Constraints On Holographic Cosmological Models From Gamma Ray Bursts

0.15 MB·

by  Alexander Bonilla Rivera

Advanced Studies inTheoretical Physics
Vol. 10,2016,no. 1, 33-43
HIKARILtd, www.m-hikari.com
http://dx.doi.org/10.12988/astp.2016.510103
6
1 Constraints on Holographic Cosmological Models
0
2
from Gamma Ray Bursts
b
e
F
AlexanderBonilla Rivera1
1
1
UnversidadDistritalFrancisco Jose´ deCaldas

]
O UnversidadDistritalFrancisco Jose´ deCaldas, Carrera 7 No. 40B-53, Bogota´,Colombia.
C
. JairoErnesto CastilloHernandez2
h
p
-
o UnversidadDistritalFrancisco Jose´ deCaldas, Carrera 7 No. 40B-53, Bogota´,Colombia.
r
t
s
a Copyright c 2015AlexanderBonillaRiveraandJairoErnestoCastilloHernandez.Thisarticle
(cid:13)
[
is distributed under the Creative Commons Attribution License, which permits unrestricted use,

3 distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
v
3
Abstract
8
1
0 We use Gamma Ray Bursts (GRBs) data from Y. Wang (2008) to put addi-
0 tional constraints on a set of cosmological dark energy models based on the
.
1 holographic principle. GRBs are among the most complex and energetic as-
0
trophysical events known in the universe offering us the opportunity to ob-
6
tain information from the history of cosmic expansion upto about redshift of
1
: z 6. Theseastrophysical objectsprovideusacomplementaryobservational
v ∼
i testtodeterminethenatureofdarkenergybycomplementingtheinformation
X
of data from Supernovas (e.g. Union 2.1 compilation). We found that the
r
a ΛCDM modelgivesthebestfittotheobservational data,althoughourstatis-
ticalanalysis(∆AIC and∆BIC)showsthatthemodelsstudiedinthiswork
(”Hubble Radius Scale” and ”Ricci Scale Q”) have a reasonable agreement
with respect to the most successful, except for the ”Ricci Scale CPL” and
”FutureEventHorizon”models,whichcanberuledoutbythepresentstudy.
However, these results reflect the importance of GRBsmeasurements to pro-
vide additional observational constraints to alternative cosmological models,
whicharemandatory toclarifythewayinwhichtheparadigm ofdarkenergy
oranyalternative modeliscorrect.
1e-mail:[email protected]
2e-mail:[email protected]
24 AlexanderBonillaRiveraandJairoErnestoCastilloHernandez
Keywords: GammaRay Bursts,HolographicPrinciple,Dark Energy.
1 Introduction
In orderto explainthe current acceleration oftheuniverse,the fine-tuningproblem
of the valueof Λ and the cosmiccoincidence problem, different alternativemodels
have been proposed. In framing the question of the nature of dark energy there are
two directions. The firts is to assume a new type of component of energy density,
whichmaybeafluidofconstantdensityordynamic. Theotherdirectionistomod-
ify the Einstein’s equations thinking that the metric is inappropriate or that gravity
works differently on large scales. The observational tests are of great importance
to discriminate between these scenarios [1]. The holographic dark energy is one
dynamical DE model proposed in the context of quantum gravity, so called holo-
graphic principle, which arose from black hole and string theories [2]. The holo-
graphicprinciplestatesthatthenumberofdegreesoffreedomofaphysicalsystem,
apartfrombeingconstrainedbyaninfraredcutoff,shouldbefiniteandshouldscale
with its bounding area rather than with its volume. Specifically, it is derived with
the help of tha entropy-area relation of thermodynamics of black hole horizons in
general relativity, which is also known as the Bekenstein-Hawking entropy bound,
