Table Of ContentNon-singularmodels ofuniverses inteleparallel theories
Jaume de Haro and Jaume Amoros
∗ †
DepartamentdeMatema`ticaAplicadaI,UniversitatPolite`cnicadeCatalunya,Diagonal647,08028Barcelona,Spain
Differentmodelsofuniversesareconsideredinthecontext ofteleparalleltheories. Assumingthattheuni-
verseisfilledbyafluidwithequationofstate(EoS)P = −ρ−f(ρ),fordifferentteleparalleltheoriesand
differentEoSwestudyitsdynamics. Twoparticularcasesarestudiedindetail: inthefirstoneweconsidera
functionf withtwozeros(twodeSittersolutions)thatmimicsahugecosmologicalconstantatearlytimesand
apressurelessfluidatlatetimes. Inthesecond one, inthecontext of loopquantum cosmology withasmall
cosmologicalconstant,weconsiderapressurelessfluid(P = 0 ⇔ f(ρ) = −ρ)whichmeansthattherearea
deSitterandanantideSittersolution.Inbothcasesoneobtainsanon-singularuniversethatatearlytimesisin
aninflationaryphase,afterleavingthisphaseitpassestroughamatterdominatedphaseandfinallyatlatetimes
itexpandsinanacceleratedway.
3
1 PACSnumbers:04.50.Kd,98.80.-k,98.80.Jk
0
2
1.Introduction.—Teleparalleltheories(F(T)theories)[1– accelerate (inflationary phase) leaving the contracting phase
n 4] are built from the scalar torsion T = 6H2, where H is to enter in the expanding one (it bounces), then it starts to
a −
J the Hubble parameter. The field equationsare second-order, decelerate and enters in a matter dominated phase. Finally,
which is a great advantage to F(R) theories, whose fourth- at late times it enters in a de Sitter phase (late time cosmic
2
2 order equations lead to pathologieslike instabilities or large acceleration).
correctionstoNewton’slaw[5,6]. Thisentailsthatthemod- In some sense our work seems a teleparallel version of
] ifiedFriedmannequationdepictsacurveintheplane(H,ρ), the early idea that F(R) cosmology could unify inflation
c
q thatis,theuniversemovesalongthiscurveanitsdynamicsis with late cosmic acceleration[7] if oneuses a function F of
- givenbythemodifiedRaychaudhuriequationandtheconser- the form F(R) = 1 R a +b(R Λ )m , with
gr vationequation. 2 − (R−Λ1)n − 2
[ This opens the possibility to built non-singularmodels of n,m,a,b,Λ1,Λ2 ≥ 0(cid:16)parameters. However, our p(cid:17)oint of
view is different from the one adopted in current dark en-
2 universes filled by a fluid with equation of state (EoS) P = ergycosmology(see[8]forarecentreview):Where,inmod-
v ρ f(ρ), being P the pressure and ρ the energy density. els of oscillating dark energy one takes an EoS of the form
6 −For−this EoS, the zeros of the function f give de Sitter and P = ω(t)ρ (see forexample[9, 10]), ora scalar field under
3
antideSittersolutions. Then,choosingafunctionf withtwo the actionof an oscillating potential[11]. Itis also different
3
5 zerosoneobtainsanon-singularmodelofuniverse. from the current F(T) or F(R) theories where, in order to
. AsexamplesofteleparalleltheorieswestudyEinsteinCos- reconstructmodelsrealizingcosmologies,theauthorshaveto
1
1 mology(EC)withandwithoutasmallcosmologicalconstant, useverycomplicatedfunctionsF (seeforexample[12–15]).
2 loopquantumcosmology(LQC)withandwithoutasmallcos- Theunitsusedinthepaperarec=~=8πG=1.
