PLEARY TALKS REVISTA MEXICANA DE FiacuteSICA 49 SCl)LI-lElO 2 85-90

Cosmology with a time dependcnt cosmological constant

SEPTIEMBRE 2003

Luis O PimentelDeparamento de Ffsica Universidad AllIoacutenoma Metropolitana-lztapalapa

Aparado postal 44-534 09340 Meacutexico DF Maicoe-mail loprxanumuammx

Luz M Diaz-RiveraPhysics Department University of Florida

l O Box 11844 Gainesville Fl 32611-8440 USAe-mail luzphysujledu

Cesar MoraDepartamento de Fiacutesica Unidad Profesional Interdisciplinaria de Hioecnologiacutea lnstillto Politeacutecnico Nacional

Av Acueducto sn Col Barrio La Laguna 7icomaacuten 07340 Meacutexico Dr Mexicoe-mail cemlxarlUmuammx

Recibido el 31 de marLOde 2002 aceptado el 21 de julio de 2(X)2

In the contcxt of the scalar-tensor theories we consider cosmological modcls with a time dependent cosmological constant Several toymodels are obtained among them thre are solutions without singularity and accelerating

Keywords Alternative theories of gravity cxact solutions singularities

En el contexto de las teonas tensoriales-escalarcs consideramos modelos con constante cosmoloacutegica dependiente del tiempo Se obtienenvanos modelos de juguete entre ellos hay soluciones sin singularidad y acelerados

DeJcripores Teorias tenso-escalares soluciones exactas singularidades

PACS 114S(+h 11420Dw 0420Jh

1 Inlroduction

Unificalion theories have a nonzero cosmologica constantlhat is about 120 orders of magnitude larger than lbe obser-ved value for A this constitutes the cosmological constantproblcm [12] In orderto explain and solve such a problemand to make compatible lbe actual observational data wilh theinf1ationary sccnario and particle physics expcclations a timedependenl cosmological conslant was proposed [3J This oldidea ha~ rcceived a lot of allemion What pcople have in mindis lo make the vaeuum energy dynarnical In such a way du-ring lbe evolution of lbe universe the energy density of lhevaeuum dccays imo particles hus leading lO lhe dccreaseof the cosmological constant obtaining a~ a result althoughsmall a creation of particles

A broad summary of cosmological models wilb time de-pendent cosmological constant is given by Overduin andCooperslock 14] re-examining lhere the evolution of the sca-le factor when is given a~ function of t a(t) H or q A fai-rly general cquation oC state is considercd and new numericalsolulions are obtained but a~ in most of the previous workslhe lime dependence of lbe cosmological tecm is introducedad hoc

An altcrnativc is an effectivc lime dependent cosmologi-cal conslant in the context of a scalar-lensor theories whichbecomes a tme conslant for t raquo 0[5) Using Jordan-Brans-Dieke lheory (JBD) in particular the gracefol exit problem

of old inftationary cosmology might be improved It remainsthe problcm of delecmining he JBD parameter w lbal accor-ding lo solar system experimenls its value is Ilwll 3000which has becn derived from light bending experimems [6]A beller estimation of lbis pararneler should be obtained frommcasure of others cosmological parameters in order to cons-train w more slrongly than by mC3ns of solar system cxpcri-mcnls 17] Howcvcr theorics of very carly universc as slringtheory are beller described in the context of JBD and showsthat w can take negative values [8]

Thus scalar-lensor lheories and in particular JBD arebeller lbeories in order to get in a natural way a time de-pendent cosmological constan Clearly recem observalionalresulls restrict lbis kind of theory eg lhe lype la super no-va (SNla) resuL~ which in 1998 show lhat OA - 06 [9)implying lha our universe is specding up Thus a modelwhich allcmpts to describe the cosmological constant beha-vior should take inlo accoum the observational evidences

In a recenl work [10] we investigated the effect of a timedependent cosmoogical constam in a farnily of scalar-tensorlheories There we gel cosmological modcls in lbe coastingperiod where the lime dependence on the cosmological cons-tant occurs in a natural way In such a models wc assurncd asimple relation (4)) = c4gt(t) (wilh c and n constanL~)

The existence of inf1ationary phaltcin scalar-tcnsor the-ories (STr) ha~ been investigated by Pimemel ami Slein-Schabes [11] finding inftalionary pha~es for a polynomial

86 LUIS O PIMENTEL LUZ M DJAZRIVERA AND CESAR MORA

cosrnological constant in a general STT which iexclneludesBrnns-Dickc model with non-zcro cosmological constanl 00lhe olher haod Guendelmao 1121 has invesligaledhe requi-rcrncnts of (he potentials in arder lO have scale invarianccThere was found the form of the pOlenlial needed by he glo-bal invariancc which in addilion it~cncrgy in the conformalEinstein [mme has the charactcristics [oc a suitablc inflatio-nary univcrsc aod A dccaying sccnario [oc the late univcrsc

MOlivaled by these ideas we shall eonsider a general STTas in our previous work [10] bul now wc shall consider a binomial funelion on ltIgt(t)in order lo oblain exaet Solulionsofhe field equalions from whieh we obtain sorne kind of in-flalionary eosmologieal models and relaled eosmologieal pa-

I

2 Field equations

ramctcrs In [3Cl we obtain in most of our solutions a powcrlaw growlh for he eosmologieal seale factor a( t) ~ fU whe-re (J ~ 1 implies inflalionary models As it is known his is agen cric feature of a elass of models hal allempt dynamieallylOsolvc the cosmological constanl problem In oue modcls (j

is a free parametcr (al Icml in most of oue ffiodcls) in ordcrto he adjustcd by physical conditions and to be in agrccmentwith rceent data for SN la lha implies (J l and whieh iseonsistenl with he nueleosynthesis [13]

Most of our solutions prcdict an accclerated expansionsuch solutions are in agrccmcnt wilh me SN la results hutflm and flA depcod on free paramelers of our model In so-rne spceifie cases we get Solulions wilh exponenlial growlhof the seale factor

We slarl wilh lhe aelioo for lhe most general sealar-tensor Oleory of gravilalion studied by Bergmann and Wagoner (BW) [141

(1)

whcrc 9 det (giexclJv) G is Ncwtons constant SNG is the action fnr me non-gravitatjonal malter We use me signature(- + + +) The arbitrary funetions w(ltIraquo aod (lt1raquodistinguish he different sealar-lensor heories of gravitalion (lt1raquoisa polenlial funetion and play s the role of a eosmologieal eonstanl w(ltIraquo ishe eoupling funetion ofhe particular heory Theiow energy lirnit of string theory givcs a scalar-tensor theory of gravitation Iike me oncs considercd herc

The explieit field equaions are

_ ( 1 ) -1 ) 81rT~G~ =(lt1raquoOg~+ wltlgt ltIgt~ltIgt- 2g~ltIgtltIgt + lt1gt (ltIgt~- g~ 0lt1gt + -lt1gt-

0lt1gt + ~ltIgt ltIgt~ In [w(ltIraquo] + ~ lt1gt( ) R + 2 d)ltIgt(ltIraquoJ= 02 dltlgt lt1gt 2wltlgt Ultf

where G~is he Einstein teosor The lasl equation can be sllbstituled by

2lt1gtdjdltlgt- 2lt1gt(lt1raquo_ 1 (81rT _ dw ~)0lt1gt+ 3+2w(ltIraquo - 3+2w(ltIraquo dltlgtltIgtA

