Phillip Helbig
February 1996
Note: Formoreinformation, seethearticle `Ageneral and
practi-cal methodfor calculatingcosmological distances'by RainerKayser,
Phillip Helbig and Thomas Schramm in Astronomy and Astrophysics,
1996.
a. General description
The ANGSIZ routine calculates the angular size distance b etween two objects
as a function of their redshifts. (In world models which have no big bang,
but contract from in nity to a nite size b efore expanding, there are two or
four indep endent distances. In these cases, the rst distance is returned by
ANGSIZ and the rest by the routine BNGSIZ .) The distance also dep ends on
thecosmological parameters
0 ,
0
and (z). Thesearenotpassedto ANGSIZ
but rather to anauxiliary routine,INICOS ,whichcalculates allz-indep endent
informationfora givenworldmodel.
The result is in units of cH 01
0
. Multiply the result by 3 210 3
(actually
2 : 99792458210 3
)togetthedistanceinMp cforH
0 =100km1s 01 1Mp c 01 . One cande nehtob eH 0 inunitsofH 0 =100km1s 01 Mp c 01 . Thus,m ultiplyingby 3210 9
corresponds toanhof1. Forothervaluesofh,simplydivide theresult
byh,forexampleby0.5forH
0
=50km1s 01
Mp c 01
. (Beawarethathcanb e
and sometimes is de ned as 1 for other values of H
0
, usually 50. Sometimes
thisisindicatedbyasubscript, i.e. h
50 ,h
100
andso on.)
Thecallingsequenceisdescribedbelowin section b.
Allroutines areinabsolutelystandard FORTRAN77andhaveb een testedon
a num b erofdi erentcom binationsofcompiler/hardw are/OSsoessentiallythe
same numericalresultsshouldb eobtained everywhere.
i. INICOS
The routine INICOS takes the cosmological parameters and `user wishes' as
inputandcalculatesquantitiesneededbyANGSIZ |whicharerelayedinternally
to ANGSIZ |and also returns these to the calling routine. In addition, as in
ANGSIZitself, anerrormessage isreturned. INICOScanb e usedindep endently
ofANGSIZ ofcourse,butnottheotherwayaround.
INICOSreturnstheINTEGERvariableWMTYPE ,worldmodeltyp e,whichgives
aqualitativ eclassi cationofthecosmologicalmodel. Thetableshowsthe
corre-sp ondenceb etweenthevalueofWMTYPEandsome standardclassi cationsfrom
theliterature,andalsogivessomebriefinformationonthetemporalandspatial
prop erties ofthecorresponding worldmodel. The gureshows thelocationof
thevariousworldmodeltyp esin the
0
-0 -plane.
max 0 0 0 0 1 no yes no no <0 01 0 0 >0 O (0) 1(iii) 2 no yes no no <0 01 >0 >0 >0 O (1) 1(iii) 3 no yes no no <0 0 >0 >0 > 1 2 O (2) 1(ii) 4 no no no no <0 +1 >0 >0 > 1 2 O (3) 3(iv) 5 no no no no MTW 0 +1 >1 >0 > 1 2 O (4) 3(iv) 6 no no no no >0 +1 > 0 ; c > 0 ; c > 1 2 O (5) 3(iii)(a)
7 yes yes no no Milne 0 01 0 0 0 M
1 (1) 1(ii) 8 yes yes no no `LCDM' 0 01 >0 >0 0<q 0 < 1 2 M 1 (2) 1(ii)
9 (yes) yes no no Einstein-deSitter 0 0 1
1 2 1 2 M 1 (3) 2(ii) 10 yes yes no no >0 01 0 0 01<q 0 <0 M 1 (0) 1(i) 11 yes yes no no >0 01 >0 >0 01<q 0 < 1 2 M 1 (4) 1(i)
12 yes yes no no deSitter 1 0 0 0 -1 S 2(i)
13 yes yes no no `3CDM' >0 0 >0 >0 01<q 0 < 1 2 M 1 (5) 2(i)
14 yes no no no Lema^tre >0 +1 note note note M
1 (6) 3(1) 15 (yes) no no no >0 +1 0 ; c 0 ; c > 1 2 A 1 3(ii)(b)
16 (yes) no (yes) (no) Einstein >0 +1
0 ; c
=1
0 ; c
=1 1 E 3(ii)(a)
17 yes no yes no Eddington >0 +1
0 ; c 0 ; c <01 A 2 3(ii)(c)
18 yes no yes yes b ounce >0 +1 0<
0 < 0 ; c 0< 0 < 0 ; c <01 M 2 (1) 3(iii)(b)
19 yes no yes yes Lanczos >0 +1 0 0 <01 M
2
(0) 3(iii)(b)
Table 1. Classi cation of cosmological models as done by INICOS . Further information is provided in the text, particularly
concerningparen theticalexpressions and when `note' appears insteadof the expressionin question.
