# Barrow Holographic Dark Energy Model with GO Cut-off - An Alternative
Perspective

Nandhida Krishnan.P<EMAIL_ADDRESS>Titus K
<EMAIL_ADDRESS>
*† Department of Physics, Cochin University of Science and Technology(CUSAT),
Kochi 682-022, India.
†Inter University Center for the studies on Kerala Legacy Astronomy and
Mathematics, CUSAT
†Center for Particle Physics, CUSAT

###### Abstract

Recently, Barrow holographic dark energy (BHDE), based on Barrow entropy, has
been proposed to describe the late acceleration of the universe. Contrary to
the earlier analysis of this model in the literature, we consider the BHDE
with the Granda–Oliveros length as IR cut-off, as a dynamical vacuum, having a
constant equation of state $\omega_{\Lambda}=-1.$ We have analytically solved
for the Hubble parameter and studied the evolution of cosmological parameters.
The model is compared with the observational data on Hubble parameter (OHD36)
and Supernovae type Ia (SN Ia), the pantheon data. In the absence of
interaction between the dark sectors, we found that the model predicts a
$\Lambda$CDM like evolution of the universe with an effective cosmological
constant. In this case, the model is found to satisfy the generalized second
law (GSL), irrespective of the value of the Barrow index. The interaction also
shows the safe validity of GSL, for the extracted value of the Barrow index,
$\Delta=0.063\pm 0.029$. The thermodynamic analysis of the model predicts an
end de Sitter phase of maximum entropy. We performed a dynamical system
analysis, which reveals that the end de Sitter phase is stable. Furthermore,
we performed the Information Criterion analysis using Akaike and Bayesian
Information Criterion to compare the statistical compatibility of the present
model with the standard $\Lambda$CDM model.

Keywords: Barrow entropy; holographic dark energy model; Granda-Oliveros cut-
off; Generalized second law of thermodynamics

##  1 Introduction

Observational results from the type Ia Supernovae (SN Ia) for different
redshift lead to one of the most pivotal discoveries in the twentieth century
that the current universe is not just expanding but follows an accelerating
expansion [1, 2, 3, 4, 5, 6, 7, 8]. This implies that the universe, in its
late epoch, can be dominated by an exotic component with negative pressure
termed as dark energy, which is responsible for the current accelerated
expansion. The best-fitting model for the present accelerating universe is the
$\Lambda$CDM, referred to as the standard model of cosmology, which adopts the
cosmological constant, $\Lambda$ as dark energy. Despite the success in
explaining the observational data [9], the $\Lambda$CDM is plagued with two
major flaws. The first one is the cosmological constant problem, which arises
due to the huge inconsistency between the observed value of the cosmological
constant and its theoretically predicted value. The latter is the coincidence
problem, the intriguing coincidence of the present value of dark energy
density with that of matter, even though both have evolved differently. To
alleviate these problems, one must consider the dark energy density as varying
or adopt suitably modified gravity theories. The first one motivates the
proposal of various dynamical dark energy models. Among these, holographic
dark energy models are potentially important to interpret the dynamical dark
energy [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 15, 25,
26, 27, 28, 29, 30, 31, 32].

The holographic dark energy models are based on the holographic principle,
[33, 34, 35] which state that the black hole entropy scales non-extensively,
[36, 37, 38, 39] in proportion to the area of their horizon. However, in the
effective quantum field theory confined to a box, the entropy behaves
extensively as, $S\sim L^{3}\Lambda^{3}$ where $L$ is the size of the box with
UV cut-off $\Lambda$. To resolve the breakdown of quantum field theory to
describe a black hole, a longer distance IR and short distance UV cut-off
relation was proposed by cohen et al. [40, 41, 42] as, $L^{3}\Lambda^{4}\leq
LM_{p}^{2},$ Where $\Lambda^{4}$ is the maximum energy density in the
effective theory and $M_{p}$ is the Planck mass. The largest length, which
saturates this inequality, enables us to define an energy density
corresponding to the holographic principle as [10],

$\rho_{{}_{\Lambda}}=3c^{2}M_{p}^{2}L^{-2}$ (1)

where $c^{2}$ is the model parameter. The energy density $\rho_{{}_{\Lambda}}$
can be adopted as the dark energy density in the cosmological context, with
$L,$ the IR cutoff, as the cosmological length scale. There are various
approaches to choose the length scale, in which the most straightforward
choice is the Hubble horizon, $L=1/H$. Even though this assumption accounts
for the present dark energy density of the universe [10], it failed badly in
explaining the current accelerated expansion of the universe with non-
interacting dark sectors since it predicts an equation of state around
$\omega_{{}_{\Lambda}}=0$ [43], which is in dispute with the condition
corresponding to accelerated expansion, $\omega_{{}_{\Lambda}}<-1/3.$ However,
using this IR cutoff and assuming a possible interaction between the dark
sectors, can solve the coincidence problem [44, 45, 46, 47, 48].
Alternatively, particle horizon or event horizon becomes the reasonable choice
for the cosmological length scale. On consideration of particle horizon as the
length scale, the corresponding model results in an equation of state
parameter $\omega_{{}_{\Lambda}}>-1/3$, thus failing to explain accelerated
expansion[49]. The choice of the future event horizon as the length scale
results in a model which predicts the recent acceleration of the universe and
is also favored by the observational data to some extent[35, 50, 51]. However,
it faces the causality problem, such that the nature of present dark energy
seems to depend on the future evolution of the scale factor[11].

Later attention has then turned to defining the holographic dark energy with
curvature scalar as the IR cutoff[52, 53, 54]. Following this, as a formal
generalization of the Ricci scalar curvature, a hybrid cut-off scale known as
the Granda-Oliveros (GO) scale has been proposed[55]. The GO cut-off scale,
defined as the combination of the Hubble parameter and its time derivative, is
used to formulate holographic dark energy density. The corresponding model
thus depends on the local quantities; hence, no causality problems arise. Much
interest has arisen in holographic dark energy with GO cut-off[56, 57, 58, 59,
60, 61, 62, 63]. Recently a new model of holographic dark energy, known as the
Barrow holographic dark energy, has been proposed based on the Barrow entropy.
Barrow has expressed that the black hole surface has an intricate, fractal
structure due to quantum gravitational effects, which leads to a finite volume
but infinite (or finite) area. This distorted geometrical structure could have
implications on the form of horizon entropy. Following this perspective,
Barrow has proposed that the black hole entropy deformed to a more general
relation [64],

$S_{h}=\left(\frac{A}{A_{0}}\right)^{1+\Delta/2}$ (2)

where $A$ and $A_{0}$ are the area of the black hole horizon and Planck area,
respectively. The exponent $\Delta$ quantifies the amount of quantum-
gravitational deformation effects. In literature the Barrow exponent (or
Barrow index), takes the range $0\leq\Delta\leq 1$. The upper limit of
$\Delta$ corresponds to the most intricate and fractal structure of the black
hole horizon, while the lower limit, $\Delta=0$ corresponds to the simplest
horizon structure, at which it reduces to the standard Bekenstein-Hawking
entropy [65, 39, 38]. The above deformation in entropy differs from the
logarithmic corrections to the black hole entropy [66, 67]. Even though the
above expression bears some resemblance to Tsallis non-extensive entropy, the
underlying principles are completely different [68, 69]. In reference[70], the
author proposed a specific holographic dark energy model by employing the
holographic principle with Barrow entropy as the horizon entropy. The standard
holographic principle can also be stated as $\rho_{{}_{\Lambda}}L^{4}\leq S.$
Using Barrow entropy in Eq. (2), the Barrow holographic dark energy (BHDE)
density can be defined as [70],

$\rho_{{}_{\Lambda}}=3CL^{\Delta-2}$ (3)

Where $C$ is the parameter with dimension $[L]^{-2-\Delta}$, the factor $3$ is
used for convenience. As $\Delta=0$ then
$\rho_{{}_{\Lambda}}=3c^{2}M_{p}^{2}L^{-2},$ which corresponds to the standard
holographic dark energy with $C=c^{2}M_{p}^{2}$. Using Eq. (3) it is possible
to construct various BHDE models by taking different length scales as IR cut-
offs in either interacting or non-interacting scenarios. In [70], authors
analyzed the thermal history of the flat universe having BHDE with future
event horizon as IR cut-off and non-relativistic matter. They have shown that
the Barrow exponent $\Delta$ has a significant role in determining the
evolutionary nature of dark energy. For $\Delta>0.5$ the dark energy shows a
phantom behavior. By using the data from Supernovae (SN Ia) Pantheon sample
and Hubble data from Cosmic Chronometer (CC), the authors in [71] constraint
the Barrow exponent in the range 0.094 - 0.095 and inferred that non-
interacting BHDE model with future event horizon as IR cut-off is compatible
with standard $\Lambda$CDM and in good agreement with observational data. The
generalized second law (GSL) of thermodynamics of the universe with Barrow
entropy as the apparent horizon entropy has been examined,[72] and has shown
that the GSL is always valid without quantum gravitational deformation.
Furthermore, their analysis showed that the generalized second law could be
conditionally violated for finite non-zero values of $\Delta$. The GSL is
usually valid only for cases where the Barrow exponent should be constrained
to a small value such that the Barrow entropy shows only a small deviation
from the standard relation. However, assuming $\Lambda$CDM-like behavior for
the Hubble parameter, the Validity of GSL can be restored independently of the
$\Delta$ values. There is a claim of strong Bayesian evidence for BHDE with
Hubble horizon as IR cut-off compared with the standard $\Lambda$CDM in
reference[73]. In [74], the author extracted the modified Friedmann equations
from the first law of thermodynamics, $-dE=TdS,$ for a (3+1)-dimensional FLRW
universe with Barrow entropy for the horizon and found that, on neglecting the
Barrow exponent the conventional form of Friedmann equation can be recovered.
In the same paper, it is argued that the dark energy equation of state
parameter shows a deviation from the $\Lambda$CDM behavior at intermediate
times. However, the model predicts a de Sitter epoch independent of the Barrow
exponent at asymptotically large times. Similarly, Sheykhi [75] obtained the
modified Friedmann equation with Barrow entropy, using the unified first law
of thermodynamics, $dE=TdS+WdV,$ where $W=(\rho-P)/2,$ the work density. In
addition, the author explored the generalized second law and has shown that
the generalized second law of thermodynamics is valid for a universe with
fractal boundaries. The interacting BHDE model with $H$ as the IR cut-off has
been examined in [76], where the authors pointed out the possibility of a
conditional violation of GSL. Authors in [77] studied the BHDE for a non-flat
FLRW universe with the apparent horizon as IR cut-off and have shown that the
dark energy has quintessence behavior. However, for a flat universe, the dark
energy exhibits quintessence and chaplygin gas behavior [78]. In reference
[79], the authors have carried out the dynamical analysis and the statefinder
diagnostic on interacting BHDE with different IR cut-offs. Recently, the
thermodynamics of the BHDE model has been discussed with a specific cut-off,
called Nojiri-Odintsov (NO) cut-off, which is a particular combination of the
event horizon scale [80]. In the same paper, the authors incorporated the bulk
viscosity in the BHDE sector and studied the evolution of the cosmological
parameters. Then showed that the model satisfies the GSL. Very recently, the
BHDE model has been studied in the non-flat universe and explains the thermal
history of the universe with the matter, and dark energy eras [81]. They also
investigated the effect of Barrow exponent $\Delta$ on the equation of state
(EoS) parameter. For $\Delta=0$, the EoS parameter lies entirely in the
quintessence region, while for $\Delta>0.03$, they claimed that the model
crosses the phantom divide.

