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Complete observational bounds on a fractal horizon holographic dark energy

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Abstract

A novel fractal structure for the cosmological horizon, inspired by COVID-19 geometry, which results in a modified area entropy, is applied to cosmology in order to serve dark energy. The constraints based on a complete set of observational data are derived. There is a strong Bayesian evidence in favor of such a dark energy in comparison to a standard ΛCDM model and that this energy cannot be reduced to a cosmological constant. Besides, there is a shift towards smaller values of baryon density parameter and towards larger values of the Hubble parameter, which reduces the Hubble tension. Introduction. Black holes and cosmological horizons are very strongly explored phenomena in physics recently because they make the link between the classical and the quantum in the context of gravity. Through the Hawking temperature [1] and Bekenstein area entropy [2], they allow thermodynamics to be related to the classical geometry of space. So, any modification of geometry will influence the entropy related to the horizons. A modification of fractal nature has been recently proposed by Barrow [3] who was inspired by the COVID-19 virus geometrical structure. The idea is to consider the core sphere of the horizon with the attached number of heavily packed smaller spheres, to each of which some smaller spheres are latched on and so on, forming a fractal. Simple calculations which apply geometrical series allow to add up all the surfaces of such a hierarchical system. The resulting surface A ef f with an effective radius r ef f = r 1+∆/2 (0 ≤ ∆ ≤ 1) is larger than the core sphere surface A of radius r. The appropriate areas are A ef f ∝ r 2 ef f and A ∝ r 2 (so r ∝ A 1/2). When ∆ = 1 one has the most intricate surface of the horizon with the COVID-19-like fractal geometry. Modification of the horizon area immediately leads to a change of the effective Bekenstein entropy, making it larger than in a smooth case, according to the formula
Complete observational bounds on a fractal horizon holographic dark energy
Mariusz P. D¸abrowski1, 2, 3, aand Vincenzo Salzano1 , b
1Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
2National Centre for Nuclear Research, Andrzeja So ltana 7, 05-400 Otwock, Poland
3Copernicus Center for Interdisciplinary Studies, Szczepa´nska 1/5, 31-011 Krak´ow, Poland
(Dated: June 30, 2020)
A novel fractal structure for the cosmological horizon, inspired by COVID-19 geometry, which
results in a modified area entropy, is applied to cosmology in order to serve dark energy. The
constraints based on a complete set of observational data are derived. There is a strong Bayesian
evidence in favor of such a dark energy in comparison to a standard ΛCDM model and that this
energy cannot be reduced to a cosmological constant. Besides, there is a shift towards smaller values
of baryon density parameter and towards larger values of the Hubble parameter, which reduces the
Hubble tension.
Introduction. Black holes and cosmological horizons
are very strongly explored phenomena in physics recently
because they make the link between the classical and the
quantum in the context of gravity. Through the Hawking
temperature [1] and Bekenstein area entropy [2], they al-
low thermodynamics to be related to the classical geome-
try of space. So, any modification of geometry will influ-
ence the entropy related to the horizons. A modification
of fractal nature has been recently proposed by Barrow
[3] who was inspired by the COVID-19 virus geometri-
cal structure. The idea is to consider the core sphere of
the horizon with the attached number of heavily packed
smaller spheres, to each of which some smaller spheres
are latched on and so on, forming a fractal. Simple cal-
culations which apply geometrical series allow to add up
all the surfaces of such a hierarchical system. The result-
ing surface Aeff with an effective radius reff =r1+∆/2
(0 1) is larger than the core sphere surface A
of radius r. The appropriate areas are Aef f r2
eff and
Ar2(so rA1/2). When ∆ = 1 one has the most
intricate surface of the horizon with the COVID-19-like
fractal geometry. Modification of the horizon area im-
mediately leads to a change of the effective Bekenstein
entropy, making it larger than in a smooth case, accord-
ing to the formula
Seff Aeff r2
eff r2+∆ A1+
2,(1)
where we can take as the smooth core sphere radius
reither the black hole Schwarzschild radius rsor the
cosmological horizon length L. The exact formula in
terms of the Planck area AP l is given by [3], Seff =
kB(A/APl )1+∆/2, where kBis the Boltzmann constant.
