Study of Baryon Acoustic Oscillations with SDSS DR13 Data and Measurements of Ωk and ΩDE(a) ()
1. Introduction
In this article, we present studies of BAO with Sloan Digital Sky Survey (SDSS) publicly released data DR13 [7] . The study has three parts:
1) We measure the BAO observables
,
, and
[8] 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 [2] [9] , and measurements of BAO distances in the Lyman-alpha (Ly
) forest with SDSS BOSS DR11 data at
[10] and
[11] .
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:
(1)
The expansion parameter
is normalized so that
at the present time
. The Hubble parameter
is normalized so that
at the present time, i.e.
(2)
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 [8] .
![]()
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(3)
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 [14] , 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
[8] :
(4)
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:
(5)
(6)
(7)
A numerical analysis obtains
for
, dropping to
for
(in agreement with the method introduced in Reference [1] 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
(8)
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 [2] [9] .
(9)
(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:
(10)
For
we do not neglect
of photons or neutrinos (we take
[2] 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 [7] , and includes the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) [15] , and the SDSS-IV Extended Baryon Oscillation Spectroscopic Survey (eBOSS) [16] 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 [7] 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.
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
![]()
where
![]()
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.
4. Uncertainties
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
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.
5. 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
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 [6] obtain
and
at
,
and
at
, and
and
at
. In the following sections we present fits with the corrections
(11)
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 [5] 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
[3] [17] [18] :
. While this gas dominates
Equation [2]
(12)
can be integrated with the result [3] [17] [18]
(13)
If
and
we obtain constant
as in the General Theory of Relativity.
3) Same as Scenario 2 with
constant, i.e.
.
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
(14)
when
is allowed to vary as in Scenario 4. There is no tension between
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
[10] and Ly
forest autocorrelation at
[11] . From the combination in Reference [11] we obtain
(15)
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
line corresponding to the central values of
and
of the fit for Scenario 4. In Figure 3 we present the results for offset bins of
(which are partially correlated with the entries in Figure 2).
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 [1] as follows:
(16)
where the speed of sound is
(17)
We can write the result for our purposes as
(18)
where
(19)
![]()
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
[2] [9] on the cosmo- logical parameters).
In this paragraph we take
corresponding to 3 flavors of neutrinos [2] . From Big-Bang nucleosynthesis,
(at 68% confi- dence) [2] . With the latest direct measurement
by the Hubble Space Telescope Key Project [19] we obtain
. An alternative choice is the Planck “TT + lowP + lensing” analysis [2] , 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 [8] is 44.8 for 18 degrees of freedom. The
between Table 3 and Table III of Reference [12] 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 [8] or Table III of Reference [12] with miss-fits
converging on background fluctuations instead of the BAO signal:
and
. The fluctuation of
can be seen in Table 1 for the northern galactic cap, but not in Table 2 for the southern galactic cap. Removing the two miss-fits from the comparisons obtains
and
respectively.
We compare Equation (14) for DR13 data, with the corresponding fits for DR12 data. From Table VIII of Reference [8] :
(20)
From Table VII of Reference [12] :
(21)
Note in Equation (14) how the DR13 data has lowered the uncertainties.
The final consensus measurements of the SDSS-III Baryon Oscillation Spectroscopic Survey [20] (an analysis of the DR12 galaxy sample), are presented in Table 7 (reproduced from Reference [12] for completness). There is agreement with the measurements of DR13 data in Table 3. The notation of Reference [20] is related to the notation of the present article as follows:
(22)
(23)
where
Mpc and
.
10. Conclusions
[12] for DR12 data.
2) From the 18 BAO measurements in Table 3, and no other input, we obtain
(24)
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
):
[2] . 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
(25)
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
.
4) Detailed measurements of
are presented in Figure 2 and Figure 3.
5) From 18 BAO plus
plus 2 Ly
measurements we obtain
(26)
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
(27)
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.
Acknowledgements
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.