Effective-one-body Hamiltonian with next-to-leading order spin-spin coupling
We propose a way of including the next-to-leading (NLO) order spin-spin coupling into an effective-one-body (EOB) Hamiltonian. This work extends [S. Balmelli and P. Jetzer, Phys. Rev. D 87, 124036 (2013)], which is restricted to the case of equatorial orbits and aligned spins, to general orbits with arbitrary spin orientations. This is done applying appropriate canonical phase-space transformations to the NLO spin-spin Hamiltonian in Arnowitt-Deser-Misner (ADM) coordinates, and systematically adding “effective” quantities at NLO to all spin-squared terms appearing in the EOB Hamiltonian. As required by consistency, the introduced quantities reduce to zero in the test-mass limit. We expose the result both in a general gauge and in a gauge-fixed form. The last is chosen such as to minimize the number of new coefficients that have to be inserted into the effective spin squared. As a result, the 25 parameters that describe the ADM NLO spin-spin dynamics get condensed into only 12 EOB terms.
Thanks to the LIGO/Virgo network of second-generation ground based interferometers, a first direct detection of gravitational waves (GW) is expected to occur in few years aas:11. Furthermore, in the next decades, space-born GW detectors vita:14 (such as the planned eLISA) will open an entire new window to astrophysics, and allow tests of General Relativity at an unprecedented level amar:13. Both types of detectors rely on coalescing (and possible spinning) black hole binaries as a primary GW source.
In order to extract the GW signal from the noise, very precise waveform templates need to be constructed. Currently, the most accurate description of coalescing black holes is provided by numerical relativity (NR) (see e.g. lov:12; buch:12; mrou:13; hind:14 for some recent advances). However, the full parameter space is too large (especially in the case of nonzero spins) for being densely covered by NR simulations . This is the main reason why semi-analytical methods, by far less computationally expensive, can turn out to be very useful. At the present time, the effective-one-body (EOB) approach (we refer to dam:12_EOB for a general review) is the only semi-analytical method that has been able to accurately describe the complete waveform of a coalescing process. After years of constant development buon:99; buon:00; dam:00; dam:02; buon:03; dam:07; dam:07_2; buon:07; buon:07_2; dam:08_4; dam:08_5; dam:08_3; dam:09; buon:09; dam:09_2; dam:12_EvsJ the EOB has now reached an excellent agreement with NR waveforms in the case of nonspinning binaries dam:13; hind:14. A lot of effort has also been put into the modeling of spins dam:01; dam:08; bar:09; bar:10; pan:10; bar:11; pan:11; pan:11_2; nag:11; tar:12; bal:13; bal:13:err; tar:14; pan:14; dam:14, leading to a good overlap with NR waveforms in the case of nonprecessing (aligned or anti-aligned) spins hind:14; tar:14; dam:14. By contrast, the description of precessing spins still needs some work before reaching a comparable performance pan:14, and is currently one of the most urgent tasks. In view of this, it may be crucial to incorporate more analytical information from the spin-orbit and spin-spin Hamiltonian computed within the post-newtonian (PN) theory.
In Ref. bal:13, a possible way of including the next-to-leading (NLO) spin-spin coupling stein:08s1s2; stein:08s^2 (see also por:08; por:10; por:08_2; por:10_2; levi:10; levi:14) into the spinning EOB model of Refs. dam:01; dam:08; nag:11 has been exposed for the special case of equatorial orbits and nonprecessing spins. A general inclusion of NLO spin-spin effects for fully precessing orbits would be a necessary step for extending the reliabilty of EOB waveforms to a significantly larger portion of the parameter space. The present paper aims at filling this gap, providing a possible implementation of the missing terms.
Recently, an improved (and calibrated) EOB model for spinning binaries has been proposed dam:14, where the NLO spin-spin coupling is only incorporated for the case of circular orbits. The present paper could be a first step for developing, at a next stage, a more general version of that improved model.
The paper has the following structure: in Sec. (II) we summarize the main concepts and the formalism of Ref. bal:13. In Sec. (III), which is the central part of the paper, we propose a way of including the NLO spin-spin terms and show the explicit result, both in a general form (with a given number of free gauge parameters) and in a gauge-fixed formulation. Finally, Sec. (IV) discusses a second, slightly different way of incorporating the wished spin-spin terms. Both approaches are compared plotting the (gauge invariant) angular frequency and binding energy at the last stable orbit (LSO). The plot also shows the prediction of the calibrated models of Refs. tar:12; tar:14.
Throughout the paper, we use geometric units with . When writing formulae in a PN expanded form, however, we will reintroduce the usage of , prepending a factor with the mere purpose of labeling the PN order .
