Relativistic cosmology number densities in void-Lemaitre-Tolman-Bondi models

0.34 MB·

by  A. Iribarrem

Astronomy&Astrophysicsmanuscriptno.LF3-arxiv (cid:13)c ESO2014
January28,2014
Relativistic cosmology number densities in
void-Lemaˆıtre-Tolman-Bondi models
A.Iribarrem1 ⋆,P.Andreani2,S.February3,C.Gruppioni4,A.R.Lopes1,M.B.Ribeiro5,andW.R.Stoeger6
1 Observato´riodoValongo,UniversidadeFederaldoRiodeJaneiro,Brazil
2 EuropeanSouthernObservatory,Karl-Schwarzschild-Straße2,85748Garching,Germany
3 Astrophysics,CosmologyandGravitationCentre,andDepartmentofMathematicsandAppliedMathematics,UniversityofCape
Town,Rondebosch7701,CapeTown,SouthAfrica
4 INAF–OsservatorioAstronomicodiBologna,ViaRanzani1,I-40127Bologna,Italy
4
5 InstitutodeF´ısica,UniversidadeFederaldoRiodeJaneiro,Brazil
1
6 VaticanObservatoryResearchGroup,StewardObservatory,UniversityofArizona,AZ85721,Tucson,USA
0
2

n
a ABSTRACT
J
Aims.Thegoalofthisworkistocomputethenumberdensityoffar-IRselectedgalaxiesinthecomovingframeandalongthepast
5
lightconeofobservationallyconstrainedLemaˆıtre-Tolman-Bondi“giantvoid”modelsandtocomparethoseresultswiththeirstandard
2
modelcounterparts.

Methods.Wederivedintegral number densitiesanddifferential number densitiesusingdifferent cosmological distance definitions
]
O in the Lemaˆıtre-Tolman-Bondi dust models. Then, we computed selection functions and consistency functions for the luminosity
functionsinthecombinedfieldsoftheHerschel/PACSevolutionaryprobe(PEP)surveyinbothstandardandvoidcosmologies,from
C
whichwederivedtheobservedvaluesoftheabove-mentioneddensities.WeusedtheKolmogorov-Smirnovstatisticstostudyboth
. theevolutionoftheconsistencyfunctionsanditsconnectiontotheevolutionofthecomovingdensityofsources.Finally,wefitted
h
thepower-lawbehaviourofthedensitiesalongtheobserver’spastlightcone.
p
Results.TheanalysisofthecomovingnumberdensityshowsthattheincreasedflexibilityoftheLemaˆıtre-Tolman-Bondimodelsis
-
o notenoughtofittheobservedredshiftevolutionofthenumbercounts,ifitisspecialisedtoarecentbest-fitgiantvoidparametrisation.
r Theresultsforthepower-lawfitsofthedensitiesalongtheobserver’spastlightconeshowgeneralagreementacrossbothcosmological
st modelsstudiedherearoundaslopeof-2.5±0.1fortheintegralnumberdensityontheluminosity-distancevolumes.Thedifferential
a numberdensitiesshowmuchbiggerslopediscrepancies.
[ Conclusions.Weconclude thatthedifferentialnumber densitiesontheobserver’spastlightconewerestillrendereddependent on

thecosmological model by thefluxlimitsof thePEPsurvey. Inaddition, weshow that anintrinsicevolution of thesources must
1
beassumedtofitthecomovingnumber-densityredshiftevolutioninthegiantvoidparametrisationfortheLemaˆıtre-Tolman-Bondi
v
modelsusedinthiswork.
2
7 Keywords.Galaxies:distancesandredshifts–Cosmology:theory–Galaxies:evolution–Infrared:galaxies
5
6
.
1 1. Introduction HubbleparametermeasurementsfromthedistancestoCepheids
0 (Freedmanetal. 2001), or from the massive, passively evolv-
4 The aim of many studies in observational cosmology is to in- ingearly-typegalaxies(Morescoetal.2012;Liuetal.2012),or
1 fer to what extent the geometry of the spacetime contributes
from the extragalacticHII regions(Cha´vezetal. 2012); the lu-
: to the formation and evolution of the galaxies, i.e. the build-
v minosity distance-redshift relation stemming from supernovae
ingblocksoftheluminousUniverse.Inpracticalterms,mostof
i type Ia (henceforth, simply SNe) surveys (Riessetal. 1998;
X whatwecaninferaboutgalaxyformationandevolutioncomes
Perlmutteretal. 1999), the power spectrum of the cosmic mi-
r from analyses of redshift surveys. Although the redshift is an crowavebackgroundradiation(CMB,e.g.Komatsuetal.2011),
a observable quantity related to the energy content and geome-
and the angular size scale obtained from baryonicacoustic os-
try of the Universe – regardless of how we model it – trans-
cillation(BAO)studies(e.g.Percivaletal.2010).Thedegreeof
lating these measurements into distance estimations cannot be
confidence on the concordance model is such that the above-
performedwithoutassumingacosmologicalmodel.Asaconse-
mentionedmodeldependencyofgalaxyformationandevolution
quence,it is clear that anystudythat involvesgalaxydistances
nowadayssoundslikeanunavoidable,butotherwiselessconse-
willbemodeldependent.Thisdependencycannotbeovercome
quentialfact.
aslongasthedistanceestimatorsusedinthesestudiesarenotdi-
rectlyobtained,butinsteadderivedfromredshiftmeasurements. Despite the sizeable constraining power of the current
The standard cosmologicalmodel, often called the concor- set of observations show, there is still room for considering
dancemodel,ispresentlyabletosimultaneouslyfitmostofthe alternative cosmologicalmodels. Even if most of these models
currentobservationsof independentcosmologicalquantities.A are disfavoured when compared to the standard model (e.g.
few examples of observations that support this model include Sollermanetal. 2009), some of them cannot be ruled out
entirely yet. Among these models, motivated mainly as alter-
⋆ [email protected] native to dark energy, various non-homogeneous cosmologies
1
A.Iribarremetal.:Numberdensitiesinvoidcosmologies
have been proposed, with an increasing interest in the past framecomputations.Theempiricalapproachofit,ifnotasam-
few years (Hellaby&Alfedeel 2009; Alfedeel&Hellaby bitious, shares the same philosophy of the ideal observational
2010; Februaryetal. 2010; Biswasetal. 2010; Sussman cosmologyprogrammeasEllisetal.(1985),wheretheaimwas
2010; Bolejkoetal. 2011b; Nadathur&Sarkar 2011; todeterminethespacetimegeometryoftheUniversewithoutas-
Clarkson&Regis2011;Clarksonetal.2012;Regis&Clarkson sumingacosmologicalmodelapriori.
