Formulae for Growth Factors In Expanding Universes Containing Matter and a Cosmological Constant

7 Pages·0.24 MB·English

by  A. J. S. Hamilton

Accepted for publication in Monthly Notices
Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2008 (MNLATEXstylefilev1.4)
Formulae for Growth Factors In Expanding Universes
Containing Matter and a Cosmological Constant
A. J. S. Hamilton
JILA and Dept. Astrophysical & Planetary Sciences, Box 440, U. Colorado, Boulder CO 80309, USA;
[email protected]; http://casa.colorado.edu/∼ajsh/
1
0 1February2008
0
2
ABSTRACT
n
a Formulae are presented for the linear growth factor D/a and its logarithmic deriva-
J tivedlnD/dlnainexpandingFriedmann-Robertson-WalkerUniverseswitharbitrary

5 matterandvacuumdensities.Theformulaepermitrapidandstablenumericalevalua-
2 tion. A fortran programis available at http://casa.colorado.edu/∼ajsh/growl/.

3 Key words: cosmology:theory – large-scale structure of Universe
v
9
8
0 1 INTRODUCTION
6
Thelineargrowthfactorg D/a,whereD istheamplitudeofthegrowingmodeandaisthecosmicscalefactor,determines
0 ≡
0 thenormalization of the amplitude of fluctuations in Large Scale Structure(LSS) relative to those in theCosmic Microwave
0 Background (CMB) (Eisenstein, Hu & Tegmark 1999 Appendix B.2.2). Its logarithmic derivative, the dimensionless linear
/ growth rate f dlnD/dlna, determines theamplitude of peculiar velocity flows and redshift distortions (Peebles 1980 14;
h ≡ §
p Willick 2000; Hamilton 1998). As such, the growth factor g and growth rate f are of basic importance in connecting theory
- and observations of LSS and theCMB.
o
r In a Friedmann-Robertson-Walker (FRW) Universe containing only matter and vacuum energy, with densities Ωm and
⋆
st ΩΛ relative to thecritical density,thelinear growth factor is given by (Heath 1977; Peebles 1980 §10)
a 1
v: g(Ωm,ΩΛ)≡ Da = 5Ω2m a3Hda(a)3 , (1)
i Z0
X
where a is the cosmic scale factor normalized to unity at the epoch of interest, H(a) is the Hubbleparameter normalized to
r unityat a=1,
a
H(a)= Ωma−3+Ωka−2+ΩΛ 1/2 , (2)
and the(cid:0)curvaturedensity Ωk is(cid:1)definedto be thedensity deficit
Ωk 1 Ωm ΩΛ , (3)
≡ − −
which is respectively positive, zero, and negative in open, flat, and closed Universes. The normalization factor of 5Ωm/2 in
equation (1) ensures that g 1 as a 0. It follows from equation (1) that thedimensionless linear growth rate f is related
→ →
to thegrowth factor g by
f(Ωm,ΩΛ) dlnD = 1 Ωm +ΩΛ+ 5Ωm . (4)
≡ dlna − − 2 2g
Thefactthattheintegrandontherighthandsideofequation(1)isarationalfunctionofthesquarerootofacubicimplies
thattheintegralcanbewritten analytically in termsofelliptic functions.However,theanalyticexpressionsarecomplicated,
⋆ Regardedasafunctionofcosmicscalefactora,thegrowthfactorevolves as
5ΩmH(a) a da′ 5Ωm(a)a2H(a)3 a da′
g(a)= =
2 a a′3H(a′)3 2 a′3H(a′)3
Z0 Z0
whereΩm(a)=Ωma−3/H(a)2.
(cid:13)c 0000RAS
2 A. J. S. Hamilton
and have yet to be implemented in any published code. Explicit analytic expressions in the special case of a flat Universe,
Ωk =0, havebeen given in terms of theincomplete Beta function byBildhauer, Buchert & Kasai (1992) and in termsof the
hypergeometric function by Matsubara (1995). Analytic expressions for the luminosity distance in the general non-flat case
are given in terms of elliptic functions by Kantowski, Kao & Thomas (2000).
