# Finetuning BERT on Partially Annotated
NER Corpora ††thanks: This work was supported by RFBR Grant 20-07-00561 A.

Viktor Scherbakov 1 2 *  —<EMAIL_ADDRESS>Vladimir Mayorov 1 *  —
1Ivannikov Institute for System Programming of the Russian Academy of Sciences
2Lomonosov Moscow State University<EMAIL_ADDRESS>
###### Abstract

Most Named Entity Recognition (NER) models operate under the assumption that
training datasets are fully labelled. While it is valid for established
datasets like CoNLL 2003 and OntoNotes, sometimes it is not feasible to obtain
the complete dataset annotation. These situations may occur, for instance,
after selective annotation of entities for cost reduction. This work presents
an approach to finetuning BERT on such partially labelled datasets using self-
supervision and label preprocessing. Our approach outperforms the previous
LSTM-based label preprocessing baseline, significantly improving the
performance on poorly labelled datasets. We demonstrate that following our
approach while finetuning RoBERTa on CoNLL 2003 dataset with only 10% of total
entities labelled is enough to reach the performance of the baseline trained
on the same dataset with 50% of the entities labelled.

###### Index Terms:

named entity recognition, partial annotation, self-supervision, BERT

##  I Introduction

The goal of Named Entity Recognition (NER) is to find mentions of real-world
entities from the set of fixed categories in unstructured text. While entity
categories could overlap [1] and be organized in complex hierarchies [2], we
will consider the categories to not overlap for the sake of simplicity.
Generally, the solutions are based on some ML sequence-to-sequence
classification model [3, 4, 5, 6]. Each text token (word, punctuation symbol,
whitespace, etc., depending on the tokenization used) is assigned a label that
carries information about whether this token belongs to an entity of a
specific category. By default, every token is labelled as “Outside of an
entity“.

Even though NER is a relatively well-studied field with many high-quality
datasets [7, 8, 9], there are still limitations to its application in
practice. Since ML models are heavily used, a lot of training data is
required, and it is often the case that existing datasets cannot satisfy this
requirement due to a mismatch of the category sets. For example, suppose there
is an extensive dataset with only a LOCATION entity category, and we need a
model that can detect LOCATION and PERSON categories. In that case, we will
not be able to use this dataset to train such NER model since there may be a
lot of unlabelled PERSON entities.

There are several ways to combat this issue. The first and the most obvious
approach is to relabel the whole dataset to introduce a new entity category.
Unfortunately, this is a very expensive process due to the significance of the
context in which a particular entity was mentioned. For instance, the word
“Washington“ may be somebody’s family name or refer to the US capital. The
expenses can be cut down with the help of crowdsourcing the annotations [10,
11]. However, not all of the entity categories can be crowdsourced. There are
some cases that require expert knowledge to obtain the correct annotations. An
example of such a case is a biomedical domain [12] with categories such as
chemical or disease. Finally, annotations could be inferred from an extensive
knowledge base such as WikiData [13] and used as distantly supervised
examples. Nevertheless, such annotations will inevitably contain a lot of
noise from incorrect or missing entity mentions.

Figure 1: Demonstration of selectively labelled person example in fully
labelled dataset of locations. “Mary“ is selectively labelled, while
“Brooklyn“ token remained labelled as “Outside of an entity“.

We propose using selective annotation for new entity category introduction to
existing corpora. When there is a need for a new category, the training
dataset is either populated with new examples mentioning entities of this
category or some part of it is relabelled so that some existing mentions are
labelled correctly. This process yields a partially labelled dataset, meaning
that there are examples of correctly annotated entities present. Yet, some
entity mentions are incorrectly labelled as “Outside of an entity“ since, by
default, every unlabelled token is considered to not be a part of an entity.
Figure 1 depicts a selectively labelled example.

Although the task of training on partially annotated datasets is similar to
Few-Shot Learning [14], the restrictions dictated by these tasks on the
available annotations are quite different. Even though Few-Shot Learning
methods require a lot less labelled data, such methods expect the available
examples to be fully annotated. Given these differences, it is unclear how to
compare Few-Shot methods with methods of training on partially annotated
datasets that are explored later in this work.

##  II Related Work

The problem of training on partially annotated datasets is not novel. There
have been a number of papers that explored various methods of modelling NER
partial annotations (Fig. 2b) by modifying the CRF loss function to account
for the label uncertainty at non-entity tokens [15, 16, 17]. The idea of
specific modelling was introduced by [18] and further expanded in later works.

Another complementary approach to partial annotations modelling is dataset
“denoising“—detection and correction of various annotation mistakes. For
example, many methods dealing with automatically annotated training data (or
distantly supervised) employ strategies that can be applied to partially
annotated datasets. This is due to a specific process of distant label
generation for NER that usually uses external knowledge bases and dictionary
matching to annotate unlabelled data. As shown in [19, 20, 16], such a process
usually yields annotations with high precision and low recall compared to
ground truth. In other words, most of the annotations are correct, yet not all
of the named entities are labelled. To overcome low recall of the training
dataset, self-supervised methods such as [19, 17] are used to iteratively
relabel the training dataset with the models trained on annotations from the
previous iterations. The following sections II-A and II-B will describe the
general methodology behind these methods.

###  II-A BOND

Liang et al. [19] proposed BOND as a training procedure to finetune BERT on
distantly labelled datasets, i.e. automatically annotated by some other model,
such as gazetteers crawled from knowledge bases. Their method consists of the
following steps:

  1. Step 1:

NER fitting stage: use early stopping to train the model without overfitting
the “noisy“ annotations.

  2. Step 2:

Define the teacher and student models as copies of the model trained in the
previous step.

  3. Step 3:

Self-training stage: use the teacher model to obtain the label distributions
(See Fig. 2c), update the student model w.r.t. to these label distributions as
soft targets for several iterations. Then replace the teacher model with the
current student model and repeat this step.

Authors of the method state that the third step allows utilising the language
understanding encoded in pre-trained BERT weights to correct any missing or
incorrect annotations of the entities.

While BOND has shown excellent performance on distantly supervised data, it
needs to be clarified how well it is suited for training on partially
annotated datasets. Additionally, it is essential to note that this training
method does require a fully labelled high-quality validation dataset to detect
overfitting during the NER fitting stage and for the model selection in the
self-training stage.

(a) Ground truth annotations

(b) Partial annotations

(c) Teacher annotations

(d) Teacher annotations corrected with partial annotations

Figure 2: Demonstration of various types of annotations. Lines indicate the
next possible labels, and lighter colours show low-confidence labels. Compared
to ground truth annotation (a), partial annotations (b) do not have any
confident “O“ (Outside of an entity) labels and have some of the entity labels
missing, e.g. “Brooklyn“ does not have any confident label. Teacher
annotations (c) are expected to be close to ground truth with some noise that
comes from using soft targets, e.g. “Brooklyn“ is assigned “LOC“ and “PER“
labels with high confidence.

###  II-B Base Distribution Estimation (BDE)

Jie et al. [17] combined partial annotation modelling and self-supervision to
correct missing annotations for the partially annotated training dataset. The
authors propose using a modification of the CRF decoder that can be trained on
label distributions (soft labels) rather than on labels and a specific
training procedure:

  1. Step 1:

Estimate the “base distribution“ in cross-validation fashion:

    1. 1.1

Partition the training dataset into $k$ equal parts: $K_{1},...,K_{k}$.

    2. 2.2

For each $K_{i}$ train a model on $\\{K_{1},...,K_{k}\\}\setminus K_{i}$ and
use it to estimate label distributions $d_{i}$ for $K_{i}$ dataset part.

  2. Step 2:

Train the final model on the whole dataset using the obtained label
distributions $d_{1},...,d_{k}$ as soft targets.

While the authors of this method used an LSTM-based architecture for the
model, in practice, such a training procedure can be applied to any model that
can be trained with soft targets.

Even though the authors never specifically mentioned it, analyzing the
codebase they have provided led us to the conclusion that they might have used
a fully labelled validation set to select the best models for the BDE stage
and during the main training procedure (Step 2). Moreover, they also utilized
an adaptive learning rate schedule based on validation metrics during
training.

##  III Our Approach

Both training procedures described in sections II-B and II-A can be used to
train models on partially annotated datasets since they employ self-supervised
methods that can recover missing annotations. Due to the fact that the BDE
method can be seen as some preprocessing of target labels, it can be utilized
to discover the missing annotations in training data before the main training
procedure. Moreover, some noise-resistant model or training method can be used
during the base distributions estimation to increase the quality of
preprocessed annotations further.

While using BOND to finetune BERT on partially annotated data is possible, the
available annotations will not be used efficiently since all annotations will
be discarded after the NER fitting stage and replaced with the soft targets of
the teacher model. To solve this issue, we implement a modification of the
BOND training procedure—GuidedBOND. During the self-training stage, we do not
discard the annotated entities $E$ but instead, use them to “guide“ the model
in the right direction by correcting the label distributions $D$ obtained from
the teacher model.