In cosmology, there has been a vivid discussion of pos-
sible explanations of the dark energy phenomenon as
contributed from cosmological horizons, leading to holo-
graphic dark energy [4]. It emerges that the fractal hori-
zon can extra contribute to the matter. This has been
first calculated in [5] and then tested in [6] using only few
probes.
In this letter we apply the full set of cosmological data
up-to-date to derive the most comprehensive bounds on
the holographic and fractal parameters. The issue is that
they strongly differ from the ones obtained in [5], as we
will show later.
Theoretical background. According to [4] the holo-
graphic dark energy is given by ρHSeff L4with the
effective Bekenstein entropy Seff Aef f L2+∆ , where
Lis the horizon length. Thus, following [5] we can ex-
press Barrow holographic dark energy (BH) as:
ρBH =3C2
8πG L2(
21),(2)
where Cis the holographic parameter with dimensions of
[T]1[L]1/2and Gthe Newton gravitational constant.
As suggested in [7], we identify the length Lwith the
future event horizon:
LaZ
t
dt0
a=aZ
a
da0
H(a0)a02,(3)
where ais the scale factor. The cosmological equation is
simply
H2=8πG
3(ρm+ρr+ρBH ),(4)
where the suffices mand rrefer respectively to mat-
ter and radiation. Note that the standard continuity
equation for matter and radiation is still valid, ˙ρm,r +
3H(ρm,r +pm,r/c2) = 0, where the pressure pi=wiρi,
with the equation of state parameter wibeing 0 for
standard pressureless matter and 1/3 for radiation. We
rewrite Eq. (4) as: 1 = Ωm(a)+Ωr(a)+ΩH(a), intro-
ducing the dimensionless density parameters Ωi(a) [8],
defined as Ωm,r(a) = H2
0/H2(a)Ωm,ra3(1+wm,r )and
BH (a) = C2
H2(a)L2(
21).(5)
Combining previous equations one can express the Hub-
ble parameter as
H(a) = H0sma3+ Ωra4
1BH (a).(6)
2
In order to find the evolution of the holographic dark
energy density, we follow [7] and [5] procedure: (1.) we
insert Eq. (6) into Eq. (3); (2.) we obtain the future
event horizon from inversion of Eq. (5); (3.) we compare
results from step (1) and step (2) and differentiate both
of them w.r.t. to a. We end up with the following differ-
ential equation for the BH dark energy (where prime is
derivative with respect to a):
a0
H(a) = 1 +
2Fr(a) (7)
+ ΩH(a) (1 H(a)) 1+2
2Fm(a)
+ (1 H(a))
/2
2(
2
1)H(a)
1
2(1
2)Q(a)#,
with:
Fr(a) = 2Ωra4
ma3+ Ωra4,(8)
Fm(a) = ma3
ma3+ Ωra4,
Q(a)=21
2H0pma3+ Ωra4
/2
1
2C
1
2
1.
It is easy to verify that in the limit Ωr0 one retrieves
Eq. (14) from [5].
Statistical Analysis. To analyze in full detail the com-
patibility of BH with cosmological data, we use the most
updated set of data available today related to the geo-
metrical global evolution of our Universe at large scales.
We consider: Type Ia Supernovae (SNeIa) from the Pan-
theon sample; Cosmic Chronometers (CC); the gravi-
tational lensing data from COSMOGRAIL’s Wellspring
project (H0LiCOW); the “Mayflower” sample of Gamma
Ray Bursts (GRBs); Baryon Acoustic Oscillations (BAO)
from several surveys; and the latest Planck 2018 release
for Cosmic Microwave Background radiation (CMB).
