A new determination of the primordial helium abundance using the analyses of HII region spectra from SDSS
O.A. Kurichin, P.A. Kislitsyn, V.V. Klimenko, S.A. Balashev, A.V. Ivanchik
MMNRAS , 1–12 (2021) Preprint 19 February 2021 Compiled using MNRAS L A TEX style file v3.0
A new determination of the primordial helium abundanceusing the analyses of H ii region spectra from SDSS. O.A. Kurichin, ★ P.A. Kislitsyn, V.V. Klimenko, S.A. Balashev, and A.V. Ivanchik Ioffe Institute, Polytekhnicheskaya 26, 194021, Saint-Petersburg, Russia Alferov University, Khlopina d.8, k.3, A, 194021, Saint-Petersburg, Russia
Accepted 2021 January 21. Received 2020 December 27; in original form 2020 October 01MNRAS v. 502, Issue 2, pp. 3045–3056, April 2021
ABSTRACT
The precision measurement of the primordial helium abundance 𝑌 𝑝 is a powerful probe ofthe early Universe. The most common way to determine 𝑌 𝑝 is analyses of observations of metal-poor H ii regions found in blue compact dwarf galaxies. We present the spectroscopic sampleof 100 H ii regions collected from the Sloan Digital Sky Survey. The final analysed sampleconsists of our sample and HeBCD database from Izotov et al. 2007. We use a self-consistentprocedure to determine physical conditions, current helium abundances, and metallicities ofthe H ii regions. From a regression to zero metallicity, we have obtained 𝑌 𝑝 = . ± . 𝑌 Planck p = . ± . 𝑌 𝑝 and the primordial deuterium abundance taken from Particle Data Group (Zyla et al.2020) we put a constraint on the effective number of neutrino species 𝑁 eff = . ± .
16 whichis consistent with the Planck one 𝑁 eff = . ± .
17. Further increase of statistics potentiallyallows us to achieve Planck accuracy, which in turn will become a powerful tool for studyingthe self-consistency of the Standard Cosmological Model and/or physics beyond.
Key words: early Universe – primordial nucleosynthesis – galaxies: abundances – cosmolog-ical parameters
The determination of the primordial element abundances is a pow-erful method for studying the physics of the early Universe. Thelight elements D, He, He, Li were produced during PrimordialNucleosynthesis which began a few seconds after the Big Bang andlasted for several minutes. Comparison of theoretical calculationsbased on the well-established knowledge of nuclear and particlephysics with observations of the light element abundances allowsone to obtain an independent estimate of one of the key cosmolog-ical parameters – the baryon-to-photon ratio 𝜂 ≡ 𝑛 𝑏 / 𝑛 𝛾 (see e.g.,Weinberg 2008; Gorbunov & Rubakov 2011). Since this quantityis measured independently for the later cosmological epoch (theepoch of Primordial Recombination), it gives us the possibility ofchecking the Standard Cosmological Model for self-consistency orstudying the effects related to the new physics (or “physics beyond’,see e.g. Cyburt et al. 2005).Direct measurements of the primordial element abundances arerather complicated by the presence of non-primordial fractions ofthese elements in the studied objects. This non-primordial fractionmostly arises from stellar nucleosynthesis (e.g. Nomoto et al. 2013)thus in order to take this fraction into account, one needs to observe ★ E-mail: [email protected] objects with a low star-formation history. The most common wayto determine the primordial helium abundance 𝑌 𝑝 (the fraction ofthe primordial helium in the total mass of baryonic matter) is theanalysis of emission lines produced in metal-deficient H ii regionslocated in blue compact dwarf (BCD) galaxies. Interstellar medium(ISM) of BCDs is chemically relatively unevolved, and thus itselemental composition may be close to the primordial one (seee.g. Izotov et al. 1994). Since the non-primordial fraction of Heis produced over time due to stellar nucleosynthesis as well asthe heavier elements, one can expect a correlation between theobserved He abundance 𝑌 and the metallicity 𝑍 (the abundanceof elements heavier than He). Thus the primordial He abundance 𝑌 p can be estimated by extrapolation of the 𝑌 − Z dependence tozero metallicity. This technique was firstly introduced by Peimbert& Torres-Peimbert (1974) and is still in use (Izotov et al. 2014; Averet al. 2015; Peimbert et al. 2016; Fernández et al. 2019; Valerdi &Peimbert 2019; Hsyu et al. 2020).According to the Peimbert & Torres-Peimbert (1974), the he-lium abundance determination requires measurements of relativefluxes of helium, hydrogen, and metal emission lines. However,there are numerous systematic effects which significantly affect ob-served fluxes: interstellar reddening, underlying stellar absorption,collisional excitation, self-absorption etc. (e.g. Izotov et al. 2014;Aver et al. 2015). In order to account for them the photoionization © a r X i v : . [ a s t r o - ph . C O ] F e b O.A. Kurichin et al. models for H ii regions are required. Currently, there are severalindependent scientific groups which developed specific models tosolve this problem. They obtained similar values of the primordialhelium abundances, however there is an inconsistency between theestimation by Izotov et al. (2014) and others. The reasons for thisinconsistency has not yet been found and therefore, for instance, theParticle Data Group (PDG, Zyla et al. 2020) reports all mentionedvalues of 𝑌 𝑝 but recommends to use the value 𝑌 p = . ± . Planck result 𝑌 Planck p = . ± . 𝑌 p in Table 1 and Fig. 1.The largest data sample of metal deficient galaxies was com-posed by Izotov et al. (1997); Thuan & Izotov (1998); Izotov &Thuan (2004). The sample contains optical spectra of H ii regionsin BCD galaxies, and is called the HeBCD database (Izotov et al.2007). For each object in the HeBCD sample, the authors (Izo-tov et al. 2014, hereafter ITG14) derived physical conditions andheavy-element abundances using so-called “direct method” (Izo-tov et al. 2006). The authors used five lines He i in the opticalband: 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝑌 estimate.Using a routine based on Monte-Carlo method Izotov et al. (2014)presented the final result 𝑌 p = . ± . 𝑌 Planck p = . ± . 𝑌 𝑝 using the same the HeBCD database and NIR observations fromITG14. AOS15 used Markov Chain Monte Carlo method (MCMC)on 8-dimensional parameter space in order to self-consistently de-termine the best-fit parameters for the photoionization model of H iiregions. AOS15 used 6 lines He i in the optical range: 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝑌 p = . ± . Planck result but differs formthe ITG14 result.Peimbert et al. (2016) (PPL16) used a different approach to adetermination of 𝑌 𝑝 . For each object they separately calculated afraction of He abundance Δ 𝑌 produced as a result of stellar nucle-osynthesis and subtracted this fraction from the observed 𝑌 . PPL16presented 𝑌 p = . ± . 𝑌 p = . ± . 𝑌 p using their own observational sample. Methodproposed by the authors implies a step by step excision of varioussystematic effects from observed spectra, including the underlyingstellar absorption and reddening. The determination of the heliumabundance and metallicity of objects was preformed via minimiza-tion of the likelihood function calculated for three observed He i Table 1.
