Fourth post-Newtonian effective-one-body Hamiltonians with generic spins

We use geometric units such that the speed of light $c$ and the Newton constant $G$ are equal to 1. We utilize various combinations of the masses $m_{1}$, $m_{2}$ of the binary’s components,

	$$\displaystyle M=m_{1}+m_{2},\quad\mu=\frac{m_{1}m_{2}}{M},\quad\nu=\frac{\mu}{M},$$
	$$\displaystyle q=\frac{m_{1}}{m_{2}},\quad X_{1}=\frac{m_{1}}{M},\quad X_{2}=\frac{m_{2}}{M}.$$		(1)

For the spins $\bm{S}_{1}$, $\bm{S}_{2}$, we define the dimensionless versions

$$\displaystyle\bm{\chi}_{1}=\frac{\bm{a}_{1}}{m_{1}}=\frac{\bm{S}_{1}}{m_{1}^{2}},\qquad\bm{\chi}_{2}=\frac{\bm{a}_{2}}{m_{2}}=\frac{\bm{S}_{2}}{m_{2}^{2}},$$

(2)

along with the intermediate ${\bm{a}}_{1}$, ${\bm{a}}_{2}$. The relative position and momentum are denoted by ${\bm{r}}$ and ${\bm{p}}$, respectively. Using an implicit Euclidean background, it holds

$${\bm{p}}^{2}=p_{r}^{2}+\frac{L^{2}}{r^{2}},\quad p_{r}=\bm{n}\cdot\bm{p},\quad\bm{L}=\bm{r}\times\bm{p},$$

(3)

where $\bm{n}=\bm{r}/r$ with $r=|{\bm{r}}|$, and ${\bm{L}}$ is the orbital angular momentum with magnitude $L$. For convenience, we also introduce rescaled dimensionless variables,

$$\hat{{\bm{r}}}=\frac{{\bm{r}}}{M},\quad\hat{{\bm{p}}}=\frac{{\bm{p}}}{\mu},\quad\hat{H}=\frac{H}{\mu},\quad\hat{{\bm{L}}}=\frac{{\bm{L}}}{M\mu},\quad\hat{{\bm{a}}}=\frac{{\bm{a}}}{M},$$

(4)

and similarly for the magnitudes $\hat{r}=|\hat{{\bm{r}}}|$, etc.; here, $H$ is any of several Hamiltonians encountered below, and ${\bm{a}}={\bm{S}}_{\text{Kerr}}/M$ is the rescaled spin of an effective Kerr black hole.

The central idea of the EOB Hamiltonian is to combine the dynamics in the test-body limit (with no restriction on the speed or field strength) with the PN dynamics (not restricted in the mass ratio). In this way, one might overcome some of the limitations of the individual approximations. This can be achieved by making an ansatz for $H^{\text{eff}}$ as a deformation of the test-body-limit Hamiltonian (deforming it such that PN results are recovered), which is the purpose of this section. Note that in the test-body limit $H^{\text{EOB}}\approx H^{\text{eff}}+\text{const}$.

Let us review the Hamiltonian of a spinning test-body in Kerr spacetime Barausse et al. (2009); Vines et al. (2016). One can easily specialize this to the nonspinning (geodesic) case, which is the basis of some SEOB models. These test-body Hamiltonians are the basis for all SEOB models. The (inverse) Kerr metric $g_{\text{Kerr}}^{\mu\nu}$ in Boyer-Lindquist coordinates $(x^{\mu})=(t,r,\theta,\phi)$ is given by the line element

	$\displaystyle-d\tau^{2}$	$\displaystyle=g_{\text{Kerr}}^{\mu\nu}\partial_{\mu}\partial_{\nu}$
		$\displaystyle=-\frac{\Lambda}{\Delta\Sigma}\partial_{t}^{2}+\frac{\Delta}{\Sigma}\partial_{r}^{2}+\frac{1}{\Sigma}\partial_{\theta}^{2}$
		$\displaystyle\quad+\frac{\Sigma-2Mr}{\Sigma\Delta\sin^{2}\theta}\partial_{\phi}^{2}-\frac{4Mra}{\Sigma\Delta}\partial_{t}\partial_{\phi},$		(8)

where $M$ is the mass of the black hole, $\sigma=Ma$ is its spin, and

	$$\displaystyle\Sigma\equiv r^{2}+a^{2}\cos^{2}\theta,\qquad\Delta\equiv r^{2}-2Mr+a^{2},$$		(9a)
	$$\displaystyle\Lambda\equiv(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta.$$		(9b)

The Hamiltonian of a spinning test-body $H^{\text{Kerr}}$ can be obtained as a solution of the mass-shell constraint (see, e.g., Ref. Vines et al. (2016))

$$\begin{split}-\mu^{2}&=g_{\text{Kerr}}^{\mu\nu}\left(p_{\mu}-\frac{1}{2}\omega_{\mu ab}S_{*}^{ab}\right)\left(p_{\nu}-\frac{1}{2}\omega_{\mu ab}S_{*}^{ab}\right)\\ &\quad+\operatorname{\mathcal{O}}(S_{*}^{2}),\end{split}$$

(10)

where $p_{\mu}=(-H^{\text{Kerr}},p_{r},p_{\theta},p_{\phi})$, $\mu$ is the mass of the test-body, $S_{*}^{ab}=-S_{*}^{ab}$ is its spin tensor in a local Lorentz frame ($e_{a}{}^{\mu}e^{a\nu}=g_{\text{Kerr}}^{\mu\nu}$), $\omega_{\mu ab}=e_{b\nu}\nabla_{\mu}e_{a}{}^{\nu}$ are the Ricci rotation coefficients, and $\nabla_{\mu}$ is the covariant derivative. The canonical spin vector of the test-body ${\bm{S}}_{*}$ is given by $S^{i}_{*}=\frac{1}{2}\epsilon^{ijk}S_{*}^{jk}$ and the components $S_{0i}$ are fixed by the supplementary condition $S^{ab}(e_{b}{}^{\mu}p_{\mu}+\mu^{2}\delta^{0}_{b})=0+\operatorname{\mathcal{O}}(S_{*}^{2})$, all in the local frame.

