Mode analysis and Ward identities for perturbative quantum gravity in de Sitter space Page: 3 of 15
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The invariant Lagrangian is:
Si.ny- [ R - (D-2)(D-1)H2J v/- (2)
where the Hubble constant is H2 = ' A, our metric has spacelike signature rnd R is
the Ricci scalar formed from Roppy = r , + r ppwa - ( ia v). Perturbation theory
derives from the expansion:
g p = 9pW + rshy (3)
where 9 v is an exact solution. We shall work in the open conformal coordinate system
where the background metric is:
= 1 (4)
9fv_(H )2 v=St1p
A peculiarity of this system is that while the spatial coordinate, Z, can take any value in
(D -1)-dimensional Euclidean space the time coordinate, u, runs only from zero to infinty.
It is also inverted with respect to physical time; that is, the far future is obtained by letting
ii approach zero while the far past is probed by taking u to +oo. The flat space limit is
obtained by substituting u = 1 - r and taking H to zero while holding the flat space
time xO fixed. Although the conformal coordinate system covers only half of the full de
Sitter manifold it is complete in the sense that nothing leaks into or out of the submanifold;
surfaces of constant u are Cauchy surfaces. An important advantage of restricting physics
to this submanifold is that one avoids the linearization instability which has frustrated all
previous attempts to formulate quantum gravity on de Sitter space .
Although interactions are most easily described using the pseudo-graviton field, Op
Q-2 hl, a slightly different rescaling gives the simplest formulation of the free theory:
kpv SfT -/iyv ( )
The indices on hpp are raised and lowered with j,. but those of both p and Xy are
raised and lowered with the Minkowski metric. It is of course completely trivial to convert
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Tsamis, N. C. & Woodard, R. P. Mode analysis and Ward identities for perturbative quantum gravity in de Sitter space, report, June 1, 1992; United States. (https://digital.library.unt.edu/ark:/67531/metadc1342280/m1/3/: accessed April 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.