Structure formation in cosmologies with oscillating dark energy

0.43 MB·English

by  F. Pace

Mon.Not.R.Astron.Soc.000,1–17(2010) PrintedNovember8,2011 (MNLATEXstylefilev2.2)
Structure formation in cosmologies with oscillating dark energy
F. Pace1,2⋆, C. Fedeli3, L. Moscardini4,5,6 and M. Bartelmann2
1
1InstituteofCosmologyandGravitation,UniversityofPortsmouth,DennisSciamaBuilding,Portsmouth,PO13FX,U.K.
1
2Zentrumfu¨rAstronomiederUniversita¨tHeidelberg,Institutfu¨rTheoretischeAstrophysik,Albert-Ueberle-Str.2,D-69120Heidelberg,Germany
0
3DepartmentofAstronomy,UniversityofFlorida,211BryantSpaceScienceCenter,Gainesville,FL32611-2055,USA
2
4DipartimentodiAstronomia,Universita`diBologna,ViaRanzani1,I-40127Bologna,Italy
v 5INFN,SezionediBologna,VialeBertiPichat6/2,I-40127Bologna,Italy
o 6INAF,OsservatorioAstronomicodiBologna,viaRanzani1,I-40127Bologna,Italy
N

7
ReceivedNovember8,2011;accepted?

]
O
ABSTRACT
C
Westudytheimprintsontheformationandevolutionofcosmicstructuresofaparticularclass
. ofdynamicaldarkenergymodels,characterizedbyanoscillatingequationofstate. Thisin-
h
p vestigationcomplementsearlierworkonthetopic,thatfocusedexclusivelyontheexpansion
- historyoftheUniverseforsuchmodels.Oscillatingdarkenergycosmologieswereintroduced
o
inanattempttosolvethecoincidenceproblem,sinceinthecourseofcosmichistorymatter
r
t and dark energy would have had periodically comparable energy densities. In this class of
s modelstheredshiftevolutionoftheequationofstateparameterw(z)fordarkenergyischar-
a
[ acterized by two parameters, describingthe amplitudeand the frequencyof the oscillations
(the phase is usually set by the boundary condition that w(z) should be close to −1 at re-

1 centtimes).We considersixdifferentoscillatingdarkenergymodels,eachcharacterizedby
v adifferentsetofparametervalues.Foroneofthesemodelsw(z)islowerthan−1atpresent
6 andlargerthan −1in the past, in agreementwith somemarginalevidencefromrecenttype
5
Iasupernovastudies.Underthecommonassumptionthatdarkenergyisnotclusteringonthe
5
scalesofinterest,westudydifferentaspectsofcosmicstructureformation.Inparticular,we
1
. self-consistentlysolvethesphericalcollapseproblembasedontheNewtonianhydrodynam-
1
icalapproach,andcomputetheresultingsphericaloverdensityasafunctionofcosmictime.
1
Wethenestimatethebehaviorofseveralcosmologicalobservables,suchasthelineargrowth
1
factor,theIntegratedSachs-Wolfe(ISW)effect,thenumbercountsofmassivestructures,and
1
thematterandcosmicshearpowerspectra.Weshowthat,independentlyoftheamplitudeand
:
v thefrequencyofthedarkenergyoscillations,noneoftheaforementionedobservablesshow
i anoscillatingbehaviorasafunctionofredshift.Thisisaconsequenceofthesaidobservables’
X
beingintegralsoversome functionsofthe expansionrateovercosmichistory,thussmooth-
r
a ing anyoscillatoryfeaturesin w(z) below detectability.We also noticethat deviationswith
respecttotheexpectationsforafiducialΛCDMcosmologyaregenericallysmall,andinthe
majorityofthecasesdistinguishinganoscillatingdarkenergymodelwouldbedifficult.Ex-
ceptionstothisconclusionareprovidedbythecosmicshearpowerspectrum,whichforsome
ofthemodelsshowsadifferenceatthelevelof∼ 10%overawiderangeofangularscales,
andtheabundanceofgalaxyclusters,whichismodifiedatthe∼ 10−20%levelatz & 0.6
forfuturewideweaklensingsurveys.
Keywords: cosmology:theory-darkenergy-methods:analytical
1 INTRODUCTION quantumfluctuationswereamplifiedtoproducetinyperturbations
in the matter distribution, whose imprints can nowadays be ob-
In recent years an increasingly large body of observations con-
servedintheCosmicMicrowaveBackground(CMB)temperature
firmed the general framework of a standard cosmological model
map.Lateron,andduetogravitationalinstability,theseseedfluc-
basedonGeneralRelativity.Accordingly,rightaftertheBigBang
tuationsgrewup,givingrisetothewebofcosmicstructuresthatwe
theUniverseexperiencedanacceleratedexpansionphase,dubbed
observetoday.Afterinflation,theUniverseexperiencedaperiodof
inflation (Guth 1981; Linde 1982; Zelnikov 1991), during which
reheating(Shtanov,Traschen&Brandenberger1995;Kaiser1996)
withthe formation of light elements. Current observations of the
CMBandoftheluminositydistancesoftypeIaSupernovae(SNe
⋆ E-mail:[email protected]
2 F.Pace etal.
Ia)show thatthegeometryoftheUniverseisspatiallyflat,inac- tosolvethecoincidenceproblem,becausethepresentaccelerated
cordancewiththepredictionsoftheinflationaryparadigm,andfur- expansion phase would just be one of the many such phases oc-
thermoreshowingthattheUniverseiscurrentlyundergoinganother curring over cosmic history, especially at early times. Moreover,
acceleratedexpansionphase. oscillationswouldmorenaturallyaccommodatethecrossingofthe
After the first detection of an accelerating expansion rate phantombarrier,w = 1asitismarginallysuggestedbyrecent
−
at low redshift (possibly z . 0.5, Shapiro&Turner 2006) by observations (Alametal. 2004; Allenetal. 2004; Dicus&Repko
Riessetal.(1998)andPerlmutteretal.(1999)manyotherdifferent 2004;Riessetal.2004;Fengetal.2005;Huterer&Cooray2005;
andindependentstudiesledtothesameconclusionsmakingthisin- Choudhury&Padmanabhan 2005). In the framework of particle
ferenceverysolid.Inparticular,evidenceforanacceleratedexpan- physics, it is possible to have an oscillating quintessence poten-
sioncomesfromtheCMB(Jaffeetal.2001;Komatsuetal.2011) tialifoneconsidersapseudo-NambuGoldstonebosonfieldwhen
andtheIntegratedSachs-Wolfe(ISW)effect(Hoetal.2008),the ithasrolledthroughtheminimum.Asmodelsfordarkenergy,os-
Large Scale Structure (LSS) and the Baryon Acoustic Oscilla- cillating scalar fields were proposed by Dutta&Scherrer (2008),
tion (BAO) (Eisensteinetal. 2005; Percivaletal. 2010), globular Johnson&Kamionkowski (2008), and Gu (2008) . Lazkozetal.