The abundance of the primordial helium 𝑌 p obtained by differentauthors and methods. 𝑌 𝑝 Paper Abbreviation0.2551 ± ± ± ± ± ± ± ± This paper0.2471 ± Y p ITG14AOS15PPL16CF18VP19FTDT19HCPB20This work C M B Figure 1.
The estimates of 𝑌 p obtained by different authors and methods(data taken from Table 1). The vertical cyan line represents the Planck
CMB+BBN prediction for 𝑌 p . Our estimate is marked by the red point. Wemark the deviating estimate of Izotov et al. (2014) by the maroon point. Theresult obtained with the corrected ITG14 procedure (see Sec. 4.4) is markedby the red triangle point. lines - 𝜆 𝜆 𝜆 𝑌 p = . ± . 𝑌 p . The authors analysed absorption producedby near-pristine intergalactic gas along the line of sight to the quasarHS 1700 + 𝑌 p estimate. Onthe other hand the precision of this method is considerably lower(up to date) than the previously discussed approaches. The authorsreported 𝑌 p = . ± . 𝑌 p is presented in Hsyu et al.(2020). Their sample consists of the Keck observations, objectsselected from Sloan Digital Sky Survey (SDSS) (Aguado et al.2019), and HeBCD+NIR objects. The authors used a method similarto the AOS15 and obtained 𝑌 p = . ± . 𝑌 p estimation proposed by Izotov et al. (2014)and Aver et al. (2015) and also present our independent result for 𝑌 p based on analyses of SDSS spectra of BCD galaxies, such a MNRAS000
CMB+BBN prediction for 𝑌 p . Our estimate is marked by the red point. Wemark the deviating estimate of Izotov et al. (2014) by the maroon point. Theresult obtained with the corrected ITG14 procedure (see Sec. 4.4) is markedby the red triangle point. lines - 𝜆 𝜆 𝜆 𝑌 p = . ± . 𝑌 p . The authors analysed absorption producedby near-pristine intergalactic gas along the line of sight to the quasarHS 1700 + 𝑌 p estimate. Onthe other hand the precision of this method is considerably lower(up to date) than the previously discussed approaches. The authorsreported 𝑌 p = . ± . 𝑌 p is presented in Hsyu et al.(2020). Their sample consists of the Keck observations, objectsselected from Sloan Digital Sky Survey (SDSS) (Aguado et al.2019), and HeBCD+NIR objects. The authors used a method similarto the AOS15 and obtained 𝑌 p = . ± . 𝑌 p estimation proposed by Izotov et al. (2014)and Aver et al. (2015) and also present our independent result for 𝑌 p based on analyses of SDSS spectra of BCD galaxies, such a MNRAS000 , 1–12 (2021) rimordial Helium Abundance possibility was discussed in our previous work (Kurichin et al.2019) . The structure of the paper is as follows. The sample of H iiregions selected from the SDSS catalog is presented in Sect. 2. InSect. 3 we present our estimates of the primordial helium abundanceand slope of 𝑌 − O / H relation. In Sect. 4 we discuss obtained results,reasons for discrepancy with Izotov et al. (2014) and further possibleimprovements, before we conclude in Sect. 5. In this paper we quote68% confidence regions on measured parameter.
In this section, we describe observational sample of BCD galaxies.The spectra are selected from the SDSS catalog. We determine thephysical properties of H ii regions and the abundance of heliumusing the approach similar to AOS15 (see Appendix A for details).
The determination of the primordial helium abundance requires theanalyses of high quality H ii region spectra. Despite the mediumresolution and signal-to-noise (S/N) ratio of the SDSS spectra, agreat amount of SDSS objects can drastically increase statistics. Weuse the spectroscopic data from the SDSS Data Release 15 (DR15,Aguado et al. 2019) which contains the flux-calibrated spectra of80 420 starburst galaxies (this objects are tagged as ’STARBURST’in the SDSS catalog). We scanned the SDSS catalogs in a searchfor metal-poor H ii regions. From the catalog we selected spectra ofH ii regions satisfying the following selection criteria: (1) the red-shift is within the range of 0 ≤ 𝑧 ≤ . 𝜆 ≥
10, which leaves 46 197 objects. Up to date we manually selectedand analysed 580 objects (hereafter the 𝑆 sample). We estimate 𝑌 p using approach, similar to AOS15. We measure thefluxes and equivalent widths of the following emission lines: He i( 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝑆 sample is presented in Fig. 2.For each emission line we construct a local continuum by fitting aspline to selected continuum regions devoiding of any absorptionand emission. We determine a flux and equivalent width of a lineby the Gaussian profile fitting. We fit the profile using Monte CarloMarkov Chain approach with affine invariant sampler to obtainthe posterior probability function. The values of fitted parameters(amplitude, centroid and variance) corresponds to the maximumposterior probability and uncertainties estimate corresponds to thatcontaining 68.3% of area. The photoionization model used for the determination of the ob-served helium mass fraction 𝑌 and metallicity is described in Ap-pendix A. Here we present a brief description of its main properties.The fluxes of hydrogen and helium emission lines are calcu-lated as functions of specific physical parameters and then com-pared with the measured fluxes. These parameters include: the ratio of number densities of the single ionized helium to hydrogen 𝑦 + ,electron density 𝑛 𝑒 , electron temperature 𝑇 𝑒 , optical depth 𝜏 , pa-rameters of He and H underlying absorption 𝑎 He and 𝑎 H , extinctioncoefficient C(H 𝛽 ), and neutral hydrogen fraction 𝜉 . Each of theparameters is related to the corresponding systematic effect whichchanges the intrinsic line fluxes (see Appendix A for details).We obtain the probability distribution functions of these pa-rameters with the Monte Carlo Markov Chain approach implement-ing the affine-invariant ensemble sampler Foreman-Mackey et al.(2019). Such a technique ensures that we confidently find the globalmaximum in the many parametric space and provides reliable es-timates of statistical errors on the parameters. Furthermore, thecorrelation between different parameters can be easily traced bythis method. Applying the procedure described above to the sample 𝑆 and usingadditional selection criteria, we form the final sample 𝑆 f to be usedfor a determination of 𝑌 𝑝 . Firstly, we exclude 72 spectra where He 𝜆 𝑎 He determi-nation, which in turn allows to significantly decrease the systematicuncertainty of the 𝑦 + . It leaves us 508 spectra from the sample 𝑆 .We exclude 11 additional spectra, where the weak O iii 𝜆 𝜒 criteria. Weassumed that 𝜒 value for a good-fit model should be in the range [ 𝜈 − √ 𝜈, 𝜈 + √ 𝜈 ] , where 𝜈 is the number of degrees of freedom(see e.g. Hogg et al. 2010). We selected 100 objects satisfying the 𝜒 criteria and form our final sample 𝑆 f for the regression analysis.For each object in the sample 𝑆 f the metallicity O/H, 𝑌 , and 𝜒 values are presented in Table 2. In addition to our final sample 𝑆 f (Table 2) we use the HeBCDdatabase, which is one of the largest databases of high quality spectraof metal deficient H ii regions. We use the optical spectra fromIzotov et al. (2007) and the NIR spectra from Izotov et al. (2014)and determine the physical properties and abundances of helium andoxygen of the HeBCD objects using the approach discussed above.Applying the selection criteria from Section 2.4 to HeBCD databasewe selected 20 objects (for comparison AOS15 selected 17 objectsand ITG14 selected 28). We present 𝑌 , O/H and 𝜒 for these objectsin Table 3. Note that, it could be seen from Fig. 3, Tab. 2 and 3 thatHeBCD objects have less uncertainty in O/H than SDSS objects,and at the same time their uncertainties in 𝑌 are comparable. Thelarge uncertainty in O/H for SDSS objects is associated with lowerS/N ratio of their spectra. The fact is that the uncertainty in O/H isdirectly determined by the error in the measured flux of the weakline [O iii] 𝜆 𝑌 for the SDSS and HeBCDobjects is comparable, since it is determined by the photoionizationmodel itself and weakly depends on the errors in fluxes of individualHe lines. MNRAS , 1–12 (2021)
O.A. Kurichin et al. Å F l u x , e r g / c m / s / Å J1403+3913 H [ O III ] H e H H e H e H + H e H e H H [ O III ] H e Figure 2.