(2)

(3)

(4)

wherc T TiexclJ is me trace of the stress-cncrgy tensor In a prcvious work [10] was oemotlstrated that the divcrgcnce-Iesseondition ofh~ stress-energy maller lensor is salisfied if lhe fiel Eq (3) is satisfied too alhollgh Ollf field cquations are givenby Eqs (2) and (4)

In what follows we shall assume w(ltIraquo= eonstaot bull = (lt1raquoThe eorresponding field equalions wih a pcrfcet fluid for thematlcr contcnt in the isotropic ami homogencous line clement

ds =-dt +a(t) [ dr +r(dO + sin OdQgt)] 1- kr

will he eonsidered Thus the field cquations are

(0) 3k 81rp w(~) o~3 - +--(lt1raquo---- - +3--=0a a lt1gt 2 lt1gt a lt1gt

2 2 iexcliexcl (O) k 81rp w(ltIraquo lt1gt a lt1gt

-2~ - ~ - a + (lt1raquo- T - 2 4gt - ~ - 2~4gt = 0

[4gt o~] (d) 811-+3-- (3+2w)-2 -lt1gt- --(p-3p)=0

lttgt a lt1gt dltlgt lt1gt

Where we have assumed lt1gt = ltIgt(t)and lhe derivatives respeel tare denoted by a do

Rev Mex Fiacutes 49 S2 (2003) 85-90

(5)

(6)

(7)

(8)

COSMOLOGY Wrn1 A TIME DEPENDE-1 COSMOLOGICAL CO--STANT H7

3 Vacuum solulions

Whcn we consider vacuum models the action of the previous scction can be wotteo a~

(9)

where N is lhe lapse function a is he scale faclor of lheUnivcrsc and the Ricci scalar iexcls

where g = det (g) R() is he sealar curvalure of lheFriedmann-Robertson-Walker heory ltIgt(t)is lhe conventio-nal real scalar gravilational field lp is lhe Planck lengh andA(ltIraquois he cosmologicallenn Thc second inlegral is a sur-faee lcnn involving thc induccd rnclric hij and scrond fun-damental fonn Kii on lhe boundary uccdcd lo cancel lhesccond dcrivalives in n(4) whcn (he aCiion iexcls varied with themetric and scalar field bul nol lheir normal derivatives fixedon lhe houndary We are inleresled lo sludy an homogene-ous and isotropic cosmological model conscqucntly we lIscdlhe FRW melric line elemenl in spherical polar coordinales(tr8ltIraquo given by

whcrc wc have chosen thc cosmological tcnn as

A(y) = Aiexcl cosh(2y) + A2sinh(2y) (18)

( 16)

( 19)

(20)

f3 = x2 sinh(2y)

a(T) - 2Aiexcl = O

f3(T) + 2A2 = O

a = x2 cosh(2y)

s = ~I [~(a2 - f32) + Ala + A2f3 - k] dT (17)

whcre Al and A2 are constanLiThe equations of motion foclhis mlxtel derived from lhe aClion (17) are

Ihe aCliou (15) takes he symmetric fonn

and with the following change of indepcndent variables

(10)[dr2 ]ds2 = _N2(t) dt2 + a2(t) 2 + r2 do2

1- kr

(11)wilh lhe Hamillonian constrainl equalion given by

a - f32 - 4Aiexcla - 4A2f3 + 4k = O (21)

whcrc dOl denotes the time dcrivativc with rcspcet lo the limet now we introduce lhe folowing new variables

substituling (11) into Eq (9) and inlegrating Wilh respCCl lOspace coordinales after simplifications we have

I[ altlgt2 a2 5=- -Nkaltlgt+-a +-altlgt2 N N

_ N~~ltIraquo a3~2 + ~ a3lt1gtA(ltIraquo]dt (12)

31 IlransDiekc IhllfY

(23)

(22)

where Cl and C2 are inlegration conslanls and satisfy Cf -Cj -4Aiexcl C2 - 4A2C + 4k = O Iu lhe folowing subsecliouswe sludy lwo simply choices ofw(ltIraquo function lhis lead liS toBrans-Dicke and Barker lhenries and also we consider lhreegeneral parametrized thcorics

The solulion of syslem (19)-(20) is

a(T) = AiexclT2 + CiexclT + C2

f3(T) = -A2T2 + C3T + C

= I (2w(ltIraquo + 3) dltlgt

Y 12 lt1gt

x = a(jJ

s = ~I [~X2 - ~y12 - Nkx + Nx3 A(Y)] dT (14)

dT = lt1gtdt( ) _ A(ltIraquo

i Y - 3lt1gt

lhenhe Bergmann-Wagoner (BW) action simplifies 10

(13) The choice w(lt1raquo= Wo=const produces an inflatiouary cos-mological lenn in lhe Brans-Dicke lheory

(24)

where a is given by a = J(2wo + 3)3 Choosing w(ltIraquo =wowe oblainhe following solution

prime denOles the time derivativc wilh rcspcct lo the ncw ti-me T In he following seclions we sludy particular gaugesN = I and N = lx sinee we obtain simple solvables wa-ve equations respcctively When we sel N = lx hen lheaction (14) Occorncs

5= I~[X2X2 - Xy2 + x2 A(y) - k] dT (15)

a(T) = [(Aiexcl + A2)T2 + (CI - C3)T + C2 - C]m

x [(Al - A)T2+(Ciexcl + C3)T+C2+C]m-n (25)

ltIgt(T)= [(Al - A2)T2 + (Ciexcl + C3)T + C + C] n (26)(Al + A2)T2 + (CI - C3)T + C2 - C

where m = (1 + p)4a n = 1217 and 17= J(2wo + 3)3

Rev Mex Fiacutes 49 S2 (2(XJ3) 85~90

ss LUIS O PIME1ltTEL LUZ M DIAZ-RIVERA AND CESAR MORA

(27)

ponds 10 a eosmologieal lerm of Ihe form A(aB) =Al vI(a + fJ)(a - fJ) and is nol singular for k = 1 wecan verify this by direet substitulion in the metrie invariants

3 Non singular soulion

Selting CI = C3 = C = O C = k Aiexcl and Aiexcl = A in Iheaboye equations wc have the solulion

a(T) = ao [~ +rand

[

] -n4gt(T)= ~J+ 1 (28)

where ao = (k Ai) t and TJ = k2A lhis solulion eorres-I

R = -6 [a~ + 24gta + a4gta4gt+ a4gta]

RI = ~ [k - a34gta4gt_ a34gta]4a4 1

_ 1 3R - - y3Riexcl

7 R3 = Riexcl

v12

(29)

(30)

(31 )

(32)

Introdueing the solutions (27) and (28) in Ihe Rieei sealar (29) we obtain

R=- T 6 (m+iexcl)k(TJ+T)+2T[kTJ+atm(6m-2n-1)(~+1)]+2atmTJ(~+1) (33)

aoTo CJ + 1)and the RI invarian take lhe form

RI= 3 ( +1)k(T3+T)+2T[krg+2a~m(2n-2m+1)(T+1)]-2a~mrg(T+1) (34) 8 (T ) m TO TO

4aoTo TJ + 1

3 J2 Singular soution

h 21 - ~ 2 Th l Ihw ere ao = 4 TO and filo = TO ese so ullOOS m eeosmie time t take the form