Let $C$ be the number of unique labels, $L$—length of the token sequence and
$N$—the number of annotated entities. Given the sequence of encoded labels
$(l_{1},...,l_{L})$, $l_{i}\in(1,...,C)$, $E$ can be simplified as a set of
non-intersecting continuous label subsequences, indicating which labels are
known to be labelled correctly:

$\displaystyle E=\\{(m_{i},n_{i})\mid 1\leq m_{i}<n_{i}\leq
L+1,~{}i=1,...,N\\}$

Teacher model label distributions $D$ are the probability vectors $d_{k}$ with
$p^{k}_{j}$ components, $\sum_{j=1}^{C}p_{j}^{k}=1$. The index $k$ indicates
that $d_{k}$ was obtained for $k$-th token.

$\displaystyle D=\\{d_{k}\mid
d_{k}=(p_{1}^{k},p_{2}^{k},...,p_{C}^{k}),~{}k=1,...,L\\}$

The corrected label distributions $\hat{D}$ are defined equivalently with
$\hat{d}_{k}$ probability vectors that are the same as corresponding $d_{k}$
vectors if no entity includes $k$-th token, and a one-hot vector of encoded
label $l_{k}$ otherwise. Formally, $\hat{p}^{k}_{j}$ component of
$\hat{d}_{k}$ can be defined as follows:

$\hat{p}_{j}^{k}=\begin{cases}\mathds{1}[j=l_{k}]&\exists(m,n)\in E\mid m\leq
k<n\\\ p_{j}^{i}&otherwise\end{cases}$

See illustrations 2c and 2d for reference. We evaluate our approach in IV-B
section.

We propose using GuidedBOND during the BDE label preprocessing in the
following fashion. First, we estimate base distributions by finetuning the
BERT model on $k$ folds with the GuidedBOND training procedure. Next, we use
these estimated distributions as soft targets to finetune a pre-trained BERT
model. We suppose that utilizing GuidedBOND during the BDE preprocessing stage
will yield better estimations on missing annotations since, as we show in
section IV-A and IV-B, GuidedBOND yields significant improvements when trained
on partial annotations compared to both the conventional supervised BERT
finetuning approach and BOND.

##  IV Experiments

In our experiments, we use the English version of the CoNLL 2003 dataset [7].
We train, validate and test all models on the official dataset split.
Following the approach in [17] on simulating the partial annotations, some
fraction of the total entities in the training dataset is removed. Entities in
the validation dataset remain fully labelled. Following [19], the base version
of RoBERTa [21] was chosen as a pre-trained BERT model. We report the results
of training 5 models with different random initializations on the training
dataset with 0.05, 0.1, 0.15, 0.2, 0.3, 0.4 and 0.5 entities labelled. Entity
sets were chosen randomly without any regard to initial class imbalances and
fixed across all runs111We were not able to fix entity sets for LSTM-CRF
baseline since we have used an implementation provided by the authors [17]..
Lines in the Figures 3, 4, 5, 6 and 7 indicate the mean of 5 runs and a
lighter coloured area around them demonstrates the standard deviation. We
release the code for the experiments as well as the reproduction
instructions.222 https://github.com/ViktorooReps/guided-bond

In the following sections, the specific naming convention is used:

Method |  Meaning
---|---
GuidedBOND / BOND |  GuidedBOND / BOND training procedure is used to finetune RoBERTa
RoBERTa |  RoBERTa is finetuned conventionally
LSTM-CRF |  LSTM-CRF is trained conventionally
BDEM1+M2 |  M2 is trained on base distributions obtained with M1

###  IV-A Baseline

First, we confirm that finetuning RoBERTa on partially annotated corpora does,
in fact, produce results that are worse than the existent BDELSTM-CRF+LSTM-CRF
approach. As demonstrated in Figure 3, more than 50% of the missing entity
annotations in the training data drastically deteriorate the performance of
the conventionally finetuned RoBERTa.

Figure 3: Comparison of conventionally finetuned RoBERTa without BDE with
LSTM-CRF model trained with BDE proposed by [17]. RoBERTa needs half of the
total entities to be annotated to reach the performance of the BDE-based
approach.

###  IV-B Effect of guidance

Next, we assess the effectiveness of guidance for the BOND training procedure.
The experiments illustrated in Figure 4 not only confirm that guidance during
the self-training stage does help to converge to a better model but also show
that on lower fractions of labelled entities, the training process becomes a
lot more stable with respect to initialization of the model.

Figure 4: Demonstration of the significance of guidance on BOND training
procedure. The guidance on lower fractions of annotated entities makes the
finetuning process more stable.

###  IV-C Combined approach

Finally, we compare our approach—estimation of base distributions with
GuidedBOND training procedure with the LSTM-CRF baseline (see Fig. 5).
Unsurprisingly—since BERT was shown to outperform LSTM-based approaches
[6]—our approach yields modest improvements when trained on datasets with
higher fractions of labelled entities. Nonetheless, on lower fractions, the
improvements become a lot more significant. Our approach reaches a 0.9 F1
score compared to the baseline with close to 5 times fewer annotated entities.

Figure 5: Comparison of RoBERTa finetuned on BDE obtained with GuidedBOND
training procedure with LSTM-CRF model trained on BDE proposed by [17]. Our
model is better overall, with the most significant improvements on the lower
fractions of annotated entities.

##  V Ablation Study

In this section we will evaluate an impact of the different training stages on
the final model performance.

###  V-A Main training procedures

Due to the fact that annotations preprocessed with BDE can be considered as
distant soft labels, and since BOND was originally designed for training with
distant supervision, we attempt to apply GuidedBOND to train models on
estimated base distributions (see the results in Fig. 6). We report no
improvements compared to the supervised approach. The reason may be that the
increase in F1 score during the self-training stage is mainly due to the
discovery of the missing annotations, and most of the missing annotations
(that are recoverable by self-supervision) are recovered during the BDE stage.

Figure 6: Comparison of different frameworks for training on base
distributions. GuidedBOND does not show any significant improvements over the
simple supervised approach.

###  V-B Annotations preprocessing

To evaluate whether using GuidedBOND during the BDE stage (and the stage
itself) is necessary, we conduct an experiment by training RoBERTa with
GuidedBOND on base distributions obtained from conventionally finetuned
RoBERTa and RoBERTa finetuned with GuidedBOND. The comparison between these
models and the model trained without the BDE stage is demonstrated in Figure
7. We conclude that BDE yields no improvements except for the cases with
extremely low fractions of entities labelled, where, interestingly enough, a
fully supervised approach of estimating base distribution leads to
exceptionally poor performance.

Figure 7: Comparison of different ways to obtain the base distributions. Out
of three approaches GuidedBOND yields the most stable results.

##  VI Conclusion and Future Work

We have proposed an approach based on a self-supervised training procedure and
preprocessing of the available annotations for finetuning BERT models on
partially annotated data. Our experiments have shown that our approach
outperforms the previous method of dealing with partial annotations, with
substantial improvements on lower fractions of annotated entities.

Nevertheless, our approach, as well as the approaches it is based on, relies
on a fully annotated development dataset for the model selection and for
determining the learning rate schedule. Due to the reasons discussed in the
previous sections, it is often not feasible to collect such a dataset under
real conditions. This complicates the application of our method to real-world
tasks.

Our future work will be focused on researching the approaches to evaluation on
partially labelled datasets, as well as on the application of our method to
training on several combined datasets with different sets of entity
categories.

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# Modified Equations of State for Dark Energyand Observational Limitations

German S. Sharov1,2* and Vasily E. Myachin  Department of Mathematics, Tver
State University, Sadovyi per. 35, Tver, Russia
2 International Laboratory for Theoretical Cosmology, Tomsk State University
of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia
<EMAIL_ADDRESS>
###### Abstract

Cosmological models with variable and modified equations of state for dark
energy are confronted with observational data, including Type Ia supernovae,
Hubble parameter data $H(z)$ from different sources, and observational
manifestations of cosmic microwave background radiation (CMB). We consider
scenarios generalizing the $\Lambda$CDM, $w$CDM, and
Chevallier–Polarski–Linder (CPL) models with nonzero curvature and compare
their predictions. The most successful model with the dark energy equation of
state $w=w_{0}+w_{1}(1-a^{2})/2$ was studied in detail. These models are
interesting in possibly alleviating the Hubble constant $H_{0}$ tension, but
they achieved a modest success in this direction with the considered
observational data.

cosmological model; dark energy; equation of state; Hubble constant

##  I Introduction

In modern cosmology based on recent observational data, our Universe includes
dominating fractions of dark energy and dark matter, whereas all kinds of
visible matter fill about 4% in total energy balance nowadays. The latest
estimations of Planck collaboration Planck15 ; Planck18  predict about 70%
fraction of dark energy, if we apply the standard $\Lambda$CDM model, where
dark energy may be represented as the cosmological constant $\Lambda$ or as a
matter with density $\rho_{x}$ and pressure $p_{x}=-\rho_{x}$. Almost all
remaining part of matter in this model is cold dark matter with close to zero
pressure. Because of the last property, it is convenient to consider cold dark
matter together with visible baryonic matter, where the unified density is
$\rho_{m}=\rho_{c}+\rho_{b}$. One should also add the radiation component
including relativistic species (neutrinos) with $p_{r}=\frac{1}{3}\rho_{r}$,
which was sufficient before and during the recombination era, but is is almost
negligible now.