An example of an H ii region spectrum from the sample 𝑆 . The spectrum has S/N ratio of 22.42 and the metallicity of 12 + log(O/H) = 7.93. The He abundances 𝑌 derived from the emission line analysis inmetal-poor starburst galaxies of the 𝑆 f sample are presented inFig.3 together with the measurements obtained from the HeBCDsubsample.We perform a regression analysis of 𝑌 versus O/H values withthe following expression: 𝑌 = 𝑌 𝑝 + 𝑑𝑌𝑑 ( O / H ) × ( O / H ) (1)We estimate 𝑌 p and the slope 𝑑𝑌 / 𝑑 ( O / H ) with the MCMCroutine in the following cases: (i) for the 𝑆 f and HeBCD samplesseparately, and (ii) their combination. The results are summarized inTable 4. One can note that the fit to the SDSS and HeBCD samplesgives similar results. However, larger number of SDSS objects anda wider range of metallicities improve the precision of the slopeestimate by a factor of about 1.5. Combining two samples gives usthe final estimate: 𝑌 𝑝 = . ± . 𝑑𝑌 / 𝑑 ( 𝑂 / 𝐻 ) = ±
13 (2)The obtained 𝑌 𝑝 is in a good agreement with Planck’s result 𝑌 𝑃𝑙𝑎𝑛𝑐𝑘𝑝 = . ± . (c) is a simplified visualisation of Fig. 3 (b) . The en-tire range of metalicities is divided into equal bins for which theweighted mean of the points was calculated. The means and theiruncertainties are plotted on a 𝑌 − (O/H) plane along with a regres-sion curve, 1 𝜎 interval and 𝑌 𝑝 which were obtained earlier. Thefigure shows that the linear correlation of 𝑌 − (O/H) is preserved atleast up to O/H ∼ × − .Theoretical and observational estimates of the increase of he-lium with the increase of oxygen were discussed in Peimbert et al.(2007). Note that the authors use the oxygen abundance O by massfraction instead of the number density abundance O/H (which weuse in this paper). These quantities are connected by the relation O = · O / H1 + 𝑦 , where 𝑦 is the helium number density. The authors showed that the linear relation 𝑌 − O is preserved up to O ∼ × − , whichin our terms corresponds to O/H ∼ × − . Additionally, the au-thors derived the slope of the relation 𝑌 − O. Rewriting our estimateof the slope 𝑑𝑌 / 𝑑 ( O / H ) = ±
13 in Peimbert’s terms, we get thevalue Δ Y/ Δ O = 3 . ± .
1, which is in a good agreement with thevalue Δ Y/ Δ O = 3 . ± . 𝑁 eff In the frame of Primordial Nucleosynthesis the primordial Heabundance 𝑌 𝑝 has a strong dependence on the effective number ofneutrino species 𝑁 eff which allows to constrain this quantity usingthe estimated 𝑌 𝑝 value. To constrain 𝑁 eff we use the followingrelation from Fields et al. (2020): 𝑌 𝑝 = . (cid:16) 𝜂 . (cid:17) . (cid:16) 𝑁 eff . (cid:17) . (cid:16) 𝐺 N 𝐺 N , (cid:17) . (cid:16) 𝜏 𝑛 . (cid:17) . × [ 𝑝 ( 𝑛, 𝛾 ) 𝑑 ] . [ 𝑑 ( 𝑑, 𝑛 ) 𝐻𝑒 ] . [ 𝑑 ( 𝑑, 𝑝 ) 𝑡 ] . (3)Here 𝜂 = 𝜂 , 𝐺 N is Newtonian gravitational constant, 𝜏 𝑛 is a neutron mean lifetime. The last three terms are the key nuclearrates which affect the 𝑌 𝑝 . Note that 𝑌 𝑝 is not a precise baryometer,but instead, 𝑌 𝑝 sets a tight constraint on the effective number ofneutrino species. To estimate 𝜂 we use a precise measurement ofthe primordial deuterium abundance 𝐷 / 𝐻 = ( . ± . ) × − from Particle Data Group (Zyla et al. 2020) which is a weightedmean of 16 measurements. To constrain 𝜂 we use the followingrelation from Fields et al. (2020): MNRAS000
1, which is in a good agreement with thevalue Δ Y/ Δ O = 3 . ± . 𝑁 eff In the frame of Primordial Nucleosynthesis the primordial Heabundance 𝑌 𝑝 has a strong dependence on the effective number ofneutrino species 𝑁 eff which allows to constrain this quantity usingthe estimated 𝑌 𝑝 value. To constrain 𝑁 eff we use the followingrelation from Fields et al. (2020): 𝑌 𝑝 = . (cid:16) 𝜂 . (cid:17) . (cid:16) 𝑁 eff . (cid:17) . (cid:16) 𝐺 N 𝐺 N , (cid:17) . (cid:16) 𝜏 𝑛 . (cid:17) . × [ 𝑝 ( 𝑛, 𝛾 ) 𝑑 ] . [ 𝑑 ( 𝑑, 𝑛 ) 𝐻𝑒 ] . [ 𝑑 ( 𝑑, 𝑝 ) 𝑡 ] . (3)Here 𝜂 = 𝜂 , 𝐺 N is Newtonian gravitational constant, 𝜏 𝑛 is a neutron mean lifetime. The last three terms are the key nuclearrates which affect the 𝑌 𝑝 . Note that 𝑌 𝑝 is not a precise baryometer,but instead, 𝑌 𝑝 sets a tight constraint on the effective number ofneutrino species. To estimate 𝜂 we use a precise measurement ofthe primordial deuterium abundance 𝐷 / 𝐻 = ( . ± . ) × − from Particle Data Group (Zyla et al. 2020) which is a weightedmean of 16 measurements. To constrain 𝜂 we use the followingrelation from Fields et al. (2020): MNRAS000 , 1–12 (2021) rimordial Helium Abundance O/H Y Y p = ± dY / d (O/H) = 50 ± Y p a ) O/H Y Y p = ± dY / d (O/H) = 46 ± Y p b ) O/H -2-0++2 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030
O/H Y Y p = ± dY / d (O/H) = 46 ± Y p c ) Figure 3. (a) “ 𝑌 - (O/H)” diagram for 100 H ii regions of the final sample 𝑆 f . (b) “ 𝑌 - (O/H)” diagram for the final sample 𝑆 f (navy colored) combined with 20H ii regions of the HeBCD subsapmle (brown colored). The panel beneath the plot (b) represents the dispersion of points around the regression line in termsof sigma intervals with ∼
75% fraction of all points falling into 1 𝜎 interval and ∼
94% falling into 2 𝜎 interval. (c) the same as (b) but the whole sample wassliced into equally-sized bins in which of those the weighted mean point was calculated for a grater visualisation and clarity (regression line on (c) is the sameas on (b) ).MNRAS , 1–12 (2021) O.A. Kurichin et al.
