A Bayesian study of the primordial power spectrum from a novel closed universe model

1.2 MB·English

by  J. Alberto Vazquez

Mon.Not.R.Astron.Soc. 000,1–11(2011) Printed16January2013 (MNLATEXstylefilev2.2)
A Bayesian study of the primordial power spectrum from a
novel closed universe model
J. Alberto Va´zquez1,2(cid:63), A.N. Lasenby1,2, M. Bridges1,2, M.P. Hobson1
2 1Astrophysics Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK.
1 2Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK.
0
2
n Accepted–.Received–;inoriginalform11December2011
a
J

3 ABSTRACT
2 Weconstraintheshapeoftheprimordialpowerspectrumusingrecentmeasurementsof
the cosmic microwave background (CMB) from the Wilkinson Microwave Anisotropy

] Probe (WMAP) 7-year data and other high-resolution CMB experiments. We also in-
O
cludeobservationsofthematterpowerspectrumfromtheluminousredgalaxy(LRG)
C subset DR7 of the Sloan Digital Sky Survey (SDSS). We consider two different mod-
. els of the primordial power spectrum. The first is the standard nearly scale-invariant
h
spectrumintheformofageneralisedpower-lawparameterisedintermsofthespectral
p
amplitude A , the spectral index n and (possibly) the running parameter n . The
- s s run
o second spectrum is derived from the Lasenby and Doran (LD) model. The LD model
r is based on the restriction of the total conformal time available in a closed Universe
t
s and the predicted primordial power spectrum depends upon just two parameters. An
a importantfeatureoftheLDspectrumisthatitnaturallyincorporatesanexponential
[
fall-off on large scales, which might provide a possible explanation for the lower-than-

3 expected power observed at low multipoles in the CMB. In addition to parameter
v estimation, we compare both models using Bayesian model selection. We find there
9 is a significant preference for the LD model over a simple power-law spectrum for a
1 CMB-only dataset, and over models with an equal number of parameters for all the
6 datasets considered.
4
. Key words: cosmologicalparameters–cosmology:observations–cosmology:theory
3
– cosmic background radiation – large-scale structure
0
1
1
:
v
i 1 INTRODUCTION 2003;Contaldietal.2003),sothat,anextravariabletoac-
X
count for a cut-off scale might be considered. It would be
Cosmological inflation not only explains the homogeneity
r feasible to continue adding parameters in this fashion un-
a of the universe on large scales, but also provides a theory
tilsomearbitraryaccuracyofmodelfitisachieved,butthis
for explaining the observed level of anisotropy (Guth 1981;
procedurewouldfailtoaccountforOccam’srazor:asimpler
Albrecht & Steinhardt 1982; Mukhanov & Chibisov 1982).
model should be preferred, unless the data require a more
Inflationary models generically predict Gaussian, adiabatic
sophisticatedone.UsingtheBayesianevidencetoselectbe-
andnearlyscale-invariantprimordialfluctuations.Todeter-
tween models is one way to include this consideration.
minetheshapeoftheprimordialpowerspectrumfromcos-
mologicalobservations,itisusualtoassumeaparameterised There have been several recent studies regarding the
formforit.Thesimplestassumptionisthattheinitialspec- shape of the primordial spectrum, some based on physical
trum has the form of a simple power-law, parameterised in models,someusingobservationaldatatoconstrainanapri-
terms of the spectral amplitude As and the spectral index ori parameterisation,andothersattemptingadirectrecon-
ns. Recent analyses have shown, however, that the spectral struction from data (Barriga et al. 2001; Bridle et al. 2003;
indexmaydeviatefromaconstantvalue(closetounity)and Hannestad 2004; Bridges et al. 2006, 2007, 2009; Verde &
soconsiderationofmodelsthatprovidesomerunning ofthe Peiris 2008; Peiris & Verde 2010; Hlozek et al. 2011; Guo
index (defined by nrun ≡dns/dlnk) is warranted. Further- etal.2011).Aprimarygoalofthisworkistofitthepredicted
more,modelsthatpredictadecrementinCMBpoweratlow form of the primordial spectrum from a physically moti-
multipolesseemtobepreferredbyobservations(Efstathiou vated model described by Lasenby & Doran (2005) (hence-
forth LD). We translate observational data into constraints
on the generalised power-law and LD forms for the primor-
(cid:63) E-mail:[email protected] dialpowerspectrum(andthestandardcosmologicalparam-
(cid:13)c 2011RAS
2 V´azquez et al.
eters), and decide which model provides the best fit to ob- index to vary as a function of scale, such that n (k). This
s
servational data using the Bayesian evidence. can be achieved by including a second order term in the
The paper is organised as follows: in Section 2 we expansion of the power spectrum
present the two different models for the power spectrum
(cid:18) k (cid:19)ns−1+(1/2)ln(k/k0)(dn/dlnk)
and in Section 3 we describe basic parameter estimation P (k)=A , (7)
and model selection. We list the datasets and the cosmo- R s k0
logical parameters considered in Section 4 and present the where dn/dlnk is termed the running parameter n and
run
resulting parameter constraints in Section 5. We compute we would expect n ≈0 for standard inflationary models.