Lahav et al. (1991) give a simple and widely used approximation to thegrowth rate†:
f(Ωm,ΩΛ)≈Ωm4/7+(1+Ωm/2)ΩΛ/70 . (5)
From this, together with relation (4), follows the approximate expression for the growth factor quoted by Carroll, Press and
Turner(1992):
g(Ωm,ΩΛ) 5Ωm . (6)
≈ 2 Ω4m/7 ΩΛ+(1+Ωm/2)(1+ΩΛ/70)
−
h i
GiventheincreasingprecisionofmeasurementsoffluctuationsintheCMB(deBernardisetal.2000;Hananyetal.2000)
andLSS(Gunn&Weinberg1995;Yorketal.2000;Colless2000) andthegrowingevidencefavouringacosmological constant
(Gunn&Tinsley 1975;Perlmutteretal.1999;Riessetal.1998; Kirshner1999),itseemstimelytopresentexactexpressions,
suitable for numerical evaluation, for the growth factor g (hence f, through formula [4]) valid for arbitrary values of the
cosmological densities Ωm in matter and ΩΛ in vacuum.
Theprocedurepresentedinthispaperistoexpandtheintegralofequation(1)asaconvergentseriesofincompleteBeta
functions, conventionally definedby
x
B(x,a,b) ta−1(1 t)b−1dt . (7)
≡ −
Z0
Ineffect, themethod can beregarded asgeneralizing Bildhauer et al.’s (1992) formula. What makestheschemeattractive is
that the Beta functions in successive terms of the series can be evaluated recursively from each other through the recursion
relations
aB(x,a,b)=xa(1 x)b+(a+b)B(x,a+1,b) (8)
−
bB(x,a,b)= xa(1 x)b+(a+b)B(x,a,b+1) (9)
− −
aB(x,a,b+1)=xa(1 x)b+bB(x,a+1,b) (10)
−
the last of which follows from the other two. In practice, each evaluation of the growth factor g involves from one to seven
calls to an incomplete Beta function, followed byelementary recursive operations.
The resulting numerical algorithm is fast (provided that Ωm and ΩΛ are not huge – see 3) and stable, and has been
| | §
implementedinafortranpackagegrowλavailableathttp://casa.colorado.edu/ ajsh/growl/. Thegrowλpackageincludes
∼
an updated version of an incomplete Beta function originally written by theauthor a decadeago.
2 FORMULAE
Figure1showscontourplotsofthegrowthfactorg(Ωm,ΩΛ)andgrowthratef(Ωm,ΩΛ)computedfromtheformulaepresented
below,asimplementedinthecodegrowλ.TheresultshavebeencheckedagainstnumericalintegrationswiththeMathematica
program.
Perhapsthemost strikingaspect oftheseplots, emphasized byLahavet al. (1991), isthat thegrowth ratef is sensitive
mainly toΩm, with only a weak dependenceon ΩΛ.
Figure 2 shows the ratio of the Lahav et al. (1991) growth rate f, equation (5), to the true growth rate. The Figure
illustrates that the Lahav et al. approximation works well except at small Ωm. The approximation (with the 4/7 exponent
advocatedbyLightman&Schechter1990)worksparticularlywellforcurrentlyfavouredcosmologies,beingaccuratetobetter
than 1% for flat Universeswith Ωm =0.2–3.9.
2.1 Limiting cases
The requirement that the Hubble parameter H(a), equation (2), be the square root of a positive quantity for all a from 0
(Big Bang) to 1(now) imposes two requirements. The first is that thematter density be positive
† Lahavetal.adoptΩ0.6 ratherthanΩ4/7,followingPeebles(1980).The0.6exponentismoreaccurateforlowΩm,but4/7(Lightman
& Schechter 1990) works better elsewhere, and in particular is more accurate for currently favoured cosmologies, flat Universes with
Ωm≥0.2.The4/7exponentisthereforecurrentlyfavoured.
