Exploring the Global Topology of the Universe
Helio V.Fagundes
InstitutodeF
isiaTeoria,UniversidadeEstadual Paulista
S~aoPaulo,SP01405-900, Brazil
Reeivedon20Deember,2001
Wereviewtheworkdonebyourgrouponosmitopology. Itrangesfromearlyattemptstosolve
afamousontroversyaboutquasarsthroughthemultipliityofimages, toquantumosmologyin
thisontextandanappliationtoQEDrenormalization.
I Introdution
The preferred osmologial models for the
desrip-tion of the universe, exept for its very rst
in-stants, are those of Friedmann-Lema ^
itre-Robertson-Walker(FLRW), popularlyknownas`Big Bang'
mod-els. See,forexample,OhanianandRuÆni[1℄.
It isgenerallyassumed that theglobal topologyof
osmispatialsetionsissimplyonneted,thatis,they
areoneofthespherial(S 3
);Eulidean E 3
,or
hyper-boli (H 3
)spaes,whihhavepositive,null,and
nega-tiveurvature,respetively.
ButspaesE 3
andH 3
haveinniteextension. This
hasledsomepioneers,likeEinstein,Shwarzshild,and
Friedmann,toonsiderthequestionofthenitevs.
in-nite sizeoftheuniverse,anditstopology.
Beginningin1971,withapaperbyEllis[2℄,researh
on the possibility of osmi spae being nite and its
topologymultiplyonneted, hasbeendonemore
sys-tematially,ifslowlyatrst.
Today there is areasonable numberof researhers
in this area(my guessis aboutahundred worldwide),
inludinghereinBrazil.
Sine this area is still little known, in this talk I
will presentasummary of work doneby myselfand a
few ollaborators at IFT/UNESP in S~ao Paulo. The
setions thatfollowlistanumberofpublished papers,
whiharerepresentativeoftheeetofnontrivial
os-mi topologyon the topis: setion II,formal
theo-retial works; III, thequasar redshiftontroversy;IV,
tting models to data; V, osmi rystallography; VI,
quantumeldtheoryandquantumosmology;andVII,
theosmimirowavebakground.
Note that most of this researh was done before
1998, so we ould with impunity assume a null
os-II Formal theoretial works
II.1 A osmologial model with ompat spae
setionsand low massdensity[3℄.
Inthis model theadopted spaetime topologywas
M 4
=R,whereRisthetimeaxisandisthespae
setion. Thisformofspaetimeholdsforallworksited
below(withdierent's,ofourse).
Here =T
g S
1
; whereT
g
isthegenus-g surfae
andS 1
istheirle. Themetriis
ds 2
=a 2
()(d 2
d 2
sinh 2
d' 2
) b 2
()d 2
;
where
t() = a
(sinh );
a() = a
(osh 1);
b() = 3a
[oth(=2) 2℄;
witha
=onstant. This metrihad beenstudied by
KantowskiandSahs[4℄,andisoftypeIIIin the
las-siationofBianhi[5℄.
AtthetimeIwasunawareoftheexisteneof
three-dimensional losedhyperboli manifolds,and this was
thebestsubstituteI ouldndforalosedFriedmann
modelofunderritialdensity.
II.2CompatiationofFriedmann'shyperboli
model[6℄
Now is a losed hyperboli manifold onstruted
by Best[7℄. I didnotgo into thedetails ofimage
for-mation, but found that the spae of images was the
overingspaeH 3
,andthathomogeneityandisotropy
should bereinterpretedin terms ofthe distributionof
multiple images in the overing spae. I also found
that we should have the relative density of matter
0
<0:964in order thatarepeated patternof images
ouldbeobserved.
HereIestablishedaorrespondenebetweenthe
ge-ometrilassiationsofBianhiandKantowski-Sahs
(BKS) on one side, and Thurston's on the other,
a-ordingtothefollowingtable:
Thurston type BKS type
T1 BIX
T2 BI,BVII(0)
T3 BV,BVII(A> 0)
T4 KS
T5 BVI(1)
T6 BVIII
T7 BII
T8 BVI(0)
It was also found that Bianhi types IV and
VI(A); 0 < A < 1; annot be ompatied - i. e.,
there are not any losed, loally homogeneous
mani-foldsofthese Bianhitypes.
