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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

A∝r2(so r∝A1/2). 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 ρH∝Seff L−4with 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(∆

2−1),(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:

L≡aZ∞

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,ra−3(1+wm,r )and

ΩBH (a) = C2

H2(a)L2(∆

2−1).(5)

Combining previous equations one can express the Hub-

ble parameter as

H(a) = H0sΩma−3+ Ωra−4

1−ΩBH (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):

aΩ0

H(a) = 1 + ∆

2Fr(a) (7)

+ ΩH(a) (1 −ΩH(a)) 1+2∆

2Fm(a)

+ (1 −ΩH(a))

∆/2

2(∆

2

−1)ΩH(a)

1

2(1−

∆

2)Q(a)#,

with:

Fr(a) = −2Ωra−4

Ωma−3+ Ωra−4,(8)

Fm(a) = Ωma−3

Ωma−3+ Ωra−4,

Q(a)=21−∆

2H0pΩma−3+ Ωra−4

∆/2

1−

∆

2C

1

∆

2

−1.

It is easy to verify that in the limit Ωr→0 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) [9–11] 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 <Ωb<Ωm<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 ·C−1

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·C−1

SN ·

∆µSN ,b≡∆µS N T·C−1

SN ·1,d≡1·C−1

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(θ−β)2−ˆ

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),

D∆t≡(1 + zL)DLDS/DLS , is generally called time-

delay distance and is constrained by H0LiCOW. The

data (Dobs

∆t,i) and the corresponding errors (σD∆t,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 D∆t,i(p)−Dobs

∆t,i2

σ2

D∆t,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

a≡∆µGT·C−1

G·∆µG,b≡∆µGT·C−1

G·1and

d≡1·C−1

G·1.

Baryon Acoustic Oscillations. For BAO we consider

multiple data sets from diﬀerent surveys. In general, the

χ2deﬁned as χ2

BAO = ∆FBAO ·C−1

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) = 100√Ωmh2DV(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 (∝H−1) 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 s−1.

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 ·C−1

CM B ·∆FCMB ,

where the vector FCM B is made of the quantities

R(p)≡pΩmH2

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

j≈3.5, 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

Ωm0.293+0.016

−0.016 0.319+0.005

−0.005 0.290+0.019

−0.019 0.317+0.007

−0.007

Ωb−0.0494+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

j∗r j∗r j∗r j∗r

(102) (10−3) (102) (10−3) (102) (10−3) (102) (10−3)

Ωm70 0.2 230 0.240.6 2 2

Ωb− − 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

[36–38] 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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