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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 two-point function will be written in terms of a mode sum for various modes on de Sitter spacetime.

Quantum field theory has been one of the greatest scientific achievements of the last centenary. The perturbative quantum theory has led to many important developments in high energy physics and condensed matter physics. These can include things like ghost condensation which causes the breaking of Lorentz symmetry [

As this rate is measured by the Hubble’s constant H, so we can write

The Hubble’s constant can be written in terms of the radius of de Sitter spacetime r,

Furthermore, we can write

So we get

Now we can also write

As we know,

So we can also write

The metric in de Sitter spacetime can be written as

Furthermore, we can write it using the metric on a three-sphere

Here

Now using

we can write

We now start with pure gravity with a cosmological constant

We adopt units, such that

We now split the metric

Now we have

and

If

where we have define

and

along with

We can now write,

So we have

This operator has a zero eigenvalue and it is not invertible. However, we can fix a gauge. This will be done by adding a ghost term and a gauge fixing term to the original action. Now if the sum of this term is given by

for perturbative quantum gravity are

We have

Furthermore, the inner product on a space-like hyper-surface

Now we have

This inner product is conserved. Now by using a complete set of solutions

So we have

We also have

and

So, we can write

Now we can write

where

Thus we have

and

In Matrix notation, this can be written as

So we have

and

In this paper, we analysed the inner product for perturbative quantum gravity in de Sitter spacetime. In doing our calculations, we proposed a general kind of inner product for modes on de Sitter spacetime. Using this general inner product, we were able to calculate the two-point function on de Sitter spacetime. The de Sitter spacetime is important as the universe is expected to be approaching de Sitter spacetime. This two-point function can be written as a mode sum of the graviton modes on de Sitter spacetime. It may be noted that quantum gravity correction to quantum field theory has been studied [

Ashaq Hussain Sofi,Shabir Ahmad Akhoon,Asloob Ahmad Rather,Anil Maini, (2015) Perturbative Quantum Gravity on de Sitter Spacetime. Open Access Library Journal,02,1-9. doi: 10.4236/oalib.1101422