Study of baryon acoustic oscillations with SDSS DR13 data and measurements of $\Omega_k$ and $\Omega_\textrm{DE}(a)$

We measure the baryon acoustic oscillation (BAO) observables $\hat{d}_\alpha(z, z_c)$, $\hat{d}_z(z, z_c)$, and $\hat{d}_/(z, z_c)$ as a function of redshift $z$ in the range 0.1 to 0.7 with Sloan Digital Sky Survey (SDSS) data release DR13. These observables are independent and satisfy a consistency relation that provides discrimination against miss-fits due to background fluctuations. From these measurements and the correlation angle $\theta_\textrm{MC}$ of fluctuations of the Cosmic Microwave Background (CMB) we obtain $\Omega_k = -0.015 \pm 0.030$, $\Omega_{\textrm{DE}} + 2.2 \Omega_k = 0.717 \pm 0.004$ and $w_1 = 0.37 \pm 0.61$ for dark energy density allowed to vary as $\Omega_{\textrm{DE}}(a) = \Omega_{\textrm{DE}} [ 1 + w_1 ( 1 - a)]$. We present measurements of $\Omega_{\textrm{DE}}(a)$ at six values of the expansion parameter $a$. Fits with several scenarios and data sets are presented. The data is consistent with space curvature parameter $\Omega_k = 0$ and $\Omega_{\textrm{DE}}(a)$ constant.


INTRODUCTION
Peaks in the density of the primordial universe are the sources of acoustic waves of the tightly coupled plasma of photons, electrons, protons and helium nuclei. These acoustic waves propagate a distance r ′ S ≈ 145 Mpc until the time of recombination and decoupling t dec [1,2]. (All distances in this article are co-moving, i.e. are referred to the present time t 0 .) The baryon acoustic oscillation (BAO) distance r ′ S corresponds to the observed correlation angle θ MC of fluctuations of the cosmic microwave background (CMB) [2]. Dark matter follows the BAO waves. The results, well after decoupling, for an initial point-like peak in the density, are two concentric shells of overdensity of radius ≈ 145 Mpc and ≈ 18 Mpc [1,3,4]. The inner spherical shell becomes reprocessed by the hierarchical formation of galaxies [5], while the outer shell is unprocessed to better than 1% [4,6] (or even 0.1% with corrections [4,6]) and therefore is an excellent standard ruler to measure the expansion parameter a(t) of the universe as a function of time t. Histograms of galaxygalaxy distances show an excess in the approximate range 145 − 11 Mpc to 145 + 11 Mpc. We denote by d ′ BAO the mean of this BAO signal. We set r ′ S = d ′ BAO f , where f is a correction factor due to the peculiar motions of galaxies (f depends on the orientation of the galaxy pair with respect to the line of sight). Measurements of these BAO signals are well established: see Refs. [3] and [4] for extensive lists of early publications.
In this article we present studies of BAO with Sloan Digital Sky Survey (SDSS) publicly released data DR13 [7]. The study has three parts: (i) We measure the BAO observablesd α (z, z c ), d z (z, z c ), andd / (z, z c ) [8] in six bins of redshift z from 0.1 to 0.7. These observables are galaxy-galaxy correlation distances, in units of c/H 0 , 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.
(ii) We measure the space curvature parameter Ω k and the dark energy density relative to the critical density Ω DE (a) as a function of the expansion parameter a with the following BAO data used as an uncalibrated standard ruler:d α (z, z c ),d z (z, z c ), andd / (z, z c ) for 0.1 < z < 0.7 (this analysis), θ MC for z dec = 1089.9 ± 0.4 from Planck satellite observations [2,9], and measurements of BAO distances in the Lyman-alpha (Lyα) forest with SDSS BOSS DR11 data at z = 2.36 [10] and z = 2.34 [11].
(iii) Finally we use the BAO measurements as a calibrated standard ruler to constrain a wider set of cosmological parameters.

