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Tensor definitions

V3[a_,b_]:=FourVector[a,b,Dimension->3];
δ[i_,j_]:=MetricTensor[i,j,Dimension->3];
ENS[x_]:=Expand[Normal[Simplify[x]]];

General definitions

κp=κ1+κ2;
κm=κ1-κ2;
SS=SP[S1,S1]+SP[S2,S2]+2SP[S1,S2];
ΣΣ=(m1+m2)^2(SP[S2,S2]/m2^2+SP[S1,S1]/m1^2-2SP[S1,S2]/(m1 m2));
SΣ=(m1+m2)(SP[S2,S2]/m2-SP[S1,S1]/m1+SP[S1,S2](1/m2-1/m1));
projSS=(SS-(SP[S1,L]SP[S1,L]+SP[S2,L]SP[S2,L]+2SP[S1,L]SP[S2,L]))/2;
projΣΣ=(ΣΣ-(m1+m2)^2(SP[S2,L]SP[S2,L]/m2^2+SP[S1,L]SP[S1,L]/m1^2-2SP[S1,L]SP[S2,L]/(m1 m2)))/2;
projSΣ=(SΣ-(m1+m2)(SP[S2,L]SP[S2,L]/m2-SP[S1,L]SP[S1,L]/m1+SP[S1,L]SP[S2,L](1/m2-1/ m1)))/2;
projSSnn=(SP[S1,n]SP[S1,n]+SP[S2,n]SP[S2,n]+2SP[S1,n]SP[S2,n]);
projΣΣnn=(m1+m2)^2(SP[S2,n]SP[S2,n]/m2^2+SP[S1,n]SP[S1,n]/m1^2-2SP[S1,n]SP[S2,n]/(m1 m2));
projSΣnn=(m1+m2)(SP[S2,n]SP[S2,n]/m2-SP[S1,n]SP[S1,n]/m1+SP[S1,n]SP[S2,n](1/m2-1/ m1));
nS=SP[n,S1]+SP[n,S2];
vS=SP[v,S1]+SP[v,S2];
nΣ=(m1+m2)(SP[n,S2]/m2-SP[n,S1]/m1);
vΣ=(m1+m2)(SP[v,S2]/m2-SP[v,S1]/m1);

η=m1 m2/(m1+m2)^2;
M=m1+m2;
(*Efac=-1/2η M;*)
δe=(m1-m2)/(m1+m2);

ω - γ relationship

Eq. (3) of arXiv : astro - ph/0504538
γ=x(1+x γ1PN +x^(3/2)((M/m1)-1/3)SP[S1,L]+...+x^2/2(SP[S1,S2]-3SP[S1,L]SP[S2,L]))

From the equation a^i = -(ω^2 r) n^i
ω^2 r = M/r^2 (1 + (Mω)^2/(M/r)^2 g1PNa + M/r g1PNb + (Mω) gSO1p5PN + M^2/r^2  + (M/r)^3 gSS3PN +...) -> x^3 = γ^3 (1 + x^3/(M/r)^2 g1PNa + M/r g1PNb + x^(3/2) gSO1p5PN + γ^2 gSS2PN + 
gSS is given by (-1 times) the rhs of (A3).n of arXiv:1501.01529

2 PN

γ1PN=(1-η/3);

From eq.(A3) of arXiv : 1501.01529

gSS2PN=-1/4 (SS (-6 κp-12)+SΣ (-6 δe κp-12 δe+6 κm)+ΣΣ (3 δe κm-3 κp+η (6 κp+12))+projSS (30 κp+60)+projSΣ (30 δe κp+60 δe-30 κm)+projΣΣ (15 κp-15 δe κm+η (-30 κp-60))+projSS (-12 κp-24)+projSΣ 2 (-6 δe κp-12 δe+6 κm)+projΣΣ (6 δe κm-6 κp+η (12 κp+24)));
gSS2PNinst=-1/4 (SS (-6 κp-12)+SΣ (-6 δe κp-12 δe+6 κm)+ΣΣ (3 δe κm-3 κp+η (6 κp+12))+nS^2 (30 κp+60)+nS nΣ (30 δe κp+60 δe-30 κm)+nΣ^2 (15 κp-15 δe κm+η (-30 κp-60))+nS^2 (-12 κp-24)+2 nS nΣ (-6 δe κp-12 δe+6 κm)+nΣ^2 (6 δe κm-6 κp+η (12 κp+24)));
(*Now use that M^2ω^2 = M^3/r^3(1-M^2/(4r^2)(χ^2+..))*)
γSS2PN=-1/3gSS2PN;
γSS2PNinst=-1/3gSS2PNinst;

Checks

From (3.32) of arXiv : 1501.01529

γSS2PNcheck=SS(-κp/2-1)+SΣ(-δe κp/2-δe +κm/2)+ΣΣ(δe κm/4-κp/4+η(κp/2+1));
ENS[(γSS2PN-γSS2PNcheck)/.SP[L,S1]^2->SP[S1,S1]/.SP[L,S2]^2->SP[S2,S2]/.SP[L,S1]SP[L,S2]->SP[S1,S2]]
ENS[(γSS2PNinst-γSS2PNcheck)/.SP[S1,n]->0/.SP[S2,n]->0]

3 PN

g1PNa=1+3η(*v^2*);
g1PNb=-2(2+η)(*γ*);
g1p5PNSO=-SP[L,S1] (3M/m1-1)-SP[L,S2](3M/m2-1);
g2PNa=3/4(12+29η)(*γ^2*);
g2PNb=η(3-4η)(*v^4*);
g2PNc=-η/2(13-4η)(*γ v^2*);
γ2PN=(1-65η/12);

γSO2p5PNm2=ENS[2/9g1p5PNSO(g1PNa-2g1PNb)];
γSO2p5PN=ENS[1/9g1p5PNSO(2g1PNa+5g1PNb)];

g3PNSSa=-1/8(projSS(12δe κm-36κp-48+η(-72 κp-144)+6δe κm-6κp+84+60κp+120+η(180κp+360))+projSΣ(24κm -24δe κp+η(-36δe κp-72δe+12κm)-24δe κp-24δe +24κm+η(-36δe κp-72δe +12κm)-12δe κp+132δe+12κm+η(-24κm)+60δe κp+120δe-60κm+η(180 δe κp+360δe-180κm))+projΣΣ(24δe κm-24κp+24+η(24δe κm+24κp)+η^2(72κp+144)+6δe κm-6κp+48+η(-6δe κm+18κp-180)+30κp-30δe κm+η(-90δe κm+30κp-120)+η^2(-180κp-360))+SS(-6δe κm-6κp-48+η(-36κp-72))+SΣ(-72δe +η(-36δe κp-72δe+60κm))+ΣΣ(-24+η(24δe κm-24κp+96)+η^2(36κp+72)));
g3PNSSb=-1/4(projSS(-24δe κm+72κp+164+η(36κp+72)+48δe κm-192κp-420+η(-96κp-192))+projSΣ(48δe κp+72δe-48κm+η(18δe κp+36δe+30κm)+48δe κp+84δe-48κm+η(18δe κp+36δe+30κm)-240δe κp-396δe+240κm+η(-96δe κp-192 δe-96κm))+projΣΣ(48κp-48δe κm+η(6δe κm-102κp-148)+η^2(-36κp-72)+120δe κm-120κp+η(240κp+372)+η^2(96κp+192))+SS(-8δe κm+40κp+72+η(20κp+40))+SΣ(48δe κp+72δe-48κm+η(20δe κp+40δe+12κm))+ΣΣ(24κp-24δe κm+η(-2δe κm-46κp-72)+η^2(-20κp-40)));
γSS3PN=ENS[-1/3(g3PNSSa+g3PNSSb)+2/9g1p5PNSO^2+1/9(4 g1PNa+7g1PNb) gSS2PN];
γSS3PNm2=ENS[2/3(g3PNSSa+g3PNSSb)-1/9g1p5PNSO^2-2/9(g1PNa+4g1PNb) gSS2PN];
γSS3PN2=ENS[-2/3(g3PNSSa+g3PNSSb)+5/9g1p5PNSO^2+2/9(5g1PNa+8g1PNb) gSS2PN];

g3PNSSinsta=-1/8(nS^2(12δe κm-36κp-48+η(-72 κp-144)+60κp+120+η(180κp+360))+nS nΣ(24κm-24δe κp+η(-36δe κp-72δe+12κm)-24δe κp-24δe+24κm+η(-36δe κp-72δe+12κm)+60δe κp+120δe-60κm+η(180 δe κp+360δe-180κm))+
nΣ^2(24δe κm-24κp+24+η(24δe κm+24κp)+η^2(72κp+144)+30κp-30δe κm+η(-90δe κm+30κp-120)+η^2(-180κp-360))+vS^2(6δe κm-6κp+84)+vS vΣ(-12δe κp+132δe+12κm+η(-24κm))+
vΣ^2(6δe κm-6κp+48+η(-6δe κm+18κp-180))+SS(-6δe κm-6κp-48+η(-36κp-72))+
+SΣ(-72δe +η(-36δe κp-72δe+60κm))+ΣΣ(-24+η(24δe κm-24κp+96)+η^2(36κp+72)));
g3PNSSinstb=-1/4(nS^2(-24δe κm+72κp+164+η(36κp+72)+48δe κm-192κp-420+η(-96κp-192))+nS nΣ(48δe κp+72δe-48κm+η(18δe κp+36δe+30κm)+48δe κp+84δe-48κm+η(18δe κp+36δe+30κm)-240δe κp-396δe+240κm+η(-96δe κp-192δe-96κm))+
nΣ^2(48κp-48δe κm+η(6δe κm-102κp-148)+η^2(-36κp-72)+120δe κm-120κp+η(240κp+372)+η^2(96κp+192))+SS(-8δe κm+40κp+72+η(20κp+40))+
+SΣ(48δe κp+72δe-48κm+η(20δe κp+40δe+12κm))+ΣΣ(24κp-24δe κm+η(-2δe κm -46κp-72)+η^2(-20κp-40)));
γSS3PNinst=ENS[-1/3(g3PNSSinsta+g3PNSSinstb)+1/9(4 g1PNa+7g1PNb) gSS2PNinst+2/9  g1p5PNSO^2];
γSS3PNm2inst=ENS[2/3(g3PNSSinsta+g3PNSSinstb)-1/9g1p5PNSO^2-2/9(g1PNa+4g1PNb) gSS2PNinst];
γSS3PN2inst=ENS[-2/3(g3PNSSinsta+g3PNSSinstb)+5/9g1p5PNSO^2+2/9(5g1PNa+8g1PNb) gSS2PNinst];

