Gravitational-Wave Memory Effects in Brans-Dicke Theory: Waveforms And

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Virgo’s observations. The results of these tests of GR Memory effects associated with changes in the super- are summarized in a number of papers, for example [4– Lorentz charges and their fluxes are called the spin and 10], which have all found the observed GWs to be con- center-of-mass (CM) GW memory effects [34, 35, 37].1 sistent with the predictions of GR, within the statistical These memory effects are related to time integrals of (and systematic) errors of the measurements. There re- the GW strain [34, 35, 37], and they could be measured main GW effects that are still too weak to be extracted by nearby freely falling observers with an initial rela- from the detector’s noise, but which hold promise for tive velocity [39]. The (generalized) BMS flux balance revealing some of the more subtle nonlinear and dynam- laws provide a useful way to approximately compute the ical effects in GR. One such class of strong-field effects GW memory effects starting from gravitational wave- (which will be the subject of this paper) go by the name of forms without the GW memory effect, and this technique gravitational-wave memory effects, because these effects has been applied for the displacement, spin, and CM GW are characterized by lasting changes in the GW strain memory effects (see [35, 40–45]). and its time integrals that develop over the full history The extent to which the relationship between symme- of the evolution of the system [11–13]. Although these ef- tries, conserved quantities, and memory effects might fects have not yet been observed, they could be detected hold in modified theories of gravity is not immediately in a population of binary-black-hole mergers measured apparent. First, modified theories of gravity often ad- by the Advanced LIGO and Virgo detectors over an ob- mit additional polarizations of GWs [46–48], and it is servation period of several years [14–16]; they could also natural to suppose there would be memory effects asso- be detected from an individual event by the space-based ciated with the additional polarizations (and there are detector LISA [17, 18], third-generation ground-based de- such effects [49–52]). Second, the asymptotic fall-off con- tectors like Einstein Telescope or Cosmic Explorer [19], ditions on the fields in the modified gravity theory (i.e., or pulsar timing arrays [20]. what one defines as an asymptotically flat solution) could In GR, gravitational-wave memory effects are closely conceivably differ from those in GR. Different boundary related to the symmetry group of asymptotically flat conditions could lead to different symmetry algebras and spacetimes, the Bondi-Metzner-Sachs (BMS) group [21– conserved quantities; this could in turn change both the 23] and its generalizations. We remind the reader that types of possible memory effects and their relations to the BMS group is a semidirect product of an infinite- fluxes of conserved quantities. It was with these consid- dimensional group of supertranslations (which are a erations in mind that we recently investigated asymptotic kind of “angle-dependent” translations around an iso- symmetries, conserved quantities, and memory effects in lated source) and the Lorentz group. The charges con- one specific modified theory of gravity: Brans-Dicke the- jugate to these symmetries are the relativistic angular ory [53]. momentum for the Lorentz symmetries [and angular mo- Brans-Dicke (BD) theory, is a so-called scalar-tensor mentum can be split into an intrinsic and a center-of- theory in which a massless scalar field is coupled nonmini- mass (CM) part, which correspond to the rotations and mally to gravity (namely, there is a product of the scalar Lorentz boots symmetries, respectively] and the super- field and Ricci scalar in the theory’s action). Scalar- momentum for the supertranslations symmetries (see, tensor theories appear in the context of string theory and e.g., [24–27]). There are more recent, and larger, symme- in phenomenological models used to explain the late-time try algebras studied in which the Lorentz symmetries are acceleration of the Universe [54–56] as well as cosmic in- extended to include all the conformal Killing vectors of flation [57, 58]. The massless scalar field leads to an the 2-sphere [28–30] (not just the globally defined ones) additional “breathing” polarization of GWs in BD the- or all smooth diffeomorphisms of the 2-sphere [31, 32]. In ory; for a freely falling ring of particles, this breathing the first case the symmetries are called super-rotations, mode produces a relative contraction and expansion in and in the second, they are called super-Lorentz trans- the plane transverse to the direction of GW propagation. formations; with the appropriate notion of supertransla- Recently, we (in [50]) and Hou and Zhu (in [49]) inde- tions, they form the extended and generalized BMS alge- pendently studied BD theory in the Bondi-Sachs frame- bras, respectively. The conserved charges corresponding work. In addition to arriving at similar boundary condi- to superrotations (and also the super Lorentz transfor- tions for asymptotically flat solutions in BD theory, we mations) were called superspin and super CM when de- both showed that the symmetry group of asymptotically composed into the two parts of opposite parities [33–35]. flat spacetimes in such a theory is the same as the (extended or generalized) BMS group. In [50], we observed As gravitational waves are radiated from a spacetime, fluxes and hence changes in the charges of the (generalized) BMS algebra generate GW memory effects. The 1 displacement memory effect is produced by changes in Note that the spin and CM GW memory effects are not related to the supermomentum charges, and the preferred shear- a super-Lorentz transformation between certain canonical refer- ence frames before and after a burst of GWs. There is a memory free frames before and after the burst of GWs are related effect dubbed a refraction (or velocity-kick) memory that does by a supertranslation (see, e.g., [33, 36]); the memory can correspond to a super-Lorentz transition between early time and be measured by nearby freely falling and comoving ob- late times [38], but such solutions do not preserve the asymptot- servers, who experience a lasting relative displacement. ically flat boundary conditions of Bondi and Sachs [21, 22]. 3 that in addition to the GW memory effects present in post-Newtonian (PN) than the tensor GW fluxes do (see, GR, there are two more memory effects in BD theory re- e.g., [67] for a review of the post-Newtonian, as well as the lated to the breathing-mode polarization. We will adopt multipolar post-Minkowskian, expansion). For a fixed the nomenclature used in [51], which calls the GW mem- value of the small (dimensionless) inverse coupling pa- ory effects in the tensor polarizations by the name “ten- rameter in BD theory, there is thus a smallest PN param- sor” memory effects and those associated with the breath- eter at which our approximation of the smaller scalar- ing polarization by “scalar” memory effects.2 We [50], as field fluxes holds. To compute GW memory effects in well as Hou and Zhu [61], derived the conserved charges BD theory at Newtonian order, we will need to include associated with the BMS symmetries using the Wald- higher-PN-order terms (in the frequency evolution and Zoupas prescription, and we determined that the charges Kepler’s law, for example) than we would need to go to included contributions coming from the scalar field. Newtonian order in the calculation in GR. In addition, Because [50] (and [49, 61]) provided the necessary we will also truncate our results at a finite, but smallest framework in which to compute and interpret the GW PN parameter, which is the smallest value for which our memory effects in BD theory, we now turn to applying approximation holds (unlike in GR, in which we can take the results of [50] to construct the GW memory wave- the PN parameter to zero).3 form from compact binary systems (as one application We computed our memory effects using the oscillatory of this formalism). We will focus on the memory effects waveforms computed in, e.g., [68–70] after verifying that that appear in the tensor polarizations of the GWs, be- we can relate the waveforms computed in harmonic coor- cause we can use the BMS flux balance laws to construct dinates in these references to our Bondi-Sachs quantities. nonlinear memory effects based on linearized (or nonlin- The memory effects that we compute in BD theory, have ear) waveforms that do not include the memory effects. small terms (proportional to the small BD parameter) The procedure in Brans-Dicke theory is closely analogous that appear at a PN order less than the leading Newto- to that used in GR [35, 40–45]. The two new memory nian order. We can relate part of our results to a part of effects in the scalar polarizations of the GWs are related the waveform computed by Lang in [68, 69] using the di- to shifts in the scalar field and its time integral. It was rect integration of the relaxed Einstein equations for the recently shown in [62] that the scalar memory effects are scalar and tensor waveforms up to 1.5PN and 2PN orders, closely related to the large gauge symmetries of 2-form respectively. Lang found no scalar GW memory effects, theory that was shown in [63] to be dual to the scalar but he computed a (hereditary) tensor GW memory ef- field theory. The symplectic flux of the scalar field is lin- fect formally at 1.5PN order that arises from the flux ear in the field and in the large gauge transformation; of energy radiated in the scalar waves. Upon integrating as a result, we cannot use the flux balance laws to con- this 1.5PN term for compact-binary source in our approx- struct a nonlinear memory of the scalar waves as one can imation, this term leads to a memory effect that depends for the tensor waves via the BMS flux-balance laws (one logarithmically on the PN parameter (this is analogous to must instead solve the scalar field equation directly to how a formally 2.5PN order term in GR, when integrated determine the scalar memory effect). Since our focus is for compact binaries, leads to a Newtonian-order effect on the application of the BMS balance laws in BD the- in the waveform [71, 72]). If we compare our result in the ory to determine the GW memory effects, we will focus Bondi-Sachs framework with Lang’s harmonic-coordinate here on computing the tensor memory effects in BD the- expression, the two terms agree. The BMS flux balance ory, which differ from those of GR due to the emission of laws are not as helpful for verifying the absence of scalar scalar radiation. memory effects at 1.5PN order. In BD theory, tensor GW memory effects are gener- Our BMS flux-balance approach allows us to com- ated by energy and angular momentum fluxes of both pute the Newtonian-order tensor waveforms—which have tensor and scalar radiation. Because solar-system exper- not been computed before, as far as we are aware—that iments [64] and pulsar observations [65, 66] have con- should appear if the work of Lang [68] were extended to strained the amount of scalar radiation in BD theory, we 2.5PN order. We find that because of the dipole emission, assume that the scalar radiation leads to energy and an- the Newtonian-order GW memory effects sourced by the gular momentum fluxes that are small compared to the tensor-GW energy flux has contributions from current leading quadrupole fluxes of tensor GWs in GR. Note, however, that the scalar field’s fluxes appear at a lower

