# A field- and time-normalized Bayesian approach to measuring the impact of a
publication

Emilio Gómez–Déniz, Pablo Dorta–González,
Department of Quantitative Methods in Economics and TiDES Institute,
University of Las Palmas de Gran Canaria, Spain
http://orcid.org/0000-0002-5072-7908 (EGD)
http://orcid.org/0000-0003-0494-2903(PDG) P. Dorta–González, Department of
Quantitative Methods in Economics, University of Las Palmas de Gran Canaria,
35017 Las Palmas de Gran Canaria, Spain. E-mail<EMAIL_ADDRESS>
###### Abstract

Measuring the impact of a publication in a fair way is a significant challenge
in bibliometrics, as it must not introduce biases between fields and should
enable comparison of the impact of publications from different years. In this
paper, we propose a Bayesian approach to tackle this problem, motivated by
empirical data demonstrating heterogeneity in citation distributions. The
approach uses the a priori distribution of citations in each field to estimate
the expected a posteriori distribution in that field. This distribution is
then employed to normalize the citations received by a publication in that
field. Our main contribution is the Bayesian Impact Score, a measure of the
impact of a publication. This score is increasing and concave with the number
of citations received and decreasing and convex with the age of the
publication. This means that the marginal score of an additional citation
decreases as the cumulative number of citations increases and increases as the
time since publication of the document grows. Finally, we present an empirical
application of our approach in eight subject categories using the Scopus
database and a comparison with the normalized impact indicator Field Citation
Ratio from the Dimensions AI database.

##### Keywords:

Normalized citation impact, field normalization, time normalization, Bayesian
score, citation obsolescence, citation potential, citation density.

##### Mathematics Subject Classification (2020):

62E10, 62P25.

#### Funding:

Emilio Gómez-Déniz was partially funded by grant PID2021-127989OB-I00
(Ministerio de Economía y Competitividad, Spain).

##  1 Introduction

There is a large literature on field-normalized citation counting. An overview
of this topic can be found in Bornmann and Marx, (2015), and Waltman, (2016).
However, the bibliography is much smaller in relation to time normalization.
In general, for simplicity, the accumulated citations of a document are
divided by the number of years since its publication. In this way, a citation
is given the same value regardless of when it occurred.

The use of research publications tends to decrease as the literature ages, and
authors tend to cite more recent documents while neglecting older ones, a
phenomenon known as literature obsolescence. Despite this, older papers still
have had more time to accumulate citations. Modeling literature obsolescence
has been studied since the 1960s, with De Solla Price, (1965) proposing a
negative exponential distribution for the decline of literature use over time,
while others suggest a lognormal distribution as a better measure Egghe and
Ravichandra Rao,, 1992; Gupta,, 1998. However, due to the ever-increasing rate
of scientific publication Bornmann and Marx,, 2015, older influential papers
may have had lower citation potential in the early stages of their publication
compared to younger influential papers.

Both because of the phenomenon of obsolescence, which causes the density of
citations to decrease as the age of the cited document increases, and because
of the increasing rate of growth of the scientific corpus, citations to older
documents should be given more weight than citations to more recent documents.
Greater weight in the sense of greater recognition when measuring the
cumulative impact of a publication.

The empirical citation data show heterogeneity in the distributions.
Therefore, the Bayesian approach could be used to address the problem of field
and time normalization. In this approach, the a priori distribution of
citations in each field is used to estimate the expected a posteriori
distribution in that field. This a posteriori distribution is then used to
normalize the citations received by a publication in that field.

Regarding the impact of publications, Pérez-Hornero et al., (2010) propose a
Bayesian approach that utilizes a weighting scheme based on previous impact
factors of specific journals. This method aims to obtain a reliable and
consistent version of the actual impact factor. The approach depends on
selecting a probability distribution for the citation process and a prior
distribution over the parameters. They suggest a Poisson-Gamma family, which
extends the negative binomial distribution and is suitable for analyzing over-
dispersed data, with the mean being the impact factor of the journal in the
previous year.

However, the Journal Impact Factor favours journals that concentrate a large
proportion of their citations in the first few years after publication, i.e.
journals in fields with high obsolescence. Thus, a proposal that gives an
increasing marginal score over time could benefit fields with less
obsolescence.

This paper adopts a Bayesian approach to tackle the issues of field and time
normalization and proposes a Bayesian Impact Score as a measure of a
publication’s impact. The score is characterized as increasing and concave
with the number of citations received and decreasing and convex with the age
of the publication. This implies that the additional value of a citation
decreases as the total number of citations increases, while it increases with
the time since the publication. Therefore, the impact score assigns less
weight to citations received in the early years after publication and
gradually reduces their influence as the total number of citations grows.

The paper is organized into several sections. Section 2 presents the
theoretical framework, while section 3 introduces the empirical data. Section
4 describes the model used in the study. Section 5 introduces the Bayesian
Impact Score, discussing its properties and elasticity. The paper then
presents in section 6 the results of numerical experiments conducted on eight
subject categories from the Scopus database and a comparison with the
normalized impact indicator Field Citation Ratio from the Dimensions AI
database. Finally, section 7 provides conclusions based on the study’s
findings.

##  2 Theoretical framework for field-normalized citation counting

Bibliometrics has a long-standing tradition of normalization, with literature
reviews on this topic available in Bornmann and Marx, (2015) and Waltman,
(2016). The problem of field-specific differences in citation counts arises in
the evaluation of research institutions. Interdisciplinary research
institutes, in particular, typically draw scholars from diverse disciplinary
backgrounds Wagner et al.,, 2001, making it challenging to compare citation
counts directly.

One of the fundamental principles of citation analysis is that the citation
counts of publications from different fields should not be compared directly,
as there are significant differences in citation density. Citation density
refers to the average number of citations per publication and is responsible
for the bias effect called "citation potential," a term introduced by
Garfield, (1979) based on the average number of references cited in a
publication. For instance, the biomedical field often has long reference lists
with over fifty references, while mathematics typically has short lists with
less than twenty references Dorta-González and Dorta-González, 2013a, .

Moreover, even within the same field, comparing the citation counts of
publications from different years directly, even after dividing by the age of
publication, is not recommended due to significant differences in the average
number of citations per year Dorta-González and Dorta-González, 2013b, . These
variations result from differences in citation habits, which influence both
the number of citations and the likelihood of being cited Leydesdorff and
Bornmann,, 2011; Zitt and Small,, 2008.

However, for practical purposes, it is necessary to compare publications from
different fields, years, or document types. In order to facilitate such
comparisons, normalized citation impact indicators have been proposed.

###  2.1 Normalized indicators based on average number of citations

Normalized indicators rely on the concept of the expected number of citations
for a publication, which is determined by calculating the average number of
citations of all publications in the same field, from the same year, and of
the same document type. Field normalization is traditionally based on a
classification system, such as the WoS journal subject categories, although
there are other options available Bornmann and Wohlrabe,, 2019. Under this
method, each publication is allocated to one or more fields, and its citation
impact is evaluated in comparison to other publications within the same field.

The relative citation rate (RCR) was among the earliest techniques developed
for normalizing citations across fields. Schubert and Braun, (1986) and
Vinkler, (1986) were the originators of this method. They determined the
average citation rate for a specific field or journal and used it as a
reference score to normalize papers published within the same field or
journal. This was accomplished by dividing the citation counts of each paper
by the reference score. The Field-Weighted Citation Impact (FWCI) is a variant
of this indicator, although it is highly correlated with the RCR Purkayastha
et al.,, 2019. However, the use of the arithmetic mean in RCR normalization
has been criticized because it is not suitable for skewed distributions with a
long tail as it is sensitive to outliers Glänzel and Moed,, 2013; Van Raan,,
2019.

A variant of the RCR can be obtained for a set of publications in two ways.
One way is by calculating the average of the normalized citation scores of the
publications in the set. The other way is by computing the ratio of the total
number of actual citations to the expected number of citations for the same
set of publications. There is no consensus in the literature as to which of
the two approaches is better. However, most researchers seem to prefer the
first approach, which is the average of ratios, over the second approach,
which is the ratio of averages. This preference is evident in the works of
Lundbert, (2007), Opthof and Leydesdorff, (2010), Thelwall, (2017) and Van
Raan et al., (2010). Nevertheless, some authors Moed,, 2010; Vinkler,, 2012
prefer the average ratio approach. Empirical studies comparing the two
approaches indicate that the differences are minor, especially at the level of
research institutions and countries Herranz and Ruiz-Castillo,, 2012;
Larivièere and Gingras,, 2011.

This paper uses a basic methodology centred on the normalization of actual
citation counts through the integration of expected citation counts.
Currently, the three major general purpose bibliographic databases provide
standardized citation indicators. Expected citation counts are defined as the
average citation counts of papers within the same field and year, giving rise
to the concept of CNCI (Category Normalised Citation Impact). CNCI is a metric
closely related to Clarivate Analytics’ InCites database. It is similar to the
Field-Weighted Citation Impact (FWCI) in Scopus’ SciVal database and the Field
Citation Ratio (FCR) in the Dimensions AI database.

