Euler-Poisson-Newton approach in Cosmology

0.2 MB·English

by  Roland Triay

Euler-Poisson-Newton approach in Cosmology
R. Triay and H. H. Fliche†
∗
CentredePhysiqueThéorique1
∗
CNRSLuminyCase907,13288MarseilleCedex9,France
†LMMT2,Fac.desSciencesetTechniquesdeStJérôme
av.Normandie-Niemen,13397MarseilleCedex20,France
7
0 Abstract. ThislectureprovidesuswithNewtonianapproachesfortheinterpretationoftwopuzzlingcosmologicalobserva-
0 tionsthatarestilldiscussedsubject:abulkflowandafoamlikestructureinthedistributionofgalaxies.Forthefirstone,we
2 modelthemotionsdescribingallplanar distortionsfromHubbleflow,inadditionoftwoclassesofplanar-axialdistortions
n withorwithoutrotation,whenspatialdistributionofgravitationalsourcesishomogenous.Thisprovidesuswithanalternative
a tomodelswhichassumethepresenceofgravitationalstructuressimilartoGreatAttractorasoriginofabulkflow.Forthe
J secondone,themodelaccountsforanisotropicuniverseconstitutedbyasphericalvoidsurroundedbyauniformdistribution
of dust. It does not correspond tothe usual embedding of avoidsolution intoacosmological background solution, but to
9
aglobal solutionof fluidmechanics. Thegeneral behavior of thevoidexpansion showsahugeinitialburst,whichfreezes
1
asymptotically up to match Hubble expansion. Whilethe corrective factor to Hubble law on the shell depends weakly on

cosmologicalconstantatearlystages,itenablesustodisentanglesignificantlycosmologicalmodelsaroundredshiftz 1.7.
1 ∼
Themagnificationofsphericalvoidsincreaseswiththedensityparameterandwiththecosmologicalconstant.Aninterest-
v
ingfeatureisthatforspatiallyclosedFriedmannmodels,theemptyregionsaresweptout,whatprovidesuswithastability
7
criterion.
0
1 Keywords: CosmicVelocitiesFields,GreatAttractor,LargeScaleMotions,Voids,CosmologicalConstant
1 PACS: 45.20.D-,45.50.Dd,95.30.Sf,98.80.Ðk,98.80.Es
0
7
0 INTRODUCTION
/
c
q Ononehand,itisamatteroffactthattheuseofNewtonstheoryofgravityforunderstandingcosmologystructuresis
- easierthangeneralrelativity.Ontheotherhand,itishoweverclearthatsuchanapproachmustbetakenwithcaution
r
g at largescales althoughit “is much closer to generalrelativity (GR) than commonlyappreciated”[39]. With thisin
: mind,wefocusonNewtonianinterpretationoftwocosmologicalstructuresthathavebeenmentionedintheliterature,
v
i namely:thepresenceofabulkflowandafoamlikestructureintheobserveddistributionofgalaxies.Forthat,wewrite
X
Euler-Poissonequationssysteminadaptedcoordinates,whatenablesustodefineinastraightforwardwaysolutions,
r asNewton-Friedmannandthevacuum(deSitter)models.Thebulkflowisanalysedwithinanan-isotropicextension
a
ofHubblemodel,see([16]).Subsequently,themodelofasinglesphericalvoidisderivedwithinacovariantapproach,
see([17]).
EULER-POISSON EQUATIONSSYSTEM
As usual, for modelling the dynamics of the cosmological expansion, the distribution of gravitational sources is
supposed to behave as dust (i.e., such a description does not accountfor shear, collision, etc...and the presence of
radiations).Themotionsofsucha pressurelessmediumaredescribedatposition~r andtimet byitsspecificdensity
r =r (~r,t)andvelocity~v=~v(~r,t).ThesefieldsareconstraintbyEulerequationssystem
¶r
+div(r ~v) = 0 (1)
¶ t
1 UnitéMixtedeRecherche(UMR6207)duCNRS,etdesuniversitésAix-MarseilleI,Aix-MarseilleIIetduSudToulon-Var.Laboratoireaffiliéà
laFRUMAM(FR2291).