i.e., S M2L2, where S is the maximum entropy of the system of length L and
≃ p
M = 1/√8πG is the reduced Planck mass. This principle can be applied to the
p
dynamicsoftheuniverse,whereLmaybeassociatedwithacosmologicalscaleand
itsenergy densityas:
3c2M2
ρ = p. (1)
H L2
If for example L is the Hubble’s radius, which represents the current size of the
universe,thenρ represents theholographicdark energy densityassociated.
H
Our analysis begin with the holographic dark energy models ( 2), following with
the Gamma-Ray Bursts model and data ( 3) and then finish with the results and
H
discussion( 4).
H
H
2 Holographic Dark Energy Models
Inthissectionwepresentsomeofthemostpopularholographicdarkenergymodels
reported intheliterature.
TheFriedmann equationsforaspatiallyflat universecan bewrittenas:
3H = 8πG(ρ +ρ ), (2)
m H
Constraintsonholographiccosmologicalmodelsfromgammaraybursts 25
where ρ is the energy density of the matter component and ρ is the holographic
m H
dark energy density. Thesecomponentsarerelated by an interactionQ termas:
dρ dρ
m +3Hρ = Q H +3H(1+w )ρ = Q, (3)
m H H
dt dt −
wherew = p /ρ istheequationofstateoftheholographicdarkenergydensity.
H H H
ThechangerateoftheHubbleparametercan bewritten as:
dH 3
= H2 1+weff , (4)
dt −2
(cid:0) (cid:1)
whereweff = w/(1+r)istheeffectiveequationofstateofthecosmicfluidandr =
ρ /ρ istheratioofenergydensities,whichisrelatedtothesaturationparameterc2
m H
asc2(1+r) = 1,whichestablishesthatenergyinaboxofsizeLshouldnotexceed
the energy of the black hole of the same size, under the condition L3ρ M2L.
H ≤ p
Different scales leadto differentcosmologicalmodels[2].
2.1 ΛCDM
We beginouranalysis withthestandard cosmologicalmodel. In thisparadigm, the
DEisprovidedbythecosmologicalconstantΛ,withanEoS,suchthatw = 1. In
−
thismodeltheFriedmmanequationE2(z,Θ) foraflat universeis givenby
E2(z,Θ) = Ω (1+z)4 +Ω (1+z)3 +Ω , (5)
r m Λ
whereΩ andΩ arethedensityparametersformatteranddarkenergyrespectively
m Λ
and Ω = Ω (1 + 0.2271N ) is the radiation density parameter, where Ω =
r γ eff γ
2.469 10−5h−2 is photon density parameter and N = 3.046 is the effective
eff
×
number of neutrinos . The free parameters are h, Ω , Ω and the best fit is shown
m Λ
in Table1,
Λ ColdDarkMattermodel
h = 0.7009 0.0035 Ω = 0.716 0.028
Λ
± ±
Ω = 0.266 0.0042
m
±
Table1: Best fit parameters withall datasetto ΛCDMmodel.
where h = H /100km.s−1Mpc−1 isdimensionlessHubbleparameter.
0
2.2 Hubble Radius Scale
In this model L = H−1 and the dark energy density ρ = 3c2M2H2 and the
H p
Friedmann equationcan bewrittenas:
26 AlexanderBonillaRiveraandJairoErnestoCastilloHernandez
1/n
E2(z) = (1 2q )+2(1+q )(1+z)3n/2 1/n 1 , (6)
0 0
− 3
(cid:18) (cid:19)
(cid:2) (cid:3)
whereq isthepresentvalueofthedecelerationparameter. Thismodelissimilarto
0
ΛCDM when n = 2. The free parameters are h, q and n, whose best fit values are
0
shownintheTable2.
HubbleRadiusScale
h = 0.7004 0.0038 n = 1.71 0.20
± ±
q = 0.569 0.047
0
±
Table2: Best fit parameters withall dataset toHRS model.
2.3 Future Event Horizon ξ = 1
With L = R the holographic DE density is ρ = 3c2M2R−2, where R is the
E H p E E
futureeventhorizon. TheFriedmannequationis givenby:
2/c
1+r (1+z) r (1+z)+1 1
E2(z) = (1+z)3/2−1/c 0 0 − , (7)
s r0 +1 "p √r0 +1+1 #
where R = c (1+r)H−1 and r = Ω /(1 Ω ). The best fit values of thefree
E 0 0 0
−
parameters h, r and care givenintheTable3.
0
p
FutureEvent Horizonξ = 1
h = 0.6799 0.0025 r = 0.322 0.032
0
± ±
c = 1.046 0.017
±
Table3: Best fit parameters withalldataset toFEH model.
2.4 Ricci Scale CPL
