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Physics Letters B 752 (2016) 66–75
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Physics Letters B
www.elsevier.com/locate/physletb
Galaxy clustering, CMB and supernova data constraints on φ CDM
model with massive neutrinos
Yun Chen a,∗ , Lixin Xu b,c
a
b
c
Key Laboratory for Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, PR China
Institute of Theoretical Physics, School of Physics & Optoelectronic Technology, Dalian University of Technology, Dalian, 116024, PR China
State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, PR China
a r t i c l e
i n f o
Article history:
Received 8 July 2015
Received in revised form 8 September 2015
Accepted 9 November 2015
Available online 12 November 2015
Editor: S. Dodelson
a b s t r a c t
We investigate a scalar field dark energy model (i.e., φ CDM model) with massive neutrinos, where
the scalar field possesses an inverse power-law potential, i.e., V (φ) ∝ φ −α (α > 0). We find that the
sum of neutrino masses mν has significant impacts on the CMB temperature power spectrum and
on the matter power spectrum. In addition, the parameter α also has slight impacts on the spectra.
A joint sample, including CMB data from Planck 2013 and WMAP9, galaxy clustering data from WiggleZ
and BOSS DR11, and JLA compilation of Type Ia supernova observations, is adopted to confine the
parameters. Within the context of the φ CDM model under consideration, the joint sample determines
the cosmological parameters to high precision: the angular size of the sound horizon at recombination,
the Thomson scattering optical depth due to reionization, the physical densities of baryons and cold dark
0.0012
+0.0266
−2
matter, and the scalar spectral index are estimated to be θ∗ = (1.0415+
−0.0011 ) × 10 , τ = 0.0914−0.0242 ,
0.0118
b h2 = 0.0222 ± 0.0005, c h2 = 0.1177 ± 0.0036, and ns = 0.9644+
−0.0119 , respectively, at 95% confidence
level (CL). It turns out that α < 4.995 at 95% CL for the φ CDM model. And yet, the CDM scenario
corresponding to α = 0 is not ruled out at 95% CL. Moreover, we get mν < 0.262 eV at 95% CL for the
φ CDM model, while the corresponding one for the CDM model is mν < 0.293 eV. The allowed scale
of mν in the φ CDM model is a bit smaller than that in the CDM model. It is consistent with the
qualitative analysis, which reveals that the increases of α and mν both can result in the suppression of
the matter power spectrum. As a consequence, when α is larger, in order to avoid suppressing the matter
power spectrum too much, the value of mν should be smaller.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
1. Introduction
Neutrino is one of the important bonds linking nuclear physics,
particle physics, astrophysics and cosmology [1]. In the Standard
Model (SM) of particle physics, it is anticipated that there are three
types, or “flavors”, of neutrinos: electron neutrino (νe ), muon neutrino (νμ ) and tau neutrino (ντ ), which are also dubbed as three
normal/active neutrinos. Besides that, neutrinos are assumed to be
massless in the SM of particle physics [2].
It was first predicted by Bruno Pontecorvo in 1957 that if neutrinos are massive the neutrino flavor should be unstable, that is
called neutrino (flavor) oscillations [3]. Briefly put, neutrino oscillation is a phenomenon that a neutrino produced in a definite
* Corresponding author.
E-mail addresses: chenyun@bao.ac.cn (Y. Chen), lxxu@dlut.edu.cn (L. Xu).
flavor is observed in a different flavor after traveling some distances. In other words, neutrinos are able to oscillate among the
three available flavors while they propagate through space. Nowadays there are compelling evidences for neutrino oscillations from
a variety of experimental data on solar, atmospheric, reactor and
accelerator neutrinos. The discovery of neutrino oscillations implies that neutrinos have small but non-zero masses, with at least
two species being non-relativistic today. However, the present experimental results on neutrino oscillations only measure the difference of two squared masses, such as m221 = m22 − m21 and
m232 = m23 − m22 , but give no hint on their absolute mass scales.
m1 , m2 and m3 are the neutrino mass eigenstates. For example,
the solar neutrino analysis supplemented by KamLAND produces
an estimate of m221 ∼ 8 × 10−5 eV2 [4], and the measurement of
atmospheric neutrino oscillation by Super-Kamiokande I indicates
m232 ∼ 3 × 10−3 eV2 [5]. If it is the case of oscillations among
three light neutrinos, only two of the three m2i j are independent,
http://dx.doi.org/10.1016/j.physletb.2015.11.022
0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP3 .
Y. Chen, L. Xu / Physics Letters B 752 (2016) 66–75
as m221 + m232 + m213 = 0, where m213 = m21 − m23 . Recent reviews on progress in both theoretical and experimental aspects of
neutrino oscillations can be found in [6].
