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Stealth Schwarzschild QNMs - 1911.07593

Expressions as a function of \ell for the QNMs of stealth Schwarzschild black holes with a non-trivial scalar profile in Horndeski gravity. QNMs calculated using the analytic 1/(\ell+1/2) series method developed by Dolan et al in 0908.0329.

The potential is of the form V = l(l+1)/r^2 - 6M/r^3 + δV(ϕ1)

σ = G_{4ϕ} / G4 i.e. measure of the conformal coupling.

NB these expressions are for the fundamental overtone, and the ω function takes \ell (spherical harmonic index) as an argument.

Generic QNM expression (very long expression within - just run this section)

ω[l_]:=((-1719783649980+99272249345 I)-(7784347580280-2226667094496 I) l-(15365674722816-784548796896 I) (l^2)-(13783231226880+7115901456384 I) (l^3)+(9501654896640-14137112881152 I) (l^4)+(33422315986944-12694994583552 I) (l^5)+(29621654028288-4231664861184 I) (l^6)+8463329722368 (l^7))/(396718580736 (Sqrt[3]) (((1+2 l))^6) M)+σ (-((756571 M (δϕ)''[3 M])/(7558272 (Sqrt[3]) ((((1/2)+l))^5)))+((235 M (δϕ)''[3 M])/(2592 (Sqrt[3]) ((((1/2)+l))^3)))-((M (δϕ)''[3 M])/(4 (Sqrt[3]) ((1/2)+l)))-((100 (4 (δϕ)'[3 M]-M (δϕ)''[3 M]-6 (M^2) Derivative[3][δϕ][3 M]))/(3 (Sqrt[3]) (((1+2 l))^5)))-((316091 (2 (δϕ)'[3 M]-2 M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/(3779136 (Sqrt[3]) ((((1/2)+l))^5)))-((85 (2 (δϕ)'[3 M]-2 M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/(324 (Sqrt[3]) ((((1/2)+l))^3)))+((2 (2 (δϕ)'[3 M]-2 M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/((Sqrt[3]) (((1+2 l))^3)))+((8258 (2 (δϕ)'[3 M]+M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/(243 (Sqrt[3]) (((1+2 l))^5)))-((4 (2 (δϕ)'[3 M]+M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/(3 (Sqrt[3]) (((1+2 l))^3)))+((21739 (-2 (δϕ)'[3 M]+2 M (δϕ)''[3 M]+3 (M^2) Derivative[3][δϕ][3 M]))/(2592 (Sqrt[3]) ((((1/2)+l))^5)))+((80 (2 (Sqrt[3]) (δϕ)'[3 M]-2 (Sqrt[3]) M (δϕ)''[3 M]-3 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]))/(3 (((1+2 l))^5)))+((80 (2 (Sqrt[3]) (δϕ)'[3 M]+4 (Sqrt[3]) M (δϕ)''[3 M]-3 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]))/(27 (((1+2 l))^5)))+((2953817 (-8 (δϕ)'[3 M]+16 M (δϕ)''[3 M]-12 (M^2) Derivative[3][δϕ][3 M]-9 (M^3) Derivative[4][δϕ][3 M]))/(15116544 (Sqrt[3]) ((((1/2)+l))^5)))-((185 (-8 (δϕ)'[3 M]+16 M (δϕ)''[3 M]-12 (M^2) Derivative[3][δϕ][3 M]-9 (M^3) Derivative[4][δϕ][3 M]))/(1296 (Sqrt[3]) ((((1/2)+l))^3)))+((9038 (16 (δϕ)'[3 M]-24 M (δϕ)''[3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/(81 (Sqrt[3]) (((1+2 l))^5)))+((16 (16 (δϕ)'[3 M]-12 M (δϕ)''[3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/(3 (Sqrt[3]) (((1+2 l))^5)))-((4 (24 (δϕ)'[3 M]-32 