PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 1. INTRODUCTION
where the quantity h k represents the amplitude of the tensor modes. Following the case
of scalars, if we setu k = ah k , then one obtains the equation satisfied byu k to be
( )
u ′′ k + k 2 − a′′
u k = 0. (1.18)
a
The inflationary scalar and tensor perturbation spectra
It is essentially the spectrum of the Bardeen potential when the modes enter the Hubble
radius during the radiation and the matter dominated epochs that determines the pattern
of the anisotropies in the CMB and the formation of the LSS [6]. The correlations in
the Bardeen potential or, equivalently, in the curvature perturbation, originate due to the
quantum fluctuations associated the scalar field. As it directly involves the perturbation
in the scalar field, it is the curvature perturbation R that is elevated to a quantum operator.
On quantization, the operator corresponding to the curvature perturbation R can be
expressed as
ˆR(η,x) =
=
∫
∫
d 3 k
(2π) ˆR 3/2 k (η) e ik·x
d 3 k
[
]
â
(2π) 3/2 k f k (η)e ik·x +â † k f∗ k (η)e−ik·x , (1.19)
where â k and â † k
are the usual creation and annihilation operators that satisfy the standard
commutation relations, while the modes f k are governed by the differential equation
(1.15). The dimensionless scalar power spectrum P S
(k) is given in terms of the correlation
function of the Fourier modes of the curvature perturbation ˆR k by the following
relation:
〈0| ˆR k ˆRp |0〉 = (2π)2
2k 3 P S
(k) δ (3) (k+p), (1.20)
where|0〉 denotes the vacuum state that is defined asâ k |0〉 = 0∀k. In terms of the modes
f k and the Mukhanov-Sasaki variable v k , the scalar power spectrum is given by
( ) 2
P S
(k) = k3
2π |f k| 2 = k3 |vk |
, (1.21)
2 2π 2 z
with the expression on the right hand side to be evaluated on super-Hubble scales
[i.e. when k/(aH) ≪ 1], as the curvature perturbation approaches a constant value 3 .
3 Earlier, we had illustrated that, if the non-adiabatic pressure perturbation δp NA can be neglected, then
the curvature perturbation is conserved on super-Hubble scales. It can be shown that the non-adiabatic
pressure perturbation associated with the inflaton [cf. Eq. (1.14)] decays exponentially on super-Hubble
scales and, hence, can be ignored. In fact, it is because of this reason that the scalar perturbations produced
by single scalar fields are termed as adiabatic.
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