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# Smart obervation method with wide field small aperture telescopes for real
time transient detection

Peng Jia  College of Physics and Optoelectronics, Taiyuan University of
Technology, Taiyuan, 030024, China  Department of Physics, Durham University,
South Road, Durham DH1 3LE, UK  Qiang Liu  College of Physics and
Optoelectronics, Taiyuan University of Technology, Taiyuan, 030024, China
Yongyang Sun  College of Physics and Optoelectronics, Taiyuan University of
Technology, Taiyuan, 030024, China  Yitian Zheng  College of Physics and
Optoelectronics, Taiyuan University of Technology, Taiyuan, 030024, China
Wenbo Liu  College of Physics and Optoelectronics, Taiyuan University of
Technology, Taiyuan, 030024, China  Yifei Zhao  College of Physics and
Optoelectronics, Taiyuan University of Technology, Taiyuan, 030024, China

###### Abstract

Wide field small aperture telescopes (WFSATs) are commonly used for fast sky
survey. Telescope arrays composed by several WFSATs are capable to scan sky
several times per night. Huge amount of data would be obtained by them and
these data need to be processed immediately. In this paper, we propose ARGUS
(Astronomical taRGets detection framework for Unified telescopes) for real-
time transit detection. The ARGUS uses a deep learning based astronomical
detection algorithm implemented in embedded devices in each WFSATs to detect
astronomical targets. The position and probability of a detection being an
astronomical targets will be sent to a trained ensemble learning algorithm to
output information of celestial sources. After matching these sources with
star catalog, ARGUS will directly output type and positions of transient
candidates. We use simulated data to test the performance of ARGUS and find
that ARGUS can increase the performance of WFSATs in transient detection tasks
robustly.

###### keywords:

wide field small aperture telescopes, transient detection, embedded device,
ensemble learning, telescope arrays, time domain astronomy, deep learning

##  1 INTRODUCTION

Discovery of transients is an important scientific aim for time domain
astronomy. Because transients could appear in any positions of the sky with
different variation rates and different magnitudes, sky coverage, observation
cadence and depth are all important for discovery of transients. However,
these requirements often contradict with each other and according to different
scientific observation requirements, we should make trade off between them to
design different telescopes with different observation strategies. Wide field
small aperture telescope (WFSAT) is a kind of telescope that can obtain images
with a large field of view in a cost effective way [1, 2, 3, 4, 5, 6, 7].
WFSATs are widely used to detect bright transits for fast sky survey. Besides,
because WFSATs are low cost, it would be possible to build a telescope array
with several WFSATs to scan sky continuously, which can further increase sky
coverage and observation cadence [8, 9, 10, 11, 12].

During observations, huge amount of observation data are obtained by WFSATs
each night. To unleash the observation ability of WFSATs and provide
information for some very important targets (such as electromagnetic
counterparts of gravitational wave, supernova, tidal disruption events or
asteroids), these data should be processed immediately to provide necessary
information for follow–up observations. Besides as large amount of transient
candidates would be detected by WFSATs each night and there are only limited
number of telescopes with large aperture to follow up these candidates, the
accuracy of detection is important. Processing large amount of observation
data to output information of transients with fast speed and high accuracy is
beyond the capacity of contemporary method. New data processing method which
do not need manual intervention and can output information of transient
candidates with high accuracy and fast speed is required.

To satisfy data processing requirements of WFSATs, a lot of smart data
processing methods have been proposed in recent years [13, 14, 15, 16, 17, 18,
19, 20, 21]. These smart data processing methods use deep learning algorithms
to detect or classify transient targets from observed images. In real
applications, these algorithms have shown superior performance than
traditional methods: they could automatically obtain positions and types of
transients. For example, in the task of detecting different types of
astronomical targets for fast sky survey, a faster–rcnn based astronomical
target detection algorithm has better performance than that of classic methods
[22].

However, although smart data processing algorithms can improve the detection
abilities of WFSATs, there are still gaps that need to be fulfilled. As we can
notice that contemporary transient detection algorithms are post-processing
methods, which means they process observed images afterwards. Therefore, some
dim astronomical targets that would be seen by WFSATs in telescope arrays
several times, but due to their low signal to noise ratio, they would not be
detected by telescope arrays. Some suspicious transients could be confirmed as
soon as they are detected by one of WFSATs. Detection astronomical targets
from stacked images is a possible method to locate transients with low signal
to noise ratio, but WFSATs in a telescope array may have different detection
abilities, which makes it hard to design robust image stacking methods.
Besides, stacked images would reduce detection abilities for transients with
fast variations.

In this paper, we report a new concept transient detection framework for
WFSATs in telescope arrays (Astronomical taRGets detection framework for
Unified telescopes – ARGUS). In ARGUS, each WFSAT in a telescope array will be
equipped with a embedded device. An astronomical target detection neural
network will be deployed in the embedded device for real–time astronomical
target detection. Detection results will be sent back to data center. A
classifier based on AdaBoost algorithm will be used to merge these results.
Finally, the ARGUS will be able to output detection results with high accuracy
for follow–up observations directly. We will report the basic structure of the
ARGUS in Section 2 and give our preliminary results in Section 3. We will make
our conclusions in Section 4.

##  2 SMART TRANSIENT DETECTION FRAMEWORK FOR WFSATS

The ARGUS includes two parts: the remote transient detection part and the data
center that are used to receive and process detection results. The structure
of the ARGUS is shown in figure 1. We will describe each part in this Section.

---
Figure 1:  The structure of the ARGUS for real applications. It includes two
different parts that are in blue and red boxes. The remote part which includes
the telescope with the embedded device is used to obtain coordinates and
confidence of different astronomical targets from each WFSATs. The detection
result will be sent to the data center part (red box). The data center will
merge detection results and cross-match the detection results with celestial
catalog. Then the ARGUS will send out transient alerts to follow-up
telescopes.

The remote part in the ARGUS stands for the detection algorithm and the
embedded device that are installed at each WFSATs. In this paper, we use
NVIDIA Jetson AGX Xavier as the embedded device, because it has 512 tensor
cores which can accelerate DNNs and its power cost (30 Watts) is low and its
size ($105mm\times 105mm$) is small. A faster-rcnn based real-time
astronomical target detection neural network is deployed in the embedded
device [22]. Meanwhile, the detection algorithm is also being developed in a
FPGA device now (F10A provided by Inspur). We have modified the structure of
the faster-rcnn to make it suitable for detection of astronomical targets in
WFSATs: we introduces Resnet-50 as backbone network and a feature pyramid
network to extract features. The detection neural network is trained in double
float accuracy in a workstation with GTX 1080 Ti Graphic Card. After training,
the neural network is directly deployed in Jetson AGX Xavier also with double
float accuracy. The neural network could achieve 0.5 frames per second for
images with $300\times 300$ pixels. The processing speed can further be
increased if we use TensorRT to implement the detection deep neural network in
the Jetson AGX Xavier and we will discuss it in our later papers.

The faster-rcnn detection deep neural network is a stand-along astronomical
target detection framework. In the last layer of a detection algorithm, there
is a classification function to output positions and types of astronomical
targets. One frame of detection images is shown in figure 2. Targets with
green bounding box stand for stars and targets with red bounding box stand for
moving targets. As shown in this figure, the detection algorithm in the remote
part of the ARGUS can automatically finish source extraction and deblending
tasks. Besides, the detection algorithm has better performance than classic
methods as we tested with real observation data[22]. For some astronomical
targets with low signal to noise ratio, the faster-rcnn would output results
with low confidence. Since astronomical targets with low signal to noise ratio
would be seen by WFSATs in a telescope array several times, it is
straightforward to think that whether we can use detection results from
several telescopes to increase detection ability of telescope arrays?

---
Figure 2:  The detection results for one frame images that are obtained by the
remote part of the ARGUS in real applications.

Therefore in the ARGUS, we try to merge detection results from each WFSATs and
we have removed the classification layer and the neural network direct outputs
the bounding box with confidence value in the format:
$(x_{1},y_{1},x_{2},y_{2},conf)$, where $x_{1},y_{1}$ stand for position of
the bottom-left pixel, $x_{2},y_{2}$ stand for position of the upper-right
pixel and $conf$ stands for the confidence of an object being a specific
target. Directly merge the detection results from several telescopes with
bootstrap aggregation (average the confidence of a detection from different
telescopes for a target) is a possible way [23]. But because image qualities
will change and will introduce different point spread functions (PSF),
detection results would be unstable and can affect final results. Besides,
because there are a lot of WFSATs, we would be also interested in the
performance of each WFSATs for astronomical targets detection tasks which
could guide us to maintain instruments and optimize our algorithms.

To meet our requirements, we propose a dynamical classification framework in
the ARGUS to merge detection results of each WFSATs. Our method is based on
the adaptive boosting (AdaBoost) algorithm. The AdaBoost algorithm is used to
increase the performance of a classifier through combining several weak
learners [24]. Although individual learners is weak, the performance of the
final classification model would be better, if weak learners have better
performance than random guess. Based on the concept of the AdaBoost algorithm,
detection algorithms from different WFSATs are assumed as a weak classifier
and the ARGUS would merge these results to output detection results with high
accuracy.

In the center part, the ARGUS includes two stages: calibration stage and
implication stage. In the calibration stage, all WFSATs will observe the same
sky area and their detection results will be sent to the data center. All
detection results will be calibrated by the celestial catalog. Then we will
modify weights of classification algorithms from each WFSATs according to the
matching results between detection results and celestial catalog. The AdaBoost
algorithm defined in the scikit-learn is used in this paper [25]. In the
implementation stage, we will directly use weights to modify the detection
results to output celestial objects. As weights of each classifier are
directly related to the performance of WFSATs, WFSATs with extremely low
weights will be further checked by human experts for possible malfunctions.
After classification, the ARGUS will further cross-match coordinates of
celestial objects with the star catalog to check for transient candidates. All
celestial objects detected by the ARGUS that can not be matched by any targets
in star catalog will be labeled as transient candidates.

##  3 PRELIMINARY RESULTS OF THE ARGUS

In this paper, we simulate a telescope array with three WFSATs and a large sky
survey telescope with the skymaker [26] to test the ARGUS. The detail
information of them is shown in table 1. Three telescopes (Tel2, Tel3 and
Tel4) are WFSATs with diameter of 1 meter and they represent a small part of
the telescope array – Sitian [12]. We set different parameters for these
telescopes to test whether the ARGUS can find telescopes with best or worst
performance through weights of each telescope in the data center. The first
telescope (Tel1) represents the Simonyi Survey Telescope which will be used
for Legacy Survey of Space and Time (LSST) [27]. Images of the same sky area
obtained by these four telescopes are shown in figure 4. The gray scale of
these figures has been stretched by log transformation.

---
Figure 3:  Images of the same sky area obtained by four telescopes. Gray
scales of all images have been stretched by log transformation.

We use two scenarios to test the performance of the ARGUS. The first scenario
is that we will use the ARGUS to merge data from both the large telescope and
that from three WFSATs. The second scenario is that we will compare the
performance of the ARGUS with 3 WFSATs in a telescope array and the
performance of the large telescope. Four DNN based celestial objects detection
algorithms are trained in this part. We simulate observation data of each
telescopes and train each DNN model with the simulated data.