(cid:13)c 0000RAS,MNRAS000,000–000
Formulae for Growth Factors In Lambda Universes 3
0 1 2 3 4 0 1 2
3.0 3.0
2.5 ound orever 2.5 ound orever
Ωm12..05 willturnarwillexpandf deceleraatcicneglerating Ωm12..05 willturnarwillexpandf deceleraatcicneglerating
1.0 1.0
..05 opcelnosed in accessible ..05 opcelnosed in accessible
−1.0 −.5 .0 .5 1.0 1.5 2.0 −1.0 −.5 .0 .5 1.0 1.5 2.0
Ω Ω
Λ Λ
Figure1.Contourplotsof(left)thegrowthfactorg(Ωm,ΩΛ),and(right)thegrowthratef(Ωm,ΩΛ),inexpandingUniversescontaining
matter and cosmological constant with densities Ωm and ΩΛ relative to the critical density. Universes below the 45◦ dashed line are
geometrically open, while those above are closed. Universes to the upper left of the long dashed line are decelerating, while those to
lowerrightareaccelerating. UniversestotheleftofthealmostverticallinenearΩΛ≈0willeventuallyturnaround,whileUniversesto
the right will expand forever. The region at the bottom right corner is physically inaccessible to Universes that expand from zero and
thatcontainonlymatterandcosmologicalconstant.Theboundarybetweenturnaroundandeternalexpansion,andtheboundaryofthe
inaccessibleregion,together form,intheclosedcase,theapproachingandrecedingpartsoftheloiterline.StartingatΩm=1,ΩΛ=0,
aUniversecanevolveupwardandrightwardalongtheapproachingpartoftheloiterlinetoEinstein’sloiterpointatΩm=2ΩΛ→∞.
After an indefinite period of hanging around, the Universe can then either recollapse along the same loiter line, or else continue into
renewedexpansiondownwardandleftwardalongtherecedingpartoftheloiterline,theboundaryoftheinaccessibleregion.Thegrowth
factor g tends to infinity at the boundary of the inaccessible region, but only contours up to g = 5 are marked. The contour plot of
g(Ωm,ΩΛ)issimilartoFigure7ofCarrolletal.(1992).
.9 1.0 1.1
3.0
2.5 ound orever
Ωm12..05 willturnarwillexpandf deceleraatcicneglerating
1.0
..05 opcelnosed in accessible
−1.0 −.5 .0 .5 1.0 1.5 2.0
Ω
Λ
Figure 2.ContourplotoftheratiooftheLahavetal.(1991)growthratef,equation(5),tothetruegrowthrate.Theplotillustrates
that the Lahav et al. approximation works well except for small Ωm. In particular, the approximation is accurate to better than 1%
for currently favoured cosmologies, flat Universes (45◦ dashed line) with Ωm ≥ 0.2. For small Ωm (and any ΩΛ) a somewhat better
approximationresultsiftheLightman&Schechter(1990)exponent4/7inequation(5)isreplacedwith0.6,asinLahavetal.’soriginal
paper. The white region in the diagram at Ωm <∼ 0.1 and ΩΛ < 1 is where the error in the Lahav et al. approximation (with the 4/7
exponent) exceeds 10%;contoursinthisregionareomittedtoavoidconfusion.
(cid:13)c 0000RAS,MNRAS000,000–000
4 A. J. S. Hamilton
Ωm 0 , (11)
≥
and thesecond can be interpreted as a condition that the Universenot be too closed
Ωk min 3Ωm, 27Ω2mΩΛ 1/3 . (12)
≥ − 2 − 4
" (cid:18) (cid:19) #
If either of the two conditions (11) or (12) were violated, then the Hubble parameter H(a) would go to zero at some finite
cosmic scale factor a, indicating that the Universe did not expand from a = 0, but turned around from a collapsing to an
expandingphase at some finitea.
The limiting values of thegrowth factor g and growth rate f as the matter density goes zero, Ωm 0, are
→
g(Ωm =0,ΩΛ)=0 , f(Ωm =0,ΩΛ)=0 (13)
providedalso thatΩΛ <1,in accordance withthecondition Ωk >0from equation(12). Physically,structurecannot grow in
a Universecontaining only vacuum.
The second condition, equation (12), is saturated when Ωk → −(27Ω2mΩΛ/4)1/3 with Ωk < −3Ωm/2. This marks the
boundary of the inaccessible region to the bottom right of Figure 1. The limiting values of the growth factor g and growth
rate f in this case are
g(Ωm,ΩΛ) , f(Ωm,ΩΛ)= 1 Ωm +ΩΛ . (14)
→∞ − − 2
Physically,thegrowthfactortendstoinfinitybecausesuchUniversesareinrenewedexpansionafterhavingspentanindefinite
period of time at Einstein’s unstable loitering point, at Ωm =2ΩΛ .
