In this article, we present studies of BAO with Sloan Digital Sky Survey (SDSS) publicly released data DR13  . The study has three parts:
1) We measure the BAO observables, , and  in six bins of redshift from 0.1 to 0.7. These observables are galaxy- galaxy correlation distances, in units of, of galaxy pairs respectively transverse to the line of sight, along the line of sight, and in an interval of angles with respect to the line of sight, for a reference (fictitious) cosmology.
2) We measure the space curvature parameter and the dark energy density relative to the critical density as a function of the expansion parameter with the following BAO data used as an uncalibrated standard ruler:, , and for (this analysis), for from Planck satellite observations   , and measurements of BAO distances in the Lyman-alpha (Ly) forest with SDSS BOSS DR11 data at  and  .
3) Finally, we use the BAO measurements as a calibrated standard ruler to constrain a wider set of cosmological parameters.
2. BAO Observables
To define the quantities being measured we write the (generalized) Friedmann equation that describes the expansion history of a homogeneous universe:
The expansion parameter is normalized so that at the present time. The Hubble parameter is normalized so that at the present time, i.e.
The terms under the square root in Equation (1) are densities relative to the critical density of, respectively, non-relativistic matter, ultra-relativistic radiation, dark energy (whatever it is), and space curvature. In the General Theory of Relativity is constant. Here, we allow be a function of to be determined by observations. Measuring and is equivalent to measuring the expansion history of the universe. The expansion parameter is related to redshift by.
The distance between two galaxies observed with a relative angle and redshifts and can be written, with sufficient accuracy for our purposes, as  .
and are the distance components, in units of, transverse to the line of sight and along the line of sight, respectively. (should not be confused with the of fits). The difference between the approximation (3) and the exact expression for, given by Equation (3.19) of Reference  , is negligible for two galaxies at the distance: the term of proportional to in Equation (3) changes by at.
We find the following approximations to and valid in the range with precision approximately for  :
Our strategy is as follows. We consider galaxies with redshift in a given range. For each galaxy pair we calculate, and with Equations (3) with the approximation (4) and fill one of three histograms of (with weights to be discussed later) depending on the ratio:
・ If fill a histogram of that obtains a BAO signal centered at. For this histogram, is a small correction relative to that is calculated with sufficient accuracy with the approximation (4), i.e. an error less than 0.2% on.
・ If fill a second histogram of that obtains a BAO signal centered at. For this histogram, is a small correction relative to that is calculated with sufficient accuracy with the approximation (4) and, i.e. an error less than 0.2% on.
・ Else, fill a third histogram of that obtains a BAO signal centered at.
Note that these three histograms have different galaxy pairs, i.e. have inde- pendent signals and independent backgrounds.
The galaxy-galaxy correlation distance, in units of, is obtained from the BAO observables, , or as follows:
A numerical analysis obtains for, dropping to for (in agreement with the method introduced in Reference  that obtains when covers all angles). The redshift in Equations (5), (6) and (7) corresponds to the weighted mean of in the interval to. The fractions in Equations (5), (6) and (7) are within of 1 for. Note that the limits of or or as are all equal to.
The independent BAO observables, , and satisfy the consistency relation
The approximations in Equations (4) obtain galaxy-galaxy correlation distances, , and of a reference (fictitious) cosmology. We emphasize that these approximations are undone by Equations (5), (6), and (7) so in the end has the correct dependence on the cosmological parameters which is different for Equations (5), (6), and (7).
The BAO observables, , and were chosen because 1) they are dimensionless, 2) they are independent, 3) they do not depend on any cosmological parameter, 4) they are almost independent of (for an optimized value of) so that a large bin may be analyzed, and 5) satisfy the consistency relation (8) which allows discrimination against fits that converge on background fluctuations instead of the BAO signal.
It is observed that fluctuations in the CMB have a correlation angle   .
(we have chosen a measurement by the Planck collaboration with no input from BAO). The extreme precision with which is measured makes it one of the primary parameters of cosmology. The correlation distance, in units of, is obtained from as follows:
For we do not neglect of photons or neutrinos (we take  corresponding to 3 neutrino flavors).
3. Galaxy Selection and Data Analysis
The present analysis is based on publicly released SDSS-IV DR13 data described in Reference  , and includes the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS)  , and the SDSS-IV Extended Baryon Oscillation Spectroscopic Survey (eBOSS)  which are designed for BAO measurements. A list of participating institutions in the SDSS-IV is given in the acknowledgment.
We obtain the following data from the SDSS DR13 catalog  for all objects identified as galaxies that pass quality selection flags: right ascension ra, declination dec, redshift, redshift uncertainty, and the absolute value of the magnitude. We require a good measurement of redshift, i.e.
