Compact binary systems in scalar-tensor gravity. III. Scalar waves and energy flux

Compact binary systems in scalar-tensor gravity. III. Scalar waves and energy flux

Ryan N. Lang lang@phys.ufl.edu Department of Physics, University of Florida, Gainesville, Florida 32611, USA
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
Abstract

We derive the scalar waveform generated by a binary of nonspinning compact objects (black holes or neutron stars) in a general class of scalar-tensor theories of gravity. The waveform is accurate to 1.5 post-Newtonian order [] beyond the leading-order tensor gravitational waves (the “Newtonian quadrupole”). To solve the scalar-tensor field equations, we adapt the direct integration of the relaxed Einstein equations formalism developed by Will, Wiseman, and Pati. The internal gravity of the compact objects is treated with an approach developed by Eardley. We find that the scalar waves are described by the same small set of parameters which describes the equations of motion and tensor waves. For black hole–black hole binaries, the scalar waveform vanishes, as expected from previous results which show that these systems in scalar-tensor theory are indistinguishable from their general relativistic counterparts. For black hole–neutron star binaries, the scalar waveform simplifies considerably from the generic case, essentially depending on only a single parameter up to first post-Newtonian order. With both the tensor and scalar waveforms in hand, we calculate the total energy flux carried by the outgoing waves. This quantity is computed to first post-Newtonian order relative to the “quadrupole formula” and agrees with previous, lower order calculations.

pacs:
04.30.Db, 04.25.Nx, 04.50.Kd

I Introduction

We are entering the age of gravitational-wave (GW) astronomy. The Advanced LIGO interferometric detectors in Louisiana and Washington Harry and LIGO Scientific Collaboration (2010) will come online in 2015, followed later by Advanced Virgo in Italy Accadia et al. (), KAGRA in Japan Aso et al. (2013), and possibly a third LIGO detector in India Iyer et al. (). This network of detectors will measure GWs in the “high-frequency” band, 1– Hz. The International Pulsar Timing Array collaboration is currently using pulsar timing residuals to search for GWs in the “very-low-frequency” band, Hz McLaughlin (2014). In between these two bands, GW detection requires interferometry in space. This science is the target of the European Space Agency “L3” mission, due to launch in 2034, most likely with an instrument resembling the well-known eLISA design Amaro-Seoane et al. (2013).

The primary sources for all of these experiments are compact binaries comprising white dwarfs, neutron stars, and black holes. We expect to learn a great deal about the astrophysics of these systems, ranging from simple event rates to very precise information on their masses, spins, orientations, and sky locations. We can also use the gravitational waves they emit to test Einstein’s theory of general relativity (GR).

General relativity is nearing its one-hundredth birthday and continues to pass all tests with flying colors. Deviations from GR have been strongly constrained by measurements of the Solar System and of binary pulsar systems Will (2014). However, GR has not yet been tested in the strong-field, dynamical regime we expect to probe with the GW observatories. In particular, as compact binaries emit gravitational radiation, they lose energy to that radiation, causing the binary orbit to decay. Eventually, the two objects merge, producing a final burst of gravitational waves before settling into a quiescent state. The combination of strong gravitational fields and short time scales involved in this inspiral and merger process represents a limit in which general relativity might break down.

One peculiarity of gravitational-wave detection is the need for very accurate template waveforms with which to compare the data. While perhaps some very strong signals in the eLISA detector could be picked out by eye, most GW signals will be heavily buried in noise. Matched filtering with a template bank is often the only way to recover the actual GW data. While it is possible to conduct searches without templates (and sometimes necessary, in the case of unmodeled burst searches), templates are necessary to fully realize the detection capability of an instrument. Furthermore, finding the best-fit template allows one to precisely determine the system parameters for purposes of astrophysics, as well as search for deviations from general relativity.

With this application in mind, we have undertaken an effort to produce waveform models for a particular alternative to general relativity, scalar-tensor theory. This theory dates back over 50 years and yet remains extremely relevant today. For instance, many so-called theories, which modify the action of general relativity to allow arbitrary functions of the Ricci scalar, can be expressed in the form of a scalar-tensor theory de Felice and Tsujikawa (2010). These theories may explain the acceleration of the Universe without resorting to dark energy. Scalar-tensor theory is also a potential low-energy limit of string theory Fujii and Maeda (2003).

