# Inflation and tachyon model of Barrow holographic dark energy

G. G. Luciano<EMAIL_ADDRESS>Dipartimento di Fisica, Università di
Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy  INFN,
Sezione di Napoli, Gruppo collegato di Salerno, Via Giovanni Paolo II, 132
I-84084 Fisciano (SA), Italy  Y. Liu<EMAIL_ADDRESS>School of
Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
Nottingham Centre of Gravity, University of Nottingham, Nottingham NG7 2RD, UK

###### Abstract

We study the connection between the tachyon dark energy model and Barrow
holographic dark energy (BHDE). The latter is a modified scenario based on the
application of the holographic principle with Barrow entropy instead of the
usual Bekenstein-Hawking one. We reconstruct the dynamics of the tachyon
scalar field both in the presence and absence of interactions between dark
energy and matter. For a flat Friedmann-Robertson-Walker Universe, it is shown
that the time derivative of the tachyon field $\dot{T}^{2}$ vanishes. This
leads to the condition $\omega_{D}=-1$ for the equation of state parameter of
BHDE, providing a potential explanation for the origin of the cosmological
constant. On the other hand, the tachyon field exhibits a non-trivial dynamics
for a non-flat Universe. In particular, in a closed Universe $\dot{T}^{2}$
decreases monotonically for increasing $\cos(R_{h}/a)$, where $R_{h}$ and $a$
are the future event horizon and the scale factor. The same behavior occurs in
an open Universe, but in this case $\dot{T}^{2}<0$, revealing that BHDE cannot
model the tachyon dark energy. We then study the inflation mechanism. We find
an analytical solution for the characteristic slow-roll parameters, the scalar
spectral index and the tensor-to-scalar ratio. We finally comment on the
Trans-Planckian Censorship Conjecture in BHDE.

##  I Introduction

Experimental evidences from Supernova SNIa, Baryon acoustic oscillations and
gravitational waves have definitely proved that our Universe is expanding at
an accelerated rate. In spite of enormous effort, a fully consistent
explanation for the origin of this behavior is missing. Among the various
mechanisms, the existence of an unknown form of energy (Dark Energy, DE)
affecting the Universe on large scales is the most widely accepted proposal.
But yet, the nature of DE remains quite elusive. The possibility that DE is
modeled by the cosmological constant acting as source of vacuum energy has
been originally considered as natural way out of the DE puzzle Sahni:1999gb ;
Peebles:2002gy . However, this scenario is at odds with our field theoretical
understanding of the quantum properties of vacuum, thus requiring further
investigation. Along this line, a plethora of DE models have been put forward
over the years Peebles:1987ek ; Ratra:1987rm ; Wetterich:1987fm ;
Frieman:1995pm ; Turner:1997npq ; Caldwell:1997ii ; Armendariz-Picon:2000nqq ;
Amendola:1999er ; Armendariz-Picon:2000ulo ; Kamenshchik:2001cp ;
Caldwell:1999ew ; Caldwell:2003vq ; Nojiri:2003vn ; Feng:2004ad ; Guo:2004fq ;
Elizalde:2004mq ; Nojiri:2005sx ; Deffayet:2001pu ; Sen:2002in .

An interesting model to account for the nature of DE is the so called
Holographic Dark Energy (HDE) Cohen:1998zx ; Horava:2000tb ; Thomas:2002pq ;
Li:2004rb ; Bamba:2012cp ; Ghaffari:2014pxa ; Wang:2016och , which emerges
within quantum gravity framework. The main ingredient of this approach is the
holographic principle, according to which the description of a volume of space
can be thought of as encoded on a lower-dimensional boundary surface to the
region. In tHooft:1993dmi ; Susskind:1994vu  it has been pointed out that
effective local quantum field theories over-count the number of independent
degrees of freedom, predicting that entropy scales extensively ($S\sim L^{3}$)
for systems of size $L$ with UV cutoff $\Lambda$. Later on, a solution to this
problem has been provided in Cohen:1998zx , where it has been argued that the
total energy of a system with size $L$ should not exceed that of an equally
sized black hole, i.e. $L^{3}\rho_{\Lambda}\leq LM_{p}^{2}$. Here,
$M_{p}=(8\pi G)^{-1/2}$ is the reduced Planck mass, while $\rho_{\Lambda}$
denotes the quantum zero-point energy density caused by the UV cutoff
$\Lambda$ (we are working in natural units $\hslash=c=1$). The inequality is
saturated for the largest value of $L$. In this context, the holographic dark
energy density is obtained as

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

where $c$ is a dimensionless constant and the factor $3$ has been introduced
for mere convenience.

Cosmological applications of the holographic principle and HDE have been
largely considered in literature. As an example, it was analyzed by
Enqvist:2004ny  that the consequence of excluding those degrees of freedom of
the system that will never be observed by the effective field theory results
into an IR cutoff $L$ at the future event horizon. In a DE dominated Universe,
such an horizon is then predicted to tend toward a constant value of the order
$H_{0}^{-1}$, with $H_{0}$ being the present Hubble radius Setare:2007hq .
Furthermore, the issue of assuming the apparent (Hubble) horizon $R_{A}=1/H$
as IR cutoff in a flat Universe has been examined in Hsu:2004ri .

Despite the intensive study, the shortcomings of the HDE in describing the
history of a flat Friedmann-Robertson-Walker (FRW) Universe have prompted
tentative changes to this approach. For instance, HDE has been used to address
the DE problem in Brans-Dicke Cosmology Gong:1999ge ; Kim:2005gk ;
Setare:2006yj ; Banerjee:2000gt ; Xu:2008sn ; Khodam-Mohammadi:2014wla  by
considering different IR cutoffs Wang:2016och ; Ghaffari:2014pxa ; Xu:2008sn
or and/or deformed entropy-area laws Tavayef:2018xwx ; Saridakis:2018unr ;
Saridakis:2020zol ; Moradpour:2020dfm ; Drepanou:2021jiv ; Hernand . In
particular, the latter path has led to promising models, such as Tsallis
Tavayef:2018xwx ; Saridakis:2018unr , Barrow Saridakis:2020zol  and Kaniadakis
Drepanou:2021jiv ; Hernand  holographic dark energy. Recently, motivated by
the analysis of Setare:2007hq , it has been shown that Tsallis holographic
description of DE (THDE) is non-trivially intertwined with tachyon dark energy
model Liu:2021heo , in which the tachyon scalar field is considered as a
source of DE. Specifically, a correspondence between the two scenarios has
been established based on the reconstruction of the dynamics of the tachyon
field in THDE.

Starting from the above premises, in this work we explore more in-depth the
connection between the tachyon dark energy model and HDE. We frame our
analysis in the context of HDE based on Barrow entropy Saridakis:2020zol ,
which arises from the attempt to incorporate quantum gravity effects on the
horizon surface Barrow:2020tzx . In this sense, our study must be intended as
a preliminary effort toward extending the paradigm of Liu:2021heo  to a fully
quantum gravity picture. We analyze the cases of flat and non-flat FRW
Universe, for both interacting and non-interacting DE. Since scalar fields are
generally conjectured to have driven inflation in the very early Universe, we
then study the inflation mechanism in our BHDE model. We find an analytical
solution for the slow-roll parameters, the scalar spectral index and the
tensor-to-scalar ratio. We also compare our findings with recent results in
the literature.

The remainder of the work is structured as follows: in the next Section we
introduce BHDE. Section III and IV are devoted to analyze the correspondence
between the tachyon dark energy and BHDE in a flat and non-flat FRW Universe,
respectively. In Sec. V we discuss inflation in BHDE, while conclusions and
outlook are summarized in Sec. VI.

##  II Barrow holographic dark energy

Let us briefly review the basics of BHDE. We consider the four-dimensional
Friedmann-Robertson-Walker (FRW) metric

$ds^{2}\,=\,-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-k\,r^{2}}+r^{2}d\Omega^{2}\right),$
(2)

of scale factor $a(t)$ and spatial curvature $k=0,1,-1$ for a flat, closed and
open Universe, respectively.

