Bayesian Evidence for a Cosmological Constant using new High-Redshift Supernovae Data

0.47 MB·English

by  Paolo Serra

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2)
Bayesian Evidence for a Cosmological Constant using
new High-Redshift Supernovae Data
Paolo Serra1, Alan Heavens2, Alessandro Melchiorri1
1Dipartimento di Fisica e Sezione INFN, Universit`a degli Studi di Roma “La Sapienza”, Ple Aldo Moro 5,00185, Rome, Italy
2SUPA, Institute for Astronomy, University of Edinburgh, Royal Observatory,Blackford Hill, Edinburgh EH9-3HJ, UK
5February2008
7
0
0 ABSTRACT
2 We carry out a Bayesian model selection analysis of different dark energy
parametrizations using the recent luminosity distance data of high redshift su-
n
pernovae from Riess et al. 2007 and from the new ESSENCE Supernova Survey.
a
J Including complementary cosmological datasets, we found substantial evidence
4 (∆ln(E) ∼ 1) against a time-varying dark energy equation of state parameter,
2 and against phantom dark energy models. We find a small preference for a stan-
dard cosmological constant over accelerating non-phantom models where w is
2 constant, but allowed to vary in the range −1 to −0.33.
v
8 Key words: cosmology, dark energy.
3
3
1
0
1 INTRODUCTION tal datasets is finally opening the possibility of falsify-
7
0 ing cosmological theories and of discriminating between
/ Over the last few years, observations of luminosity dis- different theories. There have been many proposed ex-
h tances of Type Ia supernovae (SN-Ia) have established planations for this acceleration: the Einstein’s cosmolog-
p
thattheexpansionoftheUniverseisaccelerating(seee.g. ical constant, a new fluid with negative pressure (con-
-
o Riess et al. (1998), Perlmutteret al. (1999), Riess et al. stant or varying with time) (see e.g. Peebles & Ratra
r (2004), Astieret al. (2005), Bassett et al. (2004)). This (2003)) or a modification of general relativity (see for
st result is now well confirmed and complemented by a example Dvali, Gabadadze, Porrati (2000)). However, to
a large amount of independent observations such as, for date,noneofthemissupportedbyawell-establishedfun-
v: example, the angular-diameter distance vs. redshift rela- damentaltheory.Moreover,sinceweknow(almost)noth-
i tion measured by Baryonic Acoustic Oscillations (BAO) ing about dark energy thereis in principle notheoretical
X
experiments Eisenstein et al. (2005), the distortion of limit to the numberof parameters that one might use to
r background images measured by weak lensing experi- characterize it.
a
mentsJarvis et al.(2005),thedistancetothelastscatter-
Itisthereforetimelynotonlytoconstraintheparam-
ing surface measured by Cosmic Microwave Background
etersofaspecificdarkenergymodelbutalsotoestablish
(CMB) experiments Spergel et al. (2006), galaxy clus-
reliable criteria to choose between different models.
tering (Large Scale Structure, LSS)(see Efstathiou et al.
As pointed out in Mukherjee, Parkinson, Liddle
(2002);Tegmark et al.(2006))and,finally,theIntegrated
(2006) and Liddle et al. (2006a), there is an important
Sachs Wolfe effect, correlating LSS with CMB (see e.g
difference between parameter fitting and model selection.
Giannantonio et al. (2006)).
In the first case we work in the context of a single theo-
The recent analysis of Riess et al. (2007) has fur-
retical model framework, and establish which choice of
ther confirmed in an impressive way these results, re-
model parameters gives the best fit to the data. In a
porting the discovery of 21 new SN-Ia with the Hub-
Bayesian interpretation, with flat priors the most likely
ble Space Telescope (HST). Together with a recalibra-
model parameters will simply have the maximum likeli-
tionofpreviousHST-discoveredSN-Ia,thefullsampleof hood and the lowest χ2 regardless of the degrees of free-
23 SN-Ia at z > 1 provides the highest-redshift sample
dom. More complicated models (with a larger number of
known.This dataset hasthen beenanalyzed in combina-
free parameters) will normally produce better fits. How-
tion with some of the aforementioned datasets providing
ever, in model selection, we wish to know which model
newconstraintsonseveraldarkenergyproperties(seee.g.
is favoured, regardless of the values of the parameters.