Table 1: Detailed information of WFSATs used in this paper. The Seidel aberrations include defocus, spherical aberration, coma in x and y directions, astigmatism in x and y directions. Telescopes | Tel1 | Tel2 | Tel3 | Tel4
---|---|---|---|---
Diameter (meter) | 8.4 | 1.0 | 1.0 | 1.0
Seeing FWHM (arcsec) | 0.7 | 0.5 | 0.9 | 1.2
Background (mag) | 24 | 23 | 23 | 25
Pixel Scale(arcsec) | 0.2 | 0.2 | 0.2 | 0.2
Static Seidel Aberration | 0, 0.2, 0.15 | 0.1, 0.3, 0.1 | 0.2, 0.1, 0.02 | 0, 0.2, 0.1
(in waves) | 0.1, 0.2, 0.1 | 0.05, 0, 0 | 0.05, 0.1, 0 | 0.05, 0.1, 0

In the first scenario, we merge detection results from all 4 telescopes with
the ARGUS. Because the accuracy and precision are both important for transient
detection, we use f1 score ($f1=2*(precision*accuracy)/(precision+accuracy)$)
to evaluate the performance of the detection algorithms. The detection ability
will be better, if the f1 score is closer to 1. The f1 scores of the ARGUS and
that of the detection results from different telescopes are shown in figure 4.
From these results we can find that the ARGUS could increase detection ability
of telescope arrays that are composed by different telescopes. The detection
results of the ARGUS are better than most telescopes (Tel1, Tel3 and Tel4).
For very dim targets, its performance is slightly worse that Tel2, since it
has very small FWHM (0.5 arcsec).

Besides, the weights of different telescopes in the ARGUS reflect their
detection abilities. After training of the Adaboost algorithm in the ARGUS,
the weights for each telescope are: 1.198, 1.106, 1.001, 1.018. The large
telescope (Tel1) has the largest weight, because it has very large aperture to
capture enough photons. The second telescope (Tel2) has the second largest
weight, because it has very small FWHM to make it better in detection of dim
targets. Because FWHM is defined by observation conditions of different
telescopes and we can find that WFSATs with smaller FWHM have better detection
abilities, it indicates us that if we place WFSATs in different regions, it
would be possible to compose a telescope array that has stronger and more
stable transient detection ability than any telescopes in the same region.
Therefore, we test the performance of the ARGUS with three WFSATs and that of
the large telescope.

---
Figure 4:  Detection results of the ARGUS with four telescopes for celestial
objects with different magnitudes. From this figure, we can find that the Tel2
has the best detection ability for dim targets ($f2\\_score\\_2$), while the
detection ability of large telescope is affected by the larger FWHM induced by
the atmospheric turbulence ($f1\\_score\\_2$). The ARGUS have merged all
detection results and have better performance than most WFSATs
($f1\\_score\\_ensemble$).

In the second scenario, we use the ARGUS to process images obtained from Tel2,
Tel3 and Tel4 and compare detection results with that of the large telescope.
The detection results are shown in figure 5. We can find that the ARGUS has
actively connected three WFSATs and achieves better detection ability than any
WFSATs in the telescope array. It indicates us that the detection ability of
the telescope array becomes better and more stable with the ARGUS. Considering
WFSATs in a telescope array that distribute in different regions, the ARGUS
makes the detection ability of a telescope array distributed in different
regions have more chance to get better detection results than a telescope in
the same region [28].

---
Figure 5:  Detection results of the ARGUS obtained by three 1 meter telescopes
for celestial objects with different magnitudes. From this figure, we can find
that the ARGUS has better and more stable detection ability than that of any
WFSATs ($f1\\_score\\_ensemble$ is larger than the best detection results of
all WFSATs – $f1\\_score\\_234\\_high$). With only three telescopes, the ARGUS
have better performance than that of a large telescope in detection of
transients ($f1\\_score\\_1$).

##  4 CONCLUSIONS AND FUTURE WORK

In this paper, we propose the ARGUS, a general purpose celestial objects
detection framework for WFSATs in telescope arrays. The ARGUS is successor of
DNN based celestial objects detection algorithms. Different celestial objects
detection algorithms in different WFSATs are assumed as weak learning machine.
The contribution of each WFSATs is calibrated by the calibration data and we
directly use calibrated weights to merge detection results from different
telescopes. The ARGUS dynamically links different telescopes for transients
detection and could be used for next generation sky survey projects with
WFSATs [29], such as the Sitian [12]. In the future, we will use real
observation data to optimize the ARGUS and use the ARGUS to modify observation
strategy to increase observation ability of WFSATs.

##  5 Acknowledgments

is work is supported by National Natural Science Foundation of China (NSFC)
(11503018, 61805173), the Joint Research Fund in Astronomy (U1631133) under
cooperative agreement between the NSFC and Chinese Academy of Sciences (CAS).
Authors acknowledge the French National Research Agency (ANR) to support this
work through the ANR APPLY (grant ANR-19-CE31-0011) coordinated by B. Neichel.
This work is also supported by Shanxi Province Science Foundation for Youths
(201901D211081), Research and Development Program of Shanxi (201903D121161),
Research Project Supported by Shanxi Scholarship Council of China
(HGKY2019039), the Scientific and Technological Innovation Programs of Higher
Education Institutions in Shanxi (2019L0225).

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# A new regularization of loop integral, no divergence, no hierarchy problem

Wenyu Wang1, Jian-Feng Wu1, Si-Hong Zhou2 1Institute of Theoretical Physics,
College of Applied Science,
Beijing University of Technology, Beijing 100124, China
2Institute of High Energy Physics and Theoretical Physics Center for Science
Facilities,
Chinese Academy of Sciences,Beijing 100049, China

###### Abstract

We find a new regularization scheme which is motivated by the Bose-Einstein
condensation. The energy of the virtual particle is considered as discrete.
Summing them and regulating the summation by the Riemann $\zeta$ function can
give the result of loop integral. All the divergences vanish, we can get
almost the same results as Dimensional Regularization. The prediction beyond
Dimensional Regularization is also shown in the QED. The hierarchy problem of
the radiative correction of scalar mass completely vanish.

###### pacs:

11.10.Gh, 67.85.Hj, 31.30.jr, 14.80.Bn,

##  I Introduction

It is well known that the higgs suffers from the hierarchy problem. The key
point of the mass hierarchy is the radiative loop of a scalar with no
protection from a symmetry, namely, the one propagator integration:

$\int\frac{{\rm d}^{d}l}{(2\pi)^{d}}\frac{i}{l^{2}-m^{2}+i\epsilon}\,,$ (1)

which is quadratically divergent in four dimension. This implies large fine
tuning between bare value and radiative corrections, making the higgs in the
Standard model (SM) very unnatural. Here, we should suspect on the integral in
momentum space more seriously. Always we use the Fourier transformation
between the coordinate and the momentum space in the study. The basic idea of
such transformation is periodic boundary condition in coordinate or momentum
space and orthogonality of trigonometrical functions. Summing all the partial
wave gives transformation between periodic functions. Summing changes to a
continuous integral when the period goes to infinity. This is all about
mathematics. However, it is quite obscure to do such an integral since in
general the energy of a particle in the quantum world have no reason to be
continuous, neither the momenta.

Figure 1: Sketch map of Bose-Einstein condensation.

The way from discrete summing to integral is classic in mathematics and
physics. However, such operation may go wrong in some physical systems. One
example is Bose-Einstein condensation (BEC) in statistical physics. The sketch
map of BEC process is shown in FIG. 1: Bose and Einstein worked out the Bose-
Einstein statistics in the analysis of boson system with discrete energy, then
changed it to integral when they apply the statistics to a real number density
conserved system. When the temperature decreases over the critical point (the
chemical potential $\mu$ approaches zero), finite number of particles must
condense on the ground state which can not be integrated. This is a phase
transition process and there are other physics processes implying that we must
be very careful in the integral operations. For example, we can do integral to
calculate the work of isothermal processes, but can not do it for a process
composed of infinite infinitesimal adiabatic free expansions, these two
processes can form a superficially same curve in the pressure and volume plot.

As for the integral Eq. (1) in the momentum space, we believe the quantum
nature should also dominate the propagator progress such that the integral
must be substituted by a summation. We do not allow propagating particles have
energy lower than its effective mass. This is a physical demand because under
this energy bound, there should be no such a particle. In space-time
viewpoint, it means that for a quantum scattering process triggered, the
collision time should not be too long because a long time collision implies a
classical process happens. In fact, a “point” or a “plane wave” is only a
concept in the mathematical quantum field theory, there are no such things in
the real world. By uncertainty principle, the more precise measurement on the
space-time, the more uncertainty on the momentum-energy. The increase of the
collision energy of the colliders only increases the accuracy of the space-
time, but a lower limit of the measurement always exists. Thus our demand
brings a strong constraint on the momentum space and it implies that there is
no way from the summation to integral. the infinitesimal energy of massive
propagating particle does not survive in a quantum process. There should be a
lower bound of smallest energy.

It’s very interesting that there are many regularization of divergence summing
in mathematics, especially the Riemann $\zeta(s)$ functions: (Einstein used
the finite part in BEC processes.) $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$. One
classical regularization is $\zeta(-1)$:

$\zeta(-1)=1+2+3+4+...=-\frac{1}{12}\,,$ (2)

which details can be found in Ref. zeta . Such regularization and BEC process
give us a hint on the integral in the momentum space. In this paper we find a
new regularization scheme of loop integral by using regularization of Riemann
$\zeta$ functions. The divergences of all the loop integral vanish, and they
can be considered as a condensation on the vacuum partly. Rich physics can be
found in our regularization.

The paper is organized in the following: In Sec. II, we show the detail of our
regularization. In Sec. III we show the implications of our method. The
conclusion is given in the Sec. IV.

##  II Discrete regularization of loop function

The standard method of dimensional regularization (DR) is: Feynman
parameterization, Wick rotation and integral in the Euclidean space. Here we
begin at the point after the Wick rotation. Any loop function in $d$ dimension
is

$I(d,n,S)=\int\frac{d^{d}l}{(2\pi)^{d}}\frac{1}{(l^{2}+S)^{n}}\,,$ (3)

where $S$ is the effective mass got from Feynman parameterization. Using DR,
additional $\mu$ parameter should be introduced, the result is:

$I_{D}(d,n,S,\mu)=\mu^{\epsilon}\int\frac{d^{d}l}{(2\pi)^{d}}\frac{1}{(l^{2}+S)^{n}}=\mu^{\epsilon}\frac{1}{(4\pi)^{d/2}}\frac{\Gamma(n-\frac{d}{2})}{\Gamma(n)}\biggl{(}\frac{1}{S}\biggl{)}^{n-\frac{d}{2}}\,.$
(4)

Note that, $\epsilon$ is a constant, not an input parameter. However, as
argued in the introduction, such integral in the energy direction in our
scenario should be replaced by a summation of discrete energy. Thus we
consider the virtual particle like an oscillator, which energy gap is denoted
as $l_{0}$, the energy level is $jl_{0}$. Integral Eq. (3) $I(d,n,S)$ can be
changed into another function $I_{W}(d,n,S,l_{0})$ which is

$\displaystyle I_{W}(d,n,S,l_{0})$ $\displaystyle=$
$\displaystyle\frac{l_{0}}{\pi}\sum_{j=1}^{\infty}\int\frac{d^{d-1}l}{(2\pi)^{d-1}}\frac{1}{(l^{2}+j^{2}l_{0}^{2}+S)^{n}}+\frac{l_{0}}{2\pi}\int\frac{d^{d-1}l}{(2\pi)^{d-1}}\frac{1}{(l^{2}+S)^{n}}\,.$
(5)