l
W
-4
-3
-2
-1
0
1
2
3
4
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
1
2
3
4
9
5
6
12
19
17
18
14
8
7
15
10
11
13
Figure1. Locationofvariouscosmologicalmodelsin the
0
-0
-plane.
A given worldmo del is eitheron aline orcurve segment(b ounded by at
leastonevertex), onavertexorinthespacebetweenthevariouslinesand
curves. The diagonal linecorresp onds to k=0,the verticalline to 0 =0,
the curve immediatelyto the right of this line divides mo dels whic hwill
eventually collapse(to the left) tothose whic hwill not(to the right)and
the curveatlowerrightdividesworldmo dels withabig bang(tothe left)
fromthosewithnobig bang(to the right).
WMTYPE referstotheclassi cation returnedbyINICOSas thevalueofthe
vari-ableWMTYPE .Thefollo wingcosmologicalmodelsarede nitively ruledout
byobservations: 1,7,10,12,16,17,18,19. 16isruledoutb ecause ithas
no expansion; the rest which aren't ruled out by the fact that they are
empty areruled outbythefact that a z
max
which isat leastas large as
the largestobserved redshiftimpliesa valuefor
0
which isso low as to
b e de nitivelyruledout.
t=1 answersthequestion whethertheuniverse willexpandforever or, more
precisely, is theopp osite of theanswer to thequestion `will the universe
collapse to R =0 in the future'. The subtletyarises b ecause of thefact
theymigh t`exist forever'. In theEinstein-de SittermodelWMTYPE 9 ,the
expansion rate asym totically approaches 0; the valueof the scale factor
R at t = 1 is, however, R = 1 . Thus, the universe expands forever,
although at t=1thevalueof _
R is0. TheEddington universe,WMTYPE
15 , expandsforever, as well, but at t =1 not only isthe valueof _
R 0
but R reachesa nite maxim umas well. TheEinstein universe, WMTYPE
16 , isstatic. It existsforever, but do esn'treally expandforever, sinceit
do esn't expandat all. R is xedata nite valueand _
R is0atalltimes.
R=1 answersthequestion whetherthe3-dimensionalspaceisin nitein
ex-tent. Itis in nitefor k=01 ork=0 and nitefork=+1(assuminga
simpletop ology).
9z
max If a z
max
exists, then there was no big bang. (In the de Sitter model
the big bang o ccurs at t = 01 .) Rather, the universe expands from
a minimum value of R . The static Einstein universe (WMTYPE 16 ) has
a redshift of 0 for allobjects, thus distance cannot b e determined from
redshiftinthis case.
bounce Iftherehasb eenab ounce,theuniverseisexpandingaftercontracting
from R =1 to R=R
min
. This means thatthere arefour indep endent
distancesforagivenredshift,unlessthedep endenceof onzisthesame
during thecollapsingandexpandingphases,inwhichcasethereareonly
two indep endent distances. (In theEinstein static universe,WMTYPE 16,
there areanin nitenum b erof distancesfortheonlyp ossible redshiftof
0.)
