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 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 where ∇ 2 → − k $^{2}$, in the Fourier domain. Therefore, with the help of Eqs. (16,23), one obtains 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] 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 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 18 Δ T$_{k}$ T = − Φ$_{k}$ . (28) δa a = 2 3 δt t = 2 3 Φ$_{k}$ . (29) Δ T$_{k}$ T = − Φ$_{k}$ + 2 3 Φ$_{k}$ = − Φ$_{k}$ 3 . (30) δ$_{k}$ ≡ δρ ρ ∣ k = − 2 3 ( k aH ) 2 Φ$_{k}$ , (31) δ 2 k ≡ 4 9 P $_{Φ}$( k ) = 4 9 9 25 ( H ˙ φ ) 2 ( H 2 π ) 2 , (32) δ$_{k}$ = 2 5 ( k aH ) 2 ζ$_{k}$ , (33) δ 2 k ≈ 1 75 π $^{2}$M 6 P V 3 V ′ 2 = 1 150 π $^{2}$M 4 P V ϵ , and P $_{ζ}$( k ) = 1 24 π $^{2}$M 4 P V ϵ , (34)