An Improved Semi-Analytical Spherical Collapse Model for Non-linear Density Evolution

0.2 MB·English

by  Douglas J. Shaw

Draftversion February1,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/10/03
AN IMPROVED SEMI-ANALYTICAL SPHERICAL COLLAPSE MODEL FOR NON-LINEAR DENSITY
EVOLUTION
Douglas J. Shaw1 and David F. Mota2
1 DAMTP,CentreforMathematical Sciences,UniversityofCambridge,WilberforceRoad,CambridgeCB30WA,UKand
2 InstituteforTheoretical Physics,UniversityofHeidelberg,69120Heidelberg,Germany
Draft versionFebruary 1, 2008
ABSTRACT
Wederiveasemi-analyticalextensionofthesphericalcollapsemodelofstructureformationthattakes
accountoftheeffectsofdeviationsfromsphericalsymmetryandshellcrossingwhichareimportantin
8
0 thenon-linearregime. Ourmodelisdesignedsothatitpredictsarelationbetweenthepeculiarvelocity
0 and density contrast that agrees with the results of N-body simulations in the region where such a
2 comparisoncansensiblybemade. Priortoturnaround,whentheunmodifiedsphericalcollapsemodel
is expect to be a goodapproximation,the predictions ofthe two models coincide almostexactly. The
n
effects of a late time dominating dark energy component are also taken into account. The improved
a
spherical collapse model is a useful tool when one requires a good approximation not just to the
J
evolution of the density contrast but also its trajectory. Moreover, the analytical fitting formulae
7
presented is simple enough to be used anywhere where the standardspherical collapse might be used

but with the advantage that it includes a realistic model of the effects of virialisation.
]
h Subject headings: Cosmology: Theory, miscellaneous. Relativity. Galaxies: general.
p
-
o 1. INTRODUCTION a Taylorseries in 1/δ¯to the standardequation: the idea
r
t The Spherical Collapse Model (hereafter SCM) devel- being that these additional terms would encode all the
s
effects due to shell-crossing and deviations from spheri-
a oped by Gunn & Gott (1972) is perhaps the simplest
[ model for the evolution of non-linear structure, and yet cal symmetry that occur for large δ¯. The coefficients of
the terms in Taylor series were chosen so as to provide
ithas beenshownto be remarkablysuccessfulwhencor-
2 a good approximation to the statistical density-contrast
rectly interpreted. However, despite the SCM’s many
v
found from N-body simulations. Whilst their improved
8 success, it is ultimately flawed since it predicts that any
SCM was found to agree fairly well with data from N-
6 overdensityofmattercollapsestoasingularityinafinite body simulations for δ¯ & 15, their solution for δ¯ is in-
8 time. Additionally, making the assumption of spheri-
accurate in the linear regime, where spherical symmetry
0 cal symmetry, whilst simplifying, means abandoning the
and hence also the unmodified SCM are expected to be
. manyinterestingandimportantaspects ofstructurefor-
8 good approximations.
mation that result from deviations from spherical sym-
0 In this article we take a similar approach to
metry. Indeed, these deviations play a crucial role in ul-
7
Engineer et al. (2000) and add terms to the standard
0 timately halting the collapse of the overdensity and the SCM evolution equation so that for large δ¯ the evolu-
: formation of virialized structures.
v The usual approach to virialisation in the SCM is to tion of the density contrast and the peculiar velocity
Xi put it in by hand; the collapse is simply halted once the agrees with the data from N-body simulations. Impor-
virial radius has been reached. This procedure, within tantly though, the analytical solutions, which we find,
r remainvalidas δ¯→0 and agreealmostexactlywith the
a an Einstein-de Sitter Universe, leads to the result that
bound structures are formed when the non-linear over- unmodified SCM prior to turnaround.