We use the definition (1) for the holographic dark energy in standard
Cosmology and assume

$L(t)\,=\,a(t)r(t)\,,$ (3)

where $r(t)$ is the (time-dependent) radius that is relevant to the future
event horizon of the Universe Setare:2007hq . Since

$\displaystyle\int_{0}^{r_{1}}\frac{dr}{\sqrt{1-kr^{2}}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{|k|}}\sin\mathrm{n}^{-1}(\sqrt{|k|}r_{1})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{rcl}&&\sin^{-1}(\sqrt{|k|}r_{1})/\sqrt{|k|},\hskip
17.07182ptk=1,\\\\[8.53581pt] \vspace{2mm}\hfil&&r_{1},\hskip
95.3169ptk=0,\\\\[2.84526pt]
&&\sinh^{-1}(\sqrt{|k|}r_{1})/\sqrt{|k|},\,\,\,\,\,\,\,k=-1,\end{array}\right.$

we easily obtain

$L(t)\,=\,\frac{a(t)\sin\mathrm{n}\left[\sqrt{|k|}R_{h}(t)/a(t)\right]}{\sqrt{|k|}}\,,$
(6)

where $R_{h}$ is the future event horizon given by

$R_{h}\,=\,a\int_{t}^{\infty}\frac{dt}{a}\,=\,a\int_{a}^{\infty}\frac{da}{Ha^{2}}\,,\quad\,\,H=\frac{\dot{a}}{a}\,.$
(7)

HDE relies on the holographic principle, which asserts that the number of
degrees of freedom describing the physics of any quantum gravity system _i_)
scales as the bounding surface (rather than the volume) of the system and
_ii_) should be constrained by an infrared cutoff tHooft:1993dmi ;
Susskind:1994vu . This is in tune with Bekenstein-Hawking (BH) relation
$S_{BH}=A/A_{0}$ for black holes, where $S_{BH}$ and $A$ denote the entropy
and area of the black hole, respectively, while $A_{0}=1/(4G)$ its Planck
area. Recently, deformations of this relation have been proposed to take
account of quantum Barrow:2020tzx ; Tsallis:1987eu ; Tsallis:2012js  and/or
relativistic Kaniadakis:2002zz  effects. In particular, in Barrow:2020tzx  it
has been argued that quantum gravity may introduce intricate, fractal features
on the black hole horizon, leading to the modified area law

$S_{\Delta}\,=\,\left(\frac{A}{A_{0}}\right)^{1+\Delta/2}.$ (8)

Deviations from BH entropy are quantified by the exponent $0\leq\Delta\leq 1$,
with $\Delta=0$ giving the BH limit, while $\Delta=1$ corresponding to the
maximal horizon deformation. We emphasize that although this relation
resembles Tsallis entropy in non-extensive statistical thermodynamics
Tsallis:1987eu ; Tsallis:2012js , the origin and motivation underlying Eq. (8)
are completely different. Cosmological implications of Barrow entropy have
been recently studied in the context of Big Bang Nucleosynthesis
Barrow:2020kug  and Baryogenesis Luciano:2022pzg , among others. The
possibility of a running $\Delta$ has also been considered in Running .

Strictly speaking, Eq. (8) has been formulated for black holes. However, it is
known that in any gravity theory one can consider the entropy for the Universe
horizon in the same form as the black hole entropy, the only adjustment being
the replacement of the black hole horizon radius with the apparent horizon
radius. This is at the heart of the various generalizations of HDE with
modified entropy laws (see, e.g. Tavayef:2018xwx ; Saridakis:2018unr ;
Saridakis:2020zol ; Moradpour:2020dfm ; Drepanou:2021jiv ).

Now, in Cohen:1998zx  Cohen et al. have proposed the following inequality
between the entropy, the IR ($L$) and UV ($\Lambda$) cutoffs for a given
system in an effective local quantum field theory

$L^{3}\Lambda^{3}\,\leq\,S_{max}\simeq\,S_{BH}^{3/4}\,.$ (9)

If we express $S_{BH}$ as in Eq. (8), we have

$\Lambda^{4}\,\leq\,\left(2\sqrt{\pi}\right)^{2+\Delta}\frac{L^{\Delta-2}}{A_{0}^{1+\Delta/2}}\,,$
(10)

where $\Lambda^{4}$ denotes the vacuum energy density, i.e. the energy density
of DE ($\rho_{D}$) in the HDE hypothesis Guberina:2006qh .

By using the above inequality, Barrow holographic dark energy density can be
proposed as

$\rho_{D}\,=\,CL^{\Delta-2}\,,$ (11)

where $C$ is an unknown parameter with dimensions $[L]^{-2-\Delta}$. It is
worth noticing that for $\Delta=0$, the above relation reduces to the standard
HDE (1), provided that $C=3c^{2}M_{p}^{2}$. On the other hand, in the case
where deformations effects switch on ($\Delta\neq 0$), BHDE departs from the
standard HDE, leading to different cosmological scenarios Saridakis:2020zol .

Following the standard literature, we now define the critical energy density
$\rho_{cr}$ and the curvature energy density $\rho_{k}$ as

$\rho_{cr}\,=\,3M_{p}^{2}H^{2}\,,\qquad\rho_{k}\,=\,\frac{3k}{8\pi Ga^{2}}\,.$
(12)

We also introduce the three fractional energy densities

$\displaystyle\Omega_{m}$ $\displaystyle=$
$\displaystyle\frac{\rho_{m}}{\rho_{cr}}\,=\,\frac{\rho_{m}}{3M_{p}^{2}H^{2}}\,,$
(13) $\displaystyle\Omega_{D}$ $\displaystyle=$
$\displaystyle\frac{\rho_{D}}{\rho_{cr}}\,=\,\frac{C}{3M_{p}^{2}H^{2}}L^{\Delta-2}\,,$
(14) $\displaystyle\Omega_{k}$ $\displaystyle=$
$\displaystyle\frac{\rho_{k}}{\rho_{cr}}\,=\,\frac{k}{H^{2}a^{2}}\,,$ (15)

where $\rho_{m}=3(1-c^{2})M_{p}^{2}H^{2}$ is the matter energy density. In
particular, by setting $L=H^{-1}$ we obtain111There are several choices for
the IR cutoff $L$. Following Tavayef:2018xwx , here we resort to the simplest
one $L=H^{-1}$. Other possible choices are the particle horizon, the future
event horizon, the GO cutoff GO  or combination thereof. However, in these
cases one must generally resort to numerical evaluation to study the
cosmological evolution of the model Saridakis:2018unr . Since we are
interested in extracting analytical solutions and given the degree of
arbitrariness in the selection of the best dark energy description, we leave
the analysis of dark energy models with different IR cutoffs for future
investigation.

$\Omega_{D}\,=\,\frac{C}{3M_{p}^{2}\,H^{\Delta}}\,.$ (16)

From Eqs. (3) and (II), one can derive the following expression for the time
derivative of $L$ Liu:2021heo

$\dot{L}\,=\,HL+a\dot{r}\,=\,1-\frac{1}{\sqrt{|k|}}\cos\mathrm{n}(\sqrt{|k|}R_{h}/a)\,,$
(17)

where we have defined Setare:2007hq

$\frac{1}{\sqrt{|k|}}\cos\mathrm{n}(\sqrt{|k|}x)=\left\\{\begin{array}[]{rcl}&&\cos(x)\,,\hskip
17.07182ptk=1,\\\\[8.53581pt] \vspace{2mm}\hfil&&1\,,\hskip
38.98013ptk=0,\\\\[2.84526pt] &&\cosh(x),\,\,\,\,\,\,\,\hskip
1.42262ptk=-1.\end{array}\right.$ (18)

Now, for a flat FRW Universe filled by non-interacting BHDE and pressureless
DM, the first Friedmann equation takes the form

$H^{2}\,=\,\frac{1}{3M_{p}^{2}}\left(\rho_{D}+\rho_{m}\right),$ (19)

which, by use of Eqs. (13) and (14), can be rewritten as

$\Omega_{m}+\Omega_{D}\,=\,\Omega_{D}(1+u)\,=\,1\,,$ (20)

where

$u=\frac{\rho_{m}}{\rho_{D}}=\frac{\Omega_{m}}{\Omega_{D}}\,.$ (21)

Since BHDE does not interact with other parts of cosmos (DM), the conservation
equations of dust and THDE read

$\displaystyle\dot{\rho}_{m}+3H\rho_{m}\,=\,0\,,$ (22)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+\omega_{D})\,=\,0\,,$ (23)

where we have denoted by $\omega_{D}=p_{D}/\rho_{D}$ and $p_{D}$ the equation
of state parameter and pressure of THDE, respectively.

Deriving Eq. (19) respect to time and using the continuity equations (22) and
(23), after some algebra we are led to

$\frac{\dot{H}}{H^{2}}\,=\,-\frac{3}{2}\left(1+\omega_{D}+u\right)\Omega_{D}\,.$
(24)

Likewise, by plugging Eq. (11) into (23), we find

$\frac{\dot{H}}{H^{2}}\,=\,(1+\omega_{D})\frac{3}{\Delta-2}\,,$ (25)

which gives, by comparison with Eq. (24)

$\omega_{D}\,=\,\frac{u(2-\Delta)\Omega_{D}}{2-(2-\Delta)\Omega_{D}}-1\,.$
(26)

With the aid of Eq. (20), this finally yields222Unlike THDE model, where
$\omega_{D}$ is divergent for $\delta_{T}<1$ and $\Omega_{D}=1(2-\delta_{T})$
($\delta_{T}$ is the Tsallis exponent), BHDE is well-defined for any value of
$0\leq\Delta\leq 1$.

$\omega_{D}\,=\,\frac{\Delta}{(2-\Delta)\Omega_{D}-2}\,.$ (27)

On the other hand, if there exists an interaction

$Q=3b^{2}H(\rho_{m}+\rho_{D})$ (28)

between BHDE and matter, the continuity equations (22) and (23) become

$\displaystyle\dot{\rho}_{m}+3H\rho_{m}\,=\,Q\,,$ (30)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+\omega_{D})\,=\,-Q\,.$

Following similar calculations as above, one can show that Barrow holographic
energy equation of state takes the form

$\omega_{D}\,=\,\frac{\Delta+2b^{2}/\Omega_{D}}{(2-\Delta)\Omega_{D}-2}\,,$
(31)

where $b$ is the coupling parameter that quantifies the interaction.