The $\Lambda$CDM model successfully describes numerous observations, including
Type Ia supernovae (SNe Ia) data, estimates of the Hubble parameter $H(z)$,
manifestations of baryonic acoustic oscillations (BAO), cosmic microwave
background radiation (CMB), and other data Planck15 ; Planck18 . However this
model does not explain the nature of dark energy, the small observable value
of the phenomenological constant $\Lambda$ and the approximate equality
$\rho_{x}$ and $\rho_{m}$ now (although these densities evolve differently).

Another essential problem in the $\Lambda$CDM model is the tension between
Planck estimations of the Hubble constant $H_{0}=67.37\pm 0.54$ km
/(s$\cdot$Mpc) Planck18  (2018) and measurements of SH0ES group in the Hubble
Space Telescope $H_{0}=73.2\pm 1.3$ km s-1 Mpc-1 Riess2020  (2020) or
$H_{0}=73.3\pm 1.04$ km s-1 Mpc-1 Riess2021  (2021).

The Planck estimations Planck15 ; Planck18  are based on the $\Lambda$CDM
model and the Planck satellite measurements of the CMB anisotropy and power
spectra related to the early Universe at redshifts $z\sim 1000$, whereas the
SH0ES method uses local distance ladder measurements of Cepheids in our Galaxy
Riess2020  and in nearest galaxies, in particular, in the Large Magellanic
Cloud HST19 , that implies $z$ close to 0 (the late Universe). This $H_{0}$
tension has not diminished during the last years, and now it exceeds
$4\sigma$.

Cosmologists suggested numerous scenarios for solving the mentioned problems
with dark energy and the $H_{0}$ tension; they include modifications of early
or late dark energy, dark energy with extra degrees of freedom, models with
interaction in dark sector, models with extra relativistic species, viscosity,
modified gravity including $F(R)$ theories, and other models (see reviews
BambaCNO:2012 ; NojiriO:2011 ; OdintsovO:2017 ; DiValentInter2020 ;
DiValentRealmTens2021  and papers KumarNunes2016 ; PanYDiVSCh2019 ;
DiValentMMV2019 ; PanSharov2017 ; PanSharovYang2020 ; OdintsovSGSvisc2020 ;
OdintsovSGS2017 ; OdintsovSGS2019 ; NojiriOSGS:2021 ; OdintsovSGS:2020 ).

In this paper, we consider cosmological scenarios with modified equation of
state (EoS) for dark energy; they generalize the $\Lambda$CDM model and its
simplest extensions: the $w$CDM model with EoS

$p_{x}=w\rho_{x},\qquad w=\mbox{const}$ (1)

and the models with variable EoS

$p_{x}=w(a)\,\rho_{x},$ (2)

where $w$ depends on the scale factor $a$. The class (2) includes the
following well-known dark energy equations of state: the linear model Linear

$w=w_{0}+w_{1}(a^{-1}-1)=w_{0}+w_{1}z;$ (3)

Chevallier–Polarski–Linder (CPL) parametrization ChP ; Linder

$w=w_{0}+w_{1}(1-a)=w_{0}+w_{1}\frac{z}{1+z}$ (4)

and (their generalization) Barboza–Alcaniz–Zhu–Silva (BAZS) EoS BarbozaAZS09

$w=w_{0}+w_{1}\frac{1-a^{\beta}}{\beta}=w_{0}+w_{1}\frac{1-(1+z)^{-\beta}}{\beta}.$
(5)

Here, the scale factor $a$ is normalized, so $a(t_{0})=1$ at the present time
$t_{0}$; $a$ is connected with redshift $z$: $a=(1+z)^{-1}$. Obviously, BAZS
parametrization (5) transforms into CPL EoS (4) at $\beta=1$, into the linear
EoS (3) if $\beta=-1$ and into the logarithmic EoS $w=w_{0}+w_{1}\log(1+z),$
if $\beta\to 0$.

For all mentioned equations of state, one can integrate the continuity
equation for non-interacting dark energy:

$\dot{\rho}_{x}=-3H(\rho_{x}+p_{x}).$ (6)

In this paper, we explore the above scenarios and suggest the following
generalization of BAZS EoS:

$w=w_{0}+w_{1}\frac{1-a^{\beta}}{\beta}a^{\gamma}.$ (7)

We test these models, confronting them with the following observational data:
Type Ia supernovae data (SNe Ia) from the Pantheon sample survey Scolnic17 ,
data extracted from Planck 2018 observations Planck18 ; ChenHuangW2018  of
cosmic microwave background radiation (CMB) ,and estimations of the Hubble
parameter $H(z)=\dot{a}/a$ for different redshifts $z$ (the dot means
$\frac{d}{dt}$). We use $H(z)$ data from two sources: (a) from cosmic
chronometers, that is, measured from differential ages of galaxies, and (b)
estimates of $H(z)$ obtained from line-of-sight baryonic acoustic oscillations
(BAO) data.

This paper is organized as follows. In the next section, we describe $H(z)$,
SNe Ia, and CMB observational data analyzed here. Section III is devoted to
dynamics and free model parameters for scenarios (1)–(7). In Section IV we
analyze the results of our calculations for these models, and estimate values
of model parameters including the Hubble constant $H_{0}$, and in Section V we
discuss the results and their possible applications for alleviating the Hubble
constant tension problem.

##  II Observational Data

Observational data should be described by the considered cosmological models.
For each model, we calculate the best fit for its free parameters from the
abovementioned data sources: (a) Type Ia supernovae (SNe Ia) data from
Pantheon sample Scolnic17 , (b) CMB data from Planck 2018 Planck18 ;
ChenHuangW2018 , and (c) estimates of the Hubble parameter $H(z)$ from cosmic
chronometers and line-of-sight BAO data.

The Pantheon sample database Scolnic17  for SNe Ia contains
$N_{\mbox{\scriptsize SN}}=1048$ data points of distance moduli
$\mu_{i}^{\mathrm{obs}}$ at redshifts $z_{i}$ in the range $0<z_{i}<1.92$. We
compare them with theoretical values by minimizing the $\chi^{2}$ function:

$\chi^{2}_{\mbox{\scriptsize
SN}}(\theta_{1},\dots)=\min\limits_{H_{0}}\sum_{i,j=1}^{N_{\mbox{\scriptsize
SN}}}\Delta\mu_{i}\big{(}C_{\mbox{\scriptsize
SN}}^{-1}\big{)}_{ij}\Delta\mu_{j},\qquad\Delta\mu_{i}=\mu^{\mathrm{th}}(z_{i},\theta_{1},\dots)-\mu^{\mathrm{obs}}_{i}\
,$ (8)

Here, $\theta_{j}$ are free model parameters, $C_{\mbox{\scriptsize SN}}$ is
the covariance matrix Scolnic17 , and the distance moduli $\mu^{\mathrm{th}}$
are expressed via the luminosity distance $D_{L}$ depending on the spacial
curvature fraction $\Omega_{k}$ and the Hubble parameter $H(z)$:

$\mu^{\mathrm{th}}(z)=m_{B}(z)-M_{B}=5\log_{10}\frac{D_{L}(z)}{10\mbox{pc}},\qquad
D_{L}(z)=(1+z)\,D_{M},$
$D_{M}(z)=\frac{c}{H_{0}}S_{k}\bigg{(}H_{0}\int\limits_{0}^{z}\frac{d\tilde{z}}{H(\tilde{z})}\bigg{)},\qquad
S_{k}(x)=\left\\{\begin{array}[]{ll}\sinh\big{(}x\sqrt{\Omega_{k}}\big{)}\big{/}\sqrt{\Omega_{k}},&\Omega_{k}>0,\\\
x,&\Omega_{k}=0,\\\
\sin\big{(}x\sqrt{|\Omega_{k}|}\big{)}\big{/}\sqrt{|\Omega_{k}|}.&\Omega_{k}<0,\end{array}\right.$

Here, $m_{B}$ is a supernova apparent magnitude, and $M_{B}$ is its absolute
magnitude. The distance moduli $\mu^{\mathrm{th}}$ are not Hubble-free and
depend on the Hubble constant $H_{0}=H(t_{0})$ (via the summand
$-5\log_{10}H_{0}$). On the other hand, $\mu_{i}^{\mathrm{obs}}$ is
essentially connected with the absolute magnitude $M_{B}$ and calculated with
corrections coming from deviations of lightcurve shape, SN Ia color, and mass
of a host galaxy Scolnic17 ; NesserisP2005 . In the Pantheon sample, these
corrections and the connected pair $(H_{0},M_{B})$ were considered as nuisance
parameters, in particular: “Using only SNe, there is no constraint on $H_{0}$
since $H_{0}$ and $M_{B}$ are degenerate” Scolnic17 . We cannot divide
uncertainties in the Hubble constant $H_{0}$ and possible uncertainties in
$M_{B}$ Efstathiou2021 ; CamarenaM2021 ; NunesdV2021 .