The holographic dark energy is defined by relating the IR cut-off,
corresponding to the size of the universe, to the UV cut-off, corresponding to
the scale of vacuum energy. Since the UV cut-off is related to vacuum, it is
more significant to reconsider holographic dark energy as decaying vacuum
energy with the equation of state parameter, $\omega_{\Lambda}=-1.$ So far,
all the works in the current literature on BHDE models have been done by
considering the dark energy equation of state parameter as varying with the
expansion of the universe. So it’s worth analyzing the consequences of
treating BHDE as a decaying vacuum. Motivating from this, we treat the BHDE as
a dynamical vacuum in the present work. In this study, we have accounted for
the interaction between the dark sectors phenomenally. Our study provides the
interesting result that, in the absence of interaction, the model shows an
evolution similar to that of the $\Lambda$CDM by predicting an effective
cosmological constant. However, with the presence of interaction, our model
predicts a quintessence behavior of the universe that perfectly obeys the
principles of thermodynamics, particularly the GSL of thermodynamics. We
extracted the model parameters using the latest cosmological data and found
that the universe evolves to an end de Sitter epoch. We have also performed
the dynamical system analysis and found that the end de Sitter epoch is a
stable equilibrium. Later we performed the Information criteria analysis for
model selection.

We organize the paper as follows. The next section introduces an interacting
BHDE model with GO IR cut-off, and the corresponding Hubble parameter of the
model is evaluated. The third section presents the extraction of model
parameters using chi-square statistics with corresponding error bars. Section
four consists of the study of the evolution of the various cosmological
parameters. In section the fifth, the present model is distinguished from the
standard $\Lambda$CDM model by performing statefinder diagnostics. In section
six, We extended our work to study the thermodynamics of the model and
analyzed the status of the GSL in this model. Later in section seven, we
explored the dynamical system analysis following the Information Criteria
analysis in section eight. We conclude our work in the last section.

##  2 Interacting Barrow Holographic Dark Energy

Consider a spatially flat, homogeneous, and isotropic Friedmann-
Leima$\hat{i}$tre-Robertson-Walker(FLRW) universe with background metric given
by,

$dS^{2}=-dt^{2}+a(t)^{2}[dr^{2}+r^{2}d\Omega^{2}]$ (4)

Where $a(t)$ is the scale factor at time $t$,
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ and $(r,\theta,\phi)$ are
the co-moving coordinates. The Friedmann equation for such a universe, with
$k=0$ (flat universe) having dark matter, with density $\rho_{m},$ and dark
energy, with density $\rho_{{}_{\Lambda}},$ is,

$3H^{2}=\rho_{m}+\rho_{{}_{\Lambda}},$ (5)

where $H=\dot{a}/a$ is the Hubble parameter. Here we adopt the natural units,
such that $\hbar=1=c,$ $8\pi G=1$ (unless we mentioned). In the present work,
we take the dark energy density $\rho_{{}_{\Lambda}}$ as Barrow holographic
dark energy (BHDE) with Granda-Oliveros (GO) length scale as IR cut-off. The
GO cut-off is given by [55],

$L^{-2}=(\alpha_{1}H^{2}+\beta_{1}\dot{H}).$ (6)

Here $\alpha_{1}$ and $\beta_{1}$ are dimensionless parameters, and $\dot{H}$
is the time derivative of Hubble parameter. Substituting the GO length scale
in Eq. (6) in to Eq. (3), we get the BHDE density as,

$\displaystyle\rho_{{}_{\Lambda}}=3(\alpha
H^{2}+\beta\dot{H})^{\frac{2-\Delta}{2}}$ (7)

where $\alpha$ and $\beta$ are constant parameters with dimension
$[L]^{\frac{-2\Delta}{2-\Delta}}$. In the present study we treat the BHDE as a
dynamical vacuum, so its equation of state becomes $\omega_{{}_{\Lambda}}=-1$
(i.e. $p_{{}_{\Lambda}}=-\rho_{{}_{\Lambda}}$). The second cosmic component is
non-relativistic matter, with equation of state, $\omega_{m}=0,$ (i.e.
$p_{m}=0$). We account the interaction between these sectors
phenomenologically, and hence the total energy-momentum conservation can be
bifurcated in to[82],

$\dot{\rho}_{{}_{\Lambda}}=-Q\quad;\quad\dot{\rho}_{m}+3H\rho_{m}=Q,$ (8)

where $Q$ is the interaction term, characterized by the amount of energy
transfer between the dark sectors. A negative value of $Q$ corresponds to an
energy transfer from dark matter to dark energy, while a positive value
implies an energy transfer in a reverse manner. Owing to the lack of
information about the microscopic origin of the interaction between the dark
sectors, [27, 60, 83, 84, 85, 15, 86, 87] we choose $Q=3bH\rho_{m}$[88], where
$b$ is a dimensionless coupling parameter, characterizing the strength of
energy flow [82, 76]. Using the variable, $x=lna$ , the above conservation law
in Eq. (8) can be reformulated as,

$\frac{d\Omega_{{}_{\Lambda}}}{dx}=-3b\Omega_{m}\quad;\quad\frac{d\Omega_{m}}{dx}=-3(1-b)\Omega_{m},$
(9)

where $\Omega_{m}=\rho_{m}/(3H_{0}^{2})$ and
$\Omega_{{}_{\Lambda}}=\rho_{{}_{\Lambda}}/(3H_{0}^{2})$ with $H_{0}$ is the
present value of the Hubble parameter. The present values of the mass
parameters, $\Omega_{{}_{\Lambda 0}}$ and $\Omega_{m_{0}},$ satisfying the
constrained relation, $\Omega_{{}_{\Lambda 0}}+\Omega_{m_{0}}=1$. The second
of the above equations, implies the evolution of the matter density as,
$\Omega_{m}=\Omega_{m_{0}}e^{-3(1-b)x}.$ Now Eq. (9) together with Eq. (5)
gives a second order differential equation for the weighted Hubble parameter,

$\frac{d^{2}\bar{H}^{2}}{dx^{2}}+3\frac{d\bar{H}^{2}}{dx}=-9b\Omega_{m}$ (10)

where $\bar{H}=H/H_{0},$ On solving Eq. (10), we get,

$\bar{H}^{2}=\frac{\Omega_{m}}{1-b}-\frac{\Gamma_{1}}{3}a^{-3}+\Gamma_{2}$
(11)

Here $\Gamma_{1}$ and $\Gamma_{2}$ are constant parameters, which are to be
evaluated using suitable conditions. For this we take both the above solution
and ${d\bar{H}^{2}/dx}$ obtained from the relation of dark energy density in
Eq. (7), and evaluated their values at $a=a_{0}=1,$ where $a_{0}$ is the
present scale factor of expansion and is standardized as one, we then get,

$\bar{H}^{2}\mid_{{}_{a=1}}=1\quad\quad\left.\frac{d\bar{H}^{2}}{dx}\right|_{{}_{x=0(a=1)}}=\frac{2}{\beta}\left((H_{0}^{\Delta}\Omega_{{}_{\Lambda
0}})^{\frac{2}{2-\Delta}}-\alpha\right)$ (12)

The constants are obtained as,

$\Gamma_{1}=3\Omega_{m_{0}}+\frac{2}{\beta}\left[\left((1-\Omega_{m_{0}})H_{0}^{\Delta}\right)^{\frac{2}{2-\Delta}}-\alpha\right]$
(13)
$\displaystyle\Gamma_{2}=1-\frac{b\Omega_{m_{0}}}{1-b}+\frac{2}{3\beta}\left[\left((1-\Omega_{m_{0}})H_{0}^{\Delta}\right)^{\frac{2}{2-\Delta}}-\alpha\right]$
(14)

Now we will analyze the features of the Hubble parameter in Eq. (11). In the
asymptotic limit, as $a\rightarrow 0$, the first two terms in Eq. (11)
dominates over the constant $\Gamma_{2}.$ As a result the Hubble parameter
satisfies,
$\bar{H}^{2}\to\left(\frac{\Omega_{m_{0}}}{1-b}a^{3b}-\frac{\Gamma_{1}}{3}\right)a^{-3},$
which shows a decelerated expansion for the interaction parameter in the range
$0\leq b<1$. For the future limit as $a\rightarrow+\infty$ , the constant
$\Gamma_{2}$ became the dominated term in Eq. (11), hence
$\bar{H}^{2}\rightarrow\Gamma_{2},$ which implies the end de Sitter phase. So
the model predicts a transition to the late accelerated epoch from a prior
matter dominated decelerated phase. It is interesting to note that, for $b=0$,
i.e. when the dark sectors are not interacting with each other, then Eq. (11)
reduces to the form,

$\bar{H}^{2}=\bar{\Omega}_{m_{0}}a^{-3}+\bar{\Omega}_{{}_{\Lambda 0}}$ (15)

This implies that, the present model shows a $\Lambda$CDM-like behavior, with
effective mass parameters[89] for non-relativistic matter,
$\bar{\Omega}_{m_{0}}$ and dark energy $\bar{\Omega}_{{}_{\Lambda 0}}$ as,

$\bar{\Omega}_{m_{0}}=\frac{2\alpha}{3\beta}\left[1-\frac{\left((1-\Omega_{m_{0}})H_{0}^{\Delta}\right)^{\frac{2}{2-\Delta}}}{\alpha}\right],\quad\quad\bar{\Omega}_{{}_{\Lambda
0}}=1-\bar{\Omega}_{m_{0}}$ (16)

respectively. The term $\bar{\Omega}_{{}_{\Lambda}{0}}$, is analogues to an
effective cosmological constant. The reason for this behavior is that for
$b=0$ the conservation laws in Eq. (8) reduce to separate conservation, i.e.,
matter and dark energy. Hence, they are independently conserved such that the
dark energy density will effectively become a constant.

##  3 Contrasting the Model with Observational Data

We adopted the chi-square ($\chi^{2}$) statistics method to obtain the best-
fit model parameters. Let $\xi$ is a cosmological variable, for which, there
is an observational value $\xi_{obs}$, and a corresponding theoretically
predicted value, $\xi_{theo}(P).$ The predicted value depends on the model
parameters, say $P.$ In the present model the parameters are
$P=H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,$ and $b$. Then the $\chi^{2}$
function defined as,

$\chi^{2}_{\xi}(P)=\sum_{i}\frac{\left[\xi_{theo}(P)-\xi_{obs}\right]^{2}}{\sigma_{\xi}^{2}}$
(17)

where $\sigma_{\xi}$ is the standard deviation associated with the observation
of the physical quantity. The most probable values of the model parameters can
then be extracted by statistically minimizing the $\chi^{2}$ function. In
order to perform the $\chi^{2}$ minimization, we have applied the Markov chain
Monte Carlo (MCMC) algorithm by considering the emcee python package [90]
using lmfit library [91]. We have used the observational Hubble dataset
containing 36 H(z) measurements ($OHD36$) in the redshift range $0.07\leq
z\leq 2.36$ [92, 93], in which 31 data are obtained using the Cosmic
Chronometric technique, 3 measurements are from the radial BAO signal in the
galaxy distribution, and the last 2 are measured from the BAO signal in the
Lyman forest distribution alone or cross correlated with QSOs[92]. For
observational Hubble data, the expression for $\chi^{2}$,

$\chi^{2}_{OHD36}(H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,b)=\sum_{i=1}^{36}\frac{\left[H(H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,b,z_{i})-H_{i}\right]^{2}}{\sigma_{i}^{2}}$
(18)

where $H(H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,b,z_{i})$ and $H_{i}$ are
the theoretical and observed Hubble parameter respectively and $\sigma_{i}$ is
the corresponding standard deviation. Another set of data is taken from the
frequently used cosmological probe, type Ia Supernovae (SN Ia). We have used
the SN Ia Pantheon dataset with 1048 apparent magnitude ($m$) versus redshift
measurements in the redshift range $0.01\leq z\leq 2.3$ [94]. The $\chi^{2}$
function corresponding to the Supernovae data can be expressed as,

$\chi^{2}_{SNIa}(H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,b,M)=\sum_{i=1}^{1048}\frac{\left[m(H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,b,M,z_{i})-m_{i}\right]^{2}}{\sigma_{i}^{2}}$
(19)

where $m(H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,b,M,z_{i})$ is the
theoretical value of the apparent magnitude of the Supernovae at a given
redshift and $m_{i}$ is the corresponding observed magnitude. The $\sigma_{i}$
in the denominator of the above equation is now the standard deviation in the
observation of the Supernovae. The theoretical magnitude is evaluated by
determining the luminosity distance relation,

$d_{L}(H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,b,z_{i})=c(1+z_{i})\int_{0}^{z_{i}}\frac{dz}{H(H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,b,z)}.$
(20)

Thus, the apparent magnitude can be determined by the standard relation,

$m(H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,b,M,z_{i})=M+5log_{10}\left(\frac{d_{L}(H_{0},\Omega_{m_{0}},\alpha,\beta,\Delta,b,z_{i})}{Mpc}\right)+25,$
(21)

where $M$ is absolute magnitude of the supernovae is usually treated as a
nuisance parameter. The most probable values of the model free parameters is
then estimated corresponds to the minimal $\chi_{total}^{2}$ value of the
combined dataset of observational Hubble data and Supernovae type Ia
($OHD36+SNIa$) as, $\chi_{total}^{2}=\chi^{2}_{OHD36}+\chi^{2}_{SNIa}$ with
$\chi^{2}_{min}$ =$1053.927$ and $\chi^{2}_{d.o.f}$ = $0.979$. The absolute
magnitude $M$ is extracted as $-19.378\pm 0.039$. By using the pygtc open
python package[95], the plot of two-dimensional posterior contours with
$1\sigma(68\%)$ and $2\sigma(95\%)$ confidence level and one dimensional
marginalized posterior distributions of model parameters $H_{0},\hskip
2.84544pt\Omega_{m_{0}},\hskip 2.84544pt\alpha,\hskip 2.84544pt\beta,\hskip
2.84544pt\Delta$, $b$ and $M$ are performed for the combined dataset
$OHD36+SNIa$ is given in Fig. 1. The best-estimated values of various model
parameters for 1$\sigma$ error bar is given in Table. 1.

The Present value of Hubble parameter is extracted as $69.256\pm{1.228}$
$kms^{-1}Mpc^{-1}$ which shows a slight variation from the observational value
(Planck 2018 observation 67.4$\pm$ 0.5 $kms^{-1}Mpc^{-1}$[96], and WMAP sky
survey value 71.9${}^{+2.6}_{-2.7}$ $kms^{-1}Mpc^{-1}$ [97]. In a previous
work[98], the best fit value of $H_{0},$ for non-interacting HDE model with GO
cut-off is estimated as $66.35^{+{2.38}}_{{-2.14}}.$ The improved value of
$H_{0}$ in the current model shows its significance, in the light of
observational results. In the same reference the estimated value of the model
parameters $\alpha=0.8502^{+{0.0984}}_{{-0.0875}}$ and
$\beta=0.4817^{+{0.0842}}_{{-0.0773}}$ [98] are comparatively less than that
of our model (ref.Table.1). Recently, Oliveros et al.[100] studied a non-
interacting BHDE with GO cut-off and reported the parameter values,
$\alpha=1.00^{+{0.02}}_{-0.02},\beta=0.69^{+0.03}_{-0.02}$ and
$\Delta=0.000^{+0.004}_{-0.000}$, for OHD dataset [100].
In contrast to this, our model favors a non-zero value for the Barrow index,
$\Delta,$ owing to the inclusion of the possible interaction between the dark
sectors. The positive value of the interaction implies an energy flow from
BHDE to dark matter. This flow of energy between the components of one and the
same universe is in line with the LeChatelier-Braun Principle [101]. In this
juncture, it may be interesting to note about a similar study in Tsallis HDE
model with GO cut-off [99], in which the authors have estimated the model
parameters as, $\alpha=0.960^{+0.138}_{-0.125},\beta=0.337^{+0.184}_{-0.207}$,
and the coupling parameter $b=0.023^{+0.033}_{-0.026}$, using dataset
SNIa+OHD.

Figure 1: Corner plot of two-dimensional posterior contours with $1\sigma(68\%)$ and $2\sigma(95\%)$ confidence level and one dimensional marginalized posterior distributions of model parameters for the combined dataset $OHD36+SNIa$ Table 1: Observational constraints on model parameters using $OHD36+SNIa$ dataset $H_{0}$ $kms^{-1}Mpc^{-1}$ | $\Omega{m_{0}}$ | $\alpha$ | $\beta$ | $\Delta$ | $b$
---|---|---|---|---|---
$69.256^{+{1.228}}_{-1.228}$ | $0.281^{+{0.050}}_{-0.050}$ | $1.030^{+{0.116}}_{-0.116}$ | $0.211^{+{0.063}}_{-0.063}$ | $0.063^{+{0.029}}_{-0.029}$ | $0.026^{+{0.030}}_{-0.030}$

Using the best-fit model parameters, we have compared the theoretically
estimated apparent magnitude with the corresponding observational magnitude,
as shown in Fig. 2. We have also compared the predicted Hubble parameter
values with the observed one in Fig. 3. Both figures show good agreement
between the prediction and observation.
Figure 2: Comparison plot of apparent magnitude m(z) for interacting BHDE
model with the best-estimated model parameters from the combined dataset
$OHD36+SNIa$ and observational Supernovae data.  Figure 3: Comparison plot of
Hubble parameter for interacting BHDE model with best-estimated model
parameters from the combined dataset $OHD36+SNIa$ and observational Hubble
data.

##  4 Behavior of Cosmological Parameters

The Eq. (11) shows the evolution of the Hubble parameter for the interacting
BHDE model with respect to the scale factor $a$. The background evolution of
Hubble parameter as a function of redshift z (where $a=1/(1+z)$) is plotted,
for the best-estimated model parameters in Fig. 4 reveals that the evolution
ultimately leads to an end de Sitter epoch.