A variety of cosmological tests are sensitive to the absolute
scale of neutrino mass, such as the cosmic microwave background
(CMB) radiation, galaxy surveys, and the Lyman-alpha forest [7]. In
[8], the effect of massive neutrinos on the Sunyaev–Zel’dovich and
X-ray observables of galaxy clusters are investigated with a set of
six very large cosmological simulations (8h−3 Gpc3 comoving volume). The analysis of current cosmological observations
provides
an upper bound on the total neutrino mass
mν (summed over
the three
neutrino families) of order 1 eV or less. However, the
limits on
mν from cosmology are rather model dependent and
vary strongly with the data combination adopted. For example, in
the framework of one-parameter extensions to the base CDM
model the Planck 2015 results
[9] give 95% upper limits on the
sum of neutrino masses, i.e.,
mν < 0.23
eV for a combination of
Planck TT + lowP + lensing + ext, and
mν < 0.59 eV for Planck
TT, TE, EE + lowP + lensing, where “TT” denotes the combination
of the TT likelihood at multipoles l ≥ 30 and a low-l temperatureonly likelihood, “TE” denotes the likelihood at l ≥ 30 using TE
spectra, and “EE” denotes the likelihood at l ≥ 30 using EE spectra, “lowP” denotes the low-l Planck polarization data, “lensing” is
the Planck lensing data, and “ext” represents the external data including the baryon acoustic oscillations (BAO), Type Ia supernovae
(SNe Ia), and H 0 . In [10], the power law and exponential types of
viable f ( R ) theories along with massive
neutrinos are studied. It
shows that the allowed scales of
mν in the viable f ( R ) models
are greater than that in the CDM model. The cases of fixing the
effective number of neutrino species as N eff = 3.046 and treating
N eff as a free parameters are both considered in [10]. The former
corresponds to just consider the active neutrinos without the effect
of dark radiation. The latter corresponds to include the contribution of dark radiation (represented by N eff = N eff − 3.046). For
more details on dark radiation, we refer the reader to [11]. The
model of holographic dark energy with massive neutrinos and/or
dark radiation is investigated in [12], but the computed results
from this model are not compared with those from the CDM
model. Actually, the CDM model with massive neutrinos is discussed broadly with constraints from various cosmological observations [13]. The time evolving of neutrino mass is also explored
in the literature [14]. For further details on neutrino cosmology,
the reader is referred to recent reviews such as [7,15].
In this paper,
we will discuss the constraints on the sum of
neutrino masses
mν in the framework of φ CDM model by using
a combination of the CMB data from Planck 2013 and WMAP9, the
galaxy clustering data from WiggleZ and BOSS surveys, and the JLA
compilation of SNe Ia observations. The effect of dark radiation is
not considered in this work, i.e., N eff = 3.046. We also assume that
one of the three active neutrinos is massive, and the other two are
massless. The φ CDM model — in which dark energy is modeled as
a scalar field φ with a gradually decreasing (in φ ) potential V (φ)
— is a simple dynamical model with a slowly decreasing (in time)
dark energy density. This model could resolve some of the puzzles
of the CDM model [16], such as the coincidence and fine-tuning
problems. Here we focus on the scalar field with an inverse powerlaw potential V (φ) ∝ φ −α , where α is a nonnegative constant [17,
18]. When α = 0 the φ CDM model is reduced to the corresponding CDM case. The φ CDM model with this kind of V (φ) has been
extensively investigated [19–21], but without considering the massive neutrinos.
The rest of the paper is organized as follows. In Sec. 2 we
present the background and perturbation evolutions
of the φ CDM
model with massive neutrinos. The impacts of
mν and α on the
CMB temperature power spectrum and on the matter power spec-
67
trum are also discussed. Constraints from the cosmological data
are derived in Sec. 3, and the results for φ CDM model are compared with those for the CDM model. We summarize our main
conclusions in Sec. 4.
2. The φ CDM model with massive neutrinos
2.1. Background evolution of the φ CDM model
Quintessence as one of the popular scalar field dark energy
models is a hypothetical form of dynamical dark energy to explain
the late-time cosmic acceleration. Since quintessence is described
by the scalar field φ , the corresponding dark energy model can
also be called as φ CDM model. In what follows, we will use the
terms “quintessence” and “φ CDM” essentially interchangeably. We
consider the self-interacting scalar field φ minimally coupled to
gravity on cosmological scales. The action of this φ CDM model is
given by
S=
√
−g −
m2p
16π
R + Lφ + L d4 x,
(1)
where g is the determinant of the metric g μν , R is the Ricci scalar,
√
m p = 1/ G is the Planck mass with G being the Newtonian constant of gravitation, L is the Lagrangian density for matter and
radiation, and Lφ is the Lagrangian density for the field φ , given
by
Lφ =
m2p
16π
1 μν
g ∂μ φ∂ν φ − V (φ) ,
2
(2)
where V (φ) is the field’s potential. In this work, we take a
flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric for the
background evolution, which is described by
ds2 = −dt 2 + a2 δi j dxi dx j ,
(3)
where xi is the comoving coordinate. a(t ) is the scale factor usually
normalized to unity now a0 = a( z = 0) = 1 and related to the redshift z as a/a0 = 1/(1 + z). Throughout, the subscript “0” denotes
the value of a quantity today. By the variation of the action in
Eq. (1) with respect to φ , one can obtain the Klein–Gordon equation (equation of motion) for the scalar field
φ̈ + 3
ȧ
a
φ̇ +
dV
dφ
= 0.
