2.1. Tailor made models

Careful studies of Y_P indicate that the uncertainties in most determinations are dominated by systematic errors rather than statistical errors. Increasing the number of objects in the samples used to determine Y_P will, of course, decrease the statistical errors. However, it will not decrease the systematic ones.

Some systematic errors can be diminished by a careful selection of the objects used for the determination as well as by the use of tailor-made models for each object. Normal observational procedures, like reddening correction and underlying absorption correction, include systematic errors; this occurs because both the reddening law and the underlying absorption correction for the different helium lines are not perfectly known, and any error that affects any helium (or hydrogen) line will affect systematically the determinations of each object; such systematic effects can be minimized by selecting objects with small reddening corrections and He I large equivalent widths in emission. Corrections like the ionization correction factor due to the presence of neutral helium, ICF (He), or the collisional contribution to I(Hβ) depend on the particular objects included in the sample. Since each object is unique, there is no such thing as an average ICF (He) or a typical I(Hβ) collisional correction for H II regions; the final error for these effects will be systematic in any sample, hence tailor-made models for each object are required.

For the previous reasons we consider that a better Y_P determination can be obtained by studying in depth a few H II regions, rather than by using larger sets of objects without a tailor-made model for each of them.

2.2. The new recombination coefficients of the He I Lines

To obtain a precise Y_P value it is necessary to have the most accurate atomic physics parameters attainable. ^{Porter et al. (2013)} have computed updated effective recombination coefficients for the He I lines that differ from those by ^{Porter et al. (2005)}. The new values were computed to correct small errors in the implementation of Case B calculations; they also included a finer grid of calculations, useful for high- precision determinations. The differences between both sets of coefficients are small but significant for the determination of Y_P.

From the new atomic data, we present in Table 1 the physical characteristics of our 5 favorite objects derived following the same procedure used in Paper I.

2.3. Updated Y values

To determine Y_P we have to estimate the amount of helium produced by the stars during the evolution of the galaxies in our sample. To this end we assume that the helium mass increase to oxygen mass increase ratio, ∆Y /∆Z_O, is constant. It is possible to determine this ratio self-consistently from the points in our sample, as done by ^{Aver et al. (2015)} and ^{Izotov et al. (2014)} for their samples. We consider that this procedure for a sample as small as ours increases the error in the ∆Y /∆Z_O value. Instead, we use observations of brighter objects with a metallicity not as low and high quality data, as well as chemical evolution models for galaxies of low mass and metallicity. From ^{Carigi & Peimbert (2008)} and ^{Peimbert et al. (2010)} we obtain ∆Y /∆Z = 1.75, Z_O/Z = 0.53 and ∆Y /∆Z_O = 3.3 ± 0.7.

In Table 2 we present the Y and Y_P determinations for each object of our sample as well as the Y values we determined in Paper I. For each determination we have broken down the error into its statistical and systematic components: we first present the statistical and then the systematic ones. By taking a weighted average of these 5 Y_P values we obtain the updated Y_P value of the sample. The final statistical error amounts to 0.0019, the final systematic error amounts to 0.0021; adding quadratically both components the total error adds up to 0.0029.

It can be seen from Table 2 that the extrapolation from Y to Y_P for the objects in our sample is small and amounts to ∆Y = 0.0044.

Once the He I recombination coefficients have been recomputed (^{Porter et al. 2013}), we consider that the new determinations produce an uncertainty on Y_P of about 0.0010, the value we adopted in Paper I.

A thorough discussion on the systematic and statistical errors adopted in our Y_P determination is presented in Paper I.

