\subsection{Spin-induced quadrupole and
octupole pieces in compact binary waveforms}

The waveform used in this work is a variant of the one that is presented in
Ref.~\cite{MKAF16}. These are constructed by simply making the dependences on
parameters characterising the spin-induced quadrupole moment (through
$\kappa_s$ and $\kappa_a$) and spin-induced octupole moment (through
$\lambda_s$ and $\lambda_a$) explicit in the waveform, which were set to their
respective values for Kerr BHs while writing the waveform model of
Ref.~\cite{MKAF16}. In this note we list various pieces of the waveform where
such dependences occur.\\ 

Let us first recall the schematic expression for the frequency domain amplitude
of a gravitational wave signal, $\tilde h(f)$, given in Ref.~\cite{MKAF16}.
\footnote{Pre-factor of Eq.~1 of Ref.~\cite{MKAF16} should be multiplied with a
factor ${1/\sqrt \eta}$. We have corrected this in the Eq.~\eqref{eq:hf}.} This
reads

\begin{equation} \tilde{h}(f)=\frac{M^2}{D_L}
\sqrt{\frac{5\,\pi}{48\eta}}\sum_{n=0}^{4}\sum_{k=1}^{6}
V_k^{n-7/2}\,C_{k}^{(n)}\,e^{\mathrm{i} (k\,\Psi_\mathrm{SPA}(f/k)-\pi/4)} \, .
\label{eq:hf} \end{equation}

Here, $M$, $\eta$ and $D_L$ denote the total mass, symmetric mass ratio
parameter and the distance to the binary, respectively and the indices $n$ and
$k$ denote the PN order and harmonic number, respectively. The coefficients
${\cal C}_{k}^{(n)}$ denote the amplitude corrections associated with the
contribution from $k{\rm th}$ harmonic at $n{\rm th}$ PN order. Related
expressions for each of the ${\mathcal C}_{k}^{(n)}$s can be found in
Ref.~\cite{ABFO08, MKAF16}. Here we list the only coefficient which has
explicit dependence on the parameters ($\kappa_s$ and $\kappa_a$) and
corresponds to the contributions from the $2$nd harmonic at the 2PN order
(${\cal C}_{2}^{(4)}$). In addition, $\Psi_{\rm SPA}$ represents the phase of
the first harmonic in the frequency domain as obtained under the Stationary
Phase Approximation (SPA) (see sec. VI of Ref.~\cite{ABFO08} for details on
SPA). Schematically the expression for this phase can be written as follows

\begin{equation} \Psi_\mathrm{SPA}(f) = 2\pi f t_\mathrm{c} - \phi_\mathrm{c} +
\left\{\frac{3}{128\eta\,v^{5}}
\left[\psi_\mathrm{NS}+\psi_\mathrm{SO}+\psi_\mathrm{SS}
+\psi_\mathrm{SSS}\right]\right\}_{v=V_1(f)},  \end{equation}

where $\phi_\mathrm{c}$ denotes the orbital phase at the instant $t_\mathrm{c}$
of coalescence. \\

Further, one can write the spin part of the SPA phase more explicitly as 

\begin{equation} \psi_\text{Spin}\equiv\psi_\mathrm{SO}+\psi_\mathrm{SS}
+\psi_\mathrm{SSS}
={v^3}\left[\mathcal{P}_3+\mathcal{P}_4\,v+\mathcal{P}_5\,v^2
+\mathcal{P}_6\,v^{3}+\mathcal{P}_7 v^4+\mathcal{P}_8 v^{5} +\cdots\right].
\label{eq:psi_so} \end{equation}

Again expressions for the coefficients $\mathcal{P}_n$ can be found in
Ref.~\cite{ABFO08, MKAF16} where explicit dependence on $\kappa_s$ and
$\kappa_a$ is suppressed by setting them to their respective values for Kerr
BBHs. Here we provide expressions for coefficients that contain explicit
dependence on $\kappa_s$ and $\kappa_a$. Below we list the amplitude/phase
coefficients that do contain explicit dependence on $\kappa_s$ and $\kappa_a$
and can be combined to those listed in Ref.~\cite{ABFO08, MKAF16} to write the
final waveform expression. 

