1. 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 measured by the Planck Collaboration we obtained assuming flat space and a cosmological constant . At the time, the 2016 Review of Particle Physics quoted . The new 2018 Planck “TT, TE, EE + lowE + lensing” measurement  obtains , while the “TT, TE, EE + lowE + lensing+BAO” measurement obtains . Due to the growing tension between these measurements, we decided to repeat the BAO analysis in Reference , 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 . As a result, the present analysis is based on 6 independent BAO measurements, compared to 18 in .
We assume flat space, i.e. , and constant dark energy density, i.e. , 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 . All uncertainties have 68% confidence.
The analysis presented in this article obtains 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 , where .
Combining the direct measurement with the 2018 Planck “TT, TE, EE + lowE + lensing” analysis obtains and , at the cost of an increase of the Planck from 12956.78 to 12968.64.
Finally, we update the measurement of of Reference  with the data of this Planck + Ωm combination, and two new direct measurements of , and obtain eV. This result is sensitive to the accuracy of the direct measurements of .
2. Measurement of Ωm with BAO as an Uncalibrated Standard Ruler
We measure the comoving galaxy-galaxy correlation distance , in units of , with galaxies in the Sloan Digital Sky Survey SDSS DR14 publicly released catalog  , with the method described in Reference . Briefly, from the angle between two galaxies as seen by the observer, and their red-shifts and , we calculate their distance d, in units of , assuming a reference cosmology . At this “uncalibrated” stage in the analysis, the unit of distance is neither known nor needed. The adimensional distance d has a component transverse to the line of sight, and a component along the line of sight, given by Equation (3) of . We fill three histograms of d according to the orientation of the galaxy pairs with respect to the line of sight, i.e. , , and remaining pairs. Fitting these histograms we obtain excesses centered at , , and respectively. Examples are shown in Figure 1 and Figure 2. From each BAO observable , , or we recover for any given cosmology with Equations (5), (6), or (7) of Reference . Requiring that be independent of red shift z and orientation we obtain the space curvature , the dark energy density as a function of the expansion parameter , and the matter density . Full details can be found in .
Figure 1. Fits to histograms of G-LG distances d that obtain , , or at . See Table 1 and Table 2 for 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 , , and to converge, and that the consistency relation
 be satisfied within .
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 or , where and are absolute luminosities; see  for details. In the present analysis we have off-set the bins of redshift z with respect to Reference  to obtain different background fluctuations.
Figure 2. Fits to histograms of LG-LG distances d that obtain , , or at . 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  : the radial mass profile of an initial point like adiabatic excess results, well after recombination, in peaks at radii 17 Mpc and Mpc, so we can expect the BAO signal to extend from approximately 148-17 Mpc to 148+17 Mpc, with at the mid-point. From galaxy simulations described in , the smearing of due to galaxy peculiar motions has a standard deviation approximately 7.6 Mpc at , and 8.5 Mpc at . So the observed BAO signal has an unexpected “step-up-step-down” shape, and is narrower than implied by the simulation in reference .
The selections of galaxies are as in  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  ) with a center , a half-width , and an amplitude A relative to the background. Each fit used for the final measurements is required to have a significance (in the analysis of  this requirement was , which allows more bins of z).
Successful triplets of fits are presented in Table 1. Note the redundancy of measurements with and . 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 , , or in Table 1, together with the sound horizon angle obtained by the Planck experiment , is a sensitive measurement of as shown in Table 3.
Table 1. Measured BAO distances , , and , in units of , with (see  ) 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 .
Table 2. Details of the fits selected for the final analysis (indicated by a “*” in Table 1). Note that the significance of the fitted signal amplitudes (relative to the background) A range from to 9.8.
Table 3. Calculated , , , and for and , as a function of , for and constant. is the BAO galaxy comoving standard ruler length in units of . It is calculated from , , , , , and . , , and are calculated with Equations (5), (6), and (7) of  with . The dependence on or eV is negligible compared to the uncertainties in Table 5.
The peculiar motion corrections were studied with the galaxy generator described in  . Results of these simulations are shown in Table 4, for G-G distances, for two cases: “correct ” and “correct ”. The “correct ” simulations have the predicted linear power spectrum of density fluctuations of the ΛCDM model (Equation (8.1.42) of  ), while the “correct ” simulations have a steeper input so that the generated galaxy power spectrum matches observations, see Figure 15 of . (The need for the steeper 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  that where taken from a study in . In summary, in the present analysis we apply the same peculiar motion corrections as in , i.e. we multiply the measured BAO distances , , and , by correction factors , , and , respectively, where
Table 4. Study of peculiar motion corrections to be added to the G-G measurements of , , and in Table 1, obtained from simulations.
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 , , and 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.
Four Scenarios are considered. In Scenario 1 the dark energy density is constant, i.e. . 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 . We allow the index w to be a function of a   : . Scenario 3 is the same as Scenario 2, except that w is constant, i.e. . In Scenario 4 we assume .
with 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, from the Planck “TT, TE, EE + lowE + lensing” measurement . The “angular distance” at decoupling is , with
which has negligible dependence on h or .