Check

ENS[γ1PN-(-1/3(g1PNa+g1PNb))]

Kidder 2.2 b 9506022, minus from collecting Newtonian term in front

g1PNcheck=(1+3 η)-2 (2+η);
ENS[g1PNcheck-(g1PNa+g1PNb)]
(*Kidder 2.2c 9506022, minus from collecting Newtonian term in front*)
g1p5PNSOcheck=-5 (SP[S1,L]+SP[S2,L])-3 (m1-m2)(SP[S2,L]/m2-SP[S1,L]/m1);
(*A4 of arXiv:1501.01529*)
ENS[g1p5PNSOcheck-g1p5PNSO]

ENS[γ2PN-(-1/3(g2PNa+g2PNb+g2PNc)+1/3g1PNb(g1PNa+g1PNb))]

From eq.(3) of arXiv : astro - ph/0504538

γSO1p5PN=((M/m1)-1/3)SP[S1,L]+((M/m2)-1/3)SP[S2,L];
ENS[γSO1p5PN-(-1/3g1p5PNSO)]

γSS3PNcheck=SS(-11/12δe κm-11κp/12+14/9+η(-κp/6-1/3))+ΣΣ(1+η(δe κm-κp-2)+η^2(κp/6+1/3))+SΣ(5δe/3+η(-δe κp/6-δe/3+23κm/6));
ENS[(γSS3PNinst-γSS3PNcheck)/.SP[n,x_]->0/.SP[v,_]->0/.SP[L,x_]^2->SP[x,x]/.SP[L,S1]SP[L,S2]->SP[S1,S2]]

Energy S1^2

1 PN

ENa=-x^2/γ^2+2γ/x;
E1PNa=-3/4(1-3η)(*x^4/γ^4*);
E1PNb=-(3+η)(*x/γ*);
E1PNc=-1(*γ^2/x^2*);
E1PN=E1PNa+E1PNb+E1PNc+4γ1PN;

1.5 PN

FullForm[γSO1p5PN]

E1p5PN0=-2SP[LN,S1](-1+M/m1)-2SP[LN,S2](-1+M/m2);
E1p5PN=E1p5PN0+4 γSO1p5PN;
ENS[SeriesCoefficient[E1p5PN,{SP[LN,S1],0,1}]/.m2->MT-m1](*XLALSimInspiralPNEnergy_3PNSOCoeff()*)
ENS[SeriesCoefficient[E1p5PN,{SP[LN,S2],0,1}]/.m1->MT-m2]

2 PN

3.30 of arXiv : 1501.01529 v1

Energy2PNSS0=-1/2(SS(-2κp-4)+SΣ(-2δe κp-4δe+2κm)+ΣΣ(δe κm-κp+η(2κp+4))+projSS(6κp +12)+projΣΣ(3κp-3δe κm+η(-6κp-12))+projSΣ(6δe κp+12δe -6κm));
Energy2PNSS=Energy2PNSS0+4γSS2PN;
(*The Newtonian Energy is -1/2 η M x-2γss=-1/2ηM(x+4 γSS)*)
Energy2PNSSnoquad[S1_,S2_,L_,m1_,m2_]:=(-3/η SP[L,S1]SP[L,S2]+1/η SP[S1,S2]);(*XLALSimInspiralPNEnergy_4PNS1S2CoeffAvg(), XLALSimInspiralPNEnergy_4PNS1OS2OCoeffAvg()*)
Energy2PNκ[m_,L_,S_]:=(Mt/m)^2(-3/2SP[L,S]^2+1/2SP[S,S]);(*XLALSimInspiralPNEnergy_4PNQMS1S1CoeffAvg(), XLALSimInspiralPNEnergy_4PNQMS1OS1OCoeffAvg()*)
ENS[Energy2PNSS-(Energy2PNSSnoquad[S1,S2,L,m1,m2]+κ1 Energy2PNκ[m1,L,S1]+κ2 Energy2PNκ[m2,L,S2])/.Mt->m1+m2]

Energy2PNSS0inst=-1/2(SS(-2κp-4)+SΣ(-2δe κp-4δe+2κm)+ΣΣ(δe κm-κp+η(2κp+4))+(nS)^2(6κp +12)+(nΣ)^2(3κp-3δe κm+η(-6κp-12))+nS nΣ(6δe κp+12δe -6κm));
Energy2PNSSinst=Energy2PNSS0inst+4γSS2PNinst;
Energy2PNSSinstnoquad[S1_,S2_,n_,m1_,m2_]:=(6/η SP[n,S1]SP[n,S2]-2/η SP[S1,S2]);(*XLALSimInspiralPNEnergy_4PNS1nS2nCoeff(), XLALSimInspiralPNEnergy_4PNS1S2Coeff() *)
Energy2PNinstκ[m_,n_,S_]:=(Mt/m)^2(3SP[n,S]^2-SP[S,S]);(*XLALSimInspiralPNEnergy_4PNQMS1S1Coeff(), XLALSimInspiralPNEnergy_4PNQMS1nS1nCoeff()*)
ENS[Energy2PNSSinst-(Energy2PNSSinstnoquad[S1,S2,n,m1,m2]+κ1 Energy2PNinstκ[m1,n,S1]+κ2 Energy2PNinstκ[m2,n,S2])/.Mt->m1+m2]

Checks

(3.33) of arXiv : 1501.01529v1

E2PNSaligned[S1_,S2_,m1_,m2_]:=(SS(-κp-2)+SΣ(-δe κp-2δe+κm)+ΣΣ(δe κm/2-κp/2+η(κp+2)))
E2PNSA=(-2/η)SP[S1,S2]+κ1(-(M/m1)^2)SP[S1,S1]+κ2(-(M/m2)^2)SP[S2,S2];
ENS[(E2PNSaligned[S1,S2,m1,m2])-E2PNSA]

ENS[(Energy2PN[S1,S2,m1,m2]-E2PNSA)/.SP[L,S1]^2->SP[S1,S1]/.SP[L,S2]^2->SP[S2,S2]/.SP[L,S1]SP[L,S2]->SP[S1,S2]]
ENS[(Energy2PNinst-E2PNSA)/.SP[n,x_]->0]

Eq. 5.14e of 0812.4413 (this paper consider BHs, so κ1=κ2=1)

S0S0=SP[S1,S1](1+m2/m1)^2+SP[S2,S2](1+m1/m2)^2+2SP[S1,S2](1+m2/m1)(1+m1/m2);
projS0S0=(S0S0-((1+m2/m1)^2SP[S1,L]SP[S1,L]+(1+m1/m2)^2SP[S2,L]SP[S2,L]+2(1+m2/m1)(1+m1/m2)SP[S1,L]SP[S2,L]))/2;
Energy2PNRacine=ENS[S0S0-3projS0S0+4γSS2PN];
ENS[(Energy2PN[S1,S2,m1,m2]-Energy2PNRacine)/.κ1->1/.κ2->1]

Eq.(21) of arXiv : gr - qc/9709032 v1+eq.(6) of astro-ph/0504538v2

Energy2PNPoi=SP[S1,S2]/η-3/η SP[S1,L]SP[S2,L]+κ1(M/m1)^2(-3/2SP[L,S1]^2+1/2SP[S1,S1])+κ2(M/m2)^2(-3/2SP[L,S2]^2+1/2SP[S2,S2]);
ENS[Energy2PN[S1,S2,m1,m2]-Energy2PNPoi]

E2p5PNcheck=ENS[(11-61/9η)(SP[S1,L]+SP[S2,L])+(m1-m2)(3-10/3η)(SP[S2,L]/m2-SP[S1,L]/m1)];
ENS[SeriesCoefficient[E2p5PNcheck,{SP[S1,L],0,1}]/.m2->MT-m1]

E3p5PNcheck=ENS[(135/4-367/4η+29/12η^2)(SP[S1,L]+SP[S2,L])+(m1-m2)(27/4-39η+5/4η^2)(SP[S2,L]/m2-SP[S1,L]/m1)];
ENS[SeriesCoefficient[E3p5PNcheck,{SP[S1,L],0,1}]/.m2->MT-m1]