2.4. The fluorescent contribution to the H I and He I Lines

Nonionizing stellar continua are a potential source of photons for continuum pumping of the hydrogen Lyman transitions, the so-called Case D (^{Luridiana et al. 2009}). Since these transitions are optically thick, de-excitation occurs through higher series lines, in particular excitation to n_u ≥ 3 produces transitions to n_l ≥ 2. As a result, the emitted flux in the affected lines has a fluorescent contribution in addition to the usual recombination one; consequently, Balmer emissivities are systematically enhanced above Case B predictions. Moreover, the He I lines are also enhanced by fluorescence. To a first approximation the effect of Case D on the H I lines is compensated by the effect of Case D on the He I lines. We leave for a future paper an estimate of the importance of Case D in the Y_P determination.

4.1. Determination of N_ν from Y_P and BBN

There is still no good agreement on the value of τ_n, see for example the discussion in ^{Salvati et al. (2016)}. There are three values of τ_n that are relevant: (a) five determinations based on the bottle method that yield τ_n = 879.6 ± 0.8 s (^{Pignol 2015}); (b) two determinations based on the beam method that yield τ_n = 888.0 ± 2.1 s (^{Pignol ibid.}), and (c) the average over the best seven measurements presented by the Particle Data Group (^{Olive et al. 2014}) that yield τ_n = 880.3 ± 1.1 s.

We will adopt τ_n = 880.3 ± 1.1 s, the recommended value by the Particle Data Group (^{Olive et al. 2014}), and the Y_P values derived from H II regions to determine the number of neutrino families and we will compare these numbers with that adopted by SBBN to check the validity of the adopted number of neutrino families.

Based on the production of the Z particle by electron-positron collisions in the laboratory and taking into account the partial heating of neutrinos produced by electronpositron annihilations during BBN, ^{Mangano et al. (2005)} find that N_eff = 3.046. Therefore the difference between the number of neutrino families and the SBBN number of neutrino families is given by ∆N_ν = N_eff -3.046.

Discussions on the implications for N_eff values different from 3.046 have been presented by ^{Steigman (2013)} and ^{Nollett & Steigman (2014}, ²⁰¹⁵⁾.

From the SBBN N_eff = 3.046 value and the re- lation ∆N_ν = 75∆Y (^{Mangano & Serpico 2011}), it follows that our Y_P determination implies that N_eff = 2.90 ± 0.22 and consequently that ∆N_ν amounts to −0.16 ± 0.22 (68% confidence level, CL), a result in good agreement with SBBN.

In Table 3 we present the ∆N_ν values derived from the three Y_P determinations; we also present the Y_P (H II+CMB) values that combine the Y_P (H II) and the Y_P (CMB) values as well as the ∆N_ν derived from such Y_P determinations.

^{Izotov et al. (2014)} find Y_P = 0.2551 ± 0.0022, which implies an effective number of neutrino families, N_eff = 3.58±0.25 (68% CL), ±0.40 (95.4% CL), and ±0.50 (99% CL) values. This result implies that a non-standard value of N_eff is preferred at the 99% CL, suggesting the prescence of a fourth neutrino family with a fractional contribution to N_eff at the time of decoupling.

4.2. Determination of τ_n from Y_P and BBN

It is possible from the Y_P values and the SBBN to determine τ_n. Following ^{Salvati et al. (2016)}, we present in Table 4 the τ_n values obtained from the Y_P values derived by ^{Izotov et al. (2014)}, ^{Aver et al. (2015)} and ourselves. Also in Table 4 we present the τ_n(H II+CMB) values that combine the τ_n(H II) and the τ_n(CMB) values.

The τ_n results by ^{Aver et al. (2015)} and ourselves are within 1σ from the average presented by the Particle Data Group (^{Olive et al. 2014}), and while consistent with both the bottle and the beam τ_n de- terminations, they slightly favor the determination based on the bottle method. On the other hand, the determination of ^{Izotov et al. (2014)} is more than 3σ away from both laboratory determinations.

The τ_n values from the three groups derived from Y_P are within 1σ from the result of the SBBN obtained by Planck based on the TT, TE, and EE spectra that amounts to τ_n(CMB) = 907 ± 69 s (^{Planck Collaboration 2015}). The τ_n Planck result is independent of the Y_P values derived from H II regions.