These metrics, including CNCI, FWCI and FCR, share the common goal of
providing a normalized assessment of citation impact. They do this by
accounting for the variation that naturally occurs across fields and years. In
essence, they provide a more nuanced and equitable assessment of scholarly
impact by considering the different contexts in which scholarly research is
conducted.

###  2.2 Normalized indicators based on percentiles

McAllister et al. (1983) suggested using percentiles to address skewed
distributions. Typically, a field-specific threshold is selected to identify
highly cited publications. For example, Tijssen et al., (2002) consider the
top 1% and 10% most highly cited publications, while Van Leeuwen et al.,
(2002) focus on the top 5%, and González-Betancor and Dorta-González, (2017)
estimate the top 10% most highly cited publications empirically. Bornmann and
Williams, (2015) provide a recent review of percentile measures.

However, determining the exact number of publications above a given threshold
is often impractical because many publications in a field have the same number
of citations. Pudovkin and Garfield, (2009) and Leydesdorff et al., (2011)
propose some solutions to this issue. Waltman and Schreiber, (2013) review
this topic, and Schreiber, (2013) provide an empirical comparison.

Alternatively, Leydesdorff et al., (2011) propose dividing publications into
several classes based on percentiles of the citation distribution in a field
(e.g. below the 50th percentile, between the 50th and 75th percentile, etc.),
instead of distinguishing between highly cited and not highly cited
publications. A similar approach is presented by Glänzel, (2013) and Glänzel
et al., (2014).

Finally, it is worth noting that older papers have had more time to accrue
citations. However, as the scientific literature expands at an accelerating
rate (Bornmann and Mutz,, 2015), influential papers that are older may have
had a lower potential for citations in the short term than similarly
influential but younger papers. Furthermore, calendar year-based normalization
methods can yield imprecise results for recently published papers (Ioannidis
et al.,, 2016). In a recent empirical study comparing normalized indicators,
Dunaiski and Visser, (2019) found that percentile-based citation scores are
less affected by field and time biases than mean-based citation scores.

##  3 Empirical data

The empirical application used the Scopus database as the primary data source.
Eight subject categories (subfields) were considered, including three from the
natural sciences, two from the social sciences, one from the health sciences,
one from engineering and one from the humanities: Cell Biology, Economics &
Econometrics, Electrical & Electronic Engineering, General Chemistry, General
Medicine, General Physics & Astronomy, History and Library & Information
Science.

The selection of these subfields was guided by the authors’ previous
experience to include disciplines with diverse characteristics. Within each
subfield, one disciplinary journal was selected from the top 10% of the most
cited journals, as determined by the Scopus CiteScore. Then, for each selected
journal, all research articles published in 2019 and catalogued in the
database were collected. Finally, for each research article, the publication
years of all cited references less than 150 years old were downloaded. Table
1, which was used in Dorta-González and Gómez-Déniz, (2022), shows the index
of dispersion ($\mbox{ID}=var(X)/{\rm I\kern-2.2ptE}(X)$) together with the
mean and variance of the empirical data.

Table 1: Mean, variance and index of dispersion of the different journals studied | Ag. Cell | Amer. Econ. Rev. | IEEE Comm. Mag. | Acc. Chem. Res.
---|---|---|---|---
Mean | 9.21893 | 13.6897 | 4.68677 | 8.59967
Variance | 61.9357 | 258.575 | 49.128 | 115.578
ID | 6.71832 | 18.8883 | 10.4823 | 13.4398
| BMC Med. | Adv. in Theor. & MP | Am. Ant. | IEEE Trans. Inf. Th.
Mean | 8.22895 | 17.5842 | 21.6714 | 14.6476
Variance | 69.0984 | 270.698 | 467.049 | 254.451
ID | 8.39699 | 15.3944 | 21.5514 | 17.3715
Note: Ag. Cell: Aging Cell; Amer. Econ. Rev.: American Economic Review; IEEE
Comm. Mag.: IEEE Communications Magazine;
Acc. Chem. Res.: Accounts of Chemical Research; BMC Med.: BMC Medicine;
Adv. in Theor. & MP: Advances in Theoretical and Mathematical Physics; Am.
Ant.: American Antiquity;
IEEE Trans. Inf. Th.: IEEE Transactions on Information Theory.

To empirically analyze the behavior of our Bayesian Score, we used the
Dimensions AI database and its article-level normalized impact indicator, the
Field Citation Ratio (FCR). This choice was made because this database offers
Open Metrics, which provides open access to both citation counts and the FCR.
This contrasts with similar normalized indicators in Web of Science (InCites)
and Scopus (SciVal), where a subscription to these services is required to
access such information.

The FCR serves as a metric to measure the relative citation performance of a
publication compared to other articles within its field and of similar age. An
FCR greater than 1 indicates an above-average citation impact as determined by
the journal classification system used in the Dimensions AI database,
specifically the Field of Research (FoR) subject code and publication year.
This calculation is applied to all publications in Dimensions AI that are at
least 2 years old and published after the year 2000.

We focused on the eight journals listed in Table 1. We randomly selected a
simple sample of size $N=5048$ research articles from the 2014-2021 cohort,
each with a calculated FCR. This represents approximately 32% of the total
articles published during these years. We downloaded the total number of
citations from the database up to the year 2023. In addition, we retrieved the
FCR values and calculated the corresponding Bayesian Score based on the
definition given in this paper.

##  4 The model

We consider that $X_{1},X_{2},\dots,X_{t}$ are independent and identically
distributed random variables with values in ${\cal
X}\in{\mathbb{N}}=\\{0,1,\dots\\}$. They represent the number of citations of
a journal or a collection of journals in a subject category (in general, a set
of papers from the same year and as homogeneous as possible on the topic
addressed) in the last $t$ years. Following to Burrell, (2002) we are going to
assume that $X\equiv X_{i}$ initially follows a Poisson distribution with mean
$\theta\in\Theta>0$. That is,

$\displaystyle f(x|\theta)=\Pr(X=x)=\frac{\theta^{x}}{x!}\exp(-\theta),\quad
x=0,1,\dots$ (1)

Based on empirical data, it appears that citations of papers tend to decline
over time, particularly after reaching a certain peak. This trend can be
represented through a Poisson distribution model. Nevertheless, the Poisson
distribution assumes equidispersion, where the variance is identical to the
mean, rendering it unsuitable for defining the random variable $X$. Empirical
research has demonstrated that $X$ exhibits overdispersion, whereby the
variance surpasses the mean.

Additionally, it has been proposed that over-dispersion is associated with the
heterogeneity within the population of subject categories. Under these
circumstances, the parameter $\theta$ can be treated as a random variable that
varies across distinct journals within the same subject category. This
reflects the uncertainty surrounding this parameter, with its value
fluctuating from one entity to another, following a probability density
function. In this situation, it is assumed that the parameter adheres to a
gamma distribution, characterized by a shape parameter of $\alpha>0$ and a
rate parameter of $\beta>0$, i.e.

$\displaystyle\pi(\theta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\theta^{\alpha-1}\exp(-\beta\theta),\quad\theta>0.$
(2)

Thus, the unconditional distribution results a negative binomial distribution,
$X\sim NB(\alpha,1/(\beta+1))$. It is well-known that for mixed Poisson
distributions the variance is always greater than the mean.

Other mixing distributions, such as the inverse Gaussian distribution, can be
considered in practice, in addition to the gamma distribution. In line with
this, Burrell, (2005) explored the use of the negative binomial and beta
distributions as mixing distributions, resulting in the Waring distribution.

##  5 Approaching the Bayesian Impact Score via a loss function

The number of cites of a journal or a subject category of journals is thus
specified by the random variable $X$ following the probability mass function
$f(x|\theta)$ which depends on an unknown parameter $\theta$. In our case $X$
is assumed to follow a Poisson distribution. An impact score can be considered
as a functional, say $F\in{\cal F}$, that assigns to $X$, the number of cites,
a positive real number $F(X)$. Here ${\cal F}$ is the set of possible values
of the impact score. Let now $L:X\times{\cal F}\rightarrow\mathbb{R}^{+}$ be a
loss function that assigns to any $(x,F)\in\mathbb{N}\times{\cal F}$ the loss
sustained, $L(x,F)$, when $X$ takes the value $x$ and the impact score of the
journal is $F$. The impact score we propose can be viewed as a function that
assigns to each value $x\in\mathbb{N}=\\{0,1,\dots\\}$ a value within the set
${\cal F}\in\mathbb{R}^{+}$, the action space. The impact score is assumed to
take only positive values. The impact score should be determined such that
this expected loss is minimised. Thus,

$\displaystyle F=\arg\min_{F}{\rm I\kern-2.2ptE}[L(x,F)].$

The quadratic loss function, $L(x,F)=(x-F)^{2}$, is the most commonly used
loss function. Here, the unknown $F$ is the mean of the random variable $X$,
i.e., $F(\theta)={\rm I\kern-2.2ptE}(X|\theta)$, where $F$ is dependent on
$\theta$. While other loss functions exist, the use of this loss function is
often preferred due to its symmetry, which allows for fair compensation
between gains and losses.