run
Bayesian evidences in Section 6 to decide which model pro- In what follows we will consider three Power-Law pa-
videsthebestdescriptionforcurrentobservationaldataand rameterisations.Inthefirstmodel(PL1),wewillassumea
wevalidateouranalysisbyapplyingittoaCMBsimulated simple power-law spectrum (without running) and restrict
data. Our conclusions are presented in Section 7. theuniversetobespatiallyflat.InPLmodel2,weallowthe
spatial curvature of the universe to be a free parameter. In
PLmodel3,weallowforarunningspectrum,butagainre-
2 PRIMORDIAL POWER SPECTRUM stricttheuniversetobespatially-flat.Inthisway,power-law
models 2 and 3 have the same number of parameters.
The correlation function ξ of density fluctuations δ ≡ δρ/ρ
at two separated points x and x+r is defined as
2.2 The LD model
ξ(r)≡(cid:104)δ(x)δ(x+r)(cid:105). (1)
Assumingthecosmologicalconstantistheoriginofdarken-
Becausetheassumptionofhomogeneityandisotropy,ξ isa
ergy, Lasenby & Doran (2005) provided a construction for
functiononlyofr≡|r|.ThepowerspectrumP(k)describes
embeddingclosed-universemodelsinadeSitterbackground.
the amplitude of fluctuations on different length scales and
Asaconsequenceofthisnovelapproach,aboundarycondi-
itisrelatedwiththeinverseFouriertransformofthecorre-
tiononthetotalavailableconformaltimeemerges.Defining
lation function ξ by:
the total conformal time η as
P(k)≡(cid:104)|δ |2(cid:105). (2)
k (cid:90) ∞ dt
η≡ , (8)
During the inflationary period, fluctuations in the inflaton R(t)
0
field δφ result in curvature perturbations R(k) given by
k
the LD model requires η=π/2. For more details about the
(cid:20) (cid:21)
H choice of the boundary condition, including how it can be
R(k)=− δφ , (3)
φ˙ k reinterpretedasaneigenvalueconditiononthesolutionofa
k=RH
differential equation, see Lasenby (2003); Lasenby & Doran
wherethequantitiesareevaluatedatthehorizonexitepoch
(2004, 2005). In order to understand some consequences of
k = RH. Here R is the scale factor of the universe and
H ≡R˙/RistheHubbleparameter.Inthispaper,wefollow thenewboundaryconditionwesplitthehistoryoftheUni-
verseintwomaincontributionstothetotalconformaltime:
a slow-roll approximation (e.g. Liddle & Lyth 1999). Hence
matter(radiationanddust)andinflationaryeras.Hence,we
the power spectrum of the inflaton fluctuations is constant
wanttocomputetheconformaltimeη elapsedduringthe
in time and equal to M
mattereraandaddittothatelapsedintheinflationaryera
(cid:18)H(cid:19)2 η , such that the boundary condition is satisfied:
P (k)= . (4) I
δφ 2π π
k=RH η +η = . (9)
I M 2
Thus, the primordial curvature spectrum P (k) computed
R
from (2) - (4) is It is found that this constraint leads to a ‘see-saw’ mecha-
nism linking the parameters describing the current state of
(cid:34)(cid:18)H(cid:19)2(cid:18)H(cid:19)2(cid:35) the universe with the initial conditions (Lasenby & Doran
P (k)= . (5)
R φ˙ 2π 2004).
k=RH
2.1 Power-law parameterisations 2.2.1 Matter era
Cosmological slow-roll inflation predicts the spectrum of The general description of the large scale Universe is based
curvatureperturbationstobeclosetoscale-invariant.Based on the Robertson-Walker space-time with dynamics gov-
onthis,thespectrumiscommonlyassumedtohavetheform erned by the Einstein equations. The resulting Friedmann
equations can be written as (with c=1)
(cid:18) k (cid:19)ns−1
PR(k)=As k , (6) R˙2+k Λ 8πG
0 − = ρ, (10)
R2 3 3
wherethespectral indexn isexpectedtobeclosetounity;
k0 is the pivot scale (set tso k0 = 0.05 Mpc−1 throughout). 2R¨ + R˙2+k −Λ = −8πGP.
