We perform a measurement of the Hubble constant, $H_0$, using the latest
baryonic acoustic oscillations (BAO) measurements from galaxy surveys of 6dFGS,
SDSS DR7 Main Galaxy Sample, BOSS DR12 sample, and eBOSS DR14 quasar sample, in
the framework of a flat $\Lambda$CDM model. Based on the Kullback-Leibler (KL)
divergence, we examine the consistency of $H_0$ values derived from various
data sets. We find that our measurement is consistent with that derived from
Planck and with the local measurement of $H_0$ using the Cepheids and type Ia
supernovae. We perform forecasts on $H_0$ from future BAO measurements, and
find that the uncertainty of $H_0$ determined by future BAO data alone,
including complete eBOSS, DESI and Euclid-like, is comparable with that from
local measurements.Comment: 5 pages, 2 figures, 2 tables; accepted for publication in Ap

Wang, Yuting

Xu, Lixin

Zhao, Gong-Bo

English

arXiv

Portsmouth University Research Portal (Pure)

A Measurement of the Hubble Constant Using Galaxy Redshift Surveys
Yuting Wang1,2, Lixin Xu3, and Gong-Bo Zhao1,2
1 National Astronomy Observatories, Chinese Academy of Science, Beijing, 100012, China; ytwang@nao.cas.cn, lxxu@dlut.edu.cn
2 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK; gbzhao@nao.cas.cn
3 Institute of Theoretical Physics, School of Physics & Optoelectronic Technology, Dalian University of Technology, Dalian, 116024, China
Received 2017 July 3; revised 2017 September 11; accepted 2017 September 20; published 2017 November 6
Abstract
We perform a measurement of the Hubble constant, H0, using the latest baryonic acoustic oscillation (BAO)
measurements from galaxy surveys of 6dFGS, SDSS DR7 Main Galaxy Sample, BOSS DR12 sample, and eBOSS
DR14 quasar sample, in the framework of a ﬂat ΛCDM model. Based on the Kullback–Leibler divergence, we
examine the consistency of H0 values derived from various data sets. We ﬁnd that our measurement is consistent
with that derived from Planck and with the local measurement of H0 using the Cepheids and type Ia supernovae.
We perform forecasts on H0 from future BAO measurements, and ﬁnd that the uncertainty of H0 determined by
future BAO data alone, including complete eBOSS, DESI, and Euclid-like, is comparable with that from local
measurements.
Key words: cosmological parameters – distance scale – large-scale structure of universe
1. Introduction
Determining the Hubble constant, H0, which is the present
expansion rate of the Universe, with high precision plays a
crucial role in cosmology, and H0 can be measured locally, or
derived cosmologically through measurements of the Cosmic
Microwave Background (CMB) and Baryon Acoustic Oscilla-
tions (BAO; see Freedman & Madore 2010; Freedman 2017 for
a recent review on astronomical methods of H0 measurements
and their signiﬁcance in cosmology).
Recently, a direct measurement of H0 led by Riess et al.
(2016; R16) using Cepheids and type Ia supernovae ﬁnds
H 73.24 1.74 km s Mpc0 1 1=  - - , which is a 2.4% measure-
ment. On the other hand, a recent CMB measurement of H0
using the Planck satellite (PLC15) achieved a percent level
precision, namely, H 67.27 0.66 km s Mpc0 1 1=  - - (Planck