M (δϕ)''[3 M]-12 (M^2) Derivative[3][δϕ][3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/((Sqrt[3]) (((1+2 l))^5)))-((49 (8 (δϕ)'[3 M]-16 M (δϕ)''[3 M]+12 (M^2) Derivative[3][δϕ][3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/(18 (Sqrt[3]) ((((1/2)+l))^5)))-((7084 (8 (δϕ)'[3 M]-4 M (δϕ)''[3 M]+12 (M^2) Derivative[3][δϕ][3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/(243 (Sqrt[3]) (((1+2 l))^5)))-((16 (8 (Sqrt[3]) (δϕ)'[3 M]+8 (Sqrt[3]) M (δϕ)''[3 M]+12 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]+9 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]))/(27 (((1+2 l))^5)))+(1/(839808 (Sqrt[3]) ((((1/2)+l))^5)))697801 (12 (δϕ)'[3 M]-28 M (δϕ)''[3 M]+30 (M^2) Derivative[3][δϕ][3 M]-18 (M^3) Derivative[4][δϕ][3 M]-9 (M^4) Derivative[5][δϕ][3 M])-((7 (12 (δϕ)'[3 M]-28 M (δϕ)''[3 M]+30 (M^2) Derivative[3][δϕ][3 M]-18 (M^3) Derivative[4][δϕ][3 M]-9 (M^4) Derivative[5][δϕ][3 M]))/(108 (Sqrt[3]) ((((1/2)+l))^3)))+((1754 (12 (δϕ)'[3 M]-8 M (δϕ)''[3 M]+30 (M^2) Derivative[3][δϕ][3 M]-18 (M^3) Derivative[4][δϕ][3 M]-9 (M^4) Derivative[5][δϕ][3 M]))/(81 (Sqrt[3]) (((1+2 l))^5)))+((55 (-12 (δϕ)'[3 M]+28 M (δϕ)''[3 M]-30 (M^2) Derivative[3][δϕ][3 M]+18 (M^3) Derivative[4][δϕ][3 M]+9 (M^4) Derivative[5][δϕ][3 M]))/(432 (Sqrt[3]) ((((1/2)+l))^5)))-(1/(243 (((1+2 l))^5)))1462 (76 (Sqrt[3]) (δϕ)'[3 M]-124 (Sqrt[3]) M (δϕ)''[3 M]+30 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-54 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]-27 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M])+(1/(243 (((1+2 l))^5)))532 (76 (Sqrt[3]) (δϕ)'[3 M]-164 (Sqrt[3]) M (δϕ)''[3 M]+150 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-9 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]-27 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M])+(1/(4478976 (Sqrt[3]) ((((1/2)+l))^5)))1438243 (-64 (δϕ)'[3 M]+160 M (δϕ)''[3 M]-192 (M^2) Derivative[3][δϕ][3 M]+144 (M^3) Derivative[4][δϕ][3 M]-72 (M^4) Derivative[5][δϕ][3 M]-27 (M^5) Derivative[6][δϕ][3 M])-(1/(384 (Sqrt[3]) ((((1/2)+l))^3)))(-64 (δϕ)'[3 M]+160 M (δϕ)''[3 M]-192 (M^2) Derivative[3][δϕ][3 M]+144 (M^3) Derivative[4][δϕ][3 M]-72 (M^4) Derivative[5][δϕ][3 M]-27 (M^5) Derivative[6][δϕ][3 M])-(1/(324 (Sqrt[3]) (((1+2 l))^5)))637 (64 (δϕ)'[3 M]-40 M (δϕ)''[3 M]+192 (M^2) Derivative[3][δϕ][3 M]-144 (M^3) Derivative[4][δϕ][3 M]+72 (M^4) Derivative[5][δϕ][3 M]+27 (M^5) Derivative[6][δϕ][3 M])-(1/(3 (Sqrt[3]) (((1+2 l))^5)))(144 (δϕ)'[3 M]-320 M (δϕ)''[3 M]+312 (M^2) Derivative[3][δϕ][3 M]-54 (M^3) Derivative[4][δϕ][3 M]+72 (M^4) Derivative[5][δϕ][3 M]+27 (M^5) Derivative[6][δϕ][3 M])+(1/(54 (((1+2 l))^5)))193 (48 (Sqrt[3]) (δϕ)'[3 M]-80 (Sqrt[3]) M (δϕ)''[3 M]+24 