Nonlocal-in-time effective one body Hamiltonian in scalar-tensor gravity at third post-Newtonian order

We complete the nonlocal-in-time effective-one-body (EOB) formalism of conservative dynamics for massless Scalar-Tensor (ST) theories at third post-Newtonian (PN) order. The nonlocal-in-time EOB Hamiltonian is obtained by mapping the order-reduced Hamiltonian corresponding to the nonlocal-in-time Lagrangian derived in [Phys. Rev. D 99, 044047 (2019)]. To transcribe the dynamics within EOB formalism, we use a strategy of order-reduction of nonlocal dynamics to local ordinary action-angle Hamiltonian. We then map this onto the EOB Hamiltonian to determine the nonlocal-in-time ST corrections to the EOB potentials $(A,B,Q_e)$ at 3PN order.


I. INTRODUCTION
In 2015, the direct detection of gravitational waves (GW) by the LIGO-Virgo Collaboration [1] emitted by inspiralling compact binary, opened new avenues for probing the dynamics in strong gravity regime [2][3][4][5][6][7][8]. It is expected that future GW detectors, like the Einstein Telescope [9] and Cosmic Explorer [10] will shed more light on alternative theories of gravity by constraining the parameters of such theories.
The important violations for ST theories arise through the non-perturbative strong field effects in neutron-stars such as spontaneous scalarisation [13]. Although the current constraints come from the binary pulsar observations, the future GW detections can better constraint the parameters using strong-field information and additional terms in radiation, i.e. dipolar radiation which is not present in GR, due to the scalar extension of GR [15,27,28].
The EOB formalism was introduced to construct analytical waveform templates for GR [29][30][31][32][33][34][35]. Recently, the two-body PN dynamics has also been mapped within the EOB formalism to construct waveform templates for ST theories [36][37][38]. In our previous work [38], we determined the EOB potentials for the local part of dynamics at 3PN order. The aim of this paper is to determine the complete nonlocal-in-time EOB potentials following our results of local part in Ref. [38] starting from the 3PN nonlocal-in-time Lagrangian of Ref. [20,21]. Hereafter, the companion paper [38] will be referred as Paper I. * tj317@cam.ac.uk The paper is organised as follows. In Sec. II, we give a summary of results obtained in Paper I. Then, in Sec. III we derive the conserved energy for nonlocalin-time part using two methods, (i) non-order-reduced nonlocal Hamiltonian using nonlocal phase shift, and (ii) order-reduction of nonlocal dynamics to local ordinary action-angle Hamiltonian. Finally, in Sec. IV we map the nonlocal-in-time ordinary Hamiltonian into an EOB Hamiltonian at 3PN order.

II. SUMMARY OF PREVIOUS RESULTS
We consider mono-scalar massless ST theories described by the following action in the Einstein Frame (the scalar field minimally couples to the metric), where g µν is the Einstein metric, R is the Ricci scalar, ϕ is the scalar field, Ψ collectively denotes the matter fields, g ≡ det(g µν ) and G is the bare Newton's constant [38]. As Paper I (see , Table I), we adopt the conventions and notations of Refs. [11,13]. In Einstein Frame, the dynamics of the scalar field arises from its coupling to the matter fields Ψ, and the field equations can be found in Ref.
[11] where the parameter measures the coupling between the matter and the scalar field. The scalar field is non-minimally coupled to the metric in Jordan Frame (physical frame) whereg µν is the metric in Jordan frame. We follow the approach suggested by [39] to "skeletonize" the compact, self-gravitating objects in ST theories as point particles, i.e. the total mass of each body is dependent on the local value of the scalar field. The skeletonized matter action with the scalar field dependent massm I (ϕ) is then given by where λ is the affine parameter. Sinceg µν = A(ϕ) 2 g µν , the Einstein-frame mass is defined as (2.5) In Paper I, we first derive the ordinary Hamiltonian (dependent only on the positions and momenta) using the contact transformation at 3PN order starting from the Lagrangian of Ref.
[21] only for the local-in-time part of the dynamics. The Jordan-Frame parameters of Ref.
[21] that encompass the scalar field effect are converted to the dimensionless Einstein-Frame parameters (see, Table I). The mass function m(ϕ) is used to define these dimensionless body-dependent parameters following Refs. [11,13,37] i.e. (2.9) Here, we follow the notations of Paper I for the binary parameters and use the same notation as [20, 21] to denote weak-field and strong-field parameters. Finally, we then determine the ST corrections to the EOB metric potential (A, B, Q e ) at 3PN order for the local in time (instantaneous) part of the dynamics by mapping the EOB Hamiltonian in DJS gauge [31] (2.10) wherep r ,p φ are the dimensionless radial and angular momenta, andr(= r/(G AB M ) is the dimensionless radial separation, to the ordinary two-body Hamiltonian (hereafter the superscript hat is used to denote the dimensionless variables). The three EOB potentials at 3PN are (2.13) The GR and ST corrections in coefficients (a i , b i ) are separated as (2. 16) Since there are also nonlocal-in-time and tidal contributions at 3PN order in ST theory, all the 3PN ST coefficients can thus be decomposed as Eq. (5.23) of Paper I. The complete expressions of local-in-time ST corrections at 3PN can be found in Eqs.(5.14)-(5.16) of Paper I.
In Paper I, we also derive the nonlocal-in-time (tail) and tidal corrections only for the circular orbits using the gauge invariant energy for circular orbits given in Ref. [21,22]. The complete expression for these coefficients can be found in Eqs. (5.25)-(5.27) of Paper I.

