Compute gravitational waveforms from particle trajectory

4.3 Generation of numerical kludge waveforms

4.3.2 Compute gravitational waveforms from particle trajectory

tensor of the particle motion since we are not including the energy-momentum of “the wire” along which the particle moves.

formula (hereafter, an overdot denotes a time derivative) h¯^jk(t,x)=2

r hI¨^jk(t⁰)i

t⁰=t−r, (4.17)

where

I^jk(t⁰)=Z

x^0jx^0kT⁰⁰(t⁰,x⁰)d³x⁰ (4.18) is the source’s mass quadrupole moment. Including the next order terms, we obtain the quadrupole-octupole formula (see Refs. [38, 39] for details),

h¯^jk= 2 r

I¨^jk−2niS¨^{i jk}+ni

M...^{i jk}

t⁰=t−r

, (4.19)

with⁴

S^{i jk}(t⁰) = Z

x^0jx^0kT⁰ⁱ(t⁰,x⁰)d³x⁰, (4.22) M^{i jk}(t⁰) = Z

x⁰ⁱx^0jx^0kT⁰⁰(t⁰,x⁰)d³x⁰. (4.23) Note that in both Eqs. (4.17) and (4.19), the retarded time ist−rinstead of the more complicated expression appearing in (4.16). If desired, it is a straightforward (but increasingly tedious) task to include more terms in the slow-motion expansion of (4.16). In the present work we shall make use of Eq. (4.17) (the “quadrupole formula”), Eq. (4.19) (the “quadrupole-octupole” formula) and the full Press formula Eq. (4.16).

The waveform in the standard transverse-traceless (TT) gauge is given by the TT projection of the above expressions. We define an orthonormal spherical coordinate system via

e_r= ∂

∂r, e_Θ=1 r

∂

∂Θ, e_Φ= 1 rsinΘ

∂

∂Φ. (4.24)

The angles{Θ,Φ}denote the observation point’s latitude and azimuth respectively. The waveform in transverse-

4HereM^{i jk}is the mass-octupole moment. Note that formally, the current-quadrupole moment is usually defined as (Eq. (5.19b) in Ref. [60]):

Sab = Z

apqx_p(−T0q)xbd³x STF

. (4.20)

Here the superscript “STF” stands for “symmetric and trace-free”. The corresponding current quadrupole moment contribution to the radiation field is (Eq. (4.8) in [60])

h^TT_jk =

"8xq

3r²pq(jS¨k)p

#TT

. (4.21)

It can be verified that then_iS¨^{i jk}term in Eq. (4.19), withS^{i jk}defined by Eq. (4.22), is equivalent to the right hand side of Eq. (4.21).

traceless gauge is then given by

h_{T T}^jk = 1 2



















0 0 0

0 h^ΘΘ−h^ΦΦ 2h^ΘΦ 0 2h^ΘΦ h^ΦΦ−h^ΘΘ



















, (4.25)

with

h^ΘΘ = cos²Θh

h^xxcos²Φ +h^xysin 2Φ +h^yysin²Φi

+h^zzsin²Θ−sin 2Θh^xzcosΦ +h^yzsinΦ, (4.26a) h^ΦΘ = cosΘ

"

−1

2h^xxsin 2Φ +h^xycos 2Φ +1

2h^yysin 2Φ

#

+sinΘh^xzsinΦ−h^yzcosΦ, (4.26b)

h^ΦΦ = h^xxsin²Φ−h^xysin 2Φ +h^yycos²Φ. (4.26c)

The usual “plus” and “cross” waveform polarizations are given byh₊= ¹₂ h^ΘΘ−h^ΦΦandh_×=h^ΘΦrespec- tively. The expressions (4.16), (4.17) and (4.19) are valid for a general extended source in flat-space. If we specialize to the case of a point-massµmoving along a trajectoryx⁰_p(τ), then the energy-momentum tensor in flat spacetime is given by

T^µν(t⁰,x⁰)=µZ ∞

−∞

dx^0µp

dτ dx^0ν_p

dτ δ⁴

x⁰−x⁰_p(τ)

dτ=µdτ dt⁰_p

dx^0µp

dτ dx^0ν_p

dτ δ³

x⁰−x⁰_p(t⁰)

, (4.27)

whereτis the proper time along the trajectory. It is related to the particle’s coordinate time by dτ = (1− v²/c²)^1/2dt⁰, wherev² =|dx⁰_p/dt⁰_p|². On the right hand side of Eq. (4.27) is a term dt⁰_p/dτ= 1+O(v²/c²).