2012; Meures&Bruni 2012; Humphreysetal. 2012; The present line of work started with Ribeiro&Stoeger
Nishikawaetal. 2012; Bull&Clifton 2012; Valkenburgetal. (2003), which connected the relativistic cosmology number
2012; Wang&Zhang 2012; Hellaby 2012; Hoyleetal. 2013; countsresultssummarizedinEllis(1971)tothepracticallumi-
Bulletal. 2012; dePutteretal. 2012; Keenanetal. 2012; nosity function (LF) results from galaxy redshift surveys. The
Fleuryetal. 2013). Some recent reviews include Ce´le´rier LF is a statistical tool to infer the formation and evolution of
(2007), Bolejkoetal. (2011a), Ellis (2011), and Krasin´ski galaxies from a set of photometric or spectroscopic data. To
(2013). compute the luminosities of the sources from their observed
Thesimplestnon-homogeneousmodelassumesaLemaˆıtre- fluxes,aluminositydistancemustbeobtainedfromtheredshift
Tolman-Bondi (LTB) metric coupled to a pressure-less (dust) estimation,astep thatrequiresadoptingacosmologicalmodel.
energy-momentumtensor.LTBdustmodelsyieldanalyticalso- Todealwithincompletenessduetothefluxlimits,anassumption
lutionstotheEinstein’sfieldequations,e.g.Bonnor(1972),that on the spatial homogeneity of the distribution is usually made.
canbereducedbyanappropriateparametrizationtothestandard Spatialhomogeneityisdefinedonconstanttime-coordinatehy-
modelones.It is, forexample,quiteusualto set thefree equa- persurfaces,nottobeconfusedwithobservationalhomogeneity
tionsallowedinthismodelinawaythatitreducestotheusual thatis definedonthe observer’spastlightcone.Thespatial ho-
Einstein-deSittersolutionatlargeenoughredshifts,enablingthe mogeneityassumptiondoesnotholdfornon-homogeneouscos-
model to naturally fit many CMB constraints, albeit with very mologies in general, but Iribarremetal. (2013) showed that it
lowH .Thismodelcanbeunderstoodasageneralisationofthe does not lead to significant differences in the LF results in the
0
FLRWmetric,aneffectivemodel,possiblythesimplestone,in caseoftheLTB/GBHmodelsadoptedhere.
which the observed large-scale inhomogeneities play a role in Ribeiro (2005) considers the effect of the expansion of the
thelate-timedynamicsandnearbygeometryoftheUniverse. spacetimeinthenumberdensitiesalongthepastlightconelead-
Usingthegreaterflexibilityofthesemodels,Mustaphaetal. ingtoobservationalnon-homogeneitiesinthespatiallyhomoge-
(1997)showedthatanysphericallysymmetricastronomicalob- neous Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) model.
servation, such as redshift surveys, can be fitted by the non- RangelLemos&Ribeiro (2008) expanded on those theoretical
homogeneities allowed in these models, instead of any source results,showingthathomogeneousdistributionsinthepastlight-
evolution.Thisisparticularlyrelevantforgalaxyevolutionsince cone would lead to spatial non-homogeneity, in disagreement
itimpliesthatinordertoestablishitbeyondthistheoreticalpos- withtheusualformofthecosmologicalprinciple.
sibility,itisnecessarytouseacombinationofvariousindepen- Albanietal. (2007) combined the results of the first two
dent observable quantities. Garcia-Bellido&Haugbølle (2008, previous papers to compute such lightcone distributions us-
hereafter GBH) propose a “giant void” parametrisation for the ing the luminosity functions for the CNOC2 survey from
LTB dust model that was able to reach that exactly: fit simul- Linetal.(1999),confirmingthepresenceofsuchlightconenon-
taneously the SNe Ia hubble diagram and some general fea- homogeneities. In Iribarremetal. (2012) we discussed further
tures of the CMB. Despite the many problems this alterna- ways to compare the relativistic effect of the expandingspace-
tivemodelshows(Marra&Pa¨a¨kko¨nen2010;Mossetal.2011; timetotheevolutionofthesourcesintheFLRWpastlightcone,
Zhang&Stebbins2011;Zibin2011;PlanckCollaborationetal. using the much wider redshift range in the luminosity func-
2013),arguablyitremainsthesimplestandbest-studiedwayto tionsfortheFORSDeepFieldsurveyfromGabaschetal.(2004,
allowfornon-homogeneitiesinthecosmologicalmodel.Given 2006).
thetheoreticalpossibilityoffittingtheredshiftevolutionofthe Thegoalofthepresentworkistoincludenon-FLRWspace-
number counts without any intrinsic evolution of the sources times in the past lightcone studies described above. We spe-
mentioned above, we aimed to investigate what our view of cialisedthegeneralequationsofRibeiro&Stoeger(2003)tothe
galaxyevolutionwouldbeunderthesealternativemodels. LTBmetric,combiningtheresultsfrombothRibeiro(1992),and
To address the question of how robustly the current Garcia-Bellido&Haugbølle(2008). Finally, we used the lumi-
observations render the luminosity function of galaxies, in nosityfunctionsinIribarremetal.(2013),whichwerecomputed
Iribarremetal. (2013) we used the dataset of FIR-selected byassumingtheLTB/GBHmodelfromtheirbuild-up.
sourcesinGruppionietal.(2013)forthecombinedGOODS-N, Thequestionweaimtoaddressisthefollowing:arethenum-
GOODS-S,COSMOSandECDF-SfieldsintheHerschel/PACS berdensitiesalongthepastlightconeandthecharacterisationof
evolutionary probe (PEP) galaxy redshift survey, computed in theirpowerlawbehaviour,robustamongthestandardandvoid
boththeΛCDMstandardandLTBgiantvoidmodels,thelatter cosmological models cited above? In addition, we discuss the
parameterizedasinGarcia-Bellido&Haugbølle(2008)withthe comovingnumberdensityinthevoidmodels,withpossibleim-
best-fitparametersgivenbyZumalaca´rreguietal.(2012). plicationsforempiricalmodelslikethatin,e.g.,Gruppionietal.
WhilethemotivationofIribarremetal.(2013)wastoprobe (2011);Be´therminetal.(2012).
the implications of the underlying cosmology on the redshift Thepaperis organisedas follows. In Sect. 2, we specialise
evolutionofobservablepropertiesofthesources,suchasnum- the results in Ribeiro&Stoeger (2003) and Ribeiro (2005) to
bercountandluminosity,thepresentworkaimstofurtherstudy theconstrainedGBHparametrisationoftheLTBdustmodels.In
these implications, specifically, on the characterization of the Sect.3,wecomputetheselectionfunctionsfortheFIRdatasets
number density of sources along the past lightcone of the as- in Iribarremetal. (2013). In Sect. 4 we compute the connec-
sumed cosmology.The past lightcone is a direct observable:it tionbetweenthepredictedcomovingdensityofsourcesinboth
istheonlyregionoftheUniversemanifoldthatgalaxyredshift FLRWandLTBgeometriestotheselectionfunctions.InSect.5
surveys probe. In computing the densities in the lightcone, we wecomputethenumberdensitiesofPEPsourcesdownthepast
remove the model assumption intrinsic to the usual comoving lightconeofbothcosmologies.InSect.6,wepresenttheresults,
2
A.Iribarremetal.:Numberdensitiesinvoidcosmologies
showevidenceofgalaxyevolutionintheLTB/GBHmodel,and ThelastequationisessentiallyaversionoftheresultofEllis
comparethecomovingnumberdensityevolutionoverbothcos- (1971),specialisedto thepastnullgeodesicof theLTBmodel.
mologies.InSect.7wediscussthedependencyoftherelativistic Next, we further specialised it to use the GBH parametrisation
numberdensitiesonbothfluxlimitandcosmologicalmodel.In for their constrainedmodel,and the best-fit valuesobtainedby
Sect.8wepresentourconclusions. Zumalaca´rreguietal. (2012) in a simultaneousanalysisofSNe
Ia,CMB,andBAOdata.
Itisstraightforwardtorelate f(r)tothespatialcurvaturepa-
2. TheoreticalquantitiesintheLTB/GBHmodel
rameterk(r)inGBHbywriting
In the empirical approach of Albanietal. (2007), the number
f(r)= 1−k(r). (6)
densities along the observer’s past lightcone were computed
fromLFdatausingdifferentrelativisticcosmologydistancedef- p
TheboundaryconditionequationslistedinGBHreadas
initions. These distances are defined in the same cosmological
modelastheoneassumedonthebuild-upoftheLF.Inthiscon- A˙(r,t)
text no assumptions on the redshift evolution of the sources is HT(r,t)= A(r,t), (7)
made.Themethodologyiscompletelyempirical.