Let us split $H^{\text{Kerr}}$ into a part dependent on the test-spin ${\bm{S}}_{*}$ and the remaining $S_{*}$-independent terms into parts even and odd in the Kerr spin $a$,

$$H^{\text{Kerr}}=H^{\text{Kerr}}_{\text{even}}+H^{\text{Kerr}}_{\text{odd}}+H^{\text{Kerr}}_{S_{*}}.$$

(11)

Following the procedure outlined above, and choosing the local frame from Ref. Barausse and Buonanno (2010), this leads to

	$\displaystyle H^{\text{Kerr}}_{\text{even}}$	$\displaystyle=\alpha^{\text{Kerr}}\sqrt{\mu^{2}+\gamma_{\text{Kerr}}^{\phi\phi}p_{\phi}^{2}+\gamma_{\text{Kerr}}^{rr}p_{r}^{2}+\gamma_{\text{Kerr}}^{\theta\theta}p_{\theta}^{2}},$		(12a)
	$\displaystyle H^{\text{Kerr}}_{\text{odd}}$	$\displaystyle=\beta^{\text{Kerr}}p_{\phi}$		(12b)
	$\displaystyle H^{\text{Kerr}}_{S_{*}}$	$\displaystyle=\left[\bm{F}_{t}+\left(\beta^{\text{Kerr}}+\frac{\alpha^{\text{Kerr}}\gamma_{\text{Kerr}}^{\phi\phi}p_{\phi}}{\sqrt{q^{\text{Kerr}}}}\right)\bm{F}_{\phi}\right]\cdot\bm{S}_{*}$
		$\displaystyle\quad+\frac{\alpha^{\text{Kerr}}}{\sqrt{q^{\text{Kerr}}}}\left(\gamma_{\text{Kerr}}^{rr}p_{r}\bm{F}_{r}+\gamma_{\text{Kerr}}^{\theta\theta}p_{\theta}\bm{F}_{\theta}\right)\cdot\bm{S}_{*}$
		$\displaystyle\quad+\operatorname{\mathcal{O}}(S_{*}^{2}),$		(12c)

with

	$\displaystyle\alpha^{\text{Kerr}}$	$\displaystyle=\frac{1}{\sqrt{-g_{\text{Kerr}}^{tt}}}=\sqrt{\frac{\Delta\Sigma}{\Lambda}},$		(13a)
	$\displaystyle\beta^{\text{Kerr}}$	$\displaystyle=\frac{g_{\text{Kerr}}^{t\phi}}{g_{\text{Kerr}}^{tt}}=\frac{2aMr}{\Lambda},$		(13b)
	$\displaystyle\gamma_{\text{Kerr}}^{\phi\phi}$	$\displaystyle=g_{\text{Kerr}}^{\phi\phi}-\frac{g_{\text{Kerr}}^{t\phi}g_{\text{Kerr}}^{t\phi}}{g_{\text{Kerr}}^{tt}}=\frac{\Sigma}{\Lambda\sin^{2}\theta},$		(13c)
	$\displaystyle\gamma_{\text{Kerr}}^{rr}$	$\displaystyle=g_{\text{Kerr}}^{rr}=\frac{\Delta}{\Sigma},$		(13d)
	$\displaystyle\gamma_{\text{Kerr}}^{\theta\theta}$	$\displaystyle=g_{\text{Kerr}}^{\theta\theta}=\frac{1}{\Sigma},$		(13e)
	$\displaystyle\sqrt{q^{\text{Kerr}}}$	$\displaystyle=\frac{H^{\text{Kerr}}_{\text{even}}}{\alpha^{\text{Kerr}}},$		(13f)

and with explicit expressions for the fictitious gravito-magnetic (frame-dragging) force interacting with the test-spin ${\bm{S}}_{*}$ given in Ref. Hinderer et al. (2013) in terms of the vectors ${\bm{F}}_{\mu}$ (reproduced here in Sec. IV). A simplified version of this Hamiltonian for aligned spins and motion in the equatorial plane can be found in Ref. Bini et al. (2015). Simplifications for the generic-spin case are possible by making a different choice for the local frame which may simplify the Ricci rotation coefficients, see, e.g., Appendix C of Ref. Vines et al. (2016).

The Hamiltonian above is written in terms of components instead of vectors, which is a disadvantage for some purposes. Following Ref. Balmelli and Damour (2015), we transform to a 3-vector notation (with an implicit flat Euclidean background) by treating $(r,\theta,\phi)$ as spherical coordinates, with ${\bm{r}}=(x,y,z)=r(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ and ${\bm{a}}=(0,0,a)$. This is accompanied by a transformation of the momenta $p_{r}$, $p_{\theta}$, $p_{\phi}$ to the new momenta ${\bm{p}}$,

	$$\displaystyle p_{r}=\bm{n}\cdot\bm{p},\quad p_{\phi}=L_{z}=({\bm{r}}\times{\bm{p}})_{z}$$		(14a)
	$$\displaystyle\frac{p_{\theta}^{2}}{r^{2}}={\bm{p}}^{2}-p_{r}^{2}-\frac{p_{\phi}^{2}}{r^{2}\sin^{2}\theta}$$		(14b)

which makes it an overall canonical transformation. Noting that $a^{2}p_{\phi}^{2}/r^{2}=(\bm{n}\times\bm{p}\cdot\bm{a})^{2}$, $a\cos\theta={\bm{n\cdot a}}$, and $a^{2}\sin^{2}\theta=a^{2}-({\bm{n\cdot a}})^{2}$, this results in the even-in-$a$ Hamiltonian