Here the gap $l_{0}$ is a new parameter, the second term is the ground state
contribution. We can see that this operation can remove one dimension of the
divergence, in the following we will use regularization of Riemann $\zeta$
function to regulate the divergence. Using Eq. (4) we get:

$\displaystyle I_{W}(d,n,S,l_{0})$ $\displaystyle=$
$\displaystyle\frac{l_{0}}{\pi}\sum_{j=1}^{\infty}\frac{1}{(4\pi)^{d/2-1/2}}\frac{\Gamma(n-\frac{d}{2}+\frac{1}{2})}{\Gamma(n)}\biggl{(}\frac{1}{j^{2}l^{2}_{0}+S}\biggl{)}^{n-\frac{d}{2}+\frac{1}{2}}+\frac{l_{0}}{2\pi}\frac{1}{(4\pi)^{d/2-1/2}}\frac{\Gamma(n-\frac{d}{2}+\frac{1}{2})}{\Gamma(n)}\biggl{(}\frac{1}{S}\biggl{)}^{n-\frac{d}{2}+\frac{1}{2}}$
(6) $\displaystyle=$
$\displaystyle\frac{l_{0}}{\pi}\frac{1}{(4\pi)^{d/2-1/2}}\frac{\Gamma(n-\frac{d}{2}+\frac{1}{2})}{\Gamma(n)}\biggl{[}\sum_{j=1}^{\infty}\left(j^{2}l_{0}^{2}+S\right)^{-(n-\frac{d}{2}+\frac{1}{2})}+\frac{1}{2}\biggl{(}\frac{1}{S}\biggl{)}^{n-\frac{d}{2}+\frac{1}{2}}\biggl{]}$
$\displaystyle=$
$\displaystyle\frac{l_{0}^{-2n+d}}{\pi}\frac{1}{(4\pi)^{d/2-1/2}}\frac{\Gamma(n-\frac{d}{2}+\frac{1}{2})}{\Gamma(n)}\biggl{[}\sum_{j=1}^{\infty}\left(j^{2}+\frac{S}{l_{0}^{2}}\right)^{-(n-\frac{d}{2}+\frac{1}{2})}+\frac{1}{2}(S/l_{0}^{2})^{-(n-\frac{d}{2}+\frac{1}{2})}\biggl{]}$
$\displaystyle=$
$\displaystyle\frac{4l_{0}^{-2n+d}}{(4\pi)^{d/2+1/2}}\frac{\Gamma(n-\frac{d}{2}+\frac{1}{2})}{\Gamma(n)}\left[E_{1}^{S/l_{0}^{2}}(n-\frac{d}{2}+\frac{1}{2};~{}1)+\frac{1}{2}(S/l_{0}^{2})^{-(n-\frac{d}{2}+\frac{1}{2})}\right]\,.$

All the divergences in the $\Gamma$ function now vanish in case of even number
dimension. This is just what we want. The divergences are absorbed by the
Epstein-Hurwitz function :

$E_{1}^{c^{2}}(s;~{}1)\equiv\sum_{j=1}^{\infty}(j^{2}+c^{2})^{-s}\,,$ (7)

where $c^{2}=S/l_{0}^{2}$ and $s=n-d/2+1/2$. It can be regulated by Remiann
$\zeta$ function in case of $c^{2}\leq 1$,the results depend on the parameter
$s$, which is: zeta

  1. 1.

in case of $\frac{1}{2}-s\in N$:

$\displaystyle E_{1}^{c^{2}}(s;1)$ $\displaystyle=$
$\displaystyle-\frac{(-1)^{-(s-1/2)}\pi^{1/2}}{2\Gamma(s)\Gamma(\frac{3}{2}-s)}c^{1-2s}\left[\psi(\frac{1}{2})-\psi(\frac{3}{2}-s)+\ln
c^{2}+2\gamma\right]-\frac{1}{2}c^{-2s}$ (8)
$\displaystyle-\sum_{k=0,k\neq\frac{1}{2}-s}^{\infty}(-1)^{k}\frac{\Gamma(k+s)}{k!\Gamma(s)}\zeta(2k+2s)c^{2k}\,.$

  2. 2.

in case of $\frac{1}{2}-s\notin N$ and $-s\notin N$:

$\displaystyle E_{1}^{c^{2}}(s;1)$ $\displaystyle=$
$\displaystyle\frac{\pi^{1/2}}{2\Gamma(s)}\Gamma(s-\frac{1}{2})c^{1-2s}-\frac{1}{2}c^{-2s}$
(9)
$\displaystyle-\sum_{k=0}^{\infty}(-1)^{k}\frac{\Gamma(k+s)}{k!\Gamma(s)}\zeta(2k+2s)c^{2k}\,.$

  3. 3.

in case of $-s\in N$:

$\displaystyle E_{1}^{c^{2}}(s;1)$ $\displaystyle=$
$\displaystyle-\sum_{k=0}^{-s}(-1)^{k}\frac{\Gamma(k+s)}{k!\Gamma(s)}\zeta(2k+2s)c^{2k}\,.$
(10)

where $N$ is the natural number $N=0,1,2,3,\cdots$, $\gamma$ is Euler constant
and $\psi(x)$ is Di-Gamma function. For $d=4$ and $n\leq 2$, we should use the
first formula to calculate the loop integral, for $n\geq 3$ we should use the
second formula. Also we can use the similar way like dimensional
regularization that we let $d$ approaches the even dimension number, then we
should use the second formula for all the loop integrals. One can check that
in case of $n\leq 2$, the divergences of the second formula cancel each other
when the dimension number $d$ approaches $4$. In fact, $E_{1}^{c^{2}}(s;~{}1)$
is a continuous function in the complex plane, thus all the divergences vanish
in our new method of regularization.

Putting the result of $E_{1}^{c^{2}}(s;~{}1)$ to Eq. (6), we can get the
analytical formula of loop integral in the new regularization, Note that the
second term of the Eq.(8) and Eq.(9) cancels the last term in the bracket of
Eq. (6) which is the ground state contribution, remaining only with two kinds
of terms: one is finite term composed by the product of $\Gamma$ functions,
another is a summation of a power series of $S/l_{0}^{2}$. If we set the
$l_{0}^{2}\gg S$ the first kinds of term will be the leading contribution to
the loop integral. The summation of the power series will be higher order
contributions, and it can be evaluated order by order. Interestingly, the
first terms are the exact analytical result of dimensional regularization. For
example, the result of DR of two point $B_{0}$ function is: (labeled as $D$)

$\displaystyle B_{0}^{D}$ $\displaystyle=$
$\displaystyle\triangle+\mathrm{ln}\frac{\mu^{2}}{m_{1}^{2}}-\int_{0}^{1}\mathrm{d}x\mathrm{ln}S(x)\,,$
(11)

in which $\triangle=\frac{2}{\epsilon}-\gamma+\mathrm{ln}4\pi$ is the
divergent term of DR. The result of the new regularization is: (labeled as
$W$)

$\displaystyle B_{0}^{W}$ $\displaystyle=$ $\displaystyle
2\mathrm{ln}2-2\gamma+\mathrm{ln}\frac{l_{0}^{2}}{m_{1}^{2}}-\int_{0}^{1}\mathrm{d}x\mathrm{ln}S(x)$
(12)
$\displaystyle-2\sum_{k=1}^{\infty}(-1)^{k}\frac{\Gamma(k+1/2)}{k!\pi^{1/2}}\zeta(2k+1)\int_{0}^{1}\mathrm{d}x\left(\frac{m_{1}^{2}}{l_{0}^{2}}S(x)\right)^{k}\,.$

We can see that the divergence regulated from DR are replaced by some
dimensionless constants. Other terms of the the first line of both
regularization are exactly the same. Results of other functions are similar,
one can check them in the appendix in which we show the analytical results of
A, B and C function in the two kinds of regularization.

Before using the new regularization, the first question should be where does
the divergence go. As we use the feynman rules to do the loop calculation, the
integration in the momentum space is ultraviolet divergent in some functions.
In the new regularization, we use the $\Gamma(x)$ and Riemann $\zeta(s)$
functions to eliminate the divergences mathematically. Take quadratically
divergent A function as an example, the divergence are regulated into
$\Gamma(-\frac{1}{2})$ and $\zeta(-1)$ indeed. These two functions are
superficially divergent but are regulated to be finite by mathematicians.
Therefore, we emphasize that the divergence of our new regularization is only
a mathematical problem. However, if using the ordinary dimensional
regularization, the divergences are extracted and canceled by the introducing
the counter terms. Thus they are physical problems, the variables are
renormalized by physicists and the renormalization exposes an understanding of
the physical world. Here we address that there are two levels to understand
our regularization:

  1. 1.

Level I: This regularization is only a trick by which we can get the almost
the same results of dimensional regularization in case of $l_{0}^{2}\gg S$.
The divergence are removed and the running of renormalization group (shown in
the QED) can still remain.

  2. 2.

Level II: As talked in the introduction, the way from summing to integral may
go wrong in real physical systems such as in the BEC process in which integral
should be divergent in case of phase transition. What we are doing here is
kind of an anti-BEC process in which integral is divergent but summation
became finite. We can consider that the divergences are in fact condensed in
the vacuum. If so, we should take a new look at the quantum field theory.

In addition to remove the divergences, another difference between the DR and
the new regularization is the power series term which become significantly
important when $S/l_{0}^{2}$ approaches 1. In the following section, we will
show the implication of the new regularization.

##  III implication of new regularization

The quantum field theory has achieved great successes, of which the most
important parts are the radiative corrections and the renormalizations. At
first, physicists do the renormalization just for dealing with the irritating
divergence when doing the radiative corrections. Finally they found that
renormalization exposed a deep understanding on the quantum theory. In our new
regularization, though there is no divergence in loop integrals, the
renormalization is still necessary, This is because every physical variable
will be changed by radiative corrections, and we must define a theory at a
renormalization points. Since the leading analytical terms are the same as the
dimensional regularization, the energy gap $l_{0}$ can be denoted as the
renormalization point of a theory, like the $\mu$ parameter in DR. The
renormalization will be exactly the same as the dimensional regularization
when we set $l_{0}^{2}\gg S$. Such regularization will be only a trick. Note
that though there is no divergent term, the renormalization group evolution is
still the same in this case. However, here we want to address that what we are
doing is very like an anti-BEC process, thus the energy gap $l_{0}$ is kind of
the temperature of the vacuum. The verification of our proposal will be the
power series terms of the loop integral, which will modify the ordinary loop
integral in case of the momentum of a processes is close to the energy scale
at which the theory is defined. We will see such implication on the quantum
electro-dynamics (QED) and the hierarchy problem of radiative mass of a
scalar.

###  III.1 Predications in the QED

Among all the theories in science, perhaps the QED is the most precisely
tested. Loop calculations are taken into higher and higher order, still no
significant deviation is founded. Thus the new regularization must recover the
result of the QED. Also we should use the precision test of the QED to see
that if the new regularization have some new predictions. At one loop level,
the radiative predictions of the QED come from three diagrams:peskin  the
electron self-energy, the photon self-energy and the photon-electron vertex.
By doing this we can get electron magnetic moment $a_{e}$, the Lamb shift, the
running of coupling strength $\alpha_{\rm eff}(q^{2})$, and the $\beta$
function of the QED. As talked above if we use the same terms as in the DR,
there is no new prediction of the new regularization. Here we list and check
the modification predicted from leading term of the power series terms:

  1. 1.

Electron magnetic movement $a_{e}$.