We consider two different cases: the set which we call
“full data”, where we join both early- (CMB and BAO
data from SDSS) and late-time observations (SNeIa, CC,
H0LiCOW, GRBs and BAO data from WiggleZ); and
the “late-time” data set, which includes only late-time
data. We have decided to consider these two cases sep-
arately, because while on the one hand early-time data
have much more stringent constraining power in cosmo-
logical model inference than late-time ones, on the other
hand, they seem to be biased to statistically support a
standard ΛCDM model, i.e. a cosmological constant as
dark energy. By separating data in such a way, we could
aim to have some more insight into a possible presence
of a time varying dark energy candidate.
To perform our statistical analysis, we define the to-
tal χ2as the sum of all the contributions considered,
χ2=χ2
SN +χ2
G+χ2
H+χ2
HCOW +χ2
BAO +χ2
CM B . To
minimize the χ2we use our own code implementation
of a Monte Carlo Markov Chain (MCMC) [911] and
we test its convergence using the method of [12]. Fi-
nally, we assess BH reliability using Bayesian Evidence,
E. Our reference model is the standard ΛCDM model,
analyzed with the same set of data. Then, we calculate
the Bayesian Evidence using the algorithm from [13]. To
reduce its prior dependence [14] and avoid any mislead-
ing estimation, we have used the same uninformative flat
priors on the parameters for each model while running
our MCMC codes, on a sufficiently wide range, so that a
further increasing has negligible impact on E. Such pri-
ors are mainly physically motivated: 0 <b<m<1,
0< h < 1, 0 1 [3], and C > 0 (given Eq. 2, we
cannot discriminate among positive and negative values).
After the Bayesian Evidence, we define the Bayes Factor
as the ratio of evidence between two models, Miand Mj,
Bi
j=Ei/Ej: if Bi
j>1, model Miis preferred over Mj,
given the data. As stated above, here the ΛCDM model
will play the role of the reference models Mj. Finally, in
order to state how much better is model Miwith respect
to model Mj, we have followed the Jeffreys’ Scale [15].
Type Ia Supernovae. The Pantheon compilation [16] is
made of 1048 objects spanning the redshift range 0.01 <
z < 2.26. The corresponding χ2
SN is defined as χ2
SN =
µSN ·C1
SN ·µS N , where ∆µ=µtheo µobs is
the difference between the theoretical and the observed
value of the distance modulus for each SNeIa and CSN
the total covariance matrix. Note that we do not use the
binned version as in [6], but the full one. The distance
modulus is defined as µ(z, p) = 5 log10 [dL(z, p)] + µ0,
where the dimensionless luminosity distance dL(z, p) =
(1 + z)dM(z, p) with
dM(z, p) = Zz
0
dz0
E(z0,p)(9)
the dimensionless comoving distance and θthe vector
of cosmological parameters. Because of the degeneracy
between the Hubble constant H0and the SNeIa absolute
magnitude (both included in the nuisance parameter µ0),
we marginalize the χ2
SN over µ0following [17], obtaining
χ2
SN =a+log d/(2π)b2/d, where a(∆µS N )T·C1
SN ·
µSN ,bµS N T·C1
SN ·1,d1·C1
SN ·1and
1is the identity matrix.
Cosmic Chronometers. The definition of CC is used
for Early-Type galaxies which exhibit a passive evolution
and a characteristic feature in their spectra [18], for which
can be used as clocks and provide measurements of the
Hubble parameter H(z) [19]. The sample we are going
to use in this work is from [20] and covers the redshift
range 0 < z < 1.97. The χ2
His defined as
χ2
H=
24
X
i=1
(H(zi,p)Hobs(zi))2
σ2
H(zi),(10)
where σH(zi) are the observational errors on the mea-
sured values Hobs(zi).
