The Age Problem and Growing of
Strutures for Open System Cosmology
M.de Campos and N.A. Tomimura
InstitutodeFsia
UniversidadeFederalFluminense
Av. Gal. MiltonTavaresdeSouzas/n. Æ
24210-340, Niteroi,RJ,Brasil
Reeivedon13November,2000
A new equation for the density ontrast is derived in the framework of reexamined Newtonian
osmologyifwetakeintoaountadiabatimatterreationintheuniverse.Theageoftheuniverse
and the reah of non linear regime of the density ontrast are usually treated separately in the
literatureandthis may leadto ontroversialonlusionsregardingthe mostadequate senario to
desribe the universe. We relate the age of the universe and the growing mode of the density
ontrast by introduinga variable that relates both ofthem, so that bothaspets are treated
simultaneously.WeapplythisproeduretotheFriedmanntypemodelwherethesoureofpartile
produtionis =3nH.
I Introdution
In1934,E.A.Milneanalyzedtheexpansionofthe
uni-verse using elementary Newtonian theory [1℄. In the
sameyearW.H.M CreaandE.A.Milnedemonstrated
the identity between the governing dierential
equa-tionsforrelativistiandNewtonianosmology,fornull
pressure[2℄. Threedeadesafter,E.R.Harrison[3℄
ex-tendedNewtonian osmologytoinlude pressure,
gen-eralizingMaCreaandMilne'sapproahfordust
mat-ter. TheHarrison'sapproahisbasedonthefollowing
setofequations:
t +
~
r
r (+
P
th
2
)~u=0 (1)
d~u
dt =
~
r
r
r 2
r
=4G(+ 3P
th
2
); (2)
whih are respetively the ontinuity equation,
Pois-son's equation and the motion equation. Posteriorly,
Harrison[4℄ derivedthe motionequation by
introdu-inganon-uniform pressure,namely:
~u
t +~u
~
r
r ~ u=
~
r
r
(+P
th )
1
r
r P
th
: (3)
In the above equations ;P
th
;u and denote
respe-tively thedensityof matter, thethermodynami
pres-sure,theveloityeldandthegravitationalpotential.
Theeldequationsforahomogeneousandisotropi
universe are obtainedby using equations (1), (2) and
(3). Theseequationsareidentialtotheonesobtained
in theframeworkofgeneralrelativityin theomoving
oordinatesystem.
However,itwasobservedintheendoftheappendix
IIoftheR.K.SahsandA.M Wolfeworkthatthe
per-turbed modied eld equations to the rst order are
not the samein the generalrelativity and Newtonian
osmology. Therefore,thedensityontrastequationfor
boththeoriesarequitedierent.
This ambiguity was solved by Lima, Zanhin and
Brandenberger[6℄that derivedtheontinuityequation
inthemodiedNewtonianapproahtoosmologywhen
pressureeetsareinluded. Thenewontinuity
equa-tionisompletelyonsistentwithrelativisti
perturba-tiontheoryandsolvestheontraditionpointedoutby
SahsandWolfe.
Inthispaper,we generalizetheevolutionequation
for the density ontrast obtainedby Lima et. al., for
theuniversewithpartileprodutionandobtainedthe
growinganddeayingmodesforthedensityontrast.
In our work we introdue a variable () that
de-pends of the age of the universe and of the growing
modeofthedensityontrast. Theuseof andisern
theosmologialmodelthat furnishesabetter
onor-danewiththeexperimentalevideneabouttheageof
theuniverseandthetimeofstrutureformation.
Thispaperisorganizedasfollows: weobtainin
se-tionII,inthelinearapproximation,thetimeevolution
equationforthemassdensityontrastforaNewtonian
the time dependene of the density ontrast in whih
the soure of partile prodution is = 3nH. We
obtain a range for the parameter that updates the
range obtained in the literature when the questionof
the ageoftheuniverse isinvolved. Finallyweusethe
reah of the non linearregime to establisha riterion
aboutthemostadequateosmologialsenario.