density is about 178 or, equivalently, the linear overden-
2. MODELLINGNON-LINEARSTRUCTURES
sity is approximately 1.68 1. Despite the fact that these
figures are not too far away from what is actually ob- We consider a spherical overdensity embedded in
served in N-body simulations, the ad hoc nature of this a background universe described by a Friedman-
approachmeans the SCM cannot be used to predict the Robertson-Walker (FRW) metric. In a spherically-
precise manner in which the overdensity evolves; more- symmetric system and in the absence of shell-crossing,
over, the SCM’s prediction for the peculiar velocity be- the mass, M, inside each comoving spherical shell, with
comes virtually useless shortly after turnaround. physical radius R(t), remains constant. The mean den-
Engineer et al. (2000) proposed a different and better sity contrast inside shell therefore scales as:
motivated way in which the SCM could be extended to
a3(t)
includevirialisation. Theirideawastoalterthestandard 1+δ¯=λ , (1)
R(t)3
SCM evolution equation for the mean density contrast,
δ¯,insuchawaythatstablestructureswouldform. Their
wherea(t)isthescalefactorofthebackgroundFRWuni-
modified evolution equation was constructed by adding
verse, and λ is constant on each shell. Eq. (1) can also
be used to define R(t) for shells of constantM when de-
1 Indark energy scenarios these values change slightlydepend-
viations from spherical symmetry occur. In these cases,
ingonthemodel(Percival2005;Debnath etal.2006;Baglaet al.
2005;Weinberg&Kamionkowski2003) however, the physical meaning of R(t) is less clear. Fol-
lowing Engineer et al. (2000) we continue to think of
2
R(t), as defined by Eq. (1), as the effective “radius” When deviations from spherical symmetry and the
of each shell of constant M. leading order effects of a gradient in the velocity dis-
The peculiar velocity, h , is defined by: persion are taken into account, Engineer et al. (2000)
SC
showed that one should have:
1dln(1+δ¯) R˙
h ≡ = 1− , (2) d2R GM H2R
SC 3 dlna HR =− − S (8)
! dt2 R2 3
where H = a˙/a is the Hubble parameter. In a matter GM 2
=− 1+ S ,
dominatedUniversetheunmodifiedSCMpredictshSC = R2 3(1+δ¯)
h (δ¯). Our key assumption in what follows is that this (cid:18) (cid:19)
SC
remains the case even when deviations from spherical where S(a,x) = a2(σ2 −2Ω2)+f(a,x); σ2 and Ω2 re-
symmetry are taken into account. This assumption is spectively quantify the shear and rotation of the fluid;
supported by numerical studies of structure formation f(a,x)containsthelowestordercontributionfromveloc-
such as that conducted by Hamilton et al. (1991) which ity dispersion terms. Importantly, no matter what form
we discuss below. S(a,x)takes,if,aswehaveassumedhSC =hSC(δ¯),then
For the moment we take the background universe to we must have S = S(δ¯) and by comparing Eqs. (7) and
bematterdominated,howeverinSection5wegeneralize (8) we can clearly see that:
our results to non-linear overdensities in more realistic 3(1+δ¯) T′′
universes which have recently transitioned to an epoch S(δ¯)= T′−2−1+sinη(1−cosη) .
of dark energy domination. We define a new coordinate 2 T′3
(cid:18) (cid:19)
η(δ¯) by: In principle, the form of both S(δ¯) and hence T(η) can
9 T(η)2
1+δ¯= , (3) befoundusingtheresultsofN-bodysimulations. Unfor-
2(1−cosη)3 tunately, however,making the requiredcomparisonwith
simulationsisnotasstraightforwardasonemightexpect
for some function T(η). Now M is constant on each
it to be. This is because the results of such simulations
shell and so, taking t = t , R = R and δ¯= δ at some
i i i aregivenintermsofthestatisticalpropertiesofthemat-
instance, we have
terdistributionratherthanintermsofthe meandensity
2R3(1+δ ) 2R3(1+δ¯) contrast, δ¯, and peculiar velocity, h . The statistical
M = i i = . SC
9t2 9t2 properties in question are the averaged two point corre-
i lationfunction,ξ¯,andtheaveragedpairvelocity,h(a,x),
ByconsideringthisrelationtogetherwithEq. (3)wesee which are given as defined by:
that:
3 r hv(a,x)i
R (1+δ ) ξ¯= ξ(x,a)x2dx; h(a,x)=− , (9)
R= i i (1−cos(η)) r3 a˙x
2δ j(η,R ) Z0
i i
where ξ is the two-point correlation function and is de-
3t (1+δ )
t= i i T(η). finedtobe theFouriertransformofthe powerspectrum,
4δ3/2j3/2(η,R ) P(k). The assumption that h(a,x) depends on a and x
i i
onlythroughξ¯i.e. h(a,x)=h(ξ¯(a,x))is commoninthe
where j(η,δ ) is some function of η and δ . If h =
h (δ¯) thenij(η,R ) = j(η), and without lioss of gSeCner- literature (see Engineer et al. (2000), Moe et al. (1995))
SC i anditappearstohavebeenconfirmedbynumericalsim-
ality we can set j(η)=1. This gives:
ulations (see Hamilton et al. (1991), Peacock & Dodds
1+δ¯ sinη dτ (1996)).