Due to these reasons, we have to consider $H_{0}$ in Equation (8) as a
nuisance parameter, its estimations cannot be obtained from
$\chi^{2}_{\mathrm{SN}}$, and we minimize this function over $H_{0}$
OdintsovSGSvisc2020 ; OdintsovSGS2017 ; OdintsovSGS2019 ; NojiriOSGS:2021 ;
OdintsovSGS:2020 .  However, SHe Ia data in $\chi^{2}_{\mbox{\scriptsize SN}}$
is important for fitting other model parameters.

Unlike SNe Ia Pantheon data, the CMB observations are related to the photon-
decoupling epoch near $z_{*}\simeq 1090$. We use the following parameters
extracted from Planck 2018 CMB observations Planck18 ; ChenHuangW2018 ;
NojiriOSGS:2021 :

$\mathbf{x}=\left(R,\ell_{A},\omega_{b}\right)\,,\quad
R=\sqrt{\Omega_{m}^{0}}\frac{H_{0}D_{M}(z_{*})}{c}\,,\quad\ell_{A}=\frac{\pi
D_{M}(z_{*})}{r_{s}(z_{*})}\,,\quad\omega_{b}=\Omega_{b}^{0}h^{2}$ (9)

and their estimations for the non-flat $\Lambda$CDM + $\Omega_{k}$ model
ChenHuangW2018 :

$\mathbf{x}^{\mathrm{Pl}}=\left(R^{\mathrm{Pl}},\ell_{A}^{\mathrm{Pl}},\omega_{b}^{\mathrm{Pl}}\right)=\left(1.7429\pm
0.0051,\;301.409\pm 0.091,\;0.0226\pm 0.00017\right).$ (10)

Considering the flat case ($\Omega_{k}=0$) of these models, we use the flat
$w$CDM data ChenHuangW2018 . The comoving sound horizon $r_{s}$ at $z_{*}$ is
calculated as the integral

$r_{s}(z_{*})=\int_{z_{*}}^{\infty}\frac{c_{s}(\tilde{z})}{H(\tilde{z})}\,d\tilde{z}=\frac{1}{\sqrt{3}}\int_{0}^{1/(1+z_{*})}\frac{da}{a^{2}H(a)\sqrt{1+\big{[}3\Omega_{b}^{0}/(4\Omega_{\gamma}^{0})\big{]}a}}$
(11)

with the fitting formula from Refs. ChenHuangW2018 ; HuSugiyama95  for the
value $z_{*}$. The resulting $\chi^{2}$ function is

$\chi^{2}_{\mathrm{CMB}}=\min_{\omega_{b}}\Delta\mathbf{x}\cdot
C_{\mathrm{CMB}}^{-1}\left(\Delta\mathbf{x}\right)^{T},\qquad\Delta\mathbf{x}=\mathbf{x}-\mathbf{x}^{\mathrm{Pl}}\,,$
(12)

where we minimize over the normalized baryon fraction $\omega_{b}^{0}$ to
diminish the effective number $N_{p}$ of free model parameters. The covariance
matrix $C_{\mathrm{CMB}}=\left\|\tilde{C}_{ij}\sigma_{i}\sigma_{j}\right\|$
and other details are described in papers OdintsovSGSvisc2020 ; ChenHuangW2018
.

In this paper, we use the Hubble parameter data $H(z)$ obtained from two
different sources OdintsovSGSvisc2020 ; OdintsovSGS2017 ; OdintsovSGS2019 ;
NojiriOSGS:2021 ; OdintsovSGS:2020 ; SharovSin2020 . The first one is the
cosmic chronometers (CC), in other words, estimations of $H(z)$ via
differences of ages $\Delta t$ for galaxies with close redshifts $\Delta z$
and the formula

$H(z)=\frac{\dot{a}}{a}\simeq-\frac{1}{1+z}\frac{\Delta z}{\Delta t}.$

Here, we include 31 CC $H(z)$ data points from Refs. Simon ; Stern ; Moresco12
; Zhang ; Moresco15 ; Moresco16 ; Ratsimbazafy  used earlier in papers
OdintsovSGSvisc2020 ; OdintsovSGS2017 ; OdintsovSGS2019 ; NojiriOSGS:2021  and
the recent estimate from Ref. HzBorghi2021 ; they are shown in Table 1.

Table 1: $H(z)$ data from cosmic chronometers (CC) Simon ; Stern ; Moresco12 ; Zhang ; Moresco15 ; Moresco16 ; Ratsimbazafy ; HzBorghi2021  and line-of-sight BAO Gaztanaga09 ; Blake12 ; ChuangW13 ; Chuang13 ; Busca13 ; Oka14 ; Font-Ribera14 ; Delubac15 ; Anderson14 ; Wang17 ; Alam17 ; Bautista17 ; Bourboux17 ; Zhu18 ; Zhao18 .  CC Data |  $H_{\mathrm{BAO}}$ Data
---|---
$z$ | $H(z)$ | $\sigma$ | Refs | $z$ | $H(z)$ | $\sigma$ | Refs
0.070 | 69 | 19.6 | Zhang 14 | 0.240 | 79.69 | 2.992 | Gaztañaga 09
0.090 | 69 | 12 | Simon 05 | 0.30 | 81.7 | 6.22 | Oka 14
0.120 | 68.6 | 26.2 | Zhang 14 | 0.31 | 78.18 | 4.74 | Wang 17
0.170 | 83 | 8 | Simon 05 | 0.34 | 83.8 | 3.66 | Gaztañaga 09
0.1791 | 75 | 4 | Moresco 12 | 0.350 | 82.7 | 9.13 | ChuangW 13
0.1993 | 75 | 5 | Moresco 12 | 0.36 | 79.94 | 3.38 | Wang 17
0.200 | 72.9 | 29.6 | Zhang 14 | 0.38 | 81.5 | 1.9 | Alam 17
0.270 | 77 | 14 | Simon 05 | 0.400 | 82.04 | 2.03 | Wang 17
0.280 | 88.8 | 36.6 | Zhang 14 | 0.430 | 86.45 | 3.974 | Gaztañaga 09
0.3519 | 83 | 14 | Moresco 12 | 0.44 | 82.6 | 7.8 | Blake 12
0.3802 | 83 | 13.5 | Moresco 16 | 0.44 | 84.81 | 1.83 | Wang 17
0.400 | 95 | 17 | Simon 05 | 0.48 | 87.79 | 2.03 | Wang 17
0.4004 | 77 | 10.2 | Moresco 16 | 0.51 | 90.4 | 1.9 | Alam 17
0.4247 | 87.1 | 11.2 | Moresco 16 | 0.52 | 94.35 | 2.64 | Wang 17
0.445 | 92.8 | 12.9 | Moresco 16 | 0.56 | 93.34 | 2.3 | Wang 17
0.470 | 89 | 34 | Ratsimbazafy | 0.57 | 87.6 | 7.83 | Chuang 13
0.4783 | 80.9 | 9 | Moresco 16 | 0.57 | 96.8 | 3.4 | Anderson 14
0.48 | 97 | 62 | Stern 10 | 0.59 | 98.48 | 3.18 | Wang 17
0.5929 | 104 | 13 | Moresco 12 | 0.600 | 87.9 | 6.1 | Blake 12
0.6797 | 92 | 8 | Moresco 12 | 0.61 | 97.3 | 2.1 | Alam 17
0.75 | 98.8 | 33.6 | Borghi 21 | 0.64 | 98.82 | 2.98 | Wang 17
0.7812 | 105 | 12 | Moresco 12 | 0.730 | 97.3 | 7.0 | Blake 12
0.8754 | 125 | 17 | Moresco 12 | 0.8 | 106.9 | 4.9 | Zhu 18
0.880 | 90 | 40 | Stern 10 | 0.978 | 113.72 | 14.63 | Zhao 19
0.900 | 117 | 23 | Simon 05 | 1.0 | 120.7 | 7.3 | Zhu 18
1.037 | 154 | 20.17 | Moresco 12 | 1.230 | 131.44 | 12.42 | Zhao 19
1.300 | 168 | 17 | Simon 05 | 1.5 | 161.4 | 30.9 | Zhu 18
1.363 | 160 | 33.6 | Moresco 15 | 1.526 | 148.11 | 12.75 | Zhao 19
1.430 | 177 | 18 | Simon 05 | 1.944 | 172.63 | 14.79 | Zhao 19
1.530 | 140 | 14 | Simon 05 | 2.0 | 189.9 | 32.9 | Zhu 18
1.750 | 202 | 40 | Simon 05 | 2.2 | 232.5 | 54.6; | Zhu 18
1.965 | 186.5 | 50.4 | Moresco 15 | 2.300 | 224 | 8.57 | Buska 13
|  |  |  | 2.330 | 224 | 8.0 | Bautista 17
|  |  |  | 2.340 | 222 | 8.515 | Delubac 15
|  |  |  | 2.360 | 226 | 9.33 | Font-Ribera 14
|  |  |  | 2.40 | 227.6 | 9.10 | Bourboux 17

These 32 CC data points need a covariance matrix of systematic uncertainties
connected with a choice of initial mass function, metallicity, star formation
history, stellar population synthesis models, and other factors Moresco2020 ;
Moresco2021 .