name referstoacommonlyusedtermforthegivenworldmodel. MTWrefers
to the fact that this world modelis discussed in Gravitation by Misner
et al. [ 1973]in x27.10and Box27.4as anexampleofthe typ eof
cosmo-logical model preferredby Einstein ( =0 b ecause Einstein preferred a
`simpler'universewithoutthecosmological constantafteritb ecameclear
thata)hisstaticmodelcannotdescribetherealuniverse,notevenasa0 th
approximation and b) non-static solutions with a cosmological constant
exist; k=+1 in orderto havenoproblemswith b oundary conditionsat
1 ). Thisworldmodelisnotedashavingb een`investigatedbyEinstein'in
Stab ell &Refsdal[ 1966]; thename`Einsteinuniverse',althoughp erhaps
more appropriate for this cosmological model, is historically irrevo cably
asso ciated with the static universe (WMTYPE 16 ). Milne's cosmological
model is not really our WMTYPE 7 , although it is equivalent; Milne had
G=0in hiskinematicrelativity,andeitherG=0 or=0hasthee ect
ofmakingtheexpansionindep endentofgravitation. `LCDM'referstothe
factthatthesecosmologicalparametersareusedinthetypical` l owdensity
standardcosmologicalmodel';hereCDMisanabbreviationfor` colddark
m atter',whichincludesdetailsonscalessmallenoughwherehomogeneit y
cannot b e used as an approximation and do esn't concern us here.
Nat-urally, global prop erties of cosmological modelsare indep endent of such
localstructure. Ofcourse, theparametersare thesame forLHDM(h ot)
and LHCDM (or LMDM: m ixed). `3CDM' refers to the fact that these
cosmologicalparameters areusedinthetypical`lo wdensity atstandard
cosmological model with a cosmological constant'. Note: our
`Edding-ton model' (WMTYPE 17) is referred to as the Lema^tre model in Stab ell
& Refsdal [ 1966] and as theEddington-Lema^tre model in Bondi[ 1961].
Harrison [ 1981] also usesour termEddington modelfor thiscase.
modelforthiscase. 0 therangeof 0 . k thesignof k. 0 the range of 0 . 0 ; c = 2 0 ; c . In WMTYPE 14, 0 , 0 and q 0 can take on
allp ossible values,but notallcom binationsare p ossible. Sp eci cally,for
01 < q 0 < 1 2 , 0 ; c ( 0 ; c
) has no meaning, and the range 0 <
0 ; c is
allo wed. Otherwise, we m ust have
0 > 0 ; c ( 0 > 0 ; c ). (Of course, 0 =2 0 .) 0 therangeof 0 ;=0 : 52 0 . 0 ; c is de nedas 0 ; c = 1 6 ( q 0 +1 ) q 0 +16 s (q 0 +1 ) 1 0 0 1 3 ! (SREq:12) In WMTYPE 14, 0 , 0 and q 0
can takeon all p ossible values,but notall
com binationsare p ossible. Sp eci cally, for01<q
0 < 1 2 , 0 ; c ( 0 ; c )has
no meaning,andtherange0<
0 ; c
is allo wed. Otherwise,wem usthave
0 > 0 ; c ( 0 > 0 ; c ). q 0 the sign of q 0 ; q 0 = 0 0 0 . In WMTYPE 14, 0 , 0 and q 0 can take on
allp ossible values,but notallcom binationsare p ossible. Sp eci cally,for
01 < q 0 < 1 2 , 0 ; c ( 0 ; c
) has no meaning, and the range 0 <
0 ; c is
allo wed. Otherwise,wem usthave
0 > 0 ; c ( 0 > 0 ; c ).
SR referstothedescriptionofcosmologicalmodelsinStab ell&Refsdal[ 1966,
p. 383]wheremodelsare classi edwitha schemesimilartothat usedby
INICOS .
HB Refers to the classi cation in chapter IX of Bondi [ 1961]. Since Bondi
implicitly assumes
0
>0,thisis notentirely correct inthecaseswhere
there is a qualitativ edi erence for
0
= 0, as in WMTYPE s 7, 10 and 12.
Inthiscase,q
0
isnever p ositive,so thequalitativ ecurvesgivenbyBondi
shouldb eextrap olatedbacktoR=0withoutthenegativecurvaturepart
at theb eginning.
ii. ANGSIZ
Thisroutinecalculatestheangularsizedistanceb etweentworedshifts(justone
ofthep ossibledistances intheb ouncemodels). This isdonebythenumerical
integrationofasecondorderdi erentialequation. (FordetailsseeKayser,
Hel-big&Schramm[ 1996].) Asfarasweknow,noanalyticsolutionforthegeneral
caseexists. AnadvantageofusingANGSIZevenwhenananalyticsolutionfora
sp ecial caseexistsis thatANGSIZ always givesa validresult, i.e. ,theequation
holdsforall cosmologicalmodelsand nocasesm ustb e distinguished. (ANGSIZ
is notrecommended for use in the Eddington model, WMTYPE 17. If the user
sp eci cally wantsto usetheroutines forthiscosmological model,itisp ossible
to allo wthis bydeclaringtheCOMMONblockARTHUR inthecallingroutine,
con-tainingalogicalvariable,whichshouldb eset to.TRUE.b eforecallingANGSIZ .