Aspectrumwherethetypicalamplitudeofperturbationsis R R2
identicalonalllengthscalesisknownasHarrison-Zel’dovich Herek=0,±1definesthegeometryoftheuniverse,Λisthe
spectrum(n =1).Thisparticularparameterisationinvolves cosmologicalconstant,andtherelationshipbetweendensity
s
only one free parameter, the spectral amplitude P(k)=A . ρ and pressure P is encoded in the equation of state P =
s
Afurtherextensionispossiblebyallowingthespectral γρ.Thebehaviourofthehomogeneousuniverseisgoverned
(cid:13)c 2011RAS,MNRAS000,1–11
Bayesian study of the primordial power spectrum 3
by the parameters representing its matter-energy content, with
√
namely 118 3πb2 u2 √
φ = b +b u4/3− 4u8/3− (11 3πµ2
0 0 4 99 1296π
√ √
ΩM = 83πHG2ρ, ΩΛ = 3HΛ2, (11) −54 3πΛ−216 3π3/2µ2b20+36πµ2b0),
(cid:114) 1 µ2 (cid:16) √ (cid:17)
φ = − − − 3π+36πb u2,
and its expansion history defined by the Hubble parameter 1 12π√ 216π 0
H.Moreover,ifweassumethematterdensityismadeupof 1 32 3π 2µ2 Λ 4π
H = + b u1/3+( + + µ2b2
decoupled dust and radiation, the equations governing Ω 0 3u 27 4 81 3 3 0
M √
and ΩΛ can be solved exactly and its solution is controlled +4 3πµ2b )u− 6656πb24u5/3,
by two arbitrary constants α and β given by 27 0 891
dH uΛ 8π u(cid:18)dφ (cid:19)2
Ω2 Ω H = −u 0 −uH2+ − 0
α = mo Λo , (12) 1 du 0 3 3 du
(Ω +Ω +Ω −1)3
mo ro Λo 16πφ dφ 8πφ2 4πµ2uφ2
αβ = ΩroΩΛo , − 3 1 du0 − 3u1 + 3 0. (17)
(Ω +Ω +Ω −1)2
mo ro Λo
We observe that two new free parameters b and b ,
0 4
where subscript ‘o’ denotes quantities evaluated at present appear in the series expansions in (17). Together with the
time.Thetotalconformaltimeforthistypeofuniversecan mass of the scalar field µ, they control the magnitude of
be written in terms of the dimensionless parameters α and thefieldandhowlongtheinflationaryperiodlasts.Inorder
β as (see Lasenby & Doran 2005) to decide on the priors we shall employ in our subsequent
Bayesian analysis, it is worth pointing out some features
(cid:90) ∞ dx related with these new parameters.
η = . (13)
M (βx4+x3−x2+α)1/2 • The amplitude of the perturbations is determined by
0
thescalarfieldmassµ.TomatchtheobservedlevelofCMB
anisotropies, we shall need to set it to be about
µ∼10−6. (18)
• Thenumberofe-foldingsN isprimarilydeterminedby
2.2.2 Inflationary era b0 and may be approximated as
N ≈2πb2. (19)
The computation of the conformal time in the inflationary 0
epoch is a more elaborate process. Let us consider a basic Hence, to obtain realistic models we need b to be of order
0
inflationary model where the particle responsible for this ∼ a few.
processissimplyareal,time-dependent,homogeneous,free, • The conformal time is estimated by
massive scalar field φ, described by the equations
(cid:18) |b | (cid:19)1/2(cid:18)1(cid:19)
η ≈0.92 4 . (20)
H˙ +H2− Λ + 4πG(2φ˙2−m2φ2) = 0, I µ4/3 b20
3 3
φ¨+3Hφ˙+m2φ = 0. (14) Employing the constraint (9), |b4|µ−4/3 should thus be
around unity.
• The parameter b controls the initial curvature, as can
Forcloseduniversemodels,thescalefactorisgivenexplicitly 4
be seen from (15):
by
R (cid:18) 2187 (cid:19)1/4 u1/3
≈ √ . (21)
1 = 4πG(φ˙2+m2φ2)−H2+ Λ. (15) lp 12544π −b4
R2 3 3
Thereforeb mustbenegative.Makinguseoftherestofthe
4
parameters and (20), |b | should be around 10−9.
Inordertocomputetheconformaltimeη ,itisnecessaryto 4
I
seekoutsuitableconditionsbeforetheonsetofinflationand The restriction on the values for the model parameters to-
thensolvethedynamicsforthescalarfieldencodedinequa- getherwiththeboundarycondition,severelylimitstheclass
tions (14). To do this, Lasenby & Doran (2005) developed of models allowed to reproduce current cosmological obser-
formal seriesexpansions outof theinitial singularity,t=0, vations.
intermsofdimensionlessvariablesu=t/t andµ=m/m , Once we have found the initial conditions (17), it is
p p
wherethesubscript‘p’denotesaPlanckunitsvariable.The straightforward to solve numerically the dynamics of the
series are given by scalar field φ and the expansion history H to determine
the evolution of the universe. As an example, let us con-
φ(u) = 1 (cid:88)∞ φ (u)lnn(u), sider the best-fit values for the cosmological parameters
lp n givenbyWMAP+BAO+H0(Komatsuetal.2011),inwhich
n=0 case the conformal time elapsed during the matter epoch is
∞
H(u) = 1 (cid:88)H (u)lnn(u), (16) ηM = 0.22. In order to satisfy the boundary condition we
tp n=0 n shouldchooseappropriatevalues{b0,b4,µ}suchthatbythe
(cid:13)c 2011RAS,MNRAS000,1–11
4 V´azquez et al.
1.4 Table 1. Jeffreys scale for evaluating the strength of evidence
1.35 whentwomodelsarecompared.
