This document is part of https : // dcc.ligo.org/LIGO - T1500554/public

ENS[x_]:=Expand[Normal[Simplify[x]]];

SS -Energy 2PN (spin aligned, BHs)

S1S2

-2/η

Sself^2

0

Sself^2 κ

1/2/mBM^2-3/2/mBM^2

SS Flux 2PN (spin aligned)

S1S2

41/6/η-71/24/η

Sself^2

1/12/mBM^2-1/48/mBM^2

Sself^2 κ

7/2/mBM/mBM-3/2/mBM/mBM

SS Energy 3PN (spin aligned)

S1S2

-2/η-1/3+(1/3/η)-2/9

Sself^2

4/mBM^2-3/mBM-1+1/mBM^2+4/3/mBM-1/9

Sself^2 κ

-3/2/mBM/mBM-7/2/mByM-5/6-1/mBM/mBM+1/mByM

SS Flux 3PN (spin aligned)

S1S2

-1825/56/η-12905/504+97/4/η+499/56

Sself^2

587/(112*mBM^2)-2819/(504mBM)+19/1008-937/(112mBM^2)+1861/(252mBM)+267/112

Sself^2 κ

-2245/(84mBM^2)+373/(168mBM)+359/28+3653/(168mBM^2)-659/(84mBM)-29/14

Check

eN[η_]:=-η/2;
fN[η_]:=-32/5η 2;
e1PN[η_]:=(-3/4-1/12η);
e1p5PNSO[m1M_,m2M_]:=(2/3+2/m1M)SP[S1,L]+(2/3+2/m2M)SP[S2,L];
e2PN[η_]:=-(27/8-19/8η+1/24η^2);
e2PNSS[η_,m1M_,m2M_]:=-κ1/m1M^2SP[S1,S1]-κ2/m2M^2SP[S2,S2]-2/η SP[S1,S2];
e3PNSS[η_,m1M_,m2M_]:=(5/m1M^2-5/3/m1M-10/9)SP[S1,S1]+(5/m2M^2-5/3/m2M-10/9)SP[S2,S2]+(-5/3/η-5/9)SP[S1,S2]+κ1(-5/2/m1M^2-5/2/m1M-5/6)SP[S1,S1]+κ2(-5/2/m2M^2-5/2/m2M-5/6)SP[S2,S2]
f1PN[η_]:=(-1247/336-35/12η);
f1p5PNSO[m1M_,m2M_]:=(-3/2-5/4/m1M)SP[S1,L]+(-3/2-5/4/m2M)SP[S2,L];
f2PN[η_]:=-(44711/9072-9271/504η-65/18η^2);
f2PNSS[η_,m1M_,m2M_]:=(1/16/m1M^2)SP[S1,S1]+(1/16/m2M^2)SP[S2,S2]+(31/8/η)SP[S1,S2]+κ1(2/m1M^2)SP[S1,S1]+κ2(2/m2M^2)SP[S2,S2];
f3PNSS[η_,m1M_,m2M_]:=(-25/8/m1M^2+43/24/m1M+173/72)SP[S1,S1]+(-25/8/m2M^2+43/24/m2M+173/72)SP[S2,S2]+(-467/56/η-601/36)SP[S1,S2]+κ1(-279/56/m1M^2-45/8/m1M+43/4)SP[S1,S1]+κ2(-279/56/m2M^2-45/8/m2M+43/4)SP[S2,S2];
facPsi[a_]:=40/((2a-8)(2a-5));

η=m1 m2/(m1+m2)^2;
M=m1+m2;
psi[η_,x_,m1M_,m2M_]:=(1+(2e1PN[η]-f1PN[η])facPsi[1]x+(3e2PNSS[η,m1M,m2M]-f2PNSS[η,m1M,m2M]+0*f1PN[η](f1PN[η]-2e1PN[η]))facPsi[2]x^2+(4e3PNSS[η,m1M,m2M]-f3PNSS[η,m1M,m2M]-3f1PN[η]e2PNSS[η,m1M,m2M]-2f2PNSS[η,m1M,m2M]e1PN[η]+2f1PN[η]f2PNSS[η,m1M,m2M]+f1p5PNSO[m1M,m2M]^2-5/2e1p5PNSO[m1M,m2M]f1p5PNSO[m1M,m2M])facPsi[3]x^3)

ENS[ENS[ENS[SeriesCoefficient[psi[η,x,m1/M,m2/M],{x,0,2}]/.SP[S1,L]->SP[S1,S1]^(1/2)/.SP[S2,L]->SP[S2,S2]^(1/2)/.SP[S1,S1]^(1/2)->(SP[S1,S2]SP[S2,S2]^(-1/2))]-((-5/8(M/m1)^2)SP[S1,S1]+(-5/8(M/m2)^2)SP[S2,S2]+(-395/4/η)SP[S1,S2]+κ1(-50(M/m1)^2)SP[S1,S1]+κ2(-50(M/m2)^2)SP[S2,S2])]]

ENS[ENS[SeriesCoefficient[psi[η,x,m1/M,m2/M],{x,0,3}]/.SP[S1,L]->SP[S1,S1]^(1/2)/.SP[S2,L]->SP[S2,S2]^(1/2)/.SP[S1,S1]^(1/2)->(SP[S1,S2]SP[S2,S2]^(-1/2))]-((-410825/672(M/m1)^2-1085/12M/m1+1255/36)SP[S1,S1]+(-410825/672(M/m2)^2-1085/12M/m2+1255/36)SP[S2,S2]+(32675/112/η+5575/18)SP[S1,S2]+κ1(47035/84(M/m1)^2+2935/6M/m1-120)SP[S1,S1]+κ2(47035/84(M/m2)^2+2935/6M/m2-120)SP[S2,S2])]

