Signature of the Quantum Gravity on the CMB
29
h k =
ξ k
a
,
(23)
where
ξ k (τ ) = u k (τ ) + v
∗
k (τ )
(24)
satisfies the equation of motion
ξ
k +
k
2
−
a
a
ξ k = 0.
(25)
Here prime means derivative with respect to the conformal time τ .
Since the contribution from each polarization is the same, using (19) and (23), we
get
h(x, τ ) =
B
a(τ )(2π)
3
2
+∞
−∞
d
3 k[ξ k (τ )b k + ξ
∗
k (τ )b
†
k ]e
ik.x
.
(26)
The two-point correlation function of the tensor perturbation in a vacuum state is
given by
h k h
∗
k =
2π
2
k 3 P T (k)δ
3
(k − k
),
(27)
where P T is the tensor power spectrum in the super horizon limit (k << a H).
Therefore, using (19) and (27), we get the primordial tensor power spectrum and
it can be expressed in a power law form as
P T (k) = A T (k 0 )
k
k 0
n t
,
(28)
where A T (k 0 ) = B
2
H
2π
2 is the amplitude at the pivot wavenumber k 0 = a H and
H is the Hubble parameter during the inflation.
4 Effective Theory and Inflation
The effective theory is an approach to quantum gravity. Therefore, it is interesting
to study the inflation in the light of the effective theory and its reflection on the BB
mode angular power spectrum of the CMB. The action for the scalar field coupled
to gravity in the most general effective theory is [13]