1.2
1 |lnB01| Odds Probability StrengthofZ
<1.0 <3:1 <0.750 Inconclusive
0.8
!inf 1.0 ∼3:1 0.750 Significant
0.6
2.5 ∼12:1 0.923 Strong
0.4
>5.0 >150:1 >0.993 Decisive
0.2
0
6 8 10 12 14 16 primetoolforthemodelselectionwefocusonistheBayesian
ln(u)
evidence.
Figure 1. Evolution of the conformal time ηI as a function of The Bayesian evidence provides a natural mechanism
ln(u). We observe that ηI saturates at a value of around 1.35 tobalancethecomplexityofcosmologicalmodelsandthen,
by the end of inflation. The parameters used in this model are elegantly, incorporates Occam’s razor. It applies the same
b0=2.47,b4=−17.7×10−9 andµ=1.68×10−6. type of analysis from parameter estimation but now at the
level of models rather than parameters (Trotta 2007). Let
usconsiderseveralmodelsM,eachofthemwithprobability
end ofthe inflationary periodthe achieve conformaltime is
P(M). The Bayesian evidence appears again, but now in
η ≈ 1.35, as shown on Fig 1. The expected shape of the
I
another form of the Bayes theorem:
primordial power spectrum P(k) for this model is directly
computed from (5). P(D|M)P(M)
P(M|D)= . (23)
P(D)
Theleft-handsidedenotestheprobabilityofthemodelgiven
3 BAYESIAN ANALYSIS
the data, which is exactly what we are looking for model
3.1 Parameter Estimation selection.
ItwaspreviouslymentionedthattheBayesianevidence
A Bayesian analysis provides a consistent approach to es-
is simply the normalisation over its posterior expressed by:
timate the set of parameters Θ within a model M, which
(cid:90)
best describes the data D. The method is based on the as- Z = L(Θ)π(Θ)dNΘ, (24)
signment of probabilities to all quantities of interest, and
then the manipulation of those probabilities given a series whereN isthedimensionalityoftheparameterspace.When
of rules, of which Bayes’ theorem plays the principal role, twomodelsarecompared,M andM ,thequantitytobear
0 1
(Bayes 1761). Bayes’ theorem states that in mind is the ratio of the posterior probabilities given by
P(D|Θ,M) P(Θ|M) P(M |D) Z P(M ) P(M )
P(Θ|D,M)= , (22) 0 = 0 0 = B 0 , (25)
P(D|M) P(M |D) Z P(M ) 01P(M )
1 1 1 1
where the prior probability P(Θ|M) ≡ π represents our
where P(M )/P(M ) is the prior probability ratio for two
0 1
knowledge of Θ before considering the data. This probabil-
models, usually set to unity. The evidence ratio, often
ityismodifiedthroughthelikelihoodP(D|Θ,M)≡Lwhen
termedastheBayesfactorB ,quantifieshowwellmodel0
01
theexperimentaldataDareconsidered.Thefinalobjective
mayfitdatawhenitiscomparedtomodel1.Jeffreys(1961)
for Bayesian inference is to obtain the posterior probability
provided a useful guide on which we are able to make qual-
P(Θ|D,M) which represents the state of knowledge once
itative conclusions based on this difference (see Table 1).
we have taken the new information into account. The nor-
Until recently, numerical methods such as thermody-
malisation constant is the marginal likelihood or Bayesian
namic integration (Beltra´n et al. 2005; Bridges et al. 2007)
evidence P(D|M) ≡ Z. Since this quantity is independent required around 107 likelihood evaluations to obtain accu-
of the parameters Θ, it is commonly ignored in parame-
rate estimates of the Bayesian evidence, making the pro-
terestimationbuttakesthecentralroleformodelselection
cedurehardlytractable.Thenested samplingalgorithm,in-
(Hobson et al. 2002; Liddle 2004).
ventedbySkilling(2004),hasbeensuccessfullyimplemented
for cosmological applications (Mukherjee et al. 2006; Shaw
et al. 2007; Feroz et al. 2008) requiring about a hundred
3.2 Model selection
times fewer posterior evaluations than thermodynamic in-
Acomplexmodelthatexplainsthedataslightlybetterthan tegration to achieve the same accuracy in the evidence es-
asimpleoneshouldbepenalisedfortheintroductionofextra timate. A substantially improved and fully-parallelized al-
parameters, because the additional parameters bring with gorithmcalledMultiNest(Ferozetal.2009)increasesthe
themalackofpredictability.Also,ifamodelistoosimple,it sampling efficiency for calculating the evidence and obtain-
mightnotfitthedataequallywell,thenitcanbediscarded ing posterior samples, even from distributions with multi-
(e.g. Liddle 2006, 2007; Trotta 2008). Following this line, plemodesand/orpronounceddegeneraciesbetweenparam-
many attempts have been performed to translate Occam’s eters.Inourcase,tocarryouttheexplorationofthecosmo-
razorintoamathematicallanguageformodelselection.The logical parameter space we use a modified version of both
(cid:13)c 2011RAS,MNRAS000,1–11
Bayesian study of the primordial power spectrum 5
the MultiNest (Feroz et al. 2009) and CosmoMC (Lewis
Table 2.Parametersandpriorrangesusedinouranalysis.