In whal follows we shall eonsider two imporlara aump-tioos

a4gtm=a A(4)) = AI4gtn +A4gt (40)

whcrc m o Al gt2 nI and n2 are constant~The first ai-sumption is a very well known one (see eg ReL 15 and refe-renee therein) and it has becn used as a eondition for Ihe de-eeleration parameler lo be eonslanl for nat models in Brans-Dieke lheory Furlhermore wilh Ihis eondition our field cqua-lions simpliacutefy nOloriously and allows us to obtain exael solu-tioos Thc second condition is the maio assumption of thepresenl work whieh is mOlivated by Ihe eosmologieal no-hairthcorem for sealar-tensor Iheories [11] in order to study in-flalionary solulions in a thcory of gravitation with a natura-lIy dynarnieal eosmologieal eonstanl In this work we alwayswork in the lardan reame whcrc G is variable howcvcr weeould make a conforma transformalion to Ihe Einstein framewhere G is eonstant and we have General Relativity plus aminimally eoupled sealar field Ihen our potenlial beeomes anexponential one ie bullbull VI + V ~ exp(WI4gte) + exp(w4gte)(where lt is a constan and 4gteis a canonieally defined sea-lar field) This is Ihe type of polential aceording to Guendel-mariexcl [12] Ihat is necesary lOhave scale invarianee in a Iheoryof gravilalion free of the cosmological eonstanl problem lhais one wiIh an early expanding phase and a A-decaying forlale limes Details of Ihe con formal transformation for STTcan be seen in ReL 11 In Ihe following seclions we shall findcxact solutions on diffcrenl cases acwell as cosmological pa-rarnclcrs which allow us lOcompare with actual obscrvations

(37)

(38)

(35)(36)

() ma T = 00T

I 2(1+p)a(t) = a t-=r

bull4gt(t)= 4gtt=

whcrc

Anmher e1assieal eosmologieal solulion can be obtained byehoosing Ciexcl = C3 = O C = C = k2AI and Al = Athus lhe solutions (25)-(26) are redueed to

4 Sotutions with matter

Assuming a barotropie cqualion of state p = (Y - l)p we ha-ve been able lOsolve Ihe eosmologieal equalions [16] In Ihefollowing secuao we show sorne solutions foc the dust andradiation doninatcd cosmos

Re Mex Fiacutes 4952 (2003) 85-90

COSMOLOGY wrm A TIME DEPENDENT COSMOLOGICAL CONSTA 89of today value of Hubble parameter Ha the aelual value ofdeeeleration parameler qo

41 Dus

We have found lhe following solution for dust

values whieh t can take t lt eP 2 and e gt Oas we can sce from he definition of eP In hisca he eosmologieal term in spile of being abinomial funetion on 1gt deeays with he timeThe eorresponding Rieei scalar and euevalOre in-varianl show that this solution is singular

TIte expansion takes place wilh a eonstant aeeeleration

- ( ) Where 1gt = e a = oc A = 4 2w - 5 e A2 = OP = [(1-2w)7fjeo3 ande isaninlegra-tiao constanl This is an extended inOationary solutionwih a time decaying eosmologieal eonstant and initialsingularity as it is shown by lhe eorresponding Rieeiscalar

(48)

The present values of Ihe dece1eration and Hub-ble pararnelers are given by he following expres-sioos

6 ( k)R= 61gtt - 21gt+ - (46)(l-lgtt) a

3 1 ( k)R=---- -2eP t -2eP-- (47)4 (l-ePt) al

(42)

(41)

Igt(t) = 1gtt-

a(t) = a t

A(Iraquo = AIIgt(t)

P = pa(t)-3

I k=O

Wih Ha - 65 ol 5 km 8-Mpe- we obtain to -301 Gy whieh is too big value eompared wih he glo-bular cluster age

2Ho=-to

(43)Because 1gt t~ lt 1 lhen qo gt O TItis model ex-pands from t = _1gt- until t = O lhen il eon-lraeLsuntil t = 1gt) in bolh cases with positivedeeeleration parameter TIte numerieal value forto depends on he values of lhe free eonstanLS

2 ki O

bull Fork(3+2w) gt OTIte soJutions in terms of lhe time t is given by

Igt(t) = e [1 -1gt tiexcl-

a(t) = ad1 - eP tJ

A(Iraquo = AIgt(t) + A21gt(t)

P = pa(t)-3 (44)

bull Fork(3+2w)ltOThe eorresponding solution in terms ofhe lime tis given as

Igt(t) = c [1 + 1gttiexcl-

a(t) = ad1 + 1gttJ

A(eP) = AIgt(t) + AIgt(t)

P = pa(t)-3 (49)

where

(50)

al = o el

2kAl=2o

ke 3

P=-4 o7fO

and el is an inlcgralion conslanl In this caiC wchave nol restrictions on the valucs which t can ta-ke If -IN lt (to + e) lhe expansion takesplace with non-constant accclcration and wiLhout

(45)

oal = 12e

whcrc

and e is an inlegralion eonslan Clearly a(t) andIgt(t) muSl be positive in order lo be physieallysignifieant this requirement restrieLs the range of

Re Mex F(s 4952 (2003) 85-90

90 LUIS O PIMEIEL LUZ M DlAZ-RIVERA ANO CESAR MORA

singularily ai il is shown fmm thc correspondingRicci scalar and curvalurc invariant

42 Radialon

In thc case in which the maUee contenl of he univcrsc is radialion we have found [161

R 6 ( k)= ( ) 61gt t + 21gt + 2 1 + 1gt t a

( )3 1 kR = 4 bull -21gtt +21gt-2(1+ 1gt t) a

(51)

(52)

1gt(t)= 1gt t-

a(t) = a t)(1)) = )1gt(t)

p = pa(t)- (54)

Aeeordingly R and R do nol diverge For lhissolution the corrcsponding prcscnt valucs of thedcnsity and Hubblc paramctcrs are givcn as

where 1gt = 4e a = (oe)2 and) = OThis is a singo-lar solutioT1 according 10 lhe Ricci scalar which foc this caOCis givCTl hy

(56)

(55)

1Ho = -toqo = O

and lherefme to = 1 Ho ~ 1505 Gy

R=6(1+ a~)t-The deeelcralioo parameler bceomes null instead he lodayHubble parameler is giveo as

(53)

As in lhe previous case lhe values of to dependson he frcc eonstanls of our model but in lhis ca-se fm H~ ~ 1gt = to ~ 1Ho

Acknowlcdgmcnts

LOP was parlially soppmled hy CONACYT granl 42191-F

1 S Weinbcrg Rrv Mod Phys 6t (1989) l

2 Y Ng 1m J Mod Ihys DI (1992) 145

3 f1PBronstcin Ihys eit derSoujelUfljoll3 (1933) 73

4 J Ovtrduin ami F Coopersock Fhys Nev D SH (1998)043506

5 S Capoziexcljcllo and R de Ritis Gefl Re Gral 29 (1997) 1425

6 CM Will UVifl8 Reviews iriexcl Reatilily4 (2001) No 4 iexcliexclvaila-ble al hupllwwwliving-rcvicwsorgArlic1esVolumc42(X)I-4PD Se harre and CM Will Pllys RtI ) 65 (2002) (~t(X)X(hf-qc010 10(1)