The above procedure describes the impact score for a journal when $\theta$ is
known. However, the population of journals results in practise heterogeneous,
among the collective of journals in the subject category for which the journal
belongs. This can be seen as that $\theta$ is a particular realization of a
random variable $\Theta$, with a prior distribution $\pi(\theta)$. If prior
experience is not available, the corresponding impact score can be computed by
minimising the risk function, i.e. by minimising ${{\rm
I\kern-2.2ptE}}_{\pi(\theta)}\left[L(F(\theta),F)\right]$, as $F={\rm
I\kern-2.2ptE}_{\theta}\left[F(\theta)\right]$, which is the mean of the
unknown impact score $F(\theta)$ among all the journals in the population of
journals (the subject category).

In practice, it is common to have some prior experience or information
available. In this case, we can use a sample $\tilde{x}=(x_{1},\dots,x_{t})$
from the random variable $X$ to estimate the unknown impact score $F(\theta)$
by means of the Bayes estimate $F^{\ast}$, which minimises the Bayes risk.
This involves minimising ${\rm
I\kern-2.2ptE}_{\theta}\left[L(F(\theta),\theta)\right]$, where the
expectation is taken with respect to $\pi(\theta|\tilde{x})$, the posterior
distribution of the risk parameter $\theta$ given the sample information
$\tilde{x}$.

Observe that given the sample $\tilde{x}$ from (1) and assumed (2) as the
prior distribution we get that the posterior distribution of $\theta$ given
$\tilde{x}$ is proportional to

$\displaystyle\pi(\theta|\tilde{x})\propto\theta^{\alpha+x^{+}-1}\exp\left[-(\beta+t)\theta\right],$

where $x^{+}=\sum_{i=1}^{t}x_{i}$ is the number of cites in the sample.
Therefore, this posterior distribution is again a gamma, i.e. the likelihood
and prior are a conjugate pair, but with updated parameters given by,

$\displaystyle\alpha^{\ast}$ $\displaystyle=$ $\displaystyle\alpha+x^{+},$ (3)
$\displaystyle\beta^{\ast}$ $\displaystyle=$ $\displaystyle\beta+t.$ (4)

The decision-maker must now calculate the best estimator of the expected
number of $X$ given the past experience. To achieve this, they must calculate
the Bayes estimator of $F(\theta)$, which can be attained by updating the
parameters provided in (3)-(4) in ${\rm I\kern-2.2ptE}(\theta)=\alpha/\beta$
to get

$\displaystyle F(x^{+},t)={\rm
I\kern-2.2ptE}_{\pi(\theta|\tilde{x})}(F(\theta)|\tilde{x})=\frac{\alpha+x^{+}}{\beta+t}.$
(5)

We now have all the ingredients needed to formulate an expression for the
impact score to a journal which $x^{+}$ cites in the previous $t$ years.
First, we normalize the expression given in (5) such that at the outset
$(x^{+},t)=(0,0)$, which is given by and due that the impact score should be
revised every two years, we allow $t$ to take even values including the zero.
That is,

$\displaystyle
F^{\ast}(x^{+},t)=\frac{F(x^{+},c)}{F(0,0)}=\frac{\beta}{\alpha}\frac{\alpha+x^{+}}{\beta+t},\quad
x^{+}=0,1,\dots;t=0,2,4,\dots$ (6)

The expression (6) assures us that at the beginning of the process, i.e.
$(x^{+},t)=(0,0)$, the impact score is 1. This is the impact score for a new
journal which enters into the system, since no available information exists
for it. The remaining combinations of $x^{+}$ and $t$ are expressed as a
percentage of this first value.

In practice, if the decision-maker wishes the initial rate to be a value other
than 1, it could simply be replaced (6) by

$\displaystyle
F^{\ast}(x^{+},t)=\frac{\omega\,F(x^{+},c)}{F(0,0)}=\frac{\omega\,\beta}{\alpha}\frac{\alpha+x^{+}}{\beta+t},\quad
x^{+}=0,1,\dots;t=0,2,4,\dots$ (7)

where $\omega>0$ is a constant.

###  5.1 Properties of the Bayesian Impact Score and its elasticity

The following relations of the Bayes rate suggested are compatible with the
ideas about a impact score should verify,

$\displaystyle\Delta F^{\ast}_{x^{+}}(x^{+},t)$ $\displaystyle=$
$\displaystyle
F^{\ast}(x^{+}+1,t)-F^{\ast}(x^{+},t)=\frac{\omega\,\beta}{\alpha(\beta+t)}>0,$
(8) $\displaystyle\Delta F^{\ast}_{t}(x^{+},t)$ $\displaystyle=$
$\displaystyle
F^{\ast}(x^{+},t+2)-F^{\ast}(x^{+},t)=-\frac{2\omega\beta(\alpha+x^{+})}{\alpha(\beta+t)(\beta+t+2)}<0.$
(9)

The equation (8) suggests that, all else being equal, the impact score will
increase as the number of citations increases. In contrast, according to
equation (9), the impact score will decrease as $t$ increases, while keeping
the number of citations constant.

Observe that it is possible to write (7) as

$\displaystyle
F^{\ast}(x^{+},t)=\frac{\omega\,\beta}{\alpha}\left[\gamma(t)\bar{x}+\left[1-\gamma(t)\right]{\rm
I\kern-2.2ptE}(\theta)\right],\quad 0<\gamma(t)<1,$ (10)

where $\bar{x}$ is the sample mean and $\gamma(t)=t/(\beta+t)$. That is, this
Bayesian estimate of the impact score can be written as the convex (weighted)
sum of the prior mean of $\Theta$, and the mean of the number of cites in the
sample, where the weighted factor $\gamma(t)$ can be rewritten see Ericson,,
1969 as

$\displaystyle\gamma(t)=\frac{t}{\beta+t}=\frac{t\;var\left[{\rm
I\kern-2.2ptE}(X|\theta)\right]}{t\;var\left[{\rm
I\kern-2.2ptE}(X|\theta)\right]+{\rm
I\kern-2.2ptE}\left[var(X|\theta)\right]}.$

The quantity $var[{\rm I\kern-2.2ptE}(X|\theta)]$ provides a measure of the
heterogeneity of subject categories, while ${\rm
I\kern-2.2ptE}[var(X|\theta)]$ represents a global measure of the dispersion
of these means across all journals in the population. It should be noted that
$\gamma(t)$ is a specific function that depends on the model parameters, such
as $\beta$ in this case, rather than an arbitrary constant, and is determined
by both the journal and the collective citation experiences.

Moreover, it is worth mentioning that $\gamma(t)$ is supported on the interval
$(0,1)$ and approaches 0 as $\beta\to 0$, while $\gamma(\infty)\to 1$,
although these are only limiting cases. In practice, the value of $\gamma(t)$
falls between 0 and 1, implying that $F^{\ast}(x^{+},t)$ will always depend on
the estimated parameters of the model, which will be obtained using an
empirical Bayes approach. However, these extreme values have meaningful
interpretations: when $\gamma(t)$ is close to 1, the journal’s prior
experience has a more significant impact on the final impact score, whereas
when $\gamma(t)=0$, less weight is assigned to this experience, and the a
priori information plays a more dominant role.

Table 3 presents the impact score values, as computed using the formula in
(7), for various journals and different values of $x^{+}$ and $t$, based on
the parameter estimates provided in Table 2.

In addition, expressions (7), (8), and (9) can be employed to estimate the
elasticity of the impact score, which measures the impact of changes in the
number of citations on the score for a given journal or subject category. This
estimator is obtained by

$\displaystyle\eta_{x^{+}}=\frac{\Delta F_{x^{+}}^{\ast}(x^{+},t)}{\Delta
x^{+}}\frac{x^{+}}{F^{\ast}(x^{+},t)}=\frac{x^{+}}{\alpha+x^{+}}.$ (11)

Observe that from (11), we have that $\eta_{x^{+}}<1$, i.e. an inelastic
function, being an increasing function on $x^{+}$ and decreasing with respect
to $\alpha$. Then that category (journal) with an estimated value of $\alpha$
greater than another category (journal) will have a worse response in the
posterior mean of $X$ from one period to another.

On the other hand, the elasticity of (10) with respect to $t$ results

$\displaystyle\eta_{t}=\frac{\Delta F_{t}^{\ast}(x^{+},t)}{\Delta
t}\frac{t}{F^{\ast}(x^{+},t)}=-\frac{t}{\beta+t+2},$ (12)

which results negative and again lower than 1. Furthermore, it is an
increasing function on $\beta$.

##  6 Numerical experiments

Table 2 displays the parameter estimates for the journals using both Poisson
and negative binomial distributions. Model selection was based on Akaike’s
information criterion, which is calculated as $\mbox{AIC}=2(k-\ell_{\max})$,
where $k$ represents the number of model parameters and $\ell_{\max}$
represents the maximum value of the log-likelihood function. For further
details on this criterion, see Akaike, (1974). A lower value of AIC is
indicative of a better model fit, and thus desirable for model selection.

