The CMB neutrino mass / vacuum energy degeneracy: a simple derivation of the degeneracy slopes

Abstract

It is well known that estimating cosmological parameters from cosmic microwave background (CMB) data alone results in a significant degeneracy between the total neutrino mass and several other cosmological parameters, especially the Hubble constant H_0 and the matter density parameter ${\Omega }_m$. Adding low-redshift measurements such as baryon acoustic oscillations (BAOs) breaks this degeneracy and greatly improves the constraints on neutrino mass. The sensitivity is surprisingly high, e.g. adding the $\sim 1$ percent measurement of the BAO ratio $r_s/D_V$ from the BOSS survey leads to a limit $\Sigma m_{\nu }<0.19$ eV, equivalent to ${\Omega }_{\nu }<0.0045$ at 95\% confidence. For the case of $\Sigma m_{\nu }<0.6$ eV, the CMB degeneracy with neutrino mass almost follows a track of constant sound horizon angle (Howlett et al 2012). For a $\Lambda$CDM + $m_{\nu }$ model, we use simple but quite accurate analytic approximations to derive the slope of this track, giving dimensionless multipliers between the neutrino to matter ratio ($x_{\nu }\equiv {\omega }_{\nu }/{\omega }_{cb}$) and the shifts in other cosmological parameters. The resulting multipliers are substantially larger than 1: conserving the CMB sound horizon angle requires parameter shifts $\delta \ln H_0\approx -2\delta x_{\nu }$, $\delta \ln {\Omega }_m\approx +5\delta x_{\nu }$, $\delta \ln {\omega }_{\Lambda }\approx -6.2\delta x_{\nu }$, and most notably $\delta {\omega }_{\Lambda }\approx -14\delta {\omega }_{\nu }$. These multipliers give an intuitive derivation of the degeneracy direction, which agrees well with the numerical likelihood results from the Planck team.

Comments: Latex, 9 pages, 2 figures. Accepted by MNRAS