Collaboration et al. 2016). Note that, unlike the local
measurement, the CMB measurement of the Hubble constant
is model-dependent as a cosmological model, which is ΛCDM
used for the measurement we quote here, is needed to convert
the observed angular diameter distance at z 1100~ and the
sound horizon into a measurement of H0.
These two measurements are in apparent tension at a more
than 3s level (Riess et al. 2016). The tension may imply that
the ΛCDM used in the CMB analysis needs to be extended
(Wyman et al. 2014; Wang et al. 2015; Di Valentino
et al. 2016; Pourtsidou & Tram 2016; Sola et al. 2017; Zhao
et al. 2017a), or that the measurements were contaminated by
systematics to an unknown level. In this situation, additional
independent measurements of H0, e.g., using BAO distance
measurements derived from galaxy surveys4 (Cheng &
Huang 2015), can provide the critical information that we need.
The BAO distance measurements using galaxy redshift
surveys play a key role in probing the cosmic expansion
history. The BAO characteristic scale can be measured in
both radial and transverse directions of the line of sight to
provide estimates of the Hubble parameter, H(z), and angular
diameter distance, DA(z), respectively, at redshift z. Recently,
the collaboration of the Baryon Oscillation Spectroscopic
Survey (BOSS), which is a part of the Sloan Digital Sky
Survey (SDSS)-III, performed BAO measurements in the
redshift range of z0.2 0.75< < using the completed Data
Release 12 (DR12; Alam et al. 2017; Wang et al. 2017; Zhao
et al. 2017b). The extended BOSS (eBOSS, part of SDSS-IV)
detected a BAO signal at a 4% precision at z 1.5~ using the
DR14 quasar sample (Ata et al. 2017). These new BAO
measurements can provide a H0 measurement, which is
independent of CMB and local measurements, and thus can
be highly informative.
In this paper, we determine the Hubble constant using the
BOSS DR12 and eBOSS DR14 BAO measurements, combined
with other measurements presently available, and investigate
the consistency of H0 values derived from different data sets
using the Kullback–Leibler (KL) divergence (Kullback &
Leibler 1951). We also perform a forecast for future BAO data
for a feasibility study.
This paper is organized as follows. In the next section, we
present the method and data used in this work, followed by a
section devoted to the results. We present our conclusion and
discussions in Section 4.
2. Method and Data
In the spatially ﬂat ΛCDM model, the Hubble parameter is
H z H z z1 1 , 1r m0 4 3= W + + W + + WL( ) ( ) ( ) ( )
where 1r mW + W + W =L . The present energy density of
radiation5 z1r m eqW = W +( ), with z h2.5 10 meq 4 2= ´ W
T 2.7 KCMB 4-( ) being the redshift of matter-radiation equality.
We adopt T 2.7255 KCMB = . The angular diameter distance is
D z
z
dz
H z
1
1
. 2A
z
0ò= + ¢¢( ) ( ) ( )
The Astrophysical Journal, 849:84 (4pp), 2017 November 10 https://doi.org/10.3847/1538-4357/aa8f48
© 2017. The American Astronomical Society. All rights reserved.
4 There are other methods to determine the Hubble constant using galaxies.
See Chen et al. (2017) for an example.