7 A Liddlc A Mazurndar and J HarrowIlIys Rev J) 58 (199R)027302

8 F Dahia and C Romero Phys Rev IJ (() (1998) 104014 (gr-qd9812CXll) M Suspcrrcgi and A Mazumdar Phys Rev1J5K(1998) 083511

9 S Perlmutler el al ASlrophys 1 Vol 517 (1999) 565 (as-troph9811(33)

10 LO Pimcnlcl and LM Diacutea-Rivcra Ifll J Mod Plly A 14(1999) 1513

11 LO Pimclllc1 and J Stcin-Schabes IJhys [ell B 216 (19R9)27

12 EJ Gucndc1man Mad Ihys lRIl A 14 (1999) 1()l3 (gr-qc1990 IO17) E1 Gucndclman gr-qc9901 067

13 tv1 Scthi A Balra and D Lohiya Phys Rev ) 60 (1999) 1043(astm-pIV9903084)

14 C Wil1 ll1eory afld experimclIl ifl graVilalioflllJ pllysin (Cam-bridge Univcrsity Press Cambridge 1981)

15 11 Dehncn and O Obregoacuten htropllys Space (i 17 (1972)338 V Johri and K Desikan Gen ReJ Gral 26 (1994) 1217

16 LM Diltlz-Riwra and LO pimenlc1 Pllys Nev IJ 60 (1999)11350 L

Rev Mex Fis 49 S2 (2003) 85-90

COSMOLOGY Wrn1 A TIME DEPENDE-1 COSMOLOGICAL CO--STANT H7

3 Vacuum solulions

Whcn we consider vacuum models the action of the previous scction can be wotteo a~

(9)

where N is lhe lapse function a is he scale faclor of lheUnivcrsc and the Ricci scalar iexcls

where g = det (g) R() is he sealar curvalure of lheFriedmann-Robertson-Walker heory ltIgt(t)is lhe conventio-nal real scalar gravilational field lp is lhe Planck lengh andA(ltIraquois he cosmologicallenn Thc second inlegral is a sur-faee lcnn involving thc induccd rnclric hij and scrond fun-damental fonn Kii on lhe boundary uccdcd lo cancel lhesccond dcrivalives in n(4) whcn (he aCiion iexcls varied with themetric and scalar field bul nol lheir normal derivatives fixedon lhe houndary We are inleresled lo sludy an homogene-ous and isotropic cosmological model conscqucntly we lIscdlhe FRW melric line elemenl in spherical polar coordinales(tr8ltIraquo given by

whcrc wc have chosen thc cosmological tcnn as

A(y) = Aiexcl cosh(2y) + A2sinh(2y) (18)

( 16)

( 19)

(20)

f3 = x2 sinh(2y)

a(T) - 2Aiexcl = O

f3(T) + 2A2 = O

a = x2 cosh(2y)

s = ~I [~(a2 - f32) + Ala + A2f3 - k] dT (17)

whcre Al and A2 are constanLiThe equations of motion foclhis mlxtel derived from lhe aClion (17) are

Ihe aCliou (15) takes he symmetric fonn

and with the following change of indepcndent variables

(10)[dr2 ]ds2 = _N2(t) dt2 + a2(t) 2 + r2 do2

1- kr

(11)wilh lhe Hamillonian constrainl equalion given by

a - f32 - 4Aiexcla - 4A2f3 + 4k = O (21)

whcrc dOl denotes the time dcrivativc with rcspcet lo the limet now we introduce lhe folowing new variables

substituling (11) into Eq (9) and inlegrating Wilh respCCl lOspace coordinales after simplifications we have

I[ altlgt2 a2 5=- -Nkaltlgt+-a +-altlgt2 N N

_ N~~ltIraquo a3~2 + ~ a3lt1gtA(ltIraquo]dt (12)

31 IlransDiekc IhllfY

(23)

(22)

where Cl and C2 are inlegration conslanls and satisfy Cf -Cj -4Aiexcl C2 - 4A2C + 4k = O Iu lhe folowing subsecliouswe sludy lwo simply choices ofw(ltIraquo function lhis lead liS toBrans-Dicke and Barker lhenries and also we consider lhreegeneral parametrized thcorics

The solulion of syslem (19)-(20) is

a(T) = AiexclT2 + CiexclT + C2

f3(T) = -A2T2 + C3T + C

= I (2w(ltIraquo + 3) dltlgt

Y 12 lt1gt

x = a(jJ

s = ~I [~X2 - ~y12 - Nkx + Nx3 A(Y)] dT (14)

dT = lt1gtdt( ) _ A(ltIraquo

i Y - 3lt1gt

lhenhe Bergmann-Wagoner (BW) action simplifies 10

(13) The choice w(lt1raquo= Wo=const produces an inflatiouary cos-mological lenn in lhe Brans-Dicke lheory

(24)

where a is given by a = J(2wo + 3)3 Choosing w(ltIraquo =wowe oblainhe following solution

prime denOles the time derivativc wilh rcspcct lo the ncw ti-me T In he following seclions we sludy particular gaugesN = I and N = lx sinee we obtain simple solvables wa-ve equations respcctively When we sel N = lx hen lheaction (14) Occorncs

5= I~[X2X2 - Xy2 + x2 A(y) - k] dT (15)

a(T) = [(Aiexcl + A2)T2 + (CI - C3)T + C2 - C]m

x [(Al - A)T2+(Ciexcl + C3)T+C2+C]m-n (25)

ltIgt(T)= [(Al - A2)T2 + (Ciexcl + C3)T + C + C] n (26)(Al + A2)T2 + (CI - C3)T + C2 - C

where m = (1 + p)4a n = 1217 and 17= J(2wo + 3)3

Rev Mex Fiacutes 49 S2 (2(XJ3) 85~90

ss LUIS O PIME1ltTEL LUZ M DIAZ-RIVERA AND CESAR MORA

(27)

ponds 10 a eosmologieal lerm of Ihe form A(aB) =Al vI(a + fJ)(a - fJ) and is nol singular for k = 1 wecan verify this by direet substitulion in the metrie invariants

3 Non singular soulion

Selting CI = C3 = C = O C = k Aiexcl and Aiexcl = A in Iheaboye equations wc have the solulion

a(T) = ao [~ +rand

[

] -n4gt(T)= ~J+ 1 (28)

where ao = (k Ai) t and TJ = k2A lhis solulion eorres-I

R = -6 [a~ + 24gta + a4gta4gt+ a4gta]

RI = ~ [k - a34gta4gt_ a34gta]4a4 1

_ 1 3R - - y3Riexcl

7 R3 = Riexcl

v12

(29)

(30)

(31 )

(32)

Introdueing the solutions (27) and (28) in Ihe Rieei sealar (29) we obtain

R=- T 6 (m+iexcl)k(TJ+T)+2T[kTJ+atm(6m-2n-1)(~+1)]+2atmTJ(~+1) (33)

aoTo CJ + 1)and the RI invarian take lhe form

RI= 3 ( +1)k(T3+T)+2T[krg+2a~m(2n-2m+1)(T+1)]-2a~mrg(T+1) (34) 8 (T ) m TO TO

4aoTo TJ + 1

3 J2 Singular soution

h 21 - ~ 2 Th l Ihw ere ao = 4 TO and filo = TO ese so ullOOS m eeosmie time t take the form

