Measurements of the Cosmological Parameters Ω m and H 0

From Baryon Acoustic Oscillation measurements with Sloan Digital Sky Survey SDSS DR14 galaxies, and the acoustic horizon angle * θ measured by the Planck Collaboration, we obtain 0.2724 0.0047 m Ω = ± , and 0.020 0.7038 0.0060 h mν + ⋅ = ± ∑ , assuming flat space and a cosmological constant. We combine this result with the 2018 Planck “TT, TE, EE + lowE + lensing” analysis, and update a study of mν ∑ with new direct measurements of 8 σ , and obtain 0.27 0.08 mν = ± ∑ eV assuming three nearly degenerate neutrino eigenstates. Measurements are consistent with 0 k Ω = , and ( ) de a Λ Ω = Ω constant.


Introduction and Summary
From a study of Baryon Acoustic Oscillations (BAO) with Sloan Digital Sky Survey (SDSS) data release DR13 galaxies and the "sound horizon" angle MC m Ω = ± [3].Due to the growing tension between these measurements, we decided to repeat the BAO analysis in Reference [1], this time with SDSS DR14 galaxies.The main difficulty with the BAO measurements is to distinguish the BAO signal from the cosmological and statistical fluctuations.The aim of the present analysis is to be very conservative by choosing large bins in redshift z to obtain a larger significance of the BAO signal than in [1].As a result, the present analysis is based on 6 independent BAO measurements, compared to 18 in [1].
( ) = Ω , except in Tables 6-8 that include more general cases.We assume three neutrino flavors with eigenstates with nearly the same mass, so . We adopt the notation of the Particle Data Group 2018 [4].All uncertainties have 68% confidence.
The analysis presented in this article obtains 0.2724 0.0047 m Ω = ± so the tension has increased further.We present full details of all fits to the galaxygalaxy distance histograms of the present measurement so that the reader may cross-check each step of the analysis.Calibrating the BAO standard ruler we obtain 0.020 0.7038 0.0060 , where , at the cost of an increase of the Planck 2 P χ from 12956.78 to 12968.64.
Finally, we update the measurement of m ν ∑ of Reference [5] with the data of this Planck + Ω m combination, and two new direct measurements of 8 σ , and obtain 0.27 0.08 m ν = ± ∑ eV.This result is sensitive to the accuracy of the direct measurements of 8 σ .

