_{1}

^{*}

From 21 independent Baryon Acoustic Oscillation (BAO) measurements we obtain the following sum of masses of active Dirac or Majorana neutrinos:
, where
and
. This result may be combined with independent measurements that constrain the parameters
Σm_{v}, h, and
Ω_{b}h^{2} . For
and
, we obtain
at 95% confidence.

We extend the analysis presented in “Study of baryon acoustic oscillations with SDSS DR13 data and measurements of Ω k and Ω DE ( a ) ” [

r s = d (1)

to constrain the sum of neutrino masses ∑ m ν . c is the speed of light, and H 0 ≡ 100 h 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 [

To be specific we consider three active neutrino flavors with three eigenstates with nearly the same mass m ν , so ∑ m ν = 3 m ν . This is a useful scenario to consider since our current limits on m ν 2 are much larger than the mass-squared-differences Δ m 2 and Δ m 21 2 obtained from neutrino oscillations [

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 / 94 eV 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 / ( 2 a 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 right-handed neutrinos. For a > a ν , we take the radiation density to be ( Ω γ N e q / 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 [

The data used to obtain d are 18 independent BAO distance measurements with Sloan Digital Sky Survey (SDSS) data release DR13 galaxies in the redshift range z = 0.1 to 0.7 [

As a reference we take

h = 0.678 ± 0.009 , Ω b h 2 = 0.02226 ± 0.00023 (2)

(at 68% confidence) from “Planck TT + low P + lensing” data (that does not contain BAO information) [

Due to correlations and non-linearities we obtain our final result (Equation (9) below) with a global fit. The following equations are included to illustrate the dependence of r s and d on the cosmological parameters h, Ω b h 2 and ∑ m ν in limited ranges of interest. Integrating the comoving sound speed of the photon-baryon-electron plasma until a dec = 1 / ( 1 + z dec ) with z dec = 1089.9 ± 0.4 [

r s ≈ 0.0339 × A × ( 0.28 Ω m ) 0.24 (3)

with

A ≈ 0.990 + 0.007 ⋅ δ h − 0.001 ⋅ δ b + 0.020 ⋅ ∑ m ν 1 eV , (4)

where

δ h ≡ ( h − 0.678 ) / 0.009 , (5)

δ b ≡ ( Ω b h 2 − 0.02226 ) / 0.00023 . (6)

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 Ω DE = 0.718 ± 0.003 and

d ≈ 0.0340 ± 0.0002, (7)

with χ 2 per degree of freedom 19.8/19, and correlation coefficient 0.989 between Ω DE and d (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 ⋅ δ h + 0.05 ⋅ δ b ± 0.15 eV . (8)

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 ⋅ δ h + 0.050 ⋅ δ b ± 0.063 eV , (9)

with χ 2 / d .f . = 19.9 / 19 , and correlation coefficient 0.924 between Ω DE and ∑ m ν . 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.

Setting δ h = ± 1 and δ b = ± 1 we obtain the following upper bound on the mass of active neutrinos m ν = 1 3 ∑ m ν :

m ν < 0.43 eV at 95 % confidence . (10)

Hoeneisen, B. (2018) Constraints on Neutrino Masses from Baryon Acoustic Oscillation Measurements. International Journal of Astronomy and Astrophysics, 8, 1-5. https://doi.org/10.4236/ijaa.2018.81001

Appendix 1. Removing constraints

Freeing Ω k and keeping Ω DE constant we obtain Ω k = − 0.003 ± 0.006 , Ω DE + 2.2 Ω k = 0.719 ± 0.003 , and

∑ m ν = 0.623 − 0.334 ⋅ δ h + 0.050 ⋅ δ b ± 0.191 eV , (11)

with χ 2 / d .f . = 19.6 / 18 .

Fixing Ω k = 0 and letting Ω DE ( a ) = Ω DE ⋅ { 1 + w a ⋅ ( 1 − a ) } we obtain Ω DE = 0.716 ± 0.004 , w a = 0.064 ± 0.148 , and

∑ m ν = 0.603 − 0.349 ⋅ δ h + 0.052 ⋅ δ b ± 0.257 eV , (12)

with χ 2 / d .f . = 19.7 / 18 .

Freeing Ω k and letting Ω DE ( a ) = Ω DE ⋅ { 1 + w a ⋅ ( 1 − a ) } we obtain Ω k = − 0.008 ± 0.004 , Ω DE + 2.2 Ω k = 0.718 ± 0.004 , w a = 0.227 ± 0.069 , and

0 < ∑ m ν = − 0.388 − 0.350 ⋅ δ h + 0.050 ⋅ δ b ± 0.830 eV , (13)

with χ 2 / d .f . = 17.8 / 17 .

Appendix 2. Removing data.

In this Appendix we apply the constraints Ω k = 0 and Ω DE constant. Removing the measurement of θ MC we obtain Ω DE = 0.722 ± 0.011 and

∑ m ν = 0.579 − 0.333 ⋅ δ h + 0.049 ⋅ δ b ± 0.285 eV , (14)

with χ 2 / d .f . = 19.7 / 18 .

Removing the measurement of θ MC and the two Lya measurements we obtain Ω DE = 0.716 ± 0.014 and

∑ m ν = 0.743 − 0.330 ⋅ δ h + 0.049 ⋅ δ b ± 0.366 eV , (15)

with χ 2 / d .f . = 11.2 / 16 .

Keeping only the measurement of θ MC we need to fix Ω DE in order to get zero degrees of freedom and have a unique solution. The best way to fix Ω DE is with BAO measurements, and that is the purpose of the present study.