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-48 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]+24 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M]+9 (Sqrt[3]) (M^5) Derivative[6][δϕ][3 M])+(1/(699840 (Sqrt[3]) ((((1/2)+l))^5)))36367 (400 (δϕ)'[3 M]-1040 M (δϕ)''[3 M]+1320 (M^2) Derivative[3][δϕ][3 M]-1080 (M^3) Derivative[4][δϕ][3 M]+630 (M^4) Derivative[5][δϕ][3 M]-270 (M^5) Derivative[6][δϕ][3 M]-81 (M^6) Derivative[7][δϕ][3 M])-(1/(18 (Sqrt[3]) (((1+2 l))^5)))(-400 (δϕ)'[3 M]+200 M (δϕ)''[3 M]-1320 (M^2) Derivative[3][δϕ][3 M]+1080 (M^3) Derivative[4][δϕ][3 M]-630 (M^4) Derivative[5][δϕ][3 M]+270 (M^5) Derivative[6][δϕ][3 M]+81 (M^6) Derivative[7][δϕ][3 M])+(1/(36 (((1+2 l))^5)))(-960 (Sqrt[3]) (δϕ)'[3 M]+1600 (Sqrt[3]) M (δϕ)''[3 M]-480 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]+1080 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]-630 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M]+270 (Sqrt[3]) (M^5) Derivative[6][δϕ][3 M]+81 (Sqrt[3]) (M^6) Derivative[7][δϕ][3 M])+(1/(746496 (Sqrt[3]) ((((1/2)+l))^5)))8675 (-960 (δϕ)'[3 M]+2560 M (δϕ)''[3 M]-3360 (M^2) Derivative[3][δϕ][3 M]+2880 (M^3) Derivative[4][δϕ][3 M]-1800 (M^4) Derivative[5][δϕ][3 M]+864 (M^5) Derivative[6][δϕ][3 M]-324 (M^6) Derivative[7][δϕ][3 M]-81 (M^7) Derivative[8][δϕ][3 M])+(1/(2592 (Sqrt[3]) ((((1/2)+l))^5)))(7840 (δϕ)'[3 M]-21280 M (δϕ)''[3 M]+28560 (M^2) Derivative[3][δϕ][3 M]-25200 (M^3) Derivative[4][δϕ][3 M]+16380 (M^4) Derivative[5][δϕ][3 M]-8316 (M^5) Derivative[6][δϕ][3 M]+3402 (M^6) Derivative[7][δϕ][3 M]-1134 (M^7) Derivative[8][δϕ][3 M]-243 (M^8) Derivative[9][δϕ][3 M])+(1/(221184 (Sqrt[3]) ((((1/2)+l))^5)))(-71680 (δϕ)'[3 M]+197120 M (δϕ)''[3 M]-268800 (M^2) Derivative[3][δϕ][3 M]+241920 (M^3) Derivative[4][δϕ][3 M]-161280 (M^4) Derivative[5][δϕ][3 M]+84672 (M^5) Derivative[6][δϕ][3 M]-36288 (M^6) Derivative[7][δϕ][3 M]+12960 (M^7) Derivative[8][δϕ][3 M]-3888 (M^8) Derivative[9][δϕ][3 M]-729 (M^9) Derivative[10][δϕ][3 M])+I (-((882390629 M (δϕ)''[3 M])/(4353564672 (Sqrt[3]) ((((1/2)+l))^6)))-((2075 M (δϕ)''[3 M])/(31104 (Sqrt[3]) ((((1/2)+l))^4)))-((5 M (δϕ)''[3 M])/(24 (Sqrt[3]) ((((1/2)+l))^2)))+((2 M (δϕ)''[3 M])/((Sqrt[3]) (((1+2 l))^4)))-((1756 (4 (δϕ)'[3 M]-M (δϕ)''[3 M]-6 (M^2) Derivative[3][δϕ][3 M]))/(3 (Sqrt[3]) (((1+2 l))^6)))-((74173691 (2 (δϕ)'[3 M]-2 M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/(1088391168 (Sqrt[3]) ((((1/2)+l))^6)))-((91 (2 (δϕ)'[3 M]-2 M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/(864 (Sqrt[3]) ((((1/2)+l))^4)))+((2 (δϕ)'[3 M]-2 M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M])/(6 (Sqrt[3]) ((((1/2)+l))^2)))+((2192 (2 (δϕ)'[3 M]-2 M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/((Sqrt[3]) (((1+2 