The epoch of structure formation in blue mixed dark matter models

6 Pages·0.18 MB·English

by  Stefano Borgani

Mon.Not.R.Astron.Soc.000,000–000 (2008)
The Epoch of Structure Formation in Blue Mixed Dark
Matter Models
Stefano Borgani1,2, Francesco Lucchin3, Sabino Matarrese4 and Lauro Moscardini3
1 INFN – Sezione di Perugia, c/o Dipartimento di Fisica dell’Universit`a, via A. Pascoli, I–06100 Perugia, Italy
6 2SISSA – International School forAdvanced Studies, via Beirut 2–4, I–34013 Trieste, Italy
9 3Dipartimento di Astronomia, Universit`adi Padova, vicolodell’Osservatorio 5, I–35122 Padova, Italy
9 4Dipartimento di Fisica Galileo Galilei, Universit`adi Padova, via Marzolo 8, I–35131 Padova, Italy
1

n
a 1February2008
J

8
1 ABSTRACT
Recentdataonthehigh–redshiftabundanceofdampedLyαsystemsarecomparedwith

2 theoretical predictions for ‘blue’ (i.e. n > 1) Mixed Dark Matter models. The results
v
show that decreasing the hot component fraction Ων and/or increasing the primordial
3
spectral index n implies an earlier epoch of cosmic structure formation. However, we
0
0 alsoshowthatvaryingΩν andninthesedirectionsmakesthe modelsbarelyconsistent
with the observed abundance of galaxy clusters. Therefore, requiring at the same time
6
observationalconstraintsondampedLyαsystemsandclusterabundancetobesatisfied
0
5 represents a challenge for the Mixed Dark Matter class of models.
9
1 INTRODUCTION data as the most constraining ones and we will compare
/
h them with model predictions.
p Sincealongtimeobservationsofhigh–redshiftobjectshave
Several authors (Subramanian & Padmanabhan 1994;
- become a potentially powerful constraint for models of cos-
o Mo & Miralda–Escud´e 1994; Kauffmann & Charlot 1994;
r mic structure formation. The availability of statistically re- Ma & Bertschinger 1994) have recently claimed that the
st liablesamplesofquasarsallowed toaddressthisproblemin largevalueofΩ observedatz> 3isincompatiblewithpre-
a aquantitativewayintheframework oftheCold DarkMat- dictions of thegMixed (i.e. cold∼+hot) Dark Matter (MDM)
: ter cosmogony (Efstathiou & Rees 1988; Haehnelt 1993).
v model with spectral index n=1 and Ω 0.3 contributed
Moreover,thecomparisonofpredictionsandobservationsof ν ≃
i by one species of massive neutrinos and Ω = 0.1 for the
X quasar abundance at different redshifts has been used as a b
baryon fractional density (Klypin et al. 1993; Nolthenius,
r testformodelreliability(e.g.Nusser&Silk1993;Pogosyan
Klypin & Primack 1995). Klypin et al. (1995) reached sub-
a & Starobinsky 1993).
stantially the same conclusions about this model, but em-
Recently,dampedLyαsystems (DLAS)havebeen rec- phasized two relevant points: (i) any theoretical prediction
ognized as a promising way to trace the presence of high is very sensitive to the choice of the parameters of the
redshift collapsed structures, thanks to the possibility of model needed to obtain Ω from a given power–spectrum;
g
identifying them as protogalaxies and to their detectabil- (ii) slightly lowering Ω to 20–25% keeps MDM into bet-
ν
ity at high z (see Wolfe 1993 for a comprehensive review). ter agreement with DLAS data, independently of whether
DLAS are seen as wide absorbtion features in quasar spec- thehot component is given by one or two massive neutrino
tra.Theassociatedabsorbingsystemshaveaneutralhydro- species (see also Primack et al. 1995).
