Post Newtonian emission of gravitational waves from binary systems: a gauge theory perspective

We derive a gauge inspired combinatorial formula based on localization for the Post-Newtonian expansion of the gravitational wave form luminosity of binary systems made of objects with very different masses orbiting at large distances and small velocities. The results are tested against previous formulae in the literature for Schwarschild and Kerr black holes at the 5th and 3rd Post Newtonian order respectively beyond the quadrupole approximation. Tidal effects show up in the wave form at the 5th PN order, providing a quantitative measure of the blackness/compactness properties of the heavy object.


Introduction
The detection of gravitational waves (GW) by the LIGO-Virgo collaboration [1] is definitively one of the most exciting achievements in contemporary physics.To extract the signal from the experimental data, templates of waveforms have been meticulously crafted through a sophisticated blend of theoretical and numerical calculations simulating the three phases (inspiral, merging, ring down) of the coalescence of black holes (BH) and/or neutron stars.The matching against observations provides a strong test of general relativity and confirms the existence of black hole like objects in nature.The upgrade of the LIGO-Virgo experiments and the coming of the ET experiment and the space based LISA will open a new era of precision gravitational wave (GW) astronomy allowing to test gravity beyond the realm of general relativity.
In this paper we deal with the study of the inspiral phase of binary systems.We work in the extreme mass ratio limit of the inspiral motion (EMRI) where the light particle orbits for a long time around the heavy one accumulating thousands of radians bringing tidal heating in the sensitivity band of a space based interferometer like LISA [65].In this regime the GW emission is well approximated by the the Post-Newtonian (PN) limit which assumes that the distance between the two objects is large and the relative velocity small.The theoretical studies of gravitational systems in the PN limit have a long history, and are very well described in many excellent textbooks [66][67][68][69].These results have been checked and further expanded using BH perturbation theory.In this framework, the Einstein equations are expanded to linear order in the perturbations around the background and written in terms of an ordinary second order differential equation of the Heun type resembling a Schrödinger equation.A binary system can also be described along these lines introducing in the equations a source term.The PN results can now be checked against this independent computation.Early results appeared in [70] and were later expanded in a series of papers [70][71][72][73][74][75][76][77][78] to extremely high orders for binary system orbiting circularly around Schwarzschild and Kerr BH's.
We revisit the computation of the PN expansion of the waveform using the CFT/gravity correspondence developed in [14,16,18].This line of research is founded on the so called AGT correspondence [79] which relates the results of localization for the partition functions of a N " 2 supersymmetric quiver gauge theories [80][81][82][83] with certain correlators of the Liouville CFT.The crucial point is that some correlators in the two-dimensional CFT satisfy a second order differential equation [84] which can be mapped to those of the Heun type which describe the wave form in the BH perturbation theory.As a result, the instanton gauge theory partition functions provide a combinatorial representation of the Heun function and its connection formulae [16,18].Following these ideas we relate the PN description of the wave form in the near zone to the instanton partition function of a quiver gauge theory given as a double series in the couplings which are related to the PN expansion parameters.As a result the PN expansion is given as a series with coefficients given by rational functions of the orbital quantum number ℓ.All logs and transcendental numbers that typically appear in the PN expansions are summed up into exponentials and Gamma functions.The instanton series is known to be convergent [85,86] and as a consequence so is the PN expansion.
Tidal effects are expected to show up in the gravitational wave signal at the 5th PN order (v 10 ) when the two independent solutions of the Heun equation start to mix [74,87].Their effects are proportional to the dynamical Love number defined in [18,88] and provide a quantitative measure of the compactness of the gravitational object.Along these lines we compute the static and dynamic Love numbers of Kerr-like compact objects.Recent results on this subject appeared recently in [89].This is the plan of the paper: In Section 2 we introduce the binary system and review the derivation of the wave form and luminosity of the gravitational wave signal.In section 3, we describe the CFT gravity dictionary adapting previous results [18] to the needs of the present discussion.In Section 4 and 5 we derive the PN expansion of the the wave form and its luminosity for Schwarschild and Kerr BH's at the 5th and 3rd PN order beyond the quadrupole approximation.In Section 6 we discuss the tidal interactions and point out the differences between BH's and ECO's.Appendices A, B, C, and D collect some background and technical material.