F(r)=2Ω (r)H2(r)r3, (8)
Thenumberdensitiesusedinthispaper,asdefinedinRibeiro M 0
(2005),areabletoprobethegeometricaleffectoftheexpansion k(r)=−[1−Ω (r)]H2(r)r2, (9)
M 0
ofUniverseonthehomogeneityofthedistributionofthesources
along the observer’spast lightcone. This effect dependson the withthegaugechoiceA(r,0)=rincluded,whereHT isthetrans-
distance definition used in computingthese densities, which in verseHubblerate,H0(r)= HT(r,0),andΩM(r)isthedimension-
turndependsonthelineelementofthecosmologicalmodelas- lessmatterdensityparameter.Thislastquantityisdefinedrela-
sumed. tivetotheintegratedcriticaldensityinthecomovingvolumeat
In this section we connect the key results from Ribeiro radialcoordinateras
(1992, henceforth R92) for the number count of sources 3H2(r)
in the LTB metric to the parametrisation advanced by ρ¯ = 0 . (10)
C 8πG
Garcia-Bellido&Haugbølle (2008) for that cosmology. The
goalhere is to computethe above-mentionednumberdensities Thepresent-timetransverseHubbleparameter,H (r),inthe
0
inthegiantvoidparametrisationoftheLTBmodel.Dottedquan- constrainedversionoftheGBHmodelisparametrisedas
titiesrefertotime-coordinatederivativesandprimedonesrefer
toradial-coordinatederivatives. ∞ 2[1−Ω (r)]n
H (r)= H M . (11)
0 in (2n+1)(2n+3)
Xn=0
2.1.Differentialnumbercount
Equations(6),(9),and(11)canbereadilycombinedtoyield
We started by writing the line element for the LTB model fol-
lowingBonnor(1972) ∞ 2[1−Ω (r)]n 2
A′2(r,t) f(r)= vut1+[1−ΩM(r)] (2n+1)(2Mn+3) Hin2r2. (12)
ds2 =−c2dt2+ dr2+A2(r,t)dΩ2, (1) Xn=0 
f2(r) CombiningEqs.(8)and(11),wecanwritetheF(r)function
wheredΩ2 = dθ2 +sin2θdφ2,with f(r) and A(r,t) beingarbi- fortheconstrainedmodelas
trary functions.Assuming a pressure-less(dust) matter content
∞ 2[1−Ω (r)]n
ewqiuthatρioMnpsrfoopretrhadtenlisnietye,lietmcaenntbceanshboewcnomthbaitntehdetEoinysieteldin(’Rs9fi2e)ld F(r)= Hin (2n+1)(2Mn+3)ΩM(r)r3, (13)
Xn=0 
F′ wherethe dimensionlessmatterdensity parameterΩ (r) in the
8πGρ = , (2) M
M 2A′A2 GBHmodelbecomes
where F(r) is another arbitrary function satisfying the relation 1−tanh[(r−R)/2∆r]
above,andGisthegravitationalconstant.Startingfromthegen- Ω (r)=Ω +(Ω −Ω ) . (14)
M out in out ( 1+tanh[R/2∆r] )
eralexpressionforthenumbercountofsourcesderivedbyEllis
(1971),R92obtained Here H is the transverse Hubble constant at the center of the
in
void,Ω isthedensityparameteratthecenterofthevoid,Ω the
A′A2 in out
dN =4πn dr, (3) asymptoticdensityparameteratlargecomovingdistances,R is
f thesizeoftheunder-denseregion,and∆Rthewidthofthetran-
sition between the central void and the exterior homogeneous
where n is the number density per unit proper volume, and N
region.Theseparameterscompletelydeterminethemodel.
thetotalnumberofsourcesdownthepastlightcone.Assuming
Because of the generality of the evolution of A(r,t) in the
asanorderofmagnitudeestimationoftheaveragetotalmassof
eachsourceM≈1011M⊙,R92wrote LTBmodels,thetimecoordinatetbb atwhichA(r,tbb)reducesto
zero, can, in general, assume different values for different co-
F′ movingdistances from the centre of the under-denseregion,r.
n= . (4)
16πGMA′A2 Thisleadstodifferentmeasurementsfortheelapsedtimesince
the Big Bang, t (r), dependingon the position of the observer
Combiningthelasttwoequations,weobtained bb
in the void. Setting this extra degree of freedom for the big-
dN 1 F′ bang time in order to make it simultaneous(same value for all
= . (5)
dr 4GM f observers) yields the constrained version of the GBH model,
3
A.Iribarremetal.:Numberdensitiesinvoidcosmologies
Table1.BestfitvaluesfortheLTB/CGBHmodelsfromZumalaca´rreguietal.(2012)usedinthiswork.
Modelparameter CGBH OCGBH
H 66.0±1.4 71.1±2.8
in
Ω 0.22±0.4 0.22±0.4
in
R[Gpc] 0.18+0.64 0.20+0.87
−0.18 −0.19
∆R[Gpc] 2.56+0.28 1.33+0.36
−0.24 −0.32
Ω 1 0.86±0.03
out
or CGBH model. The best-fit values we use in this work were
obtained in Zumalaca´rreguietal. (2012), considering both an 1012
asymptotically flat CGBH model with Ω = 1 and an open
out
CGBH model (OCGBH) with Ω = 0.87, which the authors
out
show better fits the CMB constraints.Table 1 reproducesthese
1011
values.
For each comoving distance r, Ω (r), f(r), and F(r) can
M
be computed through Eqs. (12), and (13). The radial deriva- z
d
tiveF′(r)canbeobtainednumericallythroughcentraldifference N/1010
quotients,F(r)≈lim ∆F/∆r,whichinturnallowthevalues d
∆r≪r
ofdN/drtobecomputedthroughEq.(5).
The redshift can be related to the radial coordinate in this 109
modelfollowingEnqvist&Mattsson(2007) ΛCDM
CGBH
dr 1 f OCGBH
dz = 1+z A˙′, (15) 108 0.1 z 0.5 1 2 3 4 56
and to the time coordinate through the incoming radial null
geodesicequationasinGBH Fig.1.Differentialnumbercountestimateswithinthepastlight-
coneofthethreecosmologicalmodelsusedinthepresentwork.
dt A′
=− . (16)
dz f(r)
Following GBH, all the first- and second-order derivatives of whereas their redshift derivatives, d(d )/dz, d(d )/dz, and
A L
A(r,t) can be written as power series expansion, which allows d(d )/dz can be computed numerically with the same method
G
Eqs. (15) and (16) to be solved numerically. This yielded the used for computing F′(r). To compare the theoretical results
look-backtimeandradialdistancetables:t(z)andr(z).Wethen forthe numberdensitiesin the pastlightconeofthe LTB/GBH
combinedEqs.(5)and(15)as modelswiththosefortheFLRWgeometrygiveninAlbanietal.
dN 1 F′[r(z)] (2007,Figs.5-6),wealsocomputetheredshiftdistancedz sim-
= , (17) plyasd =cz/H .
