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Global fitting analysis on cosmological models after BICEP2

Recently, BICEP2 collaboration has released their results on the measurements of the CMB polarizations. In the framework of the $Λ$CDM with a power law form of the scalar primordial power spectrum, this new measurement on the B-mode puts a tight constraint on tensor/scalar ratio $r=0.2^{+0.07}_{-0.05}~(1σ)$, which however is in tension with the Planck limit, $r<0.11$ (95% C.L.). In this paper, we consider various extensions of the $Λ$CDM model by introducing extra cosmological parameters such as the equation state of dark energy $w$, the curvature of the universe $Ω_k$, the running of the scalar power spectrum index $α_s$, the sum of the neutrino mass $Σm_ν$, the effective number of neutrinos $N_{eff}$, the tensor power spectrum index $n_{t}$, then perform the global fit to the BICEP2, Planck and also the SN and BAO data. Our results show the cosmological parameters $α_s$ and $n_t$ are highly and directly correlated with the tensor/scalar ratio $r$. However indirectly the parameters $Ω_k$ and $N_{eff}$ are also correlated with $r$. We will in this paper give the numerical values on the parameters introduced and show explicitly how the tension between the BICEP2 and Planck is effectively alleviated by the inclusion of the parameters $Ω_k$, $α_s$, $N_{eff}$ and $n_{t}$ separately.