	$\displaystyle H^{\text{Kerr}}_{\text{even}}=\bigg{[}$	$\displaystyle A^{\text{Kerr}}\Big{(}\mu^{2}+B_{p}^{\text{Kerr}}\bm{p}^{2}+B_{np}^{\text{Kerr}}(\bm{n}\cdot\bm{p})^{2}$
		$\displaystyle+B_{npa}^{\text{Kerr}}(\bm{n}\times\bm{p}\cdot\bm{a})^{2}\Big{)}\bigg{]}^{1/2},$		(15a)

with

	$\displaystyle A^{\text{Kerr}}$	$\displaystyle=(\alpha^{\text{Kerr}})^{2}=\frac{\Delta\Sigma}{\Lambda},$		(16a)
	$\displaystyle B_{p}^{\text{Kerr}}$	$\displaystyle=r^{2}\gamma_{\text{Kerr}}^{\theta\theta}=\frac{r^{2}}{\Sigma},$		(16b)
	$\displaystyle B_{np}^{\text{Kerr}}$	$\displaystyle=\gamma_{\text{Kerr}}^{rr}-r^{2}\gamma_{\text{Kerr}}^{\theta\theta}=\frac{r^{2}}{\Sigma}\left[\frac{\Delta}{r^{2}}-1\right],$		(16c)
	$\displaystyle B_{npa}^{\text{Kerr}}$	$\displaystyle=\frac{r^{2}}{a^{2}}\left[\gamma_{\text{Kerr}}^{\phi\phi}-\frac{\gamma_{\text{Kerr}}^{\theta\theta}}{\sin^{2}\theta}\right]=-\frac{r^{2}}{\Sigma\Lambda}(\Sigma+2Mr),$		(16d)

and

	$$\displaystyle\Sigma=r^{2}+({\bm{n}}\cdot{\bm{a}})^{2},\qquad\Delta=r^{2}-2Mr+a^{2},$$		(17a)
	$$\displaystyle\Lambda=(r^{2}+a^{2})^{2}-\Delta a^{2}+\Delta({\bm{n}}\cdot{\bm{a}})^{2}.$$		(17b)

Similarly, the odd-in-$a$ part reads

$$H_{\text{odd}}^{\text{Kerr}}=\beta^{\text{Kerr}}p_{\phi}=\frac{2Mr}{\Lambda}\bm{L}\cdot\bm{a}.$$

(18)

We now have all ingredients in order to discuss how an ansatz for the effective Hamiltonian can be built. In general, one takes the effective Hamiltonian to be a deformation of the Kerr Hamiltonian (the deformation parameter being the symmetric mass ratio $\nu$), either for a test spin or a test mass. While all EOB models agree on the identification of the masses between the test-body and comparable mass case ($M=m_{1}+m_{2}$, $\mu=m_{1}m_{2}/M$), different choices are made for mapping the spins ${\bm{a}}$ and ${\bm{S}}_{*}$ to ${\bm{S}}_{1}$ and ${\bm{S}}_{2}$. Let us consider a simple explicit example. We could write the even-in-$a$ part of the effective Hamiltonian as

	$\displaystyle H^{\text{eff}}_{\text{even}}$	$\displaystyle=\bigg{[}A\Big{(}\mu^{2}+B_{p}\bm{p}^{2}+B_{np}(\bm{n}\cdot\bm{p})^{2}$
		$\displaystyle\quad\quad+B_{npa}(\bm{n}\times\bm{p}\cdot\bm{a})^{2}+\mu^{2}Q\Big{)}\bigg{]}^{1/2},$		(19)

where the momentum-independent potentials $A$, $B_{p}$, $B_{np}$, $B_{npa}$ are the Kerr potentials given above modified by PN corrections (to be determined). The quantity $Q$ is a momentum-dependent potential introduced in Ref. Damour et al. (2000), which may accommodate PN terms that do not fit into the momentum-independent potentials. (In cases where $Q$ vanishes, the deformed Hamiltonian can be interpreted as describing geodesic motion in a $\nu$-deformed Kerr metric.) The mentioned potentials should all be of even order in spin, while terms of odd order in spin should be included via a deformation of $H^{\text{Kerr}}_{\text{odd}}$. More explicit ansätze for the PN-corrected SEOB Hamiltonians and their potentials are discussed below.

To fix the potentials in the ansatz for an effective Hamiltonian, one demands that the EOB Hamiltonian $H^{\text{EOB}}$ agrees with the Hamiltonian in the PN approximation $H^{\text{PN}}$ up to a canonical transformation. This will eventually not uniquely fix the potentials, but leave some (gauge) freedom.

Here we use the spinning PN Hamiltonian derived in the framework and gauges introduced in Ref. Levi and Steinhoff (2015a), since it is available to 4PN order in the spinning sector Levi and Steinhoff (2016a). Broken up into leading order (LO), next-to-LO (NLO), next-to-NLO (NNLO) PN parts and into powers of spin, it reads

	$\displaystyle H^{\text{PN}}_{\text{spin}}=$		(20)
	$\displaystyle\begin{array}[]{l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}l@{\,}}H^{\text{LO}}_{S}&&+H^{\text{NLO}}_{S}&&+H^{\text{NNLO}}_{S}&\\ &+H^{\text{LO}}_{S^{2}}&&+H^{\text{NLO}}_{S^{2}}&&+H^{\text{NNLO}}_{S^{2}}\\ &&&&+H^{\text{LO}}_{S^{3}}&\\ &&&&&+H^{\text{LO}}_{S^{4}}\\ &&&&&\qquad\quad\ddots\\ \operatorname{\mathcal{O}}(\frac{1}{c^{3}})&+\operatorname{\mathcal{O}}(\frac{1}{c^{4}})&+\operatorname{\mathcal{O}}(\frac{1}{c^{5}})&+\operatorname{\mathcal{O}}(\frac{1}{c^{6}})&+\operatorname{\mathcal{O}}(\frac{1}{c^{7}})&+\operatorname{\mathcal{O}}(\frac{1}{c^{8}})\end{array}$		(27)