dz 4GM (1+z)A˙′[r(z),t(z)] Withzthatwe0cancomputetherelativisticdifferentialnumber
and computed the differential number count dN/dz for each densities(γi),thenumberofsourcesperunitvolumeinaspher-
value of z in the r(z) and t(z) tables. A comparison between icalshellatredshiftz,foreachdistancedi ini= [A,G,L,z]and
the estimates for this quantity in the ΛCDM and both CGBH ineachcosmologicalmodelasinRibeiro(2005):
parametrisationsinZumalaca´rreguietal.(2012)canbefoundin
Fig.1. γ = dN 4π(d)2 d(di) −1. (19)
i dz ( i dz )
2.2.Distancesandnumberdensities Following Iribarremetal. (2012), the integralnumber den-
sities (γ∗), i.e. the number of sources per unit volume located
Since it is impossible to measure the distance to galaxies di- i
inside the observer’s past lightcone down to redshift z, can be
rectly, there are a few different ways of deriving this quantity
computedforeachdistancedefinitiond as
fromothermeasurements.Theluminositydistance,forexample, i
isbasedontherelationbetweentheemittedandreceivedfluxes 3N
of the source, whereas the angular diameter distance is based γ∗ = , (20)
i 4π(d)3
ontherelationbetweentheobservedangularsizeandtheactual i
physicalsizeofthesource.Thoserelations,however,dependon withthecumulativenumbercountN(z)obtainedsimplyby
theexpansionhistoryofthe Universeandwill, in general,lead
to differentresultssimply becausetheir dependencyon the un- N(z)= z dN(z′)dz′. (21)
derlyingcosmologicalmodelisnotthesame. Z dz
0
The angular diameter distance in LTB models can be iden-
Resultsforthevariousγ andγ∗inthedifferentcosmologies
tified as d (z) = A[r(z),t(z)]. From that we obtain the lumi- i i
A consideredin this work are plotted in figures2 and 3. We note
nosity distance d (z) and the galaxy-area distance d through
L G thatamongthesecosmologicalmodels,thedifferentialandinte-
Etherington’s reciprocity law (Etherington 1933; Ellis 1971,
gralnumberdensitiesshownoticeabledifferencesinalldistance
2007)
definitionsused,evenifapparentlysmalltotheeye.Thesedif-
d =(1+z)2d =(1+z)d , (18) ferencesmay actually be observed,dependingon the precision
L A G
4
A.Iribarremetal.:Numberdensitiesinvoidcosmologies
rest-frameluminosityfunctionsfor the combinedPEPfields in
102 Iribarremetal.(2013)as
γA
101 γγLG ψ = Lmax,z¯φ(L)dL, (22)
γz z¯ ZLlim,z¯ z¯
100
] whereL isthebrightestsourceluminosityinsidetheredshift
3− max,z¯
c interval,and L is the rest-frameluminosityassociated to the
p10-1 lim,z¯
M fluxlimitoftheobservations.
[
γ The dataset used in this work was built using combined
10-2 fields,i.eobservationscarriedoutindifferentskyregionsatdif-
ferentdepths.Thelargenumberofsourcesperredshiftinterval
10-3 ΛCDM allowed us to derive the observed quantities with better statis-
CGBH
tics,attheexpenseofanaddeddifficultyrelatedtothedefinition
OCGBH
10-4 of the luminosity limits correspondingto the actual flux limits
0.1 0.5 1 2 3 4 56
z of the observations.This difficulty stems from the fact that the
computationoftherest-frameluminosityinvolveskcorrections,
Fig.2. Redshift evolution of the relativistic differential densi-
whichinturndependonthespectralenergydistribution(SED)
tiesforthethreecosmologicalmodelsusedinthepresentwork.
ofthesource.Thatis,eachSEDtemplatedefinesaslightlydif-
Differentcurvescorrespondtothecomputationsperformedwith
ferent luminosity limit for the same flux limit. In the case of
respecttodifferentdistanceestimatorsalongtheobserver’spast
the PEP survey, each field – namely GOODS-N, GOODS-S,
lightcone(dA,dL,dG,anddz). COSMOSandECDF-S–alsohadadifferentfluxlimit,depend-
ing on the PACS passband in which the observation was done
(Lutzetal.2011).
102 To investigate how important the selection function varia-
γA∗ tionscausedbythedifferentSEDtemplatesinthedatasetsare,
101 γγLG∗∗ wvaelafisrssutmcoinmgptuhteedlotwheesstecloemctipountefdunlucmtioinnossiintyeaamchonregdtshheifstoiunrtceers-
γz∗
in that interval, and then compared the results to the averages
100
] and to the highest luminosities. The variations in the selection
3−
c functions caused by assuming different luminosity limits were
Mp10-1 found to be many times greater than the error bars propagated
[
∗ fromthe luminosityfunctionuncertainties.Forexample,atz =
γ10-2 0.2,theselectionfunctioncomputedusingthelowestmonochro-
maticluminosityinthe100-µmdatasetread(9.0±1.8)×10−3
10-3 ΛCDM Mpc−3, while the one computed using the average luminosity
CGBH read (3.7 ± 0.7) × 10−3 Mpc−3, and the one using the highest
OCGBH
10-4 luminosity read (2.6 ± 0.5) × 10−3 Mpc−3 - a variation more
0.1 0.5 1 2 3 4 56
z thanthreetimeslargerthanthecombineduncertaintiesobtained
by propagatingthe onesfromthe LFparameters.For the same
Fig.3.IntegralofthefunctionsshowninFig.2.Thecurvesshow dataset, at z = 1, the selection functions read as (2.3 ± 0.3) ×
the evolution of the relativistic integral densities for the three 10−3 Mpc−3, (1.1±0.1)× 10−3 Mpc−3,and(4.9± 0.6)×10−4
cosmologicalmodelsused in the presentwork as a functionof Mpc−3,respectively,avariationalmostsixtimeslargerthanthe
thedifferentdistances. propagateduncertainties.
Therefore,foreverysourceineachredshiftinterval,wecom-
putedtherest-frameluminosityforthefluxlimitofthefieldand
achieved by a galaxy survey. We go on to check whether this
filterwhereitwasobserved,givenitsbest-fitSEDandredshift.
purelygeometricaleffectcanbedetectedontheLFforthePEP
Next,wecomputedasetofselectionfunctionvaluesinthatred-
survey computed in Iribarremetal. (2013). This requires con-
shift interval, using the luminosity limits computed above for
sideringtheluminosityandthenumberevolutionofthegalaxies
each source in the interval. Finally, we computed the average
inthesurveyvolume.
overthissetofselectionfunctionvaluesforagivenredshiftin-
tervalandusedthisaverageasthevaluefortheselectionfunc-
3. Selectionfunctions tioninthatsameinterval.Theuncertainties,asdiscussedabove,
aredominatedbythevariationintheluminositylimitsandthere-
Selection functions are an estimate of the number density of forecanbetakensimplyasthestandarddeviationoverthesame
galaxies with luminosity above a chosen threshold, L . Once
lim set of computed selection function values for each redshift in-
computed,theselectionfunctionscanbeusedtoestimatetheob- terval.Theresultingmonochromaticrest-frame100µmand160
servednumberofobjectspercomovingvolume,derivedinSect. µmselectionfunctionsaregiveninTables2and3.
2.2,andfromthattheobserveddifferentialandintegraldensities
inthepastlightcone.
FollowingtheempiricalapproachofIribarremetal.(2012),
4. Consistencyfunctions
wecomputedselectionfunctionsψineachredshiftintervalus-
ing each of the three cosmological models (ΛCDM, CGBH, In the empirical framework of Ribeiro&Stoeger (2003),
and OCGBH) and the monochromatic 100 µm and 160 µm Albanietal. (2007), and Iribarremetal. (2012), the observed
5
A.Iribarremetal.:Numberdensitiesinvoidcosmologies
Table2.Selectionfunctionsfortherest-frame100µmdatasets. evolutionof the totalmasses ofthe sources,missing in this es-
UnitsareMpc−3. timation,areimprintedontheobservedLFandinheritedbyits
derived selection functions. By translating this purely theoreti-
z¯ ψΛCDM ψCGBH ψOCGBH caln (z¯) estimationto the correspondingselection functionsψ
C z¯
0.1 (4.0±0.8)×10−3 (5.0±1.0)×10−3 (5.6±1.1)×10−3 throughthe consistencyfunction J(z¯),we allowanytheoretical
0.3 (3.6±0.5)×10−3 (4.4±0.6)×10−3 (5.0±0.7)×10−3 quantity that assumes a n (z) built on a cosmologicalmodelto
0.5 (2.6±0.4)×10−3 (3.2±0.5)×10−3 (3.6±0.5)×10−3 usetheobservedvaluesgiCvenbytheselectionfunctions.