2 UPRESEA2596
¶ ~v ¶ ~v
+ ~v = ~g (2)
¶ t ¶~r
where~g=~g(~r,t)standsforthegravitationalfield.ItsatisfiesthemodifiedPoisson-Newtonequations
−ro→t~g = ~0 (3)
div~g = 4p Gr +L (4)
−
whereGisNewtonconstantofgravitationandL thecosmologicalconstant.
Fortheinvestigationofanexpandingmedium,itisconvenienttowritetheseequationswithnewcoordinates,herein
namedreferencecoordinates.Theyaredefinedby
~r
(t, ~x= ), a>0 (5)
a
wherea=a(t)isanonconstantmonotonicfunctionwitha =a(t )=1.Suchamethodismotivatedbytheproper
dynamicsofmedium,sothattheequationssystemwhichacc◦ountsf◦orthemotiontranslatesintoitssimplestform.
Fromnow,letdivand−ro→tdenotethedifferentialoperatorswithrespectto~x.Euler-Poisson-Newton(EPN)equations
systemreadsintermofreferencecoordinatesasfollows
¶r
c +div(r ~v ) = 0 (6)
¶ t c c
¶ ~v ¶ ~v
c c
+ ~v +2H~v = ~g (7)
¶ t ¶ ~x c c c
−ro→t~g = ~0 (8)
c
4p G L
div~g = r 3 H˙+H2 (9)
c − a3 c− − 3
(cid:18) (cid:19)
wherethedottedvariablesstandfortimederivatives,
d~x 1 a˙
r =r a3, ~v = = (~v H~r), H= (10)
c c
dt a − a
actrespectivelyasthedensityandthevelocityfieldsofmediuminthereferenceframe,and
1
~g = ~g H˙+H2 ~r (11)
c
a −
(cid:0) (cid:0) (cid:1) (cid:1)
astheaccelerationfield.Letusnotethatthismodeldependsontwodimensionlessparameter
8p G L
W = r , l = (12)
◦ 3H2 c ◦ 3H2
◦ ◦
whereH =H(t ).
◦ ◦
Newton-Friedmann and Vacuum models
Forthesemodels,themostadaptedfamilyoffunctionsa=a(t)verifyaFriedmanntypeequation
L 4p G
H˙+H2 + r a−3=0 (13)
− 3 3 ◦
wherer isanarbitraryconstantwhichstandsforaparameteroffunctiona(t).Thisdifferentialequationadmitsthe
followin◦gintegrationconstant
8p G L
K = r + H2 (14)
◦ 3 ◦ 3 − ◦
ItintegratesforprovidinguswiththefunctionH(t)from
L K 8p Gr
H2= ◦ + ◦ 0 (15)
3 − a2 3 a3 ≥
Letusdenoteitsasymptoticalvalue
L
H¥ = limH= (16)
a ¥ 3
→ r
Accordingto eq.(14),a dependsonH andontwo additionalparameterschosenamongL ,r andK . Atthisstep,
exceptedL ,thesevariablesdonotidenti◦fytocosmologicalparametersassociatedtoFriedmanns◦olution,◦theyintervene
merelyinthecoordinateschoice.Theformalexpressiont aisderivedasreciprocalmappingofaquadraturefrom
7→
eq.(15),itstandsforthereferenceframechronology.