###  II.1 The age of the Universe

We can now estimate the age of the present Universe as

$\displaystyle t$ $\displaystyle=$ $\displaystyle\int
dt\,\frac{dH}{dH}=\int\frac{H^{2}}{\dot{H}}\,\frac{dH}{H^{2}}$ (32)
$\displaystyle=$
$\displaystyle\frac{2\left(\frac{C}{3M_{p}^{2}}\right)^{1/\Delta}}{3\,\Delta}\int\frac{1-\left(1-\Delta/2\right)\Omega_{D}}{\Omega_{D}^{1-1/\Delta}\left(1-\Omega_{D}\right)}\,d\Omega_{D}\,,$

where we have used Eqs. (25) and (27) along with

$\frac{dH}{H^{2}}\,=\,-\frac{\left(\frac{C}{3M_{p}^{2}}\right)^{1/\Delta}}{\Delta\,\Omega_{D}^{1-1/\Delta}}\,d\Omega_{D}\,.$
(33)

By integrating the above relation, it follows that

$t=\frac{2-\Delta}{3H}\left[1+\left(\frac{\Delta}{2-\Delta}\right)\,{}_{2}F_{1}(1,\frac{1}{\Delta};1+\frac{1}{\Delta};\frac{C}{3M_{p}^{2}\,H^{\Delta}})\right],$
(34)

where ${}_{2}F_{1}(a,b;c;d)$ is the hypergeometric function of first kind.
This equation can be used to estimate the order of the age of the current
Universe ($z=0$) in our model. By exploiting Eqs. (25) and (32), we get

$\displaystyle t$ $\displaystyle\approx$
$\displaystyle\left(\frac{H^{2}}{\dot{H}}\right)\big{|}_{z=0}\int\frac{dH}{H^{2}}$
(35) $\displaystyle=$
$\displaystyle\frac{2-\Delta}{3H_{0}}\left(1-\frac{\omega_{D}(z=0)}{1+\omega_{D}(z=0)}\right).$

For $\omega_{D}(z=0)=-2/3$, we then have $t=1/H_{0}$ for $\Delta=1$,
corresponding to the maximal deformation of the Bekenstein-Hawking area law.
As observed in Tavayef:2018xwx , further corrections to Eq. (35) may arise due
to either different modifications of the horizon entropy or other IR cutoffs.

##  III Tachyon scalar field as Barrow holographic dark energy: flat FRW
Universe

In this Section we analyze the correspondence between the tachyon dark energy
model and the BHDE scenario in a flat FRW Universe. We consider the non-
interacting and interacting case separately. A preliminary study along this
direction has been carried out in Prelimin .

###  III.1 Non-interacting case

In Setare:2007hq  it has been shown that the energy density $\rho_{T}$ and
pressure $p_{T}$ for the tachyon scalar field take the form

$\displaystyle\rho_{T}$ $\displaystyle=$
$\displaystyle\frac{V(T)}{\sqrt{1-\dot{T}^{2}}}\,,$ (36) $\displaystyle p_{T}$
$\displaystyle=$ $\displaystyle-V(T)\sqrt{1-\dot{T}^{2}}\,,$ (37)

where $V(T)$ is the tachyon potential energy. From these relations, we derive
the equation of state parameter for the tachyon as

$\omega_{T}\,=\,p_{T}/\rho_{T}=\dot{T}^{2}-1\,.$ (38)

We now aim at describing the dynamics of the tachyon scalar field in BHDE.
Toward this end, we assume that the tachyon energy density (36) can be modeled
by Barrow holographic dark energy density (11) evaluated at the future event
horizon $R_{h}$. We then obtain

$CR_{h}^{\Delta-2}\,=\,\frac{V(T)}{\sqrt{1-\dot{T}^{2}}}\,.$ (39)

On the other hand, by equating Eqs. (27) and (38), it follows that

$\frac{\Delta}{(2-\Delta)\Omega_{D}-2}\,=\,\dot{T}^{2}-1\,,$ (40)

which gives

$\dot{T}\,=\,\sqrt{\frac{\left(2-\Delta\right)\Omega_{D}+\Delta-2}{\left(2-\Delta\right)\Omega_{D}-2}}\,.$
(41)

Let us now invert Eq. (39) respect to the potential $V(T)$

$V(T)\,=\,CR_{h}^{\Delta-2}\sqrt{1-\dot{T}^{2}}\,.$ (42)

Inserting Eq. (41), we infer

$V(T)\,=\,CR_{h}^{\Delta-2}\sqrt{\frac{\Delta}{2-(2-\Delta)\Omega_{D}}}\,.$
(43)

Similarly, from Eq. (37) we get for the tachyon pressure

$p_{T}\,=\,p_{D}\,=\,-CR_{h}^{\Delta-2}\,\frac{\Delta}{2-(2-\Delta)\Omega_{D}}\,.$
(44)

The above relations can be further elaborated by taking the time derivative of
$\rho_{D}=\rho_{T}$ in Eq. (11)

$\dot{\rho}_{D}\,=\,C(\Delta-2)L^{\Delta-3}\dot{L}\,,$ (45)

and observing that $\dot{L}=0$ in a flat Universe (see Eq. (17)), which
implies $\dot{\rho}_{D}=0$. In the absence of interactions between holographic
dark energy and matter, from Eq. (23) we obtain $\omega_{D}=-1$ and, thus,
$\dot{T}=0$. This is satisfied, provided that

$\left(2-\Delta\right)\Omega_{D}+\Delta-2\,=\,0\,,$ (46)

which admits as solution

$\Omega_{D}=1\,.$ (47)

By using the above results in Eqs. (42) and (44), we reconstruct the tachyon
potential and pressure as

$\displaystyle V(T)$ $\displaystyle=$ $\displaystyle CR_{h}^{\Delta-2}\,,$
(48) $\displaystyle p_{T}$ $\displaystyle=$ $\displaystyle-
CR_{h}^{\Delta-2}\,,$ (49)

We notice that the above picture is consistent with the cosmological constant
scenario, since $\omega_{D}=-1$ as discussed above.

###  III.2 Interacting case

Let us now extend the above analysis to the interacting case. In this context,
the continuity equations for dark energy and matter are provided by Eqs. (30)
and (30), respectively, with the coupling term being given by Eq. (28). As in
the previous case, the condition $\dot{\rho}_{D}=0$ holds true. However, by
combining with Eq. (30), we now obtain

$\displaystyle\omega_{D}$ $\displaystyle=$
$\displaystyle-b^{2}\left(\frac{\Omega_{m}}{\Omega_{D}}+1\right)-1$ (50)
$\displaystyle=$ $\displaystyle-\frac{b^{2}}{\Omega_{D}}-1\,,$

where in the second step we have used Eq. (20). By equating to Eq. (38), we
get

$\dot{T}^{2}\,=\,-\frac{b^{2}}{\Omega_{D}}\,,$ (51)

which makes only sense for $b=0$. As in Liu:2021heo , we thus find that in a
flat FRW Universe, $\dot{T}^{2}$ must always be vanishing. We can also extract
the following relations

$\displaystyle V(T)\,=\,\rho_{D}\,,$ (52) $\displaystyle
p_{D}\,=\,-\rho_{D}\,,$ (53)

which are consistent with the condition $\omega_{D}=-1$ discussed in the non-
interacting case.

From Eqs. (52) and (53) we can also explore the stability of our BHDE model
against perturbation. To this end, we compute the squared sound speed
$v_{s}^{2}$. Clearly, for $v_{s}^{2}>0$, the model is stable, otherwise it is
unstable. The squared sound speed is defined as

$v^{2}_{s}=\frac{dp_{D}}{d\rho_{D}}\,,$ (54)

which gives in our case $v^{2}_{s}=-1<0$. This means that the model is
unstable. Therefore, we cannot conclude that a Barrow holographic dark energy
dominated Universe will be the fate of the future Universe.

Moreover, we have $\omega_{D}=-1$, then $\omega^{\prime}_{D}=0$, where the
prime denotes the derivative with respect to $\ln a$. Therefore, the
$\omega$-$\omega^{\prime}$ analysis is meaningless for this model.

##  IV Tachyon scalar field as Barrow holographic dark energy: non-flat FRW
Universe

Let us explore how the connection between BHDE and tachyon dark energy model
appears in a non-flat FRW Universe. Toward this end, we consider the time
derivative of BHDE (11) and use Eq. (17) to get

$\dot{\rho}_{D}\,=\,C(\Delta-2)L^{\Delta-3}\left[1-\frac{1}{\sqrt{|k|}}\cos\mathrm{n}(\sqrt{|k|}x)\right],$
(55)

where $x\equiv R_{h}/a$. By means of the continuity equation (30), this can be
cast as

$\displaystyle-3H\rho_{D}(1+\omega_{D})-3b^{2}H(\rho_{m}+\rho_{D})$
$\displaystyle\hskip
8.53581pt=C(\Delta-2)L^{\Delta-3}\left[1-\frac{1}{\sqrt{|k|}}\cos\mathrm{n}(\sqrt{|k|}x)\right].$
(56)

We can now resort to Eq. (38) (we remind that we are imposing
$\omega_{T}=\omega_{D}$ in our model) to obtain

$\displaystyle-3H\rho_{D}\dot{T}^{2}-3b^{2}H(\rho_{m}+\rho_{D})$
$\displaystyle\hskip
8.53581pt=C(\Delta-2)L^{\Delta-3}\left[1-\frac{1}{\sqrt{|k|}}\cos\mathrm{n}(\sqrt{|k|}x)\right].$
(57)

This relation can be further manipulated by dividing both sides by
$3H\rho_{D}$ and using Eq. (21) to give

$\displaystyle-\dot{T}^{2}-b^{2}(u+1)$ $\displaystyle\hskip
8.53581pt=\frac{C(\Delta-2)L^{\Delta-3}}{3H\rho_{D}}\left[1-\frac{1}{\sqrt{|k|}}\cos\mathrm{n}(\sqrt{|k|}x)\right].$
(58)

After employing Eq. (11) and the condition $L=H^{-1}$, we finally reach

$\displaystyle-\dot{T}^{2}-b^{2}(u+1)$ $\displaystyle\hskip
8.53581pt=\frac{\Delta-2}{3}\left[1-\frac{1}{\sqrt{|k|}}\cos\mathrm{n}(\sqrt{|k|}x)\right],$
(59)

which can be equivalently written as

$\displaystyle\dot{T}^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{3}-b^{2}(u+1)-\frac{\Delta}{3}$
$\displaystyle+\,\frac{\Delta-2}{3}\frac{1}{\sqrt{k}}\cos\mathrm{n}(\sqrt{|k|}x)\,.$

Figure 1: Evolution trajectories of $\dot{T}^{2}$ for a closed ($k=1$)
Universe. We set $u=0.04$ and $b^{2}=0.01$ as in Liu:2021heo .