(4)
For the φ CDM model, there are many kinds of V (φ) which can
satisfy the requirement of the late-time accelerating expansion of
the universe [22]. In 1988, Peebles and Ratra [18] proposed a scalar
field that is slowly rolling down with a potential V (φ) = 12 κ m2p φ −α
at a large φ , where κ and α are nonnegative parameters. This
inverse power-law potential can not only lead to the late-time acceleration of the universe but also partially solve the cosmological
constant problems. The larger value of α induces the stronger time
dependence of the scalar field energy density ρφ . When α = 0, this
φ CDM model is reduced to the CDM case. What is more, the parameter κ depends on α (see [19,23] for its dependence on α ).
The Friedmann equation of the φ CDM model with massive neutrinos can be written as
H 2 ( z) =
8π
3m2p
(ρb + ρc + ρφ + ργ + ρν ),
(5)
where ρb , ρc , ρφ , ργ and ρν denote the energy densities of
baryons, cold dark matter (CDM), scalar field dark energy, photons
68
Y. Chen, L. Xu / Physics Letters B 752 (2016) 66–75
and neutrinos, and H ( z) ≡ ȧ/a is the Hubble parameter. The energy density and pressure of the scalar field dark energy are given
by
ρφ =
m2p
16π
(φ̇ 2 /2 + V (φ)),
(6)
(φ̇ 2 /2 − V (φ)).
(7)
and
Pφ =
m2p
16π
Then, one can work out the equation of state (EoS) of the field φ ,
ωφ ≡ P φ /ρφ =
φ̇ 2 − 2V (φ)
,
φ̇ 2 + 2V (φ)
(8)
which is clearly bounded in the range −1 < ωφ < 1 and usually
non-constant. One can see that if the scalar field φ rolls slowly
enough such that the kinetic energy density is much less than the
potential energy density, i.e. φ̇ 2 V (φ), the pressure P φ of the
field will become negative with ωφ → −1.
Based on Eqs. (4) and (5), along with the initial conditions described in Refs. [18,19], one can numerically compute the Hubble
parameter H ( z). We also introduce the dimensionless density parameter for each component as, X = ρ X /ρcr , where the index
“X” denotes the individual components, such as radiation (“r”),
neutrino (“ν ”) and matter (“m”). The critical energy density is expressed as ρcr = 3H 2 m2p /(8π ). m is the energy density of matter
including both baryons and CDM. ν is the total neutrino energy density which scales as ∝ a−4 at early times, and thereafter
evolves as ∝ a−3 after the non-relativistic transition. One can see
that the massive neutrinos behave like the radiation at early times
and like the matter later.
2.2. Cosmological perturbation of the φ CDM model
Let us consider perturbations of the flat FLRW metric in the
Newtonian Gauge [24]. In this gauge, the linear perturbed metric
is given by
ds2 = a2 (η) −(1 + 2)dη2 + (1 + 2)δi j dxi dx j ,
(9)
where the scalar perturbations are dominant over vector or tensor
perturbations, and η = a−1 dt is the conformal time. The Newtonian force gives rise to the dynamics of the perturbed fluids,
while the curvature perturbation measures the local energy density fluctuations. The linear perturbation theory is a good tool both
for describing the early universe at any scales, and the recent universe on the largest scales.
It has been shown in [25] that for self-interacting scalar field
dark energy models it is phenomenologically sufficient to regard
the dark energy component as a perfect fluid. We treat each component in the universe as perfect fluid, including the baryon, CDM,
photon, neutrino and scalar field dark energy. In the perfect fluid
approach, the perturbed Einstein equations lead to the following
Eqs. (10)–(13) in the Fourier space:
δ X + 3Ha(c 2s, X − ω X )δ X , = −(1 + ω X )(θ X + 3 )
ωX
θX
θ X + Ha(1 − 3ω X ) +
1 + ωX
2
= k2
c s, X
1 + ωX
δX + + σX ,
k2 = 4π Ga2 ρi δ X + 3Ha(ω X + 1)θ X /k2 ,
(10)
(11)
(12)
and
= −.
(13)
The great advantage of linear theory is to obtain independent
equations of evolution for each Fourier mode. All of the perturbed quantities (δ X , θ X , , , etc.) are functions of space x
and time t, where X denotes each perfect fluid composing the
universe. In the linear perturbations, the anisotropic stress σ X is
negligible for the perfect fluids. Note that a prime represents a
derivative with respect to the conformal time η . The spatial variation of density fluctuations is expressed by the density contrast
δ X ≡ δ ρ X /ρ̄ X = (ρ X − ρ̄ X )/ρ̄ X , and ρ̄ X is the background energy
density of component X . In the approximation of negligible irrotational flow, the divergence of the peculiar velocity v X , θ X = ∇ · v X
can be used to describe the fluid motion. In the Fourier space, we
have θ X ≡ ik · v X . While ω X ≡ P̄ X /ρ̄ X is the equation of state of
each component, and c 2s, X ≡ δ P X /δ ρ X represents the sound velocity. Eq. (10) is called as the (perturbed) continuity equation, that
states the conservation of local density. Eq. (11) is called as the
Euler equation, that represents the conservation of local energy momentum, and describes dynamics of perturbed fluids originated
by the Newtonian force . The curvature perturbation is constrained to the local inhomogeneity via the Poisson equation (12).