h =1− , (4) The results of, for example, Hamilton et al. (1991)
SC r 2 (1−cosη)1/2dT can be used to construct the fitting formula for h(ξ¯)
where τ(η)=η−sinη, and: (Engineer et al.2000). However,beforewecanmakeuse
of such a formula, we must relate the statistical quanti-
R=R(η)= Ri(1+δi)(1−cosη), (5) ties ξ¯and h(ξ¯) to δ¯ and hSC(δ¯). It is well known that
2δi (Padmanabhan (1996),Padmanabhan (2002),Peebles
3t (1+δ ) (1980),Padmanabhan & Engineer (1996),Popolo et al.
i i
t=t(η)= T(η),. (6)
4δ3/2 (2001),Padmanabhan & Ray (2006)), on scales smaller
i than the size of the collapsing objects and around high
We reduce to the standard SCM in the limit where T = density peaks:
τ(η)=η−sinη(Padmanabhan2002),anditisclearthat ρ≃ρ (1+ξ). (10)
b
h = h (δ¯) in the unmodified SCM. It is important
SC SC It follows that, in the non-linear regime, we have δ¯≈ ξ¯.
to stress that if hSC =hSC(δ¯) then Eqs. (5) and (6) are This relationship between δ¯ and ξ¯ was also used by
the most general solutions for R and t.
Engineer et al. (2000), although it was, as it is here, the
The radial acceleration of each shell is found to be:
weakestpartofthe whole analysis. We onlyrequirethat
dd2tR2 =−GRM2 T′−2+sinη(1−cosη)TT′′3′ , (7) δm¯a≈tioξ¯nhio.eld. wδ¯h&er1e5i.t Aisseδ¯xp→ect∞edwtoe absesuamgeoothdaatpδ¯pr∼oxξ¯i-.
(cid:18) (cid:19) Peebles (1980) showed that h(ξ¯) satisfies:
where T′ = dT/dτ, and T′′ = d2T/dτ2. As should be
expected, when T′ = 1 we reduce to the standard SCM 1 1 dξ¯
h= , (11)
equation for R,tt. 31+ξ¯dlna
3
thus if ξ¯≈ δ¯, then by comparing Eqs. (2) and (11), we with τ ≈ τ = η − sinη ; η is defined by
sim sim sim sim
see that h ≈ h . It must be stressed that this second Eq.(13).
SC
relationbetweenh andhisonlyvalidifξ¯≈δ¯,however We fit for the parameter A (or equivalently R /R )
SC ta vir
whenever it does hold it implies that h , is given by a by considering some important physical constraints on
SC
function of δ¯alone, i.e. h =h (δ¯). thebehaviourofT(η). Sincetheeffectofdeviationsfrom
SC SC
The assumption that ξ¯≈ δ¯ breaks down for small δ¯. spherical symmetry is to slow down the collapse of the
Fortunately, when δ¯ is small, the unmodified SCM pro- overdensity,it follows from Eq.(7) that:
vides an accurate model. For times t ≫ti, the unmodi- T′′
fied SCM predicts that h =h (δ¯). Our key assump- T′2−sinη(1−cosη) ≥1. (17)
tionthat h =h (δ¯) iSsCtherefSoCre expected to hold for T′
SC SC
almostallδ¯. Theassumptiondoesbreakdownfort∼t , Furthermore, the unaltered SCM should be a good ap-
i
however this is entirely due to the decaying mode in δ¯ proximationupto aroundturnaround(when η =π)and
so we must have that T′ ≈ 1, T′′ ≪ T′ for η . π. As
which is negligible for t≫t .
