# Holographic vacuum energy regularization and corrected entropy of de Sitter
space

S. Jalalzadeh1,2, H. Moradpour3, and H. Tebyanian4 1 Departamento de Física,
Universidade Federal de Pernambuco, Recife-PE, 50670-901, Brazil  2 Center for
Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096,
Azerbaijan  3 Research Institute for Astronomy and Astrophysics of Maragha
(RIAAM), P.O. Box 55134-441, Maragheh, Iran  4 Department of Mathematics,
University of York, Heslington, York, YO10 5DD, United Kingdom
<EMAIL_ADDRESS><EMAIL_ADDRESS>&
<EMAIL_ADDRESS>
###### Abstract

We propose that the spectrum of the surface area of the apparent horizon (AH)
of de Sitter (dS) spacetime leads to corrected temperature and entropy of the
dS spacetime, offering new insights into its thermodynamic properties. This is
done by employing the spectrum of the AH radius, acquired from the
Wheeler–DeWitt (WDW) equation, together with the Stefan–Boltzmann law, the
time-energy uncertainty relation, and the unified first law of thermodynamics.

††: Class. Quantum Grav.

  * March 2024

##  1 Introduction

In the early 1970s, Jacob Bekenstein revolutionized our understanding of black
holes (BHs) by positing that the entropy of a BH is directly proportional to
the area of its event horizon [1]. This groundbreaking hypothesis laid the
foundation for the integration of thermodynamic principles into the realm of
general relativity. A few years later, in 1975, Stephen Hawking extended this
paradigm by introducing the concept of BH evaporation, thereby linking quantum
mechanics with the thermodynamics of BHs [2]. Recent observational studies
have provided compelling evidence in support of these theoretical frameworks.
For instance, gravitational wave observations have been employed to test the
area-entropy relationship, yielding results that are consistent with the
predictions of Bekenstein and Hawking [3].

These empirical validations have invigorated the scientific community’s
interest in the intricate interplay between thermodynamics, quantum physics,
and the geometry of spacetime horizons. Over the years, theoretical physicists
have delved deeper into this nexus, uncovering a rich tapestry of
relationships that intertwine thermodynamic temperature, quantum field theory,
and the geometrical properties of horizons. These discoveries have cemented
the role of thermodynamics in the study of gravitational systems and opened up
new avenues for exploring the quantum nature of spacetime itself.

In quantum gravity, it is possible to derive the thermodynamical
characteristics associated with a specific stationary spacetime by employing
the technique of Euclidean path integral [4]. The thermal equilibrium state of
the system of spacetime, $(\mathcal{M},g)$, plus matter field, $\Phi$, for the
canonical ensemble is given by the partition function $Z$ as

$Z=\int{\mathcal{D}}g^{\text{(E)}}{\mathcal{D}}\Phi\exp\\{I^{\text{(E)}}[g^{\text{(E)}},\Phi]\\},$
(1)

where ${\mathcal{D}}g^{\text{(E)}}$ and ${\mathcal{D}}\Phi$ represent the path
integrals over the Euclidean metric and matter fields, respectively, and
$g^{\text{(E)}}$ is the Euclidean metric of signature $(++++)$ obtained from
the Lorentzian metric $g$ by the Wick rotation $t\rightarrow-i\tau$, and
$I^{\text{(E)}}[g^{\text{(E)}},\Phi]$ denotes the Euclidean action functional
111Because it enhances the convergence of path integrals, a wick rotation of
the time axis is frequently employed in quantum field theory. On the other
hand, this is not the case in quantum gravity. In general, the integral (1) is
poorly divergent, and the gravitational component of the Euclidean action is
unbounded from below. Only partially successful attempts to solve this issue
using analytical continuation have been made, and it is still unknown if an
integral like (1) can be defined in a meaningful way [5]. . In order to
consider equilibrium states of spacetime with a matter field, it is necessary
to impose the periodic boundary condition in imaginary time, $\tau$, for both
the metric and the matter field $\Phi$.

Note that the fields $g$ and $\Phi$ are not necessarily solutions of classical
field equations, however, one can expand these fields as
$g=g_{\text{cl}}+\delta g$ and $\Phi=\Phi_{\text{cl}}+\delta\Phi$ around the
dominant classical solutions when the gravitational and matter fields are
weak. As a result, the Euclidean action functional becomes

$I^{\text{(E)}}[g^{\text{(E)}},\Phi]=I^{\text{(E)}}[g^{\text{(E)}}_{\text{cl}},\Phi_{\text{cl}}]+I^{\text{(E)}}[\delta
g^{\text{(E)}}]+I^{\text{(E)}}[\delta\Phi]+\text{higher order terms},$ (2)

where $g_{\text{cl}}$ and $\Phi_{\text{cl}}$ denote the classical solutions
around which the fields are expanded, $\delta g^{\text{(E)}}$ and $\delta\Phi$
denote quantum/statistical fluctuations of metric and matter fields, and
$I^{\text{(E)}}[\delta g^{\text{(E)}}]$ and $I^{\text{(E)}}[\delta\Phi]$ are
quadratic in fluctuations.

Therefore, the partition functional (1) becomes

$\ln Z=\ln Z_{\text{cl}}+\ln\int\mathcal{D}[\delta
g^{\text{(E)}}]e^{I^{\text{(E)}}[\delta
g^{\text{(E)}}]}+\ln\int\mathcal{D}[\delta\Phi]e^{I^{\text{(E)}}[\delta\Phi]}+\text{higher
order terms},$ (3)

where $Z_{\text{cl}}$ is the classical partition functional. The first term in
the above includes only the classical solutions and consequently, the
partition function of the thermal equilibrium state of background classical
spacetime and matter is defined by $\ln
Z_{\text{cl}}=I^{\text{(E)}}[g^{\text{(E)}}_{\text{cl}},\Phi_{\text{cl}}]$.
Now, utilizing the Boltzmann’s relation, $S=\ln Z$, one obtains the entropy of
the system. This shows that the standard Bekenstein–Hawking entropy (as well
as other thermodynamical properties) of an empty background spacetime horizon
is encoded in the classical geometry itself. In addition, the correction terms
arise due to statistical fluctuations around the thermal equilibrium of the
classical geometry and matter field [6, 7]. Also, in the presence of a matter
field, one should use a proper regularization method to compute the portion of
quantum matter field in Eq. (3).

Another approach to acquiring the thermodynamic properties linked to a
particular static spacetime is through the utilization of canonical
quantization of the relevant gravitational system featuring a horizon,
developed by V.F. Mukhanov [8]. In the case of a Schwarzschild BH, the
quantization of the system results in a discrete spectrum of horizon areas.
Bekenstein pioneered the concept of BH horizon area quantization, proposing
that the area, $A_{\text{BH}}$, could be represented by discrete eigenvalues
[9]:

$A_{\text{BH}}(n)=\gamma L^{2}_{\text{P}}{\,}n,~{}~{}~{}~{}n=\\{\text{large
integers}\\},$ (4)

where $\gamma$ is a dimensionless constant of order one, depending on the
details of the quantization method [10], and $L_{\text{P}}=\sqrt{G}$ is the
Planck length. Subsequently, numerous studies (see, for example, [11] and the
references therein) have been integrated into it, all of which have played a
role in bolstering the accuracy of the mass spectrum (4).