3
H0LiCOW. H0LiCOW [21] has used 6 selected lensed
quasars [22] for which it was possible to retrieve multiple
(lensing) images. When multiple images are produced,
they can exhibit a time delay at collection given by
t(θ,β) = 1 + zL
c
DLDS
DLS 1
2(θβ)2ˆ
Ψ(θ).(11)
In a typical gravitational lensing configuration [23], zL
is the lens redshift, θthe angular position of the im-
age, βthe angular position of the source and ˆ
Ψ the ef-
fective lens potential. The distances DS,DLand DLS
are, respectively, the angular diameter distances from the
source to the observer, from the lens to the observer,
and between source and lens. The angular diameter dis-
tance is given by DA(z, p) = DM(z, p)/(1 + z), where
the comoving distance is DM(z, p) = c/H0dM(z, p).
Thus, we have: DS=DA(zS), DL=DA(zL), and
DLS = 1/(1 + zS) [(1 + zS)DS(1 + zL)DL] [24]. The
combination of distances which appears in Eq. (11),
Dt(1 + zL)DLDS/DLS , is generally called time-
delay distance and is constrained by H0LiCOW. The
data (Dobs
t,i) and the corresponding errors (σDt,i ) on
this quantity for each of the 6 considered quasars are
provided in [22]. Eventually, the χ2for H0LiCOW data
is
χ2
HC OW =
6
X
i=1 Dt,i(p)Dobs
t,i2
σ2
Dt,i
,(12)
Gamma Ray Bursts. Although the possibility to
“standardize” GRBs is still on debate, we focus on the
“Mayflower” sample, made of 79 GRBs in the redshift
interval 1.44 < z < 8.1 [25], because it has been cal-
ibrated with a robust cosmological model independent
procedure. The observational probe related to GRBs
observable is the distance modulus, so the same proce-
dure used for SNeIa is also applied here. The χ2
Gis thus
given by χ2
GRB =a+ log d/(2π)b2/d as well, with
aµGT·C1
G·µG,bµGT·C1
G·1and
d1·C1
G·1.
Baryon Acoustic Oscillations. For BAO we consider
multiple data sets from different surveys. In general, the
χ2defined as χ2
BAO = ∆FBAO ·C1
BAO ·FBAO , has
observables FBAO which change from survey to survey.
When we employ the data from the WiggleZ Dark
Energy Survey (at redshifts 0.44, 0.6 and 0.73) [26],
the relevant physical quantities are the acoustic param-
eter A(z, p) = 100mh2DV(z, p)/(c z), where h=
H0/100, and the Alcock-Paczynski distortion parame-
ter F(z, p) = (1 + z)DA(z, p)H(z, p)/c, where DAis
the angular diameter distance and DV(z, θ) = [(1 +
z)2D2
A(z, θ)cz/H(z, θ)]1/3is the geometric mean of the
radial (H1) and tangential (DA) BAO modes. Note
that this data set is independent of early-time evolution,
thus it is included in the late-time data analysis.
We also consider data from multiple analysis of SDSS-
III Baryon Oscillation Spectroscopic Survey (BOSS) ob-
servations. Each of the following data is used for the full
data analysis but not for the late-time one.
In the DR12 analysis described in [27], the follow-
ing quantities are given: DM(z , p)rfid
s(zd)/rs(zd,p) and
H(z)rs(zd,p)/rf id
s(zd), where the sound horizon evalu-
ated at the dragging redshift is rs(zd), while rf id
s(zd) is
the sound horizon calculated for a given fiducial cosmo-
logical model (in this case, it is 147.78 Mpc). The drag-
ging redshift is estimated using the analytical approxi-
mation provided in [28]. Finally, the sound horizon is
defined as:
rs(z, p) = Z
z
cs(z0)
H(z0,p)dz0,(13)
with the sound speed cs(z) = c/q3(1 + Rb(1 + z)1),
and the baryon-to-photon density ratio parameter Rb=
31500Ωbh2(TCM B /2.7)4, with TCMB = 2.726 K.
From the DR12 we also include measurements de-
rived from the void-galaxy cross-correlation [29]: DA(z=
0.57)/rs(zd)=9.383 ±0.077 and H(z= 0.57)rs(zd) =
(14.05 ±0.14)103km s1.
