PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 1. INTRODUCTION
sourced by the perturbations in the inflaton. In the absence of any anisotropic stress,
as it occurs in the context of scalar fields, the number of independent scalar degrees of
freedom describing the perturbations in the Friedmann metric reduces to one. Therefore,
upon taking into account the scalar and the tensor perturbations, the Friedmann lineelement
can be written as follows:
ds 2 = (1+2Φ) dt 2 −a 2 (t) [(1−2Φ)δ ij +h ij ] dx i dx j , (1.9)
where the quantities Φ and h ij depend on both time and space, while i and j are spatial
indices that run from one through three. The quantityΦdescribes the scalar perturbation
and is known as the Bardeen potential, while h ij , which is considered to be transverse
and traceless, represents the tensor perturbations.
We had pointed pointed out above that, at the linear order in perturbation theory, the
scalars and the tensors evolve independently. If, say, δρ and δp denote the perturbations
in the energy density and pressure, one can show that, when no anisotropic stresses are
present, the first order Einstein equations lead to the following equation of motion for the
Bardeen potential Φ:
Φ ′′ +3H ( 1+c 2 A)
Φ ′ −c 2 A ∇2 Φ+ [ 2H ′ + ( 1+3c 2 A)
H
2 ] Φ = ( 4πGa 2) δp NA , (1.10)
where the overprimes refers to differentiation with respect to the conformal time η, and
H denotes the conformal Hubble parameter that is defined asH = aH = a ′ /a. Moreover,
the quantity δp NA represents the non-adiabatic pressure perturbation which is defined by
the relation (see, for example, Refs. [36, 37])
δp NA = δp−c 2 δρ, (1.11)
A
where c 2 A ≡ p′ /ρ ′ is referred to as the adiabatic speed of the perturbations. It proves to be
convenient to introduce a quantity R, which is commonly known as the curvature perturbation
(since it is proportional to the local three curvature on the spatial hypersurface),
that is given in terms of the Bardeen potential Φ by the expression
R = Φ+ 2ρ ( ) Φ ′ +HΦ
. (1.12)
3H ρ+p
In Fourier space, the equation of motion (1.10) for the Bardeen potential then leads to the
following differential equation governing the curvature perturbation:
R ′ k =
H
H 2 −H ′ [(
4πGa
2 ) δp NA
k −c2 A k2 Φ k
]
, (1.13)
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