Thermodynamics of Horndeski Black Holes with Generalized Uncertainty Principle

Mohaddeseh<EMAIL_ADDRESS>, Akram S.
<EMAIL_ADDRESS>
†Department of Theoretical Physics, Faculty of Basic Sciences,

University of Mazandaran,

P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

Horndeski theory is the most general scalar-tensor extension of General
Relativity with second order field equations. It may be interesting to study
the effects of the Generalized Uncertainty Principle on a static and
asymptotically flat shift symmetric solutions of the Horndeski black holes.
With this motivation, here we obtain the modified black hole temperatures in
shift symmetric Horndeski gravity by employing the Generalized Uncertainty
Principle. Using the corrected temperature, the entropy and heat capacity are
calculated with details. We also investigate the tunneling probability of
particles from Horndeski black holes horizon and possible correlations between
the emitted modes (particles).
PACS: 04.20.Dw, 04.50.Kd, 04.70.-s, 04.70.Dy
Key Words: Generalized Uncertainty Principle, Quantum Gravity, Horndeski
Theory, Black Hole thermodynamics.

##  1 Introduction

At present, it is believed that Nature can be described by quantum mechanics
and general relativity. In 1915, Albert Einstein proposed General Relativity
(GR) that is able to successfully describe physical phenomena in astrophysics
and cosmology [1, 2]. Besides all the significant achievements, general
relativity can not describe some theoretical and observational issues. One of
the known failures of GR is the problem of black hole singularities that leads
to various black hole spacetimes singularities. It seems that GR is cursed
with own solutions, i.e., black holes. Also, the cosmological constant problem
and the issue of the dark matter/energy can not be explained by GR (for more
details see for instance[3, 4, 5]). During the recent decades, many efforts
have been undertaken to construct a more comprehensive theory. One way to
modify GR is reconstructing the geometric part of the Einstein field equations
[6, 7]. A special class, that is, the most general scalar-tensor theory with
second order field equations, proposed by Horndeski in 1970s [8]. Recently,
Hondeski theory has been received much attention and investigated in
astrophysics and cosmology [9, 10, 11, 12, 13, 14]. More attractively,
Horndeski black holes have been investigated. For instance, spherically
symmetric and static solutions [15, 16, 17], black hole solutions in the
presence of a cosmological constant and magnetic field [18, 19], the
observational results and gravitational lensing effects for Horndeski black
holes [20, 21, 22] are studied. Moreover, thermodynamics of Horndeski black
holes are studied, Hawking temperature and entropy and circular orbits are
investigated also in Refs. [23, 24, 25, 26].

Trying to construct a quantum theory of gravity leads to a minimal measurable
length of the order of the Planck length, $\ell_{P}\sim 10^{-35}m$. Most
quantum gravity approaches such as string theory [27, 28, 29, 30], loop
quantum gravity [31] and quantum geometry [32] predict the existence of a
minimal measurable length in spacetime [33]. Also, the existence of a minimal
measurable length can be supported from micro-black hole Gedanken experiment
[34]. In GUP concept, the Heisenberg Uncertainty Principle (HUP) is modified
to the so called Generalized Uncertainty Principle (GUP) [35, 36, 37, 38, 40,
41, 43, 44]. Incorporation of GUP effects in standard quantum mechanics’
problems reveals several novel corrections and modifies the results in high
energy regime [45, 46, 47, 48, 49]. On the other hand, Doubly Special
Relativity (DSR) proposes an upper bound for a test particle’s momentum [50,
51, 52]. In fact, the existence of a minimal measurable length restricts a
test particle’s momentum to take a maximal measurable momentum of the order of
the Planck momentum [53, 54, 55]. Several interesting and novel results are
obtained by considering both a minimal length and a maximal momentum [56, 57,
41, 42]. Moreover, when one considers curvature effects, it can be shown that
there is a nonzero minimal uncertainty in momentum measurement too [37, 38].
That is, in large distances, where the curvature of space time becomes
important, momentum cannot be precisely determined. With the path integral
formulation, such noncommutative background geometries can ultraviolet and
infrared regularize quantum field theories in arbitrary dimensions through
minimal uncertainties both in positions and in momenta (for more details, see
[37, 38]). It is important to note that natural cutoffs are essentially
related to the compactness of corresponding symplectic manifold [39]. It is
well-known that thermodynamic quantities of a black hole can be obtained by
the standard uncertainty principle. So, in this respect incorporation of the
GUP can modify the black hole physics. Recently, black holes, as a connection
between general relativity and quantum mechanics, have been investigated
widely in GUP framework. For instance, the GUP prevents black holes from total
evaporation. Also the GUP modifies Hawking temperature [58]. So, because of
the existence of a maximal temperature originating from minimal length/maximal
energy, the GUP predicts a non-radiating remnant of the order of the Planck
mass in the final stage of evaporation. So, while it provide a possible
candidate for dark matter[59], it may be also a clue for solving the black
hole information loss problem and interestingly opens a realistic door for
studying the final stage of black hole evaporation [60, 61, 62, 63, 64, 65,
66]. The importance of the subject lies in the fact that black holes are
essentially a quantum gravity object and therefore GUP as a phenomenological
aspect of quantum gravity provides a more realistic framework to study black
hole physics and thermodynamics. This feature has shown its efficiency in
recent years study of black hole physics.

With this motivations, we adopt Horndeski gravity with phenomenological
quantum gravitational effects to study the black hole thermodynamics. We
consider a generalized/extended uncertainty relation that includes a minimal
length, a minimal momentum and a maximal momentum to modify the black hole
temperature and entropy. Then, by using the modified black hole temperature,
we obtain the modified heat capacity. Finally, we consider Parikh-Wilczek
tunneling process to describe the Hawking radiation emitted from Horndeski
black holes. We compute tunneling rate and also possible correlation between
emitted modes. To be more clarified, we study possible correlations between
the emitted particles, a feature that can be used by itself to address at
least a part of the lost information in the process of black hole formation.
The motivation for performing such a study in Horndeski framework is the
existence of a gap in this respect in literature in one side, and the fact
that Horndeski theory is the most general scalar-tensor theory of gravity
where incorporation of quantum gravitational effects may bring new physics in
the realm of black hole thermodynamics in this framework.

##  2 Horndeski Theory

In the modern formulation of the Horndeski gravity, the action takes the
following form [17]

$S=\int\sqrt{-g}d^{4}x\left(\mathcal{L}_{2}+\mathcal{L}_{3}+\mathcal{L}_{4}+\mathcal{L}_{5}+\mathcal{L}_{4}^{bH}+\mathcal{L}_{5}^{bH}\right),$
(1)

where $g\equiv det(g_{\mu\nu})$ and $g_{\mu\nu}$ is the metric tensor. In our
case, we investigate a class of the Horndeski theory which posses shift
symmetry, $\phi\longrightarrow\phi+\textit{constant}$,. It includes six
arbitrary functions of the scalar field and its canonical kinetic term which
are denoted by $\phi$ and $X=-\partial^{\mu}\phi\partial_{\mu}\phi/2$,
respectively. In this notation we consider $G_{2},G_{3},G_{4},G_{5}$ for
ordinary Horndesky theory and $F_{4},F_{5}$ for beyond Horndeski (bH) theory.
These are in the following form

$\displaystyle\mathcal{L}_{2}$ $\displaystyle=G_{2},$ (2)
$\displaystyle\mathcal{L}_{3}$ $\displaystyle=-G_{3}\Box\phi,$ (3)
$\displaystyle\mathcal{L}_{4}$
$\displaystyle=G_{4}R+G_{4X}[(\Box\phi)^{2}-(\nabla_{\mu}\nabla_{\nu}\phi)^{2}],$
(4) $\displaystyle\mathcal{L}_{5}$
$\displaystyle=G_{5}G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi-\frac{1}{6}G_{5X}[(\Box\phi)^{3}-3\Box\phi(\nabla_{\mu}\nabla_{\nu}\phi)^{2}$
$\displaystyle+2(\nabla_{\mu}\nabla_{\nu}\phi)^{3}],$ (5)
$\displaystyle\mathcal{L}_{4}^{bH}$
$\displaystyle=F_{4}\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\gamma}_{\sigma}(\nabla_{\mu}\phi\nabla_{\alpha}\phi)(\nabla_{\nu}\nabla_{\beta}\phi)\nabla_{\rho}\nabla_{\gamma}\phi,$
(6) $\displaystyle\mathcal{L}_{5}^{bH}$
$\displaystyle=F_{5}\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\gamma\delta}(\nabla_{\mu}\phi\nabla_{\alpha}\phi)(\nabla_{\nu}\nabla_{\beta}\phi)(\nabla_{\rho}\nabla_{\gamma}\phi)(\nabla_{\sigma}\nabla_{\delta}\phi),$
(7)

where $R$ is the Ricci scalar, and $G_{\mu\nu}$ is the Einstein tensor. For
simplicity, in our notation, $\Box\phi\equiv g^{\mu\nu}\partial_{\mu\nu}\phi$,
$(\nabla_{\mu}\nabla_{\nu}\phi)^{2}\equiv\nabla_{\mu}\nabla_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi$,
$(\nabla_{\mu}\nabla_{\nu}\phi)^{3}\equiv\nabla_{\mu}\nabla_{\nu}\phi\nabla^{\nu}\nabla^{\rho}\phi\nabla_{\rho}\nabla^{\mu}\phi$
and $G_{X}=\partial G(X)/\partial X$. Obviously, GR and $f(R)$ gravity are the
spatial limits of the Horndeski gravity which are chosen by
$G_{2}=G_{3}=G_{5}=0,G_{4}=1/2$ and $G_{2}=G_{3}=G_{5}=0,G_{4}=f(R)$,
respectively.