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 a(t) is normalized so that a(t 0 ) = 1 at the present time t 0 . The Hubble parameter H 0 ≡ 100h km s −1 Mpc −1 is normalized so that E(1) = 1 at the present time, i.e.
The terms under the square root in Eq. (1) are densities relative to the critical density of, respectively, nonrelativistic matter, ultra-relativistic radiation, dark energy (whatever it is), and space curvature. In the General Theory of Relativity Ω DE (a) is constant. Here we allow Ω DE (a) be a function of a to be determined by observations. Measuring Ω k and Ω DE (a) is equivalent to measuring the expansion history of the universe a(t).
The expansion parameter a is related to redshift z by a = 1/(1 + z). The distance d ′ between two galaxies observed with a relative angle α and redshifts z 1 and z 2 can be written, with sufficient accuracy for our purposes, as [8] [14], is negligible for two galaxies at the distance d BAO : the term of d α proportional to Ω k in Eq. (3) changes by 0.1% at z = 0.7. We find the following approximations to χ(z) and 1/E(z) valid in the range 0 ≤ z < 1 with precision approximately ±1% for z c ≈ 3.79 [8]: (4) Our strategy is as follows: We consider galaxies with redshift in a given range z min < z < z max . For each galaxy pair we calculate d α (z, z c ), d z (z, z c ) and d(z, z c ) with Eqs. (3) with the approximation (4) and fill one of three histograms of d(z, z c ) (with weights to be discussed later) depending on the ratio d z (z, z c )/d α (z, z c ): • If d z (z, z c )/d α (z, z c ) < 1/3 fill a histogram of d(z, z c ) that obtains a BAO signal centered at d α (z, z c ). For this histogram, d 2 z (z, z c ) is a small correction relative to d 2 α (z, z c ) that is calculated with sufficient accuracy with the approximation (4), i.e. an error less than 0.2% ond α (z, z c ).
of d(z, z c ) that obtains a BAO signal centered at d z (z, z c ). For this histogram, d 2 α (z, z c ) is a small correction relative to d 2 z (z, z c ) that is calculated with sufficient accuracy with the approximation (4) and Ω k = 0, i.e. an error less than 0.2% on d z (z, z c ).
• Else, fill a third histogram of d(z, z c ) that obtains a BAO signal centered atd / (z, z c ).
Note that these three histograms have different galaxy pairs, i.e. have independent signals and independent backgrounds.
The galaxy-galaxy correlation distance d BAO , in units of c/H 0 , is obtained from the BAO observablesd α (z, z c ), d z (z, z c ), ord / (z, z c ) as follows:

.(7)
A numerical analysis obtains n = 1.70 for z = 0.2, dropping to n = 1.66 for z = 0.8 (in agreement with the method introduced in [1] that obtains n ≈ 2 when d covers all angles). The redshifts z in Eqs. (5), (6) and (7) correspond to the weighted mean of z in the interval z min to z max . The fractions in Eqs. (5), (6) and (7) are within ≈ 1% of 1 for z c = 3.79. Note that the limits of d α (z, z c ) ord z (z, z c ) ord / (z, z c ) as z → 0 are all equal to d BAO .
The BAO observablesd α (z, z c ),d z (z, z c ), andd / (z, z c ) were chosen because (i) they are dimensionless, (ii) they are independent, (iii) they do not depend on any cosmological parameter, (iv) they are almost independent of z (for an optimized value of r c ≈ 3.79) so that a large bin z max − z min may be analyzed, and (v) 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 [2,9] θ MC = 0.010410 ± 0.000005, (we have chosen a measurement by the Planck collaboration with no input from BAO). The extreme precision with which θ MC is measured makes it one of the primary parameters of cosmology. The correlation distance r S , in units of c/H 0 , is obtained from θ MC as follows: For χ(z dec ) we do not neglect Ω r ≡ Ω γ N eq /2 of photons or neutrinos (we take N eq = 3.38 [2] corresponding to 3 neutrino flavors).