Table 5. Uncertainties of , , and at 68% confidence. For “et al.” see the Appendix.
Table 6. Cosmological parameters obtained from the 6 independent galaxy BAO measurements indicated with a “*” in Table 1 in several scenarios. Corrections for peculiar motions are given by Equation (1) except, for comparison, the fit “1*” which has no correction. Scenario 1 has constant. Scenario 3 has . Scenario 4 has .
From the Planck “TT, TE, EE + lowE + lensing” measurement , . Then the comoving sound horizon at decoupling is , with
The BAO standard ruler for galaxies is larger than because last scattering of electrons occurs after last scattering of photons due to their different number densities. In the present analysis, we take with
from the Planck “TT, TE, EE + lowE + lensing” analysis, with the uncertainty from Equation (10) of Reference . Note from (4) and Equation (10) of Reference  that (5) is insensitive to cosmological parameters, so the uncalibrated analysis decouples from h or .
We can test (5) experimentally. From Table 6 we obtain . From (4) and (2) we obtain , so the measured .
To the 6 independent galaxy BAO measurements, we add the sound horizon angle , and obtain the results presented in Table 7. Note that measurements
Table 7. Cosmological parameters obtained from the 6 independent galaxy BAO measurements indicated with a “*” in Table 1, plus from the Planck experiment, in several scenarios. Corrections for peculiar motions are given by Equation (1). . Scenario 1 has constant. Scenario 2 has . Scenario 3 has . Scenario 4 has .
are consistent with flat space and a cosmological constant. Note also that the constraint on becomes tighter if is assumed constant, and that the constraint on becomes tighter if is assumed zero. In the scenario of flat space and a cosmological constant we obtain
with for 5 degrees of freedom. This is the final result of the present analysis.
Adding two measurements in the quasar Lyman-alpha forest    we obtain the results presented in Table 8. In particular, for flat space and a cosmological constant we obtain
with for 7 degrees of freedom. Note that the Lyman-alpha measurements tighten the constraints on , , , and .
As a cross-check of the z dependence, from the 4 independent fits to at different redshifts z presented in Figure 3, plus , we obtain
with for 3 degrees of freedom, for flat space and a cosmological constant.
As a cross-check of isotropy, from the 3 independent fits to at shown in Figure 4 corresponding to different regions of the sky, we obtain
with for 2 degrees of freedom, for flat space and a cosmological constant.
Additional studies are presented in the Appendix.
Figure 3. Fits to histograms of G-LG distances d that obtain at , and 0.65. The bins of z are , , , and , respectively. The fits obtain , , , and respectively, where uncertainties are statistical from the fits. A fit with these four measurements (with the total uncertainties of Table 5), plus from the Planck experiment, obtains and with for 3 degrees of freedom.
Figure 4. Fits to histograms of G-LG distances d, with z in the range 0.25 - 0.45, that obtain at . 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 , , and respectively, where uncertainties are statistical from the fits. A fit with these three measurements (with the total uncertainties of Table 5), plus from the Planck experiment, obtains and with for 2 degrees of freedom.
Table 8. Cosmological parameters obtained from the 6 galaxy BAO measurements indicated with a “*” in Table 1, plus from the Planck experiment, plus two Lyman-alpha measurements    in several scenarios. Corrections for peculiar motions are given by Equation (1). . Scenario 1 has constant. Scenario 2 has . Scenario 3 has . Scenario 4 has .
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 at . From top to bottom, they correspond to SDSS DR14 (this analysis), DR14 with galaxy selections of , and DR13 with galaxy selections of . The fits obtain , , and respectively, where uncertainties are statistical from the fits. Note that our assigned total uncertainty for is ±0.00030. This single fit for the current analysis, together with obtains and , with zero degrees of freedom. The relative amplitudes A of the fitted signals are , , and respectively. The number of galaxies (G) and large galaxies (LG) are , , and , respectively. Note that the relative amplitude is larger for the current galaxy selections.
3. 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 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 up to decoupling and obtain the “comoving acoustic horizon distance” , with
The acoustic angular scale is
in agreement with Equation (11) of .
Let us now consider the measurement of h. From the galaxy BAO measurements in Table 6 we obtain and . From Big Bang Nucleosynthesis, at 68% confidence . From this data and Equations (5) and (10), or the corresponding fit, we obtain
with for 4 degrees of freedom.
The Planck measurement of allows a more precise measurement of h. From Table 7, we obtain . Then from Big Bang Nucleosynthesis and (11), or the corresponding fit, we obtain
with for 5 degrees of freedom. Note that the uncertainties of h and are correlated through Equation (11).