In our case, we are interested in to investigate the spherically symmetric and
static black hole solutions in shift symmetric Horndeski theories. These black
holes are static and asymptotically flat with a static scalar field [17]. So,
the static and spherically symmetric ansatz for spacetime and scalar field
take the following form respectively:

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}(d\theta^{2}+sin^{2}\theta
d\varphi^{2}),$ (8) $\phi=\phi(r).$ (9)

Also, we set the $G_{i}$ functions of the Lagrangian as follows:

$\displaystyle G_{2}$ $\displaystyle=\eta X-2\Lambda,$ $\displaystyle G_{4}$
$\displaystyle=\zeta+\gamma\sqrt{-X},$ $\displaystyle G_{3}$
$\displaystyle=G_{5}=F_{4}=F_{5}=0,$ (10)

where $\eta$ and $\gamma$ are dimensionless parameters and $\Lambda$ is the
cosmological constant. The first term of $G_{2}$ is a canonical kinetic term
and the first term of $G_{4}$ is $\zeta=M^{2}_{Pl}/(16\pi)$ that yields an
Einstein-Hilbert term in the action. Finally, the action takes the following
form [17]

$S=\int
d^{4}x\sqrt{-g}\left\\{\left[\zeta+\gamma\sqrt{(\partial\phi)^{2}/2}\right]R-\frac{\eta}{2}(\partial\phi)^{2}-2\Lambda-\frac{\gamma}{\sqrt{2(\partial\phi)^{2}}}[(\square\phi)^{2}-(\nabla_{\mu}\nabla_{\nu}\phi)^{2}]\right\\}.$
(11)

The scalar field can be obtained from the metric and scalar field ansatz as
[17]

$\phi^{{}^{\prime}}=\pm\frac{\sqrt{2}\gamma}{\eta r^{2}\sqrt{f}}.$ (12)

For our particular case, the spacetime metric solution takes the following
form [17]

$f(r)=g(r)=1-\frac{\mu}{r}-\frac{\gamma^{2}}{2\zeta\eta
r^{2}}-\frac{\Lambda}{3\zeta}r^{2}.$ (13)

Explicitly, the solution has the Reissner-Nordström-de sitter (RN+$\Lambda$)
form. As a consequence of similarity to the RN+$\Lambda$ form, this solution
describes a black hole with mass $\mu/2$ where $\mu$ is a free integration
constant and electric charge $\sqrt{\frac{-\gamma^{2}}{2\zeta\eta}}$ for
spacetime. The parameters $\gamma$ and $\eta$ unavoidably share the same sign.
Additionally, this solution has singularity which can be absorbed in the
coordinate transformation. Finally, identical to other static solutions, space
and time coordinates exchange their roles in the interior of the black hole.
Also, $\phi^{{}^{\prime}}$ is real for outside of the black hole horizon,
$f>0$, and imaginary for interior of the black hole horizon, $f<0$. Further,
the solution (13) with Eq.(8), recovers the Reissner-Nordström (RN) metric and
the Schwarzschild metric in the limits $\Lambda\rightarrow 0$ and
$\gamma\rightarrow 0$, respectively. Having introduced a particular black hole
solution in shift symmetric Horndeski theory, now we study its thermodynamics
in the presence of phenomenological quantum gravitational effects encoded in a
GUP relation.

##  3 Thermodynamics of Horndeski Black Holes

To incorporate the GUP effects on the black hole thermodynamics, let us start
with the metric to obtain the location of the horizons [67]. The radii of the
horizons are determined by the equation $f(r)=0$. In general this equation has
four roots, which we can classify them as

$r_{1}>r_{2}>r_{3}>r_{4}$ (14)

The lack of cubic term in equation $f(r)=0$ leads to a negative and unphysical
root. So, we have three positive (real) roots that the outermost one is the
cosmological horizon. $r_{2}$ and $r_{3}$ are event horizon and Cauchy
horizon. In the current work, we are only interested in the event horizon and
Caushy horizon which are given by

$\displaystyle r_{+}$
$\displaystyle=-\frac{1}{2}\sqrt{Y}+\frac{1}{2}\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}},$
(15) $\displaystyle r_{-}$
$\displaystyle=\frac{1}{2}\sqrt{Y}-\frac{1}{2}\sqrt{\frac{6\zeta}{\Lambda}-Y-\frac{6\zeta\mu}{\Lambda\sqrt{Y}}},$
(16)

where

$\displaystyle X$
$\displaystyle=\bigg{[}432\zeta^{3}\eta^{3}-2592\gamma^{2}\zeta\eta^{2}\Lambda-1944\zeta^{2}\eta^{3}\Lambda\mu^{2}$
$\displaystyle+\sqrt{\left[-4\left(36\zeta^{2}\eta^{2}+72\gamma^{2}\eta\Lambda\right)^{3}+\left(432\zeta^{3}\eta^{3}-2592\gamma^{2}\zeta\eta^{2}\Lambda-1944\zeta^{2}\eta^{3}\Lambda\mu^{2}\right)^{2}\right]}\bigg{]}^{\frac{1}{3}},$
(17) $\displaystyle Y$ $\displaystyle=\frac{2\zeta}{\Lambda}-\frac{6\times
2^{\frac{1}{3}}(\zeta^{2}\eta^{2}+2\gamma^{2}\eta\Lambda)}{\eta\Lambda
X}-\frac{X}{6\times 2^{\frac{1}{3}}\eta\Lambda}.$ (18)

In the standard framework, the uncertainty principle can be used to obtain the
temperature and entropy of black hole [68]. So, in the same way the GUP is
capable to modify the temperature and entropy. The black hole thermodynamics
in the presence of the GUP are investigated extensively [60, 61, 62, 63, 64,
65]. The uncertainty in the position of an emitted particle in the Hawking
effect is given by

$\Delta
x=2r_{+}=-\sqrt{Y}+\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}}.$
(19)

Also, the uncertainty in the energy of the Hawking particle is

$\Delta E\approx c\Delta p\approx\frac{\hbar c}{\Delta x}=\hbar
c\left[-\sqrt{Y}+\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}}\right]^{-1}.$
(20)

In standard cases the black hole temperature is
$T_{BH}=\frac{\kappa}{2\pi}=T_{0}$, where $\kappa$ and $T_{0}$ are the horizon
surface gravity and the Hawking temperature, respectively. What is the black
hole temperature in Horndeski theory? For our special case, considering black
hole solution Eq.(13) yields the standard formula again and there is not any
temperature shift (for more details see [24]). So, the Hawking temperature is
associated to the black hole event horizon radius by

$T=\frac{1}{4\pi r_{+}}=\frac{1}{2\pi\Delta x}\,.$ (21)

Using the Hawking temperature, the Bekenstein-Hawking entropy related to the
black hole mass has the following well known form

$T=\frac{dE}{dS}=\frac{dM}{dS}\,.$ (22)

Using Eq.(19), Eq.(21) and Eq.(22), we find

$S=2A\left(-\sqrt{Y}+\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}}\right)^{-2}\times\int
dM\left[-\sqrt{Y}+\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}}\right]\,,$
(23)

where $A=4\pi r_{+}^{2}$ is considered as the surface area of the black hole
event horizon. Note that the solution (23) recovers $T_{Sch}=4\pi M^{2}$ in
appropriate limit where $T_{Sch}$ is the entropy of the Schwarzschild black
hole in the standard framework. To incorporate the quantum gravity effects on
the black hole thermodynamics in this shift symmetric Horndeski theory, we
take into account the GUP and develop the study in more details. We consider a
general uncertainty principle that includes a minimal length, a minimal
momentum and a maximal momentum. This GUP has the following form [35, 36, 37,
38]

$\Delta x\Delta p\geq\hbar\Big{[}1-\alpha\ell_{p}(\Delta
p)+\alpha^{2}\ell_{p}^{2}(\Delta p)^{2}+\beta^{2}\ell_{p}^{2}(\Delta
x)^{2}\Big{]},$ (24)

where $\ell_{p}$ is Planck length, $\alpha$ and $\beta$ are dimensionless
parameters which normally are of the order of unity and depend on the quantum
gravity approaches. Solving this relation for $\Delta p$ gives us the
following momentum uncertainty

$\frac{\Delta
p}{\hbar}=\frac{(\alpha\hbar\ell_{p}+2r_{+})}{2\alpha^{2}\hbar^{2}\ell_{p}^{2}}\left(1\pm\sqrt{1-\frac{4\alpha^{2}\hbar\ell_{p}^{2}(\hbar+4\beta^{2}\hbar\ell_{p}^{2}r_{+}^{2})}{(\alpha\hbar\ell_{p}+2r_{+})^{2}}}\right).$
(25)

We can show that this solution has a minimal length, $(\Delta
x)_{min}=\alpha\ell_{p}$, , a minimal momentum, $(\Delta
p)_{min}=2\beta\ell_{p}$, and a maximal momentum, $(\Delta
p)_{max}\simeq\frac{1}{\alpha\ell_{p}}$ with $c=1$. Using the series expansion

$\displaystyle\frac{\Delta p}{\hbar}$ $\displaystyle=\frac{1}{\Delta
x}\bigg{(}1-\frac{\alpha\hbar\ell_{p}}{\Delta
x}+\frac{(2\alpha^{2}\hbar^{2}+\beta^{2}\Delta x^{4})\ell_{p}^{2}}{\Delta
x^{2}}-\frac{\alpha\hbar(4\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{3}}{\Delta x^{3}}$
$\displaystyle+\frac{3\alpha^{2}\hbar^{2}(3\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{4}}{\Delta x^{4}}+O(\ell_{p}^{5})\bigg{)}\,,$ (26)

and substituting Eq.(3) into Eq.(24), we find the GUP corrected position
uncertainty as

$\displaystyle\Delta x^{{}^{\prime}}$ $\displaystyle=\Delta
x\Bigg{[}\Bigg{(}1-\frac{\alpha\hbar\ell_{p}}{\Delta
x}+\frac{(2\alpha^{2}\hbar^{2}+\beta^{2}\Delta x^{4})\ell_{p}^{2}}{\Delta
x^{2}}-\frac{\alpha\hbar(4\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{3}}{\Delta x^{3}}$
$\displaystyle+\frac{3\alpha^{2}\hbar^{2}(3\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{4}}{\Delta
x^{4}}\Bigg{)}^{-1}-\frac{\alpha\hbar\ell_{p}}{\Delta
x}+\frac{\alpha^{2}\ell_{p}^{2}\hbar^{2}}{\Delta
x^{2}}\Bigg{(}1-\frac{\alpha\hbar\ell_{p}}{\Delta
x}+\frac{(2\alpha^{2}\hbar^{2}+\beta^{2}\Delta x^{4})\ell_{p}^{2}}{\Delta
x^{2}}$ $\displaystyle-\frac{\alpha\hbar(4\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{3}}{\Delta
x^{3}}+\frac{3\alpha^{2}\hbar^{2}(3\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{4}}{\Delta x^{4}}\Bigg{)}+\beta^{2}\ell_{p}^{2}\Delta
x^{2}\Bigg{(}1-\frac{\alpha\hbar\ell_{p}}{\Delta x}$
$\displaystyle+\frac{(2\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{2}}{\Delta
x^{2}}-\frac{\alpha\hbar(4\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{3}}{\Delta
x^{3}}+\frac{3\alpha^{2}\hbar^{2}(3\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{4}}{\Delta x^{4}}\Bigg{)}^{-3}$
$\displaystyle-2\hbar\alpha\beta^{2}\ell_{p}^{3}\Delta
x\Bigg{(}1-\frac{\alpha\hbar\ell_{p}}{\Delta
x}+\frac{(2\alpha^{2}\hbar^{2}+\beta^{2}\Delta x^{4})\ell_{p}^{2}}{\Delta
x^{2}}-\frac{\alpha\hbar(4\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{3}}{\Delta x^{3}}$
$\displaystyle+\frac{3\alpha^{2}\hbar^{2}(3\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{4}}{\Delta x^{4}}\Bigg{)}^{-2}\Bigg{]}.$ (27)

So, the modified Hawking temperature for a GUP-corrected Horndeski black hole
without charge can be obtained as follows