5 Three species of massless neutrinos are included, and the energy density is
given in terms of the photon density, rg , by 3 7 8 4 11 4 3r r=n g( )( ) .
1
The sound horizon, rs, at the redshift of the drag epoch, zd,
can be calculated as
r r z
da
a H a R a3 1
, 3d s d
b0
2
zd
1
1òº = ¢¢ ¢ + ¢
+( )
( ) ( )
( )
where R h T3.15 10 2.7 Kb b4 2 CMB 4= ´ W -( ) . Note that zd is
well approximated analytically (Eisenstein & Hu 1998) as
z
h
h
b h
1291
1 0.659
1 , 4d
m
m
b
b
2 0.251
2 0.828 1
2 2= W+ W + W
( )
( )
[ ( ) ] ( )
where
b h h0.31 1 0.607 , 5m m1 2 0.419 2 0.674= W + W-( ) [ ( ) ] ( )
b h0.238 . 6m2 2 0.223= W( ) ( )
We use a ﬁxed value of the baryon density h 0.02225b 2W =
from the Planck result (Planck Collaboration et al. 2016).6 The
baryon density can also be accurately determined in a CMB-
independent way, e.g., using the primordial deuterium
abundance in the Big Bang Nucleosynthesis (BBN) theory
(Riemer-Sørensen & Sem Jenssen 2017).
Note that the quantities H z rd( ) and D z rA d( ) can be
estimated from anisotropic BAO measurements, while the
quantity,
D z r z D z
z
H z
r1 , 7V d A d
2 2
1
3º +⎡⎣⎢
⎤
⎦⎥( ) ( ) ( ) ( ) ( )
is determined by isotropic BAO measurements.
As shown above, the BAO distance measurements,
H z r D z r,d A d( ) ( ) or D z rV d( ) , are two-variable functions of
mW and H0 (once hb 2W is known) in a ﬂat ΛCDM cosmology,
and therefore the Hubble constant can be, in principle,
determined from the BAO distances with mW margin-
alized over.
In what follows, we use isotropic or anisotropic BAO
distance measurements to determine the Hubble constant with
mW marginalized over, i.e., our parameter space is simply
(assuming a ﬂatness of the Universe)
P H, . 8m 0º W{ } ( )
The sound horizon at the drag redshift rd is calculated using
Equation (3).7 We perform a Monte Carlo Markov Chain
(MCMC) global ﬁtting for parameter estimation using a
modiﬁed version of CosmoMC (Lewis & Bridle 2002).8
The BAO data sets used in this work include
1. the isotropic BAO measurements using the 6dFRS (6dF;
Beutler et al. 2011) and SDSS main galaxy sample
(MGS) (Ross et al. 2015) at effective redshifts
z 0.106eff = and z 0.15eff = , respectively;
2. the BOSS DR12 anisotropic BAO measurements at three
effective redshifts (BOSS 3zbin) in Alam et al. (2017) or
at nine effective redshifts (BOSS 9zbin) in Wang et al.
(2017) and Zhao et al. (2017b);
3. the eBOSS DR14 isotropic BAO measurement at
z 1.52eff = (Ata et al. 2017); and
4. a combination of 6dF + MGS + BOSS 3zbin + eBOSS
DR14 (All 3zbin), or a combination of 6dF + MGS +
BOSS 9zbin + eBOSS DR14 (All 9zbin).
To check the consistency of the H0 values determined from
different data sets within the ΛCDM model, we compute the
tension, T, based on the KL divergence (Kunz et al. 2006;
Paykari & Jaffe 2013; Amara & Refregier 2014; Seehars
et al. 2014, 2016; Verde et al. 2014; Grandis et al. 2016; Raveri
et al. 2016), which quantiﬁes the distance between two
probability density functions (PDFs), p1, and p2. If both p1 and
p2 are assumed to be Gaussian, the relative entropy in bits
between the two PDFs can be evaluated as
D p p d
1
2 log 2
Tr log
det
det
, 9T
2 1 1
1
2
2
1
2 1 1
1
2 1
  