clusters(Krauss&Chaboyer2003),galaxyclusters(Haimanetal. (2005)andKureketal.(2008)foundabetteragreementwithSNe
2001;Allenetal.2004,2008;Wangetal.2004)andweaklensing Iadataifanoscillatingequationofstateisusedinsteadofthecos-
(Hoekstraetal.2006;Jarvisetal.2006). mological constant or an equation of state linearly dependent on
As a homogeneous and isotropic General Relativistic model the scale factor. Further indications in the same direction come
universefilledwithmatterisunabletoreproducetheobservedac- from the study performed by Riessetal. (2007). Previous works
celerated expansion, three different explanations have been pro- onthistopicfocusedmainlyontheexpansionhistoryoftheUni-
posedtoaccountforit.Onepossibilityconsistsinputtingasidethe verseandmarginallyonthelinearperturbationtheoryintheframe-
hypothesis of homogeneity on large scales: these models are de- workofoscillatingquintessence(Fengetal.2005;Xiaetal.2005;
scribedbytheLemaˆıtre-Tolman-Bondi(LTB)metricorarebased Barenboim&Lykken 2006; Barenboimetal. 2006; Kureketal.
on the idea of backreaction (Kolbetal. 2006). A second possi- 2008,2010;Lanetal.2010).
bility is to suppose that on very large scales General Relativity Thenoveltyofthisworkliesinthefactthatweexploresig-
breaksdownandgravityismodified.Inthiscasewewillbeinthe naturesofanoscillatingequationofstatew(z)inthe(non-linear)
needforanewtheoryofgravityandGeneralRelativitywouldbe growthofcosmicstructures,thusextendingandcomplementingthe
onlythesmall-scalelimitofamoreprofoundtheory.Examplesof majorityofforegoingstudies.Themainideaweexplorehereisto
thisideaarethef(R)models(Amendolaetal.2007;Starobinsky findoutcosmologicalobservablesbasedonstructureformationthat
2007), brane models (Deffayet 2001; Dvalietal. 2000), and the canhinttowardoscillatingquintessenceeventhoughtheexpansion
f(T)models(Bengochea&Ferraro2009;Linder2010;Dentetal. ofthehomogeneousandisotropicbackgrounddoesnot.Therestof
2011;Zheng&Huang2011).Finally,onecouldassumethatGen- thispaperishenceorganizedasfollows.InSection2wedescribe
eral Relativity is correct but the low-z Universe is dominated by theformalismofthesphericalcollapseusedtoderiveimportantpa-
somekindofexoticfluidwithnegativepressure,thedarkenergy. rametersfortheformationandevolutionofstructures.InSection3
Specifically,ifdarkenergyconstitutes 70%ofthematter-energy wedescribeandmotivateseveralparametrizationsusedinorderto
∼
content of the Universe, from the second Friedmann equation it describe the oscillating dark energy and in Section 4 we present
turnsout that itsequation of stateparameter w would need tobe resultsforthedifferentobservablesweconsidered.Section5isde-
w< 1/2inordertoprovideacceleratedexpansion. votedtoourconclusions. InAppendixAwepresentimplementa-
−
In the concordance cosmological model the roleof dark en- tiondetailsofthecodeusedtoevaluatethelineargrowthfactorand
ergy is played by the cosmological costant Λ, having a redshift- theevolutionofthesphericaloverdensity.
independent equationofstateparameter w = 1andcommonly
−
interpretedastheenergydensityofthevacuum.Eventhoughthis
Λ-Cold DarkMatter (ΛCDM henceforth) model is now thestan-
2 SPHERICALCOLLAPSEMODEL
dardreferenceframeworkincosmology,itsuffersfromsomefun-
damentaltheoreticalproblems,thatcanbesummarizedbythefol- Despite its simplifying nature, the model describing the collapse
lowingquestions. ofauniformnon-rotatingsphericaloverdensityinacosmological
settingprovidesnumerousinsightsontheactualprocessofstruc-
Whyistheenergydensityimpliedbythecosmological con-
• ture formation. For instance, the linear density contrast extrapo-
stantmuchsmallerthanthetheoreticallyexpectedvacuumenergy
latedat the spherical collapse timeprovides afair approximation
density?
forthethresholdatwhichactualperturbationscancollapsetoform
Whyisthedarkenergydensitycomparabletothedarkmatter
• boundstructures.Thus,inthisSectionwesketchthederivationof
densityonlytoday?
the relevant equations for the spherical collapse model under the
The last one is also known as the coincidence problem. In order assumptionthatonlydarkmattercanformclumps,whiledarken-
to solve or at least alleviate these issues, it is possible to iden- ergyisjustpresentasbackgroundfluid.Forfurtherdetailswere-
tify dark energy with the energy density of a minimally coupled fertothecurrentliteratureonthetopic(see,e.g.Bernardeau1994;
scalar field (named quintessence), that evolves through cosmic Ohtaetal.2003,2004;Mota&vandeBruck2004;Nunes&Mota
timeasdictatedbyitsownpotential.Thisgivesrisetoaredshift- 2006;Abramoetal.2007;Paceetal.2010).
dependentequationofstateparameterw(z),hencemakingatleast Rather than studying the time evolution of the radius of the
thecoincidenceproblemlesssevere.Thesedynamicaldarkenergy collapsingsphere,westudydirectlythetimeevolutionoftheover-
models can be roughly grouped into two classes, tracking mod- density.Thisprocedureprovestobenumericallymorestableand
els (Steinhardtetal. 1999) and scaling models (Halliwell 1987; lesspronetoerrorsthantheclassicalapproachbasedontheradius
Wandsetal.1993;Wetterich1995). evolution(Paceetal.2010).Weconsideraperfectfluiddescribed
Modelswithanoscillatingequationofstatewereintroduced bytheenergy-momentumtensorTµν,satisfyingthelocalconser-
Structureformationwithoscillatingdarkenergy 3
vationlawsexpressedby νTµν = 0.Thissetoffourequations thenon-linearevolutionofthedensityperturbationuptothetime
∇
encapsulates both the continuity and the Euler equations, while of virialization, we need to evaluate the turn-around scale factor
from Einstein’s fieldequations it ispossible to derive a relativis- ata,definedasthetimewhentheradiusofthesphere reachesits
ticgeneralizationofthePoissonequation.Inamoreexplicitform maximum, detaches from the overall expansion of the Universe,
theseexpressionsread andcollapsesafterwards.Usingthenthevirialtheoremandenergy
conservation considerations, thevirialoverdensity ∆v canbede-
rivedaccordingtothediscussionofMaor&Lahav(2005).
∂ρ P
+~ (ρ~v)+ ~v=0, (1)
∂t ∇· c2∇·
3 OUTLINEOFTHECOSMOLOGICALMODELS
∂~v
+(~v ~)~v+~Φ=0, (2)
∂t ·∇ ∇ AsoutlinedinSection1,atthemomentthereisnoexplanationfor
darkenergyintermsoffundamentalphysics,thereforeallthemod-
and
elsthatweexploredinthepresentworkarepurelyphenomenolog-
icalandthevaluesoftheirparametersaregenericallyadjustedsuch
3P
2Φ 4πG ρ+ =0, (3) thatcertainclassesofcosmological observables (most commonly
∇ − (cid:18) c2 (cid:19) theluminositydistanceofSNeIaandtheCMBtemperaturepower
whereρ,P,~vandΦarethedensity,thepressure,thevelocityand spectrum)arewellreproduced.Thesemodelsaredescribedinthe
thegravitationalpotentialofthefluid. presentSection.Asafiducialreferencecosmologyweassumethe
Fortheaveragebackgroundmatterdensitythefollowingcontinuity standard flatΛCDMmodel. Thecosmological parametersareset
equationholds, toΩm,0 = 0.274,Ωq,0 = 0.726andh = 0.7inaccordancewith
WMAP-7 data (Komatsuetal. 2011; Larsonetal. 2011) and the
SupernovaLegacySurvey3data(seeSullivanetal.2011).These
P¯
ρ¯˙+3H ρ¯+ =0, (4) same parameter values are also kept intact for all the dynamical
c2
(cid:18) (cid:19) darkenergycosmologiesinvestigatedinthispaper.