→∞
2.2 Case Ωk/(1 Ωk) 1
| − |≤
For small curvaturedensity Ωk, expand theintegral in equation (1) as a power series in Ωk:
g(Ωm,ΩΛ)= 5Ω2m Z01 a3(Ωma−d3a+ΩΛ)3/2 Xn∞=0(−)nn(3!/2)n (cid:18)ΩmΩa−ka3−+2ΩΛ(cid:19)n (15)
where (x) x(x+1)...(x+n 1) is a Pochhammer symbol.
n
≡ −
Each term of the sum on the right hand side of equation (15) integrates to an incomplete Beta function. The cases of
positive and negative cosmological constant must be distinguished. For positive cosmological constant, ΩΛ >0,
g(Ωm,ΩΛ)= 56ΩΩm1Λ5//36 Xn∞=0 (−)nn(3!/2)n (cid:18)Ω2m/Ω3Ωk1Λ/3(cid:19)nB(cid:16)ΩmΩ+ΛΩΛ,56 + n3,23 + 23n(cid:17) , (16)
while for negative cosmological constant, ΩΛ <0,
g(Ωm,ΩΛ)= 65|ΩΩΛ1m|/53/6 Xn∞=0(−)nn(3!/2)n (cid:18)Ω2m/3Ω|ΩkΛ|1/3(cid:19)nB(cid:18)|ΩΩmΛ|,56 + n3,−12 −n(cid:19) . (17)
Ineachofformulae(16)and(17),incompleteBetafunctionsmustbeevaluatedforthreeterms,andthentheremainingBeta
functions can be evaluated recursively from these three, through the recursion relations (8)–(10). In equation (16) (ΩΛ >0),
the recursion is stable from n=0 upward, or from large n downward, according to whether the argument ΩΛ/(Ωm+ΩΛ) is
greater than or less than‡ 1/3. In equation (17) (ΩΛ <0), therecursion is stablefrom n=0 upward or large n downward as
theargument ΩΛ /Ωm is greater or less than 1+(3/2)[(√2 1)1/3 (√2 1)−1/3]=0.106.
| | − − −
Howfastdotheexpansions(16)and(17)converge?Convergenceisdeterminedessentiallybytheconvergenceoftheparent
expression(15)attheplacewheretheexpansionvariableΩka−2/(Ωma−3+ΩΛ)attainsitslargestabsolutemagnitudeoverthe
integration range a [0,1]. For ΩΛ >0, the expansion variable attains its largest magnitude at a=min 1,[Ωm/(2ΩΛ)]1/3 ,
while for ΩΛ <0, th∈eexpansion variableis always largest at a=1. Physically,theextremumat Ωma−3={2ΩΛ occurs wher}e
the Universe transitions from decelerating to accelerating. It follows that if Ωm 2ΩΛ (decelerating), then successive terms
≥
of the expansions (16) and (17) decrease by Ωk/(Ωm+ΩΛ), whereas if Ωm 2ΩΛ (accelerating), then successive terms
≈ ≤
‡
A somewhat lengthy calculation, confirmed by numerical experiment, shows that the asymptotic (i.e.after manyiterations) point of
neutralstabilityoftherecurrenceB(x,a,b)→B(x,a+m,a+n),wheremandnarepositiveornegativeintegers,occursatthatunique
pointx∈[0,1]satisfying
mmnn
xm(1−x)n=± .