Teukolsky equations
The dynamics of the linear perturbations of the Kerr metric with ∆ r prq " pr ´r`q pr ´r´q " r 2 ´2M r `a2 , Σ " r 2 `a2 cos 2 θ can be decomposed into spin s components, Ψ s , obtained as projections of the curvature tensors deviation from the Kerr metric.The Einstein equations for Ψ s can be separated into radial and angular components via the ansatz with χ " cos θ, and the functions R ℓm prq, s S ℓm pχq satisfying the Teukolsky equations [90] d dχ Here M is the mass, a " J{M , the angular momentum parameter, A " pℓ ´sqpℓ `s `1q Òpωaq is a separation constant and is the Fourier transform of the spin s stress-energy tensor s T pxq.We refer the reader to [90] for details.We are interested in the spin s " ´2 mode associated to the ψ 4 -perturbation.

The Green function
The radial Teukolsky equation with a non-trivial source s T ℓm prq can be solved with the method of the Green function.For simplicity, we shall omit in the following the subscripts ℓ, m.Given ℓ, m, we denote by T the source term, by R T the solution of the inhomogenous equation with incoming boundary conditions at the horizon and by R in,up the solutions of the homogenous equation, satisfying incoming boundary conditions at the horizon and outgoing ones at infinity respectively.The Green function will be written in terms of R in and R up .More explicitly, we write R T prq " ż 8 r `Gpr, r 1 q ´2T pr 1 q∆ r pr 1 q ´2dr 1 . (2.6) with Gpr, r 1 q the Green function with incoming boundary conditions at the horizon.Explicitly r `Rin pr 1 q 2iωB in ´2T pr 1 q∆ r pr 1 q ´2dr 1 (2.11)

Circular orbits
The stress energy tensor produced by the particle motion is given by x ν δ p4q rx ´x0 pτ qs (2.12) with x 0 pτ q the geodesic trajectory.In the Hamiltonian formalism, the dynamics of a particle moving along a geodetics of the Kerr geometry is governed by the separable Hamiltonian H " 1 2 pg µν P µ P ν `µ2 q " H r `Hθ " 0 (2.13) with µ the particle mass, 2ΣH r " P 2 r ∆ r ´`P t pa 2 `r2 q `aP ϕ P θ pθq " P r prq " B r P r prq " 0 (2.15) with P θ pθq, P r prq obtained by solving (2.14).It is convenient to introduce the parameters v, q to parametrize the radius r 0 of the orbit and the angular parameter a via the identifications r 0 " M v ´2 , a " qM (2.16) In terms of these variables the solutions of (2.14) and (2.15) are The geodesic orbit is determined by integrating the velocities The angular velocity is therefore (2.20)

Wave form and energy flux
Putting together (2.12), (2.5) and (2.11), one finds [75] with the A i 's given by (B.7) in Appendix B. In terms of Z ℓm , the wave form takes the form and the gravitational luminosity (the amount of energy carried away per unit time) is given by In the next section we will derive a PN expansion of these formulae for the Schwarzschild and Kerr geometries.

CFT gravity correspondence
The basic idea behind the CFT gravity correspondence [14,16,18] relies on the fact, that the type of differential equations governing the dynamics of black hole perturbations appear in the context of Liouville theory as conformal Ward identities [84], satisfied by certain five points conformal blocks in the limit of large central charge.
According to the AGT correspondence, the degenerate conformal block is in turn related to the instanton partition function of a N " 2 supersymmetric quiver gauge theory [79,91] with gauge group SU p2q 2 .The large central charge limit corresponds to the Nekrasov-Shatashvili limit [92] of the instanton partition function.In this section we collect the formulae [16,18] which will be relevent to our computation.We refer the reader to these references and Appendix A for details.

Confluent Heun equation
The homogeneous part of the Teukolsky angular and radial equations (2.3) can be both brought into the normal form with z expressed in terms of the angular or radial variable and [18] Qpzq " ´x2 2) has two regular singularities at z " 8, 1 and an irregular one at z " 0. The solution can be written in terms of confluent Heun functions with parameters expressed in terms of p 0 , k 0 , c, u and x.The parametrization (3.2) is inspired by the correspondence with a five-point correlator in the Liouville theory that will be discussed below.

The angular equation
The angular equation can be written in the normal form (3.1), by writing and identifying 1 x χ " 4aω , c χ " s , p χ 0 " The radial equation The homogenous part of the radial equation can be written in the normal form (3.1), by taking and identifying 1 The superscript χ is to distinguish this angular equation from the radial one.