$\displaystyle T^{{}^{\prime}}$ $\displaystyle=\frac{1}{2\pi\Delta
x^{{}^{\prime}}}=T\Bigg{[}\Bigg{(}1-\frac{\alpha\hbar\ell_{p}}{\Delta
x}+\frac{(2\alpha^{2}\hbar^{2}+\beta^{2}\Delta x^{4})\ell_{p}^{2}}{\Delta
x^{2}}-\frac{\alpha\hbar(4\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{3}}{\Delta x^{3}}$
$\displaystyle+\frac{3\alpha^{2}\hbar^{2}(3\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{4}}{\Delta
x^{4}}\Bigg{)}^{-1}-\frac{\alpha\hbar\ell_{p}}{\Delta
x}+\frac{\alpha^{2}\ell_{p}^{2}\hbar^{2}}{\Delta
x^{2}}\Bigg{(}1-\frac{\alpha\hbar\ell_{p}}{\Delta
x}+\frac{(2\alpha^{2}\hbar^{2}+\beta^{2}\Delta x^{4})\ell_{p}^{2}}{\Delta
x^{2}}$ $\displaystyle-\frac{\alpha\hbar(4\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{3}}{\Delta
x^{3}}+\frac{3\alpha^{2}\hbar^{2}(3\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{4}}{\Delta x^{4}}\Bigg{)}+\beta^{2}\ell_{p}^{2}\Delta
x^{2}\Bigg{(}1-\frac{\alpha\hbar\ell_{p}}{\Delta x}$
$\displaystyle+\frac{(2\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{2}}{\Delta
x^{2}}-\frac{\alpha\hbar(4\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{3}}{\Delta
x^{3}}+\frac{3\alpha^{2}\hbar^{2}(3\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{4}}{\Delta x^{4}}\Bigg{)}^{-3}$
$\displaystyle-2\hbar\alpha\beta^{2}\ell_{p}^{3}\Delta
x\Bigg{(}1-\frac{\alpha\hbar\ell_{p}}{\Delta
x}+\frac{(2\alpha^{2}\hbar^{2}+\beta^{2}\Delta x^{4})\ell_{p}^{2}}{\Delta
x^{2}}-\frac{\alpha\hbar(4\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{3}}{\Delta x^{3}}$
$\displaystyle+\frac{3\alpha^{2}\hbar^{2}(3\alpha^{2}\hbar^{2}+\beta^{2}\Delta
x^{4})\ell_{p}^{4}}{\Delta x^{4}}\Bigg{)}^{-2}\Bigg{]}^{-1}\,,$ (28)

where $\Delta x$ is given by Eq.(19) and $T$ is the standard Bekenstein-
Hawking temperature. This allows us to calculate the modified entropy from
Eq.(22) as follows

$S^{{}^{\prime}}(M)=S(M)-7\ell_{p}^{5}F(M)+O(\ell_{p}^{6}),$ (29)

where $F(M)$ is given by

$\displaystyle F(M)$ $\displaystyle=\int
dM\Bigg{[}2A\Big{(}\left(\alpha^{3}\beta^{2}\hbar^{3}\right)\left(-\sqrt{Y}+\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}}\right)^{-2}$
$\displaystyle+\left(3\alpha^{5}\hbar^{5}\right)\left(-\sqrt{Y}+\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}}\right)^{-6}\Big{)}\Bigg{]}\,,$
(30)

and the surface area of the black hole’s outer horizon is given by

$A=4\pi r_{+}^{2}\,.$ (31)

We note that the existence of maximal momentum and minimal momentum leads to
extra terms in Hawking temperature and we have both of even and odd powers of
Planck length, $\ell_{p}$, in comparison to the results reported in [67].
Further, the $F(M)$ term in the (3) only exists in the presence of the GUP.
So, in the absence of GUP, this term vanishes and Eq.(3) reduces to the
Schwarzschild entropy in the standard framework as is expected.

###  3.1 Heat Capacity

The heat capacity of black hole, in the semiclassical approach, can be
obtained by the inverse temperature, $T^{-1}=\beta=\frac{dS}{dM}$. Generally,
the heat capacity can be calculated as follows

$C=\frac{dM}{dT}$ (32) $C=\frac{1}{\pi
T^{2}}\left[\frac{1}{\sqrt{Y}}+\left(\frac{6\zeta}{\Lambda}-Y+\frac{12\zeta
M}{\Lambda\sqrt{Y}}\right)^{-\frac{1}{2}}\left(1+\frac{6\zeta M}{\Lambda
Y^{\frac{3}{2}}}\right)\right]^{-1}\frac{1}{W},$ (33)

where W and Z are given by

$W=\frac{dY}{dX}Z=\left[\frac{6\times
2^{\frac{1}{3}}\left(\zeta^{2}\eta^{2}+2\gamma^{2}\eta\Lambda\right)}{\eta\Lambda
X^{2}}-\frac{1}{6\times 2^{\frac{1}{3}}\eta\Lambda}\right]Z$ (34)

and

$\displaystyle Z$
$\displaystyle=\frac{dX}{dM}=\frac{1}{3}X^{-2}\bigg{(}15552\zeta^{2}\eta^{3}\Lambda
M+\frac{1}{2}\Big{[}X^{3}-432\zeta^{3}\eta^{3}-592\gamma^{2}\zeta\eta^{2}\Lambda$
$\displaystyle-7774\zeta^{2}\eta^{3}\Lambda
M\Big{]}^{-1}\left[2\left(432\zeta^{3}\eta^{3}-2592\gamma^{2}\zeta\eta^{2}\Lambda-7776\zeta^{2}\eta^{3}\Lambda
M\right)15552\zeta^{2}\eta^{3}\Lambda M\right]\bigg{)}.$ (35)

If we consider the GUP, we get to

$\displaystyle C^{{}^{\prime}}$
$\displaystyle=C\left[1-7\ell_{p}^{5}\left(\frac{\alpha^{3}\beta^{2}\hbar^{3}}{(-\sqrt{Y}+\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}})}+\frac{3\alpha^{5}\hbar^{5}}{(-\sqrt{Y}+\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}})^{5}}\right)\right]^{2}$
$\displaystyle\left[7\ell_{p}^{5}\left(\frac{\alpha^{3}\beta^{2}\hbar^{3}}{\left(-\sqrt{Y}+\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}}\right)^{2}}+\frac{15\alpha^{5}\hbar^{5}}{\left(-\sqrt{Y}+\sqrt{\frac{6\zeta}{\Lambda}-Y+\frac{6\zeta\mu}{\Lambda\sqrt{Y}}}\right)^{6}}\right)\right]^{-1}\,,$
(36)

where $C$ is the standard Bekenstein-Hawking heat capacity. When $\alpha$ and
$\gamma$, the GUP parameters tend to zero, Eq.(3.1) reduces to Eq.(33) as
would be expected.

###  3.2 Tunneling Process

In 1974, Stephen Hawking demonstrated [69] that black holes have an emission
spectrum of a black body, the so called Hawking radiation, and so are not
purely black. In 2000, Parikh and Wilczek exhibited [70] a semiclassical
method to derive Hawking radiation as a tunneling process from the event
horizon of black hole. In this section we calculate Hawking radiation in
Parikh-Wilczek tunneling formalism. The coordinate system should be well-
behaved for calculations at the event horizon. So, we define the Painlevé-like
coordinate transformation as follows [71, 72, 73]

$dt_{R}=dt+f^{{}^{\prime}}(r)dr,$ (37)

where $t_{R}$ is the black hole time coordinate. Substituting Eq.(37) into
Eq.(8) we have

$\displaystyle ds^{2}$ $\displaystyle=-\Delta
dt_{R}^{2}+\frac{1}{\Delta}dr^{2}+r^{2}d\Omega^{2}$ $\displaystyle=-\Delta
dt^{2}+(-\Delta f^{{}^{\prime}2}+\frac{1}{\Delta})dr^{2}-2\Delta
f^{{}^{\prime}}drdt+r^{2}d\Omega^{2}\,,$ (38)

where simplicity we have defined
$\Delta=1-\frac{\mu}{r}-\frac{\gamma^{2}}{2\zeta\eta
r^{2}}-\frac{\Lambda}{3\zeta}r^{2}$. Then $f^{{}^{\prime}}(r)$ satisfies

$f^{{}^{\prime}}=\pm\frac{\sqrt{1-\Delta}}{\Delta}$ (39)

and the Painlevé line element and the radial geodesics take the following form
respectively

$ds^{2}=-\Delta dt^{2}+dr^{2}\mp 2\sqrt{1-\Delta}drdt+r^{2}d\Omega\,,$ (40)
$\dot{r}=\frac{dr}{dt}=\pm
1\mp\sqrt{\frac{\mu}{r}+\frac{\gamma^{2}}{2\zeta\eta
r^{2}}+\frac{\Lambda}{3\zeta}r^{2}}.$ (41)

In this process, that occurs near inside the horizon, the particle with
positive energy, $\tilde{w}$, tunnels out and escapes the event horizon.
Considering the energy conservation, the mass parameter will be replaced with
$\mu\rightarrow\mu-\tilde{w}$. We can rewrite the new line element and the
radial null geodesics which are respectively as

$ds^{2}=-\tilde{\Delta}dt^{2}+dr^{2}\mp
2\sqrt{1-\tilde{\Delta}}drdt+r^{2}d\Omega\,,$ (42)

and

$\dot{r}=\frac{dr}{dt}=\pm
1\mp\sqrt{\frac{(\mu-\tilde{\omega})}{r}+\frac{\gamma^{2}}{2\zeta\eta
r^{2}}+\frac{\Lambda}{3\zeta}r^{2}},$ (43)

where
$\tilde{\Delta}=1-\frac{(\mu-\tilde{\omega})}{r}-\frac{\gamma^{2}}{2\zeta\eta
r^{2}}-\frac{\Lambda}{3\zeta}r^{2}$. To compute the tunneling rate, as a semi-
classical procedure, we consider the Wentzel-Kramers-Brillouin (WKB)
approximation. The tunneling probability is the imaginary part of the action

$\Gamma\sim\exp(-2\,\textrm{Im}\,\textit{S}).$ (44)

The imaginary part of the particle action across the event horizon, $r_{+}$,
from initial position, $r_{in}$, to the final position, $r_{out}$, is defined
as

$\textrm{Im}\,\textit{S}=\textrm{Im}\,\int_{r_{in}}^{r_{out}}p_{r}dr=\textrm{Im}\,\int_{r_{in}}^{r_{out}}\int_{0}^{p_{r}}d{\tilde{p}_{r}}dr,$
(45)

where $p_{r}$ is the canonical momentum of the outgoing particle. By using the
Hamilton’s canonical equation