Ii Summary of the previous work
In this section, we outline the method followed in Ref. bal:13, which forms the basis of the current paper. We work in EOB coordinates, with being the radial coordinate, the unit radial vector and the momentum vector. We will often use the rescaled variables and . Here, is the central EOB mass ( and being the individual masses of the two black holes), and is the reduced mass. Moreover, we denote by the symmetric mass ratio.
The starting point is the EOB model of Ref. dam:01 (which includes both spin-spin and spin-orbit coupling at leading order (LO)), together with its extensions to the NLO dam:08 and to the next-to-next-to leading order (NNLO) nag:11 spin-orbit coupling. As already mentioned, in Ref. bal:13 the NLO spin-spin Hamiltonian in ADM coordinates has been reformulated and inserted into this EOB model for the special case of two black holes whose spins are aligned (or anti-aligned) with the orbital angular momentum. In particular, it has been shown that it is sufficient to replace the spin parameter of the effective metric (see for instance Eqs. (4.7) and (4.8) of Ref. bal:13), whenever it appears as a second power, by a new, effective squared spin parameter. Using the dimensionless notation , this prescription takes the form
We recall that is a combination of the dimensionless spin parameters and () of the two bodies:
The additional term is of fractional 1PN order with respect to and carries the information for reproducing the correct NLO spin-spin coupling. It reads as
The calculation of the coefficients and is the main result of Ref. bal:13, given by Eq. (5.12) there. All of them vanish in the test mass limit , consistently with the requirement that the EOB metric must reduce to the Kerr one.
PN results in ADM coordinates can be included into an EOB model after suitable canonical transformations. We denote here by and the generating functions of the corresponding purely orbital and spin-spin transformation, respectively. The procedure for transforming the NLO spin-spin Hamiltonian from ADM into EOB coordinates is made of three steps:111Here, we treat the PN expansion under the assumption of rapidly rotating black holes (, with ), which assigns a well-defined PN order to the spin-dependent terms. As a consequence, throughout this paper, and are of 2PN order when labeled with “LO”, of 3PN order when labeled with “NLO”, and so on.
A purely orbital transformation
A LO spin-spin transformation
A NLO spin-spin transformation
Notice that, for consistency, the transformations must be performed in a well-defined order. As indicated above, we make first use of the orbital transformation , and then of the spin-spin transformation 222Since the set of canonical transformations carries a group structure, the successive evaluation of and of is a canonical transformation itself (with generating function ). Despite taking a unique generating function would avoid the necessity of fixing an evaluation order prescription, we prefer here to use two separated transformations, so as to mantain the continuity with respect to Ref. bal:13. As a second reason, the transformation of the Hamiltonian is more easily calculated here than for a single generating function, since in that last case effects quadratic in the transformation must be considered (see e.g. Eq. (6.9) of Ref. buon:99).. Notice that, in Ref. nag:11, the spin-orbit generating function is also applied after . By contrast, an evaluation order prescription between spin-orbit and spin-spin transformation would first be necessary when taking into account contributions that are cubic in the spins.
The final Hamiltonian must be equal to the corresponding term arising from the PN expansion of the EOB Hamiltonian. Since, for the moment, this is only true in the spin-aligned case, we are just allowed to write:
The NLO spin-spin transformation is given by
Notice that, in view of the generalization to precessing orbits, we have written the individual dimensionless spins as vectors. The coefficients and can be found in Eq. (5.13) of Ref. bal:13. It is also provided an expression for in the test mass limit (assuming ):
The purpose of this paper is that of generalizing the prescription (2.1) to the case of general orbits, i.e., when the scalar products and () cannot be set to zero.
Iii Including NLO spin-spin terms into the EOB for general orbits
iii.1 The prescription
In Ref bal:13, the EOB metric is written in Boyer-Lindquist-like coordinates. When spin precessions must be taken into account, however, it is necessary to switch to another system of coordinates. Following Ref. dam:01 (and reformulating the angular variable according to ), we can write the effective metric in Cartesian-like coordinates,
where . and can be found e.g. in Eq. (4.9) of Ref. bal:13 (notice that they both depend on the spin through a term ). The effective Hamiltonian (that we denote here as “old”, in order to avoid confusion with the modified version that is presented in this paper) takes the form
with a quartic-in-momenta term dam:00; dam:01 and with
has been introduced to describe higher-order spin-orbit couplings. It is defined in terms of the gyro-gravitomagnetic ratios and , see Eqs. (4.15) and (4.16) of Ref. dam:08.