gen column density 1020 cm−2. The rather large abun-
≥ A possible alternative, still in the framework of MDM,
dance of DLAS makes it possible to compile reliable statis-
is to consider ‘blue’ (n > 1) primordial spectra of density
tical samples (Lanzetta 1993; Lanzetta, Wolfe & Turnshek
fluctuations. The advantage of these models is an anticipa-
1995;Storrie–Lombardietal.1995;Wolfeetal.1995).Once
tion of the epoch of structure formation due to the higher
the parameters of the Friedmann background are specified,
small–scale power.Thechoiceofbluespectrawasoriginally
observations of DLAS can be turned into the value of the
suggestedbytheanalysisofCosmicMicrowaveBackground
cosmological density parameter Ωg contributed by theneu- anisotropiesonscaleslargerthan1◦ (e.g.Devlinetal.1994;
tralgas,whichisassociatedwithDLAS.Itturnsoutthatat
Hancock et al. 1994; Bennett et al. 1994). Possible indica-
z 3thisquantityiscomparabletothemassdensityofvis-
∼ tionsforbluespectracomesfrom largebulkflows(Lauer&
ible matter in nearby galaxies, thus suggesting that DLAS
Postman 1994; see however Riess, Press & Kirshner 1995;
trace a population of galaxy progenitors.
Branchini&Plionis1995)andlargevoidsinthegalaxydis-
Based on the APM QSO catalogue, Storrie–Lombardi tribution (Piran et al. 1993). In recent years many authors
et al. (1995) recently presented the most extended DLAS have pointed out that the inflationary dynamics can easily
sample up-to-date,covering the range 2.8<z <4.4. In the account for the origin of blue perturbation spectra (Liddle
followingofthispaperwewillconsidertheirhighestredshift & Lyth 1993; Linde 1994; Mollerach, Matarrese & Lucchin
2 S. Borgani, F. Lucchin, S. Matarrese and L. Moscardini
Figure 1.The r.m.s.fluctuation amplitude withina top–hat sphere of 8h−1Mpc, σ8, forCOBE normalization (see text). Left panel:
dependence on the hot component fraction Ων; different lines refer to n = 0.9,1,1.1,1.2,1.3,1.4 from bottom to top. Right panel:
dependence onthespectralindexn;differentlinesrefertoΩν =0,0.1,0.2,0.3,0.4,0.5fromtoptobottom.
1994;Copelandetal.1994),inparticularintheframeworkof gasconsumption intostars(e.g. Lanzettaet al.1995; Wolfe
theso–called hybridmodels.RecentlyLucchinetal.(1995), et al. 1995), the actual value of f at the high redshift we
g
using linear theory and N–body simulations, performed an are interested in is not clear. In any case, since Ω at such
g
extended analysis of the large–scale structure arising from high redshifts is quite similar to the fractional density con-
blueMDM(BMDM)models:themostinterestingadvantage tributed by visible matter in present–day normal galaxies,
of these models is theincrease of thegalaxy formation red- we expect f not to be a particularly small number. In the
g
shift: for instance, taking Ω =0.3 one has for the redshift following we will show results based both on f = 0.5 and
ν g
of non–linearity on galactic scale (M =1012 M⊙) znl 1.9 1.
≈
if n = 1.2 and z 0.6 if n = 1. The same class of mod- Taking h=0.5 and an Einstein–de Sitteruniverse, the
nl
≈
els has been tested against observational data, using linear data at z = 4.25 from Storrie–Lombardi et al. (1995) turn
theory predictions, by Dvali, Shafi & Schaefer (1994) and into Ω = (8.8 2.0) 10−2 and (4.4 1.0) 10−2 for
coll
± × ± ×
Pogosyan & Starobinsky (1995a). f =0.5 and 1, respectively.