Riess et al.(2007),Alam(2006),Gong and Wang(2006),
In choosing models, we should also look for simplicity,
Nesseris & Perivolaropoulos(2006),Barger et al.(2006)).
following somewhat a principlebased on Occam’s Razor.
This increasing quality and number of experimen- Simplicity is of course in contrast with more parameters
(cid:13)c 0000RAS
2 P. Serra et al.
and we need then to weight the “need” for extra param- 2 ANALYSIS METHOD
eters. This can be accomplished by using the Bayesian
2.1 Theoretical Framework
Evidence, defined as the probability of the model given
the data, and given by the average likelihood of a model Werestrictouranalysistoflat,Friedmann-Lemaitreuni-
over its prior parameter space: verses, with the redshift evolution of the expansion rate
given by:
E =P(D~|H)= d~θP(D~|~θ,H)P(~θ,H), (1)
where H is theZmodel considered, θ~ is the vector of the H2(z)=H02»Ωm(1+z)3+(1−Ωm)ρρXX((z0))– (5)
modelparameters,D~ isthedata,P(~θ,H)isthepriorand whereΩmistheenergydensityparameterinmatter,H(z)
P(D~|~θ,H)isthelikelihood.Forflatpriorsonthemodels, istheHubbleconstantandρX isthedarkenergydensity
given by:
theprobabilityofthemodelgiventhedataisproportional
to thIenEvgiedneenrcael., comparing two models, the term ρX(z) =exp z 3[1+w(z′)]dz′ . (6)
P(D~|~θ,H) will be larger for the more complicated one ρX(0) Z0 1+z′ ff
(which has more parameters or the same number of pa- where w(z) is the equation of state parameter defined
rameters but with larger prior space), but, at the same as the ratio of pressure over density of the dark energy
time,itwillhavealowerP(~θ,H)d~θcomparedtothesim- component PX =w(z)ρXc2.
pler model. The first term indicates how well the model We consider different parametrizations of the dark
fits data, the second one indicates how simple is the energy equation of state parameter w(z). The simplest
model. Following theequationabovewecanassume that modelistheusualflat ΛCDMmodel with fixedequation
the best model will be the one with the greatest Evi- ofstatew=−1(MODELI).Wethenlet w vary,assum-
dence.1 ing it is small enough to lead to acceleration. MODEL
Jeffreys(1961)providesausefulguidetodiscriminate II has constant w with a flat prior in the range −1 6
thedifference between two models with E and E : w 6 −0.33. In MODEL III, we expand the prior range
1 2
toallow phantomdarkenergymodels,constantw witha
1<∆ ln(E)<2.5 (substantial) (2) flat prior −26 w 6 −0.33. We also consider dynamical
dark energy models where w can depend on redshift. In
2.5<∆ ln(E)<5 (strong) (3) particularweconsideralineardependenceonscalefactor
a=(1+z)−1 as Chevallier & Polarski (2001):
5<∆ ln(E)(decisive). (4)
w(a)=w0+wa(1−a) (7)
The Bayesian Evidence can therefore help us to choose with −2 6 w0 6 −0.33 and −1.33 6 wa 6 1.33
between models because it establish a tension between (MODEL IV). The above model is a low redshift ap-
thesimplicity of a model and its power of fitting data. proximation that may break at higher redshift. In this
In this paper we aim exactly to make use of cos- respect it is useful to include a more sophisticated
mological model selection methods in order to discrim- parametrization that takes in to account the high red-
inate between dark energy models. In the next sections shift behaviour. We consider two possibilities. The first
we therefore compute Bayesian Evidence for a large set one is the one proposed by Hannestad and Mortsell (see
of models and we compare this value with the one ob- Hannestad & Mortsell (2004)), where:
tained for the concordance model: a flat universe with
aq+aq
cold matter and a cosmological constant. More specifi- w(a)=w w s . (8)
cally, in Section 2 we present our analysis method and 0 1w1aq+w0aqs
thedark energymodels considered.InSection 3weshow Inthismodel(MODELV),theequationofstatechanges
the results of our analysis and we derive our conclusions from w0 to w1 around redshift zs = 1 − 1/as with a
inSection 4.Ourpaperfollows theresearch linesalready gradient transition given by q. The priors are flat within
investigated by previous papers (see e.g. Liddle et al. −26 w 6 0, −26 w 6 0 and 06 q6 10.
