Tilt of primordial gravitational wave spectrum in a universe with sterile neutrinos

TL;DR

This study addresses the tension between Planck and BICEP2 observations of primordial gravitational waves by treating the tensor tilt $n_t$ as a free parameter in a $\Lambda$CDM+$r$+$\nu_s$+$n_t$ framework. It employs CAMB to generate theoretical spectra and CosmoMC for MCMC inference, combining data from Planck+WP, WMAP9, BAO, BICEP2, and additional $H_0$, SZ, and lensing measurements to simultaneously constrain $m_{\nu,\rm sterile}^{\rm eff}$ and $N_{\rm eff}$. A key result is a strong anticorrelation between $n_t$ and $r_{0.002}$ driven by BICEP2 around $k\sim0.01\,\mathrm{Mpc}^{-1}$, which biases the fit toward a blue-tilted spectrum but allows red tilt at larger $r_{0.002}$. With the full data set, the analysis yields $m_{\nu,\rm sterile}^{\rm eff}=0.48^{+0.11}_{-0.13}$ eV, $N_{\rm eff}=3.73^{+0.34}_{-0.37}$, $r_{0.002}<0.172$ (95% CL), and $n_t=0.96^{+0.48}_{-0.63}$, indicating a tension-relieved, nearly concordant cosmology that accommodates sterile neutrinos and a range of tensor tilts. These results illustrate how extending the neutrino sector can reconcile high-$r$ hints with CMB observations and place tight bounds on sterile-neutrino parameters.

Abstract

In this work, we constrain the spectral index $n_t$ of the primordial gravitational wave power spectrum in a universe with sterile neutrinos by using the Planck temperature data, the WMAP 9-year polarization data, the baryon acoustic oscillation data, and the BICEP2 data. We call this model the $Λ$CDM+$r$+$ν_s$+$n_t$ model. The additional massive sterile neutrino species can significantly relieve the tension between the Planck and BICEP2 data, and thus can reduce the possible effects of this tension on the fit results of $n_t$. To constrain the parameters of sterile neutrino, we also utilize the Hubble constant direct measurement data, the Planck Sunyaev-Zeldovich cluster counts data, the Planck CMB lensing data, and the cosmic shear data. We find that due to the fact that the BICEP2 data are most sensitive to the multipole $\ell\sim150$ corresponding to $k\sim0.01$ Mpc$^{-1}$, there exists a strong anticorrelation between $n_t$ and $r_{0.002}$ in the BICEP2 data, and this further results in a strongly blue-tilt spectrum. However, a slightly red-tilt tensor power spectrum is also allowed by the BICEP2 data in the region with larger value of $r_{0.002}$. By using the full data sets, we obtain $m_{ν,{\rm{sterile}}}^{\rm{eff}}=0.48^{+0.11}_{-0.13}$ eV, $N_{\rm{eff}}=3.73^{+0.34}_{-0.37}$, and $n_t=0.96^{+0.48}_{-0.63}$ for the $Λ$CDM+$r$+$ν_s$+$n_t$ model.

Figures (3)

• Figure 1: Constraint results in the $n_s$--$r_{0.002}$ plane for the $\Lambda$CDM+$r$+$\nu_s$+$n_t$ model by using the Planck+WP+BAO (left panel) and the Planck+WP+BAO+BICEP2 (right panel) data. Points are color-coded according to the values of $n_t$.
• Figure 2: Constraint results in the $n_t$--$r_{0.002}$ plane for the $\Lambda$CDM+$r$+$\nu_s$+$n_t$ model by using different data combinations. The black dashed line is plotted for the function $r_{0.002}=0.2(\frac{k=0.002}{k=0.01})^{n_t}$. The black dotted line denotes the consistency relation $r_{0.002}=-8n_t$.
• Figure 3: Cosmological constraints on the $\Lambda$CDM+$r$+$\nu_s$+$n_t$ model by using the Planck+WP+BAO+BICEP2 data and the Planck+WP+BAO+BICEP2+$H_0$+SZ+Lensing data.