1 mologicalconstant, andtheteleparallelversionofthe model
v: F(R)= R µ4,Rbeingthescalarcurvature.Forthesetele- 2. Teleparallelcosmologicaltheories.—Teleparalleltheor-
2 −2R ies are based in the Weitzenbo¨ck space-time. To built this
i parallelexamplesweconsidermodelsofEoS,whichmimica
X space-timeone choosesa globalsystem of fourorthonormal
hugecosmologicalconstantatearlytimesandapressureless
ar dfleunidsitaytolaftethteimunesiv,elriksee.P = −ρρ2i, ρi beingtheinitialenergy vtheectroerlafiteioldnse{ie=i}ewµih∂icµh. aTrheerneloanteediwntirtohdtuhceevseactcoorvsa{ri∂aµn}tdveia-
rivative that defines absolute parallelism with respect the
We will see how these simple models could solve the so- globalba∇sis e , thatis eµ = 0. From this, oneobtains
called coincidenceproblemdue to periodsof accelerationof themetricW{eitiz}enbo¨ckco∇nνneictionΓγ = eγ∂ ei. (Note
µν i ν µ
our universe, that is, they could mimic the evolution of a thatthisconnectionismetric,itsatisfies g =0).
γ µν
universe that begins in an inflationary phase, after leaving it The Weitzenbo¨ck space-time has id∇entically vanishing
passestroughamatterdominatedone,andatlatetimesenters curvature(theRiemanntensorvanishes),butitis nottorsion
inanacceleratedphase. free.OneeffectivelyhasTγ =Γγ Γγ =0. Then
µν νµ− µν 6
We also study the specific case of a universe filled by a in order to built the Lagrangian in the Weitzenbo¨ck space-
pressurelessfluid(P =0)inthecontextofLQCwithasmall time,onehastointroducethecontorsiontensor:
cosmologicalconstant.Inthatcase,atearlytimestheuniverse
1
isinanantideSitterphase,afterleavingthisphaseitstartsto Kµν = Tµν Tνµ T µν , (1)
γ −2 γ − γ − γ
(cid:0) (cid:1)
andthetensor
∗E-mail:[email protected] 1
†E-mail:[email protected] Sγ µν = 2 Kµνγ +δγµTθνθ−δγνTθµθ , (2)
(cid:16) (cid:17)
2
toconstructtheso-calledscalartorsion ThemodifiedRaychaudhuriequationis obtainedfromthe
modified Friedmann equation taking its derivative with re-
T =Sγ µνTγ µν, (3) spect to the time, givingrise to the equation H˙ = f(ρ) .
−4G′(T)
and one can define modified teleparallel theories via a Lag- Then, the dynamicsof the universe is givenby the modified
rangian of the form = VF(T) Vρ, where V is the Raychaudhuriequationandtheconservationequation,i.e. by
T
L −
volumeofthespatialpart. thesystem
What is interesting here is that if one chooses the basis
{e0 = ∂0,e1 = a∂1,e2 = a∂2,e3 = a∂3}, then for the H˙ = −4Gf(′ρ(T)) (9)
Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric one ( ρ˙ = 3Hf(ρ),
obtains[16]T = 6H2.
The problemati−cpointin teleparallelism is that it depends providedtheuniversemovesalongthecurveρ=G(T).
on the choice of the global basis. In the sense that, if one Whatisimportanttostresshereisthatthecriticalpointsof
uses a localLorentztransformationto transformthe original thesystemaregivenbythesolutionsoftheequationf(ρ) =
global basis in another one, in general, one obtains another 0,thatis,thedeSitterandantideSittersolutionsarethesolu-
Weitzenbo¨ck connection and thus T could change [17, 18], tions of the equationf(ρ) = 0. This gives us a simple way
which does not happen in modified gravity (F(R) theories) tobuiltnon-singularmodelsofuniverses: Onehastochoose
where the invariants do not depend of a global basis. How- a functionf with two zeros (two de Sitter solutions, or a de
ever,foraFLRWmetric,ifweuselocalLorentztransforma- Sitter and an anti de Sitter solutions), then the universe will
tionsthat only dependon the time, thatis, of the form¯e = movefromonetotheother.
i
Λk(t)e , then, even though the torsion tensor changes, the To finish this section, we will introduce three parameters
i k
torsionscalarT remainsconstantwithavalueofT = 6H2. whichareimportantincosmology:
−
From our point of view, this gives a consistency with tele- 1. ω P is useful to classify the fluid that fills the uni-
parallel theories in cosmology, because T is invariant from ≡ ρ
verse(whenω < 1thefluidiscalledphantom,when
“isotropicandhomogeneous”localLorentztransformations. −
ω =0dust,whenω =1/3radiation,...).