Table 2: Estimates, via maximum likelihood, of the parameters of the Poisson and negative binomial distributions and AIC values for the different journals studied Journal | Poisson |  | NB
---|---|---|---
| $\widehat{\theta}$ | AIC |  | $\widehat{\alpha}$ | $\widehat{\beta}$ | AIC
Ag.Cell | 9.21 | 61920.80 |  | 1.99 | 0.21 | 42425.50
Amer. Econ. Rev. | 13.68 | 71880.30 |  | 1.03 | 0.07 | 29571.10
IEEE Comm. Mag. | 4.68 | 21884.30 |  | 1.27 | 0.27 | 13981.00
Acc. Chem. Res. | 8.59 | 219046.00 |  | 1.08 | 0.12 | 112400.00
BMC Med. | 8.22 | 83089.30 |  | 1.45 | 0.17 | 51546.40
Adv. in Theor. & MP | 17.58 | 28589.80 |  | 1.34 | 0.07 | 12752.40
Am. Ant. | 21.67 | 40815.70 |  | 1.15 | 0.05 | 14954.80
IEEE Trans. Inf. Th. | 14.64 | 172890.00 |  | 1.14 | 0.07 | 72734.40

In all cases, it is apparent that the negative binomial distribution
outperforms the Poisson distribution for the datasets considered. This
superiority of the negative binomial distribution over the Poisson
distribution and the empirical data is further supported by Figure 1.

Figure 1: Empirical and fitted distribution using Poisson (thin line) and
negative binomial (thick line) distributions

Figure 2 shows the graphics of elasticities given in (11) and (12) for the
different journals considered. The citation elasticity (time elasticity) of
the impact score is a measure used to show the degree of response, or
elasticity, of a journal’s impact to changes in the number of citations
received (time). It gives the percentage change in the impact generated in
relation to a unit percentage change in the citations (time), considering that
time (citations) remain constant (ceteris paribus).

Figure 2: Elasticity of the impact score for the different journals considered

In general, the impact score of a journal can be considered inelastic (or
relatively inelastic) when the elasticity is less than one (in its absolute
value); this happens when changes in citations (time) have a relatively small
effect on the impact generated by the journal. As seen in Figure 2, the impact
score of the journals considered in this application is inelastic with respect
to citations and time (in absolute value), although its elasticity tends to
increase with the volume of citations and time.

The elasticity is therefore a measure of the sensitivity (or responsiveness)
of the impact score to changes in its variables. The formula returns a
positive or negative result, depending on whether the relationship between the
variable and the impact is direct or inverse. For example, if citation
increases by 10% and impact increases by 5%, the citation elasticity is 5%/10%
= 0.5. However, if time (the age of the cited reference) increases by 10% and
impact decreases by 5%, the time elasticity is –5%/10% = –0.5.

It should be noted that the conclusions drawn here are based on a small sample
of journals and not on the entire Scopus publication corpus. As can be seen in
Figure 2, the differences between the journals considered in this empirical
application in terms of elasticities are small, almost imperceptible to the
naked eye in the case of time elasticity. The elasticity of the impact score
with respect to the number of citations (citation elasticity) is higher for
journals in the social sciences and humanities (i.e. economics and
archaeology) and lower for journals in the natural sciences and health (i.e.
biology and medicine). The opposite relationships are observed for the
elasticity of the impact score with respect to time (time elasticity) in
absolute value. Except for one engineering journal (IEEE Comm. Mag.) that
exhibits the smallest absolute value of time elasticity, the impact score’s
elasticity in relation to time is less for social sciences and humanities
journals (such as economics and archaeology) and greater for natural sciences
and health journals (such as biology and medicine). Note that in the case of
negative values, the smallest numbers become the largest in absolute value.

Table 3: Estimated factor for the journals considered assuming that $\omega=1$. | Number of cites: $x^{+}$ |
---|---|---
Year: $t$ | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
2 | 0.10 | 0.57 | 1.05 | 1.53 | 2.01 | 2.48 | 2.96 | 3.44 | 3.92 | 4.39 | 4.87 | Ag. Cell
4 | 0.05 | 0.30 | 0.55 | 0.80 | 1.05 | 1.30 | 1.55 | 1.80 | 2.06 | 2.31 | 2.56 |
6 | 0.03 | 0.20 | 0.37 | 0.54 | 0.71 | 0.88 | 1.05 | 1.22 | 1.39 | 1.56 | 1.73 |
8 | 0.03 | 0.15 | 0.28 | 0.41 | 0.54 | 0.67 | 0.80 | 0.93 | 1.05 | 1.18 | 1.31 |
2 | 0.03 | 0.36 | 0.69 | 1.02 | 1.35 | 1.68 | 2.00 | 2.33 | 2.66 | 2.99 | 3.32 | Amer. Econ. Rev.
4 | 0.02 | 0.18 | 0.35 | 0.52 | 0.69 | 0.85 | 1.02 | 1.19 | 1.35 | 1.52 | 1.69 |
6 | 0.01 | 0.12 | 0.24 | 0.35 | 0.46 | 0.57 | 0.68 | 0.80 | 0.91 | 1.02 | 1.13 |
8 | 0.00 | 0.09 | 0.18 | 0.26 | 0.35 | 0.43 | 0.51 | 0.60 | 0.68 | 0.77 | 0.85 |
2 | 0.12 | 1.06 | 1.99 | 2.93 | 3.87 | 4.80 | 5.74 | 6.67 | 7.61 | 8.55 | 9.48 | IEEE Comm. Mag.
4 | 0.06 | 0.56 | 1.06 | 1.56 | 2.05 | 2.55 | 3.05 | 3.55 | 4.05 | 4.54 | 5.04 |
6 | 0.04 | 0.38 | 0.72 | 1.06 | 1.40 | 1.74 | 2.08 | 2.42 | 2.76 | 3.09 | 3.43 |
8 | 0.03 | 0.29 | 0.55 | 0.80 | 1.06 | 1.32 | 1.58 | 1.83 | 2.09 | 2.35 | 2.60 |
2 | 0.06 | 0.58 | 1.10 | 1.63 | 2.15 | 2.68 | 3.20 | 3.73 | 4.25 | 4.77 | 5.30 | Acc. Chem. Res.
4 | 0.03 | 0.30 | 0.57 | 0.84 | 1.11 | 1.38 | 1.65 | 1.92 | 2.19 | 2.46 | 2.73 |
6 | 0.02 | 0.20 | 0.38 | 0.56 | 0.75 | 0.93 | 1.11 | 1.29 | 1.47 | 1.65 | 1.84 |
8 | 0.01 | 0.15 | 0.29 | 0.43 | 0.56 | 0.70 | 0.84 | 0.97 | 1.11 | 1.25 | 1.38 |
2 | 0.08 | 0.62 | 1.16 | 1.70 | 2.24 | 2.78 | 3.32 | 3.86 | 4.40 | 4.94 | 5.48 | BMC Med.
4 | 0.04 | 0.32 | 0.60 | 0.88 | 1.17 | 1.45 | 1.73 | 2.01 | 2.29 | 2.57 | 2.85 |
6 | 0.03 | 0.22 | 0.41 | 0.60 | 0.79 | 0.98 | 1.17 | 1.36 | 1.55 | 1.74 | 1.93 |
8 | 0.02 | 0.16 | 0.31 | 0.45 | 0.59 | 0.74 | 0.88 | 1.03 | 1.17 | 1.31 | 1.46 |
2 | 0.03 | 0.29 | 0.54 | 0.79 | 1.04 | 1.30 | 1.55 | 1.80 | 2.05 | 2.31 | 2.56 | Adv. in Theor. & MP
4 | 0.02 | 0.15 | 0.27 | 0.40 | 0.53 | 0.66 | 0.79 | 0.92 | 1.04 | 1.17 | 1.30 |
6 | 0.01 | 0.10 | 0.18 | 0.27 | 0.36 | 0.44 | 0.53 | 0.61 | 0.70 | 0.79 | 0.87 |
8 | 0.00 | 0.07 | 0.14 | 0.20 | 0.27 | 0.33 | 0.40 | 0.46 | 0.53 | 0.59 | 0.66 |
2 | 0.02 | 0.24 | 0.45 | 0.66 | 0.87 | 1.08 | 1.30 | 1.51 | 1.72 | 1.93 | 2.15 | Am. Ant.
4 | 0.01 | 0.12 | 0.23 | 0.33 | 0.44 | 0.55 | 0.66 | 0.76 | 0.87 | 0.98 | 1.09 |
6 | 0.00 | 0.08 | 0.15 | 0.22 | 0.30 | 0.37 | 0.44 | 0.51 | 0.58 | 0.66 | 0.73 |
8 | 0.00 | 0.06 | 0.11 | 0.17 | 0.22 | 0.28 | 0.33 | 0.38 | 0.44 | 0.49 | 0.55 |
2 | 0.03 | 0.33 | 0.63 | 0.92 | 1.22 | 1.52 | 1.81 | 2.11 | 2.41 | 2.70 | 3.00 | IEEE Trans. Inf. Th.
4 | 0.02 | 0.17 | 0.32 | 0.47 | 0.62 | 0.77 | 0.92 | 1.07 | 1.22 | 1.38 | 1.53 |
6 | 0.01 | 0.11 | 0.21 | 0.32 | 0.42 | 0.52 | 0.62 | 0.72 | 0.82 | 0.92 | 1.02 |
8 | 0.00 | 0.08 | 0.16 | 0.24 | 0.31 | 0.39 | 0.47 | 0.54 | 0.62 | 0.69 | 0.77 |

In the interpretation at the paper level, the impact score of the paper
appears directly in Table 3 when crossing the age of the paper ($t$) and the
citations received ($x^{+}$). This normalization of the impact of a paper is
in relation to the citation frequencies observed in the past in each journal.
It is adjusted, therefore, to the obsolescence of the literature on the topic
of the journal. For reasons of space, the scores in Table 3 are shown with
only two decimals and in intervals or classes of two years and 10 citations.
However, it can be extended in a similar way to smaller intervals, and even to
one citation and one year.