It can be hypothesized that a BH emits Hawking radiation during its
spontaneous transition from state $n+1$ to the nearest lower state level,
specifically, $n$. Supposing the emitted thermal radiation’s frequency is
denoted as $\omega_{0}$, it follows that

$\omega_{0}=M^{(n+1)}-M^{(n)}=\frac{\gamma
M_{\text{P}}}{\sqrt{2n}}+\text{higher order
terms}=\frac{\gamma^{2}M^{2}_{\text{P}}}{2M}+\text{higher order terms}.$ (5)

According to Mukhanov [8] this equation bridges the gap between quantum
mechanics and classical thermodynamics in the context of BHs.

The argument employed by Mukhanov to derive the thermodynamical
characteristics of a BH, as detailed in the reference [8], can be elucidated
in the following manner: Initially, he assumed that the spectrum of a BH
adheres to the Eq. (4). Subsequently, in his analysis to determine the rate of
Hawking radiation, he proceeded to calculate the BH’s lifetime while it is in
a specific quantum state represented by the quantum number $n$, as expressed
in Eq. (5). Lastly, by incorporating the Stefan–Boltzmann law in conjunction
with the first law of thermodynamics and Mukhanov’s argument, as elaborated in
Refs. [12, 13, 14], one is able to deduce the temperature and entropy of a
Schwarzschild BH.

The advantage of this method of calculating entropy to Euclidean path integral
is its simplicity in calculating the higher-order correction terms: a Taylor
expansion of (5) at the desired order, gives us the corresponding order of
correction term of the entropy as well as the temperature of the BH. In
addition, sometimes, we just know the modified WDW equation, and the
corresponding path integral is not developed enough. For example, in Ref.
[13], the authors developed a fractional WDW equation of a Schwarzschild BH.
Also, a quantum deformed (through quantum groups) extension of the WDW
equation is represented in [14] for a Schwarzschild BH. In this method, the
corresponding extension of (5) immediately gives us the corresponding entropy
of the model under investigation.

In this article, we extend the aforementioned approach to a closed
Friedmann–Lemaître–Robertson–Walker (FLRW) universe with a positive
cosmological constant, in which the matter content is radiation. To achieve
this objective, our first task is to address the WDW equation of the model
universe to acquire the appropriate wavefunction. On the other hand, in order
to acquire a finite value for the spectrum of the cosmological constant, it is
imperative to employ an appropriate regularization technique that ensures a
finite value for the vacuum energy associated with the electromagnetic field.
As a result of the holographic regularization method, a discrete spectrum
emerges for the dS AH, mirroring the characteristics of the spectrum observed
for a BH as delineated in Eq. (4). Note that one can use other regularization
methods [15, 16] to keep vacuum energy finite. However, the benefit of
holographic regularization is that the classical universe emerged from the
quantum region as a universe without the horizon, flatness, singularity, and
cosmic coincidence problems [17].

The present paper is organized in the following manner: In Section 2, we delve
into a brief discussion regarding the WDW equation that governs the behaviour
of a closed FLRW universe in the presence of radiation as a matter field. Upon
establishing the fact that the dS AH radius undergoes quantization, we proceed
to Section 3, where we employ semiclassical arguments, to obtain the classical
emergent dS universe. In section 4 the Stefan–Boltzmann law, the time-energy
uncertainty relation, and the unified first law of thermodynamics utilized to
derive the corrected temperature and entropy of dS space emerged from quantum
cosmology. Lastly, Section 5 encompasses the conclusion that further
elucidates the findings and implications presented throughout this paper.

##  2 Quantum cosmology with holographic regularized vacuum energy

Consider a background minisuperspace, a spatially closed ($k=1$), homogeneous,
and isotropic metric. The line element is

$ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)h_{ij}dx^{i}dx^{j},$ (6)

in which $a(t)$ is the scale factor, $N(t)$ is the lapse function, and
$h_{ij}dx^{i}dx^{j}=d\chi^{2}+\sin^{2}(\chi)d\Omega_{(2)}^{2}$ is the line
element on the unit 3-sphere, $\mathbb{S}^{3}$.

Consider the action functional for gravity and a minimally coupled
electromagnetic field $F_{\mu\nu}$ with the standard form

$I=\frac{1}{16\pi
G}\displaystyle\int_{\mathcal{M}}(R-2\Lambda)\sqrt{-g}d^{4}x-\frac{1}{4}\displaystyle\int_{\mathcal{M}}F_{\mu\nu}F^{\mu\nu}\sqrt{-g}d^{4}x+S_{\text{GHY}},$
(7)

where $S_{\text{GHY}}$ is the Gibbons–Hawking–York boundary term, $g$ is the
determinant of the metric (6), and $\Lambda$ is the cosmological constant. By
Fourier expansions [15] of electromagnetic 4-vector fields $A_{\mu}$

$\begin{split}A_{0}(x,t)&=\sum_{j=0}^{\infty}\sum_{l=0}^{j}\sum_{m=-l}^{l}g_{jlm}(t)Y_{jlm}(x),\\\
A_{k}(x,t)&=\sum_{B=0}^{2}\sum_{j=j_{\text{min}}}^{\infty}\sum_{l=l_{\text{min}}}^{j}\sum_{m=-l}^{l}f^{jlm}_{(B)}(t)Y^{jlm}_{(B)k},~{}~{}~{}~{}k=1,2,3,\end{split}$
(8)

in terms of the scalar hyperspherical harmonics, $Y_{jlm}(x)$, and the vector
hyperspherical harmonics, $Y^{jlm}_{(B)k}$ into the Maxwell action functional
in the background (6), and choosing the Coulomb-type (or radiation) gauge
condition

$A_{0}=0,\hskip 14.22636pt^{(3)}\nabla^{i}A_{i}=0,$ (9)

we arrive at the Arnowitt–Deser–Misner (ADM) Lagrangian and Hamiltonian (which
may be found in [15] for computational details), respectively

$L_{\text{ADM}}=\frac{3\pi}{4G}\Big{(}-\frac{a\dot{a}^{2}}{N}+Na-\frac{\Lambda}{3}a^{3}\Big{)}+\frac{1}{2}\sum_{B=1}^{2}\sum_{j=1}^{\infty}\sum_{l=1}^{j}\sum_{m=-l}^{l}\Big{(}\frac{a}{{N}}\dot{f}^{2}_{(B)J}-\frac{N}{a}(j+1)^{2}f^{2}_{(B)J}\Big{)},$
(10)

$H_{\text{ADM}}=N\Big{\\{}-\frac{G}{3\pi
a}\Pi_{a}^{2}-\frac{3\pi}{4G}a+\frac{\pi\Lambda}{4G}a^{3}+\\\
\frac{1}{2a}\sum_{B=1}^{2}\sum_{j=j_{\text{min}}}^{\infty}\sum_{l=l_{\text{min}}}^{j}\sum_{m=-l}^{l}\left(\Pi_{(B)jlm}^{2}+(j+1)^{2}f_{(B)jlm}^{2}\right)\Big{\\}},$
(11)

where the expansion coefficients of $A_{i}$, $f^{jlm}_{(B)}(t)$, depend only
on the cosmic time $t$. Also, $\Pi_{a}=-\frac{3\pi}{2G}\frac{a\dot{a}}{N}$ and
$\Pi_{(B)J}=\frac{a}{N}\dot{f}_{(B)J}$ (we use compact notation
$J=\\{j,l,m\\}$) are the conjugate momenta of the scale factor, $a$, and
$f_{(B)J}$, respectively.