The term “scalar-tensor theory” is really a catch-all for a collection of theories, possibly including multiple scalar fields in addition to the tensor field (metric) of general relativity. We limit ourselves to a single scalar. The action we choose is

(1)

where is the spacetime metric, is its determinant, is the Ricci scalar derived from this metric, is the scalar field, and is the scalar-tensor coupling. Note that is not a constant; that is, we are not restricting our attention to Brans-Dicke theory. We do, however, restrict ourselves only to massless scalar fields (i.e., those without a potential). We have also written to represent the matter action. Note that it depends only on the matter fields and the metric; the scalar field does not couple directly to the matter. This means that (1) is expressed in the “Jordan” frame. All of our work will be done in this frame. An alternative representation is the “Einstein” frame, related to the Jordan frame by a conformal transformation Damour and Esposito-Farèse (1992). Regardless of in which frame the calculation is done, the final result—the gravitational waveform—will be the same.

The action generates the following field equations:

(2a)
(2b)

Here is the Einstein tensor, is the stress energy of matter and nongravitational fields, and is its trace. We use commas to denote ordinary derivatives. Semicolons denote covariant derivatives (taken using in the usual way), and is the d’Alembertian with indices raised by the metric. These conventions will be used throughout the paper.

The computation of the gravitational waveform in this set of scalar-tensor theories spans three papers, including this one. We focus on the inspiral phase, when the compact objects can be regarded as separate bodies, and derive the waveform using the post-Newtonian (PN) approximation. This approximation is an expansion in powers of , with each power representing half a post-Newtonian order, and is valid until just before the end of the inspiral phase. In the first paper (Mirshekari and Will (2013), hereafter paper I), Mirshekari and Will computed the equations of motion for a compact binary to order (2.5PN) beyond the leading (“Newtonian”) term. This result is a necessary precursor for computing the waveform, as well as extremely interesting in its own right. In the second paper (Lang (2014), hereafter paper II), we derived the tensor gravitational waveform to order (2PN) beyond the leading term. In both of these papers, the final results were shown to reduce to the general relativistic versions in the appropriate limit.

In this paper, we compute the final piece of the puzzle, which has no GR analog: the scalar waveform. A gravitational-wave detector will measure both tensor and scalar waves. In paper II, we showed that the separation between masses in a detector will obey the equation

(3)

where

(4)

Here is the direction from the source to the detector, is the Kronecker delta, and and are redefined tensor and scalar fields which we use in this series. They are written out explicitly in Sec. II.1 below. The subscript “TT” designates the transverse-traceless projection of the tensor field. The tensor field has the usual two polarizations expected in general relativity, and , though the exact time dependence of the waves will take the new form computed in paper II. The scalar waves, on the other hand, will appear as a transverse “breathing” mode. The detection of a third polarization state like this one would be a smoking gun that general relativity is somehow incorrect.

The scalar waveform actually begins at PN order, keeping the same definition of “0PN” that we use for the tensor waveform in both GR and scalar-tensor theory. This feature is due to the presence of a nonvanishing “scalar dipole moment,” which also creates 1.5PN radiation-reaction effects in the equations of motion and various effects in the tensor waves. (Technically, there is a monopole piece in the far field at PN order; however, it is time independent and not wavelike. The leading-order monopole contribution to the waves thus enters at 0PN order.) Because of this “shifting” of post-Newtonian orders, it is much more difficult to calculate the scalar waveform at the same order as the tensor waveform. Therefore, we compute the scalar waveform only to 1.5PN order.

The challenge in all three papers is to solve the scalar-tensor field equations (2a)–(2b). To do so, we use a method called “direct integration of the relaxed Einstein equations” (DIRE), based on the original framework of Epstein and Wagoner Epstein and Wagoner (1975) and then extended by Will, Wiseman, and Pati Wiseman (1992); Will and Wiseman (1996); Pati and Will (2000, 2002). It is easily adapted to scalar-tensor theories. In the adapted DIRE method, the scalar-tensor field equations are first rewritten in a “relaxed” form: flat-spacetime wave equations for the “gravitational field” and the modified scalar field . The wave equations are simplified by the choice of a particular coordinate system, represented by a gauge condition on . Together, the wave equations and gauge condition contain all the content of the full field equations. The wave equations can be solved formally by using a retarded Green’s function. When converting to a more useful form, we evaluate these formal solutions differently depending on whether the source and field points are close to the compact objects (in the “near zone”) or far away (in the “radiation zone”). The four different methods of evaluation are described in detail in Will and Wiseman (1996); Pati and Will (2000).