$\displaystyle
a_{e}\equiv\frac{g-2}{2}=\frac{\alpha}{2\pi}-\frac{\alpha}{2\pi}\frac{\zeta(3)}{6}\frac{m_{e}^{2}}{l_{0}^{2}}\,.$
(13)

The first term is the same result of DR and the last term is leading
modification from the new regularization. One can check that the modification
is very small if $l_{0}$ is set at electro-weak scale. More precision
measurement of $a_{e}$ can test the new regularization. Note that the last
term gives a negative contribution to magnetic movement, thus it can not
account for the 3.4$\sigma$ deviation Hagiwara:2006jt  of muon magnetic
movement by pure QED. However, the complete checking should include the
electro-weak and hadronic loop contributions which is beyond this work.

  2. 2.

Lamb shift. the Lamb shift will not be changed by the new regularization. This
is because that the Uehling potential comes from the imaginary part of photon
self energy $\hat{\Pi}_{2}(q^{2})$ which not appear in the power series terms.

  3. 3.

Gauge Invariance. Since we divide the one of the four dimension integral into
discrete summation, the new regulator will violate local U(1) gauge symmetry
of the QED, thus Ward-Takahashi Identity will no be preserved. Among the usual
regulators such as Cut-off, Pauli-Villars and DR, only DR preserves the gauge
symmetry and Lorentz symmetry. As talked in the introduction, continuous gauge
symmetry and Lorentz symmetry is based on the “point” or “plane wave”
mathematical quantum field theory, violation of these symmetry in real physics
processes can be accepted.

  4. 4.

Running of coupling strength $\alpha_{\rm eff}(q^{2})$.

$\displaystyle\alpha_{\rm
eff}(q^{2})=\frac{\alpha}{1-\frac{\alpha}{3\pi}\mathrm{ln}(-q^{2}/A^{\prime}m^{2})}\,.$
(14)

where $A^{\prime}=\mathrm{exp(\frac{5}{3}+\frac{\zeta(3)}{5})}$. The
additional $\zeta$ term is leading modification from the new regularization.
Note that, to get this formula, we use the condition $l_{0}^{2}\sim-q^{2}\gg
m^{2}$ which means that the energy scale is much higher than the mass of
electron.

  5. 5.

$\beta$ function of the QED. If the energy scale $l_{0}$ is the temperature of
the vacuum, the $\beta$ function of the coupling is kind of thermal
capacitance of a theory. Especially when the momentum $q^{2}$ approaches the
temperature $l_{0}$ ($\mathrm{ln}\frac{-q^{2}}{l_{0}^{2}}\sim\mathrm{ln}M^{2}$
e.g. $M^{2}\to 1$), then the $\beta$ function will be exactly the capacitance:

$\displaystyle\beta(\lambda)$ $\displaystyle=$ $\displaystyle
M\frac{\partial}{\partial M}(\mbox{counter terms})$ (15) $\displaystyle=$
$\displaystyle 2\frac{\partial}{\partial\mathrm{ln}M^{2}}(\mbox{counter
terms})$ $\displaystyle\simeq$ $\displaystyle 2\frac{\partial}{\partial
M^{2}}(\mbox{counter terms})\,.$

This means that the $\beta$ function will change when running to the high
energy scale ($-q^{2}\sim l_{0}^{2}$). In case of $-q^{2}$ approaches
$l_{0}^{2}$, for example, $\beta$ function of the QED is:

$\displaystyle\beta(e)=\frac{\mathrm{e}^{3}}{12\pi^{2}}-\frac{1}{3}\zeta(3)\frac{\mathrm{e}^{3}}{16\pi^{2}}\,,$
(16)

The first term is the prediction of DR. The second term is the modification of
the new regularization. Using renormalization group equation of the QED and
the Eq. (16), setting $l_{0}^{2}=Am^{2}$, we get the running of coupling
strength:

$\displaystyle\alpha_{eff}(q^{2})=\frac{\alpha}{1-(\frac{\alpha}{3\pi}-\frac{\zeta(3)\alpha}{12\pi})\mathrm{ln}(-q^{2}/Am^{2})}\,.$
(17)

Similar to the Eq. (14), the prediction also deviate from result of DR by
$\frac{\zeta(3)\alpha}{12\pi}$ in second term of denominator.

In all, for the QED, predictions of the new regularization will be the same as
DR at the low energy. However, when energy increases, the additional power
series term will become significant. When it goes close to the temperature of
the vacuum, the new regularization will give a different story.

###  III.2  Hierarchy problem

Scalar mass in quantum field theory is a big obstacle of renormalization. For
the radiative correction of a scalar mass is quadratic divergent, making
subtly tuning between bare value and the corrections. However, in the new
regularization, such tuning completely vanishes.

  * •

$\lambda\phi^{4}$ theory: the leading term of mass counter term is

$\displaystyle\delta_{m}=\frac{\lambda}{2}\frac{m^{2}}{16\pi^{2}}\left(2\mathrm{ln}2-2\gamma+\mathrm{ln}\frac{l_{0}^{2}}{m^{2}}+1+\frac{l_{0}^{2}}{3m^{2}}\right)\,.$
(18)

  * •

Yukawa theory: the leading term of mass counter term is

$\displaystyle\delta_{m}=-\frac{Y^{2}}{4\pi^{2}}\left[\frac{l_{0}^{2}}{3}+\int_{0}^{1}dx(m_{f}^{2}-x(1-x)m_{s}^{2})\left(6\ln
2-6\gamma-3\ln\frac{m_{f}^{2}-x(1-x)m_{s}^{2}}{l_{0}^{2}}+1\right)\right]+m_{s}^{2}\delta_{Z}\,,$
(19)

where $m_{f}$ is the fermion mass, $m_{s}$ is the scalar mass and $\delta_{Z}$
is the counter term of scalar field renormalization which is:

$\displaystyle\delta_{Z}=-\frac{3Y^{2}}{4\pi^{2}}\int_{0}^{1}dxx(1-x)\left(2\ln
2-2\gamma-\frac{2}{3}-\ln\frac{m_{f}^{2}-x(1-x)m_{s}^{2}}{l_{0}^{2}}\right)\,.$
(20)

Though there are quadratic mass term of $l_{0}^{2}$ in Eq. (18) and Eq. (19),
$l_{0}$ (energy gap) is not corresponding to an energy scale of new physics.
Note that the summation of $j$ in Eq. (6) is from 1 to infinity, we do not set
a cut on the integration. What we need is to set $l_{0}$ heavier than the
effective mass of the integral. In such case, no tuning is needed and no
hierarchy in our new regularization.

We know that the hierarchy problem is perhaps the biggest theoretical dilemma
of SM. It has motivated the most new physics model beyond SM, such as
supersymmetry, extra dimension et. al. np  If we use the new regularization,
hierarchy problem does not seem to be the first motivation of new physics. In
our opinion, the divergence of a radiative correction is non-physical, the
emergence of divergence is because of a wrong mathematical tools used by
physicists. Our regularization gives a clue to the right tools. Also it
exposed a deep understanding of relation between virtual particle and vacuum.
The virtual particles condensed on the vacuum do not give the radiative
corrections. Note that, the regularization by Riemann $\zeta$ function can
also calculate Casimir effect and the vacuum energy in Ref. zeta .

##  IV conclusion

In this paper, we find a new regularization scheme which is motivated by the
BEC process, the energy of the virtual particle is considered as discrete.
Summing them and regulating the summation by the Riemann $\zeta$ function can
give the result of loop integral. All the divergences vanish and we can get
almost the same results of DR. The prediction beyond DR is also shown in the
QED. The hierarchy problem of the radiative correction of scalar mass
completely vanishes. This give us very comfortable understanding on the QED
and SM in quantum field theory.

As we consider the energy of particle as discrete, one may argue that, our
regularization breaks Lorentz symmetry. However, the leading term of the new
regularization is exactly the same as DR which preserves the Lorentz symmetry.
The new regularization is very powerful if one treat it as trick. The Lorentz
symmetry violating term may expose a deep understanding of the vacuum. We want
emphasis that what we are doing is the statistics of vacuum, the implication
of the thermal physics of vacuum needs further study.

## APPENDIX

Here we list the A, B, C loop function under DR (labeled as $D$) and the new
regularization (label as $W$):

  * •

A function: $\triangle=\frac{2}{\epsilon}-\gamma+\mathrm{ln}4\pi\,,$

$\displaystyle\frac{i}{16\pi^{2}}A(m^{2})$ $\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{d}l}{(2\pi)^{d}}\frac{1}{l^{2}-m^{2}+i\epsilon}\,,$
(21) $\displaystyle A^{D}$ $\displaystyle=$ $\displaystyle
m^{2}\left[\triangle+\mathrm{ln}\frac{\mu^{2}}{m^{2}}+1\right]\,,$ (22)
$\displaystyle A^{W}$ $\displaystyle=$ $\displaystyle
m^{2}\left[2\mathrm{ln}2-2\gamma+\mathrm{ln}\frac{l_{0}^{2}}{m^{2}}+1\right.\,$
(23) $\displaystyle\left.+2\sum_{k=0,k\neq
1}^{\infty}(-1)^{k}\frac{\Gamma(k-\frac{1}{2})}{k!\pi^{1/2}}\zeta(2k-1)\left(\frac{m^{2}}{l_{0}^{2}}\right)^{k-1}\right]\,.$

  * •

B function:
$S(x)=\frac{p^{2}}{m_{1}^{2}}x^{2}-\frac{p^{2}+m_{1}^{2}-m_{2}^{2}}{m_{1}^{2}}x+1\,,$

