Waveforms in the case of circular orbit

5.3 Waveforms in the case of circular orbit

In the previous two subsections, we only considered the luminosity formulas for the energy and the angular momentum. Here, focusing on circular orbits, we review the previous calculation of the gravitational waveforms.

On the other hand, the gravitational waveforms have also been calculated. In the case of circular orbit around a Schwarzschild black hole, Poisson [83 ] derived the 1.5PN waveform and Tagoshi and Sasaki [100 ] derived the 4PN waveform. These were done by using the post-Newtonian expansion of the Regge–Wheeler equation discussed in Section 3. Recently, Fujita and Iyer [39 ] derived the 5.5PN waveform by using the MST formalism. They also discussed factorized re-summed waveforms which is useful to obtain better agreement with accurate numerical data.

In the case of circular orbit around a Kerr black hole, Poisson [83 ] derived the 1.5PN waveform under the assumption of slow rotation of the black hole. In [94 ] and [101 ], although the luminosity was derived up to 2.5PN and 4PN order respectively, the waveform was not derived up to the same order. Recently, Tagoshi and Fujita [97 ] computed the all multipolar modes $&tidle;Z lmω$ necessary to derive the waveform up to 4PN order, and the results were used to derive the factorized, re-summed, multipolar waveform in [78 ].

From Equations (46 ) and (47 ), we have

2 ∑ −2Saℓωmn iωn(r∗−t)+im φ h+ − ih × = − -- Zℓmωn -√----e r ℓmn 2π ≡ ∑ (h+ − ih × ) (182 ) ℓm ℓm ℓm

Since the formulas for the waveform are very complicated, we only show the mode for $ℓ = m = 2$ up to 4PN order in the Schwarzschild case. We define $+,× ζℓ,−m$ as [83 ]

$ζ+2,,×2$ are given as

( + 107v2 cos(2 ψ) ζ2,2 = (3 + cos(2 𝜃)) cos(2ψ ) − --------------- ( ( ) 42 ) 4 +v3 2π cos(2 ψ) + − 17-+ 4 ln 2 sin(2 ψ) − 2173-v-cos(2-ψ)- ( 3( ) ) 1512 5 − 107 π cos(2 ψ ) 1819 214 ln2 +v ----------------+ ----- − -------- sin(2ψ ) ( 21( 126 212 ) +v6 cos(2ψ ) 49928027- − 856-γ-+ 2π--+ 668-ln-2-− 8 (ln 2)2 − 856-ln-v- ( 1940400) 105 ) 3 105 105 − 254 π + --35--- + 8π ln 2 sin(2ψ ) ( ( ) ) +v7 −-2173-π-cos(2ψ-)+ 36941- − 2173-ln-2 sin(2 ψ) ( 75(6 4536 378 8 326531600453 45796 γ 107 π2 +v cos(2ψ ) − -12713500800--+ -2205---− --63-- 2 ) 35738-ln-2 428-(ln-2)- 45796-lnv- − 2205 + 21 + 2205 ( ) ) ) + 13589-π-− 428-π-ln-2- sin(2 ψ) , (184 ) 735 21 ( 107 v2 sin(2 ψ) ( ( 17 ) ) ζ×2,2 = 4 cos(𝜃) sin(2 ψ) − ---------------+ v3 cos(2ψ ) ---− 4 log(2) + 2 π sin(2ψ ) ( 42( ) 3 ) 2173v4-sin(2ψ-)- 5 1819- 214-log(2) 107-π-sin-(2-ψ)- − 1512 + v cos(2 ψ) − 126 + 21 − 21 ( ( 254 π ) +v6 cos(2ψ ) ------− 8 π log(2) ( 35 ) ) 49928027- 856-γ- 2-π2 668-log(2) 2 856-log(v) + 1940400 − 105 + 3 + 105 − 8 log(2) − 105 sin (2 ψ) ( ( 36941 2173 log(2)) 2173 π sin(2 ψ)) +v7 cos(2ψ ) − ------+ ----------- − --------------- ( ( 4536 378 ) ( 756 8 −-13589-π- 428π-log(2)- 326531600453-- 45796-γ- +v cos(2ψ ) 735 + 21 + − 12713500800 + 2205 2 2 ) ) ) − 107-π- − 35738--log-(2)+ 428-log(2)--+ 45796--log-(v-) sin(2ψ ) , (185 ) 63 2205 21 2205

where

Other modes are given in [100 ] up to 4PN order. (Note however the following errors which were pointed out in [61 , 17 , 4 , 5 , 39 ]. The signs of $+ ζℓm$ are opposite. The sign of $× ζ7,3$ is also opposite. $× ζ8,7$ and $ζ+10,6$ have errors and the corrected formulas are given in Equations (6.6) and (6.7) in [39 ].) In the literature on the post-Newtonian approximation [17 , 4 , 5 ], the post-Newtonian waveforms are express in terms of $(n∕2) H +,×$ defined as

where $x ≡ (M Ω )2∕3 φ$ . The expression of $H (n∕2) +,×$ up to 5.5PN order are given in [39 ].