ItisobviousthatEPNequationsystemeq.(6,7,8,9)showstwotrivialsolutions:
1. Theonedefinedby
r =r , ~v =~0, ~g =~0 (17)
c c c
◦
whichaccountsfora uniformdistributionofdust,hereinnamedNewton-Friedmannmodel(NF).Accordingto
eq.(15,10), a and H recovernow their usualinterpretationsin cosmologyas expansionparameter and Hubble
parameter respectively, r =r a3 identifies to the density of sources in the comoving space and K interprets
in GR as its scalar curvat◦ure. We limit our investigation to motions which do not correspond to co◦smological
bouncingsolutions,whatrequirestheconstraint
K3<(4p Gr )2L (18)
◦ ◦
to be fulfilled, according to analysis on roots of third degree polynomials. Hence, the kinematics shows two
distinctbehaviorscharacterizedbythesignofK .Indeed,H decreaseswithtimebyreachingH¥ eitherupward
(K 0)ordownward(K >0)fromaminimum◦definedby
◦≤ ◦
K3
Hm=H¥ s1−L (4p G◦r )2 <H¥ (19)
◦
at(epoch)a=4p Gr K 1,whatdefinesaloiteringperiod.
−
◦ ◦
2. Theonedefinedby
4p G
r c=0, ~vc=(H¥ −H)~x, ~gc= a3 r ◦~x (20)
whichaccountsforvacuumsolution,hereinnamedVacuummodel(V).
ANISOTROPICHUBBLE MODEL
Thismodelaccountsforcollisionlessmotionswhichsatisfythefollowingkinematics
~r=A~r , A(t )=1l (21)
◦ ◦
whereA=A(t)standsforanonvanishingdeterminantmatrix;itscoefficientscanbedeterminedbyanobserveratrest
withrespecttoCosmologicalBackgroundRadiation(CMB).OnehasdetA>0becauseitidentifiestoaunitmatrixat
timet=t .Fromeq.(21),thevelocityfield~v=~v(~r,t)reads
◦
d~r d
~v= =A˙A 1~r, A˙ = A (22)
−
dt dt
whereA 1standsfortheinversematrix3.Letuschoose
−
a(t)=√3detA (23)
3 A−1A=AA−1=1l
andwiththeuniquematrixdecomposition4
H a˙
A˙A−1=H1l+ a2◦B, H = a, H◦=H(t◦), trB=0 (24)
eq.(9,10,11)provideuswiththeequationsofmotioninthereferenceframe
H
~vc = a2◦B~x (25)
H H
~gc = a2◦ B˙ + a2◦B2 ~x (26)
(cid:18) (cid:19)
H2 4p G L
a4◦trB2 = − a3 r c−3 H˙+H2− 3 (27)
(cid:18) (cid:19)
whereastandsforthe(generalized)expansionfactor,thusH=H(t)actsastheusualHubblefactor,andB=B(t)for
atracelessmatrixhereinnameddistortionmatrix.Itcharacterisesthedeviationfromisotropyofthe(dimensionless)
velocityfield,itsamplitudeisdefinedbythematrixnorm
B = tr(BtB) (28)
k k
wherethesign”t”standsforthematrixtransposition.Accpordingtoeq.(12),eq.(27)readsasfollows
1 W 1
H2 H˙+H2 =−2a◦3+l ◦−3a4b 2 (29)
◦ (cid:0) (cid:1)
where
b (t)=tr(Bn), n=1,2,3 (30)
n
Eq(29) shows that W does not depend on spatial coordinates, i.e. the space distribution of gravitationalsources is
necessarilyhomogeno◦us
r =r (t) (31)
andhenceitstandsforaconstantparameterbecauseofeq.(6).
Bymultiplyingeachtermofeq.(29)by2a˙a,oneidentifiesthefollowingconstantofmotion
W a˙2 2 ab
k = ◦ +l a2 2da=W +l 1 (32)
◦ a ◦ −H2 −3 1 a3 ◦ ◦−
Z
◦
Itisimportanttonotethatthechronology5,whichisgivenby
1 ada
dt= (33)
H P(a)
◦
where p
2 ab
P(a)=l a4 k a2+W a a2 2da 0, P(1)=1 (34)
◦ − ◦ ◦ −3 1 a3 ≥
Z
isdistortiondependent.