Thus, for example, in the American Economic Review, a highly prestigious
publication in economics, a paper with 10 citations after two years (an
average of 5 cites/year), its impact can be quantified in a score of 0.36. If
said paper does not receive citations in the two subsequent years (third and
fourth of life), its score is reduced by half (0.18). On the other hand, if
during the third and fourth year of life it receives another 10 additional
citations (same average of 5 cites/year), then its score would be 0.35,
slightly lower than it was in the second year.

Observe in this example that with a uniform citation distribution of 5
cites/year, the marginal score of the first citations is somewhat higher than
that of the last ones. This is because in this journal the articles usually
receive fewer citations during the first two years of life than during the
following two, so the marginal score of citations in the field defined by the
topic of the journal decreases during the first years of life. However, the
opposite happens when we exceed the peak of the citation distribution, which
for this journal is between 4 and 6 years. This is observed in the rest of the
values of the diagonal that are again 0.35.

Now suppose that a different article in this journal receives 20 citations
during its first two years (at a rate of 10 cites/year). In this case, its
impact score is 0.69. If this citation rate remains constant, then its score
remains at 0.69 during the second interval (third and fourth year of life),
although it slightly decreases to 0.68 in the following intervals.

As can be seen, the same number of citations has a different impact score
depending on the expected frequencies in the topic of each journal. These
scores are comparable within the same journal, but also between journals from
different fields. Thus, for example, for similar values of citations and
years, the scores in the communication journal are much higher (almost double)
than those in the chemistry journal, for example. Moreover, scores in medicine
are somewhat higher than those in biology. The lowest scores are reached in
archaeology and physics, thus indicating that the citation density is higher
in these fields.

To empirically analyze the behavior of our Bayesian Score, we used the
Dimensions AI database and its article-level normalized impact indicator, the
Field Citation Ratio (FCR). The FCR serves as a metric to measure the relative
citation performance of a publication compared to other articles within its
field and of similar age. An FCR greater than 1 indicates an above-average
citation impact as papers in the same field and publication year. For the
eight journals listed in Table 1, we randomly selected a simple sample of size
N=5048 research articles from the 2014-2021 cohort. This represents
approximately 32% of the total articles published during these years.

The Pearson linear correlation coefficient between the FCR and the Bayesian
Score is 0.68. This value indicates a significant correlation between the two
measures, as would be expected when trying to measure the same phenomena.
However, it also indicates that there are some differences between these
normalized measures. We have therefore looked at the descriptive statistics of
the data distributions to try to identify where these observed differences
between the two indicators come from.

The coefficient of variation (CV) in Table 4 is a measure of relative
variability used to express the dispersion of a sample relative to its mean.
It is calculated by dividing the sample standard deviation (SD) by the mean.
This coefficient is useful for comparing the relative dispersion between two
data sets, especially when the scales of measurement are different, as is the
case here, as the CV is dimensionless. A low CV indicates low variability
relative to the mean, while a high CV indicates higher relative variability.

As can be seen in Table 4, the differences in CV between journals from
different disciplines are considerable. In the case of the Bayesian Score,
these differences are comparatively smaller, indicating greater consistency in
this new indicator. This can be seen in the range of variation for this
measure across journals (0.62-3.26 for the FCR and 0.67-2.31 for the Bayesian
Score). However, the discipline of the journal also appears to have a
significant impact on the CV, highlighting the importance of considering the
scientific area of application when using either of these indicators. It can
also be concluded that the ranking of journals from highest to lowest CV is
similar according to both metrics. The only difference is between the American
Economic Review and Aging Cell, where they exchange positions between sixth
and seventh place depending on the metric considered.

Table 4: Descriptive statistics for Field Citation Ratio (FCR) and Bayesian Score in a random sample. The coefficient of variation (CV) is defined as the ratio of the standard deviation (SD) to the mean (CV = SD/mean) FCR | $n$ | Mean | SE | Median | SD | CV | Kurtosis | Skewness | $\min$ | $\max$
---|---|---|---|---|---|---|---|---|---|---
Ag. Cell | 385 | 6.53 | 0.298 | 4.32 | 5.847 | 0.90 | 3.453 | 1.766 | 0 | 35.5
Amer. Econ. Rev. | 346 | 18.52 | 0.719 | 16.38 | 13.368 | 0.72 | 0.086 | 0.748 | 0.7 | 67.7
IEEE Comm. Mag. | 1072 | 4.88 | 0.253 | 1.24 | 8.290 | 1.70 | 5.880 | 2.390 | 0 | 52.4
Acc. Chem. Res. | 454 | 7.61 | 0.222 | 6.94 | 4.739 | 0.62 | 0.597 | 0.811 | 0.3 | 24.9
BMC Med. | 597 | 8.74 | 0.339 | 5.82 | 8.286 | 0.95 | 4.209 | 1.732 | 0 | 56.3
Adv. in Theor. & MP | 124 | 3.43 | 0.521 | 1.73 | 5.797 | 1.69 | 36.923 | 5.294 | 0 | 49.9
Am. Ant. | 439 | 3.13 | 0.487 | 0.00 | 10.204 | 3.26 | 116.771 | 9.602 | 0 | 144.9
IEEE Trans. Inf. Th. | 1631 | 4.14 | 0.127 | 2.57 | 5.124 | 1.24 | 11.326 | 3.043 | 0 | 44.1
Bayesian Score | $n$ | Mean | SE | Median | SD | CV | Kurtosis | Skewness | $\min$ | $\max$
Ag. Cell | 385 | 0.94 | 0.036 | 0.57 | 0.702 | 0.75 | 4.099 | 1.708 | 0 | 4.9
Amer. Econ. Rev. | 346 | 0.69 | 0.031 | 0.52 | 0.581 | 0.85 | 3.003 | 1.532 | 0 | 3.3
IEEE Comm. Mag. | 1072 | 0.75 | 0.039 | 0.12 | 1.284 | 1.70 | 10.709 | 2.868 | 0 | 9.5
Acc. Chem. Res. | 454 | 1.58 | 0.049 | 1.38 | 1.053 | 0.67 | 1.630 | 1.152 | 0 | 5.3
BMC Med. | 597 | 0.90 | 0.035 | 0.62 | 0.853 | 0.94 | 4.427 | 1.770 | 0 | 5.5
Adv. in Theor. & MP | 124 | 0.08 | 0.011 | 0.03 | 0.121 | 1.61 | 19.965 | 3.996 | 0 | 0.9
Am. Ant. | 439 | 0.05 | 0.005 | 0.02 | 0.111 | 2.31 | 45.134 | 5.592 | 0 | 1.3
IEEE Trans. Inf. Th. | 1631 | 0.17 | 0.006 | 0.11 | 0.223 | 1.33 | 17.910 | 3.322 | 0 | 2.7

The CV is therefore related to the discriminatory power of a measure. The
discriminatory power of an impact measure refers to its ability to effectively
discriminate between the impact levels of different papers. In the context of
citation-based impact measures, the CV plays a role in this discrimination. A
higher CV implies greater variability in the citation scores of different
publications, suggesting a wider range of impact levels. This can increase the
discriminatory power, as it reflects more pronounced differences in the impact
of individual papers. Researchers or evaluators may find such a measure
valuable in identifying and prioritizing highly impactful papers, taking into
account a more diverse impact landscape. Conversely, a lower CV implies a more
uniform distribution of citations, which may lead to reduced discriminatory
power. While this may provide a more consistent view of impact across papers,
it may struggle to discriminate subtle differences in impact. The
discriminatory power of an impact measure therefore depends on the specific
aims and preferences of the analysis. A higher CV might be preferred if the
aim is to capture different levels of impact, while a lower CV might be chosen
for a more uniform and stable assessment of impact.

In essence, a higher CV highlight the heterogeneity in the impact of scholarly
papers and emphasize that a few can dominate the citation landscape.
Researchers and evaluators considering impact measures need to be aware of
these dynamics and choose metrics that are consistent with their objectives,
whether they seek to capture diversity in impact (higher CV) or prioritize
stability and uniformity (lower CV).

##  7 Conclusions

A central problem in bibliometrics is how to compare the impact of
publications across different fields and years. While there is a large
literature on field-normalized citation counts, there is much less research on
time normalization.

Traditionally, citation counts are divided by the number of years since a
publication’s release to simplify the comparison process. This approach
assumes that all citations are equally valuable, regardless of when they were
made. However, as research publications age, their usage tends to decrease,
and authors tend to cite more recent publications instead of older ones,
resulting in a phenomenon known as literature obsolescence.

Furthermore, older papers have had more time to accumulate citations, but due
to the increasing rate at which the scientific corpus is expanding, older
influential papers had a lower citation potential shortly after publication
than younger influential papers. These issues highlight the importance of
developing a time normalization method that considers these factors for more
accurate impact comparisons.