& Bridle 2002) packages.
Model Parameter Priorrange
All Ω h2 0.01 0.03
b
4 DATASETS ΩDMh2 0.01 0.3
θ 1.0 1.1
Measurements of the CMB anisotropies and large-scale τ 0.01 0.3
structure play an important role in both fitting parame- Power-law log[1010As] 2.5 4.0
ters and comparison of models in cosmology. To constrain PL1 ns 0.5 1.5
the space-parameter in each model, we first use the latest PL2 Ωk -0.2 0.2
7-yeardatareleasefromWMAP(henceforthWMAP7;Lar- PL3 nrun -0.2 0.2
son et al. 2011), which includes a good measurement up to LD Ωk -0.2 -0.0001
thethirdacousticpeakinthetemperatureCMBspectrum. b0 1.0 4.0
This comprises our dataset 1. b4[10−9] -30.0 -0.1
WMAP7 data by itself cannot, however, place strong
constraints on all the parameters because of the existence
of parameter degeneracies, such as the τ −A degeneracy
and the well-known geometrical degeneracy, involving Ω ,
m
Ω and Ω . Nevertheless, when WMAP7 is combined with
Λ k
other cosmological observations, they together increase the
constraining power and considerably weaken the degenera-
cies. In addition to WMAP7, we therefore include recent 0.8 0.85 0.9n 0.95 1 1.05
s
resultsfromCMBexperimentsthatareabletoreachhigher
resolutiononsmallpatchesofthesky,suchastheArcminute
Cosmology Bolometer Array (ACBAR; Kuo et al. 2004),
Cosmic Background Imager (CBI; Readhead et al. 2004),
BallonObservationsofMillimetricExtra-galacticRadiation
and Geophysics (BOOMERang; Jones et al. 2006). We also
includeobservationsofthematterpowerspectrumfromthe 0.8 0.85 0.9ns 0.95 1 1.05 −0.15 −0.1 Ω−k0.05 0
luminousredgalaxy(LRG)subsetDR7oftheSloanDigital
Sky Survey (SDSS; Reid et al. 2010). In addition to CMB
and galaxy surveys we include the Hubble Space Telescope
(HST; Riess et al. 2009) key project for the Hubble pa-
rameterH .Togethertheseobservationsmakeupdataset2.
0
0.8 0.85 0.9 0.95 1 1.05 −0.1 −0.05 0 0.05
n n
s run
Throughout, we consider purely Gaussian adiabatic
scalar perturbations and neglect tensor contributions. We Figure2.1-Dmarginalposteriordistributionsfordifferentpower
assume a universe with a cosmological constant where the spectrumparametermodels:PL1(ns;toprow),PL2(ns+Ωk;
background cosmology is specified by the following five pa- middle row) and PL 3 (ns+nrun; bottom row), using dataset 1
(solidline)anddataset2(dashedline).
rameters: the physical baryonic matter density Ω h2, the
b
physicaldarkmatterdensityΩ h2,theratioofthesound
DM
horizontoangulardiameterdistanceθ,theopticaldepthto
reionisation τ and the curvature density Ω . the running parameter (PL 3) displaces and broadens the
k
The parameters defining each model are listed in Ta- posterior probability, such that ns =0.944±0.020.
ble 2, together with the ranges of the flat prior imposed on When the curvature is considered as a free parame-
them in our Bayesian analysis. ter, dataset 1 selects a closed universe Ωk = −0.033+−00..003401
with a Hubble constant H = 61.0+12.2 km/s/Mpc. Sim-
0 −12.5
ilarly, for a flat Universe the constraints for the running
parameter n = −0.038 ± 0.027 and H = 66.6 ± 3.8
run 0
5 PARAMETER ESTIMATION km/s/MpcareingoodagreementwithLarsonetal.(2011).
The inclusion of measurements on different scales (dataset
5.1 Power-law parameterisations
2) weakens the geometric degeneracy and tightens signifi-
In Fig 2 we plot the 1-D marginal posteriors for the power cantly the constraints, yielding a mean posterior value for
spectrumparametersineachofthepower-lawmodels,using thecurvatureΩ =−0.0026±0.0049andHubbleparameter
k
bothdataset1 and dataset2. Wenowonrefertothemean H =70.4±1.7 km/s/Mpc. On the other hand, we observe
0
oftheposteriordistributionofeachparametertogetherwith thebest-fitfortherunningparameterismovedclosertoits
its 68 % confidence interval. For model 1 (n ), we see that zero value with constraints n =−0.018±0.016
s run
scale-invariant spectrum n = 1 is ruled out, as expected, InFig3,weplotthereconstructedshapeforthepower
s
with high confidence level using either dataset. Indeed, the spectrum for the simple spectral index in a curved uni-
constraints on n for a flat universe (PL 1) and a curved verse and the running parameterisation in flat universe, us-
s
universe (PL 2) using dataset 2 are similar: n = 0.963± ing mean values of the posterior distributions found from
s
0.011 and n =0.965±0.012 respectively. The inclusion of dataset 2.
