Testing strong-field gravity with tidal Love numbers
Supplementary material

V. Cardoso, E. Franzin, A. Maselli, P. Pani and G. Raposo, “Testing strong-field gravity with tidal Love numbers,” Phys. Rev. D 95 (2017) no.8,  084014, doi:10.1103/PhysRevD.95.084014, arXiv:1701.01116 [gr-qc].

For each model, we give the quadrupolar (l=2) and octupolar (l=3) axial and polar Love numbers.
We define ξ := ((Subscript[r, 0])/(2M))-1. The first line is the full result, the second line is the expansion around ξ=0 up to order O(ξ).

Tidal Love numbers for Wormholes

Axial, l=2

(16 (((1+ξ))^3) (3+62 ξ+192 (ξ^2)+204 (ξ^3)+72 (ξ^4)+24 ξ (((1+ξ))^3) (1+3 ξ) Log[ξ]-24 ξ (((1+ξ))^3) (1+3 ξ) Log[1+ξ]))/(5 (3+22 ξ+30 (ξ^2)+12 (ξ^3)+12 ξ (((1+ξ))^3) Log[ξ]-12 ξ (((1+ξ))^3) Log[1+ξ]) (31+96 ξ+102 (ξ^2)+36 (ξ^3)+12 (((1+ξ))^3) (1+3 ξ) Log[ξ]-12 (((1+ξ))^3) (1+3 ξ) Log[1+ξ]))

(16/(5 (31+12 Log[ξ])))

Axial, l=3

(32 (((1+ξ))^3) (3+209 ξ+1947 (ξ^2)+6690 (ξ^3)+10530 (ξ^4)+7740 (ξ^5)+2160 (ξ^6)+60 ξ (((1+ξ))^3) (1+12 ξ+39 (ξ^2)+36 (ξ^3)) Log[ξ]-60 ξ (((1+ξ))^3) (1+12 ξ+39 (ξ^2)+36 (ξ^3)) Log[1+ξ]))/(7 (6+155 ξ+480 (ξ^2)+510 (ξ^3)+180 (ξ^4)+60 ξ (((1+ξ))^3) (1+3 ξ) Log[ξ]-60 ξ (((1+ξ))^3) (1+3 ξ) Log[1+ξ]) (209+1320 ξ+2730 (ξ^2)+2340 (ξ^3)+720 (ξ^4)+60 (((1+ξ))^3) (1+9 ξ+12 (ξ^2)) Log[ξ]-60 (((1+ξ))^3) (1+9 ξ+12 (ξ^2)) Log[1+ξ]))

(16/(7 (209+60 Log[ξ])))

Polar, l=2

(4/5) (1+ξ) (((4 ξ (((1+ξ))^2))/(3+2 ξ (11+3 ξ (5+2 ξ))+12 ξ (((1+ξ))^3) Log[(ξ/(1+ξ))]))+((1+2 ξ (2+ξ))/(8+3 ξ (5+2 ξ)+3 (1+ξ) (1+2 ξ (2+ξ)) Log[(ξ/(1+ξ))])))

(4/(5 (8+3 Log[ξ])))

Polar, l=3

(8 (1+ξ) (6+5 ξ (85+4 ξ (215+ξ (847+ξ (1643+30 ξ (56+ξ (29+6 ξ))))))+120 ξ (((1+ξ))^3) (1+3 ξ) (1+10 ξ (((1+ξ))^2)) Log[-1+(1/(1+ξ))]))/(35 ((21+10 ξ (11+3 ξ (5+2 ξ))) (6+5 ξ (31+6 ξ (16+ξ (17+6 ξ))))+6 (1+ξ) Log[(ξ/(1+ξ))] (6+5 ξ (85+4 ξ (215+ξ (847+ξ (1643+30 ξ (56+ξ (29+6 ξ))))))+60 ξ (((1+ξ))^3) (1+3 ξ) (1+10 ξ (((1+ξ))^2)) Log[(ξ/(1+ξ))])))

(8/(105 (7+2 Log[ξ])))

Tidal Love numbers for Gravastars

Axial, l=2

-((32 (((1+ξ))^(9/2)) (2+15 ξ-((3 ξ (4+5 ξ) ArcCoth[(Sqrt[1+ξ])])/(Sqrt[1+ξ]))))/(5 (3 ArcCoth[(Sqrt[1+ξ])] (8+ξ (112+ξ (333+2 ξ (214+3 ξ (43+10 ξ))))+12 ξ (((1+ξ))^4) (4+5 ξ) (Log[ξ]-Log[1+ξ]))-(Sqrt[1+ξ]) (86+ξ (479+6 ξ (144+ξ (109+30 ξ)))+12 (((1+ξ))^4) (2+15 ξ) Log[ξ]-12 (((1+ξ))^4) (2+15 ξ) Log[1+ξ]))))

-(32/(5 (-43+Log[4096]-18 Log[ξ])))