The ADM action of the model is given by

$I_{\text{ADM}}=\frac{1}{16\pi
G}\int_{t_{i}}^{t_{f}}\Big{\\{}\dot{a}\Pi_{a}+\sum_{B,J}\dot{f}_{(B)J}\Pi_{(B)J}-N\mathcal{H}\Big{\\}}dt,$
(12)

where $\mathcal{H}$ is the super-Hamiltonian, given by

$\mathcal{H}=-\frac{G}{3\pi
a}\Pi_{a}^{2}-\frac{3\pi}{4G}a+\frac{\pi\Lambda}{4G}a^{3}+\frac{1}{2a}\sum_{B,J}\left(\Pi_{(B)J}^{2}+(j+1)^{2}f_{(B)J}^{2}\right).$
(13)

Therefore, the corresponding phase space is the cotangent bundle

$\Omega=\\{a,f_{(B)J},N,\Pi_{a},\Pi_{(B)J},\Pi_{N}\\}.$ (14)

The ADM Lagrangian of the model (as all ADM Lagrangians in GR) is singular
because the conjugate momenta $\Pi_{N}$ weakly vanishes

$\Pi_{N}=\frac{\partial L_{\text{ADM}}}{\partial\dot{N}}\approx 0,$ (15)

meaning that the Legendre transformation

$\\{a,f_{(B)J},N,\dot{a},\dot{f}_{(B)J},\dot{N}\\}\rightarrow\\{a,f_{(B)J},N,\Pi_{a},\Pi_{(B)J},\Pi_{N}\\},$
(16)

is a surjective map and not an invertible map. Thus, we are working with a
constrained Hamiltonian system since the Legendre transform is non-invertible.
According to Dirac constraint theory, the fact that $\Pi_{N}$ is weakly vanish
means that N, the laps function, is a freely selectable and physically
irrelevant variable. Thus, the primary Hamiltonian of the model is
$H_{Pr}=H_{\text{ADM}}+\lambda\Pi_{N}$, where $\lambda$ is a Lagrange
multiplier. Now, we have

$\dot{\Pi}_{N}=\\{\Pi_{N},H_{Pr}\\}=\mathcal{H}.$ (17)

This demonstrates that the primary constraint is not preserved, necessitating
the imposition of the following secondary [11] constraint

$\mathcal{H}\approx 0.$ (18)

Utilizing the quantization map in the configuration space

$\begin{split}(\hat{a},\hat{\Pi}_{a})&=(a,-i\frac{\partial}{\partial a}),\\\
(\hat{f}_{(B)J},\hat{\Pi}_{(B)J})&=(f_{(B)J},-i\frac{\partial}{\partial{f_{(B),J}}}).\end{split}$
(19)

the super-Hamiltonian constraint (18) transforms to the WDW equation [15, 17],
where the super-Hamiltonian annihilate wavefunction

$\Bigg{\\{}\frac{1}{3\pi
aM_{\text{P}}^{2}}a^{q}\frac{d}{da}\left(a^{-q}\frac{d}{da}\right)+\frac{3\pi
M_{\text{P}}^{2}}{4}a-\frac{1}{2a}\sum_{BJ}\Big{(}-\partial_{(B)J}^{2}\\\
+(j+1)^{2}f_{(B)J}^{2}\Big{)}\Bigg{\\}}\Psi(a,f_{(B)J})=0,$ (20)

where $q$ is the Hartle–Hawking–Verlinde [18, 19, 20] factor ordering
parameter.

Note that the electromagnetic part of the above super-Hamiltonian,

$\mathcal{H}_{m}=\frac{1}{2a}\sum_{BJ}\left(-\partial_{(B)J}^{2}+(j+1)^{2}f_{(B)J}^{2}\right),$
(21)

represents the contribution of an infinite number of harmonic oscillators with
eigenfrequencies $\omega_{j}=(j+1)/a$.

If we rewrite the matter part of the super-Hamiltonian, instead of
$(f_{(B)J},\Pi_{(B)J})$, in terms of the creation, $\hat{C}^{\dagger}_{(B)J}$,
and annihilation, $\hat{C}^{\dagger}_{(B)J}$, operators

$\begin{split}\hat{C}_{(B)J}:=\frac{1}{\sqrt{2(j+1)}}\left(\partial_{(B)J}+(j+1)f_{(B)J}\right),\\\
\hat{C}^{\dagger}_{(B)J}:=\frac{1}{\sqrt{2(j+1)}}\left(-\partial_{(B)J}+(j+1)f_{(B)J}\right),\\\
\left[\hat{C}_{(B)J},\hat{C}^{\dagger}_{(B^{\prime})J^{\prime}}\right]=i\delta_{BB^{\prime}}\delta_{JJ^{\prime}},\end{split}$
(22)

then, (21) will be

$\mathcal{H}_{m}=\frac{1}{a}\displaystyle\sum_{B=1}^{2}\sum_{j=1}^{\infty}\sum_{l=1}^{j}\sum_{m=-l}^{l}(j+1)\left(\hat{N}_{(B)jlm}+\frac{1}{2}\right),$
(23)

where $\hat{N}_{(B)J}:=\hat{C}^{\dagger}_{(B)J}\hat{C}_{(B)J}$ are number
operators. According to the above Hamiltonian, the vacuum energy of the
electromagnetic radiation is

$\mathcal{H}_{m}^{\text{(vacuum)}}=\frac{1}{a}\sum_{j=1}^{\infty}j(j^{2}-1).$
(24)

All vacuum modes contribute to the zero-point energy, causing the sum in the
above relation to be divergent. Therefore, we must utilize a regularization
method to reduce it to a finite value. Inserting (24) into the WDW equation
(20) gives us the following WDW equation in the presence of the
electromagnetic vacuum energy

$\Bigg{\\{}-a^{q}\frac{d}{da}\left(a^{-q}\frac{d}{da}\right)+\left(\frac{3\pi}{2L_{P}^{2}}\right)^{2}a^{2}\left(1-\frac{a^{2}}{L_{\Lambda}^{2}}\right)-{3\pi
aM_{\text{P}}^{2}}\mathcal{H}_{m}^{\text{(vacuum)}}\Bigg{\\}}\psi(a)=0,$ (25)

The contribution of vacuum energy is boundless, posing a significant challenge
to comprehending and regulating it effectively in the WDW equation. However,
one promising approach to regularize this energy is through holographic
regularization, as recently formulated in Ref. [17]. This novel technique, in
summary, employs the Cohen–Kaplan–Nelson holographic bound [21], which
suggests that the UV cutoff of an effective quantum field theory in a box of
size $L$ should be low enough that states of characteristic energy density are
not BHs. Employing this bound the authors of [17] showed that there exist a
$j_{\text{max}}$ in which

$\sum_{j=1}^{j_{\text{max}}}j(j^{2}-1)\simeq\frac{j_{\text{max}}^{4}}{4}=\frac{1}{4}\left(\frac{L_{\Lambda}}{L_{\text{P}}}\right)^{2},$
(26)

as the cutoff on the wave number of the virtual photons. As per the findings
in [17], holographic vacuum energy regularization leads to the inception of a
classical universe at the Planck time, with a much larger scale factor than
that of the Planck length. Additionally, the emergence of such a universe can
take place without any singularity, horizon, or flatness issues. Notably, it
proposes a fresh, non-anthropic solution to the cosmic coincidence problem,
which could potentially aid in eliminating fine-tuning.