Both because of the phenomenon of obsolescence, which causes the density of
citations to decrease as the age of the cited document increases, and because
of the increasing rate of growth of the scientific corpus, citations to older
documents should be more widely recognised than citations to more recent
documents.

The empirical citation data show heterogeneity in the distributions.
Therefore, the Bayesian approach is used to solve the problem of field and
time normalization. In this approach, the a priori distribution of citations
in each field is used to estimate the expected a posteriori distribution in
that field. This a posteriori distribution is then used to normalize the
citations received by a publication in that field.

The proposed Bayesian Impact Score is increasing and concave with the number
of citations received and decreasing and convex with the age of the
publication. This means that the marginal score of an additional citation
decreases with the cumulative number of citations and increases with the time
since publication.

This impact score gives less value to citations received in the first few
years after publication and reduces its value as the accumulated volume of
citations increases. This reduction in score with citation volume attempts to
mitigate the effect whereby highly cited documents attract more attention and
tend to receive more citations simply because they are highly cited.

Some considerations can be made. The well-known Journal Impact Factor favours
journals that concentrate a large proportion of their citations in the first
few years after publication, i.e. journals in fields with high obsolescence.
However, our proposal, which gives an increasing marginal score over time, is
fairer in fields with less obsolescence.

A citation loss function is used to fit the citations received by a journal to
the expected citation distribution in its field. It is a bivariate measure and
its properties include that it increases with citations, decreases over time,
and is comparable across fields. It is also aggregable in the sense that it
can be used at the level of authors by simply adding the scores of each of
their publications.

In conclusion, the normalization of citation impact indicators is crucial for
accurate assessment of research output, and this can be achieved by assigning
publications to specific fields. Although the WoS journal subject categories
are still the most commonly utilized field classification system, questions
have emerged about the reliability of normalized indicators based on the
selection of the classification system. In addition, scholars have
investigated the feasibility of substituting alternative classification
systems for the WoS journal subject categories. It is important to note that
our proposal avoids using a journal classification system altogether, instead
utilizing the journal of publication to define expected citations within the
subject area of each paper. This approach addresses some of the concerns
raised about the use of traditional classification systems, and could
potentially offer a more robust and accurate method for normalization.

#### Funding and/or Conflicts of interests/Competing interests:

The authors have no competing interests to declare that are relevant to the
content of this article.

#### Acknowledgement

The authors would like to thank to the Editor and two anonymous referees for
valuable comments and suggestions which improved the presentation of the
paper.

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# First comprehensive constraints on the Finslerian models using cosmological
observations

Deng Wang<EMAIL_ADDRESS>Theoretical Physics Division, Chern
Institute of Mathematics, Nankai University, Tianjin 300071, China  Xin-He
Meng<EMAIL_ADDRESS>Department of Physics, Nankai University, Tianjin
300071, China

###### Abstract

We perform the first comprehensive constraints on two Finslerian models, i.e.,
the simplest Finslerian $\Lambda$CDM (F$\Lambda$) and non-flat F$\Lambda$
models using the Type Ia supernovae, baryonic acoustic oscillations, cosmic
microwave background and lensing observations. Using the most stringent
constraints we can provide, we find that the constrained typical parameters of
both Finslerian models are all consistent with zero at the $2\sigma$
confidence level (C.L.). This means that both Finslerian models just deviate
very slightly from the $\Lambda$CDM one, which is also verified by using two
geometrical diagnostics to distinguish three different models from each other.
We also find that a spatially flat universe is still preferred in the
framework of Finsler geometry, that our measured values of the spectral index
$n_{s}$ of primordial power spectrum in both Fisnlerian models exclude the
scale invariance at more than $8\sigma$ C.L., and that the current $H_{0}$
tension can be relieved from $3.4\sigma$ to $2.9\sigma$ and $2.8\sigma$ in the
F$\Lambda$ and non-flat F$\Lambda$ models, respectively.

##  I Introduction

Modern cosmology has entered a new era driven by “ big data ”. In the past
about two decades, with gradually mounting data, a number of cosmological
observations such as Type Ia supernovae (SNIa) 1 ; 2 , baryonic acoustic
oscillations (BAO) 3 , cosmic microwave background (CMB) anisotropies 4 ; 5 ,
etc., have strongly indicated that the universe is undergoing a phase of
accelerated expansion. The simplest scenario to explain the intriguing
phenomena is a combination of the cosmological constant and cold dark matter
component ($\Lambda$CDM model) 6 , which is still in remarkably good agreement
with almost all cosmological data more than ten years after the observational
discovery of the accelerated expansion rate of the universe. Nonetheless, this
very successful model still faces two unsolved and attractive puzzles 7 : (i)
Why is the value of the cosmological constant unexpectedly small with respect
to any physically meaningful scale, except the current horizon scale ? (ii)
Why this value is not only small, but also surprisingly close to another
unrelated physical quantity, the present matter density ? Since we are still
unclear about the nature of dark sector of the universe, to alleviate or even
solve these puzzles, a great deal of alternative cosmological models based on
different physical origins are proposed. In general, they can be divided into
two main classes, the dark energy model 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ;
16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 22 ; 23 ; wm ; 24 , which introduces a new fluid
or field in the universe, and the extended theory of gravity 25 ; 26 ; 27 ; 28
; 29 ; 30 ; 31 ; 32 ; 33 ; 34 ; 35 , which modifies the standard lagrangian of
Einstein’s gravity. Very interestingly, one can also address the above puzzles
by exploring the connection between gravitation and new geometry. Finsler
geometry 24 ; 25 ; 26 , which includes Riemann geometry as its special case,
is a good candidate to understand the current cosmological puzzles. This new
geometry keeps the elegant properties of Riemann geometry, i.e., the
corresponding isometric group is a Lie group on a Finslerian manifold, while
it admits less Killing vectors than a Riemannian spacetime does. Generally,
there are $n(n-1)/2+1$ independent Killing vectors in a $n$ dimensional non-
Riemannian Finslerian spacetime at most. Taking the simplest possible
asymmetrical generalization of Riemannian metric into account, G. Randers 36
proposed the so-called Randers space, a subclass of Finslerian space. In the
framework of Randers space, a generalized Friedmann-Robertson-Walker (FRW)
cosmological model based on Finsler geometry has been studied 37 , and a
modified dispersion relation of free particles has also been discussed 38 .

The gravity in a Finslerian space were studied for a long time 39 ; 40 ; 41 ;
42 . The gravitational field equations (GFEs) derived from a Riemannian
osculating metric were presented in Ref. 43 . For such a metric, the FRW-like
cosmological scenario and the anisotropies of the universe were also
investigated 37 ; 44 . However, their GFEs are derived without satisfying the
Bianchi identity and the general covariance principle of Einstein’s gravity.
Attractively, the authors in Refs. 45 ; 46 ; 47  have solved these problems
and derived the corresponding Friedmann-like equations in Ref. 45  by
constructing a Randers-Finsler space of approximate Berwald type, which is
just an extension of a Riemannian space. Following this theoretical line, we
are motivated by placing constraints on the new Finslerian cosmological model
using the current cosmological observations.

This study is organized in the following manner. In Section 2, we introduce
briefly the Finslerian models to be constrained. In Section 3, we describe the
methodology and data we use in this analysis, and exhibit the corresponding
constraining results. In Section 4, we distinguish this model from the
$\Lambda$CDM model using two geometrical diagnostics. The concluding remarks
are presented in the final section.

##  II Finslerian models

In this study, we consider two Finslerian models, i.e., the simplest
Finslerian $\Lambda$CDM (F$\Lambda$) model and its one-parameter extension,
non-flat F$\Lambda$ model. In a Finsler-Berwald FRW universe, combining the
time-component Friedmann equation and the acceleration equation 45 ; 46 ; 47 ,
the continuity equation can be expressed as

$\dot{\rho}(\frac{3\alpha}{4}+\frac{\beta}{12}+1)+\dot{p}(-\frac{3\alpha}{4}+\frac{\beta}{4})=-\frac{\dot{a}}{a}[\rho(\frac{3\alpha}{2}+\frac{\beta}{6}+2)+p(-\frac{3\alpha}{2}+\frac{\beta}{2})+(\rho+3p)(1+\frac{\beta}{3})],$
(1)

where $a$, $\alpha$ and $\beta$ are the scale factor, effective time-component
and space-component parameters, respectively. Note that here “ effective ”
means that the physical quantities are derived based on the non-Riemannian
Berwald space. Substituting the equation of state (EoS)
$p_{i}=\omega_{i}\rho_{i}$ of each independent component $i$ (where the
constant $\omega_{i}$ corresponds to the non-relativistic matter, dark energy
and effective curvature, respectively) into Eq. (1), one can obtain the
effective energy density as follows

$\rho_{i}\propto
a^{-\frac{36(1+\omega_{i})+18(1-\omega_{i})\alpha+6(1+3\omega_{i})\beta}{12+9(1-\omega_{i})\alpha+(1+3\omega_{i})\beta}}.$
(2)

Subsequently, using the time-component Friedmann equation and Eq. (2) the
square of the effective dimensionless Hubble parameter $E_{1}(a)$ of the
F$\Lambda$ model can be written as