From the extended Baryon Oscillation Spectroscopic
Survey (eBOSS) we use the point DV(z= 1.52) =
3843 ±147 rs(zd)/rf id
s(zd) Mpc [30]. Finally, we also
take into account data from eBOSS DR14 obtained from
the combination of the Quasar-Lyman αautocorrela-
tion function [31] with the cross-correlation measurement
[32]: DA(z= 2.34)/rs(zd) = 36.98+1.26
1.18 and c/[H(z=
2.34)rs(zd)] = 9.00+0.22
0.22.
Cosmic Microwave Background. As CMB data we use
the shift parameters defined in [33] and derived from
the latest Planck 2018 data release [34]. The χ2
CM B is
defined as χ2
CM B = ∆FCMB ·C1
CM B ·FCMB ,
where the vector FCM B is made of the quantities
R(p)pmH2
0r(z,p)/c,la(p)πr(z,p)/rs(z,p)
and Ωbh2. Here rs(z) is the comoving sound horizon
evaluated at the photon-decoupling redshift evaluated us-
ing the fitting formula from [35], while ris the comoving
distance at decoupling, i.e. r(z,p) = DM(z,p).
Discussion and Conclusion. We display constraints on
cosmological parameters from our MCMC analysis in Ta-
ble I, and the convergence test on MCMC runs in Ta-
ble II. The posterior distributions for each parameter are
shown in Fig. 1.
If we compare our analysis to the one in [6], we can
definitely assess that late-time-only observations are un-
able to constrain BH dark energy parameters, with the
fractal parameter ∆ spanning the full range of validity,
and the BH characteristic energy scale Cbeing well de-
fined (contrarily to what found in [6]), but with large
errors. The addition of early-time data is crucial: both
parameters’ confidence levels are now highly narrowed,
as can be seen in the left panel of Fig. 2. Note that the
4
0.20 0.25 0.30 0.35 0.40
0.0
0.2
0.4
0.6
0.8
1.0
Ωm
0.60 0.65 0.70 0.75 0.80
0.0
0.2
0.4
0.6
0.8
1.0
h
0.035 0.040 0.045 0.050 0.055
0.0
0.2
0.4
0.6
0.8
1.0
Ω
b
0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Δ
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
C
Figure 1. Normalized histograms for each cosmological parameter. In green: late-time analysis; in blue: full cosmological data
set. Dashed: ΛCDM model; solid: Barrow Holographic dark energy.
Figure 2. Joint contours for the Barrow Holographic dark energy parameters (left panel) and for the baryonic density parameter
vs Hubble constant (right panel). Colors as in previous figures. Hard colors: 68% confidence levels; soft colors: 95% confidence
levels.
value of the fractal parameter ∆ totally rejects both the
lower (∆ = 0) and the upper (∆ = 1) limit.
The most striking result is given by the Bayes ra-
tio: given Jeffreys’ scale, there is “strong evidence”,
ln Bi
j3.5, in favour of BH dark energy w.r.t. a stan-
dard ΛCDM. This is a very surprising claim taking into
account the outside of cosmological origin (i.e. COVID-
19-like) nature of this type of dark energy, reinforced by
the fact that the BH dark energy cannot be reduced to
a cosmological constant.
Moreover, if we pay more attention to the values of the
cosmological parameters, we can see that this statistical
preference is led by a shift in both the Hubble constant, h,
and the baryonic content, Ωb(this is an important point,
because Ωbis called into question only when dealing with
early-time data). As shown in the right panel of Fig. 2,
we have a shift toward smaller values of Ωb, partially
consistent with the ΛCDM scenario on the upper tail,
5
0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
a
ΩΛ(a)vs. ΩBH(a)
10-40.001 0.010 0.100 1
10-5
10-4
0.001
0.010
0.100
1
a
ΩΛ(a)vs. ΩBH(a)
Figure 3. Evolution of dark energy at late times (left panel) and at early times (right panel). Colors as in previous figures.
Table I. Results from MCMC analysis. We report 1σconfi-
dence intervals for each parameter and the Bayes Factors.