4. Studies of CMB Fluctuations
In Table 9, we present a qualitative study of the sensitivity of the CMB power spectrum to constrain and . We use the approximate analytic expression (7.2.41) of , modified to include , to compare the spectra with Planck 2018 “TT, TE, EE + lowE + lensing” parameters with the best fit spectra with fixed values and 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 , has low sensitivity to constrain or .
In view of the low sensitivity of the CMB power spectra to constrain , the Planck analysis can benefit from a combination with the direct measurement of given by Equation (6). The combination, obtained with the “base_mnu_plikHM_TTTEEE_lowTEB_lensing_*.txt MC chains” made public by the Planck Collaboration , is presented in Table 10. This combination is preliminary due to the sparseness of the MC chains at low values of .
Table 9. Cosmologies with fixed and fitted to the CMB power spectrum with the Planck 2018 “TT, TE, EE + lowE + lensing” parameters , eV, , , , , and . The approximate analytic Equation (7.2.41) of  (modified to include ) was used. Notation: .
Table 10. Combination of the Planck 2018 “TT, TE, EE + lowE + lensing” analysis  with the directly measured . Uncertainties are at 68% confidence. The Planck increases from 12,956.78 to 12,968.64 with this combination. The galaxy . Preliminary.
Figure 6. Comparison of the power spectra [μK2] for the Planck 2018 “TT, TE, EE + lowE + lensing” parameters, with the best fit spectra with and eV fixed, calculated with the approximate Equation (7.2.41) of  (modified to include ). The r.m.s. difference is 6.07 μK2, corresponding to 0.11% of the first acoustic peak, so the two spectra can not be distinguished by eye.
We consider four direct measurements: 1) by the Sh0es Team , 2) from the abundance of rich galaxy clusters  , 3) from weak gravitational lensing  , and 4) from galaxy BAO and from Planck, Equation (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 + Ωm 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 + Ωm combination reduces the tensions with the direct measurements. Note that the Planck + Ωm combination has greater than the direct measurements. This 2.7σ tension may be due to neutrino masses.
6. Update on Neutrino Masses
We consider the scenario of three neutrino flavors with eigenstates of nearly the same mass, so . Massive neutrinos suppress the power spectrum of linear density fluctuations by a factor for Mpc−1 . This suppression affects and the galaxy power spectrum , 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 .
To obtain we minimize a with four terms corresponding to , , and two parameters obtained from the Planck + Ωm combination: , and . In the fit, is obtained from Equation (11), and . is obtained from the combination of the two direct measurements presented in Section 5.
For  obtained from the Sachs-Wolfe effect measured by the COBE satellite (see list of references in  ) we obtain
with zero degrees of freedom, in agreement with  where the method is explained in detail.
Since 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 . From the Planck + Ωm combination of Table 10 we obtain , and
with zero degrees of freedom.
To strengthen the constraints from the two direct measurements of , 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 . We obtain
with for 2 degrees of freedom, and find no significant pulls on , h, or ns. These results are sensitive to the accuracy of the direct measurements of .
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  in the form of “MC chains”, and the corresponding analysis tool “GetDist GUI”.
1) Comparison with Reference 
Table 4 and Table 5 of Reference  can be compared with Table 6 and Table 7 of the present analysis. We find agreement between all measurements when d in Reference  is identified with in the present analysis. We find that d in Table 4 of Reference  is biased low with respect to in Table 6 of the present analysis. For the scenario of flat space and a cosmological constant, Table 4 of Reference  obtains and . From this and Equation (4) we obtain , in good agreement with d, so in Reference  no correction for was needed or applied.
2) Bias of BAO measurements of small galaxy samples
We have investigated the difference of between Reference  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.
As an extreme test, we divide the bin into 6 sub-samples: N, N, N, S, S, and S. We try to fit each one, and average the successful fits (only about half are successful), and obtain , , and . We also fit the sum of these six bins, and obtain , , and . 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 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 , instead of >1 for Reference . 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 and half-width . The widths of fits vary typically from to 0.0025, see Table 2. We have used the center as the BAO standard ruler, but could have used the lower edge of the signal at , or the upper edge at , or somewhere in between, i.e. . We have investigated the value of that minimizes the root-mean-square fluctuations of a representative selection of measurements. The result is , 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 standard ruler, i.e. . The r.m.s. fluctuation of the lower edge with is 0.00068, and the fluctuation of the upper edge with is 0.00091, which again illustrates the bias described in Appendix 7.2, i.e. the lower edge fluctuates less than the upper edge.
A separate open question is whether this center coincides with the of Equation (5)?
Yet another question is this: what value of would reproduce the Planck ? We obtain ranging from −0.81 for at , to for at . These large values of , and their strong dependence on z and galaxy-galaxy orientation, do not seem plausible.
Finally, how well do we understand ? The present study takes and from the Planck analysis . Note the extremely small uncertainty obtained by the Planck Collaboration. In comparison, from Equation (4) of Reference  we obtain and .
An estimate of the uncertainties due to the issues discussed in this Appendix is included in Table 5.
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