The five-point conformal block
We consider the five point conformal blocks, Ψ ˘pzq, with the insertion of a degenerate field at position z, two primaries at 8, 1, and two colliding at 0. Ψ ˘pzq are the independent solutions of (3.1)2 and are functions of the momenta, p 0 , k 0 , c, of the primary fields and are distinguished by the field exchanged in the intermediate channel.Using the AGT correspondence they can be written as with [79,81,82,91] the instanton partition functions of SU p2q and SU p2q 2 gauge theories with three flavours.The sums in (3.10) run over the pairs of Young tableau tY ˘u, tW ˘u while |Y |, |W | denote the total number boxes in each pair.The functions z bifund Y,W , z hyp Y 2 represent the contributions of a hypermultiplet transforming in the bifundamental and fundamental representation of the gauge group and are given by a product over the boxes of the Young tableau specified by s " pi, jq (see Appendix A for details) where λ Λi is the number of boxes in the i-th row and λ T Λj is the number of boxes in the j-th column of the tableau Λ.The conformal field theory variables entering in the fivepoint conformal block are related to the SU p2q 2 gauge theory couplings, Ω-background parameters and masses via the dictionary Similar relations hold between the four-point conformal block and the SU p2q gauge theory with coupling now given by q " q 1 q 2 " x.On the other hand are the momenta of the degenerate insertion and of its OPE with a primary field with momentum p s [84].We notice that each factor in (3.8) is divergent in the limit b Ñ 0 but the product is finite.The result is given by a double series expansion Finally the parameter a labels the momenta of the Liouville field exchanged in the intermediate channel and characterizes the monodromy of the gravity solutions around z " 0 In the gauge theory framework a represents the scalar vacuum expectation value (the fundamental Seiberg-Witten period) and can be determined by solving as a series in x with The result reads

Braiding and fusion connection formulae
According to the gauge gravity correspondence, the double instanton series (3.15) describes the PN expansion of the gravitational waveform in the limit where both z and x{z are small.In the gravity variables, see (3.5-3.6), this corresponds to the region where the distance from horizons are large but still much smaller than the wavelength of the perturbation ωpr ´r´q ! 1 and r " r `(3.20) We will refer to this region as the near zone.We notice that what it is typically called the near zone is the region where ωpr ´r`q ! 1.The PN expansion requires in addition that r " r `, so we could better call the region described by (3.20), the asymptotically far near zone, but for simplicity we will refer to it as the near zone.Moreover, we will further divide this zone into two patches depending on whether x{z !z (int) or z !x{z (ext).In these two extreme cases the instanton series (3.15) can be resummed and written in terms of the hypergeometric functions In these patches the effective potential can be approximated as Using the standard hypergeometric connection formulae these functions can be alternatively written as and p ψ α , r ψ α arise as limits of the conformal blocks p Ψ α and r Ψ α describing the near horizon and far zone respectively.The latters correspond to different OPE expansion of the same fivepoint correlator and are related to each other by the CFT braiding and fusion relations [14,16,18] where the fusion and braiding matrices are given in (3.25).Any solution of the homogenous differential equation can be written in the alternative ways Ψpzq " For example, the solution Ψ in pzq satisfying the incoming boundary conditions can be written in the three regions as For the PN expansion we will use On the other hand, the asymptotics at infinity is obtained by expanding the conformal blocks r Ψ α 1 pzq near z " 0. In this limit, the five-point conformal block factorizes into a four-point function depending only on x, times a function of z (see Appendix A for details).The latter is obtained by expanding the hypergeometric function in the second line of (3.24) in the limit where z !x.One finds Comparing against (3.29) one finds 4 Schwarzschild BH In this section, we deal with the case of the Schwarzschild geometry.We consider a particle of mass µ moving in a circular orbit of radius r 0 along the equatorial plane.From (2.18) and (2.17) one finds with For circular orbits

The PN expansion of R in and B in
The incoming solution R in prq and the asymptotic coefficient B in are given by the CFT formulae We notice that the PN expansion is largely dominated by the Ψ ´contribution, since Ψ ìs suppressed by an extra z 2a " r ´2ℓ´1 factor.The two solutions start to mix at order v 10 in the PN expansion for ℓ " 2 modes.The mixing is codified by the tidal function L " F ´1 ´`{F ´1 ´´computing the ratio response/source.We will study this function later in this paper.Here we just notice that the leading contribution for a Schwarschild BH's vanishes in the static limit ω Ñ 03 .

Kerr BH
In the case of the Kerr geometry both the radial and the angular equations are of the confluent Heun type and the separation constant A becomes a non trivial function of aω.