where columns correspond to PN orders counted by the inverse of the speed of light $c$ (one PN order is $\operatorname{\mathcal{O}}(c^{-2})$). Except for the self-spin-squared interactions in $H^{\text{NNLO}}_{S^{2}}$ calculated in Ref. Levi and Steinhoff (2016b), these results have been derived in different frameworks and checked against each other: $H^{\text{LO}}_{S}$ in Refs. Tulczyjew (1959); Barker and O’Connell (1970, 1975); D’Eath (1975); Barker and O’Connell (1979); Damour (1982); Thorne and Hartle (1985); Damour and Schäfer (1988), $H^{\text{NLO}}_{S}$ in Refs. Tagoshi et al. (2001); Faye et al. (2006); Damour et al. (2008b); Steinhoff et al. (2008a); Perrodin (2011); Porto (2010); Levi (2010a), $H^{\text{NNLO}}_{S}$ in Refs. Hartung and Steinhoff (2011a); Hartung et al. (2013); Levi and Steinhoff (2016c); Marsat et al. (2013); Bohe et al. (2013), $H^{\text{LO}}_{S^{2}}$ in Refs. Barker and O’Connell (1975); D’Eath (1975); Barker and O’Connell (1979); Poisson (1998); Thorne and Hartle (1985), $H^{\text{NLO}}_{S^{2}}$ in Refs. Steinhoff et al. (2008b); Porto and Rothstein (2006, 2008a); Levi (2010b); Porto and Rothstein (2008b); Steinhoff et al. (2008c); Hergt et al. (2010, 2012); Bohé et al. (2015), $H^{\text{NNLO}}_{S^{2}}$ in Refs. Hartung and Steinhoff (2011b); Levi (2012); Hartung et al. (2013); Levi and Steinhoff (2014, 2016b), $H^{\text{LO}}_{S^{3}}$ in Refs. Hergt and Schäfer (2008a, b); Levi and Steinhoff (2015b); Marsat (2015); Vaidya (2015), and $H^{\text{LO}}_{S^{4}}$ in Refs. Levi and Steinhoff (2015b); Hergt and Schäfer (2008a, b); Vaidya (2015). These Hamiltonians are valid for both black holes and neutron stars. They depend on coefficients ($\tilde{C}_{(\text{ES}^{2})}$, $\tilde{C}_{(\text{BS}^{3})}$, $\tilde{C}_{(\text{ES}^{4})}$) which are the proportionality constants between the spin-induced multipoles (quadrupole, octupole, hexadecapole) and symmetric-tracefree tensors built out of (two, three, four) spin vectors (respectively). The proportionality constants depend on the type of compact object (and on the equation of state in case of a neutron star); here they are normalized to 0 for black holes (in the original paper Levi and Steinhoff (2016a), they are normalized to 1 and denoted without a tilde; see also Appendix B). This normalization makes sense here since we base the EOB Hamiltonian on a deformation of the Kerr one. Of course, the PN Hamiltonian $H^{\text{PN}}=H^{\text{PN}}_{\text{ns}}+H^{\text{PN}}_{\text{spin}}$ must be supplemented by its nonspinning (ns) part $H^{\text{PN}}_{\text{ns}}$, which we only need to 2PN order here in order to construct the canonical transformation of the spin sector; it can be derived, e.g., from the Lagrangian in Ref. Gilmore and Ross (2008). The nonspinning part was derived to 4PN order using independent methods Damour et al. (2014); Bernard et al. (2017); Foffa and Sturani (2019); Blümlein et al. (2020a) and partial results at 5PN have already been obtained Foffa et al. (2019); Blümlein et al. (2020b); Bini et al. (2019).

The condition that the EOB Hamiltonian $H^{\text{EOB}}$ must coincide with results for the PN-approximate binary Hamiltonian $H^{\text{PN}}$ up to a canonical transformation reads

$$H^{\text{EOB}}=H^{\text{PN}}+\{\mathcal{G},H^{\text{PN}}\}+\frac{1}{2!}\{\mathcal{G},\{\mathcal{G},H^{\text{PN}}\}\}\\ +\frac{1}{3!}\{\mathcal{G},\{\mathcal{G},\{\mathcal{G},H^{\text{PN}}\}\}\}+\dots{}$$

(28)

where $\mathcal{G}$ is the generating function of the canonical transformation. If $\mathcal{G}$ is small in the PN approximation, then the series in Eq. (28) terminates after a finite number of terms at a given PN order. In practice, one makes a PN-approximate and manifestly rotation invariant ansatz for $\mathcal{G}$ in terms of the canonical variables; we provide an explicit expression for $\mathcal{G}$ as Mathematica code in the supplementary material. Equation (28) then leads to constraints on the coefficients in the ansatz for $\mathcal{G}$ and $H^{\text{eff}}$. The remaining freedom in the coefficients is a gauge freedom within the EOB formalism.

Let us note some general considerations about how part of this gauge freedom can be fixed in SEOB models. Since binaries are expected to be on almost circular orbits during their last orbits, it makes sense to fix the gauge freedom of the EOB Hamiltonian such that it simplifies for circular orbits, for which $p_{r}\equiv{\bm{n}}\cdot{\bm{p}}=0$ Damour et al. (2000). Taking the ansatz in Eq. (II.1) as an example, this means that—using the canonical transformation discussed above—one should transform as many PN terms as possible into a form such that they can be included in the potential $B^{\text{Kerr}}_{np}$, which drops out of the Hamiltonian for circular orbits. In the nonspinning case, it is additionally possible to require that the potential $Q$ depends on the momentum only via $p_{r}$, and this uniquely fixes all EOB gauge freedom Buonanno and Damour (1999); Damour et al. (2000). For the example in Eq. (II.1), following the structure of the nonspinning Hamiltonian, it is natural to require that: (i) the momentum-dependence of $H_{\text{odd}}^{\text{Kerr}}$ is expressed in terms of $p_{r}$ whenever possible Damour et al. (2008a), (ii) $B_{npa}=B_{npa}^{\text{Kerr}}$ Balmelli and Damour (2015), and (iii) terms in $Q$ have a power in $p_{r}$ that is as high as possible. The last requirement ensures that $Q$ vanishes for circular orbits, as in the nonspinning case.