AClassofpossiblemotions
AccordinglytousualHubblemodel,ifweassumearadialaccelerationfield
~g (cid:181) ~x (35)
c
4 Byusingthetrace,thedeterminantonehas ddtdetA=Tr A˙A−1 detA.
5 Accordingtoobservations,oneassumesH >0sincethecaseH <0accountsforacollapse,whatisnotenvisaged.
◦ (cid:0) (cid:1)◦
thenthedistortionmatrixsatisfiestheevolutionequation
H 1H
B˙ + ◦B2= ◦trB2 (36)
a2 3 a2
accordingtoeq.(26,27).With
dt
dt =H (37)
◦a2
eq.(36)readsinadimensionlessform
dB 1
= b 1l B2 (38)
dt 3 2 −
Witheq.(25,26,33,36),theequationsofmotionread
d~x d2~x 1
= B~x, = b ~x (39)
dt dt 2 3 2
da
dt = (40)
a P(a)
The resolution of these equations can be performed bpy mean of numerical techniques : the evolution with time of
distortionmatrixBisderivedfromeq.(38),theparticlestrajectoriest ~x(t )areobtainedbyintegratingeq.(39)and
7→
theevolutionofthegeneralizedexpansionfactorafromeq.(40).
Analysisofanalyticsolutions
Analytic solutions of eq.(39,40,38) can be obtained thanks to particular properties of distortion matrix B. The
parameters l , W and H given in eq.(12) correspond to cosmological parameters of Friedmann-Lemaître (FL)
solution. The◦cons◦traint b ◦ =0 in eq.(33,34) provides us with the FL chronology, where k in eq.(32) represents
2
thecurvatureparameterintheFLmodel(i.e.thedimensionlessscalarcurvatureW ofthecom◦ovingspace,see[49]),
k
while the flatness of (simultaneousevents) Newtonspace. Itmust be noted thatthe particle position~x as definedin
eq.(5)doesnotidentifytotheusualFLcomovingcoordinatebecausethe(generalized)expansionfactoradependson
theanisotropyunlessb =0.
2
Evolutionoffunctionsb . BecauseBisatracelessmatrix,itscharacteristicpolynomialreads
n=2,3
1 1
Q(s)=det(s1l B)=s3 b s b (41)
2 3
− −2 −3
according to Leverrier-Souriau’s algorithm [45]. With Cayley-Hamilton’s theorem (i.e. Q(B)=0) and eq.(38) we
obtainthefollowingdifferentialequationssystem
d
b = 2b (42)
dt 2 − 3
d 1
b = b 2 (43)
dt 3 −2 2
andwenotethatthediscriminantofthirdorderpolynomialQ,itisproportionalto
1
a =3b 2 b 3 (44)
3 −2 2
isaconstantofmotion(i.e.,da /dt =0).Theintegrationofeq.(42,43)gives
t =t +e √6 b2(t) dx , e = 1 (45)
◦ 2 Zb2(t ) √2a +x3 ±
◦
Hence,b isdefinedbyaquadrature,andb fromeq.(42);inadditionofthesingularsolution
2 3
b =b =0, (i.e., B3=0) (46)
2 3
definedequivalentlyeitherbyb =0orb =0,accordingtoeq.(42,43).
2 3
Therelateddynamicsdependsonrootsh ofcharacteristicpolynomialQgivenineq.(41),i.e.theeigenvalues
i=1,2,3
ofdistortionmatrixB.Theirrealvaluesidentifytodilatationratesattimet towardthecorresponding(timedependent)
eigenvectors(notnecessarilyorthogonal).Theirsumisnull(b =0)andtheirproduct(b =3detB)iseitherdecreasing
1 3
withtimeorisnull,accordingtoeq.(43).Thesignofa givenineq.(44)isusedtoclassifythesolutionsasfollows:
• ifa =0thenQhasarealdoublerooth 1=h 2 andasimpleoneh 3.Therelatedinstantaneouskinematicshows
aplanar-axialsymmetry(eitheracontractionwithinaplanewithanexpansiontowardatransversedirectionor
viceversa),seesec..Ifh =h thenbothvanishandtherelatedsolutionidentifiestothesingularonedefinedin
1 3
eq.(46);
• ifa >0thenQhasasinglerealrooth 1;
• if a <0 then Q has three distinct real roots h i=1,2,3. Their order is conserved during the evolution (since a
coincidenceofeigenvaluesmakesa =0),thelargestonemustbepositivewhilethesmallestonemustbenegative
(becauseb =0).