3 Note, of course, that we could also compute the memory effects from a PN parameter of zero up to the small PN parameter at 2 The terminology “scalar” and “tensor” memory effects also has which the fluxes of scalar and tensor radiation have comparable been used after Du and Nishizawa [51] by Satishchandran and magnitudes, if we assume that the radiated fluxes are dominated Wald [59] in the context of general relativity to refer to different by the scalar emission. This, in fact, is the approximation used classes of “ordinary” memory effects (in the sense of [60]); since in [68, 69], for example. However, because memory effects are we work in Brans-Dicke theory, and we compute “null” memory most important when the fluxes are large, this early-time (or effects throughout this paper, we think the sense in which we use small-PN-parameter) regime is not expected to produce a sig- this naming should be clear (with this footnote as an attempt to nificant GW memory effect, and we do not compute it in this dispel any potential lingering ambiguities). paper. 4 quadrupole, mass octopole, and mass hexadecapole mo- expansion of the scalar waveform in Bondi coordinates ments. These higher multipole moments produce GW and harmonic coordinates. memory waveforms that have a different dependence on We require both coordinate systems and frameworks, the inclination angle than the tensor GW memory effect because the nonlinear and null GW memory effects are in GR at the equivalent PN order. The Newtonian GW straightforward to compute through the BMS balance memory effect generated by the scalar field’s energy flux laws in the Bondi approach, but it is more challenging also has a different dependence on inclination angle from to relate the Bondi-Sachs framework to a specific solu- that sourced by the tensor GWs. The inclination-angle tion of a Cauchy initial-value problem. In the harmonic- dependence of the GW memory effect has been shown to gauge PN approach, the scalar and tensor GW waveforms be something that can be tested with second- and third- already have been computed generally and for specific generation ground-based GW detectors [73]. compact-binary sources in, e.g., [68, 69]; however, the The rest of the paper is organized as follows. In GW memory effects are of a sufficiently high PN order Sec. II, we present a few elements of BD theory in har- in PN theory that they have not been fully computed in monic and Bondi coordinates. Section III lists the os- BD theory. After relating the harmonic-gauge waveform cillatory radiative mass and current multipole moments to the shear in the Bondi-Sachs framework, we can then for a quasi-circular, nonspinning compact-binary inspi- determine the GW memory waveforms using the balance ral (Sec. IIIA); it reviews the derivation of Kepler’s law, laws (and thereby avoiding high PN-order calculations). the evolution of the orbital frequency, and the phase of Throughout this paper, we treat Brans-Dicke theory GWs in BD theory at the necessary PN orders in our in the Jordan frame [53], in which the action takes the approximation (Sec. IIIB); and it presents scalar multi- form pole moments generated by an inspiraling quasi-circular, λ ω (∂ λ) (∂ λ) nonspinning compact binary (Sec. IIIC). In Sec. IV, we S = d4x√ g gµν µ ν . (2.1) − 16π R− 16π λ compute the nonlinear displacement and spin GW mem- Z ory waveforms in BD theory from the BMS fluxes. We Here gµν is the Jordan-frame metric, is the Ricci scalar conclude in Sec. V. We give additional results in two R of gµν , λ is a massless scalar field with a nonminimal cou- appendices in which we show the coordinate transfor- pling to gravity, and ω is a coupling constant called the mations that relate the Bondi coordinates to harmonic Brans-Dicke parameter. In this section and subsequent coordinates including relations between the metric func- ones, we set the gravitational constant at infinity to unity, tions in two coordinates systems (Appendix A), and we i.e., argue that the ordinary parts of the GW memory effects are subleading compared to null memory effects in BD 4+2ω 1 G0 = =1 , (2.2) theory (Appendix B). 3+2ω λ0 Throughout this paper, we use units in which c = 1, where λ is the constant value that λ approaches in the and we also set the asymptotic value of the gravita- 0 limit of infinite distances from an isolated source. tional constant in BD theory to 1. We use Greek indices (µ, ν, . . . ) to denote four-dimensional spacetime indices, and uppercase Latin indices (A, B, C, . . . ) for indices on A. Waveform in harmonic coordinates the 2-sphere, and lowercase Latin indices (i, j, k, . . . ) for the spatial indices in quasi-Cartesian harmonic coordi- We will denote our quasi-Cartesian harmonic-gauge co- nates. ordinates by Xµ, and we will use the notation X0 = t for the time coordinate and Xi (for i =1, 2, 3) for the spatial coordinates. We will denote the Euclidean distance from II. WAVEFORM IN HARMONIC AND BONDI the origin at fixed t by R = XiXj δ . The tensor GWs COORDINATES ij in Brans-Dicke theory are described by the transverse- tracelesss (TT) componentsp of the metric perturbation, In this section, we discuss briefly the Bondi-Sachs ˜TT hij , and the scalar GWs are encapsulated in the scalar framework [21, 22] and the harmonic-gauge waveform field λ. Both fields can be obtained from the metric at in post-Newtonian theory, both of which we will use order 1/R, in an expansion in inverse R, from the spatial to compute the GW memory waveform. Specifically, in components of the spacetime metric gij . The metric is Sec. II A, we discuss BD theory in harmonic coordinates more conveniently written in terms of the metric pertur- and decompose the GW strain into radiative multipole bation h˜ij and its trace h˜ rather than the TT part. For moments. In Sec. IIB, we present BD theory in the extracting the GWs, we need only the part of the space- Bondi-Sachs framework and again perform a multipole time metric that is linear in the fields λ and h˜ij at linear decomposition of the radiative data. The last part of order 1/R. We write the metric in this approximation as this section (Sec. IIC) relates the multipole moments of in [68], the shear tensor in Bondi coordinates to the radiative mass and current multipole moments of the harmonic- 1 λ g = δ + h˜ hδ˜ 1 δ . (2.3) gauge waveform; it then does the same for the multipole ij ij ij − 2 ij − λ − ij 0 5

The scalar GWs are present in the 1/R part of λ, which B. Waveform and metric in Bondi coordinates we expand as We use (u, r, xA) to denote Bondi coordinates. The A Ξ(˜u,y ) 2 coordinate u is the retarded time, r is an areal radius, λ = λ + + O R− . (2.4) 0 R and xA are coordinates on a 2-sphere cross sections of constant u and r (where A = 1, 2). We expand λ as a We have written the scalar field in terms of u˜ = t R, series in 1/r as the retarded time in harmonic coordinates, and the− an- A gles y (ι, ϕ). The angle ι is the polar angle and λ u, xA ≡ A 1 2 ϕ is the azimuthal angle of a spherical polar coordinate λ(u, r, x )= λ0 + + O(r− ) . (2.10) 4 r system. We expand the TT projection of h˜ij in terms of second-rank electric-parity and magnetic-parity ten- The metric in Bondi gauge satisfies the conditions grr = (e),lm (b),lm sor spherical harmonics (T and T , respectively; 0, grA = 0, and the determinant of the metric on the ij ij 4 see, e.g., [74]) as [67] 2-sphere cross sections scaled by r− is independent of r (and u). We imposed a set of aysmptotic boundary con- 1 ditions on the nonzero components of the metric in Bondi h˜TT = U (˜u)T (e),lm + V (˜u)T (b),lm . (2.5) ij R lm ij lm ij gauge in [50] and postulated a Taylor series expansion of Xl,m h i the scalar field and metric on the 2-sphere cross sections in 1/r. This allowed us to solve the field equations of The sum runs over integer values of l and m with l 2 ≥ Brans-Dicke theory to obtain the following solution for and l m l. The coefficients Ulm and Vlm are two the line element [50]: sets of− radiative≤ ≤ multipole moments which are called the mass and current moments, respectively. Because h˜TT is ˙ ij 2 λ1 1 λ1 3λ1 ˙ 2 real, the mass and current moments satisfy the following ds = 1+ 2 + + 2 λ1 du − λ0 − r M λ0 2λ properties under complex conjugation: " 0 λ1 2 1 A B ¯ m ¯ m 2 1 dudr + r qAB + cAB dx dx Ulm = ( 1) Ul, m, Vlm = ( 1) Vl, m . (2.6) − − λ0r r − − − − A ð λ1 1 1 We use an overline to denote the complex conjugate. + ð cAF + 4L + c ð cBC F − λ r − A 3 AB C We will also use the complex gravitational waveform h 0 1 1 which is composed of the plus and cross polarizations as 2λ ðBc + c ðB λ ð λ2 dudxA − 3λ 1 AB AB 1 − λ A 1 follows: 0 0 + .... (2.11) h = h+ ih . (2.7) − × The ellipsis at the end of the equation indicates higher We use the conventions for the polarization tensors e+ order terms in powers of 1/r that we are neglecting (the ij terms are of order 1/r2 except for the term proportional and e× given in [72] or [75] to construct the polariza- ij to dxAdxB , which is of order unity, because of the r2 ˜ij + ˜ij tions h+ = h eij and h = h eij×. We expand h as TT × TT term multiplying the expression). In Eq. (2.11), we have in terms of spin-weighted spherical harmonics sYlm with introduced and LA which are (related to) functions spin weight s = 2: M − of integration in Brans-Dicke theory that are the ana- logues of the Bondi mass aspect and angular momentum h = hlm(˜u)( 2Y lm) . (2.8) aspect in GR [50]. The two-dimensional metric qAB is − A l,m the unit-sphere metric and ð is the covariant derivative X compatible with qAB . We will raise and lower 2-sphere AB For a nonspinning planar binary, the modes hlm are re- indices (such as A and B) with the metrics q and qAB, lated to the mass and current multipole moments by (see, respectively. The overhead dot means a partial deriva- e.g., [67]) tive with respect to the retarded time u. The symmetric trace-free tensor cAB is called the shear tensor, and is 1 U (l + m is even) , related to the GW strain. The time-derivative of cAB is √2R lm hlm = (2.9) a symmetric trace-free tensor known as the news tensor: i V (l + m is odd) . (− √2R lm NAB = ∂ucAB . (2.12)

It is not constrained by the asymptotic ﬁeld equations 4 For compact binary sources, ι is the inclination angle between the in Brans-Dicke theory, and it contains information about orbital angular momentum of the binary (assumed to be along the tensor GWs. In GR, if the news tensor vanishes it the Z axis) and ϕ is the azimuthal angle as measured from the means the corresponding region of spacetime contains no X axis. GWs [24]. 6

We will also expand cAB in spherical harmonics as The second-rank tensor spherical harmonics on the unit 2-sphere in spherical and Cartesian coordinates are (e),lm (b),lm cAB = c(e),lmTAB + c(b),lmTAB . (2.13) related by the following transformation: Xl,m (e),lm (e),lm i j TAB = Tij ∂An ∂Bn , (2.18a) The tensor spherical harmonics can be defined from the (b),lm (b),lm T = T ∂ ni∂ nj . (2.18b) scalar spherical harmonics AB ij A B Combining the expressions (2.5), (2.13), and (2.16a), we find that the multipole moments of the strain and shear (e),lm 1 2(l 2)! ð ð 2 TAB = − 2 A B qABÐ Ylm , (2.14a) are related by 2s (l + 2)! − c(e),lm = Ulm, c(b),lm = Vlm , (2.19) (b),lm 2(l 2)! ð ðC TAB = − ǫC(A B) Ylm , (2.14b) s (l + 2)! as was given in [35] (though there the relationship was derived through a different argument involving the Rie- or instead in terms of spin-weighted spherical harmonics mann tensor in linearized gravity). The relation (2.19) and a complex null dyad on the unit 2-sphere of mA and allows us to express the multipole moments of the shear its complex conjugate m¯ A (see, e.g., [74]): tensor in terms of multipole moments of the harmonic- gauge TT strain tensor, once the difference between the (e),lm 1 retarded times in harmonic and Bondi coordinates in T = ( 2YlmmAmB + 2Ylmm¯ Am¯ B) , (2.15a) AB √2 − Eq. (2.17a) is taken into account. (b),lm i T = ( 2YlmmAmB 2Ylmm¯ Am¯ B) . AB − √2 − − (2.15b) III. POST-NEWTONIAN RADIATIVE MULTIPOLE MOMENTS A The dyad is normalized such that m m¯ A =1. In this section, we compute expressions for the radiative multipole moments Ulm and Vlm, as well as the scalar C. Relation between Bondi- and harmonic-gauge multipole moments which we will define herein. We ob- quantities tain the moments for nonspinning, quasicircular binaries. We denote the total mass by M = m1 + m2 (where m1 and m2 are the individual masses), the symmetric We construct a coordinate transformation between 2 mass-ratio by η = m1m2/M , and orbital separation harmonic and Bondi gauges in Appendix A that brings by a. We also introduce the parameters ξ = 1/(2 + ω) the harmonic metric at order 1/R to a Bondi-gauge met- and x = (πMf)2/3, where f is the GW frequency. We ric at an equivalent order. The procedure used is similar will work in the approximation in which ξ x, which to that recently outlined in [76], but it is adapted to corresponds to assuming that the BD modifications≪ to Brans-Dicke theory (rather than GR) and it is accurate the waveforms and the dynamics are small corrections only to the first nontrivial order in 1/R. This coordinate to the corresponding quantities in GR. Given that the transformation leads to a simple relationships between Shapiro-delay measurement in the solar system bounds ˜TT cAB and hij , first, and λ1 and Ξ, second: the BD parameter to be ω > 4 104 [64] (a similar bound has been derived from the× pulsar triple system A ˜TT A i j cAB(u, x )= R hij (t R,y )∂An ∂Bn , (2.16a) PSR J0337+1715 [66]), this implies that our approxima- − 5 A A tion is valid when x 2.5 10− . In this paper, we λ1(u, x ) = Ξ(t R,y ) . (2.16b) − will compute GW memory≫ waveforms× through Newto- The spatial vector ni is the unit vector pointing radially nian order, and we keep BD terms that are linear in ξ. outward at fixed t in harmonic coordinates (i.e., in the We then retain only the terms in the radiative multipole direction of propagation for outgoing GWs). The full moments at the appropriate powers of x and ξ to obtain transformation between the spherical polar coordinates Newtonian-order-accurate memory waveforms. Because A we always work to linear order in ξ, we do not include R and y = (ι, ϕ) constructed from the quasi-Cartesian 2 harmonic coordinates and Bondi coordinates is given in error terms of O(ξ ) in our expressions (they should be App. A; we list here the relevant leading-order parts of considered to be implied). We do, however, include such the transformation needed to relate u to t R, r to R, error terms in the PN parameter x, because the power and xA to yA in Eq. (2.16): − of x that constitutes a “Newtonian-order quantity” is not the same for the various quantities that we consider in the next two sections. 2M 1 u = t R log(R)+ O(R− ) , (2.17a) We will need radiative mass and current multipole mo- − − λ0 0 A A 2 ments at a higher PN order than those required for com- r = R + O(R ) , x = y + O(R− ) . (2.17b) puting 0PN memory effects in GR. Specifically, in GR, 7 one only requires the mass quadrupole moment at New- expression for U22: tonian order—i.e., O(x)—to obtain the Newtonian-order memory effects. In BD theory, we will need several addi- 2π iφ U = 8 ηMxe− tional terms. First, we must compute the BD correction 22 − 5 linear in ξ to the mass quadrupole moment. Second, r ξ 2 55η 107 we will need the 1PN or O(x2) GR terms of the mass 1 G + x + O(x5/2) . × − 2 − 3 42 − 42 quadrupole moment, because they multiply 1PN terms present in the GW phase and in the frequency− evolu- (3.5a) tion to contribute to Newtonian-order memory effects. Third, we require the current quadrupole and mass oc- The term proportional to x in the square bracket is the tupole moments which begin at 0.5PN order or O(x3/2). 1PN GR term taken from [67]. Fourth, we must include the current octopole and mass As we will show in Section IV, to compute the GW hexadecapole moments at 1PN or O(x2). We do not memory waveform at Newtonian order, we need the ra- need to compute the linear in ξ BD corrections to the diative current quadrupole moment and several radiative O(x3/2) and O(x2) radiative multipole moments, how- mass octopole and hexadecapole moments. To work to ever, because the Newtonian-order GW memory effects linear order in ξ, we can use the GR amplitudes of the produced by them would be quadratic in ξ. For scalar moments (though we use the phase with the BD correc- multipole moments, we require the dipole with relative tions). This allows us to take the amplitudes from the 1PN corrections [up to O(x3/2)], the quadrupole at O(x) expressions given, e.g., in the review [67]: and the octopole, which begins at O(x3/2). The scalar field moments are all linear in ξ. 8 2π 3/2 iφ/2 5/2 V21 = ηδmx e− + O(x ) , (3.5b) 3r 5 3π 3/2 3iφ/2 5/2 A. Radiative mass and current multipole moments U33 =6i ηδmx e− + O(x ) , (3.5c) r 7 8 π 2 iφ 5/2 We first give the expression of lowest-order radiative V = i Mη(1 3η)x e− + O(x ) , (3.5d) 32 3 14 − mass multipole moment U22. We decompose U22 into r two parts 2i π 3/2 iφ/2 5/2 U = ηδmx e− + O(x ) , (3.5e) 31 − 3 35 0 1,GR r U22 = U22 + U22 , (3.1) 8 2 iφ 5/2 U42 = √2πMη(1 3η)x e− + O(x ) . (3.5f) 0 − 63 − where U22 is the Newtonian-order part of U22, which in- cludes the BD corrections to linear order in ξ, and U 1,GR 22 We use the notation δm = (m1 m2). We give an ex- consists of the 1PN terms in U22 in GR. We can obtain pression for the phase φ(x) in the− next subsection. ˜TT the moment U22 from the expression for hij in BD theory in Eqs. (7.1) and (7.2a) of Ref. [68] by contracting ˜TT ¯ij hij with T(e),22 and integrating over the 2-sphere. We find that to linear order in ξ, B. Kepler’s law, frequency evolution, and GW phase