PN expansion of R in and B in
The solution of the homogeneous radial equation with incoming boundary conditions and the asymptotic coefficient are given again by the CFT formula (5.8) with radial dictionary

5.10)
The result to order v 4 is

Tidal interactions
Tidal interactions describe the response of a gravitational object to small perturbations of the geometry.They show up in the PN expansion of the wave form at order v 10 , where the solutions Ψ ˘describing the propagation of the wave in the near zone start to mix.The strength of the tidal interactions is measured by the Love and dissipation numbers defined as the real and imaginary parts of the ratio between the coefficients of the source and response components Ψ ˘of the solution in the near zone.This ratio is sensitive to the boundary conditions defining Ψpzq and provide therefore a quantitative measure of the compactness properties of the geometry.In this section, following [18], we compute the dynamical Love and dissipation numbers for Kerr BH's and Kerr like compact objects.A spin s perturbation of a Kerr BH is described by a solution of the confluent Heun equation with incoming boundary conditions Ψ " p Ψ ´leading to R Kerr pzq " p1 ´zq ´ps`1q The dynamical Love and dissipation numbers are given as the real and imaginary parts of the ratio of the Ψ ˘coefficients with `1qp2ℓ ´1qp2ℓ `1qp2ℓ `3q `Opϵ 4 q (6.4) Here we introduced the notion of "quantum" orbital number p ℓpωq, that account for quantum corrections of the SW period apϵq.The non-trivial ω-dependence of the tidal response function L Kerr is encoded in the replacement ℓ Ñ p ℓ.The analysis above can be easily adapted to the case of a compact Kerr like geometry.This geometry is obtained by cutting out a ball of radius r ``∆r around the origin and replacing the internal region by a horizonless geometry.Since the metric in the external region is that of Kerr, the solution outside is given again by a linear combination of Ψ α .The precise combination depends on the reflectivity assumed at the boundary of the two regions.For simplicity, we assume here that ∆r !r `, so boundary conditions are imposed at r " r `, where now a partial reflection of the wave is allowed.We write Ψ " a number, parametrizing the reflectivity of the boundary.Expanding the right hand side of (6.5) in the near zone, one finds Γp p ℓ`1`s`iϵqΓp1´s`2iP `qΓp p ℓ`1´2iP `´iϵq Γ ´p ℓ`1´s´iϵ ¯Γ p1`s´2iP `q Γp p ℓ`1`2iP ``iϵq (6.9) In the static limit ω Ñ 0 one finds ´" p2ℓq!pℓ´sq!Γp1`ℓ`2iP `qΓp1`s´2iP `q ℓ!pℓ`sq!Γ p1`ℓ´2iP `q Γp1´s`2iP `q (6.11)

Summary and Discussion
In this paper we derive a combinatorial formula for the PN expansion of the wave form produced by particles orbiting around Schwarzschild and Kerr BH's.To this aim, we exploit techniques coming from Seiberg-Witten theory, localization and AGT duality.In this framework the wave form is written as the partition function of an SU p2q 2 quiver gauge theory given as a double series with expansion parameters accounting for, on the gravity side, to Post Newtonian and Post Minkowskian corrections.All log's and trascendental numbers typically appearing in the PN expansions are resummed into exponential and Gammma functions.The results are in agreeement with those appeared in the literature obtained with other computational techniques and provide a compact formula for the wave form valid for arbitrary values of the orbital quantum number ℓ.
The results show that, in the PN limit, the wave form is dominated by one of the two solutions of the Heun equation describing the gravitational perturbations at linear order.The second solution starts to contribute at the 5-th PN order beyond the quadrupole approximation.This mixing is encoded in the tidal Love function carrying the information of the boundary conditions specifying the gravitational object.We derive a formula for the tidal Love function for Kerr BH's and Kerr like ECO's.
It would be interesting to apply the techniques in this paper to the case of open orbits for which the Post Newtonian and the Post Minkoskian approximations are not directly related as in the case of circular orbits.This line of research would certainly contribute to the recent efforts to relate the results coming from scattering amplitudes, effective theories, worldline quantum field theories etc.Each one of these techniques performs better than others in particular regions providing complementary viewpoints which can be combined to improve our actual understanding.

A.2 The confluent limit
The degenerated five point conformal blocks satisfy a differential equation of the Heun type in the variable z.The confluent Heun equation is obtained by considering the limit where two singularities collide at z " 0.More precisely, we take z 2 to zero, k 2 , p 3 to infinity, keeping finite the combinations c " k 2 `p3 , x " z 2 pk 2 ´p3 q. (A.10) The confluent limits of the relevant four and five point conformal block obtained will be denoted as In this limit the last contribution to the instanton partition function in (3.11)The first few terms in the instanton expansion are given in (3.15).Similarly, the description of the solution in the near horizon zone can be obtained from Ψ α pzq with the use of braiding and fusion relations.Diagramatically with braiding and fusion matrices given by (3.25).In particular, the asymptotics at infinity is given by [18] Ψ in px, zq " lim bÑ0 zÑ0 The three point function in (A.17) can be computed by expanding (3.24) in the limit z{x ! 1, while the ratio of four point functions gives the instanton contribution.Finally comparing against (3.29) one finds (3.33).