(m+n)m+n
(cid:13)c 0000RAS,MNRAS000,000–000
Formulae for Growth Factors In Lambda Universes 5
idfeΩcrmea≥se2bΩyΛ≈, o2r2/a3Ωpkre/c(i3siΩo2mn/3oΩf1Λ≈/3[)2.2T/3hΩuks/n(3tΩer2mm/3sΩoΛ1f/3th)]eneixfpΩamns≤ion2sΩ(Λ16.)and(17)willyieldaprecisionof≈[Ωk/(Ωm+ΩΛ)]n
2.3 Case ΩΛ/(1 ΩΛ) 1
| − |≤
For small cosmological constant ΩΛ, expand theintegral in equation (1) as a power series in ΩΛ:
g(Ωm,ΩΛ)= 5Ω2m Z01 a3(Ωma−3d+aΩka−2)3/2 Xn∞=0(−)nn(3!/2)n (cid:16)Ωma−3Ω+ΛΩka−2(cid:17)n . (18)
Again,eachtermofthesumontherighthandsideofequation(18)integratestoanincompleteBetafunction.Thecases
of positive and negative curvaturedensity Ωk must be distinguished. For an open Universe,Ωk >0,
g(Ωm,ΩΛ)= 25ΩΩk52m/2 Xn∞=0 (−)nn(3!/2)n (cid:18)Ω2mΩΩ3kΛ(cid:19)nB(cid:16)ΩmΩ+kΩk,25 +3n,−1−2n(cid:17) , (19)
while for a closed Universe, Ωk <0,
g(Ωm,ΩΛ)= 25ΩΩk2m5/2 ∞ (−)nn(3!/2)n ΩΩ2mkΩ3Λ nB |ΩΩmk|,52 +3n,−12 −n . (20)
| | n=0 (cid:18) | | (cid:19) (cid:18) (cid:19)
X
HeretheBetafunctionsinsuccessivetermscanbecomputedrecursivelyfromasingleBetafunction.Inequation(19)(Ωk>0),
the recursion is stable from n=0 upward or large n downward as the argument Ωk/(Ωm+Ωk) is greater than or less than
3/4. In equation (20) (Ωk <0), the recursion is stable from n=0 upward or large n downward as the argument Ωk /Ωm is
greater than or less than (3/2)[(√2+1)1/3 (√2+1)−1/3]=0.894. | |
−
The convergence of the expansions (19) and (20) is determined by the convergence of the parent expansion (18) at the
place where the expansion variable ΩΛ/(Ωma−3+Ωka−2) attains its largest magnitude over the integration range a [0,1],
∈
which occurs at a = 1 for both positive and negative Ωk. It follows that successive terms of the expansions (19) and (20)
decrease by ΩΛ/(Ωm+Ωk), and n terms will yield a precision of [ΩΛ/(Ωm+Ωk)]n.
≈ ≈
2.4 Case Ωm/(1 Ωm) 1
| − |≤
Forsmall matterdensity Ωm,a powerseries expansion of theintegral in equation(1) isagain possible, buttheintegral must
besplit into two parts to ensureconvergence of theintegrand over thefull range a [0,1] of theintegration variable. Define
tsriehmdeuiclcaoerslmytoitchΩsamctaala−eetq3fma=cattoΩtrekraa-e−evqqa2cauitfumΩmakt‘t>eeqru0-ca,ulairtnvyda’ttuΩormebae‘e−eqqw3uh=aelritΩeyΛ|’ΩtimofabΩ−eeqΛ3w/>(hΩe0rk.eaA−e|qΩ2gm+oaoΩ−edq3Λ/p)(|rΩo=kcea|d∈Ω−equ2Λr+/e(ΩiΩsmΛt)ao|−eq=u3s+e|ΩoΩknake−eaq−e2oq/f2()tΩ|h.meTamh−eeqe3ste+hcoΩodnΛsd)o|i,tfiaotnhndes
previous two subsections, 2.2 or 2.3, to integrate up to the cosmic scale factor aeq at matter-curvature or matter-vacuum
§ §
‘equality’, whichever is later (larger aeq) since the later epoch yields a more convergent Ωm series, and then to complete the
integral using the small Ωm expansion:
g(Ωm,ΩΛ)=G(aeq)+ 5Ω2m Zae1q a3(Ωka−d2a+ΩΛ)3/2 n∞=0(−)nn(3!/2)n (cid:18)ΩkΩa−m2a+−3ΩΛ(cid:19)n . (21)
X
Thefirsttermontherighthandsideofequation(21)istheintegraluptoaeq,evaluatedbyoneofthemethodsoftheprevious
two subsections:
5Ωm aeq da aeq
G(aeq)≡ 2 Z0 a3(Ωma−3+Ωka−2+ΩΛ)3/2 = H(aeq)g[Ωm(aeq),ΩΛ(aeq)] (22)
with Ωm(a)=Ωma−3/H(a)2, Ωk(a)=Ωka−2/H(a)2, ΩΛ(a)=ΩΛ/H(a)2, and H(a)given byequation (2).
