Gaussian processes reconstruction of dark energy from observational data
Eur. Phys. J. C
Gaussian processes reconstruction of dark energy from observational data
Ming-Jian Zhang 0
Hong Li 0
0 Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Science , P. O. Box 918-3, Beijing 100049 , China
In the present paper, we investigate the dark energy equation of state using the Gaussian processes analysis method, without confining a particular parametrization. The reconstruction is carried out by adopting the background data including supernova and Hubble parameter, and perturbation data from the growth rate. It suggests that the background and perturbation data both present a hint of dynamical dark energy. However, the perturbation data have a more promising potential to distinguish non-evolution dark energy including the cosmological constant model. We also test the influence of some parameters on the reconstruction. We find that the matter density parameter m0 has a slight effect on the background data reconstruction, but has a notable influence on the perturbation data reconstruction. While the Hubble constant presents a significant influence on the reconstruction from background data.
1 Introduction
Multiple experiments have consistently approved the cosmic
late-time accelerating expansion. Observations contributing
to this pioneering discovery contain the type Ia supernova
(SNIa) [
1,2
], large scale structure [3], cosmic microwave
background (CMB) anisotropies [
4
], and baryon acoustic
oscillation (BAO) peaks [
5
]. Theoretical paradigms trying
to explain this discovery include the exotic dark energy with
repulsive gravity, or modification to general relativity [
6,7
],
or violation of cosmological principle [
8–10
]. In which, dark
energy theory attracts lots of interests. For understanding the
nature of dark energy, a crucial parameter is the equation of
state (EoS) w, which is the ratio of pressure to energy density.
Basing on the value of w, dark energy can be classified to
different categories. The cosmological constant model with
w = −1 is the most notable candidate. In addition to this
one, the time evolution model, Chevallier–Polarski–Linder
(CPL) [
11,12
] is also a potential competitor. A short review
can be seen in Ref. [
13
].
However, above understanding of the dark energy is a
parametrization on w. It is, after all, an ansatz of the dark
energy. To extract the information of EoS honestly, Huterer
and Starkman [
14
] first proposed the principal component
analysis technique in dark energy study. It is a
modelindependent way which treats the w as a piecewise constant
in each redshift bin. By extracting essential information from
multiple observational data, one can obtain a series of
orthogonal eigenfunctions to expand the EoS w. In following Refs.
[
15–18
], this method was greatly adopted and improved in
different forms.
Another effective technique, Gaussian processes (GP),
is also model-independent. Unlike the parametrization
constraint, this approach does not rely on any artificial dark
energy template. It can reconstruct the w directly via its
relationship with the observational variable. In this process, it
firstly assumes that each observational data satisfies a
Gaussian distribution. Thus, the observational data should satisfy
a multivariate normal distribution. Relationship between two
different data points is connected by a covariance function.
Using this covariance function, values of data at other
redshift points which have not be observed also can be obtained
because they all obey this probability distribution.
Moreover, derivative of these data also can be calculated using the
covariance function. Finally, with the preparation of more
data, a variable or goal function can be reconstructed at
any redshift point via their relationship with the data and
its derivatives. We note that the primary task in this Gaussian
processes is to determine the covariance function at different
redshift points using the observational data. Moreover,
determination of the covariance function has nothing to do with the
w. Thus, its understanding on the w is more faithful. In
cosmology, it has incurred a wide application in reconstructing
dark energy [
19,20
] and cosmography [21], or testing
standard concordance model [
22
] and distance duality relation
[
23
], or determinating the interaction between dark matter
and energy [
24
] and spatial curvature [
25
]. Also, this method
has been used to study the dark energy [
20,26,27
]. However,
most focus were on the background observational data, such
as supernova and Hubble parameter data. Moreover, they did
not consider the effect of matter density parameter.
In the present paper, we want to learn more about the dark
energy. The data we use are not only the background data
including supernova luminosity distance and Hubble
parameter, but also the perturbation data, growth rate of structure
f σ8. Moreover, we consider to test the influence of some
parameters including the matter density. For the perturbation
level data, they measure the redshift-space distortions. It has
been evidenced that these data can provide tight constraint
on the parameter space [
28,29
], or test the cosmic
acceleration [30], or distinguish the Galileon model from CDM
model [
31,32
], or distinguish some modified gravity
models [33]. Motivated by the advantage of growth rate data, we
expect to obtain a new model-independent constraint on the
dark energy. Another difference from previous work is that
the SNIa and H (z) data here are used as a combination of
background data, not two single ones.