Measurement of Ωm with BAO as an Uncalibrated Standard Ruler
We measure the comoving galaxy-galaxy correlation distance drag d , in units of 0 c H , with galaxies in the Sloan Digital Sky Survey SDSS DR14 publicly released catalog [6] [7], with the method described in Reference [1].Briefly, from the angle α between two galaxies as seen by the observer, and their red-shifts 1 z and 2 z , we calculate their distance d, in units of 0 c H , assuming a reference cosmology [1].At this "uncalibrated" stage in the analysis, the unit of distance 0 c H is neither known nor needed.The adimensional distance d has a component d α transverse to the line of sight, and a component z d along the line of sight, given by Equation (3) of [1].We fill three histograms of d according to the orientation of the galaxy pairs with respect to the line of sight, i.e.
1 3 , and remaining pairs.Fitting these histograms we obtain excesses centered at dα , ˆz d , and / d respectively.
Examples are shown in Figure 1 and Figure 2. From each BAO observable dα , / d , or ˆz d we recover drag d for any given cosmology with Equations ( 5), (6), or (7) of Reference [1].Requiring that drag d be independent of red shift z and orientation we obtain the space curvature k Ω , the dark energy density ( ) de a Ω as a function of the expansion parameter ( ) + , and the matter density Full details can be found in [1].International Journal of Astronomy and Astrophysics .See Table 1 and Table 2 for  0.033 i j F F d , where i F and j F are absolute luminosities; see [1] for details.In the present analysis we have off-set the bins of redshift z with respect to Reference [1] to obtain different background fluctuations.International Journal of Astronomy and Astrophysics .See Table 1 and Table 2 for details.Now consider pattern recognition.Figure 1 and Figure 2 show that the BAO signal is approximately constant from d ≈ 0.032 to ≈0.037, corresponding to ≈137 Mpc to ≈158 Mpc.This characteristic shape of the BAO signal can be understood qualitatively with reference to Figure 1 of [8]: the radial mass profile of an initial point like adiabatic excess results, well after recombination, in peaks at radii 17 Mpc and drag 148 r ≈ Mpc, so we can expect the BAO signal to extend from approximately 148-17 Mpc to 148+17 Mpc, with drag r at the mid-point.
From galaxy simulations described in [5], the smearing of drag r due to galaxy peculiar motions has a standard deviation approximately 7.6 Mpc at 0.5 z = , and 8.5 Mpc at 0.3 z = .So the observed BAO signal has an unexpected "step-up-step-down" shape, and is narrower than implied by the simulation in reference [8].The selections of galaxies are as in [1] with the added requirements for SDSS DR14 galaxies that they be "sciencePrimary" and "bossPrimary", and have a smaller redshift uncertainty zErr < 0.00025.
The fitting function has 6 free parameters, corresponding to a second degree polynomial for the background, and a "smooth step-up-step-down" function (described in [1]) with a center d , a half-width ∆ , and an amplitude A relative to the background.Each fit used for the final measurements is required to have a significance 2 A A σ > (in the analysis of [1] this requirement was 1 A A σ > , which allows more bins of z).
Successful triplets of fits are presented in Table 1.Note the redundancy of measurements with 0.250 0.425 z < < and 0.425 800 z < < . The independent triplets of fits selected for further analysis, are indicated with a "*", and are shown in Figure 1 and Figure 2, with further details presented in Table 2.We note that each measurement of dα , / d , or ˆz d in Table 1, together with the sound horizon angle * θ obtained by the Planck experiment [3], is a sensitive measurement of m Ω as shown in Table 3. (see [1]) from SDSS DR14 galaxies with right ascension 110˚ to 270˚, and declination −5˚ to 70˚, in the northern (N) and/or southern (S) galactic caps.Uncertainties are statistical from the fits to the BAO signal.No corrections have been applied.The independent measurements with a "*" are selected for further analysis.The corresponding fits are presented in Figure 1 and Figure 2, and details are presented in Table 2.
For comparison, measurements with a "&" correspond to SDSS DR13 data with the galaxy selections of [1].( ) ( ) , and ˆz d are calculated with Equations ( 5), (6), and ( 7) of [1] with . The dependence on The peculiar motion corrections were studied with the galaxy generator described in [5] [9].Results of these simulations are shown in Table 4, for G-G distances, for two cases: "correct ( ) P k is currently not understood.)All of these G-G corrections, and also the corrections for LG-LG and G-C, are in agreement, to within a factor 2, with the corrections applied in [1] that where taken from a study in [11].In summary, in the present analysis we apply the same peculiar motion corrections as in [1] We take half of these corrections as a systematic uncertainty.The effect of these corrections is relatively small as shown in Table 6.
Uncertainties of dα , / d , and ˆz d are presented in Table 5.These uncertainties are dominated by cosmological and statistical fluctuations, and are estimated from the root-mean-square fluctuations of many measurements, from the width of the distribution of Q, and from the issues discussed in the Appendix.
Fits to the two independent selected triplets dα , / d , and ˆz d indicated by a "*" in Table 1, with the uncertainties in Table 5, are presented in Table 6.
Four Scenarios are considered.In Scenario 1 the dark energy density is constant, i.e.
( ) = Ω .In Scenario 2 the observed acceleration of the expansion of the universe is due to a gas of negative pressure with an equation of state 0 w p ρ ≡ < .We allow the index w to be a function of a [12] [13]: − .Scenario 3 is the same as Scenario 2, except that w is constant, i.e. 0 a w = .In Scenario 4 we assume ( ) ( ) which has negligible dependence on h or m ν ∑ .International Journal of Astronomy and Astrophysics The BAO standard ruler for galaxies drag r is larger than * r because last scattering of electrons occurs after last scattering of photons due to their different number densities.In the present analysis, we take drag drag 0 from the Planck "TT, TE, EE + lowE + lensing" analysis, with the uncertainty from Equation (10) of Reference [3].Note from (4) and Equation (10) of Reference [3] that ( 5) is insensitive to cosmological parameters, so the uncalibrated analysis decouples from h or We can test (5) experimentally.From To the 6 independent galaxy BAO measurements, we add the sound horizon angle * θ , and obtain the results presented in Table 7.Note that measurements International Journal of Astronomy and Astrophysics with 2 1.2 χ = for 5 degrees of freedom.This is the final result of the present analysis.
Adding two measurements in the quasar Lyman-alpha forest [1] [14] [15] we obtain the results presented in Table 8.In particular, for flat space and a cosmological constant we obtain 0.2714 0.0047, with 2 10.0 χ = for 7 degrees of freedom.Note that the Lyman-alpha measurements tighten the constraints on k Ω , 0 w , 1 w , and a w .
As a cross-check of the z dependence, from the 4 independent fits to dα at different redshifts z presented in Figure 3, plus * θ , we obtain 0.2745 0.0040, with 2 3.0 χ = for 3 degrees of freedom, for flat space and a cosmological constant.
As a cross-check of isotropy, from the 3 independent fits to dα at 0.36 z = shown in Figure 4 corresponding to different regions of the sky, we obtain 0.2737 0.0043, with 2 1.1 χ = for 2 degrees of freedom, for flat space and a cosmological constant.
To check the stability of dα , / d , and ˆz d with the data set and galaxy selections, we compare fits highlighted with "*" and "&" in Table 1, and also fits in Figure 5.
Additional studies are presented in the Appendix.