Thevariablegivesthefractionofhomogeneouslydistributedmatterinthe
cos-mological model. Onlargescales,homogeneit yisassumed,in accordance with
theRobertson-Walkermetric,theCosmologicalPrinciple, andso on. However,
onsmallerscales,mattercanofcourseb eclumped. Thefraction10ofmatter
isintheseclumps,whichmeansthattheyareoutside(andsucientlyfarfrom)
theconeoflightraysb etweensourceandobserver. (Ifaclumpofmatterisnear
orintheb eamitself,onem usttakeaccountofthisexplicitlyasa gravitational
lense ect.)
Itisimportanttoremem b erthatthevalueof|fora xedrealsituation|
dep endsontheangularscaleinvolved.Forexample,ahaloofcompactMACHO
typ eobjectsarounda galaxy ina distantclusterwould b ecountedamongthe
homogeneouslydistributedmatterifone wereconcernedwith theangular size
distancetobackgroundgalaxiesfurtheraway,butwouldb econsideredclumped
onscalessuchasthoseimportantwhenconsideringmicrolensingbythecompact
objects themselves. Thus, the clumps m ust have a scale comparable to the
separation oflightraysfromanextragalacticobjectorlarger.
Sincewedon'tknowexactlyhowdarkmatterisdistributed,di erentvalues
can b eexamined togetanideas as tohowthis uncertaintya ects whatever it
isoneis interestedin.
Ofcourse,canb eafunctionofz. Iftheuserwishes tob econstantforall
z,then thevariable VARETAshould b e setto .FALSE.when callingINICOS .In
thiscase,thevalueofETApassedtoINICOSdetermines. IfVARETAist.TRUE. ,
thenthevalueof ETAisirrelevantandisgivenbytheREAL FUNCTION ETA(Z) ,
whichtheusercanmodifyas needed. IfaconstantvalueofETAisdesired,this
canofcourseb eachievedthroughde ningtheFUNCTION ETA(Z)appropriately,
but this willtypicallymak ethe computingtime2{3 timeslongerthan setting
VARETAto .FALSE.andusing ETAtodeterminethevalueof.
iv. BNGSIZ
ThisSUBROUTINEBNGSIZisessentiallythesameasANGSIZexceptthatit
calcu-latestheotherthreedistancesinb ouncemodels. Aslong asETAZdo esn'tmak e
thevalueofdep endonwhethertheuniverseisexpandingorcontracting,there
is only one additional indep endent distance, D14 ; D34is the same as D12 and
D32isthesameisD14 . D12isthedistancereturnedbyANGSIZ ,whereb oththe
startingp ointand theend p ointoftheintegrationarein theexpansion phase.
D14hasitsstartingp ointZ1intheexpansionphaseanditsendp ointZ2inthe
contraction phase,thusthe angular size distance is foundby integratingfrom
Z1to ZMAXandbackto Z2 . D34hasb oth b oundariesin thecontractionphase,
and D32 its starting p oint in the contractionphase and its end p oint in the
expansionphase.
Ifoneisnotinterestedintheb ouncemodels,thisroutineisn'tneeded. Ifone
is only interestedin the `primary' distance in b ounce models,this is returned
byANGSIZ ,soalso inthiscaseBNGSIZisn'tneeded.
BNGSIZm ustb ecalled after callingANGSIZ :therearenoinputparameters;
know
Inthedescriptionsof theinputvariablesin theparameterliststotheroutines
calledbytheuserwealsoindicatearangeofsuggestedvaluesinsquarebrackets.