In whal follows we shall eonsider two imporlara aump-tioos

a4gtm=a A(4)) = AI4gtn +A4gt (40)

whcrc m o Al gt2 nI and n2 are constant~The first ai-sumption is a very well known one (see eg ReL 15 and refe-renee therein) and it has becn used as a eondition for Ihe de-eeleration parameler lo be eonslanl for nat models in Brans-Dieke lheory Furlhermore wilh Ihis eondition our field cqua-lions simpliacutefy nOloriously and allows us to obtain exael solu-tioos Thc second condition is the maio assumption of thepresenl work whieh is mOlivated by Ihe eosmologieal no-hairthcorem for sealar-tensor Iheories [11] in order to study in-flalionary solulions in a thcory of gravitation with a natura-lIy dynarnieal eosmologieal eonstanl In this work we alwayswork in the lardan reame whcrc G is variable howcvcr weeould make a conforma transformalion to Ihe Einstein framewhere G is eonstant and we have General Relativity plus aminimally eoupled sealar field Ihen our potenlial beeomes anexponential one ie bullbull VI + V ~ exp(WI4gte) + exp(w4gte)(where lt is a constan and 4gteis a canonieally defined sea-lar field) This is Ihe type of polential aceording to Guendel-mariexcl [12] Ihat is necesary lOhave scale invarianee in a Iheoryof gravilalion free of the cosmological eonstanl problem lhais one wiIh an early expanding phase and a A-decaying forlale limes Details of Ihe con formal transformation for STTcan be seen in ReL 11 In Ihe following seclions we shall findcxact solutions on diffcrenl cases acwell as cosmological pa-rarnclcrs which allow us lOcompare with actual obscrvations

(37)

(38)

(35)(36)

() ma T = 00T

I 2(1+p)a(t) = a t-=r

bull4gt(t)= 4gtt=

whcrc

Anmher e1assieal eosmologieal solulion can be obtained byehoosing Ciexcl = C3 = O C = C = k2AI and Al = Athus lhe solutions (25)-(26) are redueed to

4 Sotutions with matter

Assuming a barotropie cqualion of state p = (Y - l)p we ha-ve been able lOsolve Ihe eosmologieal equalions [16] In Ihefollowing secuao we show sorne solutions foc the dust andradiation doninatcd cosmos

Re Mex Fiacutes 4952 (2003) 85-90

COSMOLOGY wrm A TIME DEPENDENT COSMOLOGICAL CONSTA 89of today value of Hubble parameter Ha the aelual value ofdeeeleration parameler qo

41 Dus

We have found lhe following solution for dust

values whieh t can take t lt eP 2 and e gt Oas we can sce from he definition of eP In hisca he eosmologieal term in spile of being abinomial funetion on 1gt deeays with he timeThe eorresponding Rieei scalar and euevalOre in-varianl show that this solution is singular

TIte expansion takes place wilh a eonstant aeeeleration

- ( ) Where 1gt = e a = oc A = 4 2w - 5 e A2 = OP = [(1-2w)7fjeo3 ande isaninlegra-tiao constanl This is an extended inOationary solutionwih a time decaying eosmologieal eonstant and initialsingularity as it is shown by lhe eorresponding Rieeiscalar

(48)

The present values of Ihe dece1eration and Hub-ble pararnelers are given by he following expres-sioos

6 ( k)R= 61gtt - 21gt+ - (46)(l-lgtt) a

3 1 ( k)R=---- -2eP t -2eP-- (47)4 (l-ePt) al

(42)

(41)

Igt(t) = 1gtt-

a(t) = a t

A(Iraquo = AIIgt(t)

P = pa(t)-3

I k=O

Wih Ha - 65 ol 5 km 8-Mpe- we obtain to -301 Gy whieh is too big value eompared wih he glo-bular cluster age

2Ho=-to

(43)Because 1gt t~ lt 1 lhen qo gt O TItis model ex-pands from t = _1gt- until t = O lhen il eon-lraeLsuntil t = 1gt) in bolh cases with positivedeeeleration parameter TIte numerieal value forto depends on he values of lhe free eonstanLS

2 ki O

bull Fork(3+2w) gt OTIte soJutions in terms of lhe time t is given by

Igt(t) = e [1 -1gt tiexcl-

a(t) = ad1 - eP tJ

A(Iraquo = AIgt(t) + A21gt(t)

P = pa(t)-3 (44)

bull Fork(3+2w)ltOThe eorresponding solution in terms ofhe lime tis given as

Igt(t) = c [1 + 1gttiexcl-

a(t) = ad1 + 1gttJ

A(eP) = AIgt(t) + AIgt(t)

P = pa(t)-3 (49)

where

(50)

al = o el

2kAl=2o

ke 3

P=-4 o7fO

and el is an inlcgralion conslanl In this caiC wchave nol restrictions on the valucs which t can ta-ke If -IN lt (to + e) lhe expansion takesplace with non-constant accclcration and wiLhout

(45)

oal = 12e

whcrc

and e is an inlegralion eonslan Clearly a(t) andIgt(t) muSl be positive in order lo be physieallysignifieant this requirement restrieLs the range of

Re Mex F(s 4952 (2003) 85-90

90 LUIS O PIMEIEL LUZ M DlAZ-RIVERA ANO CESAR MORA

singularily ai il is shown fmm thc correspondingRicci scalar and curvalurc invariant

42 Radialon

In thc case in which the maUee contenl of he univcrsc is radialion we have found [161

R 6 ( k)= ( ) 61gt t + 21gt + 2 1 + 1gt t a

( )3 1 kR = 4 bull -21gtt +21gt-2(1+ 1gt t) a

(51)

(52)

1gt(t)= 1gt t-

a(t) = a t)(1)) = )1gt(t)

p = pa(t)- (54)

Aeeordingly R and R do nol diverge For lhissolution the corrcsponding prcscnt valucs of thedcnsity and Hubblc paramctcrs are givcn as

where 1gt = 4e a = (oe)2 and) = OThis is a singo-lar solutioT1 according 10 lhe Ricci scalar which foc this caOCis givCTl hy

(56)

(55)

1Ho = -toqo = O

and lherefme to = 1 Ho ~ 1505 Gy

R=6(1+ a~)t-The deeelcralioo parameler bceomes null instead he lodayHubble parameler is giveo as

(53)

As in lhe previous case lhe values of to dependson he frcc eonstanls of our model but in lhis ca-se fm H~ ~ 1gt = to ~ 1Ho

Acknowlcdgmcnts

LOP was parlially soppmled hy CONACYT granl 42191-F

1 S Weinbcrg Rrv Mod Phys 6t (1989) l

2 Y Ng 1m J Mod Ihys DI (1992) 145

3 f1PBronstcin Ihys eit derSoujelUfljoll3 (1933) 73

4 J Ovtrduin ami F Coopersock Fhys Nev D SH (1998)043506

5 S Capoziexcljcllo and R de Ritis Gefl Re Gral 29 (1997) 1425

6 CM Will UVifl8 Reviews iriexcl Reatilily4 (2001) No 4 iexcliexclvaila-ble al hupllwwwliving-rcvicwsorgArlic1esVolumc42(X)I-4PD Se harre and CM Will Pllys RtI ) 65 (2002) (~t(X)X(hf-qc010 10(1)