0 2π iφ ξ 2 U = 8 ηMxe− 1 G , (3.2) 22 − 5 − 2 − 3 Before computing the phase, we ﬁrst give an expres- r sion for Kepler’s law, which we will need to compute the where scalar multipole moments and the frequency evolution as well as the phase. To obtain a Newtonian-order accurate

G = ξ(s1 + s2 2s1s2) . (3.3) GW memory waveform, we need to have an expression for − Kepler’s law at 1PN order. This higher order is needed, The variables s1 and s2 denote the sensitivities (see, because when evaluating the integrals involved in com- e.g., [70]) of the binary components. The quantity G puting the GW memory effect, there are 1PN terms − is related to the modified gravitational constant in BD arising from dipole radiation in the energy flux, which theory by multiply 1PN terms in the GW frequency’s evolution and give rise to Newtonian-order terms in the waveform. =1 G . (3.4) The two-body equations of motion of nonspinning com- G − pact objects in Brans-Dicke theory has been computed The GW phase is denoted φ(x), and it differs from the in Ref. [77]. For circular orbits, the relative acceleration phase of the l =2, m = 2 modes of the waveform in GR; is proportional to the orbital frequency squared, Ω2, and we give the expression for± the phase in Eq. (3.18) below. the relative separation to 1PN order. Working to linear We use the 1PN GR terms from the review article [67]. order in ξ, the results of Eqs. (1.4) and (1.5a) of [77] show Putting these two results together, we have the following that Kepler’s law in BD theory (in this approximation) 8 is The rate of change of energy radiated in GWs in BD theory through Newtonian order is given by a 1PN term M M M − Ω2 = 1 G (1 2G )(3 η) γ¯ . plus a Newtonian term. If we define a3 − − a − − − a G (3.6) S = s1 s2 (3.12) We have introduced the parameter −

1 and we make use of expressions (6.16) and (6.19) given γ¯ = − ξ (1 2s ) (1 2s ) , (3.7) −G − 1 − 2 in [77], then linearizing their expression in ξ, we have in the equation above and is defined in Eq. (3.4). G 32 5ξS2 7 5 Let us now compute the evolution of the GW fre- E˙ = η2x5 +1 G + γ¯ GW 5 48x − 3 12G quency. We again need a 1PN-order-accurate expression, which, in full generality, contains many terms. Because 5 ξS2(3+2η) + O(x11/2) . (3.13) we work to linear order in ξ, the only 1PN terms that we − 72 need to obtain a Newtonian-order expression for the GW memory waveforms are the GR terms in f˙ (i.e., the 1PN Imposing energy balance E˙ = E˙ (the change in 5 b GW terms without ξ). We can then write the expression for the binding energy is equal to the energy− radiated by the ˙ f = df/dt in the following form: GWs) and using the chain rule to write f˙ = (df/dEb)E˙ b, we can write the Newtonian-order frequency derivative ˙ ˙ ˙ f = f0 + f1,GR , (3.8) f˙0 as a function of the PN parameter x as where f˙ is the BD expression to Newtonian order and 2 0 ˙ 96η 11/2 5S ˙ f0 = x 1+ ξ + F , (3.14) linear order in ξ, and f1,GR is ξ =0 (or GR) limit of the 5πM 2 48x 1PN terms. We first compute f˙0 using results from [77], and then we add to it the terms f˙1,GR taken from [78]. To where we defined compute f˙0, we first use the binding energy of a binary F 5 5 5 in BD theory from Eq. (6.14) of [77], which is valid to = (s1 + s2)+ (51+7η)s1s2 − 12 − 6 144 1PN order: 5 (3+7η)(s2 + s2) . (3.15) 1 M 3 − 288 1 2 E = µv2 µG + µ(1 3η)v4 b 2 − a 8 − 2 Finally, including the GR frequency evolution at 1PN [78] 1 M 2 1 M ˙ + µG (3+2¯γ + η)v + µ(1 2G ) . to f0, we find 2 a 2 − a (3.9) 96ηx11/2 5S2 743 11 f˙ = 1+ ξ + F + η x 5πM 2 48x − 336 4 We used µ = ηM to denote the reduced mass. We will next express the binding energy in terms of the PN pa- + O(x7) . (3.16) rameter x = (πMf)2/3. To do this it is useful to have the expressions for M/a and v2 written in terms of x: We previously introduced a waveform phase variable φ(x), which we will now compute explicitly. For comput- M 1 1 1 1 ing the GW memory waveforms, we will again need an = x 1+ G + 1 G 1 η x + γx¯ a 3 − 3 − 3 3G expression for the GW phase through 1PN order; how- + O(x3) , (3.10a) ever, because we are working to linear order in ξ, we will only need the terms without ξ at 1PN order in the phase 1 1 1 ˙ v = √x 1 G (1 G ) 1 η x γx¯ (analogously to our calculation of f). The GW phase is − 3 − − − 3 − 3G typically obtained by integrating the GW frequency with + O(x5/2) . (3.10b) respect to time from some appropriate starting time. For calculations of the GW memory waveform, it is more use- We can then substitute Eq. (3.10) into Eq. (3.9) to obtain ful to write the phase as a function of x. By using the chain rule, we can then write the time integral of the fre- 1 2 1 4 2 quency in terms of an integral with respect to the PN E = µx 1 G 1 G (9 + η)x γx¯ b − 2 − 3 − 12 − 3 − 3G parameter x as follows: 3 + O(x ) . (3.11) x f df φ(x)=2π dx′ , (3.17) f˙ dx′ Zxi

5 The 1PN term that multiplies the 1PN term in this calculation where the frequency f and the derivatives f˙ and df/dx ˙ is linear− in ξ, which implies that the BD modification to f at 1PN are functions of x. We have also introduced an initial PN enters at higher order in ξ in the GW memory waveforms. parameter xi that should be greater than ξ, so that our 9 approximation of ξ x holds. From f˙ in Eq. (3.16), we velocity vector take the form find that the GW phase≪ is given by ni = (sin ι cos ϕ, sin ι sin ϕ, cos ι) , (3.22a) 2 1 25S i φ(x) φ = 1 ξ F + n12 = cos[φ(u)/2], sin[φ(u)/2], 0 , (3.22b) − c −16ηx5/2 − 336x { } vi = v sin[φ(u)/2], v cos[φ(u)/2], 0 . (3.22c) 5 1 2 743 11 1 {− } + x ξS + η + O(x− ) . 3 − 8 336 4 For the magnitude of the velocity, v, we need only the (3.18) GR limit of the expression (zeroth-order in ξ) at 1PN order in x in Eq. (3.10). We defined a constant φc, the phase at coalescence, which We wrote the phase as φ(u) as a shorthand for φ[x(u)] is chosen such that the phase at xi vanishes. The terms in Eq. (3.18), so as to emphasize the retarded-time de- in the second line of Eq. (3.18) come from the product of pendence of the phase. The multipole moments can be a 1PN term multiplying a 1PN term, which produces extracted through the integral a Newtonian-order− effect on the phase (specifically, it is the scalar dipole radiation in BD theory that gives rise to 2 λ = d Ω λ1Y¯lm(ι, ϕ) . (3.23) the 1PN-order effects). We discuss the scalar radiation 1(lm) Z in more− detail in the next subsection. Using Eqs. (3.19)–(3.22) and the GR limit of Eq. (3.10), we find that the harmonic components of λ1 can be writ- C. Scalar Multipole Moments ten as

2π iφ/2 λ = 2i λ ξηM√xe− We expand λ1 in terms of the scalar spherical harmon- 1(11) 0 − r 3 ics x δm S 12Γ + S (15 + 34η) + O(x2), lm × − 15 M λ1 = λ1(lm)Y , (3.19) (3.24a) Xl,m 2π iφ 2 and the corresponding coefficients in the expansion are λ1(22) = 2 λ0ξΓηMxe− + O(x ) , (3.24b) − 15 the scalar multipole moments λ1(l,m). Specifically, we r i π δm 3/2 iφ/2 compute expressions for the scalar moments λ1(11), λ1(22) λ = λ ξ Γ +2ηS ηMx e− 1(31) − 10 21 0 M and λ1(31) in terms of the PN parameter x, which we will r then use to derive the tensor GW memory effect sourced + O(x2) . (3.24c) by the fluxes of the scalar field. These three moments are those needed to compute the GW memory effects at Equations (3.5) and (3.24) are all the sets of radiative Newtonian order. The relevant part of the scalar field λ1 moments that we will need to compute the GW memory had been computed previously in [69, 70], and we give effects in the next section. the expression to 1PN order above the leading dipole radiation and to linear order in ξ: IV. MEMORY EFFECTS