1
Planarkinematics. ThesingularsolutionB3=0showsaFLchronologyandthedistortionmatrix
B= B2t +B , B3=0 (47)
− ◦ ◦ ◦
is solely defined by its initial value B , according to eq.(38). It is neither symmetric nor asymmetric (otherwise it
vanishes),seeeq.(28).Hence,eq.(39)◦transforms
d~x
= B2t +B ~x (48)
dt − ◦ ◦
(cid:0) (cid:1)
whichaccountsforeternalmotions
t 2
~x=exp B2 +B t ~x =(1+B t )~x (49)
− ◦ 2 ◦ ◦ ◦ ◦
(cid:18) (cid:19)
The trajectory of a particle located at initial position~x identifies to a straight line toward the direction B~x . The
analysisofB range(i.e.,itsimage)providesuswithch◦aracteristicsoftrajectoriesflow.Thenilpotentproper◦ty◦ofB
showsthatits◦kernelisnotemptyKer(B )=0/.Itsdimensiondim(Ker(B ))=mcharacterizesthekinematics,which◦
iseitherplanar(m=1)ordirectional(m◦=6 2),i.e.abulkflow.Conversel◦y,ifthekernelofdistortionmatrixBisnot
emptythenb =3det(B)=0,andthusb =0,seetoeq.(42,43).Therefore,allplanarkinematicscanbedescribed
3 2
bysuchamodel.
Planar-Axialkinematics. Ifb =0thenthechronologydifferentiatesfromFLone.Letusfocusonthea =0with
2
twodistincteigenvaluesh =h cl6assofsolutions.Witheq.(45),eq.(42,43)integrate
1 3
6
6
b n= (t t⋆)n, (n=2,3), t⋆=t◦+√6b 2−1/2(t◦) (50)
−
whichshowsasingularityatdatet =t >0thatsplitsthemotionintworegimest <t andt >t .Thecomplete
⋆ ⋆ ⋆
investigationofthissingularityproblemdemandstosolveanintegro-differentialequation,seeeq.(33,34).Theroots
ofQread
1
h = , h = 2h (51)
1 (t t ) 3 − 1
⋆
−
where h stands for the doubleroot. Among others, two class of solutionsare defined by mean of a constant (time
1
independent)matrixP,theprojectorassociatedtoh ,see[45],
1
P2=P, trP=2 (52)
TheydescribedistinctkinematicsdependingonwhethermatrixBisdiagonalizable.
• Irrotationalmotions:IfBisdiagonalizablethen
B=h (3P 21l) (53)
1
−
Fromeq.(39,53),onehas
d~x
=h (3P 21l)~x (54)
dt 1 −
andthesolutionreads
1
~x= h P~x + (1l P)~x (55)
− 1 h 2 −
1
where~x isconstant.Iftheeigenvectorsareorthogonalthenthekinematicsaccountsforirrotationalmotions.
• Rotationalmotions :IfBisnotdiagonalizablethen
1
B = h (3P 21l)+ N (56)
1 − h 2
1
d~x 1
= h (3P 21l)+ N ~x (57)
dt 1 − h 2
(cid:18) 1 (cid:19)
whereNisaconstantnilpotentmatrix,whichaccountsforrotationalmotionsontheeigenplaneofP.Thesolution
reads
1 1
~x= h P~x + (1l P)~x + N~x (58)
− 1 h 2 − h 2
1 1
where~x isconstant.