II Equation for the density
on-trast
In theontext of matter reationthe thermodyamial
onservationlawreads:
d dV + 1 V (P th + 2 )+ h nV d dV
(nV)=0; (4)
where n is the partile number density, h = +P
th
is the enthalpy per unity volume. Consequently the
ontinuityequationbeomes
t +r
r
(~u)+ P th 2 r r ~u=
h
n
; (5)
where is the soure of partile prodution. The
otherhydro-dynamialequationsthatdesribethe
os-miuidarethemomentumonservationequationand
Poisson'sequation,respetively:
~u
t +~ur
r ~ u= r
r (+ P 2 ) 1 r r
P; (6)
and
r 2
r
=4G(+ 3P
2
); (7)
isthegravitationalpotentialandP isthetotal
pres-sure that inludes the thermodynamial pressureand
thereationpressure ~
P , so
P = ~
P+P
th
: (8)
Thereationproesswillbeonsideredadiabati,
on-sequentlythepressurereationisgivenby[7℄:
~
P = h
n
: (9)
Inorderto obtainaosmologialsenariowith matter
reationonestillneedstoprovidethestateequation
P
th
=: (10)
Using the eld equations for FRW model [8℄ with
partile prodution and the state equation above, we
anwriteadierentialequationforthesalefator:
R
R+
3+1
2
(+1)
2nH ( _ R 2
+)=0 (11)
Ourmain aiminthissetionisto ndthe
dieren-tialequation that governstheevolutionof thedensity
ontrastinauniverse wherethepartileprodutionis
givenby =3nH,where isaonstant. Notethat,
put =0inequation(11)implies =0andwereover
thestandardFriedmannmodel.
Tostudy the evolution of small utuations in an
expandinguniverseweonsiderthestandard
perturba-tionansatz:
=
b
(t)f1+Æ(r;t)g; (12)
P
th =P
thb
(t)+ÆP
th
(r;t); (13)
=
b
(r;t)+'(r;t); (14)
~ u=u~
b
(t)+~v(r;t); (15)
~
P = ~
P
b (t)+Æ
~
P(r;t): (16)
Thequantities arryingthe subsript b representthe
homogeneous solution to the unperturbed equations.
Insertingthe aboveexpressions into eqs. (5), (6) and
(7)wegettorstorderinperturbations
_ v+ _ R R v= 1 R r'
(+1)
(+1)(1 ) rÆ
R
; (17)
r 2
'=4G
b (t)R
2
Æf1+3 3(+1)g; (18)
_
Æ= +1
R
rv(1 ): (19)
Change to omoving oordinates following standard
lines [9℄, eliminating the peuliar veloity from
equa-tions(17)and(19)andusing(18),isreadythat
Æ +2 _ R R _
Æ+4G
b
(t)Æf1+3 3(+1)g
= (+1)( 1)( (+1)) r 2 Æ R 2 : (20)
Thisisthedierentialequationthatgovernsthe
evolu-tionof densityontrast in thepresene of matter
re-ation,whenwedesribeamatterdistributionwith
uni-formpressureusingthemodiedNewtonianequations.
Itisspatiallyhomogeneous,so weexpettondplane
wavesolutions. ThediÆultyto applyNewtonian
os-mologyin the study of salarperturbations is related
withtheimpossibilityoftheNewtoniansenarioto
de-sribelongwavelengths perturbations. The
osmolog-ialperturbationsanbeofthetwokindaordingto
the wavelength , so that > d
H
or < d
H
, where
d
h
is the Hubble sphere. Perturbations with
wave-lengthslargerthanHubblesalerequiressomeformof
ageneralrelativistitheoryofperturbations. For
wave-lengthssmallerthanHubblesaletheevolutionofmass
density an be studied using Newtonian theory.
to non-relativistimatterandannot beused for
rela-tivistiomponentsevenforsalessmallerthanHubble
radius.
The reexamined Newtonian equations indiate a
way of obtaining the same evolution equation for the
densityontrastasouldbebythefull relativisti
ap-proah. Inthis way oneanextendthe domainof
va-lidityofequation(20)inordertoanalyzeperturbations
evenintheradiationdominatedphaseandtoapplythe
largewavelengthlimit.
Next,wederivethesolutionofthedensityontrast
when the soureof partile prodution is = 3nH,
onentratingourattentiononthegrowingmode. The
prinipal advantageofthisapproahisthe relative
fa-ility to obtaintheperturbed equationforthe density
ontrast whih is the sameequation asin general
rel-ativity when =0 in the synhronous and omoving
oordinates.
Althoughthegaugeinvariantapproahis
oneptu-allymoreattrativesinethereisnoneedtoidentifythe
physial and unrealisti spae-times, nevertheless it is
more ompliated and the physial meaning of gauge
independent variables do not in general possess any
simple interpretation. Besides, the gauge mode that
emerges from the dierentialequation for the density
ontrastisgenerallyrelatedwithsomekindofdeaying
mode whentheuniverse ishomogeneousand isotropi
[10℄.
III The growing mode and the
age of the universe
If the spae time has uniform urvature, the line
ele-mentisgivenby
ds 2
=dt 2
R 2
(t)
dr 2
1 r
2 +r
2
d 2
+r 2
sin 2
()d 2
:
(21)
Theevolutionofthesalefatorisobtainedfrom
solu-tionsofdierentialequation[8℄
R
R+(+ _
R 2
)=0; (22)
where
= 3
2
(+1)(1 ) 1: (23)
Expliit solutions for dierential equation (22) when
=1for anyvalueof arenoteasy to obtain.
Be-sides,<0brokentheseondlawofthermodynamis
and >1brokenthedominant energyondition. We
havemanagedtondalassofexatsolutionsforthe
followingases:
=06=1 ! R=R
0 (
t
t )
1
+1
; (24)
=0=1 ! R=R
0 e
t
t
0
; (25)
=1=1 ! R= 1
2 f
+e 2C
1
2
1
(t+C
2 )
e C
1
2
1
(t+C
2 )C
1
2
1
g: (26)
Thesubsript0alludestothepresenttimeandC
1 and
C
2
areintegrationonstants.
Whether the universe turns out to be spatially
losed,openorpreiselyatremainsanempirial
ques-tion. ThereentresultsoftheBoomerangprovide
on-vining evidene in favor of the standard paradigm:
The universe is at, however the possibility that the
universe mightbespatially loseddonotbedisarded
[11℄. Consequently,weonsidertheuniverseasatand
6=1forsubsequentalulation.
Using the sale fator (24) then the mass-energy
densityisgivenby:
8G
b =
1
(+1) 2
t 2
: (27)
Using equations (27), (24) and the long wavelength
limitintoequation(20)wenallyobtain
Æ+
t _
Æ+( 2) Æ
t 2
=0; (28)
where
=
4
3(+1)(1 )
: (29)
Solutionsforequation(28)are
Æ =C
1 t
1
; (30)
Æ
+ =C
2 t
2 4
3(1 )(+1)
: (31)
In the absene of partile prodution, = 0,
rela-tions (30)and (31)beome theusual density ontrast
forFriedmannmodelswithoutreation[12℄. Although
thedereasingmodeanbeimportantinsome
irum-stanes,theinreasingmodeisresponsibleforthe
for-mationof osmi struturesin thegravitational
insta-bilitypiture. Takingintoaountthedereasingmode
theuniversewouldnothavebeenhomogeneousin the
past. Besides,Peeblesarguedthatagrowingmodethat
startsto growjust aftertheendof radiationerahasa
negligibleomponentofdeayingsolution[13℄.
OneoftheproblemsofthestandardFRWmodelis
relatedtotheageoftheuniverse;itisyoungerthanit
should beaordingtothe experimental values.
How-ever,intheframeworkofpartileprodutionthevalue
experimental evidenes. To show this we dene the
quotient givenby:
= H
H
F
; (32)
at the sametime oordinate. The subsript Fand C
referrespetivelytotheFriedmannstandardmodeland
themodel withpartilereation.
