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Fisher matrix components for “Forecasts for Low Spin Black Hole Spectroscopy in Horndeski Gravity”

Terms involving only gravitational or scalar parameters given by expressions in Section IV A of Berti, Cardoso & Will arXiv:gr-qc/0512160

Gravity/Scalar cross terms and SNR given below

NB all terms should also include a 1/S_h factor for noise spectral density of detector.

Angular definitions

Solve[{Cos[2 phips]==βs+(Nxs^2) Cos[2 (phi0s+phips)],Cos[2 phip]==βg+(Nx^2) Cos[2 (phi0+phip)],Cos[phip-phips]==β1-Nx Nxs Cos[phi0-phi0s+phip-phips],Cos[phip+phips]==β2+Nx Nxs Cos[phi0+phi0s+phip+phips]},{β1,β2,βs,βg}]

Solve[{Sin[phip-phips]==α1-Nx Nxs Sin[phi0-phi0s+phip-phips],Sin[phip+phips]==α2+Nx Nxs Sin[phi0+phi0s+phip+phips]},{α1,α2}]

SNR

snr=(1/(40 (π^2)))((((As^2) (1+(Nxs^2)) Qs)/f0s)+(4 A As Q Qs (f0s Q+f0 Qs) ((f0^2) (1+4 (Q^2)) (Qs^2) (β1+β2)+(f0s^2) (Q^2) (1+4 (Qs^2)) (β1+β2)+2 f0 f0s Q Qs (β1+4 Q Qs β1+β2-4 Q Qs β2)))/(((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0-f0s))^2) (Q^2)) (Qs^2)) ((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0+f0s))^2) (Q^2)) (Qs^2)))+(((A^2) ((1+(Nx^2)) (Q+4 (Q^3))+Q βg))/(f0+4 f0 (Q^2)))+(((As^2) Qs βs)/(f0s+4 f0s (Qs^2))));

ΓAAs

gammaAAs=(Q Qs (f0s Q+f0 Qs) ((f0^2) (1+4 (Q^2)) (Qs^2) (β1+β2)+(f0s^2) (Q^2) (1+4 (Qs^2)) (β1+β2)+2 f0 f0s Q Qs (β1+4 Q Qs β1+β2-4 Q Qs β2)))/(20 (π^2) ((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0-f0s))^2) (Q^2)) (Qs^2)) ((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0+f0s))^2) (Q^2)) (Qs^2)));

ΓAphips

gammaAphis=(As Q Qs (f0s Q+f0 Qs) ((f0^2) (1+4 (Q^2)) (Qs^2) (α1-α2)+(f0s^2) (Q^2) (1+4 (Qs^2)) (α1-α2)+2 f0 f0s Q Qs (α1+4 Q Qs α1+(-1+4 Q Qs) α2)))/(20 (π^2) ((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0-f0s))^2) (Q^2)) (Qs^2)) ((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0+f0s))^2) (Q^2)) (Qs^2)));

ΓAf0s

gammaAf0s=(1/(20 (π^2)))As (Q^2) Qs ((f0s^2) (Q^2) (1+4 (Qs^2)) (-((β1)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0-f0s))^2) (Q^2)) (Qs^2)))^2))-((β2)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0+f0s))^2) (Q^2)) (Qs^2)))^2)))+2 f0 f0s Q Qs (1+4 (Qs^2)) (-((β1)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0-f0s))^2) (Q^2)) (Qs^2)))^2))-((β2)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0+f0s))^2) (Q^2)) (Qs^2)))^2)))+(f0^2) (Qs^2) ((((-1+4 Q (Q+2 Qs)) β1)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0-f0s))^2) (Q^2)) (Qs^2)))^2))+(((-1+4 Q (Q-2 Qs)) β2)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0+f0s))^2) (Q^2)) (Qs^2)))^2))));

ΓAQs

gammaAQs=(1/(20 (π^2)))As f0s (Q^2) ((f0^2) (-1+4 (Q^2)) (Qs^2) (-((β1)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0-f0s))^2) (Q^2)) (Qs^2)))^2))-((β2)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0+f0s))^2) (Q^2)) (Qs^2)))^2)))+(f0s^2) (Q^2) (-1+4 (Qs^2)) (-((β1)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0-f0s))^2) (Q^2)) (Qs^2)))^2))-((β2)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0+f0s))^2) (Q^2)) (Qs^2)))^2)))+2 f0 f0s Q Qs (((β1+4 Q Qs β1)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0-f0s))^2) (Q^2)) (Qs^2)))^2))+((β2-4 Q Qs β2)/((((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0+f0s))^2) (Q^2)) (Qs^2)))^2))));

ΓϕpAs

gammaphiAs=-((A Q Qs (f0s Q+f0 Qs) ((f0^2) (1+4 (Q^2)) (Qs^2) (α1+α2)+(f0s^2) (Q^2) (1+4 (Qs^2)) (α1+α2)+2 f0 f0s Q Qs (α1+4 Q Qs α1+α2-4 Q Qs α2)))/(20 (π^2) ((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0-f0s))^2) (Q^2)) (Qs^2)) ((f0s^2) (Q^2)+2 f0 f0s Q Qs+((f0^2)+4 (((f0+f0s))^2) (Q^2)) (Qs^2))));

Γϕpϕps

gammaphiphis=(A As Q Qs (f0s Q+f0 Qs) ((f0^2) (1+4 (Q^2)) (Qs^2) (β1-β2)+(f0s^2) (Q^2) (1+4 (Qs^2)) (β1-β2)+2 f0 f0s Q Qs (β1+4 Q Qs β1+(-1+4 Q Qs) β2)))/(20 (π^2) ((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))) ((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))));