$E_{1}^{2}(a)=\frac{9\alpha+\beta+12}{4(\alpha+1)(\beta+3)}\Omega_{m}a^{-\frac{6(3\alpha+\beta+6)}{9\alpha+\beta+12}}+\frac{9\alpha-\beta+6}{2(\alpha+1)(\beta+3)}\Omega_{de}a^{-\frac{6(3\alpha-\beta)}{9\alpha-\beta+6}},$
(3)

where $\Omega_{m}$ and $\Omega_{de}$ denote the effective matter and dark
energy density parameters today, respectively. Note that this model reduces to
the $\Lambda$CDM model when $\alpha=\beta=0$. Since we are of interest in
investigating the evolution of the late universe, we ignore the contribution
from the radiation component. Furthermore, considering the spatial curvature
in the Finslerian scenario, the square of the effective dimensionless Hubble
parameter $E_{2}(a)$ of the non-flat F$\Lambda$ model is expressed as

$E_{2}^{2}(a)=\frac{3}{3+\beta}\Omega_{k}a^{-2}+\frac{9\alpha+\beta+12}{4(\alpha+1)(\beta+3)}\Omega_{m}a^{-\frac{6(3\alpha+\beta+6)}{9\alpha+\beta+12}}+\frac{9\alpha-\beta+6}{2(\alpha+1)(\beta+3)}\Omega_{de}a^{-\frac{6(3\alpha-\beta)}{9\alpha-\beta+6}},$
(4)

where $\Omega_{k}$ is the effective curvature density parameter today. One can
also find that the contribution from curvature based on Finsler geometry
depends only on the space-component parameter $\beta$. We use units $8\pi
G=c=1$ throughout this work.

Figure 1: The one-dimensional marginalized probability distribution on the
individual parameter and $2$-dimensional contours of the F$\Lambda$ model by
using SBC data sets. Figure 2: The one-dimensional marginalized probability
distribution on the individual parameter and $2$-dimensional contours of the
non-flat F$\Lambda$ model using SBC data sets. Figure 3: The one-dimensional
posterior probability distributions of $H_{0}$ values derived from the
F$\Lambda$ (solid) and non-flat F$\Lambda$ (dashed) models using SBC data
sets, respectively.

##  III Constraints

We employ the Markov Chain Monte Carlo (MCMC) method to place constraints on
the above two Finslerian models by using the current cosmological
observations. More specifically, we modify carefully the publicly MCMC code
CosmoMC 48 ; 49  and Boltzmann code CAMB 50  to infer the posterior
probability distributions of parameters, and analyze the correlated MCMC
samples using the publicly code GetDist 51 . Meanwhile, we choose flat priors
to the model parameters and marginalize the foreground nuisance parameters
provided by Planck. For the F$\Lambda$ model, we consider the following
8-dimensional parameter space

$\mathbf{P_{1}}=\\{\Omega_{b}h^{2},\quad\Omega_{c}h^{2},\quad
100\theta_{MC},\quad\tau,\quad\alpha,\quad\beta,\quad\mathrm{ln}(10^{10}A_{s}),\quad
n_{s}\\},$ (5)

where $\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ denote, respectively, the
present-day baryon and CDM densities, $100\theta_{MC}$ is the approximation to
$100\times$ angular size of sound horizon at the redshift of last scattering
$z_{*}$, $\tau$ represents the Thomson scattering optical depth due to
reionization, $\alpha$ and $\beta$ are two typical model parameters of
F$\Lambda$ model, $\mathrm{ln}(10^{10}A_{s})$ and $n_{s}$ denote the amplitude
and the spectral index of the primordial scalar perturbation power spectrum at
the pivot scale $K_{0}=0.05$ Mpc-1, respectively. Here $h\equiv H_{0}/100$ and
$H_{0}$ is the Hubble constant. Similarly, the 9-dimensional parameter space
for the non-flat F$\Lambda$ model is expressed as

$\mathbf{P_{2}}=\\{\Omega_{b}h^{2},\quad\Omega_{c}h^{2},\quad
100\theta_{MC},\quad\Omega_{k},\quad\tau,\quad\alpha,\quad\beta,\quad\mathrm{ln}(10^{10}A_{s}),\quad
n_{s}\\},$ (6)

where $\Omega_{k}$ denotes the effective curvature density today.
Subsequently, we exhibit the cosmic probes used in this analysis, including
the SNIa, BAO, CMB and Planck lensing data.

$\bullet$ The SNIa data: Since the absolute magnitudes of all the SNIa are
considered to be the same based on the nearly same explosion mass, the SNIa is
regarded as a standard candle to explore the background evolution of the
universe in theory. We use the largest “ Joint Light-curve Analysis ” (JLA)
sample containing 740 SNIa data points, which covers the redshift range
$z\in[0.01,1.3]$ 52 . The JLA compilation consists of 118 low-$z$ SNe in the
range $z\in[0,0.1]$ from 53 ; 54 ; 55 ; 56 ; 57 ; 58 , 374 SNe in the range
$z\in[0.3,0.4]$ from the Sloan Digital Sky Survey (SDSS) SNe search 59 , 239
SNe in the range $z\in[0.1,1.1]$ from the Supernova Legacy Survey (SNLS)
project 60 , and 9 high-$z$ SNe in the range $z\in[0.8,1.3]$ from the Hubble
Space Telescope (HST) 61 .

$\bullet$ The BAO data: To break the parameter degeneracy efficiently, we take
four BAO measurements consisting of the 6dFGS (six-degree-field galaxy survey)
sample at effective redshift $z_{eff}=0.106$ 62 , the SDSS MGS (main galaxy
sample) sample at $z_{eff}=0.15$ 63 , and the LOWZ at $z_{eff}=0.32$ and CMASS
$z_{eff}=0.57$ samples of the SDSS-III BOSS (Baryon Oscillation Spectroscopic
Survey) DR12 sample 64 .

$\bullet$ The CMB data: We also use the Planck 2015 temperature and
polarization data in our numerical analysis 65 , including the likelihoods of
temperature (TT) at $30\leqslant\ell\leqslant 2500$, the cross correlation of
temperature and polarization (TE), the polarization (EE) power spectra and the
Planck low-$\ell$ temperature and polarization likelihood at
$2\leqslant\ell\leqslant 29$.

$\bullet$ The lensing data: We utilize the Planck lensing data 66 , which
gives the most powerful measurement to date with a 2.5$\%$ constraint on the
amplitude of the lensing potential power spectrum (or alternatively, a
40$\sigma$ detection of lensing effects).

Table 1: The prior ranges of different model parameters used in the Bayesian analysis. Parameter | Prior
---|---
$\Omega_{b}h^{2}$ | $[0.005,0.1]$
$\Omega_{c}h^{2}$ | $[0.001,0.99]$
$100\theta_{MC}$ | $[0.5,10]$
$\tau$ | $[0.01,0.8]$
$\Omega_{k}$ | $[-0.05,0.05]$
$\alpha$ | $[-3,3]$
$\beta$ | $[-3,3]$
$\mathrm{ln}[10^{10}A_{s}]$ | $[2,4]$
$n_{s}$ | $[0.8,1.2]$
$H_{0}$ | $[20,100]$
Table 2: The $1\sigma$ ($68\%$) uncertainties of different parameters in the F$\Lambda$ and non-flat F$\Lambda$ models using the data combinations S, SB and SBC, respectively. Model |  | F$\Lambda$ |  |  | non-flat F$\Lambda$ |
---|---|---|---|---|---|---
Data | S | SB | SBC | S | SB | SBC
$\Omega_{b}h^{2}$ | $0.0232^{+0.0016}_{-0.0014}$ | $0.02239\pm 0.00084$ | $0.02232\pm 0.00012$ | $0.0235^{+0.0023}_{-0.0026}$ | $0.02183\pm 0.00052$ | $0.02222\pm 0.00016$
$\Omega_{c}h^{2}$ | $0.1169^{+0.0027}_{-0.0037}$ | $0.1193^{+0.0043}_{-0.0071}$ | $0.11817\pm 0.00081$ | $0.110^{+0.011}_{-0.014}$ | $0.1204^{+0.0036}_{-0.0040}$ | $0.1193^{+0.0016}_{-0.0013}$
$100\theta_{MC}$ | $1.04016\pm 0.00097$ | $1.0416^{+0.0047}_{-0.0065}$ | $1.04100^{+0.00028}_{-0.00031}$ | $1.04273^{+0.00095}_{-0.0024}$ | $1.0401^{+0.0017}_{-0.0012}$ | $1.04085\pm 0.00031$
$\tau$ | $0.130\pm 0.029$ | $0.153^{+0.082}_{-0.031}$ | $0.0810^{+0.0052}_{-0.0072}$ | $0.148\pm 0.046$ | $0.076^{+0.027}_{-0.052}$ | $0.0755^{+0.0047}_{-0.0042}$
$\Omega_{K}$ | — | — | — | $-0.011^{+0.022}_{-0.019}$ | $0.0028^{+0.0043}_{-0.0048}$ | $0.0013\pm 0.0019$
$\alpha$ | $0.0131\pm 0.0058$ | $0.0013^{+0.0012}_{-0.0091}$ | $0.0027^{+0.0029}_{-0.0047}$ | $-0.001^{+0.062}_{-0.022}$ | $-0.0024^{+0.0057}_{-0.0065}$ | $-0.0036\pm 0.0019$
$\beta$ | $0.0000\pm 0.0032$ | $-0.0011^{+0.0089}_{-0.0061}$ | $0.0016\pm 0.0048$ | $-0.002^{+0.016}_{-0.026}$ | $0.0022\pm 0.0038$ | $-0.0051^{+0.0076}_{-0.0069}$
${\rm{ln}}(10^{10}A_{s})$ | $3.104^{+0.015}_{-0.010}$ | $3.102^{+0.013}_{-0.031}$ | $3.094^{+0.011}_{-0.023}$ | $3.088^{+0.016}_{-0.010}$ | $3.0994^{+0.0084}_{-0.0064}$ | $3.0848^{+0.0071}_{-0.0061}$
$n_{s}$ | $0.949^{+0.017}_{-0.020}$ | $0.969^{+0.011}_{-0.017}$ | $0.9690\pm 0.0033$ | $0.971\pm 0.067$ | $0.9468\pm 0.0088$ | $0.9655\pm 0.0042$
$H_{0}$ | $68.9^{+2.1}_{-1.9}$ | $67.95^{+0.87}_{-1.0}$ | $67.99^{+0.38}_{-0.30}$ | $67.4\pm 4.9$ | $67.84\pm 0.81$ | $68.12\pm 0.61$