Table 2.
The derived parameters: metallicity O/H, current helium abundance 𝑌 , and 𝜒 value for the objects of the sample 𝑆 f .No. Object O/H · Y 𝜒 No. Object O/H · Y 𝜒 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 𝐷𝐻 = . × − (cid:16) 𝜂 . (cid:17) − . (cid:16) 𝑁 eff . (cid:17) . (cid:16) 𝐺 N 𝐺 N , (cid:17) . × (cid:16) 𝜏 𝑛 . (cid:17) . [ 𝑝 ( 𝑛, 𝛾 ) 𝑑 ] − . [ 𝑑 ( 𝑑, 𝑛 ) 𝐻𝑒 ] − . [ 𝑑 ( 𝑑, 𝑝 ) 𝑡 ] − . × [ 𝑑 ( 𝑝, 𝛾 ) 𝐻𝑒 ] − . [ 𝐻𝑒 ( 𝑛, 𝑝 ) 𝑡 ] . [ 𝐻𝑒 ( 𝑑, 𝑝 ) 𝐻𝑒 ] − . (4)Using MCMC procedure we get the following estimates of 𝜂 and 𝑁 eff : 𝜂 = . ± .
09 and 𝑁 eff = . ± .
16. This estimates are in a good agreement with the Planck results (PlanckCollaboration et al. 2020) 𝜂 = . ± .
04 and 𝑁 eff = . ± . We find a good agreement of our estimate of 𝑌 p with the previousones (except for the ITG14 estimate), see Table 1 and Fig. 5. Herewe discuss systematic effects that could have caused the observedshift of the ITG14 result. MNRAS000
04 and 𝑁 eff = . ± . We find a good agreement of our estimate of 𝑌 p with the previousones (except for the ITG14 estimate), see Table 1 and Fig. 5. Herewe discuss systematic effects that could have caused the observedshift of the ITG14 result. MNRAS000 , 1–12 (2021) rimordial Helium Abundance Table 3.
The derived parameters: metallicity O/H, current helium abundance 𝑌 , and 𝜒 value for the objects of the HeBCD sample.No. Object O/H · Y 𝜒 ± ± ± ± ± ± ± ± ± ± ± ± № ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± № ± ± № ± ± ± ± ± ± ± ± Table 4.
Estimates of the 𝑌 p and 𝑑𝑌 / 𝑑 ( 𝑂 / 𝐻 ) .Sample 𝑌 p 𝑑𝑌 / 𝑑 (O/H)HeBCD 0 . ± . ± 𝑆 f . ± . ± 𝑆 f + HeBCD 0 . ± . ± 𝑦 𝑤𝑚 The physical parameters of H ii regions in the ITG14 approach(the electron density 𝑛 𝑒 , electron temperature 𝑇 𝑒 , and optical depth 𝜏 at 3889Å) are determined via the MCMC minimization of thefollowing 𝜒 likelihood function: 𝜒 = ∑︁ 𝑖 ( 𝑦 + 𝑖 − 𝑦 𝑤𝑚 ) 𝜎 ( 𝑦 + 𝑖 ) . (5)where 𝑦 + 𝑖 is the ratio of number densities of the single ionizedhelium and hydrogen derived for the He emission line labeled by 𝑖 , 𝜎 ( 𝑦 + 𝑖 ) is its uncertainty, 𝑦 𝑤𝑚 is a weighted mean, which defined as 𝑦 𝑤𝑚 = (cid:205) 𝑗 𝜔 𝑗 𝑦 + 𝑗 (cid:205) 𝑗 𝜔 𝑗 (6)where the statistical weight 𝜔 𝑗 = / 𝜎 ( 𝑦 + 𝑗 ) . In the Eq. (5) the sum-mation over 𝑖 is carried out for six He lines: 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝑗 is carried out only for four lines: 𝜆 𝜆 𝜆 𝜆 𝑦 + estimate. In the ITG14 approach the lines 𝜆 𝜆 𝑦 𝑤𝑚 calculation since they show higher disper-sion of the 𝑦 + 𝑗 / 𝑦 𝑤𝑚 ratio around a value of 1 (see Fig. 3 in Izotovet al. 2014). However, this dispersion is not a correct characteristicof the “goodness” of the 𝑗 line, but it only demonstrates how the 𝑗 line affects the determination of 𝑦 𝑤𝑚 . For a correct estimate of the“goodness” of the 𝑗 line, one have to consider the relative deviation ( 𝑦 + 𝑗 − 𝑦 𝑤𝑚 )/ 𝜎 ( 𝑦 + 𝑗 ) instead of the ratio 𝑦 + 𝑗 / 𝑦 𝑤𝑚 . Table 5.