i ′
τ → η∞−sinη∞, we must have T → ∞ and so T > 0
3. CONSTRUCTINGANIMPROVEDSCM as η → η∞. Eq. (17) implies that, for η > π, we can-
′ ′′
Ifh =h (δ¯)thenallpropertiesofamodifiedSCM not have 0 < T < 1 and T < 0, and so it follows
SC SC ′
that T ≥ 1 for η > π. It follows that T ≥ τ every-
areencodedinasinglefunctionT(η). Ouraimistocom-
where. This condition implies that we should choose
bine the unmodified SCM and data from N-body simu-
R /R , and hence A, so that T is always greater
lations to find a fitting formula for T(η) that results in vir ta sim
accurate predictions for h (δ¯) in all regimes. than τsim ≡ ηsim −sinηsim for ξ¯& 15, which roughly
SC
corresponds to η & 3.8, τ & 4.4. Moreover, since
Prior to turnaround we expect the standard SCM to sim sim
we want T → τ as η → 0, we choose R /R so that,
beaccurateandhenceT(η)≈η−sinη. Furthermore,for vir ta
at the minimum of T , T =τ . This requirement
the unmodified SCM to be accurate,at leading order,in sim sim sim
the linear regime we must have T(η)∼η−sinη+o(η5) gives:
for small η. When δ¯ & 15, we expect hSC(δ¯) ≈ h(ξ¯) Rvir =0.5896.
and ξ¯≈ δ¯, and we may use h(ξ) to extract the large δ¯ Rta
form of T(η). We describe how this is done below. In We now use T to find a fitting formula for T(η) that
sim
a matter dominated universe, the linearly extrapolated agreeswiththeunmodifiedSCMatearlytimes(i.e. T ∼
meantwo-pointcorrelationfunction,ξ¯lin,scalesasξ¯lin ∝ η−sinη for small η).
a2 ∝T4/3, therefore as δ¯→∞:
4. RESULTS
9 T(η)2
(1+ξ¯)≈(1+δ¯)= , (12) We find that the fitting formula:
2(1−cos(η))3
3.468(τ −τ)−1/2exp −15(τf−τ)
and so f τ
T(η(τ))=τ + ,
Aξ¯1/2 (1+0.8(τf −τ)1/2−(cid:16)0.4(τf −τ))(cid:17)
(1−cosη)≈(1−cosη )≡ lin , (13) (18)
sim (1+ξ¯)1/3
whereτ =5.516,providesanexcellentfittotheformof
f
where A is a constant and we treat it as a parameter T(η(τ)) derived from the simulations of Hamilton et al.
to be fitted. Hamilton et al. (1991) found the following (1991), i.e. T ≈ Tsim, in the range δ & 15. It also
fitting formula for ξ¯ (ξ¯): provides an evolution of δ¯that matches up smoothly to
lin
the one predicted by the standard SCM in the region
1+0.0158ξ¯2+0.000115ξ¯3 1/3 whereweexpectittoprovideanaccurateapproximation,
ξ¯ =ξ¯ . (14)
lin 1+0.926ξ¯2−0.0743ξ¯3+0.0156ξ¯4 i.e. prior to turnaround δ .5.
(cid:18) (cid:19) In FIG 1 we plot T against τ , and our fitting
We define η∞ = limδ¯→∞η. By taking ξ¯→ ∞ we find formula for T(η(τ)) agsaiimnstτ. We sesiemthat, in the range
that: δ¯&15,τ &4.4whereweexpectξ¯≈δ¯andhenceT ≈
sim
2R T, the fit is indeed very good. The parameters in the
vir
A=2.2668(1−cosη∞)=2.2668(cid:18) Rta (cid:19), (15) ftohremleualadifnogr Tor(dηe(rτ)t)erhmavseinbetehnecahsoysmenptsootitcheaxtpaasnτsio→nsτfof,
wherewehaveusedcosη∞ =1−2Rvir/Rtawhichfollows both T(η(τ)) and Tsim(ηsim(τ)) match.