Inserting (24) into the WDW equation (25) conducts us to the scale factor part
of the WDW equation, which is expressed as

$-a^{q}\frac{d}{da}\left(a^{-q}\frac{d\psi}{da}\right)+\left(\frac{3\pi}{2L_{P}^{2}}\right)^{2}a^{2}\left(1-\frac{a^{2}}{L_{\Lambda}^{2}}\right)\psi(a)=\frac{L_{\Lambda}^{2}}{L_{P}^{4}}\psi(a).$
(27)

Figure 1: The ‘particle’s’ potential energy as a function of scale factor in
the WDW Eq. (28). The plot of $V(a)$ for $q>0$ or $q<-2$ is shown by the dot-
line. The potential energy plot for $-2<q<0$ is shown by the dashed line,
while the $V(a)$ plot for $q=0$ is represented by the red line. The dotted
line shows the harmonic potential term of $V(a)$.

Let us redefine the wavefunction as $\pi=a^{\frac{q}{2}}\Phi(a)$. Then WDW
equation (27) will reduce to

$\frac{1}{2M_{\text{P}}}\frac{d^{2}\Phi}{da^{2}}+(E-V(a))\Phi(a)=0,$ (28)

where realizes a Schrodinger equation of a ‘particle’ with Planck mass, and
the following potential energy, and energy, respectively

$V(a)=\frac{1}{2M_{\text{P}}}\left\\{\frac{\frac{q}{2}(\frac{q}{2}+1)}{a^{2}}+\left(\frac{3\pi}{2L^{2}_{\text{P}}}\right)^{2}a^{2}\left(1-\frac{a^{2}}{L_{\Lambda}^{2}}\right)\right\\},$
(29)
$E=\frac{1}{2}\left(\frac{L_{\Lambda}}{L_{\text{P}}}\right)^{2}M_{\text{P}}.$
(30)

The particle energy, $E$, is always less than the maximum of the potential
barrier

$V_{\text{max}}=\frac{9\pi^{2}}{32}\left(\frac{L_{\Lambda}}{L_{\text{P}}}\right)^{2}M_{\text{P}}=\left(\frac{3\pi}{4}\right)^{2}E.$
(31)

In addition, the classical turning points, see Fig. 1, are given by

$a_{\pm}=\sqrt{\frac{1}{2}\pm\frac{1}{2}\sqrt{1-\frac{16}{9\pi^{2}}}}L_{\Lambda}.$
(32)

It is straightforward to verify that at the turning point $a_{-}$, the ratio
of the anharmonic term to the harmonic term of the potential is equal to
$a_{-}^{2}/L_{\Lambda}^{2}\simeq 0.05$. This suggests that the anharmonic part
of the potential energy in (31) can be ignored in the area (I) of Fig. 1 and
in the semi-classical approximation. This is seen in Fig. 1, where the
potential energy coincides with the quadratic potential (indicated with dots)
up to the classical turning point for all values of the ordering parameter
$q$.

Ignoring the anharmonic term of potential in (29) in the region (I), the
solution of the WDW equation (28) is a confluent hypergeometric (Kummer)
function. Thus,

$\psi(a)=a^{\frac{q}{2}}\Phi(a)=Ce^{-\frac{3\pi
a^{2}}{4L_{P}^{2}}}a^{q+1}~{}_{1}F_{1}\left(-\frac{1}{6\pi}\left(\frac{L_{\Lambda}}{L_{P}}\right)^{2};\frac{q+3}{2};\frac{3\pi
a^{2}}{2L_{P}^{2}}\right),$ (33)

where $C$ is a constant of normalization.

In order for the wavefunction to be square-integrable, it is imperative that
the hypergeometric series ${}_{1}F_{1}(\alpha;\beta;\xi)$ be a finite
polynomial of degree $n$. This can be accomplished if and only if a non-
negative integer $n$ exists, such as $\alpha=-n$. In consequence, one arrives
at the quantization of the dS AH radius

$L_{\Lambda}=\sqrt{6\pi\left(n+\frac{3+q}{4}\right)}L_{\text{P}}\simeq\sqrt{6\pi
n}L_{\text{P}}.$ (34)

Since we are interested in semiclassical approximation ($n\gg 1$), we omitted
$(3+q)/4$. Refs. [22, 23] yields a similar conclusion for quantization of dS
radius; however, there, the coefficient of $n$ is $2$ rather than $6\pi$.

The energy scale of inflation is typically expected to be around
$10^{-6}M_{\text{P}}$. Therefore, $L_{\Lambda}\simeq 10^{6}L_{\text{P}}$,
which shows that $n\simeq 10^{11}$.

##  3 Emerged de Sitter universe

In the present paper, we focus on a semiclassical approximation, disregarding
operator ordering. Consequently, we set the ordering parameter $q$ to zero.
The WKB, Linde–Vilenkin or Hawking no-boundary, wavefunction of the one-
dimensional WDW equation mentioned above for $a\geq a_{+}$, (region III of
Fig. 1), is

$\psi(a)_{\pm}\propto\frac{1}{\sqrt{\Pi_{a}}}\exp\left\\{\pm
i\int_{a_{+}}^{a}\Pi_{a}da\mp i\frac{\pi}{4}\right\\},$ (35)

where $\Pi_{a}$ is the classical conjugate momenta of the scale factor,
obtained from the classical equation

$\Pi_{a}=-\sqrt{2M_{\text{P}}(E-V)}\simeq-\frac{L_{\Lambda}}{L_{\text{P}}^{2}}\sqrt{1+\left(\frac{3\pi}{2L_{\Lambda}^{2}}\right)^{2}a^{4}}.$
(36)

Note that in this relation we ignored the factor ordering and the harmonic
potential terms in potential (31) for $a\gg a_{+}$.

For the leading order in the WKB approximation, it is easy to verify that the
wavefunctions (35) are also the eigenvectors of the operator $\Pi_{a}$ [24]

$\hat{\Pi}_{a}\psi_{\pm}(a)=\pm\Pi_{a}\psi_{\pm}(a),$ (37)

Therefore, the wavefunction described above exhibits an oscillatory pattern,
indicating an infinitely expanding, $\psi_{-}(a)$, (also a contracting,
$\psi_{+}(a)$) and the flat universe. The definition of the conjugate momenta
of the scale factor $\Pi_{a}=-\frac{3\pi}{2G}\frac{a\dot{a}}{N}$ in (11), and
(36) result

$a^{2}\dot{a}^{2}=\left(\frac{2L_{\Lambda}}{3\pi}\right)^{2}\left\\{1+\left(\frac{3\pi}{2L_{\Lambda}}\right)^{2}a^{4}\right\\},$
(38)

leading to the Friedmann equation

$H^{2}=\frac{4}{3\pi^{2}\Lambda}\frac{1}{a^{4}}+\frac{\Lambda}{3},$ (39)

where we used the definition of $L_{\Lambda}=\sqrt{3/\Lambda}$.

