Constraints from Cosmography in various parameterizations

0.09 MB·English

by  Alejandro Aviles

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CONSTRAINTS FROM COSMOGRAPHY IN VARIOUS
PARAMETRIZATIONS
3 ALEJANDROAVILES1,2,CHRISTINEGRUBER3∗,ORLANDOLUONGO1,4,5,HERNANDO
1 QUEVEDO1,4
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1Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de M´exico, AP70543,
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M´exico, DF 04510, Mexico

n 2Departamento de F´ısica, InstitutoNacional de InvestigacionesNucleares, AP 70543,
a M´exico, DF 04510, Mexico
J 3Institut fu¨r Theoretische Physik, Freie Universita¨tBerlin, Arnimallee 14,

7 D-14195 Berlin, Germany
1 4Dipartimento di Fisica and Icra, Universita`di Roma ”LaSapienza”, Piazzale Aldo Moro 5,
I-00185, Roma, Italy
] 5Dipartimento di Scienze Fisiche, Universita`di Napoli ”Federico II”, Via Cinthia,
O I-80126, Napoli, Italy
C
. We use cosmography to present constraints on the kinematics of the Universe without
h
postulating any underlying theoretical model a priori. To this end, we use a Markov
p
ChainMonteCarloanalysistoperformcomparisonstothesupernovaIaunion2compi-
-
o lation,combinedwiththeHubbleSpaceTelescopemeasurementsoftheHubbleconstant,
r and the Hubble parameter datasets. The cosmographic approach to our analysis is re-
t
s visitedandextendedfornewnotionsofredshiftpresentedasalternativestotheredshift
a z.Furthermore,weintroduceanewsetoffittingparametersdescribingthekinematical
[ evolution of the Universe in terms of the equation of state of the Universe and deriva-