Let us now distinguish between the cases $k=\pm 1$ (one can verify that $k=0$
correctly reproduces the model in Sec. III.2). For $k=1$, we get from the
definition (18)

$\dot{T}^{2}\,=\,\frac{2}{3}-b^{2}(u+1)-\frac{\Delta}{3}+\frac{\Delta-2}{3}\cos(x)\,.$
(61)

In order for $\dot{T}$ to be zero (i.e. $\omega_{D}=-1$), we must have

$\cos(x)\,=\,\frac{3b^{2}(u+1)}{\Delta-2}+1\,.$ (62)

This provides the condition for which tachyon model of BHDE can be used to
explain the origin of the cosmological constant in a closed FRW Universe. By
contrast, in Liu:2021heo  it is shown that $\dot{T}^{2}$ cannot be zero in
Tsallis holographic dark energy in a non-flat Universe. The evolution
trajectory of $\dot{T}^{2}$ in Eq. (61) is plotted in Fig. 1 for fixed $b$ and
$u$ and various $\Delta$. One can see that $\dot{T}^{2}$ is positive and
decreases monotonically for increasing $\cos(x)$.

Similarly, we can consider the dynamics of the tachyon field in an open
($k=-1$) Universe. Following the same reasoning as above, we now obtain

$\dot{T}^{2}\,=\,\frac{2}{3}-b^{2}(u+1)-\frac{\Delta}{3}+\frac{\Delta-2}{3}\cosh(x)\,,$
(63)

which vanishes provided that

$\cosh(x)\,=\,\frac{3b^{2}(u+1)}{\Delta-2}+1\,.$ (64)

However, since $\cosh(x)>1$, we infer that this model cannot explain the
cosmological constant, in agreement with the result of Liu:2021heo . Indeed,
there is no value of $0\leq\Delta\leq 1$, such that the right side of Eq. (64)
is higher than unity. The evolution of $\dot{T}^{2}$ in Eq. (63) is plotted in
Fig. 2 for different $\Delta$. As before, we notice that $\dot{T}^{2}$
decreases monotonically for increasing $\cosh(x)$, but in this case it is
always smaller than zero, which is not a physically valid situation. In
passing, we mention that the same behavior has been found in Liu:2021heo  in
the context of HDE based on Tsallis entropy. Therefore, we conclude that BHDE
is not suitable to model the tachyon dark energy in an open Universe.

Figure 2: Evolution trajectories of $\dot{T}^{2}$ for an open ($k=-1$)
Universe. We set $u=0.04$ and $b^{2}=0.01$ as in Liu:2021heo .

##  V Inflation in Barrow Holographic Dark Energy

In this Section we discuss inflation in BHDE. For reasons that will appear
clear below and following Inft , here we consider the more general expression
for the length scale $L^{-2}=\alpha H^{2}+\beta\dot{H}$, where $\alpha$ and
$\beta$ are dimensionless constant. Assuming that the expansion of the
Universe is driven by BHDE (11) and neglecting the matter contribution due to
the rapid inflationary expansion, Eq. (19) becomes

$H^{2}\,=\,\frac{C}{3M_{p}^{2}}\left(\alpha
H^{2}+\beta\dot{H}\right)^{1-\Delta/2}\,,$ (65)

from which we infer

$\dot{H}\,=\,\frac{H^{2}}{\beta}\left[\left(\frac{3M_{p}^{2}}{C}\right)^{\frac{2}{2-\Delta}}(H^{2})^{\frac{\Delta}{2-\Delta}}-\alpha\right].$
(66)

From this relation, it is clear that setting the IR cutoff $L\simeq H^{-1}$
(i.e. $\beta=0$) as in the previous study would give rise to technical issues
in the present framework.

To simplify the resolution of Eq. (66), we introduce the e-folds variable
$N=\log\left(a/a_{i}\right)$, where $a_{i}$ is the initial value of the scale
factor $a$. By observing that $dN=Hdt$ and
$\dot{H}=\frac{1}{2}\frac{dH^{2}}{dN}$, integration of Eq. (66) gives

$\log\left\\{\tilde{H}^{2}\left[\gamma\left(\tilde{H}^{2}\right)^{\frac{\Delta}{2-\Delta}}\right]^{1-2/\Delta}-\alpha\right\\}\Bigg{|}_{\tilde{H}_{i}}^{\tilde{H}_{f}}\,=\,-\frac{2\alpha
N}{\beta}\,,$ (67)

where $\tilde{H}=H/M_{p}$ is the dimensionless Hubble parameter and

$\gamma=\left(\frac{3M_{p}^{2}}{C}\right)^{\frac{2}{2-\Delta}}\,M_{p}^{\frac{2\Delta}{2-\Delta}}\,.$
(68)

Here we have denoted the Hubble parameter at the end of inflation by
$\tilde{H}_{f}$.

From Eq. (65) we can now compute the characteristic parameters of slow-roll
inflation. Specifically, the first slow-roll parameter is given by

$\epsilon_{1}\,=\,-\frac{\dot{H}}{H}=-\frac{1}{\beta}\left[\gamma\left(\tilde{H}^{2}\right)^{\frac{\Delta}{2-\Delta}}-\alpha\right].$
(69)

The other slow-roll parameters can be derived by using the definition
$\epsilon_{n+1}=d\log(\epsilon_{n})/dN$. For the second parameter
$\epsilon_{2}$ we get

$\epsilon_{2}\,=\,\frac{\dot{\epsilon}_{1}}{H\epsilon_{1}}=\frac{2\gamma}{\beta}\left(\frac{\Delta}{2-\Delta}\right)\left(\tilde{H}^{2}\right)^{\frac{\Delta}{2-\Delta}}\,.$
(70)

Let us now evaluate the Hubble parameter at the end of inflation. This phase
is characterized by $\epsilon_{1}=1$. By straightforward calculations, we
obtain

$\tilde{H}_{f}^{2}\,=\,\left(\frac{\gamma}{\alpha-\beta}\right)^{1-2/\Delta}\,.$
(71)

On the other hand, at the beginning of inflation (including the horizon
crossing time) Eq. (67) gives

$\tilde{H}_{i}^{2}\,=\,\left[\frac{\gamma}{\alpha}\left(1+\frac{\beta\,e^{2\alpha
N/\beta}}{\alpha-\beta}\right)\right]^{1-2/\Delta}\,,$ (72)

which can be used to calculate the slow-roll parameters for earlier time by
direct substitution in Eqs. (69) and (70).

In order to derive the scalar spectral index $n_{s}-1$ and the tensor-to-
scalar ratio $r$, we follow PLBSar ; Inft  and make use of the usual
perturbation procedure. We are led to

$n_{s}-1\,=\,-2\epsilon_{1}-2\epsilon_{2}\,,\qquad r=16\epsilon_{1}\,.$ (73)

Clearly, a full perturbation analysis is needed to obtain the exact
expressions of $n_{s}-1$ and $r$.

Two comments are in order here: first, we notice that the constant $\gamma$
does not intervene in the calculation of the slow-roll parameters at the
horizon crossing time, which means that neither $n_{s}-1$ nor $r$ depend on
it. As explained in Inft , this constant can be estimated by considering the
amplitude of the scalar perturbation. Furthermore, it is worth mentioning that
a similar analysis of inflation and correspondence between BHDE and tachyon
field has been proposed in Prelimin . However, in that case the authors
consider values of Barrow parameter $\Delta$ higher than unity, which are
actually forbidden in Barrow model. This somehow questions the results
exhibited in Prelimin .

###  V.1 Trans-Planckian Censorship Conjecture

The large-scale structures we currently see in the Universe originated from
matter and energy quantum fluctuations produced during inflation. Such
fluctuations cross the Hubble radius during the early phase, are stretched out
and classicalize, and finally re-enter the Hubble horizon to produce the CMB
anisotropies. The key point is that if inflation lasted longer than the
supposed minimal period, then it would be possible to observe length scales
originated from modes smaller than the Planck length at inflation Martin .
This problem is usually referred to as “Trans-Planckian problem”. To avoid
inconsistencies, it has been conjectured that this problem cannot arise in any
consistent model of quantum gravity (“Trans-Planckian Censorship Conjecture”,
TCC) Bedroya .