Measurement of H0 with BAO as a Calibrated Standard Ruler
We consider the scenario of flat space and a cosmological constant.It is useful to International Journal of Astronomy and Astrophysics present approximate analytic expressions, tho all final calculations are done directly with fits to the measurements marked with a "*" in Table 1 and numerical integrations to obtain correct uncertainties for correlated parameters.
To calibrate the BAO measurements, we integrate the comoving photon-electron-baryon plasma sound speed from The acoustic angular scale is ( ) in agreement with Equation ( 11) of [3].
Let us now consider the measurement of h.From the galaxy BAO measurements in Table 6 [4].From this data and Equations ( 5) and ( 10), or the corresponding fit, we obtain 0.026 0.716 0.027, with 2 1.0 χ = for 4 degrees of freedom.
The Planck measurement of * θ allows a more precise measurement of h.

Studies of CMB Fluctuations
In Table 9, we present a qualitative study of the sensitivity of the CMB power spectrum ( )

∑
eV.We find that the differences in spectra range from 0.11% to 0.3% of the first acoustic peak, see Figure 6.So the CMB power spectrum, while being very sensitive to constrain * θ , has low sensitivity to constrain m Ω or In view of the low sensitivity of the CMB power spectra to constrain m Ω , the Planck analysis can benefit from a combination with the direct measurement of m Ω given by Equation (6).The combination, obtained with the "base_mnu_plikHM_TTTEEE_lowTEB_lensing_*.txtMC chains" made public by the Planck Collaboration [3], is presented in Table 10.This combination is preliminary due to the sparseness of the MC chains at low values of m Ω .International Journal of Astronomy and Astrophysics  eV fixed, calculated with the approximate Equation (7.2.41) of [10] (modified to include m ν ∑ ).The r.m.s.difference is 6.07 μK 2 , corresponding to 0.11% of the first acoustic peak, so the two spectra can not be distinguished by eye.