II.4 Closedspaes in osmology[9℄
This paperimproveson theprevious one,and
dis-usses thequestion of loal vs. global homogeneityof
theonstanturvaturemodels.
II.5NumerialstudyofaperturbedEinstein-de
Sitterosmologial model[10℄
Hereis the3-torusT 3
. Ourpurposewastond
howaninhomogeneityofthematterdensity,writtenas
(x;); =1 t;aetsthemetri,whihwasassumed
tohavetheform
ds 2
= 2
d 2
a 2
(x;)dx 2
b 2
(x;)(dy 2
+dz 2
);
wherea 2
(x;),b 2
(x;)reduetoa
EdS
(t)=(t=t
0 )
2=3
if
weremovetheperturbation. Theresultwasthat,fora
perturbationof ofabout 80%,themetri utuated
at most5%, in agreementwith aheuristiestimateof
Barrow[11℄
III The quasar redshift
ontro-versy
III.1Quasar-galaxyassoiationswithdisordant
redshiftsasa topologialeet. II.Alosed
hy-perboli model[12℄
isoneofBest's[7℄manifoldswitharegular
iosa-hedronasfundamentalpolyhedron (FP).Themetriis
thesameasinFriedmann'sopenmodel.
Thesearhforimagesinonjuntiontosimulatethe
ontroversialpairswasdonebyomputer,bysanning
alldiretionsthatmettheobserver. Theobtainedpairs
were notrealisti: e. g., redshifts 0.12 for the galaxy
and4.3forthequasar.
III.2Smallestuniverseofnegativeurvature[13 ℄
isthesmallestknownCHM,withnormalized
vol-ume 0.9427... and 18 faes, disovered independently
by Weeks [14℄ and by Matveev and Fomenko[15℄. It
wasassumed
0
=0:1:HereIgotbetterresultsforthe
quasar-galaxy assoiations than in the previous ase:
manyofthesimulatedpairshadrealistiredshifts,like
0.002 for thegalaxyand 1.31 forthe quasaror
quasi-stellarobjet(QSO).
However,omparisonwithBurbidgeetal.'s[16℄
at-alog of assoiations, for example, would still not be
reliable: the model parameters are arbitrary, only 31
soureswereusedinthesimulation,andofoursemany
onjuntionsareline-of-sightonidenes.
Themodelisalsointerestingonaountofthesmall
volume: multiple images would allow astronomers to
see objets at dierentepohs of their evolution; and
it hasalargerprobabilitythan alargermodel for the
spontaneousreationoftheuniverse.
IV Fitting models to data
IV.1 A searh for QSO's to t a osmologial
modelwith at, losed spatial setions[17℄
Based on a atalog with about 1500 quasars, we
lookedforpairsequidistantfromEarth,andalongthree
orthogonalaxis. wasthe3-torus,witharetangular
parallepiped as FP, whose sides were tted to 3591,
2966,and2792Mp.
IV.2 A suggestion on the pair of QSO triplets
1130+106fB,A,Cg,fX,Y,Zg[18℄
Thequasarsinthetitle aretwoalignedtriplets
dis-overed by Arp and Hazard [19℄, with similar
orre-sponding redshifts, whih are f2.1, 0.54, 1.6g for the
rst triplet, and f2.1, 0.51, 1.7g for the seond one.
TheywereadjustedtoanEinstein-deSittermodelwith
aubeofside387MpasFP,whihisanunrealistially
smallsize.