0.7 (2.3±0.4)×10−3 (2.9±0.5)×10−3 (3.2±0.6)×10−3
Forthepurposeofobtainingtherelativisticnumberdensities
0.9 (1.3±0.2)×10−3 (1.9±0.3)×10−3 (1.9±0.3)×10−3
1.1 (1.1±0.3)×10−3 (1.6±0.4)×10−3 (1.6±0.4)×10−3 empirically,thisapproachissufficient.Itminimizesthenumber
1.4 (4.3±1.1)×10−4 (3.7±1.0)×10−4 (4.8±1.2)×10−4 of theoretical assumptions about the evolution of the sources,
1.6 (4.1±1.4)×10−4 (3.6±1.2)×10−4 (4.6±1.5)×10−4 such as including a Press&Schechter (1974) formalism, and
2.0 (5.1±1.6)×10−4 (5.9±1.9)×10−4 (6.0±1.9)×10−4 considers as much information as possible from the observa-
2.4 (4.9±1.7)×10−4 (5.6±2.0)×10−4 (5.7±2.0)×10−4 tions.SinceaPress-Schechter-likeformalismisstillnotimple-
2.8 (3.3±1.1)×10−4 (1.7±0.5)×10−4 (2.5±0.8)×10−4
mentedonLTBmodels,theempiricalapproachdescribedabove
3.2 (2.7±1.1)×10−4 (1.4±0.6)×10−4 (2.1±0.8)×10−4
is the simplest way to work with both standard and void cos-
mologiesinaconsistentway.
Table3.Selectionfunctionsfortherest-frame160µmdatasets.
UnitsareMpc−3.
5. Observednumberdensitiesinthepastlightcone
z¯ ψΛCDM ψCGBH ψOCGBH
0.1 (6.0±1.4)×10−3 (3.7±0.9)×10−3 (3.7±0.9)×10−3
We computedthe relativistic numberdensities using the J ob-
0.3 (5.4±0.9)×10−3 (3.3±0.5)×10−3 (3.3±0.5)×10−3 z¯
tainedinsection4.Bydefinition,theLFisthenumberofsources
0.5 (2.7±0.5)×10−3 (2.6±0.4)×10−3 (3.0±0.5)×10−3
0.7 (2.5±0.5)×10−3 (2.3±0.5)×10−3 (2.7±0.6)×10−3 perunitluminosity,perunitcomovingvolume.Thisallowsusto
0.9 (1.0±0.2)×10−3 (1.3±0.2)×10−3 (1.8±0.3)×10−3 identifytheselectionfunctionsinEq.(22)asthedifferentialco-
1.1 (9.8±2.7)×10−4 (1.2±0.3)×10−3 (1.8±0.4)×10−3 movingdensityofgalaxiesstemmingfromtheobservationsand
1.4 (4.6±1.2)×10−4 (4.4±1.2)×10−4 (4.2±1.1)×10−4 torewriteEq.(24)as
1.6 (4.5±1.3)×10−4 (4.3±1.3)×10−4 (4.1±1.2)×10−4
2.0 (1.1±0.3)×10−4 (3.0±0.8)×10−4 (2.4±0.7)×10−4
2.4 (1.3±0.3)×10−4 (3.4±0.9)×10−4 (2.8±0.8)×10−4 ψ = dN (z¯)= J n (z¯)= J dN(z¯), (25)
2.8 (5.9±1.9)×10−5 (1.5±0.4)×10−4 (1.4±0.4)×10−4 z¯ "dV # z¯ C z¯ dV
3.2 (3.5±1.6)×10−5 (9.2±4.3)×10−5 (8.6±4.0)×10−5 c obs c
whichcanstillbewrittenas
quantitiescomputedinthe pastlightconeofa givencosmolog-
dN dN
ical model were obtained from their predicted values through = J , (26)
z¯
whatwascalledacompletenessfunctionJ(z).Thiscanbesome- "dz# dz
obs
what confusing from the observer’s point of view, since J(z)
is not necessarily related to incompleteness like that of miss- since the purely geometrical term dV/dz cancels out on both
c
ingsourcesinasurvey,buttotherelationbetweenatheoretical sidesoftheequation.TogetherwithEqs.(24)and(17),Eq.(26)
predictionforthenumbercountsandtheactualmeasurementof allows us to obtain the differential number counts [dN/dz]
obs
thatquantity(Iribarremetal. 2012). In this sense, this quantity from the selection function of a given dataset in the different
would be better named as consistency function, a term we use comovingdensities n (z) defined by the void modelparametri-
C
fromnowon.TheconsistencyfunctionJ(z)wasobtainedbyre- sationsconsideredhere.
latingthepredictionforthecomovingnumberdensityn given Because the quantity inside the parentheses in Eq. (19) is
C
bythecosmologicalmodel alsoapurelygeometricalterm,wecansimplywrite
ρ (z) Ω [r(z)]
n (z)= M = M ρ , (23) [γ] (z¯)= J γ(z¯), (27)
C M M C i obs z¯ i
totheselectionfunctionsforagivengalaxysurveyψ inagiven which allows us to obtain the observed differential densities
z¯
redshiftintervalz¯asIribarremetal.(2012) [γ] inthelightconesofthecosmologiesconsideredhereonce
i obs
J iscomputed.
ψ z¯
J = z¯ . (24) To obtain the observed integraldensities [γ∗] (z¯), we sub-
z¯ n (z¯) i obs
C stitutedN/dzinEq.(21)with[dN/dz] fromEq.(26)andcom-
obs
The values for J can be obtained numerically for the CGBH putetheobservedcumulativenumbercounts[N] (z¯).Thenwe
z¯ obs
voidmodelsusingEqs.(10)and(14)combinedwiththeappro- cansubstitutethisresultbackintoEq.(20),sincethecosmologi-
priate r(z) table as described in §2 and the selection functions caldistancetoredshiftrelationd(z)isalsofixedbythegeometry
i
computedin§3. ofthegivencosmology.
AsdiscussedinIribarremetal.(2012),theestimationofthe
comovingnumberdensityn assumesaconstant,averagemass
C
value M for normalisation purposes– that is, getting a correct 6. Results
order-of-magnitudeestimationofthenumberdensity.Suchesti-
mationisnotsupposedtoprovideadetaileddescriptionofn (z), Withthequantitiesobtainedinthelasttwosections,wecanpro-
C
butrathertoconveytheinformationontheredshiftevolutionof ceedtoinvestigatethedifferencesinthenumberdensitiesonthe
the comovingnumber density caused solely by the cosmologi- comoving frame and on the past lightcone of the standard and
calmodel,throughρ (z),inEq.(23).Thedetailsoftheredshift voidcosmologies.