l))^6)))+((38 (2 (δϕ)'[3 M]-2 M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/((Sqrt[3]) (((1+2 l))^4)))+((30322 (2 (δϕ)'[3 M]+M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/(729 (Sqrt[3]) (((1+2 l))^6)))+((848 (2 (δϕ)'[3 M]+4 M (δϕ)''[3 M]-3 (M^2) Derivative[3][δϕ][3 M]))/(9 (Sqrt[3]) (((1+2 l))^6)))+((474469 (-2 (δϕ)'[3 M]+2 M (δϕ)''[3 M]+3 (M^2) Derivative[3][δϕ][3 M]))/(31104 (Sqrt[3]) ((((1/2)+l))^6)))-((32 (2 (Sqrt[3]) (δϕ)'[3 M]+(Sqrt[3]) M (δϕ)''[3 M]-3 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]))/(9 (((1+2 l))^4)))-((408114131 (-8 (δϕ)'[3 M]+16 M (δϕ)''[3 M]-12 (M^2) Derivative[3][δϕ][3 M]-9 (M^3) Derivative[4][δϕ][3 M]))/(8707129344 (Sqrt[3]) ((((1/2)+l))^6)))-((17159 (-8 (δϕ)'[3 M]+16 M (δϕ)''[3 M]-12 (M^2) Derivative[3][δϕ][3 M]-9 (M^3) Derivative[4][δϕ][3 M]))/(62208 (Sqrt[3]) ((((1/2)+l))^4)))+((-8 (δϕ)'[3 M]+16 M (δϕ)''[3 M]-12 (M^2) Derivative[3][δϕ][3 M]-9 (M^3) Derivative[4][δϕ][3 M])/(48 (Sqrt[3]) ((((1/2)+l))^2)))+((425863 (16 (δϕ)'[3 M]-24 M (δϕ)''[3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/(486 (Sqrt[3]) (((1+2 l))^6)))+((104 (Sqrt[3]) (16 (δϕ)'[3 M]-12 M (δϕ)''[3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/(((1+2 l))^6))-((130 (Sqrt[3]) (24 (δϕ)'[3 M]-32 M (δϕ)''[3 M]-12 (M^2) Derivative[3][δϕ][3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/(((1+2 l))^6))-((252563 (8 (δϕ)'[3 M]-16 M (δϕ)''[3 M]+12 (M^2) Derivative[3][δϕ][3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/(10368 (Sqrt[3]) ((((1/2)+l))^6)))+((8 (δϕ)'[3 M]-16 M (δϕ)''[3 M]+12 (M^2) Derivative[3][δϕ][3 M]+9 (M^3) Derivative[4][δϕ][3 M])/(24 (Sqrt[3]) ((((1/2)+l))^4)))-((8 (8 (δϕ)'[3 M]-16 M (δϕ)''[3 M]+12 (M^2) Derivative[3][δϕ][3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/((Sqrt[3]) (((1+2 l))^6)))-((152 (8 (δϕ)'[3 M]-10 M (δϕ)''[3 M]+12 (M^2) Derivative[3][δϕ][3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/((Sqrt[3]) (((1+2 l))^6)))-((349081 (8 (δϕ)'[3 M]-4 M (δϕ)''[3 M]+12 (M^2) Derivative[3][δϕ][3 M]+9 (M^3) Derivative[4][δϕ][3 M]))/(2916 (Sqrt[3]) (((1+2 l))^6)))-((2 (16 (Sqrt[3]) (δϕ)'[3 M]-24 (Sqrt[3]) M (δϕ)''[3 M]+9 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]))/(((1+2 l))^4))+((256 (12 (Sqrt[3]) (δϕ)'[3 M]-20 (Sqrt[3]) M (δϕ)''[3 M]+6 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]+9 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]))/(3 (((1+2 l))^6)))+((8 (Sqrt[3]) (δϕ)'[3 M]-4 (Sqrt[3]) M (δϕ)''[3 M]+12 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]+9 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M])/(((1+2 l))^4))-((142 (8 (Sqrt[3]) (δϕ)'[3 M]+8 (Sqrt[3]) M (δϕ)''[3 M]+12 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]+9 