$\dot{r}=\frac{dH}{dp_{r}}=\frac{d(\mu-\tilde{\omega})}{dp_{r}}\,,$ (46)

and by substituting Eq.(46) into Eq.(45) we get

$\textrm{Im}\,S=\textrm{Im}\,\int_{r_{in}}^{r_{out}}\int_{0}^{\omega}\frac{(-d\tilde{\omega})dr}{\dot{r}}=Im\int_{r_{in}}^{r_{out}}\int_{0}^{\omega}\frac{(-d\tilde{\omega})dr}{1-\sqrt{\frac{(\mu-\tilde{\omega})}{r}+\frac{\gamma^{2}}{2\zeta\eta
r^{2}}+\frac{\Lambda}{3\zeta}r^{2}}}.$ (47)

The commutation relation, from the GUP expression, in the presence of minimal
length, minimal momentum and maximal momentum is

$\left[r,p_{r}\right]=i\left(1-\alpha\ell_{p}p+\alpha^{2}\ell_{p}^{2}p^{2}+\beta^{2}\ell_{p}^{2}r^{2}\right).$
(48)

In the classical limit we can rewrite this relation between the radial
coordinate and the conjugate momentum by poisson bracket

$\\{r,p_{r}\\}=\left(1-\alpha\ell_{p}p+\alpha^{2}\ell_{p}^{2}p^{2}+\beta^{2}\ell_{p}^{2}r^{2}\right).$
(49)

So, we obtain the deformed Hamiltonian equation as follows

$\dot{r}=\\{r,H\\}=\\{r,p_{r}\\}\frac{dH}{dr}.$ (50)

Finally, we can rewrite the imaginary part of the action in the presence of
GUP as follows

$\displaystyle\textrm{ Im }\,S$
$\displaystyle=\textrm{Im}\,\int_{r_{in}}^{r_{out}}\int_{0}^{\omega}\frac{\hbar(1-\alpha\ell_{p}\tilde{\omega}+\alpha^{2}\ell_{p}^{2}\tilde{\omega}^{2})}{1-\sqrt{\frac{(\mu-\tilde{\omega})}{r}+\frac{\gamma^{2}}{2\zeta\eta
r^{2}}+\frac{\Lambda}{3\zeta}r^{2}}}(-d\tilde{\omega})dr$
$\displaystyle+\textrm{Im}\,\int_{r_{in}}^{r_{out}}\int_{0}^{\omega}\frac{\hbar(\gamma^{2}\ell_{p}^{2}r^{2})}{1-\sqrt{\frac{(\mu-\tilde{\omega})}{r}+\frac{\gamma^{2}}{2\zeta\eta
r^{2}}+\frac{\Lambda}{3\zeta}r^{2}}}(-d\tilde{\omega})dr.$ (51)

The integral takes the following form

$\displaystyle\textrm{Im}\,S$
$\displaystyle=\textrm{Im}\,\int_{r_{in}}^{r_{out}}\bigg{[}2\pi
r-2\alpha\ell_{p}\pi\left(\mu r-r^{2}+Q^{2}+\frac{\Lambda
r^{4}}{3\zeta}\right)+2\pi\alpha^{2}\ell^{2}\Big{[}\left(\mu^{2}-2Q^{2}\right)r-2\mu
r^{2}$ $\displaystyle+2\mu Q^{2}+\frac{2\mu\Lambda
r^{4}}{3\zeta}+\left(1+\frac{2Q^{2}\Lambda}{3\zeta}\right)r^{3}+Q^{4}\frac{1}{r}+\left(\frac{\Lambda}{3\zeta}\right)^{2}r^{7}+\frac{2\Lambda
r^{5}}{3\zeta}\Big{]}+2\pi\beta^{2}\ell_{p}^{2}r^{3}\bigg{]}dr\,,$ (52)

where $Q=\frac{\gamma^{2}}{2\zeta\eta}$. Therefore, the imaginary part of the
action takes the following form

ImS
$\displaystyle=-\pi\left(r_{out}^{2}-r_{in}^{2}\right)\left[-1+\alpha\ell_{p}\mu-\alpha^{2}\ell_{p}^{2}\mu^{2}+2\alpha^{2}\ell_{p}^{2}Q^{2}\right]-2\pi\frac{\left(r_{out}^{3}-r_{in}^{3}\right)}{3}\left[-\alpha\ell_{p}+2\alpha^{2}\ell_{p}^{2}\mu\right]$
$\displaystyle-2\pi\left(r_{out}-r_{in}\right)\left[2\alpha\ell_{p}Q^{2}-2\alpha^{2}\ell_{p}^{2}\mu
Q^{2}\right]-2\pi\frac{\left(r_{out}^{5}-r_{in}^{5}\right)}{5}\left[\alpha\ell_{p}\frac{\Lambda}{3\zeta}-2\mu\Lambda\alpha^{2}\ell_{p}^{2}\right]$
$\displaystyle+2\pi\alpha^{2}\ell_{p}^{2}\frac{\Lambda}{9\zeta}\left(r_{out}^{6}-r_{in}^{6}\right)-\pi\frac{\left(r_{out}^{4}-r_{in}^{4}\right)}{2}\left[\beta^{2}\ell_{p}^{2}-2\alpha^{2}\ell_{p}^{2}\frac{Q^{2}\Lambda}{3\zeta}\right]+2\pi\alpha^{2}\ell_{p}^{2}Q^{4}\left(\ln
r_{out}-\ln r_{in}\right)$
$\displaystyle+2\pi\alpha^{2}\ell_{p}^{2}\left(\frac{\Lambda}{3\zeta}\right)^{2}\frac{\left(r_{out}^{8}-r_{in}^{8}\right)}{8}.$
(53)

Substituting Eq.(3.2) into Eq.(44), we obtain the tunneling rate at the
horizon as follows

$\displaystyle\Gamma$
$\displaystyle\approx\exp\Bigg{\\{}2\pi\left(r_{out}^{2}-r_{in}^{2}\right)\left[-1+\alpha\ell_{p}\mu-\alpha^{2}\ell_{p}^{2}\mu^{2}+2\alpha^{2}\ell_{p}^{2}Q^{2}\right]+4\pi\frac{\left(r_{out}^{3}-r_{in}^{3}\right)}{3}\left[-\alpha\ell_{p}+2\alpha^{2}\ell_{p}^{2}\mu\right]$
$\displaystyle+4\pi\left(r_{out}-r_{in}\right)\left[2\alpha\ell_{p}Q^{2}-2\alpha^{2}\ell_{p}^{2}\mu
Q^{2}\right]+4\pi\frac{\left(r_{out}^{5}-r_{in}^{5}\right)}{5}\left[\alpha\ell_{p}\frac{\Lambda}{3\zeta}-2\mu\Lambda\alpha^{2}\ell_{p}^{2}\right]$
$\displaystyle-4\pi\alpha^{2}\ell_{p}^{2}\frac{\Lambda}{9\zeta}\left(r_{out}^{6}-r_{in}^{6}\right)+2\pi\frac{\left(r_{out}^{4}-r_{in}^{4}\right)}{2}\left[\beta^{2}\ell_{p}^{2}-2\alpha^{2}\ell_{p}^{2}\frac{Q^{2}\Lambda}{3\zeta}\right]-4\pi\alpha^{2}\ell_{p}^{2}Q^{4}\left(\ln
r_{out}-\ln r_{in}\right)$
$\displaystyle-4\pi\alpha^{2}\ell_{p}^{2}\left(\frac{\Lambda}{3\zeta}\right)^{2}\frac{\left(r_{out}^{8}-r_{in}^{8}\right)}{8}\Bigg{\\}}=\exp\left(\Delta\textit{S}_{BH}\right),$
(54)

where $\Delta\textit{S}_{BH}=\textit{S}_{BH}(\mu-\omega)-\textit{S}_{BH}(\mu)$
is the difference in Bekenstein-Hawking entropy before and after the particles
emission at the event horizon. When $\gamma=0$ and $\Lambda=0$, the result
reduces to the Schwarzschild black hole’s result [70, 71]. Because of the
extra terms in comparison to the results of [66], the emission spectrum is not
purely thermal.

Finally, we calculate the possible correlation between the emitted particles
(modes) that can be obtained by the following relation

$\chi(E_{1}+E_{2};E_{1},E_{2})\equiv\ln[\Gamma(E_{1}+E_{2})]-\ln[\Gamma(E_{1})\Gamma(E_{2})]\,,$
(55)

where $\ln[\Gamma(E_{1})]$ and $\ln[\Gamma(E_{2})]$ are the emission rates for
the first and second emitted particles and $\ln[\Gamma(E_{1}+E_{2})]$ is the
emission rate for a single, composed particle with energy $E=E_{1}+E_{2}$. The
emission rate for the first quanta that carries out the energy $E_{1}$ is
given by

$\displaystyle\ln[\Gamma(E_{1})]$ $\displaystyle=2\pi
r^{2}\left[-1+\alpha\ell_{p}(\mu-E_{1})-\alpha^{2}\ell_{p}^{2}(\mu-
E_{1})^{2}+2\alpha^{2}\ell_{p}^{2}Q^{2}\right]+\frac{4\pi
r^{3}}{3}\left[-\alpha\ell_{p}+2\alpha^{2}\ell_{p}^{2}(\mu-E_{1})\right]$
$\displaystyle+4\pi r\left[2\alpha\ell_{p}Q^{2}-2\alpha^{2}\ell_{p}^{2}(\mu-
E_{1})Q^{2}\right]+\frac{4\pi
r^{5}}{5}\left[\alpha\ell_{p}\frac{\Lambda}{3\zeta}-2(\mu-
E_{1})\Lambda\alpha^{2}\ell_{p}^{2}\right]$
$\displaystyle-4\pi\alpha^{2}\ell_{p}^{2}\frac{\Lambda}{9\zeta}r^{6}+\frac{2\pi
r^{4}}{2}\left[\beta^{2}\ell_{p}^{2}-2\alpha^{2}\ell_{p}^{2}\frac{Q^{2}\Lambda}{3\zeta}\right]-4\pi\alpha^{2}\ell_{p}^{2}Q^{4}\ln
r-\frac{4\pi\alpha^{2}\ell_{p}^{2}r^{8}}{8}\left(\frac{\Lambda}{3\zeta}\right)^{2}.$
(56)

Similarly, the emission rate for the second quanta that carries out the energy
$E_{2}$ is as follows

$\displaystyle\ln[\Gamma(E_{2})]$ $\displaystyle=2\pi
r^{2}\left[-1+\alpha\ell_{p}((\mu-E_{1})-E_{2})-\alpha^{2}\ell_{p}^{2}((\mu-
E_{1})-E_{2})^{2}+2\alpha^{2}\ell_{p}^{2}Q^{2}\right]+$
$\displaystyle\frac{4\pi
r^{3}}{3}\left[-\alpha\ell_{p}+2\alpha^{2}\ell_{p}^{2}((\mu-
E_{1})-E_{2})\right]+4\pi
r\left[2\alpha\ell_{p}Q^{2}-2\alpha^{2}\ell_{p}^{2}((\mu-
E_{1})-E_{2})Q^{2}\right]+$ $\displaystyle\frac{4\pi
r^{5}}{5}\left[\alpha\ell_{p}\frac{\Lambda}{3\zeta}-2((\mu-
E_{1})-E_{2})\Lambda\alpha^{2}\ell_{p}^{2}\right]-4\pi\alpha^{2}\ell_{p}^{2}\frac{\Lambda}{9\zeta}r^{6}+\frac{2\pi
r^{4}}{2}\left[\beta^{2}\ell_{p}^{2}-2\alpha^{2}\ell_{p}^{2}\frac{Q^{2}\Lambda}{3\zeta}\right]$
$\displaystyle-4\pi\alpha^{2}\ell_{p}^{2}Q^{4}\ln
r-\frac{4\pi\alpha^{2}\ell_{p}^{2}r^{8}}{8}\left(\frac{\Lambda}{3\zeta}\right)^{2}.$
(57)