g
InthisworkwewillcompareBMDMmodelpredictions, Fromthetheoreticalside,thePress&Schechter(1974)
for different valuesof Ων and n (for the sake of comparison approach gives a recipe to compute the contribution to the
wealsoconsiderthe0.9 n<1tiltedmodels),withtheob- cosmicdensityduetothematterwithincollapsedstructures
≤
served DLAS abundance. Furthermore, we will also discuss of mass M at redshift z:
the implications of the observed abundance of galaxy clus-
δ
ters for BMDM models. In fact, increasing n for a fixed Ων Ωcoll(M,z) = erfc c . (2)
and fixed normalization to COBE rises up the value of σ8, (cid:18)√2σM(z)(cid:19)
ther.m.s. fluctuation within a top–hat sphere of 8h−1Mpc TheaboveexpressionassumesGaussian fluctuationsandδ
c
radius, which is constrained by the observed abundance of isthelinearly extrapolateddensitycontrastforthecollapse
galaxy clusters to be σ8 0.6 (White, Efstathiou & Frenk ofa perturbation;σ is ther.m.s. fluctuationat themass–
≃ M
1993). scale M, where
M = (2πR2)3/2ρ (3)
2 THE METHOD
for theGaussian window that we will assume in thefollow-
Inorderto connect ourmodelpredictions toDLASobserv- ing.Here,ρistheaveragematterdensity,which istakento
ables, let us define Ω (z) as the fractional matter density havethecritical value.
coll
within collapsed structuresat redshift z. Therefore As for the mass of thestructures hosting DLAS,it has
been argued that, since the high column density of the ab-
Ω (z)
Ω (z) = g , (1) sorber is typical of large disks of luminous galaxies, DLAS
coll Ωbfg should be located within massive structures of 1012 M⊙.
∼
where Ω is the fractional baryon density (since h = 0.5 is However, it is not clear at all whether the properties of
b
assumed throughout the paper, we take Ω = 0.05 accord- present–day galaxies can be extrapolated to their high–
b
ingtostandardprimordialnucleosynthesis;see,e.g.,Reeves redshift progenitors. Therefore, we prefer to leave open the
1994) andf isthefraction oftheHIgas, whichisinvolved possibility that DLASare hosted within smaller structures.
g
in DLAS. Although the observed decrease of Ω with red- It is clear that, when a model is in trouble in accounting
g
shift for z < 3.5 is usually considered as an indication of for theDLASabundanceif thehosting structureis adwarf
∼
The Epoch of Structure Formation in Blue Mixed Dark Matter Models 3
Figure 2. The fractional matter density within collapsed structures Ωcoll at redshift z =4.25, when δc =1.5 and M =1011 M⊙ are
assumed. Left panel: dependence on the hot component fraction Ων; different lines refer to n = 0.9,1,1.1,1.2,1.3,1.4 from bottom to
top.Rightpanel:dependence onthespectralindexn;differentlinesrefertoΩν =0,0.1,0.2,0.3,0.4,0.5fromtoptobottom. Errorbars
show the effect of taking M =1010 M⊙ and 1012 M⊙. The horizontal lines refer to the observational data by Storrie–Lombardi et al.
(1995)withthecorrespondinguncertainties.
galaxy (M 1010 M⊙), it would certainly be ruled out if 0 0.5 with steps of 0.1. As expected, σ8 is an increasing
∼ −
more massive protogalaxies are required. function of n, while it decreases with Ω .
ν
Linear theory for the top–hat spherical collapse pre-
dicts δ = 1.69. However, effects of non–linearity as well
c
3 DISCUSSION
as asphericity of the collapse could cause significant devia-
tionsfromthisvalue.Klypinetal.(1995)estimatedthehalo The results of our analysis on DLAS are summarized in
abundance at different redshifts from high mass resolution Figure 2, where we plot Ω , estimated at z = 4.25, as
coll
N–body simulations. By using the Gaussian window, they a function of Ω (left panel) and of n (right panel), after
ν
found a good agreement with the Press–Schechter expres- assuming δc = 1.5 and M = 1011 M⊙. In each panel, dif-
sion for valuesas lowasδc =1.3–1.4 (seealso Efstathiou & ferent curves are for the same choice of parameters as in
Rees 1988). On the other hand, Ma & Bertschinger (1994) Figure1.Upperandlowererrorbarsshowtheeffectoftak-
found that for Ων = 0.3 and a top–hat window δc ≃ 1.8 ingM =1010 M⊙and1012M⊙,respectively.Thehorizontal
is always required, which corresponds to δc 1.7 for the solid line is the observational result with thecorresponding
≃
Gaussianwindow.Inthefollowingweprefertoshowresults uncertainties(dottedlines),whichisobtainedbyconverting
basedonδc =1.5but,duetothepreviousuncertainties,we theΩg value,asreported byStorrie–Lombardi etal. (1995)
will discuss also the effect of different choices for δc. at z=4.25, to Ωcoll according toeq.(1) with fg =1.
Figure 3 shows in the Ω – n plane the models which
FortheMDMtransferfunctionwetakethefitobtained ν
reproducetheobservedΩ ,takingf =1(leftpanel)and
by Pogosyan & Starobinsky (1995a), which provides a con- coll g
f = 0.5 (right panel). The heavy solid curve corresponds
tinuousdependenceonthefractionaldensityΩ contributed g
by one massive neutrino. As for the cold partνof the trans- to δc = 1.5 and M = 1011 M⊙, with the lighter curves
delimitingtheobservational uncertainties.Upperandlower
fer function, we use the Cold Dark Matter expression by
dashed lines show the effect of varying δ to 1.3 and 1.7,
Efstathiou, Bond & White (1992), with the shape param- c
respectively. Upper and lower dotted curves refer to M =
PeteearcoΓck=&ΩD◦hodexdps((−1929Ω4b)),,taoccaocrcdoiunngttfoortthheepbraersycorinpitciocnomo-f 1010 and 1012 M⊙, respectively. The overall result that we
getisthatdecreasingthehotcomponentfractionΩ and/or
ponent. We varied Ω in the interval 0 Ω 0.5. We as- ν
ν ≤ ν ≤ increasingtheprimordialspectralindexnimpliesanearlier
sumefortheprimordial(postinflationary)power–spectrum
formation of cosmic structures.
P(k) kn with 0.9 n 1.4. Each modelisnormalized to
the9–∝thmultipolec≤omp≤onentoftheCOBEDMRtwo–year It should be noted that realistic observational uncer-
tainties should be larger than the error bars reported by
data, a9 =8.2, which has been shown to be independent of
Storrie–Lombardi et al. (1995), since they do not include
then value toa good accuracy (G´orski et al. 1994).
any systematic observational bias. Recently, Bartelman &
InFigure1weplottheresultingσ8value.AsfortheΩν Loeb (1995) emphasized the role of the amplification bias,
dependence,resultsareplottedintheleftpanelforn=0.9 due to DLAS gravitational lensing of QSOs, in the DLAS
−
1.4 going from lower to uppercurves with steps of 0.1. In a detection. They pointed out that (a) lensing effects bias
similarfashion,intherightpanelweplotthendependence. upwards the Ω (z) value by an amount depending on the
g
Going from highertolower curves,we plot resultsforΩ = parameters of theFriedmann background,as well as on the
ν
4 S. Borgani, F. Lucchin, S. Matarrese and L. Moscardini
Figure 3.The models inthe Ων – n plane reproducing the observed Ωcoll, taking fg =1(left panel) and fg =0.5 (right panel). The
heavysolidcurveisforδc=1.5andM =1011 M⊙;lightersolidcurvesrefertotheobservationaluncertainties.Dashedanddottedlines
showtheeffect ofvaryingδc andM,respectively(seethetextfortheassumedvalues).