Since scalar-tensor theories do not obey the strong equivalence principle, the motion and gravitational-wave emission of a binary depend on the internal composition of its constituent bodies. To handle this effect, we have adopted the approach of Eardley Eardley (1975). We treat the matter stress-energy tensor as a sum of delta functions located at the position of each compact object. However, instead of assigning each body a constant mass, we let the mass be a function of the scalar field, . This gives the matter action an indirect dependence on , even though we still work in the Jordan frame. This is the origin of the term in (2b). In the final waveform, we express the varying mass using the “sensitivity,”

(5)

as well as derivatives of this quantity. (The subscript 0 means that the derivative should be evaluated using the asymptotic value of the scalar field, .) In the weak-field limit, the sensitivity is proportional to the Newtonian self-gravitational energy per unit mass of the body. For neutron stars, the sensitivity depends on the mass and equation of state of the star, with typical values 0.1–0.3 Will and Zaglauer (1989); Zaglauer (1992). For black holes, , and all derivatives vanish. Since we assume that the sensitivities are constant in time, our work does not capture any “dynamical scalarization” effects Barausse et al. (2013); Palenzuela et al. (2014); Shibata et al. (2014); however, we expect such effects to be relevant only at the end of inspiral, when the post-Newtonian approximation breaks down.

The scalar waves we find depend on the same relatively small set of parameters that characterizes the equations of motion and tensor waves. These parameters are defined in terms of the coupling function (or more precisely, its Taylor expansion coefficients) and the sensitivities and sensitivity derivatives of the compact objects. For black hole–black hole binaries, the scalar waveform vanishes completely. This result is not unexpected. Papers I and II showed that black hole binaries in scalar-tensor theory cannot be distinguished from their general relativistic counterparts by studying their motion or tensor wave emission. For mixed (black hole–neutron star) binaries, the scalar waves simplify considerably. After a mass rescaling, the waves up to 1PN order only depend on two parameters: the same parameter which characterizes the equations of motion and the tensor waves for mixed binaries, plus an overall amplitude factor. At 1.5PN order, the expression becomes more complicated.

Having computed both the tensor and scalar radiation from the system, we also calculate the total energy flux carried off by the waves. This energy loss backreacts on the system, causing the radius of the orbit to shrink and the frequency of the orbit and GWs to increase (“chirp”). The expression we derive can be used to generate useful template waveforms for GW detectors. Because of the strange nature of post-Newtonian counting in scalar-tensor theory, specifically the existence of a negative-order dipole contribution to the scalar waves, computing the energy flux at th post-Newtonian order actually requires knowing at least some pieces of the scalar waveform at th order. For this reason, we stop our calculation of the flux at first post-Newtonian (1PN) order, where “0PN” is defined as the lowest order tensor flux (i.e., the “quadrupole formula”). We plan to continue the calculation to higher order in future work.

The outline of the paper is as follows: Section II reviews necessary results from papers I and II, including the relaxed field equations, the Eardley approach to the matter source, and the definition of various “potentials” with which we express intermediate results. Section III describes the calculation of the near-zone contribution to the scalar waveform. We first describe the general technique, which requires the computation of “scalar multipole moments.” We then review the construction of the source in the near zone and give explicit expressions for it. Next, we calculate the scalar multipole moments. The techniques for doing so are identical to those used in calculating the “Epstein-Wagoner moments” of paper II. We then simplify the scalar moments to two-body systems and convert them to relative coordinates. We also discuss how the equations of motion from paper I are used to expand time derivatives of the moments.

Section IV presents the radiation-zone contribution to the scalar waves. We first discuss the formalism for finding this contribution and review results from paper II which are needed to construct in the radiation zone. We then compute the waveform produced by this source. These terms begin at 1PN order, half an order lower than in the tensor case. They include scalar “tail” terms which depend on the entire history of the binary. We call such terms “hereditary” terms. Unlike the tensor case, there are no nontail hereditary terms.

In Sec. V, we present the full 1.5PN scalar waveform for a nonspinning compact binary in massless scalar-tensor theory. Finally, in Sec. VI, we compute the rate at which energy is carried away from the system by both tensor and scalar waves.

