Eccentric self-forced inspirals into a rotating black hole

In the absence of any perturbing force, the secondary will follow a geodesic, i.e.,

$$\begin{equation}{u}^{\beta }{\nabla }_{\beta }{u}^{\alpha }=0.\end{equation} \tag{ 4 }$$

The symmetries of the Kerr spacetime allow for the identification of integrals of motion $\vec{\mathcal{P}}=\left\{\mathcal{E},\mathcal{L},\mathcal{K}\right\}$ given by

$$\begin{equation}\mathcal{E}=-{u}_{t},\qquad \mathcal{L}={u}_{\phi },\qquad \mathcal{K}={\mathcal{K}}^{\alpha \beta }{u}_{\alpha }{u}_{\beta },\end{equation} \tag{ 5a-c }$$

where ${\mathcal{K}}^{\alpha \beta }$ is the Killing tensor, $\mathcal{E}$ is the orbital energy per unit rest mass μ, $\mathcal{L}$ is the z-component of the angular momentum divided by μ and $\mathcal{K}$ is the Carter constant divided by μ² [39]. This definition of the Carter constant is related to another common definition of the Carter constant, $\mathcal{Q}$, by

$$\begin{equation}\mathcal{Q}=\mathcal{K}\;-\;{(\mathcal{L}\;-a\mathcal{E})}^{2}.\end{equation} \tag{ 6 }$$

The geodesic equation can be written explicitly in terms of these integrals of motion [40]:

$$\begin{equation}\begin{aligned}\hfill {\left(\frac{\mathrm{d}r}{\mathrm{d}\lambda }\right)}^{2}& ={\left(\mathcal{E}{\varpi }^{2}-a\mathcal{L}\right)}^{2}-{\Delta}\left({r}^{2}+\mathcal{K}\right)\hfill \\ \hfill & =(1-{\mathcal{E}}^{2})({r}_{1}-r)(r-{r}_{2})(r-{r}_{3})(r-{r}_{4}){:=}{V}_{r},\hfill \end{aligned}\end{equation} \tag{ 7a }$$

$$\begin{equation}\begin{aligned}\hfill {\left(\frac{\mathrm{d}z}{\mathrm{d}\lambda }\right)}^{2}& =\mathcal{Q}-{z}^{2}\left({a}^{2}(1-{\mathcal{E}}^{2})(1-{z}^{2})+{\mathcal{L}}^{2}+\mathcal{Q}\right)\hfill \\ \hfill & =({z}^{2}-{z}_{-}^{2})\left({a}^{2}(1-{\mathcal{E}}^{2}){z}^{2}-{z}_{+}^{2}\right){:=}{V}_{z},\hfill \end{aligned}\end{equation} \tag{ 7b }$$

$$\begin{equation}\frac{\mathrm{d}t}{\mathrm{d}\lambda }=\frac{{\varpi }^{2}}{{\Delta}}\left(\mathcal{E}{\varpi }^{2}-a\mathcal{L}\right)-{a}^{2}\mathcal{E}(1-{z}^{2})+a\mathcal{L}{:=}{s}_{t},\end{equation} \tag{ 7c }$$

$$\begin{equation}\frac{\mathrm{d}\phi }{\mathrm{d}\lambda }=\frac{a}{{\Delta}}\left(\mathcal{E}{\varpi }^{2}-a\mathcal{L}\right)+\frac{\mathcal{L}}{1-{z}^{2}}-a\mathcal{E}{:=}{s}_{\phi },\end{equation} \tag{ 7d }$$

where r₁ > r₂ > r₃ > r₄ are the roots of the radial potential V_r , z₊ > z₋ are the roots of the polar potential V_z , and λ is Mino(–Carter) time that decouples the radial and polar equations [41]. This time is related to the proper time of the particle, τ, by

$$\begin{equation}\mathrm{d}\tau ={\Sigma}\mathrm{d}\lambda .\end{equation} \tag{ 8 }$$

The two largest roots of V_r correspond to the apoapsis and periapsis of the orbit, respectively. Explicit expressions for the other roots are derived in reference [42] and given for completeness in appendix A. Rather than parameterize an orbit by the set $\left\{\mathcal{E},\mathcal{L},\mathcal{K}\right\}$ it is useful to instead use more geometric, quasi-Keplerian constants $\vec{P}=\left\{p,e,x\right\}$. Here p is the semi-latus rectum, e is the orbital eccentricity and x measures the orbital inclination. These are related to the radial and polar roots via

$$\begin{equation}{r}_{1}=\frac{pM}{1-e},\qquad {r}_{2}=\frac{pM}{1+e},\qquad {z}_{-}=\sqrt{1-{x}^{2}}.\end{equation} \tag{ 9a-c }$$

The explicit relation between the integrals of motion $\left\{\mathcal{E},\mathcal{L},\mathcal{K}\right\}$ and {r₁, r₂, z₋} can be found in, e.g., appendix B of reference [43]. We note that one advantage of using x over other common choices for inclination is that x smoothly parameterizes orbits between prograde equatorial motion with x = 1 to retrograde equatorial motion with x = −1 orbits. Not all values of {p, e, x} correspond to bound geodesics and we denote the value of p at the last stable orbit by p_LSO(a, e, x) [44, 45].

In order to later apply the near-identity transformations it will be useful to employ action-angle formulation to parameterize the orbital motion [42, 43, 46]. In this description the orbital phases $\vec{q}=\left\{{q}_{r},{q}_{z}\right\}$ are such that the geodesic equations can written in the form

$$\begin{equation}\dot{{P}_{j}}=0\quad \text{and}\quad \dot{{q}_{i}}={{\Upsilon}}_{i}(\vec{P}),\end{equation} \tag{ 10a-b }$$

where an overdot denotes derivative with respect to Mino time and the ϒ_i are the Mino time fundamental frequencies [42]. Note that the right-hand side of the ${\dot{q}}_{i}$ equation depends only on $\vec{P}$ and not also on the orbital phases. Semi-analytic solutions to equation (7) in terms of $\vec{q}$ can be found in references [42, 46] and we present the key equations in appendix A.

At this point we note that it is also common in the literature [47, 48] to express the radial and polar motion in terms of quasi-Keplerian angles, ψ and χ, via

$$\begin{equation}r(\psi )=\frac{pM}{1+e\mathrm{cos}(\psi )}\quad \text{and}\quad z(\chi )={z}_{-}\mathrm{c}\mathrm{o}\mathrm{s}(\chi ).\end{equation} \tag{ 11a-b }$$

With this parameterization the evolution equations for ψ and χ depend on both $\vec{P}$ and $\vec{q}$. This makes them inconvenient for deriving averaging transformations, though it is not an insurmountable challenge [35]. For this reason we prefer the action-angle formulation.

We now wish to describe the forced motion of a body obeying equation (3). To do so, we make use of the method of osculating orbital elements, or OGs when applied to the relativistic context [33, 34]. We first identify a set of orbital elements that uniquely identify a given geodesic, such as the integrals of motion $\vec{P}$ along with the initial values of the orbital phases of the geodesic orbit ${\vec{q}}_{0}$ and designate them as a set of 'orbital elements' $\vec{I}=\left\{\vec{P},{\vec{q}}_{0}\right\}$. For accelerated orbits, these orbital elements are promoted from constants to functions of Mino time. Note that now the orbital elements ${\vec{q}}_{0}(\lambda )$ are different quantities from the values of $\vec{q}$ evaluated at λ = 0, i.e., $\vec{q}(0)$. We assume that the worldline and four-velocity of the secondary at each instant can be described as the worldline and four-velocity of a test body on a tangent geodesic, i.e., ${x}^{\alpha }(\lambda )={x}_{\mathrm{G}}^{\alpha }({I}^{A}(\lambda ),\lambda )$ and ${u}^{\alpha }(\lambda )={u}_{\mathrm{G}}^{\alpha }({I}^{A}(\lambda ),\lambda )$. From these assumptions, one can derive the OG equations [33, 34]:

$$\begin{equation}\frac{\partial {x}_{\mathrm{G}}^{\alpha }}{\partial {I}^{A}}{\dot{I}}^{A}=0\quad \text{and}\quad \frac{\partial {u}_{\alpha }^{\mathrm{G}}}{\partial {I}^{A}}{\dot{I}}^{A}={\Sigma}{a}_{\alpha }.\end{equation} \tag{ 12a-b }$$

From these, evolution equations for each of the orbital elements can be calculated, which we have done in appendices B and C. This was first done for generic Kerr orbits in reference [34] which lays out four different formulations of the equations. Three of these formulations use the quasi-Keplerian angles ψ and χ as the orbital phases, and two of these were numerically implemented and shown to agree with each other. We make use of the final formulation where one instead uses the geodesic actions angels q_r and q_z as the orbital phases.