We can get Eq. (13) under the assumption that the perturbed fluid
remains a perfect fluid. These equations (10)–(13) completely determine the dynamical evolution of large scale structure (LSS) of
the universe, within a given expansion history H .
2.3. Matter power spectrum and CMB power spectrum in the φ CDM
model
In the framework of φ CDM model, we qualitatively investigate
the impacts of parameters α and mν on the matter power spectrum and on the CMB power spectrum. The analyses are performed
with the CAMB Boltzmann code [26].
Neutrinos rarely interact with matter after thermal decoupling,
so they are treated as free streaming particles. Massive neutrinos
are the only particles that present the transition from radiation to
matter. Before the non-relativistic transition the neutrinos behave
like radiation. Thus, when the neutrino mass mν increases, the
time of radiation/matter equality is postponed gradually, and aeq
increases. The value of mν can affect the matter power spectrum
and the CMB power spectrum mainly resulting from a change in
the time of equality, that provides a potential way to constrain it
through CMB and LSS observations [7,12,27]. In Fig. 1, we show
the impacts of neutrino mass mν on the matter power spectrum
P (k) and the CMB temperature anisotropy spectrum C lTT . The upper panels show the cases for φ CDM model with varying values of
mν , where α is fixed as α = 1, and other parameters are fixed
based on the recent Planck results [9]. For comparison, we also
display the cases for CDM model in the lower panels. For both
φ CDM and CDM models, the matter power spectrum is gradually
suppressed with the increase of mν , however, the effect is more
significant on small scales than that on large scales. One possible reason is that the neutrino perturbations do not contribute to
gravitational clustering on scales smaller than the free-streaming
scale, while on the very large scales neutrino perturbations are
never affected by free streaming, and they become indistinguishable from CDM perturbations in the non-relativistic regime [7].
The CMB temperature anisotropy spectrum C lTT is insensitive to the
variation of mν in both CDM and φ CDM models.
The parameter α indicates the dynamics of dark energy, and
then it can affect the expansion history of the universe and the
redshift of matter/dark energy equality. When α increases, the expansion of the universe occurs more rapidly, and the epoch of dark
Y. Chen, L. Xu / Physics Letters B 752 (2016) 66–75
69
Fig. 1. Impacts of the sum of neutrino masses mν on the matter power spectrum P (k) and on the CMB temperature power spectrum C lTT in the φ CDM (upper panels) and
CDM (lower panels) models. mν is varied, and other parameters are kept fixed.
energy domination begins earlier [19]. For these reasons, the variation of α can have signatures in the CMB map and the matter
clustering. The impacts of α on P (k) and C lTT are presented in
Fig. 2. We choose α = 0, 1, and 10 as examples, where α = 0 corresponds to the CDM scenario. The values of other parameters
are kept fixed. We find that P (k) is slightly suppressed with the
increase of α , and the effect is a bit significant on large scales
than that on small scales. The CMB temperature anisotropy spectrum C lTT is a little sensitive to the variation of α on the low-l tail,
which may arise from the late Integrated Sachs–Wolfe (ISW) effect.
Anyhow, CMB and LSS observations are efficient to distinguish between CDM and φ CDM models.
3. Observational constraints
The observational data sets used to constrain the cosmological
parameters are described as follows, including the galaxy clustering, CMB and SNe Ia measurements.
3.1. Cosmological data sets
3.1.1. Galaxy clustering measurements
Galaxy clustering distilled from the galaxy redshift survey is
powerful as cosmological probe [28], that can allow us to measure
the cosmic expansion history through the measurement of BAO,
and the growth history of cosmic large scale structure through
70
Y. Chen, L. Xu / Physics Letters B 752 (2016) 66–75
Fig. 2. Impacts of the parameter α on the matter power spectrum P (k) and on the CMB temperature power spectrum C lTT in the framework of φ CDM model. α is varied,
and other parameters are kept fixed.
measurements of redshift-space distortions. The length scale of
BAO, the comoving sound horizon at the baryon drag epoch r s ( zd ),
can be applied as a cosmological ruler and accurately calibrated by
observations of the CMB radiation. The position of the BAO peak in
the angle-averaged galaxy clustering pattern is usually quantified
in term of the volume averaged distance [29]
D V = (1 + z)2 D 2A cz/ H ( z)
1/3
.