Γϕpf0s

gammaphif0s=(1/(20 (π^2)))A As (Q^2) Qs ((f0^2) (Qs^2) ((((1-4 (Q^2)-8 Q Qs) α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+(((1-4 (Q^2)+8 Q Qs) α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+(f0s^2) (Q^2) (((α1+4 (Qs^2) α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+((α2+4 (Qs^2) α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+2 f0 f0s Q Qs (((α1+4 (Qs^2) α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+((α2+4 (Qs^2) α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))));

ΓϕpQs

gammaphiQs=(1/(20 (π^2)))A As f0s (Q^2) ((f0^2) (-1+4 (Q^2)) (Qs^2) (((α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+((α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+(f0s^2) (Q^2) (-1+4 (Qs^2)) (((α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+((α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+2 f0 f0s Q Qs (-(((1+4 Q Qs) α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+(((-1+4 Q Qs) α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))));

Γf0As

gammaf0As=(1/(20 (π^2)))A Q (Qs^2) (2 f0 f0s Q (1+4 (Q^2)) Qs (-((β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))-((β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+(f0^2) (1+4 (Q^2)) (Qs^2) (-((β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))-((β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+(f0s^2) (Q^2) ((((-1+8 Q Qs+4 (Qs^2)) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))-(((1+8 Q Qs-4 (Qs^2)) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))));

Γf0ϕps

gammaf0phis=(1/(20 (π^2)))A As Q (Qs^2) ((f0s^2) (Q^2) ((((-1+8 Q Qs+4 (Qs^2)) α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+(((1+8 Q Qs-4 (Qs^2)) α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+2 f0 f0s Q Qs (-(((1+4 (Q^2)) α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+((α2+4 (Q^2) α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+(f0^2) (Qs^2) (-(((1+4 (Q^2)) α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+((α2+4 (Q^2) α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))));

Γf0f0s

gammaf0f0s=(1/(10 (π^2)))A As (Q^2) (Qs^2) (3 f0 (f0s^2) (Q^2) (1+4 (Q^2)) Qs (1+4 (Qs^2)) (((β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+((β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+3 (f0^2) f0s Q (1+4 (Q^2)) (Qs^2) (1+4 (Qs^2)) (((β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+((β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+(f0^3) (Qs^3) (-(((1+4 (Q^2)) (-1+4 (Q^2)+8 Q Qs) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+(((1-16 (Q^4)+8 Q Qs+32 (Q^3) Qs) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+(f0s^3) (Q^3) (1+4 (Qs^2)) ((((1-8 Q Qs-4 (Qs^2)) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+(((1+8 Q Qs-4 (Qs^2)) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))));

Γf0Qs

gammaf0Qs=(1/(10 (π^2)))A As f0s (Q^2) Qs ((f0^3) (-1+16 (Q^4)) (Qs^3) (((β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+((β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+3 f0 (f0s^2) (Q^2) (1+4 (Q^2)) Qs (-1+4 (Qs^2)) (((β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+((β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+3 (f0^2) f0s Q (1+4 (Q^2)) (Qs^2) (-(((1+4 Q Qs) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+(((-1+4 Q Qs) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+(f0s^3) (Q^3) (-(((1-12 (Qs^2)+4 Q Qs (-3+4 (Qs^2))) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+(((-1+12 (Qs^2)+4 Q Qs (-3+4 (Qs^2))) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))));

ΓQAs

gammaQAs=(1/(20 (π^2)))A f0 (Qs^2) ((f0^2) (Qs^2) (((β1-4 (Q^2) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+((β2-4 (Q^2) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+2 f0 f0s Q Qs (((β1+4 Q Qs β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+((β2-4 Q Qs β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+(f0s^2) (Q^2) (((β1-4 (Qs^2) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+((β2-4 (Qs^2) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))));

ΓQϕps

gammaQphis=(1/(20 (π^2)))A As f0 (Qs^2) ((f0^2) (Qs^2) (((α1-4 (Q^2) α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+(((-1+4 (Q^2)) α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+2 f0 f0s Q Qs (((α1+4 Q Qs α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+(((-1+4 Q Qs) α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2)))+(f0s^2) (Q^2) (((α1-4 (Qs^2) α1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))+(((-1+4 (Qs^2)) α2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^2))));

ΓQf0s

gammaQf0s=(1/(10 (π^2)))A As f0 Q (Qs^2) (3 (f0^2) f0s Q (-1+4 (Q^2)) (Qs^2) (1+4 (Qs^2)) (((β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+((β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+(f0s^3) (Q^3) (-1+16 (Qs^4)) (((β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+((β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+3 f0 (f0s^2) (Q^2) Qs (1+4 (Qs^2)) (-(((1+4 Q Qs) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+(((-1+4 Q Qs) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+(f0^3) (Qs^3) (-(((1-12 (Q^2)-12 Q Qs+16 (Q^3) Qs) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+(((-1+12 (Q^2)-12 Q Qs+16 (Q^3) Qs) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))));

ΓQQs

gammaQQs=(1/(10 (π^2)))A As f0 f0s Q Qs (3 (f0^2) f0s Q (Qs^2) ((((1-4 (Q^2)+8 Q Qs) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+(((1-4 (Q^2)-8 Q Qs) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+3 f0 (f0s^2) (Q^2) Qs ((((1+8 Q Qs-4 (Qs^2)) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+(((1-8 Q Qs-4 (Qs^2)) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+(f0^3) (Qs^3) (((β1-12 (Q^2) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+((β2-12 (Q^2) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3)))+(f0s^3) (Q^3) (((β1-12 (Qs^2) β1)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1-4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))+((β2-12 (Qs^2) β2)/((((f0^2) (1+4 (Q^2)) (Qs^2)+2 f0 f0s Q Qs (1+4 Q Qs)+(f0s^2) (Q^2) (1+4 (Qs^2))))^3))));