where ρ¯ = 3H2Ωm/8πG is the background matter density, H Theamplitudeoftheprimordialmatterpowerspectruminthe
istheHubbleparameter,andΩm isthematterdensityparameter. fiducial ΛCDM cosmology is selected in order to attain a given
valueofthequadraticdeviationonacomovingscaleof8Mpc/h,
Sinceforordinarymatteranddarkmatterthepressurecontribution
isnegligible,fromnowonwewillsetP =0. σ8 = 0.8.Inalltheotherdynamical darkenergymodelsthatwe
consideredthenormalizationisscaledaccordingto
Assuming spherical symmetry and perturbing the physical
quantitiesappearingintheprevioussetofequations(density,veloc-
iotbytaainndthgerafvoiltlaotwioinnaglepxoatecnttnioaln)-alirnoeuanrdditfhfeeirrebnaticaklgerqouuantdionva,ldueessc,rwibe- σ8,DE = δcδ,Λc,CDDEM(z(z==0)0)σ8,ΛCDM , (7)
ingtheevolutionofthematterdensityperturbationδasafunction
ofthecosmictime, whereδcisthelinearoverdensityparameterextrapolatedatspher-
ical collapse (see Section 2 above). In this way, the exponential
tailof the darkmatter halo massfunction at redshift zero iscon-
4 δ˙2 served, and hence theabundance of massive structures at present
δ¨+2Hδ˙ 4πGρ¯δ(1+δ)=0. (5)
− 31+δ − times,whichisarguablywelldefinedfromtheobservationalpoint
of view, is the same for all models. We show the values for the
WestressthatEq.(5)isvalidalsoforlargedensitycontrasts,deep
normalizationofthedifferentmodelsinTable1.
inthenon-linearregime,aslongassphericalsymmetryissatisfied.
Inthepresentworkweanalyzedsixdifferentdarkenergycos-
Byrestrictingtoδ 1instead,atfirstorderEq.(5)reads
≪ mologieswithanoscillatingequationofstateparameterw(z).In
the first five of them w(z) has the same functional form but dif-
δ¨+2Hδ˙ 4πGρ¯δ=0, (6) ferentvaluesofthefreeparameters.Thesixthmodelhasinsteada
−
differentfunctionalformforw(z),althoughstillpresentingoscil-
and it coincides with the differential equation commonly used to
lations.Thefunctionalformforthefirstfivecosmologiesis(Linder
determinethelineargrowthfactor.
2006;Lazkozetal.2010;Fengetal.2006)
As explained in detail in Paceetal. (2010), in order to de-
terminethelineardensityperturbationthresholdforsphericalcol-
lapse,δc,oneshouldsolveEq.(6)withsuitableinitialconditions, w(a)=w0−Asin(Blna+θ), (8)
namelytheinitialoverdensityandvelocityoftheperturbation.In
wherea = 1/(1+z)isthescalefactor,Adeterminestheampli-
ordertofindtheinitialoverdensitywetakeintoaccountthatatthe
tudeoftheoscillations,Bgivestheirfrequencywhileθisaphase
timeofthecollapseoftheobjectallthematterisconcentratedin
shift.Ascanbeeasilyseen,thevalueoftheequationofstatepa-
onepoint,thereforeformallyδ + .Hencebyfixingthetime,
orscalefactor,ofcollapseac,w→itha∞root-searchmethoditispos- irfamtheeteprhtaosdeayisiθsw=(a0.=Mo1d)el=siwx0is−mAeansitnt(oθ)g,enwehriaclhizeeqtuhaelsCwPL0
sibletodeterminetheinitialoverdensityδisuchthatthesolutionof parametrization (Chevallier&Polarski 2001; Linder 2003) in or-
thenon-linearEq.(5)divergesatac.Oncetheinitialoverdensityis der to avoid the future unphysical divergence of the dark energy
found,wecomputetheinitialvelocityasdetailedinAppendixA,
equation of state typical of this model. In this case, the function
andusethembothasinitialconditionsforthelinearEq.(6).When
w(z)canbewrittenas
integrateduptoac thelatterreturnsusthelineardensitycontrast
correspondingtothetimeofsphericalcollapse,δc.
Inordertodeterminethevirialoverdensity∆v,representing w(a)=w0 A(aBsin(1/a)+θ). (9)
−
4 F.Pace etal.
Table1.Valuesofthefreeparametersforthedarkenergyequationofstate
andforthematterpowerspectrumnormalization σ8.Thenumbersinthe -0.2 Model 1
Model 2
last column corrspond to the following references: (1) : Linder (2006),
Model 3
(2) : Lazkozetal. (2010), (3) : Fengetal. (2006), (4) : Ma&Zhang Model 4
-0.4
(2011). Model 5
Model 6
-0.6
z)
w(
Model w0 A B θ σ8 Reference -0.8
1 -0.9 0.07 5.72 0.0 0.7989 (1)
2 -0.9 0.07 2.86 0.0 0.7986 (1) -1
3 -0.9 0.15 1.0 0.0 0.7983 (1)
4 0.0 1.0 0.06 π/2 0.7999 (2) -1.2
5 -1.0 1.5 0.032 5π/18 0.8012 (3)
6 -1.061 0.041 1.0 −sin(1) 0.8001 (4) 100 101 102 103 104 105 106
1+z
Figure1.Theredshiftevolutionoftheequationofstateparameterforthe
oscillatingdarkenergycosmologiesanalyzedinthiswork.Thereddashed,
Goingintothedistantfuture,wehavea andtakingthelimit
→∞ blueshort-dashedandcyandottedcurvesshowthemodels1,2and3respec-
ofthepreviousequationweobservethatitasymptotesto
tively.Theorangedot-dashedcurveshowsmodel4,whilethedark-green
dot-short-dashedandthelight-greendot-dottedlinesrepresentmodels5and
w(a)=w0 A(B+θ), (10) 6,respectively(seeTable1formoredetails).