3 PN

(*-1/2η M x factored out*)
Energy3PN[S1_,S2_,m1_,m2_]:= -γSS3PNm2+2γSS3PN-4E1PNa γSS2PN-E1PNb γSS2PN+2E1PNc γSS2PN+E1p5PN0*γSO1p5PN+Energy2PNSS0 3γ1PN+(-1/4(SS((-2δe κm+8κp-28)+η(2κp+4))+SΣ((10δe κp-52δe-10κm)+η(2δe κp+4δe+6κm))+ΣΣ((-5δe κm+5κp-24)+η(δe κm-11κp+76)+η^2(-2κp-4))+projSS(60-18κp+η(-6κp-12)+6δe κm-6κp+28)+projΣΣ(9δe κm-9κp+48+η(3δe κm+15κp-156)+η^2(6κp+12)+6δe κm-6κp+24+η(-6δe κm+18κp-76))+projSΣ(-18δe κp+108δe+18κm+η(-6δe κp-12δe+6κm)-12δe κp+52 δe+12κm+η(-24κm)))-1/2(SS(-3δe κm+5κp+8)+SΣ(8δe κp+8δe-8κm+η(12κm))+ΣΣ(4κp-4δe κm+η(3δe κm-11κp-10))+projSS(9δe κm-15κp-36)+projΣΣ(12δe κm-12κp+η(-9δe κm+33κp+30))+projSΣ(-24δe κp-32δe+24κm+η(-36κm))));
Energy3PNnoquad[S1_,S2_,m1_,m2_]:=((6(M/m1)^2-3/2(M/m1)-11/18)SP[L,S1]^2+(-(M/m1)^2-1/6M/m1-1/2)SP[S1,S1]+(6(M/m2)^2-3/2(M/m2)-11/18)SP[L,S2]^2+(-(M/m2)^2-1/6M/m2-1/2)SP[S2,S2]+(-11/3/η+23/18)SP[L,S1]SP[L,S2]+(2/η-11/6)SP[S1,S2]);
Energy3PNk1=ENS[SeriesCoefficient[Energy3PN[S1,S2,m1,m2],{κ1,0,1}]];
Energy3PNk2=ENS[SeriesCoefficient[Energy3PN[S1,S2,m1,m2],{κ2,0,1}]];

Energy3PNSS0inst=(-1/4(SS(-2δe κm+8κp-28+η(2κp+4))+SΣ(10δe κp-52δe-10κm+η(2δe κp+4δe+6κm))+ΣΣ(-5δe κm+5κp-24+η(δe κm-11κp+76)+η^2(-2κp-4))+nS^2(60-18κp+η(-6κp-12))+
vS^2(6δe κm-6κp+28)+nΣ^2(9δe κm-9κp+48+η(3δe κm+15κp-156)+η^2(6κp+12))+vΣ^2(6δe κm-6κp+24+η(-6δe κm+18κp-76))+nS nΣ(-18δe κp+108δe+18κm+η(-6δe κp-12δe+6κm))+vS vΣ(-12δe κp+52 δe+12κm+η(-24κm)))-1/2(SS(-3δe κm+5κp+8)+SΣ(8δe κp+8δe-8κm+η(12κm))+ΣΣ(4κp-4δe κm+η(3δe κm-11κp-10))+nS^2(9δe κm-15κp-36)+nΣ^2(12δe κm-12κp+η(-9δe κm+33κp+30))+nS nΣ(-24δe κp-32δe+24κm+η(-36κm))));
Energy3PNSSinst= -γSS3PNm2inst+2γSS3PNinst-4E1PNa γSS2PNinst-E1PNb γSS2PNinst+2E1PNc γSS2PNinst+E1p5PN0*γSO1p5PN+Energy2PNSS0inst 3γ1PN+Energy3PNSS0inst;
(*XLALSimInspiralPNEnergy_6PNS1S1Coeff(), XLALSimInspiralPNEnergy_6PNS1nS1nCoeff(),
XLALSimInspiralPNEnergy_6PNS1vS1vCoeff(), XLALSimInspiralPNEnergy_6PNS1OS1OCoeff(), XLALSimInspiralPNEnergy_6PNS1S2Coeff(), XLALSimInspiralPNEnergy_6PNS1nS2nCoeff(),XLALSimInspiralPNEnergy_6PNS1vS2vCoeff(), XLALSimInspiralPNEnergy_6PNS1OS2OCoeff()*)
Energy3PNinstnoquad[S1_,S2_,m1_,Mt_,n_,v_,L_]:=((2(Mt/m1)^2-(Mt/m1)-1)SP[S1,S1]+(-8(Mt/m1)^2+11/3(Mt/m1)+1)SP[n,S1]^2+(2(Mt/m1)^2-2(Mt/m1))SP[v,S1]^2+(3(Mt/m1)^2-2/3(Mt/m1)-1/9)SP[L,S1]^2+1/2((-7/η-1/3)SP[S1,S2]+(13/η-3)SP[n,S1]SP[n,S2]+(5/η)SP[v,S1]SP[v,S2]+(16/3/η-2/9)SP[L,S1]SP[L,S2]));
(*XLALSimInspiralPNEnergy_6PNQMS1S1Coeff(), XLALSimInspiralPNEnergy_6PNQMS1nS1nCoeff(), XLALSimInspiralPNEnergy_6PNQMS1vS1vCoeff()*)
Energy3PNSSκ[S_,m_,Mt_,n_,v_,L_]:=(-5/2 (Mt/m)^2-5/2 Mt/m-5/6)SP[S,S]+(13/2(Mt/m)^2+17/2 Mt/m+5/2)SP[n,S]^2+((Mt/m)^2- Mt/m)SP[v,S]^2;
ENS[Energy3PNSSinst-(Energy3PNinstnoquad[S1,S2,m1,Mt,n,v,L]+Energy3PNinstnoquad[S2,S1,m2,Mt,n,v,L]+κ1 Energy3PNSSκ[S1,m1,Mt,n,v,L]+κ2 Energy3PNSSκ[S2,m2,Mt,n,v,L])/.η->m1 m2/Mt^2/.Mt->m1+m2]

Check

Check E1 .5 PN

ENS[E1p5PN-((2/3+2M/m1)SP[L,S1]+(2/3+2M/m2)SP[L,S2])]

Energy3PNcheck=SS(-5δe κm/3-25κp/6+50/9+η(5κp/6+5/3))+SΣ(-5δe κp/2+25δe/3+5/2κm+η(5δe κp/6+5δe/3+35κm/6))+ΣΣ(5δe κm/4-5κp/4+5+η(5δe κm/4+5κp/4-10)+η^2(-5κp/6-5/3));
ENS[ENS[Energy3PNSSinst-Energy3PNcheck]/.SP[L,S1]^2->SP[S1,S1]/.SP[L,S2]^2->SP[S2,S2]/.SP[L,S1]SP[L,S2]->SP[S1,S2]/.SP[n,x_]->0/.SP[v,x_]->0]

ENS[(Energy3PNcheck)-((5(M/m1)^2-5/3M/m1-10/9)SP[S1,S1]+(5(M/m2)^2-5/3M/m2-10/9)SP[S2,S2]+(-5/3/η-5/9)SP[S1,S2]+κ1(-5/2(M/m1)^2-5/2M/m1-5/6)SP[S1,S1]+κ2(-5/2(M/m2)^2-5/2M/m2-5/6)SP[S2,S2])]

Energy SO 3.5 PN -- Eq.(4.6) of arXiv:12125520

E3p5PNSO=ENS[(135/4-367/4η+29/12η^2)(SP[S1,LN]+SP[S2,LN])+(27/4-39η+5/4η^2)(m1-m2)(SP[S2,LN]/m2-SP[S1,LN]/m1)]

ENS[(SeriesCoefficient[E3p5PNSO,{SP[S1,LN],0,1}]-(-75/4+27/(4m1/MTot)+53 m1/MTot/2+67((m1/MTot)^2)/6+17(m1/MTot)^3/12-(m1/MTot)^4/12))/.MTot->m1+m2](*XLALSimInspiralPNEnergy_7PNSOCoeff(*)

Flux S1^2

2 PN

Qtriple[i_,j_]:=(V3[b,i]V3[x,j]+3V3[a,i]V3[v,j]+3V3[v,i]V3[a,j]+V3[x,i]V3[b,j]);
ENS[Contract[Module[{i,j},Qtriple[i,j]Qtriple[i,j]]]/.Pair[Momentum[x,3],Momentum[v,3]]->0/.Pair[Momentum[x,3],Momentum[b,3]]->0/.Pair[Momentum[v,3],Momentum[a,3]]->0/.Pair[Momentum[x,3],Momentum[x,3]]->r^2/.Pair[Momentum[x,3],Momentum[a,3]]->γ/.Pair[Momentum[v,3],Momentum[v,3]]->ω^2r^2/.Pair[Momentum[a,3],Momentum[a,3]]->γ^2/r^2/.Pair[Momentum[v,3],Momentum[b,3]]->ω^2γ/.Pair[Momentum[b,3],Momentum[b,3]]->ω^2γ^2/r^2]