fromEq.(5);Rvir istheradiusoftheshellatvirialisation, In FIG 2 we use Eqs. (3) and (4) to plot h (δ¯),
SC
and R is its radius at turnaround. If spherical symme-
ta and the fitting formula found by Hamilton et al. (1991)
try is assumed then the virial theorem in an Einstein to plot h(ξ¯). We also plot the unmodified SCM pre-
de Sitter Universe gives R = R /2. This relation
vir ta diction for h (δ¯). It is clear from this plot that our
is generally used when virialisation is placed by hand SC
fitting formula forT givesanh (δ¯)that is anexcellent
into the SCM. For comparsion, Hamilton et al. (1991) SC
approximation to h(ξ¯) (provided ξ¯ ≈ δ¯) in the region
found that R /R ≈ 1.8 from their simulations. We
treat Rvir/Rttaaas avirfitting parameter. Eq. (13) provides δ¯& 20. As δ¯,ξ¯→ ∞, the curves hSC(δ¯) and h(ξ¯) have
η = η(ξ¯ ≈ δ¯), and T(η) may now be found using Eq. the same leading order asymptotic behaviour. In the re-
gion δ¯ & 15, our model gives an h (δ¯) that is always
(12): SC
within 3% of the fitting formula for h(ξ¯) derived from
1/2
2 simulations(Hamilton et al.1991). Additionally,ourfit-
T(η(ξ¯))≈Tsim ≡ 9 A3/2ξ¯l3i/n4(ξ¯), (16) ting formula for hSC(δ¯) agrees almost exactly with the
(cid:18) (cid:19)
4
10 104
Simulation Data
9
Improved SCM
8
103
7
6
τT/ 5 δ1+ 102
4
3 Improved SCM
101
Standard SCM with virialisation
2
Standard SCM w/o virialisation
1 R = 1.8 R
ta vir
0 100
3 3.5 4 4.5 5 5.5 0 1 2 3 4 5
τ δ
lin
Fig. 1.— Plot of T/τ versus τ. The solid red line is T/τ = Fig. 3.— Plot of how the non-linear mean density contrast, δ¯,
Tsim/τsim againstτ =τsim,whereasthedashedblacklineshows dependsonthelinearone,δ¯lin. Thesolidredlineistheimproved
T(τ)/τ inourimprovedSCMandusesthefittingformulaforT(τ), SCM developed here, the thick dashed green line is the standard
Eq. (18), to evaluate T/τ. As required, we see that inthe region SCM withvirialisationput inby hand and the thindashed black
τ &4.4⇒δ¯&15,ourimprovedSCMisaverygoodapproximation lineisthestandardSCMwithoutvirialisation. Thedottedblueline
tothesimulationdata. Forsmallvaluesofτ,weseethatT/τ →1, shows the asymptotic behaviour of δ¯(δ¯lin) that we would expect
asisrequired,inourimprovedSCM. if Rta/Rvir = 1.8 as seen in the simulations of Hamiltonetal.
(1991).
h (δ) − Improved SCM ally used in the spherical collapse model, despite the
SC
2 h(ξ) − Simulations fact that we did not constrained it to be so. For com-
h (δ) − Standard SCM parison, the improved spherical collapse developed by
SC
Engineer et al.(2000)gaveR /R ≈0.65,andN-body
vir ta
simulationshavebeen foundtosupportR /R ≈0.56
vir ta
1.5
(Hamilton et al. 1991). Our value of R /R is there-
vir ta
foreabetterapproximationtothevaluefoundfromsim-
h ulationsthanthatgenerallyusedintheunmodifiedSCM
1 andthatfoundinthemodeldevelopedbyEngineer et al.
(2000).
In our improved SCM the linear density contrast is
given by:
0.5
2/3 2/3
3 t 3 3
δ = δ = T(η)2/3.
lin i
5 t 5 4
0 (cid:18) i(cid:19) (cid:18) (cid:19)
100 101 102 103 104 When δ =1.6865,which correspondsto the instant of
1+δ, 1+ξ lin
collapseintheunmodifiedSCM,wefindδ¯=54.65rather
Fig. 2.— Relationship between h and δ¯ in different models. than the SCM’s value of 178. We find that δ¯=200 cor-
The solid red line shows hSC(δ¯) in our improved spherical col- respondstoδ =2.286. Wecomparetheformofδ¯(δ¯ )
lapse model. The dotted black line is hSC(δ¯) in the standard found in ourlminodel to that predicted by the unmodifiliend
SCM, and the dashed blue line shows h(ξ¯) as seen in simulations
SCM model, with virialisation put in by hand, in FIG.