We describe these uncertainties as corrections $\Delta C_{\mathrm{H}}$ to the
diagonal covariance matrix
$C^{\mathrm{d}}_{\mathrm{H}}=\mbox{diag}\\{\sigma_{i}^{-2}\\}$ (from Table 1)
taking into account their diagonal terms in the form Moresco2020

$(\Delta C_{\mathrm{H}})_{ii}=\big{[}\eta(z_{i})\,H(z_{i})\big{]}^{2}.$

Here, $\eta(z)$ is a mean percentage bias depending on redshift $z$. We
consider “the best-case scenario” from the paper Moresco2020  for $\eta(z)$
and include these contributions of stellar population synthesis and
metallicity omitting the non-diagonal terms $\Delta(C_{\mathrm{H}})_{ij}$
(they are negligible for metallicity Moresco2021 ).

The second source of $H(z)$ estimates is the baryon acoustic oscillation (BAO)
data along the line-of-sight direction. We use here 36 $H_{\mathrm{BAO}}(z)$
data points from Refs. Gaztanaga09 ; Blake12 ; ChuangW13 ; Chuang13 ; Busca13
; Oka14 ; Font-Ribera14 ; Delubac15 ; Anderson14 ; Wang17 ; Alam17 ;
Bautista17 ; Bourboux17 ; Zhu18 ; Zhao18  (see Table 1). They were considered
earlier in Ref. OdintsovSGS:2020 .

Some of the $H(z)$ measurements Gaztanaga09 ; Blake12 ; ChuangW13 ; Chuang13 ;
Busca13 ; Oka14 ; Font-Ribera14 ; Delubac15 ; Anderson14 ; Wang17 ; Alam17 ;
Bautista17 ; Bourboux17 ; Zhu18 ; Zhao18  in Table 1 used the same or
overlapping large-scale structure data, so these $H$ estimates for close
redshifts $z$ may be in duplicate. It concerns, for example, measurements of
Delubac et al. Delubac15 , Font-Ribera et al. Font-Ribera14 , uses quasars
with Lyman-$\alpha$ forest from Data Release 11 SDSS-III survey; estimates of
Alam et al. Alam17 , Wang et al. Wang17 , Bautista et al. Bautista17 , and
Bourboux et al. Bourboux17  were made with data from or DR12 of SDSS-III etc.
To avoid this doubling, we multiply the errors $\sigma_{i}$ by $\sqrt{2}$ for
$H_{\mathrm{BAO}}$ estimates in Table 1 with close $z$, data, and methods.

Note that $H_{\mathrm{BAO}}$ estimates in Table 1 should be multiplied by the
factor $r^{\mathrm{fid}}_{d}/r_{d}$, where fiducial values
$r^{\mathrm{fid}}_{d}$ of the sound horizon size $r_{d}=r_{s}(z_{d})$ at the
drag epoch vary from 147.33 Mpc to 157.2 Mpc for different authors Gaztanaga09
; Blake12 ; ChuangW13 ; Chuang13 ; Busca13 ; Oka14 ; Font-Ribera14 ; Delubac15
; Anderson14 ; Wang17 ; Alam17 ; Bautista17 ; Bourboux17 ; Zhu18 ; Zhao18 . We
include this correction to errors $\sigma_{i}$ quadratically, comparing
deviations of $r^{\mathrm{fid}}_{d}$ with $r_{s}(z_{d})$ calculated with
Formula (11) for a considered model.

For any cosmological model we calculate the $\chi^{2}$ function

$\chi_{H}^{2}(\theta_{1},\dots)=\sum_{j=1}^{N_{H}}\bigg{[}\frac{H(z_{j},\theta_{1},\dots)-H^{obs}(z_{j})}{\sigma_{j}}\bigg{]}^{2}$
(13)

by using (a) only CC $H(z)$ data and (b) the full set CC \+ $H_{\mathrm{BAO}}$
data. Note that $H_{\mathrm{BAO}}$ data points are correlated with BAO angular
distances considered in the previous papers OdintsovSGS2017 ; OdintsovSGS2019
; NojiriOSGS:2021 ). Thus, here, we do not use data with BAO angular
distances, to avoid any correlation.

##  III Models

We explore all considered models in a homogeneous isotropic universe with the
Friedmann–Lemaître–Robertson–Walker metric

$ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right)\right],$

where $k$ is the sign of spatial curvature. In this case, the Einstein
equations are reduced to the system of the Friedmann equation

$3\frac{\dot{a}^{2}+k}{a^{2}}=8\pi G\rho$ (14)

and the continuity equation

$\dot{\rho}+3H(\rho+p)=0.$ (15)

Here, the total density $\rho$ includes densities of the abovementioned cold
pressureless matter (dark matter unified with baryonic matter), radiation, and
dark energy:

$\rho=\rho_{m}+\rho_{r}+\rho_{x},\qquad\rho_{m}=\rho_{c}+\rho_{b}.$ (16)

We suppose here that dark energy and the mentioned components do not interact
in the form PanYDiVSCh2019 ; DiValentMMV2019 ; PanSharov2017 ;
PanSharovYang2020  and independently satisfy the continuity Equation (6) or
(15). We integrate this equation for cold and relativistic matter:

$\rho_{m}=\rho_{m}^{0}a^{-3},\qquad\rho_{r}=\rho_{r}^{0}a^{-4}$ (17)

(the index “0” corresponds to the present time $t_{0}$) and substitute these
relations into the Friedmann equation (14) that can be rewritten as

$H^{2}=H_{0}^{2}\Big{(}\Omega_{m}^{0}a^{-3}+\Omega_{r}^{0}a^{-4}+\Omega_{k}a^{-2}+\Omega_{x}(a)\Big{)}$
(18)

or

$H(z)=H_{0}\sqrt{\Omega^{0}_{m}(1+z)^{3}+\Omega_{r}^{0}(1+z)^{4}+\Omega_{k}(1+z)^{2}+\Omega_{x}(z)}.$
(19)

Here,

$\Omega^{0}_{j}=\frac{8\pi G\rho^{0}_{j}}{3H^{2}_{0}},\quad
j=m,r,\qquad\Omega_{k}=\frac{-k}{a^{2}_{0}H^{2}_{0}},\qquad\Omega_{x}(a)=\frac{8\pi
G\rho_{x}(a)}{3H^{2}_{0}}.$

The dark energy fraction $\Omega_{x}(a)$ results from the continuity Equation
(6) $\dot{\rho}_{x}+3H(\rho_{x}+p_{x})=0$, that, for the variable EoS (2)
${}_{x}=w(a)\,\rho_{x}$, is reduced to the form

$\log\Omega_{x}(a)=-3\int\big{[}1+w(a)\big{]}\,a^{-1}da.$ (20)

In particular, for BAZS parametrization (5) $w=w_{0}+w_{1}(1-a^{\beta})/\beta$
BarbozaAZS09 , the expression (20) is

$\Omega_{x}(a)=\Omega_{x}^{0}\,a^{-3(1+w_{0}+w_{1}/\beta)}\exp\bigg{[}\frac{3w_{1}(a^{\beta}-1)}{\beta^{2}}\bigg{]}.$
(21)

One should substitute it into Equation (18).

In the above–mentioned particular cases, the BAZS formula (21) takes the form

$\Omega_{x}(a)=\Omega_{x}^{0}\,a^{-3(1+w_{0}-w_{1})}e^{3w_{1}z},$ (22)

for the linear model (3) $w=w_{0}+w_{1}z$ (if $\beta=-1$) and

$\Omega_{x}(a)=\Omega_{x}^{0}\,a^{-3(1+w_{0}+w_{1})}e^{3w_{1}(a-1)}.$ (23)

for CPL EoS (4) $w=w_{0}+w_{1}(1-a)$ (if $\beta=1$). In the case $w_{1}=0$
(and $w_{0}\equiv w$), both models (22) and (23) transform into the $w$CDM
model with

$\Omega_{x}(a)=\Omega_{x}^{0}\,a^{-3(1+w)},$ (24)

Its particular case at $w=-1$ is the $\Lambda$CDM model where
$\Omega_{x}=\Omega_{x}^{0}=\Omega_{\Lambda}={}$const.