V Cosmi rystallography
V.1 Onlosed Einstein-de Sitteruniverses[20℄
Themethod of osmi rystallography(CC)
intro-dued by Lahieze-Rey et al. [21℄ was applied to a
fewmodels,allwithEinstein-deSittermetribut
vary-ing topologies. The idea was to verify that the CC
method yieldsobservable(inpriniple) results,even if
the FP'ssides are ofthe order of thehorizon's radius
R
H
= 2=H
0
, where H
0
is Hubble's onstant. Good
resultswere obtainedwiththeFP'ssidesequal0:7H
0 ;
V.2Cosmirystallographyinaompat
hyper-boli universe [22℄
andFParethesameasin[12℄. WeappliedtheCC
methodtothisgeometriallyinhomogeneousmodeland
didnotobtainanysharppeaks- asexpetedfrom the
independent works ofLehouqet al. [23℄and Gomero
et al. [24℄.
But we did get a smaller utuation that is not
present in the ontrol open osmology. We are at
presenttentativelyattributing thesesmallutuations
to typeIIpairs,asdened in[23℄.
VI Quantum eld theory and
quantum osmology
VI.1Aosmilattieasthesubstratumof
quan-tum elds[25℄
This work is a lowest-order attempt to avoid the
innitiesofquantumeletrodynamisrenormalization.
Bothongurationandmomentumspaesareassumed
tohavethetopologyofa3-torusT 3
;withubesofsides
a==H
0
=8000Mp=2:710 28
m,andP =2~=a
10 20
Gev/;respetively.
Lorentz invariane is abandoned for very large,
presentlyinaessibleenergies.
For harge and mass renormalization at the
one-loop order, I got Z
3
= 0:925, e = p
Z
3 e
0
= 0:962e
0 ;
m=1:185m
0 ;andZ
2
=1:160,inthenotationof
Itzyk-sonandZuber[26℄.
VI.2 Casimir energy in a small volume,
multi-ply onneted,stati hyperboli preinationary
universe[27℄
ThisworkwasorallyommuniatedatthisMeeting,
andappearselsewherein theseProeedings.
VI.3 Birth ofa losed universe of negative
spa-tial urvature [28℄;On the birthof a losed
hy-perboli universe[29℄
is the rst the lens spae L(50;1), then Weeks
manifoldasin[13℄.
Starting with the spontaneous reation, from a
spherialorbifold,ofadeSitteruniversewiththe
topol-ogy ofalens spae,weproeeded asDe Loreniet al.
[30℄, to obtain a quantum topology hange into a de
Sitter universe with Weeks manifold asspae setion.
Afterinationitbeomesalosedhyperboliuniverse,
withthemetriofanopenFLRWuniverse.
VII The osmi mirowave
bak-ground radiation
VII.1 The quadrupole omponent of the reli
model[31℄
This was an appliation of the model in [3℄ to a
quadrupole moment of the osmi mirowave
bak-groundreported by Fabbri and Melhiorri[32℄.
Com-paringtheobtainedrelation
1+Z= a(
observ )
a(
emission )
(1 "os 2
observ )
with the result of [32℄, I obtained " = 4Q=2:7K =
0:0013;
0
=1 3"=2=0:998:
VII.2 Fitting hyperboli universes to Cay
on-Smoot spots in COBE maps[33℄; Soures of
CMB spots in losed hyperboli universes [34℄
Cayon and Smoot [35℄ identied several spots in
NASA'ssatelliteCOBE'smapsoftheosmimirowave
bakground as physial (rather than noise), hene as
smallutuationsofthematterdensity.
InthesepapersIsimulatedtheposition ofsixold
andeighthotCSspotsin losedhyperboli manifolds,
and interpreted them as having evolved into today's
galaxysuperlustersandgalaxyvoids,respetively.
Present atalogs [36℄ only list superlusters up to
Z = 0:12, whih is ompletely inadequate for at of
thesimulatedmodels: typially,inoneofthelatterthe
14redshiftsareintherange0:361to1:370.
I thank the Brazilian ageny CNPq for partial
-nanialsupport.
Referenes
[1℄ H.Ohanian and R.RuÆni,Gravitation andSpaetime
(W.W.Norton,New York,1994),Chapter9.
[2℄ G.F.R.Ellis,Gen.Relat.Grav.2,7(1971).