−
thusremainingfinite.Formoredetailsonthismodelwereferthe
readertoMa&Zhang(2011). the corresponding redshift evolution for the dark energy density
In Table 1 we summarize the values for the free parameters parameters.Asonecouldnaivelyexpect,theamount ofmatterat
characterizingeachcosmology.Wealsoquotethepaperwherethe earlytimesisthesameinallmodels,consequenceofthefactthatat
expansionhistoryoftheUniverseresultingfromthatspecificdark highredshiftthedarkenergycontributionbecomesnegligible,and
energymodelhasbeenstudiedindetail.Thequotedvaluesarethe hencetheUniversealwaysbehavesasanEinstein-deSitter(EdS)
best fit to certain classes of cosmological observables considered cosmology.Differencesbecomesignificantatz.5andareatmost
by those authors. In the literatureother parametrizations of w(z) atthelevelof 10 15%.Itisworthnoticingthatinnocircum-
∼ −
showing an oscillatory behavior can be found (see for example stancesweseefulloscillationsinthedensityparameters,implying
Kureketal.2008,2010).Howevertheyrepresentonlylocalfitsto thatoneintegraloverthecosmichistoryisenough tosmoothout
theexpansionhistory,givingunphysicaldivergencesinthedistant mostfeaturesofw(z).
past.Forthisreasonwedidnotincludetheminouranalysis. Itisworthnoting that model 6and, especially, model 4dis-
InFigure1weshowtheredshiftevolutionoftheequationof playverylittledifferencewithrespecttotheconcordancecase,an
stateparameterw(z)forthedifferentmodelsstudiedinthiswork. instance that will show up time and again throughout the discus-
Werefertothecaptionforthedescriptionofthelinestylesadopted. sion of our results. The other models instead, with the exception
AsitappearsevidentfromtheFigure,models1and2havethesame ofmodel5thatseesareductionintheabundanceofdarkenergy,
amplitudebutthefrequencyofoscillationschangesbyafactorof show asubstantially higher amount of dark energyat earlytimes
twobetweeneachother.Model3isqualitativelythesame,butthe thanthecosmological constant case.Atz 20thedifferencein
∼
amplitudeistwiceasbigandtheoscillationsonethirdlessfrequent thedarkenergydensityparameterisofoneorderofmagnitudeor
withrespecttomodel1.Thesemodelsallowacomparativestudy more.Itshouldberecalledhoweverthatatsuchhighredshiftsthe
ontheinfluenceoftheamplitudeandthefrequencyoftheoscilla- contributionofdarkenergytotheexpansionhistoryoftheUniverse
tions.Models4and5,despitebeingdescribedbythesamefunc- isnegligibleanyway.
tionalform,haveaverylongperiodandoscillationsarenoteven Adifferentperspectiveonthesameresultsisgivenbyexam-
visiblethroughoutcosmichistory.Models3and5arealsocharac- iningtheHubbleparameters(thatare,theexpansionrates)forthe
terized by a crossing of the phantom barrier, w = 1, a feature variouscosmologies,shownintheupperpanelofFigure3.Forflat
thatismarginallyfindtobestfittheluminositydistan−ceofSNeIa. universes,theHubbleparametercanbewrittenas
Finally,model6showstinyoscillationsonlyatrecenttimes,while
fmoordze&l61w0itthhethfuenoctthioernsww(ezc)aanppdrroawachceosncalucosinosntasnatb.oBuytcthoemipmaprionrg- H(a)=H0E(a)=H0 Ωam3,0 +Ωq,0g(a), (11)
r
tanceoftheoscillationsatearlytimes.
whereg(a)isdefinedas
a 1+w(a′)
g(a)=exp 3 da′ , (12)
3.1 Homogeneousbackgroundanalysis − a′
(cid:18) Z1 (cid:19)
We now explore in detail the redshift evolution of functions re- andΩq isthedarkenergydensity.Thebehavioroftheexpansion
lated to the homogeneous background for the various oscillating ratesisverysimilartothatofthematterdensityparameters,which
quintessence cosmologies presented above. In the upper panel of isexpectedbecausedarkenergycomestodominatetheevolution
Figure2weshowtheratiobetweenthematterdensityparameters oftheHubblefunctiononlyatverylowredshift,wheredifferences
inthesixdynamical darkenergymodelsconsidered hereandthe betweendifferentmodelstendtovanish.
same function in the concordance ΛCDM cosmology, as a func- Sinceoscillationsarenotpresent intheredshiftevolutionof
tionofredshift.InthelowerpanelofthesameFigurewedisplay thematteranddarkenergydensityparameters,thesameholdstrue
Structureformationwithoscillatingdarkenergy 5
1.15
Model 1 1.04
Model 2
Model 3
1.1
Model 4 1.02
Model 5
Λm,CDM 1.05 Model 6 ΛCDM 1
Ω H
/E /E 0.98
D 1 D
m, H Model 1
Ω 0.96 Model 2
Model 3
0.95
0.94 Model 4
Model 5
Model 6
0.9 0.92
0 5 10 15 20 0 5 10 15 20
z z
7
Model 1
Model 2 1.2
6
Model 3
Model 4
5 Model 5 1
M Model 6
D M
ΛC 4 CD
Ωq, ’HΛ 0.8
/E 3 /E
D D
Ωq, ’H Model 1
2 0.6 Model 2
Model 3
Model 4
1
0.4 Model 5
Model 6
0
0 5 10 15 20 0 2 4 6 8 10
z z
Figure 2. The redshift evolution of the density parameters. In the upper Figure3.Upper(lower)panel.RatiooftheHubbleparameter(derivativeof
(lower)panelweshowtheratioofthematter(darkenergy)densityparame- theHubbleparameterwithrespecttothescalefactor)inthesixoscillating
terinthesixoscillatingdarkenergycosmologiesstudiedinthisworktothe darkenergymodelsconsideredinthepresentpapertothesamefunctionin
correspondingfunctioninthefiducialΛCDMmodel.Linetypesandcolors thefiducialΛCDMcosmology.LinestylesandcolorsareasinFigure1.
areasinFigure1.
measurementsaswellasoftheintrinsicscatteraroundthebestfit.
This,togetherwiththeveryslightdeviationsinvariouscosmolog-
fortheexpansionrateinourquintessencecosmologies.However,
icalfunctions showninprevious Figuresleadustoconclude that
considering the derivative of the expansion rate with respect to
the oscillating quintessence cosmologies studied in this work are
the scale factor (dubbed deceleration parameter, see Dunajski &
not distinguishable from the concordance model by current geo-
Gibbons 2008 and references therein), shown in the lower panel
metricalprobes. Thisisperfectlyconsistent withprevious works,
ofFigure3,weobservesomepartialindicationofoscillations,in
sincetheparametervaluesthatweadoptedareindeedchosensoas
thattheoscillatingdarkenergydecelerationparametercrossesthe
toreproducesomeparticulargeometricaltests.
ΛCDM one at least once for models 1, 2, and 5. However, since
theoverallpatternlooksthesameforallmodels,itislikelynotdi-
rectlyconnectedwithdarkenergyoscillations.Moreover,oscillat-
3.2 Redshiftdrift
ingquintessence modelsintroduceabsolutedifferencesinthede-
celeration parameter of at least 20 30% with respect to the Animportantcosmologicaltestrelatedwiththeexpansionhistory
∼ −
fiducialcase,whichisaquitesignificanteffect. thathasnotbeenconsideredinthepast,butmightrevealitselfvalu-
Intheupper(lower)panelofFigure4wepresenttheratioof ableinthenearfutureisthesocalledredshiftdrift,thatrepresents
theluminositydistance(ageoftheUniverse)inthevariousdynam- the variation of the cosmological redshift of a source due to the
ical dark energy cosmologies to the same quantity in the ΛCDM expansion of theUniverse (Balbi&Quercellini 2007; Liskeetal.
model, asafunction of redshift.Inboth cases weseethat differ- 2008; Uzanetal. 2008; Jain&Jhingan 2010; Arau´jo&Stoeger
encesareatmostoftheorderof∼4%andpredominantlylocated 2010).Letusindicate withts thetimeof emissionof anelectro-
atrelativelylowredshifts(z.2),althoughtheluminositydistance magneticsignalfromasource,andwitht0thetimeofobservation
shows a 2% deviation even at arbitrarily high redshifts. This ofthesamesignal.Thecosmologicalredshiftofthesourceisthen
∼
factisexpected,sinceboththeluminositydistanceandtheageof definedas
theUniversearesuitableintegralsoversomefunctionoftheHub-
bleparameter,whichalsoshowsmostdifferencesatlowredshift.