For the generalization (7) (with the factor $\gamma$) of BAZS parametrization
(5), the expression (20) takes the form

$\Omega_{x}(a)=\Omega_{x}^{0}\,a^{-3(1+w_{0})}\exp\bigg{[}3\frac{w_{1}}{\beta}\bigg{(}\frac{1-a^{\gamma}}{\gamma}+\frac{a^{\beta+\gamma}-1}{\beta+\gamma}\bigg{)}\bigg{]}.$
(25)

If $\gamma\to 0$ it transforms into Equation (21), and in the case $\beta\to
0$ we have

$\Omega_{x}(a)=\Omega_{x}^{0}\,a^{-3(1+w_{0}-w_{1}a^{\gamma}/\gamma)}\exp\bigg{(}3w_{1}\frac{1-a^{\gamma}}{\gamma^{2}}\bigg{)}.$
(26)

For all considered models, the dark energy fraction $\Omega_{x}^{0}$ and other
$\Omega_{j}$ satisfy the equality

$\Omega^{0}_{m}+\Omega^{0}_{r}+\Omega_{k}+\Omega^{0}_{x}=1,$

resulting from Equation (18) or (19). Further, if we fix the ratio
OdintsovSGS2017 ; OdintsovSGS2019 ; NojiriOSGS:2021 ; Planck13

$X_{r}=\frac{\rho^{0}_{r}}{\rho^{0}_{m}}=\frac{\Omega^{0}_{r}}{\Omega^{0}_{m}}=2.9656\cdot
10^{-4}$ (27)

to diminish the number $N_{p}$ of free model parameters. We will work with the
following five free parameters in the linear (22) and CPL (23) models:

$\Omega^{0}_{m},\quad\Omega_{k},\quad H_{0},\quad w_{0},\quad w_{1}.$ (28)

Here, we consider $\omega_{b}$ in $\chi^{2}_{\mathrm{CMB}}$ (12) as a nuisance
parameter. In the $\Lambda$CDM model the number of parameters $N_{p}=3$
($\Omega^{0}_{m}$, $\Omega_{k}$, $H_{0}$), in the $w$CDM model (24) $N_{p}=4$
with $w_{0}\equiv w$, in the generalized model (7), (25) we have $N_{p}=7$
free parameters with additional $\beta$ and $\gamma$ to the set (28).

In the next sections, we compare predictions of these models with the
observational data from Section II.

##  IV Results

We evaluate how the considered models fit the observations taking into account
the $\chi^{2}$ functions for SNe Ia (8), CMB (12), and $H(z)$ data (13) in the
form

$\chi^{2}_{\mathrm{tot}}=\chi^{2}_{\mathrm{SN}}+\chi^{2}_{H}+\chi^{2}_{\mathrm{CMB}}.$
(29)

For the Hubble parameter data, we separately use (a) only 32 data points with
cosmic chronometers (CC) $H(z)$ data and (b) the full set CC +
$H_{\mathrm{BAO}}$ data (see Table 1).

Figure 1: For the linear model (3), $1\sigma$, $2\sigma$ contour plots in the
left panels (“all” means all $H(z)$ data); one-parameter distributions
$\chi^{2}_{\mathrm{tot}}(\Omega_{m}^{0})$ and likelihood functions ${\cal
L}(w_{1})$ in the right panels are compared with the $w$CDM model. The stars
and circles denote positions of $\chi^{2}_{\mathrm{tot}}$ minima points.

When we compare two models with different number $N_{p}$ of free parameters,
we can expect that the model with larger $N_{p}$ achieves more success in
minimizing $\chi^{2}_{\mathrm{tot}}$. However, some models are not successful
in this sense. In particular, the linear model (3), (22) with $N_{p}=5$
parameters (28), with our set of observational data, yields the same minimal
value $\min\chi^{2}_{\mathrm{tot}}\simeq 1092.34$ (for CC +
$H_{\mathrm{BAO}}$) as the $w$CDM model (24) with $N_{p}=4$. The reason is the
following: the best fitted value $w_{1}$ for the linear model (3) is very
close to zero ($w_{1}\simeq-0.0006$). In this case, the linear model works as
the $w$CDM model (24) and yields the same $\min\chi^{2}_{\mathrm{tot}}$.

Such a behavior of the model (3) is shown in Figure 1, where $1\sigma$ and
$2\sigma$ filled contour plots in the left panels correspond to the full set
of $H(z)$ data (here and below, “all” denotes CC + $H_{\mathrm{BAO}}$). The
$1\sigma$, $2\sigma$ contours for the $w$CDM model (red lines) in the
$\Omega_{m}^{0}-\Omega_{k}$ plane behave similarly and closely; positions of
$\chi^{2}_{\mathrm{tot}}$ minima point (shown as the star and the circle)
practically coincide.

Here, the contours are drawn for $\chi^{2}$ functions minimized over all other
parameters, in particular, for the linear model (3) in the top-left panel:

$\chi^{2}_{\mathrm{tot}}(\Omega_{m}^{0},\Omega_{k})=\min\limits_{H_{0},w_{0},w_{1}}\chi^{2}_{\mathrm{tot}}.$
(30)

A similar picture also takes place for only CC $H(z)$ data, where
$\min\chi^{2}_{\mathrm{tot}}\simeq 1074.73$ for both models. Equality of these
minima for only CC and all $H(z)$ data is illustrated with one-parameter
distributions $\chi^{2}_{\mathrm{tot}}(\Omega_{m}^{0})$ in the top-right
panel. In one-parameter distributions, we also minimize over all other
parameters.

In the $\Omega_{m}^{0}-w_{1}$ plane (the bottom-left panel of Fig. 1) we see
that for both models, minima of $\chi^{2}_{\mathrm{tot}}$ are achieved near
$w_{1}=0$. It is also shown in the bottom-right panel, where the likelihood
functions

${\cal L}(w_{1})\sim\exp\big{[}-\chi^{2}_{\mathrm{tot}}(w_{1})/2\big{]}$ (31)

are depicted for the linear model.

Figure 2: CPL model (23) in comparison with the $w$CDM model: $1\sigma$,
$2\sigma$ contours, likelihoods ${\cal L}(\theta_{j})$, and one-parameter
distributions $\chi^{2}_{\mathrm{tot}}(H_{0})$.

For the same observational data we can observe in Figure 2 more successful
behavior of the Chevallier–Polarski–Linder (CPL) model (23) ChP ; Linder .
Here, $1\sigma$ and $2\sigma$ contours are drawn for
$\chi^{2}_{\mathrm{tot}}(\theta_{i},\theta_{j})$ of the type (30) in all
planes of two parameters $(\theta_{i}-\theta_{j})$ in notation of Figure 1. We
consider four cases: for two models (CPL and $w$CDM) we calculate
$\chi^{2}_{\mathrm{tot}}$ for CC and all $H(z)$ data, positions of all
$\chi^{2}_{\mathrm{tot}}$ minima points are shown. Naturally, in the panels
with $w_{1}$, only the CPL model is presented.

The likelihood functions ${\cal L}(\theta_{i})$ of the type (31) are shown in
Figure 2 for all five model parameters (28). They are used for estimating the
best fits and $1\sigma$ errors for these parameters, summarized below in Table
2.

The CPL model achieves lower values of $\min\chi^{2}_{\mathrm{tot}}$ in
comparison with the $w$CDM and $\Lambda$CDM models. It can be seen in Table 2
and in the top-right panel of Figure 2, where one-parameter distributions
$\chi^{2}_{\mathrm{tot}}(H_{0})$ of these models are compared. The graphs
$\chi^{2}_{\mathrm{tot}}(H_{0})$ and the correspondent likelihoods ${\cal
L}(H_{0})$ in the top-left panel demonstrate that the best fitted values
$H_{0}$ are very close for the $w$CDM and CPL models and differ more
essentially when we compare CC and all $H(z)$ data.

The best fits of $\Omega_{m}^{0}$ depend stronger on the chosen model and vary
from $\Omega_{m}^{0}\simeq 0.206$ for CPL, all $H(z)$ to $\Omega_{m}^{0}\simeq
0.289$ for $w$CDM, CC. The best fits of $\Omega_{k}$ behave similarly, but
with the maximal estimate for CPL, all $H(z)$. The best CPL fits for $w_{1}$
are close to $0.66$ for both variants of $H(z)$ data. This value is far from
zero; in other words, the CPL model with the considered observational data
behaves differently to the $w$CDM model.

Figure 3: The generalized model (25) with all $H(z)$ data: one-parameter
distributions for $\beta$ and $\gamma$.

The CPL model achieves good results, but it is not the most successful
scenario in the frameworks of the generalized model (7), (25). However, this
generalized model has $N_{p}=7$ free parameters, including $\beta$ and
$\gamma$ in addition to five parameters (28). This large number is the serious
disadvantage of the generalized model (25) if we keep in mind informational
criteria OdintsovSGS2019 ; NojiriOSGS:2021 ; OdintsovSGS:2020  and
difficulties in calculations.

Following these reasons, we calculated the $\chi^{2}_{\mathrm{tot}}$ function
(29) for the generalized model (18), (25) with all $H(z)$ data searching its
minimum in the $\beta-\gamma$ plane. The results are presented in Figure 3 as
one-parameter distributions $\chi^{2}_{\mathrm{tot}}(\beta)$ and
$\chi^{2}_{\mathrm{tot}}(\gamma)$ (minimized over all other parameters).