Expressions (4.17) and (4.19) are slow-motion expansions, and at the order of these expansions we should replace dt⁰_p/dτby 1 for consistency. The Press formula (4.16) is a fast-motion expression, and therefore we do include this term. The presence of aδ-function inT^µνfacilitates the simplification of the various moments in Eqs. (4.17) and (4.19):

I^jk = µx^0j_px^0k_p , (4.28)

S^{i jk} = vⁱI^jk = µvⁱx^0jpx^0k_p , (4.29) M^{i jk} = x⁰ⁱ_p I^jk=µx⁰ⁱ_px⁰_p^jx^0k_p . (4.30) Herev^a ≡dx^0a/dt⁰_p. The Press formula similarly simplifies to

h¯^jk(t,x)=2µ r

d² dt²

"

(1−nav^a)x^0j_px^0k_p dt⁰_p dτ

!#

t⁰_p=t−|x−x⁰_p|

. (4.31)

The square-bracketed expression is to be evaluated at a timet⁰_pgiven implicitly byt=t⁰_p+|x−x⁰p(t⁰_p)|. In a numerical implementation, we evaluate the expression (4.31) as a function of the timet⁰_palong the particle’s

path. In order to obtain a time series that is evenly spaced intand compute the right hand side of Eq. (4.31) by finite differencing, we adjust the spacing of the sampling at the particle,δt⁰_p, such that

δt=(1−nav^a)δt⁰_p. (4.32)

With a delta function source (4.27), the retarded-time solution (4.15) can be evaluated directly [34]:

h¯^jk(t,x)=







 4µ

r dt⁰_p

dτ

!dx⁰p^j

dt⁰_p dx^0k_p

dt⁰_p 1 1−nav^a









t⁰_p=t−|x−x⁰_p|

. (4.33)

Naively, one could suppose that (4.33) will perform better than (4.16), since it is derived using only one of the two (invalid) flat-space equations, rather than both. In fact, we find that the retarded integral expression (4.33) performs much worse than either (4.17), (4.19) or (4.16) when compared to TB waveforms. The reason appears to be that the manipulations which lead to the quadrupole, quadrupole-octupole, and Press formulae ensure that the actual source terms—mass motions—are on the right hand side. This is to be contrasted with the retarded integral expression (4.15), which identifies the dominant GW on the left hand side with weak spatial stresses terms inT^{i j}.

We must emphasize that the NK prescription is clearly inconsistent—we are binding the particle motion to a Kerr geodesic while assuming flat spacetime for GW generation and propagation. This is manifested by the fact that the energy-momentum tensor (4.27) is not flat-space conserved,∂_νT^µν , 0. However, the spirit of this calculation is not a formal and consistent approximation to EMRI waveforms; it is rather a

“phenomenological” approach which takes into account those pieces of physics we believe are the most crucial—in particular, the exact Kerr geodesic motion. By including the exact source trajectory, we ensure that the spectral components of the kludge waveforms are at the correct frequencies, although their relative amplitudes will be inaccurate. As we shall see, this line of thinking is validatedpost factoby the remarkable agreement between the kludge and TB waveforms (see Section 4.4).

It is important to underline the physical assumptions that have been made in the derivation of the quadrupole formula (4.16), the quadrupole-octupole formula (4.17) and the Press formula (4.19), in order to understand their generic limitations. First of all, the assumed absence of any background gravitational field means that our kludge waveforms are unable to capture any features related to back-scattering. This effect is known to first appear at 1.5PN level (i.e., atO(v³); see, e.g., Ref. [61]). Such “tails” of waves are particularly promi- nent in the strong-field TB equatorial “zoom-whirl” waveforms [16] and in the waveforms from plunging or parabolic orbits [62]. In all these cases, the hole’s quasinormal mode ringing, produced by back scattering, adorns the emitted signal.

The slow-motion nature of expressions (4.17) and (4.19) suggests that they might be bad models for waveforms generated by orbits venturing deep inside the central BH’s spacetime. Indeed, these formulas poorly reproduce the rich multipole structure of the true waveform from such orbits, as they (and any other

slow-motion approximation) essentially truncate the sum over multipoles. The Press formula (4.16) includes contributions at all multipoles and so might be expected to handle these contributions quite well. However, it turns out not to perform much better than the quadrupole-octupole waveform in this regime. While it includes contributions at all multipoles, the lack of background curvature in the waveform modelmightcause the amplitudes of the higher modes to be much lower than for true EMRI waveforms.

Fortunately, these two deficiencies become important for the same class of orbits—those that allow the body to approach very close to the black hole. As a rule of thumb we shall find that NK waveforms are reliable (in terms of the overlap discussed in the next section) as long as the closest orbital approach (periastron) is rp&5M.