Eq.(4.12) of arXiv : 1501529 v1

Ffac=1/96;(*The leading order is assumed 1 as 32/5 η^2 is factored out*)
Flux2PNSS0=Ffac(SS(288κp+576)+SΣ(288δe κp+576δe-288κm)+ΣΣ((-144δe κm+144κp+6)+η(-288κp-576))+
projSS((-1008κp-2016)+(144κp+288))+
projΣΣ((504δe κm-504κp)+η(1008 κp+2016)+(-72δe κm+72κp+2+η(-144κp-288)))+
projSΣ((-1008δe κp-2016δe+1008κm)+(144δe κp+288δe-144κm)));
Flux2PNSS=Flux2PNSS0+2γSS2PN;
Flux2PNSSselfnoquad[S_,L_,m_]:=1/96(Mt/m)^2(7SP[S,S]-SP[S,L]SP[S,L]);
(*XLALSimInspiralPNFlux_4PNS1S2CoeffAvg(), XLALSimInspiralPNFlux_4PNS1OS2OCoeffAvg()*)
Flux2PNSSnoquad[S1_,S2_,L_,m1_,m2_]:=-103/48/η SP[S1,S2]+289/48/η SP[L,S1]SP[L,S2]+Flux2PNSSselfnoquad[S1,L,m1]+Flux2PNSSselfnoquad[S2,L,m2];
Flux2PNκ[m_,L_,S_]:=3(M/m)^2SP[L,S]^2-(M/m)^2 SP[S,S];(*XLALSimInspiralPNFlux_4PNQMS1S1CoeffAvg(), XLALSimInspiralPNFlux_4PNQMS1OS1OCoeffAvg()*)
ENS[Flux2PNSS-(Flux2PNSSnoquad[S1,S2,L,m1,m2]+κ1 Flux2PNκ[m1,L,S1]+κ2 Flux2PNκ[m2,L,S2])/.Mt->m1+m2]

Flux2PNSS0inst=Ffac(SS(288κp+576)+SΣ(288δe κp+576δe-288κm)+ΣΣ((-144δe κm+144κp+6)+η(-288κp-576))+
(nS)^2(-1008κp-2016)+
(vS)^2(144κp+288)+
(nΣ)^2((504δe κm-504κp)+η(1008 κp+2016))+(vΣ)^2((-72δe κm+72κp+2)+η(-144κp-288))+nS nΣ(-1008δe κp-2016δe+1008κm)+
vS vΣ(144δe κp+288δe-144κm));
Flux2PNSSinst=Flux2PNSS0inst+2γSS2PNinst;
(*XLALSimInspiralPNFlux_4PNS1S1Coeff(), XLALSimInspiralPNFlux_4PNS1vS1vCoeff()*)
Flux2PNSSselfinstnoquad[S_,n_,v_,m_]:=1/16(Mt/m)^2SP[S,S]+1/48( Mt/m)^2SP[S,v]^2;
(*LALSimInspiralPNFlux_4PNS1S2Coeff(), XLALSimInspiralPNFlux_4PNS1nS2nCoeff(), XLALSimInspiralPNFlux_4PNS1vS2vCoeff()*)
Flux2PNSSinstnoquad[S1_,S2_,n_,v_,m1_,m2_]:=31/8/η SP[S1,S2]-15/η SP[n,S1]SP[n,S2]+71/24/η SP[v,S1]SP[v,S2]+
Flux2PNSSselfinstnoquad[S1,n,v,m1]+Flux2PNSSselfinstnoquad[S2,n,v,m2];
(*XLALSimInspiralPNFlux_4PNQMS1S1Coeff(), XLALSimInspiralPNFlux_4PNQMS1nS1nCoeff(),XLALSimInspiralPNFlux_4PNQMS1vS1vCoeff()*)
Flux2PNκInst[S_,n_,v_,m_]:=2(Mt/m)^2SP[S,S]-15/2(Mt/m)^2 SP[n,S]^2+3/2(Mt/m)^2 SP[S,v]^2;
ENS[Flux2PNSSinst-(Flux2PNSSinstnoquad[S1,S2,n,v,m1,m2]+κ1 Flux2PNκInst[S1,n,v,m1]+κ2 Flux2PNκInst[S2,n,v,m2])/.Mt->m1+m2]

Checks

Eq. 4.14 of arXiv:1501.01529v1

Flux2PNcheck[S1_,S2_,m1_,m2_]:=(SS(2κp+4)+SΣ(2δe κp+4δe-2κm)+ΣΣ(-δe κm+κp+1/16+η(-2κp-4)))
F2PNSA=((1/16(M/m1)^2)SP[S1,S1]+(1/16(M/m2)^2)SP[S2,S2]+(31/8/η)SP[S1,S2]+κ1(2(M/m1)^2)SP[S1,S1]+κ2(2(M/m2)^2)SP[S2,S2]);
ENS[Flux2PNcheck[S1,S2,m1,m2]-F2PNSA]

ENS[(Flux2PNSS-F2PNSA)/.SP[L,S1]^2->SP[S1,S1]/.SP[L,S2]^2->SP[S2,S2]/.SP[L,S1]SP[L,S2]->SP[S1,S2]]
ENS[(Flux2PNSSinst-F2PNSA)/.SP[n,x_]->0/.SP[v,x_]->0]

Eq. 5.4 of 0812.4413

Flux2PNRacine[S1_,S2_,m1_,m2_]:=2/5/(32/5)(48S0S0-8*21projS0S0+24projS0S0+ΣΣ+1/3projΣΣ)+(2γSS2PN/.κ1->1/.κ2->1)
ENS[(Flux2PN[S1, S2, m1, m2]/.κ1->1/.κ2->1)-Flux2PNRacine[S1,S2,m1,m2]]

Eq.(33) of arXiv : gr - qc/9709032

Flux2PNk1Poi[m_,S_]:=((M/m)^2(3SP[L,S]^2-SP[S,S]))
ENS[Flux2PNk1-Flux2PNk1Poi[m1,S1]]
ENS[Flux2PNk2-Flux2PNk1Poi[m2,S2]]

3 PN

fac1PN=1/336;
Flux1PNa=fac1PN(785-852η)(*x^6*);
Flux1PNb=fac1PN(-160(17-η))(*x^3γ^3*);
Flux1PNc=fac1PN(16(1-4η))(*γ^6*);
Flux1PN=Flux1PNa+Flux1PNb+Flux1PNc+2γ1PN;
fac1p5PN=1/12;
Flux1p5PN0a=fac1p5PN((-43M/m1+6)SP[L,S1]+(-43M/m2+6)SP[L,S2])(*γ^2*);
Flux1p5PN0b=fac1p5PN((4M/m1-16)SP[L,S1]+(4M/m2-16)SP[L,S2])(*γ^5*);
Flux1p5PN=Flux1p5PN0a+Flux1p5PN0b+2γSO1p5PN;

Avged

Flux3PN0=1/(32*21)(SS(366δe κm+5034κp+18276+η(-4584κp-9168))+SΣ(4668δe κp+20812δe-4668κm+η(-4584δe κp-9168δe+3120κm))+ΣΣ(-2334δe κm+2334κp+3796+η(1926δe κm-6594κp-23336)+η^2(4584κp+9168))+projSS(570δe κm-14778κp-6546+η(13632κp+27264)-1668δe κm-324κp-6478+η(120κp+240))+projΣΣ(7674δe κm-7674κp+3090+η(-7386δe κm+22734κp+10884)+η^2(-13632κp-27264)-672δe κm+672κp-2242+η(1608δe κm-2952κp+9788)+η^2(-120κp-240))+projSΣ(-15348δe κp -8664δe+15348κm+η(13632δe κp+27264δe-15912κm)+1344δe κp-8188δe-1344κm+η(120δe κp+240δe+6552κm))+SS(2572δe κm-21028κp-41832+η(720κp+1440))+SΣ(-23600δe κp-48872δe+23600κm+η(720δe κp+1440δe-11008κm))+ΣΣ(11800δe κm-11800κp-4408+η(-2932δe κm+26532κp+56288)+η^2(-720κp-1440))+projSS(-12380δe κm+77736κp+182752+η(-1992κp-3984)+4664δe κm-14652κp-27240+η(-168κp-336))+projΣΣ(-45058δe κm+45058κp+9700+η(13376δe κm-103492κp-185532)+η^2(1992κp+3984)+9658δe κm-9658κp+4784+η(-4580δe κm+23896κp+9808)+η^2(168κp+336))+projSΣ(90116δe κp+183716δe-90116κm+η(-1992δe κp-3984δe+51512κm)-19316δe κp-18480δe+19316κm+η(-168δe κp-336δe-18488κm))+SS(16δe κm+144κp+368+η(-576κp-1152))+SΣ(128δe κp+224δe-128κm+η(-576δe κp-1152δe+512κm))+ΣΣ(-64δe κm+64κp+24+η(272δe κm-400κp-384)+η^2(576κp+1152))+projSS(-48δe κm-432κp-936+η(1728κp+3456))+projΣΣ(192δe κm-192κp-16+η(-816δe κm+1200κp+928)+η^2(-1728κp-3456))+projSΣ(-384δe κp-784δe+384κm+η(1728δe κp+3456δe-1536κm)));
Flux3=γSS3PN2+(Flux1PNb 3+6Flux1PNc) γSS2PN+Flux1p5PN0a*2γSO1p5PN+Flux1p5PN0b*5γSO1p5PN+Flux2PN0*4 γ1PN+Flux3PN0;
Flux3PNnoquad=(189/16(M/m1)^2-35/144(M/m1)+47/144)SP[S1,S1]+(189/16(M/m2)^2-35/144(M/m2)+47/144)SP[S2,S2]+(-5647/168/η-2023/72) SP[L,S1]SP[L,S2]+(2123/84/η+821/72)SP[S1,S2]+(-239/16 (M/m1)^2+293/144M/m1+299/144)SP[L,S1]^2+(-239/16 (M/m2)^2+293/144M/m2+299/144)SP[L,S2]^2;
Flux3PNk1=(-837/112(M/m1)^2-135/16(M/m1)+129/8)SP[L,S1]^2+(279/112(M/m1)^2+45/16(M/m1)-43/8)SP[S1,S1];
Flux3PNk2=(-837/112(M/m2)^2-135/16(M/m2)+129/8)SP[L,S2]^2+(279/112(M/m2)^2+45/16(M/m2)-43/8)SP[S2,S2];