The Ricci scalar R = 6(2H2 + H˙ ) is relate to the cutoff-scale through L2 = 6/R
and theenergy:
R
ρ = 3c2M2 = α 2H2 +H˙ , (8)
H p 6
(cid:16) (cid:17)
whereα = 3c2/8πG. IfweusetheCPL parameterizationw(a) = w +(1 a)w ,
0 1
−
theFriedmannequationcan bewrittenas:
Constraintsonholographiccosmologicalmodelsfromgammaraybursts 27
−11+r0+3w0
E2(z,Θ) = (1+z)321+1r+0r+0w+03+w41w1 1+r0 +3w1 1+zz 21+r0+3w1 . (9)
1+r
" 0 (cid:0) (cid:1)#
The free parameters of this model are h, r , w and w . The best fit is given in the
0 0 1
Table4.
Ricci ScaleCPL
h = 0.6518 0.0021 r = 9.39+0.51
w = 2.64±+0.49 w0 = 0.46−+00.4.374
0 − −0.55 1 −0.33
Table4: Best fit parameters withalldataset toRSCPL model.
2.5 Ricci Scale Q
Iftheinteractionterm isgivenby Q = 3Hβρ , then
H
1 w˙
β = rw (10)
1+r − H
(cid:20) (cid:21)
and theEoSisgivenby:
1u s (u+s)Aas
w = − − (11)
−6 1 Aas
−
whereu r 3w +3β,v r +3w +3β,s (u2 12β(1+r 3w ))1/2 y
0 0 0 0 0 0
≡ − ≡ ≡ − −
A (v s)/(v+s). TheFriedmannequationcan bewriten as:
≡ −
E2(z,Θ) = n(1+z)−s −m 23lmm−nksn (1+z)32(1−mk), (12)
n m
(cid:20) − (cid:21)
suchthatm 1+r 1/2(v s),n [1+r 1/2(v +s)]A,k 1/6(u s)y
0 0
≡ − − ≡ − ≡ −
l 1/6(u+s)A. The free parameters are h, r , w and β, whose best fit is shown
0 0
≡
in theTable5.
RicciScaleQ
h = 0.6999 0.0038 r = 0.201+0.025
w = 0.84±2+0.055 β0= 0.01−1+0.00.20215
0 − −0.056 − −0.019
Table5: Best fit parameters withalldataset toRSCPL model.
28 AlexanderBonillaRiveraandJairoErnestoCastilloHernandez
3 Gamma-Ray Bursts
InthissectionwepresentabriefintroductionofGRBs asastrophysicalobjectsand
itslateruseincosmology.
3.1 GRBs Model
Gamma-ray bursts (GRBs) are the most luminous astrophysical events observable
today, because they are at cosmological distances. The duration of a gamma-ray
burst is typically a few seconds, but can range from a few milliseconds to several
minutes. The initial burst at gammay-ray wavelengths is usually followed by a
longer lived afterglow at longer wavelengths (X-ray, ultraviolet, optical, infrared,
and radio). Gamma-ray bursts have been detected by orbiting satellites about two
to three times per week. Most observed GRBs appear to be collimated emissions
caused by the collapse of the core of a rapidly rotating, high-mass star into a black
hole. Atleastonceaday,apowerfulsourceofgammaraystemporarilyappearsinto
theskyinanunpredictablelocationandlaterdisappears,whichlastformilliseconds
to minutes. In the location of the gamma ray event is usually observed a dominant
afterglow inX-rays, opticaland radioafter longdecays.
3.2 GRBs Data
WeuseGRBdataintheformofthemodel-independentdistancefromWang(2008)
[6], which were derived from the data of 69 GRBs with 0.17 z 6.6 from
≤ ≤
Schaefer(2007). TheGRBdataareincludedinouranalysisbyaddingthefollowing
term tothegivenmodel:
χ2 = [∆r¯ (z )].(C−1 ) .[∆r¯ (z )], (13)
GRB p i GRB ij p i
where ∆r¯ (z ) = r¯data(z ) r¯ (z ) andr¯ (z ) is givenby
p i p i − p i p i
r (z)
r¯ (z ) = p (14)
p i
r (0.17)
p
where
(1+z)1/2 H
0
r (z) = r(z) (15)
p
z ch
and r(z) is thecomovingdistanceat z. Thecovariancematrixisgivenby:
Cgrb = σ(r¯ (z ))σ(r¯ (z ))C¯grb (16)
ij p i p j ij
where C¯grb is thenormalizedcovariancematrix:
ij
Constraintsonholographiccosmologicalmodelsfromgammaraybursts 29
Datapoint (z) r¯ (z )dat σ(r¯ (z ))+ σ(r¯ (z ))−
p i p i p i
0 0.17 1.0000 - -
1 1.036 0.9416 0.1688 0.1710
2 1.902 1.0011 0.1395 0.1409
3 2.768 0.9604 0.1801 0.1785
4 3.634 1.0598 0.1907 0.1882
5 4.500 1.0163 0.2555 0.2559
6 6.600 1.0862 0.3339 0.3434
Table6: DistancesindependentmodelGRBs.
1.0000
0.7056 1.0000
 