redshift; (b) the observed absorber sample may be biased where
toward larger values of their internal line–of–sight velocity δ ∞ η(R) δ2 dR
dispersions leading toan overestimate of thetotal absorber n(M)dM = c exp c (5)
α√2π Z σ(R) (cid:18)−2σ2(R)(cid:19) R2
mass. Therefore, both effects go in the direction of allevi- R
ating the galaxy formation redshift problem. On the other istheaverageclusternumberdensitywithmassintherange
hand,Fall&Pei(1995)detailedtheconsequencesofdustab- [M,M+dM]. In theaboveexpression, thequantities
sorbtioninDLAS.Theyarguedthatdustobscurationcauses
incompletenessintheopticallyselectedquasarsamples,and, σ2(R) = 1 k2P(k)W2(kR)dk ,
therefore,in theDLASsamplesaswell.Inthiscase, there- 2π2 Z
sultingΩg(z)isbiaseddownwardsbyanamountdepending η(R) = 1 k4P(k)dW2(kR) dk (6)
on themodel for theDLASchemical evolution. 2π2σ2(R)Z d(kR) kR
It is however clear that, even taking the observational
convey the information about the power–spectrum. As be-
results at face value with their small error bars, the rather
fore, we use a Gaussian window for W(kR), so that α =
poor knowledge of the parameters entering in the Press– (2π)3/2 in eq.(5) and the mass M is related to the scale
Schechter prediction for Ω (i.e. δ , M and f ) makes it
g c g R according to eq.(3). Klypin & Rhee (1994) found that
difficult toput stringent constraints on Ω and n.
byseFvoerrailnNst–abnocde,yisfimonuelattaiokness(1e.3.g.∼<Eδfscta∼<tνh1io.5u,&asRseuegsg1es9t8e8d; δΩcν≃=10..53afonrdtGheaiurssMiaDnMfiltcelrus(tseereBNo–rbgoadnyiestimalu.l1a9t9io5n,sf,orwtihthe
dependence of the cluster mass function on δ for different
Klypin et al. 1995) and analytical considerations on the c
dark matter models).
Press–Schechter approach (e.g. Jain & Bertschinger 1994),
1hΩ0aν1n1d≃,Mb0⊙l.u2,eiuannngldetshnsefs=gpei1cstrwsueomnuslitdbolybneb=eall1olo.w2wieundncrifteoyar.seOMsnth∼tehae1ll0oo1tw0hee−dr mmaastseAedssaefxoccrleuesodtbeisnregravbMautino=dnaan8lc.4dea×otfa1,a0b1W4ouhMtit⊙e5×eutsi1na0gl−.7X(1M–9r9pa3yc)−d3easfttoair-.
haroettfarakcetni.onIntoorΩdeνr∼>to0m.4o,ruentliegshstδlyc ≃con1s.7traoirnMthe≃m1o0d12elMs,⊙a Bt7e.i5rvivaenl1oo0ce−itt7yadMl.isp(p1ce9−r93s3io)fnobraansceldudsottbehtreasiirneeaxdncaeaelnydsiainsbguonnthdoeabnasceberovovefedambclouaussst-
betterunderstandingofgalaxyformation throughhydrody- ×
limit.
namical simulationswould beneededtoclarify what DLAS
InFigure4weplottheΩ –nrelationsfordifferentval-
Mactuaanldlyfagr.e.Thiswouldprovidemorereliable valuesfor δc, uQeusitoefNre(m>arMka)b(lsye,efocrapδtio=n)1a.ν5nd(lteaftkipnagnMel)=no8.v4a×lu1e0s14ofMΩ⊙.