3. Generalfeatures in teleparalleltheories.—Forthe flat
FLRW space-time filled by a perfect fluid with equation of 2. ω 1 1 R = 1 2H˙ isrelatedtotheex-
eff ≡ 3 − 3H2 − − 3H2
state(EoS)P(ρ) = ρ f(ρ), teleparalleltheories,areob- pansionoftheuniverse. Actually,whenω < 1/3
− − (cid:0) (cid:1) eff −
tainedfromtheLagrangian (respectively ω > 1/3) the universe accelerates
eff
−
(respectivelydecelerates).
=VF(T) Vρ(V), (4)
L −
3. Ω ρ givestheamountofmatterintheuniverse.
whereρ(V)isobtainedfromtheconservationequation(first ≡ 3H2
principleofthermodynamics) For a general teleparallel theory ρ = G(T) one has ω =
1 f(ρ) and
dρ f(ρ) − − ρ
d(ρV)= PdV = ρ˙=3Hf(ρ). (5)
− ⇐⇒ dV V ⇐⇒
f(ρ) 2ρ
ω = 1 , Ω= . (10)
The conjugate momentum is then given by p = ∂ = eff − − TG(T) −T
V ∂VL˙ ′
4HF (T),andthustheHamiltonianis
− ′ 4. Examplesofteleparalleltheories.—In thisSection we
=V˙pV =(2TF′(T) F(T)+ρ)V. (6) will study three important examples of teleparallel theories:
H −L − Einstein cosmology, loop quantum cosmology [19] and the
In general relativity the Hamiltonian is constrained to be teleparallelversionoftheF(R)= R µ4 model[20,21].
zero. (Inf(T)gravitytheHamiltonianisalsoconstrainedto 2 − 2R
be zero, this comesfrom the equationgivenby the variation
oftheactionwithrespectthevierbieneµ. Actually,thecon- Einsteincosmology
i
straincomesfromthe (0,0)componentofthisequation(see
eq. (9)of[15])). ThisconstrainleadstothemodifiedFried- ThefirstexampleofteleparalleltheoryisEinsteinCosmo-
mannequation logy(EC),whichisobtainedfromtheLagrangian
ρ= 2F′(T)T +F(T) G(T), (7) 1
− ≡ = RV ρ(V)V (11)
E
L 2 −
whichisacurveintheplane(H,ρ).
Conversely,givenacurveoftheform ρ = G(T)forsome whereR=6(H˙ +2H2)isthescalarcurvature.
functionG,itcouldbeobtainedfromthemodifiedFriedmann Notethat,theLagrangiancanbewrittenasfollows =
E
equationbychoosing 3dV˙ + ¯ ,where L
dt LE
√ T G(T) 1
F(T)= − dT. (8) ¯ = TV ρV. (12)
− 2 T√ T LE 2 −
Z −
3
Thensince and ¯ differbyatotalderivative,onecan space where quantum states are represented by almost peri-
E E
L L
concludethattheyareequivalent. odic functions of the dynamical part of the connection [22–
InECequation(7)becomestheFriedmannequation 24]. Unfortunately, the connection variable doesn’t corres-
pondtoawelldefinedquantumoperatorinthisHilbertspace,
1
T +ρ=0 H2 =ρ/3, (13) whatleadstore-expressthegravitationalpartoftheHamilto-
2 ⇐⇒
nian in terms of almost periodic functions, which could be
whichdepictsaparabolaintheplane(H,ρ),thatis,theevol- done from a process of regularization. This new regular-
utionof the universefollowsthis parabola,and its dynamics ized (effective) Hamiltonian introduces a quadratic modific-
isgivenbythesystem ation(ρ2)intheFriedmannequationathighenergies[25,26],
whichgivesrisetoabouncewhentheenergydensitybecomes
H˙ = f(ρ)
2 (14) equaltoacriticalvalueoftheorderofthePlanckenergydens-
ρ˙ = 3Hf(ρ).