Now, the emission rate for a single quanta that carries out the energy
$E_{1}+E_{2}$ is given by

$\displaystyle\ln[\Gamma(E_{1}+E_{2})]$ $\displaystyle=2\pi
r^{2}\left[-1+\alpha\ell_{p}(\mu-E_{1}-E_{2})-\alpha^{2}\ell_{p}^{2}(\mu-
E_{1}-E_{2})^{2}+2\alpha^{2}\ell_{p}^{2}Q^{2}\right]+$
$\displaystyle\frac{4\pi
r^{3}}{3}\left[-\alpha\ell_{p}+2\alpha^{2}\ell_{p}^{2}(\mu-
E_{1}-E_{2})\right]+4\pi
r\left[2\alpha\ell_{p}Q^{2}-2\alpha^{2}\ell_{p}^{2}(\mu-
E_{1}-E_{2})Q^{2}\right]+$ $\displaystyle\frac{4\pi
r^{5}}{5}\left[\alpha\ell_{p}\frac{\Lambda}{3\zeta}-2(\mu-
E_{1}-E_{2})\Lambda\alpha^{2}\ell_{p}^{2}\right]-4\pi\alpha^{2}\ell_{p}^{2}\frac{\Lambda}{9\zeta}r^{6}+\frac{2\pi
r^{4}}{2}\left[\beta^{2}\ell_{p}^{2}-2\alpha^{2}\ell_{p}^{2}\frac{Q^{2}\Lambda}{3\zeta}\right]$
$\displaystyle-4\pi\alpha^{2}\ell_{p}^{2}Q^{4}\ln
r-\frac{4\pi\alpha^{2}\ell_{p}^{2}r^{8}}{8}\left(\frac{\Lambda}{3\zeta}\right)^{2}.$
(58)

The non-zero statistical correlation function can be calculated as

$\displaystyle\chi(E_{1}+E_{2};E_{1},E_{2})$ $\displaystyle=(-2\pi
r^{2}-4\pi^{2}r^{4})+\Big{(}8\pi Q^{2}r\alpha-\frac{4}{3}\pi
r^{3}\alpha+\frac{4\pi r^{5}\alpha\Lambda}{15\zeta}+2\pi r^{2}\alpha(\mu-
E_{1}-E_{2})$
$\displaystyle+\frac{4\pi^{2}r^{3}(120Q^{2}\alpha\zeta-20r^{2}\alpha\zeta+4r^{4}\alpha\Lambda+30r\alpha\zeta\mu-30r\alpha\zeta
E_{1}-15r\alpha\zeta E_{2})}{15\zeta}\Big{)}\ell_{p}$ $\displaystyle+(-4\ln
r\pi Q^{4}\alpha^{2}-\frac{4\pi r^{6}\alpha^{2}\Lambda}{9\zeta}-\frac{\pi
r^{8}\alpha^{2}\Lambda^{2}}{18\zeta^{2}}+\pi
r^{4}(\beta^{2}-\frac{2Q^{2}\alpha^{2}\Lambda}{3\zeta})$ $\displaystyle+2\pi
r^{2}(-4\ln r\pi Q^{4}\alpha^{2}-\frac{4\pi
r^{6}\alpha^{2}\Lambda}{9\zeta}-\frac{\pi
r^{8}\alpha^{2}\Lambda^{2}}{18\zeta^{2}}+\pi
r^{4}(\beta^{2}-\frac{2Q^{2}\alpha^{2}\Lambda}{3\zeta})$ $\displaystyle+2\pi
r^{2}(2Q^{2}\alpha^{2}-\alpha^{2}(\mu-E_{1})^{2})-\frac{8}{5}\pi
r^{5}\alpha^{2}\Lambda(\mu-E_{1})+\frac{8}{3}\pi r^{3}\alpha^{2}(\mu-E_{1})$
$\displaystyle+8\pi Q^{2}r\alpha^{2}(-\mu+E_{1}))\Big{(}8\pi
Q^{2}r\alpha-\frac{4}{3}\pi r^{3}\alpha+\frac{4\pi
r^{5}\alpha\Lambda}{15\zeta}+2\pi r^{2}\alpha(\mu-E_{1})\Big{)}$
$\displaystyle\Big{(}8\pi Q^{2}r\alpha-\frac{4}{3}\pi r^{3}\alpha+\frac{4\pi
r^{5}\alpha\Lambda}{15\zeta}+2\pi r^{2}\alpha(\mu-
E_{1}-E_{2})\Big{)}-\frac{8}{5}\pi r^{5}\alpha^{2}\Lambda(\mu-E_{1}-E_{2})$
$\displaystyle+\frac{8}{3}\pi r^{3}\alpha^{2}(\mu-E_{1}-E_{2})+8\pi
Q^{2}r\alpha^{2}(-\mu+E_{1}+E_{2})+2\pi r^{2}\alpha^{2}(2Q^{2}-(\mu-
E_{1}-E_{2})^{2})$ $\displaystyle+2\pi r^{2}\Big{(}-4\ln r\pi
Q^{4}\alpha^{2}-\frac{4\pi r^{6}\alpha^{2}\Lambda}{9\zeta}-\frac{\pi
r^{8}\alpha^{2}\Lambda^{2}}{18\zeta^{2}}+\pi
r^{4}\big{(}\beta^{2}-\frac{2Q^{2}\alpha^{2}\Lambda}{3\zeta}$
$\displaystyle-\frac{8}{5}\pi r^{5}\alpha^{2}\Lambda(\mu-
E_{1}-E_{2})+\frac{8}{3}\pi r^{3}\alpha^{2}(\mu-E_{1}-E_{2})+8\pi
Q^{2}r\alpha^{2}(-\mu+E_{1}+E_{2})$ $\displaystyle+2\pi
r^{2}\alpha^{2}(2Q^{2}-(\mu-
E_{1}-E_{2})^{2})\big{)}\Big{)}\ell_{p}^{2}+O(\ell_{p}^{3}).$ (59)

Obviously, the statistical correlation function is not zero. So, black hole
radiation is not purely thermal. Also, existence of the non-zero correlation
means that information can come out during the evaporation process. Since
these correlations can store some sort of information, so these correlations
are capable to address at least part of the lost information in essence.

##  4 Conclusion

While the issue of black hole thermodynamics in the framework of the
generalized/extended uncertainty relations has been studied vastly in
literature, the issue of Horndeski black holes’ thermodynamics in the
framework of GUP/EUP has been overlooked in literature. On the other hand,
Horndeski theory provides the most general framework for scalar-tensor
theories of gravity. For these reasons, in this paper we have focused on the
thermodynamics of shift symmetric Horndeski black hole solutions in the
framework of phenomenological quantum gravity corrections encoded in a class
of generalized/extended uncertainty relation. We obtained in details the
temperature and then the heat capacity of such a black hole that recovers the
standard Schwarzschild, Reissner-Nordström or Reissner-Nordström-de sitter
solutions in the appropriate limits. Then the issue of Hawking radiation as a
semi-classical tunneling from the event horizon has been studied in details.
For this purpose, the imaginary part of the classical action has been
calculated within the WKB approximation. The issue of possible correlations
between the emitted modes (particles) has been treated carefully and it is
shown that these correlations are not vanishing, leading to the conclusion
that part of the lost information may be stored in these quantum gravitational
correlations. It is important to note that we focused mainly on the near,
“event” horizon calculations. There is in fact some correlations between the
various horizons of these multi-horizon geometry [74] and these correlations
should be taken into account in a more realistic and concrete study. We leave
this issue for our forthcoming study.

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# A neural net architecture based on principles of neural plasticity and
development evolves to effectively catch prey in a simulated environment

Addison Wood<EMAIL_ADDRESS>Neuroscience Program, Michigan State University
Jory Schossau<EMAIL_ADDRESS>BEACON Center, Michigan State University  Nick
Sabaj  Richard Liu  Mark Reimers<EMAIL_ADDRESS>Neuroscience Program,
Michigan State University

###### Abstract

A profound challenge for A-Life is to construct agents whose behavior is
‘life-like’ in a deep way. We propose an architecture and approach to
constructing networks driving artificial agents, using processes analogous to
the processes that construct and sculpt the brains of animals. Furthermore,
the instantiation of action is dynamic: the whole network responds in real-
time to sensory inputs to activate effectors, rather than computing a
representation of the optimal behavior and sending off an encoded
representation to effector controllers. There are many parameters and we use
an evolutionary algorithm to select them, in the context of a specific prey-
capture task. We think this architecture may be useful for controlling small
autonomous robots or drones, because it allows for a rapid response to changes
in sensor inputs.

††preprint: APS/123-QED

##  I introduction

One of the most profound challenges of research into artificial life (A-Life)
is to construct agents whose behavior is generated in ways deeply analogous to
the way an animal’s brain drives its motions. Many researchers argue for using
neural networks to control behavior [1, 2, 3, 3] and some favor an approach
that builds an ersatz nervous system, whose operations mimic aspects of an
animal’s nervous system [4, 5, 6].

However, there is no consensus about how closely construction must hew to
biology Furthermore, for large neural networks, the number of possible network
configurations is so large that it is difficult to search for network
constructions that drive adaptive or effective behavior. Evolutionary
algorithms have a solid track record for solving problems with a large search
space [7]. We introduce here an evolutionary approach to constructing a neural
network controller for a small animat in a 2-dimensional world.

Our approach is distinctive in several ways. First, behavior will emerge
through dynamic equilibria of the network and continuous feedback from the
surroundings carried by a rich array of simple sensors, rather than having the
network compute a representation of behavior, which is then implemented by
mechanical systems. The network architecture will be recurrent feedback loops,
analogous to loops between cortex and subcortical structures, such as thalamus
and basal ganglia. Second, neurons will compete and oppose the actions of
other neurons, so that the dynamic equilibrium may be rapidly altered as
circumstances change. Third, each instance of a controller will be constructed
by methods modeled on developmental processes in animal brains, as detailed
below. To our knowledge, this is the most life-like approach yet taken to
constructing A-life simulations.

##  II Methods

###  II.1 Animat Construction

Our system builds neural networks in three stages, modeled on neurobiology:

#### Node definition

Nodes are assigned random pseudo-locations in a 2D, 3D or 4D virtual space.
Parameters define the probabilities of specific node classes at various
locations within the virtual space, and within a class, the specific signals
employed for further stages.