These considerations still leave some remaining gauge freedom in the spinning case, which we fix such to simplify the EOB Hamiltonian also for aligned spins. For example, it is possible to choose PN corrections in the potential $B_{p}$ such that it only depends on terms of the form $\bm{n}\cdot\bm{S}$ but not $\bm{S}\cdot\bm{S}$. Any remaining gauge freedom beyond that may be chosen arbitrarily.

Reference Balmelli and Damour (2015) was the first to construct an SEOB Hamiltonian with NLO spin-squared terms for generic spin orientations (but omitting a subtle contribution, included here, see Appendix A). Here we extend the Hamiltonian to include NNLO spin-squared and LO spin-cubed terms, and add multipole constants to make it applicable to generic bodies like neutron stars.

For the even-in-spin part of the SEOB${}_{\text{TM}}^{r_{c}}$ Hamiltonian, we use the ansatz in Eq. (II.1),

	$\displaystyle{H}^{\text{eff}}_{\text{even}}$	$\displaystyle=\bigg{[}A\Big{(}\mu^{2}+B_{p}\bm{p}^{2}+B_{np}(\bm{n}\cdot\bm{p})^{2}$
		$\displaystyle\quad\quad+B_{npa}(\bm{n}\times\bm{p}\cdot\bm{a})^{2}+\mu^{2}Q\Big{)}\bigg{]}^{1/2},$		(30)

where the Kerr spin is mapped according to

$${\bm{a}}={\bm{a}}_{1}+{\bm{a}}_{2}\,.$$

(31)

This ensures that the Hamiltonian reproduces leading-order PN results at all even orders in spin Vines and Steinhoff (2018). The effective Hamiltonian further uses the “centrifugal radius” $r_{c}$, which was introduced in Ref. Damour and Nagar (2014) and is defined such that the Kerr Hamiltonian for aligned spins and equatorial orbits can be written as $H_{\text{even}}^{\text{Kerr}}=\sqrt{A^{\text{Kerr}}\left(\mu^{2}+p_{\phi}^{2}/r_{c}^{2}+p_{r}^{2}/B(r)\right)}$, which implies the definition

$$r_{c}=\sqrt{r^{2}+a^{2}+\frac{2Ma^{2}}{r}}\,.$$

(32)

The centrifugal radius was generalized to generic spin orientations in Ref. Balmelli and Damour (2015). In terms of $r_{c}$, the Kerr potential $A^{\text{Kerr}}$ from Eq. (16a) can be written equivalently as

$$A^{\text{Kerr}}=\left(1-\frac{2M}{r_{c}}\right)\frac{\left(1+\frac{2M}{r_{c}}\right)}{\left(1+\frac{2M}{r}\right)}\frac{1+\frac{(\bm{n}\cdot\bm{a})^{2}}{r^{2}}}{1+\Delta\frac{(\bm{n}\cdot\bm{a})^{2}}{r^{2}r_{c}^{2}}}\,.$$

(33)

In the nonspinning limit, only the first term above remains, which reduces to the Schwarzschild $A$-potential. This is the reason why Ref. Balmelli and Damour (2015) adds the zero-spin PN corrections to $1-2M/r_{c}$. However, in this paper we intend to investigate spin effects across different Hamiltonian descriptions, so we need to make sure that the nonspinning Hamiltonians are identical. That is, we need to choose a method for adding zero-spin corrections that can be applied to all four EOB Hamiltonians considered here. We simply multiply the Kerr potential $A^{\text{Kerr}}$ by zero-spin PN corrections denoted $A^{0}$ below (without performing a Padé or $\log$ resummation³³3The justification for the Padé or $\log$ resummations is that they improve agreement with NR in some models and may hence be seen as an implicit calibration. In this paper, however, we consider EOB Hamiltonians with no calibration to NR, so we try to avoid such resummations, in particular in the nonspinning part. of $A^{0}$). For the spin-squared corrections, we follow Ref. Balmelli and Damour (2015) and add spin-squared corrections of the form $\bm{n}\cdot\bm{S}$ to the term $1+(\bm{n}\cdot\bm{a})^{2}/r^{2}$, and add corrections of the form $\bm{S}\cdot\bm{S}$ to $1+2M/r_{c}$, since it has an expansion of the form $1+2M/r-Ma^{2}/r^{3}+\dots$. One employs similar considerations for adding PN corrections to the $B$-potentials in Eq. (III.1), leading to the following ansatz:

	$\displaystyle A$	$\displaystyle=\left(1-\frac{2M}{r_{c}}\right)\frac{\left(1+\frac{2M}{r_{c}}+A^{SS}+A^{S^{4}}\right)}{\left(1+\frac{2M}{r}\right)}\frac{\left(1+\frac{(\bm{n}\cdot\bm{a})^{2}}{r^{2}}+A^{nS}\right)}{\left(1+\Delta\frac{(\bm{n}\cdot\bm{a})^{2}}{r^{2}r_{c}^{2}}\right)}A^{0}(r_{c})\,,$		(34a)
	$\displaystyle B_{p}$	$\displaystyle=\left[1+\frac{(\bm{n}\cdot\bm{a})^{2}}{r^{2}}+B_{p}^{nS}\right]^{-1},$		(34b)
	$\displaystyle B_{np}$	$\displaystyle=\frac{1}{1+\frac{(\bm{n}\cdot\bm{a})^{2}}{r^{2}}}\left[\left(1-\frac{2M}{r}+\frac{a^{2}}{r^{2}}\right)\left(A^{0}(r_{c})\,D^{0}(r_{c})+B_{np}^{SS}+B_{np}^{nS}\right)-1\right],$		(34c)
	$\displaystyle B_{npa}$	$\displaystyle=B_{npa}^{\text{Kerr}}\,,$		(34d)
	$\displaystyle Q$	$\displaystyle=Q^{0}(r_{c})+Q^{S^{2}}.$		(34e)

Note that we use the gauge choice from Ref. Balmelli and Damour (2015), i.e., there are no corrections of the form $\bm{S}\cdot\bm{S}$ in the potential $B_{p}$, which simplifies the Hamiltonian for aligned spins and circular orbits.