0 1
(2006b),Saini et al.(2004),Elgaroy & Multamaki(2006) Thesecondoneistheparametrizationintroducedby
and Szydlowski et al. (2006)) which we will complement Upadhyeet al. (2005) where
and extend by using a new and independent algorithm
for computing evidence (which we will illustrate in Sec- w(z)=w0+w1z (9)
tion 2.3), a larger set of dark energy models and finally
for z<1 and
by considering more recent SN-Iadatasets.
w(z)=w +w (10)
0 1
forz>1(MODELVI).Inthiscasewechooseflatpriors
in−26 w 6 −0.2and−46 w 6 2.Inanalyzingeach
0 1
model,thepriorsfortheotherparametersareflatwithin
1 We want to stress that the “best” model might not be in theranges
generalthe“truemodel”;Naturedecideswhatistrue,notour 0.16 Ωm 6 0.5
personalaestheticbehaviour insimplicity. 566 H0/(kms−1Mpc−1)6 72.
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Bayesian Evidence for a Cosmological constant 3
2.2 Cosmological Datasets f2(~θ)
gi(θi)∝ dd−1θj (15)
Thedarkenergymodelsarethencomparedwiththedata sZj6=i j6=igj(θj)
following the approach described in Wang& Mukherjee
where f(~θ) is the inteQgrand to be sampled. An initial
(2004); Liddle et al. (2006b). In particular we compare
(e.g. random) sampling estimates f roughly, from which
theluminosity distance at redshift z of each model given
by an initial set gi can be constructed. Subsequent samples
drawn from the gi can then be used to refine the gi. See
dL(z)=c(1+z)Z0z Hd(zz′′) (11) PGrAesSsiesttahl.at(1i9t86m)afyornofutrtdhoerwdeelltawilhs.enAtphreobinletmegroafnVdEis-
with the SN-Ia luminosity distances from the latest concentrated in one-dimensional (or higher) curved tra-
catalogue of Riess et al. (2007). This includes 182 SN- jectories (or hypersurfaces), unless these happen to be
Ia, “flux-averaged” with a ∆z = 0.05 binning as in oriented close to thecoordinate directions.
Wanget al. (2005) to reduce possible systematic effects To solve this problem we have generalised the algo-
from weak lensing. We also consider the new 57 su- rithm,andcalculate,forafirstandpreliminarysampling,
pernovae coming from the ESSENCE Supernova Suvey thecovariancematrixoftheLikelihoodfunction.Thiswe
of Wood-Vasey et al. (2007) in combination with the diagonalise and use the eigenvectors to define new pa-
38 nearby supernovae (with < 0.023 < z < 0.15) of rameters.Intermsofthesenewparameters,thelikelihood
Riess et al. (2007). We also consider the CMB shift pa- shouldbeclosertoseparable.NotethatthismodifiedVE-
rameter R measured by the three-year WMAP experi- GASalgorithmshouldbeparticularlyefficientifthelike-
ment, R=1.70±0.03 Spergel et al. (2006), in combina- lihood is single-peaked. For multimodal likelihoods with
tion with the BAO measurement of the distance param- moresignificantpeaks,itwillbecomeincreasinglylessef-
eter at redshift z = 0.35, dV(z = 0.35) = 1.300±0.088 ficient, depending on the number, shape, relative height
Gpc(seeEisenstein et al.(2005)).TheshiftparameterR and orientation of thepeaks.
is definedas Givenacosmologicaldatasetandatheoreticalframe-
1 zCMB dz′ work,theLikelihoodfunctionwillbeclearlyafunctionof
R=Ωm2 H(z′), (12) θ~:
Z0
wherezCMB =1089 istheredshift of recombination. For L(~θ)∝ exp −χ2(~θ) (16)
theBAO measurement, the distance parameter is: " 2 #
1
dV(zBAO)= r2(zBAO) czBAO 3 (13) and the algorithm calculates
» H(zBAO)– I = d~θL(~θ). (17)
where r(z) is the comoving distance at redshift z and
Z
zBAO = 0.35. We decide not to use data coming from Wealwaysconsiderflatpriorsinouranalysis;thisimplies
weak gravitational lensing and galaxy clustering because
that the priors will be constant so the Evidence will be
they have the largest systematics and we prefer to be
given by:
conservativein our analysis.