7 A Liddlc A Mazurndar and J HarrowIlIys Rev J) 58 (199R)027302

8 F Dahia and C Romero Phys Rev IJ (() (1998) 104014 (gr-qd9812CXll) M Suspcrrcgi and A Mazumdar Phys Rev1J5K(1998) 083511

9 S Perlmutler el al ASlrophys 1 Vol 517 (1999) 565 (as-troph9811(33)

10 LO Pimcnlcl and LM Diacutea-Rivcra Ifll J Mod Plly A 14(1999) 1513

11 LO Pimclllc1 and J Stcin-Schabes IJhys [ell B 216 (19R9)27

12 EJ Gucndc1man Mad Ihys lRIl A 14 (1999) 1()l3 (gr-qc1990 IO17) E1 Gucndclman gr-qc9901 067

13 tv1 Scthi A Balra and D Lohiya Phys Rev ) 60 (1999) 1043(astm-pIV9903084)

14 C Wil1 ll1eory afld experimclIl ifl graVilalioflllJ pllysin (Cam-bridge Univcrsity Press Cambridge 1981)

15 11 Dehncn and O Obregoacuten htropllys Space (i 17 (1972)338 V Johri and K Desikan Gen ReJ Gral 26 (1994) 1217

16 LM Diltlz-Riwra and LO pimenlc1 Pllys Nev IJ 60 (1999)11350 L

Rev Mex Fis 49 S2 (2003) 85-90

ss LUIS O PIME1ltTEL LUZ M DIAZ-RIVERA AND CESAR MORA

(27)

ponds 10 a eosmologieal lerm of Ihe form A(aB) =Al vI(a + fJ)(a - fJ) and is nol singular for k = 1 wecan verify this by direet substitulion in the metrie invariants

3 Non singular soulion

Selting CI = C3 = C = O C = k Aiexcl and Aiexcl = A in Iheaboye equations wc have the solulion

a(T) = ao [~ +rand

[

] -n4gt(T)= ~J+ 1 (28)

where ao = (k Ai) t and TJ = k2A lhis solulion eorres-I

R = -6 [a~ + 24gta + a4gta4gt+ a4gta]

RI = ~ [k - a34gta4gt_ a34gta]4a4 1

_ 1 3R - - y3Riexcl

7 R3 = Riexcl

v12

(29)

(30)

(31 )

(32)

Introdueing the solutions (27) and (28) in Ihe Rieei sealar (29) we obtain

R=- T 6 (m+iexcl)k(TJ+T)+2T[kTJ+atm(6m-2n-1)(~+1)]+2atmTJ(~+1) (33)

aoTo CJ + 1)and the RI invarian take lhe form

RI= 3 ( +1)k(T3+T)+2T[krg+2a~m(2n-2m+1)(T+1)]-2a~mrg(T+1) (34) 8 (T ) m TO TO

4aoTo TJ + 1

3 J2 Singular soution

h 21 - ~ 2 Th l Ihw ere ao = 4 TO and filo = TO ese so ullOOS m eeosmie time t take the form

In whal follows we shall eonsider two imporlara aump-tioos

a4gtm=a A(4)) = AI4gtn +A4gt (40)

whcrc m o Al gt2 nI and n2 are constant~The first ai-sumption is a very well known one (see eg ReL 15 and refe-renee therein) and it has becn used as a eondition for Ihe de-eeleration parameler lo be eonslanl for nat models in Brans-Dieke lheory Furlhermore wilh Ihis eondition our field cqua-lions simpliacutefy nOloriously and allows us to obtain exael solu-tioos Thc second condition is the maio assumption of thepresenl work whieh is mOlivated by Ihe eosmologieal no-hairthcorem for sealar-tensor Iheories [11] in order to study in-flalionary solulions in a thcory of gravitation with a natura-lIy dynarnieal eosmologieal eonstanl In this work we alwayswork in the lardan reame whcrc G is variable howcvcr weeould make a conforma transformalion to Ihe Einstein framewhere G is eonstant and we have General Relativity plus aminimally eoupled sealar field Ihen our potenlial beeomes anexponential one ie bullbull VI + V ~ exp(WI4gte) + exp(w4gte)(where lt is a constan and 4gteis a canonieally defined sea-lar field) This is Ihe type of polential aceording to Guendel-mariexcl [12] Ihat is necesary lOhave scale invarianee in a Iheoryof gravilalion free of the cosmological eonstanl problem lhais one wiIh an early expanding phase and a A-decaying forlale limes Details of Ihe con formal transformation for STTcan be seen in ReL 11 In Ihe following seclions we shall findcxact solutions on diffcrenl cases acwell as cosmological pa-rarnclcrs which allow us lOcompare with actual obscrvations

(37)

(38)

(35)(36)

() ma T = 00T

I 2(1+p)a(t) = a t-=r

bull4gt(t)= 4gtt=

whcrc

Anmher e1assieal eosmologieal solulion can be obtained byehoosing Ciexcl = C3 = O C = C = k2AI and Al = Athus lhe solutions (25)-(26) are redueed to

4 Sotutions with matter

Assuming a barotropie cqualion of state p = (Y - l)p we ha-ve been able lOsolve Ihe eosmologieal equalions [16] In Ihefollowing secuao we show sorne solutions foc the dust andradiation doninatcd cosmos

Re Mex Fiacutes 4952 (2003) 85-90

COSMOLOGY wrm A TIME DEPENDENT COSMOLOGICAL CONSTA 89of today value of Hubble parameter Ha the aelual value ofdeeeleration parameler qo

41 Dus

We have found lhe following solution for dust

values whieh t can take t lt eP 2 and e gt Oas we can sce from he definition of eP In hisca he eosmologieal term in spile of being abinomial funetion on 1gt deeays with he timeThe eorresponding Rieei scalar and euevalOre in-varianl show that this solution is singular

TIte expansion takes place wilh a eonstant aeeeleration

- ( ) Where 1gt = e a = oc A = 4 2w - 5 e A2 = OP = [(1-2w)7fjeo3 ande isaninlegra-tiao constanl This is an extended inOationary solutionwih a time decaying eosmologieal eonstant and initialsingularity as it is shown by lhe eorresponding Rieeiscalar

(48)

The present values of Ihe dece1eration and Hub-ble pararnelers are given by he following expres-sioos

6 ( k)R= 61gtt - 21gt+ - (46)(l-lgtt) a

3 1 ( k)R=---- -2eP t -2eP-- (47)4 (l-ePt) al

(42)

(41)

Igt(t) = 1gtt-

a(t) = a t

A(Iraquo = AIIgt(t)

P = pa(t)-3

I k=O

Wih Ha - 65 ol 5 km 8-Mpe- we obtain to -301 Gy whieh is too big value eompared wih he glo-bular cluster age

2Ho=-to

(43)Because 1gt t~ lt 1 lhen qo gt O TItis model ex-pands from t = _1gt- until t = O lhen il eon-lraeLsuntil t = 1gt) in bolh cases with positivedeeeleration parameter TIte numerieal value forto depends on he values of lhe free eonstanLS