III. TAIL CONTRIBUTION TO THE 3PN DYNAMICS
The nonlocal-in-time two-body 3PN Lagrangian for massless ST theory obtained in Ref.
[21] is in harmonic coordinates, i.e. it depends (linearly) on the acceleration of the two bodies. In this section, we will use two different methods to derive the Noetherian conserved energy for the tail contributions. First, we will remove the acceleration dependence from the Lagrangian (hence, the Hamiltonian) and stay within the non-order-reduced nonlocal framework (as done in Refs. [40,41] for GR). Second, we will derive the order-reduced, local Hamiltonian using the action-angle variables (see, Ref. [34] for GR).

A. Non-order-reduced Ordinary Hamiltonian
In Paper I, we derived the ordinary (dependent only on positions and momenta) Hamiltonian for local-intime contribution using contact transformation (see, Appendix A of Paper I for the contact transformation). Now, concerning the nonlocal-in-time part we need to find the nonlocal shift that removes the acceleration dependence from the tail part of the Lagrangian of Ref.
The tail part of the Lagrangian at 3PN order reads [21], where Pf is the Hadamard partie finie function, Hadamard scale r AB (= r) is the relative separation of two bodies, and I (2) s,i is the second time derivative of the dipole moment. Here, we find the shift that transforms this Lagrangian into the same expression but with the derivatives of the dipole moment evaluated using the Newtonian equations of motion. In the centre of mass (COM) frame in notations of Ref. [20,21] it is, where s A , s B are the sensitivity of two bodies. As the nonlocal contribution starts at 3PN order, the ordinary Lagrangian is where L tail ord is given by the same expression as Eq. (3.1) but with second time derivative of the dipole moment replaced by its on-shell value given in Eq. (3.2), and the nonlocal shift, ξ J,j , The ordinary Hamiltonian is then derived using the ordinary Legendre transformation, where the local contribution H loc ord is derived in Paper I (see, Appendix C) and the tail contribution is The tail part of the Hamiltonian is just opposite to tail part of Lagrangian. As shown in Ref. [35,41] for the non-order-reduced, nonlocal framework the Noetherian conserved energy (E cons ) is not given by the Hamiltonian but is given by, E cons = H tail ord + δH. This additional term δH consists of purely a constant term (DC type) and time oscillating term with zero average value (AC type) and is same as given in Eq. (4.10) of Ref. [21].

B. Order-reduced Ordinary Hamiltonian
The second method to derive the conserved energy for tail part is to work in the order-reduced, local framework as given in Ref. [34,35] for GR.
The tail part of the Hamiltonian in ST theory is, As mentioned in Ref. [41], in the action-angle form there should be an additional term (second term in Eq. (3.6)) which is local and accounts for dependence of Hadamard Partie finie function on the radial separation (r) at time t i.e.,r = a(1 − e cos(u)) in action-angle variables. The basic methodology we use to order-reduce the nonlocal dynamics of the above form is based on Refs. [34,35] for GR, and consists of four main steps: (i) Re-express the Hamiltonian in terms of action angle variables, (ii)"order-reduce" the nonolocal dependence on action angle variable, (iii) expand it in powers of eccentricity, and (iv) eliminate the periodic terms in order-reduced Hamiltonian by a canonical transformation. All of these steps lead to the order-reduced ordinary local Hamiltonian for the tail part in terms of action-angle variables.
Let us consider the expression of nonlocal-in-time piece of Eq. (3.6), i.e. (3.7) To order reduce the nonlocal piece, we use the equations of motion to express the phase-space variables at shifted time t + τ in terms of the phase-space variables at time t.
As the zeroth order equations are Newtonian equations, it will be convenient to use the action-angle form of the Newtonian equations of motion, Here, the variable L is conjugate to the "mean anamoly" l and G is conjugate to argument of periastron g. In terms of the Keplerian variables, semi-major axis a, and eccentricity e, these are (3.9) From Eq. (3.8), the variables L, G and g are independent of time, and l varies linearly with time, hence it will be sufficient to use Using the Fourier decomposition of dipole moment given in Eq. (A11), we find the structure of nonlocal-intime expression K(t,τ ) and hence the Hamiltonian. As shown in [34] for GR, all the periodically varying terms can be eliminated by a suitable canonical transformation. Hence, the order-reduced Hamiltonian can be further simplified by replacing H tail with its l-average valuē (3.12) Using the result where γ E is the Euler's constant, and inserting the expression of r from Eq. (A7), the Hamiltonian, Eq. (3.12), reads (3.14) Now, inserting the Fourier-Bessel expansion of scalar dipole moment from Eqs. (A13)-(A14) (see, Appendix A for derivation) in Eq. (3.14), the real two-body nonlocalin-time Hamiltonian in order-reduced, local framework is (in notations of Paper I)  To map the real two-body dynamics to EOB, we express the nonlocal effective Hamiltonian,Ĥ nonloc eff , in action-angle variables L, l, G, and g (hence the Keple-rian variables a and e) and compute its l-averaged value, (4.8) The explicit expression ofĤ nonloc eff depends on l-average monomials involving powers of 1/r andp r (and also ln(r) from Eqs. (4.3), (4.4), and (4.5)). These computations can be performed by expanding Eq. (4.7) in terms of eccentricity upto e 5 using the Newtonian equations of motion in action-angle form recalled in Sec. III B. The l-averaged value we obtain iŝ H The final step is then to map the real two-body dynamics to EOB metric by the nontrivial map, between the EOB Hamiltonian (Ĥ eff ) and real two-body Hamiltonian (Ĥ real ). The quadratic map relating the two Hamiltonians is proven at all PN orders in GR and ST within the Post-Minkowskian scheme in Ref. [42]. However, it can be seen that only for the nonlocal contributions at 3PN order, the map relating the two nonlocal Hamiltonians isĤ nonloc eff =Ĥ II real,nonloc . (4.11) The unique nonlocal ST contributions at 3PN from this matching are δa ST 4,nonloc,0 = 4 3 ν 2δ + +γ AB (γ AB + 2) 2 (2 ln 2 + 2γ E ), (4.12)