Inst

Flux3PNSS0insta=1/(32*21)(SS(366δe κm+5034κp+18276+η(-4584κp-9168))+SΣ(4668δe κp+20812δe-4668κm+η(-4584δe κp-9168δe+3120κm))+ΣΣ(-2334δe κm+2334κp+3796+η(1926δe κm-6594κp-23336)+η^2(4584κp+9168))+nS^2(570δe κm-14778κp-6546+η(13632κp+27264))+vS^2(-1668δe κm-324κp-6478+η(120κp+240))+nΣ^2(7674δe κm-7674κp+3090+η(-7386δe κm+22734κp+10884)+η^2(-13632κp-27264))+vΣ^2(-672δe κm+672κp-2242+η(1608δe κm-2952κp+9788)+η^2(-120κp-240))+nS nΣ(-15348δe κp -8664δe+15348κm+η(13632δe κp+27264δe-15912κm))+vS vΣ(1344δe κp-8188δe-1344κm+η(120δe κp+240δe+6552κm)));
Flux3PNSS0instb=1/(32*21)(SS(2572δe κm-21028κp-41832+η(720κp+1440))+SΣ(-23600δe κp-48872δe+23600κm+η(720δe κp+1440δe-11008κm))+ΣΣ(11800δe κm-11800κp-4408+η(-2932δe κm+26532κp+56288)+η^2(-720κp-1440))+nS^2(-12380δe κm+77736κp+182752+η(-1992κp-3984))+vS^2(4664δe κm-14652κp-27240+η(-168κp-336))+nΣ^2(-45058δe κm+45058κp+9700+η(13376δe κm-103492κp-185532)+η^2(1992κp+3984))+vΣ^2(9658δe κm-9658κp+4784+η(-4580δe κm+23896κp+9808)+η^2(168κp+336))+nS nΣ(90116δe κp+183716δe-90116κm+η(-1992δe κp-3984δe+51512κm))+vS vΣ(-19316δe κp-18480δe+19316κm+η(-168δe κp-336δe-18488κm)));
Flux3PNSS0instc=1/(32*21)(SS(16δe κm+144κp+368+η(-576κp-1152))+SΣ(128δe κp+224δe-128κm+η(-576δe κp-1152δe+512κm))+ΣΣ(-64δe κm+64κp+24+η(272δe κm-400κp-384)+η^2(576κp+1152))+nS^2(-48δe κm-432κp-936+η(1728κp+3456))+nΣ^2(192δe κm-192κp-16+η(-816δe κm+1200κp+928)+η^2(-1728κp-3456))+nS nΣ(-384δe κp-784δe+384κm+η(1728δe κp+3456δe-1536κm)));
Flux3PNSSinst=γSS3PN2inst+(Flux1PNb 3+6Flux1PNc) γSS2PNinst+Flux1p5PN0a*2γSO1p5PN+Flux1p5PN0b*5γSO1p5PN+Flux2PNSS0inst*4 γ1PN+Flux3PNSS0insta+Flux3PNSS0instb+Flux3PNSS0instc;
(*XLALSimInspiralPNFlux_6PNS1S1Coeff(),XLALSimInspiralPNFlux_6PNS1OS1OCoeff(),XLALSimInspiralPNFlux_6PNS1nS1nCoeff(),XLALSimInspiralPNFlux_6PNS1vS1vCoeff(),
XLALSimInspiralPNFlux_6PNS1S2Coeff(), XLALSimInspiralPNFlux_6PNS1OS2OCoeff(), XLALSimInspiralPNFlux_6PNS1nS2nCoeff(),XLALSimInspiralPNFlux_6PNS1vS2vCoeff()*)
Flux3PNSSinstnoquad[S1_,S2_,Mt_,m_,n_,v_,L_]=(-21/8(Mt/m)^2+215/24(Mt/m)-1/24)SP[S1,S1]+(2353/112(Mt/m)^2-647/168(Mt/m)+227/336)SP[n,S1]^2+(881/112(Mt/m)^2-3667/252(Mt/m)+61/1008)SP[v,S1]^2+(-1/2(Mt/m)^2-43/6(Mt/m)+22/9)SP[L,S1]^2+1/2((3499/42/η+11759/168) SP[n,S1]SP[n,S2]+(-389/12/η-2027/504) SP[v,S1]SP[v,S2]+(-49/6/η+44/9) SP[L,S1]SP[L,S2]+(-29/168/η-259/12)SP[S1,S2]);
(*XLALSimInspiralPNFlux_6PNQMS1S1Coeff(),XLALSimInspiralPNFlux_6PNQMS1nS1nCoeff(), XLALSimInspiralPNFlux_6PNQMS1vS1vCoeff()*)
Flux3PNSSinstκ[S_,m_,Mt_,n_,v_]:=(1541/42(Mt/m)^2+1517/168(M/m)-961/28)SP[n,S]^2+(-279/56(Mt/m)^2-45/8 Mt/m+43/4)SP[S,S]+(-3653/168(Mt/m)^2+659/84 Mt/m+29/14)SP[v,S]SP[v,S];
ENS[Flux3PNSSinst-(Flux3PNSSinstnoquad[S1,S2,Mt,m1,n,v,L]+Flux3PNSSinstnoquad[S2,S1,Mt,m2,n,v,L]+κ1 Flux3PNSSinstκ[S1,m1,Mt,n,v]+κ2 Flux3PNSSinstκ[S2,m2,Mt,n,v])/.η->m1 m2/Mt^2/.Mt->m1+m2]

Checks

ENS[Flux1PN-(-(1247/336+η 35/12))]

ENS[Flux1p5PN-((-5/4M/m1-3/2)SP[L,S1]+(-5/4M/m2-3/2)SP[L,S2])]

Flux2p5PNSO=ENS[(-9/2+272/9η)(SP[S1,L]+SP[S2,L])+(-13/16 +43/4η)(m1-m2)(SP[S2,L]/m2-SP[S1,L]/m1)];
ENS[SeriesCoefficient[Flux2p5PNSO,{SP[S1,L],0,1}]/.m2->MT-m1]

Flux3PNSO=ENS[-16 Pi(SP[S1,L]+SP[S2,L])+(-31/6 Pi)(m1-m2)(SP[S2,L]/m2-SP[S1,L]/m1)];
ENS[SeriesCoefficient[Flux3PNSO,{SP[S1,L],0,1}]/.m2->MT-m1]

Flux3p5PNSO=ENS[(476645/6804+6172/189η-2810/27η^2)(SP[S1,L]+SP[S2,L])+(9535/336+1849/126η-1501/36η^2)(m1-m2)(SP[S2,L]/m2-SP[S1,L]/m1)];
ENS[SeriesCoefficient[Flux3p5PNSO,{SP[S1,L],0,1}]/.m2->MT-m1]

Flux4PNSO=ENS[Pi((-3485/96+13789/72η)(SP[S1,L]+SP[S2,L])+(-7163/672+130583/2016η)(m1-m2)(SP[S2,L]/m2-SP[S1,L]/m1))];
ENS[SeriesCoefficient[Flux4PNSO,{SP[S1,L],0,1}]/.m2->MT-m1]

Flux3PNcheck=SS(41/16δe κm-271/112κp-5239/504+η(-43/4κp-43/2))+SΣ(-279/56δe κp-817/56δe+279/56κm+η(-43/4δe κp-43/2δe+κm/2))+ΣΣ(279/112δe κm-279/112κp-25/8+η(45/16δe κm+243/112κp+344/21)+η^2(43κp/4+43/2));
ENS[ENS[Flux3PNSS-Flux3PNcheck]/.SP[S1,L]^2->SP[S1,S1]/.SP[S2,L]^2->SP[S2,S2]/.SP[S1,L]SP[S2,L]->SP[S1,S2]/.SP[n,x_]->0/.SP[v,x_]->0]

ENS[Flux3PNcheck-((-25/8(M/m1)^2+43/24M/m1+173/72)SP[S1,S1]+(-25/8(M/m2)^2+43/24M/m2+173/72)SP[S2,S2]+(-467/56/η-601/36)SP[S1,S2]+κ1(-279/56(M/m1)^2-45/8M/m1+43/4)SP[S1,S1]+κ2(-279/56(M/m2)^2-45/8M/m2+43/4)SP[S2,S2])]

wdotTT5

En1PN[η_]:=-(3/4+η/12);
En1p5PNSO[mM_]:=(2/3+2/mM)SpOr;
En2PN[η_]:=-(27/8-19/8η+1/24η^2);
En2p5PNSO[mM_]:=(5/3+3/mM+29mM/9+mM^2/9)SpOr;
En3PN[η_]:=-(675/64-(34445/576-205/96*Pi^2)*η+155/96*η^2+35/5184*η^3);
En3p5PNSO[mM_]:=(-75/4+27/(4mM)+53mM/2+67mM^2/6+17mM^3/12-mM^4/12)SpOr;
En5PNTidal[mM_]:=-9mM^4(1-mM)Td;
En6PNTidal[mM_]:=(-33/2+11/2mM-11/2mM^2+33/2mM^3)mM^4 Td;