Table 2: The best fit values with $1\sigma$ estimates of model parameters and $\min\chi^{2}_{\mathrm{tot}}$ for SNe Ia + CMB and CC or all $H(z)$ data for the models: $\beta=2$ (32), CPL (23), $w$CDM, and $\Lambda$CDM.  Model | Data | $\min\chi^{2}_{\mathrm{tot}}/d.o.f$ | $H_{0}$ | $\Omega_{m}^{0}$ | $\Omega_{k}$ | $w_{0}\equiv w$ | $w_{1}$
---|---|---|---|---|---|---|---
$\beta=2$ | CC | $1072.30/1079$ | $68.40^{+1.86}_{-1.91}$ | $0.177^{+0.113}_{-0.072}$ | $0.080^{+0.067}_{-0.076}$ | $-0.967^{+0.12}_{-0.14}$ |  $1.33^{+0.42}_{-1.56}$
CPL | CC | $\\!\\!1072.45/1079\\!\\!$ | $\\!\\!68.53^{+1.84}_{-1.88}\\!\\!$ | $\\!\\!0.218^{+0.102}_{-0.074}\\!\\!$ | $\\!\\!0.044^{+0.047}_{-0.059}\\!\\!$ | $\\!-0.955^{+0.14}_{-0.14}\\!$ |  $\\!\\!0.65^{+0.25}_{-1.04}\\!\\!$
$\\!\\!w$CDM | CC | $\\!\\!1072.97/1080\\!\\!$ | $\\!\\!68.58^{+1.85}_{-1.89}\\!\\!$ | $\\!\\!0.283^{+0.055}_{-0.058}\\!\\!$ | $\\!\\!0.003^{+0.024}_{-0.021}\\!\\!$ | $\\!\\!-0.946^{+0.14}_{-0.14}\\!\\!$ | \-
$\\!\\!\Lambda$CDM | CC | $\\!\\!1073.81/1081\\!\\!$ | $\\!\\!68.52^{+1.89}_{-1.94}\\!\\!$ | $\\!\\!0.305^{+0.029}_{-0.029}\\!\\!$ | $\\!\\!0.0052^{+0.0018}_{-0.0018}\\!\\!$ | $-1$ | \-
$\beta=2$ | all | $\\!\\!1084.03/1115\\!\\!$ | $\\!\\!67.65^{+1.28}_{-1.26}\\!\\!$ | $\\!\\!0.190^{+0.071}_{-0.072}\\!\\!$ | $\\!\\!0.068^{+0.072}_{-0.053}\\!\\!$ | $\\!\\!-0.96^{+0.087}_{-0.067}\\!\\!$ |  $\\!\\!1.12^{+0.40}_{-1.09}\\!\\!$
CPL | all | $\\!\\!1084.28/1115\\!\\!$ | $\\!\\!67.76^{+1.15}_{-1.01}\\!\\!$ | $\\!\\!0.206^{+0.054}_{-0.025}\\!\\!$ | $\\!\\!0.052^{+0.027}_{-0.036}\\!\\!$ | $\\!\\!-0.922^{+0.07}_{-0.06}\\!\\!$ |  $\\!\\!0.62^{+0.18}_{-0.52}\\!\\!$
$\\!\\!w$CDM | all | $\\!\\!1085.19/1116\\!\\!$ | $\\!\\!67.98^{+1.11}_{-1.11}\\!\\!$ | $\\!\\!0.257^{+0.019}_{-0.019}\\!\\!$ | $\\!\\!0.017^{+0.008}_{-0.008}\\!\\!$ | $\\!\\!-0.89^{+0.05}_{-0.05}\\!\\!$ | \-
$\\!\\!\Lambda$CDM | all | $\\!\\!1089.42/1117\\!\\!$ | $\\!\\!69.02^{+1.04}_{-1.06}\\!\\!$ | $\\!\\!0.272^{+0.013}_{-0.012}\\!\\!$ | $\\!\\!0.004^{+0.0017}_{-0.0017}\\!\\!$ | $-1$ | \-

We see in Figure 3 that the absolute minimum for the generalized model (25)
$\min\chi^{2}_{\mathrm{tot}}\simeq 1084.0$ is achieved near the point
$\beta=2$, $\gamma=0$. One may conclude that the Barboza–Alcaniz–Zhu–Silva
(BAZS) model BarbozaAZS09  (corresponding to $\gamma=0$) with $\Omega_{x}(a)$
(21) and $\beta=2$ appeared to be the most successful for the considered
observational data Ia + CMB + CC + $H_{\mathrm{BAO}}$. When we fix $\beta=2$,
the BAZS model with

$w=w_{0}+w_{1}(1-a^{2})/2$ (32)

will have five free model parameters (28). We investigate this model (denoted
below as “$\beta=2$”) in detail; the results are presented in Table 2 and in
Figure 4.

Figure 4 illustrates the BAZS model with $\beta=2$ (32): $1\sigma$ and
$2\sigma$ contour plots are shown for all $H(z)$ data (filled contours) and
for only CC data. They are compared with the corresponding $1\sigma$ contours
of the CPL model (shown also in Figure 2). One can compare the related one-
parameter distributions $\chi^{2}_{\mathrm{tot}}(H_{0})$ in the top-right
panel and likelihood functions (31) ${\cal L}(H_{0})$, ${\cal
L}(\Omega_{m}^{0})$, etc.

Figure 4: BAZS model $\beta=2$ (32) with $1\sigma$, $2\sigma$ contours in
comparison with CPL model ($1\sigma$ contours), likelihoods ${\cal
L}(\theta_{j})$, and one-parameter distributions
$\chi^{2}_{\mathrm{tot}}(H_{0})$.

Figure 4 and Table 2 demonstrate that the $\beta=2$ BAZS model (32) is more
successful in minimizing $\chi^{2}_{\mathrm{tot}}$ if we compare it with the
CPL model (and, naturally, with the $w$CDM and $\Lambda$CDM models). For only
CC data and for all $H(z)$ data, the best fit values of
$\chi^{2}_{\mathrm{tot}}$ for the $\beta=2$ model (32) are achieved at lower
values of $\Omega_{m}^{0}$ ($\Omega_{m}^{0}=0.177^{+0.113}_{-0.072}$ for CC
and $0.19^{+0.071}_{-0.072}$ for all $H$ data) and at larger values of
$\Omega_{k}$ in comparison with CPL model. However, the best fit values of
$H_{0}$ approximately coincide for all considered models; they depend on a
chosen dataset: only CC data or all $H(z)$ data.

The best fitted values of $w_{0}$ are close for the $\beta=2$, CPL, and $w$CDM
models in the CC case and slightly differ for all $H(z)$ data. The optimal
values of $w_{1}$ are larger for the $\beta=2$ model
($w_{1}=1.33^{+0.42}_{-1.56}$ for CC and $1.12^{+0.40}_{-1.09}$ for all $H$
data) if we compare with the CPL model; however, one should take into account
the factor $\frac{1}{2}$ in EoS (32): $w=w_{0}+w_{1}(1-a^{2})/2$.

To compare models with different number $N_{p}$ of free model parameters, we
use here the Akaike information criterion OdintsovSGS2019 ; NojiriOSGS:2021 ;
Akaike74 :

$\mathrm{AIC}=\min\chi^{2}_{tot}+2N_{p}.$ (33)

This criterion emphasizes the advantage of models with small number of
$N_{p}$. It can be seen in Table 3 for the mentioned models, where, for only
CC $H(z)$ data, the minimal Akaike values (33) are achieved for the
$\Lambda$CDM ($\mathrm{AIC}=1079.81$) and the $w$CDM ($\mathrm{AIC}=1080.97$)
models. However, for all $H(z)$ data, the $\beta=2$ model (32) with $N_{p}=5$
parameters appeared to be more successful than $\Lambda$CDM not only in
$\min\chi^{2}_{\mathrm{tot}}\simeq 1084.03$, but also with Akaike information
(33): $\mathrm{AIC}=1094.03$. The lowest $\mathrm{AIC}=1093.19$ is achieved
here for the $w$CDM model.

On the right side of the Table 3, we present the predictions of the same
models on their flat case $k=0$ ($\Omega_{k}=0$). In this case, the values
$\min\chi^{2}_{\mathrm{tot}}$ and AIC appeared to be larger, and the $\beta=2$
model (32) loses its advantage in comparison with CPL scenario for all $H(z)$
data.

