content_image/349125.jpg content_image/349126.jpg
with the primordial tensor signal from inflation [267]. 

The current CMB fluctuations generated from strings involve the simplest Nambu-Goto strings, the presence of currents inside cosmic strings can possibly affect their precise signa- ture in CMB [268]. More generally, the properties of cosmic strings arising from non-SUSY theories, from SUSY F -and D -term or N = 2 SUSY P-term hybrid inflation, or from brane inflation are different. By adding only 1 parameter to fit the data ( f$_{10}$ ) and confronting all models to the posterior probabilities might not be the best strategy in such a case (see Sec. III G). 

## 3. Isocurvature perturbations 

The isocurvature perturbations measure the deviation from the adiabaticity of the primor- dial fluctuations, denoted by the quantity S (in the context of cold dark matter in Eq. (54), it was denoted by S$_{c}$ ). The isocurvature perturbations arise if there are light scalar fields fluctuating during inflation, which do not thermalize with the inflaton decay products, i.e. the SM degrees of freedom, after the end of inflation. Usually this deviation is measured by the parameter α related to the ratio between the entropy power spectrum, P $_{S}$, over the curvature perturbation, P $_{ζ}$, via the Eq. (55). There could be some correlations between S and ζ , the parameter: 

where P S,ζ is the cross-correlated power spectrum between S and ζ , distinguish between the totally correlated case ( β = − 1) and the totally anti-correlated case ( β = 0). Two parameters $^{α}$− 1 (for β = − 1) and α$_{0}$ (for β = 0) are commonly used to describe each case, which are typically encountered in the curvaton scenario and in the axion dark matter scenario, respectively. 

<img src='content_image/349125.jpg'>

The most recent observations from WMAP 5-years data lead to the values for $^{α}$− 1 and α$_{0}$ compatible with zero, and respectively, slightly and strongly degenerate with the spectral index [13]. Marginalizing over other parameters, it was found that, at 2 σ : 

<img src='content_image/349126.jpg'>

when the WMAP data were combined with BAO and SN data. These constraints suggest that the deviation from S = 0 is smaller than 2 . 1% and 8 . 9% respectively at 95% confidence 