s
(cid:13)c 2011RAS,MNRAS000,1–11
6 V´azquez et al.
40
35
30
]
10 25
-
0
[1 20
)
k 36 38 40 42 44 46 48 50 52 54 56
( 15 N
P
10
5
35
30
]
10 25
-10 20 65 66 67 68 H0 69 70 71 72
[
)
(k 15 Figure5.Marginalised1Dprobabilitydistributionsforthenum-
P berofe-foldingsproducedduringinflation(top)andHubblecon-
10
stant(bottom),obtainedfromtheLDmodelusingdataset2.
5
0.0001 0.001 0.01 0.1
k[Mpc-1]
intimately linked to the number of e-folds N during the in-
Figure 3.Primordialpowerspectrumreconstructedusingmean
flationary epoch through (19). We also find the constraint
valuesoftheposteriordistributionsfoundfromdataset2,forPL
b [10−9] = −17.74+2.92 and we note the effect occurring
model2(ns+Ωk;toppanel)andPLmodel3(ns+nrun;bottom th4rough (21): a gr−ea2t.7e8r initial curvature (increased |b |)
panel),with1σ errorbands. 4
would drive the universe today closer to be flat (decrease
|Ω −1|). Finally, the constraint on the scalar field mass is
k
µ[10−6]=1.79+0.17.
−0.16
The LD model requires a closed universe, then using
dataset 2 a very small value for the curvature parameter is
obtained Ω = −0.43+0.27 ×10−2, together with a Hubble
k −0.28
constant of H = 68.9±1.3 km/s/Mpc, both of which are
0
compatiblewithallexistingobservations.Wenote,however,
−25 −20 −15 −10
thatmanyauthorshavearguedonthedifficultytoconstruct
realisticclosed-universemodelswiththerequirednumberof
2.6
2.5 e-foldsandalsothatafinetuningmightarisefromtheinitial
2.4 conditionstoobtainanhomogeneousuniverseafterinflation
b02.3 (Ellisetal.2002;Linde2003;Uzanetal.2003).Inthissense,
2.2
theLDmodelprovidesthecorrectorderofmagnitudeinthe
2.1
prediction of the number of e-folds given by N = 48.1+3.3
−25 −20 −15 −10 2.2 2.4 2.6 −4.2
(see Fig 5).
2.6 2.6 Finally, we plot in Fig. 6 the reconstructed primor-
2.4 2.4 dial power spectrum for the LD model together with 1σ
−6µ [10]2.22 2.22 earurtoormbaatnicdasl.lyItciosnwtaoirntshapocilnetairngcuotu-toffthaattltohwe LkD-vaslpueecstraunmd
1.8 1.8
1.6 1.6 an additional fall-off at high wavenumber. A low-k trunca-
−25 −20 −15 −10 2.2 2.4 2.6 1.5 2 2.5 tion of the primordial spectrum of this type was proposed
b4 [10−9] b0 µ [10−6] phenomenologically by Efstathiou (2003). This kind of be-
haviour could be responsible for the small CMB power cur-
Figure 4.Marginalised1Dand2Dprobabilitydistributionsfor
the power spectrum parameters b4, b0 and µ in the LD model, rently observed at low l multipoles.
using dataset 2. 2D constraints are plotted with 1σ and 2σ con-
fidencecontours.
5.2 LD model
6 MODEL SELECTION
We choose to describe the LD spectrum in terms of the pa-
rameters b and b , letting µ be a derived parameter such We now investigate which model provides the best descrip-
0 4
that the conformal time constraint is fulfilled. The result- tionofthedatabyperformingmodelselectionbasedonthe
ing constraints from dataset 2 on the parameters describ- value of the Bayesian evidence. Before performing this pro-
ing the LD power spectrum are shown in Fig 4. In par- cessontherealdatasets,however,wefirsttestourapproach
ticular we obtain the constraint b = 2.45+0.90, which is by applying it to an idealised dataset.
0 −0.11
(cid:13)c 2011RAS,MNRAS000,1–11
Bayesian study of the primordial power spectrum 7
40
35
30 0.02160.02180.0220.02220.0224 0.108 0.11 0.112 0.114 1.03551.0361.03651.037
Ω h2 Ω h2 θ
b DM
0] 25
1
-
0
[1 20
)
k
P( 15 0.08 0.09 0.1 −0.03 −0.02 −0.01 −16−14−12−10 −8
τ Ωk b4 [10−9]
10
5
2.3 2.4 2.5 2.6
0.0001 0.001 0.01 0.1 b
k[Mpc-1] 0
Figure6.PrimordialpowerspectrumreconstructedfromtheLD Figure8.1-Dmarginalisedparameterconstraintsforasimulated
model using mean values of the posterior distributions obtained model.Thesetofparametersusedtoconstructthemockdatais
fromdataset2,with1σ errorbands. representedbytheverticallines.They-axisisrelativeprobability.