We first find the evolution equations for the integrals of motion $\vec{P}$. We do so by finding evolution equations for $\vec{\mathcal{P}}$ and relating these to evolution equations for the roots r₁, r₂ and z₋. We derive these relations in appendix B. From here, one can invert equations (9) to find

$$\begin{equation}\frac{\mathrm{d}p}{\mathrm{d}\lambda }=\frac{2}{M{({r}_{1}+{r}_{2})}^{2}}\left({r}_{2}^{2}\frac{\mathrm{d}{r}_{1}}{\mathrm{d}\lambda }+{r}_{1}^{2}\frac{\mathrm{d}{r}_{2}}{\mathrm{d}\lambda }\right){:=}{F}_{p},\end{equation} \tag{ 13a }$$

$$\begin{equation}\frac{\mathrm{d}e}{\mathrm{d}\lambda }=\frac{2}{{({r}_{1}+{r}_{2})}^{2}}\left({r}_{2}\frac{\mathrm{d}{r}_{1}}{\mathrm{d}\lambda }-{r}_{1}\frac{\mathrm{d}{r}_{2}}{\mathrm{d}\lambda }\right){:=}{F}_{e},\end{equation} \tag{ 13b }$$

$$\begin{equation}\frac{\mathrm{d}x}{\mathrm{d}\lambda }=-\frac{{z}_{-}}{x}\frac{\mathrm{d}{z}_{-}}{\mathrm{d}\lambda }\;{:=}{F}_{x}.\end{equation} \tag{ 13c }$$

The orbital phases $\vec{q}$ still evolve with their respective Mino-time frequencies, but now pick up a correction due to the evolution of the initial values, i.e.,

$$\begin{equation}\frac{\mathrm{d}{q}_{i}}{\mathrm{d}\lambda }={{\Upsilon}}_{i}+\frac{\mathrm{d}{q}_{i,0}}{\mathrm{d}\lambda }.\end{equation} \tag{ 14 }$$

To find the evolution equations for the initial values for the orbital phases, we can re-arrange the first osculating condition (12a) and exploit the fact that the evolution of r is independent of q_z , and the evolution z is independent of q_r , to get

$$\begin{equation}\frac{\mathrm{d}{q}_{i,0}}{\mathrm{d}\lambda }=-\frac{1}{\partial {x}_{\mathrm{G}}^{i}/\partial {q}_{i}}\left(\frac{\partial {x}_{\mathrm{G}}^{i}}{\partial {P}_{j}}\frac{\mathrm{d}{P}_{j}}{\mathrm{d}\lambda }\right){:=}{f}_{i}^{(1)},\end{equation} \tag{ 15 }$$

where ${x}_{\mathrm{G}}^{i}$ is the geodesic expression for r or z given by equations (A3) and (A4) respectively. Unfortunately, this expression is difficult to evaluate numerically at the orbital turning points where both $\partial {x}_{\mathrm{G}}^{i}/\partial {q}_{i}$ and the term in the parentheses go to zero. However, one can derive an alternative expression for this quantity that is regular at the turning points [34]:

$$\begin{equation}\frac{\mathrm{d}{q}_{i,0}}{\mathrm{d}\lambda }=\frac{2{{\Upsilon}}_{i}}{\partial {V}_{i}/\partial {x}_{\mathrm{G}}^{i}}\left({{\Sigma}}^{2}{a}^{i}-\left(\frac{\partial {\dot{x}}_{\mathrm{G}}^{i}}{\partial {P}_{j}}\dot{{P}_{j}}\right)\right){:=}{f}_{i}^{(1)}.\end{equation} \tag{ 16 }$$

See appendix C for details on the derivation. This expression instead has a singularity whenever $\partial {V}_{i}/\partial {x}_{\mathrm{G}}^{i}=0$. Thus, for our numerical implementation, we use equation (15) for the majority of the orbital cycle and switch to equation (16) in the vicinity of turning points.

Finally, we also require evolution equations for 'extrinsic quantities' that do not show up on the right-hand side of the equations of motion, but are necessary to compute the trajectory and the waveform. These are the time and azimuthal coordinates of the secondary which, as a set, we denote by $\vec{S}=\left\{t,\phi \right\}$. Their evolution is given by the geodesic equations for t and ϕ, i.e., equations (7c) and (7d). Putting it all together, the equations of motion take the form:

$$\begin{equation}\dot{{P}_{j}}={F}_{j}(\vec{P},\vec{q}),\end{equation} \tag{ 17a }$$

$$\begin{equation}\dot{{q}_{i}}={{\Upsilon}}_{i}(\vec{P})+{f}_{i}(\vec{P},\vec{q}),\end{equation} \tag{ 17b }$$

$$\begin{equation}\dot{{S}_{k}}={s}_{k}(\vec{P},\vec{q}).\end{equation} \tag{ 17c }$$

These equations of motion are valid for generic inspirals under the influence of an unspecified perturbing force. We find that the action angle implementation produces inspiral trajectories that are identical to inspirals calculated using the 'null tetrad' formulation described in reference [34]. We have implemented both the action angle and null tetrad osculating element equations into a Mathematica package that will be publicly available on the Black Hole Perturbation Toolkit [49]. We find numerically that the null tetrad formulation is more computationally efficient as it does not have any singular equations that necessitate switching between different formulations twice during each orbital period. As such, for direct comparisons between OG and NIT inspirals and waveforms we make use of the null tetrad formulation, but use the action angle formulation as the starting point for our averaging procedure.

For the rest of this work, we specialize to the case of eccentric inspirals in the equatorial plane (i.e., spin aligned) under the influence of the first-order ratio gravitational self force (GSF). This corresponds to setting x = ±1 for prograde and retrograde orbits, respectively. Due to symmetry, motion in the equatorial plane will stay in the equatorial plane, and thus $\dot{x}=0$. As such, we only need to track the evolution of $\vec{P}=\left\{p,e\right\}$. Similarly, the equations of motion no longer depend on the polar phase q_z , and so we only need to evolve the radial phase $\vec{q}=\left\{{q}_{r}\right\}$. The GSF scales with the small mass ratio = μ/M, meaning that the secondary's four-acceleration can be expressed as ${a}^{\alpha }={\epsilon}{a}_{\text{GSF}}^{\alpha }$. Factoring out this scaling, the equations of motion for equatorial inspirals become

$$\begin{equation}\frac{\mathrm{d}p}{\mathrm{d}\lambda }={\epsilon}{F}_{p}(a,p,e,{q}_{r}),\end{equation} \tag{ 18a }$$

$$\begin{equation}\frac{\mathrm{d}e}{\mathrm{d}\lambda }={\epsilon}{F}_{e}(a,p,e,{q}_{r}),\end{equation} \tag{ 18b }$$

$$\begin{equation}\frac{\mathrm{d}{q}_{r}}{\mathrm{d}\lambda }={{\Upsilon}}_{r}(a,p,e)+{\epsilon}{f}_{r}(a,p,e,{q}_{r}),\end{equation} \tag{ 18c }$$

$$\begin{equation}\frac{\mathrm{d}t}{\mathrm{d}\lambda }={s}_{t}(a,p,e,{q}_{r}),\end{equation} \tag{ 18d }$$

$$\begin{equation}\frac{\mathrm{d}\phi }{\mathrm{d}\lambda }={s}_{\phi }(a,p,e,{q}_{r}).\end{equation} \tag{ 18e }$$

The NIT variables, ${\tilde{P}}_{j}$, ${\tilde{q}}_{i}$ and ${\tilde{S}}_{k}$, are related to the OG variables P_j , q_i and S_k via

$$\begin{equation}{\tilde{P}}_{j}={P}_{j}+{\epsilon}{Y}_{j}^{(1)}(\vec{P},\vec{q})+{{\epsilon}}^{2}{Y}_{j}^{(2)}(\vec{P},\vec{q})+\mathcal{O}({{\epsilon}}^{3}),\end{equation} \tag{ 19a }$$

$$\begin{equation}{\tilde{q}}_{i}={q}_{i}+{\epsilon}{X}_{i}^{(1)}(\vec{P},\vec{q})+{{\epsilon}}^{2}{X}_{i}^{(2)}(\vec{P},\vec{q})+\mathcal{O}({{\epsilon}}^{3}),\end{equation} \tag{ 19b }$$

$$\begin{equation}{\tilde{S}}_{k}={S}_{k}+{Z}_{k}^{(0)}(\vec{P},\vec{q})+{\epsilon}{Z}_{i}^{(1)}(\vec{P},\vec{q})+\mathcal{O}({{\epsilon}}^{2}).\end{equation} \tag{ 19c }$$

Here, the transformation functions ${Y}_{j}^{(n)}$, ${X}_{i}^{(n)}$, and ${Z}_{k}^{(n)}$ are required to be smooth, periodic functions of the orbital phases $\vec{q}$. At leading order, equation (19) are identity transformations for P_k and q_i but not for S_k due to the presence of a zeroth order transformation term ${Z}_{k}^{(0)}$. The inverse transformations can be found for P_k and q_i by requiring that their composition with the transformations in equation (19) must give the identity transformation. Expanding order by order in , this gives us

$$\begin{equation}\begin{array}{c} {P}_{j}=\tilde{{P}_{j}}-{\epsilon}{Y}_{j}^{(1)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})\\ \quad -{{\epsilon}}^{2}\left({Y}_{j}^{(2)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})-\frac{\mathrm{\partial }{Y}_{j}^{(1)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})}{\mathrm{\partial }\tilde{{P}_{k}}}{Y}_{k}^{(1)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})-\frac{\mathrm{\partial }{Y}_{j}^{(1)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})}{\mathrm{\partial }\tilde{{q}_{k}}}{X}_{k}^{(1)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})\right)\\ \quad +\mathcal{O}({{\epsilon}}^{3}),\end{array}\end{equation} \tag{ 20a }$$

$$\begin{equation}\begin{array}{c} {q}_{i}=\tilde{{q}_{i}}-{\epsilon}{X}_{i}^{(1)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})\\ \quad -{{\epsilon}}^{2}\left({X}_{i}^{(2)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})-\frac{\mathrm{\partial }{X}_{i}^{(1)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})}{\mathrm{\partial }\tilde{{P}_{j}}}{Y}_{j}^{(1)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})-\frac{\mathrm{\partial }{X}_{i}^{(1)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})}{\mathrm{\partial }\tilde{{q}_{k}}}{X}_{k}^{(1)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})\right)\\ \quad +\mathcal{O}({{\epsilon}}^{3}).\end{array}\end{equation} \tag{ 20b }$$