. The present study is limited to galaxies with right ascension in the range to, declination in the range to, and redshift in the range 0.1 to 0.7. The galactic plane divides this data set into two independent sub-sets: the northern galactic cap (N) and the southern galactic cap (S) defined by dec.
We calculate the absolute luminosity of galaxies relative to the absolute luminosity of a galaxy with at, and calculate the corre- sponding magnitude. We consider galaxies with (G). We define “luminous galaxies” (LG) with, for example, , and “clusters” (C). Clusters C are based on a cluster finding algorithm that starts with LG’s as seeds, calculates the total absolute luminosity of all G’s within a distance 0.006 (in units of), and then selects local maximums of these total absolute luminosities above a threshold, e.g..
A “run” is defined by a range of redshifts, a data set, and a definition of galaxy and “center”. For each of 6 bins of redshift from 0.10 to 0.70, and each of 5 offset bins of from 0.15 to 0.65, and for each data set N or S, and for each choice of galaxy-center G-G, G-LG, LG-LG, or G-C (with several absolute luminosity cuts), we fill histograms of galaxy-center distances and obtain the BAO distances, , and by fitting these histograms.
Histograms are filled with weights or, where and are the absolute luminosities of galaxy and center respectively. We obtain histograms with = 3.79, 3 and 5. The reason for this large degree of redundancy is the difficulty to discriminate the BAO signal from the background with its statistical fluctuations and cosmological fluctuations due to galaxy clustering. Pattern recognition is aided by multiple histograms with different background fluctuations, and by the characteristic shape of the BAO signal that has a lower edge at approximately 0.031 and an upper edge at approximately 0.036 as shown in Figure 1.
Figure 1. Fits to histograms of that obtain the BAO distances, , and in the northern galactic cap, and distribution of the consistency parameter for the 25 N or S successful runs.
The fitting function is a second degree polynomial for the background and, for the BAO signal, a step-up-step-down function of the form
A run is defined as “successful” if the fits to all three histograms converge with a signal-to-background ratio significance greater than 1 standard deviation (raising this cut further obtains little improvement due to the cosmological fluctuations of the background), and the consistency parameter Q is in the range 0.97 to 1.03 (if Q is outside of this range then at least one of the fits has converged on a fluctuation of the background instead of the BAO signal). We obtain 13 successful runs for N and 12 successful runs for S which are presented in Table 1 and Table 2 respectively. The histogram of the consistency parameter Q for these 25 runs is presented in Figure 1.
Table 1. Measured BAO distances, , and in units of with (see text) from SDSS DR13 galaxies with right ascension to, and declination to in the northern galactic cap, i.e.
Table 2. Measured BAO distances, , and in units of with (see text) from SDSS DR13 galaxies with right ascension to, and declination to in the southern galactic cap, i.e. dec
Histograms of BAO distances have statistical fluctuations, and fluctu- ations of the background due to the clustering of galaxies as seen in Figure 1. These two types of fluctuations are the dominant source of the total uncertainties of the BAO distance measurements. These uncertainties are independent for
Table 3. Independent measured BAO distances, , and in units of with (see text) obtained by selecting, for each bin of, the entry with least in Table 1 or Table 2. Each BAO distance has an independent total uncertainty 0.00030 for and, or 0.00060 for. No corrections have been applied.
each entry in Table 3. We present several estimates of the total uncertainties of the entries in Tables 1-3 extracted directly from the fluctuations of the numbers in these tables. All uncertainties in this article are at 68% confidence level.
We neglect the variation of, , and between adjacent bins of with respect to their uncertainties. The root-mean-square (r.m.s.) differences divided by between corresponding rows in Table 1 and Table 2 for, , and are 0.00055, 0.00093, and 0.00054 respectively. We assign these numbers as total uncertainties of each entry in Table 1 and Table 2.
The 18 entries in Table 3 are independent. The r.m.s. differences for rows 1 - 2, 3 - 4 and 5 - 6 divided by are 0.00030, 0.00052, and 0.00020 for
, , and respectively.
The average and standard deviation of the columns, , and in Table 3 are respectively 0.03342, 0.00021; 0.03355, 0.00051; and 0.03348, 0.00023.
The r.m.s. of for Table 1 and Table 2 is 0.0111. The average of all entries in Table 1 and Table 2 is 0.03383. From the above estimates we take the uncertainties of, , and to be in the ratio. From these numbers, we calculate the independent total uncertainties of, , and to be 0.00026, 0.00052, and 0.00026 respectively.
From these estimates, we take the following independent total uncertainties for each entry of, , and in Table 3: 0.00030, 0.00060, and 0.00030 respectively.