The 4PN corrections to the nonspinning effective Hamiltonian were obtained in Ref. Damour et al. (2015). Since we factor the PN corrections in $A^{0}(r_{c})$, we choose it such that the PN expansion of the $A$ potential agrees, in the nonspinning limit, with the results of Ref. Damour et al. (2015). Writing the PN corrections using scaled variables (4) to simplify notation, we obtain

	$\displaystyle A^{0}(r_{c})$	$\displaystyle=1+\nu\bigg{[}\frac{2}{\hat{r}_{c}^{3}}+\left(\frac{106}{3}-\frac{41}{32}\pi^{2}\right)\frac{1}{\hat{r}_{c}^{4}}+\left(\frac{1}{20}+\frac{41}{32}\pi^{2}\nu-\frac{221}{6}\nu+\frac{963}{512}\pi^{2}+\frac{128}{5}\gamma_{E}+\frac{256}{5}\ln 2+\frac{64}{5}\ln\frac{1}{\hat{r}_{c}}\right)\frac{1}{\hat{r}_{c}^{5}}\bigg{]}$		(35a)
	$\displaystyle D^{0}(r_{c})$	$\displaystyle=1+6\nu\frac{1}{\hat{r}_{c}^{2}}+\left(52\nu-6\nu^{2}\right)\frac{1}{\hat{r}_{c}^{3}}$
		$\displaystyle\quad+\left[\left(\frac{123}{16}\pi^{2}-260\right)\nu^{2}+\nu\left(-\frac{23761}{1536}\pi^{2}-\frac{533}{45}+\frac{1184}{15}\gamma_{E}-\frac{6496}{15}\ln 2+\frac{2916}{5}\ln 3\right)+\frac{592}{15}\nu\ln\frac{1}{\hat{r}_{c}}\right]\frac{1}{\hat{r}_{c}^{4}},$		(35b)
	$\displaystyle Q^{0}(r_{c})$	$\displaystyle=\left[2(4-3\nu)\nu\frac{1}{\hat{r}_{c}^{2}}+\left(\left(-\frac{5308}{15}+\frac{496256}{45}\ln 2-\frac{33048}{5}\ln 3\right)\nu-83\nu^{2}+10\nu^{3}\right)\frac{1}{\hat{r}_{c}^{3}}\right]\hat{p}_{r}^{4}$
		$\displaystyle\quad+\left[\left(-\frac{827}{3}-\frac{2358912}{25}\ln 2+\frac{1399437}{50}\ln 3+\frac{390625}{18}\ln 5\right)\nu-\frac{27}{5}\nu^{2}+6\nu^{3}\right]\frac{\hat{p}_{r}^{6}}{\hat{r}_{c}^{2}}\,,$		(35c)

where the corrections are expressed in terms of the centrifugal radius $r_{c}$, with $\hat{r}_{c}=r_{c}/M$, and $\hat{p}_{r}=p_{r}/\mu$. Note that here and in the SEOB models discussed below, we are using Taylor-expanded and not resummed versions of these potentials—we want to compare the different ansätze of the Hamiltonians irrespective of possible resummations for the potentials (see also footnote 3).

Spin-squared contributions, up to NNLO, are added to the Hamiltonian using the following ansatz

	$\displaystyle A^{SS}$	$\displaystyle=\frac{c_{n}}{\hat{r}_{c}^{3}}\bm{\chi}_{i}\cdot\bm{\chi}_{j}+\frac{c_{n}}{\hat{r}_{c}^{4}}\bm{\chi}_{i}\cdot\bm{\chi}_{j}+\frac{c_{n}}{\hat{r}_{c}^{5}}\bm{\chi}_{i}\cdot\bm{\chi}_{j},$		(36a)
	$\displaystyle A^{nS}$	$\displaystyle=\frac{c_{n}}{\hat{r}_{c}^{3}}(\bm{n}\cdot\bm{\chi}_{i})(\bm{n}\cdot\bm{\chi}_{j})+\frac{c_{n}}{\hat{r}_{c}^{4}}(\bm{n}\cdot\bm{\chi}_{i})(\bm{n}\cdot\bm{\chi}_{j})$
		$\displaystyle\quad+\frac{c_{n}}{\hat{r}_{c}^{5}}(\bm{n}\cdot\bm{\chi}_{i})(\bm{n}\cdot\bm{\chi}_{j}),$		(36b)
	$\displaystyle B_{p}^{nS}$	$\displaystyle=\frac{c_{n}}{\hat{r}_{c}^{3}}(\bm{n}\cdot\bm{\chi}_{i})(\bm{n}\cdot\bm{\chi}_{j})+\frac{c_{n}}{\hat{r}_{c}^{4}}(\bm{n}\cdot\bm{\chi}_{i})(\bm{n}\cdot\bm{\chi}_{j}),$		(36c)
	$\displaystyle B_{np}^{nS}$	$\displaystyle=\frac{c_{n}}{\hat{r}_{c}^{3}}(\bm{n}\cdot\bm{\chi}_{i})(\bm{n}\cdot\bm{\chi}_{j})+\frac{c_{n}}{\hat{r}_{c}^{4}}(\bm{n}\cdot\bm{\chi}_{i})(\bm{n}\cdot\bm{\chi}_{j}),$		(36d)
	$\displaystyle B_{np}^{SS}$	$\displaystyle=\frac{c_{n}}{\hat{r}_{c}^{3}}\bm{\chi}_{i}\cdot\bm{\chi}_{j}+\frac{c_{n}}{\hat{r}_{c}^{4}}\bm{\chi}_{i}\cdot\bm{\chi}_{j},$		(36e)
	$\displaystyle Q^{S^{2}}$	$\displaystyle=\frac{\hat{p}_{r}^{4}}{\hat{r}_{c}^{3}}\left[c_{n}\bm{\chi}_{i}\cdot\bm{\chi}_{j}+c_{n}(\bm{n}\cdot\bm{\chi}_{i})(\bm{n}\cdot\bm{\chi}_{j})\right]$
		$\displaystyle\quad+\frac{\hat{p}_{r}^{3}}{\hat{r}_{c}^{3}}c_{n}(\bm{p}\cdot\bm{\chi}_{i})(\bm{n}\cdot\bm{\chi}_{j}),$		(36f)