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]))/(9 (((1+2 l))^6)))-(1/(9 (Sqrt[3]) (((1+2 l))^6)))268 (76 (δϕ)'[3 M]-64 M (δϕ)''[3 M]+30 (M^2) Derivative[3][δϕ][3 M]-54 (M^3) Derivative[4][δϕ][3 M]-27 (M^4) Derivative[5][δϕ][3 M])+(74490793 (12 (δϕ)'[3 M]-28 M (δϕ)''[3 M]+30 (M^2) Derivative[3][δϕ][3 M]-18 (M^3) Derivative[4][δϕ][3 M]-9 (M^4) Derivative[5][δϕ][3 M]))/(90699264 (Sqrt[3]) ((((1/2)+l))^6))-(1/(3888 (Sqrt[3]) ((((1/2)+l))^4)))1445 (12 (δϕ)'[3 M]-28 M (δϕ)''[3 M]+30 (M^2) Derivative[3][δϕ][3 M]-18 (M^3) Derivative[4][δϕ][3 M]-9 (M^4) Derivative[5][δϕ][3 M])+(1/(1458 (Sqrt[3]) (((1+2 l))^6)))302665 (12 (δϕ)'[3 M]-8 M (δϕ)''[3 M]+30 (M^2) Derivative[3][δϕ][3 M]-18 (M^3) Derivative[4][δϕ][3 M]-9 (M^4) Derivative[5][δϕ][3 M])-((2 (12 (δϕ)'[3 M]-8 M (δϕ)''[3 M]+30 (M^2) Derivative[3][δϕ][3 M]-18 (M^3) Derivative[4][δϕ][3 M]-9 (M^4) Derivative[5][δϕ][3 M]))/(3 (Sqrt[3]) (((1+2 l))^4)))+(1/(1296 (Sqrt[3]) ((((1/2)+l))^6)))9725 (-12 (δϕ)'[3 M]+28 M (δϕ)''[3 M]-30 (M^2) Derivative[3][δϕ][3 M]+18 (M^3) Derivative[4][δϕ][3 M]+9 (M^4) Derivative[5][δϕ][3 M])+(1/(9 (((1+2 l))^6)))88 (116 (Sqrt[3]) (δϕ)'[3 M]-164 (Sqrt[3]) M (δϕ)''[3 M]-30 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-54 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]-27 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M])-(1/(8748 (((1+2 l))^6)))849589 (76 (Sqrt[3]) (δϕ)'[3 M]-124 (Sqrt[3]) M (δϕ)''[3 M]+30 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-54 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]-27 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M])+(1/(9 (((1+2 l))^4)))(76 (Sqrt[3]) (δϕ)'[3 M]-124 (Sqrt[3]) M (δϕ)''[3 M]+30 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-54 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]-27 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M])-(1/(3 (((1+2 l))^6)))8 (116 (Sqrt[3]) (δϕ)'[3 M]-204 (Sqrt[3]) M (δϕ)''[3 M]+90 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-9 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]-27 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M])+(1/(243 (((1+2 l))^6)))23372 (76 (Sqrt[3]) (δϕ)'[3 M]-164 (Sqrt[3]) M (δϕ)''[3 M]+150 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-9 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]-27 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M])+(1/(9 (((1+2 l))^6)))16 (76 (Sqrt[3]) (δϕ)'[3 M]-104 (Sqrt[3]) M (δϕ)''[3 M]+150 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-9 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]-27 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M])+(1/(9 (((1+2 l))^6)))184 (4 (Sqrt[3]) (δϕ)'[3 M]+4 (Sqrt[3]) M (δϕ)''[3 M]+10 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-6 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]-3 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M])+(390438365 (-64 (δϕ)'[3 M]+160 M (δϕ)''[3 M]-192 (M^2) Derivative[3][δϕ][3 M]+144 (M^3) Derivative[4][δϕ][3 M]-72 (M^4) Derivative[5][δϕ][3 M]-27 (M^5) Derivative[6][δϕ][3 M]))/(483729408 (Sqrt[3]) ((((1/2)+l))^6))-(1/(3456 (Sqrt[3]) ((((1/2)+l))^4)))205 (-64 (δϕ)'[3 M]+160 M (δϕ)''[3 M]-192 (M^2) Derivative[3][δϕ][3 M]+144 (M^3) Derivative[4][δϕ][3 M]-72 (M^4) Derivative[5][δϕ][3 M]-27 (M^5) Derivative[6][δϕ][3 M])-(1/(1152 (Sqrt[3]) ((((1/2)+l))^6)))35 (64 (δϕ)'[3 M]-160 M (δϕ)''[3 M]+192 (M^2) Derivative[3][δϕ][3 M]-144 (M^3) Derivative[4][δϕ][3 M]+72 (M^4) Derivative[5][δϕ][3 M]+27 (M^5) Derivative[6][δϕ][3 M])-(1/(729 (Sqrt[3]) (((1+2 l))^6)))34051 (64 (δϕ)'[3 M]-40 M (δϕ)''[3 M]+192 (M^2) Derivative[3][δϕ][3 M]-144 (M^3) Derivative[4][δϕ][3 M]+72 (M^4) Derivative[5][δϕ][3 M]+27 (M^5) Derivative[6][δϕ][3 M])-(1/(9 (Sqrt[3]) (((1+2 l))^6)))8 (64 (δϕ)'[3 M]+80 M (δϕ)''[3 M]+192 (M^2) Derivative[3][δϕ][3 M]-144 (M^3) Derivative[4][δϕ][3 M]+72 (M^4) Derivative[5][δϕ][3 M]+27 (M^5) Derivative[6][δϕ][3 M])-(1/(9 (Sqrt[3]) (((1+2 l))^6)))643 (144 (δϕ)'[3 M]-320 M (δϕ)''[3 M]+312 (M^2) Derivative[3][δϕ][3 M]-54 (M^3) Derivative[4][δϕ][3 M]+72 (M^4) Derivative[5][δϕ][3 M]+27 (M^5) Derivative[6][δϕ][3 M])+(1/(486 (((1+2 l))^6)))65209 (48 (Sqrt[3]) (δϕ)'[3 M]-80 (Sqrt[3]) M (δϕ)''[3 M]+24 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-48 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]+24 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M]+9 (Sqrt[3]) (M^5) Derivative[6][δϕ][3 M])+(1/(3 (((1+2 l))^6)))8 (48 (Sqrt[3]) (δϕ)'[3 M]-40 (Sqrt[3]) M (δϕ)''[3 M]+24 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-48 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]+24 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M]+9 (Sqrt[3]) (M^5) Derivative[6][δϕ][3 M])+(1/(81 (((1+2 l))^6)))595 (136 (Sqrt[3]) (δϕ)'[3 M]-328 (Sqrt[3]) M (δϕ)''[3 M]+372 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-252 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]+18 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M]+27 (Sqrt[3]) (M^5) Derivative[6][δϕ][3 M])-(1/(3 (((1+2 l))^6)))2 (224 (Sqrt[3]) (δϕ)'[3 M]-320 (Sqrt[3]) M (δϕ)''[3 M]-48 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-144 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]+72 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M]+27 (Sqrt[3]) (M^5) Derivative[6][δϕ][3 M])-(1/(162 (Sqrt[3]) (((1+2 l))^6)))1633 (960 (δϕ)'[3 M]-1600 M (δϕ)''[3 M]+480 (M^2) Derivative[3][δϕ][3 M]-1080 (M^3) Derivative[4][δϕ][3 M]+630 (M^4) Derivative[5][δϕ][3 M]-270 (M^5) Derivative[6][δϕ][3 M]-81 (M^6) Derivative[7][δϕ][3 M])+(1/(8398080 (Sqrt[3]) ((((1/2)+l))^6)))2324023 (400 (δϕ)'[3 M]-1040 M (δϕ)''[3 M]+1320 (M^2) Derivative[3][δϕ][3 M]-1080 (M^3) Derivative[4][δϕ][3 M]+630 (M^4) Derivative[5][δϕ][3 M]-270 (M^5) Derivative[6][δϕ][3 M]-81 (M^6) Derivative[7][δϕ][3 M])-(1/(5184 (Sqrt[3]) ((((1/2)+l))^4)))19 (400 (δϕ)'[3 M]-1040 M (δϕ)''[3 M]+1320 (M^2) Derivative[3][δϕ][3 M]-1080 (M^3) Derivative[4][δϕ][3 M]+630 (M^4) Derivative[5][δϕ][3 M]-270 (M^5) Derivative[6][δϕ][3 M]-81 (M^6) Derivative[7][δϕ][3 M])-(1/(486 (Sqrt[3]) (((1+2 l))^6)))2273 (-400 (δϕ)'[3 M]+200 M (δϕ)''[3 M]-1320 (M^2) Derivative[3][δϕ][3 M]+1080 (M^3) Derivative[4][δϕ][3 M]-630 (M^4) Derivative[5][δϕ][3 M]+270 (M^5) Derivative[6][δϕ][3 M]+81 (M^6) Derivative[7][δϕ][3 M])-(1/(486 (((1+2 l))^6)))65 (904 (Sqrt[3]) (δϕ)'[3 M]-2216 (Sqrt[3]) M (δϕ)''[3 M]+2580 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-1836 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]+252 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M]-270 (Sqrt[3]) (M^5) Derivative[6][δϕ][3 M]-81 (Sqrt[3]) (M^6) Derivative[7][δϕ][3 M])+(1/(81 (((1+2 l))^6)))205 (320 (Sqrt[3]) (δϕ)'[3 M]-720 (Sqrt[3]) M (δϕ)''[3 M]+720 (Sqrt[3]) (M^2) Derivative[3][δϕ][3 M]-150 (Sqrt[3]) (M^3) Derivative[4][δϕ][3 M]+210 (Sqrt[3]) (M^4) Derivative[5][δϕ][3 M]-90 (Sqrt[3]) (M^5) Derivative[6][δϕ][3 M]-27 (Sqrt[3]) (M^6) Derivative[7][δϕ][3 M])+(17947453 (-960 (δϕ)'[3 M]+2560 M (δϕ)''[3 M]-3360 (M^2) Derivative[3][δϕ][3 M]+2880 (M^3) Derivative[4][δϕ][3 M]-1800 (M^4) Derivative[5][δϕ][3 M]+864 (M^5) Derivative[6][δϕ][3 M]-324 (M^6) Derivative[7][δϕ][3 M]-81 (M^7) Derivative[8][δϕ][3 M]))/(134369280 (Sqrt[3]) ((((1/2)+l))^6))-(1/(4608 (Sqrt[3]) ((((1/2)+l))^4)))(-960 (δϕ)'[3 M]+2560 M (δϕ)''[3 M]-3360 (M^2) Derivative[3][δϕ][3 M]+2880 (M^3) Derivative[4][δϕ][3 M]-1800 (M^4) Derivative[5][δϕ][3 M]+864 (M^5) Derivative[6][δϕ][3 M]-324 (M^6) Derivative[7][δϕ][3 M]-81 (M^7) Derivative[8][δϕ][3 M])-(1/(648 (Sqrt[3]) (((1+2 l))^6)))397 (960 (δϕ)'[3 M]-320 M (δϕ)''[3 M]+3360 (M^2) Derivative[3][δϕ][3 M]-2880 (M^3) Derivative[4][δϕ][3 M]+1800 (M^4) Derivative[5][δϕ][3 M]-864 (M^5) Derivative[6][δϕ][3 M]+324 (M^6) Derivative[7][δϕ][3 M]+81 (M^7) Derivative[8][δϕ][3 M])+(1/(324 (Sqrt[3]) (((1+2 l))^6)))115 (7360 (δϕ)'[3 M]-12160 M (δϕ)''[3 M]+3360 (M^2) Derivative[3][δϕ][3 M]-8640 (M^3) Derivative[4][δϕ][3 M]+5400 (M^4) Derivative[5][δϕ][3 M]-2592 (M^5) Derivative[6][δϕ][3 M]+972 (M^6) Derivative[7][δϕ][3 M]+243 (M^7) Derivative[8][δϕ][3 M])-(1/(36 (Sqrt[3]) (((1+2 l))^6)))(7360 (δϕ)'[3 M]-16640 M (δϕ)''[3 M]+16800 (M^2) Derivative[3][δϕ][3 