Eur. Phys. J. C (2016) 76 :588 Page 3 of 15 588
more favored by data. Among many models, one can choose
the model with minimal value of AIC as a reference model.
Roughly speaking, the models with 0 <AIC < 2have
substantial support, the models with 4 <AIC < 7have
considerably less support, and the models with AIC > 10
have essentially no support, with respect to the reference
model.
The BIC [58], also known as the Schwarz information
criterion, is given by
BIC =−2lnL
max
+ k ln N, (5)
where N is the number of data points used in the fit. The
same as AIC, the relative value between different models
can be written as BIC = χ
2
min
+ k ln N . A difference
in BIC of 2 is considerable positive evidence against the
model with higher BIC, while a BIC of 6 is considered to
be strong evidence. The model comparison needs to choose a
well justified single model, so in our work, the same as Refs.
[59–61], we use the CDM model to play this role. Thus,
the values of AIC and BIC are measured with respect to
the CDM model.
The AIC only considers the factor of parameter number
but does not consider the factor of data point number. Thus,
once the data point number is large, the result would be in
favor of the model with more parameters. In order to further
penalize models with more parameters, the BIC also takes the
number of data points into account. Considering both AIC
and BIC could provide us with more reasonable perspective
to the model comparison.
3 The observational data
We use the combination of current various observational data
to constrain the dark energy models chosen in this paper.
Using the fitting results, we make a comparison for these dark
energy models and select the good ones among the models.
In this section, we describe the cosmological observations
used in this paper. Since the smooth dark energy affects the
growth of structure only through the expansion history of the
universe, different smooth dark energy models yield almost
the same growth history of structure. Thus, in this paper, we
only consider the observational data of expansion history, i.e.,
those describing the distance-redshift relations. Specifically,
we use the JLA SN data, the Planck CMB distance prior data,
the BAO data, and the H
0
measurement.
3.1 The SN data
We use the JLA compilation of type Ia supernovae [62]. The
JLA compilation is from a joint analysis of type Ia super-
nova observations in the redshift range of z ∈[0.01, 1.30].
It consists of 740 Ia supernovae, which collects several low-
redshift samples, obtained from three seasons from SDSS-II,
three years from SNLS, and a few high-redshift samples from
the HST. According to the observational point of view, we
can get the distance modulus of a SN Ia from its light curve
through the empirical linear relation [62],
ˆμ = m
∗
B
− (M
B
− α × X
1
+ β ×C), (6)
where m
∗
B
is the observed peak magnitude in the rest frame
B band, M
B
is the absolute magnitude which depends on the
host galaxy properties complexly, X
1
is the time stretching
of the light curve, and C is the supernova color at maximum
brightness. For the JLA sample, the luminosity distance d
L
of a supernova can be given by
d
L
(z
hel
, z
cmb
) =
1 + z
hel
H
0
z
cmb
0
dz
E(z
)
, (7)
where z
cmb
and z
hel
denote the CMB frame and heliocen-
tric redshifts, respectively, H
0
= 100h km s
−1
Mpc
−1
is the
Hubble constant, E(z) = H (z)/H
0
is given by a specific cos-
mological model. The χ
2
function for JLA SN observation
is written as
χ
2
SN
= ( ˆμ − μ
th
)
†
C
−1
SN
( ˆμ − μ
th
), (8)
where C
SN
is the covariance matrix of the JLA SN observa-
tion and μ
th
denotes the theoretical distance modulus,
μ
th
= 5log
10
d
L
10pc
. (9)
3.2 The CMB data
The CMB data alone cannot constrain dark energy well,
because the main effects constraining dark energy in the
CMB anisotropy spectrum come from a angular diameter
distance to the decoupling epoch z 1100 and the late inte-
grated Sachs–Wolfe (ISW) effect. The late ISW effect cannot
be accurately measured currently, and so the only important
information for constraining dark energy in the CMB data
actually comes from the angular diameter distance to the
last scattering surface, which is important because it pro-
vides a unique high-redshift (z 1100) measurement in the
multiple-redshift joint constraint. In this work, we focus on
the smooth dark energy models, in which dark energy mainly
affects the expansion history of the universe. Thus, for an
economical reason, we do not use the full data of the CMB
anisotropies, but decide to use the compressed information
of CMB, i.e., the CMB distance priors.
We use the “Planck distance priors” from the Planck 2015
data [63]. The distance priors contain the shift parameter R,
the “acoustic scale”
A
, and the baryon density ω
b
≡
b
h
2
,
123
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