Discussion
As a result, this anisotropicgeneralizationrequiresan homogeneousdistributionsof matter. Because of the pres-
ence of strong density inhomogeneities in the sky distribution of galaxies catalogs, one is forced to ask whether it
describescorrectlythe dynamicsof observedcosmic structures. In principle,such a remarkshould be also sensible
forquestioningHubblelawwhen,regardlesstheisotropy,itisafactthatperturbationsarenotsodominantotherwise
itwouldneverhavebeenhighlighted.Actually,homogeneityisimplicitlyassumedinthestandardcosmologyforthe
interpretationofCMBisotropyandtheredshiftofdistantsources,whichprovidesuswithanexpandingbackground6.
ItiswithsuchaschemainmindthatthisanisotropicHubblelawprovidesuswithanhintonthebehaviorofthecosmic
flowfromdecouplingerauptopresentdateinordertoanswerwhethertheobservedbulkflowisdueexclusivelyto
tidalforces.
Cosmicflow offlatLSS (B3 =0)
TheB3=0classofsolutionshasinterestingpropertieswithregardtothestabilityoflargescalestructuresthatshow
a flat spatialdistribution.To answerthe questionof whetherobservationsdefineunambiguouslythe kinematics,the
distortionmatrixBisdecomposedasfollows
B=S+j(w~), trS=0 (59)
whereSandj(w~)standforitssymmetricanditsasymmetric7component,and
a2 1
w~ = ~s , ~s = −ro→t~v (60)
H 2
◦
accountsforthemotionrotation,~s beingtheswirlvector.Hence,eq.(47)gives
Bw~ =Sw~ (61)
6 NamelythecomovingspaceofFLworldmodelontowhichthegravitational instability theoryisapplied forunderstanding theformation of
cosmicstructures.
7 Theoperatorjstandsforthevectorproduct,~u w~ =j(~u)(w~).
×
Theevolutionoftheanisotropywithtimeisdefinedby
S = S2+j(w~ )j(w~ ) t +S (62)
j(w~) = −(S◦j(w~ )+◦ j(w~◦)S )t +◦j(w~ ) (63)
−(cid:0) ◦ ◦ ◦ (cid:1)◦ ◦
whichcouplesthesymmetricandtheantisymmetricpartsofthedistortionmatrix.Theswirlmagnitudereads
1
w = w~,w~ = trS2 (64)
h i 2
r
q
accordingto eq.(60), since b =0. Its orientationcannotbe determinedfrom the data because the aboveequations
2
describetwodistinctkinematicscorrespondingto w~ thatcannotbedisentangle.Accordingtoeq.(59),if(andonly
if)therotationw =0thenthedistortionvanishesS±=0◦sinceBiseitherasymmetricorantisymmetric.Inotherwords,
aplanardistortionhasnecessarilytoaccountforarotation.
Constantdistortion
Amongabovesolutionswhichshowplanarkinematics,letusinvestigatethe(simplest)onedefinedbyB2=0.In
suchacase,~k (cid:181) w~ andSw~ =~0.Accordingtoeq.(62,63,64),linearcalculusshowsthatthedistortioniscons◦tant
◦
S=S , w =w (65)
◦ ◦
Such a distortion in the Hubble flows produces a rotating planar velocities field with magnitude (cid:181) H a 2. In the
−
presentcase,themodelparameterscanbeeasilyevaluatedfromdata.Theobservedcosmicvelocityfields◦arepartially
determinedbytheirradialcomponent
~r
v = ~v, =cz, ~r=~r(m)=r~u=a~x, r=ct (66)
r
h ri
wherem,z,~u,t standrespectivelyfortheapparentmagnitude,theredshift,thelineofsight,thephotonemissiondate
ofthegalaxyandcthespeedofthelight.Accordingtoeq.(22,24,59),theradialvelocityofagalaxylocatedatposition
~risgivenby
H
vr= H+H˜~u r, H˜~u= a2◦~u·S~u (67)
BecausetrS=0,itisclearthatthesumofthre(cid:0)eradialv(cid:1)elocitiesv correspondingtogalaxieslocatedintheskytoward
r
orthogonaldirectionsand atsame distance r providesuswith the quantityH. Hence, simple algebrashowsthat the
sampleaverageofradialvelocitieswithinaspherearadiusrisequalto
v =H~r (68)
r
h i h i
Therefore,formotionsdescribedbyeq.(21),thestatisticsgivenineq.(68)providesuswithagenuineinterpretation
ofHubbleparameterH=H(t).Hence,accordingtoeq.(24,33,34),oneobtainsthe(generalized)expansionfactor
t
a(t)=exp H(t)dt (69)
t
Z
◦
Hence,thecosmologicalparameterscanbeestimatedbyfittingthedatatothefunction
y (t)l ,k ,W = P(a)=a2H/H , l +k +W =1 (70)
◦ ◦ ◦ ◦ ◦ ◦ ◦
ThecomponentofmatrixScanbeestimatedbpysubstitutingH ineq.(67),andw isobtainedfromeq.(65).