In this paper, we use units in which . We do not set ; as we shall see, the effective Newtonian gravitational constant depends on the asymptotic value of the scalar field. Greek indices run over four spacetime values (0, 1, 2, 3), while Latin indices run over three spatial values (1, 2, 3). We use the Einstein summation convention, in which repeated indices are summed over. We use a multi-index notation for products of vector components: . A capital letter superscript denotes a product of that dimensionality: . Angular brackets around indices denote symmetric, trace-free (STF) products (see Appendix B of paper II for details). Finally, we use standard notation for symmetrized indices, e.g. .

Ii Preliminaries

In this section, we review some key concepts from papers I and II with which the reader should be familiar before continuing on to the next sections. For brevity, we omit many details. They can be found in papers I and II.

ii.1 Reduced field equations

The adapted DIRE method solves the scalar-tensor field equations (2a)–(2b) by rewriting them in an equivalent, but more easily manageable, form. We first assume that far away from the compact binary, the metric reduces to the Minkowski metric, , and the scalar field tends to a constant . We introduce a rescaled scalar field,

(6)

Calculating the form of at a distant gravitational-wave detector is the primary goal of this paper. We also define a new tensor field of interest, the “gravitational field” ,

(7)

The “gothic” metric is given by

(8)

where

(9)

is a conformally transformed version of the metric and is its determinant. In general relativity, the DIRE method defines a gravitational field in much the same way. However, since there is no scalar field, the regular metric and its determinant take the place of the conformally transformed versions. For convenience, we define the components of the gravitational-field tensor to be

(10a)
(10b)
(10c)
(10d)

We are free to pick coordinates in which the field equations take on a simpler form. We choose the Lorenz gauge condition

(11)

Then the field equation (2a) reduces to

(12)

where is the flat-spacetime wave operator and the source is

(13)

Here is the stress energy of matter and nongravitational fields; in our case, it is generated by the compact source. We will discuss it further in Sec. II.2. The two terms and depend on the gravitational (tensor) and scalar fields. Exact expressions for them can be found in paper II. Our choice of variables ensures that maintains the same basic form as in general relativity; compare Eqs. (3.4) in paper I with Eqs. (4.4) in Pati and Will (2000). The term is new to scalar-tensor theory. The gauge condition (11) implies a conservation law for the source,

(14)

The scalar field equation (2b) can also be written as a flat-spacetime wave equation,

(15)

with source

(16)

Recall that is the trace of .

The wave equations (12) and (15) can be solved formally in all spacetime by using a retarded Green’s function,

(17a)
(17b)

The integrations take place over the past flat-spacetime null cone emanating from the field point . To obtain explicit solutions, we divide the spacetime into two regions. We define the characteristic size of the source as and assume the bodies move at velocities . Then the “near zone” is defined as the area with , where is the characteristic wavelength of gravitational radiation from the system. (We use capital to denote the distance from the binary’s center of mass to a field point in order to avoid confusion later with , the orbital separation of the binary.) Everything outside the near zone () is the “radiation zone.”

There are a total of four different ways to evaluate (17a)–(17b), depending on which of the two zones contains the field and source points. For instance, in paper I, the integrals were evaluated for field points in the near zone. The integration was split into two pieces, one piece for source points in the near zone and one piece for source points in the radiation zone. (The latter turned out not to contribute at the post-Newtonian order considered.) In paper II, (17a) was evaluated for field points in the radiation zone, i.e., far away from the source where gravitational waves are measured. The integral was again split into two pieces, one for source points in the near zone and one for source points in the radiation zone. In this paper, we evaluate (17b) for field points in the radiation zone. Integration over the near-zone source points is described in Sec. III. Integration over the radiation-zone source points is described in Sec. IV.

Because the radius of the boundary between zones is completely arbitrary, no terms in the final expressions for the fields should depend on it. It was shown in Will and Wiseman (1996); Pati and Will (2000) that, for general relativity, the terms dependent on which arise from the near-zone integral cancel exactly with the terms dependent on which are generated by the radiation-zone integral. In our work, we simply assume that this property holds and throw away all terms which depend on .

ii.2 Matter source and potentials

Following paper I, the compact source can be described in terms of “ densities” Blanchet and Damour (1989),

(18a)
(18b)
(18c)
(18d)

From these, we can define a number of Poisson-like potentials. For example, given a generic Poisson integral for a function ,

(19)

the simplest potentials are

(20a)
(20b)

The integrals are taken over a constant-time hypersurface at time out to radius , the boundary of the near zone. The subscript clarifies that these potentials use the densities. Expressions for all other -density potentials can be found in paper I, Eqs. (3.12)–(3.13). Note that the generic Poisson integral has the property

(21)

This will be very useful throughout the calculation. Using the densities and the associated potentials, we can solve the field equations for a generic system. To get answers specific to a system of compact objects, we must study the compact stress energy more closely.