It is worth noting that the Friedmann equation (39) shows that the classical
flat universe with radiation and cosmological constant emerged after the
quantum tunnelling of the universe for $a>a_{+}$. On the other hand, the same
equation shows that the ratio of the radiation density,
$\rho_{\gamma}=\frac{4}{3\pi^{2}\Lambda}\frac{1}{a^{4}}$, to the density
cosmological constant, $\rho_{\Lambda}=\frac{\Lambda}{8\pi G}$ at $a_{+}$ is

$\frac{\rho_{\gamma}}{\rho_{\Lambda}}\Big{|}_{a=a_{+}}=\frac{8}{9\pi^{2}}\frac{1}{1+\sqrt{1-\frac{16}{9\pi^{2}}}}\simeq
0.05.$ (40)

This means that the emerged universe is already cosmological constant
dominated just after tunnelling. Therefore, for $a>a_{+}$ (39) gives us the dS
spacetime $a=a_{0}\exp\left(\frac{t}{L_{\Lambda}}\right)$ with the Hubble
constant

$H=\frac{1}{L_{\Lambda}}=\sqrt{\frac{\Lambda}{3}}.$ (41)

The aforementioned Hubble parameter is responsible for the expansion of dS at
a later stage.

Note that in quantum cosmology, usually the assumption is tunnelling of the
Universe from ‘nothing’ into dS universe [25, 26, 27, 28]. Therefore, in these
approaches, we have only regions 2 and 3 of Fig. 1. However, in the model
studied here, we have a finite vacuum energy of the electromagnetic field,
which is proportional to the square of the dS radius. This energy generates
region I of Fig. 1.

Note that in quantum cosmology, the quantum universe is often assumed to
tunnel from “nothing” into the dS space. As a result, only regions II and III
of Fig. 1 are present in these models. On the other hand, the electromagnetic
field in the model under study has a finite vacuum energy that is proportional
to the square of the dS radius. The energy in question produces region I of
Fig. 1 and is responsible for the quantisation of $L_{\Lambda}$ in (34).

At this juncture, we are poised to synthesize our findings. This synthesis
will serve to elucidate our route in subsequent sections aimed at deriving the
temperature and entropy of dS spacetime. Our observations indicate that the
model universe tunnels through a potential barrier and emerges in a dS
universe, characterized by a quantized cosmological constant. Consequently, as
delineated in Eq. (34), the resultant system resembles a simplistic model akin
to the hydrogen atom. In the hydrogen atom, the interaction with
electromagnetic radiation leads to the spontaneous emission and absorption of
electromagnetic radiation. This process entails emitted radiation energy
equating the energy differential between the atom’s energy levels. It is
noteworthy that the atomic spectrum exhibits non-negligible width [29] and the
emitted radiation adheres to the Stefan–Boltzmann law when in a state of
equilibrium.

Similarly, we propose that the energy of emitted Hawking radiation from the
apparent horizon of dS space is equal to the Misner–Sharp–Hernandez (MSH)
energy difference of the simple quantized dS system between two states $n+1$
and $n$. This, in combination with the energy-time uncertainty relation, leads
us to calculate the time variation of MSH energy of dS spacetime due to
thermal radiation. Assuming the system is in thermodynamic equilibrium, the
emitted thermal radiation follows the Stefan-Boltzmann law, similar to the
hydrogen atom example. This assumption allows us to determine the dS
temperature. Finally, by using the unified first law of thermodynamics, we can
calculate the entropy of dS space.

##  4 de Sitter entropy

Numerous studies, including [30, 31] as well as other referenced works, have
extensively researched the relationship between geometry and thermodynamics in
FLRW spaces. In particular, researchers have modified thermodynamic equations
applicable to the dS event and AH for the non-static AH of FLRW space. This AH
is distinct from the event horizon, which may not even exist, and is often
considered a causal horizon linked to gravitational temperature, entropy, and
surface gravity in dynamic spacetimes. References such as [32, 33, 34] support
the argument that these properties also apply to cosmological horizons.

The dS horizon has thermal properties, including temperature and entropy,
similar to those of the Schwarzschild event horizon, which was discovered
through the utilization of Euclidean field theory techniques [35]. In their
study, Gibbons and Hawking [35] evaluated the thermal bath perceived by a
timelike geodesic observer in dS space while carrying a (scalar) particle
detector that is restricted to a small tube surrounding the observer’s
worldline.

Cai and Kim [31] originally showed the equivalence between the Friedmann
equation and the first law of thermodynamics on the AH in the framework of
general relativity, as well as in the Gauss–Bonnet and Lovelock gravity
theories. Later, by Akbar and Cai [36], this analysis was expanded to
incorporate $f(R)$ gravity and scalar-tensor gravity theories. The first law
of thermodynamics was therefore derived by the use of the Friedmann equations.
This connection between thermodynamics and cosmology has a significant impact
on our understanding of the universe [37, 38, 39].

In this part, let us review the Hamilton–Jacobi tunnelling proposition for BHs
and the FLRW model, a topic that is likely to captivate the audience. The
Hawking radiation computation on the FLRW AH was executed in the Refs. [40,
41]. The calculation was replicated by the authors of Refs. [39, 42] using the
Hamilton–Jacobi method within the Parikh–Wilczek framework, initially designed
for BH horizons. In this specific scenario, the emission rate of particles
under the WKB approximation signifies the probability of tunnelling for
trajectories that are classically prohibited, transitioning from the interior
to the exterior of the horizon. This rate of emission can be delineated as
$\Gamma\sim\exp{(-2\text{Im}(I))}\simeq\exp{(-\frac{\omega}{T})}$, where $I$
indicates the Euclidean action with an imaginary part $\text{Im}(I)$, $\omega$
denotes the angular frequency of the emitted particles, and the Hawking
temperature is deduced from the Boltzmann factor formula,
$T=\omega/2\text{Im}(I)$. The energy of the emitted particles, labelled as
$\omega$, is defined in a manner that remains unaltered as
$\omega=-K^{a}\nabla_{a}I$, where $K^{a}$ symbolizes the Kodama vector and the
action $I$ complies with the Hamilton–Jacobi equation. Although the definition
of energy is coordinate-invariant [43], it depends on the choice of time.

In the present article, we employ the irreducible energy (mass) through the
utilization of equation (34) as referenced in [8, 12] to determine the
temperature and entropy of the dS spacetime. This provides a simple and
consistent interpretation of entropy and energy for dS spacetime. The AH’s
volume of dS spacetime, $V_{\text{AH}}$, and the MSH mass, $M_{\text{AH}}$,
emanating [44]

$V_{\text{AH}}=\frac{4\pi}{3}L_{\Lambda}^{3},~{}~{}~{}M_{\text{AH}}^{(n)}=\frac{\Lambda}{8\pi
G}V_{\text{AH}}=\gamma\sqrt{n}M_{\text{P}},$ (42)

where, in obtaining the last equality, we used (34), and
$\gamma=\sqrt{\frac{3\pi}{2}}$. The above equation shows that the MSH mass of
dS spacetime is quantized, and its spectrum is identical to the BH mass given
by Eq. (4).