Ofcourse,othersarep ossible,buttheoneswesuggestcorrespondto
cosmolog-icalmodelswhicharenot rmlyruledoutbyotherarguments(andsomewhich
are) and have b een tested. Physically meaningless values should result in an
errormessage. Wedidn'tthinkitmeaningfultocheck theroutines for
correct-nessfor values outsidethese ranges. For output variables, therangeindicates
p ossible values. The mathematical and physical meaning of the variablescan
b efoundin Kayseretal.[ 1996].
i. INICOS
Theinterfaceto INICOSis
SUBROUTINE INICOS(LAMBDA,OMEGA,ETA,VARETA,DEBUG, C ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ C in C $ WMTYPE,MAXZ,ZMAX,BOUNCE,ERROR) C ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ C out
Theinputvariablesare:
REAL LAMBDA isthenormalisedcosmological constant. [ 010, :::,+10]
REAL OMEGA isthedensityparameter. [0, :::,10]
REAL ETA isthehomogeneit yparameter. [0, :::,1]
LOGICAL VARETA if.TRUE. meansthat isnotgivenbyETAbut bythevalue
returnedbytheREAL FUNCTION ETAZ .
LOGICAL DEBUG if.TRUE.meansthaterrormessagesshouldb ewrittento
stan-dardoutput.
Theoutputvariablesare:
INTEGER WMTYPE givessomeinformationab outthecosmologicalmodel(seethe
table). [1, :::,19]
LOGICAL MAXZ if.TRUE.indicatesthatamaxim umredshiftexists;onlyin this
caseisthevariableZMAXcorrectlyde nedandonlyinthiscaseshouldthe
variableZMAXb eused.
REAL ZMAX givesthemaxim um redshiftinthe cosmologicalmodel, ifLOGICAL
MAXZ is.TRUE. ;otherwise,themaxim umredshiftisin nite,andfor
con-veniencesetto zero.
LOGICAL BOUNCE if .TRUE. indicates that we have a `bounce' model (see the
table); onlyinthiscasecantheotherdistancesb e computedbyBNGSIZ .
INTEGER ERROR indicatesifanerrorhaso ccurredandwhatthismeans(see
Theinterfaceto ANGSIZis:
SUBROUTINE ANGSIZ(Z1,Z2, D12,ERROR)
C ^^^^^ ^^^^^^^^^
C in out
Theinputvariablesare:
REAL Z1 isa redshift;setZ1to 0 forthecommoncaseofthe`distance froma
normalobserver'. [0, :::,5]
REAL Z2 is a redshift; set Z2 to the redshift of the objectif one is interested
in the angular size distance from a `normal observer' to the object and
correspondingly Z1hasb eenset to0. [0, :::,5]
Theoutputvariablesare:
REAL D12 Is theangular size distance b etween Z1 and Z2 . In b ounce models
(and ofcoursein bigbangmodels)this corresponds tothedistance such
that the universe has always b een in a state of expansion b etween the
timesof lightemissionandreception.
INTEGER ERROR indicatesifanerrorhaso ccurredandwhatthismeans(see
sec-tionvi.). [0, :::,13]
ANGSIZisnotrecommendedforWMTYPE17,sincedistancescanb ecome
arbitrar-ily large, and thenecessary overhead would complicatethe routines to such a
degreethat `normal'p erformancewouldb e signi cantlyhampered. Iftheuser
sp eci cally wantsto usetheroutines forthiscosmological model,itisp ossible
toallo wthisbydeclaringtheCOMMONblockARTHURwithalogicalvariable,which
should b e set to .TRUE. in the calling routine. This will override the default
b ehaviour.
iii. BNGSIZ
Theinterfaceto BNGSIZis:
SUBROUTINE BNGSIZ(D14,D34,D32,ERROR)
C ^^^^^^^^^^^^^^^^^
C out
REAL D14 istheangularsize distanceb etweenZ1and Z2 . Thiscorresponds to
the distancesuch that theuniverse isnot in a stateofcontraction at Z1
and not in stateofexpansion atZ2 .
REAL D34 istheangularsize distanceb etweenZ1and Z2 . Thiscorresponds to
thedistancesuchthattheuniverseisinastateofexpansionneitheratZ1
nor atZ2 .