Comparing the upper panel of Figure 4 with the luminosity dis- 1+zs = a(t0) . (13)
tancesinferredbySNIaUnion2data(Amanullahetal.2010)we a(ts)
seethatdifferencesinducedbytheoscillatingdarkenergymodels Afteratimeinterval∆t0haspassedfortheobserver,correspond-
atlowredshiftsareatthesamelevelofthesystematicerrorsinthe ingtoaninterval∆tsforthesource,thechangeinthesourcered-
6 F.Pace etal.
1.12 0.1
Model 1
1.1 Model 2 0.05
Model 3
1.08 Model 4 0
Model 5
M Model 6
D 1.06 -0.05
/dΛEL,C 1.04 +z-E(z) -0.1
dL,D 1.02 1 -0.15 Model 1
Model 2
1 -0.2 Model 3
Model 4
0.98 -0.25 Model 5
Model 6
0.96 -0.3
0 5 10 15 20 0 2 4 6 8 10
z z
1.05 Figure5.Thedifference betweentheredshiftdriftsinthesixoscillating
Model 1 quintessencecosmologiesconsideredinthisworkandthesomequantityin
1.04 Model 2
Model 3 thefiducialΛCDMmodel,asafunctionofredshift.Linestylesandcolors
1.03 Model 4 arethesameasinFigure1.
Model 5
1.02 Model 6
M
CD 1.01 InFigure5weshowthedifferencebetweentheredshiftdrifts
Λ
/tE 1 intheoscillatingdarkenergymodelsandthesamefunctioninthe
tD concordanceΛCDMcosmology,perunitofobservedtimeandnor-
0.99
malizedbytheHubbleconstant.Wedecidedtoplotthedifferences
0.98 insteadofratiosinthiscaseinordertoavoiddivergences, dueto
0.97 the fact that ∆zs goes to zero when 1+zs = E(zs). Similarly
topreviousFigures,alsointhiscasemodels4and6behavevery
0.96
0 5 10 15 20 similarlytotheΛCDMcosmology. Theredshiftdriftissystemat-
z ically higher for the model 5 while it is systematically lower for
Figure4.Upper(lower)panel.Ratiooftheluminositydistances(agesof thefirstthreeoscillatingcosmologies.Athighredshiftsthediffer-
theUniverse)intheoscillatingdarkenergymodelsconsideredheretothe encesbetweenthemodelsdecreasesincealltheHubbleparameters
samequantityforthefiducialΛCDMcosmology,asafunctionofredshift. convergetotheEdSbehavior.Sinceallthecosmologicalinforma-
LinestylesandcolorsareasinFigure1. tionisencoded intheHubble expansion function, no oscillations
appear inthiscaseaswell,although aslightwiggleisvisiblefor
model 1atz 2.Asfor theperspective of realisticallymeasur-
shiftcanbeestimatedbyexpandingatfirstorderthepreviousequa- ∼
ingtheredshiftdrift,accordingtoLiskeetal.(2008),peculiarmo-
tion, tions arenegligible ( 10−3 cm s−1) and with atemporal base-
∼
line of 20 years it will be possible to determine the existence of
∆zs≃∆t0(cid:20)a˙(t0a)(−ts)a˙(ts)(cid:21) . (14) tlhoweicnogsmBoallobgii&calQcuoenrcstealnlitnait(32.0107σ)owbesearvlsiongndoitsictaentthqautavsaarrisa.tiFoonls-
in the recession velocity of the sources are bigger than the error
BysubstitutingH(a)=a˙/aweobtaintheexpressionforthered-
barsforecastedbyMontecarlosimulations,thereforewithasuffi-
shiftdrift
cientlylongbaseline,itshouldbepossibletodiscriminatebetween
oscillatingquintessenceandcosmologicalconstant,atleastforthe
∆zs H0∆t0[1+zs E(zs)] , (15) modelsshowingmoresignificantdeviations.
≃ −
whereE(zs)=H(zs)/H0.
Usingthevariationinthecosmologicalredshift,itisalsopos-
4 RESULTS
sible to determine the variation in the recession velocity of the
source, InthisSectionwepresent resultsconcerning thestructureforma-
tionintheoscillatingquintessencecosmologiesdescribedinSec-
∆υs = c∆zs . (16) ttiiocunl3ar.Wweefsotucduiseeddsoeuvrearattleanstpioenctosnofthsetrgurcotuwrtehffoarcmtoart,iothnealnindeianrpaanrd-
1+zs
non-linearoverdensitiesderivedfromthesphericalcollapsemodel,
Wecanthereforewrite
themassfunctionofcosmicstructures,thepowerspectrumofcos-
micshearandtheISWeffect.Wenowproceedtodescribeeachone
υ˙s= cH0 [1+zs E(zs)]. (17) oftheseobservablesindetail.
1+zs −
Sinceallthecosmologicalpropertiesofthemodelathandareen-
4.1 Growthfactor
coded into the Hubble function, we see that we can use the time
variationoftheredshiftinordertoreconstruct theexpansionhis- In Figure 6 we show the growth factor normalized by the scale
toryoftheUniverse. factor, D+(a)/a, as a function of redshift for the six oscillating
Structureformationwithoscillatingdarkenergy 7
darkenergycosmologiesdescribedabove,plusthefiducialΛCDM
1.45
model. Cosmological observables sensitive to the growth factor
1.4
include cosmic shear, ISW effect and the Rees-Sciama effect, all
discussed lateron.InFigure6thegrowthfactor isnormalizedto 1.35
unityatz = 0.Ascanbeseen, differences betweentheoscillat- 1.3
ing quintessence models and the ΛCDM cosmology (solid black
curve) are at most of 10% (for model 3), while, inagreement a)/a 1.25
withpreviousFigures,∼model4doesnotshowanyappreciabledif- D(+ 1.2 ΛCDM
ferencefromtheconcordancemodel.Itisinterestingtonotethat, 1.15 Model 1
Model 2
whilew(z)formodel4showsindeedverylittlevariationuptothe
1.1 Model 3
lastscatteringredshiftduetotheverylargeperiodofitsoscillations Model 4
1.05 Model 5
(seeFigure1),model6showsevensmallertimeevolution,yetits
Model 6
effectsoncosmologicalfunctionsarelarger.Thisimpliesthathigh 1
0 5 10 15 20
frequencyoscillationsinw(z),albeitwithaverysmallamplitude
z
andlimitedtimeextenthavemoreofaneffectontheexpansionhis-
tory(andstructureformationtoo,seelaterSubsections)thanlarger Figure6.Thegrowthfactorasafunctionoftheredshift.Theblacksolid
oscillations with a low frequency. This is because low-frequency line represents the concordance ΛCDMcosmology, the reddashed, blue
short-dashedandcyandottedcurves showthemodels1,2and3respec-
oscillationscancelintegralcontributionsmoreeffectively.
tively.Theorangedot-dashedcurveshowsmodel4,whilethedark-green
Figure6alsoshowsthatformodels5and6thegrowthfactor
dot-short-dashedandthelight-greendot-dottedrepresentmodels5and6,
issmallerthanforthecosmologicalconstantcase.Thiscanbeun-
respectively.
derstoodbythefollowingargument.FromthelowerpanelofFig-
ure2weseethatforthesetwomodelstheamountofdarkenergy
issmallerthanforthecosmological constant caseatallredshifts.