With the reduced quantities , , and , we can write, more explicitly:
The spin-squared term , generated by the contraction of , can be expressed through the simple scalars , , , and according to
Let us now do some considerations:
The prescription (2.1) acts selectively - leaving all non-squared spins untouched - and cannot be truly considered as a redefinition of the effective spin of the EOB metric. It is, rather, a direct modification of the effective Hamiltonian.
One can say the same for the inclusions of spin-orbit terms done in Refs. dam:01; dam:08; nag:11. Adding the spin-orbit coupling requires a modification of all terms in the metric that are linear in the spin - or, equivalently, a direct modification of the Hamiltonian through an additional quantity (Eq. (4.16) of Ref. dam:08).
Changing the Hamiltonian itself rather than the metric is of course not unreasonable. Since the motion of a spinning particle is non-geodesic, there is no reason to believe that the dynamics of spinning bodies can be accurately described by geodesics in an effective metric. One should in principle not worry about intervening on the structure of the effective Hamiltonian itself.
In view of these remarks, and noting, in addition, that the effective Hamiltonian depends on the spin squared only through the scalars , , and , we see a natural way to generalize (2.1). We treat the spin differently whether it is contracted with itself, or , and propose the following type of replacements in Eq. (3.7):
the symbol corresponding to , , or . Recall that the functions and depend on , and thus need to be transformed according to (3.10a). The effective Hamiltonian that results applying (3.10) onto will be simply denoted as .
The coefficients , and are already known, their explicit expression being given by Eq. (5.12) of Ref. bal:13 (where the label had not been used). In particular, since the ’s vanish, (3.10a) is consistent with Eq. (II).
Determining the ’s from the Hamiltonian in ADM coordinates cannot be done without finding, simultaneously, the generating function of the corresponding canonical transformation (see Eq. (2.7)). We are looking for a sufficiently general ansatz that implements its already known test-mass limit (2.10), and that mantains, in addition, the symmetry under exchange of the labels 1 and 2. The searched canonical tranformation may have the following form:
for and . The ’s are, instead, constant coefficients. Notice that a canonical transformation of this type applies an infinitesimal rotation on the spins according to
In our specific case, the spins are left invariant under the constraint of aligned spins and equatorial orbits:
Thus, nonprecessing orbits are preserved under the transformation given by (III.1), which is a consistency requirement for the approach we are following.
iii.2 The general solution
In order to determine all coefficients, we first explicitly calculate the transformations given by Eqs. (2.4)-(2.7). All needed expressions are already collected in Ref. bal:13, and specifically: can be found in Eq. (2.3) there, in Eq. (2.4); in Eq. (2.7), in Eq. (2.9); in Eq. (3.3), and in Eq. (3.5). We need to rearrange two of these formulae, namely and , that had been originally written in a form that is not suitable for our purpose. In Eq. (2.9b) of Ref. bal:13, it appears the scalar . Using the identity (III.1) (but with the vector instead of ) it is easy to show that it can be decomposed as
Eq. (2.9b) of Ref. bal:13 then becomes
Furthermore, using basic vector identities, Eq. (3.5) of Ref. bal:13 is simplifed as follows:
The expansion can be done simply replacing , and performing a Taylor series in the small number . is then defined as the part proportional to which is quadratic in the spins. Finding a solution for the equation
is equivalent to solving an inhomogeneous system of 57 linear equations (18 for the spin(1)-spin(1) combination, 18 for the spin(2)-spin(2) and 21 for the spin(1)-spin(2) one), with 72 variables. This means that, if the system admits a solution, there will be at least 15 undetermined variables, that, as we shall see, will play the role of gauge coefficients. Notice that, because of the symmetry under exchange of the particle label 1 and 2, the system can be reduced to 39 equations and 50 variables. The general set of equations is solved by:
As already mentioned, the coefficients , , , , and directly follow from the solution above exchanging the particle labels 1 and 2. In regard to this point, it is worth discussing the Kerr limit . In this case, indeed, one has to choose which one of the two spins vanishes - thus somehow breaking the symmetry between them. For (and ) all coefficents , and must vanish, with the only exception of , and . The reason of that lies in the form of the canonical transformation (III.1) and its limit (2.10). The price of having enforced the formal symmetry between spin(1) and spin(2), by adding -independent terms also to the spin(2)-spin(2) contribution of , is that of generating coefficients that do not tend to zero in the Kerr limit (and specifically: , and , which can be easily verified taking into account that ).
By contrast, all coefficients , and vanish for , as required by consistency. In particular, the non-zero limit of , and is responsible for the convergence towards zero of , , , and , which might not have seemed immediately obvious from Eq. (III.2). Alternatively, to make this convergence more explicit, one could redefine
which absorb the -independent terms in the spin(2)-spin(2) part of (III.1), and satisfy , ,. Now we can do the following reformulation: