_{1}

^{*}

From Baryon Acoustic Oscillation measurements with Sloan Digital Sky Survey SDSS DR14 galaxies, and the acoustic horizon angle
measured by the Planck Collaboration, we obtain
Ω_{m}=0.2724±0.0047, and
*h*+0.020⋅∑*m*_{v}=0.7038±0.0060, 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
∑*m*_{v} with new direct measurements of
σ_{8}, and obtain
∑*m*_{v}=0.27±0.08 eV assuming three nearly degenerate neutrino eigenstates. Measurements are consistent with
Ω_{k}=0, and
Ω_{de}(*a*)=Ω_{Λ} constant.

From a study of Baryon Acoustic Oscillations (BAO) with Sloan Digital Sky Survey (SDSS) data release DR13 galaxies and the “sound horizon” angle θ MC measured by the Planck Collaboration we obtained Ω m = 0.281 ± 0.003 assuming flat space and a cosmological constant [

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 [

We assume flat space, i.e. Ω k = 0 , and constant dark energy density, i.e. Ω de ( a ) = Ω Λ , except in Tables 6-8 that include more general cases. We assume three neutrino flavors with eigenstates with nearly the same mass, so ∑ m ν ≈ 3 m ν . We adopt the notation of the Particle Data Group 2018 [

The analysis presented in this article obtains Ω m = 0.2724 ± 0.0047 so the tension has increased further. We present full details of all fits to the galaxy-galaxy 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 h + 0.020 ⋅ ∑ m ν = 0.7038 ± 0.0060 , where H 0 ≡ 100 h km ⋅ s − 1 ⋅ Mpc − 1 .

Combining the direct measurement Ω m = 0.2724 ± 0.0047 with the 2018 Planck “TT, TE, EE + lowE + lensing” analysis obtains Ω m = 0.2853 ± 0.0040 and h = 0.6990 ± 0.0030 , at the cost of an increase of the Planck χ P 2 from 12956.78 to 12968.64.

Finally, we update the measurement of ∑ m ν of Reference [_{m} combination, and two new direct measurements of σ 8 , and obtain ∑ m ν = 0.27 ± 0.08 eV. This result is sensitive to the accuracy of the direct measurements of σ 8 .

We measure the comoving galaxy-galaxy correlation distance d drag , in units of c / H 0 , with galaxies in the Sloan Digital Sky Survey SDSS DR14 publicly released catalog [

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 d ^ α , d ^ / , and d ^ z to converge, and that the consistency relation

Q = d ^ / / ( d ^ α 0.57 d ^ z 0.43 ) = 1 [

Regarding redundancy, 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 0.033 2 / d 2 or 0.033 2 F i F j / d 2 , where F i and F j are absolute luminosities; see [

Now consider pattern recognition.

The selections of galaxies are as in [

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 [

Successful triplets of fits are presented in

z | z min | z max | Galaxies | Centers | Type | 100 d ^ α | 100 d ^ / | 100 d ^ z | Q |
---|---|---|---|---|---|---|---|---|---|

0.53 | 0.425 | 0.725 | 614,724 | 614,724 | G-G, N+S | 3.488 ± 0.015 | 3.504 ± 0.019 | 3.466 ± 0.032 | 1.007 |

0.53 | 0.425 | 0.725 | 614,724 | 13,960 | G-C, N+S | 3.381 ± 0.030 | 3.401 ± 0.033 | 3.395 ± 0.035 | 1.004 |

0.53 | 0.475 | 0.575 | 180,696 | 53,519 | G-LG, N | 3.424 ± 0.015 | 3.314 ± 0.018 | 3.242 ± 0.018 | 0.991 |

0.53 | 0.475 | 0.575 | 53,519 | 53,519 | LG-LG, N | 3.451 ± 0.030 | 3.447 ± 0.059 | 3.351 ± 0.022 | 1.012 |

0.53 | 0.475 | 0.575 | 180,696 | 5045 | G-C, N | 3.427 ± 0.031 | 3.331 ± 0.030 | 3.316 ± 0.033 | 0.986 |

0.56 | 0.425 | 0.800 | 230,841 | 230,841 | G-G, S | 3.441 ± 0.027 | 3.422 ± 0.017 | 3.497 ± 0.040 | 0.988 |

0.56 | 0.425 | 0.800 | 355,737 | 120,499 | G-LG, N | 3.425 ± 0.015 | 3.465 ± 0.016 | 3.351 ± 0.025 | 1.021 |

*0.56 | 0.425 | 0.800 | 120,499 | 120,499 | LG-LG, N | 3.424 ± 0.021 | 3.461 ± 0.018 | 3.424 ± 0.039 | 1.011 |

&0.56 | 0.425 | 0.800 | 143,778 | 143,778 | LG-LG, N | 3.424 ± 0.014 | 3.478 ± 0.015 | 3.451 ± 0.026 | 1.012 |

0.56 | 0.425 | 0.800 | 586,578 | 13,206 | G-C, N+S | 3.453 ± 0.038 | 3.365 ± 0.044 | 3.354 ± 0.028 | 0.987 |

0.52 | 0.425 | 0.575 | 236,693 | 236,693 | G-G, N | 3.437 ± 0.031 | 3.423 ± 0.026 | 3.432 ± 0.025 | 0.997 |

0.52 | 0.425 | 0.575 | 236,693 | 72,297 | G-LG, N | 3.416 ± 0.017 | 3.441 ± 0.012 | 3.385 ± 0.018 | 1.011 |

0.52 | 0.425 | 0.575 | 72,297 | 72,297 | LG-LG, N | 3.456 ± 0.033 | 3.447 ± 0.022 | 3.392 ± 0.060 | 1.006 |

0.48 | 0.425 | 0.525 | 151,938 | 4143 | G-C, N | 3.424 ± 0.051 | 3.383 ± 0.026 | 3.343 ± 0.062 | 0.998 |

0.36 | 0.250 | 0.450 | 114,597 | 114,597 | G-G, N | 3.456 ± 0.018 | 3.386 ± 0.015 | 3.318 ± 0.056 | 0.997 |

0.36 | 0.250 | 0.450 | 114,597 | 65,130 | G-LG, N | 3.455 ± 0.010 | 3.358 ± 0.015 | 3.293 ± 0.032 | 0.992 |

0.36 | 0.250 | 0.450 | 65,130 | 65,130 | LG-LG, N | 3.462 ± 0.016 | 3.352 ± 0.025 | 3.307 ± 0.039 | 0.988 |

0.34 | 0.250 | 0.425 | 92,321 | 92,321 | G-G, N | 3.439 ± 0.013 | 3.473 ± 0.015 | 3.423 ± 0.076 | 1.012 |

0.34 | 0.250 | 0.425 | 149,849 | 149,849 | G-G, N+S | 3.437 ± 0.014 | 3.367 ± 0.013 | 3.444 ± 0.042 | 0.979 |

*0.34 | 0.250 | 0.425 | 92,321 | 55,980 | G-LG, N | 3.449 ± 0.008 | 3.471 ± 0.013 | 3.450 ± 0.034 | 1.006 |

&0.34 | 0.250 | 0.425 | 133,729 | 94,873 | G-LG, N | 3.431 ± 0.011 | 3.469 ± 0.014 | 3.383 ± 0.024 | 1.017 |

0.34 | 0.250 | 0.425 | 55,980 | 55,980 | LG-LG, N | 3.467 ± 0.019 | 3.477 ± 0.015 | 3.459 ± 0.045 | 1.004 |

Observable | z | Relative amplitude A | Half-width Δ |
---|---|---|---|

d ^ α | 0.56 | 0.00290 ± 0.00100 | 0.00169 ± 0.00022 |

d ^ / | 0.56 | 0.00422 ± 0.00069 | 0.00164 ± 0.00020 |

d ^ z | 0.56 | 0.00505 ± 0.00226 | 0.00250 ± 0.00041 |

d ^ α | 0.34 | 0.00632 ± 0.00064 | 0.00225 ± 0.00008 |

d ^ / | 0.34 | 0.00269 ± 0.00044 | 0.00197 ± 0.00013 |

d ^ z | 0.34 | 0.00341 ± 0.00162 | 0.00238 ± 0.00035 |

Ω m | 100 d drag | 100 d ^ α | 100 d ^ / | 100 d ^ z | 100 d ^ α | 100 d ^ / | 100 d ^ z |
---|---|---|---|---|---|---|---|

z = 0.56 | z = 0.34 | ||||||

0.25 | 3.628 | 3.535 | 3.510 | 3.477 | 3.560 | 3.538 | 3.510 |

0.27 | 3.519 | 3.457 | 3.444 | 3.427 | 3.471 | 3.457 | 3.440 |

0.28 | 3.468 | 3.421 | 3.414 | 3.405 | 3.429 | 3.420 | 3.408 |

0.29 | 3.420 | 3.386 | 3.385 | 3.384 | 3.390 | 3.385 | 3.377 |

0.31 | 3.330 | 3.323 | 3.333 | 3.346 | 3.317 | 3.319 | 3.321 |

0.33 | 3.248 | 3.265 | 3.285 | 3.311 | 3.251 | 3.259 | 3.271 |

The peculiar motion corrections were studied with the galaxy generator described in [

z | Simulation | Δ d ^ α | Δ d ^ / | Δ d ^ z |
---|---|---|---|---|

0.5 | correct P ( k ) | 0.000062 | 0.000080 | 0.000112 |

0.5 | correct P gal ( k ) | 0.000096 | 0.000125 | 0.000175 |

0.3 | correct P ( k ) | 0.000063 | 0.000080 | 0.000111 |

0.3 | correct P gal ( k ) | 0.000084 | 0.000107 | 0.000148 |

f α − 1 = 0.00320 ⋅ a 1.35 , f / − 1 = 0.00350 ⋅ a 1.35 , f z − 1 = 0.00381 ⋅ a 1.35 . (1)

We take half of these corrections as a systematic uncertainty. The effect of these corrections is relatively small as shown in

Uncertainties of d ^ α , d ^ / , and d ^ z are presented in

Fits to the two independent selected triplets d ^ α , d ^ / , and d ^ z indicated by a “*” in

Four Scenarios are considered. In Scenario 1 the dark energy density is constant, i.e. Ω de ( a ) = Ω Λ . 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 w ≡ p / ρ < 0 . We allow the index w to be a function of a [

Note in

Ω m = 0.288 ± 0.037 , (2)

with χ 2 = 1.0 for 4 degrees of freedom.

Final calculations are done with fits and numerical integrations. Never-theless, it is convenient to present approximate analytical expressions obtained from the numerical integrations for the case of flat space and a cosmological constant. At decoupling, z * = 1089.92 ± 0.25 from the Planck “TT, TE, EE + lowE + lensing” measurement [

χ ( z * ) = 3.2675 ( h + 0.35 ∑ m ν 0.7 ) 0.01 ( 0.28 Ω m ) 0.4 , (3)

which has negligible dependence on h or ∑ m ν .

d ^ α | d ^ / | d ^ z | |
---|---|---|---|

Method | ±0.00003 | ±0.00004 | ±0.00008 |

Peculiar motion correction | ±0.00004 | ±0.00004 | ±0.00005 |

Cosmological et al. + | |||

statistical fluctuations | ±0.00029 | ±0.00055 | ±0.00070 |

Total | ±0.00030 | ±0.00055 | ±0.00071 |

Scenario 1* | Scenario 1 | Scenario 1 | Scenario 3 | Scenario 4 | Scenario 4 | |
---|---|---|---|---|---|---|

Ω k | 0 fixed | 0 fixed | 0.267 ± 0.362 | 0 fixed | 0 fixed | 0.262 ± 0.383 |

Ω de + 0.6 Ω k | 0.712 ± 0.037 | 0.712 ± 0.037 | 0.738 ± 0.050 | 0.800 ± 0.364 | 0.760 ± 0.151 | 0.745 ± 0.148 |

w 0 | n.a. | n.a. | n.a. | − 0.76 ± 0.65 | n.a. | n.a. |

w 1 | n.a. | n.a. | n.a. | n.a. | 0.71 ± 2.00 | 0.13 ± 2.77 |

100 d drag | 3.48 ± 0.06 | 3.487 ± 0.052 | 3.48 ± 0.06 | 3.43 ± 0.16 | 3.42 ± 0.19 | 3.48 ± 0.21 |

χ^{2}/d.f. | 0.9/4 | 1.0/4 | 0.4/3 | 0.9/3 | 0.9/3 | 0.4/2 |

From the Planck “TT, TE, EE + lowE + lensing” measurement [

d * = θ * χ ( z * ) = 0.03401 ( 0.28 Ω m ) 0.4 . (4)

The BAO standard ruler for galaxies r drag 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 r drag ≡ d drag c / H 0 with

d drag d * = 1.0184 ± 0.0004 , (5)

from the Planck “TT, TE, EE + lowE + lensing” analysis, with the uncertainty from Equation (10) of Reference [

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

Scenario 1 | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | Scenario 4 | |
---|---|---|---|---|---|---|

Ω k | 0 fixed | 0.008 ± 0.018 | 0 fixed | 0 fixed | 0 fixed | − 0.007 ± 0.101 |

Ω de + 2.1 Ω k | 0.7276 ± 0.0047 | 0.724 ± 0.009 | 0.708 ± 0.080 | 0.724 ± 0.008 | 0.723 ± 0.011 | 0.723 ± 0.011 |

w 0 | n.a. | n.a. | − 0.77 ± 1.47 | − 0.95 ± 0.10 | n.a. | n.a. |

w a or w 1 | n.a. | n.a. | − 0.91 ± 4.53 | n.a. | 0.19 ± 0.41 | 0.35 ± 2.20 |

100 d * | 3.443 ± 0.024 | 3.42 ± 0.06 | 3.35 ± 0.04 | 3.41 ± 0.07 | 3.41 ± 0.09 | 3.39 ± 0.20 |

χ^{2}/d.f. | 1.2/5 | 1.0/4 | 0.9/3 | 1.0/4 | 1.0/4 | 1.0/3 |

are consistent with flat space and a cosmological constant. Note also that the constraint on Ω k becomes tighter if Ω de ( a ) is assumed constant, and that the constraint on Ω de ( a ) becomes tighter if Ω k is assumed zero. In the scenario of flat space and a cosmological constant we obtain

Ω m = 0.2724 ± 0.0047 , (6)

with χ 2 = 1.2 for 5 degrees of freedom. This is the final result of the present analysis.

Adding two measurements in the quasar Lyman-alpha forest [

Ω m = 0.2714 ± 0.0047 , (7)

with χ 2 = 10.0 for 7 degrees of freedom. Note that the Lyman-alpha measurements tighten the constraints on Ω k , w 0 , w 1 , and w a .

As a cross-check of the z dependence, from the 4 independent fits to d ^ α at different redshifts z presented in

Ω m = 0.2745 ± 0.0040 , (8)

with χ 2 = 3.0 for 3 degrees of freedom, for flat space and a cosmological constant.

As a cross-check of isotropy, from the 3 independent fits to d ^ α at z = 0.36 shown in

Ω m = 0.2737 ± 0.0043 , (9)

with χ 2 = 1.1 for 2 degrees of freedom, for flat space and a cosmological constant.

To check the stability of d ^ α , d ^ / , and d ^ z with the data set and galaxy selections, we compare fits highlighted with “*” and “&” in

Additional studies are presented in the Appendix.

Scenario 1 | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | Scenario 4 | |
---|---|---|---|---|---|---|

Ω k | 0 fixed | − 0.011 ± 0.008 | 0 fixed | 0 fixed | 0 fixed | − 0.022 ± 0.010 |

Ω de + 2.1 Ω k | 0.7286 ± 0.0047 | 0.734 ± 0.006 | 0.703 ± 0.028 | 0.726 ± 0.008 | 0.723 ± 0.011 | 0.720 ± 0.011 |

w 0 | n.a. | n.a. | − 0.70 ± 0.33 | − 0.96 ± 0.09 | n.a. | n.a. |

w a or w 1 | n.a. | n.a. | − 1.18 ± 1.37 | n.a. | 0.24 ± 0.40 | 0.80 ± 0.49 |

100 d * | 3.449 ± 0.024 | 3.48 ± 0.04 | 3.32 ± 0.13 | 3.42 ± 0.07 | 3.40 ± 0.08 | 3.34 ± 0.09 |

χ^{2}/d.f. | 10.0/7 | 7.7/6 | 8.0/5 | 9.2/6 | 9.0/6 | 4.6/5 |

We consider the scenario of flat space and a cosmological constant. It is useful to present approximate analytic expressions, tho all final calculations are done directly with fits to the measurements marked with a “*” in

d * = 0.03407 ( h + 0.026 ∑ m ν 0.7 ) 0.513 ( 0.28 Ω m ) 0.244 ( 0.0225 Ω b h 2 ) 0.097 . (10)

The acoustic angular scale is

θ * ≡ d * χ ( z * ) = 0.010427 ( h + 0.020 ∑ m ν 0.70 ) 0.503 ( Ω m 0.28 ) 0.156 ( 0.0225 Ω b h 2 ) 0.097 , (11)

in agreement with Equation (11) of [

Let us now consider the measurement of h. From the galaxy BAO measurements in

h + 0.026 ∑ m ν = 0.716 ± 0.027 , (12)

with χ 2 = 1.0 for 4 degrees of freedom.

The Planck measurement of θ * allows a more precise measurement of h. From

h + 0.020 ∑ m ν = 0.7038 ± 0.0060 , (13)

with χ 2 = 1.2 for 5 degrees of freedom. Note that the uncertainties of h and Ω m are correlated through Equation (11).

In

In view of the low sensitivity of the CMB power spectra to constrain Ω m , the Planck analysis can benefit from a combination with the direct measurement of Ω m given by Equation (6). The combination, obtained with the “base_mnu_plikHM_TTTEEE_lowTEB_lensing_*.txt MC chains” made public by the Planck Collaboration [

Ω m | 0.2854 | 0.2854 | 0.2854 | 0.2854 | 0.2854 | 0.2854 |
---|---|---|---|---|---|---|

∑ m ν [eV] | 0.06 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |

h | 0.6980 | 0.6976 | 0.6965 | 0.6954 | 0.6942 | 0.6931 |

100 Ω b h 2 | 2.282 | 2.288 | 2.306 | 2.324 | 2.343 | 2.362 |

n s | 0.9692 | 0.9699 | 0.9716 | 0.9735 | 0.9754 | 0.9774 |

10 10 N 2 | 1.730 | 1.729 | 1.725 | 1.722 | 1.716 | 1.713 |

τ | 0.0774 | 0.0778 | 0.0787 | 0.0797 | 0.0799 | 0.0809 |

r.m.s. [μK^{2}] | 6.07 | 6.98 | 9.29 | 11.66 | 14.06 | 16.49 |

Planck | Planck + Ω_{m} | |
---|---|---|

Ω b h 2 | 0.02237 ± 0.00015 | 0.02265 ± 0.00012 |

Ω c h 2 | 0.1200 ± 0.0012 | 0.1155 ± 0.0005 |

100 θ * | 1.04092 ± 0.00031 | 1.04125 ± 0.00022 |

τ | 0.0544 ± 0.0073 | 0.078 ± 0.006 |

ln 10 10 A s | 3.044 ± 0.014 | 3.102 ± 0.020 |

n s | 0.9649 ± 0.0042 | 0.9726 ± 0.0017 |

Ω Λ | 0.6847 ± 0.0073 | 0.7147 ± 0.0040 |

Ω m | 0.3153 ± 0.0073 | 0.2853 ± 0.0040 |

h | 0.6736 ± 0.0054 | 0.6990 ± 0.0030 |

σ 8 | 0.8111 ± 0.0060 | 0.8346 ± 0.0054 |

χ P 2 | 12,956.78 | 12,968.64 |

χ G 2 | 83.31 | 7.53 |

χ tot 2 | 13,040.09 | 12,976.17 |

We consider four direct measurements: 1) h = 0.7348 ± 0.0166 by the Sh_{0}es Team [_{m} combination (right hand column of _{m} combination reduces the tensions with the direct measurements. Note that the Planck + Ω_{m} combination has σ 8 greater than the direct measurements. This 2.7σ tension may be due to neutrino masses.