0.5
ThismeansthattheHubbledragislesseffectiveintheformermod- Model 1
Model 2
els,andhencethegrowthofstructures(atleastatthelinearstage) 0.4 Model 3
is easier. Sincethe amplitude of density fluctuations at z = 0 is Model 4
almost the same amongst all the dark energy models considered DM 0.3 MMooddeell 56
C
here (itdiffersby afactor proportional tothe criticaloverdensity WΛ 0.2
forsphericalcollapse,thathoweverisonlyslightlychangedinthe S
caseofdynamicaldarkenergy,seebelow),thegrowthfactormust -IE 0.1
D
W
besmallerinordertomatchtheamplitudeoffluctuationsatearly
S
times. I 0
The growth factors depicted in Figure 6 do not present any -0.1
signofoscillations,notevenifweconsidertheirratioswithrespect
totheconcordancecosmologycase.ByrewritingEq.(6)usingthe -0.2
0 5 10 15 20
scalefactorinsteadofcosmictimeastheindependentvariable,we
z
obtain
0.5
Model 1
δ′′+ 3 + E′ δ′ 3 Ωm,0δ=0. (18) 0.4 MMooddeell 23
a E − 2a5E2 Model 4
(cid:18) (cid:19) 0.3 Model 5
As shown above, the derivative of the Hubble function presents M Model 6
D
mildsignsofoscillations,howeversincethisisasecondorderdif- ΛC 0.2
S
ferentialequation,thesolutioninvolvesadoubleintegraloverthe R
-E 0.1
scalefactor,thatefficientlysmoothesoutanyfluctuationintheco- D
S
efficients. R
0
Finally, we also estimated the logarithmic derivative of the
growth factor with respect to the scale factor, f(Ωm(a)) = -0.1
dlnD+(a)/dlna.Ithasbeenshownthatinabroadrangeofcos-
mologies f(Ωm(a)) ∼ Ωγm(a), an empirical relation that we re- -0.2 0 5 10 15 20
trievefortheoscillatingdarkenergycosmologiesaswell.Devia- z
tionswithrespect totheΛCDMexpectation however aresmaller
Figure 7. Redshift evolution for the quantities describing the ISW (up-
thanforthegrowthfactoritself.
perpanel)andtheRees-Sciama(lowerpanel)effects.FortheISWeffect
weplot the function d(D+(a)/a)/da, while for the RS effect we plot
4.2 ISWandRees-Sciamaeffects d(cid:0)D+2(a)/a(cid:1)/da.Inbothcases,theresultforthefiducialmodelissub-
tractedfromthecorrespondingquantitiesforeachofthesixoscillatingdark
energycosmologies.LinetypesandcolorsarethesameasinFigure1.
TheSachs-Wolfe(SW)effect(Sachs&Wolfe1967)describesthe
effectofgravitationalpotentialsontheCMBanisotropyatthelast-
scatteringsurface.Photonstravellingtoanobserverencountervari-
ationsinthegravitationalpotentialcausedbyvariationsinthemat-
terdensity. Photonsclimbingout apotential wellwillbegravita-
tionallyredshiftedandthiswillmaketheregiontheycomefromap-
8 F.Pace etal.
pearcolder.Togetherwiththisgravitationaleffect,onehastotake
1.7
intoaccountthetime-dilationeffectinducedbytheperturbations:
weseethephotonsascomingfromadifferentspatialhypersurface
1.69
(labeledbyadifferentscalefactora(t)).
TheIntegratedSachs-Wolfe(ISW)effectisbasedonthesame
principle, only it is given by the gravitational redshift occurring 1.68
asphotonstravelthroughthelargescalestructuretoreachanob- z)
server at present time. TheISWeffect arisesonly recently inthe δ(c 1.67 ΛCDM
cosmichistory,asdarkenergystartsdominatingtheexpansionof Model 1
Model 2
theUniverse.Thismeansthatanon-vanishingISWeffectindicates
Model 3
1.66
by itself presence of dark energy if the model is spatiallyflat, as Model 4
Model 5
indeeditisassumedinourcase.TheISWeffectissensitivetothe
Model 6
derivativeofthegrowthfactor,d(D+(a)/a)/dathatvanishesfor 1.65
0 2 4 6 8 10
an EdS universe where D+(a) = a. It was detected for the first
z
timebyBoughn&Crittenden(2004)usingX-rayclustercatalogs.
TheRees-Sciama(RS)effect(Rees&Sciama1968)isverysimilar 180
totheISWeffect,onlyitreferstothegravitationalredshiftinduced 170
bynon-linearstructuresonly,andhenceitisactiveonmuchsmaller
160
scales.Itismainlysensitivetothefunctiond D2(a)/a /da.
+
InFigure7weshowthedifferenceofthefunctionsprobedby 150
(cid:0) (cid:1)
the ISW effect (upper panel) and by the RS effect (lower panel) z) 140
forthesixoscillatingdarkenergymodelsstudiedheretothesame ∆(v 130
quantitiesevaluatedinthereferenceΛCDMcosmology,asafunc- ΛCDM
120 Model 1
tion of redshift. As can be seen, the ISW effect is preferentially Model 2
modifiedatlowredshifts,eitherpositivelyornegatively,exceptfor 110 Model 3
Model 4
models4wherenodifferencesfromtheΛCDMmodelareseen.At 100 Model 5
highredshiftsdeviationswithrespecttothecosmologicalconstant Model 6
90
casetendtodisappear,sinceallthemodelsareverywellapproxi- 0 2 4 6 8 10
matedbyanEdSuniverse.Themodelsshowingthelargesteffect z
aremodels 3and 5, which arethe ones having the largest differ-
Figure8. Theredshift evolution ofthelinear density contrast parameter
ences in the growth factor. On the other hand, hints of an oscil-
δc(upperpanel)andofthevirialoverdensityparameter∆v(lowerpanel)
latorybehavior withredshiftareseenfor model 1, whichhasthe for the six dynamical dark energy models and for the reference ΛCDM
highestfrequencyinw(z)amongthoseconsideredhere.Allinall, cosmology.LinetypesandcolorsforthedifferentmodelsareasinFigure6.
sincedifferencesbetweendifferentmodelscanbequitesubstantial,
high redshift observations could be used in principle to discrim-
inate oscillating quintessence cosmologies. Examples come from
cross-correlatinggalaxies,radiosourcesorhardX-raysourcesand
CMBtemperaturefluctuations,(seeFosalbaetal.2003;Noltaetal.