Fl1PN[η_]:=-(1247/336+35/12η);
Fl1p5PN=4Pi;
Fl1p5PNSO[mM_]:=(-3/2-5/4/mM )SpOr;
Fl2PN[η_]:=-(44711/9072-9271/504η-65/18η^2);
Fl2p5PNSO[mM_]:=(63/8-13/(16mM)-73mM/36-157mM^2/18)SpOr;
Fl2p5PN[η_]:=-(8191/672+583/24η)Pi;
Fl3PN[η_]:=
(6643739519/69854400+16/3Pi^2-1712/105*EulerGamma-1712/105Log[4]+(41/48Pi^2-134543/7776)η-94403/3024η^2-775/324η^3);
Fl3PNSO[mM_]:=Pi(-17/3-31/(6mM))SpOr;
Fl3p5PN[η_]:=-(16285/504-214745/1728η-193385/3024η^2)Pi;
Fl3p5PNSO[mM_]:=(380647/13608+9535/(336mM)-40115mM/756+3742mM^2/63-35mM^3/108-1117mM^4/54)SpOr;
Fl5PNTidal[mM_]:=6(3-2mM)mM^4 Td;
Fl6PNTidal[mM_]:=(-176/7-1803/28mM+643/4mM^2-155/2mM^3)mM^4 Td;

Clear[η];
wdotm1=ENS[(1+2v^2 En1PN[η]+5/2 v^3En1p5PNSO[mM]+3v^4 (En2PN[η]+Energy2PNSS )+7/2v^5En2p5PNSO[mM]+4v^6(En3PN[η]+)+9/2v^7En3p5PNSO[mM]+6v^10En5PNTidal[mM]+7v^12En6PNTidal[mM])/(1+v^2 Fl1PN[η]+v^3(Fl1p5PN+Fl1p5PNSO[mM])+v^4(Fl2PN[η]+Flux2PNSS)+v^5(Fl2p5PN[η]+Fl2p5PNSO[mM])+v^6(Fl3PN[η]+Fl3PNSO[mM])+v^7(Fl3p5PN[η]+Fl3p5PNSO[mM])+v^10Fl5PNTidal[mM]+v^12Fl6PNTidal[mM])];

wdotm1inst=ENS[(1+2x
 En1PN[η]+5/2 x^(3/2)(En1p5PNSO[m1/Mt]SP[L,S1]+En1p5PNSO[m2/Mt]SP[L,S2])/SpOr+3x^2 (En2PN[η]+Energy2PNSSinst)+7/2x^(5/2)En2p5PNSO[mM]+4x^3(En3PN[η]+Energy3PNSSinst)+9/2x^(7/2)En3p5PNSO[mM]+6x^5En5PNTidal[mM]+7x^6En6PNTidal[mM])/(1+x Fl1PN[η]+x^(3/2)(Fl1p5PN+(Fl1p5PNSO[m1/Mt]SP[L,S1]+Fl1p5PNSO[m2/Mt]SP[L,S2])/SpOr)+x^2(Fl2PN[η]+Flux2PNSSinst)+x^(5/2)(Fl2p5PN[η]+Fl2p5PNSO[mM])+x^3(Fl3PN[η]+Flux3PNSSinst+Fl3PNSO[mM])+x^(7/2)(Fl3p5PN[η]+Fl3p5PNSO[mM])+x^5Fl5PNTidal[mM]+x^6Fl6PNTidal[mM])];

1 PN

wdot1PNm1=ENS[SeriesCoefficient[wdotm1,{v,0,2}]]
wdot1PNm1Check=ENS[2En1PN[η]-Fl1PN[η]];
ENS[wdot1PNm1-wdot1PNm1Check]

1.5 PN

wdot1p5PNm1=ENS[SeriesCoefficient[wdotm1,{v,0,3}]]

(*XLALSimInspiralTaylorT2dtdv_3PNSOCoeff()*)
ENS[113/12((SpOr1+SpOr2)/2(1-76 η/113)+(m1-m2)/M(SpOr1-SpOr2)/2)/.SpOr2->0/.η->m1/M(1-m1/M)/.m2->M-m1]

2 PN

wdot2PNm1=ENS[SeriesCoefficient[wdotm1,{v,0,4}]];
η=m1 m2 /(m1+ m2)^2;
(*XLALSimInspiralTaylorT2dtdv_4PNS1S2CoeffAvg(), XLALSimInspiralTaylorT2dtdv_4PNS1OS2OCoeffAvg()
XLALSimInspiralTaylorT2dtdv_4PNS1S1CoeffAvg(), XLALSimInspiralTaylorT2dtdv_4PNS1OS1OCoeffAvg()*)
wdot2PNm1check=3058673/1016064+5429 η/1008+617 η^2/144-721/48/η SP[L,S1]SP[L,S2]+247/48/η SP[S1,S2]+1/96((M/m1)^2 (SP[L,S1]^2-7SP[S1,S1])+(M/m2)^2(SP[L,S2]^2-7SP[S2,S2]))+κ1(M/m1)^2(5/2SP[S1,S1]-15/2SP[L,S1]^2)+κ2(M/m2)^2(5/2SP[S2,S2]-15/2SP[L,S2]^2);
(*XLALSimInspiralTaylorT2dtdv_4PNQMS1S1CoeffAvg(), XLALSimInspiralTaylorT2dtdv_4PNQMS1OS1OCoeffAvg()*)
ENS[wdot2PNm1-wdot2PNm1check]

(*XLALSimInspiralTaylorT2dtdv_4PNS1S1Coeff(),XLALSimInspiralTaylorT2dtdv_4PNS1vS1vCoeff(),
XLALSimInspiralTaylorT2dtdv_4PNS1S2Coeff(),XLALSimInspiralTaylorT2dtdv_4PNS1vS2vCoeff(),XLALSimInspiralTaylorT2dtdv_4PNS1nS2nCoeff()
*)
wdot2PNm1inst=ENS[SeriesCoefficient[wdotm1inst,{x,0,2}]];
wdot2PNm1instcheck=3058673/1016064+5429 η/1008+617 η^2/144+(M/m1)^2(-1/48 SP[v,S1]^2-1/16 SP[S1,S1])+(M/m2)^2(-1/48 SP[v,S2]^2-1/16 SP[S2,S2])+33/η SP[n,S1]SP[n,S2]-71/24/η SP[v,S1]SP[v,S2]-79/8/η SP[S1,S2]+κ1(M/m1)^2(33/2 SP[n,S1]^2-3/2 SP[v,S1]^2-5 SP[S1,S1])+κ2(M/m2)^2(33/2 SP[n,S2]^2-3/2 SP[v,S2]^2-5 SP[S2,S2]);
(*XLALSimInspiralTaylorT2dtdv_4PNQMS1S1Coeff(), XLALSimInspiralTaylorT2dtdv_4PNQMS1nS1nCoeff(),XLALSimInspiralTaylorT2dtdv_4PNQMS1vS1vCoeff()*)
ENS[wdot2PNm1inst-wdot2PNm1instcheck]

2.5 PN

wdot2p5PNm1=ENS[SeriesCoefficient[wdotm1,{v,0,5}]];
wdot2p5PNm1/.SpOr->0
ENS[SeriesCoefficient[wdot2p5PNm1,{SpOr,0,1}]/.η->mM(1-mM)]

ENS[1/2(SpOr1- SpOr2)(m1/M-m2/M)(7η/2+146597/2016)+(SpOr1+SpOr2)/2(-17η^2/2-1213η/18+146597/2016)/.m2->M-m1/.η->m1/M(1-m1/M)/.SpOr2->0]

3 PN

(*XLALSimInspiralTaylorT2dtdv_6PNS1OS1OCoeff(),XLALSimInspiralTaylorT2dtdv_6PNS1S1Coeff(),
XLALSimInspiralTaylorT2dtdv_6PNS1nS1nCoeff(),XLALSimInspiralTaylorT2dtdv_6PNS1vS1vCoeff(),
XLALSimInspiralTaylorT2dtdv_6PNS1S2Coeff(), XLALSimInspiralTaylorT2dtdv_6PNS1OS2OCoeff(), XLALSimInspiralTaylorT2dtdv_6PNS1nS2nCoeff(),
XLALSimInspiralTaylorT2dtdv_6PNS1vS2vCoeff(),*)
wdot3PNm1instNoκ[S1_,S2_,Mt_,m_,L_,n_,v_]:=(27565/2688(Mt/m)^2-213/16 Mt/m-173/48)SP[S1,S1]+(325/16(Mt/m)^2+107/6Mt/m+67/36)SP[L,S1]^2+(-5937/112(Mt/m)^2+1037/56 Mt/m+1117/336)SP[n,S1]^2+(85/8064(Mt/m)^2+6485/1008 Mt/m+29/504)SP[v,S1]^2+1/2((-98173/1344/η-461/24)SP[S1,S2]+(1403/24/η+67/18)SP[L,S1]SP[L,S2]+(373/3/η+9325/168)SP[n,S1]SP[n,S2]+(140699/4032/η-3211/252)SP[v,S1]SP[v,S2]);
(*XLALSimInspiralTaylorT2dtdv_6PNQMS1S1Coeff(), XLALSimInspiralTaylorT2dtdv_6PNQMS1nS1nCoeff(), 
XLALSimInspiralTaylorT2dtdv_6PNQMS1vS1vCoeff()*)
wdot3PNm1instκ[S_,Mt_,m_,n_,v_]:=(-9407/336(Mt/m)^2-587/24 Mt/m+6)SP[S,S]+(5639/84(Mt/m)^2+15745/168Mt/m+-171/7)SP[n,S]^2+(5665/336 (Mt/m)^2-1709/84 Mt/m+45/7)SP[v,S]^2;

wdot3PNm1inst=ENS[SeriesCoefficient[wdotm1inst,{x,0,3}]];

tmp=ENS[wdot3PNm1inst-(wdot3PNm1instNoκ[S1,S2,Mt,m1,L,n,v]+wdot3PNm1instNoκ[S2,S1,Mt,m2,L,n,v]+κ1 wdot3PNm1instκ[S1,Mt,m1,n,v]+κ2 wdot3PNm1instκ[S2,Mt,m2,n,v])/.κ1->0/.κ2->0/.η->m1 m2/Mt^2/.Mt->m1+m2];
ENS[SeriesCoefficient[tmp,{SP[S1,S1],0,1}]]
ENS[SeriesCoefficient[tmp,{SP[L,S1],0,2}]]
ENS[SeriesCoefficient[tmp,{SP[n,S1],0,2}]]
ENS[SeriesCoefficient[tmp,{SP[v,S1],0,2}]]
Print["***"];
ENS[SeriesCoefficient[tmp,{SP[S1,S2],0,1}]]
ENS[SeriesCoefficient[tmp,{SP[L,S1],0,1}]]
ENS[SeriesCoefficient[tmp,{SP[n,S1],0,1}]]
ENS[SeriesCoefficient[tmp,{SP[v,S1],0,1}]]
ENS[tmp]