Let us assume that Hawking radiation is emitted from the dS horizon
spontaneously as the system transitions unpremeditatedly from state $n+1$ to
the nearest lower state, $n$. We can then refer to the frequency of the
resulting thermal radiation as

$\omega_{0}=M_{\text{AH}}^{(n+1)}-M_{\text{AH}}^{(n)}=\\\ \gamma
M_{\text{P}}\sum_{i=1}^{\infty}\binom{\frac{1}{2}}{i}\left(\frac{\gamma
M_{\text{P}}}{M_{\text{AH}}}\right)^{2i-1}\frac{\gamma^{2}M^{2}_{\text{P}}}{2M_{\text{AH}}}\left(1-\frac{\gamma^{2}M_{\text{P}}^{2}}{4M_{\text{AH}}^{2}}\right)+\mathcal{O}\left(\frac{M_{\text{P}}}{M_{\text{AH}}}\right)^{5}.$
(43)

This shows that dS space radiates with a characteristic temperature $T\propto
1/M_{\text{AH}}$, matching the dS Hawking temperature.

The lifetime of a quantum system in state $M^{(n+1)}$ until its decay into the
lower state $M^{(n)}$, known as the characteristic time, has been defined as
$\tau_{n}=\omega_{0}/\dot{M}_{\text{AH}}$, where a dot denotes time
derivative, and $\dot{M}_{\text{AH}}$ is the mass loss of the dS space due to
its evaporation. Due to the fluctuations of the vacuum of the quantum fields
in the vicinity of the AH, it is observed that the states’ width, denoted by
$W_{n}$, is not negligible. The estimation of the width of state $n$ can be
obtained through the utilization of established methods [8] and expressed as
$W_{n}=\beta\omega_{0}$, where $\beta$ is a numerical dimensionless factor.
Utilizing the uncertainty relation, $W_{n}\tau_{n}\simeq 1$, and the above
relations, one can easily find

$\dot{M}_{\text{AH}}=\frac{\beta\gamma^{4}}{4}\frac{M_{\text{P}}^{4}}{M^{2}_{\text{AH}}}\left(1-\frac{\gamma^{2}M_{\text{P}}^{2}}{8M_{\text{AH}}^{2}}\right).$
(44)

In the context of BH thermodynamics, Bekenstein [45] utilized the above
argument to draw a parallel between a BH and an atomic system by highlighting
the presence of a quantum number and its associated mass level, much like in
atomic structures. The phenomenon of photon emission bears resemblance to
atomic transitions, showcasing similarities in the underlying processes.
Bekenstein goes as far as delving into the discussion of Einstein coefficients
pertaining to the spontaneous emission and absorption of a Schwarzschild BH,
further solidifying the comparison to atomic behaviours. A common formula is
often applied to gauge the duration of the excited state of an atomic nucleus
in spectroscopy, mirroring the methods used in estimating aspects of BH
dynamics.

At this point, we utilize the Stefan–Boltzmann law to introduce an effective
temperature for AH. This serves as an extension of the concept applied in
establishing the temperature of a Schwarzschild BH [12, 13, 14]. If we make
the additional assumption that the source of the Hawking radiation arises from
the highly blue-shifted modes just inside the AH and simultaneously consider
the dS horizon as a blackbody, then the emitted power can be expressed using
the well-established Stefan–Boltzmann law

$\dot{M}_{\text{AH}}=\sigma A_{\text{AH}}T^{4},$ (45)

where $\sigma=\pi^{2}/60$ is the Stefan–Boltzmann constant, and
$A_{\text{AH}}=4\pi L_{\Lambda}^{2}$ is the surface area of the AH. Equating
the mass loss terms from Eqs. (44) and (45), and selecting
$\beta=\frac{\sigma}{\pi\gamma^{2}}=\frac{2}{45}$, we obtain the dS
temperature

$T_{\text{AH}}=\frac{1}{2\pi
L_{\Lambda}}\left(1-\frac{\gamma^{2}L_{\text{P}}^{2}}{2L_{\Lambda}^{2}}\right).$
(46)

It should be noted that as the authors of [46] demonstrated, the temperature
(46) is only well defined in the cosmic rest frame, in which the background
acts as a physical heat bath whose energy density obeys the Stefan–Boltzmann
law. This justifies the application of (45) for the AH of dS space.
Furthermore, the Stefan–Boltzmann law may be used to derive the
thermodynamical characteristics of a variety of BHs, as demonstrated by Refs.
[47, 14, 13, 48, 49] and others.

In this article, we employ the first law of thermodynamics to obtain the
entropy of AH. The first law of thermodynamics for AH of FLRW universe, under
the name of ‘unified first law,’ [44] is given by

$T_{\text{AH}}\dot{S}_{\text{AH}}=\dot{M}_{\text{AH}}+\frac{p-\rho}{2}\dot{V}_{\text{AH}},$
(47)

where $S_{\text{AH}}$ is the entropy of dS AH, and
$p=-\rho=\frac{\Lambda}{8\pi G}$ represent the pressure and the energy density
of the vacuum in our study. Generally, $\rho$ and $p$ are the energy density
and the pressure of the cosmic fluid. It is important to note that the
equation mentioned above represents the differential form of the Friedmann
equation [36]. Although the expansion law varies in different gravity theories
for both flat and non-flat universes as demonstrated in Refs. [50, 51], the
unified first law remains consistent in any gravity theory. Substituting the
mass and volume change rates–both of which are intrinsically negative–into the
unified first law,
$\dot{M}_{\text{AH}}=-\frac{\dot{L}_{\text{AH}}}{2L_{\text{P}}^{2}}$,
$\dot{V}_{\text{AH}}=-4\pi L^{2}_{\text{AH}}\dot{L}_{\text{AH}}$ into unified
first law (47) gives us the entropy of the AH

$S_{\text{AH}}=S_{\text{dS}}-\frac{5\pi\gamma^{2}}{2}\ln(S_{\text{dS}})+\text{const.},$
(48)

where $S_{\text{dS}}=\pi L_{\Lambda}^{2}/L_{\text{P}}^{2}=A_{\text{AH}}/(4G)$
is the well-known entropy of the dS horizon.

This shows that the entropy is quantized evenly, with the spacing between the
entropy spectrum given by $\Delta S=6\pi^{2}$. Furthermore, the reading of
$\Delta A_{\text{AH}}=24\pi^{2}$ reveals the quantum of the horizon area. The
formula for entropy (48) resembles the Bekenstein–Hawking entropy formulation
for BHs [52]. However, it is important to note that in this particular
instance, the horizon’s properties depend on the observer, leading to
ambiguities with regard to which BH concepts can be extended to dS space.