Since a compact object is gravitationally bound, its total mass depends on its internal gravitational energy. This, in turn, depends on the effective local value of the gravitational coupling. In scalar-tensor theory, the coupling is controlled by the value of the scalar field in the vicinity of the body (scaling like ). To deal with this complication, we use the approach of Eardley Eardley (1975). In his method, we consider the compact objects to be point masses, with a mass that is a function of the scalar field. The stress-energy tensor is then given by

(22)

Here is the four-velocity of body and is the proper time measured along its world line. (This is the only instance in which we use the symbol for this purpose.) This construction assumes that the dynamical time scale of the body is short compared to an orbital time scale.

We expand about the asymptotic value of the scalar field, ,

(23)

where . We define the sensitivity and its derivatives as

(24a)
(24b)
(24c)

and so on. Note that has the opposite sign of the equivalent quantity in Will (1993); Alsing et al. (2012). For later convenience, we also define the quantities

(25a)
(25b)
(25c)

If we define a new density

(26)

the stress energy becomes

(27)

where is the ordinary velocity. The various velocities and the sensitivity technically should have body labels, but they will each pick one up when multiplied by the delta function in . As shown in paper I, (27) can be used to rewrite the densities in terms of the density as a post-Newtonian expansion.

We can also define new potentials based on the density. For instance,

(28a)
(28b)

Note that (21) implies that and . More generally,

(29a)
(29b)
(29c)
(29d)
(29e)
(29f)

so that and . The other potentials we use in this paper are

Equations (5.13)–(5.22) in paper I show how to convert between many -density and -density potentials (e.g., and ).

Iii Near-zone contribution to the scalar waveform

In this section, we calculate the near-zone contribution to the scalar waveform. This is, by far, the more difficult and time consuming of the two contributions. We begin by discussing the general formalism for evaluating the near-zone integral, namely the calculation of scalar multipole moments. We also discuss the counting of post-Newtonian orders, which is a bit more subtle than in the tensor wave case. Next, we discuss the construction of the source . Third, we present the calculation of the scalar multipole moments. We provide some details but refer the reader to paper II for more. Finally, we conclude this section by rewriting the scalar multipole moments explicitly for two-body systems in relative coordinates. We hold off on presenting the final scalar waveform generated by the near-zone integral until Sec. V.

iii.1 General structure of near-zone calculation

The goal of this entire section is to evaluate (17b) for near-zone source points and radiation-zone field points. We can simplify the problem even further by considering only a subset of the radiation zone, the “far-away zone.” In the far-away zone, , so we only need to keep the leading part of the field. A distant gravitational-wave detector measuring the scalar waves from the system lies in the far-away zone. With this simplification, the near-zone contribution to the waveform is given by

(31)

Here is the three-dimensional hypersurface representing the intersection of the past null cone and the near-zone world tube. The actual integration takes place over , the intersection of the near-zone world tube with a hypersurface of constant retarded time . The direction from the source to the detector is . Finally, we define the scalar multipole moments as

(32)

In paper II, these were also known as ; here, we will only use the notation. The role of these moments in finding the near-zone contribution to the scalar waveform is analogous to the role of the Epstein-Wagoner moments [paper II, Eqs. (2.22)] in finding the near-zone contribution to the tensor waveform. The Epstein-Wagoner construction is a bit more complicated due to a convenient rearrangement using the conservation law (14). That approach requires the calculation of “surface moments” for the two- and three-index case. No such complications exist in the computation of the scalar moments.

In the tensor case, the lowest order moment is the two-index, or quadrupole, moment. Its contribution to the tensor waves is given by

(33)

where

(34)

(The second term is the surface moment, defined in paper II.) In the scalar case, the lowest order moment is the zero-index, or monopole, moment. Note that the sources that enter these two moments are of the same post-Newtonian order, . But because of the two time derivatives in (33), the lowest order tensor field contains a factor of that the lowest order scalar field does not. This means that the lowest order tensor field is one post-Newtonian order higher than the lowest order scalar field. However, in paper II, we defined the lowest order tensor field to be “0PN” order. While it would be a simple matter to redefine the post-Newtonian scale to start with the lowest order of all the waves, we instead keep the previous definition. In this way, our usage of “0PN” matches the usual usage in general relativity (as well as all other studies of scalar-tensor theory).