ENS[SeriesCoefficient[wdot3PNm1inst,{κ1,0,1}]-wdot3PNm1instκ[S1,Mt,m1,n,v]/.η->m1 m2 /(m1+m2)^2/.m2->Mt-m1]
ENS[SeriesCoefficient[wdot3PNm1inst,{κ2,0,1}]-wdot3PNm1instκ[S2,Mt,m2,n,v]/.η->m1 m2 /(m1+m2)^2/.m1->Mt-m2]

3.5 PN

wdot3p5PNm1=ENS[SeriesCoefficient[wdotm1,{v,0,7}]];
wdot3p5PNm1/.SpOr->0
ENS[SeriesCoefficient[wdot3p5PNm1,{SpOr,0,1}]/.η->mM(1-mM)]
ENS[SeriesCoefficient[wdot3p5PNm1,{SpOr,0,2}]]
ENS[SeriesCoefficient[wdot3p5PNm1,{SpOr,0,3}]]
ENS[SeriesCoefficient[wdot3p5PNm1,{SpOr,0,4}]]
ENS[-Fl1p5PN(3En2PNSS-2Fl2PNSS)]
ENS[-Fl1p5PNSO[mM](3 En2PNSS-2Fl2PNSS)-5/2En1p5PNSO[mM] Fl2PNSS]

linfit=a*x+b;
x={0.1,0.2,0.3,0.4,0.5,0.6,0.7};
yw={-44.53594134653,-32.52717259115,-20.51840383577,-8.50963508039,3.49913367499,15.50790243037,15.50790243037};
yef={-44.75649466685,-33.39624524743,-22.44425090614,-11.88080070548,-1.68618370797,8.15931102391,17.67539442765};

(yw[[1]]-yw[[2]])/0.1
(yw[[2]]-yw[[3]])/0.1
(yw[[3]]-yw[[4]])/0.1
(yef[[1]]-yef[[2]])/0.1
(yef[[2]]-yef[[3]])/0.1
(yef[[3]]-yef[[4]])/0.1

Tidal

wdotTd5PNm1=ENS[SeriesCoefficient[SeriesCoefficient[wdotm1,{v,0,10}],{Td,0,1}]]
wdotTd6PNm1=ENS[SeriesCoefficient[SeriesCoefficient[wdotm1,{v,0,12}],{Td,0,1}]/.η->mM(1-mM)]

wdotTT4

2 PN

wdot2PNcheck=ENS[SeriesCoefficient[(1+x*x Flux2PNSS)/(1+3x*x Energy2PNSS),{x,0,2}]];
(*XLALSimInspiralTaylorT4wdot_4PNS1S1CoeffAvg(),XLALSimInspiralTaylorT4wdot_4PNS1OS1OCoeffAvg(),XLALSimInspiralTaylorT4wdot_4PNS1S2CoeffAvg(), XLALSimInspiralTaylorT4wdot_4PNS1OS2OCoeffAvg()*)
wdot2PNSSnoquad=(M/m1)^2(-1/96SP[L,S1]^2+7/96SP[S1,S1])+(M/m2)^2(-1/96SP[L,S2]^2+7/96SP[S2,S2])+721/48/η SP[L,S1]SP[L,S2]-247/48/η SP[S1,S2];
(*XLALSimInspiralTaylorT4wdot_4PNQMS1S1CoeffAvg(),
XLALSimInspiralTaylorT4wdot_4PNQMS1OS1OCoeffAvg()*)
wdot2PNk1=ENS[SeriesCoefficient[wdot2PNcheck,{κ1,0,1}]];
wdot2PNk2=ENS[SeriesCoefficient[wdot2PNcheck,{κ2,0,1}]];
ENS[(wdot2PNcheck/.κ1->0/.κ2->0)-wdot2PNSSnoquad]
ENS[wdot2PNk1-5/2(M/m1)^2(3SP[L,S1]^2-SP[S1,S1])]
ENS[wdot2PNk2-5/2(M/m2)^2(3SP[L,S2]^2-SP[S2,S2])]

wdot2PNinstcheck=ENS[SeriesCoefficient[(1+x*x Flux2PNSSinst)/(1+3x*x Energy2PNSSinst),{x,0,2}]];
(*XLALSimInspiralTaylorT4wdot_4PNS1S2Coeff(), XLALSimInspiralTaylorT4wdot_4PNS1nS2nCoeff(), XLALSimInspiralTaylorT4wdot_4PNS1vS2vCoeff(),
XLALSimInspiralTaylorT4wdot_4PNS1S1Coeff(),XLALSimInspiralTaylorT4wdot_4PNS1vS1vCoeff()*)
wdot2PNSSinstnoquad=(M/m1)^2(1/16 SP[S1,S1]+1/48SP[v,S1]^2)+(M/m2)^2(1/16 SP[S2,S2]+1/48SP[v,S2]^2)+79/8/η SP[S1,S2]-33/η SP[n,S1]SP[n,S2]+71/24/η SP[v,S1]SP[v,S2];
ENS[(wdot2PNinstcheck/.κ1->0/.κ2->0)-wdot2PNSSinstnoquad]
wdot2PNk1=ENS[SeriesCoefficient[wdot2PNinstcheck,{κ1,0,1}]];
wdot2PNk2=ENS[SeriesCoefficient[wdot2PNinstcheck,{κ2,0,1}]];
(*XLALSimInspiralTaylorT4wdot_4PNQMS1S1Coeff(),XLALSimInspiralTaylorT4wdot_4PNQMS1nS1nCoeff(),XLALSimInspiralTaylorT4wdot_4PNQMS1vS1vCoeff()*)
ENS[wdot2PNk1-(M/m1)^2(5SP[S1,S1]-33/2 SP[n,S1]^2+3/2SP[v,S1]^2)]
ENS[wdot2PNk2-(M/m2)^2(5SP[S2,S2]-33/2 SP[n,S2]^2+3/2SP[v,S2]^2)]

Checks

ENS[wdot2PN-wdot2PNcheck]

Eqs. 7 - 9 of arXiv : gr - qc/0504538 v2

wdot2PNHun=(721/48/η SP[L,S1]SP[L,S2]-247/48/η SP[S1,S2]-1/96(M/m1)^2SP[L,S1]^2-1/96(M/m2)^2SP[L,S2]^2+7/96(M/m1)^2SP[S1,S1]+7/96(M/m2)^2SP[S2,S2]+κ1(M/m1)^2(15/2SP[L,S1]^2-5/2SP[S1,S1])+κ2(M/m2)^2(15/2SP[L,S2]^2-5/2SP[S2,S2]));
ENS[(wdot2PNcheck-wdot2PNHun)]

Eqs 5.15 - 5.17 of arXiv : 0812 : 4413 v2

wdot2PNRacine=(721/48/η SP[L,S1]SP[L,S2]-247/48/η SP[S1,S2]+1/96(M/m1)^2(719SP[L,S1]^2-233SP[S1,S1])+1/96(M/m2)^2(719SP[L,S2]^2-233SP[S2,S2]));
ENS[(wdot2PNHun/.κ1->1/.κ2->1)-wdot2PNRacine]