\begin{eqnarray} \mathcal{C}_2^{(4)}&=& \frac{1}{\sqrt{2}}
\Biggl\{F_+\Bigg[\frac{113419241}{40642560}+\frac{152987}{16128}\,\eta
-\frac{11099}{1152}\,\eta^2 +\left(\frac{165194153}{40642560}
-\frac{149}{1792}\,\eta+\frac{6709}{1152}\,\eta ^2\right)c_{\iota}^2 
+\left(\frac{1693}{2016}-\frac{5723}{2016}\,\eta
+\frac{13}{12}\,\eta^2\right)c_{\iota}^4
\nonumber\\&-\frac{1}{24}&\left(1-5\,\eta+5\,\eta ^2\right) c_{\iota}^6
+(1+c_{\iota}^2)\biggl[ (\boldsymbol{\chi}_\mathrm{s}\cdot
\hat{\boldsymbol{L}}_\mathrm{N})^2 \left(\frac{1}{32}+\frac{23 \,\eta
}{8}+\frac{3 \delta  \kappa _a}{2}+\frac{3}{2}\left(1-2 \,\eta \right) \kappa
_s\right)
+(\boldsymbol{\chi}_\mathrm{a}\cdot \hat{\boldsymbol{L}}_\mathrm{N})^2
\left(\frac{1}{32}-3 \,\eta +\frac{3 \delta  \kappa _a}{2}
\right.\nonumber\\&+&\left. \frac{3}{2}\left(1-2 \,\eta \right) \kappa_s\right)
+(\boldsymbol{\chi}_\mathrm{a}\cdot
\hat{\boldsymbol{L}}_\mathrm{N})(\boldsymbol{\chi}_\mathrm{s} \cdot
\hat{\boldsymbol{L}}_\mathrm{N}) \left(\frac{\delta }{16}+3\,(1-2 \,\eta )
\kappa _a+3 \delta  \kappa _s\right) \biggr]\Bigg]
+\mathrm{i}\,c_{\iota}\,F_{\times}\Bigg[\frac{114020009}{20321280}
+\frac{133411}{8064}\,\eta-\frac{7499}{576}\,\eta^2 \nonumber \\ &+&
\left(\boldsymbol{\chi}_\mathrm{s}\cdot
\hat{\boldsymbol{L}}_\mathrm{N}\right){}^2 \left(\frac{1}{16}+\frac{23}{4}\eta
+3 \delta  \kappa _a+3\,(1-2 \,\eta ) \kappa _s\right)+
\left(\boldsymbol{\chi}_\mathrm{a}\cdot
\hat{\boldsymbol{L}}_\mathrm{N}\right){}^2 \left(\frac{1}{16}-6 \,\eta +3
\delta  \kappa _a+3\,(1-2 \,\eta ) \kappa _s\right)
+(\boldsymbol{\chi}_\mathrm{a}\cdot
\hat{\boldsymbol{L}}_\mathrm{N})(\boldsymbol{\chi}_\mathrm{s}\cdot
\hat{\boldsymbol{L}}_\mathrm{N})\left(\frac{\delta }{8}
\right.\nonumber\\&+&\left.6\,(1-2 \,\eta ) \kappa _a+6 \delta  \kappa
_s\right)
+\left(\frac{5777}{2520}-\frac{5555}{504}\,\eta+\frac{34}{3}\,\eta^2\right)c_{\iota}^2
-\frac{1}{4}\left(1-5\,\eta+5\,\eta^2\right) c_{\iota}^4\Bigg]\Biggr\}\,
\Theta(2 F_\mathrm{cut} - f) \label{eq:C24} \end{eqnarray}