REAL D32 istheangularsize distanceb etweenZ1and Z2 . Thiscorresponds to
the distance such that theuniverse is not in a state of expansion at Z1
and not in astateofcontraction at Z2 .
INTEGER ERROR indicatesifanerrorhaso ccurredandwhatthismeans(see
Theinterfaceto ETAZis:
REAL FUNCTION ETAZ(Z)
which,b eing a functionwithno sidee ects, hasnooutput variables. Theone
inputvariableis:
REAL Z this is the redshift; is a function of z and is given by this function
andnot bythevalueofETAsuppliedtoINICOSifVARETAis.TRUE. ;only
in thiscasewillETAZb eused.
TheCOMMONblockWHICHcontainstheINTEGERvariableCHOICE .If thisCOMMON
blockisalsointhecallingroutine,thenadecisioncanb emadethereastowhich
dep endencyof onz istob e used;seetheuseofCHOICEin ourexampleETAZ .
This is initialised in our BLOCK DATA COSANG and so should notb e initialised
bytheuser.
For b ounce models, it isof course p ossible that the dep endence of on z
dep ends on whether the universe is expanding or contracting. Since VALUE2
is set to.TRUE. during thephaseof contraction and otherwiseto .FALSE.by
ANGSIZandBNGSIZ ,whichcanaccessVALUE2throughtheCOMMONblockCNTRCT ,
thisvariablecanb eusedinETAZordertohavetwodi erentdep endencies,asin
ourexample. Additionally,inthiscase, asinourexample,the valueof CHOICE
must benegative andotherwise mustbe positive. This distinctionisneededby
BNGSIZ inorder to avoidcalculatingD34andD32ifthese arenotindep endent
oftheotherdistances.
v. Calling sequence
Thus, one should call INICOS anew for each cosmological model tested,
indi-cating the cosmological model, whether or not is to b e a constant or is to
b e calculatedby ETAZ andwhether or not errormessages are tob e displayed.
(ERRORisofcoursealwaysappropriatelyset.) Ifonewantstocalculatedistances,
thenANGSIZshouldb ecalledwithtworedshifts,andtheangularsizedistanceis
returned. OneshouldusetheoutputvariablesofINICOStodeterminewhether
or nottocall ANGSIZ .Ifdesired, ETAZ canb e used; it canalso b emodi edby
the user. The calling routinecan alsomak euseof theCOMMON block WHICH to
enabledi erentz dep endenciesof to b ecalculatedwithouthavingtochange
theco de,recompile andrelink foreach case. BNGSIZcanb e calledaftercalling
ANGSIZifone isinterestedintheadditionaldistancesinthecaseoftheb ounce
models.
vi. Error messages
There are fourteen p ossible `error states', namely the values [0, :::,13] of the
INTEGERvariableERROR .Errors1{3cano ccurinINICOS ,4{12inANGSIZ ,12{13
inBNGSIZ .(Thus,onlyoneerrorvariableisneededinthecallingroutine.) ERROR
isalways appropriatelyset; thevariable DEBUGmerelycontrols whether ornot
messages are written to standard output. Withthe exception of ERROR .EQ.
12 ,themessageisjusttheinformation containedin thefollo wingoverview.
0 Noerrorwasdetected.
whichcasethevalueofETAis irrelevant.) ETAZ ,however,checksto seeif
ETAiswithintheallo wedrange.
3 The world model is so close to the A2 curve that the calculation of ZMAX
is numericallyunstable. Thisusually happ ens|whenitdo es|very near
the de Sitter model. However, it should b e rare in practice, as noted
b elow. Even if one were able to calculate a goo d value for ZMAX , this
wouldprobablyleadto anerrorin ANGSIZifZ1orZ2isnearZMAX .
4 TheworldmodelisWMTYPE17. Inthisworldmodel,theuniverseisin nitely
old, and there is a nite ZMAX .This meansthat distances (especially for
=0) can b ecome arbitrarilylarge. The routinesare notrecommended
forcalculationinthiscosmological model,sinceenablingthedetectionof
over owwould unnecessarilycomplicatetheroutines. Also,thepractical
value of ZMAX , used internally, would have to b e appreciable less than
the real ZMAX . If the usersp eci cally wants to use the routines for this
cosmological model, it is p ossible to allo w this by declaring the COMMON
blockARTHURwithalogicalvariable,whichshouldb esetto.TRUE.inthe
calling routine. Thiswilloverridethedefaultb ehaviour.