7000 Table 3.Differenceoflog-evidencesforthedifferentparameter-
6000 l(l+1)C T/2! isationsrelativetosimplepower-lawflatmodelforthesimulated
l CMBdataset.
5000
Model ∆lnZ
4000
PL1(ns) 0.0 ± 0.4
3000 PL2(ns +Ωk) -1.0 ± 0.4
PL3(ns +nrun) -3.1 ± 0.4
2000 LDmodel +3.7 ± 0.4
1000
areshowninFigure8.Weobservethatourapproachyields
0
constraints that appear consistent, within statistical error,
50
40 with the input values used in the simulation, which are in-
30 l(l+1)C E/2! dicated by the vertical lines in each plot. Moreover, these
l
20 results give us an indication of how accurately the param-
10
eters of the LD model can be constrained with an optimal
0
CMB dataset. In particular, we note that the behaviour of
10 100 1000
theposteriorforΩ isreflectedinthatobtainedforb ,that
l k 4
is because of the link between them through the see-saw
Figure 7.Top(bottom)panelshowsasimulatedCMBtemper- mechanism.
ature (polarisation) power spectrum, convolved with chi-square The evidence values of the models considered for the
noiseandlimiteddatauptol of2000.Themodelfromwhichit primordial spectrum are given in Table 3. Based on Jef-
wasproducedisdrawnbytheblueline. freys’ criterion we observe significant difference in the log-
evidences for each model. Thus, there is a clear distinction
between models, with the data clearly indicating a prefer-
6.1 Application to simulated data
ence, as expected, for the LD model used to generate the
To test our model selection (and parameter estimation) simulateddata.Giventhesuccessofmodelselectiononsim-
method, we implement the LD spectrum in a modified ver- ulateddata,wenowturntorealdatawithsomeconfidence.
sion of the CAMB package (Lewis et al. 2000) to simulate
theCMBpowerspectrumpredictedbytheLDmodelusing
standard values for the cosmological parameters, i.e. those 6.2 Application to real data
obtainedfromWMAP7.ThepredictedtemperatureandE-
Before performing the Bayesian selection between our dif-
mode polarisation CMB spectra are plotted as the dashed
ferent models for the primordial power spectrum, we will
line in Fig. 7. We simulate an idealised process where only
momentarily use a frequentist method and compute the
cosmic variance noise was added to the spectra such that
goodness-of-fit of each of the models using real data. Us-
eachC valuebecomesarandomvariabledrawnfromaχ2
l 2l+1 ing the best-fit parameters obtained for each model with
distribution with variance dataset 1, we may calculate χ2 , defined as −2lnL , at
min max
(∆C)2 = 2 C2. (26) the maximum point in each case. These values are given in
l 2l+1 l Table4fromwhichweseethatallthreemodelsfitthedata
The 1D marginalised parameter constraints for the LD almost equally as well.
model resulting from the analysis of this simulated dataset TurningnowtoBayesianmodelselection,thedifference
(cid:13)c 2011RAS,MNRAS000,1–11
8 V´azquez et al.
can be described in terms of matter-energy content of the
Table 4.Log-likelihoodvaluesatthebest-fitpointforthreedif-
ferentmodelsoftheprimordialpowerspectrumusingdataset1. universe: baryons (Ωb,), photons (Ωr), Cold dark matter
(Ω )anddarkenergy(Ω ),aswellasonitsexpansionhis-
DM Λ
Model Nparams -lnLmax tory: Hubble time (H0). The inflationary period is encoded
PL2(ns +Ωk) 7 7475.3 in three parameters that have emerged in the construction
PL3(ns +nrun) 7 7473.5 of the appropriate initial conditions. They are related to
LDmodel 7 7473.0 the initial curvature (b ), the initial expansion size of the
4
universe (b ) and the amplitude of perturbations (µ). Such
0
parameters are not predicted a priori, rather we have fit
Table 5. Difference of log-evidences for each of our models for
themusingcurrentobservationaldata.Noticethatwehave
theprimordialpowerspectrum,relativetothesimplepower-law
thefreedomtopicktwoofthemandcomputethethirdpa-
model.
rameter through the constraint imposed on the conformal
time.Inthispaperwehavefocusedonb andb todescribe
Model Nparams Dataset1 Dataset2 0 4
theLDspectrum.Thecomparisonofthebest-fitprimordial
PL1(ns) 6 0.0±0.3 0.0±0.3 spectra for our different models is shown on Fig 9.
