content_image/337886.jpg content_image/337887.jpg content_image/337888.jpg
## 1. Fluctuations in de Sitter 

By solving the Klein-Gordon equation for a light scalar field in a conformal metric: ds 2 = g$_{µν}$dx $^{µ}$dxν = a $^{2}$( τ, x )( dτ 2 − dx $^{2}$), one can find the plane wave solution, φ ( x , τ ) = 1 (2 π ) 3 / 2 ∫ d $^{3}$k ( φ$_{k}$ ( τ ) e ik · x + h . c . ) , for the mode function: [177, 185–191]: π 1 / 2 3 / 2 (1) (2) 

where m is the mass of the scalar field, H (1) ν and H (2) ν are the Hankel functions and c$_{1}$, c$_{2}$ are constants. By using a point splitting regularization scheme, it is possible to obtain a Bunch- Davies vacuum for a de Sitter background which actually corresponds to taking c$_{1}$ = 0, and c$_{2}$ = 1. 

Generically, in a de Sitter phase, the main contribution to the two point correlation function comes from the long wavelength modes; k | τ | ≪ 1 or k ≪ H exp( Ht ), determined by the Hubble expansion rate [177, 188]. 

The integration yields an indefinite increase in the variance with time 

<img src='content_image/337886.jpg'>

<img src='content_image/337887.jpg'>

This result can also be obtained by considering the Brownian motion of the scalar field [146, 192–195]. For a massive field with m ≪ H , and ν ̸ = 3 / 2, one does not obtain an indefinite growth of the variance of the long wavelength fluctuations, but [178, 186–188, 196]: 

In the limiting case when m → H , the variance goes as 〈 φ $^{2}$〉 ≈ H $^{2}$. In the limit m ≫ H , the variance goes as 〈 φ $^{2}$〉 ≈ ( H $^{3}$/ 12 π $^{2}$m ). Only in a massless case 〈 φ $^{2}$〉 can be treated as a homogeneous background field with a long wavelength mode. 

## 2. Adiabatic perturbations and the Sachs-Wolfe effect 

<img src='content_image/337888.jpg'>

Let us consider small inhomogeneities, φ ( x , t ) = φ ( t ) + δφ ( x , t ), such that δφ ≪ φ . Perturbations in matter densities automatically induce perturbations in the background 