2004; Scrantonetal. 2003; Boughn&Crittenden 2004; Afshordi
2004). However, one should keep in mind that at high redshift, pectations, the function δc(z) does not perfectly converge to the
wheredifferencesaremoremarked,theISWeffectitselftendsto EdSvalueofδc 1.686athighredshift.Thisisaproblemofnu-
disappear. mericalconverge≃ncerelatedtotheoscillatorynatureofw(z),and
AsfortheRSeffect,deviationswithrespecttothecosmolog- thatisbetterexploredinAppendixA.Thisfactobviouslyimplies
ical constant expectations are of the same order of magnitude as thatweshouldnot usethisδc(z)atz & 5 6.However, allthe
theISWeffectanddifferencesdonotvanishathighredshifts,but cosmologicalteststhatweproposeinthefol−lowingthatmakeuse
reachasomewhat constantvalue, theexact onedepending onthe ofthisfunctionarelimitedtosubstantiallylowerredshifts,hence
specificmodel.Thereishoweveraveryspecificredshiftpatternac- theyshouldbeunaffectedbythisissue.
cordingtowhichthedifferencewithrespecttothefiducialΛCDM Asmentionedabove,differencesbetweentheδc(z)computed
cosmologygetsreversedatz 2(exceptformodel4,thatisbasi- indifferentcosmologiesareverymild,beingatmostof 1%at
callyidenticaltotheconcorda∼ncecosmology). Hence,combining z . 2. It isapparently ageneric featureof cosmologica∼l models
RS effect observations at low and high redshift can improve the displaying a dynamical evolution of the dark energy density that
discriminationbetweenthemodels. the spherical collapse parameters are only slightly modified with
respect to the fiducial ΛCDM case (Paceetal. 2010). The lower
panel of Figure 8 shows results for the virial overdensity param-
4.3 Thecharacteristicoverdensitiesδcand∆v eter ∆v(z). In order to evaluate it we used the prescription of
InthisSubsectionwepresentresultsregardingthetimeevolution Wang&Steinhardt(1998).Inthiscaseforz &6 8allthemod-
−
of the linear density contrast parameter for spherical collapse δc els behave almost exactly as the ΛCDM cosmology. Differences
andofthevirialoverdensity∆v.Themainresultsarereportedin betweendifferentmodelsareoftheorderofafewpercentandare
Figure8.Intheupperpanelweshowthetimeevolutionofδcwhile mostlyevidentatz .3.Wetriedtoevaluate∆v(z)adoptingadif-
inthelowerpanelwepresentthetimeevolutionof∆v.Linetypes ferentprescription,detailedinWang(2006).Asitturnsout,differ-
and colors are the same as in Figure 6, to which we refer for a encesbetweendifferentmodelsareverysimilartothoseobtained
detailedexplanation.Thefirstthingtonoteisthat,contrarytoex- byusingotherrecipes.
Structureformationwithoscillatingdarkenergy 9
4.4 Massfunction cludethat theimpact of oscillatingquintessence on thecounts of
cosmic structure can be quite substantial, especially at high red-
An observable quantity depending crucially on the growth factor
shift,thusimplyingthatadetectionmightbepossiblewithfuture
D+(z)andonthelinearoverdensity thresholdforcollapseδc(z) largeclustersurveys.
isthemassfunctionofcosmicstructuresn(M,z),representingthe
Inordertoestablishamoredirectlinkwithobservations,we
numberofdarkmatterhalosperunitmassandperunitcomoving
forecasttheredshiftdistributionofgalaxyclustersthat,ineachof
volumepresentatacertainredshift.Integralsofthemassfunction
thecosmologicalmodelsconsideredinthiswork,willbeobserved
overmasscanbeobserveddirectlybyusinglargecosmologicalsur-
byupcoming widefieldsurveyswithclusterselectionbasedboth
veys,providedtheirselectionfunctioniswellunderstood.Specifi-
onXrayandweak-lensingdata.Inordertodothatweneedtodefine
cally,thecumulativenumberdensityofcosmicstructuresabovea
aredshiftdependentminimummassfortheobservedobjects,and
certainlimitingmassMmin (that willdepend onthespecificsur- integratethe mass function above that threshold. Weassume that
vey)atredshiftzissimplygivenby
preciseestimatesforthemassesoftheseobjectswillbeavailable,
whichisrealisticallyexpectedifarobustmultiwavelengthfollow-
∞ upwillbeperformed.
N(>Mmin,z)= dMn(M,z). (19) The first survey that we consider is a wide field X-ray sur-
ZMmin vey on the model of the upcoming eROSITA1 one. In order to
Themassvarianceisanotherkeyingredientforthemassfunc- determine the minimum mass of clusters that will compose the
tionformalism,andisidentifiedby X-ray catalog we need a scaling relation between the observable
at hand (in this case the X-ray flux) and the true mass. First of
1 +∞ all, knowing the redshift of the cluster, the measured bolomet-
σM2 = 2π2 k2T2(k)WR2(k)P0(k)dk. (20) ric flux can be related to the intrinsic bolometric luminosity as
Z0 L(M,z) = 4πf(M,z)d2L(z),wheredL(z)istheluminositydis-
InEq.(20)P0(k)representstheprimordialmatterpowerspectrum, tance(seeupperpanelofFigure4).InrealitytheX-raybolometric
T(k) isthe matter transfer function, while WR(k) is the Fourier fluxisalmostnevermeasured,rathertheX-rayphotoncountsina
transformoftherealspacetop-hatwindowfunction.Sincetheonly certainenergybandareused.ForthespecificcaseofaneROSITA-
differenceinducedbyoscillatingdarkenergyintheprimordialmat- like X-ray survey, we adopted the band [0.5,2.0] keV, where the
terpowerspectrumisgivenbythedifferentnormalizationσ8,that thresholdfluxisexpectedtobefmin=3.3×10−14erg/(scm2).In
asweshownaboveisveryslight,weexpectonlyminordifferences ordertoestimatethebandfluxforaclusterofagivenmassplaced
in the mass variance as well. In this work, to evaluate the mass atagivenredshiftwemodeledtherelatedintra-clustermediumus-
function, weusedtheSheth-Tormenexpression(Sheth&Tormen ingaRaymond-Smithplasmamodel(Raymond&Smith1977)as
1999). implemented intheXspecsoftwarepackage (Arnaud 1996),with
InFigure9weshowthecumulativemassfunctionforthesix metal abundance Z = 0.3Z⊙ (Fukazawaetal. 1998; Schindler
oscillatingquintessencecosmologiesconsideredinthisworkatdif- 1999). The plasma model has been normalized so as to repro-
ferent redshifts, divided by the corresponding quantity evaluated ducethebolometricluminosityexpectedfromthescalingrelation
forthereferenceΛCDMmodel.FromthisFigurewenotefirstof adoptedbyFedeli,Moscardini&Bartelmann(2009),namely
allthattherearenodifferencesbetweendifferentmodelsatz=0.