5 (Z1 .LT. 0.0)
6 (Z2 .LT. 0.0)
7 (Z1 .GT. ZMAX)
8 The valueof ZMAX returned byINICOS is thecalculated value; however,
in-ternally a somewhat smaller ZMAX is sometimes necessary for numerical
stability. ThisinternalvalueiscloseenoughtotherealZMAXforall
prac-tical purp oses (at worst it has an error of ab out 10 03
). This message
meansthatZ1isto olargefornumericalstability,b eingverynearthereal
ZMAX butslightlysmaller.
9 (Z2 .GT. ZMAX)
10 Thevalueof ZMAXreturnedbyINICOSisthecalculatedvalue;however,
in-ternally a somewhat smaller ZMAX is sometimes necessary for numerical
stability. ThisinternalvalueiscloseenoughtotherealZMAXforall
prac-tical purp oses (at worst it has an error of ab out 10 03
). This message
meansthatZ2isto olargefornumericalstability,b eingverynearthereal
ZMAX butslightlysmaller.
11 Over ow error is p ossible. Calculation can b e continued but the result
migh tnot b e as exact as usual or subsequent calculation with thesame
cosmological parametersmigh tleadtoan`unhandledexception'.
12 Very rare. This means an error haso ccurred deep down in the numerical
integration. IfDEBUGis.TRUE.thenamessagewillb eprintedidentifying
where.
13 ThismeansthatBNGSIZwas calledfora non-b ounceworldmodel.
When an error is returned, with the exception of 11, then none of the other
tines
Here isa shortdescriptionofalloftheSUBROUTINEsandFUNCTIONs:
BLOCK DATA COSANG initialisesCOMMONblockvariables.
SUBROUTINE INICOS isdescribedab ove.
REAL FUNCTION QQ calculatesQ 2
(z).
SUBROUTINE ANGSIZ isdescribedab ove.
SUBROUTINE BNGSIZ isdescribedab ove.
SUBROUTINE LOWLEV Performsthelow-levelintegration,mainlycallingODEINT .
SUBROUTINE ASDRHS calculatestherighthandsidesofthedi erentialequation
forthea ngulars izedistance
SUBROUTINE ODEINT isa`driver'routineforthenumericalintegration.
SUBROUTINE MMM isfor themodi edmidpointmethodused byODEINT .
SUBROUTINE POLEX p erforms p olynomialextrap olationusedbyODEINT .
SUBROUTINE BSSTEP p erformsone Bulirsch-Sto erintegrationstep.
REAL FUNCTION ETAZ isdescribedab ove.
i. Call tree [COSANG] [INICOS] | [QQ] [ANGSIZ] | [LOWLEV] | | [ODEINT] | | | [ASDRHS] | | | | [ETAZ] | | | [BSSTEP] | | | | [MMM] | | | | | [ASDRHS] | | | | | | [ETAZ] | | | | [POLEX] | [QQ] [BNGSIZ] | [LOWLEV] | | [ODEINT] | | | [ASDRHS] | | | | [ETAZ] | | | [BSSTEP] | | | | [MMM] | | | | | [ASDRHS] | | | | | | [ETAZ] | | | | [POLEX] | [QQ]
Thefollo wingCOMMONblocknames areusedand, sincetheseare globalnames,
should notb eused inacon ictingwayinanyotherroutines: COSMOL ,ANGINI ,
WHICH ,BNCSIZ ,LOWSIZ ,CNTRCT ,ARTHUR ,MERRORandPATH .Withtheexception
of the optional WHICH , all of these are used internally and need not further
concern the user. The same is true of the SUBROUTINE and FUNCTION names
listedab ove(exceptforINICOS ,ANGSIZ ,BNGSIZ andETAZ ,ofcourse.)
Therestofthissectionshouldb eofabsolutelynoconcerntoalmostallusers,
since `incorrect' resultsdue to the reasons discussed b elow should only o ccur
duetoroundo errorcorrespondingtoaridiculousaccuracyinthecosmological
parameters; in most of these cases this would onlyb e in world models which
areuninterestingb ecause theyareruledoutconclusively(seethetable).