tives of the total pressure. Our results are consistent with the ΛCDM model, although
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alternative models, with nearlyconstant pressureand no cosmological constant, match
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theresultsaccuratelyaswell.
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4 Keywords: Cosmography;Parametrizationsofz;Equationofstate.
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1. Introduction
.
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0 In recent years, the wide success of the generally accepted cosmological concor-
3 dance model has been overshadowed by some inconsistencies, one of which being
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the problem of dark energy, addressing the unexplained positive accelerated ex-
:
v pansion of the Universe.1 Various efforts, mostly in the form of modifications or
i
X extensions of the standard model of cosmology, have been introduced to under-
r stand the physicalnature of dark energy.Many models have been developed in the
a
literature, but unfortunately none of them has managed to clarify the origin and
nature of dark energy satisfactorily.2 Most of these models are based on the notion
of a homogeneous and isotropic Universe, described by the Friedmann-Robertson-
Walker (FRW) metric, ds2 =−c2dt2+a(t)2(dr2/(1−kr2)+r2sin2θdφ2+r2dθ2).
Given the considerable amountof proposalsto resolvethe issue of dark energy and
the difficulties in distinguishing fairly between models and evaluating precisely the
degree of accordance between a model and the data, it is desirable to develop an
analysis which describes solely the kinematics of the Universe without relying im-
plicitlyonaparticularmodel.3 Thepurposeofthisworkistwofold.Wefirstdiscuss
∗Email:[email protected]
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the concept of cosmography,a technique of data analysis able to fix bounds on the
observable Universe from a model-independent point of view; giving particular re-
gard to developing a viable cosmographic redshift parametrization, which reduces
the systematic errors on the fitting coefficients. In addition to that, we derive con-
straintsontheequationofstate(EoS)parameteroftheUniversedirectlyfromdata,
alleviating the degeneracy problem between cosmologicalmodels.
2. The experimental techniques of Cosmography
In this section, we present the basic principles of cosmography and illustrate the
wayofperformingthe cosmographicanalysis.By involvingthe cosmologicalprinci-
ple only, and correspondingly the FRW metric, it is possible to infer in which way
dark energy or alternative components are influencing the cosmological evolution,
withoutimplicitlypresuminganyspecificpropertiesornatureofthesecomponents.
TheideaistoexpandthemostrelevantobservablessuchastheHubbleparameteror
cosmological distances into power series, and introducing cosmological parameters
directlyrelatedtotheseobservablequantities.3 Indoingso,itispossibletoappraise
whichmodelsarewellinaccordancewithdataandwhichonesshouldbediscardedas
aconsequenceofnotsatisfyingthebasicdemandsofcosmography.Inexpandingthe
luminositydistancedLintoaTaylorseriesintermsofthecosmologicalredshiftz,we
z
introduce two further notions of redshift, defined as y1 = 1+z and y4 = arctanz.
These parameterizations are designed to reduce some disadvantages of the com-
monly usedand well-knownnotionofthe redshift z forthe analysis,as e.g.the loss
of convergence of the power series for values of z > 1. In particular, while y1 was
previously introduced in the literature,4 we propose to use y4, which has been ob-
tainedbyrequiringabetterconvergencebehaviorofdL.Ourrecipefordetermining
a redshift variable consists in satisfying three considerations: a) the luminosity dis-
tanceshouldnotbehavetoosteeplyintheintervalz <1,b)theluminositydistance
should not exhibit sudden flexes and c) the curve should be one-to-one invertible.
It turns out that the newly introduced y4 is more suitable for a cosmographic
analysis than y1. For y4, the parametrizationof the luminosity distance is given by
dL =c/H0·hy4+y42·(cid:16)1/2−q0/2(cid:17)+y43·(cid:16)1/6−j0/6+q0/6+q02/2(cid:17)+O(y44)i.Further-
more,tocounteracttheproblemofhighinaccuracieswhichiscreatedbycuttingthe
powerseriesexpansionstooearly,wehaveexpandedallquantitiesuptosixthorder.
For the numerical fits we made use of the recent data of Union 2 supernovae Ia, of
the Hubble Space Telescope (HST) measurements of the Hubble parameter,and of
the H(z) compilations,5 using a Markov Chain Monte Carlo method by modifying
thepubliclyavailablecodeCosmoMC.6 Inadditiontoourgeneralizationsregarding
different notions of redshift, we also include a parametrization of the cosmological
distance in terms of the EoS parameter ω of the Universe and of the derivatives
Pi of the total pressure. This allows us to directly fit the EoS parameter of the
Universe from data without having to undergo disadvantageous error propagation
in calculating the values from the cosmographic series. This procedure gives clear
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constraintsonthe EoSparameterandonthe pressurederivativesinthe framework
of GeneralRelativity, and thus providesa direct way to compare the predictions of
a model for the EoS to observationaldata.7 The parametrization of the luminosity
distance in terms of the EoS parameter set and as a function of y4 is given by
dL(y4)=c/H0·hy4+y42/4·(cid:16)1−3ω(cid:17)+y43·(cid:16)5/24−P1/4H02+ω+9ω2/8(cid:17)+O(y44)i.
The numerical results for the parameters of the cosmographic series, i.e. H0,q0,j0
etc.,usingthenewlyintroducedredshifty4,canbefoundinTable1.Thenumerical
results show a good agreement with ΛCDM, although they seem to be compatible
with dark energy possessing constant pressure and an evolving equation of state as
well. The corresponding cosmologicalmodel8 appears to be quite indistinguishable
from ΛCDM.
Table 1. Table of best fits and their likelihoods (1σ) for redshift y4, for the three sets of parameters A ≡ {H0,q0,j0,s0},
B≡{H0,q0,j0,s0,l0}andC≡{H0,q0,j0,s0,l0,m0}.Set1ofobservationsisUnion2+HST.Set2ofobservationsisUnion
2+HST+H(z).
Parameter A,Set1 A,Set2 B,Set1 B,Set2 C,Set1 C,Set2
χ2 530.3 544.8 529.7 544.6 529.9 544.5
min
H0 74.55+−77..5543 73.71+−55..2294 73.95+−77..9292 73.43+−65..0754 74.12+−87..2778 73.27+−65..8961
q0 −0.7492+−00..56829298 −0.6504+−00..43237053 −0.4611+−00..56472120 −0.7230+−00..54855815 −0.4842+−20..79122860 −0.7284+−00..64086328
j0 2.558+−78..494113 1.342+−11..379810 −3.381−+120.1.64193 2.017+−33..104292 −1.940+−82..014418 2.148+−34..401346
s0 9.85−+7246..6699 3.151+−31..972701 −37.67−+8690..5110 5.278−+1134..077362 −13.48−+7311..6258 2.179−+4325..192169
l0 – — N.C. −0.13−+9665..7857 N.C. −11.60−+119837..8986
m0 – – – – N.C. 70.9−[phone]..85
Note: H0 is given in Km/s/Mpc. N.C. means the results are not conclusive - the data do not constrain the parameters
sufficiently.
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