Constraints on neutrino masses from Baryon Acoustic Oscillation measurements

From 21 independent Baryon Acoustic Oscillation (BAO) measurements we obtain the following sum of masses of active Dirac or Majorana neutrinos: $\sum m_\nu = 0.711 - 0.335 \cdot \delta h + 0.050 \cdot \delta b \pm 0.063 \textrm{ eV,}$ where $\delta h \equiv (h - 0.678) / 0.009$ and $\delta b \equiv (\Omega_b h^2 - 0.02226) / 0.00023$. This result may be combined with independent measurements that constrain the parameters $\sum m_\nu$, $h$, and $\Omega_b h^2$. For $\delta h = \pm 1$ and $\delta b = \pm 1$, we obtain $m_\nu<0.43$ eV at 95\% confidence.

We extend the analysis presented in Ref. [1] to include neutrino masses. The present analysis has three steps: (1) we calculate the distance of propagation r s , in units of c/H 0 , referred to the present time, of sound waves in the photon-electron-baryon plasma until decoupling by numerical integration of Eqs. (16) and (17) of Ref. [1]; (2) we fit the Friedmann equation of evolution of the universe to 21 independent Baryon Acoustic Oscillation (BAO) distance measurements listed in [1] used as uncalibrated standard rulers and obtain the length d of these rulers, in units of c/H 0 , referred to the present time; and (3) we set to constrain the sum of neutrino masses m ν . c is the speed of light, and H 0 ≡ 100h km s −1 Mpc −1 is the present day Hubble expansion parameter.
The main body of this article assumes: (1) flat space, i.e. Ω k = 0, and (2) constant dark energy density relative to the critical density, i.e. Ω DE independent of the expansion parameter a. These constraints are in agreement with all observations to date [1,2]. Results without these constraints are presented in the appendix.
To be specific we consider three active neutrino flavors with three eigenstates with nearly the same mass m ν , so m ν = 3m ν . This is a useful scenario to consider since our current limits on m 2 ν are much larger than the mass-squared-differences ∆m 2 and ∆m 2 21 obtained from neutrino oscillations [2]. These neutrinos become non-relativistic at a neutrino temperature T ν = m ν /3.15 or a photon temperature T = m ν (11/4) 1/3 /3.15. The corresponding expansion parameter is The matter density relative to the present critical density is Ω m /a 3 for a > a ν . Ω m includes the density Ω ν = h −2 m ν /94eV of Dirac or Majorana neutrinos that are non-relativistic today. Note that for Dirac neutrinos we are considering the scenario in which right-handed neutrinos and left-handed anti-neutrinos are sterile and never achieved thermal equilibrium. Our results can be amended for other specific scenarios. For a < a ν we take the matter density to be (Ω m − Ω ν )/a 3 . The radiation density is Ω γ N eq /(2a 4 ) for a < a ν , where N eq = 3.36 for three flavors of Dirac (mostly) left-handed neutrinos and right-handed anti-neutrinos. We also take N eq = 3.36 for three active flavors of Majorana left-handed and righthanded neutrinos. For a > a ν , we take the radiation density to be (Ω γ N eq /2 − a ν Ω ν )/a 4 = Ω γ /a 4 . The present density of photons relative to the critical density is Ω γ = 2.473 × 10 −5 h −2 [2].
To obtain d we minimize the χ 2 with 21 terms, corresponding to the 21 BAO observables, with respect to Ω DE and d, and obtain O DE = 0.718 ± 0.003 and d ≈ 0.0340 ± 0.0002, with χ 2 per degree of freedom 19.8/19, and correlation coefficient 0.989 (this high correlation coefficient is due to the high precision of θ MC ). Setting r s = d we obtain m ν ≈ 0.73 − 0.35 · δh + 0.05 · δb ± 0.15 eV.
A more precise result is obtained with a global fit by minimizing the χ 2 with 21 terms varying Ω DE and m ν directly. We obtain Ω DE = 0.7175 ± 0.0023 and m ν = 0.711 − 0.335 · δh + 0.050 · δb ± 0.063 eV, (9) with χ 2 /d.f. = 19.9/19, and correlation coefficient 0.924. This is our main result. Equation (9) is obtained from BAO measurements alone, and is written in a way that can be combined with independent constraints on the cosmological parameters m ν , h and Ω b h 2 , such as measurements of the power spectrum of density fluctuations P (k), the CMB, and direct measurements of the Hubble parameter.