ThevalueofWMTYPEcandi erfromthe`real'valueascalculatedanalytically
from
0 and
0
butonlyifoneissoclosetotheb oundaryb etweentworegionsof
parameterspace thatthis happ ensdue to ordinaryinexactnessin the internal
representation of `real' num b ers. The same goes for the values of MAXZ and
BOUNCE ,sincetheseare triviallyrelatedtoWMTYPE .TheA1andA2curves(see
Stab ell&Refsdal[ 1966]),correspondingtoWMTYPE s15and17,haveb een`drawn
with anumerically thick p encil'but thisshould onlyb e noticeablefor LAMBDA
and OMEGA values precise to 10 05
. This is also true of the endp oints of the
curves,WMTYPE s9 and12. WMTYPE16cannotb ereturnedbyINICOS ,sincethis
corresponds to thestaticEinstein cosmological model,in which
0 and
0 are
b oth =1 . The valueof ZMAX is probably exact in the third gure after the
decimalp oint forworldmodelsextremelynear the A1orA2curve; otherwise,
itisas goo dasanynumericallycalculatedquantity.
ThevaluesofD12andD34returnedshouldb ecorrectinthethirddigitafter
the decimal p oint and shouldn'tb e o by more than one digit in the fourth.
The values of D14and D32 areprobably somewhatless precise, correct in the
second decimaldigit.
e. Development and tests
ThedevelopmentandmosttestingoftheroutineswasdoneonaDigitalVAXStation
3100 Mo del 76 running VMS 5.5-2and with DEC FORTRAN. To mak esure
that numericalaccuracyis monitored ina robustway,comparisonsweremade
tothefollo wingsystems:
DEC ALPHA4000/710,VMS, DECFORTRAN(FORTRAN 77 )
DEC ALPHA4000/710,VMS, DECFORTRAN90
CrayC916,UNICOS8,nativeFORTRAN 77compiler
ConvexC3840andC220, ConvexOS11.0,nativeFORTRAN 77
DSM In nity8000,SCO-UNIX,GreenHillsFortranVersion1.8.5
Fujitsu/SiemensS100,UXP/M,UXP/MFortran77EX/VP(V12)
Hewlett-Packard9000/735,HPUX,nativeFORTRAN 77andFortran90
com-pilers
IBMRS/6000,AIX,xlf90
IBMRS/6000,AIX,xlf90,optimisation-qarch=pwr2
`genericInTelPC', Linux,f2c+gcc
IntelPentium,MS-DOS, LaheyFortran90
Apple PowerMac 6100/60, MacintoshOSD 7.5, LanguageSystems
For-tran PPC
IBM9121-440,MVS,VSFORTRAN
f. Updates
Corrections, up dates andso onwill b e p ostedunder thesubject`ANGSIZ' in
the newsgroup sci.astro.research. Commen ts, bug rep orts, etc. should b e sent
byemailtophelbig@hs.uni-hamburg.de .Pleaseput`ANGSIZ'inthesubject
line.
g. Disclaimer
We'vetested the routines to a greater degree than usual for serious scienti c
work,thoughofcoursenotasextensively asfor(serious)commercialsoftware.
To the b est of our knowledge, they work as described, but of course we can
takenoresponsibilit yfortheconsequences ofanyerrorsin theco de. Asfar as
p ossiblewewillcorrectanyerrors;inanycasewewillmak ethemknowninthe
newsgroupp osting.
References
[1986] Berry, M.V.: CosmologyandGravitation
Bristol: AdamHilger,1986
[1961] Bondi,H.: Cosmology
Cam bridge: Cam bridgeUniversityPress,1961
[1981] Harrison,E. R.: Cosmology,the science ofthe universe
Cam bridge: Cam bridgeUniversityPress,1981
[1996] Kayser, R., P. Helbig, T. Schramm: `A general and practical
methodfor calculatingcosmologicaldistances'
A&A(accepted)
[1973] Misner,C.W.,K.S. Thorne,J.A.Wheeler: Gravitation
NewYork: Freeman,1973
[1966] Stab ell,R., S.Refsdal: `Classi cationofgeneralrelativisticworld
models'