Tensions
We consider four direct measurements: 1) by the Sh 0 es Team [16] 6) of this analysis.Comparing these measurements with Planck (left hand column of Table 10) we obtain differences of 3.5σ, 2.5σ, 1.8σ, and 4.9σ, respectively.Comparing these measurements with the Planck + Ω m combination (right hand column of Table 10) we obtain differences of 2.1σ, 2.3σ, 1.5σ, and 2.1σ, respectively.In conclusion, the Planck + Ω m combination reduces the tensions with the direct measurements.Note that the Planck + Ω m combination has 8 σ greater than the direct measurements.This 2.7σ tension may be due to neutrino masses.

Update on Neutrino Masses
We consider the scenario of three neutrino flavors with eigenstates of nearly the same mass, so ∑ [5].
To obtain m ν ∑ we minimize a 2 χ with four terms corresponding to with zero degrees of freedom, in agreement with [5] where the method is explained in detail.
with zero degrees of freedom.
To strengthen the constraints from the two direct measurements of 8 σ , we add to the fit measurements of fluctuations of number counts of galaxies in spheres of radii 16/h, 32/h, 64/h, and 128/h Mpc, as explained in [5].We obtain 0.27 0.08 eV, with 2 1.6 χ = for 2 degrees of freedom, and find no significant pulls on 2 N , h, or n s .These results are sensitive to the accuracy of the direct measurements of International Journal of Astronomy and Astrophysics Appendix 1) Comparison with Reference [1] Table 4 and Table 5 of Reference [1] can be compared with Table 6 and Table 7 of the present analysis.We find agreement between all measurements when d in Reference [1] is identified with * d in the present analysis.We find that d in Table 4 of Reference [1] is biased low with respect to drag d in Table 6 of the present analysis.For the scenario of flat space and a cosmological constant, 2) Bias of BAO measurements of small galaxy samples We have investigated the difference of drag d between Reference [1] and the present analysis.This difference is not due to the change of data set from SDSS DR13 to SDSS DR14: we have compared the coordinates of selected galaxies and have found no changes in calibrations.The fluctuation is not caused by the tighter galaxy selection requirements of the present analysis: compare the entries with "&" and "*" in Table 1, and see Figure 5.So there is evidence that fits become biased low as the number of galaxies is reduced and the significance of the fitted relative amplitude A of the BAO signal becomes marginal.The reason is that the observed BAO signal has a sharper and larger lower edge at 0.032 d ≈ compared to the upper edge at ≈0.037, so the upper edge tends to get lost in the background fluctuations as the number of galaxies is reduced.
To reduce this bias, in the present analysis we require the significance of the fitted relative amplitudes 2 A A σ > , instead of >1 for Reference [1].The price to pay is that we obtain only 2 independent bins of z, instead of 6.

3) A study of the BAO signal
The BAO signal has a "step-up-step-down" shape with center at d and half-width ∆ .The widths of fits vary typically from 0.0017 ∆ = to 0.0025, see Table 2.We have used the center d as the BAO standard ruler, but could have used the lower edge of the signal at d − ∆ , or the upper edge at d + ∆ , or somewhere in between, i.e. d + ∆  .We have investigated the value of  that minimizes the root-mean-square fluctuations of a representative selection of measurements.The result is 0.17 = −  , and the difference in the r.m.s.values is negligible (0.00037 vs. 0.00039) so we keep the center of the signal as our TE, EE + lowE + lensing+BAO" measurement obtains 0.3111 0.0056

Figure 1 .
Figure 1.Fits to histograms of G-LG distances d that obtain dα , / d , or ˆz d at