M
6
A.Iribarremetal.:Numberdensitiesinvoidcosmologies
6.1.Comovingnumberdensityevolution
−1.0
As mentioned above, Mustaphaetal. (1997) showed that any
ΛCDM
spherically symmetric observation alone, e.g. redshift estima- CGBH
tions or number counts, can be fit purely by a general enough −1.5 OCGBH
LTB dust model, with no evolution of the sources required. In
the contextof the presentdiscussiontheir argumentcan be un- −2.0
derstoodbycombiningEqs.(23)and(24)towrite
J
g−2.5
M o
J = ψ . (28) l
z¯ z¯ ρ Ω (z¯)
C M −3.0
Looking at the right-hand side of the equation above, one can
easily separate its two terms into an observed quantity to the −3.5
left and a fraction between theoretically obtained quantities to
the right. Moreover, one can identify the constant M with the
−4.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
lackofamodelforthesecularevolutionoftheaveragemassof z
the sources, and Ω (z¯) with the evolutionof the matter density
M
parameter on a given cosmology. By using the extra degree of Fig.4.Consistencyfunctionsforthemonochromatic100µmlu-
freedomin settingthematterdensityprofileintheLTBmodel, minosity functionscomputedin the three cosmologicalmodels
Ω (z),onecanobtainanumberdensityn (z)thatmatchesthese- usedinthepresentwork.Thesefunctionsarerelatedtothered-
M C
lectionfunctionsperfectly,withouttheneedtoassumeanevolv- shiftevolutionof thebaryonto totalmassfraction(seetextfor
ing averagemass M(z).To constrainthisdegreeoffreedom,it detail).
is necessary to have differentsets of independentobservations.
Thisis preciselywhatthe GBH parametrisationofa giantvoid
in an LTB dust model yields: a matter density profile that is
−1.0
parametrised to best fit the combination of different, indepen- ΛCDM
dentobservations. CGBH
−1.5 OCGBH
Wepresenttheevolutionoftheconsistencyfunctionsforthe
differentPACSfiltersandcosmologicalmodelsinFigs.4and5.
−2.0
It is clear from these plots that this quantity evolves with red-
shift in both standard and void cosmologies. The behaviour of J
theconsistencyfunctionsinallmodelsstudiedherearewellfit g−2.5
o
l
byapowerlawdecreasingwithredshift.Thebest-fittingslopes
aregivenin Table4.These resultssuggestthatnotconsidering −3.0
the evolutionof the sourcesleads to a systematic trendthatin-
creasestheinconsistencywithz.Thisisnotasobviousasitmay
−3.5
seemgiventheflexibilityoftheLTBmodelsasdiscussedabove.
It is the parametrisationof Ω (z) constrained by the combined
M −4.0
independentobservationsof SNe + CMB + BAO that requires 0.0 0.5 1.0 1.5 z 2.0 2.5 3.0 3.5
M to evolve with z in the LTB models studied here. Only by
allowingMtoevolvewithzcanEq.(28)yieldaconstantcon- Fig.5.AsFig.4forthemonochromatic160µmluminosityfunc-
sistency function that indicates that the theoretical term on the tions.Fromtheseplots,onecanseethatthedifferencebetween
rightoftheright-handsideofEq.(28)isproportionaltotheob- thestandardmodelconsistencyfunction(blackcircles)andthe
servedtermontheleft. void model ones (colour circles) is not very significant, since
We used a Kolmogorov-Smirnoff (KS) two-sample test to most of the points lie within the error bars. The p values dis-
Λ
check,onastatisticalsense,howdifferenttheconsistencyfunc- cussedin§6.1quantifythisconclusionandagreewithit.
tionsarewhenassumingthedifferentcosmologicalmodelsstud-
iedherefromahypotheticallyconstantconsistencyfunction.By
design, a constant J(z) indicatesthat the modelforthe comov- consistencyfunctionforagivenvoidmodelandtheoneforthe
ing number density used here, assuming M(z) = M, matches standardmodelarecomputedfromthesameunderlyingdistribu-
the observed number density. The KS test is a non-parametric, tionor,inotherwords,thattheirdifferencesarenotstatistically
distribution-freewaytocomparetwo datasets,anditassignsto significant.Toestablishwhethersuchadifferenceissignificant,
whichlevelofconfidencewecanrefutethehypothesisthatthey at a 3-σ confidence level, for example, the p value should be
Λ
wereobtainedfromthesameunderlyingdistribution.Theresult- lowerthanapproximately0.0027.Giventhelisted p results,we
Λ
ingp-valueofthistestcanbeunderstoodastheprobabilitythat foundthatthecomputedconsistencyfunctionsinvoidandstan-
bothdatasetscomefromthesamedistribution.Forallcosmolo- dard cosmologies were not significantly different. This means
gies studied here, the no-evolution hypothesis, M(z) = M, is thatgalaxyevolutionproceedsmostlyinthesameway,regard-
rejected at over5-σ confidencelevel, with p-valueslower than less of the differences in the geometry and the composition of
10−5. Thismeansthat an evolvingaveragemass of the sources theUniversestudiedhere.
isalsorequiredbytheLTB/GBHmodelsstudiedhere.
Next, we use the same KS test to determine how different
6.2.Lightconeinhomogeneities
theconsistencyfunctionsinthevoidmodelsarefromtheirstan-
dard modelcounterparts.We presentthe computedp-valuesof For this section, we used the observed differential and integral
thesetests,p ,inTable4.Thep valueistheprobabilitythatthe densitiesobtainedinSect.5toinvestigatetheeffectsofthecen-
Λ Λ
7
A.Iribarremetal.:Numberdensitiesinvoidcosmologies
Table4.Comoving,differentialandintegraldensitiesstatistics. Thesignificanceofthis differencein termsof itsuncertaintyis
thenobtainedfrom∆ /(δ∆ ).
a,b a,b
Dataset Model p logJ(z)slope logγ slope logγ∗slope
Λ L L
ΛCDM 1.0 -0.39±0.05 -2.4±0.2 -2.31±0.03
L100µm CGBH 0.43 -0.51±0.06 -3.1±0.6 -2.50±0.04 7. Discussion
OCGBH 0.43 -0.51±0.05 -3.0±0.4 -2.52±0.03
Nextweusetheapproachabovetoobtaincomparisonsbetween
ΛCDM 1.0 -0.75±0.03 -4.5±0.3 -2.54±0.06
standardvs.voidcosmologies,100µmvs.160µmdatasets,and
L160µm OCGCGBHBH 00..7799 --00..5630±±00..0043 --33..77±±00..34 --22..5498±±00..0065 differentialdensitiesγvs.integraldensitiesγ∗.
tralunderdensityprescribedbytheGBH modelsintheredshift 7.1.ComparisonwiththeLTB/CGBHmodels
distortionscausedbytheexpandingspacetimeontheobserver’s
Results forthe significanceof the differencebetweenslopesof
pastlightcone.
the high-redshift power-law fits to the differential densities γ
Our previous studies of these relativistic number densities andtheintegraldensitiesγ∗,givenafixedcosmologicalmodel,
(Ribeiro 2005; Albanietal. 2007; Iribarremetal. 2012) sug-
werehighlydependentonthedataset.Theywerefairlyinsignif-
gested a high-redshift power-law behaviour for both differen-
icant with the 100 µm data for all cosmologies (ΛCDM: 0.3-
tialandintegralradialdistributions.Inotherwords,thatexpan-
σ, CGBH: 0.9-σ, OCGBH: 1.1-σ); whereas they were all at
siondistributesthesourcesalongthelightconeinanincreasingly
leastmarginallysignificantinthe160µm data(ΛCDM:5.2-σ,
self-similar manner.This is a purely geometricaleffect, as dis-
CGBH:3.0-σ,OCGBH:3.4-σ).
cussedinSect.2,thatmayormaynotbedominantbecausethe
Such differences in the slopes between the high-redshift
hierarchical build-up of galaxies also plays a role in how the
power-lawfitstoγandtoγ∗canbeunderstoodbycheckingEqs.