The TCC states that no length scales which cross the Hubble horizon could ever
have had a wavelength smaller than the Planck length. This is imposed by
requiring that

$\frac{L_{p}}{a_{i}}\,<\,\frac{H_{f}^{-1}}{a_{f}}\,,$ (74)

where $L_{p}=1/M_{p}$ is the Planck length and we have denoted by $a_{f}$ the
scale factor at the end of inflation. By using Eq. (71) for the Hubble
parameter at the final time, the TCC (74) becomes

$\left(\frac{\gamma}{\alpha-\beta}\right)^{1-2/\Delta}<(8\pi\hskip
0.56905pte^{N})^{2}\,,$ (75)

the validity of which can be examined by comparison with future observational
data.

##  VI Conclusions and outlook

The origin of the accelerated expansion of the Universe is an open problem in
modern Cosmology. To date, the most reliable explanation is provided by the
existence of an enigmatic form of energy - the Dark Energy - affecting the
Universe on large scales. Several candidates have been considered to account
for this phenomenon. In particular, the holographic dark energy has been
largely studied, also in connection with different real scalar field theories,
such as quintessence Peebles:1987ek ; Ratra:1987rm ; Wetterich:1987fm ;
Frieman:1995pm ; Turner:1997npq ; Caldwell:1997ii , K-essence Armendariz-
Picon:2000nqq , phantom Caldwell:1999ew ; Caldwell:2003vq ; Nojiri:2003vn ,
braneworld Amendola:1999er , interacting models Deffayet:2001pu  and tachyon
Sen:2002in ; Setare:2007hq . Recently, the interest has been extended to the
Tsallis holographic dark energy Tavayef:2018xwx ; Saridakis:2018unr ;
Saridakis:2020zol ; Moradpour:2020dfm ; Drepanou:2021jiv  and the possibility
of using it to describe the dynamics of the tachyon field Liu:2021heo .

In this work we have considered the further scenario of tachyon model as
Barrow holographic dark energy. Barrow entropy arises from the effort to
include quantum gravity effects on the black hole horizon. In this sense, the
present analysis must be intended as a preliminary step toward a fully quantum
gravity extension of Liu:2021heo . We have established a correspondence
between BHDE and the tachyon field model in a FRW Universe, both in the
presence and absence of interactions between dark energy and matter. For a
flat FRW Universe, we have found that $\dot{T}^{2}=0$, i.e. $\omega_{D}=-1$,
providing a possible explanation for the cosmological constant. On the other
hand, the dynamics of the tachyon field turns out to be much more complicated
for a curved Universe. Specifically, for a closed Universe $\dot{T}^{2}$
decreases monotonically for increasing $\cos(R_{h}/a)$ and vanishes when Eq.
(62) is satisfied. By contrast, $\dot{T}^{2}$ is always negative for an open
Universe, which shows that BHDE cannot be used to model tachyon field dynamics
in this case. We have finally investigated an inflationary scenario described
by a Universe filled with BHDE and commented on the Trans-Planckian Censorship
Conjecture. Comparison with future observational data will enable us to
establish whether such model is empirically consistent.

Further aspects remain to be addressed. For instance, we can look at the
correspondence between the tachyon field and other dark energy scenarios, in
particular stable dark energy models. Furthermore, it is interesting to extend
the above framework to the case of Kaniadakis holographic dark energy
Drepanou:2021jiv , which is based on a relativistic self-consistent
generalization of the classical Boltzmann-Gibbs entropy Kaniadakis:2002zz .
Finally, it is essential to examine to what extent our effective model
reconciles with predictions of more fundamental candidate theories of quantum
gravity, such as String Theory or Loop Quantum Gravity. Work along these
directions is presently under active investigation and will be presented
elsewhere.

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# Topic Ontologies for Arguments

Yamen Ajjour
Lebiniz University Hannover
<EMAIL_ADDRESS>
Johannes Kiesel
Bauhaus-Universität Weimar
<EMAIL_ADDRESS>
Benno Stein
Bauhaus-Universität Weimar
<EMAIL_ADDRESS>
Martin Potthast
Leipzig University
<EMAIL_ADDRESS>

###### Abstract

Many computational argumentation tasks, like stance classification, are topic-
dependent: the effectiveness of approaches to these tasks significantly
depends on whether the approaches were trained on arguments from the same
topics as those they are tested on. So, which are these topics that
researchers train approaches on? This paper contributes the first
comprehensive survey of topic coverage, assessing 45 argument corpora. For the
assessment, we take the first step towards building an argument topic
ontology, consulting three diverse authoritative sources: the World Economic
Forum, the Wikipedia list of controversial topics, and Debatepedia. Comparing
the topic sets between the authoritative sources and corpora, our analysis
shows that the corpora topics—which are mostly those frequently discussed in
public online fora—are covered well by the sources. However, other topics from
the sources are less extensively covered by the corpora of today, revealing
interesting future directions for corpus construction.

##  1 Introduction

The term “topic” refers to a text’s subject matter. A text can be about one or
more topics; the relation underlying topics and texts is called “aboutness”
Yablo (2014). Topics play a central role in argumentation, since they
constrain or guide strategies and rhetorical devices by providing the accepted
and expected universe of discourse. Also, the view of pragma-dialectics in
argumentation emphasizes that argumentation is topic-dependent van Eemeren
(2015): “The basic aspects of strategic maneuvering […] are making an
expedient selection from the ‘topical potential’ available at a certain
discussion.” Though debaters often use commonplace arguments across topics
Bilu et al. (2019), this is only possible for related topics: a black-market
argument, for example, applies to topics like banning drugs or banning guns.
When developing computational models to extract, analyze, or generate
arguments, however, one should thus ensure a wide topic coverage in model
training to improve the model’s generalizability (e.g., as recently shown by
Reuver et al., 2021).

A set of topics may be organized as a graph, sometimes called “topic space”.
Information theorists and library scientists map hierarchical topic relations
within ontologies Hjørland (2001). Here, topics are labeled with a subject
heading, i.e., a phrase from a controlled vocabulary which concisely and
discriminatively describes a topic. Library ontologies are not designed with
argumentation tasks in mind, but other ontology efforts specifically address
argumentative topic spaces. We identified and harnessed three authoritative
sources of ontologic knowledge that cover global issues, controversies, and
popular debates: the World Economic Forum’s “Strategic Intelligence” site,
Wikipedia’s list of controversial topics, and Debatepedia’s debate
classification system (cf. Section 4).

We contribute a comprehensive overview of argument corpora and their topic
coverage as per the mentioned ontologies. The coverage of corpora that provide
topic labels is manually assessed by aligning each label to the ontologies’
topics, computing the proportion of ontology topics covered by a corpus, and
the distribution of corpus arguments in an ontology. Our analyses show that
existing corpora focus on a subset of possible topics (cf. Section 5). For the
corpora without topic labels, we categorize their argumentative texts by
measuring the semantic relatedness of corpus documents to ontology topics.
Given the 748 topic, this is a challenging classification, for which we
achieve a remarkable F1 of 0.59 (cf. Section 6).111Anonymized data at
https://zenodo.org/record/3928096.

Altogether, we lay the foundation for the study and systematic exploration of
controversial topics within computational argumentation analysis. The
identified authoritative resources already capture quite comprehensively their
respective domains. Future work will have to extend our approach to other
topic spaces, such as business, domestic, historic, and scientific
argumentation spaces.

##  2 Related Work

Our review of related work focuses on the role of the variable “topic” in
computational argumentation. Moreover, we briefly review topic ontologies and
hierarchical topic classification.

###  2.1 Topics in Computational Argumentation

In computational argumentation, arguments are typically modeled as
compositions of argument units, where an argument unit is represented as a
span of text. Habernal and Gurevych (2016a) adopts Toulmin (1958)’s (1958)
model, which defines six unit types, among which are “claim” and “data”.
Wachsmuth et al. (2017) employ a more basic model of two units, which defines
an argument as a claim or conclusion supported by one or more premises. These
models capture arguments without explicitly identifying the topic they
address. Levy et al. (2014) consider claims to be topic-dependent and study
their detection in the context of a random selection of 32 topics from
idebate.org. This work raises the question why topic-dependence has not been
addressed more urgently until now.

Key tasks for computational argumentation include the mining of arguments from
natural language Moens et al. (2007); Al-Khatib et al. (2016), classifying
their stances with regard to a thesis Bar-Haim et al. (2017), and analyzing
which arguments are more persuasive Tan et al. (2016); Habernal and Gurevych
(2016a). Current approaches to these tasks rely on supervised classification.
Daxenberger et al. (2017) show that supervised classifiers fail to generalize
across domains ($\sim$ topics). More recently, Stab et al. (2018) tweak BiLSTM
Graves and Schmidhuberab (2005) to integrate the topic while jointly detecting
(1) whether a sentence is an argument and (2) its stance to the topic. The
designed neural network outperforms BiLSTM without topic integration in both
tasks; the approach gives further evidence for the topic-dependence of
argument mining and stance classification. Whether model transfer between more
closely related topics works better is unknown. As a first step, Reuver et al.
(2021) show that cross-topic stance-classification with BERT Devlin et al.
(2018) produces mixed results depending on the topics, but misses the
relations between the topics. Gu et al. (2018) show that integrating the topic
of an argument helps assessing its persuasiveness.