Thisisduetoourchoiceofthenormalization,namelythatthera- L(M,z)=1.097 1045erg/s(ME(z))1.554, (21)
tioδc(z = 0)/σ8 shouldbeconservedforallmodels.Significant ×
differencesstartinsteadtoappearathigherredshifts,whereweno- wherethemassmustbeinsertedinunitsof1015M⊙/h.Thisrela-
tice that the six models can be broadly divided into two groups. tionresultsfromthecombinationoftwoscalinglaws,onerelating
Models1,2,and3showmoreobjectswithrespecttotheΛCDM themasstotheX-raytemperature,andtheotherrelatingthetem-
case,whilemodels4,5,and6showlessobjects.Thesedifferences perature to the luminosity. See Fedeli,Moscardini&Bartelmann
are consistent with the latter group having a lower growth factor (2009)foradditionaldetails.
withrespecttotheformerandahigher criticallinearoverdensity The second case that we consider is representative of clus-
δc(z),whichmakesmoredifficultfordensityperturbationstocol- ter catalogs selected through their weak-lensing signal. Massive
lapseintoboundstructures.Asonecouldnaivelyexpect,deviations galaxyclusterscanbeselectedashighS/Npeaksintheweaklens-
increasewithincreasingredshiftandaremostnotableinthevery ingmapproducedbyweak-lensingsurveys.TheS/Nstrengthwill
high mass tail, since rare events are very sensitive even to small also be used as a proxy for their mass, although a robust multi-
fluctuationsintheexpansionhistory.Asanexample,differencesin wavelengthfollow-upprogramwillbenecessary inorder tohave
the abundance of massive galaxy clusters M & 5 1014M⊙/h morepreciseestimates.Fortheminimumclustermassenteringthis
×
range from 5 10% at z = 0.5 up to 30% at z = 1. catalog,weadoptedthecalculationsofBerge´,Amara&Re´fre´gier
∼ − ∼
Large galaxy groups that can be expected to be found at z = 2 (2010).Inparticular,werefertotheirFigure1,wheretheypresent
(M &5 1013M⊙/h)are 20 30%moreabundantinmodels the selection function for a Euclid-like survey2 (Laureijs 2009;
× ∼ −
belongingtothefirstgroup,andupto 40 50%lessfrequent Laureijsetal.2011)inthemass-redshiftplane,assuminganumber
∼ −
inmodelsofthesecondgroup.Wealsonotethatmodelsshowing densityofbackgroundgalaxiesofn¯g =40arcmin−2.Weconsid-
an enhancement in the abundance of cosmic structures are those ered the contour referring to a S/N threshold of 5, threshold that
havingmoredarkenergyathighredshift. wasshowntobeagoodchoicetominimizespuriousdetectionsin
Itshouldbenotedthatuptonowweonlyconsideredtheco- theweaklensingmaps(seePaceetal.2007).
movingnumberdensityofobjects,thatiswedidnottakeintoac-
countpossibleeffectsderivingfromchangesinthecosmicvolume
inducedbythepresenceofoscillatingdarkenergy.Wewillinclude 1 http://www.mpe.mpg.de/erosita/
thisadditionalingredientshortly.Forthetimebeing,wecancon- 2 http://sci.esa.int/science-e/www/area/index.cfm?fareaid=102
10 F.Paceet al.
1.2 1.2
M 1.1 M 1.1
D D
C C
Λ Λ
z) z)
M, 1 M, 1
> >
N( z=0 N(
/E /E
D 0.9 D 0.9
z) Model 1 z) Model 1
M, Model 2 M, Model 2 z=0.5
> Model 3 > Model 3
N( 0.8 Model 4 N( 0.8 Model 4
Model 5 Model 5
Model 6 Model 6
0.7 0.7
1e+11 1e+12 1e+13 1e+14 1e+15 1e+11 1e+12 1e+13 1e+14 1e+15
M [M /h] M [M /h]
sun sun
1.2 1.2
Model 1
Model 2
M 1.1 M 1.1 Model 3
D D Model 4
C C
Λ Λ z=2 Model 5
z) z) Model 6
M, 1 M, 1
> >
N( N(
/E /E
D 0.9 D 0.9
z) Model 1 z)
M, Model 2 z=1 M,
> Model 3 >
N( 0.8 Model 4 N( 0.8
Model 5
Model 6
0.7 0.7
1e+11 1e+12 1e+13 1e+14 1e+15 1e+11 1e+12 1e+13 1e+14 1e+15
M [M /h] M [M /h]
sun sun
Figure9.Cumulative comovingnumberdensityofobjects withmassexceeding M atdifferent redshifts.Ratios withrespecttoΛCDMexpectations are
shown.Selectedredshiftsarez=0(upperleftpanel),z=0.5(upperrightpanel),z=1(bottonleftpanel)andz=2(bottomrightpanel).Linestylesand
colorsforthedifferentmodelsareasinFigure1.Theupperleftpanelisunitybynormalization,asexplainedinthetext.
InFigure10wepresentourresultsfortheminimummassof
thecatalogs.Intheupperpanelweshowtheminimummassforthe
adoptedweak-lensingsurvey(solidline)andX-raysurvey(dashed dV(z)
line) for the ΛCDM model. We see that as expected both mini- N(z)= dz N(>Mmin(z),z), (22)
mummassesincreasewithredshiftinordertohavethesameflux
orS/Nratio.TheX-raymassincreasesfromfewtimes1013M⊙/h where Mmin(z) isthe minimumobserved mass for the survey at
atz 0till 6 1014M⊙/hatz=2.Asevident,sincetheflux hand and dV(z)/dz is the comoving volume element contained
≃ ≃ ×
limitisconstantinredshift,theredshiftdependenceofthemasscan in the unit redshift. The first thing to note is that the deviations
beverywellapproximatedbyaparabola,thereforecompensating intheredshiftdistributionsinducedbythepresenceofoscillating
the increase of the luminosity distance (entering quadratically in darkenergyarequitesubstantiallydifferentforanX-raysurveyand
therelationbetweenfluxandluminosity).Theminimummassfor andacosmicshearsurvey.Thisislikelyduetothefactthatthese
aweak-lensingsurveyisincreasingmuchfasterwithredshiftsince surveyshaveremarkablydifferentselectionfunctions,thatsample
thelensingefficiencydropsveryfasttozeroifthelensapproaches quitedistinctregionsofthemass-redshiftplane,wheretheimpact
thesources. ofoscillatingdarkenergyisnecessarilydifferent.Letusconsider
InthelowerpanelweshowtheminimumclustermassfortheX-ray firsttheX-rayeROSITA-likesurvey.Inthiscasetheredshiftdistri-
catalogineachdarkenergycosmologyconsideredhere,dividedby butionscomputedfordifferentcosmologiesarealmostidenticalat
thesamequantityestimatedforthereferenceΛCDMmodel(lower verylowredshift,whilesubstantialdeviationsarevisibleathigher
panel).Weseethatdifferencesintheminimumobservedmassare redshift.Specifically,theabundanceofclustersisincrementedby
stronglyrelatedtothedifferencesintheHubblefunctionandinthe upto 20%atz 0.8formodels1,2,and3,anddecreasedby
∼ ∼
luminositydistance, asonemight naivelyhaveexpected. Specifi- thesameamountormorefortheothermodels.
cally,sincetheminimummassdependsonthesquareofthelumi- Fortheweaklensingsurveythesituationistotallydifferent.
nositydistance,evensmallvariationsofthelatterturnouttoaffect Cosmologicalmodelsdifferfromeachotherbyupto 5 10%
theformeratthelevelof 10%. alreadyatrelativelylowredshift.Deviationsfromthe∼ΛCD−Mcu-
∼
riouslyvanishatz 0.6forallmodels,andthengrowagain,but
∼
InFigure11wepresenttheredshiftdistributionsexpectedfor with opposite sign, for higher redshifts. In addition to the differ-
theweak-lensingsurvey(upperpanel)andtheX-raysurvey(lower entselectionfunction,oneadditionaldifferencebetweentheX-ray
panel)weconsidered,respectively.Theredshiftdistributionisde- surveyandthecosmicshear survey consideredhereisthat inthe
finedas formercasetheminimummassdependoncosmology,whileinthe