ItisclearthattheabovemodelisderivedintheGalileanreferenceframe,wheretheEuler-Poissonequationssystem
can be applied. Hence, a non vanishing velocity of the observer with respect to this frame an produces a bipolar
harmonicsignalintheH˜ distributionofdatainthesky,whichcanbe(identifiedandthen)subtracted.
~u
Discussion
The dynamics of a homogenous medium and anisotropic moving under Newton gravity was already studied by
describingtheevolutionofanellipsoid[52].Thecurrentapproachenablesustoidentifycharacteristicsofthedynamics
ofthedeformationfromisotropicHubblelawinamoresystematicwaybymeanofthedistortionmatrix.
Atfirstglance,iftheplanaranisotropicofthespacedistributionofgalaxieswithintheLocalSuperCluster(LSC)
isstablethentheabovesolutioncanbeusedforunderstandingitscosmicvelocityfields,~k beingorthogonaltoLSC
plane. It is well known however that the distribution of galaxies is not so homogenousas◦ that, whereas this model
describesmotionsofanhomogenousdistributionofgravitationalsources.Ontheotherhand,suchanapproximation
levelissimilartotheonewhichprovidesuswiththeobservedHubblelaw,thatisincludedinthismodel(S=0).
SPHERICALVOIDSINNEWTON-FRIEDMAN UNIVERSE
Formodellingthedynamicsofasphericalvoidinauniformdustdistribution,weuseacovariantformulationofEuler-
Poisson equationssystem (see Appendix).The modelis obtainedby stickingtogetherthe localsolutionsV and NF
of Euler-Poisson equations system as defined previously in the Newton-Friedmannand Vacuum models, where the
functionastandsfortheFriedmannexpansionparameter.Suchamodelaccountsforthedynamicsoftheircommon
border(i.e. boundariesconditions),whichis a materialshell. For conveniencein writing, the symbolsS, V and NF
denotes both the medium and the related dynamical model. A qualitative analysis of solutions is performed and a
generaldiscussionisgivensubsequently
Dynamicalmodel
Weconsiderthreedistinctmedia:amaterialshellwithnullthickness(S),anemptyinside(V)andoutsideauniform
dustdistribution(NF).ThesemediabehavesuchthatSmakesthejunctureofVwithNFasgivenbyeq.(17,20).The
tension-stressonSisassumedtobenegligible,whatischaracterizedbya(symmetriccontravariant)mass-momentum
tensordefinedasfollows
T00=(r ) , T0j=(r ) vj, Tjk=(r ) vjvk (71)
S S c S S c c S S c c c
Thebackgroundisdescribedbythefollowingmass-momentumtensor
T00=r , T0j =0, Tjk =0 (72)
NF c NF NF
AccordingtoAppendix,sincetheeulerianfunction(al)
T(x g )= Tmn gmn dtdS+ Tmn gmn dtdV (73)
7→ S NF
Z Z
vanisheswheng readsintheformgmn = 1 ¶ˆm x n +¶ˆn x m ,onehas
2
(cid:16) (cid:17)
¶ x +gjx +(¶ x +¶ x 2Hx )vj+vjvk¶ x (r ) dtx2dW
S 0 0 c j j 0 0 j− j c c c j k S c
Z
(cid:16)(cid:0) (cid:1) (cid:17)
= r ¶ x dtx2dxdW (74)
c 0 0
− NF
