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Second-­‐order actions of scalar perturbations
We consider scalar metric perturbations
, ⇥, R
with the ADM metric
ds 2 = [(1 + ) 2 a(t) 2 e 2R (⇤⇥) 2 ] dt 2 +2⇤ i ⇥ dt dx i + a(t) 2 e 2R dx 2
We choose the uniform field gauge: =0
The second-­‐order action of scalar perturbations in the Horndeski’s theory reduces to
S 2 = dtd 3 xa 3 Q Ṙ2 c 2 s
a 2 ( R)2
Q>0 and c 2 s > 0 are required to avoid
ghosts and Laplacian instabilities.
where
Q = w 1(4w 1 w 3 +9w 2 2)
3w 2 2
, c 2 s = 3(2w2 1w 2 H w 2 2w 4 +4w 1 ẇ 1 w 2 2w 2 1ẇ 2 )
w 1 (4w 1 w 3 +9w 2 2 )
w 1 = MplF 2 4XG 4,X 2HX ˙G 5,X +2XG 5,
w 2 =2MplHF 2 2X ˙G 3,X 16H(XG 4,X + X 2 G 4,XX )+2˙(G 4, +2XG 4, X )
2H 2 ˙(5XG5,X +2X 2 G 5,XX )+4HX(3G 5, +2XG 5, X )
w 3 = 9MplH 2 2 F + 3(XP ,X +2X 2 P ,XX ) + 18H ˙(2XG 3,X + X 2 G 3,XX ) 6X(G 3, + XG 3, X )
+18H 2 (7XG 4,X + 16X 2 G 4,XX +4X 3 G 4,XXX ) 18H ˙(G 4, +5XG 4, X +2X 2 G 4, XX )
+6H 3 ˙(15XG5,X + 13X 2 G 5,XX +2X 3 G ,5XXX ) 18H 2 X(6G 5, +9XG 5, X +2X 2 G 5, XX )
w 4 = MplF 2 2XG 5, 2XG 5,X ¨