Topic plays a central role in argument retrieval and generation since it
defines what arguments are relevant. Argument retrieval aims at delivering pro
and con arguments on a given topic query. A major challenge in argument
retrieval is the grouping of arguments that address common aspects of a topic.
As shown by Reimers et al. (2019) and Ajjour et al. (2019a), integrating the
topic is an important step while clustering arguments. For argument
generation, Bilu et al. (2019) introduce an approach that matches an input
topic against a list of topics that are paired with sets of topic-adjustable
commonplace arguments (e.g., black-market arguments). In a similar vein, Bar-
Haim et al. (2019) identify consistent and contrastive topics for a given
topic with the goal of expanding the topic in a new direction (e.g., fast food
versus obesity). Both approaches show the merit of utilizing argument topic
ontologies in argument generation. Perhaps only abstract argumentation can be
conceived of as topic independent, since it studies the structure and
relations among arguments more than their language.

###  2.2 Topic Ontologies

In information science, an ontology is defined as “an explicit specification
of a conceptualization” Gruber (1993). Topic ontologies are a specific type of
ontologies which specify topics as nodes of a directed acyclic graph. An edge
in the graph then implies an “is part of”-relation between the topics Xamena
et al. (2017). The effort in creating topic ontologies ranges from ad-hoc
decisions (e.g., tags for blog posts) to extensive classification schemes for
libraries. The oldest classification scheme that is still used today in
libraries is the Dewey Decimal Classification. It has been translated into
over 30 languages, and it contains several tens of thousands of classes. Most
topic ontologies focus on a specific domain, such as a the ACM Computing
Classification System for computer science, or DMOZ for web
pages.222https://dl.acm.org/ccs and https://dmoz-odp.org/ The only topic
ontology directly linked to arguments is that of Debatepedia.

###  2.3 Hierarchical Text Classification

Hierarchical text classification aims at classifying a document into a class
hierarchy. Depending on how the hierarchical structure is exploited,
classification can be done top-down (from higher classes downwards), bottom-
up, or flat (ignoring hierarchical relations) Silla and Freitas (2011).
Researchers usually train supervised classifiers for each class in the
hierarchy Sun and Lim (2001).

##  3 Survey of Argument Corpora

Corpus | Authors | Source | Unit granularity | Units | Topics | Exp.
---|---|---|---|---|---|---
Manual selection
Arguing Subjectivity | Conard et al. (2012) | Editorials | Editorial/blog | 84 | 1 | 1
Argumentative Sentences | Eyal et al. (2020) | Wikipedia | Arguments | 700 | 20 | 1
Argument Facet Similarity | Misra et al. (2016) | Debate portals | Argument | 6,188 | 3 | 8
Claim and Evidence 1 | Aharoni et al. (2014) | Wikipedia | Wikipedia article | 315 | 33 | 18
Claim and Evidence 2 | Rinott et al. (2015) | Wikipedia | Wikipedia article | 547 | 58 | 12
Claim Generation | Gretz et al. (2020) | Generated text | Argument Unit | 2,839 | 136 | 1
Claim Stance | Bar-Haim et al. (2017) | Wikipedia | Argument Unit | 2,394 | 55 | 10
Claim Sentence Search | Levy et al. (2018) | Wikipedia | Argument unit | 1,492,077 | 150 | 3
COMARG | Boltuz̆ić and Šnajder (2014) | Debate portals | Argument pair | 2,298 | 2 | 3
Evidence Sentences | Schnarch et al. (2018) | Wikipedia | Argument unit | 5,783 | 118 | 5
Evidence Sentences 2 | Ein-Dor et al. (2020) | Wikipedia | Argument unit | 29,429 | 221 | 3
Evidence Quality | Gleize et al. (2019) | Wikipedia | Argument pair | 5,697 | 69 | 1
ICLE Essay Scoring | Persing et al. (2010) | Essays | Essay | 1,000 | 10 | 11
Ideological Debates Reasons | Hasan and Ng (2014) | Debate portals | Argument | 4,903 | 4 | 10
Internet Argument Corpus v2 | Abbott et al. (2016) | Web | Discussion | 16,555 | 19 | 18
Key Point Analysis | Bar-Haim et al. (2020) | Wikipedia | Argument | 24,093 | 28 | 2
Micro Text v1 | Peldszus and Stede (2015) | Essays | Essay | 112 | 18 | 7
Micro Text v2 | Skeppstedt et al. (2018) | Essays | Essay | 171 | 35 | 1
Multilingual Argument Mining | Toledo-Ronen et al. (2020) | Wikipedia | Argument unit | 65,708 | 347 | 2
Political Argumentation | Menini et al. (2018) | Presidential debate | Argument pair | 1,462 | 5 | 3
Record Debating Dataset 2 | Mirkin et al. (2018) | Debating | Speech | 200 | 50 | 5
Record Debating Dataset 3 | Lavee et al. (2019) | Debating | Speech | 400 | 199 | 1
Record Debating Dataset 4 | Orbach et al. (2019) | Debating | Speech | 200 | 50 | 1
Record Debating Dataset 5 | Orbach et al. (2020) | Debating | Speech | 3,562 | 397 | 1
Sci-arg | Lauscher et al. (2018) | Scientific papers | Paper | 40 | 1 | 3
UKP Sentential | Stab et al. (2018) | Web | Argument | 25,492 | 8 | 13
UKP Aspect | Reimers et al. (2019) | Web | Argument pair | 3,595 | 28 | 3
UKPConvArg1 | Habernal and Gurevych (2016c) | Debate portals | Argument pair | 11,650 | 16 | 10
UKPConvArg2 | Habernal and Gurevych (2016b) | Debate portals | Argument pair | 9,111 | 16 | 3
WebDiscourse | Habernal and Gurevych (2016a) | Web | Document | 340 | 6 | 7
Webis-debate-16 | Al-Khatib et al. (2016a) | Debate portals | Debate | 445 | 14 | 3
Source-driven: greedy within a time-span
AIFdb | Bex et al. (2013) | Web | Argument unit | 67,408 | n/a | 7
Args-me | Ajjour et al. (2019b) | Debate portals | Argument | 387,692 | n/a | 3
ChangeMyView | Tan et al. (2016) | Discussion forum | Post/comment | 14,066 | n/a | 21
DebateSum | Roush and Balaji (2020) | Debating | Debate | 187,386 | n/a | 1
Intelligence Squared Debates | Zhang et al. (2016) | Debate portals | Debate | 108 | n/a | 3
Kialo | Kialo (2020) | Debate portals | Argument unit | 331,684 | n/a | 3
Political Speech | Lippi and Torroni (2016) | Ministerial debate | Argument unit | 152 | n/a | 1
USElecDeb60To16 | Haddadan et al. (2019) | Presidential debate | Debate | 42 | n/a | 1
Source-driven: sampled
Argument Annotated Essays | Stab and Gurevych (2017) | Essays | Essay | 402 | n/a | 28
E-rulemaking | Park and Cardie (2018) | Discussion forum | Argument | 731 | n/a | 3
ECHR | Poudyal et al. (2020) | Law Case | Argument | 743 | n/a | 1
Editorials | Al-Khatib et al. (2016b) | Editorials | Editorial | 300 | n/a | 8
GAQCorpus | Ng et al. (2020) | Web | Argument | 6,424 | n/a | 1
IDebate Persuasiveness | Persing and Ng (2017) | Debate portals | Argument | 1,205 | n/a | 1
Table 1: Survey of argument corpora indicating data source, unit granularity,
and size in terms of units and topics (if authors remarked on it). The unit
granularity is the one in the corpus’ files, using premises and conclusions as
one unit each and the best context-preserving unit for corpora featuring
multiple granularities. We presume these topic selection directives from the
corpus description: either manual selection by the authors, or source-
driven—i.e., the topics in the selected source(s)—from the units of a specific
time-span or by random sampling. Experiments (Exp.) denotes the count of
papers that use the corpus in an experiment among those papers that cite the
corpus’ paper.

To study arguments and computational argumentation tasks, researchers compile
corpora with argumentative texts. To the best of our knowledge, Table 1 lists
all corpora dedicated to argumentation to 2020. We review these corpora and
their associated publications with regard to what are the sources of
arguments, what is the granularity of the corpus, what is the size of the
corpora in terms of their units, and which and how many different topics are
covered in them. Reviewing all papers citing a corpus, we also analyzed how
many experiments were carried out using them.

The most elaborate discussion of topic selection is given in Habernal and
Gurevych (2016a), who chose six topics (homeschooling, public versus private
schools, redshirting, prayers in schools, single sex education, mainstreaming)
to focus on different education-related aspects. The broadest selection of
topics is reported by the researchers of IBM
Debater,333https://www.research.ibm.com/haifa/dept/vst/debating_data.shtml who
obtain arguments from Wikipedia. The only other work mentioning their source
of topics stems from Stab et al. (2018), who randomly select 8 topics from two
lists of controversial topics that originate from an online library and the
debate portal ProCon.org, respectively. Peldszus and Stede (2015) predefine a
set of topics and give writers the freedom to choose which one to write about,
but nothing is said about where the set of predefined topics originate from.
Conard et al. (2012) and Hasan and Ng (2014) explicitly select topics (1 and
4, respectively). For all other corpora with topic labels, their authors do
not argue on choosing topics, nor selection or sampling criteria. Neither do
the authors of corpora without topic labels.