M δm i λ1 = ηMλ0ξ 2S + 3Γ + 10Sη vin − 2a M In this section, we compute the displacement and M δm spin GW memory effects produced by a quasicircular +Γ (v ni)2 (ni n )2 (Γ +2ηS) i − a 12 i − M compact-binary inspiral. The displacement and spin memory effects are both constructed from the shear ten- 7 M (v ni)3 (v ni)(ni n )2 . (3.20) sor cAB, and they have sky patterns with opposite pari- × i − 2 a i 12 i ties. It is then useful to first decompose the shear tensor into electric- and magnetic-parity parts as follows: Above we introduced the quantities 1 ð ð 2 ð ðC Γ=1 2(m s + m s )/M , (3.21) cAB = (2 A B qABÐ )Θ + ǫC(A B) Ψ , (4.1) − 1 2 2 1 2 − i 2 A the unit vector n pointing radially outward in the direc- where Ð ð ðA is the Laplacian operator on the tion of the GW’s propagation, the unit separation vector unit 2-sphere.≡ We compute the displacement and spin i n12 between the binary’s components, and the relative GW memory effects using the BMS flux-charge balance i i i velocity vector v = v1 v2 of the binary’s masses. In laws that were computed in Brans-Dicke theory in [50]. terms of ι and ϕ (the polar− and the azimuthal angles, re- We focus on the nonlinear GW memory effects and the spectively, in the center-of-mass frame of the binary) and null memory associated with the stress-energy tensor of the GW phase φ, the two unit vectors and the relative the scalar waves. We can compute these effects using 10 low-PN-order oscillatory waveforms and the BMS bal- shown, e.g., in [79] (though using the conventions of [35]): ance laws, whereas if we were to try to compute them directly through the relaxed Einstein equations in har- ′ ′′ l+l +l (2l′ + 1)(2l′′ + 1) (s′,l′,m′; s′′,l′′,m′′) = ( 1) monic gauge, we would need to compute the gravitational Bl − 4π(2l + 1) waveform at a higher PN order in BD theory than has s been completed thus far. We also argue that the ordinary l′,s′; l′′,s′′ l,s′ + s′′ l′,m′; l′′,m′′ l,m′ + m′′ . ×h | ih | i parts of the GW memory effects will be of a higher PN (4.3) order than the nonlinear and null parts in Appendix B. The multipolar expansion of the nonlinear memory effects in terms of the radiative moments Ulm and Vlm has the same form as in GR, which is given in [35]. However, we will need to perform a new multipolar expansion of the null memory effects from the stress-energy tensor of the scalar field. For this expansion, we will need the vec- A. Spherical harmonics and angular integrals tor spherical harmonics

(e),lm 1 ð We will compute multipole moments of the GW mem- TA = AYlm , (4.4a) l(l + 1) ory eﬀects, starting from the oscillatory tensor and scalar waves expanded in terms of the multipole moments in (b),lm 1 B T = p ǫABð Ylm . (4.4b) Eqs. (3.5) and (3.24), respectively. Evaluating these mul- A l(l + 1) tipole moments involves computing angular integrals in- In terms of the spin-weightedp spherical harmonics and a volving products of three spherical harmonics of diﬀerent A types (scalar, vector, and tensor). We instead follow the complex dyad m on the unit 2-sphere, we can write the strategy in, e.g., [34, 35], in which the vector and tensor vector spherical harmonics as harmonics are recast in terms of spin-weighted spherical (e),lm 1 harmonics. The angular integrals then involve products TA = ( 1YlmmA 1Ylmm¯ A) , (4.5a) √2 − − of three spin-weighted spherical harmonics (we use the (b),lm i conventions for the spherical harmonics in [35]). We also TA = ( 1YlmmA + 1Ylmm¯ A) , (4.5b) use the notation for the integral of three spin-weighted √2 − spherical harmonics in [34] A A A where m mA =m ¯ m¯ A =0 and m m¯ A =1.

l(s′,l′,m′; s′′,l′′,m′′) B. Displacement memory eﬀects B ≡ 2 d Ω ( ′ Y ′ ′ )( ′′ Y ′′ ′′ )( ′ ′′ Y¯ ′ ′′ ) , (4.2) s l m s l m s +s l(m +m ) Supermomentum conservation requires that the change Z in the “potential” Θ that is associated with the electric part of the shear tensor, ∆Θ, must have its change be- which can be written in terms of Clebsch-Gordan co- tween two retarded times and satisfy the following rela- eﬃcients (denoted by l′,m′; l′′,m′′ l,m′ + m′′ ) as was tionship [50]: h | i

6+4ω 1 d2Ω α Ð2(Ð2 + 2)∆Θ = du d2Ω α N N AB + (∂ λ )2 +8 d2Ω α ∆ Ð2∆λ . (4.6) AB (λ )2 u 1 M− 4λ 1 Z Z 0 Z 0

The supermomentum is the charge conjugate to a BMS we will derive separately the contributions from the en- supertranslation symmetry and α(xA) is the function ergy flux of tensor and scalar waves, respectively. We that parametrizes the supertranslation symmetry. The denote the nonlinear (tensor) part by ∆ΘT and the null first two terms inside the square brackets on the right- part from the scalar field by ∆ΘS. The full memory effect hand side of Eq. (4.6) produce the null memory (i.e., the is then the sum of the two components: memory sourced by massless fields) with the first term being the nonlinear (Christodoulou) memory. Both ∆ ∆Θ = ∆ΘT + ∆ΘS . (4.7) 2 M and Ð ∆λ1 generate ordinary memory [50], but we argue in App. B that the ordinary memory is a higher-PN-order While ∆Θ is the quantity most straightforwardly effect. We will then focus on just the null memory, and constrained by supermomentum conservation, it is the change in the strain ∆h that is perhaps the more typical 11 gravitational-wave observable. It is thus useful to relate sufficiently large that the PN approximation (at the or- the potential ∆Θ to the strain ∆h. To do this, we will der at which we work) starts to require higher-PN-order first introduce the following notation for just the electric terms to remain accurate. part of the change in the shear: The multipolar expansion of ∆ΘT proceeds exactly as in GR (and we note just a few features of the calculation 1 2 here; see [35] for further details). We can first replace ∆cAB,(e) = (2ðAðB qAB Ð )∆Θ . (4.8) 2 − the function α(xC ) with the complex conjugate of a scalar ¯ We expand ∆Θ in scalar spherical harmonics spherical harmonic, Ylm. We then use Eqs. (2.12), (2.13), (2.15), and (4.9), and the expression for the moments T lm ∆Θ in terms of the radiative moments U˙ lm and V˙lm is ∆Θ = ∆ΘlmY (ι, ϕ) , (4.9) lm the same as that derived in GR in [35]: Xl,m T 1 (l 2)! where l 2 (and l m l); the l 1 harmonics ∆Θlm = − l ( 2,l′,m′;2,l′′,m′′) ≥ − ≤ ≤ ≤ 2 2 (l + 2)! ′ ′′ ′ ′′ B − are in the kernel of the operator 2ð ð q Ð . By l ,l ,m ,m A B AB X − uf substituting (4.9) into Eq. (4.8) and using the definition l,(+) (e),lm du s ′ ′′ U˙ ′ ′ U˙ ′′ ′′ + V˙ ′ ′ V˙ ′′ ′′ of T in Eq. (2.14a), we can relate ∆Θ to ∆c l ;l l m l m l m l m AB lm (e),lm × u Z i h via Eq. (2.13), l,( ) ˙ ′ ′ ˙ ′′ ′′ +2isl′;l−′′ Ul m Vl m . (4.14) (l + 2)! i ∆c = ∆Θ . (4.10) We however, introduced the coefficients (e),lm 2(l 2)! lm s ′ ′′ − l,( ) l+l +l s ′ ±′′ =1 ( 1) (4.15) The above equation will be necessary when we construct l ;l ± − the waveform from ∆Θlm. Specifically, we can compute that were used in [34] to make the notation more com- the waveform by combining Eqs. (2.9), (2.19), and (4.10) pact. As in [35], the sum runs over l′,l′′ 2 and l ≥ in Eq. (2.8) to obtain must be in the range l′ l′′ l l′ + l′′ so that | − | ≤ ≤ | | the coefficients ( 2,l′,m′;2,l′′,m′′) given in Eq. (4.3) Bl − (disp) 1 (l + 2)! are nonzero. The azimuthal indices must be related by ∆h = ∆Θlm 2Y lm . (4.11) √ 2(l 2)! − m = m′ + m′′ for the coefficients l( 2,l′,m′;2,l′′,m′′) 2R l,m s B − X − to be nonzero. Because we focus on the leading GW (disp) memory effects in the nonoscillatory (m =0) part of the We will denote the memory waveform ∆h as a waveform, this will further restrict m and m to have sum of the tensor-sourced, ∆h(disp,T), and scalar-sourced ′ ′′ (disp,S) equal magnitudes and opposite signs: m′ = m′′. While ∆h contributions as follows: T − the abstract expression for ∆Θlm in terms of radiative multipole moments has exactly the same form as that ∆h(disp) = ∆h(disp,T) + ∆h(disp,S) . (4.12) in GR, the time-derivatives of the radiative multipole ˙ ′ ′ ˙ ′ ′ We first compute ∆h(disp,T) followed by ∆h(disp,S). moments Ul m and Vl m in BD theory differ from the corresponding moments in GR. This leads to a number of order ξ terms in the expression for the GW memory 1. Displacement memory effect from the energy flux of effect that we will compute below. tensor GWs Next, we will summarize how we compute the memory waveforms, including which radiative multipoles we need The expression for the “moments” of ∆ΘT with respect and at what PN-order accuracy we require these mul- to α(xC ) have the same general form as in GR, tipole moments. For concreteness, let us first focus on products of the mass moments U˙ l′m′ U˙ l′′m′′ in Eq. (4.14); the arguments will apply to products of current moments 2 C 2 2 T d Ω α(x )Ð (Ð + 2)∆Θ and to products of mass and current moments. We per-

Z uf form the integral over u by using the chain rule to recast 2 C AB = du d Ω α(x )NABN , (4.13) the integral over u in terms of an integral over x ui Z Z d d duU˙ ′ ′ U˙ ′′ ′′ = U ′ ′ U ′′ ′′ xdx˙ , (4.16) but there is a subtlety related to the limits of integration l m l m dx l m dx l m (ui and uf ) in the retarded-time integral over u. Be- Z Z cause we work in an approximation in which ξ x, the as was outlined in, e.g., [72]. In GR, the Newtonian-order ≪ lower limit ui must start at a PN parameter xi for which memory waveform can be calculated from just Ul′m′ = xi ξ. This differs from the corresponding convention U22 and Ul′′m′′ = U2, 2 (and similarly Ul′m′ = U2, 2 and ≫ − − in GR, in the limit ui is often taken (in which it is Ul′′m′′ = U22), with U22 evaluated at Newtonian order, as assumed that x 0).→ The −∞ upper limit, u is a retarded well. In BD theory, however, both x˙ and dφ/dx (the lat- i → f time at which the corresponding PN parameter xf , is ter term coming from dUlm/dx) have contributions from 12 the dipole moments of the scalar field, and these effects of the other radiative multipole moments in Eq. (3.5) to enter at 1PN order relative to the GR result (and they obtain a Newtonian-order-accurate result; in GR, these are proportional− to ξ). To obtain the full result at the moments all give rise to higher-PN corrections to the GW Newtonian order requires the 1PN contributions to U22, memory effect. x˙ and dφ/dx. Because the 1PN terms of x˙ and dφ/dx Considering the radiative multipoles described above, − T are proportional to ξ, we only need the parts of the 1PN we can then compute the multipole moments ∆Θl0 of terms in U22, x˙ and dφ/dx that are independent of ξ to the GW memory effect from Eq. (4.14). Because the T compute ∆Θlm. Hence, the BD corrections to the 1PN memory is electric-type, only even l moments are nonva- terms in Eqs. (3.5a), (3.16) and (3.18) were not given. nishing for nonspinning compact binaries. When written The 1PN term in x˙ also requires that we compute the in terms of the relevant radiative moments Ulm and Vlm, − T part of the GW memory waveform sourced by products the moments ∆Θl0 are given by

uf T 1 5 ˙ 2 ˙ 2 ¯˙ ˙ ¯˙ ˙ ¯˙ ˙ ∆Θ20 = du 2 U22 V21 + √7 √2U31V21 + √5U22V32 + √5 U42U22 , (4.17a) 168 π u | | − | | ℑ ℜ r Z i h i h i uf √ T 1 11 ˙ 2 44 ˙ 2 ˙ 2 ˙ 2 324 5 ¯˙ ˙ ∆Θ40 = du U22 V21 21 U33 7 U31 + U42U22 23760√π ui 7 | | − 7 | | − | | − | | 7 ℜ Z h i 22 + 2√5U¯˙ V˙ +5√2V¯˙ U˙ , (4.17b) √ ℑ 22 32 21 31 7 uf h i T 1 ˙ 2 ˙ 2 ¯˙ ˙ ∆Θ60 = du U33 + 15 U31 4√5 U42U22 . (4.17c) − 36960√13π u | | | | − ℜ Z i h i