Itishoweverclearthatthosemodelswhichfitthedata c ν
and n in the considered ranges give a cluster abundance
at z 4 need also to be tested against present–day observ-
≃ as low as the observational ones. For instance, assuming
ables.Oneofthesetestsisrepresentedbytheabundanceof (Ω ,n)=(0.2,1) it is N(>M) 3.4 10−6 Mpc−3, while
galaxy clusters, which has been shown to represent a pow- N(ν>M) 7.1 10−6Mpc−3for≃(Ω ,n×)=(0.3,1.2).Ingen-
erful constraint for dark matter models (e.g., White et al. ≃ × ν
eral, lowering Ω and/or increasing n, as suggested by the
1993). In the Press & Schechter(1974) approach, the num- ν
DLASanalysis,makesthedisagreementevenworse.Thisre-
ber of density of clusters with mass above M is given by
sultagrees withtheexpectationthatσ8 0.6(Whiteetal.
∞ ≃
N(>M) = n(M′)dM′, (4) 1993)toreproducetheobservedclusterabundance,whilein
Z generallargernormalizationsarerequiredbyourmodels(cf.
M
The Epoch of Structure Formation in Blue Mixed Dark Matter Models 5
Figure 4. The models in the Ων – n plane corresponding to different values for the cluster abundance, N(> M), taking M = 8.4×
1014 M⊙.Leftandrightpanelscorrespondtoassumingδc=1.5andδc=1.7,respectively.Solid,dotted,short–dashed,long–dashedand
dot–dashed curves (partly absent in the left panel) are for logN(>M)=−5.25,−5.5,−5.75,−6.,−6.25, respectively. For comparison,
theobservational resultsbyWhiteetal.(1993) andbyBivianoetal.(1993) givelogN(>M)≃−6.3and−6.1,respectively.
Figure 1). Increasing δ to 1.7 alleviates tosome extent the Shafi 1995). The subsequent variation of the neutrino free–
c
disagreement. However,evenin thiscase, in orderto repro- streaming has been shown to decrease σ8 to an adequate
duce the observational N(> M) one should have Ω > 0.2 level, without significantly affecting results at the galactic
ν
and n< 0.95, with the limiting values (Ω ,n) = (0.2∼,0.9) scale, which is relevant for DLAS.
ν
beingo∼nlymarginallyconsistentwiththeDLASconstraints As a general conclusion we would stress the effective-
corresponding to most optimistic choice of the parameters nessofputtingtogetherdifferentkindsofobservationalcon-
(cf. the upper dotted curve in the left panel of Figure 2). straints torestrict therange of allowed models. Aswe have
Thisgeneralproblemofthemodelsinreproducingtheclus- shown, the effect of blueing the primordial perturbation
terabundancewasalsorecognizedbyPogosyan&Starobin- spectrum goes in the direction of increasing the redshift of
sky (1995a). The point is also discussed by Lucchin et al. structure formation. However, this also increases the r.m.s.
(1995). fluctuation on the cluster mass scale to a dangerous level.
Although systematic observational uncertainties could Decidingwhetherthenarrowingoftheallowedregionofthe
well affect the determination of cluster masses from both Ων –nplanepointstowardstheselection ofthebest model
X–ray and velocity dispersion data, it is not clear whether or towards ruling out the entire class of models requires a
they can justify theorder of magnitude(or even more) dis- clarification of both the observational situation and of how
crepancy between the data and those models which would models haveto be compared to data.
havebeen preferred on theground of DLAS constraints.
A firstpossibility toalleviate thisproblem would beto
increase the baryon fraction Ω . On one hand, this has the
b Acknowledgments
effectofloweringthesmall–scalefluctuationamplitudeand,
therefore, σ8. On the other hand, this fluctuation suppres- The Italian MURST is acknowledged for partial financial
sion should not damage too seriously DLAS predictions on support.SBwishestothankJoelPrimackforusefuldiscus-
Ω , since this effect is partly compensated by a larger de- sions.
g
nominator in eq.(1). For instance, taking δ = 1.5 for the
c
(Ω ,n) = (0.2,0.9) model the cluster abundance changes
ν
from n(>M) 1.4 10−6 Mpc−3 ton(>M) 8.2 10−7
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