GALAXY SELECTION AND DATA ANALYSIS
We obtain the following data from the SDSS DR13 catalog [7] for all objects identified as galaxies that pass quality selection flags: right ascension ra, declination dec, redshift z, redshift uncertainty zErr, and the absolute value of the magnitude r. We require a good measurement of redshift, i.e. zErr < 0.001. The present study is limited to galaxies with right ascension in the range 110 0 to 270 0 , declination in the range −5 0 to 70 0 , and redshift in the range 0.10 to 0.70. 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 ≷ 27.
We calculate the absolute luminosity F of galaxies relative to the absolute luminosity of a galaxy with r = 19.0 at z = 0.35, and calculate the corresponding magnitude r 35 . We consider galaxies with 17.0 < r 35 < 23.0 (G). We define "luminous galaxies" (LG) with, for example, r 35 < 19.2, 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 c/H 0 ), and then selects local maximums of these total absolute luminosities above a threshold, e.g. r 35 < 16.6.
A "run" is defined by a range of redshifts (z min , z max ), a data set, and a definition of galaxy and "center". For each of 6 bins of redshift z from 0.10 to 0.70, and each of 5 offset bins of z 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 d(z, z c ) and obtain the BAO distancesd α (z, z c ),d z (z, z c ), andd / (z, z c ) by fitting these histograms.
Histograms are filled with weights (0.033/d) 2 or F i F j (0.033/d) 2 , where F i and F j are the absolute luminosities F of galaxy i and center j respectively. We obtain histograms with z c = 3.79, 3.0 and 5.0. The reason for this large degree of redundancy is the difficulty to discriminate the BAO signal from the background with its statistical 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  The fitting function is a second degree polynomial for the background and, for the BAO signal, a step-up-stepdown 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 cos- mological 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 Tables  I and II respectively. The histogram of the consistency parameter Q for these 25 runs is presented in Fig. 2.
For each bin of redshift z we select from Tables I or II the run with least |Q − 1| and obtain the 18 independent BAO distances listed in Table III. This Table III is the main result of the present analysis, and superceeds the corresponding tables for DR12 in Refs. [8] and [12].

UNCERTAINTIES
Histograms of BAO distances d(z, z c ) have statistical fluctuations, and fluctuations of the background due to the clustering of galaxies as seen in Fig. 1. These two types of fluctuations are the dominant source of the total uncertainties of the BAO distance measurements. These uncertainties are independent for each entry in Table III. We present several estimates of the total uncertainties of the entries in Tables I, II, and III 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 ofd α (z, z c ),d z (z, z c ), and d / (z, z c ) between adjacent bins of z with respect to their uncertainties. The root-mean-square (r.m.s.) differences divided by √ 2 between corresponding rows in Tables I  and II ford α (z, z c ),d z (z, z c ), andd / (z, z c ) are 0.00055, 0.00093, and 0.00054 respectively. We assign these numbers as total uncertainties of each entry in Tables I and  II. The 18 entries in Table III are independent. The r.m.s. differences for rows 1-2, 3-4 and 5-6 divided by √ 2 are 0.00030, 0.00052, and 0.00020 ford α (z, z c ),d z (z, z c ), and d / (z, z c ) respectively.
The r.m.s. of (1 − Q) for Tables N and S is 0.0111. The average of all entries in Tables N and S is 0.03383. From the above estimates we take the uncertainties of d α (z, z c ),d z (z, z c ), andd / (z, z c ) to be in the ratio 1 : 2 : 1. From these numbers we calculate the independent total uncertainties ofd α (z, z c ),d z (z, z c ), andd / (z, z c ) to be 0.00026, 0.00052, and 0.00026 respectively.
From these estimates, we take the following independent total uncertainties for each entry ofd α (z, z c ), d z (z, z c ), andd / (z, z c ) in Table III: 0.00030, 0.00060, and 0.00030 respectively.

CORRECTIONS
Let us consider corrections to the BAO distances due to peculiar velocities and peculiar displacements of galaxies towards their centers. A relative peculiar velocity v p towards the center causes a reduction of the BAO distanceŝ d α (z, z c ),d z (z, z c ), andd / (z, z c ) of order 0.5v p /c. In addition, the Doppler shift produces an apparent shortening ofd z (z, z c ) by v p /c, and somewhat less ford / (z, z c ).