It should be noted that the same leading order logarithmic correction term has
been obtained using other methods, as evidenced by Refs. [53, 54]. In Ref.
[53], the authors showed that by applying the generalized uncertainty
principle (GUP) to the AH of dS space, one can obtain the logarithmic term
with a positive coefficient $\pi\alpha/4$, where $\alpha$ is a dimensionless
constant realizing GUP. In Ref. [54], by applying the tunnelling method, the
author obtained the logarithmic term with a positive coefficient
$4\pi\alpha_{1}/3$, where $\alpha_{1}$ is an unknown constant. On the other
hand, the authors of Ref. [55] find the same correction term with a negative
unknown coefficient, by arguing that the entropy of dS space corresponds to
the entanglement between disconnected regions. Our research has shown that the
coefficient of this term is determined by the dimensionless positive constant
$\gamma=\sqrt{3\pi/2}$, which is, in turn, dependent on the regularization
technique utilized in Eq. (26). Furthermore, incorporating higher-order
correction terms from expansion (43) leads to additional entropy terms
proportional to the inverse powers of $S_{\text{dS}}$. These extra terms and
the logarithmic term are typically viewed as quantum correction terms for the
semiclassical analysis [52]. The logarithmic correction completely
predominates over other corrections in the large limit of the dS radius.
Furthermore, since the logarithmic entropy correction is inextricably linked
to the structure of all quantum gravity models, regardless of the methods
used, it appears to be universal.

##  5 Conclusion

In this investigation, we conducted a rigorous thermodynamic analysis of dS
space, particularly emphasizing its AH. Employing the unified first law of
thermodynamics as our analytical framework, we derived an entropy formula for
the dS AH incorporating quantum corrections. Our finding of a logarithmic
correction term enriches the complexity of dS AH thermodynamics and challenges
the conventional understanding. Significantly, we unveiled that the entropy of
dS AH exhibits quantized spectral spacing, a characteristic reminiscent of
black hole entropy. However, it’s crucial to acknowledge the observer-
dependent attributes of the dS horizon when attempting to extend black hole
thermodynamic concepts to dS spaces. The implications of our research are
manifold: they deepen the scientific comprehension of dS space thermodynamics,
open avenues for exploring the intriguing but nuanced relationship between dS
space and black holes, and lay a foundation for potential advancements in
cosmological applications.

S.J. acknowledges financial support from the National Council for Scientific
and Technological Development – CNPq, Grant no. 308131/2022-3.

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11institutetext:  Technology and Innovation Management, Siemens Healthineers,
Erlangen, Germany
Pattern Recognition Lab, FAU Erlangen-Nürnberg, Erlangen, Germany

# Effect of Random Histogram Equalization on Breast Calcification Analysis
Using Deep Learning

AdarshBhandary Panambur 1122 PrathmeshMadhu 22 AndreasMaier 221122
<EMAIL_ADDRESS>
###### Abstract

Early detection and analysis of calcifications in mammogram images is crucial
in a breast cancer diagnosis workflow. Management of calcifications that
require immediate follow-up and further analyzing its benignancy or malignancy
can result in a better prognosis. Recent studies have shown that deep
learning-based algorithms can learn robust representations to analyze
suspicious calcifications in mammography. In this work, we demonstrate that
randomly equalizing the histograms of calcification patches as a data
augmentation technique can significantly improve the classification
performance for analyzing suspicious calcifications. We validate our approach
by using the CBIS-DDSM dataset for two classification tasks. The results on
both the tasks show that the proposed methodology gains more than 1% mean
accuracy and F1-score when equalizing the data with a probability of 0.4 when
compared to not using histogram equalization. This is further supported by the
t-tests, where we obtain a _p_ -value of _p_ <0.0001, thus showing the
statistical significance of our approach.

##  1 Introduction

With an estimated 2.26 million new cases in 2020, breast cancer is the leading
cause of malignancy incidence worldwide [1]. Periodic screening of women based
on their age groups using mammography has proven to reduce the late-stage
incidence and mortality rates [1]. Breast Imaging Reporting and Database
System (BI-RADS) is a scoring system used by radiologists to assign categories
to the suspicious region of interest on screening mammography based on the
type, density, morphology, and distribution of the findings [2]. In addition
to observing essential findings such as mass, density, architectural
distortions, and asymmetries in the breast, the clinicians also search for
bright well-circumscribed findings known as calcification. Calcifications are
calcium deposits, generally found inside the ducts, lobules, connective stroma
tissue, and vessels of the breast [3]. The analysis of calcification on
mammograms is of high importance as its morphology and distribution can be
representative of precancerous or malignant cells [3]. As the sensitivity of
calcification detection during the initial mammography screening is low [4],
computer-aided diagnosis (CAD) methods can potentially support clinicians for
better decision making for early detection and analysis of calcifications.
Deep learning (DL)-based CAD algorithms have proven to provide effective and
robust solutions for automated breast cancer analysis in mammography [5], with
many recent approaches showing promising results in calcification analysis
tasks [6, 7].

Contrast enhancement techniques have been vastly studied as a pre-processing
image technique in whole mammogram image analysis [8]. However, applying these
techniques to smaller calcification patches might lead to a loss of class-
specific information. This is evident due to the appearance of calcification
which generally appears as a bright spot or as a group of scattered bright
spots, and equalizing the histograms of such patches might lead to noisy
images. Adding noise to the input data is one of the data augmentation
techniques used for training DL networks [9]. Data augmentation is the most
commonly used strategy in DL to increase the quality of the training data by
applying transformations to the input enforcing the network to learn
meaningful, class-specific representations [10]. In this work, we investigate
the usage of random histogram equalization (RHE) as a data augmentation
technique for calcification analysis. Histogram equalization (HE) is a
contrast enhancement technique that non-linearly scales the input to a uniform
distribution of intensities based on the image’s histogram.

The main contributions of our research are: a) We present that using RHE as a
data augmentation technique with a probability value of 0.4 can lead to a
significant performance gain compared to not using RHE; b) We also present and
compare Gradient-weighted Class Activation Mapping (Grad-CAM) [10] for models
that were trained using no HE and full HE to show that RHE outperforms both of
them in localizing class-specific calcification features.

The remaining organization of the paper is as follows: Section 2 provides a
brief description of the data and methods used in our work. Next, we present
the quantitative and qualitative results for two and three-class
classification tasks in Section 3. Finally, in Section 4, we provide a brief
discussion on the results.

##  2 Materials and methods

###  2.1 Data

Curated Breast Imaging Subset of Digital Database for Screening Mammography
(CBIS-DDSM) is a public dataset consisting of scanned film mammography from
multiple institutions across the United States of America [11]. In this
research, we utilize a small subset of the dataset consisting of cropped
regions of calcifications observed on craniocaudal and/or mediolateral oblique
views in the mammogram images.

| Follow-up | No follow-up | Malignant | Benign | Benign w/o callback
---|---|---|---|---|---
Training | 855 | 382 | 445 | 410 | 382
Validation | 217 | 92 | 99 | 118 | 92
Test | 259 | 67 | 129 | 130 | 67
Total | 1331 | 541 | 673 | 658 | 541
Table 1: The distribution of class samples in the CBIS-DDSM dataset for the
two-class (Follow-up versus No follow-up) and three-class problems (Malignant,
benign and benign without callback).

