7.2 Tensor perturbations

Tensor perturbations hij have two polarization states, which are generally written as λ = +,× [391]. In terms of polarization tensors e+ij and e×ij they are given by
hij = h+e+ij + h×e ×ij. (7.51 )
If the direction of a momentum k is along the z-axis, the non-zero components of polarization tensors are given by e+xx = − e+yy = 1 and e×xy = e×yx = 1.

For the action (6.2View Equation) the Fourier components h λ (λ = +, ×) obey the following equation [314]

(a3F )˙ k2 h¨λ + ------h˙λ + --hλ = 0. (7.52 ) a3F a2
This is similar to Eq. (7.37View Equation) of curvature perturbations, apart from the difference of the factor F instead of Qs. Defining new variables √ -- zt = a F and √ ------ uλ = zthλ∕ 16πG, it follows that
( z′′) u ′′λ + k2 − -t- uλ = 0. (7.53 ) zt
We have introduced the factor 16πG to relate a dimensionless massless field h λ with a massless scalar field uλ having a unit of mass.

If ˙𝜖i = 0, we obtain

z′′t- ν2t −-1-∕4 2 1- (1-+-𝜖3)(2-−-𝜖1-+-𝜖3) z = η2 , with νt = 4 + (1 − 𝜖 )2 . (7.54 ) t 1
We follow the similar procedure to the one given in Section 7.1. Taking into account polarization states, the spectrum of tensor perturbations after the Hubble radius crossing is given by
16 πG 4 πk3 16 ( H )2 1 ( Γ (ν) )2 ( |kη|)3 −2νt 𝒫T = 4 × --2-------3-|uλ |2 ≃ --- ---- -- (1 − 𝜖1)----t-- ---- , (7.55 ) a F (2π ) π mpl F Γ (3∕2) 2
which should be evaluated at the Hubble radius crossing (k = aH). The spectral index of 𝒫T is
nT = 3 − 2νt, (7.56 )
where νt is given in Eq. (7.54View Equation). If |𝜖i| ≪ 1, this reduces to
nT ≃ − 2𝜖1 − 2𝜖3. (7.57 )
Then the amplitude of tensor perturbations is given by
16 ( H )2 1 𝒫T ≃ --- ---- --. (7.58 ) π mpl F

We define the tensor-to-scalar ratio

𝒫T 64π Qs r ≡ ----≃ --2----. (7.59 ) 𝒫 ℛ m plF
For a minimally coupled scalar field ϕ in Einstein gravity, it follows that nT ≃ − 2𝜖1, 𝒫T ≃ 16H2 ∕ (πm2pl), and r ≃ 16𝜖 1.
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