In what follows, for simplicity, we denote the data combinations SNIa,
CMB+lensing, SNIa+BAO and SNIa+BAO+CMB+lensing as S, C, SB and SBC,
respectively. Specifically, we perform constraints on the F$\Lambda$ and non-
flat F$\Lambda$ models by using S, SB and SBC, respectively. The prior ranges
of different model parameters considered in this analysis are shown in Tab. 1.

By implementing a Bayesian analysis, the constraining results of the above two
models using different data combinations are exhibited in Tab. 2. Utilizing
the joint constraints from SBC, the corresponding one-dimensional marginalized
probability distribution on the individual parameter and $2$-dimensional
contours for the F$\Lambda$ and non-flat F$\Lambda$ models are shown in Figs.
1-2, respectively. For the F$\Lambda$ model, we find that the values of
typical model parameters $\alpha$ and $\beta$ are well consistent with zero at
the $1\sigma$ C.L. and prefer slightly the positive best-fit values being very
close to zero when using the data combination SBC. Being different from the
F$\Lambda$ model, for the non-flat F$\Lambda$ one, the value of $\beta$ is
well consistent with zero at the $1\sigma$ C.L., but the value of $\alpha$ is
not. Attractively, adding a curvature parameter into the F$\Lambda$ model, the
values of $\alpha$ and $\beta$ prefer slightly negative best-fit values being
very close to zero when using SBC. We also find that the value of the spatial
curvature $\Omega_{k}=0.0013\pm 0.0019$ from SBC in the non-flat F$\Lambda$
model is consistent with zero at the $1\sigma$ C.L., which means that a flat
universe is still preferred in the framework of Finsler geometry. It is worth
noting that, using SBC, our measured values of the spectral index $n_{s}$ of
the primordial scalar perturbation power spectrum in the above two Fisnlerian
scenarios both rule out the scale invariance at more than $8\sigma$ C.L.,
which is very compatible with the predictions from Planck temperature and
polarization data and Planck lensing data 65 ; 66  (see Tab. 2).

Recently, the directly local measurement $H_{0}=73.24\pm 1.74$ km s-1 Mpc-1
from Riess et al. 2016 (hereafter R16) 67  using the improved SNIa calibration
techniques exhibits a strong tension with the indirectly global measurement
$H_{0}=66.93\pm 0.62$ km s-1 Mpc-1 derived by Planck collaboration (hereafter
P15) 68  under the assumption of the base six-parameter $\Lambda$CDM model at
the $3.4\sigma$ level. We are also of much interest in addressing this issue
using the Finslerian settings. Using SBC data sets, the one-dimensional
posterior probability distributions of $H_{0}$ values derived from the
F$\Lambda$ and non-flat F$\Lambda$ models are shown in Fig. 3. Interestingly,
we find that, using SBC, the current $H_{0}$ tension can be alleviated from
$3.4\sigma$ to $2.9\sigma$ and $2.8\sigma$ in the F$\Lambda$ and non-flat
F$\Lambda$ models, respectively. By opening an extra parameter $\Omega_{k}$,
we also find, as expected, a mild relaxation on $H_{0}$ limits at the
$1\sigma$ C.L. in the non-flat F$\Lambda$ model.

Figure 4: The left and right panels correspond to the relations between the
scale factor $a$ and the acceleration parameter $q(a)$ and $Om(a)$ diagnostic,
respectively. The red (solid), blue (dash-dotted) and magenta (short-dashed)
lines correspond to the best-fit F$\Lambda$, non-flat F$\Lambda$ and
$\Lambda$CDM models, respectively. The shaded yellow, purple and green regions
represent the $1\sigma$ regions of the F$\Lambda$, non-flat F$\Lambda$ and
$\Lambda$CDM models, respectively. The horizontal orange (long-dashed) line
corresponds to the zero-acceleration universe.

##  IV Diagnostics

In this section, we employ two simple geometrical diagnostics, i.e., the
deceleration parameter and $Om(a)$ diagnostic 69  to distinguish the
F$\Lambda$ and non-flat F$\Lambda$ models from the standard cosmological model
and one from the other. The deceleration parameter for a given dark energy
model can be expressed as

$q(a)=-\frac{a}{E(a)}\frac{dE(a)}{da}-1,$ (7)

where $E(a)$ of the above two Finslerian models can be found in Eqs. (3-4).

The $Om(a)$ diagnostic is an very useful geometrical method to distinguish
different cosmological models from each other, which is written as

$Om(a)=\frac{E^{2}(a)-1}{a^{-3}-1}.$ (8)

It is easy to verify that, for a flat $\Lambda$CDM model, the $Om(a)$
diagnostic is fixed to be $\Omega_{m}$, namely $Om(a)=\Omega_{m}$ (see also
Ref. 69 ).

Utilizing $\Omega_{m}=0.308\pm 0.012$ measured by Planck collaboration 65  for
the $\Lambda$CDM model and error propagations of the constrained model
parameters $\alpha$, $\beta$, $\Omega_{m}$ and $\Omega_{k}$ using SBC data
sets, we present the diagnostic results in Fig. 4. We find that the
evolutional behaviors of two Finslerian models are very close to that of the
$\Lambda$CDM one, and consequently, they cannot be distinguished from the
$\Lambda$CDM model at the $1\sigma$ C.L.. By simple calculations, we also
obtain the deceleration-acceleration redshift
$z_{t1}=0.6558^{+0.0130}_{-0.0103}$ for the F$\Lambda$ model and
$z_{t2}=0.6517^{+0.0121}_{-0.0102}$ for the non-flat F$\Lambda$ model at the
$1\sigma$ C.L., respectively.

##  V Concluding remarks

To understand accurately the current cosmic acceleration from a new
geometrical perspective, we have performed the first constraints on the
simplest Finslerian model, the F$\Lambda$ model and its one-parameter
extension, the non-flat F$\Lambda$ model by using the current cosmological
observations. Utilizing the most stringent constraints SBC we can provide, we
find that: For the F$\Lambda$ model, the values of typical model parameters
$\alpha$ and $\beta$ are well consistent with zero at the $1\sigma$ confidence
level and prefer slightly the positive best-fit values being very close to
zero; For the non-flat F$\Lambda$ one, nonetheless, the values of $\alpha$ and
$\beta$ prefer slightly negative best-fit values being very close to zero and
the value of $\beta$ is well consistent with zero at the $1\sigma$ C.L., but
the value of $\alpha$ is not; A spatially flat universe is still preferred in
the framework of Finsler geometry; Our measured values of the spectral index
$n_{s}$ of primordial power spectrum in both Fisnlerian scenarios rule out the
scale invariance at more than $8\sigma$ C.L.; The current $H_{0}$ tension can
be relived from $3.4\sigma$ to $2.9\sigma$ and $2.8\sigma$ in the F$\Lambda$
and non-flat F$\Lambda$ models, respectively.

Using two popular geometrical diagnostics, we find that both Finslerian models
cannot be distinguished from the $\Lambda$CDM model during the evolutional
process of the universe. It is noteworthy that we obtain this conclusion just
at the background evolution level without considering the perturbation
effects. Furthermore, in this analysis, we just test primarily the abilities
of two Finslerian models in explaining the current cosmological phenomena
without consider other interesting extensions. In addition, more data except
SBC can also be applied in constraining the Finslerian models. In total, our
investigations on using the Finsler geometry to understand the evolution of
the universe and reconcile the cosmological tensions among various probes are
just in the beginning stage. The remaining issues will be addressed carefully
in a forthcoming study 70 .

## ACKNOWLEDGMENTS

Deng Wang warmly thanks Jing-Ling Chen and F. Canfora for very useful
communications. Xin-He Meng appreciates S. D. Odintsov and B. Ratra for
helpful discussions on cosmology. This study is partly supported by the
National Science Foundation of China.

##  Appendix A u

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