In Table 1, we show the distribution of the class samples in the training,
validation, and test dataset. The dataset consists of 1,546 images divided
into training and validation datasets with an 80:20 ratio. In addition, we use
an independent test dataset from CBIS-DDSM with 326 images as the test
dataset. The verified pathology information available in this dataset is used
as the labels for the two classification tasks. The first task involves the
binary classification of Follow-up versus No follow-up calcification patches.
The calcifications labeled as malignant and benign are used for the Follow-up
class as these patches would require additional clinical investigations. No
follow-up class consists of benign without callback calcifications indicating
the lesions are worth tracking but does not require further investigations.
Finally, we use the original malignant , benign , and benign without callback
labels for the three-class classification.

###  2.2 Experimental setup

Based on our initial experiments with various state-of-the-art convolutional
neural networks (CNN), we chose a pre-trained ResNet50 [12] as the standard
CNN architecture for all the experiments conducted in this research. The final
fully connected layer (head) in the CNN is refactored to represent the number
of classes, i.e., 2 and 3, respectively. The input images are resized with
bilinear interpolation into 224 x 224 pixels, and the pixel values are
normalized to the range _[0,1]_. Four fixed data augmentations are used to
enhance the quality and size of the training data. Random horizontal flip,
random vertical flip, random rotations are used with a probability value of
0.5, and random erasing is used with a probability value of 0.1. The
probability value indicates the possibility of the current image being
transformed. As this research aims to investigate the effect of RHE as a data
augmentation technique in calcification analysis, different experiments with
probability values (_P_) of 0, 0.2, 0.4, 0.6, 0.8, and 1 are conducted. All
the CNNs are trained for 30 epochs with a batch size of 16. A weighted binary
cross-entropy loss function is optimized using a standard Adam optimizer with
a learning rate of $3.2e^{-6}$ and a weight decay of $1e^{-4}$.

Accuracy and F1-score are the two performance evaluation metrics used for the
classification task. We further utilize a two-tailed unpaired t-test in order
to determine the statistical significance of the classification performances.
Finally, the qualitative performance of the CNN is analyzed by localizing the
regions in the calcification patch that influenced the model’s decision by
using the Grad-CAM approach [10]. The entire DL pipeline was developed using
PyTorch.

##  3 Results

Table 2: Classification performance of the trained CNN on the test dataset. Different values of _P_ indicate the probability of RHE on a patch during training. The best results are in _italics_. | Two-Class Task | Three-Class Task
---|---|---
Accuracy | F1-Score | Accuracy | F1-Score
_P_ = 0 | 0.9215 (0.0066) | 0.8838 (0.0078) | 0.6632 (0.0126) | 0.6855 (0.0131)
_P_ = 0.2 | 0.9159 (0.0117) | 0.8759 (0.0117) | 0.6755 (0.0191) | 0.6985 (0.0221)
_P_ = 0.4 | _0.9325 (0.0085)_ | _0.8973 (0.0124)_ | _0.6840 (0.0193)_ | _0.7071 (0.0189)_
_P_ = 0.6 | 0.9196 (0.0092) | 0.8770 (0.0119) | 0.6687 (0.0058) | 0.6969 (0.0036)
_P_ = 0.8 | 0.9252 (0.0074) | 0.8869 (0.0109) | 0.6601 (0.015) | 0.6839 (0.0159)
_P_ = 1 | 0.8724 (0.0179) | 0.8103 (0.0111) | 0.6252 (0.017) | 0.6322 (0.0228)

In Table 2, we summarize the performance of the trained CNNs on the test
dataset for both the two-class and three-class classification tasks. The
different values of _P_ in the results denote the probability of a
calcification patch being equalized during the training (Tab. 2). The
probability value of 0 (_P_ =0) indicates that no HE is used during training
or testing, whereas a value of 1 (_P_ =1) indicates all the images are
equalized during both training and testing phases. For all the other
intermediate values (_P_ =0.2, 0.4, 0.6, 0.8), RHE is used only during
training. Each experiment for each probability value was run five times to
ensure stability in the final reported results. The mean accuracy and F1-score
with the standard deviation over five validation runs are reported on the test
dataset. In Figure 1, the Grad-CAM visualizations comparing the network
outputs for best-performing models for _P_ =0 and _P_ =0.4 are shown for both
the tasks. Figure 2 shows the Grad-CAM visualizations when all the input
training calcification patches are histogram equalized.

##  4 Discussion

As observed in the results summarized in Table 2, the network learns better
representations for both the classification tasks using RHE during training
with a probability of 0.4 compared to not using HE (_P_ =0). For the two-class
task, i.e., Follow-up vs. No follow-up, we achieve a mean accuracy and
F1-score of 0.9215 and 0.8838, respectively, on the test dataset, when no HE
is applied during training. When we increase the probability to a value of
0.4, we reach a mean accuracy of 0.9325 and a mean F1-score of 0.8973. A
performance boost of more than 1% is observed in both the mean accuracy and
F1-score. The results are then tested for statistical significance of accuracy
and F1-score using a t-test, where a two-tailed _p_ -value of _p_ <0.0001 is
obtained. This clearly shows that using RHE with a probability of 0.4 results
in a statistically significant performance. A similar trend is observed in
malignant, benign, and benign without callback classification. In this case,
we achieve more than 2% gain in both the metrics by using the probability
value of 0.4, with mean accuracy increasing from 0.6632 to 0.6840 and the
F1-score increasing from 0.6855 to 0.7071. A _p_ -value of _p_ <0.0001 is
obtained, showing a statistically significant classification performance
during the t-test.

Figure 1: Input image with true class label, Grad-CAM image for _P_ =0 and _P_
=0.4 with the predicted class labels. The localized regions responsible for
misclassifications can be observed while using _P_ =0. However, using _P_
=0.4, the network is able to learn robust class-discriminative features.

(a) Malignant, Benign, Malignant.

(b) Follow-up, No follow-up, Follow-up.

Figure 2: Input image with true class label, histogram equalized image and the
Grad-CAM image with the predicted class label for _P_ =1. The network is not
able to extract the class-relevant features even though the it is able to
localize the suspicious region of calcification.

(a) Malignant, Benign.

(b) Follow-up, No follow-up.

The increase in performance can be attributed to the randomly injected noise
in the form of RHE during training, forcing the network to concentrate on the
calcification regions rather than the surrounding anatomical structures. This
behavior can be observed using Grad-CAM visualizations in Figure 1. The models
trained with random probability of _P_ =0.4 are inherently more robust and
tend to focus more towards the high-intensity spots indicative of
calcifications. The results in Table 2 also show a significant reduction in
the performance while using _P_ =1, i.e., equalizing the histograms of all
images during training and testing. This is due to the enhancement of the
contrast in the brighter parts of the image leading to corruption of the
class-specific features in the data. In Figure 2, we can observe that the
essential features such as the morphology and the distribution of the
calcifications are corrupted due to the equalization of the histogram values
in the image. Even though the attention of the network is focused towards the
region of calcifications, we observe misclassifications.

As the current work involves calcification patches detected on scanned film
mammography, we plan to validate the proposed technique using digital
mammography images as a part of future work. None of the research involving
patch-level calcification analysis has exclusively explored the effect of data
augmentations on model performance. We aim to perform additional ablation
studies to examine the impact of other data augmentations techniques on
calcification analysis.

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