We consider the scenario of three neutrino flavors with eigenstates of nearly the same mass, so ∑ m ν ≈ 3 m ν . Massive neutrinos suppress the power spectrum of linear density fluctuations P ( k ) by a factor 1 − 8 Ω ν / Ω m for k ≫ 0.018 ⋅ Ω m 1 / 2 ( ∑ m ν / 1 eV ) 1 / 2 h Mpc^{−1} [

To obtain ∑ m ν we minimize a χ 2 with four terms corresponding to N 2 , σ 8 , and two parameters obtained from the Planck + Ω_{m} combination: h = 0.6990 ± 0.0030 , and n s = 0.9726 ± 0.0017 . In the fit, Ω m is obtained from Equation (11), and Ω b h 2 = 0.02265 ± 0.00012 . σ 8 is obtained from the combination of the two direct measurements presented in Section 5.

For N 2 = ( 2.08 ± 0.33 ) × 10 − 10 [

∑ m ν = 0.45 ± 0.20 eV , (14)

with zero degrees of freedom, in agreement with [

Since ∑ m ν < 1.7 eV, neutrinos 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 N 2 . From the Planck + Ω_{m} combination of

∑ m ν = 0.26 ± 0.08 eV , (15)

with zero degrees of freedom.

To strengthen the constraints from the two direct measurements of σ 8 , 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 [

∑ m ν = 0.27 ± 0.08 eV , (16)

with χ 2 = 1.6 for 2 degrees of freedom, and find no significant pulls on N 2 , h, or n_{s}. These results are sensitive to the accuracy of the direct measurements of σ 8 .

We have used data in the publicly released Sloan Digital Sky Survey SDSS DR14 catalog.

Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the US Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington.

We have also used data publicly released by the Planck Collaboration [

The author declares no conflicts of interest regarding the publication of this paper.

Hoeneisen, B. (2018) Measurements of the Cosmological Parameters Ω_{m} and H_{0}. International Journal of Astronomy and Astrophysics, 8, 386-405. https://doi.org/10.4236/ijaa.2018.84027

1) Comparison with Reference [

2) Bias of BAO measurements of small galaxy samples

We have investigated the difference of d drag between Reference [

As an extreme test, we divide the bin 0.425 < z < 0.725 into 6 sub-samples: 0.425 < z < 0.525 N, 0.525 < z < 0.625 N, 0.625 < z < 0.725 N, 0.425 < z < 0.525 S, 0.525 < z < 0.625 S, and 0.625 < z < 0.725 S. We try to fit each one, and average the successful fits (only about half are successful), and obtain d ^ α = 0.03358 ± 0.00015 , d ^ / = 0.03415 ± 0.00027 , and d ^ z = 0.03335 ± 0.00033 . We also fit the sum of these six bins, and obtain d ^ α = 0.03496 ± 0.00015 , d ^ / = 0.03459 ± 0.00010 , and d ^ z = 0.03464 ± 0.00034 . 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 d ≈ 0.032 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 A / σ A > 2 , instead of >1 for Reference [

3) A study of the BAO signal

The BAO signal has a “step-up-step-down” shape with center at d ^ and half-width Δ . The widths of fits vary typically from Δ = 0.0017 to 0.0025, see

A separate open question is whether this center d ^ coincides with the d drag of Equation (5)?

Yet another question is this: what value of ϵ would reproduce the Planck Ω m ? We obtain ϵ ranging from −0.81 for d ^ α at z = 0.34 , to ϵ = − 0.43 for d ^ z at z = 0.56 . These large values of | ϵ | , and their strong dependence on z and galaxy-galaxy orientation, do not seem plausible.

Finally, how well do we understand d drag / d * ? The present study takes z drag = 1059.94 ± 0.30 and d drag / d * = 1.0184 ± 0.0004 from the Planck analysis [

An estimate of the uncertainties due to the issues discussed in this Appendix is included in