In this paper, we will analyse the scalar harmonics on a four sphere using a associated Legendre function. Then, we will use these modes to construct two types of vector harmonics on a four sphere. Finally, we will also construct three types of tensor harmonics on a four sphere. As there is a relation between de Sitter spacetime and four sphere, these modes are related to the modes on de Sitter spacetime.
We will analyse perturbative quantum gravity on de Sitter spacetime. We propose a new type of inner product for modes on de Sitter spacetime. This inner product is used to mode decompose perturbations of the metric on de Sitter spacetime. Using this inner product, it is possible to calculate the two-point function for perturbative quantum gravity on de Sitter spacetime. This twopoint function will be written in terms of a mode sum for various modes on de Sitter spacetime.
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