2 ki O

bull Fork(3+2w) gt OTIte soJutions in terms of lhe time t is given by

Igt(t) = e [1 -1gt tiexcl-

a(t) = ad1 - eP tJ

A(Iraquo = AIgt(t) + A21gt(t)

P = pa(t)-3 (44)

bull Fork(3+2w)ltOThe eorresponding solution in terms ofhe lime tis given as

Igt(t) = c [1 + 1gttiexcl-

a(t) = ad1 + 1gttJ

A(eP) = AIgt(t) + AIgt(t)

P = pa(t)-3 (49)

where

(50)

al = o el

2kAl=2o

ke 3

P=-4 o7fO

and el is an inlcgralion conslanl In this caiC wchave nol restrictions on the valucs which t can ta-ke If -IN lt (to + e) lhe expansion takesplace with non-constant accclcration and wiLhout

(45)

oal = 12e

whcrc

and e is an inlegralion eonslan Clearly a(t) andIgt(t) muSl be positive in order lo be physieallysignifieant this requirement restrieLs the range of

Re Mex F(s 4952 (2003) 85-90

90 LUIS O PIMEIEL LUZ M DlAZ-RIVERA ANO CESAR MORA

singularily ai il is shown fmm thc correspondingRicci scalar and curvalurc invariant

42 Radialon

In thc case in which the maUee contenl of he univcrsc is radialion we have found [161

R 6 ( k)= ( ) 61gt t + 21gt + 2 1 + 1gt t a

( )3 1 kR = 4 bull -21gtt +21gt-2(1+ 1gt t) a

(51)

(52)

1gt(t)= 1gt t-

a(t) = a t)(1)) = )1gt(t)

p = pa(t)- (54)

Aeeordingly R and R do nol diverge For lhissolution the corrcsponding prcscnt valucs of thedcnsity and Hubblc paramctcrs are givcn as

where 1gt = 4e a = (oe)2 and) = OThis is a singo-lar solutioT1 according 10 lhe Ricci scalar which foc this caOCis givCTl hy

(56)

(55)

1Ho = -toqo = O

and lherefme to = 1 Ho ~ 1505 Gy

R=6(1+ a~)t-The deeelcralioo parameler bceomes null instead he lodayHubble parameler is giveo as

(53)

As in lhe previous case lhe values of to dependson he frcc eonstanls of our model but in lhis ca-se fm H~ ~ 1gt = to ~ 1Ho

Acknowlcdgmcnts

LOP was parlially soppmled hy CONACYT granl 42191-F

1 S Weinbcrg Rrv Mod Phys 6t (1989) l

2 Y Ng 1m J Mod Ihys DI (1992) 145

3 f1PBronstcin Ihys eit derSoujelUfljoll3 (1933) 73

4 J Ovtrduin ami F Coopersock Fhys Nev D SH (1998)043506

5 S Capoziexcljcllo and R de Ritis Gefl Re Gral 29 (1997) 1425

6 CM Will UVifl8 Reviews iriexcl Reatilily4 (2001) No 4 iexcliexclvaila-ble al hupllwwwliving-rcvicwsorgArlic1esVolumc42(X)I-4PD Se harre and CM Will Pllys RtI ) 65 (2002) (~t(X)X(hf-qc010 10(1)

7 A Liddlc A Mazurndar and J HarrowIlIys Rev J) 58 (199R)027302

8 F Dahia and C Romero Phys Rev IJ (() (1998) 104014 (gr-qd9812CXll) M Suspcrrcgi and A Mazumdar Phys Rev1J5K(1998) 083511

9 S Perlmutler el al ASlrophys 1 Vol 517 (1999) 565 (as-troph9811(33)

10 LO Pimcnlcl and LM Diacutea-Rivcra Ifll J Mod Plly A 14(1999) 1513

11 LO Pimclllc1 and J Stcin-Schabes IJhys [ell B 216 (19R9)27

12 EJ Gucndc1man Mad Ihys lRIl A 14 (1999) 1()l3 (gr-qc1990 IO17) E1 Gucndclman gr-qc9901 067

13 tv1 Scthi A Balra and D Lohiya Phys Rev ) 60 (1999) 1043(astm-pIV9903084)

14 C Wil1 ll1eory afld experimclIl ifl graVilalioflllJ pllysin (Cam-bridge Univcrsity Press Cambridge 1981)

15 11 Dehncn and O Obregoacuten htropllys Space (i 17 (1972)338 V Johri and K Desikan Gen ReJ Gral 26 (1994) 1217

16 LM Diltlz-Riwra and LO pimenlc1 Pllys Nev IJ 60 (1999)11350 L

Rev Mex Fis 49 S2 (2003) 85-90

COSMOLOGY wrm A TIME DEPENDENT COSMOLOGICAL CONSTA 89of today value of Hubble parameter Ha the aelual value ofdeeeleration parameler qo

41 Dus

We have found lhe following solution for dust

values whieh t can take t lt eP 2 and e gt Oas we can sce from he definition of eP In hisca he eosmologieal term in spile of being abinomial funetion on 1gt deeays with he timeThe eorresponding Rieei scalar and euevalOre in-varianl show that this solution is singular

TIte expansion takes place wilh a eonstant aeeeleration

- ( ) Where 1gt = e a = oc A = 4 2w - 5 e A2 = OP = [(1-2w)7fjeo3 ande isaninlegra-tiao constanl This is an extended inOationary solutionwih a time decaying eosmologieal eonstant and initialsingularity as it is shown by lhe eorresponding Rieeiscalar

(48)

The present values of Ihe dece1eration and Hub-ble pararnelers are given by he following expres-sioos

6 ( k)R= 61gtt - 21gt+ - (46)(l-lgtt) a

3 1 ( k)R=---- -2eP t -2eP-- (47)4 (l-ePt) al

(42)

(41)

Igt(t) = 1gtt-

a(t) = a t

A(Iraquo = AIIgt(t)

P = pa(t)-3

I k=O

Wih Ha - 65 ol 5 km 8-Mpe- we obtain to -301 Gy whieh is too big value eompared wih he glo-bular cluster age

2Ho=-to

(43)Because 1gt t~ lt 1 lhen qo gt O TItis model ex-pands from t = _1gt- until t = O lhen il eon-lraeLsuntil t = 1gt) in bolh cases with positivedeeeleration parameter TIte numerieal value forto depends on he values of lhe free eonstanLS

2 ki O

bull Fork(3+2w) gt OTIte soJutions in terms of lhe time t is given by

Igt(t) = e [1 -1gt tiexcl-

a(t) = ad1 - eP tJ

A(Iraquo = AIgt(t) + A21gt(t)

P = pa(t)-3 (44)

bull Fork(3+2w)ltOThe eorresponding solution in terms ofhe lime tis given as

Igt(t) = c [1 + 1gttiexcl-

a(t) = ad1 + 1gttJ

A(eP) = AIgt(t) + AIgt(t)

P = pa(t)-3 (49)

where

(50)

al = o el

2kAl=2o

ke 3

P=-4 o7fO

and el is an inlcgralion conslanl In this caiC wchave nol restrictions on the valucs which t can ta-ke If -IN lt (to + e) lhe expansion takesplace with non-constant accclcration and wiLhout