Ω DA (a) is arbitrary and needs to be measured at every a.
Note that BAO measurements can constrain Ω DE (a) for 0.3 a ≤ 1 where Ω DE (a) contributes significantly to E(a).
Let us try to understand qualitatively how the BAO distance measurements presented in Table III constrain the cosmological parameters. In the limit z → 0 we obtain d BAO =d α (0, z c ) =d z (0, z c ) =d / (0, z c ), so the first row with z = 0.14 in Table III approximately determines d BAO . This d BAO and the measurement of, for example,d z (0.3, z c ) then constrains the derivative of Ω m /a 3 + Ω DE + Ω k /a 2 with respect to a at z ≈ 0.3, i.e. constrains approximately Ω DE + 0.5Ω k . We need an additional constraint for Scenario 1. d BAO and θ MC constrain the last two factors in Eq. (10), i.e. approximately constrain Ω DE + 2.1Ω k . The additional BAO distance measurements in Table III then also constrain w 0 and w a , or w 1 .
In Table IV we present the cosmological parameters obtained by minimizing the χ 2 with 18 terms corresponding to the 18 independent BAO distance measurements in Table III for several scenarios. We find that the data is in agreement with the simplest cosmology with Ω k = 0 and Ω DE (a) constant with χ 2 per degree of freedom (d.f.) 11.2/16, so no additional parameter is needed to obtain a good fit to this data. For free Ω k we obtain Ω DE + 0.5Ω k = 0.710 ± 0.016 for constant Ω DE (a), or 0.732 ± 0.052 if Ω DE (a) is allowed to depend on a as in Scenario 4. We present the variable Ω DE + 0.5Ω k instead of Ω DE because it has a smaller uncertainty. The constraints on Ω k are weak.
In Table V we present the cosmological parameters obtained by minimizing the χ 2 with 19 terms corresponding to the 18 BAO distance measurements listed in Table  III plus the measurement of the correlation angle θ MC of the CMB given in Eq. (9). We present the variable Ω DE + 2.2Ω k instead of Ω DE because it has a smaller uncertainty. We obtain Ω k = −0.015 ± 0.030, Ω DE + 2.2Ω k = 0.717 ± 0.004, when Ω DE (a) is allowed to vary as in Scenario 4. There is no tension between the data and the case Ω k = 0 and constant Ω DE (a): with these two constraints we obtain Ω DE = 0.719 ± 0.003 with χ 2 /d.f. = 11.2/17. We now add BAO measurements with SDSS BOSS DR11 data of quasar Lyα forest cross-correlation at z = 2.36 [10] and Lyα forest autocorrelation at z = 2.34 [11]. From the combination in Fig. 13 Table III for Ω k = 0, and the corresponding d BAO and Ω DE from the fit for Scenario 4 in We obtain Ω DE (a) from the 6 independent measurements ofd z (z, z c ) in Table III Tables I or II. These measurements are partially correlated with those of Fig. 3.
cosmological parameters h and Ω b h 2 drop out of such an analysis, and the dependences of the results on N eq are not significant. Ω b ≡ ρ b0 /ρ crit 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 Ω k , Ω DE (a), N eq , h and Ω b h 2 . The sound horizon is calculated from first principles [1] as follows: where the speed of sound is .
We can write the result for our purposes as  [2,9] on the cosmological parameters).
In this paragraph we take N eq = 3.38 corresponding to 3 flavors of neutrinos [2]. From Big-Bang nucleosynthesis, Ω b h 2 = 0.0225 ± 0.0008 (at 68% confidence) [2]. With the latest direct measurement h = 0.720 ± 0.030 by the Hubble Space Telescope Key Project [17] we obtain A = 1.000 ± 0.021. An alternative choice is the Planck "TT+lowP+lensing" analysis [2], that assumes Ω k = 0 and a ΛCDM cosmology, that obtains Ω b h 2 = 0.02226 ± 0.00023, h = 0.678 ± 0.009 and A = 0.973 ± 0.007. The cosmological parameters that minimize the χ 2 with 22 terms (18 BAO measurements from Table III plus θ MC from Eq. (9) plus 2 Lyα measurements from Eq. (15) plus A) are presented in Table VI. Note that the addition of the external constraint from A slightly reduces the uncertainties of Ω k and w 1 if N eq = 3.38 is fixed. Note in Table VI that the data is consistent with the constraints Ω k = 0 and constant Ω DE (a) for both values of A.
In this paragraph we let N eq be free. We turn the problem around: from 18 BAO measurements from Table  III for Ω DE (a) allowed to vary as in Scenario 4. See Tables V  for fits when Ω DE (a) is allowed to vary as in Scenario 4. Note the constraint on A defined in Eq. (19). The corresponding constraint on N eq for Ω b h 2 = 0.0225 ± 0.0008, and h = 0.720 ± 0.030 is N eq = 3.92 ± 0.40 corresponding to N eff = 4.2 ± 0.9 neutrino flavors.
(vi) From 18 BAO plus θ MC plus 2 Lyα plus A measurements with N eq = 3.38 fixed we obtain the results shown in Table VI