E =P · I, (18)
2.3 A new algorithm for computing Bayesian where P is simply the product of the various priors for
Evidence theparameters, P = di=1Pi.
Inthefollowingwedescribetheprincipalstepsofour
Givenasetofcosmologicaldata,weevaluatetheBayesian
Q
algorithm:
Evidence by integrating the likelihood distribution with
a method based on a modified version of the VEGAS al-
• Wedoafirstsamplingofthelikelihoodfunctionwith
gorithm.IntroducedbyLepage(1978),VEGASiswidely NCOV sampling points~θ andwecalculate thecovariance
used for multidimensional problems which occur in el- matrix C = h(~θ−¯~θ)(~θ−¯~θ)Ti, which is symmetric and
ementary particle physics. VEGAS is an importance-
positive definite
sampling algorithm, where regions where the integrand
• Asquarematrixwhichissymmetricandpositivedef-
has large absolute value are sampled with a higher den-
inite can be written as the product of a lower triangular
sity of points than regions where it is low. The key ele-
and an upper triangular matrices (Cholesky decomposi-
mentoftheVEGASalgorithm isthatsamplesaredrawn
tion):
from a probability distribution which is separable in the
coordinates. This reduces complexity in two ways. In a C =Q· QT. (19)
d-dimensional problem, the probability distribution p is
specified by d one-dimensional distributions, rather than We use the CHOLDC routine (seePress et al. (1986),
oned-dimensional distribution: par.2.9)tocalculatethematrixQ.Ingeneralwecanwrite
thelikelihood function as:
p(~θ)∝g (θ )g (θ )...g (θ ). (14)
1 1 2 2 d d
1
L(~θ)∝ exp − θ~T[C−1]~θ , (20)
Secondly, the generation of the samples is simplified by 2
successively drawing from each of the d probability dis- „ «
tributions. The optimal weight function can beshown to where C is the covariance matrix; if ~y = Q−1θ~ we now
be(Lepage 1978) have:
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4 P. Serra et al.
Figure1.Standarddeviationinfunctionofthenumberofiter- Figure2.ThesameofFigure1butthestandarddeviationis
ationsmused(withN =5000foreachiteration)inthecalcula- plotted infunction of the number N of samplingpoints (with
tionoftheLikelihoodfunctionwith3parameters(Ωm,w,H0). m=5)
Thevalueoftheintegralis:I=0.04915
1
L(~y)∝ exp − ~yT~y (21)
2
„ «
• We now choose to sample ~y rather than ~θ, so our
sampling should be very efficient. In fact, most sampling
pointswillbegeneratedinthesubspaceoftheparameter
space where the likelihood function is not zero.
It is clear that, changing our variables, the integral of
theLikelihood will be given by:
L(~θ)d~θ= L(~y)|det(Q−1)|d~y. (22)
ZD ZD′
Weperformmstatisticallyindependentevaluationsofthe
new function using N sampling points for each iteration.
The iterations are independent but they do assist each
otherbecausethealgorithmuseseachonetoimprovethe
Figure 3. Density of sampling points in the 2-dimensional
samplinggridforthenextone.Theresultsofmiterations spaceΩm−H0
are combined into a single best answer and its estimated
error, by standard inverse variance weighting. We also
compute χ2 to check that the best-fitting solutions are the fact that it’s important to achieve a good estimate
acceptable statistical fits. of the covariance matrix, to have good “principal direc-
tions” for thenext samplings, especially when we handle
Our results are clearly dependent on two parame-
with several parameters.
ters, the number m of iterations and the number N of
samplingpointsforeachiteration.Themoreiterationsor
sampling points are used, the more accuracy is reached.
In Figures 1-2 we show how much the standard devia- 3 RESULTS
tion depends on m and N. In Figure 3 we can see the
Let us first analyze the full Riess et al. (2007) data in
densityofsamplingpointsintheΩm−H0 plane;theden- combination with CMB and BAO. The main results of
sity will be greater in the subspace where theLikelihood
this analysis are reported in Table 1 and in Figures 4-8.
functionlivesandwecanalsoseetheprincipaldirections
Thebest-fittingparametervaluesarethemeansobtained
of the function. In general, implementing the routine for
from the full posterior probability distribution, rather
the calculation of the covariance matrix, the relative er-
than the maximum likelihood values. The standard de-
ror σI in the calculation of the integral is lowered by a
I viations are similarly obtained from integration over the
factor 10, for the same number of function evaluations.
posterior.