To proceed it is useful to decompose various functions into Fourier series where we use the convention:

$$\begin{equation}A(\vec{P},\vec{q})=\sum\limits _{\vec{\kappa }\in {\mathbb{Z}}^{N}}{A}_{\vec{\kappa }}(\vec{P}){\text{e}}^{\text{i}\vec{\kappa }\cdot \vec{q}},\end{equation} \tag{ 21 }$$

where N is the number of orbital phases. Based on this, we can split the function into an averaged piece

$$\begin{equation}\left\langle A\right\rangle (\vec{P})={A}_{\vec{0}}(\vec{P})=\frac{1}{{(2\pi )}^{N}}\int \cdots {\int }_{\vec{q}}A(\vec{P},\vec{q})d{q}_{1}\dots d{q}_{N},\end{equation} \tag{ 22 }$$

and an oscillating piece

$$\begin{equation}\breve{A}(\vec{P},\vec{q})=A(\vec{P},\vec{q})-\left\langle A\right\rangle (\vec{P})=\sum\limits _{\vec{\kappa }\ne \vec{0}}{A}_{\vec{\kappa }}(\vec{P}){\text{e}}^{\text{i}\vec{\kappa }\cdot \vec{q}}.\end{equation} \tag{ 23 }$$

Using the above transformations along with the equations of motion, and working order by order in , we can chose values for the transformation functions ${Y}_{j}^{(1)},{Y}_{j}^{(2)},{X}_{i}^{(1)},{X}_{i}^{(2)},{Z}_{k}^{(0)}$ and ${Z}_{k}^{(1)}$ such that the resulting equations of motion for $\tilde{{P}_{j}},\tilde{{q}_{i}}$ and $\tilde{{S}_{k}}$ take the following form

$$\begin{equation}{\dot{\tilde{P}}}_{j}=0+{\epsilon}{\tilde{F}}_{j}^{(1)}(\overrightarrow{\tilde{P}})+{{\epsilon}}^{2}{\tilde{F}}_{j}^{(2)}(\overrightarrow{\tilde{P}})+\mathcal{O}({{\epsilon}}^{3}),\end{equation} \tag{ 24a }$$

$$\begin{equation}{\dot{\tilde{q}}}_{i}={{\Upsilon}}_{i}(\overrightarrow{\tilde{P}})+{\epsilon}{\tilde{f}}_{i}^{(1)}(\overrightarrow{\tilde{P}})+{{\epsilon}}^{2}{\tilde{f}}_{i}^{(2)}(\overrightarrow{\tilde{P}})+\mathcal{O}({{\epsilon}}^{3}),\end{equation} \tag{ 24b }$$

$$\begin{equation}{\dot{\tilde{S}}}_{k}={\tilde{s}}_{k}^{(0)}(\overrightarrow{\tilde{P}})+{\epsilon}{\tilde{s}}_{k}^{(1)}(\overrightarrow{\tilde{P}})+\mathcal{O}({{\epsilon}}^{2}).\end{equation} \tag{ 24c }$$

Crucially, these equations of motion are now independent of the orbital phases $\vec{q}$. Deriving the relationship between the transformed forcing functions (${\tilde{F}}_{j}^{(1{\backslash}2)},{\tilde{f}}_{i}^{(1{\backslash}2)}$ and ${\tilde{s}}_{k}^{(0{\backslash}1)}$) to the original forcing functions is quite an involved process with several freedoms and choices, each with their own merits and drawbacks. This is discussed at length in [36], so for brevity we will summarize the results and the particular choices we have made in this work.

The transformed forcing functions are related to the original functions by

$$\begin{equation}{\tilde{F}}_{j}^{(1)}=\left\langle {F}_{j}^{(1)}\right\rangle ,\end{equation} \tag{ 25a }$$

$$\begin{equation}{\tilde{f}}_{i}^{(1)}=\left\langle {f}_{i}^{(1)}\right\rangle ,\end{equation} \tag{ 25b }$$

$$\begin{equation}{\tilde{s}}_{k}^{(0)}=\left\langle {s}_{k}^{(0)}\right\rangle ,\end{equation} \tag{ 25c }$$

$$\begin{equation}{\tilde{F}}_{j}^{(2)}=\left\langle {F}_{j}^{(2)}\right\rangle +\left\langle \frac{\partial {\breve{Y}}_{j}^{(1)}}{\partial {\tilde{q}}_{i}}{\breve{f}}_{i}^{(1)}\right\rangle +\left\langle \frac{\partial {\breve{Y}}_{j}^{(1)}}{\partial {\tilde{P}}_{k}}{\breve{F}}_{k}^{(1)}\right\rangle ,\end{equation} \tag{ 25d }$$

$$\begin{equation}{\tilde{f}}_{i}^{(2)}=0,\;\text{and}\end{equation} \tag{ 25e }$$

$$\begin{equation}{\tilde{s}}_{k}^{(1)}=\left\langle {s}_{k}^{(1)}\right\rangle -\left\langle \frac{\partial {\breve{s}}_{k}^{(0)}}{\partial {\tilde{P}}_{j}}{\breve{Y}}_{j}^{(1)}\right\rangle -\left\langle \frac{\partial {\breve{s}}_{k}^{(0)}}{\partial {\tilde{q}}_{i}}{\breve{X}}_{i}^{(1)}\right\rangle .\end{equation} \tag{ 25f }$$

In deriving these equations of motion, we have constrained the oscillating pieces of the NIT transformation functions to be

$$\begin{equation}{\breve{Y}}_{j}^{(1)}\equiv \sum\limits _{\vec{\kappa }\ne \vec{0}}\frac{i}{\vec{\kappa }\cdot \vec{{\Upsilon}}}{F}_{j,\vec{\kappa }}^{(1)}{\text{e}}^{\text{i}\vec{\kappa }\cdot \vec{q}},\end{equation} \tag{ 26 }$$

$$\begin{equation}{\breve{X}}_{i}^{(1)}\equiv \sum\limits _{\vec{\kappa }\ne \vec{0}}\left(\frac{i}{\vec{\kappa }\cdot \vec{{\Upsilon}}}{f}_{i,\vec{\kappa }}^{(1)}+\frac{1}{{(\vec{\kappa }\cdot \vec{{\Upsilon}})}^{2}}\frac{\partial {{\Upsilon}}_{i}}{\partial {P}_{j}}{F}_{j,\vec{\kappa }}^{(1)}\right){\text{e}}^{\text{i}\vec{\kappa }\cdot \vec{q}},\end{equation} \tag{ 27 }$$

and ${\breve{Z}}_{k}^{(0)}$ is found by solving

$$\begin{equation}{\breve{s}}_{k}^{(0)}+\frac{\partial {\breve{Z}}_{k}^{(0)}}{\partial {\tilde{q}}_{i}}{{\Upsilon}}_{i}=0.\end{equation} \tag{ 28 }$$

This equation is satisfied by the oscillating pieces for the analytic solutions for the geodesic motion of t and ϕ in equation (A2), i.e.,

$$\begin{equation}{\breve{Z}}_{k}^{(0)}=-{\breve{S}}_{k,r}({q}_{r})-{\breve{S}}_{k,z}({q}_{z}).\end{equation} \tag{ 29 }$$

We have chosen the averaged pieces such that $\left\langle {Y}_{j}^{(1)}\right\rangle =\left\langle {Y}_{j}^{(2)}\right\rangle =\left\langle {X}_{i}^{(2)}\right\rangle =\left\langle {Z}_{k}^{(0)}\right\rangle =\left\langle {Z}_{k}^{(1)}\right\rangle =0$ but have used $\left\langle {X}_{i}^{(1)}\right\rangle$ to cancel out the contributions of ${\tilde{f}}_{i}^{(2)}$. In order to generate waveforms, one only needs to know the transformations in equation (19) to zeroth order in the mass ratio, i.e.,

$$\begin{equation}{P}_{j}=\tilde{{P}_{j}}+\mathcal{O}({\epsilon}),\end{equation} \tag{ 30a }$$

$$\begin{equation}{q}_{i}=\tilde{{q}_{i}}+\mathcal{O}({\epsilon}),\end{equation} \tag{ 30b }$$

$$\begin{equation}{S}_{k}=\tilde{{S}_{k}}-{Z}_{k}^{(0)}(\overrightarrow{\tilde{P}},\overrightarrow{\tilde{q}})+\mathcal{O}({\epsilon}).\end{equation} \tag{ 30c }$$

Furthermore, to be able to directly compare between OG and NIT inspirals, we will need to match their initial conditions to sufficient accuracy. To maintain an overall phase difference of $\mathcal{O}({\epsilon})$ in the course of an inspiral, this requires known the transformation of the P_j 's (19a) to linear order in , while it is sufficient to know the rest of equation (19) to zeroth order. In particular, we will never need an explicit expression for $\left\langle {X}_{i}^{(1)}\right\rangle$, which would require solving a PDE to obtain.