Next, we use Eqs. (3.5a), (3.5b)–(3.5f), and (3.16)–(3.18) are equal to the equivalent results in GR. To simplify to perform the integral over u and to write the moments the notation, we do not include the PN error terms in in terms of x. With the identity (δm/M)2 = 1 4η, we Eqs. (4.18) or (4.20), which are both are O[∆(x3/2)], − can write the moments as where ∆(x3/2)= x3/2 x3/2. f − i 2√5π 4 5ξS2 x Finally, we will construct the displacement memory ∆ΘT = Mη∆x 1 G ln f 20 21 − 3 − 48∆x x waveform from the ∆Θl0 in Eq. (4.18). To do this, it i is helpful to have the expressions for the spin-weighted 1915 665η ξ 1+ F + S2 + , spherical harmonics − 96768 1152 (4.18a) 2 1 15 2 T √π 4 5ξS xf 2Y20(ι, ϕ)= sin ι , (4.19a) ∆Θ = Mη∆x 1 G ln − 4 2π 40 1890 − 3 − 48∆x x r i 3 10 2 2 2 737045 143395η 2Y40(ι, ϕ)= sin ι(7 cos ι 1) , (4.19b) ξ 1+ F + S + , − 16 π − − −709632 25344 r (4.18b) 1 1365 2 4 2 2Y60(ι, ϕ)= sin ι(33 cos ι 18 cos ι + 1) . − 64 π − π 5MηS2ξ∆x r ∆ΘT = ( 839 + 3612η) . (4.18c) (4.19c) 60 − 13 178827264 − r

We use the notation ∆x = xf xi, where xi and xf Substituting Eqs. (4.18) and (4.19) into Eq. (4.11), we correspond to the PN parameter−x evaluated at an early obtain the displacement memory waveform due to the retarded time ui and a final time uf during the inspi- energy flux from tensor GWs in BD theory. We find ral, respectively. The terms outside the curly braces the waveform only contains the plus polarization, and at T T in Eq. (4.18) in the expressions for ∆Θ20 and ∆Θ40 Newtonian order, it is given by

ηM∆x 4 5ξS2 x 81145 65465 ∆h(disp,T) = sin2 ι(17 + cos2 ι) 1 G ln f ξ(1 + F ) ξS2 η + 48R − 3 − 48∆x x − − 73728 − 18432 i 13

20975 1075 MηS2ξ∆x 783875 35575 +ξS2 η cos2 ι + sin2 ι η . (4.20) 172032 − 2048 R 2064384 − 24576

The expression in front of the square brackets on the We will now discuss which scalar multipole moments first line of Eq. (4.20) is the same as the Newtonian- contribute to the displacement memory waveform and order waveform for the memory effect in GR. The terms the accuracies at which we need the different moments to within the square bracket highlight a number of correc- obtain a Newtonian-order-accurate GW memory wave- tions introduced into the memory waveform amplitude in form, at linear order in ξ. The 1PN scalar multipole BD theory. These include effects related to the change moments λ1(lm) that are computed from Eq. (3.20) are in the amplitude of the l =2, m = 2 modes (the ξ and at least (ξ); thus, to linear order in ξ, we can treat G terms) and changes in the frequency± evolution (the F 6+4ω asO4/ξ. We will evaluate the integral over u in term). In particular, there is a change in the scaling of Eq. (4.22) by converting it to an integral over x (as was the memory with the PN parameter that is proportional done in Sec. IVB1), but unlike in Sec. IVB1 we need to ln(xf /xi), which arises because of scalar dipole radi- to keep only the GR contribution in x˙, which scales as ation. At the end of the first and on the second line of x5. Similarly, when computing dφ/dx, we again need to 7/2 Eq. (4.20) are a number of terms arising from 1PN-order retain just the GR contribution that goes as x− . The products of multipole moments coupling to the 1PN scalar field has a radiative dipole moment, which from − 1/2 term in the frequency evolution (or x˙); the terms on the Eq. (3.24a), goes as x− . The leading-order part of the ˙ ˙ second line lead to a small (order ξ) difference to the sky integrand (proportional to λ1(11)λ1(1, 1)) scales as 1/x − pattern of the memory effect between BD theory and GR. rather than O(x0) as in GR; in this sense, the integrand is a 1PN order.6 This product of dipole terms will also contribute− to the waveform at 0PN order because of the 2. Displacement memory effect from the energy flux of 3/2 O(x ) terms in λ1(11); see Eq. (3.24a). To work to lin- scalar radiation ear order in ξ, we do not need to go to a higher PN order for λ1(11). Similar arguments show that the remaining The displacement memory effect also has a contribu- scalar moments in Eq. (3.24) (namely, λ1(22) and λ1(31)) tion from the integral of the energy flux of the scalar are the ones that are needed to compute Newtonian-order radiation. Its effect can be computed from the terms accurate moments of ∆ΘS . 2 l0 proportional to (∂uλ1) in Eq. (4.6): We then first list the integrals of the relevant moments S λ1(lm) that contribute to ∆Θl0 at Newtonian order: d2Ω α(xC )Ð2(Ð2 + 2)∆ΘS uf S 1 2 2 Z u ∆Θ = du 7 λ˙ + 10 λ˙ 6+4ω f 20 2 1(11) 1(22) 2 C 2 − 42√5πλ0ξ ui | | | | = 2 du d Ω α(x )(∂uλ1) . Z (λ0) Zui Z ˙ ¯˙ (4.21) 6√14 λ1(11)λ1(31) , (4.23a) − ℜ u h i S 1 f Expanding λ1 and ∆Θ in scalar spherical harmonics as S ˙ 2 ∆Θ40 = 2 du λ1(22) in Eqs. (3.19) and (4.9), respectively, and choosing α = 630√πλ0ξ ui | | ¯ S Z Ylm, we can determine the multipole moments ∆Θlm in 2√14 λ˙ λ¯˙ . (4.23b) terms of the multipole moments λ1(lm) and the integrals − ℜ 1(11) 1(31) of three spherical harmonics defined in Eq. (4.2). The h i result is As we did with the the memory sourced by the tensor energy flux, we first substitute the expressions for the S (l 2)! 6+4ω ∆Θlm = − 2 Bl(0,l′,m′;0,l′′,m′′) scalar moments in Eq. (3.24) into Eq. (4.23) and evaluate (l + 2)! (λ0) ′ ′ ′′ ′′ l ,mX,l ,m the integrals in terms of x by using Eqs. (3.16)–(3.18). uf This gives the following results: ˙ ′ ′ ˙ ′′ ′′ du λ1(l m )λ1(l m ) , × ui Z S Mηξ√5π 2 xf (4.22) ∆Θ20 = S ln − 144 x i where l 2 and l′,l′′ 1 (and m, m′, and m′′ must ≥ ≥ satisfy m = m′ + m′′). Because λ1 is proportional to ξ, S 2 one might be concerned that ∆Θ will be an O(ξ ) effect 6 Note, however, when the integrand is integrated, it will again and be negligible in our approximation. Note, however, give rise to a logarithm in x rather than being proportional to x, S that 3+2ω =2/ξ 1, which implies that ∆Θ is an (ξ) as in GR. We will refer to this effect sometimes as a 1PN term, effect; thus, the integrand− can be one order higherO in ξ since it comes from such an effect in the integrand, and− since log and still produce an effect at linear order in ξ. terms do not enter into the PN order counting of a term. 14

8 23 δm 71 157 +∆x Γ2 SΓ + S2 η , BMS balance law for the super angular momentum was 7 − 14 M 336 − 84 given in [50], and analogously to the computation in GR (4.24a) (see, e.g., [33]), a term involving Eq. (4.26) was needed Mηξ√π∆x δm to ensure that the balance law was satisfied. The form ∆ΘS = 8Γ2 Γ +2S2η . (4.24b) of the balance law can be written as 40 30240 − M 64π d2Ω γÐ2(Ð2 + 2)∆Σ = ∆Q + ∆ , The PN error terms in Eqs. (4.24) or (4.25) are again − λ (γ) J(γ) O[∆(x3/2)], which we dropped, to simplify the expres- Z 0 (4.27a) sions. A AB where Y = ǫ ðB γ is a smooth, magnetic-parity vector We then substitute (4.24) into Eq. (4.11), and with field on the 2-sphere, and where we have defined the expressions for the spin-weighted spherical harmonics in Eq. (4.19), we arrive at the following equation for λ ∆ = 0 du d2Ω ǫADð γ ð (c N BC) the displacement memory waveform sourced by the scalar J(γ) 64π D A BC energy flux: Z +2N BCð c 4ð (c N BC ) A BC − B AC (disp,S) 5ηMξ 2 2 xf 4ω +6 ∆h+ = sin ι S ln + (∂uλ1ðAλ1 λ1ðA∂uλ1) , − 192R xi (λ )2 − 0 6 33 δm (4.27b) + ∆x Γ2 SΓ 5 − 20 M 2 71 113 + S η λ0 2 ADð 336 − 60 ∆Q(γ) = d Ω ǫ Dγ [ 3∆LA 8π − 1 δm Z 8Γ2 SΓ 2S2η cos2 ι . 1 ðB ðB −20 − M − ∆(cAB λ1 λ1 cAB) . (4.27c) − 4λ0 − (4.25) The net flux is denoted by ∆ , the change in the J(γ) There are terms in Eq. (7.2e) of [68] which, after perform- charges is denoted by ∆Q(γ), and the left-hand side of ing the integration over time in our approximation, pro- Eq. (4.27a) (which is related to the spin memory effect) duce a 1PN term in the waveform; this term agrees with − is the additional term required for the balance law to the first line of Eq. (4.25). The Newtonian-order terms be satisfied. Analogously to the displacement memory, require going to a higher PN order than was computed the contribution of ∆Q(γ) to Eq. (4.27a) is referred to as in [68], but the BMS balance laws allow us to determine ordinary spin memory, and ∆J(γ) is the null spin mem- these expressions. ory (which contains a nonlinear part). We will focus on disp Because the total GW memory ∆h is a sum of the the null contribution to Eq. (4.27a) here, as we argue in scalar and tensor contributions, as given in Eq. (4.12), App. B that the ordinary contribution to the spin mem- then the scalar-sourced contribution will produce an ad- ory is likely to be a higher PN effect than the null memory ditional correction to the amplitude and the sky pattern is. beyond the corrections given in Eq. (4.20) for the tensor- As we did with the change in ∆Θ related to the dis- sourced part of the memory effect. placement memory effect, we will split ∆Σ into a sum of its contributions from the angular momentum flux of tensor and scalar radiation. We denote these two contri- C. GW spin memory effect butions by