$\displaystyle\frac{i}{16\pi^{2}}B_{0}(p,m_{1}^{2},m_{2}^{2})$
$\displaystyle=$
$\displaystyle{\int\frac{\mathrm{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m_{1}^{2}+i\epsilon][(l-p)^{2}-m_{2}^{2}+i\epsilon]}}\,,$
(24) $\displaystyle B_{0}^{D}$ $\displaystyle=$
$\displaystyle\triangle+\mathrm{ln}\frac{\mu^{2}}{m_{1}^{2}}-\int_{0}^{1}\mathrm{d}x\mathrm{ln}S(x)\,,$
(26) $\displaystyle B_{0}^{W}$ $\displaystyle=$ $\displaystyle
2\mathrm{ln}2-2\gamma+\mathrm{ln}\frac{l_{0}^{2}}{m_{1}^{2}}-\int_{0}^{1}\mathrm{d}x\mathrm{ln}S(x)\,$
(27)
$\displaystyle-2\sum_{k=1}^{\infty}(-1)^{k}\frac{\Gamma(k+1/2)}{k!\pi^{1/2}}\zeta(2k+1)\int_{0}^{1}\mathrm{d}x\left(\frac{m_{1}^{2}}{l_{0}^{2}}S(x)\right)^{k}\,.$
$\displaystyle\frac{i}{16\pi^{2}}B_{\mu}(p,m_{1}^{2},m_{2}^{2})$
$\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{d}l}{(2\pi)^{d}}\frac{l_{\mu}}{(l^{2}-m_{1}^{2}+i\epsilon)[(l-p)^{2}-m_{2}^{2}+i\epsilon]}\,,$
(28) $\displaystyle B_{\mu}^{D}$ $\displaystyle=$
$\displaystyle\frac{1}{2}p_{\mu}\left[\triangle+\mathrm{ln}\frac{\mu^{2}}{m_{1}^{2}}-2\int_{0}^{1}\mathrm{d}xx\mathrm{ln}S(x)\right]\,,$
(29) $\displaystyle B_{\mu}^{W}$ $\displaystyle=$
$\displaystyle\frac{1}{2}p_{\mu}\left[2\mathrm{ln}2-2\gamma+\mathrm{ln}\frac{l_{0}^{2}}{m_{1}^{2}}-2\int_{0}^{1}\mathrm{d}xx\mathrm{ln}S(x)\right.\,$
(30)
$\displaystyle\left.-4\sum_{k=1}^{\infty}(-1)^{k}\frac{\Gamma(k+1/2)}{k!\pi^{1/2}}\zeta(2k+1)\int_{0}^{1}\mathrm{d}xx\left(\frac{m_{1}^{2}}{l_{0}^{2}}S(x)\right)^{k}\right]\,.$
$\displaystyle\frac{i}{16\pi^{2}}B_{\mu\nu}(p,m_{1}^{2},m_{2}^{2})$
$\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{d}l}{(2\pi)^{d}}\frac{l_{\mu}l_{\nu}}{[l^{2}-m_{1}^{2}+i\epsilon][(l-p)^{2}-m_{2}^{2}+i\epsilon]}\,,$
(31) $\displaystyle B_{\mu\nu}^{D}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left\\{p_{\mu}p_{\nu}\left[\Delta+\mathrm{ln}\frac{\mu^{2}}{m_{1}^{2}}-3\int_{0}^{1}\mathrm{d}xx^{2}\mathrm{ln}S(x)\right]\right.\,$
(32)
$\displaystyle+\frac{g_{\mu\nu}}{d}\left[(3m_{1}^{2}+3m_{2}^{2}-p^{2})(\Delta+\mathrm{ln}\frac{\mu^{2}}{m_{1}^{2}}+1)\right.\,$
$\displaystyle\left.\left.-\frac{3m_{1}^{2}+3m_{2}^{2}-p^{2}}{2}-6m_{1}^{2}\int_{0}^{1}\mathrm{d}xS(x)\mathrm{ln}S(x)\right]\right\\}\,,$
$\displaystyle B_{\mu\nu}^{W}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left\\{p_{\mu}p_{\nu}\left[2\mathrm{ln}2-2\gamma+\mathrm{ln}\frac{l_{0}^{2}}{m_{1}^{2}}-3\int_{0}^{1}\mathrm{d}xx^{2}\mathrm{ln}S(x)\right]\right.\,$
(33)
$\displaystyle+\frac{g_{\mu\nu}}{d}\left[(3m_{1}^{2}+3m_{2}^{2}-p^{2})(2\mathrm{ln}2-2\gamma+\mathrm{ln}\frac{l_{0}^{2}}{m_{1}^{2}}+1)\right.\,$
$\displaystyle\left.-\frac{3m_{1}^{2}+3m_{2}^{2}-p^{2}}{2}-6m_{1}^{2}\int_{0}^{1}\mathrm{d}xS(x)\mathrm{ln}S(x)\right]\,$
$\displaystyle-6p_{\mu}p_{\nu}\left[\sum_{k=1}^{\infty}(-1)^{k}\frac{\Gamma(k+1/2)}{k!\pi^{1/2}}\zeta(2k+1)\int_{0}^{1}\mathrm{d}xx^{2}\left(\frac{m_{1}^{2}}{l_{0}^{2}}S(x)\right)^{k}\right]\,$
$\displaystyle-6\frac{g_{\mu\nu}}{d}\left[-l_{0}^{2}\sum_{k=0,k\neq
1}^{\infty}(-1)^{k}\frac{\Gamma(k-1/2)}{k!\pi^{1/2}}\zeta(2k-1)\int_{0}^{1}\mathrm{d}x\left(\frac{m_{1}^{2}}{l_{0}^{2}}S(x)\right)^{k}\right.\,$
$\displaystyle+\left.\left.\sum_{k=1}^{\infty}(-1)^{k}\frac{\Gamma(k+1/2)}{k!\pi^{1/2}}\zeta(2k+1)\int_{0}^{1}\mathrm{d}xm_{1}^{2}S(x)\left(\frac{m_{1}^{2}}{l_{0}^{2}}S(x)\right)^{k}\right]\right\\}\,.$

  * •

C function: $S(x,y)=(x^{2}p^{2}+y^{2}p^{\prime
2}+2xypp^{\prime})-x(p^{2}+m_{1}^{2}-m_{2}^{2})-y(p^{\prime
2}+m_{1}^{2}-m_{3}^{2})+m_{1}^{2}\,,$

$\displaystyle\frac{i}{16\pi^{2}}C_{0}(p,p^{\prime},m_{1}^{2},m_{2}^{2},m_{3}^{2})$
$\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{d}l}{(2\pi)^{d}}\frac{1}{(l^{2}-m_{1}^{2}+i\epsilon)[(l-p)^{2}-m_{2}^{2}+i\epsilon][(l-p^{\prime})^{2}-m_{3}^{2}+i\epsilon]}\,,$
(34) $\displaystyle C_{0}^{D};C_{\mu}^{D}$ $\displaystyle=$
$\displaystyle-\int_{0}^{1}\mathrm{d}x\int_{0}^{1-x}\mathrm{d}y(1;xp_{\mu}+yp_{\mu}^{\prime})\frac{1}{S(x,y)}\,,$
(35) $\displaystyle C_{0}^{W};C_{\mu}^{W}$ $\displaystyle=$
$\displaystyle-\int_{0}^{1}\mathrm{d}x\int_{0}^{1-x}\mathrm{d}y(1;xp_{\mu}+yp_{\mu}^{\prime})\left[\frac{1}{S(x,y)}\right.\,$
(36)
$\displaystyle-\left.\sum_{k=0}^{\infty}(-1)^{k}\frac{2\Gamma(k+\frac{3}{2})}{k!\pi^{1/2}}\zeta(2k+3)l_{0}^{-2}\left(\frac{S(x,y)}{l_{0}^{2}}\right)^{k}\right]\,.$
$\displaystyle\frac{i}{16\pi^{2}}C_{\mu\nu}(p,p^{\prime},m_{1}^{2},m_{2}^{2},m_{3}^{2})$
$\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{d}l}{(2\pi)^{d}}\frac{l_{\mu}l_{\nu}}{(l^{2}-m_{1}^{2}-i\epsilon)[(l-p)^{2}-m_{2}^{2}+i\epsilon][(l-p^{\prime})^{2}-m_{3}^{2}-i\epsilon]}\,,$
(37) $\displaystyle C_{\mu\nu}^{D}$ $\displaystyle=$
$\displaystyle\frac{g_{\mu\nu}}{4}\triangle-\int_{0}^{1}\mathrm{d}x\int_{0}^{1-x}\mathrm{d}y\left[\frac{x^{2}p_{\mu}p_{\nu}+xy(p_{\mu}p_{\nu}^{\prime}+p_{\mu}^{\prime}p_{\nu})+y^{2}p_{\mu}^{\prime}p_{\nu}^{\prime}}{S(x,y)}+\frac{g_{\mu\nu}}{2}\mathrm{ln}\frac{S(x,y)}{\mu^{2}}\right]\,,$
(38) $\displaystyle C_{\mu\nu}^{W}$ $\displaystyle=$
$\displaystyle\frac{g_{\mu\nu}}{4}(2\mathrm{ln}2-2\gamma-\frac{1}{2})\,$ (39)
$\displaystyle-\int_{0}^{1}\mathrm{d}x\int_{0}^{1-x}\mathrm{d}y\left[\frac{x^{2}p_{\mu}p_{\nu}+xy(p_{\mu}p_{\nu}^{\prime}+p_{\mu}^{\prime}p_{\nu})+y^{2}p_{\mu}^{\prime}p_{\nu}^{\prime}}{S(x,y)}+\frac{g_{\mu\nu}}{2}\mathrm{ln}\frac{S(x,y)}{l_{0}^{2}}\right]\,$
$\displaystyle+\frac{g_{\mu\nu}}{2}\left[-2\sum_{k=1}^{\infty}(-1)^{k}\frac{\Gamma(k+1/2)}{k!\pi^{1/2}}\zeta(2k+1)\int_{0}^{1}\mathrm{d}x\right.\int_{0}^{1-x}\mathrm{d}y\left(\frac{S(x,y)}{l_{0}^{2}}\right)^{k}\,$
$\displaystyle+\left.\frac{1}{2}\sum_{k=0}^{\infty}(-1)^{k}\frac{2\Gamma(k+\frac{3}{2})}{k!\pi^{1/2}}\zeta(2k+3)\int_{0}^{1}\mathrm{d}x\int_{0}^{1-x}\mathrm{d}y\left(\frac{S(x,y)}{l_{0}^{2}}\right)^{k+1}\right]\,$
$\displaystyle+\sum_{k=0}^{\infty}(-1)^{k}\frac{2\Gamma(k+\frac{3}{2})}{k!\pi^{1/2}}\zeta(2k+3)\int_{0}^{1}\mathrm{d}x\int_{0}^{1-x}\mathrm{d}y\,$
$\displaystyle\times\left[x^{2}p_{\mu}p_{\nu}+xy(p_{\mu}p_{\nu}^{\prime}+p_{\mu}^{\prime}p_{\nu})+y^{2}p_{\mu}^{\prime}p_{\nu}^{\prime}\right]l_{0}^{-2}\left(\frac{S(x,y)}{l_{0}^{2}}\right)^{k}\,.$
$\displaystyle\frac{i}{16\pi^{2}}C_{\mu\nu\sigma}(p,p^{\prime},m_{1}^{2},m_{2}^{2},m_{3}^{2})$
$\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{d}l}{(2\pi)^{d}}\frac{l_{\mu}l_{\nu}l_{\sigma}}{(l^{2}-m_{1}^{2}+i\epsilon)[(l-p)^{2}-m_{2}^{2}+i\epsilon][(l-p^{\prime})^{2}-m_{3}^{2}+i\epsilon]}\,,$
(40) $\displaystyle C_{\mu\nu\sigma}^{D}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{12}\left[g_{\mu\nu}(p+p^{\prime})_{\sigma}+g_{\nu\sigma}(p+p^{\prime})_{\mu}+g_{\sigma\mu}(p+p^{\prime})_{\nu}\right]\,$
(41)
$\displaystyle-\int_{0}^{1}\mathrm{d}x\int_{0}^{1-x}\mathrm{d}y\biggl{\\{}\left[x^{3}p_{\mu}p_{\nu}p_{\sigma}+x^{2}y(p_{\mu}p_{\nu}p_{\sigma}^{\prime}+p_{\mu}p_{\nu}^{\prime}p_{\sigma}+p_{\mu}^{\prime}p_{\nu}p_{\sigma})\right.\,$
$\displaystyle+\left.xy^{2}(p_{\mu}^{\prime}p_{\nu}^{\prime}p_{\sigma}+p_{\mu}^{\prime}p_{\nu}p_{\sigma}^{\prime}+p_{\mu}p_{\nu}^{\prime}p_{\sigma}^{\prime})+y^{3}p_{\mu}^{\prime}p_{\nu}^{\prime}p_{\sigma}^{\prime}\right]/S(x,y)\,$
$\displaystyle+\frac{1}{2}\left[g_{\mu\nu}(xp+yp^{\prime})_{\sigma}+g_{\nu\sigma}(xp+yp^{\prime})_{\mu}+g_{\sigma\mu}(xp+yp^{\prime})_{\nu}\right]\mathrm{ln}\frac{S(x,y)}{l_{0}^{2}}\biggl{\\}}\,,$
$\displaystyle C_{\mu\nu\sigma}^{W}$ $\displaystyle=$
$\displaystyle\frac{2\mathrm{ln}2-2\gamma-\frac{1}{2}}{12}[g_{\mu\nu}(p+p^{\prime})_{\sigma}+g_{\nu\sigma}(p+p^{\prime})_{\mu}+g_{\sigma\mu}(p+p^{\prime})_{\nu}]\,$
(42)
$\displaystyle-\int_{0}^{1}\mathrm{d}x\int_{0}^{1-x}\mathrm{d}y\biggl{\\{}\left[x^{3}p_{\mu}p_{\nu}p_{\sigma}+x^{2}y(p_{\mu}p_{\nu}p_{\sigma}^{\prime}+p_{\mu}p_{\nu}^{\prime}p_{\sigma}+p_{\mu}^{\prime}p_{\nu}p_{\sigma})\right.\,$
$\displaystyle+\left.xy^{2}(p_{\mu}^{\prime}p_{\nu}^{\prime}p_{\sigma}+p_{\mu}^{\prime}p_{\nu}p_{\sigma}^{\prime}+p_{\mu}p_{\nu}^{\prime}p_{\sigma}^{\prime})+y^{3}p_{\mu}^{\prime}p_{\nu}^{\prime}p_{\sigma}^{\prime}\right]/S(x,y)\,$
$\displaystyle+\frac{1}{2}\left[g_{\mu\nu}(xp+yp^{\prime})_{\sigma}+g_{\nu\sigma}(xp+yp^{\prime})_{\mu}+g_{\sigma\mu}(xp+yp^{\prime})_{\nu}\right]\mathrm{ln}\frac{S(x,y)}{l_{0}^{2}}\biggl{\\}}\,$
$\displaystyle-\int_{0}^{1}\mathrm{d}x\int_{0}^{1-x}\mathrm{d}y\left\\{\frac{1}{2}\left[g_{\mu\nu}(xp+yp^{\prime})_{\sigma}+g_{\nu\sigma}(xp+yp^{\prime})_{\mu}+g_{\sigma\mu}(xp+yp^{\prime})_{\nu}\right]\right.\,$
$\displaystyle\times\left[2\sum_{k=1}^{\infty}(-1)^{k}\frac{\Gamma(k+1/2)}{k!\pi^{1/2}}\zeta(2k+1)\left(\frac{S(x,y)}{l_{0}^{2}}\right)^{k}-\frac{1}{2}\sum_{k=0}^{\infty}(-1)^{k}\frac{2\Gamma(k+\frac{3}{2})}{k!\pi^{1/2}}\zeta(2k+3)\left(\frac{S(x,y)}{l_{0}^{2}}\right)^{k+1}\right]\,$
$\displaystyle+\left[x^{3}p_{\mu}p_{\nu}p_{\sigma}+x^{2}y(p_{\mu}p_{\nu}p_{\sigma}^{\prime}+p_{\mu}p_{\nu}^{\prime}p_{\sigma}+p_{\mu}^{\prime}p_{\nu}p_{\sigma})+xy^{2}(p_{\mu}^{\prime}p_{\nu}^{\prime}p_{\sigma}+p_{\mu}^{\prime}p_{\nu}p_{\sigma}^{\prime}+p_{\mu}p_{\nu}^{\prime}p_{\sigma}^{\prime})+y^{3}p_{\mu}^{\prime}p_{\nu}^{\prime}p_{\sigma}^{\prime}\right]\,$
$\displaystyle\times\sum_{k=0}^{\infty}(-1)^{k}\frac{2\Gamma(k+\frac{3}{2})}{k!\pi^{1/2}}\zeta(2k+3)\left.l_{0}^{-2}\left(\frac{S(x,y)}{l_{0}^{2}}\right)^{k}\right\\}\,.$