0.7965 0.5653 1.0000
C¯grb = (17)
ij  0.6928 0.6449 0.5521 1.0000 
 
 0.5941 0.4601 0.5526 0.4271 1.0000 
 
 0.5169 0.4376 0.4153 0.4242 0.2999 1.0000 
 
 
and
σ(r¯ (z ))+, if r¯ (z ) r¯ (z )dat
p i p i p i
σ(r¯ ) = ≥ (18)
p ( σ(r¯p(zi))−, if r¯p(zi) < r¯p(zi)dat
where σ(r¯ (z ))+ and σ(r¯ (z ))−, are 68% C.L errors given in the Table 6. As
p i p i
complementarytests we useSNIa (580-Data point),CMB (1-Data point)and BAO
(1-Datapoint)[4](See Appendix).
4 Results and Discussion
In this section we perform thestatistical analysis, where we implementedthe max-
imum likelihood criterion to get the best settings for each model and used the AIC
and BIC criterionto discriminatebetween thedifferentmodels.
Themaximumlikelihoodestimateforthebestfit parameters is:
1
= exp χ2 (19)
Lmax −2 min
(cid:20) (cid:21)
If has a Gaussian errors distribution, then χ2 = 2ln , So, for our
Lmax min − Lmax
analysis:
χ2 = χ2 +χ2 +χ2 +χ2 . (20)
min GRBs SNIa CMB BAO
30 AlexanderBonillaRiveraandJairoErnestoCastilloHernandez
Model χ2 AIC BIC ∆AIC ∆BIC
min
ΛCDM 608.5 614.5 627.6 0.0 0.0
HubbleRadiusS. 609.7 615.7 628.8 1.2 1.2
FutureEvent H. 657.3 663.3 676.4 48.8 48.8
Ricci scaleCPL 917.3 925.3 924.8 310.8 315.2
Ricci scaleQ 609.8 617.8 635.3 3.3 7.7
Table7: AICand BIC analysistodiferent dark energy modelsusingall datasets.
In theFigure 1 we show thediagrams of statisticalconfidence at 1σ, 2σ and 3σ for
differentcosmologicalmodelsandseveralparameterspace,fromajointanalysisof
69 GRBs (independent-model6-Data point), SNIa (580-Datapoint), CMB (1-Data
point),BAO(1-Datapoint)[4][2].
In this paper we use the Akaike and Bayesian information criterion (AIC, BIC),
which allow to compare cosmological models with different degrees of freedom,
with respect to the observational evidence and the set of parameters [5]. The AIC
and BIC can becalculated as:
AIC = 2ln +2k, (21)
max
− L
BIC = 2ln +klnN, (22)
max
− L
where is the maximum likelihood of the model under consideration, k is the
max
L
number of parameters. BIC imposes a strict penalty against extra parameters for
any set with N data. The prefered model is that which minimizes AIC and BIC,
however,onlytherelativevaluesbetweenthedifferentmodelsisimportant[3]. The
resultsare shownin theTable7.
5 Conclusion
WeimplementGRBs datamodel-independentfromY. Wang(2008)to complement
SNIa Union2.1 sample to high redshift. We found that model-independent GRBs
provideadditional observationalconstraints to holographicdark energy models. In
addition to the data of GRBs we use data from SNIa, BAO and CMB as usual cos-
mologicaltests.
Our analysis shows that the ΛCDM model is preferred by the ∆AIC and ∆BIC
criterion. The ”Hubble Radius Scale” and ”Ricci Scale Q” models show an inter-
esting agreement with the observationaldata. The”Ricci ScaleCPL” and ”Future
Event Horizon”modelscan beruledout bythepresentanalysis.
Constraintsonholographiccosmologicalmodelsfromgammaraybursts 31
Finally we want to highlight the importance of deepening in the development of
unified models of GRBs, given the obvious importance of these objects in the era