#### Coarse initial wiring

Projections are assigned with synaptic weights determined by the match between
paired transmitter and receiver signals, in a manner intended to be analogous
to how axons find distant targets through molecular interactions in the
developing brain [8, 9]. In our prototype system the signals are determined by
gradients in 2D virtual space determined by parameters.

#### Fine mapping

Cells sort out connections through periods of spontaneous activity and Hebbian
plasticity (“cells that fire together wire together”) [10, 11].

#### Learning

Experiences induce changes in synapses, in a Hebbian manner, contingent upon
attaining food, mediated by neuromodulator (NM) signals. The intrinsically
unstable Hebbian process is further stabilized in two ways: i) saturation of
the plasticity signal, and ii) longer-term homeostatic mechanisms, both
modeled on neurobiology [12, 13]. The processes of synapse generation at each
stage are determined by a small number of parameters. During the subsequent
learning stage, synaptic weights are periodically adjusted according to a
pseudo-Hebbian learning rule, modulated by reward signals, whose parameters
are set by genes.

###  II.2 Evaluation Task

The simulated world is a two dimensional plane, populated by two species of
prey items, both of which move according to a Brownian Motion with a strong
drift away from the animat, when the animat is within striking distance; their
top speed is slightly less than the animat’s top speed. One prey species,
comprising 1/3 of all prey, has more calories than the other; which species is
to have more calories is chosen randomly in each generation, so that the
genome cannot evolve to encode that value, but the animat must learn it in
each generation. The fitness function for the animat is how many calories are
captured per lifetime.

###  II.3 Simulation

The evolutionary algorithm was implemented by highly efficient special purpose
open source software written in C++ that has been developed at MSU: MABE
(Modular Agent Based Evolution framework) [14]. MABE allows researchers to
combine different forms of cognitive substrates, tasks, selection methods and
data collection methods to quickly develop digital neuroevolution experiments.
MABE is ideally suited for this work since it allows parts of an experiment to
be swapped without affecting other parts, which allows us to easily compare
the behavior of the animat with other more popular methods of computational
neuroevolution. Since the parameters that define the construction process and
learning characteristics are continuous or integer-valued (with a large range
of values), our mutation operator works by perturbing numerical values, rather
than swapping symbols in a string. The size of perturbation is initially set
to a plausible value about 10% of the range of values considered reasonable.
See Appendix item I for the initial values of mutation and range used for the
simulations described here.

##  III Results

In our simulations, agents started by capturing an average of 2 or 3 food
items in generation 1 and usually attained an average of 25 food items per 5
minutes simulated lifetime after 50 or 100 generations at which point their
performance stabilized until several thousands of generations, at which point
there typically was a rapid increase in fitness, which then plateaued at
several hundred food items captured, while showing a strong preference for the
high-calorie prey. After the second larger jump in most runs, in the stable
population regime, agents ate an average of 282 prey items, with a standard
deviation of 151. After performance of the top 10One issue for assessing
performance, is how much the measure reflects the genome and how much it
reflects (random) starting conditions. To test this we extracted 10 successful
agents within the stable population regime, and then placed them in 100 random
worlds, in which the starting location of the prey is randomly generated, we
see that the distribution of food eaten for each agent throughout the random
worlds (4) begins to look similar to the histogram of food eaten for all
agents (successful or not) in the stable population regime. Looking at the
histograms visualizing these count numbers, it’s easy to spot the pattern: the
lackluster agents consume next to nothing, there is a relatively large gap in-
between, then there is a larger group of well-performing agents arranged in a
(left-skewed) normal distribution. The most successful agents were exceptional
about identifying and chasing prey. These agents seemed to set and follow a
trajectory for the nearest cluster of prey items, then after momentum carried
the agent out of the group, they could loop back around or find the new
nearest cluster of prey, which is an impressive behavior, as the agent moves
quite quickly. The agent can traverse the entire range encompassing all 50
prey in 1 second of simulated time. This is in contrast to the behavior of
agents before the jump, whose movement was characterized by a large,
consistent looping pattern - a relatively successful strategy, but largely
inefficient, which we termed “filter feeding.” Evolutionary theorists often
ask how inevitable a specific evolutionary outcome is. Although so far the
large majority of our simulations have found a stable high-performing
population, the distributions of construction parameters (‘genes’) are very
different from each other, any pair overlapping less than 1%. This indicates
that even in this simple 22-parameter system a great many distinct possible
optima, with comparable performance, may be found.

Figure 1: Typical example of fitness (foods eaten) among top 10% of animats as
a function of generation. The solid section at the end represents the ‘stable
population’ period: where we narrowed the effect of the mutation operator for
genes that had a narrow range of values for successful agents, in order to
keep most offspring within plausible range of viability.

###### Acknowledgements.

Figure 2:  Visualization of the animat (large round beige object) and prey -
link to movie:  animat_video_01.mov The Roomba-like object is the animat,
where the black area in the top-right of the agent represents its front. The
blue shapes are low-calorie prey, while the yellow shapes are high-calorie
prey. It is important to note when viewing the full video that it is
$0.3\times$ the normal speed, so that the behavior of the agent can be more
easily tracked. Figure 3:  Distribution of foods eaten during lifetime (5 min)
by animat agents during 100 generations of the stable population ($N=100$ per
generation) Figure 4:  Distribution of foods eaten by 10 successful agents,
each evaluated in 100 different randomly-generated worlds. Full folder of
videos, plots, and histograms for all 10 runs:  Animat Videos & Figures

## References

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License Attribution 4.0 International (CC BY 4.0).

ITASEC 2024: Italian Conference on CyberSecurity, April 08–11, 2024, Salerno,
Italy

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# Unveiling Online Conspiracy Theorists:
a Text-Based Approach and Characterization

Alessandra Recordare  Insitute of Informatics and Telematics (IIT), National
Research Council (CNR), Via G. Moruzzi 1, 56124, Pisa, Italy  Guglielmo Cola
Tiziano Fagni  Maurizio Tesconi

(2024)

###### Abstract

In today’s digital landscape, the proliferation of conspiracy theories within
the disinformation ecosystem of online platforms represents a growing concern.
This paper delves into the complexities of this phenomenon. We conducted a
comprehensive analysis of two distinct X (formerly known as Twitter) datasets:
one comprising users with conspiracy theorizing patterns and another made of
users lacking such tendencies and thus serving as a control group. The
distinguishing factors between these two groups are explored across three
dimensions: emotions, idioms, and linguistic features. Our findings reveal
marked differences in the lexicon and language adopted by conspiracy theorists
with respect to other users. We developed a machine learning classifier
capable of identifying users who propagate conspiracy theories based on a rich
set of 871 features. The results demonstrate high accuracy, with an average F1
score of 0.88. Moreover, this paper unveils the most discriminating
characteristics that define conspiracy theory propagators.

###### keywords:

conspiracy theoristdisinformation fake news zero-shot learning social media

##  1 Introduction

In the era of social networks, where the proliferation of misinformation and
conspiracy theories has become a growing concern, the need to identify users
responsible for creating misleading content has become imperative. In
addressing disinformation, it is crucial to consider the role of social
networks, as they have been shown to act as significant amplifiers [1]. That
is why examining the spread of disinformation within social networks has
become an area of growing research interest [2, 3, 4]. In particular, in the
aftermath of the COVID-19 pandemic, there has been an increased focus on the
study and understanding of conspiracy theories in general. This interest stems
from the awareness of the significant impact that these theories can have on
public health, social cohesion, and the dissemination of accurate information.
In response to this challenge, this study aims to provide a contribution by
explaining an approach to identifying and profiling individuals who promote
conspiracy theories on social media [5].

This study builds upon prior research [6] where a technique was introduced to
collect two datasets: one consisting of apparent conspiracy theorists and the
other of generic users, all sourced from X (Twitter). Additionally, in that
work a classification study was conducted to differentiate between conspiracy
and generic users, using a combination of psycholinguistic features and
platform-specific profile characteristics, including Following Count, Follower
Count, Bio Sentences, Retweet Ratio, and more. In our research, we seek to
characterize conspiracy theorists solely based on their writing style, moving
away from dependencies on social network-related features. We explore three
distinct categories of features: emotions, idioms, and linguistic attributes.
Moreover, we aim to identify the specific features that prove to be crucial in
making this distinction. Given the definition of “conspiracy user” as someone
who believes in conspiracy theories (conspiracy theorist), our research
questions are:

RQ1 – Is it possible to identify a conspiracy user through text alone?

RQ2 – What are the features that differentiate a conspiracy user from a
generic user?

Our investigation involved training various classifiers using three distinct
classes of text-based features, ensuring that our insights could be applied
independently of the specific social networking platform. The results show
that the two groups of users exhibit divergent writing styles, underscoring
distinct attitudes. Moreover, we identified which types of features are most
effective in revealing the tendency to adhere to conspiracy theories.

The paper is organized as follows. In the following Section we briefly present
some of the most relevant studies in the field of fake news and conspiracy
theories detection. In Section 3, we describe the X dataset and the
preprocessing steps required for our analysis. Next, in Section 4.1, we show
and describe the adopted features. Section 5 shows the analysis we performed
on the dataset. Finally, in Section 6, we summarize our findings and suggest
avenues for future research.

##  2 Related work

Social media have greatly facilitated the dissemination of unverified
information and misleading content [1, 3, 7]. There has been extensive
research on fake news detection to enable the analysis of fake news spreaders.
A relevant example is the study in [8], which revealed that there are
characteristics (most dependent on social media) that differ between users who
share fake news and those who share real news. Giachanou et al. in [9]
proposed a system that exploits a set of psycholinguistic characteristics and
personality traits inferred by users to discriminate between potential
spreaders of fake news and fact-checkers.

Recent years have seen an increasing focus on the study of conspiracy theories
among fake news and disinformation, especially in response to global events
such as the COVID-19 pandemic [1, 10]. Alternative explanations for historical
or ongoing events, which claim that individuals or groups with malevolent
intentions are involved in occult conspiracies, have infiltrated online
communication, popular culture, political discourse, and various other areas
[11]. Researchers are studying how these theories spread across different
social platforms, analyzing the mechanisms that lead to their adoption by
individuals and trying to identify and characterize conspiracy users in
different ways [12, 13, 14, 15, 16, 17]. While conspiracy theories have
recently been associated with vaccines, their scope extends to several other
realms. For example, Marcellino et al. [5] collected and analyzed online
discussions related to four specific conspiracy theories. Klein et al. [18]
examined users posting a variety of conspiracy theories on Reddit, analyzing
differences in the language used by conspiracy theorists compared to other
users. The work of Fong et al. [19] analyzed conspiracy theories posted by
influencer users on Twitter. Bessi et al. [20] examined the differences
between Facebook users who adhere to conspiracy theories and those who do not,
characterizing the personalities of the two groups.