sourcesare distributedalong the lightcone.Given that some of
(19)and(20).WenotethatγisproportionaltodN/dz,whereas
thealternativevoidmodelsarebuiltassuminganLTBlineele-
γ∗ isproportionalto N.Since N isacumulativequantity,itcan
mentinstead of the standard modelFLRW one, we aim in this
only increase or remain constantwith increasingredshift. That
sectiontobettercharacterisethosepowerlawsandtoinvestigate
is, even if there are regions in the volume where N is not de-
how they are affected by the secular evolution of the sources
fined, which have a lower density of detected sources, N itself
andbythedirecteffectoftheexpansionofthedifferentgeome-
will remain constant. On the other hand, dN/dz is sensitive to
tries. We focus the discussion on the densities computedusing
suchlocaldensityvariations,whichaddsuptoahigherdegreeof
the luminosity distance, since the results for the other distance
non-homogeneityinthehigherredshiftpartofthepastlightcone
definitionsareallqualitativelythesame.
probedbythesurveyandasaconsequenceasteeperpower-law
As can be seen in Figs. 2 and 3, the geometrical effect in-
slopeofγ.
creaseswiththeredshift,deviatingboththetheoreticaldifferen-
The differences between the slopes of the high-redshift
tialandthecumulativenumberdensitiesawayfromaconstant,
power-law fits to the relativistic densities computed using the
homogeneous behaviour at high redshifts. At redshifts lower
100 µm, and the 160 µm PEP blind-selected datasets were
thanz≈0.1,thisgeometricaleffectislesspronounced,andboth
fairlyinsignificantwhenassumingthevoidmodelsstudiedhere
γ (z)andγ∗(z)shownfollowtheconstantaveragegalacticmass
L L (CGBH:0.7-σforγ,and0.3forγ∗;OCGBH:1.3-σforγ, and
Massumedintheircomputation,asdiscussedinSect.3.
1.0 for γ∗), whereas, those differences showed a moderate-to-
It follows that there must be a region in redshift space at
strongsignificanceiftheΛCDMmodelwasassumed(4.6-σfor
whicha transitionoccursbetweenthe power-lawbehaviourin-
γ,and3.0-σforγ∗).
duced by the expansion at high redshift and the behaviour de-
The differences in the slopes obtained using different
finedbytheunderlyingdensityparameterΩ (z)combinedwith
M datasets can be understood as the effect of different redshift-
the evolution of the sources stemming from the LF. However,
evolving luminosity limits on the past lightcone distributions.
theexactsizeandrangeofthisregioncannotbepredictedwith
The results above indicated that the distributions on the past
theempiricalapproachforthecomovingnumberdensitiesused
lightcone of the void models were significantly less different,
inthiswork.Therefore,ifwearetocharacterisethepower-law
hence less affected by the detection limits of the PEP datasets,
behaviourof the high-redshiftend of ournumberdensities, we
than their counterparts computed on the past lightcone of the
mustallowourfittingproceduretosearchforthebestredshiftat
standardmodel.
whichthepower-lawbehaviourbeginstodominate.
Finally,wecomputedthesignificanceofthedifferencesob-
Todoso,wecomputethebestlinearfittothelogγ vs.logd
L L tainedbycomparingthesameslopescomputedintwodifferent
tablesinaniterativeway.StartingwiththeLFvaluesderivedfor
cosmological models. The only marginally significant (≥ 3-σ)
thehighestthreeredshiftbins,weperformthesamefit,includ-
differencewefoundisforthecomparisonbetweenstandardand
ingoneextraLFvalueineachiteration,indecreasingorder,until
voidmodelsfortheintegraldensityγ∗slopes,usingthe100µm
wehaveincludedallpoints.Then,weselectthefitwiththelow-
est reduced χ2 value. The selected fits for the different dataset dataset:3.7-σfortheΛCDMvs.CGBHcomparison,and4.3-σ
fortheΛCDMvs.OCGBHone.Giventhestrikingconcordance
andcosmologycombinationsare plottedin figure6. Thelisted
of the other γ∗ slopes around a tentative value of -2.5 ± 0.1 it
uncertaintiesareformallyobtainedtakingthesquarerootofthe
is possible that the oddly low values of the slope of γ∗ in the
linearterminthecovariancematrixofthefittedpowerlaw.
100µmdatasetanditsuncertaintyareanartefactcausedbyour
To quantify how significant the differencesin the slopes of
fittingprocedure.
the power laws are, we can start by first computing the uncer-
taintyofthedifference∆ ,betweentheα slopeoflogγ using How do we combine the results from all of those differ-
a,b a L ent comparisons in a coherent picture? Iribarremetal. (2013)
agivendataset/cosmologycombinationaandthatofadifferent
showed that the corrections needed to build the LF using a
combinationbas
flux-limited survey made the results sensitive to the differ-
δ∆ = (δα)2+(δα)2. (29) ences caused by changing the underlying cosmologicalmodel.
a,b a b
p
8
A.Iribarremetal.:Numberdensitiesinvoidcosmologies
Detectionlimitsseemtoplayamajorroleinthefully-relativistic galaxies.TheyshowthatalltheirmodelsmimictheFLRWd (z)
L
analysis used here as well. On one hand, we found signif- to sub-percent level, which should lead to the changes in the
icant differences caused by the kind of statistics used (γ or maximumredshiftestimatesandintheexpectednumberdensi-
γ∗) with the 160 µm blind-selected catalogue. On the other tiescausedbyassumingsuchmodelscomparabletotheoneswe
hand, we found that the cosmology assumed (FLRW/ΛCDM observehere.
or LTB/GBH) caused significant differences on both γ and γ∗, The authors show in their Fig. 9 that up to redshift z ≈ 1,
using the 100 µm dataset instead. Some of these discrepancies their best-fit models follow closely the distance modulus, and
could be caused by the way we fit the power laws to the high- thusthe d (z) relationofΛCDM. Lookingatthesame plot,we
L
redshiftpartsofthedistributions.Futureobservationswithlower cansee thattheluminositydistancesgrowincreasinglysmaller
fluxlimitswillhelpuscheckwhichofthesediscrepancieswere when compared to their standard model counterparts at higher
causedbythepresentfluxlimits. redshifts.LookingatEq.(20),wecanexpectthatthiscouldpo-
tentiallyleadtohighervaluesofγ∗and,consequentially,alower
valueforitsbest-fitpowerlaw.
7.2.ComparisonwithotherLTBmodels
Also,theirbest-fitvoidsaremuchlargerthantheonesused
It is important to notice that the results in both Sects. 6.1 and here,ascanbeseen bycomparingtheupperleft-handpanelof
6.2,togetherwiththediscussionpresentedabove,areonlyvalid their Fig. 7 versus the upper panel of Fig. 1 in Iribarremetal.
for a very special case of LTB models, namely, the parametri- (2013), which shows the best-fit voids of Zumalaca´rreguietal.
sation for the CGBH model obtained by Zumalaca´rreguietal. (2012).Mockcataloguetestsshouldbeusedtomakesurethese
(2012). It is not in the scope of the present work to present a largervoidsdonotbiasthe1/V methodusedhere.However,
max
completeanalysis of otherLTB modelsin the literature,which as argued in Iribarremetal. (2013), if the large under-dense
would require first recomputing the luminosity functions pre- profiles used here varied smoothly enough not to significantly
sented in Iribarremetal. (2013) from the start, but given some biastheLFestimator,wedonotexpectthatthelargervoidsof
recentadvancementsin betterexploitingthe flexibilityofthese Februaryetal.(2010)will.