\begin{subequations} \begin{eqnarray} \label{eq:P4} {\mathcal P}_4 &=&
-\frac{5}{8}(\boldsymbol{\chi}_\mathrm{s}\cdot\hat{\boldsymbol{L}}_\mathrm{N})^2\,
\Bigl[1+156 \,\eta +80 \,\delta  \,\kappa _a+80 (1-2 \,\eta ) \kappa _s\Bigr]
+(\boldsymbol{\chi}_\mathrm{a}\cdot\hat{\boldsymbol{L}}_\mathrm{N})^2
\left[-\frac{5}{8}-50 \,\delta  \,\kappa _a-50 \kappa _s+100 \,\eta
\left(1+\kappa _s\right)\right] \nonumber\\
&-&\frac{5}{4}(\boldsymbol{\chi}_\mathrm{a}\cdot\hat{\boldsymbol{L}}_\mathrm{N})(\boldsymbol{\chi}_\mathrm{s}\cdot\hat{\boldsymbol{L}}_\mathrm{N})
\Bigl[\delta +80\,(1-2 \,\eta )\,\kappa _a+80 \,\delta  \,\kappa _s\Bigr]\,,\\
{\mathcal P}_6 &=& \pi\,\Biggl[\frac{2270}{3}\,\delta
\,\boldsymbol{\chi}_\mathrm{a}\cdot\hat{\boldsymbol{L}}_\mathrm{N}
+\left(\frac{2270}{3}-520\,\eta\right)
\boldsymbol{\chi}_\mathrm{s}\cdot\hat{\boldsymbol{L}}_\mathrm{N}\Biggr]
+(\boldsymbol{\chi}_\mathrm{s}\cdot\hat{\boldsymbol{L}}_\mathrm{N})^2
\left[-\frac{1344475}{2016}+\frac{829705}{504}\,\eta+\frac{3415}{9}\,\eta ^2
+\delta \left(\frac{26015}{28}-\frac{1495}{6}\,\eta\right) \kappa_a
\right.\nonumber\\&+&\left.  \left(\frac{26015}{28}-\frac{44255}{21} \,\eta
-240 \,\eta ^2\right) \kappa _s\right]
+(\boldsymbol{\chi}_\mathrm{a}\cdot\hat{\boldsymbol{L}}_\mathrm{N})^2
\left[-\frac{1344475}{2016}+\frac{267815}{252} \,\eta -240 \,\eta ^2+\delta
\left(\frac{26015}{28} -\frac{1495}{6} \,\eta \right) \kappa
_a+\left(\frac{26015}{28} \right.\right.\nonumber\\&-&\left.\left.
\frac{44255}{21} \,\eta -240 \,\eta ^2\right) \kappa _s\right]
+(\boldsymbol{\chi}_\mathrm{a}\cdot\hat{\boldsymbol{L}}_\mathrm{N})(\boldsymbol{\chi}_\mathrm{s}\cdot\hat{\boldsymbol{L}}_\mathrm{N})
\left[\left(\frac{26015}{14}-\frac{88510}{21} \,\eta -480 \,\eta ^2\right)
\kappa _a+\delta\, \biggl[-\frac{1344475}{1008}+\frac{745}{18} \,\eta
+\left(\frac{26015}{14} \right.\right.\nonumber\\&-&\left.\left.  \frac{1495
}{3} \,\eta\right) \kappa _s\biggr]\right]\,, \label{eq:P6} \\
{\mathcal P}_7 &=& \delta\,\boldsymbol{\chi}_\mathrm{a}\cdot
\hat{\boldsymbol{L}}_\mathrm{N}\left(-\frac{25150083775 }{3048192}
+\frac{26804935}{6048}\,\eta-\frac{1985}{48}\,\eta^2\right)
+\boldsymbol{\chi}_\mathrm{s}\cdot\hat{\boldsymbol{L}}_\mathrm{N}\left(-\frac{25150083775}{3048192}+\frac{10566655595}{762048}\,\eta
-\frac{1042165}{3024}\,\eta^2
\right.\nonumber\\&+&\left.\frac{5345}{36}\,\eta^3\right)+
(\boldsymbol{\chi}_\mathrm{s}\cdot\hat{\boldsymbol{L}}_\mathrm{N})^3\,
\Biggl[\frac{265}{24}+\frac{4035 \,\eta }{2}-\frac{20 \,\eta
^2}{3}+\left(\frac{3110}{3}-\frac{10250 }{3}\,\eta +40 \,\eta ^2\right) \kappa
_s+\delta \,\biggl[\left(\frac{3110}{3}-\frac{4030 \,\eta }{3}\right) \kappa _a
\nonumber\\&-&440\,(1-\eta) \lambda _a\biggr] -440\,(1-3 \,\eta ) \lambda
_s\Biggr] +(\boldsymbol{\chi}_\mathrm{a}\cdot\hat{\boldsymbol{L}}_\mathrm{N})^3
\Biggl[\left(\frac{3110}{3}-\frac{8470}{3} \,\eta \right) \kappa _a-440\,(1-3
\,\eta ) \lambda _a+\delta\,\biggl[\frac{265}{24}-2070 \,\eta
\nonumber\\&+&\left(\frac{3110}{3}-750 \,\eta \right) \kappa _s -440(1-\eta )
\lambda _s\biggr]\Biggr]
+(\boldsymbol{\chi}_\mathrm{s}\cdot\hat{\boldsymbol{L}}_\mathrm{N})^2
(\boldsymbol{\chi}_\mathrm{a}\cdot\hat{\boldsymbol{L}}_\mathrm{N})
\Biggl[\left(3110-\frac{28970}{3} \,\eta +80 \,\eta ^2\right) \kappa
_a-1320\,(1-3 \,\eta ) \lambda _a \nonumber\\&+&\delta
\biggl[\frac{265}{8}+\frac{12055 }{6}\,\eta +\left(3110-\frac{10310
}{3}\,\eta\right) \kappa _s-1320\,(1-\eta) \lambda _s\biggr]\Biggr]
+(\boldsymbol{\chi}_\mathrm{a}\cdot\hat{\boldsymbol{L}}_\mathrm{N})^2
(\boldsymbol{\chi}_\mathrm{s}\cdot\hat{\boldsymbol{L}}_\mathrm{N})
\Biggl[\frac{265}{8}-\frac{6500}{3} \,\eta +40 \,\eta ^2
\nonumber\\&+&\left(3110-\frac{27190 \,\eta }{3}+40 \,\eta ^2\right) \kappa _s
+\delta  \biggl[\left(3110-\frac{8530 \,\eta }{3}\right) \kappa
_a-1320\,(1-\eta ) \lambda _a\biggr]-1320\,(1-3 \,\eta ) \lambda _s\Biggr] \,.
\label{eq:P4} \end{eqnarray} \end{subequations}

 
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