SeriesCoefficient[(1+2a1 x+5/2a1p5 x^(3/2)+3a2 x^2+4a3 x^3)/(1+b1 x+b1p5 x^(3/2)+b2 x^2+b3 x^3),{x,0,3}]

From eq. 6 a of 1601.05588

TF2PhasingSpinSquared=ENS[(*((2270*Pi/3-520*Pi*η)*χsdl+δ*2270*Pi/3*χadl*)δ(75515/144 - (8225*η)/18)*χadl*χsdl+(75515/288 - (263245*η)/252 - 480*η^2)*χadl^2 +(75515/288 - (232415*η)/504 + (1255*η^2)/9)*χsdl^2/.χadl^2->(SP[S1,S1]+SP[S2,S2]-2SP[S1,S2])/4/.χsdl^2->(SP[S1,S1]+SP[S2,S2]+2SP[S1,S2])/4/.χadl*χsdl->(SP[S1,S1]-SP[S2,S2])/4/.δ->(m1-m2)/(m1+m2)];
ENS[SeriesCoefficient[TF2PhasingSpinSquared*(m1+m2)^4/(m1 m2)^2,{SP[S1,S2],0,1}]]
ENS[SeriesCoefficient[TF2PhasingSpinSquared*(m1+m2)^4/m1^4,{SP[S1,S1],0,1}]]

ENS[((-410825/672(M/m1)^2-1085/12M/m1+1255/36)SP[S1,S1]+(-410825/672(M/m2)^2-1085/12M/m2+1255/36)SP[S2,S2]+(32675/112/η+5575/18)SP[S1,S2]+κ1(47035/84(M/m1)^2+2935/6M/m1-120)SP[S1,S1]+κ2(47035/84(M/m2)^2+2935/6M/m2-120)SP[S2,S2])/.κ1->1/.κ2->1/.η->m1 m2/(m1+m2)^2/.δ->(m1-m2)/(m1+m2)]

TF28Phasing = (3*(1 + v^2*(3715/756 + (55*η)/9) + 
      v^3*(-16*Pi + (113*δ*χadl)/3 + (113/3 - (76*η)/3)*
         χsdl) + v^4*(15293365/508032 + (27145*η)/504 + 
        (3085*η^2)/72 + (-405/8 + 200*η)*χadl^2 - 
        (405*δ*χadl*χsdl)/4 + (-405/8 + (5*η)/2)*
         χsdl^2) + v^7*((77096675*Pi)/254016 + (378515*Pi*η)/1512 - 
        (74045*Pi*η^2)/756 + (-25150083775/3048192 + 
          (10566655595*η)/762048 - (1042165*η^2)/3024 + 
          (5345*η^3)/36 + (14585/8 - 7270*η + 80*η^2)*
           χadl^2)*χsdl + (14585/24 - (475*η)/6 + 
          (100*η^2)/3)*χsdl^3 + δ*
         ((-25150083775/3048192 + (26804935*η)/6048 - 
            (1985*η^2)/48)*χadl + (14585/24 - 2380*η)*
           χadl^3 + (14585/8 - (215*η)/2)*χadl*χsdl^2)) + 
      v^6*(11583231236531/4694215680 - (640*Pi^2)/3 - (6848*γE)/21 - 
        (15737765635*η)/3048192 + (2255*Pi^2*η)/12 + 
        (76055*η^2)/1728 - (127825*η^3)/1296 + 
        (75515/288 - (263245*η)/252 - 480*η^2)*χadl^2 + 
        ((2270*Pi)/3 - 520*Pi*η)*χsdl + 
        (75515/288 - (232415*η)/504 + (1255*η^2)/9)*χsdl^2 + 
        δ*((2270*Pi*χadl)/3 + (75515/144 - (8225*η)/18)*
           χadl*χsdl) - (6848*Log[4])/21 - (6848*Log[v])/21) + 
      v^8*((233915*Pi*δ*χadl)/168 - 
        (99185*Pi*δ*η*χadl)/252 + (233915*Pi*χsdl)/
         168 - (3970375*Pi*η*χsdl)/2268 + 
        (19655*Pi*η^2*χsdl)/189 - (233915*Pi*δ*χadl*
          Log[v])/56 + (99185*Pi*δ*η*χadl*Log[v])/84 - 
        (233915*Pi*χsdl*Log[v])/56 + (3970375*Pi*η*χsdl*
          Log[v])/756 - (19655*Pi*η^2*χsdl*Log[v])/63) + 
      v^5*((38645*Pi)/756 - (65*Pi*η)/9 + 
        δ*(-732985/2268 - (140*η)/9)*χadl + 
        (-732985/2268 + (24260*η)/81 + (340*η^2)/9)*χsdl + 
        ((38645*Pi)/252 - (65*Pi*η)/3 + δ*(-732985/756 - 
            (140*η)/3)*χadl + (-732985/756 + (24260*η)/27 + 
            (340*η^2)/3)*χsdl)*Log[v])))/(256*v^5*η)