(Hamiltonetal. 1991). As required, our improved SCM gives an
hSC thatagreeswithh(ξ¯)verywellintheregionwhereweexpect 3. We also show the divergent behaviour of δ¯(δ¯lin) in
δ¯≈ξ¯i.e. δ¯&15. WealsonotethathSC(δ¯),intheimprovedSCM, the standardSCMwithout virialisation,andthe asymp-
has the same asymptotic behaviour, to leading order, as h(ξ¯≈δ¯) totic behaviour of δ¯(δ¯lin) if Rta = 1.8Rvir as suggested
in the limit δ¯→ ∞. Importantly, our improved SCM model also by simulations (Hamilton et al. 1991). The two major
agrees with the standard SCM in the region where it is expected advantages of our improved SCM over the unmodified
to provide a very good approximation i.e. prior to the epoch of
turnaround,δ¯.5.6. version are clearly visible in this plot:
• At late times the improved SCM provides a bet-
terapproximationtothebehaviourseeninN-body
simulations than the unmodified SCM does.
predictions of the unmodified SCM prior to turnaround • δ¯(δ¯ )issmoothintheimprovedSCM.Thisisnot
lin
δ¯<5.6. We found a best fit value of R /R =0.5896, the case inthe unmodified SCM when virialisation
vir ta
which is fairly close to the value of 0.5 that is gener- is put in by hand. The improved SCM therefore
5
providesnotonlyagoodapproximationtotheevo- lution of overdensities of matter in a realistic universe
lution of density contrast, δ¯, but also to that of provided that δ¯ & 100 today. For smaller values of δ¯
dδ¯/dt and hence to that of the peculiar velocity. our results are only accurate for Ω ≈ 1. However for
m
δ¯.15theeffectsofdeviationsfromsphericalsymmetry,
5. INCLUDINGDARKENERGY which have been our primary concern in this article, are
We have constructed an improved, semi-analytical smallenoughto be ignoredandthe resultsderivedusing
SCM whose prediction for the dependence of peculiar the unmodified SCM can be used with confidence. No-
velocity on the mean density contrast concurs with that tice also that such fitting formulae and assumptions are
extrapolated from simulations in the non-linear regime. strictly speaking only applicable to dark energy models
However, as mentioned above, we have done this for a that are not coupled to matter. If such a matter cou-
matter dominated universe. It is well known, however, pling is allowed then the whole process of structure for-
that the universe today is not matter dominated and a mation, both in the linear and non-linear regimes, may
significant fraction of its total content is believed to be change (see e.g. Brookfield et al. (2006); Lahav et al.
in the form of dark energy. Fortunately though, the ef- (1991); Mota & van de Bruck (2004); Maor & Lahav
fects of this dark energy on the backgrounduniverse are (2005); Nunes & Mota (2006); Mota & Shaw (2006,
generallybelievedtoonlyhavebecomenon-negligiblerel- 2007); Koivisto & Mota (2007a,b)). This said, in many
atively recently z . 1.8. This means the local evolution dark energy models, the matter coupling is constrained
of overdensities with δ¯ & 100 today has been matter- byexperimentstobeverysmall(relativetothecoupling
dominated,to a verygoodapproximation,rightup until between matter and gravity), and as a result any alter-
the epoch of matter-radiation equality. Therefore the ationsto the processofstructureformationaresimilarly
evolution of the density, ρ, and radial velocity, v, of a small.
shellofmatter thatis wellinto the non-linearregime to-
day is, to a good approximation, the same in a matter 6. CONCLUSIONS
dominated background as it is in a background where
InthisarticlewehaveextendedtheSphericalCollapse
today Ω ≈0.27 and Ω ≈0.73 for which the equation
m de
Model so that it takes account of the effects of devia-
of state parameter for the dark energy, w, satisfies the
tions from spherical symmetry and shell crossing which
current astronomical bound of w = −1±0.1 for z < 1
areimportantinthenon-linearregime. Thekeyassump-
(Riess et al. 2006). It is therefore a fairly straightfor-
ward task to generalize our formulae for δ¯ and h to tionsthatweusedwhenconstructingourmodelwasthat
SC h =h (δ¯) and that for δ¯&15, δ¯≈ξ¯. The latter as-
included background universes where Ω ≈ 0.27 pro- SC SC
m sumptionisprobablytheweakestlink inthe wholeanal-
vided that these are only applied to overdensities that
ysis, although both assumptions are found commonly
are large today. We then find that:
in the literature (see Engineer et al. (2000); Moe et al.