Figure 1: Example for an assignment of arguments (bottom) to topics of a two-
leveled ontology. Level 2 topics are subtopics of their linked Level 1 topics.
Arguments linked to a Level 2 topic also pertain to its Level 1 upper-topics.

Altogether, it appears that the best practices in argumentation do not as of
yet consider topic sampling as a prerequisite task to ensure coverage of a
certain domain of interest, and diversity. Based on our review, we presume
three basic topic selection directives are in use today: (1) Manual selection.
Topics are manually defined or selected. Although the process may be random,
when aiming for controversial topics, one may often end up with commonplace
topics in Western culture (e.g., abortion, death penalty, gay marriage),
despite them them being still relevant and important today. (2) Source-driven
(greedy within a time-span). A source of argument ground truth is either
exploited in its entirety, or a maximum subset fulfilling desired properties
is used. Since argument-related ground truth is hard to come by, it is
understandable that all available sources are being exploited. (3) Source-
driven (sampled). A source or argument ground truth is exploited and a subset
is sampled. Here, it may be infeasible to exploit a source in its entirety.
Al-Khatib et al. (2016b) randomly select 300 documents from three websites.
Park and Cardie (2018) and Stab and Gurevych (2017) do not mention anything
about their sampling process. In general, both source-driven corpus
construction approaches inevitably incurs the source’s idiosyncracies of topic
selection, both in terms of skew towards certain topics. Scaling up may or may
not be a remedy for this problem.

We assess how many experiments have been reported on each of the corpora by
collecting the publications referring to a corpus as per Google Scholar,
focusing on conference and journal papers, but excluding books and web pages.
We then check whether the cited corpus is mentioned in its data, experiment,
or results section. As can be seen in Table 1, corpora with fewer topics tend
to be used more often in experiments than those with larger amounts. In total,
82 experiments were carried out on argument corpora with no clearly defined
topic selection directive. The skew towards smaller-scale experiments may
affect generalizability.

##  4 Acquiring Argument Topic Ontologies

Topic ontologies provide for a knowledge organization principle, and,
especially if widely accepted, also a standard. They are typically modeled as
directed acyclic graphs, where nodes correspond to topics and edges indicate
“is part of” relations; topics that are part of other topics are called their
subtopics. A topic ontology is often displayed in levels, starting with the
topics that are not subtopics of others, continuing recursively with each
lower level of subtopics. Figure 1 shows an excerpt of a two-level topic
ontology for arguments.

The identification of the topics to be included in an argument topic ontology,
as well as their relations, requires domain expertise. Building an all-
encompassing ontology thus requires experts from every top-level domain where
argumentation of scientific interest is expected. In the following, we suggest
and outline three authoritative sources of relevant topic ontologies, which
comprise a wide selection of important argumentative topics.

World Economic Forum (WEF) The World Economic Forum is a not-for-profit
foundation that coordinates organizations from both the public and the private
sector to work on economical and societal issues. As part of their efforts,
their “Strategic Intelligence” platform444https://intelligence.weforum.org
strives to inform decision makers on domestic and global topics, specifically
global issues (e.g., artificial intelligence and climate change), industries
(e.g., healthcare delivery and private investors), and economies (e.g., Africa
and ASEAN). Domain experts for each topic curate a stream of relevant news
articles which they each tag with 4-9 subtopics of their topic (e.g., the
continuous monitoring of mental health).

Wikipedia Wikipedia strives for a neutral point of view, but many topics of
public interest are discussed controversially. Some editors thus curate a list
of such controversial articles to highlight where special care is needed,
grouped into 14 top-level topics (e.g., environment and philosophy) and 4-176
subtopics (e.g., creationism and
pollution).555https://en.wikipedia.org/wiki/Wikipedia:List_of_controversial_issues
Omitted is the “People” topic and articles on countries; their
controversiality is not universal.

Debatepedia The debate portal’s goal is to create an encyclopedia of debates
which are organized as “pro” and “con” arguments. A list of 89 topics helps
visitors to browse the debates. The debates are contributed by anonymous web
users, which makes the covered topics easily accessible. Topics in Debatepedia
tend to address issues of Western culture. For example, the topic “United
States” covers 306 debates while “Third World” covers 12 debates. The project
is no longer maintained, but can be accessed through the Wayback
Machine.666https://web.archive.org/web/20180222051626/http://www.debatepedia.org/en/index.php/Welcome_to_Debatepedia%21

The three ontologies are publicly accessible, and two of them are actively
maintained and updated. Acquiring the ontologies is straightforward—not
straightforward is to make use of them. A key task associated with every topic
ontology is to categorize a given document. Having just a short string label
describing a (potentially multifaceted) topic, such as “The Great Reset”,
renders this task exceedingly difficult. Fortunately, domain experts have been
pre-categorizing documents into the aforementioned ontologies. In particular,
regarding the WEF, invited domain experts categorize news articles for every
topic, regarding Wikipedia, the text of the associated wiki articles is
available, as are the associated debates for Debatepedia.

Articles that are categorized into Level 2 topics are propagated up to their
respective Level 1 topics. Table 2 shows the large differences between the
ontologies. The WEF ontology contains the most topics and links the most
documents, which contain the most tokens overall. The topics at Wikipedia
Level 2 are just linked to a single article each, so every topic’s amount of
text is smaller. The number of authors reflects the number of editors.

##  5 Topic Coverage

To assess the topic coverage of the argument corpora in light of the three
ontologies, we map the topic labels of those corpora providing them to their
matching ontology topics.

Figure 2: Proportion of ontology topics covered by at least $n$ corpus topics
(per ontology level and per corpus).

###  5.1 Topic Label Normalization

Table 1 lists 31 argument corpora that provide topic labels. Altogether 2,117
different labels have been assigned. They are concise descriptions of the main
issues of an argument and have been provided by the corpus authors. The labels
follow the style of the text register of the respective corpus: In essays, for
instance, topics are usually thesis statements, while Wikipedia-derived
corpora use article titles, and the topics of debate corpora include clichés
such as “This house should”. Often, topic labels express a stance towards a
target issue, e.g., “ban guns”. Five types of topic labels can be
distinguished: concept, comparison of concepts, conclusion (includes claim and
thesis), question, and imperative. We normalize the topic labels by converting
all concepts to singular form, removing clichés, and dropping stance-
indicating words such as “legalize”. Our normalization aims at retaining only
the central target issue of a topic label and leads to 748 unique topic
labels.

###  5.2 Mapping Topic Labels to Ontology Topics

Using the preprocessed topic labels as queries, we retrieve for each topic
label the 50 top-most relevant topics in each level of the three ontologies.
To facilitate the retrieval of ontology topics, we employ a BM25-weighted
Robertson et al. (2004) index of the concatenated documents for each topic.
BM25 is a widely used modified version of TF-IDF Croft et al. (2009). This
enables us to narrow down the mapping of a topic label to a manageable size.
Except for a handful of cases, 50 ontology topics can be retrieved for each
topic label. The topic labels were then manually mapped to an ontology topic,
if they form synonyms, or if the former is a subtopic of the latter—which thus
indicates that all arguments in the corpus with that topic label are about the
ontology topic. A topic label can thus be mapped to multiple ontology topics.
For example, the topic label “plastic bottles” is mapped to “pollution” and
“recycling” in Wikipedia Level 2.

Figure 3: Distribution of corpora units over the top matching topics in an
ontology (31 labeled corpora).

###  5.3 Analysis of Topic Coverage

Table 2 shows general statistics of this mapping of corpora topic labels to
ontology topics. Most of the topic labels (2,002 out of 2,117) are mapped to
at least one Debatepedia topic while only 355 labels are mapped to WEF Level 2
topics. For Wikipedia Level 2, only 285 out of the 748 topics are actually
covered by argument corpora. Already this first analysis suggests that
existing argument corpora cover typically a small subset of possible
argumentative topics that people are trained to debate. For those topic labels
that can be mapped are mapped on average to 2.8 topics in Debatepedia, to 1.24
topics in Wikipedia Level 1, and to 1.48 topics in WEF Level 1. As discussed
in Section 4, topics in Debatepedia focus on the Western culture and are
easily accessible, whereas topics in WEF require deeper domain knowledge and
have more global relevance. The broad coverage of Debatepedia’s topics
indicates that the studied argument corpora focus on common topics that are
easily approachable while global issues or those that need domain knowledge
lack coverage.

For a more fine-grained analysis, Figure 2 illustrates the differences
regarding the number of ontology topics covered by a corpus: while topics in
Wikipedia Level 1 are covered well by some argument corpora, topics in
Wikipedia and WEF Level 2 are covered only marginally. Note that topic
coverage varies significantly between the corpora: the Claim Sentence Search
dataset’s topics cover 93% of the Wikipedia Level 1 topics, while the
Ideological Debates Reasons dataset covers only 14%. The colors show the topic
granularity of the corpus; especially the Record Debating Dataset 3 dataset is
fine-grained: as the highest value, 36 of its topics are mapped to the
Wikipedia Level 1 category “Politics and Economics”.