(45)

oal = 12e

whcrc

and e is an inlegralion eonslan Clearly a(t) andIgt(t) muSl be positive in order lo be physieallysignifieant this requirement restrieLs the range of

Re Mex F(s 4952 (2003) 85-90

90 LUIS O PIMEIEL LUZ M DlAZ-RIVERA ANO CESAR MORA

singularily ai il is shown fmm thc correspondingRicci scalar and curvalurc invariant

42 Radialon

In thc case in which the maUee contenl of he univcrsc is radialion we have found [161

R 6 ( k)= ( ) 61gt t + 21gt + 2 1 + 1gt t a

( )3 1 kR = 4 bull -21gtt +21gt-2(1+ 1gt t) a

(51)

(52)

1gt(t)= 1gt t-

a(t) = a t)(1)) = )1gt(t)

p = pa(t)- (54)

Aeeordingly R and R do nol diverge For lhissolution the corrcsponding prcscnt valucs of thedcnsity and Hubblc paramctcrs are givcn as

where 1gt = 4e a = (oe)2 and) = OThis is a singo-lar solutioT1 according 10 lhe Ricci scalar which foc this caOCis givCTl hy

(56)

(55)

1Ho = -toqo = O

and lherefme to = 1 Ho ~ 1505 Gy

R=6(1+ a~)t-The deeelcralioo parameler bceomes null instead he lodayHubble parameler is giveo as

(53)

As in lhe previous case lhe values of to dependson he frcc eonstanls of our model but in lhis ca-se fm H~ ~ 1gt = to ~ 1Ho

Acknowlcdgmcnts

LOP was parlially soppmled hy CONACYT granl 42191-F

1 S Weinbcrg Rrv Mod Phys 6t (1989) l

2 Y Ng 1m J Mod Ihys DI (1992) 145

3 f1PBronstcin Ihys eit derSoujelUfljoll3 (1933) 73

4 J Ovtrduin ami F Coopersock Fhys Nev D SH (1998)043506

5 S Capoziexcljcllo and R de Ritis Gefl Re Gral 29 (1997) 1425

6 CM Will UVifl8 Reviews iriexcl Reatilily4 (2001) No 4 iexcliexclvaila-ble al hupllwwwliving-rcvicwsorgArlic1esVolumc42(X)I-4PD Se harre and CM Will Pllys RtI ) 65 (2002) (~t(X)X(hf-qc010 10(1)

7 A Liddlc A Mazurndar and J HarrowIlIys Rev J) 58 (199R)027302

8 F Dahia and C Romero Phys Rev IJ (() (1998) 104014 (gr-qd9812CXll) M Suspcrrcgi and A Mazumdar Phys Rev1J5K(1998) 083511

9 S Perlmutler el al ASlrophys 1 Vol 517 (1999) 565 (as-troph9811(33)

10 LO Pimcnlcl and LM Diacutea-Rivcra Ifll J Mod Plly A 14(1999) 1513

11 LO Pimclllc1 and J Stcin-Schabes IJhys [ell B 216 (19R9)27

12 EJ Gucndc1man Mad Ihys lRIl A 14 (1999) 1()l3 (gr-qc1990 IO17) E1 Gucndclman gr-qc9901 067

13 tv1 Scthi A Balra and D Lohiya Phys Rev ) 60 (1999) 1043(astm-pIV9903084)

14 C Wil1 ll1eory afld experimclIl ifl graVilalioflllJ pllysin (Cam-bridge Univcrsity Press Cambridge 1981)

15 11 Dehncn and O Obregoacuten htropllys Space (i 17 (1972)338 V Johri and K Desikan Gen ReJ Gral 26 (1994) 1217

16 LM Diltlz-Riwra and LO pimenlc1 Pllys Nev IJ 60 (1999)11350 L

Rev Mex Fis 49 S2 (2003) 85-90

90 LUIS O PIMEIEL LUZ M DlAZ-RIVERA ANO CESAR MORA

singularily ai il is shown fmm thc correspondingRicci scalar and curvalurc invariant

42 Radialon

In thc case in which the maUee contenl of he univcrsc is radialion we have found [161

R 6 ( k)= ( ) 61gt t + 21gt + 2 1 + 1gt t a

( )3 1 kR = 4 bull -21gtt +21gt-2(1+ 1gt t) a

(51)

(52)

1gt(t)= 1gt t-

a(t) = a t)(1)) = )1gt(t)

p = pa(t)- (54)

Aeeordingly R and R do nol diverge For lhissolution the corrcsponding prcscnt valucs of thedcnsity and Hubblc paramctcrs are givcn as

where 1gt = 4e a = (oe)2 and) = OThis is a singo-lar solutioT1 according 10 lhe Ricci scalar which foc this caOCis givCTl hy

(56)

(55)

1Ho = -toqo = O

and lherefme to = 1 Ho ~ 1505 Gy

R=6(1+ a~)t-The deeelcralioo parameler bceomes null instead he lodayHubble parameler is giveo as

(53)

As in lhe previous case lhe values of to dependson he frcc eonstanls of our model but in lhis ca-se fm H~ ~ 1gt = to ~ 1Ho

Acknowlcdgmcnts

LOP was parlially soppmled hy CONACYT granl 42191-F

1 S Weinbcrg Rrv Mod Phys 6t (1989) l

2 Y Ng 1m J Mod Ihys DI (1992) 145

3 f1PBronstcin Ihys eit derSoujelUfljoll3 (1933) 73

4 J Ovtrduin ami F Coopersock Fhys Nev D SH (1998)043506

5 S Capoziexcljcllo and R de Ritis Gefl Re Gral 29 (1997) 1425

6 CM Will UVifl8 Reviews iriexcl Reatilily4 (2001) No 4 iexcliexclvaila-ble al hupllwwwliving-rcvicwsorgArlic1esVolumc42(X)I-4PD Se harre and CM Will Pllys RtI ) 65 (2002) (~t(X)X(hf-qc010 10(1)

7 A Liddlc A Mazurndar and J HarrowIlIys Rev J) 58 (199R)027302

8 F Dahia and C Romero Phys Rev IJ (() (1998) 104014 (gr-qd9812CXll) M Suspcrrcgi and A Mazumdar Phys Rev1J5K(1998) 083511

9 S Perlmutler el al ASlrophys 1 Vol 517 (1999) 565 (as-troph9811(33)

10 LO Pimcnlcl and LM Diacutea-Rivcra Ifll J Mod Plly A 14(1999) 1513

11 LO Pimclllc1 and J Stcin-Schabes IJhys [ell B 216 (19R9)27

12 EJ Gucndc1man Mad Ihys lRIl A 14 (1999) 1()l3 (gr-qc1990 IO17) E1 Gucndclman gr-qc9901 067

13 tv1 Scthi A Balra and D Lohiya Phys Rev ) 60 (1999) 1043(astm-pIV9903084)

14 C Wil1 ll1eory afld experimclIl ifl graVilalioflllJ pllysin (Cam-bridge Univcrsity Press Cambridge 1981)

15 11 Dehncn and O Obregoacuten htropllys Space (i 17 (1972)338 V Johri and K Desikan Gen ReJ Gral 26 (1994) 1217

16 LM Diltlz-Riwra and LO pimenlc1 Pllys Nev IJ 60 (1999)11350 L

Rev Mex Fis 49 S2 (2003) 85-90