Many of these studies have suggested the possibility of distinguishing the two
user groups, yet they lack detailed insights into the extent of this
differentiation. Our aim is therefore to distinguish between “conspiracy
users” and other users on social media and to give them a characterization.
Our work differs from those mentioned above in that we determine whether a
user is a conspiracy user by using only the text of posted tweets,
independently of other dynamics of the social networking platform. In this
context, a relevant study is presented in [21], where a classifier for
conspiracy users is described. However, unlike their approach, we do not seek
a distinction between users who support conspiracy theories and users who
refute them, but instead compare apparent conspiracy theorists with generic
users discussing the same topics. Additionally, while their focus was on a
narrow range of conspiracies, our study considers a broader set of conspiracy
theories.

##  3 Dataset description

In this section, we present a detailed account of the initial dataset sourced
from [6] as well as the specific preprocessing steps that were executed to
adapt the dataset to the objectives of our research.

This initial dataset includes two distinct sets, each consisting of 7,394 X
users. The first set, the “conspiracy group”, comprises users identified as
conspiracy theorists. The second set, the “control group”, includes users not
exhibiting apparent conspiracy theory patterns. Conspiracy users were
identified by analyzing likes and follows of well-known conspiracy pages or
accounts. Instead, the control group consists of users who neither explicitly
liked nor followed such content, but still engaged in discussions on the same
controversial topics as the conspiracy group and were created around the same
time. For each user, the last 3,200 tweets were collected, as of June 13,
2022. The dataset is publicly available111https://zenodo.org/records/8239530.

To ensure the dataset’s relevance and reliability for our research objectives,
a series of preprocessing steps were undertaken:

  * •

Removal of Retweets: To enhance the dataset’s suitability for profiling users,
we chose to exclude retweets. Retweets, being reposts of other’s content,
introduce redundancy. By excluding them, we ensured that the dataset primarily
consists of original content, aligning with our goal of characterizing users
based on their own tweets.

  * •

Language Filter: Our analysis focused exclusively on tweets composed in the
English language. Implementing this filter was crucial for the subsequent
phases of our research and ensured linguistic coherence in our dataset.

  * •

User Tweet Count Threshold: To ensure the inclusion of users who have a
sufficient presence on X, we implemented a per-user tweet count threshold.
Specifically, we excluded users with fewer than 10 tweets within the data
collection period. This helped improve the accuracy and reliability of the
user profiling we aimed to achieve.

  * •

Selection of Latest 100 Tweets per User: Obtaining a large number of tweets
from a single user is often challenging in practice. To address this, we
focused our analysis on the most recent 100 tweets for each user. This
approach reflects more closely real-world scenarios, where several users do
not have a high volume of tweet activity.

These preprocessing steps transformed the dataset into a more suitable form
for our research objectives. As a result, our dataset contained 547,724 tweets
from conspiracy users and 592,927 tweets from the control group, posted by a
total of 14,568 users. We then balanced the dataset using a Random
Undersampling technique, achieving a total of 7,210 conspiracy users and 7,210
control group users.

##  4 Method

The objective of our study is to characterize conspiracy users through a
series of steps: identifying suitable features that are dependent solely on
the text of the tweet and are not influenced by the platform, conducting a
classification task to distinguish between the two groups, and analyzing the
most significant features using feature importance metrics. This analysis aims
to discern the stylistic differences between the two user groups while
ensuring the exclusion of platform-related factors.

In this section we present the features used to characterize users based on
their tweets as well as the classification methods employed to discriminate
between conspiracy theorists and other users.

###  4.1 Features

We opted to employ three distinct feature groups, all centered around the text
content of each individual tweet:

  1. 1.

Emotions: We included this feature group to partially implement a sentiment
analysis on the dataset. The emotions we have chosen are Anger, Fear, Joy,
Sadness, Disgust, Surprise, Anticipation, and Trust. These emotions align with
Robert Plutchik’s model of basic emotions, which is widely recognized in the
field of psychology [22]. To assess the emotional content of each tweet, we
employed zero-shot learning techniques. Specifically, we used the
facebook/bart-large-mnli model available on Hugging Face 222
https://huggingface.co/facebook/bart-large-mnli as a sentiment classifier.
This pre-trained model provides a score on a scale from 0 to 1, where a score
of 0 indicates no agreement between the emotion and the tweet, while values
approaching 1 indicate a strong agreement between them. The facebook/bart-
large-mnli model is well known for its accuracy in discerning emotional
content in text data [23]. This agreement calculated between the emotion and
the tweet will be the feature used for our work.

  2. 2.

Idioms of conspiracy theorists: 44 sentences were generated by chatGPT-3.5
using the following prompt:

What are the typical idioms of a conspiracy theorist?Some sayings that come to
mind are:> \- think/reason/… with your head> \- they won’t tell you any of
this> \- they don’t tell us> \- nobody talks about it> \- wake up!> \- strong
powers> \- they make fun of us> \- that’s enoughDo you know any other
interesting ones?

These idioms are detailed in Table 1 and they aim to represent the typical
language used by conspiracy theorists on social media. The agreement between
tweets and idioms was assessed using the zero-shot learning capability of the
facebook/bart-large-mnli model, similarly to the method used for emotions. The
agreement score, ranging from 0 to 1, was utilized as a feature for subsequent
analyses.

  3. 3.

Linguistic features: We have identified five sets of linguistic features for a
total of 72: lexical (e.g., num_words),syntactical (e.g. num_sentences),
semantic (e.g., num_named_entities), structural (e.g., avg_sentence_length),
and subject-specific features (e.g., flesch_reading_ease). The full list is
reported in Table 2.

Idioms
---
Behind closed doors |  They want to keep us in the dark
Don’t let them catch you |  They will not tell you anything about this
Don’t let the cat out of the bag |  They’re cooking up something nefarious
Follow the money |  They’re out to get us
It’s a cover-up |  They’re planning something behind our backs
It’s a deep state conspiracy |  They’re plotting something sinister
It’s all part of the plan |  They’re trying to cover up their tracks
Nobody talks about it |  They’re trying to distract us from the real issue
Now enough! |  They’re trying to divide us
Pulling the strings |  They’re trying to silence us
Pulling the wool over our eyes |  Thinking with your head
Question everything |  Trust no one
Strong powers |  Wake up!
The conspiracy runs deep |  Watch your back
The enemy is among us |  We have to be prepared
The truth is hidden |  We have to stay one step ahead of them
The truth is out there |  We have to stick together
The truth is suppressed |  We have to watch our backs
The truth will set us free |  We need to be careful who we trust
They don’t tell us |  We need to dig deeper and uncover the truth
They don’t want us to know the truth |  We need to stay one step ahead of them
They make fun of us |  We need to uncover their secrets
Table 1: List of idioms used among conspiracy theorists Class |  Features |  Description
---|---|---
Lexical |  num_words; num_unique_words; num_chars; num_unique_chars; avg_word_length; num_stop_words; num_punct; num_digits; num_upper_case_words; num_lower_case_words; num_title_case_words; num_proper_nouns; num_nouns; num_verbs; num_adjectives; num_adverbs; num_pronouns; num_named_entities; num_noun_chunks; num_exclamation_marks; num_question_marks; num_spaces |  Word-level characteristics and properties of text. They include various measurements related to the vocabulary and composition of words within a given text.
Syntactical |  nominal_forms; voc_rich; num_sentences; avg_num_words_per_sentence; num_noun_phrases; num_verb_phrases; num_adj_phrases; num_adv_phrases; num_prep_phrases; num_coord_conj; num_subord_conj; num_coord_clauses; num_subord_clauses; punctuation_freq; num_capitalized_sentences; num_caps_word_freq; num_participial; num_present_tense; num_complementation; num_relative_clause |  Grammatical structure and syntax of sentences within a text. They capture the organization and relationships of words and phrases in terms of syntactic rules.
Semantic |  num_personal_pronouns; num_impersonal_pronouns; num_possessive_pronouns; num_reflexive_pronouns; num_reciprocal_pronouns; num_quantifiers; num_determiners; num_prepositions; num_aux_verbs; num_modal_verbs; num_negations; num_synonym; num_antonymy; 1st_person_pronouns; 2nd_person_pronouns; num_passive_verbs |  Meaning and interpretation of words and phrases within a text. They capture the underlying semantics and context of language.
Structural |  avg_sentence_length; avg_word_length; avg_noun_phrases_per_sentence; avg_verbs_per_sentence; proper_noun_ratio |  Overall organization and composition of the text at a higher level, such as sentence and paragraph structure. They provide insights into the textual coherence and complexity.
Subject-specific |  flesch_reading_ease; smog_index; flesch_kincaid_grade; coleman_liau_index; automated_readability_index; dale_chall_readability_score; difficult_words; linsear_write_formula; gunning_fog |  Specialized indicators relevant to specific domains or topics within the text.
Table 2: List of linguistic features divided by class Emotions | Idioms | Linguistic Features
---|---|---
|  | Lexical | Syntactical | Semantic | Structural | Subject-specific
8 | 44 | 22 | 20 | 16 | 5 | 9
Table 3: Number of features used per group

Table 3 shows the number of features per each group.

After selecting these features to characterize users, we decided to aggregate
tweets by individual users. This transformation resulted in a dataset where
each row represents an individual user, rather than an individual tweet. To
achieve this, we computed 7 descriptive statistics, namely the mean, median,
standard deviation, minimum, maximum, lower quartile, and uppr quartile for
the values associated with the tweets of the same user. For instance, when
considering the feature ’num_sentences’, the statistics were computed by
aggregating values of ’num_sentences’ across all tweets authored by an
individual user. As a result, each feature was expanded into 7 distinct
statistical measures. This aggregation process provided us with a more
holistic perspective on the characteristics of each user, facilitating the
summarization of tweet-level attributes into user-level attributes. After this
step, our final dataset comprised 14,420 rows (users) and 868 features.

###  4.2 Classification

The dataset was split into training (85%) and test (15%) sets. Classification
was conducted using either the three groups of features individually or a
combined set incorporating all of them. We evaluated a variety of classifiers,
including Logistic Regression, K-Nearest Neighbours (K-NN), Naive Bayes,
Support Vector Machine (SVM), Decision Trees, Random Forest, Gradient
Boosting, such as XGBoost and LightGBM, Quadratic Discriminant Analysis (QDA),
Multilayer Perceptron (MLP), Ridge Classifier, and Linear Discriminant
Analysis (LDA). For each classification algorithm, stratified k-fold cross-
validation was utilized on the training set to fine-tune the parameters.

##  5 Result and Discussion

We first report the results achieved in recognizing conspiracy users from the
text contained in their tweets (RQ1). Subsequently, we explore the feature
importance within the three groups defined in Section 4.1, in order to unveil
the key features that characterize conspiracy theorists (RQ2).

