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 H₀ and the matter density parameter Ω_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, for example, adding the ∼1 percent measurement of the BAO ratio r_s/D_V from the BOSS survey leads to a limit Σ m_ν < 0.19 eV, equivalent to Ω_ν < 0.0045 at 95 per cent confidence. For the case of Σ m_ν < 0.6 eV, the CMB degeneracy with neutrino mass almost follows a track of constant sound horizon angle (Howlett et al. 2012). For a ΛCDM + m_ν 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_ν ≡ ω_ν/ω_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 δln H₀ ≈ -2 δx_ν, δln Ω_m ≈ +5 δx_ν, δln ω_Λ ≈ -6.2 δx_ν, and most notably δω_Λ ≈ -14 δω_ν. These multipliers give an intuitive derivation of the degeneracy direction, which agrees well with the numerical likelihood results from the Planck team.