PL2(ns +Ωk) 7 −0.5±0.3 −2.6±0.3 We test the LD spectrum by building a simple toy
PL3(ns +nrun) 7 −0.8±0.3 −1.7±0.3 modelfromachosenCMBspectrumwithsimulatedcosmic
LDmodel 7 +1.2±0.3 −0.9±0.3
variance noise. A full Bayesian analysis was performed
for this toy model and various parameterisations of the
in the log-evidences for each of the models are given in Ta- primordial power spectrum. We have included not only
ble 5, as derived from dataset 1 and dataset 2 respectively. the estimation of cosmological and spectral parameters
Thelog-evidencedifferenceshouldbeinterpretedinthecon- but also the Bayesian evidence. For real data, we com-
textoftheJeffreys’scalegiveninTable1.Intheanalysiswe pared the χ2 for the three different models and concluded
haveconsideredawideconservativepriorfortheparameters that the goodness-of-fit for the LD model is about the
in each model. The parameter estimation results in Section same as the well-studied power-law models. We also
5 showed that the constraints obtained lie well within our pointed out that the constraints on the cosmological
chosenpriorandassuchanyincreaseinthepriorwouldnot parameters in the LD model are consistent within 1σ error
include very much more posterior mass within the evidence barswiththepower-lawmodels,asshowninFigs10and11.
integral.Thuswewouldnotexpectreasonablemodifications
of the prior range to alter the results makedly. The Lasenby & Doran model predicts a primordial
For dataset 1, the similarity in log-evidences and their scalarspectrumthatatlargescalesnaturallyincorporatesa
statisticaluncertaintiesmakeitdifficulttoobtaindefinitive dropoffwithouttheneedtoparameteriseit,while,atsmall
conclusions.Whentheevidencevaluesareranked,however, scalesitautomaticallyincorporatesadegreeofnegativerun-
we observe a slight preference in favour of the LD spec- ning compared with the standard power law parameterisa-
trum over the three power-law parameterisations, whereas tion (see e.g. Fig 9 below or Fig. 17 in Lasenby & Doran,
model3(ns+nrun)issignificantlythemostdisfavoured.The 2005, for a visual comparison). Given that the primordial
inclusion of additional information on different scales with parametersb andb /µ4/3 arewellconstrainedandoforder
0 4
dataset 2 significantly reduces the evidence for the three 7- unity, no fine tuning is needed for the construction of the
parameter models. We notice that a curved universe with a initialconditions.Moreover,theLDmodelhastherequired
simple power-law, PL 2 (ns +Ωk), is strongly disfavoured orderofe-foldingstoobtainrealisticmodelsofaclosedinfla-
relative to a flat universe with a power-law spectrum (ns). tionaryuniverse.Finally,eventhoughtheBayesianevidence
When we perform a comparison of models which contain fromeachmodelshowedthatitisdifficulttomakedefinitive
the same number of parameters, we observe the LD model conclusions because of the presence of statistical uncertain-
is 1.7 units of log-evidence above the PL model 2, which ties,animportantresulttoemphasiseisthatforthemodels
is a significant difference, but only 0.8 log-units above PL with the same number of parameters, the preferred model
model3.TheΛCDMmodelwithapower-lawspectrumhas toexplaincurrentcosmologicalobservationsingivenbythe
the largest evidence in this case. We observe from Table 5, LDmodel.Thismaybeseenbynotingthat,formodelswith
that from all of the models considered with equal numbers 7parameters,Table5showsthattheLDmodelispreferred
of parameters (with either dataset) that the LD model is significantly (by 1.7-2.0 log units of evidence) in all cases
currently preferred. except relative to n +n with dataset 2, for which it is
s run
preferred by 0.8 log units.
7 DISCUSSION AND CONCLUSIONS
ThenovelcloseduniversemodelproposedbyLasenby&Do-
rangivesrisetoaboundaryconditiononthetotalavailable
conformal time:
π
ηM +ηI = 2. ACKNOWLEDGMENTS
Herewehavesplittheuniverseevolutionintwomainepochs This work was carried out mainly on the Cambridge High
that contribute to the amount of conformal time: matter- Performance Computing Cluster Darwin. JAV is supported
dominatedepochandinflationaryperiod.Thematterepoch by CONACYT M´exico.
(cid:13)c 2011RAS,MNRAS000,1–11
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0.02 0.022 0.024 0.1 0.15 1.03 1.04 0.05 0.1 0.15
Ω h2 Ω h2 θ τ
b DM
3 3.1 3.2 0.8 0.9 1
log[1010A] n
s s
3 3.1 3.2 0.8 0.9 1 −0.15 −0.1 −0.05 0
log[1010A] n Ω
s s k
3 3.1 3.2 0.8 0.9 1 −0.1 −0.05 0 0.05
log[1010A] n n
s s run
−20 −15 −10 −5 2 2.2 2.4 2.6 −0.06 −0.04 −0.02
b [10−9] b Ω
4 0 k
Figure 10. Marginalisedparameterconstraintscorrespondingtons (dottedline),ns+Ωk (dash-dotline),ns+nrun (dashedline)and
L&Dmodel(solidline),usingdataset1.
(cid:13)c 2011RAS,MNRAS000,1–11