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In the matter dominated era the photon sees this potential well created by the primordial fluctuation and the redshift in the emitted photon is given by 

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At the same time, the proper time scale inside the fluctuation becomes slower by an amount δt/t = Φ$_{k}$. Therefore, for the scale factor a ∝ t 2 / $^{3}$, decoupling occurs earlier with 

By virtue of T ∝ a − 1 this results in a temperature which is hotter by 

This is known as the Sachs-Wolfe effect [199]. 

## 3. Spectrum of adiabatic perturbations 

Now, one can immediately calculate the spectrum of the metric perturbations. For a critical density universe 2 

where ∇ 2 → − k $^{2}$, in the Fourier domain. Therefore, with the help of Eqs. (16,23), one obtains 2 2 

where the right hand side can be evaluated at the time of horizon exit k = aH . In fact the above expression can also be expressed in terms of curvature perturbations [25, 158] 

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and following Eq. (22), we obtain δ 2 k = (4 / 25) P $_{ζ}$( k ) = (4 / 25)( H/ ˙ φ ) $^{2}$( H/ 2 π ) $^{2}$, exactly the same expression as in Eq. (32). With the help of the slow-roll equation 3 H ˙ φ = − V ′ , and the critical density formula 3 H $^{2}$M2 P = V , one obtains 

<img src='content_image/369380.jpg'>

where we have used the slow-roll parameter ϵ ≡ ( M 2 $_{P}$/ 2)( V ′ /V ) $^{2}$. The COBE satellite measured the CMB anisotropy and fixes the normalization of P $_{ζ}$( k ) on a very large scales. For 