[3℄ H.V.Fagundes,Lett.Math.Phys.6,417(1982).
[4℄ R.Kantowski and R. K.Sahs, J. Math. Phys.7, 443
(1966).
[5℄ L.Bianhi,Mem.Mat.Fis.So.Ital.Si.11,267(1898).
[6℄ H.V.Fagundes,Phys.Rev.Lett.51,517(1983).
[7℄ L.A.Best,Can.J.Math.23,451(1971).
[8℄ H.V.Fagundes,Phys.Rev.Lett.54,1200 (1985).
[9℄ H.V.Fagundes,Gen.Relat.Grav.24,199(1992);
Ad-denduminGen.Relat.Grav.30,1437(1998).
[10℄ H. V. Fagundes and S.F. Kwok, Astrophys. J. 368,
337(1991).
[11℄ J.D.Barrow,Quart.J.Roy.Astr.So.30,163(1989).
[12℄ H.V.Fagundes,Astrophys.J.338,615(1989);Errata
[14℄ J. R. Weeks, Ph. D. Thesis, Prineton University
(1985);SnapPea: Aomputerprogram forreatingand
studying three-manifols, freely available from the site
www.northnet.org/weeks.
[15℄ S.V.MatveevandA.T.Fomenko,RussianMath.
Sur-veys43,3(1988).
[16℄ G. Burbidge, A. Hewitt, J. V. Narlikar, and P. Das
Gupta,Astrophys.J.(Suppl.)74,675(1990).
[17℄ H.V.FagundesandU.F.Wihoski,Astrophy.J.322,
L5(1987).
[18℄ H. V. Fagundes, Rev. Mex. Astron. Astrof. 21, 79
(1990).
[19℄ H.ArpandC.Hazard,Astrophys.J.240,726(1980).
[20℄ H.V.FagundesandE.Gausmann,Phys.Lett. A238,
235(1998).
[21℄ M. Lahieze-Rey, J.-P. Luminet, and J.-P. Uzan,
As-tron.Astrophys.313,339(1996).
[22℄ H. V. Fagundes and E. Gausmann,Pro. XIX Texas
Symp. Relat. Astrophys. Cosmol. (Paris, 1998), Nul.
Phys.B(Pro.Suppl.)80,CD-ROM.
[23℄ R. Lehouq, J.-P. Luminet, and J.-P. Uzan, Astron.
Astroph.344,735(1999).
[24℄ G. Gomero, A. F. Teixeira, M. Rebouas, and A.
Bernui,preprintgr-q/9811038(1998).
[25℄ H.Fagundes,Rev.Bras.F
is.(nowBraz.J.Phys.)19,
[26℄ C. Itzykson and J.-B. Zuber, Quantum Field Theory
(Addison-Wesley,NewYork,1980),Chapter7.
[27℄ D.Muller,H.V.Fagundes,andR.Opher,Phys.Rev.
D63,123508(2001).
[28℄ S.S.e Costa and H.V. Fagundes, Gen. Relat.Grav.
30,863(1999).
[29℄ S.S.e Costa and H.V. Fagundes, Gen. Relat.Grav.
33,1489(2001).
[30℄ V.A.DeLoreni,J.Martin,N.Pinto-Neto,andI.D.
Soares,Phys.Rev.D56,3329(1997).
[31℄ H. V. Fagundes, Astrophys. Lett. (now Astrophys.
Lett.Comm.)23,161(1983).
[32℄ R.FabbriandF.Melhiorri,Gen.Relat.Grav.13,201
(1981).
[33℄ H.V.Fagundes,Astrophys.J.470,494(1996).
[34℄ H.V.Fagundes,preprintastro-ph/0007443;toappear
inPro.IXMarelGrossmannMeetingonGeneral
Rel-ativity,eds.R.T.Jantzen,V.Gurzadyan,andR.RuÆni
(WorldSienti,Singapore,2002).
[35℄ L.CayonandG.Smoot,Astrophys.J.452,494(1996).
[36℄ M. Einasto, E. Tago, J. Jaaniste, J. Einasto, and H.