where we followed Ref. Balmelli and Damour (2015) in expressing the corrections in terms of $r_{c}$. We employ notation such that, e.g., ${c_{n}\bm{\chi}_{i}\cdot\bm{\chi}_{j}}\equiv c_{n}\bm{\chi}_{1}^{2}+c_{n}\bm{\chi}_{1}\cdot\bm{\chi}_{2}+c_{n}\bm{\chi}_{2}^{2}$. Each $c_{n}$ stands for an independent undetermined coefficient in our ansatz, i.e., we use the same symbol $c_{n}$ for all coefficients to simplify notation. The full expressions after matching to PN results are provided in Appendix B. Note that we added LO S² corrections to the $A$-potential above (which vanish for black holes) to account for the multipole constants of neutron stars.

The NLO S² contributions were included in the effective Hamiltonian in Ref. Balmelli and Damour (2015), however the authors missed a contribution in matching the EOB Hamiltonian to PN results, namely from the LO S² generating function applied to the LO SO Hamiltonian, i.e., from the Poisson bracket $\{\mathcal{G}^{\text{LO}}_{S^{2}},H^{\text{LO}}_{S}\}$. In Appendix A, we write the matching results for NLO S², using the notation of Ref. Balmelli and Damour (2015), after taking into account the missing Poisson bracket.

The leading-order quartic-in-spin terms $A^{S^{4}}$ are zero for black holes, since the Kerr Hamiltonian, with the mapping ${\bm{a}}={\bm{a}}_{1}+{\bm{a}}_{2}$, automatically reproduces them, but they are nonzero for other types of compact objects. We take the most generic expression for the S⁴ corrections

	$\displaystyle A^{S^{4}}$	$\displaystyle=\frac{1}{\hat{r}_{c}^{5}}\bigg{[}c_{n}(\bm{\chi}_{i}\cdot\bm{\chi}_{j})(\bm{\chi}_{k}\cdot\bm{\chi}_{l})$
		$\displaystyle\quad\qquad+c_{n}(\bm{\chi}_{i}\cdot\bm{\chi}_{j})(\bm{n}\cdot\bm{\chi}_{k})(\bm{n}\cdot\bm{\chi}_{l})$
		$\displaystyle\quad\qquad+c_{n}(\bm{n}\cdot\bm{\chi}_{i})(\bm{n}\cdot\bm{\chi}_{j})(\bm{n}\cdot\bm{\chi}_{k})(\bm{n}\cdot\bm{\chi}_{l})\bigg{]},$		(37)

where a summation over the spins of the two bodies is implied, and terms symmetric under the exchange of the two bodies’ labels are only included once.

The spin-orbit and spin-cubed PN corrections are added to the odd-in-spin part of the Kerr Hamiltonian $H_{\text{odd}}^{\text{Kerr}}$. For the SO part we use the ansatz in Refs. Balmelli and Damour (2015); Damour and Nagar (2014), and we add to it S³ corrections,

	$\displaystyle\hat{H}_{\text{odd}}^{\text{eff}}$	$\displaystyle=\frac{G_{S}}{\hat{r}\hat{r}_{c}^{2}\left(1+\Delta\frac{(\bm{n}\cdot\bm{a})^{2}}{{r}^{2}{r}_{c}^{2}}\right)}\left(X_{1}^{2}\hat{\bm{L}}\cdot\bm{\chi}_{1}+X_{2}^{2}\hat{\bm{L}}\cdot\bm{\chi}_{2}\right)$
		$\displaystyle\quad+\frac{G_{S^{*}}}{\hat{r}_{c}^{3}}\nu\left(\hat{\bm{L}}\cdot\bm{\chi}_{1}+\hat{\bm{L}}\cdot\bm{\chi}_{2}\right)$
		$\displaystyle\quad+\frac{G_{S^{3}}}{\hat{r}_{c}^{4}}\hat{\bm{L}}\cdot\bm{\chi}_{1}+\frac{\tilde{G}_{S^{3}}}{\hat{r}_{c}^{4}}\hat{\bm{L}}\cdot\bm{\chi}_{2}\,,$		(38a)

