Mode analysis and Ward identities for perturbative quantum gravity in de Sitter space Page: 2 of 15
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Quantum gravity has beet, little considered in the search for an explanation of the
smallness of the cosmological constant. The reasons for this seem to be the theory's well
known ultraviolet problems and the widespread belief that de Sitter space, the natural
background for A > 0, is such a strange environment that we can never hope to understand
physics on it. This is a great pity because the far infrared sector of quantum gravity is in
many ways the natural place to look for a resolution to what is, after all, a gravitational
We shall elsewhere establish that Einstein's theory can be used reliably in the far
infrared; here we seek to dispel the notion that it is significantly more difficult to understand
linearized gravitons on a de Sitter background than in flat space. We make this point
by solving the theory, in toto, using the same methods that are employed in flat space.
(See, for example, chapter 10 of ref. .) The analysis is so simple that we have been
able to carry it out generally in D dimensions. The result is simple too: in conformal
coordinates and with a suitably resealed field variable the modes can be written as plane
waves characterized by their spatial (D - 1)-momenta and by polarization tensors which
are identical to the polarization tensors of flat space. The only complicating feature is a
slightly different time dependence. In four dimensional flat space the plane wave solutions
are oscillatory; in four dimensional de Sitter space this oscillatory function acquires a time
dependent prefactor and a phase:
ei . _ 1 F--) exp[-F k - 7T) (1)
where u is our time variable and k kII is the Euclidean norm of the 3-momenta. (In D
dimensions the time dependence is proportional to V'7 times a v = I- Hankel function.)
As a bonus we apply our technology to determine all possible vacua for which the ghost
and graviton propagators obey the Ward identity relating them.
This paper is based on a previous one [21 whose notational conventions we shall follow.
el si p
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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/m1/2/?rotate=90: accessed May 27, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.