This paper is organized as follows: In Sect. 2, we
introduce some theoretical basis and the GP approach. And in
Sect. 3 we introduce the relevant data we use. We present the
reconstruction result in Sect. 4. Finally, in Sect. 5 conclusion
and discussion are drawn.
2 Methodology
In this section, we introduce some theoretical basis and the
GP approach.
2.1 Theoretical basis
On background level: In the Friedmann–Robertson–Walker
universe, the luminosity distance function dL (z) of SNIa is
c
dL (z) = H0 (1 + z)
z H0dz
0 H (z )
In the GP reconstruction, it is very convenient to define a
dimensionless comoving luminosity distance
(1)
(2)
(3)
On perturbation level: In the general relativity and a
background universe filled with matter and unclustered dark
energy, the evolution of matter density contrast, δ(z) ≡
δρm (z), at scales much smaller than the Hubble radius should
ρm
obey the following second order differential equation
δ¨ + 2H δ˙ − 4π Gρm δ = 0,
where ρm is the background matter density, δρm
represents its first-order perturbation, the dot denotes derivative
with respect to cosmic time t . Basing the relation d/dt =
a H (d/da), we can change the argument of Eq. (4) from
cosmic time to scale factor. Subsequently, according to the
relation between scale factor and redshift, Hubble parameter in
Eq. (4) can be expressed as an integral over the perturbation
and its derivative [
34,35
]
(1 + z)2
E 2(z) = 3 m0 δ (z)2
z
∞
δ
1 + z (−δ )d z,
where m0 is the matter density parameter today and the
prime denotes derivative with respect to redshift z.
Conversely, the solution of perturbation also can be solved as
an integral of Hubble parameter. From the Eq. (5), we find
that the Hubble parameter E 2(z) tends to zero when the
redshift in integral z → ∞. When the redshift z = 0, we have
the initial condition
3 m0
1 = δ (z = 0)2 0
∞
δ
1 + z (−δ )d z.
For the integral in Eq. (5), it usually can be calculated by the
relation z∞ f (z)d z = 0∞ f (z)d z − 0z f (z)d z. In previous
work, they usually substitute the matter density parameter
m0 in Eqs. (6) into (5). The Hubble parameter is, thus,
expressed as a ratio of two integrals. However, perturbation
δ at higher redshift z 5 maybe cannot be determined from
observation [
34,35
]. Therefore, in practice, it may be
difficult to calculate the integral 0∞ 1 +δz (−δ )d z. In the present
paper, we deal with it in a diametrically opposite way. That is,
we replace the integral 0∞ 1 +δz (−δ )d z using the parameter
m0. Consequently, Eq. (5) can be written as
E 2(z) = (1 + z)2 δ (z = 0)2
δ (z)2
(1 + z)2
−3 m0 δ (z)2
z
δ
0 1 + z (−δ )d z.
Observationally, current cosmological surveys cannot
provide direct measurement of perturbation δ(z), but can
provide a related observation, the growth rate measurement f σ8
from redshift-space distortions (RSD) caused by the peculiar
motions of galaxies [
36
]. Here, the growth rate f is defined by
the derivative of the logarithm of perturbation δ with respect
to logarithm of the cosmic scale
d lnδ d lnδ
f ≡ d lna = −(1 + z) d z
δ
= −(1 + z) .
δ
(4)
(5)
(6)
(7)
(8)
D(z) ≡
H0 dL (z)
c 1 + z
Obviously, combining Eqs. (2) and (1), taking derivative with
respect to redshift z, it is easy for us to obtain the relation
between Hubble parameter and distance D(z)
where E (z) is the dimensionless Hubble parameter, and the
prime denotes derivative with respect to redshift z.
While the function
σ8(z) = σ8(z = 0)
δ(z)
δ(z = 0)
is the linear theory root-mean-square mass fluctuation within
a sphere of radius 8h−1Mpc, where h is the dimensionless
Hubble constant. Because RSD measurements are sensitive
to the product of these two functions, they have been used in a
wide range to constrain the evolution of universe and directly
test the general relativity, thereby providing an insight on the
fundamental physics.
In the light of above two definitions, the growth rate of
structure is written as
σ8(z = 0)
f σ8 = − δ(z = 0) (1 + z)δ .
It is easy for us to have
δ(z = 0) f σ8
δ = − σ8(z = 0) 1 + z
.