Regression analyses for HeBCD and HeBCD + 𝑆 𝑓 samples pro-cessed with different methods. The first column: the method used for theanalysis, the second and the third columns: the results for the correspondingsample. Method HeBCD HeBCD + 𝑆 f This paper 0.2464 ± ± ± ± corr ± ± To check this point we calculate the distributions of ( 𝑦 + 𝑖 − 𝑦 𝑤𝑚 )/ 𝜎 ( 𝑦 + 𝑖 ) for all HeBCD objects. As can be seen from Fig.4 all of the distributions have a comparable dispersion with only 𝜆 𝜆 𝜆 𝑦 𝑤𝑚 calculation. Thisexclusion in the ITG14 analyses might bias the 𝑌 estimate. 𝑎 He The value of underlying absorption of the He lines is given in termsof the equivalent width and normalized to the value 𝑎 He for the 𝜆 𝑎 He value is fixed to be 0.4 Å (while reala He varies from 0 to 1 Å. The 𝑎 He for other He lines are recalculatedusing the coefficients presented in Izotov et al. (2014). Since the realvalue of underlying absorption can differ from 0.4 Å, it can shift theintrinsic line fluxes. Therefore (similar to the AOS15 approach) weintroduce 𝑎 He as a free parameter in the equation 5. 𝑇 𝑒 determination In ITG14 electron temperature was randomly varied within therange ( . × (cid:101) 𝑇 𝑒 , . × (cid:101) 𝑇 𝑒 ) where (cid:101) 𝑇 𝑒 = 𝑇 𝑒 ( O iii )×( . × − 𝑇 𝑒 ( O iii )+ . + / 𝑇 𝑒 ( O iii )) (7)here the electron temperature 𝑇 𝑒 ( O iii ) was derived from the ratioof [O iii] emission lines fluxes 𝜆 /( 𝜆 + 𝜆 ) . This strictprior artificially over-constrains the range where MCMC routinesearches for the best-fit value of 𝑇 𝑒 . It leads to the concentration ofthe determined electron temperatures on lower or upper bounds ofthe prior. This effect was noted by Izotov et al. (2014) (see Fig. 4band 4d therein). We suggest to remove this prior constraint as it candirectly bias the value of 𝑦 𝑤𝑚 . The baseline of ITG14 method is presented in Appendix B. Apply-ing ITG14 to HeBCD database we obtain the following estimate of 𝑌 𝑝 = . ± . 𝑌 𝑝 = . ± . 𝑆 𝑓 and HeBCD samples and obtain 𝑌 𝑝 = . ± . 𝑌 𝑝 , we have corrected the ITG14 procedure and appliedit to the HeBCD and 𝑆 𝑓 samples. We obtain the following results: 𝑌 𝑝 = . ± . 𝑌 𝑝 = . ± . MNRAS , 1–12 (2021)
O.A. Kurichin et al. ( y i y w m ) / ( y i ) O / H -4-20+2+4 ( y i y w m ) / ( y i ) O / H -4-20+2+4 O / H -4-20+2+4 Figure 4.
The distributions of ( 𝑦 + 𝑖 − 𝑦 𝑤𝑚 )/ 𝜎 ( 𝑦 + 𝑖 ) derived for six He lines: 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝑦 𝑤𝑚 . HeBCD + 𝑆 𝑓 . It can be seen that after applying the discussed correc-tions to the ITG14 procedure, the estimate of 𝑌 𝑝 becomes consistentwith other estimates (see Fig. 1). The results of the calculations aresummarized in Table 5. We scan the SDSS DR15 catalog for spectra of H ii regions in bluecompact dwarf galaxies (BCD). Such objects are marked as the “STARBURST” in the SDSS catalog. We choose objects with theobservational signal-to-noise ratio 𝑆 / 𝑁 higher than 10. In total, weanalysed 580 such objects. After processing the spectra with theapproach similar to Aver et al. (2015) we apply the 𝜒 selectioncriteria to the database. 100 objects satisfy the criteria and makeup the final sample (Tab. 2). Using this sample in combine withthe HeBCD+NIR sample (20 objects) from Izotov et al. (2014) wereport a new estimate of the primordial helium abundance 𝑌 𝑝 = . ± . 𝑌 − O / H relation 𝑑𝑌 / 𝑑 ( O / H ) = ±
13. This slope is determined on 3 . 𝜎 confidence level which issignificantly higher compared to the previous studies.Using our value of 𝑌 𝑝 and the primordial deuterium abundanceD/H from Zyla et al. (2020) we constrain the effective number ofneutrino species 𝑁 eff = . ± .
16 and baryon to photon ratio 𝜂 = . ± .
09. The results are in good agreement with thePlanck results of 𝜂 = . ± .
04 and 𝑁 eff = . ± . 𝑌 𝑝 determination we suppose that we have found thereason for 𝑌 𝑝 overestimation. Further improvements
The SDSS catalog contains a large number of H ii region spectra inBCDs. We show that the data can be used to increase the regressionstatistics and thus significantly improve the accuracy of the estimate b Y p C M B Izotov et al. 2014Aver et al. 2015This work b h Figure 5.
The primordial abundance of He as a function of the baryon-to-photon ratio 𝜂 𝑏 . The navy curve represents the calculation of PrimordialNucleosyntesis with our code (Orlov et al. 2000). Orange, blue and greenboxes represent the estimates of 𝑌 𝑝 obtained by Izotov et al. (2014), Averet al. (2015), and in this work. The vertical cyan line is the measurementof the baryon density based on the analysis of the CMB anisotropy (PlanckCollaboration et al. 2020). of 𝑌 𝑝 . Unfortunately, manual data processing is difficult and time-consuming due to a large amount of spectral data. We plan to developan automated procedure of SDSS spectra processing in order tosignificantly increase the statistics. Further increase of statisticspotentially allows us to achieve Planck accuracy, which in turn willbecome a powerful tool for studying the self-consistency of theStandard Cosmological Model and/or physics beyond.Beyond further enlarging of the data set with SDSS observa-tions, the accuracy of the determination of 𝑌 p can be improved byadding more high quality spectra to the analysis and by refiningspectra processing. High quality spectra can be obtained by obser-vations of certain SDSS objects, for instance, from the final sample 𝑆 f . In addition, high quality spectra can allow us to use additionalweak H and He lines in the analysis, it will increase the reliability MNRAS000
The primordial abundance of He as a function of the baryon-to-photon ratio 𝜂 𝑏 . The navy curve represents the calculation of PrimordialNucleosyntesis with our code (Orlov et al. 2000). Orange, blue and greenboxes represent the estimates of 𝑌 𝑝 obtained by Izotov et al. (2014), Averet al. (2015), and in this work. The vertical cyan line is the measurementof the baryon density based on the analysis of the CMB anisotropy (PlanckCollaboration et al. 2020). of 𝑌 𝑝 . Unfortunately, manual data processing is difficult and time-consuming due to a large amount of spectral data. We plan to developan automated procedure of SDSS spectra processing in order tosignificantly increase the statistics. Further increase of statisticspotentially allows us to achieve Planck accuracy, which in turn willbecome a powerful tool for studying the self-consistency of theStandard Cosmological Model and/or physics beyond.Beyond further enlarging of the data set with SDSS observa-tions, the accuracy of the determination of 𝑌 p can be improved byadding more high quality spectra to the analysis and by refiningspectra processing. High quality spectra can be obtained by obser-vations of certain SDSS objects, for instance, from the final sample 𝑆 f . In addition, high quality spectra can allow us to use additionalweak H and He lines in the analysis, it will increase the reliability MNRAS000 , 1–12 (2021) rimordial Helium Abundance of determining the physical parameters of H ii regions. The spectraprocessing improvement could include measurements of line fluxesseparate from underlying absorption features. To account for theunderlying absorption two additional parameters are included intothe photoinization model. The direct measurement of the underlyingabsorption features will remove the associated parameters from themodel, that in turn may increase the accuracy of determining otherparameters. DATA AVAILABILITY
The observational data used in this paper is available from the publicdata archives of SDSS (SDSS SkyServer).
ACKNOWLEDGEMENTS
The authors thank the anonymous referee for useful suggestions.The authors are grateful to P. Shternin for discussions on statisticalmethods of data analysis, and E. Aver for useful comments andproviding data on He emissivities. The work was supported by theRussian Science Foundation (grant 18-12-00301).