As in the previous two subsections, each term of the sum in the second term on the right hand side of equation (21)
integrates to an incomplete Beta function. The cases of positive and negative curvature and vacuum densities must be
distinguished. For an open Universewith a positive cosmological constant, Ωk >0 and ΩΛ >0,
g(Ωm,ΩΛ)=G(aeq)+ 4Ω5kΩΩm1Λ/2 n∞=0(−)nn(3!/2)n (cid:18)ΩΩm3kΩ/1Λ2/2(cid:19)n(cid:20)B(cid:18)ΩkaΩ−k2a−+2ΩΛ,1+ 32n,12 − n2(cid:19)(cid:21)1a=aeq , (23)
X
while for an open Universewith a negative cosmological constant, Ωk >0 and ΩΛ <0,
g(Ωm,ΩΛ)=G(aeq)+ 4Ωk5|ΩΩmΛ|1/2 Xn∞=0(−)nn(3!/2)n (cid:18)ΩmΩ|Ω3k/Λ2|1/2(cid:19)n(cid:20)B(cid:18)Ω|kΩaΛ−|2,21 − n2,−12 −n(cid:19)(cid:21)1a=aeq . (24)
(cid:13)c 0000RAS,MNRAS000,000–000
6 A. J. S. Hamilton
The expression for a closed Universe with positive cosmological constant, Ωk < 0 and ΩΛ >0, turns out never to be useful,
but for reference it is
g(Ωm,ΩΛ)=G(aeq)+ 4|Ω5kΩ|Ωm1Λ/2 Xn∞=0 (−)nn(3!/2)n (cid:18)Ω|ΩmkΩ|31Λ//22(cid:19)n(cid:20)B(cid:18)|ΩkΩ|Λa−2,1+ 32n,−12 −n(cid:19)(cid:21)1a=aeq . (25)
The fourth option, a closed Universe with negative cosmological constant, Ωk < 0 and ΩΛ < 0, does not yield a convergent
expansion in Ωm. The small Ωm expressions (23)–(25) are more complicated than the small Ωk and small ΩΛ expressions
obtained in the previous two subsections, since equations (23)–(25) each involve a sum of not one but three expansions, one
to evaluate the first term on the right hand side, one to evaluate the second term at a = aeq, and a third to evaluate the
second term at a=1.
It will now beargued that a small Ωm expansion is advantageous only in thecase where matter-vacuum equality occurs
after matter-curvatureequality.Inparticular, thishas theconsequencethat theexpansion (25) is neveruseful. Supposethat
matter-curvatureequality,|Ωma−eq3/(Ωka−eq2+ΩΛ)|=|Ωka−eq2/(Ωma−eq3+ΩΛ)|, occurs after matter-vacuumequality.Then the
first term in equation (21), G(aeq), would be evaluated by the small Ωk method, equation (16) or (17). However, if matter-
curvatureequalityoccursaftermatter-vacuumequality,thenitisalsotruethatmatter-curvatureequalityoccursafter(larger
cosmic scale factor a) the transition, at Ωma−3 =2ΩΛ, from a decelerating to an accelerating Universe. As discussed in the
last paragraph of 2.2, the convergence of the small Ωk expansions (16) and (17) are then determined by the convergence
§
of the parent equation (15) at the deceleration-acceleration transition, not by its convergence at a later epoch. Thus, if
matter-curvature equality occurs after matter-vacuum equality, then there is no point in splitting the integral at aeq as in
equation (21); one might as well use the small Ωk expressions (16) or (17) all the way to a= 1, since they converge just as
fast at a = 1 as at a = aeq. In fact the small Ωk expressions (16) and (17) are computationally faster than the small Ωm
expansions (23)–(25), because the former involvea single sum whereas thelatter involvethree.
TheconclusionfromthepreviousparagraphisthatasmallΩm expansionisadvantageousonlyinthecasewherematter-
vacuum equality occurs after matter-curvature equality. Examination of evolution in the Ωm–ΩΛ plane reveals that matter-
vacuumequalityhappensaftermatter-curvatureequalityonlyifΩk >0(conversely,matter-curvatureequalityhappensafter
matter-vacuum equality only if ΩΛ >0). Thus, in the cases where the small Ωm expansion is useful, the relevant expressions
to useare (23) or (24) with thefirst term, G(aeq),being evaluated bythe small ΩΛ expansion (19) with Ωk >0.