2 0.033 d or 2 2
details.The challenge with these BAO measurements is to distinguish the BAO signal from the cosmological and statistical fluctuations of the background.Our strategy is three-fold: 1) redundancy of measurements with different cosmological fluctuations, 2) pattern recognition of the BAO signal, and 3) requiring all three fits for dα , / d , and ˆz d to converge, and that the consistency relation we repeat the fits for the northern (N) and southern (S) galactic caps; we repeat the measurements for galaxy-galaxy (G-G) distances, galaxy-large galaxy (G-LG) distances, LG-LG distances, and galaxy-cluster (G-C) distances; and we fill histograms of d with weights 2

Figure 2 .
Figure 2. Fits to histograms of LG-LG distances d that obtain dα , / d , or ˆz d at P k " and "correct ( ) gal P k ".The "correct ( ) P k " simulations have the predicted linear power spectrum of density fluctuations ( ) P k of the ΛCDM model (Equation (8.1.42) of [10]), while the "correct ( ) gal P k " simulations have a steeper ( ) P k input so that the generated galaxy power spectrum ( ) gal P k matches observations, see Figure 15 of [5].(The need for the steeper ( )

Figure 4 .
Figure 4. Fits to histograms of G-LG distances d, with z in the range 0.25 -0.45, that obtain dα at 0.36 z = .From top to bottom, they correspond to the northern galactic cap with right ascension < 180˚ (NW), to the northern galactic cap with right ascension > 180˚ (NE), and to the southern galactic cap (S).The fits obtain ˆ0.03468 0.00012 d α = ± , 0.03447 0.00012 ±

Figure 5 .
Figure 5. Fits to histograms of G-LG distances d, with z in the range 0.25 -0.45 for the northern galactic cap (N), that obtain dα at 0.36 z =.From top to bottom, they correspond to SDSS DR14 (this analysis), DR14 with galaxy selections of[1], and DR13 with galaxy selections of[1].The fits obtain ˆ0.03455 0.00010 d α = ± , 0.03416 0.00010 ± of freedom.Note that the uncertainties of h and m Ω are correlated through Equation(11).

Figure 6 .
Figure 6.Comparison of the power spectra ( ) ( ) , 1 2 S TT l l l C but does not affect the Sachs-Wolfe effect at low k.So, by comparing fluctuations at large and small k it is possible to constrain or measure m ν

2 N
are still ultra-relativistic at decoupling.Then there is no power suppression of the CMB fluctuations, and we can use the entire spectrum to fix the amplitude .From the Planck + Ω m combination of

As an extreme test, we divide the bin 0 .
try to fit each one, and average the successful fits (only about half are successful), and obtain ˆ0.03358 0We also fit the sum of these six bins, and obtain

Table 1
c z =

Table 2 .
Details of the fits selected for the final analysis (indicated by a "*" in Table1).Note that the significance of the fitted signal amplitudes (relative to the background) A 004 International Journal of Astronomy and Astrophysics k Ω = and ( ) de a Λ Ω ≡ Ω constant.drag d is the BAO galaxy comoving standard ruler length in units of 0 c H .It is calculated from drag * 1.0184 d d = ,

Table 4 .
, i.e. we multiply the measured BAO distances dα , Study of peculiar motion corrections to be added to the G-G measurements of dα , / d , and ˆz d in Table1, obtained from simulations.
/ d , and ˆz d , by correction factors f α , / f , and z f , respectively, where

Table 6 that
k Ω is consistent with zero, and ( )

Table 6 .
Cosmological parameters obtained from the 6 independent galaxy BAO measurements indicated with a "*" in Table1in several scenarios.Corrections for peculiar motions are given by Equation (1) except, for comparison, the fit "1*" which has no correction.Scenario 1 has m d z

Table 7 .
Cosmological parameters obtained from the 6 independent galaxy BAO measurements indicated with a "*" in Table1, plus * θ from the Planck experiment, in several scenarios.Corrections for peculiar motions are given by Equation (1). m

Table 8 .
Cosmological parameters obtained from the 6 galaxy BAO measurements indicated with a "*" in Table1, plus * International Journal of Astronomy and Astrophysics

Table 9 .
Cosmologies with fixed