In our analysis we always use m=5 iterations and from
N = 5000 to 30000 sampling points (depending on the
dimension of the parameter space and on the size of the
priorspace),exceptforMODELVforwhichN =100000. As we can see a cosmological constant is preferred
We reach an uncertainty of ∼ 10−3 in ∆ ln(E). We also by the data and is always compatible with it, indepen-
used NCOV = 20000 in the first iteration for calculating dently of the model considered. The constraints we ob-
the covariance matrix: this large number is justified by tain on the equation of state parameter, assumed as a
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Bayesian Evidence for a Cosmological constant 5
Table1.Theparameterconstraintstogetherwiththemeanvalueof∆ln(E)andtheminimumchisquared
forthesixmodelsconsideredusingthe182supernovae byRiessetal.2007Riessetal.(2007)binnedwith
∆z = 0.05 up to z = 1.7 (about 34 bins). The (unnormalized) ln(E) for the ΛCDM model is ln(E) =
−16.2466±0.0002. The uncertainty in the value of ∆ln(E) is calculated making use of the usual formula
for the propagation of the uncertainty of a variable x = x(y~) → σx = rPNi=1“∂∂yxi”2σx2i. In our case
∆ln(E)=ln(EE10)→σ∆ln(E)=r“σEE00”2+“σEE11”2.
Constraints ∆lnE χ2 Model
Min
Ωm=0.28±0.03 0.0 24.39 I
H0=64.5±0.09
Ωm=0.27±0.03 −0.222±0.005 22.43 II
H0=63.4±1.1
w<−0.84at1σ
w<−0.73at2σ
Ωm=0.27±0.03 −1.027±0.002 22.43 III
H0=63.4±1.1
w=−0.86±0.1
Ωm=0.28±0.04 −1.118±0.015 21.47 IV
H0=63.8±1.4
w0=−1.03±0.25
wa=0.76+−−0.−91
Ωm=0.27±0.03 −1.059±0.008 21.38 V
H0=63.5±1.1
w0=−0.85±0.12
w1=−0.81±0.21
as unconstrained
q unconstrained
Ωm=0.30±0.05 −1.834±0.006 21.52 VI
H0=63.5+−11..82
w0=−1.08+−00..2340
w1=0.78+−00..8537
Figure 4. Likelihood contours at 68% and 95% (two- Figure 5. Likelihood contours at 68% and 95% for MODEL
parameter)forMODELI II
constant, are w < −0.84 in the case of w > −1 and same order of magnitude of the previous constraints re-
w = −0.86±0.1 at 68% c.l. when models with w < −1 ported by Liddleet al. (2006b) butwhere thenewSN-Ia
areincluded.Thoseconstraintsarecompatibleandofthe datasetofRiess et al.(2007)wasnotconsidered.However
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6 P. Serra et al.
theevidenceforthew<−1caseisworseby∆ln(E)∼1,
i.e. there is no indication from the data that we should
extendtheparameterspacetothesephantomdarkenergy
models.
Thesamehappenswhenweconsidermodelswithan
equation of state parameter which varies with redshift.
For the models I-IV, which have the best Evidences
andthesameparametrization(withw0 =−1andwa =0
for MODEL I and wa = 0 for MODEL II-III) we have
also considered results coming from the Bayesian model
averaging; as explained in Liddle et al. (2006b), because
of our ignorance about the true cosmological model, we
maythinkthattheprobabilitydistributionoftheparam-
eters is a superposition of its distributions in different
models, weighted by therelative model probability, as in
quantummechanics,wherethestateofaphysicalsystem
isa superposition of itspossibilities untila measurement
determines thecollapse in a single eigenstate.
Figure 6. Likelihood contours at 68% and 95% for MODEL
Ifweconvertthe∆ ln(E)intoposteriorprobabilities,
III
assumingequalpriorprobabilities,wehave40.2%,32.2%,
14.4%, 13.2% for models I-II-III-IV.