As a consequence of this choice, the lowest order piece of the scalar monopole moment generates a PN scalar field in the far-away zone. (As it turns out, this piece is time independent and therefore uninteresting; however, higher order pieces of the monopole moment generate 0PN and higher order scalar waves.) The dipole moment generates PN and higher order scalar waves. The quadrupole moment generates 0PN and higher order waves, and the pattern continues to higher-index moments. Therefore, in order to generate a final waveform at th post-Newtonian order, the -index moment, if it contributes to the PN term at all, must be calculated to th post-Newtonian order beyond its own leading-order term.

Because of this complication, we compute the scalar waveform only to 1.5PN order. Computing it to 2PN order, as we did for the tensor waveform in paper II, would require first constructing the monopole moment to 3PN order beyond its leading-order term. The source has several times as many terms at 3PN order than at 2.5PN order. In addition, many of these terms become quite complicated to integrate. For instance, many include “triangle potentials” and “quadrangle potentials” Pati and Will (2002), which are difficult to integrate even when multiplied by a compact source (usually the easiest case, as we shall see below). Higher order pieces of the scalar waveform will be considered in future work.

iii.2 Source

To calculate the scalar multipole moments, we first need an expression for the source to 2.5PN relative order, or . Here we have introduced a post-Newtonian counting parameter , where is the total mass of the binary, is the orbital separation, and is the magnitude of the relative velocity. To simplify some of the discussion, we divide into three pieces, one compact and two noncompact,

(35)

corresponding to the first, second, and third terms in (16). We label the noncompact terms “F” for “field.” Writing out each term, we find

(36a)
(36b)
(36c)

We have defined the quantity

(37)

where . The definition ensures that for a perfect fluid with no internal gravitational binding energy (i.e., zero sensitivities), the metric component as in general relativity. We do not set equal to 1, since it depends on the asymptotic value of the scalar field , which could potentially vary in time over the history of the Universe. The other parameters in (36a)–(36c) are

(38a)
(38b)
(38c)

The expression (36c) for is exact. Equations (36a)–(36b) have been kept to the PN order necessary for the near-zone integral. These general expressions will also be useful when we compute the radiation-zone integral in Sec. IV.

We wish to convert (36a)–(36c) to a more explicit form valid in the near zone. This requires finding the fields , , , and in the near zone. This calculation was the focus of much of paper I. Finding the near-zone fields typically requires integrations over both the near and radiation zones. However, the radiation-zone integrals do not contribute at the order to which we work in this series of papers. The near-zone integrals are given by

(39a)
(39b)

Here is the intersection of the near-zone world tube with the hypersurface (i.e., not the retarded time). Note that in the near zone, the slow-motion approximation means that each time derivative corresponds to an increase of one-half post-Newtonian order.

In the near zone, it turns out that , , , and . Equations (39a) and (39b) are solved in an iterative manner. At lowest order, and are determined purely in terms of the compact piece of the sources and . At the next order, these fields are plugged into the field pieces of the sources. The process continues until we have the fields (and thus the sources) to the desired post-Newtonian order. Paper I presents expressions for , , , and to high post-Newtonian order. All are expressed in terms of -density potentials.

We can find to the order we need in this paper by plugging these expressions into (36a)–(36c). We must also convert to and the -density potentials to the -density potentials using the relations in paper I. The final results are

(40a)
(40b)
(40c)

The highest order (2.5PN) pieces of these expressions contain “regular” and scalar multipole moments. Since we are constructing for the purpose of calculating the scalar multipole moments, the process is naturally iterative. The regular multipole moments are given by

(41)

In particular, we need the quadrupole moment . Because it only appears in the highest order terms, we only require its lowest order form,

(42)

Similarly, we only need the lowest order forms of the scalar moments and . However, the scalar monopole moment is time independent to lowest order, so we need it to relative 1PN order. All of the necessary expressions are written out below. Note that begins at . Relative to this, begins at 1PN order, or , and begins at 2PN order.

iii.3 Zero-index moment

We begin with the zero-index moment,

(43)

To evaluate it to the necessary order, we require all the way to . The lowest order piece of the moment will generate a PN scalar field, while the highest order piece we calculate will generate 1.5PN scalar waves.

Since [Eq. (26)] contains delta functions, the compact moment can be written down by inspection,