Figure 3 shows how the set of the units of the 31 labeled corpora distribute
over the top matching topics in Debatepedia, Wikipedia Level 1, and WEF Level
1. Distributions over Level 2 are omitted for brevity and can be found in the
supplementary material. The distribution is significantly skewed: while the
top ten topics in Debatepdia are matched by 340k to 150k corpora units, the
top ten topics in WEF Level 1 are matched by 340k to 20k corpora units. The
comparison between the three ontologies supports our previous finding that
argument corpora cover easily accessible topics (e.g., “Media and
Entertainment” and “Society”).

##  6 Unit Categorization

Ontology |  Acquired ontologies (Section 4) |  Topic coverage (Section 5) |  Unit categorization (Section 6) |
---|---|---|---|---
| Topics | Topic statistics |  | Covered ontology topics | Direct match | Semantic interpretation | Text2vec-SI
|  | Authors | Docs | Tokens |  Mapped [-0.5ex]topic [-0.5ex]labels | All | Min | Mean | Max | P | R | F | Policy | P | R | F | Policy | P | R | F
WEF L1 | 137 |  0,0334.1 | 940.7 | 490,576.6 | 1,239 | 87 | 1 | 1.48 |  05 | 0.38 | 0.23 | 0.29 | ${k=12}$ | 0.22 | 0.75 | 0.34 | $k=7$ | 0.19 | 0.53 | 0.28
WEF L2 | 822 |  0,0216.8 | 550.3 | 310,229.7 |  0355 | 77 | 1 | 1.32 |  04 | 0.59 | 0.11 | 0.19 | ${k=33}$ | 0.21 | 0.70 | 0.33 | $\theta=0.93$ | 0.15 | 0.49 | 0.23
WP L1 |  014 | 78,013.7 | 68.0 | 339,088.0 | 1,539 | 14 | 1 | 1.24 |  03 | 0.12 | 0.04 | 0.06 | ${k=3}$ | 0.32 | 0.65 | 0.43 | $k=2$ | 0.41 | 0.55 | 0.47
WP L2 | 748 |  01,929.5 | 1.0 |  006,149.1 | 1,453 | 285 | 1 | 1.76 | 16 | 0.47 | 0.34 | 0.40 | ${\theta=0.05}$ | 0.54 | 0.64 | 0.59 | $\theta=0.89$ | 0.22 | 0.52 | 0.31
DP |  089 |  0,0145.0 | 61.7 |  084,787.6 | 2,002 | 87 | 1 | 2.80 | 10 | 0.49 | 0.37 | 0.42 | ${\theta=0.02}$ | 0.52 | 0.61 | 0.56 | $k=23$ | 0.36 | 0.80 | 0.50
Table 2: Statistics for each topic ontology level: for topics and topic
documents (Section 4), Count of mapped topic labels of the analyzed corpora
for each ontology level, Count of all covered ontology topics by the topic
labels and the min, max, and mean count of covered ontology topics per topic
label (Section 5), and the effectiveness of the approaches and baseline in
unit categorization (in terms of precision, recall, and F1-score) (Section 6).

The previous analysis is done on those argument corpora which contain topic
labels. About a third of the argument corpora are thus excluded from that
analysis. As a step toward assessing their topic coverage, we map the ontology
topics for a unit (cf. Table 1) in an argument corpus by treating the unit as
a (long) query in a standard information retrieval setup, where ontology
topics are the retrieval targets. The documents categorized into each topic
have been concatenated and used as the topic’s representation. Though the
documents associated with a topic are not necessarily argumentative, they
cover the salient aspects of the topic.

To retrieve topics for a corpus unit, we implement and evaluate the following
approaches: Semantic Interpretation (SI) and SI with Text Embeddings (T2V-SI).
The Semantic interpretation approach computes the semantic similarity of a
unit and a topic as follows: it uses the cosine similarity of the TF-IDF
vectors for the unit and the concatenated topic’s documents. This corresponds
to the semantic interpretation step that is at the core of the well-known ESA
model Gabrilovich and Markovitch (2007). In Text2vec-SI, the similarity of
topics and corpus units is calculated using BERT embeddings Devlin et al.
(2018). We follow the common approach to generate text embeddings, which is to
take the dimension-wise average of the word embeddings for all tokens in the
text.777For efficiency, we limited the embeddings to 10,000 randomly sampled
sentences for the topics that had more sentences associated with them. As a
baseline, we implement a direct match approach, which assigns a unit an
ontology topic if the topic’s text appears in the unit text (ignoring case).

For evaluation, we collect 34,638 pooled query relevance judgments (0.53
inter-annotator agreement as per Krippendorff’s $\alpha$) on 104 randomly
selected argument units as queries from 26 corpora. The annotation process is
detailed in the appendix.

Based on the similarity scores of the approach, we derive Boolean labels that
indicate whether a unit is or is not about one of the ontologies’ topics using
two policies. The threshold policy labels a unit as about a topic if their
similarity is above a threshold $\theta$. The top-$k$ policy labels a unit as
about a topic if the topic is among the top-$k$ topics with the highest
similarity to the unit. We report the parameter of policy with which the
approach achieved the highest F1-score on the pooled judgments.

Table 2 shows the results of this evaluation. The baseline produces different
results across ontologies—it performs poorly for both the abstract topics in
Wikipedia Level 1 and the specific topics in WEF Level 2. The semantic
interpretation approach clearly outperforms the baseline for all ontologies in
terms of the F1-score. The Text2vec approach outperforms the baseline and
semantic interpretation on the most abstract topics (Wikipedia Level 1), but
its performance is supbar to that of semantic interpretation on the other
ontology levels.

##  7 Conclusion

The computational argumentation community faces a dilemma: Either carry on as
before and risk topic bias, or go the extra mile to ensure topic
representativity and create extra work for corpus authors. The latter option
is further complicated by the fact that the space of controversial topics is
not well explored to date, and that there are no widely accepted argument
topic ontologies as of yet. In this paper, we give a glimpse into a future in
which the argument topic space has been mapped and is accessible to corpus
construction and designing experiments: We identified three authoritative
sources of ontologic knowledge with respect to argument topics. For each
ontology, we reveal the topic coverage of 31 argument corpora that are
provided with topic labels by aligning the topic labels of a corpus to the
ontologies topics. To assess the topic coverage of non-labeled corpora, we
introduce an approach that identifies the ontology topics of an argumentative
text, reaching an F1 of 0.59.

Our analyses show that the topic coverage of the studied argument corpora is
both limited to only a subset of the ontologies topics and skewed. The
majority of topics that require domain knowledge such as those on mental
health, philosophy, or international security are marginally covered in the
analyzed argument corpora. This renders existing argumentation technologies
more suitable to teach people how to construct arguments than to support them
taking decisions about complex topics. For a mature development of
argumentation technology, a careful sampling and controlling of the topics
should be employed while constructing corpora, designing experiments, or
applying classifiers.

In future work, major tasks are to further explore the argument topic space
regarding matters not yet covered by the existing topic ontologies and to
unify the different ontologies. Besides “is part of”-relations between topics,
other relation types may be considered as well, thereby inducing a topic
knowledge base. However, already the work presented here can assist in
selecting arguments for corpus construction and model training.

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##  8 Limitations

The three topic ontologies which we used to assess the topic coverage of
argument corpora come from recognized sources and cover different domains.
Nevertheless, these topic ontologies might not fully cover all possible
controversial topics that are relevant to argumentation (e.g., those topics
related to private life). Having said that, we do believe that our paper sets
a cornerstone for studying topic bias in argument corpora, which researchers
can extend.

Another limitation of this study is the moderate effectiveness achieved by our
approaches for unit categorization. Our approaches for unit categorization
achieved moderate effectiveness because of the large space of controversial
topics (about 742 for Wikipedia). Future research can improve upon our
approach by utilizing the structure of the topic ontology using hierarchical
classifiers. Hierarchical classifiers first map a document to one topic in the
upper level and then consider only the subtopics of this topic for
classification in the lower levels. In this way, the space of controversial
topics in the lower levels can be largely reduced.

##  9 Appendix

###  9.1 Corpus Topic Labels Mapping to Level 2 Topics

For completeness, we Figure 4 show the two graphs that are omitted from Figure
2 of the paper as their fine-grained topics are less relevant for the
discussion in Section 5.3.

Figure 4: Omitted graphs from Figure 2, Section 5.3

###  9.2 Annotation Procedure for Unit Categorization

In order to assess the effectiveness of the approaches and baseline outlined
in the paper, we employ a pooled evaluation, as it is standard for information
retrieval evaluations, where there are too many instances for a complete
manual annotation. We randomly sampled four units from 26 corpora, which were
all annotated by three expert annotators. The annotators were instructed to
label a topic as about the unit if they could imagine a discussion on the
topic for which the unit would be relevant. For each unit, we annotated for
aboutness only those topics which are among the five topics with the highest
similarity to this unit according to at least one of the approaches. The
employed assessment interface (see Figure 5) shows the unit (top left), the
current topic (top right), as well as all topics in the pool for that unit
(bottom; the current topic is marked blue, whereas already annotated topics
are marked green (about) and red (not about). The same interface has been used
for the topic label annotations.

Figure 5: Assessment interface for topic labeling.

To reduce biases, both the units and the topics were shown in a different and
random order to each assessor. The annotation took about 40 hours. The
annotation process resulted in an inter-annotator agreement of 0.53 in terms
of Krippendorff’s $\alpha$ and produced a total of 34,638 annotations of
topic-unit pairs, about 2% of what would have been needed for a complete
annotation.