models,aqualitativediscussionoftheirpossibleimpactonour Clarkson&Regis (2011) discussthe implicationsofallow-
resultsispertinenthere. ing for inhomogeneityin the early time radiation field and the
Assuming different non-homogeneouscosmologies can af- way it affects the constrainingpower of CMB results for inho-
fectthepresentanalysesin threedifferentways. First, in terms mogeneousmodels. Theyshow, for example,that even models
of building the LF from the observations, a modelwith a mat- that are asymptotically flat at the CMB can be made to fit if
ter densityprofileΩ (r)thatisdifferentenoughfromtheones the matter density profile includes a low-density shell around
M
studied here could possibly make the average homogeneityas- the central void. An under-dense shell like this could possibly
sumptionattheheartofthe1/Vmax LFestimatorinvalid.Using lead the 1/Vmax methodto over-estimatethe volume correction
mock catalogues, Iribarremetal. (2013) showed that the void of sourceswith maximumobservablecomovingvolume inside
shapes of the CGBH models used in their LF does not signif- or beyond the shell, because such correctionsare based on the
icantly affect the ability of the 1/V estimator to recover the assumption of an average homogeneousdistribution.To test in
max
underlyingLF. detail how accurately this LF estimator would recover the un-
Second,differentdistance-to-redshiftrelationscouldalsoaf- derlying distribution, we would need to build mock catalogues
fect the LF results, and thus its redshift evolution, through the thatassume thematter distributionof these models,butwe ex-
computationofthemaximumobservableredshiftofeachsource pect thatsuch differences,if present, would show morepromi-
z ,whichisusedincomputingofthe1/V valueoftheLF nentlyatmidtohighredshift,accordingtothesizeoftheshell,
max max
ineachredshiftbin.AsdiscussedinIribarremetal.(2013),this and make the faint-endslope steeper by assigninghigherover-
comes as a result of the fainter part of the galaxies in a sur- estimatedcorrectionstothefaintersources.Dependingonhow
vey having fluxes near the limit of observationin the field and non-homogeneousthematterprofileis,anover-estimationofthe
is affected mainly by differencesin the luminosity distance-to- characteristic number density parameter φ∗ might also be de-
redshift relation. Also, at luminosities L ≈ L∗, the z of the tected.
max
sources is safely hitting the higher z limit of the redshift bin, Varying bang-time functions t (r) were also studied
bb
leadingtheirvolumecorrectiontobeindependentoftheirlumi- (Cliftonetal.2009)andshowntobeanextradegreeoffreedom
nosity,henceofthed (z)relation. that helps LTB models to reconciliate CMB + H0 constraints.
L
Finally,theexpectednumberdensitiesinthepast-lightcone Bulletal.(2012),however,showthatmodelswithvaryingt (r)
bb
γ and γ∗, as computed in Sect. 2.2 are both sensitive to the constrainedbythecombinationofSNe+CMB+H0shouldpro-
diistance-ito-redshift relations, as shown in Figs. 2 and 3. The duceakineticSunyaev-Zel’dovicheffectthatisordersofmag-
differencescausedonthenumberdensitiesbythedifferentdis- nitudestrongerthanitsexpectedupperlimits.Itisnotclearhow
tance relationsmightbe largeenoughto be detected,giventhe allowingforthisextradegreeoffreedomwouldchangethekey
observationaluncertaintiesstemmingfromtheLF.Considering quantities shown to affect our results: the d (z) and r(z) rela-
L
thepresentresults,suchdifferencesintheexpectednumberden- tions and the shape of the central under-dense region. Only a
sities shouldbe greater than the CGBH-to-FLRW onesstudied full-fledgedanalysis,startingfromthebuildingupoftheLFand
here, if we were to detect a significant effect on the observed assuminga modelwith a varyingbang-timefit by the observa-
numberdensitiescausedbydifferencesintheirexpectedvalues tions,wouldallow ustomakeanyreasonablecomparisonwith
alone. theresultspresentedinthiswork.
Februaryetal.(2010)arguethatCMBandBAOconstraints
may be significantly distorted by differences in the evolution
8. Conclusions
of primordial perturbations caused by the curvature inside the
voids.Theygoontofitanumberofdifferentshapesforthein- In this paper we derivedthe theoretical results needed to com-
nermatterprofile,consideringonlylocalUniversedata,namely, pute the differential and integral densities along the past light-
SNeIaandthereconstructionofH(z)throughpassivelyevolving cone of LTB dust models. We applied these results to comput-
9
A.Iribarremetal.:Numberdensitiesinvoidcosmologies
100 µm 160 µm
100 100
3c]− 1100--21 3c]− 1100--21 Λ
p p C
M 10-3 M 10-3 D
γ[∗L 1100--54 γL ∝dL−2.4 γ[∗L 1100--54 γL ∝dL−4.5 M
, ,
γL 10-6 γL∗ ∝dL−2.3 γL 10-6 γL∗ ∝dL−2.5
10-7 10-7
103 104 103 104
d [Mpc] d [Mpc]
L L
100 100
3c]− 1100--21 3c]− 1100--21 C
Mp 10-3 Mp 10-3 G
γ[∗L 1100--54 γL ∝dL−3.1 γ[∗L 1100--54 γL ∝dL−3.7 BH
, ,
γL 10-6 γL∗ ∝dL−2.5 γL 10-6 γL∗ ∝dL−2.5
10-7 10-7
103 104 103 104
d [Mpc] d [Mpc]
L L
100 100
3pc]− 1100--21 3pc]− 1100--21 OC
M 10-3 M 10-3 G
γ[∗L 1100--54 γL ∝dL−3.0 γ[∗L 1100--54 γL ∝dL−3.7 BH
, ,
γL 10-6 γL∗ ∝dL−2.5 γL 10-6 γL∗ ∝dL−2.6
10-7 10-7
103 104 103 104
d [Mpc] d [Mpc]
L L
Fig.6.Best-fitpowerlawstothedifferentialdensityγ (dots),andtheintegraldensityγ∗(diamonds).Theleftpanelsshowtheplots
L L
forthe100µmLF,whiletherightonesshowtheplotsforthe160µmLF,inthethreecosmologicalmodels/parametrisationsstudied
here.Fulllinesshowthebest-fithigh-redshiftpowerlawforthedifferentialdensities,whilethedashedlinesshowthebest-fitpower
lawsfortheintegraldensities,asdiscussedinSect.6.2.
ingthe theoreticalpredictionsforsuch quantitiesin the CGBH iedhere.Ontheotherhand,thedifferentialdensitieswerefound
parametrisationsofZumalaca´rreguietal.(2012). to be sensitive to a change in cosmological model assumed in
their computation. These results confirmed the power-law be-
Wecomputedtheselectionfunctionsstemmingfromthefar-
haviour of the galaxy distribution on the observer’s past light-
IRLFsofIribarremetal.(2013),whichallowedustoestablish,
cone of the LTB/GBH modelsand allowed a tentative value of
atanover5-σconfidencelevel,thatgeometryaloneisnotable
-2.5±0.1tobeobtainedforthecumulativeradialstatisticsγ∗of
to fit their behaviour,given the LTB/CGBH parametrisation of i
thisdistribution,regardlessofthecosmologicalmodelassumed.
Zumalaca´rreguietal. (2012). Thisfindingconfirmsthe needto
allowforevolutionofthesources(eitherinnumberorinlumi-
nosity)inthisparticularclassofvoidmodelsaswell.Wefound
Aknowledgements
no strong evidence of any dependenceof this evolution on the
cosmologicalmodelsstudiedhere.Inotherwords,thecombined
This work was jointly supported by Brazil’s CAPES and ESO
mergertreeandbarionicprocessesneededtoreproducethered-
studentships. SF is supported by the South African Square
shiftevolutionoftheFIRLFintheCGBHvoidmodelsarenot
KilometreArray(SKA)Project.
significantly differentfrom the hierarchicalbuild up and astro-
physicalprocessesinthestandardmodel.
Finally, we computed the observed differential and inte- References
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10