Figure 4: Evolution of Hubble parameter for the model with best-fit model
parameters. Figure 5: Evolution of curvature scalar with scale factor for the
best estimate parameters

The curvature scalar $R$ is useful for probing the early epoch of universe and
is defined in terms of Hubble parameter [102],

$R=6(\dot{H}+2H^{2})$ (22)

The evolution of curvature scalar parameter is as shown in Fig. 5. It is
evident that, as $a\rightarrow 0$, curvature scalar approaches to infinity,
which indicates the existence of the Big-bang singularity.
Now we will determine the age of the universe in the present model. The age
obtained using the standard relation,

$t_{0}-t_{B}=\int_{0}^{1}\frac{1}{aH(a)}da$ (23)

Where $t_{0}$ is the present time, $t_{B}$ is the big bang time, which can be
set to zero by convention. Evaluating the above integral by substituting the
Hubble parameter of the present model, we found that the age of the universe
corresponds to the best-estimated parameters is around $13.958$ Gyr. This is
compatible with the various standard determination of age of the universe,
like from WMAP data analysis, $13.70{\pm{0.2}}$ Gyr [3], $13.72\pm 0.12$ Gyr
from the combined WMAP+BAO+SN Ia analysis[97] and $14\pm{0.5}$ Gyr [103] from
CMB anisotropic data. Also not too much different from the age obtained from
the study based on the 42 high redshift Supernovae, $14.90^{+1.4}_{-1.1}$ Gyr
in reference[2]. It is to be noted that the extracted age in the present model
is near to the upper age limit, 14.56 Gyr, set by observation of the oldest
globular clusters. Also, our result above the lower limit proposed around
12.07 Gyr [104]. The current model is also explain the coincidence problem.
The evolution of both matter density and dark energy density are obtained as
follows,

$\Omega_{m}=\Omega_{m_{0}}e^{-3(1-b)x},\quad\Omega_{{}_{\Lambda}}=\frac{1}{H_{0}^{2}}\left(\alpha
H^{2}+\frac{\beta}{2}\frac{dH^{2}}{dx}\right)^{\frac{2-\Delta}{2}}$ (24)

The co-evolution of these mass density parameters in logarithmic scale is
shown in Fig. 6. It shows the dominance of matter over dark energy density
prior to the transition and the dominance of dark energy in the later epoch.
This adequately solves the coincidence problem. The present value of matter
density parameter is estimated as $0.281^{+{0.050}}_{-0.050}$ which is
agreeing with the current observational results, $\Omega_{m_{0}}$ $\sim
0.234\pm 0.035-0.29\pm 0.007$[6]. Additionally it is in agreement with the
estimated value $\Omega_{m_{0}}\sim 0.28$ for the combined dataset of
Supernovae type Ia and Cosmic Chronometer for a non-interacting BHDE model
with future event horizon as IR cut-off [71].

Figure 6: Evolution of density parameter corresponds to non-relativistic
matter and interacting Barrow holographic dark energy in logarithmic scale.

The evolution of the deceleration parameter in the model is presented below.
The deceleration parameter $q$, describes the nature of cosmic expansion,
whether it is decelerating or accelerating. The evolution of this parameter
can be obtained using the expression,

$q=-1-\frac{\dot{H}}{H^{2}}$ (25)

Precisely the expansion will be accelerating for $q<0$ and it is decelerating
if $q>0$. After substituting the Hubble parameter, $q$ takes the form,

$q=-1+\frac{3\Omega_{m_{0}}a^{-3(1-b)}-\Gamma_{1}a^{-3}}{2\left(\frac{\Omega_{m_{0}}}{1-b}a^{-3(1-b)}-\frac{\Gamma_{1}}{3}a^{-3}+\Gamma_{2}\right)}.$
(26)

As $a\rightarrow 0$, the second term on the r.h.s. of the above equation
dominates hence $q>0,$ implies the prior deceleration era, while the second
term vanishes as $a\rightarrow\infty$, hence $q\rightarrow-1$ corresponding to
the end de Sitter epoch. For $a=1$, we get the present value, $q_{0}$ of the
parameter and it turns out that, the universe is accelerating for the current
epoch, thus

$q_{0}=-1+\frac{3\Omega_{m_{0}}-\Gamma_{1}}{2\left(\frac{\Omega_{m_{0}}}{1-b}-\frac{\Gamma_{1}}{3}+\Gamma_{2}\right)}$
(27)

in the present model. The evolutionary behavior of the deceleration parameter
$q$ as a function of redshift $z$ is plotted for the best-fit model parameters
obtained from $OHD36+SNIa$ dataset, in Fig. 7. The value of deceleration
parameter for the current epoch ($z=0$) is estimated as $-0.533$, which is in
close agreement with the observational results $q_{0}=-0.63\pm 0.12$[105],
$-0.644\pm 0.22$ for latest union2SN data [106] and $-0.48\pm 0.11$ for SN192
sample, $-0.65\pm 0.53$ for the radio galaxy sample [107]. The transition
redshift characterizing, $q=0,$ the onset of the late acceleration is found to
be around $z_{T}=0.660$ and is comparable with the observational constraint
using SN Ia+CMB+LSS joint analysis $z_{T}=0.61$[109] and 0.72 $\pm$ 0.05 [108]
from SN Ia+BAO/CMB dataset. But slightly higher compared to constraints using
SN Ia+CMB data, $z_{T}$=0.57$\pm$0.07 [105]. However it should be noted that,
the extracted values of $q_{0}$ and $z_{T}$ in the present model are
comparable to the corresponding values obtained for the standard $\Lambda$CDM
model ($z_{T}$=0.74 and $q_{0}$=-0.59) [108].

Figure 7: Behavior of q parameter vs z with transition redshift $z_{T}=0.660$

##  5 Statefinder Diagnostics

As there are a variety models of dark energy existing in the current
literature, it is needed to compare the model with the standard model. Sahni
et al. [110], have introduced the statefinder diagnostics, an effective tool
in contrasting a model with the standard model and also with different dark
energy models in a model independent way. The statefinder parameters ${r,s}$
are formulated from the scale factor $a(t)$ and its derivatives with respect
to cosmic time up to the third order. Following the evolutionary trajectory of
the model in $r-s$ plane, it can be classified as quintessence, if $r<1,s>0$
and chaplygin gas model, if $r>1$ and $s<0$ [111, 112, 110]. For the standard
$\Lambda$CDM model, the statefinder parameters are { ${r,s}$}=$\\{{1,0}\\}.$
The statefinder parameter pair $\\{{r,s}\\}$ takes the form[110, 112],

$r=\frac{\dddot{a}}{aH^{3}}$ (28) $s=\frac{r-1}{3(q-\frac{1}{2})}$ (29)

In terms of $\bar{H}$ the above equations can be expressed as,

$r=1+\frac{1}{2\bar{H}^{2}}\frac{d^{2}\bar{H}^{2}}{dx^{2}}+\frac{3}{2\bar{H}^{2}}\frac{d\bar{H}^{2}}{dx}$
(30)
$s=\frac{r-1}{{\frac{3}{2\bar{H}^{2}}}\frac{d\bar{H}^{2}}{dx}+\frac{9}{2}}$
(31)

Figure 8: Evolutionary trajectory of the interacting BHDE model in r-s plane
with best-estimated model parameters. Current value $\\{{r_{0},s_{0}}\\}$
=$\\{0.968,0.010\\}$

.

After substituting the respective terms from Eq. (10) and Eq. (11), the
parameters becomes,

$r=1-\frac{9b\Omega_{m_{0}}a^{-3(1-b)}+2\Gamma_{1}a^{-3})}{2\left(\frac{\Omega_{m_{0}}}{1-b}a^{-3(1-b)}-\frac{\Gamma_{1}}{3}a^{-3}+\Gamma_{2}\right)}$
(32)
$s=\frac{3b\Omega_{m_{0}}a^{-3(1-b)}}{\frac{b}{1-b}\Omega_{m_{0}}a^{-3(1-b)}+\Gamma_{2}}$
(33)

At the asymptotic limit $a\rightarrow\infty$ the statefinder parameters takes
the values, $r\rightarrow 1$ and $s\rightarrow 0,$ shows that the present
model will tends to the $\Lambda$CDM in the far future of the evolution. The
trajectory of the model in the $r-s$ plane with the best-fit parameters is
depicted in Fig. 8. The current value of $r$ and $s$ are determined from the
two-dimensional parametric plane as
$\\{{r_{0},s_{0}}\\}$=$\\{{0.968,0.010}\\}$ . This trajectory in the $r-s$
plane reveals that the model is arguably different from the standard
$\Lambda$CDM. It also shows that the parameters obey the condition $r<1,s>0$
throughout the evolution, and hence the proposed model has a quintessence
behavior.

##  6 Thermodynamics of Interacting BHDE Model

In this section we analyze the thermal evolution of the model, concentrating
on the evolution of the entropy. The universe is evolving as an ordinary
macroscopic system, hence advances towards an equilibrium state of maximum
entropy. The aforementioned statement is valid only if it satisfies the
appropriate constraints given by [113, 114, 115],

$\displaystyle\dot{S}\geq 0\textrm{\hskip 14.22636ptfor\hskip 2.84544pt
always\hskip 14.22636pt};\hskip 14.22636pt\ddot{S}<0\hskip
14.22636pt\textrm{for\hskip 2.84544pt atleast \hskip 2.84544ptlater\hskip
2.84544pt time\hskip 2.84544pt of\hskip 2.84544pt evolution.}$

where the over-dot represents the derivative with respect to cosmic time. The
first one represents the condition for the validity of the generalized second
law of thermodynamics, and the second one, which state that the second
derivative of the entropy should be negative at least at the end stage of the
universe, is the condition to be satisfied for having an upper bound to the
entropy of the system. According to GSL, total entropy comprises of matter
entropy enclosed by the horizon, $S_{m}$, and that of horizon $S_{h},$ should
always a non-decreasing function of cosmic time [116, 117, 118, 65, 113, 119,
120]. That is,

$\dot{S}_{m}+\dot{S}_{h}\geq 0$ (34)

Using conservation law and integrability condition[102], the entropy becomes,

$S=\frac{(\rho+P)V}{T},$ (35)

where $V$ is the volume of the horizon and $T$ is the temperature. The entropy
of the non-relativistic matter within the horizon of volume $V=4\pi
c^{3}/3H^{3}$ is,

$S_{m}=\frac{\pi\Omega_{m}H_{0}^{2}c^{5}}{\hbar GH^{4}}k_{{}_{B}},$ (36)

where we took $T,$ as the Hawking temperature, $T=\hbar H/(2\pi k_{{}_{B}})$.
For horizon entropy we have used the Barrow entropy relation such that,

$S_{h}=\left(\frac{\pi c^{5}}{\hbar GH^{2}}\right)^{1+\Delta/2}k_{B}$ (37)

Here $c$ is the speed of light in vacuum, $\hbar=h/2\pi$ with the Planck’s
constant, and $G$ is the gravitational constant. The rate of change of the
total entropy with respect to the variable, $x=lna$ is given as,

$S_{tot}^{\prime}=S_{m}^{\prime}+S_{h}^{\prime}$ (38)
$S_{tot}^{\prime}=\frac{\pi H_{0}^{2}c^{5}\Omega_{m}}{\hbar
G}k_{{}_{B}}\left(\frac{-3(1-b)}{H^{4}}-\frac{2}{H^{6}}\frac{dH^{2}}{dx}\right)-\left(\frac{\pi
c^{5}}{\hbar
G}\right)^{1+\Delta/2}\frac{2+\Delta}{2H^{4+\Delta}}\frac{dH^{2}}{dx}$ (39)

The evolution of $S_{tot}^{\prime}$ with scale factor is plotted in Fig. 9,
and is always positive. This shows that the entropy of the horizon plus that
of matter within the horizon is non-decreasing throughout the evolution of the
universe. So, the generalized second law of thermodynamics (GSL) is valid in
the universe, which comprises Barrow holographic dark energy and pressure-less
matter. It is to be noted that some recent works claim a conditional violation
of the GSL in the BHDE model for non-zero Barrow exponent, $\Delta.$ In
reference [76], the authors have considered a spatially flat FLRW universe,
which constitutes pressure-less matter and BHDE that interact with each other.
The authors extracted the time variation of total entropy and figure out the
possibility of conditional violation of GSL. Such a conditional violation of
the generalized second law is noted in reference [72], in which the authors
investigated the validity of GSL by considering Barrow entropy for the
apparent horizon along with the entropy contributions from matter and dark
energy fluids. They found that GSL may be conditionally violated depending on
the evolution of the universe. If the background evolution of the Hubble
function is similar to that in $\Lambda$CDM cosmology, then the generalized
second law is always satisfied irrespective of the value of $\Delta$.
Otherwise, the GSL be satisfied only for relatively low values of the Barrow
exponent, $\Delta$ or it violates the laws of thermodynamics for a large value
of $\Delta.$ So, the Barrow exponent is constrained such that the
corresponding entropy is very close to the standard Bekenstein entropy. For
$b=0,$ that is, dark sectors are non-interacting with each other, the Hubble
parameter in the present model behave like that of the $\Lambda$CDM with the
extracted mass density parameters, $\bar{\Omega}_{m_{0}}=0.290$ and
$\bar{\Omega}_{{}_{\Lambda 0}}=0.710$. In this situation, the GSL is perfectly
valid irrespective of the value of $\Delta.$ For $b\neq 0,$ we estimated model
parameters and found that the present model satisfies the GSL with the
estimated model parameters. However, it should be noted that the extracted
value of the Barrow exponent, $\Delta=0.063\pm 0.029,$ is much less. Hence,
the conclusion drawn in reference[72] that the GSL is generally valid for only
low values of the Barrow exponent is true. In this line, It has been reported
that the generalized second law is valid for interacting BHDE with Hubble
horizon and event horizon as IR cut-off in DGP (Dvali-Gabadadze-porrati)
braneworld cosmology [121]. Also, GSL was found to be valid in Barrow
holographic dark energy with NO (Nojiri-Odintsov) cut-off [122].

We further extended our study to check whether the present model predicts a
universe that evolves to a maximum entropy state corresponding to an upper
bound to the entropy at the end stage of the evolution. To check this, we have
obtained the second derivative of the total entropy as,

$S_{tot}^{\prime\prime}=\frac{3\pi H_{0}^{2}c^{5}\Omega_{m}}{\hbar
GH^{4}}k_{{}_{B}}\left[3(1-b)^{2}+\frac{4(1-b)}{H^{2}}\frac{dH^{2}}{dx}-\frac{2}{3H^{2}}\frac{d^{2}H^{2}}{dx^{2}}+\frac{2}{H^{4}}\left(\frac{dH^{2}}{dx}\right)^{2}\right]\\\
+\left(\frac{\pi c^{5}}{\hbar
G}\right)^{1+\Delta/2}\frac{2+\Delta}{2}\left[\frac{4+\Delta}{2H^{(6+\Delta)}}\left(\frac{dH^{2}}{dx}\right)^{2}-\frac{1}{H^{(4+\Delta)}}\frac{d^{2}H^{2}}{dx^{2}}\right]$
(40)

Since the relation (40) seems to be slightly tricky to analyze, the second
derivative of the total entropy for best-estimate model parameters is depicted
in Fig. 10. Following this, we obtained that $s_{tot}^{\prime\prime}<0$ at
least at the later time of evolution, guarantees the convexity of the
function, i.e., in an asymptotic limit $a\rightarrow\infty$ the second
derivative of the entropy approaches zero from below. So the figure points out
that the model satisfies the entropy maximization condition; thus, the entropy
will never grow unbounded. To a great extent, the analysis shows that the
universe constitutes interacting BHDE, and pressure-less matter shows a pretty
good behavior with the GSL of thermodynamics as well as evolves to a maximum
entropy state analogous to the ordinary macroscopic system.

Figure 9: Evolution of $S_{tot}^{\prime}$ vs scale factor $a$. The values of
model parameters is considered from the combined dataset of $OHD36+SNIa$.
Figure 10: $S_{tot}^{\prime\prime}$ vs scale factor $a$. The values of model
parameters is considered from the combined dataset of $OHD36+SNIa$.

##  7 Dynamical System Analysis

Phase space analysis has been widely used to obtain the asymptotic evolution
of the universe in various cosmological models [123, 124, 125]. The method
involves the formulation of a set of autonomous differential equations of
appropriately chosen dimensionless phase space variables $y_{i}$ as,

$y_{i}^{\prime}=f_{i}(y_{j})\hskip 28.45274pti,j=1,2,3,..$ (41)

Where the prime represents the derivative with respect to a suitably chosen
variable. The equilibrium solutions are referred as the critical points
($y_{i}=y_{i}^{*}$), obtained by considering $y_{i}^{\prime}=0$ for all $i$.
The stability of the model corresponding to these solutions can then be
obtained by analyzing the sign of the eigenvalues of the Jacobian matrix at
the critical points (or equilibrium points). If the eigenvalues of the
coefficient matrix is negative, the corresponding critical point is an
attractor, and the neighboring trajectories will always converge to that
point, thus a stable equilibrium. If eigenvalues are positive, then it is a
source point where the neighboring trajectories seem to be diverging from the
critical point; it is unstable. Suppose the eigenvalues have different signs,
then the critical point is said to be a saddle, and the stability depends on
the initial conditions[126].

We chose the dimensionless phase space variable $u$ and $v$ defined as,

$u=\frac{\rho_{m}}{3H^{2}}\hskip
85.35826ptv=\frac{\rho_{{}_{\Lambda}}}{3H^{2}}$ (42)

The autonomous differential equations corresponding to these phase space
variables, obtained using the Friedmann equation and the conservation
principle are,

$u^{\prime}=f(u,v)=3(b-1)u+u^{2}-2uv+2u$ (43)
$v^{\prime}=g(u,v)=\frac{(2-\Delta)v}{2}\frac{\alpha(2v-u-2)-\beta(\frac{9}{2}bu+\frac{5}{2}u+2v-2)}{\alpha-\beta(\frac{u}{2}-v+1)}-(2v^{2}-vu-2v)$
(44)

Here the differentiation is with respect to the variable $x=\ln a.$ By
equating $u^{\prime}=0$ and $v^{\prime}=0,$ we obtained physically feasible
critical points and are,

$(u^{*},v^{*})=(0,1),(1-3b,0).$ (45)

In order to study the nature of these critical points, consider a small
perturbation to the phase variables, say $u=u^{*}+\delta u$ and
$v=v^{*}+\delta v,$

$\left[\begin{array}[]{c}\delta u^{\prime}\\\ \delta
v^{\prime}\end{array}\right]=\begin{bmatrix}(\frac{\delta f}{\delta
u})^{*}&(\frac{\delta f}{\delta v})^{*}\\\ (\frac{\delta g}{\delta
u})^{*}&(\frac{\delta g}{\delta
v})^{*}\end{bmatrix}\times\left[\begin{array}[]{c}\delta u\\\ \delta
v\end{array}\right]$

The $2\times 2$ matrix in the above equation is the general form of the
Jacobian matrix. Diagonalizing the above matrix gives the corresponding
eigenvalues. Eigenvalues, along with the inference regarding stability
conditions of these critical points, are given in Table.2. The phase space
plot is given in Fig. 11.

Table 2:  Critical points and Eigenvalues  Critical points | Eigenvalues | Nature
---|---|---
$(0.923,0)$ | $(0.923,0.072)$ | unstable
$(0,1)$ | $(-2.926,-0.463)$ | stable
Figure 11: Phase space trajectories of the model in u-v plane with best-
estimated model parameters from combined dataset $OHD36+SNIa$.

The eigenvalues corresponding to the critical point $(0.923,0)$ are both
positive hence it is unstable. The figure reveals that trajectories around
this critical point are diverging. This point, in fact, corresponds to the
prior matter-dominated epoch, which is decelerating. On the other hand, the
eigenvalues corresponding to the critical point $(0,1)$ are such that both
values are negative. This critical point corresponds to the dominance of
Barrow holographic dark energy, hence corresponding to the end de Sitter
epoch. The phase plot shows that the trajectories are converging to this
critical point. Hence it can be an attractor. So the dynamical system analysis
of the interacting BHDE model holds the result that the model approaches a
stable end de Sitter state in the far future. A similar dynamical system
analysis was performed for an interacting BHDE model with dark energy density
having varying equation of state and Hubble horizon as IR cut-off [79]. The
authors performed the dynamical system analysis for the model with various
interacting terms and figured out that the end de sitter phase is stable.

##  8 Information Criteria and Model Selection

The introduction of more model parameters improves the fitting with
observational data, regardless of the relevance of extra parameters. In order
to analyze the statistical significance of any model, one can perform the
Information Criteria (IC) analysis such as Akaike information
criterion(AIC)[127] and Bayesian Information Criterion(BIC) or Schwarz’s
Bayesian criterion [128]. Such analysis will enable us to decide which one
among the competing models is preferred by the observational data[129, 130,
131]. AIC is derived by considering an approximate minimization of the
Kullback-Leibler information entropy, which measures the separation between
two probability distributions. The general expression for AIC is given as[133,
134, 135],

$AIC=-2ln(\mathcal{L}_{max})+2k+\frac{2k(k+1)}{N-k-1}$ (46)

Where $\mathcal{L}_{max}$ is the maximum likelihood of the considered dataset
contains $N$ number of data points and $k$ is the number of model parameters.
For large number of data points, the Eq. (46) reduces to,

$AIC=-2ln(\mathcal{L}_{max})+2k$ (47)

BIC is derived by approximating the Bayes factor [136], defined as,

$BIC=-2ln(\mathcal{L}_{max})+klnN$ (48)

On interpreting the obtained values of IC components, the most admitted model
with observational support is the one that minimizes the IC. Moreover, the
difference in IC has more significance, is defined as,

$\Delta IC_{model}=IC_{model}-IC_{min}$ (49)

where $IC_{min}$ is the minimum IC value in competing models. The criteria for
determining the strength of evidence against the model with minimum IC is
discussed in [134, 127, 132]. Accordingly, if $\Delta IC\leq 2$ the given
model has very strong evidence against the model with minimum $IC$, and for
the range $2-6$, there is evidence for the given model, but not strong.
Suppose the parameter $\Delta IC_{model}$ is in the range $6-10$; in that
case, the given model can claim only less evidence. For $\Delta$IC $>$10, the
model has essentially no evidence against the model with $IC_{min}$. We have
compared the AIC and BIC values of the present interacting BHDE model with the
standard $\Lambda$CDM model by evaluating the difference $\Delta AIC$ and
$\Delta BIC$, and are depicted in Table.3. The comparison of $\Delta$IC for
both AIC and BIC analysis leads to the following conclusions. The interacting
BHDE has considerably less evidence against $\Lambda$CDM in AIC analysis;
however, the model has essentially no support in BIC analysis. This is because
of the fact that the BIC substantially penalizes the cosmological model with a
larger number of free parameters.[130].

Table 3:  Information criteria AIC and BIC of interacting BHDE model and $\Lambda$CDM model along with the difference $\Delta$IC  Model | AIC | $\Delta AIC$ | BIC | $\Delta BIC$
---|---|---|---|---
Interacting BHDE | 1067.927 | 7.457 | 1102.846 | 27.411
$\Lambda$CDM | 1060.470 | 0 | 1075.435 | 0

##  9 Conclusion

The Quantum gravitational effect leads to a new area-entropy relation that
deviates from the ordinary Bekenstein-Hawking relation. In this line, Barrow
proposed a new relation for the horizon entropy, $S=(A/A_{0})^{1+\Delta/2}.$
Following the holographic principle in cosmology, a new dark energy model,
Barrow holographic dark energy, has been proposed in the recent literature. We
have analyzed a cosmological model with BHDE, with the Granda-Oliveros scale
as the IR cutoff, and non-relativistic matter as components in the present
work to explain the recent acceleration of the universe. The novelty is that
in the present model BHDE is treated as a dynamical vacuum with a constant
equation of state, $\omega_{\Lambda}=-1.$ We accounted for the interaction
between BHDE and the matter by assuming a phenomenological form for the
interaction. By assuming the interaction term as $Q=3bH\rho_{m},$ we have
analytically solved the Friedmann equations for finding the Hubble parameter
evolution. The Hubble parameter exhibits acceptable asymptotic behaviors in
its evolution. As the scale factor, $a\to 0$ the Hubble parameter reduces to a
suitable form, representing the prior matter dominated decelerated epoch. As
$a\to\infty,$ the Hubble function tends to a constant, implies an end de
Sitter epoch. Interestingly, the present model shows an exact $\Lambda$CDM
like behavior for the interaction parameter $b=0,$ in which the model predicts
effective mass density parameters for both matter and dark energy.

We extracted best-fit values of the model parameters at the $1\sigma$
confidence interval by contrasting the model with the cosmological
observational dataset $OHD36+SNIa$. The evolution of cosmological parameters
such as Hubble parameter, curvature scalar, density, deceleration parameter,
etc., have been analyzed. The obtained present value of Hubble parameter is
$69.256^{+{1.228}}_{-1.228}$ $kms^{-1}Mpc^{-1}$, present value of matter
density parameter $\Omega_{m_{0}}=0.281^{+{0.050}}_{-0.050}$ and the age of
the universe is around $13.958$ Gyr. These are in concordance with the
observational results. The behavior of the density parameter solves the
coincidence problem. The transition from a prior decelerated phase to an
accelerating phase is found to occur at transition redshift $z_{T}=0.660$, and
the present value of deceleration parameter $q_{0}=-0.533$ and is moderately
consistent with the observational results.

The thermal evolution of the model has been analyzed. We found that the model
satisfies the generalized second law of thermodynamics and the condition for
entropy maximization. Hence the end de Sitter turns out to be a state of
maximum entropy. Regarding the thermal behavior, it is to be noted that if
there is no interaction between the dark sectors, the model predicts an
evolution exactly similar to the standard $\Lambda$CDM and hence the validity
of GSL is automatically satisfied, for any value of the Barrow index,
$\Delta.$ It has been found that, with the interaction between the dark
sectors also, the GSL is satisfied. However, this time, it seems that the
validity of the GSL is guaranteed due to the low value of the $\Delta$
parameter extracted in comparison of the model with the observational data. In
this regard, the results of our model is in line with an earlier results in
the literature that,in BHDE model, the GSL is valid only for a much less value
of $\Delta.$ The dynamical system behavior of the present model predicts an
asymptotically stable de Sitter epoch. This stability of the end phase is a
ratification of the thermal evolution of the model, in which the end stage
corresponds to a state of maximum entropy. Finally, we extended our work to
perform the information criteria analysis to measure the significance of the
present model in comparison with the standard $\Lambda$CDM. Our analysis shows
that, apart from the interacting BHDE model, however, has significantly less
evidence against $\Lambda$CDM in the AIC analysis. Meantime, the model
essentially lacks support from BIC analysis.

To summarize, treating Barrow holographic dark energy as a dynamical vacuum
with $\omega_{{}_{\Lambda}}=-1$ has some advantages over the models in which
it is treated as dark energy of varying equation of state. The foremost thing
is that, at non-interaction, the model mimics a $\Lambda$CDM behavior with an
effective cosmological constant. Hence the model satisfies the GSL,
irrespective of the value of the Barrow exponent, which is results in
agreement to some previous works related to BHDE. The much less value
estimated for the Barrow exponent $\Delta$ shows that the new area-entropy
relation proposed by Barrow is very close to the Bekenstein-Hawking relation
for horizon entropy , and it generally respect GSL. Even though the model
predicts reasonably good background evolution, the AIC analysis shows less
evidence in favor of the present model. In contrast, BIC analysis shows no
evidence at all. However, it needs further detailed study regarding the
evidence of this model against the standard one, which we will reserve for
future work.

## Acknowledgments

We are thankful to the reviewers for the valuable and insightful comments
which helped us to improve the manuscript. We are thankful to Hassan Basari V
T, Sarath N and Dr. Jerin Mohan N D for their valuable suggestions in data
analysis. Nandhida Krishnan. P is thankful to CUSAT and Govt. of Kerala for
financial assistance.

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# Historical Ink: 19th Century Latin American Spanish Newspaper
Corpus with LLM OCR Correction

Laura Manrique-Gómez2 Tony Montes1 Rubén Manrique1
1 Systems and Computing Engineering Department, Universidad de los Andes
2 History and Geography Department, Universidad de los Andes
Bogotá D.C.
{l.manriqueg, t.montes<EMAIL_ADDRESS>

###### Abstract

This paper presents two significant contributions: first, a novel dataset of
19th-century Latin American press texts, which addresses the lack of
specialized corpora for historical and linguistic analysis in this region.
Second, it introduces a framework for OCR error correction and linguistic
surface form detection in digitized corpora, utilizing a Large Language Model.
This framework is adaptable to various contexts and, in this paper, is
specifically applied to the newly created dataset.

Historical Ink: 19th Century Latin American Spanish Newspaper
Corpus with LLM OCR Correction

Laura Manrique-Gómez2 Tony Montes1 Rubén Manrique1 1 Systems and Computing
Engineering Department, Universidad de los Andes 2 History and Geography
Department, Universidad de los Andes Bogotá D.C. {l.manriqueg, t.montes,
<EMAIL_ADDRESS>

##  1 Introduction

The computational processing of old press texts is an undertaking frequently
addressed. Newspapers, as key historical resources, contain a diverse range of
information about political, economic, and cultural processes and are abundant
due to focused efforts to preserve them within national archives. Indeed, the
discipline of Digital Humanities, which emphasizes the incorporation of
digital tools in humanities and social sciences research, has spent much of
the past three decades on the task of digitization, resulting in a wealth of
curated digital collections Berry and Fagerjord (2017); Dobson (2019).
However, digitizing these corpora has brought plenty of challenges in
transcribing the images into machine-readable texts.

A significant obstacle in this process is the accuracy of Optical Character
Recognition (OCR) technology, especially when dealing with historical
documents that often have degraded quality or non-standardized fonts.
Traditional OCR methods frequently produce errors that hinder subsequent text
analysis and research. Addressing these challenges requires advanced
techniques for error correction to ensure the reliability of digitized texts.

To overcome these challenges, we employed a Large Language Model (LLM),
specifically GPT-3.5, to perform OCR error correction and enhance the quality
of transcribed texts. This approach leverages the sophisticated natural
language understanding capabilities of GPT-3.5 to detect and correct errors
that traditional OCR systems might miss. By incorporating this LLM as part of
our framework, accuracy and readability improvement can be observed in the
digitized texts.