where

	$\displaystyle G_{S}$	$\displaystyle=2\left[1+\frac{c_{n}}{\hat{r}_{c}}+c_{n}\hat{p}_{r}^{2}+\frac{c_{n}}{\hat{r}_{c}^{2}}+c_{n}\frac{\hat{p}_{r}^{2}}{\hat{r}_{c}}+c_{n}\hat{p}_{r}^{4}\right]^{-1},$
	$\displaystyle G_{S^{*}}$	$\displaystyle=\frac{3}{2}\left[1+\frac{c_{n}}{\hat{r}_{c}}+c_{n}\hat{p}_{r}^{2}+\frac{c_{n}}{\hat{r}_{c}^{2}}+c_{n}\frac{\hat{p}_{r}^{2}}{\hat{r}_{c}}+c_{n}\hat{p}_{r}^{4}\right]^{-1}$		(39a)
	$\displaystyle G_{S^{3}}$	$\displaystyle=\frac{1}{\hat{r}_{c}}\Big{[}c_{n}\bm{\chi}_{1}\cdot\bm{\chi}_{1}+c_{n}\bm{\chi}_{2}\cdot\bm{\chi}_{2}+c_{n}\bm{\chi}_{1}\cdot\bm{\chi}_{2}$
		$\displaystyle\quad\qquad+c_{n}(\bm{n}\cdot\bm{\chi}_{1})^{2}+c_{n}(\bm{n}\cdot\bm{\chi}_{2})^{2}$
		$\displaystyle\quad\qquad+c_{n}(\bm{n}\cdot\bm{\chi}_{1})(\bm{n}\cdot\bm{\chi}_{2})\Big{]}$
		$\displaystyle\quad+\hat{p}_{r}^{2}\Big{[}c_{n}(\bm{n}\cdot\bm{\chi}_{1})^{2}+c_{n}(\bm{n}\cdot\bm{\chi}_{2})^{2}$
		$\displaystyle\quad\qquad+c_{n}(\bm{n}\cdot\bm{\chi}_{1})(\bm{n}\cdot\bm{\chi}_{2})\Big{]}$
		$\displaystyle\quad+\frac{\hat{L}^{2}}{\hat{r}^{2}}\Big{[}c_{n}(\bm{n}\cdot\bm{\chi}_{1})^{2}+c_{n}(\bm{n}\cdot\bm{\chi}_{2})^{2}$
		$\displaystyle\quad\qquad+c_{n}(\bm{n}\cdot\bm{\chi}_{1})(\bm{n}\cdot\bm{\chi}_{2})\Big{]},$
	$\displaystyle\tilde{G}_{S^{3}}$	$\displaystyle=G_{S^{3}}\quad\text{with}\quad 1\leftrightarrow 2.$		(39b)

Note that an inverse-Taylor resummation is used for $G_{S}$ and $G_{S^{\ast}}$, which improves the description of the binary dynamics for aligned spins Damour and Nagar (2014). In the spin-cubed corrections $G_{S^{3}}$ and $\tilde{G}_{S^{3}}$, a gauge freedom exists which we chose such that terms of the form $p_{r}^{2}\bm{\chi}_{i}\cdot\bm{\chi}_{j}$ or $L^{2}\bm{\chi}_{i}\cdot\bm{\chi}_{j}$ are not included. Explicit results after matching at 4PN can be found in Appendix B.1.

Since it is important to have fast and simple EOB waveform models, in this section, we consider a simplified version of the SEOB${}_{\text{TM}}^{r_{c}}$ Hamiltonian that uses $r$ instead of $r_{c}$ for the PN corrections. In order to assess the effect of this simplification, we also avoid resummations that are not motivated by the structure of the interactions, i.e., we factorize spin corrections to the Kerr potentials and do not use an inverse-Taylor resummation for the spin-orbit part.

The potentials of the effective Hamiltonian are simply taken to be

	$\displaystyle A$	$\displaystyle=A^{\text{Kerr}}\left(A^{0}+A^{SS}+A^{nS}+A^{S^{4}}\right),$		(40a)
	$\displaystyle B_{p}$	$\displaystyle=B_{p}^{\text{Kerr}}\left(1+B_{p}^{nS}\right),$		(40b)
	$\displaystyle B_{np}$	$\displaystyle=\frac{\left(1-\frac{2}{\hat{r}}+\frac{\hat{a}^{2}}{\hat{r}^{2}}\right)\left(A^{0}D^{0}+B_{np}^{SS}+B_{np}^{nS}\right)-1}{1+(\bm{n}\cdot\hat{\bm{a}})^{2}/\hat{r}^{2}},$		(40c)
	$\displaystyle B_{npa}$	$\displaystyle=B_{npa}^{\text{Kerr}},$		(40d)
	$\displaystyle Q$	$\displaystyle=Q^{0}+Q^{S^{2}},$		(40e)

where the zero-spin corrections $A^{0}(r),~{}D^{0}(r)$ and $Q^{0}(r)$ are given by Eq. (35) but in terms of $r$ instead of $r_{c}$. The ansätze for the S² and S⁴ corrections are given by the corresponding expressions from the previous section, i.e., Eqs. (36) and (III.1), but using $r$ instead of $r_{c}$ (and with different coefficients $c_{n}$). For $H_{\text{odd}}^{\text{Kerr}}$, we modify the odd-in-$a$ part of the Kerr Hamiltonian by the SO and S³ PN corrections, that is

	$\displaystyle\hat{H}_{\text{odd}}^{\text{eff}}$	$\displaystyle=\frac{1}{\hat{r}\hat{r}_{c}^{2}\left(1+\Delta\frac{(\bm{n}\cdot\bm{a})^{2}}{{r}^{2}{r}_{c}^{2}}\right)}\Big{[}G_{S}\left(X_{1}^{2}\hat{\bm{L}}\cdot\bm{\chi}_{1}+X_{2}^{2}\hat{\bm{L}}\cdot\bm{\chi}_{2}\right)$
		$\displaystyle\qquad\quad+G_{S^{*}}\nu\left(\hat{\bm{L}}\cdot\bm{\chi}_{1}+\hat{\bm{L}}\cdot\bm{\chi}_{2}\right)$
		$\displaystyle\qquad\quad+\frac{G_{S^{3}}}{\hat{r}}\hat{\bm{L}}\cdot\bm{\chi}_{1}+\frac{\tilde{G}_{S^{3}}}{\hat{r}}\hat{\bm{L}}\cdot\bm{\chi}_{2}\Big{]},$		(41a)

with the ansätze for the coefficients given in Eqs. (39) and (39b), but again written with $r$ instead of $r_{c}$ (and different $c_{n}$). Explicit results after matching at 4PN can be found in Appendix B.2.