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

TL;DR

This paper derives simple analytic formulas to explain the degeneracy slopes between neutrino mass and other cosmological parameters in CMB data, clarifying how parameter shifts compensate to preserve the sound horizon angle.

Contribution

It provides an intuitive, analytic derivation of the degeneracy slopes in the neutrino mass and cosmological parameters, aligning with numerical results.

Findings

Derived multipliers for parameter shifts due to neutrino mass

Confirmed analytic slopes match Planck likelihood results

Quantified how parameter shifts preserve the sound horizon angle

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.