28
P. K. Suresh
spectrum we begin with the following perturbed flat FLRW metric in the conformal
time (defined as dτ =
dt
a
), given by
ds
2
= a
2
(τ )
−dτ
2
+ (δ i j + h i j )dx
i dx
j
,
(18)
where δ i j means the flat space metric and h i j is the tensor perturbation with the
conditions |h i j | | δ i j , ∂ i h
i j
= 0, and δ
i j h i j = 0.
The tensor perturbation field h i j (x, τ ) can be written in Fourier mode
h i j (x, τ ) =
B
(2π)
3
2
+∞
−∞
d
3 k
√
2k
p=+,×
h
( p)
k (τ )b
( p)
k e
ik.x
χ
( p)
i j (k)
+h
( p)∗
k
(τ )b
( p)†
k
e
−ik.x
χ
( p)∗
i j (k)
,
(19)
where B =
√
16π G , k is the wave vector, and χ
( p)
i j (where p = +, ×) are the two
linear polarization states of gravitational wave that satisfy χ
( p)
i j δ
i j
= 0, χ
( p)
i j k
i
=
0, χ
( p)
i j χ
( p
)i j
= 2δ pp , χ
( p)
i j (-k) = χ
( p)
i j (k), and are, respectively, called plus (+)
polarization and cross (×) polarization. In (19), (b
( p)†
k
) and (b
( p)
k ) are, respectively,
the creation and annihilation operators and are governed by the Heisenberg equation
with Hamiltonian H for each component, given by
d
dτ
b
( p)†
k
(τ ) = −i
b
( p)†
k
(τ ), H
,
d
dτ
b
( p)
k (τ ) = −i
b
( p)
k (τ ), H
,
(20)
and further the operators satisfy the conditions [b
( p)
k , b
( p
)†
k
] = δ pp δ
3
(k − k
) and
[b
( p)
k , b
( p
)
k ] =[b
( p)†
k
, b
( p
)†
k
] = 0.
The initial vacuum state can be defined with respect to the annihilation operator
as
b
( p)
k |0 = 0.
Since the contribution from each polarization to the gravitational wave is the same,
hereafter we drop the superscript ( p) for convenience.
The value of the operators b
†
k (0) and b k (0) at some initial time, say t = 0, to
its later time, say t = τ , can be connected through the Bogoliubov transformation
given by
b
†
k (τ ) = u
∗
k (τ )b
†
k (0) + v
∗
k (τ )b k (0),
(21)
b k (τ ) = u k (τ )b k (0) + v k (τ )b
†
k (0),
(22)
where the complex functions u k (τ ) and v k (τ ) hold the condition |u k |
2
− |v k |
2
= 1.
The coupling of the tensor perturbation h k (τ ) with the scale factor a(τ ) gives
P. K. Suresh
spectrum we begin with the following perturbed flat FLRW metric in the conformal
time (defined as dτ =
dt
a
), given by
ds
2
= a
2
(τ )
−dτ
2
+ (δ i j + h i j )dx
i dx
j
,
(18)
where δ i j means the flat space metric and h i j is the tensor perturbation with the
conditions |h i j | | δ i j , ∂ i h
i j
= 0, and δ
i j h i j = 0.
The tensor perturbation field h i j (x, τ ) can be written in Fourier mode
h i j (x, τ ) =
B
(2π)
3
2
+∞
−∞
d
3 k
√
2k
p=+,×
h
( p)
k (τ )b
( p)
k e
ik.x
χ
( p)
i j (k)
+h
( p)∗
k
(τ )b
( p)†
k
e
−ik.x
χ
( p)∗
i j (k)
,
(19)
where B =
√
16π G , k is the wave vector, and χ
( p)
i j (where p = +, ×) are the two
linear polarization states of gravitational wave that satisfy χ
( p)
i j δ
i j
= 0, χ
( p)
i j k
i
=
0, χ
( p)
i j χ
( p
)i j
= 2δ pp , χ
( p)
i j (-k) = χ
( p)
i j (k), and are, respectively, called plus (+)
polarization and cross (×) polarization. In (19), (b
( p)†
k
) and (b
( p)
k ) are, respectively,
the creation and annihilation operators and are governed by the Heisenberg equation
with Hamiltonian H for each component, given by
d
dτ
b
( p)†
k
(τ ) = −i
b
( p)†
k
(τ ), H
,
d
dτ
b
( p)
k (τ ) = −i
b
( p)
k (τ ), H
,
(20)
and further the operators satisfy the conditions [b
( p)
k , b
( p
)†
k
] = δ pp δ
3
(k − k
) and
[b
( p)
k , b
( p
)
k ] =[b
( p)†
k
, b
( p
)†
k
] = 0.
The initial vacuum state can be defined with respect to the annihilation operator
as
b
( p)
k |0 = 0.
Since the contribution from each polarization to the gravitational wave is the same,
hereafter we drop the superscript ( p) for convenience.
The value of the operators b
†
k (0) and b k (0) at some initial time, say t = 0, to
its later time, say t = τ , can be connected through the Bogoliubov transformation
given by
b
†
k (τ ) = u
∗
k (τ )b
†
k (0) + v
∗
k (τ )b k (0),
(21)
b k (τ ) = u k (τ )b k (0) + v k (τ )b
†
k (0),
(22)
where the complex functions u k (τ ) and v k (τ ) hold the condition |u k |
2
− |v k |
2
= 1.
The coupling of the tensor perturbation h k (τ ) with the scale factor a(τ ) gives