V. CONCLUSIONS
In Paper I, building upon the results of [21] for massless scalar-tensor theory, we determined the EOB coefficients at 3PN order though restricting ourselves to local-in-time part of the dynamics and nonlocal-in-time and tail contributions only for the circular case. In the present paper, we derived the complete nonlocal-in-time EOB coefficients starting from the nonlocal-in-time Lagrangian of Ref. [21]. First, we derived the two-body conserved ordinary Hamiltonian (dependent only on positions and momenta) for nonlocal-in-time part by two methods: (i) non-order-reduced nonlocal Hamiltonian using nonlocal phase shift (see, Ref. [40,41] for GR), and (ii) orderreduction of nonlocal dynamics to local ordinary actionangle Hamiltonian [34]. We then expressed the effective Hamiltonian in Delaunay variables to recast the orderreduced ordinary action-angle Hamiltonian into equivalent, 3PN-accurate, nonlocal part of EOB potentials (A, B, Q e ), see Eqs. (4.12)-(4.17).
By combining the results of Paper I and the present work, we could transcribe the two-body Hamiltonian into equivalent 3PN-accurate EOB potentials (A, B, Q e ) for both local-in-time and nonlocal-in-time part of dynamics.
Note: During the preparation of the final manuscript of this work, the author became aware of the independent effort which recently arrived on arXiv [43].

ACKNOWLEDGMENTS
The author is grateful to P. Rettegno, M. Agathos and A. Nagar for useful discussions and suggestions during the preparation of this work. The author is jointly funded by the University of Cambridge Trust, Department of Applied Mathematics and Theoretical Physics (DAMTP), and Centre for Doctoral Training, University of Cambridge.

Appendix A: Fourier Coefficients of dipole moment in ST theory
In this appendix, we will determine the explicit expressions of Newtonian dipole moment in ST theory using the known Fourier decomposition of the Keplerian motion (see, Refs. [44,45] for GR).
The dipole moment, I s,i (t), in COM frame is where x i = (Z A − Z B ) i is the relative separation vector and Z A,B indicate the positions of the two bodies.
Since the motion is planar, we can choose the coordinate system (x, y, z) such that it coincides with the xy-plane. Using the polar coordinates (r, φ a ), x =r cos(φ a ), y =r sin(φ a ). (A2) The coordinates (x, y) are the coordinates of the dimensionless relative separation,r = x A − x B with x J = x J /(G AB M ) denoting the position of two bodies.
As mentioned in Ref. [34,44,45] for GR, for leading order contributions it is convenient to use the Delaunay (action-angle) form of the Newtonian equations of motion. In terms of the action-angle variables (L, l, G, g), the Cartesian coordinates (x, y) are given by (Here, we follow the notations of [46]) where a is the semi-major axis, e is the eccentricity, f is the "true anamoly" and the "eccenteric anamoly" u in terms of Bessels functions is given by The Bessel-Fourier expansion of cos(u) and sin(u), which directly enters x 0 , y 0 are: with The Fourier coefficients of the scalar dipole moment at the Newtonian order are derived using Eq. (A12) in terms of combinations of Bessel Functions.
Inserting the expression of Cartesian coordinates in terms of action-angle variables using Eqs. (A3)-(A10), we find the Fourier-Bessel coefficients of the scalar dipole moment are