Consideringthesalefator(24)oneobtains
= 1
1
: (33)
If weexpet that theopen systemmodel furnishes an
oldest universe, then > 1 implying 0 < < 1.
Using the oldest globular lusters, B. Chaboyer [14℄
infers a range for the age of the universe, namely
9:6Gyr<t
universe
<15:4Gyr. Consequently,the
Hub-bleonstantforthestandardmodelliesintheinterval
43:3<H
0
<69:4: (34)
This rangedoesnotagree withthe intervalforH
0
es-timatedreentlybyWillikandBatra [15℄,namely:
80<H
0
<90: (35)
One an easilyverifythat thereis asmall overlap
be-tween (34) and the values obtained by Rihtler and
Drenkhahn[16℄,givenby
68<H
0
<76: (36)
Usingthesalefator(24)fordustmatteruniverse,
the Hubble funtion for a universe with partile
pro-dutionisgivenby
H= 2
3(1 )t
: (37)
We an oniliatethe ranges (35) and (36) using the
Hubble funtion above, where the reationparameter
liesin theinterval
0:13<<0:52: (38)
This result updates the interval for determined by
Lima etal[8℄(0:34<<0:60).
Now,westudythereahofnonlinearregime,
iden-tifyingtheepohforthereahofthenonlinearregime
with theepohfortheformationofthesuperlusters.
Inaddition,wesupposethatthestarformationis
pos-teriortotheformationofthesuperlusters. First,
be-ause the density of the super luster in the universe
isapproximatelyequaltotheuniversedensity. Seond,
onsideringtheuniversetoform\fromtopdown"is
a-eptable forsometypeofmodels, forexamplethehot
dark matter model [17℄. In other words, we want to
inferthat the star formation is posteriorto thesuper
lusterformationand,onsequently,tothereahofthe
non linearregime. The growingmodefor the density
ontrast for the FRW standard model an bewritten
as
Æ
+ =Æ
d f
t
t
d g
2=3
; (39)
wherethe subsript drefersto deouplingtime.
Typ-iallywe have( Æ
)
de < (10
2
10 3
) [17℄, fora
de-oupling time of the order 10 5
ys . Substituting the
deouplingtime andtheorrespondinganisotropy,the
reahforthenonlinearregimeforFRWstandardmodel
ours in therange 10 1
Gyr <t <3:16Gyr. Note
that, thestandardmodel furnishesasubestimateage
for the universe, around 9 Gyr. Now, if we onsider
thedierenebetweentheageof theuniverseand the
reah of the non-linear regime we obtain the interval
5:8Gyr <t<8:9Gyr forthe ageof thesuperluster.
Intheliteratureitisknownthattheagefortheoldest
starsinourgalaxyisaround12Gyr[14℄.
Theopen systemosmologyfurnishes areasonable
solutions for the age of the universe. What happens
withthereahofthenonlinearregimeforthedensity
ontrast?
Tosolvethisquestionsweintrodueavariablethat
relatesthe age of the universe and the growingmode
forthedensityontrastinordertodisernamongmore
suitableosmologialmodel,itisgivenby:
= Æ
+
Æ
+F
: (40)
Aosmologialmodelwithbetterharaterististo
representtheuniversemust besatisedif >1. T
ak-ing into aount the growing mode (31) and relation
(33),the ondition(40) implies that <0. Negative
valuesfor the parameter violates the seond law of
thermodynamis. Thenweonludethattherespetive
osmologialmodelto the souregiven by =3nH
doesnotgiveabetterresultthantheusualFRWmodel
whentheage problemand thereahof thenon-linear
regimearetakingtogether. Ontheanotherhandifwe
examinethedierene ofthe ageforthe universe and
the reah for the non linear regime in the ontext of
theopensystemosmologywithapartilesouregiven
by =3nH, wenote that thisdierene issmallest
thantheresultobtainedin theFRW framework.
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