We now apply the near identity averaging transformation procedure to the equations of motion for equatorial Kerr inspirals to obtain:

$$\begin{equation}\frac{\mathrm{d}\tilde{p}}{\mathrm{d}\lambda }={\epsilon}{\tilde{F}}_{p}^{(1)}(a,\tilde{p},\tilde{e})+{{\epsilon}}^{2}{\tilde{F}}_{p}^{(2)}(a,\tilde{p},\tilde{e}),\end{equation} \tag{ 31a }$$

$$\begin{equation}\frac{\mathrm{d}\tilde{e}}{\mathrm{d}\lambda }={\epsilon}{\tilde{F}}_{e}^{(1)}(a,\tilde{p},\tilde{e})+{{\epsilon}}^{2}{\tilde{F}}_{e}^{(2)}(a,\tilde{p},\tilde{e}),\end{equation} \tag{ 31b }$$

$$\begin{equation}\frac{\mathrm{d}\tilde{{q}_{r}}}{\mathrm{d}\lambda }={{\Upsilon}}_{r}(a,\tilde{p},\tilde{e})+{\epsilon}{\tilde{f}}_{r}^{(1)}(a,\tilde{p},\tilde{e}),\end{equation} \tag{ 31c }$$

$$\begin{equation}\frac{\mathrm{d}\tilde{t}}{\mathrm{d}\lambda }={\tilde{s}}_{t}^{(0)}(a,\tilde{p},\tilde{e})+{\epsilon}{\tilde{s}}_{t}^{(1)}(a,\tilde{p},\tilde{e}),\end{equation} \tag{ 31d }$$

$$\begin{equation}\frac{\mathrm{d}\tilde{\phi }}{\mathrm{d}\lambda }={\tilde{s}}_{\phi }^{(0)}(a,\tilde{p},\tilde{e})+{\epsilon}{\tilde{s}}_{\phi }^{(1)}(a,\tilde{p},\tilde{e}).\end{equation} \tag{ 31e }$$

The leading order terms in each equation of motion are simply the original function averaged over a single geodesic orbit, i.e.,

$$\begin{equation}{\tilde{F}}_{p}^{(1)}=\left\langle {F}_{p}\right\rangle ,\qquad {\tilde{F}}_{e}^{(1)}=\left\langle {F}_{e}\right\rangle ,\qquad {\tilde{f}}_{r}^{(1)}=\left\langle {f}_{r}\right\rangle ,\end{equation} \tag{ 32 }$$

$$\begin{equation}{\tilde{s}}_{t}^{(0)}=\left\langle {s}_{t}\right\rangle ={{\Upsilon}}_{t},\qquad {\tilde{s}}_{\phi }^{(0)}=\left\langle {s}_{\phi }\right\rangle ={{\Upsilon}}_{\phi },\end{equation} \tag{ 33 }$$

where ϒ_t and ϒ_ϕ are the Mino-time t and ϕ fundamental frequencies. The remaining terms are more complicated and require Fourier decomposing the original functions and their derivatives with respect to the orbital elements (p, e). To express the result, we define the operator

$$\begin{align}\hfill \mathcal{N}(A)& =\sum\limits _{\kappa \ne 0}\frac{-1}{{{\Upsilon}}_{r}}\left[{A}_{\kappa }{f}_{r,-\kappa }-\frac{i}{\kappa }\left(\frac{\partial {A}_{\kappa }}{\partial \tilde{p}}{F}_{p,-\kappa }+\frac{\partial {A}_{\kappa }}{\partial \tilde{e}}{F}_{e,-\kappa }\right.\right.\hfill \\ \hfill & \quad -\left.\left.\frac{{A}_{\kappa }}{{{\Upsilon}}_{r}}\left(\frac{\partial {{\Upsilon}}_{r}}{\partial \tilde{p}}{F}_{p,-\kappa }+\frac{\partial {{\Upsilon}}_{r}}{\partial \tilde{e}}{F}_{e,-\kappa }\right)\right)\right].\hfill \end{align} \tag{ 34 }$$

With this in hand, the remaining terms in the equations of motion are found to be

$$\begin{equation}{\tilde{F}}_{p}^{(2)}=\mathcal{N}({F}_{p}),\qquad {\tilde{F}}_{e}^{(2)}=\mathcal{N}({F}_{e}),\qquad {\tilde{s}}_{t}^{(1)}=\mathcal{N}({s}_{t}),\qquad {\tilde{s}}_{\phi }^{(1)}=\mathcal{N}({s}_{\phi }).\end{equation} \tag{ 35 }$$

The GSF approach is reviewed extensively elsewhere [52, 53] and so we just give a brief overview of the calculations that we employ. The GSF approach starts by expanding the metric of the binary around the metric of the primary, i.e., ${g}_{\mu \nu }={\bar{g}}_{\mu \nu }+{\epsilon}{h}_{\mu \nu }^{(1)}+{{\epsilon}}^{2}{h}_{\mu \nu }^{(2)}+\dots \,$ where ${\bar{g}}_{\mu \nu }$ is the Kerr metric and the h⁽ⁿ⁾ are the nth order perturbations to the spacetime due to the presence of the secondary. The interaction between these metric perturbations and the motion of the secondary can be derived through a matched asymptotic expansion analysis [53]. In this work we use only first-order (in the mass ratio) results, as second order results are still emerging [30]. The first order metric perturbation generated by a compact object can be found by solving the linearized Einstein field equations with a point particle source moving on a geodesic of ${\bar{g}}_{\mu \nu }$. The matched asymptotic expansion analysis identifies a regular part of this (divergent) metric perturbation that is responsible for the backreaction on the compact object [53–56]. The result is a (self-)force that appears on the right-hand side of equation (3) computed from derivatives of the regular metric perturbation [53].

Solving the perturbation equations requires picking a gauge, and the resulting self-force is gauge dependent [57]. The self-force was first computed in the Lorenz gauge where the procedure for obtaining the regular part was best understood. Numerical calculations of the Lorenz gauge self-force have been made in both the frequency—[25, 26, 58] and time-domains [22–24]. All these results have been for motion in Schwarzschild spacetime, with one exception in Kerr [59].

Calculating perturbations of the Kerr spacetime is hampered by the lack of separability of the linearized Einstein field equations on this background. This difficulty can be circumvented by using the Teukolsky formalism for describing perturbations to the Weyl scalars [60], which is fully separable in the frequency domain. From the Weyl scalars, the metric perturbation can be reconstructed in a radiation gauge [61–63]. There has also been recent progress understanding how to reconstruct the metric in the Lorenz gauge [64]. Regularization of the metric perturbation in radiation gauges is more subtle [65], but self-force calculations in the radiation gauge are now routine [27, 28, 66, 67].

In this work we primarily use the self-force computed using the code of references [27, 28, 67]. This code uses the Mano–Suzuki–Takasugi methods [68–71] to compute the perturbations to the Weyl scalars in the frequency domain. The metric is then reconstructed into an outgoing radiation gauge (including mass and angular momentum perturbations [72, 73] and gauge completion contributions [74]). The metric perturbation is then projected onto a basis of spherical harmonics before regularization is carried out using the mode-sum approach [65, 75]. Depending on the eccentricity of the orbit the code must compute the metric perturbation by summing over thousands to tens of thousands of Fourier harmonic modes. With the current Mathematica implementation, the self-force for an, e.g., a = 0.9M, p = 3.375, e = 0.5 orbit takes approximately 90 CPU hours to compute.

The self-force can be split into dissipative (time anti-symmetric) and conservative (time-symmetric) contributions [76].

The dissipative pieces causes the orbit to shrink until the secondary plunges into the primary. It also generally causes the orbit to circularize, with the exception being just before the transition to plunge where the orbit gains eccentricity [44, 77–79]. To produce adiabatic waveforms, we only require knowledge of the orbit averaged dissipative pieces of the first-order self-force. These can be related, via balance laws, to the fluxes of GWs to infinity and down the event horizon. Since calculating fluxes avoids regularization of the metric perturbation, adiabatic inspirals are typically calculated via flux balance laws [44, 78–83]. The conservative pieces have more subtle effects on the inspiral, such as altering the rate of periapsis advance and the location of the innermost stable circular orbit (ISCO) [50, 84–89].

To compute post-adiabatic inspirals requires knowledge of both the dissipative and conservatives pieces of the first-order self-force and the orbit average piece of the second-order self-force [19]. There are as yet no calculations of the latter so we will make do with just the first-order self-force information in this work. This will allow us to explore some of the effects of the conservative self-force on equatorial Kerr inspirals for the first time.