9f(a)T(η)2 (1995); Peebles (1980)). Our improved SCM predicts a
1+δ¯(Ωm,t)= 2(1−cosη)3, (19) form for hSC(δ¯) that is consistent with the results of N-
body simulations in the regime where a comparison can
1+δ¯ dτ sensiblybemade(δ¯&15)andwiththeunmodifiedSCM
hSC(Ωm,t)=1−Ω0m.5s2(1−cosη)sinηdT (20) prior to turnaround. Analytical formulae for δ¯and hSC
intheimprovedmodelhavebeenpresented,andtheyes-
where τ = η−sinη as before, T(η) is still given by Eq. sentially differ from the comparable formulae in the un-
(18) and modified SCM only by the replacement of τ = η−sinη
4 with T(η), which is given by Eq. (18). The improved
f(a)= .
9t2Ω H2 SCM is therefore simple enough to be used anywhere
m
where the unmodified SCM might be used but with the
The equations for R(η) and t(η) remain unchanged.
advantagethatitincludes a realisticmodelofthe effects
If dark energy has behavedsimilarly to a cosmological
of virialisation.
constant in the recent past (z <1.8) then:
1−Ω
f(a)≈ m ≈Ω−0.4.
Ω (sinh−1( (1−Ω )/Ω ))2 m DJS acknowledges support from PPARC. DFM ac-
m m m
knowledges support from the A. Humboldt Foundation
Eqs. (19) and (20) compbined with the fitting formula and the Research Council of Norway through project
Eq. (18) provide a very good approximation to the evo- number 159637/V30.
REFERENCES
BaglaJ.,PrasadJ.andRayS.,Mon.Not.Roy.Astron.Soc.360, KoivistoT.andMotaD.F.,(2007b) Phys.Rev.D75,023518;
194(2005) LahavO., LiljeP.,PrimackJ.andReesM.,Mon.Not.Roy.Astron.
Brookfield A.W. et al. , Phys. Rev. Lett. 96, 061301 Soc.251,128(1991).
(2006)[arXiv:astro-ph/0503349]. Moe,H.,Jain,B.,White,S.M.,1995,MNRAS,276,L25
DebnathU.,NathS.,andChakrabortyS.,Mon.Not.Roy.Astron. MaorI.andLahavO.,JCAP0507,003(2005)
Soc.369,1961(2006) MotaD.F.andvandeBruckC.,Astron.Astrophys.421,71(2004)
EngineerS.,KanekarN.&PadmanabhanT.,2000,MNRAS,314, MotaD.F.andShawD.J.,Phys.Rev.Lett. 97,151102 (2006);
279 MotaD.F.andShawD.J.,Phys.Rev.D75,063501(2007).
GunnJ.E.&GottJ.R.,1972, ApJ,176,1 Nunes N. and Mota D.F., Mon. Not. Roy. Astron. Soc. 368, 751
Hamilton, A. J. S., Kumar, P., Lu E., Mathews, A., 1991, ApJ., (2006)
374,L1 Padmanabhan, T.,1996, MNRAS,278,L29
KoivistoT.andMotaD.F.,(2007a) Phys.Lett. B644,104; PadmanabhanT.andEngineerS.,Astrophys.J.493,509(1998)
6
Padmanabhan T. and Ray S., Mon. Not. Roy. Astron. Soc. Lett. PercivalW.,Astron.Astrophys.443,819(2005)
372,L53(2006) DelPopoloA.,ErcanE.andXiaZ.,Astron.J.122,487(2001)
Padmanabhan, T., 2002, Theoretical astrophysics. III - Galaxies WeinbergN.and Kamionkowski M.,Mon.Not. Roy. Astron.Soc.
andcosmology,CambridgeUniversityPress 341,251(2003)
Peacock, J.A.,Dodds,S.J.,1996,MNRAS,280,L19 RiessA.G.etal.,toappear inAstrophys.J.(2006).
Peebles, P. J. E., 1980, The large scale structure of the Universe,
PrincetonUniversityPress