ΛCDM BH
late full late full
m0.293+0.016
0.016 0.319+0.005
0.005 0.290+0.019
0.019 0.317+0.007
0.007
b0.0494+0.0004
0.0004 0.045+0.002
0.002
h0.713+0.013
0.013 0.673+0.003
0.003 0.715+0.014
0.013 0.705+0.015
0.015
>0.60 0.53+0.11
0.07
C 3.67+1.90
1.79 2.13+0.63
0.30
Bi
j1 1 0.48+0.05
0.05 32.604+0.001
0.001
ln Bi
j0 0 0.74+0.03
0.03 3.48+0.03
0.04
Table II. Convergence test for MCMC. Convergence is
achieved when all parameters have j>20 and r < 0.01
[12].
ΛCDM BH
late full late full
jr jr jr jr
(102) (103) (102) (103) (102) (103) (102) (103)
m70 0.2 230 0.240.6 2 2
b 10 0.2 0.8 3
h10 0.2 10 0.2 10 0.4 1 3
−−−−0.720.8 3
C−−−−12 1 0.4 5
and larger values of h, thus reducing the Hubble tension
[3638] to .2.75σ. In Fig. 3we additionally show how
the BH dark energy differs from a standard cosmological
constant mainly for early-times behaviour.
M.P.D. thanks John Barrow for the constructive feed-
back.
amariusz.dabrowski@usz.edu.pl
bvincenzo.salzano@usz.edu.pl
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Article
Sampling, Statistics and Computer Code Error Analysis for Independent Random Variables Markov Chain Monte Carlo Error Analysis for Markov Chain Data Advanced Monte Carlo Parallel Computing Conclusions, History and Outlook.
Best known in our circles for his key role in the renaissance of low- density parity-check (LDPC) codes, David MacKay has written an am- bitious and original textbook. Almost every area within the purview of these TRANSACTIONS can be found in this book: data compression al- gorithms, error-correcting codes, Shannon theory, statistical inference, constrained codes, classification, and neural networks. The required mathematical level is rather minimal beyond a modicum of familiarity with probability. The author favors exposition by example, there are few formal proofs, and chapters come in mostly self-contained morsels richly illustrated with all sorts of carefully executed graphics. With its breadth, accessibility, and handsome design, this book should prove to be quite popular. Highly recommended as a primer for students with no background in coding theory, the set of chapters on error-correcting codes are an excellent brief introduction to the elements of modern sparse-graph codes: LDPC, turbo, repeat-accumulate, and fountain codes are de- scribed clearly and succinctly. As a result of the author's research on the field, the nine chapters on neural networks receive the deepest and most cohesive treatment in the book. Under the umbrella title of Probability and Inference we find a medley of chapters encompassing topics as varied as the Viterbi algorithm and the forward-backward algorithm, Monte Carlo simu- lation, independent component analysis, clustering, Ising models, the saddle-point approximation, and a sampling of decision theory topics. The chapters on data compression offer a good coverage of Huffman and arithmetic codes, and we are rewarded with material not usually encountered in information theory textbooks such as hash codes and efficient representation of integers. The expositions of the memoryless source coding theorem and of the achievability part of the memoryless channel coding theorem stick closely to the standard treatment in (1), with a certain tendency to over- simplify. For example, the source coding theorem is verbalized as: " i.i.d. random variables each with entropy can be compressed into more than bits with negligible risk of information loss, as ; conversely if they are compressed into fewer than bits it is virtually certain that informa- tion will be lost." Although no treatment of rate-distortion theory is offered, the author gives a brief sketch of the achievability of rate with bit- error rate , and the details of the converse proof of that limit are left as an exercise. Neither Fano's inequality nor an operational definition of capacity put in an appearance. Perhaps his quest for originality is what accounts for MacKay's pro- clivity to fail to call a spade a spade. Almost-lossless data compres- sion is called "lossy compression;" a vanilla-flavored binary hypoth-
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