###  5.1 Conspiracy users classification

In the classification task, as mentioned above, we evaluated various
classifiers using a single group of features (emotions, idioms, or linguistic)
or all of them combined. For emotions, the best performance was achieved with
Logistic Regression. For idioms, the best results were obtained through
Logistic Regression, Ridge Classifier, and Linear Discriminant Analysis (LDA).
On the other hand, for linguistic features and for the combined features, the
best performances were achieved using the Light Gradient Boosting Machine
(LGBM) algorithm. Classification results are shown in Table 4.

Classifier | Emotion | Idioms | Linguistic features | All features
---|---|---|---|---
| Prec. | Recall | F1 | Prec. | Recall | F1 | Prec. | Recall | F1 | Prec. | Recall | F1
Logistic regression | 0.74 | 0.81 | 0.77 | 0.80 | 0.85 | 0.82 | 0.79 | 0.83 | 0.81 | 0.83 | 0.85 | 0.84
K-NN | 0.70 | 0.73 | 0.72 | 0.77 | 0.69 | 0.73 | 0.70 | 0.76 | 0.73 | 0.76 | 0.75 | 0.75
Naive Bayes | 0.61 | 0.95 | 0.74 | 0.65 | 0.92 | 0.76 | 0.54 | 0.98 | 0.69 | 0.60 | 0.96 | 0.73
SVM | 0.75 | 0.78 | 0.76 | 0.78 | 0.69 | 0.73 | 0.76 | 0.78 | 0.77 | 0.82 | 0.85 | 0.83
MLP | 0.73 | 0.74 | 0.73 | 0.81 | 0.80 | 0.81 | 0.80 | 0.78 | 0.79 | 0.85 | 0.84 | 0.85
Ridge Classifier | 0.73 | 0.82 | 0.77 | 0.80 | 0.85 | 0.82 | 0.78 | 0.83 | 0.80 | 0.81 | 0.87 | 0.84
LDA | 0.73 | 0.82 | 0.77 | 0.80 | 0.85 | 0.82 | 0.78 | 0.83 | 0.80 | 0.81 | 0.87 | 0.84
DT | 0.68 | 0.67 | 0.67 | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 | 0.72 | 0.70 | 0.71
RF | 0.73 | 0.79 | 0.76 | 0.76 | 0.84 | 0.79 | 0.77 | 0.77 | 0.77 | 0.78 | 0.84 | 0.81
XGBoost | 0.73 | 0.77 | 0.75 | 0.78 | 0.85 | 0.81 | 0.83 | 0.87 | 0.85 | 0.85 | 0.88 | 0.86
LightGBM | 0.74 | 0.79 | 0.76 | 0.78 | 0.85 | 0.81 | 0.84 | 0.87 | 0.86 | 0.86 | 0.89 | 0.87
Mean | 0.703 | 0.802 | 0.752 | 0.756 | 0.818 | 0.780 | 0.731 | 0.751 | 0.716 | 0.766 | 0.779 | 0.752
Table 4: Precision, Recall and F1 score on test sets

###  5.2 Feature importance

Figure 1: SHAP values for the 20 most important features considering all
features

From these results, it is apparent that the feature group which excels at
distinguishing between conspiracy users and control group is the set of
linguistic features. Figure 1 shows the 20 most important features for
discriminating between control group users (on the left) and conspiracy users
(on the right). Blue points represent low feature values, while red points
indicate high values. The SHAP value (the distance from the central vertical
axis) indicates the importance of that feature for classification. The
analysis of the 20 most crucial features for classification shows that the top
10, in terms of importance, originate from the linguistic feature group, with
the remaining 10 linked to idioms. Notably, none of the top 20 features are
related to emotions, suggesting that emotional features have relatively
limited discriminatory power between the two user groups. The most
discriminative feature is mean(num_coord_clauses), showing lower values for
conspiracy users, followed by mean(num_reflexive_pronouons) and
mean(num_possessive_pronouons), both showing higher values for conspiracy
users.

Figure 2: F1 score based on the number of features used for classification
(ordered by feature importance) considering all features

Figure 2 depicts the variation of the F1 score as a function of the number of
features employed in the classification, arranged according to their order of
importance. We can see that by utilizing the first 30 features, the maximum F1
score is achieved, and notably, even with just the first 14 features, an F1
score of 0.85 is attained.

In the following subsections, we provide a detailed analysis on the relevance
of each group of features in recognizing conspiracy users, directly addressing
our second research question (RQ2).

###  5.3 Emotions

Figure 3: SHAP values for the 20 most important features in the emotions set

We conducted an in-depth analysis of emotion-based feature importance in the
LGBM classifier and observed that the most prominent distinguishing emotion
between the two user groups is “disgust”, followed by “joy”, “sadness”, and
“anticipation”. Figure 3 shows the 20 most important emotional features for
discriminating between control group users and conspiracy users.

For the “disgust” emotion, the average, median, standard deviation, and 75th
percentile values were significantly higher for conspiracy users. In the
control group, the mean and seventy-fifth percentile of the “joy” emotion
exhibited higher values. As for the “sadness” emotion, conspiracy users showed
higher mean and 75th percentile values compared to the control group.
Interestingly, for the “anger” emotion, both the mean and median were higher
in the control group.

###  5.4 Idioms of conspiracy theorists

Figure 4: Descriptive statistics relative to the analyzed idioms, for control
group users and conspiracy users

Figure 4 is a heatmap showing the average values of several descriptive
statistics for the majority of idioms in our analysis, divided by user group
(control group and conspiracy users). The average of each descriptive
statistic was computed among all the users in a group. Cells with higher
values tend towards yellow, whereas lower values are represented by violet
blue cells. We excluded the least discriminating idioms and statistics for
better readability.

Most of the idioms identified by ChatGPT tend to align more closely, on
average, with tweets from conspiracy theorists, except for We have to stick
together, Strong powers, It’s all part of the plan, The truth will set us
free, and Follow the money, which align more with tweets from the control
group. Some idioms exhibit a strong agreement with both user groups, like We
have to be prepared, while others show little agreement for either group, such
as Trust no one, They’re plotting something nefarious, and The enemy is among
us. The average standard deviations are consistently higher for conspiracy
theorists, suggesting that this group has more diverse data among themselves.
There are substantial differences in the averages of the 75th percentiles, for
example, in phrases like Pulling the wool over our eyes, Question everything,
and The conspiracy runs deep, where quite higher values are noted for
conspiracy users. This indicates that agreement values for these idioms tend
to be higher for this class of users.

###  5.5 Linguistic features

Figure 5: SHAP values for the 20 most important features in the linguistic set

As previously mentioned, linguistic features have proven to be the most
effective in classifying conspiracy and control group users. To further
explore this, we divided these features into five groups: lexical,
syntactical, semantic, structural, and subject-specific features. Our goal was
to ascertain which of these groups contributed most significantly to the
differentiation between the two user classes. Regarding their utility for
classification, we found that semantic features ranked the highest, followed
by syntactical, lexical, subject-specific, and, lastly, structural features.
This observation is corroborated by Figure 5, which illustrates that the top
20 features contributing to classification predominantly belong to the
semantic, syntactical, or lexical categories.

Among the most significant semantic features are the average count of
reflexive pronouns, the average count of possessive pronouns (in both cases,
the number of pronouns mentioned is higher for conspiracy users), and the
average count of named entities (conspiracy users tend to mention fewer
entities). As for syntactical features, the mean and standard deviation of the
number of coordinating clauses, along with the standard deviation of the
number of subordinate clauses, were identified as the most important. Both of
these features exhibited higher values among conspiracy theorists. Among the
prominent lexical features, the mean count of digits (higher for conspiracy
users) and the standard deviation of title case word count (also higher for
conspiracy users) were found to be the most influential. Other noteworthy
features include the higher count of question marks among conspiracy users, as
well as vocabulary richness, indicating a more sophisticated word choice in
their tweets. Furthermore, all readability indices suggest a greater level of
reading difficulty (and therefore lower readability) in tweets of conspiracy
users. This is likely attributed to the usage of acronyms or hashtags typical
of the movement they support.

##  6 Conclusions and future work

In this study, we introduced a method for profiling users who endorse
conspiracy theories, focusing specifically on characterizing their writing
style. This characterization was achieved by analyzing textual content of
tweets, intentionally excluding platform-dependent metrics such as likes,
retweets, and comments.

We selected a dataset consisting of 14,420 users, evenly split between two
categories: 7,210 conspiracy users and 7,210 control group users, who did not
exhibit explicit conspiracy theory behavior patterns. For each user, we
analyzed between 10 to 100 of their most recent tweets, calculating scores
based solely on textual content. These scores were subsequently aggregated for
each user, using statistical measures like mean and median to capture the
essence of each user’s textual patterns. We also implemented and tested
classification algorithms, with the Light Gradient Boosting Machine classifier
yielding the most promising results. This classifier enabled us to effectively
differentiate between conspiracy and control users, achieving an F1 score of
0.87.

Responding to RQ1, this research has shown that users can be categorized based
solely on the characteristics of their writing style. Furthermore, in response
to RQ2, this study identified specific linguistic traits that can be
considered characteristic of conspiracy theorists, thus shedding light on the
distinct markers of this group within the digital landscape. We found that the
features that best characterize conspiracy users from the control group are
linguistic features, in particular the number of coordinate clauses, the
number of possessive and reflexive pronouns. Our work shows that conspiracy
theorists use fewer coordinate clauses than the control group but more
reflexive and possessive pronouns, use more digits, name fewer entities, use a
richer vocabulary and have worse readability. Regarding sentiment analysis,
the tweets from conspiracy users show a higher agreement with disgust and
sadness, while the tweets of the control group are more akin to joy and anger.
Considering the set of conspiracy idioms generated via chat-GPT, it turns out
that most of them have a higher agreement with conspiracy users.

In future work, we plan to extend our classifier’s application to other
platforms like Telegram. This will help assess the model’s generalizability
and robustness across diverse social media. Furthermore, we plan to improve
the accuracy of our model by incorporating a wider range of text-only
features, enhancing our understanding of user behavior and the overall ability
of recognizing conspiracy theorists. Additionally, we are interested in
exploring different time windows to capture evolving trends and emerging
patterns in the propagation of conspiracy theories. Lastly, an interesting
avenue for future research is examining the implications of our findings on
disinformation mitigation strategies. This could lead to more effective
methods to counter the spread of disinformation and promote digital literacy.

###### Acknowledgements.

We acknowledge the support provided by:

  * •

Project SoBigData.it, which receives funding from European Union –
NextGenerationEU – National Recovery and Resilience Plan (Piano Nazionale di
Ripresa e Resilienza, PNRR) – Project: "SoBigData.it – Strengthening the
Italian RI for Social Mining and Big Data Analytics" – Prot. IR0000013 –
Avviso n. 3264 del 28/12/2021;

  * •

Project SERICS (PE00000014) under the NRRP MUR program funded by the EU –
NGEU;

  * •

Regione Toscana, which partially funded this work within the framework of the
“Por Fse 2014-2020 Investimenti a favore della crescita, dell’occupazione e
del futuro dei giovani Fondo SocialeEuropeo”. Decreto di finanziamento DD
21607 del 29/11/2021.

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