q q q q
= - -
+ - -
-
-
⎡
⎣⎢( ∣∣ ) ( )
( ) ( )] ( )
where iq is the best-ﬁt parameter vector, i is the corresponding
covariance matrix, and d denotes the dimensions of the
parameter space (e.g., d= 2 in our case where both H0 and
mW are free parameters). If data are assumed to be more
informative than the priors, one can compute the expected
relative entropy, Dá ñ, with its standard deviation, Σ, via
D
1
log 2
Tr
1
2
log
det
det
, 102 1
1 2
1
  

á ñ -- ⎡⎣⎢
⎤
⎦⎥( ) ( )
D
1
2 log 2
Tr , 111
1
2
2  S +-( ) ( ) ( )
S D p p D , 122 1º - á ñ( ∣∣ ) ( )
where the Surprise, S, is deﬁned as the difference between the
relative entropy and its expectation value. The tension, T, is
deﬁned as the signal-to-noise ratio of the Surprise, i.e.,
T S . 13º S ( )
If T 1 , then p1 and p2 are consistent with each other, while
otherwise the two PDFs are in tension (Seehars et al. 2016).
We also perform forecasts on the uncertainty of H0
using ongoing and upcoming redshift surveys, including
eBOSS9 (Dawson et al. 2016; Zhao et al. 2016), Dark Energy
Spectroscopic Instrument (DESI10; DESI Collaboration
et al. 2016a, 2016b), and ESA’s Euclid satellite11 (Laureijs
et al. 2011). We use a ﬂat, ΛCDM cosmology derived from the
Planck mission as our ﬁducial model (Planck Collaboration
et al. 2016), take the forecasted BAO data for galaxy surveys of
a complete eBOSS from Zhao et al. (2016; i.e., the BAO result
from eBOSS luminous red galaxies, high-density emission-line
galaxies, and clustering quasars in Table 4 from Zhao et al.
2016), DESI (i.e., the BAO result from DESI luminous red
galaxies, emission-line galaxies, and clustering quasars in
Table 2.3 from DESI Collaboration et al. (2016a) and DESI
bright galaxies in Table 2.5 from DESI Collaboration
6 We have tested to marginalize over hb 2W with a Gaussian prior derived from
the Planck measurement and ﬁnd that the result is largely unchanged.
7 Except for 6dF, the value of rd is rescaled by a factor r rd d˜ , where rd˜ value
is calculated from CAMB (Lewis et al. 2000, available athttp://camb.info/) in
the same ﬁducial cosmology (Bennett et al. 2014).
8 http://cosmologist.info/cosmomc/
9 We use eBOSS throughout for the complete 5-year eBOSS sample, while
we use eBOSS DR14 to denote the eBOSS DR14 quasar sample. More
information on the eBOSS survey is available athttp://www.sdss.org/
surveys/eboss/.
10 http://desi.lbl.gov/
11 http://www.euclid-ec.org/
2
The Astrophysical Journal, 849:84 (4pp), 2017 November 10 Wang, Xu, & Zhao
et al. 2016a), and Euclid-like Font-Ribera et al. (2014; i.e.,
Table 6 in Font-Ribera et al. 2014), respectively, and perform
parameter estimation using the MCMC method, in the same
way as we did for the current data sets.
3. Results
We present the joint constraint on H0 and mW , and the
posterior probability distribution of H0 from various BAO data
sets, including the latest eBOSS DR14 quasar sample, in
Figure 1. As shown, the contours derived from different data
sets show different degeneracy between H0 and mW . This is
expected as the degeneracy is largely determined by the
effective redshift at which the BAO measurement is performed.
Hence having tomographic BAO measurements at a large
number of redshifts helps to break the degeneracy. This can be
seen by comparing the “All 3zbin” to the “All 9zbin” results.
The only difference in these two data sets is that the BOSS
DR12 galaxies were subdivided into more redshift slices in the
9zbin sample to gain more light-cone information. As shown in
the upper part of Table 1, the improvement on the uncertainty
of H0 is signiﬁcant, namely, the error of H0 reduces from 3.05
to 2.34 km s Mpc1 1- - , which is a 23% improvement.
We quantify the (in)consistency among the derived mW and
H0 from BAO data and PLC15 using the quantity deﬁned in
Equation (13). We also calculate the KL divergence between
the PDFs for H0 with mW marginalized over from various data
sets, including those from PLC15 and R16. The result is
presented in Table 2, including the relative entropy, D, its
expected value, Dá ñ, the Surprise, S, in bits, and the tension, T,
with 1s error. As shown, except for the PLC15 and R16 pair,