Table 3: The best fits of $\min\chi^{2}_{\mathrm{tot}}$, AIC, and $H_{0}$ for the models $\beta=2$ (32), CPL (23), $w$CDM, $\Lambda$CDM, and these models in the spatially flat case ($k=0$).  Model | Data | $\min\chi^{2}_{\mathrm{tot}}$ | AIC | $H_{0}$ | Flat Model | $\min\chi^{2}_{\mathrm{tot}}/d.o.f$ | AIC | $H_{0}$
---|---|---|---|---|---|---|---|---
$\beta=2$ | CC | $1072.30$ | $1082.30$ | $68.40^{+1.86}_{-1.91}$ |  flat $\beta=2$ | $!1072.99/1080$ | $1080.99$ |  $68.46^{+1.82}_{-1.81}$
CPL | CC | $\\!\\!1072.45\\!\\!$ | $\\!\\!1082.45\\!\\!$ | $\\!\\!68.53^{+1.84}_{-1.88}\\!\\!$ | flat CPL | $\\!\\!1072.98/1080\\!\\!$ | $\\!\\!1080.98\\!\\!$ |  $\\!\\!68.47^{+1.81}_{-1.80}\\!\\!$
$\\!\\!w$CDM | CC | $\\!\\!1072.97\\!\\!$ | $\\!\\!1080.97\\!\\!$ | $\\!\\!68.58^{+1.85}_{-1.89}\\!\\!$ |  flat $w$CDM | $\\!\\!1073.01/1081\\!\\!$ | $\\!\\!1079.01\\!\\!$ |  $\\!\\!68.50^{+1.81}_{-1.79}\\!\\!$
$\\!\\!\Lambda$CDM | CC | $\\!\\!1073.81\\!\\!$ | $\\!\\!1079.81\\!\\!$ | $\\!\\!68.52^{+1.89}_{-1.94}\\!\\!$ |  flat $\Lambda$CDM | $\\!\\!1073.93/1082\\!\\!$ | $\\!\\!1077.93\\!\\!$ |  $\\!\\!69.21^{+1.92}_{-1.91}\\!\\!$
$\beta=2$ | all | $\\!\\!1084.03\\!\\!$ | $\\!\\!1094.03\\!\\!$ | $\\!\\!67.65^{+1.28}_{-1.26}\\!\\!$ |  flat $\beta=2\\!\\!$ | $\\!\\!1087.84/1116\\!\\!$ | $\\!\\!1095.84\\!\\!$ |  $\\!\\!67.81^{+1.12}_{-1.06}\\!\\!$
CPL | all | $\\!\\!1084.28\\!\\!$ | $\\!\\!1094.28\\!\\!$ | $\\!\\!67.76^{+1.15}_{-1.01}\\!\\!$ | flat CPL | $\\!\\!1087.56/1116\\!\\!$ | $\\!\\!1095.56\\!\\!$ |  $\\!\\!67.99^{+1.11}_{-1.16}\\!\\!$
$\\!\\!w$CDM | all | $\\!\\!1085.19\\!\\!$ | $\\!\\!1093.19\\!\\!$ | $\\!\\!67.98^{+1.11}_{-1.11}\\!\\!$ |  flat $w$CDM | $\\!\\!1089.90/1117\\!\\!$ | $\\!\\!1095.90\\!\\!$ |  $\\!\\!68.16^{+1.12}_{-1.14}\\!\\!$
$\\!\\!\Lambda$CDM | all | $\\!\\!1089.42\\!\\!$ | $\\!\\!1095.42\\!\\!$ | $\\!\\!69.02^{+1.04}_{-1.06}\\!\\!$ |  flat $\Lambda$CDM | $\\!\\!1089.96/1118\\!\\!$ | $\\!\\!1093.96\\!\\!$ |  $\\!\\!68.35^{+0.56}_{-0.55}\\!\\!$

##  V Discussion

We considered different cosmological models with variable equations of state
(EoS) for dark energy of the type (2) $p_{x}=w(a)\,\rho_{x}$, more precisely,
models with EoS (7),

$w(a)=w_{0}+w_{1}\frac{1-a^{\beta}}{\beta}a^{\gamma},$

generalizing the $\Lambda$CDM, $w$CDM, Chevallier–Polarski–Linder (CPL), and
Barboza–Alcaniz–Zhu–Silva (BAZS) models ChP ; Linder ; BarbozaAZS09 . These
scenarios with nonzero spatial curvature and with $\Omega_{k}=0$ were
confronted with observational data described in Section II and including SNe
Ia, CMB data, and two classes of the Hubble parameter estimates $H(z)$: from
cosmic chronometers (CC) and from line-of-sight baryonic acoustic oscillations
($H_{\mathrm{BAO}}$) data.

The results of our calculations for different models are presented in Tables 2
and 3, including minima of $\chi^{2}_{\mathrm{tot}}$, Akaike information
criterion (33), and the best fitted values with $1\sigma$ estimates of model
parameters. We also investigated the linear model (3), (22) with $N_{p}=5$
parameters (28), however it appeared to be unsuccessful because it achieved
the best fitted value of $\chi^{2}_{\mathrm{tot}}$ at $w_{1}$ very close to
zero (see Figure 1). In other words, the linear model (22) with the considered
observational data was reduced to the $w$CDM model with only $N_{p}=4$
parameters, but both models have the same $\min\chi^{2}_{\mathrm{tot}}$.

Unlike the linear model (22) CPL scenario (4), (23) with the same $N_{p}=5$
parameters (28) appeared to be more successful, in particular, for all $H(z)$
data, the CPL model yields $\min\chi^{2}_{\mathrm{tot}}\simeq 1084.28$ in
comparison with $1085.19$ for the $w$CDM and $1089.42$ for the $\Lambda$CDM
models. In this case, the best fitted value $w_{1}\simeq 0.62$ is far from
zero, so CPL is not reduced to the $w$CDM model.

We should remember that a large number $N_{p}$ of free model parameters is a
drawback of any model, and when we use, in Table 3, the Akaike information
criterion (33), the $w$CDM model with $\mathrm{AIC}=1093.19$ will have the
advantage over CPL with $\mathrm{AIC}=1094.28$ (and more essential advantage
over the $\Lambda$CDM model with $1095.42$). If we consider the generalized
model (7), (25) with $N_{p}=7$ and additional parameters $\beta$ and $\gamma$,
the Akaike expression (33) becomes too large and the model looks worse in
comparison with others.

However, our analysis and Figure 3 showed that the minimum
$\min\chi^{2}_{\mathrm{tot}}\simeq 1084.03$ of the generalized model (25) may
be achieved if we fix $\beta=2$, $\gamma=0$; the resulting model “$\beta=2$”
(32) has the same $N_{p}=5$ parameters (28) and absolutely minimal
$\min\chi^{2}_{\mathrm{tot}}$ for all $H(z)$ data (its $\mathrm{AIC}=1094.03$
is behind only the $w$CDM AIC). The behavior of one-parameter distributions
$\chi^{2}_{\mathrm{tot}}(H_{0})$ and their minima for all models with all
$H(z)$ data are shown in Figure 5.

As mentioned above, if we use only CC $H(z)$ data, we observe smaller
differences between minima of $\chi^{2}_{\mathrm{tot}}$ for the considered
models in Table 2. In this case, the Akaike criterion (33) gives advantage to
the $\Lambda$CDM model with minimal $N_{p}$.

Table 2 and Figures 2 and 4 demonstrate that the success of the CPL scenario
is achieved at lower (best fitted) values of $\Omega_{m}^{0}$ and at larger
values of $\Omega_{k}$, if we compare these results with the $w$CDM model.
This tendency is strengthened for the $\beta=2$ model (32). It looks natural,
because the CPL model (23) is the particular case of the BAZS model (21) with
$\beta=1$.

Figure 5: One-parameter distributions $\chi^{2}_{\mathrm{tot}}(H_{0})$ and
likelihood functions for models with all $H(z)$ data (the flat cases $k=0$ are
shown with dotted lines); $H_{0}$ estimates in the right panel are drawn as
whisker plots.

One can see in Tables 2 and 3 and Figures 2, 4, and 5 that the best fitted
values of the Hubble parameter $H_{0}$ are very close for models $w$CDM, CPL,
and $\beta=2$ in general and spatially flat cases. However, the predictions of
these models diverge with that of the $\Lambda$CDM model for all $H(z)$ data.
The last result is illustrated in Figure 5; it becomes more clear if we look
at Figure 2, keeping in mind that the $\Lambda$CDM model is the particular
case of the $w$CDM model when $w=-1$. One should add the observed difference
between $H_{0}$ estimates of all models when we compare CC and all $H(z)$
data.

In the left panels of Figure 5 we draw one-parameter distributions
$\chi^{2}_{\mathrm{tot}}(H_{0})$ and likelihood functions (31) ${\cal
L}(H_{0})$ for the considered models with all $H(z)$ data to clarify their
best fits of $H_{0}$. For the considered four models we demonstrate here the
results for the flat cases $k=0$ ($\Omega_{k}=0$) as dotted lines.

One can see that in the flat case $k=0$, the mentioned models yield
appreciably larger minima $\min\chi^{2}_{\mathrm{tot}}$, but the best fitted
values of $H_{0}$ change unessentially for the $w$CDM, CPL, and $\beta=2$
models. For the flat $\Lambda$CDM model, the graph
$\chi^{2}_{\mathrm{tot}}(H_{0})$ is inscribed between the correspondent graphs
of the general $\Lambda$CDM and flat $w$CDM models; hence, the $H_{0}$
estimate for the flat $\Lambda$CDM model lies between two estimates of the
mentioned scenarios.

All $H_{0}$ predictions of the $w$CDM, CPL, and $\beta=2$ models with all
$H(z)$ data fit the Planck 2018 estimation of the Hubble constant Planck18
(see Figure 5), but they are far from the SH0ES 2021 estimation Riess2021 .
All $H_{0}$ predictions with CC $H(z)$ data are very close for the four
considered models and their flat variants: they are larger but overlap the
Planck 2018 value; however, they cannot describe the tension with the SH0ES
data.

We can see from Table 3 that the best values of AIC (33) for all $H(z)$ data
demonstrate the $w$CDM and $\beta=2$ (32) models, but for CC $H(z)$ data, the
best AIC has the flat $\Lambda$CDM model.

###### Acknowledgements.

G.S.S. is grateful to Sergei D. Odintsov for useful discussions and support.

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