Let us consider corrections to the BAO distances due to peculiar velocities and peculiar displacements of galaxies towards their centers. A relative peculiar velocity towards the center causes a reduction of the BAO distances
, , and of order. In addition, the Doppler shift produces an apparent shortening of by, and somewhat less for.
We multiply the measured BAO distances, , and
by correction factors, and respectively. Simulations in Reference  obtain and at, and at
, and and at
. In the following sections we present fits with the corrections
The effect of these corrections can be seen by comparing the first two fits in Table 4 below. An order-of-magnitude estimate of this correction can be obtained by calculating the r.m.s. corresponding to modes with
with Equation (11) of Reference  and normalizing the result to, i.e. to the r.m.s. density fluctuation in a volume.
6. Measurements of and from Uncalibrated BAO
We consider five scenarios:
1) The observed acceleration of the expansion of the universe is due to the cosmological constant, i.e. is constant.
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 be a function of    :. While this gas dominates Equation 
can be integrated with the result   
If and we obtain constant as in the General Theory of Relativity.
3) Same as Scenario 2 with constant, i.e..
Table 4. Cosmological parameters obtained from the 18 independent BAO measurements in Table 3 in several scenarios. Corrections for peculiar motions are given by Equation (11) except, for comparison, the fit “1*” which has no correction. Scenario 1 has constant. Scenario 3 has. Scenario 4 has.
4) We assume.
5) is arbitrary and needs to be measured at every.
Note that BAO measurements can constrain for where contributes significantly to.
Let us try to understand qualitatively how the BAO distance measurements presented in Table 3 constrain the cosmological parameters. In the limit we obtain, so the first row with
in Table 3 approximately determines. This and the measurement of, for example, then constrains the derivative of
with respect to at, i.e. constrains approxi- mately. We need an additional constraint for Scenario 1. and constrain the last two factors in Equation (10), i.e. approximately constrain. The additional BAO distance measurements in Table 3 then also constrain and, or.
In Table 4, we present the cosmological parameters obtained by minimizing the with 18 terms corresponding to the 18 independent BAO distance measurements in Table 3 for several scenarios. We find that the data is in agreement with the simplest cosmology with and constant with per degree of freedom (d.f.), so no additional parameter is needed to obtain a good fit to this data. For free we obtain
for constant, or if
is allowed to depend on as in Scenario 4. We present the variable instead of because it has a smaller uncertainty. The con- straints on are weak.
In Table 5 we present the cosmological parameters obtained by minimizing the with 19 terms corresponding to the 18 BAO distance measurements listed in Table 3 plus the measurement of the correlation angle of the CMB given in Equation (9). We present the variable instead of because it has a smaller uncertainty. We obtain
when is allowed to vary as in Scenario 4. There is no tension between
Table 5. Cosmological parameters obtained from the 18 BAO measurements in Table 3 plus from Equation (9) in several scenarios. Corrections for peculiar motions are given by Equation (11). Scenario 1 has constant. Scenario 2 has. Scenario 3 has. Scenario 4 has.
the data and the case and constant: with these two constraints we obtain with.
We now add BAO measurements with SDSS BOSS DR11 data of quasar Ly forest cross-correlation at  and Ly forest autocorrelation at  . From the combination in Reference  we obtain
From the 18 BAO plus plus 2 Ly measurements, for free, and allowed to vary as in Scenario 4, we obtain, , and. The is. Note that the Ly measurements reduce the uncertainties of and. Requiring and constant raises the to, so we observe no tension between the data and these two requirements, and obtain
7. Detailed Measurement of
We obtain from the 6 independent measurements of in Table 3, and Equations (1) and (6) for the case. The values of and are obtained from the fit for Scenario 4 in Table 5. The results are presented in Figure 2. To guide the eye, we also show the straight
Figure 2. Measurements of obtained from the 6 in Table 3 for, and the corresponding and from the fit for Scenario 4 in Table 5. The straight line is from the central values of this fit. The uncertainties correspond only to the total uncertainties of. To illustrate correlated uncertainties we present results for (squares), (triangles), (inverted tri- angles), and (circles). For clarity some offsets in have been applied.
8. Measurements of, and from Calibrated BAO
Up to this point, we have used the BAO distance as an uncalibrated standard ruler. The cosmological parameters and drop out of such an analysis, and the dependences of the results on are not significant. is the present density of baryons relative to the critical density. In this section we consider the BAO distance as a calibrated standard ruler to constrain the cosmological parameters, , , and.
The sound horizon is calculated from first principles  as follows:
where the speed of sound is
We can write the result for our purposes as
Figure 3. Same as Figure 2 for offset bins of with least in Table 1 or Table 2. These measurements are partially correlated with those of Figure 2.
(we have neglected the dependence of   on the cosmo- logical parameters).