Z
wheredW standsforthesolidangleelement.Theradialsymmetryofsolutionsenablesustowritethereducedpeculiar
velocityandaccelerationofatestparticlelocatedontheshellasfollows
~v =a ~x, ~g =b ~x (75)
c c
wherethefunctionsa =a (t)andb =b (t)havetobedetermined.Abypartintegrationofeq.(74)providesuswith
t2(¶ (r ) +3(r ) a r a x)x2x dt (76)
0 S c S c c 0
Zt1 −
= t2 ¶ 0((r S)ca )+4(r S)ca 2+2H(r S)ca (r S)cb x−1 x3x˜dt
Zt1 −
(cid:0) (cid:1)
wherex= ~x standsfortheradiusofSandx˜ = x 2+x 2+x 2.Thisequalitymustbefulfilledforallboundedtime
k k 1 2 2
intervalandcompactsupport1-form.Hence,weeqasilyderivetheconservationequationsforthemass
¶ (r ) +(3(r ) r x)a =0 (77)
0 S c S c c
−
andforthemomentum
da r b
+ 1+ c x a 2+2Ha + =0 (78)
dt (r ) x
(cid:18) S c (cid:19)
With eq.(5), the calculationof the gravitationalforce fromthe entire shell acting on a particularpoint8 providesus
with
4p G r (r )
b = c S c (79)
a3 3 − 2x
(cid:18) (cid:19)
Aboutmassconservation,itisnoticeablethat
1
(r ) = r x (80)
S c c
3
issolutionofeq.(77),whatensuresthattheamountofmatterwhichformstheshellcomesfromitsinterior.Hence,
eq.(78)transforms
da 2p Gr
+4a 2+2Ha c =0 (81)
dt − 3 a3
Itisconvenienttousethedimensionlessvariable
a
c =4 a2 (82)
H
◦
where the ratio a H 1 stands for the expansion rate of S in the reference frame. Hence, eq.(81) transforms into a
−
Riccatiequation ◦
dc c 2 1
= W (83)
da ◦− a P(a)
(cid:18) (cid:19)
where
p K
P(a)=l a4 k a2+W a, k = ◦, P(1)=1 (84)
◦ − ◦ ◦ ◦ H2
◦
wherethedimensionlessparameters9aredefinedineq.(12,14).Accordingtoeq.(10,82),theevolutionoftheradius
ofSisgivenby
a c da
x=x exp (85)
i
Zai 4a P(a)!
wherex anda standsfortheinitialvaluesattimet.
i i i p
Qualitativeanalysis
Theshellexpansionisanalysedintermofdimensionlessquantities:themagnificationX andtheexpansionrateY
x a
X = , Y = (86)
x H
i
◦
Their evolution with the expansion parameter a are obtained from eq.(83, 85) by numerical integration10. Let us
focusourinvestigationaroundthegenerallyacceptedvaluesl =0.7andW =0.3.Theinitialconditionslieonthe
expansionrateY andtheformationdatet,asexpressedbymea◦nsofa =a(t)◦.Thevaluesa =0.003andY =0(void
i i i i i i
initiallyexpandingwith Hubbleflow) areusedasstandardinourdiscussion.Hereafter,we simplyprovidesuswith
synthesisresults,moreinformationcanbefoundin[17].
8 Themodifiednewtoniangravitationfieldreads~g= L Gm ~r
3− r3
9 ThesenotationsarepreferredtotheusualW L =l a(cid:16)ndW K=(cid:17) k foravoidingambiguitiesontheinterpretationofcosmologicalparameters,see
e.g.,[13],[14]. ◦ − ◦
10 Becausethemappingt a(t)=1/(1+z)isamonotonicfunctioninthepresentinvestigation,theirevolutionwithcosmictimetorwithredshift
7→
zcanbestraightforwardlyderived.