The GW spin memory effect is a lasting offset in the ∆Σ = ∆ΣT + ∆ΣS , (4.28) time integral of the magnetic part of the shear tensor. It can be constrained through the evolution equation for and we compute these two contributions separately be- the Bondi angular momentum aspect, or equivalently the low. magnetic part of the flux of the super Lorentz charges. In addition, while the spin-weight-zero quantity ∆Σ To compute the spin memory effect, it is helpful to denote is the most convenient quantity to compute from the the time integral of the potential Ψ, which gives rise to balance law (4.27a), the shear cAB or strain h are the the magnetic part of cAB in Eq. (4.1), as ∆Σ: more commonly used quantities in gravitational waveform modeling and data analysis. We thus relate ∆Σ to a time integral of the shear; specifically, we denote the ∆Σ = Ψdu . (4.26) change in the time integral of the magnetic-parity part Z of the shear tensor by We leave off the limits of integration for convenience, though we will later restore these limits when we com- ð ðC pute the spin memory in the PN limit. The generalized cAB,(b)du = ǫC(A B) ∆Σ . (4.29) Z 15

Expanding ∆Σ in terms of scalar spherical harmonics Equation (4.31) allows us to compute the time integral of the strain from ∆Σlm. lm ∆Σ = ∆ΣlmY (ι, ϕ) (4.30) Xl,m (with l 2 and l m l), then we can relate the ≥ − ≤ ≤ 1. Spin memory effect from the angular momentum flux of multipole moments ∆Σlm to the time integrals of the tensor GWs radiative current moments Vlm via Eqs. (4.29), (2.14b), and (2.13). The result is that The null part of the spin memory effect in Eqs. (4.27a) (l + 2)! and (4.27b), that is sourced by the tensor GWs can be V du = ∆Σ . (4.31) computed from the following expression: lm 2(l 2)! lm Z s −

uf d2Ω γÐ2(Ð2 + 2)∆ΣT = du d2Ω ǫADð γ ð (c N BC)+2N BCð c 4ð (c N BC) . (4.32) D A BC A BC − B AC Z Zui Z

It has the same form as the analogous expression in GR. concise: It can then be recast into the same form as in Eq. (3.23) l of [33] by using the identities in Appendix C of [33]. This c ′ ′ ′′ ′′ =3 (l′ 1) (l′ + 2) l ( 1,l′,m′;2,l′′,m′′) l m ;l m − B − expression was then the starting point for the multipolar +p (l′′ 2) (l′′ + 3) l ( 2,l′,m′;3,l′′,m′′) . expansion of the spin memory effect given in [35]. We − B − (4.33) reproduce the result from [35] below; however, we first p l,( ) introduce, in addition to sl′;l±′′ in Eq. (4.15), the following The expression for the moments ∆Σlm can then be de- coefficients (defined in [34]) to make the expression more rived through a lengthy calculation outlined in [35], and the result is given by

uf 1 (l 2)! l,( ) T l ′ ′ ˙ ′′ ′′ ˙ ′ ′ ′′ ′′ ′ ′ ˙ ′′ ′′ ˙ ′ ′ ′′ ′′ ∆Σlm = − cl′m′;l′′m′′ du isl′;l−′′ Ul m Ul m Ul m Ul m + Vl m Vl m Vl m Vl m 4 l(l + 1) (l + 2)! u − − l′,l′′, Z i mX′,m′′ h p l,(+) s ′ ′′ U ′ ′ V˙ ′′ ′′ + V˙ ′ ′ U ′′ ′′ U˙ ′ ′ V ′′ ′′ V ′ ′ U˙ ′′ ′′ . − l ;l l m l m l m l m − l m l m − l m l m (4.34)i

To compute the spin memory effect at Newtonian or- compute the spin memory effect following the same pro- der, we need precisely the same set of radiative multi- cedure in Sec. IVB1 by first writing the needed moments pole moments Ulm and Vlm that were used to compute of ∆Σlm in terms of integrals of the relevant radiative the displacement memory effect in Sec. IVB1. We then moments Ulm and Vlm:

1 uf ∆ΣT = du 9 U¯˙ U 2V¯˙ V +7 U¯˙ U U¯˙ U + 11√5 U¯˙ U + U¯˙ U 30 √ 22 22 21 21 31 31 33 33 22 42 42 22 −720 7π ui (ℑ − − Z h i ˙ ˙ 7 ˙ ˙ + 5√35 V¯32U22 U¯22V32 5 V¯21U31 U¯31V21 , (4.35a) ℜ " − − 2 − #) r uf T 1 ¯˙ ¯˙ 38 ¯˙ ¯˙ ∆Σ50 = du 5 U33U33 +5U31U31 U22U42 + U42U22 5040√11π ui (ℑ − √5 Z 7 ˙ ˙ ˙ ˙ + 2 U¯22V32 V¯32U22 +2√14 V¯21U31 U¯31V21 . (4.35b) ℜ " 5 − − #) r 16

As in Sec. IVB1, we then use Eqs. (3.5a), (3.5b)– remaining terms in ∆Σ30 and the entire expression for (3.5f), and (3.16)–(3.18) to evaluate the integrals in ∆Σ50 appear in the BD-theory waveform at Newtonian Eqs. (4.35a)–(4.35b) and to write the expression for the order because of the 1 PN term in dφ/dx and the ad- − moments ∆Σ30 and ∆Σ50 in terms of x. Unlike in ditional radiative multipole moments that contribute in Sec. IVB1, the integrand does not depend on x˙, when BD theory, but not in GR. written as an integral over x, because there is only one Finally, we construct the time-integrated strain from time derivative of the radiative multipole moments. The the moments ∆Σl0 in Eqs. (4.36a)–(4.36b). Using result of this integration is given by Eqs. (2.8), (2.9), (2.19), and (4.31), we can write the relation between the time integral of h and a general ∆Σlm 2 2 T π ηM 5ξS 3/2 1/2 4 as ∆Σ = ∆(x− ) + ∆(x− ) 1 G 30 7 10 − 144 − 3 r uf 2 (spin) i (l + 2)! 5(47796η + 5003)S ξ h du = − ∆Σlm( 2Y lm) . (1 + F )ξ + , 2R (l 2)! − − 435456 Zui l,m s X − (4.36a) (4.37) 2 2 Because the modes ∆Σl0 that produce the time- T π ηM ξS 1/2 (spin) ∆Σ = (21588η 4117)∆(x− ) , integrated strain h in Eq. (4.36a) and (4.36b) are 50 − 11 12192768 − r real, then from Eq. (2.7) it follows that the spin mem- (4.36b) ory enters in the cross mode polarization of gravitational waves (as it does in GR [35]). Finally substituting 1/2 1/2 1/2 where we have deﬁned ∆(x− ) = xf− xi− and Eqs. (4.36a)–(4.36b) into (4.37), and using the expres- 3/2 − similarly for ∆(x− ). The PN errorterms in Eqs. (4.36) sions for the spin-weighted spherical harmonics and (4.39) are both expected to be of order log(xf /xi), which would arise from O(1/x) terms in the u integral. 1 105 2 2Y30(ι, ϕ)= sin ι cos ι , (4.38a) We did not write these terms out explicitly, so as to sim- − 4r 2π plify the notation. 1 1155 2 2 2Y50(ι, ϕ)= sin ι cos ι(3 cos ι 1) , (4.38b) In GR, the only term that appears in the Newtonian- − 1/2 8 2π − order spin-memory waveform is ∆(x− ) times the co- r eﬃcient outside the curly braces in Eq (4.35a). The we obtain for the time-integrated strain

uf 2 2 (spin,T) 3ηM 2 5ξS 3/2 4 2 3365 1915 1/2 du h = sin ι cos ι ∆(x− )+ 1 G (1 + F )ξ ξS η + ∆(x− ) × 8R − 144 − 3 − − 6912 27648 Zui 2 20585 1285 1/2 2 +ξS η ∆(x− )cos ι . (4.39) 580608 − 6912

The expression for the u integral of h(spin,T) is writ- 2. Spin memory effect from the angular momentum flux of ten such that the angular dependence× and coefficient the scalar radiation 3ηM 2/(8R) outside of the curly braces coincides with the expression in GR at Newtonian order. Within the The angular momentum flux from the scalar radiation curly braces there are several sorts of terms: (i) the first produces a second contribution to the spin memory ef- 3/2 term proportional to ∆(x− ) is a 1PN term arising − fect. Its contribution can be obtained from the scalar from the dipole term in the phase, but which has the field terms in the balance law in Eq. (4.27a), same angular dependence as the spin memory effect in GR; (ii) the terms in the square bracket (aside from the uf 2 4 2 S 6+4ω 2 ABð factor of unity that reproduces the GR expression for the d ΩγÐ (Ð + 2)∆Σ = 2 d Ωduǫ B γ − (λ0) spin memory) are small BD corrections (proportional to Z Zui (λ˙ ð λ λ ð λ˙ ) . ξ) that modify the amplitude of the waveform without × 1 A 1 − 1 A 1 changing its x or ι dependence; (iii) the final terms on (4.40) the second line are those which have the same x dependence, but a different angular dependence from the GR The multipolar expansion of ∆Σ can be obtained by as- ¯ expression (and are again proportional to ξ). suming γ is equal to the spherical harmonic Ylm, and then using the multipolar expansion of λ1 in Eq. (3.19). Af- ter relating the gradients and curls of spherical harmonics in this expansion to the electric- and magnetic-parity 17 vector harmonics in Eqs. (4.4a)–(4.4b) and then employ- V. CONCLUSIONS ing the relationship between these vector harmonics and the spin-weighted spherical harmonics in Eqs. (4.5a)– In this paper, we computed the displacement and spin (4.5b),we can derive the moments ∆Σlm in terms of mo- GW memory effects generated by nonprecessing, quasi- ments λ1(lm) (and their time derivatives and complex circular binaries in BD theory. We worked in the PN conjugates) and the coefficients (s′,l′,m′; s′′,l′′,m′′) Bl approximation, and the expressions that we computed given in Eq. (4.2). The resulting expression is given be- are accurate to leading Newtonian order in the PN pa- low: rameter x, and they include the leading-order corrections in the BD parameter ξ. Our calculations relied upon us- S i(2ω + 3) (l 2)! l,( ) ∆Σ = − s ′ −′′ ing the BMS balance laws associated with the asymptotic lm 2 (l + 2)! l ;l λ0 l(l + 1) l′,m′,l′′,m′′ symmetries in BD theory (which are the same as in GR). X These balance laws permit us to determine the tensor pl (l + 1) (0,l ,m ;1,l ,m ) ′′ ′′ l ′ ′ ′′ ′′ GW memory effects, but the scalar GW memory effects × u B f associated with the breathing polarization are not con- p du(λ˙ ′ ′ λ ′′ ′′ λ ′ ′ λ˙ ′′ ′′ ) . × 1(l m ) 1(l m ) − 1(l m ) 1(l m ) strained through these balance laws. We further focused Zui (4.41) on the null contributions to the tensor GW memory effects, because we estimated that the ordinary (linear) l,( ) memory effects would contribute at a higher PN order The coefficient s ′ −′′ is defined in Eq. (4.15). l ;l than the null (including the nonlinear) memory effects. The moments of λ that contribute to the 1(lm) Using the BMS balance laws has the advantage that Newtonian-order spin memory effect are the same mo- we can determine the memory effects in BD theory to a ments needed to compute the scalar-sourced displace- higher PN order than had been previously done through ment memory effect in Sec. IVB2. The only nonvan- the direct integration of the relaxed Einstein equations ishing moment of ∆Σ at this order then is ∆Σ , and to 30 in harmonic gauge. However, the balance laws take as linear order in ξ it is given by input radiative and nonradiative data at large Bondi radius and this data must be obtained through some other 1 1 uf S ˙ ¯ method. Specifically, in the context of this paper in PN ∆Σ30 = 2 du λ1(22)λ1(22) −30ξ(λ0) √7π ℑ Zui theory, we needed to take as input the oscilliatory scalar 7 and tensor GWs computed in harmonic gauge in BD the- λ˙ λ¯ λ λ¯˙ . (4.42) − 2 1(11) 1(31) − 1(11) 1(31) ory in [69]. This required us to compute a coordinate r transformation between harmonic and Bondi gauges at We can then use the expressions for the scalar-field multi- leading order in the inverse distance to the source, so that pole moments in Eq. (3.24) to evaluate the integrals and we could relate the radiative GW data in these two dif- write them in terms of x (analogously to what was done ferent coordinate choices (and formalisms). There were for the tensor-GW part of the spin memory effect), and relatively simple transformations that allowed us to re- we find it is given by late the Bondi shear tensor to the transverse-traceless components of GW strain, and these relations were par- 2 ticularly simple when expressed in terms of the radia- S √πηM ξ 2 δm 2 1/2 ∆Σ = 2ηS +ΓS 8Γ ∆ x− . tive multipole moments of the shear and strain tensors. 30 1440√7 M − There were similar expressions relating the scalar field (4.43) at leading order in inverse distance in the two coordinate It is then straightforward to use Eq. (4.37) to write the systems and the corresponding multipole moments of the retarded-time integral of the spin memory waveform gen- scalar field. With these relationships between the multi- erated by scalar angular momentum flux as polar expansions of the scalar field and the shear tensor in harmonic and Bondi gauges, we could then use the uf ηM 2ξ δm duh(spin,S) = 2ηS2 +Γ S 8Γ2 BMS balance laws to compute the GW memory effects. × 384R M − Zui The tensor GW memory effects in BD theory have sev- 1/2 2 eral noteworthy differences from the corresponding ef- ∆ x− sin ι cos ι . (4.44) × fects in GR at the equivalent PN order. First, because of scalar dipole radiation in BD theory, there are relative The PN error terms in Eqs. (4.43) and (4.44) are both 1PN-order terms in the memory effects in BD theory. expected to be of order log(xf /xi), which would arise −The 1PN term in the displacement memory waveform from O(1/x) terms in the u integral. comes− from two sources in the supermomentum balance The full retarded-time integral of the spin memory law: (i) directly from the energy flux of scalar radiation waveform is h(spin) = h(spin,S) + h(spin,T). It then adds and (ii) indirectly from the energy flux in tensor radia- small correction× linear in×ξ to Eq. (4.39× ) that changes the tion (specifically through dipole contributions to the fre- amplitude but does not alter the x or ι dependence of the quency evolution and the GW phase). The spin memory effect. waveform, however, has a relative 1PN correction from − 18