(cid:26) ity. ThismodifiedFriedmannequationdepictsthe following
NotethatinEConealwayshasω =ω (ρ)= 1 f(ρ) ellipseintheplane(H,ρ)
eff − − ρ
and Ω 1. Once we have obtainedthe dynamicsin EC we ρ ρ H2 (ρ ρc)2
≡ H2 = 1 + − 2 =1, (16)
can study some examplesof EoS. The simplest one is given
3 − ρ ⇐⇒ ρ /12 ρ2/4
by a fluid with ω = ω constant, which means that f(ρ) = (cid:18) c(cid:19) c c
0
−(0(,10)+. Iωn0)thρa,tacnadseth,uits,isthterivoinallytocrsithicoawlpthoainttfoorfωthe>syste1mthies 0w.h2e3r7e5ρbcei≡ngγt2h3λe2Bisartbheeroso-I-mcamllierdzicpriatricaamledteernsaintyd, wλitahpγar∼=a-
0
universemovesfromρ= toρ=0(BigBangsingu−larity). meterwithdimensionsoflength,whichisdeterminedinvok-
Andforaphantomfield(ω∞< 1)theuniversemovesfrom ingthequantumnatureofthegeometry,thatis,identifyingits
0
ρ=0toρ= (BigRipsingul−arity). square with the minimum eigenvalue of the area operator in
∞
To finish EC we will built a cosmological model without LQG,whichgivesasaresultλ √3γ (see[26]).
singularities. Onecoulddothisintroducingasmallcosmolo- Thedynamicsisnowgivenin≡LqQC4bythesystem
gicalconstant, namelyµ4, which couldbe done changingin
the Lagrangian(12) ρ by ρ+µ4. Once we have introduced H˙ = f(ρ) 1 2ρ
2 − ρc (17)
this smallcosmologicalconstant, we will choosea fluid that ( ρ˙ = 3H(cid:16)f(ρ), (cid:17)
for large values of ρ mimics a huge cosmological constant
where the first equation is the modified Raychaudhuriequa-
(Λ=ρ ),andforsmallvaluesofρbecomespressureless. An
i tioninLQC.Theparametersω andΩbecome
exampleofthiskindoffluidsisgivenbyanEoS eff
f(ρ)ρ 2ρ ρ
ρ2 ρ ρ ω = 1 c− , Ω= c . (18)
eff
P = ω = f(ρ)= ρ 1 . (15) − − ρ ρ ρ ρ ρ
c c
−ρ ⇔ −ρ ⇔ − − ρ − −
i i (cid:18) i(cid:19) As a first example, we can study an EoS with f(ρ) < 0
Then,thesystemhastwocriticalpointsaf = (µ2/√3,0) (non-phantom fluid because ω > 1). Then one obtains
−
and a = ( µ4+ρ /√3,ρ ). At the pointa one hasω = a cyclic universe moving in an clockwise sense along the
i i i i
ωeff = −1pandΩ ∼= 1 (deSitterphase),at af onehasω = ellipse. This universe bounces at points a1 = (0,0) and
ω = 1 and Ω = 0 (de Sitter phase), and when µ4 a =(0,ρ ).Ina theuniverseentersinthecontractingphase
eff 2 c 1
− ≪
ρ ≪ρi onehasω =ωeff ∼=0andΩ ∼=1(matterdominated and in a2 the universe enters in the expanding one. On the
phase). other hand, if one considers a phantom fluid f(ρ) > 0, one
Thismodelshowsauniverseevolvingfromanearlyinfla- obtains a cyclic universe moving in an anti-clockwise sense
tionary phase (point a ) to a late time accelerated expansion alongtheellipse.
i
(pointa )passingtroughamatterdominatedphase. As a second example, we consider a fluid with ω = ω
f 0
constant,whichmeansthatf(ρ)= (1+ω )ρ,andthus,the
Remark .1. Note that asymptotically, this model converge − 0
onlycriticalpointofthesystem is (0,0). Inthissituation, it
to the LCDM model, because at late times ρ 0 and con-
sequentlythe cosmologicalconstantµ4 domin→ates. Thenthe is easy to show that for ω0 > −1 the universe moves from
(0,0) to itself in a clockwise sense, and for a phantom fluid
deceleration,thejerkandsnarkparameters(see[12]forthe
(ω < 1)theuniversemovesfrom(0,0)toitselfinananti-
definition of these parameters) converge respectively to 1, 0 −
− clockwisesense.
