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 \citep howlett12 .
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 Planck team .