Signature of the Quantum Gravity on the CMB
27
by the tensor power spectrum and can be written in terms of the inflation potential
as follows:
P T =
2
3π 2 m
4
pl
V | k=a H .
(11)
It can be observed that the tensor and scalar power spectra are not independent of
the first slow-roll parameter. Therefore, the tensor-to-scalar ratio can be written as
r ≡
P T (k)
P S (k)
= 16.
(12)
This ratio means the amplitude of the tensor perturbation at the CMB scale.
In reality, the tensor and scalar spectra are not necessarily scale invariant because
the scalar field varies slowly during inflation. The variation is usually accounted with
a quantity known as the spectral index and can be defined, respectively, for the scalar
and tensor power spectrum as follows:
n s − 1 =
d ln P S
d ln k
(13)
n t =
d ln P T
d ln k
,
(14)
where n s is known as the scalar spectral index and n t is the tensor spectral index.
The spectral indices can also be written in terms of the slow-roll parameters, hence
n s = 1 + 2η V − 6 V
(15)
and
n t = −2 V ,
(16)
therefore comparing with (12) leads to the consistency relation
r = −8n t ,
(17)
Which is, in principle, purely related to the observable quantities.
In the present work, we mainly need only the tensor power spectrum to examine
the imprint of quantum gravity on the BB mode angular power spectrum of the CMB.
3 Tensor Power Spectrum
To study the quantum gravity effect on the BB mode spectrum of the CMB, knowledge
of the tensor power spectrum is essential. Therefore, to compute the tensor power