(14)
It is common to report the BAO distance measurements as combinations of the angular diameter distance, D A ( z), and the Hubble
parameter, H ( z), such as
A ( z) ≡
0
H 0 m
D V ( z)
cz
,
(15)
or
d z ≡ D V ( z)/r s ( zd ).
away from low-density regions, such that the galaxy clustering in
redshift space is enhanced in the LOS direction compared to the
transverse direction [32]. The RSD effect on large scales can be described by linear theory [32,33], while the “finger of God” effect is
a non-linear phenomenon. On large scales where the gravitational
growth is linear, measuring the relative clustering in both LOS
and transverse directions leads to measurements of the parameter
combination f ( zeff )σ8 ( zeff ), where zeff is the effective redshift. f is
the growth rate of cosmic structure, which is associated with the
evolution of matter density perturbations δm via the relation f ≡
d ln δm /d ln a. In the linear regime, the linear growth rate can be
expressed as f = d ln D ( z)/d ln a, where D ( z) = δm ( z)/δm ( z = 0) is
the linear growth factor normalized such that D ( z = 0) = 1. σ8 ( z)
is the root–mean–square amplitude of the matter fluctuations in
spheres of 8h−1 Mpc, and σ8 ( z = 0)/σ8 ( z) = D ( z = 0)/ D ( z). Thus,
one can figure out
f ( z)σ8 ( z) = σ80
(16)
The redshifts of galaxies include indistinguishable contributions
from both the Hubble recession and the peculiar velocity of the
galaxies themselves, so that there are errors in the distances we
assign to galaxies. The differences between the redshift-inferred
distances and true distances are known as redshift-space distortions (RSD) [30]. In another word, the RSD are introduced in
the observed clustering pattern by galaxy peculiar motions. As
a consequence, the correlation function and the power spectrum
measured in the redshift space are different from those in the
real space, which have to be corrected to be expressed in real
space. Because the effects of RSD couple the density and velocity
fields, the RSD signals within the correlation function are difficult
to model. On different scales, peculiar motions produce different
types of distortions to the power spectrum. On small scales, i.e.,
in the cluster cores, the peculiar velocities of galaxies are almost
randomly oriented, that cause the structures to appear elongated
along the line of sight (LOS) when viewed in redshift space (i.e.,
the so called “finger of God” effect) [31], leading to a damping
of the clustering. On large scales, because of gravitational growth,
the galaxies tend to fall towards high-density regions, and flow
dD ( z)
d ln a
,
(17)
where σ80 = σ8 ( z = 0). In linear theory, the galaxy bias b and the
growth rate f are degenerate with σ8 , so the RSD measurements
are better presented in terms of b( z)σ8 ( z) and f ( z)σ8 ( z), rather
than f ( z). Currently, the bias-independent parameter combination
f ( z)σ8 ( z) measured by RSD are widely used [20,34,35].
The Alcock–Paczynski (AP) test [36] is proved to be a significant
link between BAO and RSD. AP test states that if an astrophysical structure is spherically symmetric or isotropic, then it should
possess equal comoving transverse and radial sizes. An AP measurement is carried out by comparing the observed transverse and
radial dimensions of objects. While the AP test is equally valid for
an isotropic process such as the two-point statistics of galaxy clustering. The apparent anisotropy of the two-dimensional correlation
function of galaxies mainly arises from the geometry and expansion of the universe which should be correctly embodied in the
fiducial cosmological model, and the RSD effect which is supposed
to be marginalized by using appropriate RSD model (see [30] for
a recent review of RSD models). According to the requirement of
AP test, the signature of BAO should have identical comoving sizes
(i.e., r s ) in transverse and radial dimensions. The observed transverse dimension is the angular projection θ = r s /[(1 + z) D A ( z)].
Y. Chen, L. Xu / Physics Letters B 752 (2016) 66–75
The radial one is the redshift projection z = r s H ( z)/c. The relative radial/transverse distortion depends on the value of
F ( z) ≡
71
The galaxy clustering (GC) measurements from WiggleZ and
BOSS DR11 are both employed in this study. Thus, the corresponding chi-squared statistic is expressed as
z/ θ
= (1 + z) D A ( z) H ( z)/c ,
(18)
where F ( z) is dubbed as the AP distortion parameter.
By combining the BAO peak, AP test and RSD effect, one can
report the galaxy clustering effectively as joint measurements
of ( A , F , f σ8 ) or (d z , F , f σ8 ). These joint measurements are extremely good at helping to constrain basic cosmological parameters
and distinguish between the dark energy models. By using largescale structure measurements from the WiggleZ Dark Energy Survey [37], Blake et al. (2012) [38] have performed joint constraints
of ( A , F , f σ8 ) in three overlapping redshift slices with effective
redshifts zeff = (0.44, 0.6, 0.73). Utilizing these data, it is straightforward to put constraints on the model parameters by calculating
2
the corresponding χWiggleZ
, given by
2
χWiggleZ
= ( X obs − X th )C −1 ( X obs − X th )T .
(19)
The observational data vector is
X obs = [ A 1 , A 2 , A 3 , F 1 , F 2 , F 3 , f σ8,1 , f σ8,2 , f σ8,3 ],
(20)
i.e., X obs = [0.474, 0.442, 0.424, 0.482, 0.650, 0.865, 0.413, 0.390,
0.437] by using the maximum likelihood estimations of ( A , F , f σ8 )
listed in Table 1 of [38]. The vector of theoretical values is
(21)
where [ z1 , z2 , z3 ] = [0.44, 0.6, 0.73], and the corresponding theoretical values of ( A , F , f σ8 ) can be obtained with Eqs. (15), (18)
and (17), respectively. C is a 9 × 9 covariance matrix between parameters and redshift slices, and the value of 103 C is listed in
Table 2 of [38], that is achieved by generating 400 lognormal realizations for each WiggleZ survey region and redshift slice with the
methods described in [34]. In the analysis of [38], the fitting formulae provided by Jennings et al. (2011) [39] have been taken as
the fiducial RSD model. The effect of different choices of the RSD
model is also considered in Section 3.4 of [38]. It turns out that the
systematic error induced from modeling RSD is much lower than
the statistical error in the measurement.
Joint measurements of (d z , F , f σ8 ) at an effective redshift of
zeff = 0.57 are provided in Samushia et al. (2014) [40] by utilizing
the observed anisotropic clustering of galaxies in the Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 11 (DR11) CMASS
sample [41]. We employ this data set in our analysis with the chisquared statistic
2
χBOSS
= (Y obs − Y th )Cov−1 (Y obs − Y th )T .
(23)
2
2
where χWiggleZ
and χBOSS
are given by Eqs. (19) and (22), respectively.
3.1.2. CMB power spectrum measurements
The CMB radiation deemed as the afterglow of the big bang can
supply us with some information of the very early universe. The
observations of CMB provide another independent test for the existence of dark energy. The recent precise measurements of the CMB
radiation from Planck and Wilkinson Microwave Anisotropy Probe
(WMAP) projects can efficiently improve the accuracy of constraining the cosmological parameters. Currently, the Planck 2015 results
have come out [9], but the Planck 2015 likelihoods are not yet
available. Given this, we use the low multipoles (2 ≤ l ≤ 49) and
high multipoles (50 ≤ l ≤ 2479) temperature power spectrum likelihoods from Planck 2013 [44], together with the low multipoles
(l ≤ 23) polarization power spectrum likelihoods from nine-year
WMAP (WMAP9) [45]. To employ the previously mentioned CMB
2
power spectrum data in the analysis, we compute the χCMB
statistic
2
χCMB
=
(C lobs − C lth )Mll−1 (C lobs − C lth ),
(24)
ll
X th = [ A ( z1 ), A ( z2 ), A ( z3 ), F ( z1 ), F ( z2 ), F ( z3 ),
f σ8 ( z1 ), f σ8 ( z2 ), f σ8 ( z3 )],
2
2
2
χGC
= χWiggleZ
+ χBOSS
,
(22)
The observational data vector is Y obs = [d z , F , f σ8 ], i.e., Y obs =
[13.85, 0.6725, 0.4412] by using the mean values presented in
Eq. (30) of [40]. The vector of theoretical values is Y th = [d z ( zeff ),
F ( zeff ), f σ8 ( zeff )], where the corresponding theoretical values of
(d z , F , f σ8 ) can be obtained with Eqs. (16), (18) and (17), respectively. The covariance matrix Cov of measurements is listed
in Eq. (31) of [40]. A suite of 600 PTHalo simulations are used
to estimate the covariance matrix (see [42] for details of mock
generation). In the analysis of [40], the “streaming model”-based
approach developed in [43] has been adopted to model the RSD
signal, that has been demonstrated to fit the monopole and
quadrupole of the galaxy correlation function with better than percent level precision to scales above 25h−1 Mpc, for galaxies with
bias of b ≃ 2.
where C lobs is the observational value of the related power spec-
trum, C lth is the corresponding theoretical value in the framework
of the cosmological model under consideration, and M is the covariance matrix for the best-fit data spectrum.
3.1.3. Magnitude-redshift measurements of Type Ia supernovae
The first direct evidence for the cosmic acceleration came from
SNe Ia observations, which provide the measurement of the cosmic expansion history through the measured luminosity distance
as a function of redshift, d L ( z) = (1 + z)r ( z). In the spatially flat
universe, the comoving distance r ( z) from the observer to redshift
z is given by
r ( z; p) =
c
H0
z
0
dz
E ( z ; p)
(25)
,
wherein p denotes the parameter space of the considered cosmological model, and E = H / H 0 is the dimensionless Hubble parameter.
Here, we use the “joint light-curve analysis” (JLA) compilation
of SNe Ia [46], which is a joint analysis of SNe Ia observations including several low-redshift samples (z < 0.1), all three seasons
from the SDSS-II (0.05 < z < 0.4), three years from SNLS (0.2 <
z < 1), and 14 very high redshift (0.7 < z < 1.4) from the Hubble
Space Telescope (HST) observations. It totals 740 spectroscopically
confirmed SNe Ia with high quality light curves. In Ref. [46], SALT2
light-curve model [47,48] have been used to fit the supernova light
curves of the JLA sample. From the observational point of view, the
distance modulus of a SN Ia can be yielded from its light curve
with an empirical linear relation:
μobs
B = m B − (M B − α × X1 + β × C )
(26)
The light-curve parameters (m , X , C ) result from the fit of the
B
1
SALT2 light-curve model to the photometric data, where mB corresponds to the observed peak magnitude in rest-frame B band,
72
Y. Chen, L. Xu / Physics Letters B 752 (2016) 66–75
Table 1
Fitting results from the joint sample. We present the best-fit values (i.e., the parameters that maximize the overall likelihood), and the mean values with 95% confidence
limits for the parameters of interest. Where mν is in unit of eV, and H 0 is in unit of km/s/Mpc. The top block contains parameters with uniform priors that are varied in
the MCMC chains. The lower block defines various derived parameters.
Parameters
CDM model
b h2
c h2
100θMC
φ CDM model
Best-fit values
95% limits
Best-fit values
95% limits
0.0221
0.1197
1.0412
0.0221 ± 0.0005
0.0036
0.1180+
−0.0037
1.0414 ± 0.0011
0.0222
0.1176
1.0414
0.0263
0.0904+
−0.0255
0.0222 ± 0.0005
0.1177 ± 0.0036
0.0012
1.0415+
−0.0011
0.0846
τ
0.0921
ln(1010 A s )
3.0939
ns
0.9601
mν
0.043
α
...
m
0.312
σ8
0.834
H0
67.51
0.0513
3.0854+
−0.0467
0.0118
0.9644+
−0.0119
< 0.293
...
0.038
2.482
< 0.262
< 4.995
0.023
0.311+
−0.022
0.310
0.023
0.313+
−0.021
(27)
where μ0 = 42.38 − 5 log10 h, which is also treated as a nuisance
parameter, and the Hubble-free luminosity distance is given by
D L ( z; p) ≡
H0
c
d L ( z) = (1 + z)
0
dz
E ( z ; p)
.
(28)
The best-fit cosmological parameters from SNe Ia data are determined by minimizing
2
χSNe
=
740
,i
μobs
(α , β, M B ) − μth,i ( zi ; p, μ0 )
B
i , j =1
1
× Cov−
μB
ij
obs, j
(α , β, M B ) − μth, j ( z j ; p, μ0 ) ,
(29)
where Cov is the covariance matrix of data vector μobs
B . The values of the covariance matrix Cov and the SALT2 fit parameters
(mB , X 1 , C ) are available from Ref. [46].
3.2. Results and analysis
In our analysis, the likelihood is assumed to be Gaussian, thus
we have the total likelihood
L ∝ e −χtot /2 ,
2
where χ
(30)
2
tot is constructed as
2
2
2
2
χtot
= χGC
+ χCMB
+ χSNe
,
0.0516
3.0869+
−0.0475
0.9607
0.044
0.806+
−0.049
1.76
67.47+
−1.79
μth (z; p, μ0 ) = 5 log10 [ D L (z; p)] + μ0 ,
0.0266
0.0914+
−0.0242
0.0113
0.9636+
−0.0112
X 1 describes the time stretching of the light-curve, and C describes the supernova color at maximum brightness. α , β and
M B are nuisance parameters in the distance estimate, which are
estimated simultaneously with the cosmological parameters and
then marginalized over when obtaining the parameters of interest, wherein M B is the absolute B-band magnitude. The theoretical
(predicted) distance modulus is
z
3.0758
(31)
2
2
2
wherein χGC
, χCMB
and χSNe
are given by Eqs. (23), (24) and (29),
respectively, and denote the contributions from galaxy clustering,
CMB and SNe Ia data sets described above.
We derive the posterior probability distributions of parameters
with Markov Chain Monte Carlo (MCMC) exploration using the
February 2015 version of CosmoMC [49]. The parameter space of
the CDM model is
0.806
67.33
0.043
0.805+
−0.046
1.79
67.11+
−1.81
P CDM ≡ {b h2 , c h2 , 100θMC , τ , ln(1010 A s ), ns , mν },
2
(32)
2
where b h and c h , respectively, stand for the baryon and CDM
densities today, θMC is an approximation to θ∗ = r s ( z∗ )/ D A ( z∗ ) (i.e.,
the angular size of the sound horizon at the time of decoupling)
that is used in CosmoMC and is based on fitting formulae given
in [50], τ refers to the Thomson scattering optical depth due to
reionization, ns and A s are the power-law index and amplitude of
the power-law scalar primordial power spectrum of curvature perturbations, and mν is the sum of neutrino masses. The parameter
space of the φ CDM model is
Pφ CDM ≡ {b h2 , c h2 , 100θMC , τ , ln(1010 A s ), ns , mν , α }, (33)
which has one more parameter than that of CDM model, where
α determines the steepness of the scalar field potential in the
framework of φ CDM model.
The one-dimensional (1D) probability distributions and twodimensional (2D) contours for the cosmological parameters of interest are shown in Fig. 3 for CDM model and in Fig. 4 for φ CDM
model. It shows that constraints from the joint sample are quite restrictive, though there are degeneracies between some parameters,
such as the degeneracies in the m − H 0 and σ8 − mν planes. In
addition, the differences between the marginalized likelihoods and
the mean likelihoods are modest in 1D and 2D plots. It implies
that the distributions of the parameters are almost Gaussian. We
also present best-fit values and mean values with 95% confidence
limits for the parameters of interest in Table 1 both for CDM and
φ CDM models. We find α < 4.995 at 95% CL for the φ CDM model,
while the CDM scenario corresponding to α = 0 is not ruled out
at this confidence level. The constraints on b h2 , c h2 , 100θMC , τ ,
ln(1010 A s ), ns , m , σ8 and H 0 are consistent at 95% CL for these
two models.
Here, we pay attention to the constraints on mν . Note that
mν is in unit of eV. The best-fit vale is mν = 0.038(0.043)
with mν < 0.262(0.293) at 95% CL in the framework of φ CDM
( CDM) model. The allowed neutrino mass scale in the φ CDM
model is a bit smaller than that in the CDM model. As it is
shown in Figs. 1 and 2, the increases of α and mν both can result
in the suppression of the matter power spectrum. Therefore, when
α is larger, in order to avoid suppressing the matter power spectrum too much, the value of mν should be smaller. Consequently,
the case with α > 0, i.e., φ CDM model, has smaller mν ; correspondingly, the case with α = 0, i.e., CDM scenario, has larger
mν . Additionally, in Ref. [10], they obtained mν < 0.200 eV at
95% CL in the CDM model, that is consistent with our result.
With the results presented in Table II of [10], one can see that our
Y. Chen, L. Xu / Physics Letters B 752 (2016) 66–75
73
Fig. 3. The 1D and 2D probability distributions of parameters of interest in the CDM model constrained with the joint sample. In the 1D plots, the solid lines denote the
marginalized likelihoods and the dotted lines correspond to the mean likelihoods. In the 2D plots, the contours refer to the marginalized likelihoods while the colors refer
to the mean likelihoods. The contours correspond to 68%, 95% and 99% confidence levels. (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
Fig. 4. The 1D marginalized distribution and 2D contours of parameters of interest in the φ CDM model constrained from the joint sample. The implications of line styles and
colors are the same as those in Fig. 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
74
Y. Chen, L. Xu / Physics Letters B 752 (2016) 66–75
constraints on b h2 , c h2 , τ and ns are also consistent with theirs
at 95% CL for the CDM model.
4. Conclusion
We have concentrated on a quintessence model (or called as
φ CDM model) of dark energy with massive neutrinos. In the φ CDM
model under consideration, the scalar field φ is taken as a candidate of dark energy to drive the late-time acceleration of the
universe with an inverse power-law potential V (φ) ∝ φ −α (α > 0).
The larger value of α corresponds to the stronger time dependence
of the scalar field energy density. When α = 0, it is reduced to
the corresponding CDM scenario. The linear perturbation theory
is employed in the framework of this model. Through qualitative
analyses, we find that the increases of the sum of neutrino masses
mν and the parameter α both can gradually suppress the matter power spectrum P (k). It implies that when the value of α is
bigger, in order to avoid suppressing the matter power spectrum
too much, mν should be smaller. It is in accordance with the
results from the observational constraints. The variations of these
two parameters also can have signatures in the CMB temperature
anisotropy spectrum C lTT . In order to make a comparison, the impacts of mν on P (k) and C lTT in the context of CDM model have
also been presented.
A combination of the CMB data from Planck 2013 and WMAP9,
the galaxy clustering data from WiggleZ and BOSS DR11, and the
JLA compilation of the SNe Ia observations is used to constrain the
parameters. The results indicate that constraints on the cosmological parameters from this joint sample are quite restrictive. It turns
out that mν < 0.262 eV (95% CL) in the framework of φ CDM
model and mν < 0.293 eV (95% CL) in the CDM model. The
allowed neutrino mass scale in the φ CDM model is a little shrunk
comparing to that in the CDM model. In Ref. [10], it is concluded
that the allowed neutrino mass scales in the viable f ( R ) models
are bigger than that in the CDM model. Given this, we can infer that the allowed scale of mν in our φ CDM model must be
smaller than those in the viable f ( R ) models. In addition, we get
α < 4.995 at 95% CL for the φ CDM model, meanwhile, the CDM
scenario corresponding to α = 0 is not ruled out. Consequently, the
observational data that we have employed here still cannot distinguish whether dark energy is a time-independent cosmological
constant or a time-varying dynamical component.
Acknowledgements
Yun Chen would like to thank Bharat Ratra, Jie Liu, Chung-Chi
Lee, Qiao Wang and Yuting Wang for useful discussions. Y.C. was
supported by the National Natural Science Foundation of China
(Nos. 11133003 and 11573031), the Strategic Priority Research Program “The Emergence of Cosmological Structures” of the Chinese
Academy of Sciences (No. XDB09000000), the China Postdoctoral
Science Foundation (No. 2015M571126), and the Young Researcher
Grant of National Astronomical Observatories, Chinese Academy of
Sciences. L.X. was supported by the National Natural Science Foundation of China (No. 11275035), and the Open Project Program of
State Key Laboratory of Theoretical Physics, Institute of Theoretical
Physics, Chinese Academy of Sciences (No. Y4KF101CJ1).
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