1 −2(1 + z)D
w(z) = 3 D −
m0(1 + z)3 D 3
− 3D
for the background data, and
1 (1 + z)E 2(z) − 3E 2(z)
w(z) = 3 E 2(z) − m0(1 + z)3
for the RSD data.
Obviously, derivative of the perturbation δ can be easily
transferred or reconstructed from the observational RSD data.
Taking an integral to the two sides of Eq. (11) over redshift,
we have
δ(z = 0) z f σ8 d z.
δ = δ(z = 0) − σ8(z = 0) 0 1 + z
For the constant δ(z = 0), it was commonly fixed as the
normalization value δ(z = 0) = 1 [
37,38
] or a fiducial value.
In this paper, we would like to test the influence of different
δ(z = 0) on the reconstruction, at the normalization value
δ(z = 0) = 1 and a fiducial value δ(z = 0) = 0.7837. We
also intend to test the influence of different σ8(z = 0) on the
reconstruction.
We consider a spatial flat Friedmann–Robertson–Walker
universe with dark matter and dark energy
E 2(z) =
m0(1 + z)3
+ (1 −
m0) exp 3
d z .
(13)
0
z 1 + w(z )
1 + z
Obviously, the EoS of dark energy can be obtained by taking
the derivative with respect to z on the two sides of above
equation. Substituting Eqs. (3) and (7) into (13) respectively,
we have
(9)
(10)
(11)
(12)
(14)
(15)
2.2 Gaussian processes
To reconstruct the goal function f (z), a parametrization
constraint or a model-independent technique should be carried
out. For the former method, a prior form on the constrained
function f (z) is usually restricted. For example, to
understand the dark energy, EoS w(z) is assumed to be the CPL
model with two artificial parameters w0 and wa . Instead, a
model-independent method such as the Gaussian processes,
does not limit to a particular parametrization form. It only
needs a probability on the goal function f (z). A Gaussian
process is a generalization of the Gaussian probability
distribution. Assuming the observational data, such as the distance
D, obeys a Gaussian distribution with mean and variance, the
posterior distribution of goal function f (z) can be expressed
via the joint Gaussian distribution of different data of
distance D. In this process, the key ingredient is the covariance
function k(z, z˜) which correlates the values of different
distance D(z) at points z and z˜. Commonly, the covariance
function k(z, z˜) has several types, and most associated with two
hyperparameters σ f and which can be determined by the
observational data via a marginal likelihood. With the trained
covariance function, the data can be extended to any redshift
points. Using the relation between the goal function f (x )
and distance data D, the goal function can be reconstructed.
Due to its model-independence, this method has been widely
applied in the reconstruction of dark energy EoS [
19,20,39
],
or in the test of the concordance model [
22,26
], or
determination to the dynamics of dark energy by dodging the matter
degeneracy [40].
For the covariance function k(z, z˜), many templates are
available. The usual choice is the squared exponential
k(z, z˜) = σ 2f exp[−|z − z˜|2/(2 2)]. Analysis in Ref. [
41
]
shows that the Matérn (ν = 9/2) covariance function is a
better choice to present suitable and stable result. It thus has
been widely used in previous work [
22,24
]. It is read as
k(z, z˜) = σ 2f exp
−
3 |z − z˜|
3 |z − z˜|
× 1 +
+
18 |z − z˜|3
7 3
+
27(z − z˜)2
7 2
+
27(z − z˜)4
35 4
With the chosen Matérn (ν = 9/2) covariance function, we
can reconstruct the EoS of dark energy by modifying the
publicly available package GaPP developed by [
20
]. We refer
the reader to Ref. [
20
] for more details on the GP method.
3 Observational data
In this section, we report the related observational data.
(16)
For the supernova data, we use the Union2.1
compilations [
42
] released by the Hubble Space Telescope Supernova
Cosmology Project and the JLA datasets [
43
]. Usually, they
are presented as tabulated distance modulus with errors. For
the Union2.1 data, they contain 580 dataset. Their redshift
regions are able to span over z < 1.414. For the JLA sample,
it spans a range at redshift 0.01 < z < 1.3. It consists of
740 SNIa datasets, including three-season data from
SDSSII (0.05 < z < 0.4), 3-year data from SNLS (0.2 < z < 1),
HST data (0.8 < z < 1.4), and several low-redshift
samples (z < 0.1). According to their test, the binned JLA data
have a same constraint power as the full version of the JLA
likelihood on the cosmological model. In our calculation, we
use the 31 binned distance modulus with covariance matrix,
which is issued in their Ref. [
43
].
The distance modulus of each supernova can be estimated
as
μ(z) = 5log10dL (z) + 25,
(17)
where dL is the luminosity distance in Eq. (1). In our
calculation, we set the same prior of H0 as the following H (z) data
and include the covariance matrix with systematic errors in
our calculation. To obtain the dimensionless comoving
luminosity distance D(z), we should make a transformation from
the distance modulus via Eq. (2). Moreover, the theoretical
initial conditions D(z = 0) = 0 and D (z = 0) = 1 are also
taken into account in the calculation.
For the H (z) data, they were not direct products from a
tailored telescope, but can be acquired via two ways. One
is to calculate the differential ages of galaxies [
61–63
],
usually called cosmic chronometer. The other is the deduction
from the BAO peaks in the galaxy power spectrum [
64,65
]
or from the BAO peak using the Lyα forest of QSOs [66].
In the present paper, we use the 30 cosmic chronometer data
points which compiled in Table 1 of Ref. [
67
], because the
latter method is model-dependent. An underlying cosmology
is needed to calculate the sound horizon in the latter method.
After the preparation of H (z) data, we should normalize them
to obtain the dimensionless one E (z) = H (z)/H0.
Obviously, the initial condition E (z = 0) = 1 should be taken
into account in our calculation. Considering the error of
Hubble constant, we can calculate the uncertainty of E (z)
2 σ H2 H 2
σE = H 2 + H 4 σ H20 .
0 0
H0 = 71.00 ± 2.80 km s−1Mpc−1 from the latest
determination [
69
].
To probe the growth of structure, several promising types
of cosmological measurements were proposed, such as the
clustering of galaxies in spectroscopic surveys, counts of
galaxy clusters, and weak gravitational lensing.
For the RSD data, they are in fact effects due to the
differences between the observed distance and true distance on
the galaxy distribution in redshift space. These differences
are caused by the velocities in the overdensities deviation
from the cosmic smooth Hubble flow expansion. Anisotropy
of the radial direction relative to transverse direction in the
clustering of galaxies is correlated with the cosmic
structure growth. Smaller deviation from the General
Relativity implies a smaller anisotropic distortion in the redshift
space. The RSD data is a very promising probe to
distinguish the cosmological models, because different
cosmological models may have similar background evolution, but
cosmic growth of the structure in their models can be very
distinct. Till now, the RSD data have been used extensively in
previous literatures. In this paper, we utilize the most recent
RSD data from 2dF, 6dF, BOSS, GAMA, WiggleZ, galaxy
surveys. We also consider the four very recent measurements
from the eBOSS DR14 data [
59
]. We collect the compilation
in Table 1, which includes the survey, RSD data with errors,
the corresponding references and year.
For the reconstruction using RSD, the Eq. (12) indicates
that an integral should be calculated to obtain the perturbation
δ(z). Moreover, covariance of the f σ8 should be propagated
into the uncertainty of δ(z). However, covariance propagation
in the integral at all times is a difficult thing. For a simplicity,
we only consider the uncertainties of f σ8. For a Gaussian
error propagation, the uncertainty of Hubble parameter can
be expressed as
2
σE2 =
∂ E 2 2
∂δ
σδ2 +
∂ E 2 2
∂ I
σI2,
where function I = 0z 1 +δz (−δ )d z. Finally, uncertainty of
the EoS can be calculated via
σw2 =
∂w
∂ E 2
2
2
σE2 +
∂w
∂ E 2(z)
2
2
σE2(z) .
(19)
(20)
(18)
In fact, it only influences the uncertainty of reconstruction. It
does not impact the mean values of reconstruction of w. From
the following w reconstruction, we find that it is reasonable
to do this manipulation.
We utilize the same prior of H0 as the supernova data.
Different from previous most work, we do not use the H (z)
data alone. We combine them with the supernova data as
a derivative of distance D, using the relation D = E1(z) .
To test the influence of Hubble constant, we respectively
consider two priors on the w reconstruction, namely, H0 =
73.24 ± 1.74 km s−1Mpc−1 with 2.4% uncertainty [
68
] and
To map the w(z) of dark energy, we can reconstruct it from
the Eq. (14) for background data, and Eq. (15) for the
perturbation data.
We report corresponding GP reconstruction in this
section. In order to test the influence of some parameters on the
reconstruction, we consider the effect of dark matter
density parameter m0 and Hubble constant H0 for the
background data, and effect of initial value δ(z = 0), σ8(z = 0)
and parameter m0 for the RSD data. For the matter density
parameter m0, we consider it in three cases: m0 = 0.25,
0.279 ± 0.025 from WMAP-9 [
70
] and 0.308 ± 0.012 from
Planck 2015 [
71
]. For the initial value δ(z = 0), we consider
it as δ(z = 0) = 1 and 0.7837 in fiducial cosmology from
Planck 2015 [
71
]. For the parameter σ8(z = 0), we consider
it as σ8(z = 0) = 0.821 ± 0.023 from WMAP-9 [
70
] and
0.8149 ± 0.0093 from Planck 2015 [
71
].
4.1 Test on the Hubble parameter
Before the reconstruction of w, we can perform a test on
the Hubble parameter, in order to test whether they present a
consistent background information.
In Fig. 1, we plot the Hubble parameter
reconstruction from the background data and perturbation data. From
the upper panel, we find that Hubble parameter from the
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Year
Union2.1 + H (z) data at low redshift is consistent with that
from the JLA + H (z). For high redshift, they present a
different Hubble parameter, which indicates that they may give
a slightly different w reconstruction.
For the perturbation data, we find that they give a quite
different E 2(z) for different parameter m0. On the one hand,
the perturbation data present a smaller Hubble parameter E 2
at high redshift, when compared with the background data.
Therefore, w reconstruction from the perturbation data may
be different from the background data. On the other hand, we
note that the square of Hubble parameter E 2(z) for different
parameter m0 is also different from each other. Especially,
Hubble parameter E 2(z) for m0 = 0.308 ± 0.012 at high
redshift is negative, which indicates that the parameter m0
should not be too bigger. Thus, we deem that the perturbation
data are sensitive to the parameter m0. They should be able
to provide a tighter constraint on parameter m0. Of course,
it may be also highly influenced by the parameter m0.
In short, we find a tension between the Hubble parameter
from background data and perturbation data. And this tension
may influence the dark energy reconstruction, leading to a
different w.
4.2 Reconstruction from the Union2.1 and H (z) data
We show the GP reconstructions for combination data in Figs.
2 and 3. The dashed lines and shaded region correspond to
the mean values and errors of reconstructions, respectively.
We find that the reconstructions are consistent with the
observational data, as shown in the first two panels of Fig.
2. We also note that D and D in this figure are quite
different from the previous work using the supernova data
alone. For the supernova data alone, their derivatives D and
D change smoothly and softly. In contrast, the derivatives
change acutely in this paper. This is because the input H (z)
data as a prior of the derivative D change the covariance
function k(z, z˜) at different points. Thus, they present a
different GP reconstruction.
To test the impact of matter density parameter m0, we
plot the w(z) reconstruction for different m0 in Fig. 3.
Comparison shows that the parameter m0 produces slight
influence on the reconstruction. First, they present a similar
estimation of dark energy w(z) over the redshift. Current EoS of
dark energy at all cases is w0 < −1 at 68% C.L., as shown
in Table 2. Second, the cosmological constant model with
w = −1 at 95% C.L. cannot be ruled out by the background
observational data, especially for high redshift. Third, mean
values of reconstruction hint a dynamical dark energy, but
they cannot rule out the non-evolution model at 95% C.L.
Last, uncertainty of reconstruction at redshift z > 0.5 has
becomes very large. It indicates that a precise evaluation on
w(z) at high redshift is still a luxury from current background
data.
4.3 Reconstruction from the JLA and H (z) data
We test the influence of parameter m0 and H0 on the w
reconstruction from JLA and H (z) data.
4.3.1 Effect of the parameter
m0
We show the test of parameter m0 on w reconstruction from
JLA and H (z) data in Fig. 4. From the comparison with Fig.
3, we find that this combination presents a similar w
reconstruction as the Union2.1 and H (z) data. For low redshift,
EoS w < −1 can be highlighted, which can be seen from
the w0 in Table 2. We also note that errors of w from the
JLA combination is smaller. This is because the JLA data
have more samples with high precision, which can present a
tighter constraint. Same as the Union2.1 data combination,
the JLA combination also hint a dynamical dark energy.
However, because of its smaller errors, the cosmological constant
model is not so consistent with the reconstruction from JLA
and H (z) data.
4.3.2 Effect of the parameter H0
We present the test of influence of different Hubble constant
in Fig. 5. Firstly, we find that JLA data combination in this
case give a similar w as above reconstruction. A dynamical
w is also presented. Secondly, The Hubble constant has a
notable influence on the w, as shown in Table 2 on the
current EoS w0. Moreover, errors of w in the upper panel is
smaller, due to its smaller uncertainty of Hubble constant.
The cosmological constant model in the lower panel cannot
be distinguished from the reconstruction.
4.4 Reconstruction from the RSD data
Using the GP method, we obtain the current growth rate
f σ8(z = 0) = 0.3854 ± 0.0239 and f σ8(z = 0) =
0.1660 ± 0.0706, as shown in Fig. 6. We should emphasize
that this estimation about the current growth rate is
modelindependent. We also note that the growth rate was
decreasing. Within 95% C.L., the derivative f σ8 gradually decreases
to nagative value for redshift z 0.5.
For the perturbation δ and EoS w(z), they are dependent
of the initial values δ(z = 0), σ8(z = 0) and matter density
parameter m0. In the following text, we intend to test their
influences on the reconstruction.
4.4.1 Effect of the parameter δ(z = 0)
The initial value δ(z = 0) is unknown for us. Generally, it
is taken as the normalization value δ(z = 0) = 1 [
37,38
]
or a fiducial value. Using the GP approach, we plot the
related reconstructions in Fig. 7. Assuming the constant
σ8(z = 0) = 0.8149, we reconstruct the perturbation δ and
its derivatives. We find that δ decreases with the
increasing redshift. To higher redshift, it may decrease to zero.
For the δ , investigation about it was absent. From the GP
method, we obtain that δ (z = 0) = −0.4729 ± 0.0294 and
δ (z = 0) = 0.2691 ± 0.0915. The first derivative δ with
95% C.L. is negative, although it increases with the redshift.
To test the influence of different initial values δ(z = 0),
we reconstruct the dark energy w at two cases, normalization
value and fiducial value δ(z = 0) = 0.7837, under the same
assumption of matter density parameter m0 = 0.308. We
plot them in Fig. 8. From the comparison, we find that they
both present a dynamical w, almost without any difference.
This reconstruction is much different from the background
data. From the list in Table 2, we find that the current EoS of
dark energy w0 in two cases are the same. It indicates that the
initial values δ(z = 0) has no influence on the reconstruction.
Therefore, it is safe for us to take the initial value δ(z = 0) =
1 in future calculation.
4.4.2 Effect of the parameter σ8(z = 0)
To test the influence of σ8(z = 0), we respectively consider
its value in the WMAP-9 prior and Planck 2015 prior. The
initial value and density parameter are fixed to be δ(z = 0) = 1
and m0 = 0.308. We show the reconstruction in Fig. 9.
From the comparison, firstly, we find that the RSD data
in these cases also present a dynamical w, similar as the
constraint in Fig. 8. Moreover, the cosmological constant
model can be demonstratively distinguished from the
reconstruction. Secondly, we find that current EoS of dark energy
are w0 = −0.8653 ± 0.2496 for the WMAP-9 prior and
w0 = −0.8980 ± 0.2477 for the Planck 2015 prior,
respectively. It indicates that the initial value σ8(z = 0) can
influence the w reconstruction slightly, which can be evidenced
in Table 2.
4.4.3 Effect of the parameter
m0
For the background data, we consider three types of matter
density parameter m0 on the reconstruction. It implies that
the parameter m0 does not produce drastic effect on the w.
In this subsection, we also test this effect for different
parameter m0. By fixing the initial value δ(z = 0) = 1 and
σ8(z = 0) = 0.8149, we perform this test, and make a series
of comparisons in Fig. 10. Firstly, we find that effect of matter
density parameter m0 is very significant. Specifically, in the
first panel, RSD data for m0 = 0.25 present a very
different w, when compared with two other reconstructions. In this
case, the reconstruction favors dark energy with a constant
w. Moreover, the cosmological constant model cannot be
ruled out. However, we also note that this w is different from
the one obtained by the background data, even though the
latter also cannot exclude the cosmological constant model.
We think that the reason can be understood from the Hubble
parameter in Fig. 1. From the comparison in that figure, we
find that the RSD data present a quite different Hubble
parameter at three different parameter m0, which implies that RSD
data are sensitive to the parameter m0. From the definition
of w in Eq. (15), we note that it inevitably depends on the
Hubble parameter. Therefore, it should not difficult to
understand why the w reconstruction for different parameter m0
are different. We also examine the definition of w in Eqs. (14)
and (15). We find that denominator of Eq. (15) crosses the
zero, when the parameter m0 0.30. While for the
background data, it happens only when parameter m0 0.45.
Therefore, a big steep slop in w from perturbation data is
presented. Secondly, the w in two other cases is also suggested
to be a dynamical one. This is same as the reconstructions in
above several scenarios. The cosmological constant model is
highly discordant with these model-independent
reconstructions.
In short, the RSD data are strongly influenced by the matter
density parameter m0, and can present a dynamical w
reconstruction, which is different from the reconstruction using
background data.
5 Conclusion and discussion
In this paper, we carry out a model-independent
reconstruction on the dark energy w, using the Gaussian processes
approach. The observational data we use are supernova data,
H (z) parameter and growth rate data.
Different from previous work, the background data we
use are the combination of Union2.1 supernova data, JLA
data and H (z) data. The H (z) parameter data here are used
together with the supernova data, and as the derivative of
distance D. Their inclusion can provide more information
about the cosmic evolution. Moreover, we find that inclusion
of them can significantly change the cosmic evolution, as the
reconstruction in Fig. 2.
Using the combination of supernova and H (z) data, we
find that the background data present a hint of dynamical w.
However, it cannot exclude the CDM model within 95%
C.L.. Recently, Zhao et al. [
72
] investigated the dark energy
using the latest data including CMB temperature and
polarization anisotropy spectra, supernova, BAO from the
clustering of galaxies and from the Lyman-α forest, Hubble constant
and H (z). They found that the dynamical dark energy can
relieve the Hubble constant tension and is preferred at a 3.5σ
significance level. Moreover, the upcoming dark energy
survey DESI++ would be able to provide a decisive Bayesian
evidence. Comparing with their reconstruction, this work
presents a similar evolution on the w(z), which is consistent
with their determination. At the same time, we test the effect
of matter density parameter m0 and Hubble constant. The
comparisons indicate that parameter m0 has a slight
influence on the w reconstruction from background data.
However, the Hubble constant presents a notable influence on the
reconstruction.
Another work we have done is to investigate the dark
energy using growth rate data at a perturbation level. We
obtain f σ8(z = 0) = 0.3854 ± 0.0239 and f σ8(z = 0) =
0.1660 ± 0.0706, which are model-independent, as shown
in Fig. 6. For the perturbation δ, its objective estimation was
absent. From the GP reconstruction, we obtain its derivatives
δ (z = 0) = −0.4729 ± 0.0294 and δ (z = 0) = 0.2691 ±
0.0915, assuming the initial value σ8(z = 0) = 0.8149.
We also test the effect of three parameters on the w
reconstruction from perturbation data. We find that the initial value
δ(z = 0) has no effect on the w reconstruction. It is safe for
us to take the normalization value δ(z = 0) = 1. While for
the parameter σ8(z = 0), it presents a slight influence on
the reconstruction. However, importantly, the matter density
parameter m0 has a notable influence on the w
reconstruction. Similar as the background data, the growth rate data
also provide a dynamical dark energy w. However, an
obvious difference between them is that the perturbation data have
a more promising potential to distinguish the CDM model.
One improvement of our work concerns the GP method, a
model-independent approach, allowing us to break the
limitation of specific model. Another potential difference of our
work is the extension of data types. In previous analysis,
luminosity distance and H (z) data were often used alone. In
this work, we not only use them together, but also compare
their reconstruction with the perturbation data.
In addition, our consideration on the effect of some
parameters also presents a full complement to previous study on this
subject. Especially, we find that the matter density parameter
m0 has a notable influence on the perturbation data, but no
important influence on the background data.
Acknowledgements We thank the anonymous referee whose
suggestions greatly helped us improve this paper. H. Li is supported by the
Youth Innovation Promotion Association Project of CAS. M.-J. Zhang
is funded by China Postdoctoral Science Foundation under grant No.
2015M581173. The research is also supported in part by NSFC under
Grant Nos. 11653001, Pilot B Project of CAS (No. XDB23020000) and
Sino US Cooperation Project of Ministry of Science and Technology
(No. 2016YFE0104700).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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