GR arising from only the energy flux of the tensor waves ACKNOWLEDGMENTS (the scalar-sourced part of the energy flux gives rise to a contribution at the same leading order as in GR, and S.T. and K.Y. acknowledge support from the Owens it comes from products of dipole and octupole moments, Family Foundation. K.Y. also acknowledges support as well as quadrupole moments with themselves). The from the NSF Grant No. PHY-1806776, the NASA Grant absence and presence of the 1PN term from the scalar − 80NSSC20K0523, and a Sloan Foundation Research Fel- radiation in the spin and displacement memory effects, lowship. K.Y. would like to also thank the support by the respectively, arises because of the different lowest multi- COST Action GWverse CA16104 and JSPS KAKENHI pole term in the sky pattern of the effects: the spin mem- Grants No. JP17H06358. D.A.N. acknowledges support ory effect begins with the l = 3, m = 0 mode, whereas from the NSF Grant No. PHY-2011784. the displacement memory has an l =2, m =0 mode. A second noteworthy feature is that the computation of Newtonian-order memory effects in BD theory requires Appendix A: Coordinate transformations from radiative multipole moments of the strain at higher PN harmonic gauge to Bondi gauge orders than are required in GR (in which the leading- order effects can be computed from just the l = 2, In this appendix, we construct a coordinate transfor- m = 2 radiative mass quadrupole moments. Because mation between harmonic coordinates to linear order in there± are 1PN terms in the GW phase and the evolu- − 1/R and Bondi coordinates at an equivalent order in 1/r tion of the GW frequency, computing the Newtonian GW for a radiating spacetime in Brans-Dicke theory. The memory effects requires higher-PN-order radiative mul- procedure is similar to that recently described in [76] in tipole moments (including the current quadrupole, the GR, but we do not work to all orders in 1/r as in [76]. mass and current octupoles, and the mass hexadecapole). Rather, we work only to an order in 1/r so that we can These higher-order mass and current multipole moments ˜TT relate the radiative data in harmonic gauge (hij and Ξ) also give rise to a sky pattern of the GW memory effect to the corresponding radiative data in Bondi gauge (c in BD theory that differs from the sky pattern of the ef- AB and λ1) and compute the GW memory effects from PN fect in GR. In addition, the presence of dipole radiation waveforms in harmonic gauge. required us to include 1PN corrections to the GW phase (H) and frequency time derivative to compute the memory We will denote the harmonic-gauge metric by gµν which we write in quasi-Cartesian coordinates Xµ = effects to Newtonian order (though we only required the 0 i 0 i GR limit of these 1PN corrections when working to linear (X ,X ), where X = t and X = (X,Y,Z). We i j order in the BD parameter ξ). find it convenient to define R = X X δij to be the harmonic-gauge distance from the origin, and yA = (ι, ϕ) Finally, let us conclude by commenting on potential p to be spherical polar coordinates with cos ι = Z/R and applications of the calculations of GW memory effects tan ϕ = Y/X. The components of the metric can then given in this paper. Because the calculations herein have be written in the form shown that the memory effect in BD theory differs from that in GR, it is natural to ask if these differences could 2AM 1 g(H) = 1+ + H (t R,yA) , (A1a) be detected. Given the challenges of detecting the mem- 00 − R R 00 − ory effect in GR with LIGO and Virgo [15, 16, 80] and 1 g(H) = H (t R,yA) , (A1b) the fact that the PN corrections are small, it would 0i R 0i − be more natural to consider whether next-generation (H) 2BM 1 A gravitational-wave detectors—such as the space-based g = δij + δij + Hij (t R,y ) . (A1c) ij R R − detector LISA [81] or ground-based detectors like the Einstein Telescope or Cosmic Explorer [82, 83]—could We have written the metric in the center-of-mass rest constrain BD theory through a measurement of the GW frame of the system, and we have split the O(1/R) part memory effect. The constraints could come from search- of the metric in terms of the constant mass monopole mo- ing for differences in the leading-order amplitude of the ment M and all the time-dependent, higher-order mul- effect, in the time dependence of the accumulation of the tipole moments in Hµν . We have also introduced con- memory effect (through the different dependence on the stants A and B [defined in Eq. (A3)] which are needed PN parameter x), or in the sky pattern of memory effects so that the metric satisfies the modified Einstein’s equa- in BD theory (the latter being similar to the hypothesis tions in the static limit (at the leading nontrivial order in test described in [73]). Because memory effects accumu- 1/R). To specify a solution, we also need an expression late most rapidly during the merger of compact binaries, for the scalar field, which we give in the Jordan frame, having waveforms that go beyond the PN approximation and which we denote by λ. We will again expand it in and include the merger and ringdown would be important terms of a static monopole moment and time-dependent, for producing the most accurate constraints. We thus higher-order multipole moments as follows: postpone quantitatively assessing the detection prospects until inspiral-merger-ringdown waveforms in BD theory ψ(t R,yA)+2CM λ = λ 1+ − . (A2) are available. 0 R 19

The monopole term is the O(1/R) piece proportional to nonzero components of the inverse Bondi metric are given 2λ0CM and ψ contains the higher-order, time-dependent by moments. The third coefficient C is again needed to satisfy the field equations in the static limit. The ex- ur λ1 2 g(B) = 1 + O r− , (A7a) pressions for the constants A, B, and C were previously − − λ0r determined in [68, 70, 84] and are given by ∂ λ 1 λ λ ∂ λ grr =1+ u 1 + 1 2 + 1 u 1 (B) λ r λ − M 2λ2 ω +2 κ 0 0 0 A = − , (A3a) 2 ω +2 + r− , (A7b) O ω +1+ κ ðA rA 1 AB λ1 3 B = , (A3b) g = ð c + O r− , (A7c) ω +2 (B) 2 B 2r − λ0 1 2κ C = − . (A3c) AB 1 AB 1 AB 4 g = q c + O(r− ) (A7d) 2ω +4 (B) r2 − r3 We introduced the variable κ = (m s + m s )/M in the 1 1 2 2 (and where guu = guA = 0). For simplicity, we will equations above. The coefficients also satisfy the rela- (B) (B) tionships A B =2C and A + B =2/λ0. summarize Eq. (A7) as The harmonic− gauge conditions lead to relationships µν µν 1 µν between the components of the quantities Hµν and ψ. g(B) = η(B) h(B) , (A8) These relationships are more conveniently expressed in − r terms of a quantity H˜µν , which is related to Hµν and ψ µν 0 where the quantity η(B) consists of the O(r ) pieces of by ur rr 1 rA g(B) and g(B), the O(r− ) part of g(B) (which vanishes) 1 and the O(r 2) part of gAB; the quantity hµν consists Hµν = H˜µν Hη˜ µν ψηµν . (A4) − (B) (B) − 2 − of the coefficients of the relative 1/r corrections to the µν The gauge conditions are given by components of η(B). We perform the coordinate transformation in two H˜ = ninj H˜ , H˜ = njH˜ . (A5) 00 ij 0i − ij stages for illustrative purposes (one could perform it in i i We have defined n = X /R = ∂iR above, and it reduces one stage as in [76], but the two-stage process here allows to the expression ~n (sin ι cos ϕ, sin ι sin ϕ, cos ι) when us to highlight the different roles of the different terms written in terms of ι ≡and ϕ.7 in the transformation more easily). The first stage im- Our procedure for transforming from harmonic gauge poses the gauge conditions on the inverse Bondi metric uu uA to Bondi gauge follows some aspects of the work [76] that g(B) and g(B) vanish to the accuracy in 1/r at which in GR, in which Bondi coordinates were determined in we work, it also makes r an areal radius, and finally, terms as functions of harmonic-gauge quantities. In our it relates the Bondi-coordinate retarded time u to the case, however, we consider BD theory, we work only to harmonic-gauge retarded time t R. It can be thought − linear order in 1/R in harmonic coordinates, and we de- of as a “finite” gauge transformation, in the sense that it µν µν termine the corresponding Bondi coordinates in an ex- is needed to relate the background metrics η(B) and η(H), pansion in 1/R, such that Bondi gauge is imposed to one which differ by a large (not perturbative in 1/r) coor- order in 1/r beyond the leading-order metric. To per- dinate transformation. The second stage, which can be form the coordinate transformation, it is useful to work treated as perturbative in 1/r, then sets the metric fol- with the components of the inverse metric. In harmonic lowing the first transformation into a Bondi-gauge metric gauge, these components are given by that satisfies the modified Einstein equations of BD theory. 1 µν µν µν µ ν µ ν ij 2 The first stage, the finite part of the coordinate trans- g(H) = η(H) H +2M(Aδ0 δ0 + Bδi δj δ ) +O(R− ) , −R formation expresses a set of coordinates xα = (u, r, xA) (A6) α i µν in terms of harmonic gauge coordinates X = (t,X ), where Hµν is related to H by raising indices with η . µν (H) though, it is expressed more easily in terms of the spher- We aim to put the metric in Bondi form, in which the ical polar coordinates (t,R,yA) constructed from harmonic coordinates as follows: 2M 7 u = t R ln R, (A9a) The conditions in Eq. (A5) can be derived by first making the − − λ0 definitions given, e.g., in [70, 77], of a conformally rescaled metric ˜µν ˜µν µν µν ψ g˜µν = λgµν , then defining h to be h = η √ g˜g˜ , and r = R + BM , (A9b) ˜µν − − imposing the harmonic gauge condition ∂ν h = 0. When the − 2 harmonic gauge condition is imposed at leading order in 1/R, xA = yA . (A9c) then the conditions in (A5) can be obtained up to integration constants that we set to zero [so as to maintain the static solution The coordinates (u, r, xA) resemble, but are not precisely of Einstein’s equations in Eqs. (A1) and (A2)]. Bondi coordinates at the order in 1/r at which we work, 20