1 and s = 0. Note also that the model may be solved ana-
Thirdexampleismoreinterestingbecausewewillbuilda
lyticallyandthenonecouldevaluatetheseparametersinthe
model without singularities. To do this we perform a small
differentphasesinwhichouruniverseevolves.
modificationinthemodifiedFriedmannequation,introducing
a small cosmological constant µ4 (changing ρ by ρ+µ4 in
equation (16)) satisfying µ4 ρ . Then, equation (16) be-
Loopquantumcosmology c
≪
comestheellipse
The main idea of LQC is thatit assumes a discrete nature ρ+µ4 ρ+µ4
H2 = 1 . (19)
ofspacewhichleads, atquantumlevel, toconsideraHilbert 3 − ρ
(cid:18) c (cid:19)
4
Forapressurelessfluid(P =0 f(ρ)= ρ),thesystem toa dynamicalsystem withtwo criticalpoints(twodeSitter
⇔ −
has two critical points af ≡ (√µ23 1− µρc4 ,0) and ai ≡ sNoolutetiothnast),aftorafthi≡s m(o√d36eµl,2,w0e)daondnoati n≡eed(ρtio+i√ntρ6r2io+d3uµc4e,ρani)y.
r
(cid:16) (cid:17)
( µ2 1 µ4 ,0). ThefirstoneisadeSittersolutionand smallcosmologicalconstantbecauseitisimplicitlycontained
−√3 − ρc inthemodel.
r
the secon(cid:16)d one is(cid:17)an anti de Sitter solution. In that case the
At the de Sitter points ω equals to 1 and Ω take the
eff
universe moves along the ellipse from a to a in a clock- −
i f values
wisesense. Whenitarrivesatthepointa = ( ρ /4,ρ /4)
1 ∼ − c c 2ρ
the universe starts to accelerate (inflationary phase) because Ω(a )=0andΩ(a )= i =1, (22)
aa¨ = H˙ +H2 ≥ 0 when ρ & ρc/4, then it bounces at the f i ρi+ ρ2i +3µ4 ∼
topoftheellipse,i.e.,atthepointa = (ρ µ4,0)theuni-
2 c− andwhen √3µ2 ρ ρ , onehpasω = 0 andΩ = 1,
verse leaves the contracting phase and enters the expanding ≪ ≪ i eff ∼ ∼
that is, the universeis in the matterdominatedphase, which
one where the energy density starts to decrease. When the
meansthattheuniversemovesfromthepointa tothepoint
uanndivewrsheenartrhiveedseantsiaty3 s∼=ati(sρfice/s4ρ,cρc/4)ρit starµts4ttohedeucneilveerarstee af along the curve ρ = −T2 + 32µT4 passing troiugh a matter
≪ ≪ dominatedphase.
enters in a matter dominatedphase (ω = 0 and Ω = 1).
eff ∼ ∼
TheauthorswouldliketothankProfessorSergeiD.Odint-
Finally,afterleavingthisphaseitgoesasymptotically,atlate
sov for his valuable and useful comments. This investiga-
times,tothepointa (deSitterphasethatmimicsthelatetime
f
tion has been supported in part by MICINN (Spain), pro-
acceleratedcosmicexpansion).
jects MTM2008-06349-C03-01 and MTM2009-14163-C02-
Note that, a modelofuniversemovingfroma inflationary
02, and by AGAUR (Generalitat de Catalunya), contracts
phasetowardsalatetimeacceleratingphasepassingtrougha
2009SGR345and994.
matter dominated phase, could also obtained in LQC with a
smallcosmologicalconstantbychoosingasEoSofthemodel
givenin(15),withρ ρ /2.ForthisEoS,theuniversemoves
c
≈
from the de Sitter solution a = ( ρi+µ4 1 ρi+µ4 ,ρ )
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