TheBetafunctionsinsuccessivetermsoftheexpansionsinthesecondtermontherighthandsidesofequations(23)–(25)
can be computed recursively from two Beta functions. In equation (23) (Ωk > 0 and ΩΛ > 0), the recursion is stable from
n = 0 upward or large n downward as the argument Ωka−2/(Ωka−2+ΩΛ) (with a = aeq or 1) is greater than or less than
(3/2)[(√2+1)1/3 (√2+1)−1/3]=0.894,thesameasforequation(20).Inequation(24)(Ωk >0andΩΛ <0),therecursion
−
isstable from n=0upwardin all cases. In equation(25) (Ωk<0andΩΛ >0),therecursion isstable from n=0upward or
large n downward as theargument Ωk a−2/ΩΛ is greater than or less than 3/4, the same as for equation (19).
| |
Theconvergence of theexpansions (23)–(25) is determinedby theconvergenceof theparent expansion (21) at theplace
wheretheexpansionvariableΩma−3/(Ωka−2+ΩΛ)attainsitslargestmagnitudeovertheintegrationrangea [aeq,1],which
oanccdunrstaetrmas=waiellqyiniealdllacapsreesc.iIstiofnolloofw≈st[Ωhamtas−euq3c/c(eΩsskivae−eq2te+rmΩsΛi)n]nt.heexpansions(23)–(25)decreaseby≈Ωma−eq3∈/(Ωka−eq2+ΩΛ),
2.5 Which formula to use?
Thevariousformulae(16),(17),(19),(20),and(23)–(25)convergeoveroverlappingrangesofΩm andΩΛ.Asensiblestrategy
would beto choose theexpression that converges most rapidly.
If Ωk 0 (closed Universe),then use thesmall Ωk ( 2.2) or small ΩΛ ( 2.3) methods as Ωk/(1 Ωk) or ΩΛ/(1 ΩΛ)
≤ § § | − | | − |
is smallest.
If Ωk >0 (open Universe), then use the small Ωk ( 2.2), small ΩΛ ( 2.3), or small Ωm ( 2.4) methods as Ωk/(1 Ωk),
§ § § | − |
ΩΛ/(1 ΩΛ),or Ωm/(1 Ωm) issmallest.If Ωm/(1 Ωm) issmallest,thendeterminewhichoccurslater(largeraeq),matter-
|c|ΩuΛrv/a(tΩu−mrea−eeqq3u|+alΩitky|a|−eΩq2m)|a.−e−Iqf3/th(Ωek|foar−emq2e+r,ΩthΛe)n| =re|v|Ωerktate−oq2−/u(sΩinmga|t−ehq3e+smΩaΛll)|Ω,kormmetahtotedr,-v§a2c.2u.uImftehqeulaaltitteyr,|Ωthmean−equ3s/e(Ωoknae−eoq2f+thΩeΛsm)|a=ll
Ωm expressions (23) or (24) with the first term, G(aeq), equation (22), evaluated bythe small ΩΛ expression (19).
3 COLLAPSING UNIVERSE
TheprocedureproposedinthispaperfailsforUniversesthatarecollapsing ratherthanexpanding.AsaUniverseapproaches
turnaround, the Hubble parameter, and consequently the critical density, approaches zero, causing one or more of Ωm, Ωk,
and ΩΛ to tend to . In such cases theexpansions presented herein converge evermore slowly.
±∞
(cid:13)c 0000RAS,MNRAS000,000–000
Formulae for Growth Factors In Lambda Universes 7
It is not clear how to adapt the method to deal with Universes near turnaround and thereafter, or indeed whether this
is possible.
4 SUMMARY
Formulae suitable for numerical evaluation have been presented for the linear growth factor g(Ωm,ΩΛ) D/a and its
≡
logarithmic derivative f(Ωm,ΩΛ) dlnD/dlna in expanding FRW Universes with arbitrary matter and vacuum densities
≡
ΩmandΩΛ.Afortranpackagegrowλimplementingtheseformulaeisavailableathttp://casa.colorado.edu/ ajsh/growl/.
∼
ACKNOWLEDGEMENTS
The author thanks Daniel Eisenstein for helpful information and references, Gil Holder and Ioav Waga for pointing out an
erroneous, now eliminated, section in the original manuscript, Matthew Graham for finding an important bug, and Paul
Schechterfor urging themerits of 4/7. This work was supported by NASAATPgrant NAG5-7128.
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