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APPENDIX A: PHOTOINIZATION MODEL
Here we describe our photoionization model of the H ii region. Weuse the approach similar to one described by Aver et al. (2015).The helium mass fraction ( 𝑌 = 𝑚 He / 𝑚 gas ) of the H ii regionis derived by the analytic expression 𝑌 = 𝑦 + 𝑦 ( − 𝑍 ) (A1)where 𝑦 is the ratio of the total number densities of helium andhydrogen 𝑦 = 𝑛 He 𝑛 H , (A2) 𝑍 is the total metallicity, which is connected with the total oxygenabundance (O/H) with the following equation (Aver et al. (2010)): 𝑍 = × O / H . (A3)The helium abundance is given by a sum of abundances of itsdifferent ionization states: 𝑦 = 𝑦 + 𝑦 + + 𝑦 ++ = 𝐼𝐶𝐹 × (cid:0) 𝑦 + + 𝑦 ++ (cid:1) (A4)here 𝑦 , 𝑦 + and 𝑦 ++ are abundances of neutral, single and doubleionized helium, the 𝐼𝐶𝐹 is the ionization correction factor account-ing a part of neutral helium. We assumed this contribution to benegligible and set the
𝐼𝐶𝐹 = 𝑦 + and 𝑦 ++ we fit the observed fluxes ofthe helium and hydrogen emission lines using analytic functions. Wedetermine O / H using the direct method from Izotov et al. (2006).The relative fluxes of helium and hydrogen recombination linesare given by 𝐹 He ( 𝜆 ) 𝐹 ( H 𝛽 ) theor = 𝑦 + × 𝐸 He ( 𝜆 ) 𝐸 ( H 𝛽 ) × 𝑓 𝜏 ( 𝜆 ) + 𝐶𝑅 ( H 𝛽 ) ×× 𝐸𝑊 ( H 𝛽 ) + 𝑎 H ( H 𝛽 ) 𝐸𝑊 ( H 𝛽 ) 𝐸𝑊 ( 𝜆 ) 𝐸𝑊 ( 𝜆 ) + 𝑎 He ( 𝜆 ) × − 𝑓 ( 𝜆 ) 𝐶 ( H 𝛽 ) (A5)and 𝐹 H ( 𝜆 ) 𝐹 ( H 𝛽 ) theor = 𝐸 H ( 𝜆 ) 𝐸 ( H 𝛽 ) × + 𝐶𝑅 ( 𝜆 ) + 𝐶𝑅 ( H 𝛽 ) ×× 𝐸𝑊 ( 𝐻𝛽 ) + 𝑎 H ( H 𝛽 ) 𝐸𝑊 ( H 𝛽 ) 𝐸𝑊 ( 𝜆 ) 𝐸𝑊 ( 𝜆 ) + 𝑎 H ( 𝜆 ) × − 𝑓 ( 𝜆 ) 𝐶 ( H 𝛽 ) (A6)Here 𝐸 He , H ( 𝜆 ) are the helium and hydrogen emissivity functionsof 𝑛 𝑒 and 𝑡 𝑒 , 𝑓 𝜏 ( 𝜆 ) is optical depth function, 𝐶𝑅 ( 𝜆 ) is the correctionfor collision excitation of hydrogen lines, 𝑓 ( 𝜆 ) is the correction forinterstellar reddening. The factors containing EWs and 𝑎 He , H ( 𝜆 ) MNRAS , 1–12 (2021) O.A. Kurichin et al.
Table A1.
Coefficients for the hydrogen emissivities 𝐴 𝑖 𝑗 .Line i ↓ j → 𝛼 − − − 𝛾 − − − 𝛿 − − − − 𝛾 − − − − − − − − are responsible for the helium and hydrogen underlying stellar ab-sorption. These terms are discussed in details below.The helium emissivity 𝐸 He ( 𝜆 ) is calculated using the bilinearinterpolation in the fine grid of the electron density and temperaturepresented by Aver et al. (2013). The hydrogen emissivity 𝐸 H ( 𝜆 ) iscalculated using data from Hummer & Storey (1987). The emissivityof H( 𝛽 ) line is calculated using the following expression: 𝐸 ( H 𝛽 ) = (cid:20) 𝑎 − 𝑏 ( ln ( 𝑡 𝑒 )) + 𝑐 ln ( 𝑡 𝑒 ) + 𝑑 ln ( 𝑡 𝑒 ) (cid:21) × 𝑡 − 𝑒 , (A7)where 𝑎 = − . × , 𝑏 = . 𝑐 = . 𝑑 = . × . The emissivity of other hydrogen lines is givenby 𝐸 H ( 𝜆 ) = ∑︁ 𝑖 𝑗 𝐴 𝑖 𝑗 ( log ( 𝑡 𝑒 )) 𝑖 ( log ( 𝑛 𝑒 )) 𝑗 (A8)Coefficients 𝐴 𝑖 𝑗 are presented in Tab. A1.Following Aver et al. (2010) we now define the 𝐶𝑅 ( 𝜆 ) factor 𝐶𝑅 ( 𝜆 ) = 𝜉 ∑︁ 𝑖 𝑎 𝑖 ( 𝜆 ) exp (cid:18) − 𝑏 𝑖 ( 𝜆 ) 𝑡 𝑒 (cid:19) 𝑡 𝑐 𝑖 ( 𝜆 ) 𝑒 (A9)where 𝜉 = 𝑛 H / 𝑛 H + is the ratio of densities of neutral and ionizedhydrogen, 𝑎 𝑖 , 𝑏 𝑖 and 𝑐 𝑖 are coefficients presented in Tab. A2.Then optical depth function 𝑓 𝜏 is used to make a correction forphotons that are reabsorbed or scattered out inside the H ii region.The corrections for each helium line is calculated individually usingexpression from Benjamin et al. (2002): 𝑓 𝜏 ( 𝜆 ) = + 𝜏 ( 𝑎 + 𝑡 𝑒 × ( 𝑏 + 𝑏 𝑛 𝑒 + 𝑏 𝑛 𝑒 )) (A10)Coefficients 𝑎 , 𝑏 , 𝑏 and 𝑏 are presented in Tab. A3.The observed helium and hydrogen line fluxes are also affectedby the underlying stellar absorption and interstellar reddening (thirdfactor in Eq. A5 and Eq. A6). First, the underlying absorption pa-rameters can be expressed as: 𝑎 He ( 𝜆 ) = 𝐴 ( 𝜆 ) × 𝑎 He ( ) 𝑎 H ( 𝜆 ) = 𝐵 ( 𝜆 ) × 𝑎 H ( 𝐻𝛽 ) (A11)where coefficients 𝐴 ( 𝜆 ) and 𝐵 ( 𝜆 ) presented in Tab. A4. Then, the correction of the interstellar reddening can be accounted by usingof the combination of the logarithmic correction factor 𝐶 ( H 𝛽 ) andthe reddening function 𝑓 ( 𝜆 ) , which are described in Cardelli et al.(1989).Secondly, we take into account that the He 𝜆 𝐹 ( He 𝜆 ) 𝐹 ( H 𝛽 ) = 𝐹 ( H8 + He 𝜆 ) 𝐹 ( H 𝛽 ) 𝐸𝑊 ( H8 ) + 𝑎 H ( H8 ) 𝐸𝑊 ( H8 ) − 𝐸 ( H8 ) 𝐸 ( H 𝛽 ) − 𝑓 ( H8 ) 𝐶 ( H 𝛽 ) (A12)Therefore our model have 8 fitting parameters (the abundanceof the single ionized helium 𝑦 + , electron density 𝑛 𝑒 , electron temper-ature 𝑡 𝑒 , optical depth 𝜏 , underlying stellar H and He absorptionparameters 𝑎 𝐻 and 𝑎 𝐻𝑒 , reddening parameter 𝐶 ( 𝐻𝛽 ) and the frac-tion of neutral hydrogen 𝜉 ) which are determined by minimizing ofthe likelihood function 𝜒 = ∑︁ 𝑖 (cid:16) 𝐹 ( 𝜆 𝑖 ) 𝐹 ( 𝐻 𝛽 ) theor − 𝐹 ( 𝜆 𝑖 ) 𝐹 ( 𝐻 𝛽 ) obs (cid:17) 𝜎 𝑜𝑏𝑠 ( 𝜆 𝑖 ) (A13)where the summation is over the sample of 7 helium ( 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝛼 , H 𝛾 , H 𝛿 ). The parameters are varied in the range of 0 . <𝑦 + < .