###  1.1 Related Work

One notable achievement in this realm is the "Chronicling America" initiative.
Produced as part of the American National Digital Newspaper Program and funded
by the National Endowment for the Humanities and the Library of Congress, this
project represents a major stride in the digitization of historical press
Humanities . Another substantial project is the "Digging into Data Challenge".
A part of the Transatlantic Partnership for Social Sciences and Humanities
2016, this initiative yielded a vast collection of 19th-century press
materials known as "Atlas - Oceanic Exchanges. Tracing Global Information
Networks in Historical Papers" Exchanges . Other significant works include
“Viral Texts: Mapping Networks of Reprinting in 19th-Century Newspapers and
Magazines” Cordell and Smith , a project that investigates 19th-century
journalistic reports to understand the culture of reprinting in the United
States before the Civil War, and the European project “Project Impresso: Media
Monitoring the Past” Impresso , which provides significant insights into the
specific requirements of the OCR tasks, necessary for transcribing old texts
in English and other Germanic languages.

Despite these advancements, there’s a lack of specialized corpora for the
Latin American press of the 19th century that allows for an understanding of
the region’s unique historical, cultural, and Spanish linguistic
specificities. To address this gap, our research presents a novel dataset of
Latin American press texts written in old Spanish from this period, with the
first version predominantly including newspapers from the region formerly
known as Nueva Granada, which encompassed Colombia, Panama, Venezuela, and
Ecuador. This dataset has been enriched with OCR-LLM models, aiding the
detection of character recognition errors and distinguishing them from
historical linguistic surface forms111The dataset is available at
https://huggingface.co/datasets/Flaglab/latam-xix in its three versions
”original”, ”cleaned”, and ”corrected”.

##  2 Sourcing

The dataset was constituted from the digital catalog of Colombia’s most
important newspaper archives, the Colombian National Library and the Luis
Ángel Arango Library. The objective of the collection focuses on those
publications with prints or illustrations for subsequent multimodal modeling.
Given that public collections did not contain metadata with information about
illustrations or cartoons, a manual revision was carried out that extended to
the physical collections in situ, since the physical collection is only
digitized to approximately 50%. 64 newspaper titles were identified (7% of the
total of the 1,655 publications of the collections) whose geographic origin is
mostly Bogotá but also includes publications from other cities of the Nueva
Granada, as well as Guayaquil, Panamá City, and Lima. This corpus constitutes
the version of the dataset used for the research in this paper. However, the
intention is to continue broadening the collection to include other countries
in Latin America.

The dataset consists of 4,032 pages of scanned images that were processed with
a layout model, trained specifically for the task, that was able to separate
the images from the texts. The texts were subsequently transcribed with the
Azure AI Vision Model which provides an OCR service222Model available through
Azure cloud services at https://learn.microsoft.com/en-us/azure/ai-
services/document-intelligence/concept-read. From a sample of 2,500 text
strings of 1,000 characters manually supervised, 8.5% of the total proved
unreadable. However, most of the texts contained multiple transcription errors
due to the highly artisanal printing techniques and the use of the epoch’s
different grammatical and lexical forms. Thus, impacting the readability of
the texts and adding a bias for its use as input to NLP-LLM models.

###  2.1 Structuring the data

Once the source newspaper text pictures are processed through Azure’s OCR, the
resulting JSON files contain the extracted text. These texts were manually
complemented with the newspaper metadata essential for future analysis. This
data was structured into a single parquet file with the following column
structure: newspaper id, text id within the newspaper (in the format {file
number}-{page}-{chunk number}), newspaper title, year, city, and the text
itself. For example, the first dataset row id is PD168, 1-page_0-0, for the
newspaper El oso from Lima, Peru.

##  3 Processing

The dataset contains samples of newspapers that were written by hand, or using
carving machines. These machines normally wear out with use and therefore some
features of the text turn out to be easily confused with backward accent
marks, unwanted punctuations, or misplaced characters between words. This
misreading interrupts the continuity of the text and doesn’t add any
semantical meaning to it.

Detecting these errors automatically is a challenge due to the language change
between modern Spanish and XIX-Century Spanish. There’s a lack of OCR models
trained on those types of texts and historical semantic and orthographic
shifts, thus not necessarily an error; instead, it’s a surface form of the
word e.g. the addition connector "y" (and) used to be written as "i".

There were other types of texts that, in addition to being simply poorly read,
were completely unintelligible for OCR (and also very difficult for humans to
understand) due to the fonts of some newspapers. Also, the multiples
newspapers’ layouts resulted in texts that contained lots of scores or
numbers, or samples containing only chapter titles or numbering such as "III
IV V" which add noise to the dataset.

It is possible to detect some of these errors with hardcoded rules that cover
them, addressed in an initial cleaning and filtering step, but some of them
such as surface form extraction or more complex OCR errors, are particularly
hard to express in coding rules. On the other hand, OCR correction using LLMs
is a very useful technique for this task as shown in Langlais (2024), but it
must be used with caution due to the semantic change of text between periods,
and the fact that LLMs are mostly trained on modern data, which of course, may
bias their output for this correction task.

###  3.1 Cleaning and filtering

Some of the most common cleaning steps for text data consist of removing
duplicates and removing noisy data. In particular, in this case, and for
purposes of subsequent analysis, these steps are fundamental.

  1. 1.

Remove duplicates and empty texts. $1.06\%$ of rows were removed.

  2. 2.

Filter out rows where over 50% of the characters are non-alphabetic. $1.28\%$
of rows were removed.

  3. 3.

Remove the rows with 4 or fewer tokens. For this, a new tokenizer was trained
with a vocabulary size of 52,000, trained from the BETO pre-trained tokenizer
Cañete et al. (2020). $0.81\%$ of rows were removed.

###  3.2 LLM-made Correction

On the other hand, OCR errors from newspapers are very difficult to detect
automatically and correct. Still, they are a huge source of noise as mentioned
in Lopresti (2008), and much more in old sources such as newspapers from the
19th century, where these errors tend to appear more due to the wear and tear
of newspaper paper or writing methodologies that are very different from the
modern ones.

In this paper, we use a technique for detecting OCR errors and correcting them
using GPT and taking advantage of the fact that LLMs were trained mostly on
modern language, using manually-checked rules, it is possible to classify
corrections between errors, word surface forms or none of both
(hallucinations). These rules, explained in the following section, were
revised and selected by a field expert who served as well as an evaluator for
these corrections testing their precision for this case.

To date, most of the LLMs are not capable of effectively returning in a
machine-readable format, all the corrections of a text, especially when texts
are very long, which is the case. This is the reason why a diff algorithm was
required to use the maximum capability of LLMs to correct the text and later
with algorithm-based approaches detect the differences between the original
text and the correct one to later be able to classify them. An example of an
old original text, a corrected one, and the differences returned by the
algorithm can be found in Appendix A, as well as the parameters chosen for
this step.

Figure 1: Distribution of newspapers and years between the whole dataset

###  3.3 Corrections Classification

Once the corrections are detected and isolated through the diff algorithm, the
last step is to classify them. Still, first, it is important to state the main
differences between the possible labels for each correction:

  * •

Surface form: In linguistics, the term surface form (or word form) denotes the
specific appearance of a word in a given context, contrasting with its lexical
form, which pertains to its meaning Sarveswaran et al. (2019). During the 19th
century in Latin America, certain words were documented with variant spellings
reflecting language shifts over time. It’s important to note that changes in
surface forms do not necessarily alter the semantic content of the word, but
rather represent orthographic modifications.

  * •

OCR error: An OCR error, on the other hand, refers to every possible misread
text from the real newspaper text. The OCR errors must be corrected but must
be carefully selected from the real newspaper linguistic "errors" that
contribute to the linguistics of the time.

  * •

Hallucinations: If none of the above is the case, the correction is an LLM
hallucination or a translation to modern Spanish, which would be wrong, so
these corrections must be omitted.

To enhance the analysis of classification rules, corrections were noted
alongside their frequency across the entire dataset to assess their relevance.
Additionally, all corrections were converted to lowercase to effectively group
them. During this process, many corrections were reviewed and consolidated
into a set of linguistic rules for categorization. This framework can be
applied to identify and analyze similar changes and classification rules in
other languages and specific contexts. In particular, this paper presents a
collection of carefully validated rules and exceptions standardized for
classifying corrections in the LatamXIX dataset.

####  3.3.1 Accent changes

When there are only accent changes (add or removal) between the original text
and the corrected one, the correction refers mostly to a surface form, because
Spanish accent rules in the 19th century tended to be very different from the
modern ones Montgomery (1966), and in particular there was a lack of many
accent rules, which allowed a very diverse set of accents expressions for the
same meaning word, such as the word "antes", which in some cases used to be
written as "ántes", with the accent. These kinds of surface forms mean a
problem for some NLP tasks, because, in Spanish, some words without the accent
may have a different meaning, specifically for the past forms of some verbs,
such as "acepto" (present, e.g. I accept) and "aceptó" (past, He accepted).
Therefore, for certain NLP tasks, it may be preferable to focus solely on the
surface forms without accent changes, which is another outcome presented in
this paper.

####  3.3.2 Specific changes

A particular set of changes was extracted to encapsulate an important set of
surface words and another set of changes that encapsulates some common OCR
errors. For example, the most common changes for surface words tend to be
related to the usage of "y" instead of "i" or "g" instead of "j", for example
in the words "mui", "jeneral" y "geroglíficos" (currently the words "muy",
"general" and "jeroglíficos"); in fact, the connector "y" used to be written
as "i" in most of the early 19th century texts Bouzouita and Gutiérrez (2015).
On the other hand, the most common OCR errors tend to be accent misreading or
number confusion such as "ó" being read as "6" or "i" being read as "1". A
more detailed list of examples and a list of surface form changes is available
in Appendix B and a list of OCR error changes in Appendix C.

####  3.3.3 Other letter-to-letter changes

Different from the previously stated changes, when the number of letters of
the original and the corrected one coincide, in general, tend to refer to OCR
errors, for example, "la" was misread as "In" or "señor" as "sefor".