Theconstraintsonthecosmologicalparametersfrom
the Bayesian model averaging of models I-II-III-IV are:
Ωm =0.27±0.03, H0=63.2+−11..82,w0 =−1.0±0.1 at 1σ,
wa =0.0+−00..0023 at2σ.Theconfidencelimitsinwa areexa-
clyzeroat1σ;thisisbecausetheprobabilitydistribution
for this parameter is a delta function for models I-II-III
and it is superimposed to theextended tails of model IV
(as explained in Liddle et al. (2006b)).
It is also interesting to consider the effects on the
cosmological parameters if we remove from the dataset
the supernovae with z > 1. In Table 2 we report our
principal results.
As we can see, there are no significant differencies
on the mean values however the error bars are generally
reduced by a ∼ 30%. Models with varying-with redshift
equationofstatehaveaslightlybetterevidencebutwith
a cosmological constant is still favoured.
Figure 7. Likelihood contours at 68% and 95% for MODEL It’s also useful to check if our results are the same
IV when we consider a different dataset. To this extent, we
use the 57 supernovae coming from the ESSENCE Su-
pernova Suvey Wood-Vasey et al. (2007) in combination
withthe38nearbysupernovae(with0.023<z <0.15)of
Riessetal.2007(Riess et al.(2007)).Wedonotconsider
modelswithevolvingdarkenergy,becauseinthisanalysis
we limit our redshift range to z<0.670.
The results are reported in Table 3. The results are
fully compatible with those from the previous analysis
and,again,theyprovideasubstantialevidenceforacon-
stant w, with a cosmological constant being preferred.
4 CONCLUSIONS
In this paper we have carried out a Bayesian model
selection analysis of several dark energy models using
the new data of high redshift supernovae of Riess et al.
2007 (Riess et al. (2007)) and from the ESSENCE sur-
vey (Wood-Vasey et al. (2007)), together with Baryonic
Figure 8. Likelihood contours at 68% and 95% for MODEL
AcousticOscillationsandCosmicMicrowaveBackground
VI
Anisotropies.Tothisextent,wehavedevelopedanewal-
gorithm to calculate the Bayesian Evidence which is fast
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Bayesian Evidence for a Cosmological constant 7
Table2.Theparameterconstraintstogetherwiththemeanvalueof∆ln(E)andtheminimumchisquared
forthe sixmodels considered when z <1, usingthe 166supernovae with z<1ofRiess etal.2007 binned
with∆z=0.05(about 20bins).
Constraints ∆lnE χ2 Model
Min
Ωm=0.27±0.04 0.0 16.43 I
H0=64.6±0.09
Ωm=0.27±0.04 −0.223±0.004 14.54 II
H0=63.4±1.3
w<−0.82at1σ
w<−0.73at2σ
Ωm=0.27±0.04 −1.022±0.004 14.54 III
H0=63.4±1.2
w=−0.86±0.1
Ωm=0.29±0.04 −1.090±0.011 13.33 IV
H0=63.6±1.2
w0=−0.98+−00..2263
wa=0.72+−−1.−08
Ωm=0.28±0.04 −1.020±0.008 13.00 V
H0=63.4±1.0
w0=−0.88±0.12
w1=−0.63+−00..227
as=(unconstrained)
q=(unconstrained)
Ωm=0.30±0.05 −1.691±0.01 12.95 VI
H0=64.3±1.2
w0=−1.17±0.37
w1=(unconstrained)
Table 3. Parameter constraints and Evidence for MODEL I-II-III, using the 95 supernovae of
ESSENCE+nearbyRiessetal.(2007)binnedwith∆z=0.05uptoz=0.65(15bins)
Constraints ∆lnE χ2 Model
Min
Ωm=0.25±0.04 0.0 10.73 I
H0=64.7±0.09
Ωm=0.27±0.04 −0.258±0.004 9.28 II
. H0=63.6±1.3
w<−0.80at1σ
w<−0.67at2σ
Ωm=0.27±0.04 −1.027±0.003 9.28 III
H0=63.5±1.2
w=−0.86±0.11
(less than 1 hour for each calculation of the Evidence on be the “true” model. In this way, until we have a the-
a 2GHz CPU) and very accurate (a relative uncertainty oretical explanation of the accelerated expansion of the
σ∆ln(E) <10−2 in 105 likelihood evaluations). Universe, one should keep an open mind to all the alter-
∆ln(E)
nativestotheΛCDMscenario, evenif,atthemoment,it
We find that with current observational data the
seems thesimplest description of our Universe.
usual ΛCDM model is slightly preferred with respect to
dark energy models with equations of state in the range
−1 6 w 6 0 and substantially preferred to dark energy
models with −26 w60 or to dark energy models with
an equation of state which evolves with time.