## References

  * (1) E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko and S. Zerbini, “Zeta regularization techniques with applications,” Singapore, Singapore: World Scientific (1994) 319 p; E. Elizalde, “Ten physical applications of spectral zeta functions,” Lect. Notes Phys. 855, 1 (2012); K. Kirsten, “Basic zeta functions and some applications in physics,” MSRI Publ. 57, 101 (2010) [arXiv:1005.2389 [hep-th]].
  * (2) K. Hagiwara, A. D. Martin, D. Nomura and T. Teubner, “Improved predictions for g-2 of the muon and alpha(QED) (M**2(Z)),” Phys. Lett. B 649, 173 (2007) [hep-ph/0611102].
  * (3) M. E. Peskin, D. V. Schroeder ”An Introduction to Quantum Field Theory”, ISBN 0-201-50397-2, Beijing World Publishing Corp.
  * (4) G. L. Kane, “Perspectives on issues beyond the standard model,” hep-ph/0403040.


2022

Mario A. Acero∗

# Barrow holographic dark energy with Granda-Oliveros cut-off

A. Oliveros<EMAIL_ADDRESS>ORCID M. A. Sabogal
msabogal@est.uniatlantico.edu.co,ORCID<EMAIL_ADDRESS>ORCID Programa de Física, Universidad del Atlántico, Carrera 30 No. 8-49,
Puerto Colombia, Atlántico, Colombia

###### Abstract

A study on the effects of implementing the Granda–Oliveros infrared cutoff in
the recently introduced Barrow Holographic Dark Energy model is presented, and
its cosmological evolution is investigated. We find how the deformation
parameter, $\Delta$, affects the evolution of $H(z)$, and that from this model
it is possible to obtain an accelerated expansion regime of the universe at
late times. We also observe that increasing $\Delta$ causes a transition of
the EoS parameter from quintessence to phantom regimes. In addition, we show
that the model can be used to describe the know eras of dominance. Finally,
after studying the stability of the proposed model, a fit of the corresponding
parameters is preformed, utilizing the measurements of the expansion rate of
the universe, $H(z)$. The best fit of the parameters is found to be
$(\alpha,\,\beta,\,\Delta)=(1.00^{+0.02}_{-0.02},\,0.69^{+0.03}_{-0.02},\,0.000^{+0.004}_{-0.000})$
at $1\sigma$ C.L., for which the Bekenstein-Hawking relation is favored.

###### keywords:

Holographic, Barrow, Dark Energy, Granda–Oliveros cutoff

##  1 Introduction

Nowadays, there is a general consensus among cosmologists about of the
existence the so-called dark energy (DE), which is behind the accelerated
expansion of the universe at late times. This phenomenon has been supported by
a huge amount of cosmological observations, but it was originally reported in
1998 by two independent research groups, the High-z Supernova Search Team and
the Supernova Cosmology Project SupernovaSearchTeam:1998fmf ;
SupernovaCosmologyProject:1998vns . Yet, there is currently no adequate
solution to the DE problem and its explanation from fundamental theories of
physics is still unknown.

Within the wide spectrum of proposals to handle the DE problem, there is a
hypothesis known as the _Holographic principle_ tHooft:1993dmi ;
Susskind:1994vu , which plays an important role in quantum gravity. The main
idea behind the holographic principle is that the entropy of a system does not
scale with its volume, but with its surface area Bousso:1999xy . Inspired by
this proposal, Cohen et al. Cohen:1998zx , suggested that, in a quantum field
theory, a short distance cutoff is related to a long distance cutoff due to
the limit set by the formation of a black hole, namely, if $\rho$ is the
quantum zero-point energy density caused by a short distance cutoff, the total
energy in a region of size $L$ should not exceed the mass of a black hole of
the same size, hence, $L^{3}\rho\leq LM_{pl}^{2}$. The largest (infrared
cutoff) $L_{{}_{IR}}$ allowed is the one saturating this inequality, thus

$\rho=3M_{p}^{2}c^{2}L_{{}_{IR}}^{-2},$ (1)

where $c$ is an arbitrary parameter, and $M_{p}$ is the reduced Planck mass.
This holographic consideration has been widely applied in cosmology,
especially for the description of late time dark energy era, and it is
commonly known as holographic dark energy (HDE) (see Wang:2016och  for an
extensive review). From this point of view, the infrared cutoff,
$L_{{}_{IR}}$, has a cosmological origin and the authors of Nojiri:2005pu ;
Nojiri:2017opc ; Nojiri:2021iko ; Nojiri:2020wmh  introduced the most general
form for this cutoff (so-called generalized HDE), which includes some
combination of the FRW parameters, e.g., the Hubble constant, the particle and
future horizons, the cosmological constant and the universe life-time –if
finite.

A new proposal that has recently caught the attention of the community is the
so-called Barrow holographic dark energy (BHDE) Saridakis:2020zol , which has
its roots in the idea introduced by Barrow in Barrow:2020tzx , inspired by the
COVID-19 virus illustrations. Barrow proposed that due to quantum
gravitational effects, the black hole Bekenstein-Hawking entropy
Bekenstein:1973ur ; Hawking:1975vcx  must be modified introducing a fractal
structure for the horizon geometry:

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

where $A$ is the standard horizon area, $A_{0}$ the Planck area, and $\Delta$
is the deformation parameter, which indicates the amount of the quantum
gravitational deformation effects to the horizon structure. Notice that, for
$\Delta=0$, the standard Bekenstein-Hawking entropy is recovered, while
$\Delta=1$ corresponds to the most intricate fractal structure. When the BHDE
is implemented using the holographic principle and the new Barrow entropy
proposal, Eq. (2), the resulting holographic energy density becomes

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

where $C$ is a parameter having dimensions $[L]^{-2-\Delta}$ Barrow:2020tzx .
Clearly, for $\Delta=0$, Eq. (3) reduces to the standard HDE, i.e.,
$\rho_{\Lambda}=3M_{p}^{2}c^{2}L^{-2}$, where $C=3M_{p}^{2}c^{2}$ with $c^{2}$
the corresponding model parameter.

The infrared cutoff $L$ used in Barrow:2020tzx  was the future event horizon
$R_{h}$ and from this, the author shows that under this scenario it is
possible to describe the thermal history of the universe, with the sequence of
matter and dark-energy eras. Additionally, the new Barrow exponent, $\Delta$,
significantly affects the dark-energy equation of state (EoS), and according
to its value it can lead the EoS to lie in the quintessence regime, in the
phantom regime, or to experience the phantom-divide crossing during its
evolution. Furthermore, in Anagnostopoulos:2020ctz  the authors used
observational data from Supernovae (SNIa) Pantheon sample, as well as from
direct measurements of the Hubble parameter from the cosmic chronometers (CC)
sample, to impose constraints on the Barrow holographic dark energy scenario.
It is important to notice that the BHDE has been widely studied in the
literature, e.g., Mamon:2020spa ; Dabrowski:2020atl ; Srivastava:2020cyk ;
Pradhan:2021cbj ; Adhikary:2021xym ; Leon:2021wyx ; Asghari:2021bqa ;
Jusufi:2021fek ; P:2021edm ; Nojiri:2021jxf ; Nojiri:2022aof ; Huang:2021zgj ;
Luciano:2022pzg ; Wang:2022hun ; DiGennaro:2022ykp ; Nojiri:2022dkr ;
Farsi:2022dvd .