15, 0 < 𝑛 𝑒 < . < 𝑡 𝑒 < .
2, 0 . < 𝑎 𝐻𝑒 < . . < 𝑎 𝐻 < .
0, 0 . < 𝜏 < .
0, 0 . < 𝐶 ( 𝐻𝛽 ) < .
99, 0 < 𝜉 < 𝑡 𝑒 and C(H 𝛽 ) to estimate thevalue of 𝑦 ++ with the following equation: 𝑦 ++ = . 𝑡 . 𝑒 𝐹 ( 𝜆 ) 𝐹 ( H 𝛽 ) C ( H 𝛽 ) 𝑓 ( 𝜆 ) (A14)Following Aver et al. (2011) we add the prior distribution forthe electron temperature: 𝜒 𝑇 = ( 𝑡 𝑒 − 𝑡 ( O iii )) ( . 𝑡 ( O iii )) (A15)where 𝑡 (O iii) is the electron temperature derived from the analysisof [O iii] emission line fluxes.Following Izotov et al. (2006) we determine the temperature 𝑡 ( O iii )) = − × 𝑇 (O iii) using: 𝑡 = . (cid:16) 𝜆 + 𝜆 𝜆 (cid:17) − log 𝐶 𝑇 (A16)Here and after letter 𝜆 with the specific wavelength denotes themeasured flux of corresponding ion line normalized to the H 𝛽 flux.The 𝐶 𝑇 term is given by 𝐶 𝑇 = ( . − . 𝑡 + . 𝑡 − . 𝑡 ) + . 𝑥 + . 𝑥 (A17)where 𝑥 = − 𝑛 𝑒 √ 𝑡 .Temperature 𝑡 from A16 is determined via iterative processstarting with 𝑡 = .
0. Such iteration gives correct results after 6iteration steps. To determine 𝑛 𝑒 needed for the 𝐶 𝑇 term calculationwe use fitting formula from Proxauf et al. (2014):log ( 𝑛 𝑒 ) = . × tan (− . 𝑅 + . ) + . −− . 𝑅 + . 𝑅 − . 𝑅 (A18)here 𝑅 = 𝜆 / 𝜆 MNRAS000
0. Such iteration gives correct results after 6iteration steps. To determine 𝑛 𝑒 needed for the 𝐶 𝑇 term calculationwe use fitting formula from Proxauf et al. (2014):log ( 𝑛 𝑒 ) = . × tan (− . 𝑅 + . ) + . −− . 𝑅 + . 𝑅 − . 𝑅 (A18)here 𝑅 = 𝜆 / 𝜆 MNRAS000 , 1–12 (2021) rimordial Helium Abundance Table A2.
Coefficients for the hydrogen collisional to recombination correction 𝐶 / 𝑅 ( 𝜆 ) .H 𝛼 a 0.4155 2.4965 2.4063 0.2914 0.3685 4.6426b 14.80 14.30 14.03 14.80 14.80 14.03c 0.4209 0.5853 0.6187 0.6766 0.7076 0.7788H 𝛽 a 0.2384 0.6964 0.1991 0.1409 0.2201 1.9228 1.4845 2.8179b 15.15 14.80 15.15 15.15 15.15 14.80 14.80 14.80c 0.3082 0.4978 0.6017 0.6765 0.7293 0.7535 0.7845 0.9352H 𝛾 a 0.3629 0.8351 2.0044 1.4757 2.7947b 15.15 15.15 15.15 15.15 15.15c 0.3598 0.6533 0.7281 0.7809 0.8582H 𝛿 a 0.3629 0.8351 2.0044 1.4757 2.7947b 15.34 15.34 15.34 15.34 15.34c 0.3598 0.6533 0.7281 0.7809 0.8582 Table A3.
Coefficients of the optical depth function.Line a 𝑏 𝑏 𝑏 − × − × − − × − × − × − × − × − ...4471 2.74 × − × − − × − ...5876 4.70 × − × − − × − ...6678 0 0 0 07065 3.59 × − − × − − × − × − × − × − − × − × − Table A4.
Wavelength dependence coefficients for the underlying heliumand hydrogen absorption.Line 𝐴 ( 𝜆 ) Line 𝐵 ( 𝜆 ) 𝜆 𝜆 𝛾 𝜆 𝛿 𝜆 𝛽 𝜆 𝛼 𝜆 𝛾 𝜆 for 𝑇 (S ii) = 10 K. The electron density for different 𝑇 (S ii) can becalculated with following scaling factor: 𝑛 𝑒 ( 𝑇 ( S ii )) = 𝑛 𝑒 ( 𝐾 ) × √︂ 𝑇 ( S ii ) 𝐾 (A19)According to Izotov et al. (2006) we consider 𝑇 ( S ii ) = 𝑇 ( O ii ) which is calculated using following relation: 𝑇 ( O ii ) = − . + 𝑡 × ( . − . 𝑡 ) , A(O/H) < 7.2 𝑇 ( O ii ) = − . + 𝑡 × ( . − . 𝑡 ) , ≤ A(O/H) ≤ 𝑇 ( O ii ) = . + 𝑡 × (− . + . 𝑡 ) , A(O/H) > 7.6 (A20)here 𝐴 ( O / H ) = + log ( O / H ) .The oxygen abundance O/H is determined by expression:OH = O + H + O ++ H (A21)The ionic abundances are determined using following formulas from Izotov et al. (2006):12 + log (cid:16) O ++ H (cid:17) = log (cid:16) 𝜆 + 𝜆 𝜆 (cid:17) + . + . 𝑡 −− .
55 log 𝑡 − . 𝑡 (A22)12 + log (cid:16) O + H (cid:17) = log ( 𝜆 + 𝜆 ) + . + . 𝑡 −− .