Axial, l=3

(32 (((1+ξ))^(9/2)) (((1+ξ (31+5 ξ (25+21 ξ)))/(Sqrt[1+ξ]))-15 ξ (1+ξ (6+7 ξ)) ArcCoth[(Sqrt[1+ξ])]))/(7 (307+5 ξ (1085+ξ (5031+ξ (10439+6 ξ (1847+5 ξ (197+42 ξ)))))+60 (((1+ξ))^4) (1+ξ (31+5 ξ (25+21 ξ))) Log[ξ]-15 (((1+ξ))^(5/2)) ArcCoth[(Sqrt[1+ξ])] (4+5 ξ (1+2 ξ) (35+6 ξ (13+7 ξ))+60 ξ (((1+ξ))^2) (1+ξ (6+7 ξ)) (Log[ξ]-Log[1+ξ]))-60 (((1+ξ))^4) (1+ξ (31+5 ξ (25+21 ξ))) Log[1+ξ]))

(32/(7 (307-60 Log[2]+90 Log[ξ])))

Polar, l=2

-((8 (1+ξ) (4 (1+ξ) (6+ξ (61+88 ξ+30 (ξ^2)))+3 ξ (Sqrt[1+ξ]) (12+ξ (39+2 ξ (18+5 ξ))) (Log[2+ξ-2 (Sqrt[1+ξ])]-Log[2+ξ+2 (Sqrt[1+ξ])])))/(5 (4 (69+ξ (373+3 ξ (203+ξ (133+30 ξ))))+3 (1+ξ) Log[ξ] (4 (1+ξ) (6+ξ (61+88 ξ+30 (ξ^2)))+3 ξ (Sqrt[1+ξ]) (12+ξ (39+2 ξ (18+5 ξ))) (Log[2+ξ-2 (Sqrt[1+ξ])]-Log[2+ξ+2 (Sqrt[1+ξ])]))-3 (1+ξ) Log[1+ξ] (4 (1+ξ) (6+ξ (61+88 ξ+30 (ξ^2)))+3 ξ (Sqrt[1+ξ]) (12+ξ (39+2 ξ (18+5 ξ))) (Log[2+ξ-2 (Sqrt[1+ξ])]-Log[2+ξ+2 (Sqrt[1+ξ])]))+3 (((1+ξ))^(3/2)) (6+ξ (73+93 ξ+30 (ξ^2))) (Log[2+ξ-2 (Sqrt[1+ξ])]-Log[2+ξ+2 (Sqrt[1+ξ])]))))

-(16/(5 (-23+Log[64]-9 Log[ξ])))

Polar, l=3

(16 ξ (((1+ξ))^3) (-7 (3+2 ξ (9+ξ (13+5 ξ)))+2 (6+ξ (42+5 ξ (15+7 ξ))) Hypergeometric2F1[1,(3/2),(9/2),(1/(1+ξ))]))/(7 (-7 (1+ξ) (6+5 ξ (52+ξ (227+2 ξ (181+3 ξ (41+10 ξ))))+30 ξ (((1+ξ))^2) (3+2 ξ (9+ξ (13+5 ξ))) Log[(ξ/(1+ξ))])+Hypergeometric2F1[1,(3/2),(9/2),(1/(1+ξ))] (24+5 ξ (220+ξ (1308+ξ (3087+2 ξ (1762+15 ξ (65+14 ξ)))))+60 ξ (((1+ξ))^3) (6+ξ (42+5 ξ (15+7 ξ))) Log[(ξ/(1+ξ))])))

-(16/(35 (-31+Log[64]-9 Log[ξ])))

Tidal Love numbers for Perfect Mirrors

Axial, l=2

((32 (((1+ξ))^4))/(5 (25+52 ξ+42 (ξ^2)+12 (ξ^3)+12 (((1+ξ))^4) Log[ξ]-12 (((1+ξ))^4) Log[1+ξ])))

(32/(5 (25+12 Log[ξ])))

Axial, l=3

((32 (((1+ξ))^4) (1+6 ξ))/(7 (197+1010 ξ+1770 (ξ^2)+1320 (ξ^3)+360 (ξ^4)+60 (((1+ξ))^4) (1+6 ξ) Log[ξ]-60 (((1+ξ))^4) (1+6 ξ) Log[1+ξ])))

(32/(7 (197+60 Log[ξ])))

Polar, l=2

((8 (7+2 ξ (12+ξ (9+2 ξ))))/(5 (49+12 ξ (4+ξ)+3 (7+2 ξ (12+ξ (9+2 ξ))) Log[ξ]-3 (7+2 ξ (12+ξ (9+2 ξ))) Log[1+ξ])))

(8/(5 (7+3 Log[ξ])))

Polar, l=3

(8 (13+10 ξ (12+ξ (27+22 ξ+6 (ξ^2)))))/(35 (10 (13+3 ξ (18+ξ (19+6 ξ)))+3 (13+10 ξ (12+ξ (27+22 ξ+6 (ξ^2)))) Log[ξ]-3 (13+10 ξ (12+ξ (27+22 ξ+6 (ξ^2)))) Log[1+ξ]))

(8/(35 (10+3 Log[ξ])))