ofcosmologyaccuracy, whichhelp toshed lighton thedark energy paradigm.
Acknowledgements. A. Bonilla and J. Castillo wish to express their gratitude
to Universidad Distrital FJDC for the academic support and funding. A. Bonilla
wishes to thank to professor Arthur Kosowsky (University of Pittsburgh, U.S.A.)3
for his comments and suggestions to this work within the framework of Centenary
Celebration of General RelativityTheory: Andean School on Gravityand Cosmol-
ogy 4. A. Bonilla also wish give thanks to Dr. Gae¨l Foe¨x 5 for their appropriate
commentsandreviewofthiswork.
References
[1] A. Albrecht, L. Amendola, G. Bernstein, D. Clowe, D. Eisenstein, L. Guzzo,
C. Hirata and D. Huterer et al., Findings of the Joint Dark Energy Mission
FigureofMeritScience WorkingGroup,2009,[arXiv:0901.0721].
[2] V. H. Ca´rdenas, A. Bonilla, V. Motta, S. del Campo, Constraints on holo-
graphic cosmologies from strong lensing systems, J. Cosmology Astropart.
Phys.,(2013), no.11, 053.http://dx.doi.org/10.1088/1475-7516/2013/11/053
[3] A.R.Liddle,Howmanycosmologicalparameters?,2004,MNRAS351,L49-
L53,[astro-ph/0401198v3].
[4] S. Nesseris, L. Perivolaropoulos, Crossing the phantom divide: theoretical
implications and observational status, J. Cosmology Astropart.Phys., (2007),
no.1, 018.http://dx.doi.org/10.1088/1475-7516/2007/01/018
[5] G.Schwarz, EstimatingtheDimensionofaModel,TheAnnalsofStatistics,6
(1978),461-464.http://dx.doi.org/10.1214/aos/1176344136
[6] Y. Wang, Model-independent distance measurements from gamma-ray
bursts and constraints on dark energy, Phys. Rev. D., 78 (2008), 123532.
http://dx.doi.org/10.1103/physrevd.78.123532
3http://www.physicsandastronomy.pitt.edu/people/arthur-kosowsky
4http://gravityschool.uniandes.edu.co/index.php/en/speakers
5http://ifa.uv.cl/index.php/es/2-uncategorised/83-dr-gael-foeex
32 AlexanderBonillaRiveraandJairoErnestoCastilloHernandez
Appendix
SNIa
Here, we use the Union 2.1 sample which contains 580 data points. The SNIa
data give the luminosity distance d (z) = (1 + z)r(z). We fit the SNIa with the
L
cosmologicalmodelby minimizingtheχ2 valuedefined by
580 [µ(z ) µ (z )]2
χ2 = i − obs i , (23)
SNIa σ2
i=1 µi
X
where µ(z) 5log [d (z)/Mpc] + 25 is the theoretical value of the distance
10 L
≡
modulus,and µ theobservedone.
obs
CMB
WealsoincludeCMBinformationbyusingtheWMAPdata. Theχ2 fortheCMB
cmb
dataisconstructed as:
2
(1.7246 R)
χ2 = − . (24)
cmb 0.032
HereR is“shift parameter”, defined as:
√Ω
R = m D (z). (25)
L
c(1+z∗)
where dL(z) = DL(z)/H0 and the redshift of decoupling z∗ is z∗ = 1048[1+
0.00124(Ω h2)−0.738][1+g (Ω h2)g2]and
b 1 m
0.0783(Ω h2)−0.238 0.560
g = b ,g = . (26)
1 1+39.5(Ω h2)0.763 2 1+21.1(Ω h2)1.81
b b
BAO
Similarly,for theDR7 BAO data,theχ2 can beexpressedas:
2
d 0.469
χ2 = z − . (27)
6dFGS 0.017
(cid:18) (cid:19)
whered = r (z )/D (z)denotesthedistanceratio. Here,r (z )isthecomov-
z s d V s d
ing sound horizon at the baryon drag epoch (z = 0.35) and D (z) is the acoustic
d V
scale.
Received: November 6, 2015;Published: January21, 2016