In order to drive inspirals, we need the self-force to be rapidly computed across the EMRI parameter space. To achieve this we tile the parameter space with GSF data which we can then interpolate. This approach has been implemented for eccentric Schwarzschild inspirals [31, 32]. In these works the data was interpolated using standard cubic spline methods, which required computing the self-force at tens of thousands of points in the parameter space. While this might not pose much of a problem for the 2D parameter space of eccentric, Schwarzschild inspirals, these approaches would not scale well to the 4D parameter space for generic Kerr inspirals. Motivated by this, as well as the computational expense of the eccentric Kerr self-force code, we build an interpolation model based on Chebyshev polynomials that is accurate to percent level across a 2D slice of the EMRI parameter space using only a few hundred points.

We start by fixing the value of the spin parameter of the primary, which we choose to be a = 0.9M for Kerr inspirals or a = 0 for Schwarzschild inspirals and set the inclination x to be either 1 or −1 for prograde orbits or retrograde orbits respectively. This reduces our parameter space to two parameters; the semilatus rectum p and the eccentricity e. We then define a parameter y using the p and the position of the last stable orbit p_LSO. For Kerr orbits, we chose y to be

$$\begin{equation}{y}_{\text{Kerr}}=\sqrt{\frac{{p}_{\text{LSO}}(a,e,x)}{p}}.\end{equation} \tag{ 36 }$$

With this parameterization we found that the accuracy of the Chebyshev interpolation is limited by the appearance of cusps at the LSO in the data. To ameliorate their impact we instead used a parameter y given by

$$\begin{equation}{y}_{\text{Schwarz}}=1-{\left(1-\frac{{p}_{\text{LSO}}(0,e,x)}{p}\right)}^{1/3}\end{equation} \tag{ 37 }$$

for later runs in Schwarzschild spacetime. In either case, tiling the parameter space in y instead of p will concentrate more points near the separatrix where the self force varies the most.

We let y range from y_min = 0 (0.01 for Schwarzschild) to y_max = 1 and e range from e_min = 0 to e_max = 0.5 for Kerr and e_min = 0 to e_max = 0.3 for Schwarzschild. We define parameters u and v which cover this parameter space as they range from (−1, 1)

$$\begin{equation}u{:=}\frac{y-({y}_{\mathrm{min}}+{y}_{\mathrm{max}})/2}{({y}_{\mathrm{min}}-{y}_{\mathrm{max}})/2}\quad \text{and}\quad v{:=}\frac{e-({e}_{\mathrm{min}}+{e}_{\mathrm{max}})/2}{({e}_{\mathrm{min}}-{e}_{\mathrm{max}})/2}.\end{equation} \tag{ 38a-b }$$

This parameterization is convenient when using Chebyshev polynomials of the first kind, where the order n polynomial is defined by T_n (cos φ) := cos(nφ). The Chebyshev nodes are the roots these polynomials, and the location of the kth root of nth polynomial is given by

$$\begin{equation}{N}_{k}=\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{2k-1}{2n}\pi \right).\end{equation} \tag{ 39 }$$

We then calculate the GSF on a 15 × 7 grid of Chebyshev nodes, with the u values given by the roots of the 15th order polynomial and the v values given by the roots of the 7th order polynomial. At each point on our grid, we Fourier decompose each component of the force with respect to the radial action angle q_r . We then multiply the data for each Fourier coefficient by a factor of (1 − y)/(1 − e²), as we find that this smooths the behaviour of the force near the separatrix and improves the accuracy of our interpolation. Next, we use Chebyshev polynomials to interpolate each Fourier coefficient across the (u, v) grid. We then resum the modes to reconstruct our interpolated GSF model:

$$\begin{equation}{a}_{\alpha }=\frac{1-{e}^{2}}{1-y}\sum\limits _{\kappa =0}^{15}{A}_{\alpha }^{\kappa }(y,e)\mathrm{cos}(\kappa {q}_{r})+{B}_{\alpha }^{\kappa }(y,e)\mathrm{sin}(\kappa {q}_{r}),\end{equation} \tag{ 40 }$$

where

$$\begin{align}\hfill {A}_{\alpha }^{\kappa }(y,e)& =\sum\limits _{i=0}^{14}\,\sum\limits _{j=0}^{6}\,{A}_{\alpha }^{\kappa ij}{T}_{i}\left(u\right){T}_{j}\left(v\right)\quad \text{and}\hfill \\ \hfill {B}_{\alpha }^{\kappa }(y,e)& =\sum\limits _{i=0}^{14}\,\sum\limits _{j=0}^{6}\,{B}_{\alpha }^{\kappa ij}{T}_{i}\left(u\right){T}_{j}\left(v\right).\hfill \end{align} \tag{ 41 }$$

Using this procedure forces each component to become singular at the last stable orbit. While the GSF changes rapidly as one approaches the last stable orbit, we do not expect the components of the self force to diverge at the LSO. Understanding the analytic structure of the self-force in this region would likely improve future interpolation models.

We note that the GSF should satisfy the orthogonality condition with the geodesic four-velocity, i.e., a_α u^α = 0. Interpolation will bring with it a certain amount of error which can cause this condition to be violated. We find empirically that we can reduce this interpolation error by projecting the force so that this condition is always satisfied, i.e.,

$$\begin{equation}{a}_{\alpha }^{\perp }={a}_{\alpha }+{a}_{\beta }{u}^{\beta }{u}_{\alpha }.\end{equation} \tag{ 42 }$$

This procedure allows us to create a smooth, continuous model for the GSF with relative errors less than 5 × 10⁻³ in the strong field—see figure 1. The variation in the accuracy of the model is primarily a by-product of how close a given test point (green cross) is to the data points (white dots) used to create the model. We note that this level of precision would not be sufficient for production grade waveforms for LISA, as we would need the relative error of the orbit averaged dissipative self-force to be less than $\sim 1/{\epsilon}$, whereas the oscillatory pieces of the self-force only need to be interpolated to an accuracy of a fraction of a percent [32]. Our present interpolation model already likely reaches the latter criteria and a future hybrid method that combines flux and self-force data, similar to the one constructed in reference [32], can likely reach the overall accuracy goal. Nonetheless, our present model is more than sufficient to test our averaging procedure and to explore the effects of the GSF for eccentric Kerr inspirals. This will now be treated as the underlying forcing model for both the OG and NIT inspirals.

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To make the offline calculation we complete the following steps.

• (a)  
  We start by selecting a grid to evaluate the NIT functions upon. We chose y values between 0.2 and 0.998 in 320 equally spaced steps and e values from 0.001 to 0.5 in 500 equally spaced steps (160 000 points) in the case of Kerr, or use the same spacing in y but only grid in e from 0.001 to 0.3 in 300 equally spaced steps (96 000 points) in Schwarzschild. ³
• (b)  
  For each point in the parameter space (a, y, e) we evaluate the functions F_p\e , f_r and s_t\ϕ along with their derivatives with respect to p and e for 30 equally spaced values of q_r from 0 to 2π.
• (c)  
  We then perform a fast Fourier transform on the output data to obtain the Fourier coefficients of the forcing functions and their derivatives.
• (d)  
  With these, we then use equations (32)–(35) to construct ${\tilde{F}}_{p{\backslash}e}^{(1{\backslash}2)}$, ${\tilde{f}}_{r}^{(1)}$ and ${s}_{t{\backslash}\phi }^{(1)}$ for that point in the parameter space.
• (e)  
  We also use equations (26) and (27) to construct the Fourier coefficients of the first order transformation functions ${Y}_{p{\backslash}e}^{(1)}$ and ${X}_{r}^{(1)}$.
• (f)  
  We then repeat this procedure across the parameter space for each point in our grid.
• (g)  
  Finally we interpolate the values for ${\tilde{F}}_{p{\backslash}e}^{(1{\backslash}2)},{\tilde{f}}_{r}^{(1)}$ and ${\tilde{s}}_{t{\backslash}\phi }^{(1)}$ along with the coefficients of ${Y}_{p{\backslash}e}^{(1)}$ and ${X}_{r}^{(1)}$ across this grid using Hermite interpolation and store the interpolants for future use.

We implemented the above algorithm in Mathematica 12.2 and find, parallelized across 20 CPU cores takes, the calculation takes about one day to complete. This is a small price to pay, since these offline steps need only be completed once.

The online steps are required for every inspiral calculation, but are comparatively inexpensive. The online steps for computing an NIT inspiral are as follows.

• (a)  
  We load in the interpolants for ${\tilde{F}}_{p{\backslash}e}^{(1{\backslash}2)}$ and ${\tilde{f}}_{r}^{(1)}$ and ${\tilde{s}}_{t{\backslash}\phi }^{(1)}$, define the NIT equations of motion.
• (b)  
  In order to make comparisons between NIT and OG inspirals, we also load interpolants of the Fourier coefficients of ${\breve{Y}}_{p/e}^{(1)}$ and ${\breve{X}}_{r}^{(1)}$ and equation (19) to construct first order near-identity transformations. ⁴
• (c)  
  We state the initial conditions of the inspiral (p₀, e₀, q_r,0) and use the NIT to leading order in the mass ratio to transform these into initial conditions for the NIT equations of motion, i.e., $({\tilde{p}}_{0},{\tilde{e}}_{0},{\tilde{q}}_{r0})$.
• (d)  
  We then evolve the NIT equations of motion using an ODE solver (in this case Mathematica's NDSolve).