483 log 𝑡 − . 𝑡 + log ( − . 𝑥 ) (A23) APPENDIX B: PHOTOINIZATION MODEL ITG14
Here we describe the ITG14 method of the determination of physicalproperties of observed H ii regions in blue compact dwarf galaxies.The crucial difference with the AOS15 self-consistent approach isthat ITG14 involves a step by step correction for the systematiceffects.The determination of 𝑌 𝑝 with the Izotov et al. (2014) methodinvolves calculation of current helium and oxygen abundances, 𝑌 and O/H.The current helium mass fraction 𝑌 is determined via sameequation as presented in A: 𝑌 = 𝑦 + 𝑦 ( − 𝑍 ) (B1)Here 𝑦 is total helium to hydrogen abundance ratio and 𝑍 = 𝐵 × O / H.Unlike Aver et al. (2015), where the authors set 𝐵 =
20, Izotov et al.(2014) determine 𝐵 using following relation: 𝐵 = . ( + log ( O / H )) − .
44 (B2) 𝑦 is determined via the following equation: 𝑦 = 𝐼𝐶𝐹 ( He )( 𝑦 + + 𝑦 ++ ) (B3)The following steps of systematic effects corrections involvethe determined O/H. The oxygen abundance alongside the temper-ature 𝑇 (O iii) and the electron density 𝑛 𝑒 is determined in the sameway presented in A.The method is focused in the pure recombination value of hy-drogen intensity, and thus hydrogen line intensities should be cor-rected for collisional and fluorescent excitation. This is done with MNRAS , 1–12 (2021) O.A. Kurichin et al. the coefficient ( 𝐶 + 𝐹 )/ 𝐼 , where 𝐶 and 𝐹 are collisional and flu-orescent contributions to the intensity. This quantity is determinedvia equations (16 - 27) from Izotov et al. (2013) as a function ofoxygen abundance.The fluxes are corrected for both interstellar extinction andhydrogen underlying absorption using an interative procedure fromIzotov et al. (1994). Reddening parameter 𝐶 ( H 𝛽 ) and underlyingabsorption parameter 𝑎 H are determined via comparison of ob-served and theoretical Balmer decrement values with the followingequations: 𝐼 ( 𝜆 ) 𝐼 ( H 𝛽 ) = 𝐹 ( 𝜆 ) 𝐹 ( H 𝛽 ) 𝐸𝑊 ( 𝜆 ) + 𝑎 H 𝐸𝑊 ( 𝜆 ) 𝐸𝑊 ( H 𝛽 ) 𝐸𝑊 ( H 𝛽 ) + 𝑎 H 𝑓 ( 𝜆 ) 𝐶 ( H 𝛽 ) (B4)Here 𝐼 ( 𝜆 ) and 𝐹 ( 𝜆 ) denote intrinsic and observed line flux respec-tively, 𝐸𝑊 is the equivalent width of a line, 𝑓 ( 𝜆 ) is the reddeningfunction from Cardelli et al. (1989). It is assumed that 𝑎 H is thesame for all hydrogen lines. The intrinsic Balmer decrement is cal-culated using Hummer & Storey (1987). After this parameters aredetermined the whole observed spectrum is corrected for these ef-fects.The He line fluxes are corrected for underlying stellar absorp-tion. This is done using the following equation: 𝐼 He ( 𝜆 ) 𝐼 ( H 𝛽 ) = 𝐹 He ( 𝜆 ) 𝐹 ( H 𝛽 ) 𝐸𝑊 He ( 𝜆 ) + 𝑎 He ( 𝜆 ) 𝐸𝑊 He ( 𝜆 ) (B5)The absorption line equivalent width 𝑎 He ( ) is fixed on 0.4 Å.The equivalent widths of the other absorption lines were fixed ac-cording to the ratios (one can compare them with coefficients pre-sented below: 𝑎 He ( )/ 𝑎 He ( ) = . 𝑎 He ( )/ 𝑎 He ( ) = . 𝑎 He ( )/ 𝑎 He ( ) = . 𝑎 He ( )/ 𝑎 He ( ) = . 𝑎 He ( )/ 𝑎 He ( ) = . 𝜒 = ∑︁ 𝑖 ( 𝑦 + 𝑖 − 𝑦 𝑤𝑚 ) 𝜎 ( 𝑦 + 𝑖 ) (B7)Here 𝑦 + 𝑖 is a single ionized He number density derived from theintensity of the 𝑖 He emission line (see eq. B8), 𝜎 ( 𝑦 + 𝑖 ) is the un-certainty propagated from the measured error of 𝑖 -line flux, 𝑦 𝑤𝑚 is the weighted mean of 𝑦 + 𝑖 . The summation over 𝑖 is carried forthe following He lines: 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝑦 𝑤𝑚 calculation. 𝑦 + 𝑖 = 𝑅 𝑖 𝐸 H 𝛽 ( 𝑛 𝑒 , 𝑇 𝑒 ) 𝐸 𝑖 ( 𝑛 𝑒 , 𝑇 𝑒 ) 𝑓 𝑖 ( 𝑛 𝑒 , 𝑇 𝑒 , 𝜏 ) (B8)Here 𝑅 𝑖 is the 𝑖 He line flux corrected for reddening, underlying Hand He absorption and hydrogen non-recombination contribution, 𝐸 H 𝛽 ( 𝑛 𝑒 , 𝑇 𝑒 ) is the emissivity of H 𝛽 line taken from Hummer &Storey (1987), 𝐸 𝑖 ( 𝑛 𝑒 , 𝑇 𝑒 ) is the 𝑖 He line emissivity calculatedusing analytical fits from Izotov et al. (2013). The optical depthfunction 𝑓 𝑖 ( 𝑛 𝑒 , 𝑇 𝑒 , 𝜏 ) is taken from Benjamin et al. (2002).The likelihood function B7 is minimized over 𝑛 𝑒 , 𝑇 𝑒 , and 𝜏 which are varied within the following ranges:0 < 𝑛 𝑒 < . 𝑇 𝑝𝑟 < 𝑇 𝑒 < . 𝑇 𝑝𝑟 < 𝜏 < 𝑇 𝑝𝑟 is derived using the following relation from Izotov et al.(2013): 𝑇 𝑝𝑟 = (cid:0) . × − 𝑇 𝑒 ( O iii ) + . + / 𝑇 𝑒 ( O iii ) (cid:17) × 𝑇 𝑒 ( O iii ) (B10)Finally 𝑦 𝑤𝑚 is determined using the best-fit parameters 𝑇 𝑒 , 𝑛 𝑒 and 𝜏 . In case the He ii 𝜆 𝑦 ++ is calculated.The total helium abundance 𝑦 is determined with the followingequation: 𝑦 = 𝐼𝐶𝐹 ( He ) × ( 𝑦 𝑤𝑚 + 𝑦 ++ ) (B11)Unlike AOS15, ITG14 use the ionization correction factor 𝐼𝐶𝐹 ( He ) which is calculated as a function of oxygen excitationparameter 𝑥 = O + /( O + + O + ) . The analytical fits for this quantityare presented in Izotov et al. (2013). This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS000