# Oscillating dark energy in light of the latest observations and it’s impact
on the Hubble tension

Mehdi Rezaei Iran Meteorological Organization
Hamedan Research Center for Applied Meteorology
Hamedan, Iran

###### Abstract

In this paper we have performed a comparative study of different types of
oscillating dark energy models using the Metropolis algorithm of MCMC. Eight
different oscillating parameterizations being examined herein which have
demonstrated considerable ability to fit the overall cosmological observations
including Pantheon sample of SnIa, Baryon Acoustic Oscillations, Cosmic
Chronometer Hubble data and distance priors of Planck CMB. In order to compare
the consistency of these models with observations, we have used both Akaike
and Deviation information criteria. Although, the values of Akaike Information
criteria for different models indicate that there is no support for
oscillating DE models, Deviation Information criteria showed that there’s
significant support for some of these models. Our results showed that these
models are capable of solving cosmic coincidence problem and alleviating the
Hubble tension. Comparing the $H_{0}$ values obtained for different
oscillating scenarios, with that of $\Lambda$CDM, we observed that our
oscillating models led to $\bar{H_{0}}=69.78$ which is $0.29$ greater than
$H_{0,\Lambda}$ and thus reduce the Hubble tension. Among all of models,
Model(1), with $H_{0}=70.00\pm 0.71$ is the most capable of alleviating the
$H_{0}$ tension. Forthermore, we examined our models assuming $H_{0}=73.0\pm
1.4$ from SHoES measurements. We find that adding this data point to our data
combination, led to a $\Delta\bar{H_{0}}=0.95$ increase in $H_{0}$ value for
different models.

Cosmology (343) — Cosmological models (337) — Cosmological parameters (339) —
Dark energy (351) — Hubble constant (758)

##  1 Introduction

Recent observations from the SHoES collaboration (Riess et al., 2022) indicate
that there is more than a $4-\sigma$ tension in the measurements of the Hubble
parameter from Planck Cosmic Microwave Background (CMB) data and direct local
distance ladder measurements. We have $H_{0}=67.27\pm 0.60$ km/s/Mpc from the
$\Lambda$CDM based Planck measurements, while the results of SH0ES, using 70
long-period Cepheids in the Large Magellanic Cloud, give $H_{0}=74.03\pm 1.42$
km/s/Mpc (Aghanim et al., 2020a, b; Riess et al., 2019). Another reanalysis of
the Planck data using the Pearson correlation coefficient of T and E modes
$R_{l}^{TE}$ leads to a new determination of Hubble constant as $H_{0}=67.5\pm
1.3$ km/s/Mpc (La Posta et al., 2021)

Besides these measurements, other independent measurements were done by the
South Pole Telescope (SPT-3G) and Atacama Cosmology Telescope (ACT), two
ground-based CMB experiments, have obtained $H_{0}=68.8\pm 1.5$ and
$H_{0}=68.4\pm 1.5$ respectively (Dutcher et al., 2021; Aiola et al., 2020).
Furthermore, the other direct probes at lower redshift ranges have inferred
lower values of $H_{0}$ which are in agreement with those predicted by
$\Lambda$CDM based CMB data. One of these probes is based on a calibration of
the Tip of the Red Giant Branch applied to SNIa (Freedman et al., 2019) which
found $H_{0}=69.8\pm 0.8$ and the other is the latest strong lensing time
delay data which found $H_{0}=67.4^{+4.1}_{-3.2}$ was reported in Birrer et
al. (2020). On the other hand, the HoLiCOW collaboration, using strong
gravitational lensing time delays, inferred $73.3^{+1.7}_{-1.8}$ km/s/Mpc as
the value of the Hubble constant (Wong et al., 2020). Additionally, a
measurement using the distance ladder method with SNIa and Mira variable stars
found $H_{0}=73.3\pm 4.0$ km/s/Mpc (Huang et al., 2020). Given the importance
of the $H_{0}$ tension, it is crucial, to investigate other cosmological
models addressing this tension. These investigations are looking for the
cosmologies that increase the $H_{0}$ value inferred from CMB data to resolve
the tension(Solà et al., 2017; Solà Peracaula et al., 2019, 2021; Sabla &
Caldwell, 2021; Freese & Winkler, 2021; Bargiacchi et al., 2023; Briffa et
al., 2023). These attempts may be helpful in solving or alleviating the
mentioned problem, as well as other problems that suffer the concordance
model, namely $\sigma_{8}$ tension and the fine-tuning and cosmic coincidence
problems. To this end, different cosmological models have been proposed. Many
of these models have been studied in the literature including dynamical DE,
running vacuum DE, different parameterizations for the equation of state (EoS)
of DE, Holographic DE, Agegraphic DE, etc (Perico et al., 2013; Rezaei et al.,
2017, 2020, 2021; Rezaei & Sola Peracaula, 2022; Rezaei, 2019a; Malekjani et
al., 2018; Rezaei & Malekjani, 2017; Moreno-Pulido & Sola Peracaula, 2022;
Moreno-Pulido et al., 2023). One of the choices for solving the problems of
$\Lambda$CDM, especially the coincidence problem is considering
nonmonotonicity in the EoS of dark energy, same as an oscillating form for EoS
of dark energy (Linder, 2006; Jain et al., 2007; Pace et al., 2012; Schmidt,
2017; Tamayo & Vazquez, 2019). Assuming an oscillating behavior for $w_{de}$,
the accelerated expansion phase of the universe can be considered as one of
the many such phases occurring over cosmic history (Rezaei, 2019b; Smith et
al., 2020; Xu et al., 2019). This property of oscillating DE helps us to
overcome the coincidence problem. Adding, results of (Liu et al., 2023) using
the latest Pantheon+ type Ia supernova sample and the Hubble parameter
measurements indicate that a slightly oscillating EoS of dark energy around
the $w_{de}=-1$ is favored. These motivate us to have a new look on
oscillating DE models in the light of the latest observations and from a new
point of view; those impact on the Hubble tension. An oscillating Eos
parameter was studied in Feng et al. (2006) in which the EoS gets across $-1$.
Their results show that this form of dark energy, dubbed oscillating quintom,
can unify the early inflation and current acceleration of the universe,
leading to the oscillations of the Hubble constant and a recurring universe.
Some oscillating scalar fields have been proposed in Dutta & Scherrer (2008)
as DE models. Authors of this work examine proposed models, with particular
emphasis on the evolution of the ratio of the oscillation frequency to the
expansion rate, and find that a subset of all three different classes of
proposed oscillating models can be consistent with the observational data. The
possibility of detecting oscillating patterns in the EoS of the dark energy
has been studied in (Lazkoz et al., 2011). Their results show that data cannot
discriminate between a cosmological constant and an oscillating equation of
state. A new dark energy model has been proposed in Tián (2020) for solving
the cosmological fine-tuning and coincidence problems. The default assumption
of this model is that the fine-tuning problem disappears if one does not
interpret DE as vacuum energy and the coincidence problem will be solved if
the Universe has several acceleration phases across the whole cosmic history.
They study a quintessence model with approximately repeated double exponential
potential, which only introduces one Planck scale parameter and three
dimensionless parameters of order unity. Their results show that the
oscillating model is able to explain the observed cosmic late-time
acceleration. Authors of Tamayo & Vázquez (2019), provide a general
description for $w(z)$ in the form of a Fourier series. Their description
generalizes some oscillating DE models and is in agreement with the $w(z)$
reconstructions. Using Markov chain Monte Carlo algorithm and different
observational data they have compared the proposed model with $\Lambda$CDM,
$w$CDM, and the standard Taylor approximation. They find that, even though
there are extra parameters, there is a slight preference for the Fourier
series compared with the $\Lambda$CDM model. This first analysis considering
$w(z)$ as a Fourier series has its success in having a natural oscillatory
behavior and being compatible with model-independent reconstructions(Tamayo &
Vázquez, 2019).

In Tian (2020), authors revisit the cosmological dynamics of an oscillating
dark energy model and find that this model allows the Universe to evolve as an
oscillating scaling solution (OSS) in the radiation era and as a chaotic
accelerating solution (CAS) in the matter era. Mathematically, the transition
from OSS to CAS is a route of period-doubling bifurcation to chaos. Upon their
results, the global cosmological parameter constraints are practicable if the
Universe evolves as an OSS in the radiation era and the late-time Universe
described by a CAS can successfully explain the observed cosmic acceleration
at low redshifts. The above oscillating DE model has been investigated in
other studies and led to good results in predicting the tensor-to-scalar ratio
$r$ and also as a new solution to the possible deuterium problem(Tian & Zhu,
2021; Tian, 2022).

Following the above studies in this work, we wish to focus on Oscillating dark
energy models as a proposal for solving the problems of $\Lambda$ cosmology,
especially the $H_{0}$ tension. To this end, we have selected eight different
oscillating DE parametrizations that have been proposed in the literature.
Applying the MCMC algorithm we confront these DE models with the latest
observations from different surveys. Adding to a comparative study between
different models, we have examined the ability of this type of DE models to
solve the $H_{0}$ tension.

The paper is organized as follows: In Sec.2, we have reviewed eight different
oscillating DE models as well as data sets we have used in our analysis. Sec.3
gives the results of our analysis for different models under study and related
discussions. Finally, we conclude our study in the Sec.4

##  2 DE Models and Observations

In the first part of this section we describe different oscillating DE
parameterizations. In the second part we introduce different data samples that
we have used in our analysis.

###  2.1 Oscillating DE parameterizations

Considering the universe including non-relativistic matter and dark energy in
a spatially flat geometry, we have the Hubble parameter as follows:

$H^{2}=(\dot{a}/a)^{2}=\frac{8\pi
G}{3}(\rho_{m}+\rho_{de})=\Omega_{m0}a^{-3}+\Omega_{de0}f(a).$ (1)

where $\rho_{i}$ is the energy density of fluid $i$, $\Omega_{i}=\frac{8\pi
G\rho_{i}}{3H^{2}}$ is the density parameter for fluid $i$ and $f(a)$ has the
following form:

$f(a)=\exp{(-3\int_{1}^{a}{\frac{1+w(a^{\prime})}{a^{\prime}}da^{\prime}})}.$
(2)

Determining EoS $w_{d}(a)$ for each of the DE scenarios and using the above
equations, we can obtain the evolution of the Hubble parameter in the whole of
cosmic history. Oscillating DE parametrizations which we have studied in this
work are as follows:

  * •

Model(1): We have taken the second parametrization from (Linder, 2006) in
which the authors examine the periodicity of the cosmic expansion in units of
the e-folding scale $\ln{a}$ and select the following form for $w_{de}$:

$w_{de}(a)=w_{0}-A\sin{(B\ln{a})}.$ (3)

where $w_{0}$ is the present value of $w_{de}$ and $A$ and $B$ are the
amplitude and frequency of oscillation respectively.

  * •

Model(2): We select the third parametrization from (Ma & Zhang, 2011). This
parameterization is a generalization of the CPL parametrization (Chevallier &
Polarski, 2001) in order to avoid the future unphysical divergence of this
model.

$w_{de}(a)=w_{0}-A(aB\sin{(1/a)}).$ (4)

where $w_{0},A$ and $B$ are the present value of $w_{de}(a)$, amplitude and
frequency of oscillation respectively.

  * •

Model(3): As the first parametrization in this work we consider a simple
oscillating DE parametrization which was proposed by (Feng et al., 2006) with
an EoS given by:

$w_{de}(a)=w_{0}+w_{1}\sin{(\frac{1}{a}-1)}.$ (5)

The model has two free parameters $w_{0}$ and $w_{1}$, in which $w_{0}$ is the
current value of EoS. At $a\rightarrow 1$ this model reduces to a linear
parameterization while at $a<<1$ the $w_{de}$ oscillates and will differ
substantially from the model with linear parameterization.

  * •

Model(4): As the fourth parametrization in our study, we have investigated the
following parametrization from (Kurek et al., 2010):

$w_{de}(a)=-1+A\sin{(B\ln{(1/a)})}.$ (6)

The above parametrization in which oscillating behavior could not be detected
nowadays, has been compared with different linear and oscillating DE
parametrizations in (Kurek et al., 2010) and obtained the best results between
those DE scenarios.

  * •

Model(5): The other parametrization we have studied in this work, is an
oscillating ansatz for $H(z)$ which is studied in (Nesseris &
Perivolaropoulos, 2004). Comparing this oscillating DE scenario with
observations leads to the best results between 14 different DE models. In this
scenario, $H(z)$ has the following form:

$H^{2}(a)=H_{0}^{2}[\Omega_{m0}a^{-3}+a_{1}\cos{(a_{2}(1/a-1)^{2}+a_{3})}+F_{0}],$
(7)

where $F_{0}=1-a_{1}\cos{a_{3}}-\Omega_{m0}$. This parametrization has
$a_{1},a_{2}$, and $a_{3}$ as the model parameters and those values will
determine the behavior of the Hubble function at different epochs.

  * •

Model(6): Same as Model (4), another parametrization has been investigated in
(Kurek et al., 2010) as a purely oscillating dark energy with the following
form:

$w_{de}(a)=w_{0}\cos{(\alpha\ln{(1/a)})}.$ (8)

In this parametrization, we have two model parameters $w_{0}$ and $\alpha$
where the first one determines the value of EoS at the present time,
$w_{de}(a=1)=w_{0}$.

  * •

Model(7): Another oscillating parametrization which investigated in (Feng et
al., 2006; Pan et al., 2018), has two model parameters as following:

$w_{de}(a)=w_{0}+\alpha(1-\cos{(\ln{1/a})}).$ (9)

where $w_{0}$ is the current value of $w_{de}$ and $\alpha$ quantifies the
dynamical character of the model. For $\alpha=0$ we acquire $w_{da}=w_{0}$,
while $\alpha\neq 0$ corresponds to a deviation from $\Lambda$CDM.

  * •

Model(8): The last parametrization which proposed in (Pan et al., 2018) is:

$w_{de}(a)=w_{0}+\alpha(1-a)\cos{1/a}.$ (10)

where $w_{0}$,the present value of EoS and $\alpha$ are the model parameters.

Assuming each of the above parametrizations, we will study the ability of
oscillating DE scenarios to fit the latest observational data.

###  2.2 Observational data

In our analysis we have used the following data samples:

  * •

SnIa: We take the Pantheon sample of SnIa including 1048 data points within
the redshift range $0.01<z<2.26$ from Scolnic et al. (2018).

  * •

Baryon acoustic oscillations (BAO): Concerning this data sample we have used
the radial component of the anisotropic BAOs have found from the measurements
of the power spectrum and bispectrum from the BOSS Data Release 12 galaxy from
Gil-Marín et al. (2017), the complete SDSS III Ly$\alpha$-quasar from du Mas
des Bourboux et al. (2017) and the SDSS-IV extended BOSS DR14 quasar sample
from Gil-Marín et al. (2018).

  * •

Cosmic Chronometers Hubble data(CCH): This data sample was obtained from the
spectroscopic techniques applied to passively–evolving galaxies, i.e.,
galaxies with old stellar populations and low star formation rates. In this
work we have used CCH sample including 37 data points from the following
references: Jimenez et al. (2003); Simon et al. (2005); Stern et al. (2010);
Moresco et al. (2012); Zhang et al. (2014); Moresco (2015); Moresco et al.
(2016); Ratsimbazafy et al. (2017).

  * •

$H_{0}$ from SHoES measurements: In order to study the effects of SHoES
measurements on our analysis, we have used $H_{0}=73.0\pm 1.4$ from Riess et
al. (2021).

  * •

Cosmic microwave background(CMB): We have used the CMB distance prior from the
final Planck 2018 release instead of full CMB data. The distance priors that
are derived from the base $\Lambda$CDM model can be used to replace the global
fitting of full data released by Planck in 2018 for the other DE models (L.
Chen & Wang, 2019).

##  3 results and discussions

Here we present the results of our analysis in two different parts. In the
first part, we report our numerical results and compare different DE scenarios
under study in fitting observations. In the second part, we compare the
ability of oscillating DE models to solve cosmological problems, especially
the Hubble tension.

###  3.1 Oscillating DE and observations

In order to put constraints on the model parameter we have used the Metropolis
algorithm of MCMC (Metropolis et al., 2004). As the data set, we have used a
data combination including 1094 data points of SN+BAO+CCH+CMB. To search all
the parameter space and prevent the MCMC from selecting a local best fit, in
our algorithm we did not set any limitations on the priors. In each of the
analyses, we have 150000 chains which provides sufficient iterations in order
to obtain the best fit parameters. Traceplot corresponding to a Markov chain,
provides a visual way to inspect sampling behavior and convergence. It is one
intuitive and easily implemented diagnostic tool that plots the parameter
value against the iteration number. When the traceplot moves around the mode
of distribution, we find that our model has converged. It is easy to see in
Figs.2 & 3 that in our analysis we obtained traceplots with normal
distribution which means our models have converged. In order to examine the
effect of choosing initial values in MCMC, we have repeated our algorithm with
different initial values for the $\Lambda$CDM model. Our results indicate that
changing initial values may lead to a change in the best fits up to $1\%$.
Although choosing convenient initial values led to saving time in reaching the
convergence, it hasn’t significant effect on the numerical results. The best
value of model parameters and their 68% CL constraints are reported in Tab.1.
In this table, we only presented those parameters which are common between all
models. In different plots of Fig.2, we showed One-dimensional marginalized
posterior distributions of $\Omega_{m0}$ and $h$, and the two-dimensional
joint contours between these model parameters ($\Omega_{m0}$ and $h$) at
$68\%,95\%$ and $99.7\%$ CL of DE models. There are several methods for
selecting the best model among different probable models. The simplest way is
to compare the $\chi^{2}$ values obtained from MCMC chains. In this way, the
model with the smallest value of $\chi^{2}$ is selected as the best one. But
in this manner, we did not assume the extra model parameters. In order to
assume the effect of extra model parameters, different information criteria
have been proposed in literature; Akaike Information Criteria (AIC) (Akaike,
1974), Corrected Akaike Information Criteria ($AIC_{c}$) (Cavanaugh, 1997),
Bayes Information Criteria (BIC) (Schwarz, 1978), Deviance Information
Criteria (DIC) (Spiegelhalter et al., 2002), etc [see detailed discussions in
Refs. Rezaei & Malekjani (2021)]. Among these criteria in this work, we have
used the most popular information criterion AIC and DIC which is understood as
a Bayesian version of AIC. Like AIC, it trades off a measure of model adequacy
against a measure of complexity and is concerned with how hypothetically
replicated data predict the observed data (Li et al., 2020). DIC is an
asymptotically unbiased estimator of the expected Kullback-Leibler divergence
between the data-generating process and the plug-in predictive distribution.

The AIC is defined as:

$\displaystyle{\rm AIC}=\chi^{2}_{\rm min}+2K\;.$ (11)

in which $K$ is the number of model parameters. Assuming this information
criterion, the preferred model is the one that has the minimum value of AIC.
The other criterion, DIC is defined as (Liddle, 2007)

$\displaystyle{\rm DIC}=D(\bar{{\bf p}})+2C_{\rm B}\;.$ (12)

where $C_{\rm B}=\overline{{D({\bf p})}}-D(\bar{{\bf p}})$ is the Bayesian
complexity and over-lines imply the standard mean value. The Bayesian
deviation, $D({\bf p})$ is expressed as $D({\bf p})=\chi^{2}_{\rm t}({\bf p})$
in the case of an exponential class of distributions (Rezaei & Malekjani,
2021). We have reported the AIC and DIC values for all of the models under
study. In order to have a comparison between different models, we calculated
$\Delta$AIC ($=$ AIC (model) $-$ AIC ($\Lambda$CDM)) and $\Delta$DIC ($=$ DIC
(model) $-$ DIC ($\Lambda$CDM)). These resulting differences are used to
determine the “level of support for” each model with respect to the
concordance model [see more details in (Rezaei et al., 2019)].

Table 1: Summary of the 68% CL constraints on various free parameters of the
Oscillating DE scenarios and $\Lambda$CDM for the combined dataset
SN+BAO+CCH+CMB. We note that all of $\Delta$ICs were calculated with respect
to $IC_{\Lambda}$.  SN+BAO+CCH+CMB
---
Model | $\Omega_{m,0}$ | $h$ |  | $\chi^{2}_{min}$ | $AIC$ | $DIC$ | $\Delta AIC$ | $\Delta DIC$
Model(1) | $0.2792\pm 0.0086$ | $0.7000\pm 0.0071$ |  | $1062.2$ | $1074.2$ | $1075.0$ | $6.8$ | $2.0$
Model(2) | $0.2808\pm 0.0086$ | $0.6947\pm 0.0077$ |  | $1063.6$ | $1075.6$ | $1075.4$ | $8.2$ | $2.4$
Model(3) | $0.2792^{+0.0083}_{-0.0095}$ | $0.6980^{+0.0086}_{-0.0077}$ |  | $1063.6$ | $1073.6$ | $1078.8$ | $6.2$ | $5.8$
Model(4) | $0.2819\pm 0.0073$ | $0.6998\pm 0.0071$ |  | $1063.8$ | $1073.8$ | $1074.2$ | $6.4$ | $1.2$
Model(5) | $0.2820\pm 0.0083$ | $0.6979^{+0.0074}_{-0.0067}$ |  | $1064.1$ | $1076.1$ | $1072.3$ | $8.7$ | $-0.7$
Model(6) | $0.2797\pm 0.0080$ | $0.6987^{+0.0081}_{-0.0067}$ |  | $1064.4$ | $1074.4$ | $1074.4$ | $7.0$ | $1.4$
Model(7) | $0.2765\pm 0.0086$ | $0.6942\pm 0.0082$ |  | $1062.4$ | $1072.4$ | $1076.0$ | $5.0$ | $3.0$
Model(8) | $0.2801\pm 0.0078$ | $0.6991\pm 0.0074$ |  | $1064.1$ | $1074.1$ | $1075.3$ | $6.7$ | $2.3$
$\Lambda$CDM | $0.277\pm 0.0084$ | $0.6949\pm 0.0082$ |  | $1061.4$ | $1067.4$ | $1073.0$ | $0.0$ | $0.0$

As one can see in Tab.1, $\Lambda$CDM with $AIC=1067.4$ is the best model,
while for all of the oscillating models $\Delta$AIC$>5$. This means that from
AIC point of view, there is ”Considerably less support for” these models
compared with the concordance model. This result was predictable because our
oscillating DE models have $2-3$ more extra parameters, which led to an
increment in the AIC value. On the other hand upon DIC results, the
performance of oscillating models is good. In this case, Model (5) with
$\Delta$DIC$=-0.5$ is the best model followed by $\Lambda$CDM. Also, we have
found ”Significant support” for models (1), (4), and (6) because of
$\Delta$DIC$<2$. For models (2), (7) and (8) we have ”Less support” and
finally we have ”Considerably less support” for Model (3).

###  3.2 Oscillating DE and cosmological problems

The considerable fact that the energy density of DE and that of dark matter
are of the same order around the present time, is called the “Cosmological
Coincidence Problem” (CCP). As a simple result, one can conclude from this
fact, that we are living in a very special time of cosmic history. In
$\Lambda$ cosmology where the DE density $\Lambda$ is constant while the
density of dark matter scales with the $a^{-3}$, this appears to be a
coincidence (Velten et al., 2014). Because it requires extreme fine-tuning in
the initial conditions of the early Universe. This point becomes more
interesting knowing that in the very early and in the far-future Universe,
both of the energy densities differ by many orders of magnitude. The $\Lambda$
cosmology offers no explanation for the peculiar circumstance that
$\Omega_{\Lambda}\approx\Omega_{m}$ at the present time. On the other hand,
oscillating DE scenarios offer a new response to the CCP. Upon these
scenarios, the current state of the Universe, with
$\Omega_{\Lambda}\approx\Omega_{m}$, has happened previously and will happen
again. Therefore, no fine-tuning is required, so the CCP will resolve in DE
scenarios with oscillating EoS parameter. In Fig.1 we showed the evolution of
$w_{de}$ for models under study. Although at present time $(a\approx 1)$,
$w_{de}$ for all of oscillating models, come close to $w_{\Lambda}=-1$, but at
smaller scale factor, oscillating model deviate from concordance models. This
deviation from $w_{\Lambda}=-1$ changes behavior of $\Omega_{de}$ in these
models which finally led to solving CCP. As we mentioned in section 1, the
observed tension in the value of the Hubble parameter obtained from Planck
measurements and that of Cepheid-calibrated SnIa distances, has reached a
statistical significance of $5-\sigma$. We note that the Planck measurement
was done upon the $\Lambda$ cosmology, while the SH0ES measurement does not.
This situation motivated the science community to develop new extensions of
the $\Lambda$ cosmology attempting to resolve or alleviate this tension by
increasing the $H_{0}$ value. As one of the main goals of this work, we have
examined the ability of oscillating DE models to resolve $H_{0}$ tension. In
this way, we selected $h(=\frac{H_{0}}{100})$ as one of the free parameters of
our models and then we put constraints on it using observations. In Fig.3 we
showed One-dimensional marginalized posterior distributions of the $h$ for
different DE models. It is easy to see that for all of the oscillating models
except Model (2) and Model (7), the best value of $h$ is greater than $0.6949$
obtained for $\Lambda$CDM. In order to have a quantifying view on the ability
of models to alleviate Hubble tension, we will compare the results of our
models with those of $\Lambda$ based Planck measurements and SHoES
measurements from Riess et al. (2021). While there is a 4.1-$\sigma$ tension
between the results of Planck and SHoES, we obtained less values of tensions
upon oscillating models. As one can see in Tab. 1, the best value of
$H_{0}(=100h)$ for all the oscillating models is greater than 67.27 which
obtained by Planck measurements. In Model (1), as the more capable model in
solving Hubble tension, we have $H_{0}=70.00\pm 0.71$ that indicate a
$2.14-\sigma$ tension with SHoES results. After Model (1), we have Model (4)
which decreases the Hubble tension to 2.16-$\sigma$ with respect to SHoES
result. Model (8), Model (6), Model (3), Model (5) and Model (2) stand in the
lower position in our ranking as the cosmological models which can alleviate
the Hubble tension. Finally, we have Model (7) as the worst model which
decreases the Hubble tension to 2.56-$\sigma$. From another point of view, we
can compare the ability of oscillating models in alleviating Hubble tension
with $\Lambda$CDM. As we observe in Tab.1, in our analysis we obtained
$H_{0}=69.49\pm 0.82$ for concordance model which shows a 2.51-$\sigma$
tension with respect to SHoES result. So, we can claim that oscillating DE
scenario in 6 cases lead to better results compared to concordance model. In
the other meaning, just for Model (2) and Model (7) we observed tensions
greater than 2.51-$\sigma$, while for other oscillating models Hubble tension
falls below 2.51-$\sigma$. Summarizing all of the above, we found that
oscillating DE models reduces the observed tension between $\Lambda$ based
Planck measurements and SHoES, from 4.1-$\sigma$ to less than 2.56-$\sigma$.
In comparison with $\Lambda$CDM, all of oscillating models except Model (2) &
(7), lead to smaller value of Hubble tension between 2.14 to 2.29-$\sigma$.
Therefore, Oscillating DE models can reduce the Hubble tension. In order to
have a comparison with SHoES measurements, we have plotted the $H_{0}$ values
obtained for studied models and their uncertainties, with those obtained from
the SHoES measurements, based on the Cepheid calibrations for the LMC, NGC
4258, and the Milky Way. From these measurements, we have $H_{0}=74.22\pm
1.82$ (Riess et al., 2019), $H_{0}=72.0\pm 1.9$ (Riess et al., 2019), and
$H_{0}=73.0\pm 1.4$ (Riess et al., 2021) for the LMC, NGC 4258, and the Milky
Way respectively. The results can be seen in the upper panel of Fig.4. We
observe that assuming obtained $H_{0}$ values and their $1-\sigma$
uncertainty, all of the oscillating DE models investigated in this work, can
reduce the $H_{0}$ tension. These results indicate that oscillating DE
scenario can be an alternative candidate for alleviating Hubble tension in
cosmology. In order to see the effect of including SHoES measurements in our
data sets, we added $H_{0}=73.0\pm 1.4$ from (Riess et al., 2021) to our data
sets. We reported the results of this part of the analysis in Tab.2. In this
table, we compared the best fit values of $h$ obtained using SN+BAO+CCH+CMB
and those obtained using SN+BAO+CCH+CMB+$H_{0}$. Our results indicate that
adding SHoE data to the data samples leads to an increase in the best fit of
parameter $h$. This increment varies between $\Delta h=0.005$ to $\Delta
h=0.0144$ for different models which means a $0.5$ to $1.44[km/s/Mpc]$
increase in $H_{0}$ value. The maximum change ($\Delta h$)occurred in Model
(7), while the minimum change occurred in Model (1). In the case of
$\Lambda$CDM we observed a $0.0109$ increase in the best value of $h$ after
adding $H_{0}$ from SHoE to our data. In the lower panel of Fig.4, we compared
our results for different DE models after assuming SHoES in our data sets.
Differences between the upper panel (data samples without SHoES) and the lower
panel (data samples with SHoES) show the effect of including $H_{0}$ in our
data sets clearly.

Table 2: Summary of the 68% CL constraints on parameters $h$ of the Oscillating DE scenarios and $\Lambda$CDM for two different combined datasets, SN+BAO+CCH+CMB (data sample I) and SN+BAO+CCH+CMB+$H_{0}$ (data sample II). Note that $\Delta h=h_{datasampleII}-h_{datasampleI}$.  Model | data sample I |  | data sample II | $\Delta h$
---|---|---|---|---
Model(1) | $0.7000\pm 0.0071$ |  | $0.7050^{+0.0076}_{-0.0068}$ | $0.0050$
Model(2) | $0.6947\pm 0.0077$ |  | $0.7085^{+0.0077}_{-0.0069}$ | $0.0138$
Model(3) | $0.6980^{+0.0086}_{-0.0077}$ |  | $0.7072^{+0.0081}_{-0.0078}$ | $0.0092$
Model(4) | $0.6998\pm 0.0071$ |  | $0.7071\pm 0.0072$ | $0.0073$
Model(5) | $0.6979^{+0.0074}_{-0.0067}$ |  | $0.7079\pm 0.0071$ | $0.0100$
Model(6) | $0.6987^{+0.0081}_{-0.0067}$ |  | $0.7057^{+0.0079}_{0.0064}$ | $0.0070$
Model(7) | $0.6942\pm 0.0082$ |  | $0.7086\pm 0.0065$ | $0.0144$
Model(8) | $0.6991\pm 0.0074$ |  | $0.7066^{+0.0086}_{0.0075}$ | $0.0075$
$\Lambda$CDM | $0.6949\pm 0.0082$ |  | $0.7058\pm 0.0065$ | $0.0109$
Figure 1: The evolution of EoS parameter $w_{de}$ obtained for different
Oscillating DE models as well as $\Lambda$CDM upon their best fit parameters.

Figure 2: Contour plots for the confidence levels of the model parameters
$\Omega_{m0}$ and $h$ up to $3\sigma$, obtained for different oscillating DE
scenarios investigated in this work. Figure 3: One-dimensional marginalized
posterior distributions of the $h$ obtained for different Oscillating DE
models as well as $\Lambda$CDM.

Figure 4: The values of $h(=H_{0}/100)$, obtained for $\Lambda$CDM and
Oscillating models with their $1-\sigma$ uncertainties. The results have
compared with those obtained from the SHoES measurements, based on the Cepheid
calibrations for the LMC, NGC 4258, and the Milky Way. The upper panel shows
the results which obtained using SN+BAO+CCH+CMB, while the lower panel shows
the results were obtained using SN+BAO+CCH+CMB+$H_{0}$.

##  4 Conclusions

Although the concordance model has been successful in describing cosmological
observations,but we can’t justify the mismatch between several early and late-
time inferences of the Hubble constant $H_{0}$ using $\Lambda$CDM, which
called Hubble tension. Depending on the dataset considered for local
measurements, the significance of this tension falls between $4-6\sigma$. In
order to solve this tension, as well as the other challenge against
$\Lambda$CDM, in this paper we focused on oscillating DE parameterizations.
Applying the Metropolis algorithm of MCMC and using the latest observations
include SnIa, BAO, CCH and CMB, we have constrained our DE models. To compare
the ability of models in fitting observations, we have used AIC as a simple
information criteria and DIC as a generalized form of AIC using the
Kullback–Leibler divergence, which has been developed based on Bayesian
statistic (Spiegelhalter et al., 2002; Mootoovaloo et al., 2020; Salas et al.,
2022). Although, DIC is really suitable for prediction, but it has also been
used for selecting the correct model. Upon AIC results, $\Lambda$CDM selected
as the best model while because of $\Delta$AIC$>5$ with respect to
$\Lambda$CDM, we could not observe any significant support for oscillating
dark energy models. These models have $2-3$ more extra parameters with respect
to concordance model which led to an increment in those AIC value. Upon DIC
results, we observed good performance for oscillating models. It is easy to
see in Tab.1 that Model (5) with $\Delta$DIC$=-0.7$ is the best model which
followed by $\Lambda$CDM. Moreover, we observed $\Delta$DIC$<2$ for models
(1), (4), and (6) which means ”Significant support” for these models. For
models (2), (7) and (8) we have ”Less support” and finally ”Considerably less
support” for Model (3). Upon these results we conclude that some of
oscillating DE models are consistent with data as well as concordance
$\Lambda$ cosmology. Furthermore, we examine the ability of oscillating models
in solving the problems against $\Lambda$CDM. In the case of CCP, as one can
see in Fig.1, because of oscillating behavior of these models at early time,
there’s no need to fine tunings and CCP will resolve. In the case of Hubble
tension, comparing with $\Lambda$CDM, our models can reduce the tension (see
Fig.4). Comparing with $\Lambda$ based Planck measurements, our oscillating
models led to a reduction of Hubble tension from 4.1-$\sigma$ to
2.14-2.56-$\sigma$ with respect to SHoES results. On the other hand, using
$\Lambda$CDM in our analysis we found a 2.51-$\sigma$ tension with SHoES
results which indicate that 6 models of our 8 models have better advantages
with respect to concordance model in alleviating Hubble tension. In order to
find the effect of adding SHoES measurements on the results of our analysis,
we repeated the analysis after adding $H_{0}=73.0\pm 1.4$ from SHoES (Riess et
al., 2021) to our datasets. Assuming this data point led to an increase in the
best value of parameter $h$ for all of the models. These increases were
relatively significant, averaging $\bar{\Delta h}=0.0095$ occurring mostly in
the Model(7), Model(2) and $\Lambda$CDM (see Tab.2 and the lower panel of
Fig.4). These results indicate that, independent of the DE models under study,
assuming a single data point from SHoES measurements can change the $H_{0}$
results significantly. we can not relate these extra changes in $H_{0}$ to the
ability of models in solving Hubble tension. This is in full agreement with
the results of (Rezaei et al., 2024). Summarizing the results, we can conclude
that oscillating DE scenario can be assumed as an alternative for $\Lambda$CDM
model. Although couldn’t fit the observations as well as $\Lambda$CDM, it can
solve the coincidence problem and also reduces the Hubble tension. These
abilities also can be investigated in future studies using other observations
at perturbation level, such as LSS data.

## References

  * Aghanim et al. (2020a) Aghanim, N., et al. 2020a, Astron. Astrophys., 641, A1, doi: 10.1051/0004-6361/201833880
  * Aghanim et al. (2020b) —. 2020b, Astron. Astrophys., 641, A6, doi: 10.1051/0004-6361/201833910
  * Aiola et al. (2020) Aiola, S., et al. 2020, JCAP, 12, 047, doi: 10.1088/1475-7516/2020/12/047
  * Akaike (1974) Akaike, H. 1974, IEEE Transactions on Automatic Control, 19, 716, doi: 10.1109/TAC.1974.1100705
  * Bargiacchi et al. (2023) Bargiacchi, G., Dainotti, M. G., Nagataki, S., & Capozziello, S. 2023, doi: 10.1093/mnras/stad763
  * Birrer et al. (2020) Birrer, S., et al. 2020, Astron. Astrophys., 643, A165, doi: 10.1051/0004-6361/202038861
  * Briffa et al. (2023) Briffa, R., Escamilla-Rivera, C., Levi Said, J., & Mifsud, J. 2023, Mon. Not. Roy. Astron. Soc., 522, 6024, doi: 10.1093/mnras/stad1384
  * Cavanaugh (1997) Cavanaugh, J. E. 1997, Statistics & Probability Letters, 33, 201
  * Chevallier & Polarski (2001) Chevallier, M., & Polarski, D. 2001, Int. J. Mod. Phys. D, 10, 213, doi: 10.1142/S0218271801000822
  * du Mas des Bourboux et al. (2017) du Mas des Bourboux, H., et al. 2017, Astron. Astrophys., 608, A130, doi: 10.1051/0004-6361/201731731
  * Dutcher et al. (2021) Dutcher, D., et al. 2021, Phys. Rev. D, 104, 022003, doi: 10.1103/PhysRevD.104.022003
  * Dutta & Scherrer (2008) Dutta, S., & Scherrer, R. J. 2008, Phys. Rev. D, 78, 083512, doi: 10.1103/PhysRevD.78.083512
  * Feng et al. (2006) Feng, B., Li, M., Piao, Y.-S., & Zhang, X. 2006, Phys. Lett. B, 634, 101, doi: 10.1016/j.physletb.2006.01.066
  * Freedman et al. (2019) Freedman, W. L., et al. 2019, Astrophys. J., 882, 34, doi: 10.3847/1538-4357/ab2f73
  * Freese & Winkler (2021) Freese, K., & Winkler, M. W. 2021, Phys. Rev. D, 104, 083533, doi: 10.1103/PhysRevD.104.083533
  * Gil-Marín et al. (2017) Gil-Marín, H., Percival, W. J., Verde, L., et al. 2017, Mon. Not. Roy. Astron. Soc., 465, 1757, doi: 10.1093/mnras/stw2679
  * Gil-Marín et al. (2018) Gil-Marín, H., et al. 2018, Mon. Not. Roy. Astron. Soc., 477, 1604, doi: 10.1093/mnras/sty453
  * Huang et al. (2020) Huang, C. D., Riess, A. G., Yuan, W., et al. 2020, The Astrophysical Journal, 889, 5, doi: 10.3847/1538-4357/ab5dbd
  * Jain et al. (2007) Jain, D., Dev, A., & Alcaniz, J. S. 2007, Phys. Lett. B, 656, 15, doi: 10.1016/j.physletb.2007.09.023
  * Jimenez et al. (2003) Jimenez, R., Verde, L., Treu, T., & Stern, D. 2003, Astrophys. J., 593, 622, doi: 10.1086/376595
  * Kurek et al. (2010) Kurek, A., Hrycyna, O., & Szydlowski, M. 2010, Phys. Lett. B, 690, 337, doi: 10.1016/j.physletb.2010.05.061
  * L. Chen & Wang (2019) L. Chen, Q. G. H., & Wang, K. 2019, Journal of Cosmology and Astroparticle Physics, 2019, 28
  * La Posta et al. (2021) La Posta, A., Louis, T., Garrido, X., Tristram, M., & Henrot-Versillé, S. 2021, Phys. Rev. D, 104, 023527, doi: 10.1103/PhysRevD.104.023527
  * Lazkoz et al. (2011) Lazkoz, R., Salzano, V., & Sendra, I. 2011, Phys. Lett. B, 694, 198, doi: 10.1016/j.physletb.2010.10.002
  * Li et al. (2020) Li, Y., Yu, J., & Zeng, T. 2020, Journal of Econometrics, 216, 450, doi: https://doi.org/10.1016/j.jeconom.2019.11.002
  * Liddle (2007) Liddle, A. R. 2007, Mon. Not. Roy. Astron. Soc., 377, L74, doi: 10.1111/j.1745-3933.2007.00306.x
  * Linder (2006) Linder, E. V. 2006, Astropart. Phys., 25, 167, doi: 10.1016/j.astropartphys.2005.12.003
  * Liu et al. (2023) Liu, Y., Wang, B., Yu, H., & Wu, P. 2023.  https://arxiv.org/abs/2305.19634
  * Ma & Zhang (2011) Ma, J.-Z., & Zhang, X. 2011, Phys. Lett. B, 699, 233, doi: 10.1016/j.physletb.2011.04.013
  * Malekjani et al. (2018) Malekjani, M., Rezaei, M., & Akhlaghi, I. A. 2018, Phys. Rev. D, 98, 063533, doi: 10.1103/PhysRevD.98.063533
  * Metropolis et al. (2004) Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. 2004, The Journal of Chemical Physics, 21, 1087, doi: 10.1063/1.1699114
  * Mootoovaloo et al. (2020) Mootoovaloo, A., Heavens, A. F., Jaffe, A. H., & Leclercq, F. 2020, Monthly Notices of the Royal Astronomical Society, 497, 2213–2226, doi: 10.1093/mnras/staa2102
  * Moreno-Pulido & Sola Peracaula (2022) Moreno-Pulido, C., & Sola Peracaula, J. 2022, Eur. Phys. J. C, 82, 1137, doi: 10.1140/epjc/s10052-022-11117-y
  * Moreno-Pulido et al. (2023) Moreno-Pulido, C., Sola Peracaula, J., & Cheraghchi, S. 2023, Eur. Phys. J. C, 83, 637, doi: 10.1140/epjc/s10052-023-11772-9
  * Moresco (2015) Moresco, M. 2015, Mon. Not. Roy. Astron. Soc., 450, L16, doi: 10.1093/mnrasl/slv037
  * Moresco et al. (2012) Moresco, M., et al. 2012, JCAP, 08, 006, doi: 10.1088/1475-7516/2012/08/006
  * Moresco et al. (2016) Moresco, M., Pozzetti, L., Cimatti, A., et al. 2016, JCAP, 05, 014, doi: 10.1088/1475-7516/2016/05/014
  * Nesseris & Perivolaropoulos (2004) Nesseris, S., & Perivolaropoulos, L. 2004, Phys. Rev. D, 70, 043531, doi: 10.1103/PhysRevD.70.043531
  * Pace et al. (2012) Pace, F., Fedeli, C., Moscardini, L., & Bartelmann, M. 2012, Mon. Not. Roy. Astron. Soc., 422, 1186, doi: 10.1111/j.1365-2966.2012.20692.x
  * Pan et al. (2018) Pan, S., Saridakis, E. N., & Yang, W. 2018, Phys. Rev. D, 98, 063510, doi: 10.1103/PhysRevD.98.063510
  * Perico et al. (2013) Perico, E. L. D., Lima, J. A. S., Basilakos, S., & Sola, J. 2013, Phys. Rev. D, 88, 063531, doi: 10.1103/PhysRevD.88.063531
  * Ratsimbazafy et al. (2017) Ratsimbazafy, A. L., Loubser, S. I., Crawford, S. M., et al. 2017, Mon. Not. Roy. Astron. Soc., 467, 3239, doi: 10.1093/mnras/stx301
  * Rezaei (2019a) Rezaei, M. 2019a, Mon. Not. Roy. Astron. Soc., 485, 4841, doi: 10.1093/mnras/stz733
  * Rezaei (2019b) —. 2019b, Mon. Not. Roy. Astron. Soc., 485, 550, doi: 10.1093/mnras/stz394
  * Rezaei & Malekjani (2017) Rezaei, M., & Malekjani, M. 2017, Phys. Rev. D, 96, 063519, doi: 10.1103/PhysRevD.96.063519
  * Rezaei & Malekjani (2021) —. 2021, Eur. Phys. J. Plus, 136, 219, doi: 10.1140/epjp/s13360-021-01200-w
  * Rezaei et al. (2017) Rezaei, M., Malekjani, M., Basilakos, S., Mehrabi, A., & Mota, D. F. 2017, Astrophys. J., 843, 65, doi: 10.3847/1538-4357/aa7898
  * Rezaei et al. (2019) Rezaei, M., Malekjani, M., & Sola, J. 2019, Phys. Rev. D, 100, 023539, doi: 10.1103/PhysRevD.100.023539
  * Rezaei et al. (2020) Rezaei, M., Naderi, T., Malekjani, M., & Mehrabi, A. 2020, Eur. Phys. J. C, 80, 374, doi: 10.1140/epjc/s10052-020-7942-6
  * Rezaei et al. (2024) Rezaei, M., Pan, S., Yang, W., & Mota, D. F. 2024, JCAP, 01, 052, doi: 10.1088/1475-7516/2024/01/052
  * Rezaei & Sola Peracaula (2022) Rezaei, M., & Sola Peracaula, J. 2022, Eur. Phys. J. C, 82, 765, doi: 10.1140/epjc/s10052-022-10653-x
  * Rezaei et al. (2021) Rezaei, M., Solà Peracaula, J., & Malekjani, M. 2021, Mon. Not. Roy. Astron. Soc., 509, 2593, doi: 10.1093/mnras/stab3117
  * Riess et al. (2021) Riess, A. G., Casertano, S., Yuan, W., et al. 2021, Astrophys. J. Lett., 908, L6, doi: 10.3847/2041-8213/abdbaf
  * Riess et al. (2019) Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, Astrophys. J., 876, 85, doi: 10.3847/1538-4357/ab1422
  * Riess et al. (2022) Riess, A. G., et al. 2022, Astrophys. J. Lett., 934, L7, doi: 10.3847/2041-8213/ac5c5b
  * Sabla & Caldwell (2021) Sabla, V. I., & Caldwell, R. R. 2021, Phys. Rev. D, 103, 103506, doi: 10.1103/PhysRevD.103.103506
  * Salas et al. (2022) Salas, J., Bautista, F., & Chaparro, G. 2022, Monthly Notices of the Royal Astronomical Society, 518, 75–86, doi: 10.1093/mnras/stac3112
  * Schmidt (2017) Schmidt, F. 2017.  https://arxiv.org/abs/1709.01544
  * Schwarz (1978) Schwarz, G. 1978, The Annals of Statistics, 6, 461 , doi: 10.1214/aos/1176344136
  * Scolnic et al. (2018) Scolnic, D., et al. 2018, Astrophys. J., 859, 101, doi: 10.3847/1538-4357/aab9bb
  * Simon et al. (2005) Simon, J., Verde, L., & Jimenez, R. 2005, Phys. Rev. D, 71, 123001, doi: 10.1103/PhysRevD.71.123001
  * Smith et al. (2020) Smith, T. L., Poulin, V., & Amin, M. A. 2020, Phys. Rev. D, 101, 063523, doi: 10.1103/PhysRevD.101.063523
  * Solà et al. (2017) Solà, J., Gómez-Valent, A., & de Cruz Pérez, J. 2017, Phys. Lett. B, 774, 317, doi: 10.1016/j.physletb.2017.09.073
  * Solà Peracaula et al. (2019) Solà Peracaula, J., Gomez-Valent, A., de Cruz Pérez, J., & Moreno-Pulido, C. 2019, Astrophys. J. Lett., 886, L6, doi: 10.3847/2041-8213/ab53e9
  * Solà Peracaula et al. (2021) Solà Peracaula, J., Gómez-Valent, A., de Cruz Perez, J., & Moreno-Pulido, C. 2021, EPL, 134, 19001, doi: 10.1209/0295-5075/134/19001
  * Spiegelhalter et al. (2002) Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van Der Linde, A. 2002, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64, 583, doi: https://doi.org/10.1111/1467-9868.00353
  * Stern et al. (2010) Stern, D., Jimenez, R., Verde, L., Kamionkowski, M., & Stanford, S. A. 2010, JCAP, 02, 008, doi: 10.1088/1475-7516/2010/02/008
  * Tamayo & Vazquez (2019) Tamayo, D., & Vazquez, J. A. 2019, Mon. Not. Roy. Astron. Soc., 487, 729, doi: 10.1093/mnras/stz1229
  * Tamayo & Vázquez (2019) Tamayo, D., & Vázquez, J. A. 2019, Monthly Notices of the Royal Astronomical Society, 487, 729, doi: 10.1093/mnras/stz1229
  * Tián (2020) Tián, S. X. 2020, Phys. Rev. D, 101, 063531, doi: 10.1103/PhysRevD.101.063531
  * Tian (2020) Tian, S. X. 2020, Phys. Rev. D, 102, 063509, doi: 10.1103/PhysRevD.102.063509
  * Tian (2022) —. 2022, Phys. Rev. D, 106, 043524, doi: 10.1103/PhysRevD.106.043524
  * Tian & Zhu (2021) Tian, S. X., & Zhu, Z.-H. 2021, Phys. Rev. D, 103, 123545, doi: 10.1103/PhysRevD.103.123545
  * Velten et al. (2014) Velten, H. E. S., vom Marttens, R. F., & Zimdahl, W. 2014, Eur. Phys. J. C, 74, 3160, doi: 10.1140/epjc/s10052-014-3160-4
  * Wong et al. (2020) Wong, K. C., et al. 2020, Mon. Not. Roy. Astron. Soc., 498, 1420, doi: 10.1093/mnras/stz3094
  * Xu et al. (2019) Xu, Y., Zhan, H., & Cheung, Y.-K. E. 2019, JCAP, 10, 006, doi: 10.1088/1475-7516/2019/10/006
  * Zhang et al. (2014) Zhang, C., Zhang, H., Yuan, S., Zhang, T.-J., & Sun, Y.-C. 2014, Res. Astron. Astrophys., 14, 1221, doi: 10.1088/1674-4527/14/10/002