As with the offline steps we implement the online steps in Mathematica. Note that steps (b) and (c) are only necessary because we want to make direct comparisons between NIT and OG inspirals with the same initial conditions. In general, the difference between the NIT and OG variables will always be $\mathcal{O}({\epsilon})$, and so performing the NIT transformation or inverse transformation to greater than zeroth order in mass ratio will not be necessary when producing waveforms to post adiabatic order, i.e. with phases accurate to $\mathcal{O}({\epsilon})$.

Before computing inspirals, we perform a series of consistency checks on the NIT equations of motion. A useful feature of the NIT is how it separates adiabatic and post-adiabatic effects of the GSF. At first order in the mass ratio, this corresponds to the dissipative and conservative pieces respectively. We note that when we substitute ${a}^{\alpha }\to {a}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}^{\alpha }$, we find that ${\tilde{F}}_{p{\backslash}e}^{(2)}$, ${\tilde{f}}_{r}^{(1)}$ and ${\tilde{s}}_{t{\backslash}\phi }^{(1)}$ are numerically consistent with zero, while ${\tilde{F}}_{p{\backslash}e}^{(1)}$ remains unchanged. Similarly, when we substitute ${a}^{\alpha }\to {a}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}}^{\alpha }$, ${\tilde{F}}_{p{\backslash}e}^{(1)}$ and ${\tilde{F}}_{p{\backslash}e}^{(2)}$ become consistent with zero, while ${\tilde{f}}_{r}^{(1)}$ and ${\tilde{s}}_{t{\backslash}\phi }^{(1)}$ remain the same as before. The functions ${\tilde{F}}_{p{\backslash}e}^{(2)}$ only becomes non-zero when both dissipative and conservative effects of the first order self-force are present.

From ${\tilde{F}}_{p{\backslash}e}^{(1)}$, one can calculate the average rate of change of energy and angular momentum via the following relation:

$$\begin{equation}\left\langle \frac{\mathrm{d}\mathcal{E}}{\mathrm{d}t}\right\rangle =\frac{{\epsilon}}{{{\Upsilon}}_{t}}\left(\frac{\partial \mathcal{E}}{\partial p}{\tilde{F}}_{p}^{(1)}+\frac{\partial \mathcal{E}}{\partial e}{\tilde{F}}_{e}^{(1)}\right)\end{equation} \tag{ 43a }$$

$$\begin{equation}\left\langle \frac{\mathrm{d}\mathcal{L}}{\mathrm{d}t}\right\rangle =\frac{{\epsilon}}{{{\Upsilon}}_{t}}\left(\frac{\partial \mathcal{L}}{\partial p}{\tilde{F}}_{p}^{(1)}+\frac{\partial \mathcal{L}}{\partial e}{\tilde{F}}_{e}^{(1)}\right).\end{equation} \tag{ 43b }$$

We compared these to the energy and angular momentum fluxes at infinity tabulated in the Black Hole Perturbation Toolkit [49] and generated with a variant of the Gremlin code [80, 81] and found that the balance laws were upheld up to relative errors $< 1{0}^{-3}$ throughout the parameter space which is consistent with the interpolation error of our self-force model.

From all of this, we can infer the significance of each of the terms in equation (31): ϒ_r , ϒ_t and ϒ_ϕ capture the background geodesic motion, ${\tilde{F}}_{p}^{(1)}$ and ${\tilde{F}}_{e}^{(1)}$ capture the adiabatic effects due to the first order dissipative self-force, ${\tilde{f}}_{r}^{(1)}$, ${\tilde{s}}_{t}^{(1)}$, and ${\tilde{s}}_{\phi }^{(1)}$ capture the post-adiabatic effects due to the first order conservative self-force, and ${\tilde{F}}_{p}^{(2)}$, ${\tilde{F}}_{e}^{(2)}$ capture the interplay between the first order dissipative and conservative self-force, as well as the effect of the orbit averaged contribution from the second order self-force.

In order to test the accuracy of our implementation, we compare inspirals calculated using the OG equations of motion found in reference [34] to those calculated using the near-identity transformed equations of motion. To demonstrate these results, we choose a binary with a primary of mass M = 10⁶ M_⊙ and a secondary of mass μ = 10M_⊙ for a typical EMRI mass ratio of = 10⁻⁵. To push our procedure to the limit, we chose the initial conditions of our prograde inspiral to be deep in the strong field and highly eccentric with p₀ = 7.1 and e₀ = 0.48 such that the resulting inspiral would take approximately 1 year to plunge. We also set q_r,0 = t₀ = ϕ₀ = 0 for simplicity.

Figure 2 shows the evolution of p and e over time. The trajectories calculated with the OG equations of motion have order oscillations on the orbital timescale which requires the numerical integrator to take small time steps to accurately resolve. The NIT trajectory does not have these oscillations so the numerical integrator can take much larger steps and still faithfully track the averaged trajectory throughout the entire inspiral. The inverse NIT given in equation (20a) through $\mathcal{O}({\epsilon})$ can be used to add the oscillations back on to the NIT trajectory. We find that while this is unnecessary for computing accurate waveforms, it demonstrates that the NIT trajectory remains in phase with the OG trajectory—see the insets of figure 2.

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The accuracy of our NIT model is further demonstrated by figure 3 which shows the absolute difference in the orbital phase q_r and the extrinsic quantities t and ϕ between the NIT and OG evolutions. Over the course of the year long inspiral, $\vert t-(\tilde{t}-{Z}_{t}^{(0)})\vert \leqslant 5\times 1{0}^{-3},\vert \phi -(\tilde{\phi }-{Z}_{\phi }^{(0)})\vert \leqslant 1{0}^{-5}$ and $\vert {q}_{r}-{\tilde{q}}_{r}\vert \leqslant 1{0}^{-3}$ with the differences only spiking to $\leqslant 1{0}^{-2}$ just as the trajectories reach the separatrix where the adiabatic approximation breaks down.

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Finally, we test the effect the NIT procedure has on the waveform. In principle, we could use our averaged equations of motion in conjunction with the FastEMRIWaveforms (FEW) framework to rapidly compute waveforms with relativistic amplitudes. However, currently, the FEW framework only has amplitude data for Schwarzschild inspirals. As such, we make use of the same procedure as the numerical kludge [10] by mapping the Boyer–Lindquist coordinates {t, r, θ, ϕ} to flat space coordinates and using the quadrupole formula to generate the waveform. The resulting waveforms are only an approximation to the true waveforms, but since both inspiral trajectories are being fed through the same waveform generation scheme this should not bias the results when finding the difference in the waveform as a result of using the NIT trajectory instead of the OG trajectory.

From figure 4, we can see that the waveforms generated by each evolution scheme, sampled every t = 1M ≈ 5s, are almost identical by eye. We can further quantify this by calculating the waveform mismatch using the WaveformMatch function from the SimulationTools [90] Mathematica package and assuming a flat noise curve. From figure 5, we see that the mismatch remains below 5 × 10⁻⁸ throughout the inspiral. At this level of mismatch the two waveforms would be completely indistinguishable for EMRIs with SNR of up to (at least) 3000 [91–93].

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Next, the difference between the OG and NIT quantities should scale linearly with the mass ratio. This is illustrated in figure 6, where starting with initial conditions p₀ = 4 and e₀ = 0.2 we evolved the inspiral until it reached p = 3 for mass ratios ranging from 10⁻¹ to 10⁻⁵. While working with only machine precision arithmetic we found that for smaller mass ratios the numerical error of the solver of the OG inspiral became dominant over the difference with the NIT. To rectify this, we increased the working precision of our solver to 30 significant digits and found that the difference does, in fact, scale linearly with the mass ratio. This requirement for higher precision only affected the OG solver, the NIT equations of motion can be solved with machine precision arithmetic without introducing any significant error.

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Since the difference between OG and NIT quantities scales with the mass ratio, it is natural to ask how large can the mass ratio be before the NIT and OG waveforms differ enough to affect data analysis. Following the procedure outlined in reference [38], we used our fast NIT inspiral code along with a root-finding algorithm to find the initial value of p that corresponds to a year long inspiral for a given value of the mass ratio and initial eccentricity, and assuming a primary mass of 10⁶ M_⊙. We use these initial conditions to calculate the overlap between year-long NIT and OG waveforms. This calculation is repeated with mass ratios = {1, 3, 5, 7, 9} × 10⁻³ and initial eccentricities e₀ ranging from 0.05 to 0.45 in equally spaced steps of 0.05. The result of this analysis can be seen in figure 7. This demonstrates that NIT and OG waveforms have overlaps larger than the benchmark of 0.97 [94] for mass ratios less than $\approx \;3\times 1{0}^{-3}$, but these overlaps decrease substantially for mass ratios larger than this. We also see that the overlap generally decreases as the initial eccentricity increases, though this effect is not as strong as the effect demonstrated by a similar analysis in reference [38] for NITs applied to highly eccentric inspirals in Schwarzschild. They also found that the mismatch between NIT and OG waveforms became substantial for mass ratios larger than 2 × 10⁻⁴. These differences between the two analyses are most likely the result of our inspirals being deeper in the strong field and driven by a self-force computed in a different gauge (reference [38] uses the Lorenz gauge self-force). Such mismatches should not be an issue for EMRI data analysis as EMRIs have mass ratios that range from 10⁻⁷ to 10⁻⁴. However, these mismatches become significant for intermediate mass ratio inspirals, with mass ratios between 10⁻⁴ to 10⁻¹. Since both the OG and NIT equations of motion are formally valid to the same order in the mass ratio, it is not clear a priori which of the two would be closer to the true inspiral. When completed at one-post-adiabtatic (1PA) order the two sets of equations represent different resummations of the 1PA equations of motion, differing only in their higher order (2+) PA terms. The fact that we are seeing a significant difference between these two resummations for intermediate mass ratios suggests that such higher order PA terms might become relevant. However, in this case it might just be signalling the importance of the missing orbit-averaged dissipative self-force term at 1PA order.

