RevModPhys.84.671

There consistent theories with cosmological

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Unformatted text preview: review. This work is supported in part by NSF Grant No. PHY-0930521 and by funds provided by the University of Pennsylvania. APPENDIX: TOTAL DERIVATIVE COMBINATIONS Define the matrix of second derivatives Å ¼ @ @ : (A.1) At every order in , there is a unique (up to overall constant) contraction of Å ’s that reduces to a total derivative18, LTD ðÅÞ ¼ ½ÅŠ; 1 (A.2) LTD ðÅÞ ¼ ½ÅŠ2 À ½Å2 Š; 2 (A.3) LTD ðÅÞ ¼ ½ÅŠ3 À 3½ÅŠ½Å2 Š þ 2½Å3 Š; 3 (A.4) LTD ðÅÞ ¼ ½ÅŠ4 ; À6½Å2 Š½ÅŠ2 þ 8½Å3 Š½ÅŠ 4 þ 3½Å2 Š2 À 6½Å4 Š; . .; . (A.5) where the brackets are traces. LTD ðhÞ is just the Fierz-Pauli 2 term, and the others can be thought of as higher order generalizations of it. They are characteristic polynomials, terms in the expansion of the determinant in powers of H , 1 1 detð1 þ ÅÞ ¼ 1 þ LTD ðÅÞ þ LTD ðÅÞ þ LTD ðÅÞ 1 22 3! 3 1 (A.6) þ LTD ðÅÞ þ Á Á Á : 4! 4 The term LTD ðÅÞ vanishes identically when n > D, with D n the spacetime dimension, so there are only D nontrivial such combinations, those with n ¼ 1; . . . ; D. 18 The proof of this fact is the same as the proof showing the uniqueness of the Galileons in Nicolis, Rattazzi, and Trincherini (2008). See also Creminelli et al. (2005). Kurt Hinterbichler: Theoretical aspects of massive gravity 708  For spatial indices i, j and time index 0, They can be written explicitly as X LTD ðÅÞ ¼ ðÀ1Þp 1 pð1 Þ 2 pð2 Þ Á Á Á n pðn Þ n ðn Xij Þ has at most two time derivatives; p  ðÅ1 1 Å2 2 Á Á Á Ån n Þ: (A.7) ðÀ1Þp The sum is over all permutations of the  indices, with the sign of the permutation. They satisfy a recursion relation LTD ðÅÞ ¼ À n n X ðÀ1Þm m¼ 1 ð n À 1Þ ! ½Åm ŠLTD m ðÅÞ; nÀ ðn À m Þ! (A.9) The first few are ð0 Þ X ¼  ; (A.10) ð1 Þ X ¼ ½ÅŠ À Å ; (A.11) ð2 Þ X ¼ ð½ÅŠ2 À ½Å2 ŠÞ À 2½ÅŠÅ þ 2Å2 ;  (A.12) ð3 Þ X ¼ ð½ÅŠ3 À 3½ÅŠ½Å2 Š þ 2½Å3 ŠÞ À 3ð½ÅŠ2 À ½Å2 ŠÞÅ þ 6½ÅŠÅ2 À 6Å3 ;   . .: . (A.13) The following is an explicit expression: ðn Þ X ¼ n X ðÀ1Þm m¼ 0 n! Åm LTD ðÅÞ: ðn À mÞ!  nÀm (A.14) They satisfy the recursion relation ðn Þ ð nÀ ðn À X ¼ ÀnÅ X  1Þ þ Å X 1Þ  : (A.15) ðn Þ Since LTD ðÅÞ vanishes for n > D, X vanishes for n ! n D. ðn Þ The X satisfy the following important properties.  They are symmetric and identically conserved and are the only combinations of Å at each order with these properties: ðn Þ @ X ¼ 0: (A.16) ðnÞ Note that our definition of the X used here differs by a factor of 2 from that of de Rham, Gabadadze, and Tolley (2010). 19 Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 ðn X00Þ has no time derivatives: ð1Þ E  ð Þ ¼ ÀðD À 2ÞX ; with LTD ðÅÞ ¼ 1. 0 ðn Þ In addition, there are tensors X that we construct out of 19 Å as fol...
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