M]-3600 (M^3) Derivative[4][δϕ][3 M]+5400 (M^4) Derivative[5][δϕ][3 M]-2592 (M^5) Derivative[6][δϕ][3 M]+972 (M^6) Derivative[7][δϕ][3 M]+243 (M^7) Derivative[8][δϕ][3 M])+(1/(5598720 (Sqrt[3]) ((((1/2)+l))^6)))61709 (7840 (δϕ)'[3 M]-21280 M (δϕ)''[3 M]+28560 (M^2) Derivative[3][δϕ][3 M]-25200 (M^3) Derivative[4][δϕ][3 M]+16380 (M^4) Derivative[5][δϕ][3 M]-8316 (M^5) Derivative[6][δϕ][3 M]+3402 (M^6) Derivative[7][δϕ][3 M]-1134 (M^7) Derivative[8][δϕ][3 M]-243 (M^8) Derivative[9][δϕ][3 M])-(1/(108 (Sqrt[3]) (((1+2 l))^6)))(-7840 (δϕ)'[3 M]+1120 M (δϕ)''[3 M]-28560 (M^2) Derivative[3][δϕ][3 M]+25200 (M^3) Derivative[4][δϕ][3 M]-16380 (M^4) Derivative[5][δϕ][3 M]+8316 (M^5) Derivative[6][δϕ][3 M]-3402 (M^6) Derivative[7][δϕ][3 M]+1134 (M^7) Derivative[8][δϕ][3 M]+243 (M^8) Derivative[9][δϕ][3 M])+(1/(72 (Sqrt[3]) (((1+2 l))^6)))(-21280 (δϕ)'[3 M]+34720 M (δϕ)''[3 M]-8400 (M^2) Derivative[3][δϕ][3 M]+25200 (M^3) Derivative[4][δϕ][3 M]-16380 (M^4) Derivative[5][δϕ][3 M]+8316 (M^5) Derivative[6][δϕ][3 M]-3402 (M^6) Derivative[7][δϕ][3 M]+1134 (M^7) Derivative[8][δϕ][3 M]+243 (M^8) Derivative[9][δϕ][3 M])+(1/(35831808 (Sqrt[3]) ((((1/2)+l))^6)))16865 (-71680 (δϕ)'[3 M]+197120 M (δϕ)''[3 M]-268800 (M^2) Derivative[3][δϕ][3 M]+241920 (M^3) Derivative[4][δϕ][3 M]-161280 (M^4) Derivative[5][δϕ][3 M]+84672 (M^5) Derivative[6][δϕ][3 M]-36288 (M^6) Derivative[7][δϕ][3 M]+12960 (M^7) Derivative[8][δϕ][3 M]-3888 (M^8) Derivative[9][δϕ][3 M]-729 (M^9) Derivative[10][δϕ][3 M])+(1/(331776 (Sqrt[3]) ((((1/2)+l))^6)))29 (80640 (δϕ)'[3 M]-224000 M (δϕ)''[3 M]+309120 (M^2) Derivative[3][δϕ][3 M]-282240 (M^3) Derivative[4][δϕ][3 M]+191520 (M^4) Derivative[5][δϕ][3 M]-102816 (M^5) Derivative[6][δϕ][3 M]+45360 (M^6) Derivative[7][δϕ][3 M]-16848 (M^7) Derivative[8][δϕ][3 M]+5346 (M^8) Derivative[9][δϕ][3 M]-1458 (M^9) Derivative[10][δϕ][3 M]-243 (M^10) Derivative[11][δϕ][3 M])+(1/(1474560 (Sqrt[3]) ((((1/2)+l))^6)))(-896000 (δϕ)'[3 M]+2508800 M (δϕ)''[3 M]-3494400 (M^2) Derivative[3][δϕ][3 M]+3225600 (M^3) Derivative[4][δϕ][3 M]-2217600 (M^4) Derivative[5][δϕ][3 M]+1209600 (M^5) Derivative[6][δϕ][3 M]-544320 (M^6) Derivative[7][δϕ][3 M]+207360 (M^7) Derivative[8][δϕ][3 M]-68040 (M^8) Derivative[9][δϕ][3 M]+19440 (M^9) Derivative[10][δϕ][3 M]-4860 (M^10) Derivative[11][δϕ][3 M]-729 (M^11) Derivative[12][δϕ][3 M])));

Examples

No scalar hair - i.e. recover GR spectrum

Clear[δϕ]

δϕ[r_]:=0

ω[2]//N
ω[3]//N
ω[4]//N

1/r scalar hair

Clear[δϕ]

δϕ[r_]:=Q1*(2M/r)

ω[2]//N
ω[3]//N
ω[4]//N

1/r^2 scalar hair

Clear[δϕ]

δϕ[r_]:=Q1*(2M/r)^2

ω[2]//N
ω[3]//N
ω[4]//N

1/r^(3/4) scalar hair (a la confluent Heun functions)

Clear[δϕ]

δϕ[r_]:=Q1*(2M/r)^(3/4)

ω[2]//N
ω[3]//N
ω[4]//N