It is well known that a model using $R_{h}$ as the length scale predicts the
observed current acceleration of the universe; however, this kind of models
also display the causality problem, i.e., DE at present appears to depend on
the future evolution of the scale factor, which in turns violates causality
Li:2004rb . Hence, here we analyze the cosmological evolution at late times in
the framework of the BHDE model using the Granda-Oliveros (G-O) cutoff which,
in addition to the square of the Hubble parameter, also includes the time
derivative of the Hubble parameter, namely,

$L_{IR}=(\alpha H^{2}+\beta\dot{H})^{-1/2},$ (4)

where $\alpha$ and $\beta$ are arbitrary dimensionless parameters. This cutoff
was proposed by the authors of Refs. Granda:2008dk ; Granda:2008tm ,
considering purely dimensional arguments, and succeed to avoid the causality
problem. Also, we perform an analysis aimed to fit the parameters in the BHDE
model with the G-O cutoff, using updated measurements from the dynamics of the
expansion of the universe, $H(z)$. Recently in P:2021edm , the authors have
considered the Barrow holographic dark energy with Granda-Oliveros length as
IR cutoff, but in a scenario where the BHDE is a dynamical vacuum, and taking
into account an interaction between the matter and dark energy sectors.

The article is organized as follows: In Section 2 we describe the details of
the proposed model and the resulting evolution equations for the most
important quantities. The change of these quantities with the redshift is
presented also shown in this Section, looking at the effects of different and
interesting values of the parameters; the stability of the model under
consideration is also studied in this section. The constraints on the
parameter imposed by the requirement of stability allow us to set specific
ranges which are then implemented in Section 3 to perform a fit of the
parameters considering measurements of the Hubble parameter as a function of
the redshift, $H(z)$. Finally, the conclusions are posed in Section 4.

##  2 The model

As stated in Sec. 1, we are implementing the BHDE density in Eq. (3) with the
G-O IR cutoff, Eq. (4). In such a model, the holographic DE density becomes

$\rho_{\Lambda}=3M_{p}^{2}\left(\alpha
H^{2}+\beta\dot{H}\right)^{1-\frac{1}{2}\Delta}.$ (5)

In this case, contrary to the original HDE model with the G-O cutoff (for
$\Delta=0$), $\alpha$ and $\beta$ are parameters with dimension
$[L]^{\frac{2\Delta}{\Delta-2}}$, for the sake of preserving the dimensions of
$\rho_{\Lambda}$. Furthermore, note that in this modified version of
$\rho_{\Lambda}$ the parameter $C$ in Eq. (3) has been replaced by
$3M_{p}^{2}$ (considering $c=1$), since in our case, the role played by $c$ in
Eq. (3) via $C=3M_{p}^{2}c^{2}$ is performed now by $\alpha$ and $\beta$.

Since observations suggest that the universe is homogeneous and isotropic at
large scales, we consider a flat Friedmann-Robertson-Walker (FRW) geometry
with metric

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

where $a(t)$ is the scale factor, and the content of the universe at large-
scale is taken to be a perfect fluid. The first Friedmann equation for a
universe made of non relativistic matter, radiation and DE is given by

$3M_{p}^{2}H^{2}=\rho_{m}+\rho_{r}+\rho_{\Lambda}.$ (7)

Considering these components as barotropic fluids ($p_{i}=w_{i}\rho_{i}$, with
$i=m,r$), matter ($w_{m}=0$) and radiation ($w_{r}=1/3$) behave as

$\rho_{m}=\rho_{m_{0}}a^{-3},\qquad\rho_{r}=\rho_{r_{0}}a^{-4},$ (8)

which together with Eq. (5) for $\rho_{\Lambda}$, Eq. (7) takes the form

$H^{2}=\Omega_{m_{0}}H_{0}^{2}a^{-3}+\Omega_{r_{0}}H_{0}^{2}a^{-4}+\left(\alpha
H^{2}+\beta\dot{H}\right)^{1-\frac{1}{2}\Delta},$ (9)

where $\Omega_{i_{0}}H_{0}^{2}=\frac{\rho_{i_{0}}}{3M_{p}^{2}}$. In the
following calculations, we shall perform a change of variable in order to
align better our study with the late-time dynamics, i.e., we use the redshift
$z$ instead of the cosmic time $t$ as the dynamical parameter. From its
definition,

$1+z=\frac{1}{a(t)},$ (10)

where we assumed that $a(t=0)=a_{0}=1$. From the above, the time derivatives
can be expressed in terms of derivatives with respect to the redshift, using
the following rule:

$\frac{d}{dt}=-H(1+z)\frac{d}{dz}.$ (11)

Implementing such a change, Eq. (9) takes the form

$(1+z)\frac{\beta}{2}\frac{dH^{2}}{dz}-\alpha
H^{2}+\left[H^{2}-\Omega_{m_{0}}H_{0}^{2}(1+z)^{3}-\Omega_{r_{0}}H_{0}^{2}(1+z)^{4}\right]^{\frac{2}{2-\Delta}}=0,$
(12)

where it is required that
$\Omega_{m_{0}}+\Omega_{r_{0}}+\Omega_{\Lambda_{0}}=1$111Specifically, we take
$\Omega_{m_{0}}=0.315$ , $\Omega_{r_{0}}=3\times 10^{-4}$,
$\Omega_{\Lambda_{0}}=0.6847$, and also $H_{0}=67.37$ km/s/Mpc Planck:2018vyg
., with the ‘0’ index indicating the corresponding present ($z=0$) value for
each quantity i.e. matter density $\Omega_{m_{0}}$ and the Hubble parameter
$H_{0}$.

The complexity of Eq. (12) prevents us to find an analytic solution, but it is
still possible to solve it numerically and study how the Hubble parameter
varies with $z$. This evolution is shown in Fig. 1, where $\alpha$ and $\beta$
are selected close to the ones reported in Ref. Wang:2010kwa , which
considered a model with $\Omega_{r_{0}}=0$ and without the effect of the
deformation parameter, $\Delta$. The left panel of Fig. 1 shows that the
evolution of $H(z)$ is barely modified when different values of $\Delta$ are
considered, specially for $z\geq 0$, though increasing $\Delta$ makes $H(z)$
steadily smaller, as can be seen in the zoomed region included in this figure.
Notwithstanding, notice that the impact of the deformation parameter becomes
more clear in the future ($z<0$), and the opposite effect is observed, i.e.,
for larger values of $\Delta$, the Hubble parameter increases. This behavior
is compatible with the one obtained by other authors; for instance, see
Huang:2021zgj ; Rani:2021hvh .

Figure 1: (Left) Evolution of the Hubble parameter, $H(z)$, as a function of
the redshift $z$, for different values of $\Delta$. (Right) Comparison of the
proposed model against $\Lambda$CDM, for different values of $\Delta$, using
the dimensionless Hubble parameter through Eq. (13).

It is possible to compare the model proposed here against the $\Lambda$CDM
standard model using the dimensionless Hubble parameter $E(z)=H(z)/H_{0}$, and
calculating

$\Delta E(z)=100\times\left[\frac{E(z)}{E(z)_{\Lambda\rm{CDM}}}-1\right],$
(13)

which, by definition, is zero for the $\Lambda$CDM model (black-dashed line in
the right plot of Fig. 1).

The evolution of $\Delta E(z)$ is shown in the right plot of Fig. 1. One can
see that our model gives $\Delta E(z)>0$, indicating that $H(z)$ is larger
than the $\Lambda$CDM prediction. However, increasing the deformation
parameter leads negative values of $\Delta E(z)$ near the present epoch,
$z=0$. In particular, for $\Delta=2\times 10^{-4}$, there is a difference of
the order of $\sim-0.5\%$ until $z\sim 0.2$, an epoch during which we observe
a phantom regime with $w_{\Lambda}<-1$ (see Fig. 3 and the discussion about
it, later in this section). As depicted in the inner plot, the biggest
deviation from $\Lambda$CDM occurs up to early times around $z\sim
10^{2}-10^{3}$. For even earlier times ($z>10^{4}$), the difference reduces
reaching a plateau for some $\Delta$’s at around a few percents, and going
down to $<1\%$ for large values of the deformation parameter.

Two other important quantities to study are the deceleration parameter,
$q(z)$, and the DE equation of state (EoS), $w_{\Lambda}(z)$. The first one,
the deceleration parameter, which is defined as

$q=-1-\frac{\dot{H}}{H^{2}}\ \ \ \Rightarrow\ \ \
q(z)=-1+\frac{(1+z)}{2H^{2}}\frac{dH^{2}}{dz},$ (14)

(where we have performed the change of variable as described earlier,
considering Eqs. (10) and (11)), is also computed numerically by means of
solving Eq. (12). The obtained evolution with the redshift is presented in
Fig. 2, with the same values of the parameters ($\alpha,\beta$) used before
and for different values of the deformation parameter, $\Delta$.

Figure 2: Evolution of the deceleration parameter, $q(z)$, as a function of
the redshift $z$, for different values of $\Delta$.

As clearly visible in Fig. 2, the deformation parameter changes markedly the
evolution of the deceleration parameter. For all the cases, the transition
from a decelerated ($q>0$) to an accelerated ($q<0$) expansion of the universe
is continuous and smooth, and the different values for the transition
redshift, $z_{T}$ (shown in the inner panel) are compatible with those
obtained from fits of recent observations Jesus:2019nnk . We have noticed that
fixing $\Delta$ and allowing the other parameters ($\alpha$ or $\beta$) to
vary once at a time, produces similar results, though $z_{T}$ exhibits
significantly larger changes in those cases, pointing to stringent
restrictions to those parameters.

Regarding to the EoS parameter $w_{\Lambda}$, using the same change of
variable described above, we have

$w_{\Lambda}=-1-\frac{2}{3}\frac{\dot{H}}{H^{2}}\ \ \ \Rightarrow\ \ \
w_{\Lambda}(z)=-1+\frac{(1+z)}{3H^{2}}\frac{dH^{2}}{dz}.$ (15)

Looking at the evolution of the $w_{\Lambda}$ plotted for different values of
$\Delta$ (Fig. 3), one can see that it has an asymptotic behavior towards a
radiation-type equation of state for $\Delta=0$ at early times (left panel),
though increasing $\Delta$ reduces the value of the asymptote (see the zoomed
region inside the left plot). Also, at present time ($z=0$), as $\Delta$
increases, the DE EoS exhibits (right panel of Fig. 3) a transition from a
quintessence regime ($w_{\Lambda}>-1$) to a phantom regime ($w_{\Lambda}<-1$),
crossing the cosmological constant case ($w_{\Lambda}=-1$). Interestingly, for
$z<0$ (the future), there is a quintessence-phantom transformation for certain
values of the parameters.

Figure 3: Evolution of the EoS, $w_{\Lambda}(z)$, as a function of the
redshift $z$. See text for a detailed description.

For $w_{\Lambda}$ we have also studied the effect of fixing $\Delta$ and
varying $\alpha$ or $\beta$, at a time. The results indicate that the passage
from a quintessence regime to a phantom one occurs when $\alpha$ ($\beta$)
decreases (increases), a behavior that is characteristics of a kind of models
known as Quintom Feng:2004ff .

In the left plot of Figure 4, the evolution of the densities involved in the
model is shown for different values of $\Delta$, as a function of the number
of e-folds, $x=ln(a)$. Our model exhibits an initial era of radiation
dominance, followed by the non-relativistic matter and the current era of DE
dominance. As can be seen, our model exhibits a non negligible contribution of
the DE component at early times, when compared against radiation; this is an
opposite behavior of that predicted by the $\Lambda$-CDM model as shown in the
left panel of Figure 4, where the dark energy component, $\Omega_{\Lambda}$,
is nonzero for different values of $\Delta$ in the distant past, behavior
associated with the extra terms of (effective) matter and (effective)
radiation added through the G-O IR cutoff to $H(z)$. According to the
Holographic principle the energy density is proportional to $L^{\Delta-2}$.
Since the length scale is expected to be very small during early times, the
energy density generated by our model should be large enough to support the
inflationary scenario. In fact, it is proposed as a possible candidate for
inflation Maity:2022gdy .