In this paragraph we take corresponding to 3 flavors of neutrinos  . From Big-Bang nucleosynthesis, (at 68% confi- dence)  . With the latest direct measurement by the Hubble Space Telescope Key Project  we obtain. An alternative choice is the Planck “TT + lowP + lensing” analysis  , that assumes and a cosmology, that obtains, and. The cosmological parameters that minimize the with 22 terms (18 BAO measurements from Table 3 plus from Equation (9) plus 2 Ly measurements from Equation (15) plus) are presented in Table 6. Note that the addition of the external constraint from slightly reduces the uncertainties of and if is fixed. Note in Table 6 that the data is consistent with the constraints and constant for both values of.
In this paragraph we let be free. We turn the problem around: from 18 BAO measurements from Table 3 plus from Equation (9) plus 2 Ly measurements from Equation (15) we constrain. The results are
for free and allowed to vary as in Scenario 4, for fixed and allowed to vary as in Scena- rio 4, and for fixed and constant. For free, allowed to vary as in Scenario 4,
, and we obtain corresponding to neutrino flavors. For fixed, constant, , and we obtain
corresponding to neutrino flavors.
9. Comparison with Previous Measurements
Let us compare the results obtained with SDSS DR13 data with DR12 data. The between Table 3 and Table III of Reference  is 44.8 for 18 degrees of freedom. The between Table 3 and Table III of Reference  is 25.9 for 17 degrees of freedom. The disagreement in both cases is due to the same two entries in Table III of Reference  or Table III of Reference  with miss-fits
Table 6. Cosmological parameters obtained from the 18 BAO measurements in Table 3 plus from Equation (9) plus 2 Ly measurements in Equation (15) plus in several scenarios. Corrections for peculiar motions are given by Equation (11).. Scenario 1 has constant. Scenario 4 has.
converging on background fluctuations instead of the BAO signal:
We compare Equation (14) for DR13 data, with the corresponding fits for DR12 data. From Table VIII of Reference  :
From Table VII of Reference  :
Note in Equation (14) how the DR13 data has lowered the uncertainties.
The final consensus measurements of the SDSS-III Baryon Oscillation Spectroscopic Survey  (an analysis of the DR12 galaxy sample), are presented in Table 7 (reproduced from Reference  for completness). There is agreement with the measurements of DR13 data in Table 3. The notation of Reference  is related to the notation of the present article as follows:
where Mpc and.
Table 7. Final consensus “BAO + FS” measurements of the SDSS DR12 data set  (uncertainties are statistical and systematic), and the corresponding BAO parameters and with. These measurements include the peculiar motion corrections.
 for DR12 data.
2) From the 18 BAO measurements in Table 3, and no other input, we obtain
for allowed to vary as in Scenario 4. For and constant we obtain, which may be compared to the independent Planck “TT + lowP + lensing” result (which assumes a cosmology with):  . Note that these two results are based on independent cosmological measurements. See Table 4 for fits in several scenarios.
3) From 18 BAO measurements plus from the CMB we obtain
for allowed to vary as in Scenario 4. See Table 5 for fits in several scenarios. The cosmological parameters, and drop out of this analysis. Imposing the constraints and constant obtains
5) From 18 BAO plus plus 2 Ly measurements we obtain
when is allowed to vary as in Scenario 4. Note the constraint on defined in Equation (19). The corresponding constraint on for
, and is corre- sponding to neutrino flavors.
For and constant we obtain. The cor- responding constraint on for, and
is corresponding to neutrino flavors.
6) From 18 BAO plus plus 2 Ly plus measurements with
fixed we obtain the results shown in Table 6. For allowed to vary as in Scenario 4 and we obtain
7) For all data sets, we obtain no tension with the constraints and constant.
The SDSS has brought the measurements of with free to a new level of precision.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the US Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org.
SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard- Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.
 Eisenstein, D.J., Seo, H.-J. and White, M. (2007) On the Robustness of the Acoustic Scale in the Low-Redshift Clustering of Matter. The Astrophysical Journal, 664, 660-674.
 Alam, S., et al. (2015) The Eleventh and Twelfth Data Releases of the Sloan Digital Sky Survey: Final Data from SDSS-III. Astrophysical Journal Supplement Series, 219, 12. arXiv:1501.00963
 Dawson, K.S., Kneib, J.-P., Percival, W.J., et al. (2016) The SDSS-IV Extended Baryon Oscillation Spectroscopic Survey: Overview and Early Data. Astronomical Jour-nal, 151, 44.
 Humphreys, E.M.L., Reid, M.J., Moran, J.M., Greenhill, L.J. and Argon, A.L. (2013) Toward a New Geometric Distance to the Active Galaxy NGC 4258. III. Final Results and the Hubble Constant. Astrophysical Journal, 775, 13.