µν because they do not enforce all of the required properties transformation will take the part of the metric h(I)/r of the Bondi-gauge metric. We will thus write this “in- µν µν and bring it to the Bondi-Sachs form h(B)/r, through termediate” metric as g(I) , and it can be computed from the transformation the harmonic-gauge metric using the transformation law 1 1 for the components of a rank-two contravariant tensor: hµν = hµν + ηµν , (A14) r (B) r (I) Lξ~ (B) ∂xµ ∂xν gµν = gαβ . (A10) (I) (H) ∂Xα ∂Xβ where is the Lie derivative along ξ~. To solve for the Lξ~ perturbative gauge vector that generates this transfor- A somewhat lengthy, but otherwise straightforward cal- µ culation shows that this metric can be written in the mation, we first write the components of ξ as form µ 1 u r A 1 ξ = (ξ(1), ξ(1), ξ(1)/r) , (A15) gµν = ηµν hµν , (A11) r (I) (B) − r (I) µ A (B) where the functions ξ(1) depend on u and x (not r). where ηµν is the leading order part of the inverse Bondi These lead to a set of partial differential equations for metric in Eq. (A7) when ∂uΞ= ∂uλ1. The coefficients of the ξµ that can be integrated by requiring that hµν be the relative 1/r corrections to ηµν we denoted by hµν , (1) (I) (B) (I) transformed into Bondi-Sachs form. Before giving the full and they are given by details of this procedure, it is worth noting that given the ˜ form of the components of the vector ξµ in Eq. (A15), the rr 2M 2 H h(I) = (∂uψ) 2BM + ψ ∂uψ radiative data λ1 and cAB can be related to harmonic- λ0 − 2 − ! gauge data and the finite coordinate transformation (A9) i j 1 µ +Hij n n +2BM + O r− , (A12a) without needing the full expression for ξ in (A15). ˜ For the scalar field, because in both harmonic and ur M Ξ H 1 h = ∂uψ + + + O r− , (A12b) Bondi gauges, the field has an expansion of the form (I) 1 − λ0 λ0 2 λ = λ0 +O(r− ) [where λ0 is constant and the coefficient 1 A rA ðA i 1ðA 1 of the O(r− ) term is denoted Ξ(t R,y ) in harmonic rh(I) = H0i n ψ + O r− ,(A12c) A − − − 2 coordinates and λ1(u, x ) in Bondi coordinates], then the µ uu 1 transformation parametrized by ξ in Eq. (A15) will not h(I) = O r− , (A12d) change Ξ or λ1. Thus, one must have that uA 2 h(I) = O r− , (A12e) A A 2 AB ðA iðB j AB 1 λ1(u, x ) = Ξ(t R,y ) , (A16) r h(I) = Hij n n + ψq + O r− . (A12f) − where u is related to t R (and xA to yA) by the transfor- To arrive at Eq. (A12), we used the conditions in Eq. (A5) − mations in Eq. (A9). For cAB, a direct calculation shows and we expressed ∂tψ in terms of ∂uψ (and other terms) AB 4 µ that η is of order r− for ξ in (A15). This implies using the derivatives of the first two lines in Eq. (A9) and Lξ~ (B) the chain rule: that AB A i B j AB M 2 c = Hij ð n ð n + ψq . (A17) ∂tψ = ∂uψ (∂uψ) . (A13) − rλ0 Using the definition of H˜ in Eq. (A4), the harmonic Note that nonlinear terms involving ψ appear here and µν gauge conditions in (A5), and the fact that qAB = elsewhere in hrr, because the coordinate transformations I δ ðAniðBnj , this equation can be recast as in (A9a)–(A9c) involve ψ at order O(R0). ij The metric is similar to a Bondi-Sachs form to the AB ˜ ðA iðB j 1 ˜ AB order in 1/r at which we work, in the sense that the Bondi c = Hij n n Hq . (A18) uu uA − 2 gauge conditions g(I) = g(I) = 0 are satisfied at this order and the right-hand side of Eq. (A12f) is traceless After using Eq. (A5) again, it follows that cAB is related with respect to qAB (thereby being consistent with the to just the transverse-traceless (TT) part of H˜ij as determinant condition of Bondi gauge). Note, however, that the ur, rr, and rA components of hµν do not satisfy cAB(u, xA)= H˜ TT(t R,yA)ðAniðBnj , (A19) (I) ij − the modified Einstein equations in Bondi-Sachs gauge, as they are not consistent with the form of the inverse metric where again u is related to t R and xA to yA by the in Eq. (A7). The metric can be put into Bondi gauge with transformations in (A9). − a perturbative (in 1/r) coordinate transformation, as we For our purposes of relating the radiative data in Bondi describe next. coordinates to that in harmonic coordinates, Eqs. (A16) We will parametrize the perturbative coordinate trans- and (A19) provide the solution. However, for complete- formation in terms of a vector ξµ which effects the coor- ness we do compute the form of the required gauge vector dinate transformation xµ xµ + ξµ. This coordinate ξµ needed to complete the transformation from harmonic → 21 to Bondi coordinates. The components of the gauge vec- i.e.: tor in Eq. (A15) can be constrained from the ur, rA, and 2 2 2 O 2 1 2 rr components of Eq. (A14), which state, respectively, d Ω α Ð (Ð + 2)∆Θ = 8∆ d Ω α Ð λ1 . M− 4λ 0 ∂ ξu = hru hru , (A20a) Z Z u (1) (B) − (I) (B1) A rA rA ∂uξ = r(h h ) , (A20b) O (1) (B) − (I) Expanding ∆ , ∆Θ and ∆λ1 in spherical harmonics, M O 2∂ ξr + ξu (∂ )2(λ /λ )= hrr hrr . (A20c) the moments ∆Θ are given by u (1) (1) u 1 0 (B) − (I) By substituting the relationships in Eqs. (A16) and (A19) O (l + 2)! 2 ∆Θlm = 8∆ lm + l(l + 1)∆λ1(lm) . into Eq. (A7), extracting the relevant components of (l 2)! M λ0 µν µν − h(B), and using the expressions for h(I) in Eq. (A12), (B2) O it is straightforward to integrate the first two lines in We would like to estimate if the quantity ∆Θlm is of u A a similar PN order to the nonlinear and null parts of Eq. (A20) to obtain expressions for ξ(1) and ξ(1). Once u the memory ∆ΘT and ∆ΘS that were computed in ξ(1) has been determined, integrating the final line of lm lm Eq. (A20) to determine ξr is also, in principle, straight- Sec. IVB for any of the specific values of l = 2, 4, or (1) 6 and m = 0. To do so, we will focus on the moment forward. There is one subtlety in this procedure: hrr in- (B) ∆ΘO for simplicity (the other three moments will have volves the Bondi mass aspect , which has not yet been 20 M the same, or a higher PN order). determined from metric quantities in harmonic gauge. One natural way to compute the ordinary memory Because the mass aspect satisfies the conservation equa- would be to directly evaluate the moments ∆λ and tion [50] 1(20) ∆ 20. Using Eq. (3.20), we can show that ∆λ1(20) is M 2 1 AB 1 AB at least (x ); thus, the scalar field’s contribution to the ∂u = NABN + ðAðBN O M − 8 4 memory effect is of a higher PN order than the Newtonian 1 2 1 2 order at which we work. We do not have an independent (3+2ω) 2 (∂uλ1) + Ð ∂uλ1 , (A21) − 4λ 4λ0 expression for ∆ 20 that would allow us to directly com- 0 O M pute ∆Θ20 (although we already verified that the contri- it is possible to integrate this equation and express bution from the ∆λ is of a higher PN order). Instead, in terms of harmonic-gauge quantities using Eqs. (A16M) 1(20) we can compute ∆ΘO directly from the waveform that and (A19). (To simplify the notation, however, we will 20 were already computed in Eqs. (7.1) and (7.2a) of [68] not write this out in detail below, and we will write this to verify that ∆ would not contribute at Newtonian quantity just as .) The result of performing these inte- M20 M order. Specifically, we contract the Newtonian-order ex- grations and using the harmonic gauge conditions in (A5) ij ij µ pression with the polarization tensors e+ ie , multiply is that the components of the vector ξ are given by − × by the spin-weighted spherical harmonic 2Y¯20 to obtain 1 M O − ξu = dtH˜ + Ξ , (A22a) U20 and then rescale it to obtain ∆Θ20. We find that −2r λ2r the Newtonian-order result vanishes, and there is thus Z 0 1 ξu 2M no Newtonian-order ordinary displacement memory. ξr = dt (1) ∂2Ξ+ (∂ Ξ)2 2 + H˜ −2r λ u λ3 u − M 00 Z 0 0 1 Ξ 1 2M ∂uΞ 2. Ordinary spin memory effect ∂ Ξ H˜ 1+ , −2 λ2 u − 2 − λ λ 0 0 0 (A22b) The ordinary part of the spin memory effect can be 1 1 computed from Eq. (4.27a) with just the term (4.27c) on ξA = dt ð H˜ TTðAniðBnj + H niðAnj . −r2 2 B ij ij the right-hand side: Z (A22c) 2 2 2 0 2 AD d Ω γÐ (Ð + 2)∆Σ = 8∆ d Ω ǫ ðDγ [ 3LA This transformation, along with the finite transformation − − Z Z in Eq. (A9), brings the metric into the form in Eq. (A7), 1 B B (cAB ð λ1 λ1ð cAB) . in which λ1 and cAB are related to the harmonic-gauge − 4λ − ˜ TT 0 quantities Ξ and Hij by Eqs. (A16) and (A19). (B3)

B B The terms cABð λ1 λ1ð cAB on second line of the Appendix B: Estimates of ordinary memory effects equation will not contribute− at Newtonian order for the in Brans-Dicke theory 1/2 spin memory effect (i.e, at order x− ), because both cAB and λ1 involve non-negative powers of x in the PN 1. Ordinary displacement memory effect expansion, so their product will not be a negative power of x. The only term that could contribute to the spin The contribution to the ordinary memory effect comes memory effect comes from the change in the angular mo- from the charge rather than the flux terms in Eq. (4.6), mentum aspect, LA. 22

We do not have an expression for LA in terms of zone, we can estimate the size of the effect in linearized harmonic-gauge metric functions, which (analogously to theory. the case of the mass aspect and ordinary displacement The ordinary spin memory effect would arise at the O memory effect) prohibits a direct calculation of the or- lowest PN order from changes in ∆Σ30, which is pro- dinary spin memory effect. In addition, it is not possi- portional to the retarded time integral of the radiative ble to directly check the time integral of the waveform, moment V30. Because at leading PN order, V30 is related because the Newtonian-order terms in the spin memory to three time derivatives of the source current octopole O ¨ effect arise from formally 2.5PN-order terms in the wave- J30, then ∆Σ30 should be proportional to J30. By di- 3 form that are then integrated with respect to retarded mensional analysis, J30 is proportional to Mva (or see, 2 time; however, the waveform has only been computed to e.g., [67]); thus, J¨30 scales as Mvaa˙ . This scales with 2PN order in [68]. While we cannot then be certain that the PN parameter as ξx9/2 +x11/2, which would be a 6PN the ordinary memory terms do not contribute at the same correction to the nonlinear and null effects. We thus an- order because of additional nonlinear terms in the near ticipate from these arguments in linearized theory that the ordinary part of the spin memory will be small.

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