3 PN

wdot3PN=ENS[SeriesCoefficient[(1+x Flux1PN+x^(3/2) Flux1p5PN+x^2 Flux2PN[S1,S2,m1,m2]+x*x*x Flux3PN[S1,S2,m1,m2])/(1+2x E1PN+5/2x^(3/2)E1p5PN+3x*x Energy2PN[S1,S2,m1,m2]+4x*x*x Energy3PN[S1,S2,m1,m2]),{x,0,3}]];

wdot3PNnoquad=(-493/64(M/m1)^2+19747/576M/m1+5645/576)SP[L,S1]^2+(-493/64(M/m2)^2+19747/576M/m2+5645/576)SP[L,S2]^2+(10879/672/η+7525/288)SP[S1,S2]+(1019/64(M/m1)^2+251/576M/m1+1333/576)SP[S1,S1]+(1019/64(M/m2)^2+251/576M/m2+1333/576)SP[S2,S2]+(16225/224/η-12931/288)SP[L,S1]SP[L,S2];
ENS[(wdot3PNcheck/.κ1->0/.κ2->0)-wdot3PNnoquad]
wdot3PNk1=ENS[SeriesCoefficient[wdot3PN,{κ1,0,1}]];
wdot3PNk2=ENS[SeriesCoefficient[wdot3PN,{κ2,0,1}]];
ENS[wdot3PNk1-((1977/224(M/m1)^2-73/16M/m1+129/4)SP[L,S1]^2+(-659/224 (M/m1)^2+73/48M/m1-43/4)SP[S1,S1])]
ENS[wdot3PNk2-((1977/224(M/m2)^2-73/16M/m2+129/4)SP[L,S2]^2+(-659/224 (M/m2)^2+73/48M/m2-43/4)SP[S2,S2])]

wdot4PNSO=ENS[SeriesCoefficient[SeriesCoefficient[(1+x F1PN+x^(3/2) (F1p5PN+λ F1p5PNSO)+x^2F2PN+x^(5/2)(F2p5PN+λ F2p5PNSO)+x^3λ F3PNSO+x^4λ F4PNSO)/(1+2x E1PN+5/2x^(3/2)λ E1p5PNSO+3x*x E2PN+7/2x^(5/2)λ E2p5PNSO+4x^3 (E3PN+λ E3PNSO)+5x^4λ E4PNSO),{x,0,4}],{λ,0,1}]]

wdot3PN=ENS[SeriesCoefficient[(1+x Flux1PN+x^(3/2) Flux1p5PN+x^2 Flux2PNSSinst+x^3 Flux3PNSSinst)/(1+2x E1PN+5/2x^(3/2)E1p5PN+3x^2 Energy2PNSSinst+4x^3 Energy3PNSSinst),{x,0,3}]];
wdot3PNcheck=ENS[Flux3PNSSinst-4Energy3PNSSinst+(-3Flux1PN Energy2PNSSinst-2Flux2PNSSinst E1PN+12E1PN Energy2PNSSinst+25/4E1p5PN^2-5/2E1p5PN Flux1p5PN)];
ENS[wdot3PN-wdot3PNcheck]

(*XLALSimInspiralTaylorT4wdot_6PNS1S2Coeff(),XLALSimInspiralTaylorT4wdot_6PNS1OS2OCoeff(),
XLALSimInspiralTaylorT4wdot_6PNS1nS2nCoeff(),XLALSimInspiralTaylorT4wdot_6PNS1vS2vCoeff()
XLALSimInspiralTaylorT4wdot_6PNS1S1Coeff(),XLALSimInspiralTaylorT4wdot_6PNS1OS1OCoeff()
XLALSimInspiralTaylorT4wdot_6PNS1nS1nCoeff(),XLALSimInspiralTaylorT4wdot_6PNS1vS1vCoeff()*)
wdot3PNSSnoκ[S1_,S2_,Mt_,m_,L_,n_,v_]:=1/2((9869/336/η-1685/48)SP[S1,S2]+(3631/168/η+21167/168)SP[n,S1]SP[n,S2]+(-2303/48/η-3557/1008)SP[v,S1]SP[v,S2]+(237/4/η+49/3)SP[L,S1]SP[L,S2])+(-337/32(Mt/m)^2+415/32Mt/m+379/96)SP[S1,S1]+(5937/112(Mt/m)^2-1037/56Mt/m-1117/336)SP[n,S1]^2+(75/4(Mt/m)^2+87/4Mt/m+49/6)SP[L,S1]^2+(-23/224(Mt/m)^2-13201/2016Mt/m+115/2016)SP[v,S1]^2;
(*XLALSimInspiralTaylorT4wdot_6PNQMS1S1Coeff(),XLALSimInspiralTaylorT4wdot_6PNQMS1nS1nCoeff(),
XLALSimInspiralTaylorT4wdot_6PNQMS1vS1vCoeff()*)
wdot3PNSSκ[S_,Mt_,m_,n_,v_]:=(659/112(Mt/m)^2-73/24 Mt/m+43/2)SP[S,S]+(1963/336(Mt/m)^2-499/168 Mt/m-1857/28)SP[n,S]^2+(-3947/168(Mt/m)^2+254/21 Mt/m+51/28)SP[v,S]^2;
ENS[wdot3PN-(wdot3PNSSnoκ[S1,S2,Mt,m1,L,n,v]+wdot3PNSSnoκ[S2,S1,Mt,m2,L,n,v]+κ1 wdot3PNSSκ[S1,Mt,m1,n,v]+κ2 wdot3PNSSκ[S2,Mt,m2,n,v])/.η->m1 m2/Mt^2/.Mt->m1+m2]

wdotTT2

2 PN

dtdv2PNcheck=ENS[SeriesCoefficient[(1+3x*x Energy2PNSS)/(1+x*x Flux2PNSS),{x,0,2}]];
dtdv2PNnoquad=(M/m1)^2(1/96SP[L,S1]^2-7/96SP[S1,S1])+(M/m2)^2(1/96SP[L,S2]^2-7/96SP[S2,S2])-721/48/η SP[L,S1]SP[L,S2]+247/48/η SP[S1,S2];
dtdv2PNκ1=ENS[SeriesCoefficient[dtdv2PNcheck,{κ1,0,1}]];
dtdv2PNκ2=ENS[SeriesCoefficient[dtdv2PNcheck,{κ2,0,1}]];

dtdv2PNinstcheck=ENS[SeriesCoefficient[(1+3x*x Energy2PNSSinst)/(1+x*x Flux2PNSSinst),{x,0,2}]];
dtdv2PNinstnoquad=(M/m1)^2(-1/48 SP[v,S1]^2-1/16 SP[S1,S1])+(M/m2)^2(-1/48 SP[v,S2]^2-1/16 SP[S2,S2])+33/η SP[n,S1]SP[n,S2]-71/24/η SP[v,S1]SP[v,S2]-79/8/η SP[S1,S2];
dtdv2PNinstκ1=ENS[SeriesCoefficient[dtdv2PNinstcheck,{κ1,0,1}]];
dtdv2PNinstκ2=ENS[SeriesCoefficient[dtdv2PNinstcheck,{κ2,0,1}]];

Check

ENS[(dtdv2PNcheck/.κ1->0/.κ2->0)-dtdv2PNnoquad]
ENS[dtdv2PNκ1-5/2(M/m1)^2(-3SP[L,S1]^2+SP[S1,S1])]
ENS[dtdv2PNκ2-5/2(M/m2)^2(-3SP[L,S2]^2+SP[S2,S2])]

ENS[(dtdv2PNinstcheck/.κ1->0/.κ2->0)-dtdv2PNinstnoquad]
ENS[dtdv2PNinstκ1-(M/m1)^2(33/2 SP[n,S1]^2-3/2 SP[v,S1]^2-5 SP[S1,S1])]
ENS[dtdv2PNinstκ2-(M/m2)^2(33/2 SP[n,S2]^2-3/2 SP[v,S2]^2-5 SP[S2,S2])]

dtdv2PNSA=ENS[SeriesCoefficient[(1+3x^2 E2PNSA)/(1+x^2 F2PNSA),{x,0,2}]];
ENS[dtdv2PNSA-dtdv2PNcheck/.SP[L,S1]^2->SP[S1,S1]/.SP[L,S2]^2->SP[S2,S2]/.SP[L,S1]SP[L,S2]->SP[S1,S2]]
ENS[dtdv2PNSA-dtdv2PNinstcheck/.SP[n,x_]->0/.SP[v,x_]->0]

3 PN v. T5

dtdv3PN=ENS[SeriesCoefficient[(1+2x E1PN+5/2x^(3/2)E1p5PN+3x*x Energy2PN[S1,S2,m1,m2]+4x *x*x Energy3PN[S1,S2,m1,m2])/(1+x Flux1PN+x^(3/2) Flux1p5PN+x*x Flux2PN[S1,S2,m1,m2]+x*x*x Flux3PN[S1,S2,m1,m2]),{x,0,3}]];
dtdv3PNcheck=ENS[4Energy3PN[S1,S2,m1,m2]-Flux3PN[S1,S2,m1,m2]+(-3Flux1PN Energy2PN[S1,S2,m1,m2]-2Flux2PN[S1,S2,m1,m2]E1PN+2Flux1PN Flux2PN[S1,S2,m1,m2]+Flux1p5PN^2-5/2E1p5PN Flux1p5PN)+(2E1PN*Flux1PN^2-Flux1PN^3)];
ENS[dtdv3PN-dtdv3PNcheck]

dtdv3PNnoquad=((754979/16128(M/m1)^2+1543/288M/m1+49/288)SP[L,S1]^2+(754979/16128(M/m2)^2+1543/288M/m2+49/288)SP[L,S2]^2+(52973/8064/η+313/144)SP[S1,S2]+(-37427/2304(M/m1)^2-241/288M/m1-551/288)SP[S1,S1]+(-37427/2304(M/m2)^2-241/288M/m2-551/288)SP[S2,S2]+(-170603/8064/η-2543/144)SP[L,S1]SP[L,S2]);
ENS[(dtdv3PN/.κ1->0/.κ2->0)-dtdv3PNnoquad-(2E1PN*Flux1PN^2-Flux1PN^3)]
dtdv3PNk1=ENS[SeriesCoefficient[dtdv3PN,{κ1,0,1}]];
dtdv3PNk2=ENS[SeriesCoefficient[dtdv3PN,{κ2,0,1}]];
ENS[dtdv3PNk1-((-9407/224(M/m1)^2-587/16M/m1+9)SP[L,S1]^2+(9407/672(M/m1)^2+587/48M/m1-3)SP[S1,S1])]
ENS[dtdv3PNk2-((-9407/224(M/m2)^2-587/16M/m2+9)SP[L,S2]^2+(9407/672(M/m2)^2+587/48M/m2-3)SP[S2,S2])]