5 ACKNOWLEDGMENTS
However we would like also to stress that it may be
prematuretoreject models only on thebasis of Bayesian Paolo Serra thanks University of Edinburgh for support
model selection; in general, thesimplest models may not during his visit and members of the Royal Observatory
(cid:13)c 0000RAS,MNRAS000,000–000
8 P. Serra et al.
of Edinburgh for their kind ospitality. Wethank Andrew Y.Wangand M. Tegmark, Phys.Rev.Lett. 92,241302
Liddle for useful comments. (2004); Y.Wang, JCAP 03, 005 (2005).
Y. Wang,ApJ, 536, 531 (2000).
Y. Wang,and P. Mukherjee, ApJ,606, 654 (2004)
Y. Wang,Astrophys.J. 536 (2000) 531
W. M. Wood-Vasey et al., astro-ph/0701041, (2007).
REFERENCES
U. Alam, V. Sahni and A. A. Starobinsky,
arXiv:astro-ph/0612381, (2006).
P.Astier et al., astro-ph/0510447, (2006).
V. Barger, Y. Gao and D. Marfatia,
arXiv:astro-ph/0611775, (2006).
B.A.Bassett,P.S.CorasanitiandM.Kunz,Astrophys.
J. 617 (2004) L1 [arXiv:astro-ph/0407364].
M.ChevallierandD.Polarski, Int.J.Mod. Phy.D.,10,
213 (2001)
G. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B
485, 208 (2000).
G. P.Efstathiou et al., MNRAS,348, L29 (2002).
D.Eisenstein et al., Astrophys.J. 633, 560 (2005).
O. Elgaroy and T. Multamaki, JCAP 0609, 002 (2006)
[arXiv:astro-ph/0603053].
T. Giannantonio et al., Phys. Rev. D 74 (2006) 063520
[arXiv:astro-ph/0607572].
Y. G. Gong and A. z. Wang, arXiv:astro-ph/0612196,
(2006).
S. Hannestad and E. Mortsell, JCAP 0409 (2004) 001
[arXiv:astro-ph/0407259].
M. Jarvis, B. Jain, G. Bernstein and D. Dolney, Astro-
phys.J. 644 (2006) 71 [arXiv:astro-ph/0502243].
H.Jeffreys,“Theory ofProbability”,Third edition,Ox-
ford University Press (1961).
G.P.Lepage,JournalofComputationalPhysics,27,192.
A.Liddle,P.Mukherjee,D.Parkinson,A&G,47,4.30,
2006.
A. R. Liddle, P. Mukherjee, D. Parkinson, Y. Wang,
Pyhs. Rev.D., 74, 123506, (2006).
P. Mukherjee, D. Parkinson, A. R. Liddle, PpJL, 638,
L51, (2006).
S. Nesseris and L. Perivolaropoulos,
arXiv:astro-ph/0612653, (2006).
P.J.E.PeeblesandB.Ratra,Rev.Mod.Phys.75(2003)
559-606 (2003).
S. Perlmutter et al. Astrophys. J. 517, 565 (1999),
astro-ph/9812133.
W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P.
Flannery “Numerical Recipes for Fortran 77: The Art
ofScientificComputing”(CambridgeUniversityPress,
Cambridge, England, 1986).
A. G. Riess et al. Astron. J. 116, 1009 (1998),
astro-ph/9805201.
A.G. Riess et al., Astrophys.J. 607, 665 (2004).
A.G. Riess et al.,arXiv:astro-ph/0611572.
V.Sahni, Class. Quant.Grav. 19, 3435 (2002).
V.Saini,J.WellerandS.Bridle,Mon.Roy.Astron.Soc.
348, 603 (2004)
M. Szydlowski, A. Kurek, A. Krawiec Phys.Lett. B642
(2006) 171 astro-ph/0604327
D.N. Spergel et al., astro-ph/0603449, (2006).
M. Tegmark et al., astro-ph/0608632, (2006).
A. Upadhye,M. Ishak and P. J. Steinhardt, Phys. Rev.
D 72 (2005) 063501 [arXiv:astro-ph/0411803].
(cid:13)c 0000RAS,MNRAS000,000–000