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Finally, we note that using the NIT equations of motion produces a substantial speed-up over using the OG equations. From table 1, we see the typical computation time for an inspiral starting at p₀ = 7.1 and e₀ = 0.48 and evolved until the inspiral reaches the last stable orbit for different values of the mass ratio. We see that as we decrease the mass ratio by an order of magnitude, the OG inspiral takes roughly an order of magnitude longer to compute, as it would have to resolve an order of magnitude more orbital cycles before reaching last stable orbit. The NIT inspirals all take roughly the same amount of time to evolve to the last stable orbit, regardless of the mass ratio. Using our current Mathematica implementation, the NIT inspirals can be computed in less than a second. This time could be further reduced tens of milliseconds if one uses a compiled language such as C/C++, as was done in paper I [36]. We see that using the NIT equations of motion is most advantageous for long inspirals with small mass ratios. Another benefit of using the NIT is that the inspiral requires taking fewer time steps, which results in less numerical error, making it easier achieve a given target accuracy.

Table 1. Computational time required to evolve an inspiral from its initial conditions of p₀ = 7.1 and e₀ = 0.48 to the last stable orbit for different values of the mass ratio, as calculated in Mathematica 12.2 on an Intel Core i7 @ 2.2 GHz. The computational time for the OG inspiral scales inversely with the mass ratio, whereas the computational time for NIT inspirals is independent of the mass ratio. This demonstrates how the smaller the mass ratio of the inspiral, the greater speed-up one obtains from using the NIT equations of motion.

	OG Inspiral	NIT Inspiral	Speed-up
10−2	44 s	0.85 s	$\sim 37$
10−3	6 m 48 s	0.78 s	$\sim 491$
10−4	54 m 12 s	0.81 s	$\sim 3782$
10−5	6 h 16 m	0.76 s	$\sim 29\,655$

The only disadvantage of our formulation is that our final trajectory is parameterized in terms Mino time λ, whereas LISA data analysis applications will need waveforms parameterized Boyer–Lindquist retarded time t. Since our formulation also outputs t(λ), we can numerically invert this to get λ(t) which allows us to resolve this issue at the cost of additional computation time. This was also a problem with the NIT formulation in Schwarzschild where the final trajectory is outputted as a function of the quasi-Keplerian angle χ [36, 38]. This problem might be circumvented entirely by performing an additional transformation to our NIT equations of motion which would produce averaged equations of motion parameterized by t as outlined in [35].

Since we are now satisfied that our formulation can produce fast and accurate self-force driven trajectories, we can now use this procedure to explore the phenomenology of eccentric, equatorial Kerr inspirals.

With the ability to generate fast and accurate inspirals, we can survey the physics of equatorial Kerr inspirals and examine how this differs from the Schwarzschild case. From figure 8(a), we see the familiar effect of gravitational radiation reaction on the semilatus rectum, $\tilde{p}$, and eccentricity, $\tilde{e}$, whereby $\tilde{p}$ and $\tilde{e}$ both decrease over the inspiral with $\tilde{e}$ growing a little as the last stable orbit is approached [44, 78, 79]. As the inspiral approaches the last stable orbit adiabaticity breaks down and the inspiral undergoes a transition to plunge [95–98]. As such, we stop our inspirals just before the last stable orbit. Our results are the first inspirals to include conservative self-force corrections to the equations of motion in Kerr spacetime. The initial phase q_r,0 only evolves secularly when conservative self-force corrections are present and so we use this as a measure of the influence of these corrections [31]. This is illustrated by the dashed orange curves in figure 8(a), which mark the number of radians ${\tilde{q}}_{r,0}$ will evolve from a given pair of initial conditions $({\tilde{p}}_{0},{\tilde{e}}_{0})$ until the last stable orbit. For retrograde Kerr (and Schwarzschild orbits in figure 10), we find that ${\tilde{q}}_{r,0}$ increases throughout the inspiral, whereas for prograde Kerr ${\tilde{q}}_{r,0}$ decreases during the inspiral before increasing slightly just before plunge. This is consistent with the change of sign in the correction to the rate of periapsis advance induced by the conservative self force as a function of spin in the circular orbit limit [88]—see appendix D for further details.

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As discussed in section 6.1, one can readily calculate adiabatic inspirals using the NIT equations of motion by simply neglecting the post-adiabatic terms. However, when trying to determine how post-adiabatic corrections effect the inspiral, one must be mindful of how one matches up an adiabatic inspiral with its post-adiabatic counterpart. Following the argument found in references [31, 32], matching the initial conditions $({\tilde{p}}_{0},{\tilde{e}}_{0})$ results in an error in the orbital phases that grows linearly in t as the conservative self-force changes the orbital frequencies [86]. Instead, one should instead match the Boyer–Lindquist time fundamental frequencies Ω_r and Ω_ϕ . For an adiabatic inspiral, these are directly related to the Mino-time fundamental frequencies via ${{\Omega}}_{r{\backslash}\phi }^{\mathrm{A}\mathrm{d}}=\frac{{{\Upsilon}}_{r{\backslash}\phi }}{{{\Upsilon}}_{t}}$ [42]. To calculate these frequencies as perturbed by the conservative self-force, one can either follow the method outlined in reference [32], or one can calculate them directly from the NIT equations of motion:

$$\begin{equation}{{\Omega}}_{r}^{\mathrm{S}\mathrm{F}}=\frac{{{\Upsilon}}_{r}+{\epsilon}{\tilde{f}}_{r}^{(1)}}{{{\Upsilon}}_{t}+{\epsilon}{\tilde{s}}_{t}^{(1)}}+\mathcal{O}({{\epsilon}}^{2})\quad \text{and}\quad {{\Omega}}_{\phi }^{\mathrm{S}\mathrm{F}}=\frac{{{\Upsilon}}_{\phi }+{\epsilon}{\tilde{s}}_{\phi }^{(1)}}{{{\Upsilon}}_{t}+{\epsilon}{\tilde{s}}_{t}^{(1)}}+\mathcal{O}({{\epsilon}}^{2}).\end{equation} \tag{ 44 }$$

We find that both approaches give the same result up to an error that scales as ². With this in hand, we can now choose a value for our initial conditions $({\tilde{p}}_{0}^{\mathrm{S}\mathrm{F}},{\tilde{e}}_{0}^{\mathrm{S}\mathrm{F}})$ for our self-forced inspiral, and then root find for initial conditions $({\tilde{p}}_{0}^{\mathrm{A}\mathrm{d}},{\tilde{e}}_{0}^{\mathrm{A}\mathrm{d}})$ that satisfy the simultaneous equations

$$\begin{equation}{{\Omega}}_{r}^{\mathrm{S}\mathrm{F}}({\tilde{p}}_{0}^{\mathrm{S}\mathrm{F}},{\tilde{e}}_{0}^{\mathrm{S}\mathrm{F}})-{{\Omega}}_{r}^{\mathrm{A}\mathrm{d}}({\tilde{p}}_{0}^{\mathrm{A}\mathrm{d}},{\tilde{e}}_{0}^{\mathrm{A}\mathrm{d}})=0,\end{equation} \tag{ 45a }$$

$$\begin{equation}{{\Omega}}_{\phi }^{\mathrm{S}\mathrm{F}}({\tilde{p}}_{0}^{\mathrm{S}\mathrm{F}},{\tilde{e}}_{0}^{\mathrm{S}\mathrm{F}})-{{\Omega}}_{\phi }^{\mathrm{A}\mathrm{d}}({\tilde{p}}_{0}^{\mathrm{A}\mathrm{d}},{\tilde{e}}_{0}^{\mathrm{A}\mathrm{d}})=0.\end{equation} \tag{ 45b }$$

Using this procedure to match the initial frequencies we find that the linear-in-t growth of the difference in the orbital phases is removed and the phase difference grows quadratically in t as expected—see figure 9.

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We now turn our attention to the special case of Schwarzschild (a = 0), where we now have interpolated GSF models calculated in two different gauges. In addition to our outgoing radiation gauge self-force model, we make use of an interpolated Lorenz gauge self-force from reference [31], which is valid in the domain 6 ⩽ p ⩽ 12 and 0 ⩽ e ⩽ 0.2. We apply the same NIT procedure to inspirals driven by this force model, and find agreement with inspirals calculated in paper I, up to the precision of the numerical solver.