####  3.3.4 Remaining changes

When the correction does not fit in any of the previous examples, the
correction itself won’t refer to surface forms, but it’s a challenging task to
automatically differentiate in this case between OCR errors and hallucinations
because there may be multiword corrections. For this, a ratio of text
similarity was computed based on the coincidence of positional characters
between the original and the corrected texts, and based on this ratio, the
number of words within the corrected text and the frequency of the correction,
was categorized between OCR errors and hallucinations. For illustration, the
OCR error detected "ascripeión" changed to "suscripción" returned a ratio of
0.76, but the hallucination "que" changed to "como" returned a ratio of 0.0
which effectively allows differentiating in most of the cases.

##  4 Results

Feature | Value
---|---
Size | $\sim 26MB$
Rows | $10,176$
Words | $\sim 4.4M$
Tokens | $\sim 5.5M$
Newspapers | 58
Years Range | 1845 - 1899
Surface Forms | 11,397
Non-Accent Surface Forms | 2,231
Table 1:  Final Historical Ink: LatamXIX corrected dataset

With the execution of all the mentioned steps, we end up with a set of very
useful tools more than just the LatamXIX dataset333There are 59 newspapers in
total, but only one has the period 1806-1809, so it was excluded from the
dataset overview. One of the most important outputs of this paper is the LLM
OCR correction framework which was designed to be easily exchanged between
datasets or LLMs so that it can be applied for further research on other
datasets. Also, an important output of the mentioned process is the list of
surface forms from 19th-century Latin American Spanish from newspapers, but it
also brings a general framework to detect these surface forms in a wide range
of contexts where it may play an important role.

In particular, Old Spanish surface forms benefit semantic change detection
tasks. These forms capture the semantic meaning variations of particular words
and can help to compare their historical evolution in different periods or
among different Spanish-speaking regions 444The dataset, surface forms, and
processing steps are available in https://github.com/historicalink/LatamXIX.

##  5 Future Work

This initial version of the dataset primarily includes newspapers from the
region formerly known as Nueva Granada, encompassing Colombia, Panama,
Venezuela, and Ecuador. Future dataset extensions will aim to incorporate
newspapers from a broader range of Latin American countries to ensure a more
comprehensive representation of the region. By extending the dataset to
include diverse newspapers from various Latin American countries, we aim to
provide a richer resource for historical and linguistic research.

Future work will also involve analyzing the semantic changes between 19th-
century Spanish in Latin America and modern Spanish. Additionally, comparing
these changes to the overall evolution of the Spanish language globally will
provide valuable insights into linguistic shifts over time and across
different regions.

Moreover, while the OCR correction using LLMs has moved closer to a fully
automatic pipeline, a significant portion of rule-defining in the presented
framework still requires manual professional effort. An automated method for
evaluating OCR accuracy is currently missing, as most of the evaluations and
rule definitions were performed manually by an expert. Future work should
focus on developing a comprehensive automatic evaluation method for OCR, as
well as further automating the rule-defining process, to enhance the
efficiency and accuracy of the OCR correction framework.

## References

  * Berry and Fagerjord (2017) David M. Berry and Anders Fagerjord. 2017.  _Digital Humanities: Knowledge and critique in a Digital age_.  Polity Press.
  * Bouzouita and Gutiérrez (2015) Miriam Bouzouita and Mara Fuertes Gutiérrez. 2015.  Spanish studies: Language and linguistics.  _The Year’s Work in Modern Language Studies_ , 75:171–185.
  * Cañete et al. (2020) José Cañete, Gabriel Chaperon, Rodrigo Fuentes, Jou-Hui Ho, Hojin Kang, and Jorge Pérez. 2020.  Spanish pre-trained bert model and evaluation data.  In _PML4DC at ICLR 2020_.
  * (4) Ryan Cordell and David Smith.  The viral texts project.  _Viral Texts: Mapping Networks of Reprinting in 19th-Century Newspapers and Magazines (2022)_.
  * Dobson (2019) James E. Dobson. 2019.  _Critical Digital Humanities: The search for a methodology_.  University of Illinois Press.
  * (6) Oceanic Exchanges.  The atlas.  _Mapping the Histories and Metadata of Digitised Newspapers Collections Around the World. (2021)_.
  * (7) National Endowment for the Humanities.  Chronicling america: Library of congress.  _News about Chronicling America RSS_.
  * (8) Impresso.  Media monitoring of the past.  _Impresso_.
  * Langlais (2024) Pierre-Carl Langlais. 2024.  Post-ocr-correction: 1 billion words dataset of automated ocr correction by llm.  _Hugging Face_.
  * Lopresti (2008) Daniel Lopresti. 2008.  Optical character recognition errors and their effects on natural language processing.  In _Proceedings of the Second Workshop on Analytics for Noisy Unstructured Text Data_ , page 9–16. Association for Computing Machinery.
  * Montgomery (1966) Thomas Montgomery. 1966.  On the development of spanish y from "et".  _Romance Notes_ , 8(1):137–142.
  * Sarveswaran et al. (2019) Kengatharaiyer Sarveswaran, Gihan Dias, and Miriam Butt. 2019.  Using meta-morph rules to develop morphological analysers: A case study concerning Tamil.  In _Proceedings of the 14th International Conference on Finite-State Methods and Natural Language Processing_ , pages 76–86, Dresden, Germany. Association for Computational Linguistics.

##  Appendix A Example LLM Correction

The LLM response was successful for most of the texts except for some cases
where Azure’s Content Policy was triggered due to text content, and for very
long texts where the model started to hallucinate the whole text. An example
of an original text, its retrieved LLM correction, and all the changes
detected by the diff algorithm is the following (surface forms and OCR errors)
is:

  * •

Original: La publicacion del Oso se harà dos veces cada se mana, y constará de
un pliego en cuarto ; ofreciendo à mas sus redactores, dar los gravados
oportunos, siempre que loexija el asuntode que trate. Redactado por un Num. 8.
TEMA del Periodico. POLITICA MILITAR. OCTAVA SESION. Abierta la sesion á las
dore y un minuto de la noche , 25 de Febrero de 1845 , con asistencia de todos
los Señores Representantes, se leyó y aprobó la acta de la Asamblea anterior ,
ménos en lo tocante à la torre del Convento de Santo Domingo, punto que quedó
para ventilarse en mejor ocasion. En seguida se dió cuenta de una nota del
Ejecutivo , referente à que urjía la necesidad de organizar un Ejército ; pues
decia el Excmo. Decano: - "Un poder sin bayonetas vale tanto como un cero
puesto á la izquierda."

  * •

Corrected: La publicación del Oso se hará dos veces cada semana, y constará de
un pliego en cuarto; ofreciendo además sus redactores, dar los grabados
oportunos, siempre que lo exija el asunto de que trate. Redactado por un Num.
8. TEMA del Periódico. POLÍTICA MILITAR. OCTAVA SESIÓN. Abierta la sesión a
las dos y un minuto de la noche, 25 de Febrero de 1845, con asistencia de
todos los Señores Representantes, se leyó y aprobó la acta de la Asamblea
anterior, menos en lo tocante a la torre del Convento de Santo Domingo, punto
que quedó para ventilarse en mejor ocasión. Enseguida se dio cuenta de una
nota del Ejecutivo, referente a que urgía la necesidad de organizar un
Ejército; pues decía el Excmo. Decano: - "Un poder sin bayonetas vale tanto
como un cero puesto a la izquierda."

##  Appendix B Specific Surface Form Changes

For the surface form extraction from the texts and its differentiation from
OCR errors and LLM hallucinations, a set of surface form changes was
constructed for 19th-century Latin American Spanish. The complete set of known
changes with an example for each case is presented in Table B1.

(Previous, Corrected) | Example
---|---
`(á, a) & (a, á)` | hara $\rightarrow$ hará
`(é, e) & (e, é)` | fué $\rightarrow$ fue
`(í, i) & (i, í)` | decia $\rightarrow$ decía
`(ó, o) & (o, ó)` | ocasion $\rightarrow$ ocasión
`(ú, u) & (u, ú)` | ningun $\rightarrow$ ningún
`(i, y) & (y, i)` | mui $\rightarrow$ muy
`(j, g) & (g, j)` | jente $\rightarrow$ gente
`(v, b) & (b, v)` | gravado $\rightarrow$ grabado
`(s, x) & (x, s)` | espiró $\rightarrow$ expiró
`(j, x) & (x, j)` | méjico $\rightarrow$ méxico
`(c, s) & (s, c)` | faces $\rightarrow$ fases
`(s, z) & (z, s)` | dies $\rightarrow$ diez
`(z, c)` | doze $\rightarrow$ doce
`(q, c)` | quatro $\rightarrow$ cuatro
`(n, ñ)` | senor $\rightarrow$ señor
`(ni, ñ)` | senior $\rightarrow$ señor
`(k, qu)` | nikel $\rightarrow$ níquel
`(k, c)` | kiosko $\rightarrow$ quiosco
`(ou, u)` | boulevar $\rightarrow$ bulevar
`(s, bs)` | suscriciones $\rightarrow$ subscripciones
`(c, pc)` | suscriciones $\rightarrow$ subscripciones
`(s, ns)` | trasportar $\rightarrow$ transportar
`(t, pt)` | setiembre $\rightarrow$ septiembre
`(rt, r)` | libertar $\rightarrow$ liberar
`(r, rr) & (rr, r)` | vireinato $\rightarrow$ virreinato
`(...lo, lo ...)` | cambiólo $\rightarrow$ lo cambió
`(...se, se ...)` | acercóse $\rightarrow$ se acercó
Table B1: Set of Surface Form Changes for its extraction from dataset (2)

##  Appendix C OCR Errors Form Changes

There’s also a set of known changes for OCR errors, presented in Table C1.

(Previous, Corrected) | Observation
---|---
`(6, ó)` |
`(6, o)` |
`(1, y)` | 1 $\rightarrow$ i $\rightarrow$ y
`(4, a)` |
Table C1: Set of OCR Error Changes for its classification

Note that these known changes don’t cover the whole dataset’s OCR error cases,
and are just a predefined set for quickly detecting some of the most common
cases.