content_image/355564.jpg content_image/355565.jpg content_image/355566.jpg content_image/355567.jpg
If again one assumes that the perturbations in δφ$_{a}$ have random phases with an amplitude ( H/ 2 π ) $^{2}$, one obtains: 

<img src='content_image/355564.jpg'>

For a single component ∂N/∂φ ≡ ( $_{M}$− 2 P V /V ′ ), and then Eq. (51) reduces to Eq. (34). By using slow-roll equations we can again define the spectral index 

<img src='content_image/355565.jpg'>

where V$_{,a}$ ≡ ∂V /∂φ$_{a}$ , and similarly N$_{,a}$ ≡ ∂N/∂φ$_{a}$ . For a single component we recover Eq. (37) from Eq. (52). 

## 3. Isocurvature perturbations and CMB 

One may of course simply assume a purely isocurvature initial condition. For any species the entropy perturbation is defined by 

<img src='content_image/355566.jpg'>

Thus, if initially there is a radiation bath with a common radiation density contrast δ$_{r}$ , a baryon-density contrast δ$_{b}$ = 3 δ$_{r}$/ 4, and a CDM density contrast δ$_{c}$ , then 

<img src='content_image/355567.jpg'>

where we have used the isocurvature condition δρ$_{r}$ + δρ$_{c}$ = 0, and the last equality holds in a radiation dominated universe. Similarly the baryon isocurvature is given by: S$_{B}$ = δ$_{B}$ − (3 / 4) δ$_{r}$ and the neutrino (or any other relativistic species) isocurvature component is given by: S$_{ν}$ = (3 / 4) δ$_{ν}$ − (3 / 4) δ$_{r}$ . 

However a pure isocurvature perturbation gives five times larger contribution to the Sachs- Wolfe effect compared to the adiabatic case [25, 221, 222]. This result can be derived very easily in a matter dominated era with an isocurvature condition δρ$_{c}$ = − δρ$_{r}$ , which gives a contribution ζ$_{k}$ = (1 / 3) S$_{k}$ . Therefore from Eqs. (27,30), we obtain Δ T$_{k}$/T = − S$_{k}$/ 15. There is an additional contribution from radiation because we are in a matter dominated era, see Eq. (54), S ≈ δ$_{c}$ ≡ − (3 / 4) δ$_{r}$ . The sum total isocurvature perturbation Δ T$_{k}$/T = − S/ 15 − S/ 3 = − 6 S/ 15, where S is measured on the last scattering surface. The Sachs-Wolfe effect for isocurvature perturbations fixes the slope of the perturbations, rather than the amplitude 