To assess the accuracy of the dissipative self-force, we calculate the orbit averaged energy and angular momentum fluxes, and find that they agree with values from the literature with a relative error less than 10⁻³ for both models across the parameter space. To assess the accuracy of the conservative self-force, we calculate the periapsis advance in the circular orbit limit as outlined in [85] using the formula found in [88]. We find that both models show good agreement with the literature across the Lorenz gauge model's domain of validity, with the Lorenz gauge model producing errors less than 10⁻³ and the radiation gauge model producing relative errors less than 10⁻².

While we find good agreement between the two results for gauge invariant quantities, we see from figure 10 that the inspirals experience dramatically different conservative effects, depending on the gauge used. While in both cases, the conservative self-force acts against geodesic periapsis advance, we see that the evolution of q_r,0 depends heavily on the gauge involved, while the trajectories through p and e space are less affected. This is to be expected as the leading order averaged rates of change of p and e are related to the gauge invariant asymptotic fluxes, while the change in q_r,0 is induced entirely by the (gauge dependent) conservative self-force [27].

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Just as when comparing adiabatic and self-forced inspirals it is important to match the initial frequencies (rather than the initial (p, e) values). We note that for the Lorenz gauge model, we must account for the fact that the perturbed time coordinate, $\hat{t}$, is not asymptotically flat [99]. We can define an asymptotically flat time coordinate for Lorenz gauge inspirals via the following rescaling

$$\begin{equation}t=(1+{\epsilon}\alpha )\hat{t},\end{equation} \tag{ 46 }$$

where α is given by

$$\begin{equation}\alpha (p,e)=-\frac{1}{2}{h}_{tt}^{(1)}(r\to \infty ).\end{equation} \tag{ 47 }$$

We make use of a code provided to us by Akcay to numerically calculate this quantity for Lorenz gauge values of p and e [25, 100]. Equation (46) means the perturbed Boyer–Lindquist frequencies must also be rescaled by:

$$\begin{equation}{{\Omega}}_{r}^{(\mathrm{L}\mathrm{G})}=(1-{\epsilon}\alpha )\frac{{{\Upsilon}}_{r}+{\epsilon}{\tilde{f}}_{r(LG)}^{(1)}}{{{\Upsilon}}_{t}+{\epsilon}{\tilde{s}}_{t(LG)}^{(1)}}\quad \text{and}\quad {{\Omega}}_{\phi }^{(\mathrm{L}\mathrm{G})}=(1-{\epsilon}\alpha )\frac{{{\Upsilon}}_{\phi }+{\epsilon}{\tilde{s}}_{\phi (LG)}^{(1)}}{{{\Upsilon}}_{t}+{\epsilon}{\tilde{s}}_{t(LG)}^{(1)}}.\end{equation} \tag{ 48 }$$

In the radiation gauge model, the corresponding subtleties have been dealt with by including the gauge completion corrections, so the frequencies can be calculated using equation (44) as before. Thus, we can choose a value for ${\tilde{p}}_{0}^{(\mathrm{L}\mathrm{G})}$ and ${\tilde{e}}_{0}^{(\mathrm{L}\mathrm{G})}$ in Lorenz gauge and root find for values of ${\tilde{p}}_{0}^{(\mathrm{R}\mathrm{G})}$ and ${\tilde{e}}_{0}^{(\mathrm{R}\mathrm{G})}$ in radiation gauge that satisfy:

$$\begin{equation}{{\Omega}}_{r}^{(\mathrm{R}\mathrm{G})}({\tilde{p}}_{0}^{(\mathrm{R}\mathrm{G})},{\tilde{e}}_{0}^{(\mathrm{R}\mathrm{G})})-{{\Omega}}_{r}^{(\mathrm{L}\mathrm{G})}({\tilde{p}}_{0}^{(\mathrm{L}\mathrm{G})},{\tilde{e}}_{0}^{(\mathrm{L}\mathrm{G})})=0,\end{equation} \tag{ 49a }$$

$$\begin{equation}{{\Omega}}_{\phi }^{(\mathrm{R}\mathrm{G})}({\tilde{p}}_{0}^{(\mathrm{R}\mathrm{G})},{\tilde{e}}_{0}^{(\mathrm{R}\mathrm{G})})-{{\Omega}}_{\phi }^{(\mathrm{L}\mathrm{G})}({\tilde{p}}_{0}^{(\mathrm{L}\mathrm{G})},{\tilde{e}}_{0}^{(\mathrm{L}\mathrm{G})})=0.\end{equation} \tag{ 49b }$$

This allows us to make comparisons between inspirals driven by self-force models calculated in different gauges. We use an inspiral driven by the Lorenz-gauge force model with initial conditions (p₀, e₀) = (11, 0.18), mass ratio = 10⁻⁵ as our reference inspiral which should last just over two and a half years for a 10⁶ M_⊙ primary.

In figure 11, we see the difference in the phase of the waveform Φ as a function of time between the Lorenz gauge NIT inspiral, and a number of reference models. We make use of the relations between the NIT quantities and the waveform phases derived in reference [38] to find

$$\begin{equation}{{\Phi}}_{r}={\tilde{q}}_{r}-{{\Omega}}_{r}^{\mathrm{S}\mathrm{F}}{Z}_{t}^{(0)}+\mathcal{O}({\epsilon})\quad \;\text{and}\quad {{\Phi}}_{\phi }=\tilde{\phi }-{{\Omega}}_{\phi }^{\mathrm{S}\mathrm{F}}{Z}_{t}^{(0)}+\mathcal{O}({\epsilon}).\end{equation} \tag{ 50 }$$

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We then feed the solutions for $\left\{\tilde{p}(t),\tilde{e}(t),{{\Phi}}_{r}(t),{{\Phi}}_{\phi }(t)\right\}$ into the FEW package to generate these eccentric Schwarzschild waveforms [17]. Finally, we make use of the $\mathtt{S}\mathtt{i}\mathtt{m}\mathtt{u}\mathtt{l}\mathtt{a}\mathtt{t}\mathtt{i}\mathtt{o}\mathtt{n}\mathtt{T}\mathtt{o}\mathtt{o}\mathtt{l}\mathtt{s}$ Mathematica package to calculate the mismatches and decompose the waveforms into a single evolving amplitude A(t) and phase Φ(t). This allows us to find the difference in the waveform phase ΔΦ(t) between the Lorenz gauge inspiral and the other inspiral calculations. We use this as our point of comparison as the waveform phase is an observable and thus a gauge invariant quantity.

We note that in each case, we see constant error which gives way to quadratic growth with t just as in figure 9. As we discussed in section 6.3, this shows that the initial frequencies were correctly matched. From the blue curve, we see that the NIT radiation gauge inspiral quickly goes out of phase with the Lorenz gauge NIT inspiral, resulting in a very large mismatch of 0.93. We found that the largest source of error here is due to interpolation error for in the adiabatic pieces of the NIT. Since these are related to the gauge invariant fluxes, these pieces should be identical in both models. As such, we can rectify this error by using the Lorenz gauge functions for the adiabatic pieces and continue to use the radiation gauge functions for the conservative pieces of the NIT equations of motion. The improvement is evident in the green curve, which shows much better agreement with the Lorenz gauge NIT inspiral, with the mismatch falling to 0.83. However, it is only slightly better than matching an adiabatic inspiral (orange curve) using equation (45) resulting in a mismatch of 0.86. Both radiation gauge and adiabatic inspirals go out of phase by almost 100 radians by the time they reach the last stable orbit.

In order to rule out the possibility of interpolation error of the conservative effects being the primary cause of this difference, we repeat the Lorenz gauge inspiral, but this time we manually add a relative error of δ = 0.1 to all of the conservative pieces of both the NIT equation of motion and our matching procedure for the initial conditions, e.g., $\dot{\tilde{{q}_{r}}}={{\Upsilon}}_{r}+{\epsilon}{\tilde{f}}_{r}^{(1)}\to {{\Upsilon}}_{r}+{\epsilon}\left({\tilde{f}}_{r}^{(1)}+\delta {\tilde{f}}_{r}^{(1)}\right)$ etc. We note that this is an order of magnitude larger than the 10⁻² error produced by the radiation gauge model when calculating the gauge invariant quasi-circular periapsis advance. From the red curve, we see that manually adding a constant 10% relative error results in phase difference and a mismatch (0.54) which is significantly smaller than what we observe between the two self-forced inspirals. This gives us confidence that this difference is not dominated by numerical error.

From these investigations, we infer that the trajectories driven using only the first order self-force are gauge dependent, and thus, so too are their waveforms. Since post-adiabatic waveforms are an observable quantity, this leads us to conclude that incorporating the orbit-averaged dissipative second-order self-force will be necessary to obtain gauge invariant, post-adiabatic waveforms. Moreover, since the difference between the radiation and Lorenz gauge self-forced inspirals is of the same magnitude as the difference with the adiabatic inspiral, we further conclude that the impact of the orbit-averaged dissipative second-order self-force must be of a similar magnitude in at least one of the two gauges.