Figure 4: (Left) Evolution of radiation, matter and dark energy densities as
function of the number of e-folds for different values of $\Delta$. (Right)
Evolution of the DE density $\Omega_{\Lambda}$ for different values of
$\Delta$ and $\Lambda$-CDM, as a function of the redshift $z$.

###  2.1 Stability of the model

Studying the stability of this kind of models is of paramount relevance. To do
so, one can examine the square of the speed of sound, $v_{s}^{2}$, as a
function of the redshift, for some values of the parameters of the model.
Defined as

$v_{s}^{2}=\frac{d\,p}{d\,\rho},$ (16)

where $p$ is the pressure and $\rho$ the density of the fluid under
investigation, a model would reveal an unstable behavior if $v_{s}^{2}<0$. For
the model considered in this work, the evolution of $v_{s}^{2}$ is shown in
Fig. 5 for different values of $\Delta$ (fixing the other parameters), and in
Fig. 6 when $\alpha$ (left) or $\beta$ (right) change (fixing $\Delta$).

Figure 5: Stability of the model under consideration trough the evolution of
$v_{s}^{2}(z)$ as a function of the redshift, for different values of the
deformation parameter, $\Delta$. The sharp peak in a curve indicates the
presence of instabilities as $v_{s}^{2}$ becomes negative.

Figure 6: As in Fig. 5, but changing $\alpha$ (left) and $\beta$ (right),
while the other parameters are fixed as indicated on the top of each plot.

Figure 7: Parameter space $(\alpha,\beta)$ regions for a stable model (blue)
for $\Delta=0$ (top left), $\Delta=1\times 10^{-3}$ (top right),
$\Delta=20\times 10^{-3}$ (bottom left), $\Delta=50\times 10^{-3}$ (bottom
right). The stability criterion is $0\leq v_{s}^{2}\leq 1$.

Because of the logarithmic scale of the vertical axis of Figs. 5 and 6, the
curves with a sharp peak correspond to those cases for which $v_{s}^{2}(z)$
becomes negative. That is the case for the model when $\alpha$ and $\beta$ are
fixed and $\Delta$ increases, and also when $\alpha$ ($\beta$) decreases
(increases), while fixing the other parameters. Notice that, nevertheless, as
can be seen in Figure 7, most of the considered values for the three
parameters are well within the stable region (dark blue). It is also worth
mentioning that the stability zones display a reduction and displacement when
$\Delta$ increase.

##  3 Observational constraints

The thorough study described in Sec. 2.1 allows us to find that the stability
of the model is preserved when the parameters take the values in the ranges
presented in Table 1. This parameter space (see Fig. 8) is obtained by further
requiring that $w_{0}<-1/3$ and $q_{0}<0$, so that the physical results are
consistent with current observations. Negative values for $\alpha$ and $\beta$
were also excluded for not complying with the imposed restrictions.

Figure 8: Parameter space $(\alpha,\beta)$ regions for a stable model (blue) for $\Delta=0$ (left) and $\Delta=50\times 10^{-3}$ (right). The stability criterion is $0\leq v_{s}^{2}\leq 1$ and we also require that $w_{0}<-1/3$ and $q_{0}<0$. Table 1: Parameter ranges considered for the fit. Parameter | Considered range
---|---
$\alpha$ | [0.7 , 1.0]
$\beta$ | [0.3 , 0.9]
$\Delta$ | [0.0 , 30 $\times 10^{-3}$ ]
|

The analysis reported in this section is performed by comparing the values of
$H(z)$ as calculated with our model against recent observations, in order to
fit the parameters of the model. The specific observational data consists of
36 measurements as reported in Ref. Cao:2021uda  and shown here in Fig. 12.
The statistical analysis is implemented through a standard $\chi^{2}$ function
defined as

$\chi^{2}_{H(z)}(\theta)=\sum_{i=1}^{36}\frac{\left(H_{th}(\theta,z_{i})-H_{obs}(z_{i})\right)^{2}}{\sigma^{2}_{i}},$
(17)

where $\theta=(\alpha,\beta,\Delta)$ is the vector of parameters to be fitted,
$H_{th}$ and $H_{obs}$ are the predicted and measured values of the Hubble
parameter, for different $z_{i}$, respectively, and $\sigma_{i}$ is the error
associated to observation $i$.

In Figures 9, 10 and 11 we show the results of the analysis. In each Figure,
the central 2D contours correspond to the allowed regions at $1\sigma$,
$2\sigma$, $3\sigma$222Corresponding to 68.27%, 95.45% and 99.73% confidence
level (C.L.), respectively ParticleDataGroup:2020ssz . of the respective (2D)
parameter space (i.e., a pair of parameters of the model), resulting from the
marginalization of the third parameter (not shown in each case). The 1D
allowed intervals for each parameter are also shown in the top and left plots.

Figure 9: Allowed regions for the parameter space ($\alpha,\beta$) at 68,27%,
95,45%, 99,73% C.L., as obtained from the fit. Figure 10: Allowed regions for
the parameter space ($\alpha,\Delta$) at 68,27%, 95,45%, 99,73% C.L., as
obtained from the fit. Figure 11: Allowed regions for the parameter space
($\beta,\Delta$) at 68,27%, 95,45%, 99,73% C.L., as obtained from the fit.

The three set of Figures are consistent among each other in the sense that the
obtained allowed regions for each parameter are clearly compatible, and the
best fit (i.e., the set of parameters that minimize $\chi^{2}_{H(z)}$, Eq.
(17)) is found at $(\alpha,\,\beta,\,\Delta)=(1.0,\,0.69,\,0.0)$, with the
overall allowed intervals as shown Table 2. The contour plots suggest that the
combined observational data of $H(z)$ provide a strong constraint on $\beta$,
while the most likely value of $\Delta$ is the minimum ($\Delta=0$) within the
region of $1\sigma$, in agreement to what was recently reported in Ref.
Anagnostopoulos:2020ctz , for which the entropy-area relation is consistent
with the Bekenstein-Hawking entropy.

Table 2: Best fit and allowed regions for the parameters of the model. Parameter | Best fit | $1\sigma$ | $2\sigma$ | $3\sigma$
---|---|---|---|---
$\alpha$ | 1.00 | (0.98, 1.02) | (0.94, 1.06) | (0.89, 1.11)
$\beta$ | 0.69 | (0.67, 0.72) | (0.62, 0.75) | (0.56, 0.78)
$\Delta$ | 0.00 | (0.00, 0.004) | (0.00, 0.016) | (0.00, 0.029)
|  |  |  |

For comparison, the Hubble parameter as a function of the redshift predicted
by our model is represented by the black-dashed line in Fig. 12, where the
observational data are also drawn, together with the prediction of the
$\Lambda$CDM model (blue-full line). The shadowed (orange) region represents
the prediction within the $3\sigma$ variation of the parameters of our model
(see Table 2). Our proposal not only presents a similar behavior compared to
$\Lambda$CDM, but more importantly, it fits very well to the observational
data, with $\chi^{2}/dof=22.6/33$333$dof$ stands for _degrees of freedom_ ,
calculated as the difference between the number of data points and the number
of free parameters..

Figure 12: Comparison of $H(z)$ predicted by our model using the best fit
values of the parameters shown in Table 2 (black-dashed line), against the
observational data from Cao:2021uda . The shadowed-orange area gives the
prediction obtained when the parameters are varied within their $3\sigma$
allowed region, and the blue-full line is the $\Lambda$CDM prediction.

In addition, implementing our model with the best fit values for the
parameters (Table 2), we can compute the current value of the EoS parameter,
obtaining $w_{0}=-1.016^{+0.009}_{-0.005}$, which is consistent with the
cosmological constant ($w_{0}=-1$) at $1.8\sigma$. An estimate of the
transition redshift is also computed, and we get
$z_{t}=0.658^{+0.020}_{-0.012}$, as well as for the deceleration parameter at
present times, $q_{0}=-0.543^{+0.009}_{-0.005}$. These results are comparable
to previously reported ones, $z_{t}=0.65^{+0.10}_{-0.07}$ Mamon:2017rri  and
$q_{0}=-0.6401\pm 0.187$ Capozziello:2018jya , respectively. Furthermore, we
estimate the age of the Universe to be $t_{0}=14.24^{+0.11}_{-0.06}$ Gyr,
relatively close to the value reported by Planck 2018 Planck:2018vyg ,
$t_{0}=13.79\pm 0.02$ Gyr.

##  4 Conclusions

In this work we have studied the effects on the cosmological evolution of the
universe for late times, taking into account the Granda–Oliveros infrared
cutoff in the recently introduced Barrow Holographic Dark Energy model. First,
the evolution of $H(z)$ was analyzed (see left panel of Fig. 1) and we found
that the evolution of $H(z)$ is hardly distinguishable when different values
of $\Delta$ are used, specially for $z\geq 0$; however, it was noted that,
increasing $\Delta$ makes $H(z)$ smaller. Also, the differences against the
$\Lambda$CDM prediction (right panel of Fig. 1) are significantly reduced at
earlier times. Secondly, studying the deceleration parameter (Fig. 2) we
showed that our model predicts an accelerated expansion regime of the universe
at late times. We also verified that increasing $\Delta$ causes the EoS
parameter to exhibit a transition from quintessence to phantom regimes, both
at late and future times (see right panel of Fig. 3), while at early times the
asymptotic behavior towards a radiation-type EoS is reduced for larger
$\Delta$ (see left panel of Fig. 3). Moreover, our model exhibits the know
eras of dominance (radiation, cold matter and DE), in which at early times the
DE component may not be negligible compared to radiation (see Fig. 4).

We studied the zones of stability before perturbations (see Fig. 5 and 6) from
the early period to the present time, however, we found that these regions
exhibit a reduction and displacement when $\Delta$ increases (see Fig. 7), and
excludes the negative values of $(\alpha,\beta)$ as that part of the parameter
space would not be consistent with physical results (see Fig. 8).

At last, we set constraints to the model parameters (Table 2) from a fit to
observational data of $H(z)$, which suggests that the growth of $\Delta$ may
not favor the model, and provide a rather strict restriction on $\beta$ (see
Figs. 9 and 11). From the fit, $\Delta=0.0$ is found to be the most likely
value (see Fig. 10 and 11), for which the Bekenstein-Hawking relation is
favored. However, it would be necessary to use a more complete set of
observational data to obtain a best fit on the parameters of the model, e.g.,
using updated measurements from the dynamics of the expansion of the universe,
$H(z)$, and the growth rate of cosmic structures, $[f\sigma_{8}](z)$, or
$H(z)$ and data from Supernovae (SNIa) Pantheon sample, among others, but that
kind of analysis is beyond the scope of this work and could be addressed
later.

Acknowledgments A.O. acknowledges financial support from Patrimonio
Autónomo–Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la
Innovación Francisco José de Caldas (MINCIENCIAS-COLOMBIA) Grant No.
110685269447 RC-80740-465-2020, projects 69723 and 69553. This work has been
partially performed using the software Mathematica.

## Statements and Declarations

  * •

Availability of data and materials. All data that have been used in our
analysis have already been freely released and have been published by the
corresponding research teams. In our text we properly give all necessary
References to these works, and hence no further data deposit is needed.

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