ${\tilde {Q}}_{1}={Q}_{1}{G}_{4\phi }/{G}_{4}$, and absorbing the book-keeping parameter into the definition of our hairy parameters (such that a₁ → a₁ etc), we can write the oscillation frequency and damping time of the perturbed ℓ = 2 mode (for example) as follows:

$$\begin{equation}2\pi f=2\pi {f}_{0}-0.0126{\tilde {Q}}_{1}/M-0.0267{a}_{1}/M\end{equation} \tag{ 40a }$$

$$\begin{equation}{\tau }^{-1}={\tau }_{0}^{-1}-0.0032{\tilde {Q}}_{1}/M-0.0621{a}_{1}/M.\end{equation} \tag{ 40b }$$

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 for ${\tilde {Q}}_{1}$ and a₁:

$$\begin{equation}{\sigma }_{{\tilde {Q}}_{1}}^{2}=\frac{1}{2{\rho }^{2}{q}^{2}}\frac{{f}^{\prime 2}{q}^{2}\left(1+4{q}^{2}\right)-2fq{f}^{\prime }{q}^{\prime }+{f}^{2}{q}^{\prime 2}}{{\left(\dot {f}{q}^{\prime }-{f}^{\prime }\dot {q}\right)}^{2}}\end{equation} \tag{ 41a }$$

$$\begin{equation}{\sigma }_{{a}_{1}}^{2}=\frac{1}{2{\rho }^{2}{q}^{2}}\frac{{\dot {f}}^{2}{q}^{2}\left(1+4{q}^{2}\right)-2fq\dot {f}\dot {q}+{f}^{2}{\dot {q}}^{2}}{{\left(\dot {f}{q}^{\prime }-{f}^{\prime }\dot {q}\right)}^{2}}\end{equation} \tag{ 41b }$$

where q = πfτ 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 ℓ = 2 mode, we can use the expressions given in equations (40a) and (40b) to calculate the errors. Additionally setting ${\tilde {Q}}_{1}={a}_{1}=0$, we interpret the following errors as 'detectability' limits on the parameters:

$$\begin{equation}\rho {\sigma }_{{\tilde {Q}}_{1}}\approx \;12,\quad \rho {\sigma }_{{a}_{1}}\approx \;2.\end{equation} \tag{ 42 }$$

With an SNR of ρ ∼ 10², which could be typical of LISA events, we thus have that ${\sigma }_{{\tilde {Q}}_{1}}\approx 0.1$ whilst ${\sigma }_{{a}_{1}}\approx 0.02$ (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 a₁ 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 a₁ 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.