where T is larger than 1 at about a 2s level, all others are
consistent with each other (the tension T are all less than unity).
Given that the best measurement of H0 to date using
BAO alone (i.e., the “All 9zbin” result) has a worse precision
than that from R16 or PLC15, we investigate the constraining
capability of future BAO surveys, including the complete
eBOSS, DESI, and Euclid-like, on H0. The joint constraint on
H0 and mW , and the marginalized constraint on H0, from these
surveys are shown in Figure 2 and in the lower part of Table 1,
respectively. As shown, future galaxy surveys, especially for
DESI or Euclid-like alone, are able to provide a better
constraint on H0 than that of the current CMB constraint,
which is promising.
4. Conclusion and Discussions
In this paper, we determine the Hubble constant using BAO
measurements from galaxy redshift surveys in a ﬂat ΛCDM
cosmology. A combination of recent BAO measurements from
Figure 1. Left panel: the 68% and 95% conﬁdence level (CL) contour plots of mW and H0 derived from various BAO measurements. Right panel: the probability
distribution of H0 derived from various BAO measurements.
Table 1
The Mean and 68% CL Constraint on H0 Using Various Data Sets
Data Set H km s Mpc0 1 1- -( ) Precision
All 3zbin 71.75±3.05 4.25%
All 9zbin 69.13±2.34 3.38%
R16 73.24±1.74 2.38%
PLC15 67.27±0.66 0.98%
eBOSS 67.27±1.55 2.30%
DESI 67.27±0.33 0.49%
Euclid-like 67.27±0.21 0.31%
Note. The upper part of the table (above the horizontal line) is for current data
sets, while the lower part shows the forecast result based on a ﬁducial model
derived from PLC15.
Table 2
Quantiﬁcation of the (in) Consistency among Various Data Sets
Data Set D Dá ñ S T Ts
2D: H,m 0W{ }
All 3zbin → All 9zbin 0.87 2.46 −1.59 −0.71 0.14
All 3zbin → PLC15 7.91 7.87 0.04 0.02 1.12
All 9zbin → PLC15 8.58 8.31 0.27 0.09 1.54
1D: H0{ }
All 3zbin ↔ All 9zbin 0.62 1.23 −0.61 −0.38 0.19
All 3zbin ↔ PLC15 3.08 2.28 0.80 0.75 1.74
All 9zbin ↔ PLC15 1.62 1.94 −0.32 −0.29 0.94
All 3zbin ↔ R16 0.50 1.28 −0.78 −0.58 0.26
All 9zbin ↔ R16 2.33 1.23 1.10 0.70 0.42
PLC15 ↔ R16 61.91 8.63 53.29 6.57 2.73
Note.Top section: the KL divergence between the PDFs for Ωm and H0 using
BAO data and PLC15. Bottom Section: the KL divergence between the PDFs
for H0 with mW marginalized over from various data sets. The tension, T 1 ,
illustrates the relevant pairs of data sets that are consistent with each other.
3
The Astrophysical Journal, 849:84 (4pp), 2017 November 10 Wang, Xu, & Zhao
6dF, MGS, BOSS DR12 (with 9 redshift slices), and eBOSS
DR14 quasar samples yield a measurement of the Hubble
constant, namely, H 69.13 2.34 km s Mpc0 1 1=  - - , which is
a 3.4% measurement. Given the level of the uncertainty, this
measurement is consistent with both R16 and PLC15, which
are in tension between themselves.
Based on a forecast, we ﬁnd that future galaxy surveys,
including DESI and Euclid-like, will be able to provide
competitive constraints on H0, compared with current local or
CMB measurements.
We thank Will Percival for discussions and comments. We
also thank Chao Liu for discussions. Y.W. is supported by the
NSFC grant No. 11403034, and by the Young Researcher Grant
of National Astronomical Observatories, Chinese Academy of
Sciences. L.X. is supported by the NSFC grant No. 11275035,
Grant No.11675032, and by “the Fundamental Research Funds
for the Central Universities” under grant No. DUT16LK31.
G.B.Z. is supported by an NSFC grant No. 11673025, and by a
Royal Society-Newton Advanced Fellowship.
This research used resources of the SCIAMA cluster
supported by University of Portsmouth, and the ZEN cluster
supported by NAOC.
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Figure 2. The 68% and 95% CL contour plots of parameters mW and H0 in a
ﬂat ΛCDM model, derived from the complete eBOSS (red dashed), DESI (blue
dash-dotted), and Euclid-like (magenta solid). The black cross corresponds to
the ﬁducial model.
4
The Astrophysical Journal, 849:84 (4pp), 2017 November 10 Wang, Xu, & Zhao


A measurement of the Hubble constant using galaxy redshift surveys

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