Unified Model for Inflation and the Dark Side of the Universe

0.05 MB·English

by  Gabriel Zsembinszki

Unified Model for Inflation and the Dark Side of
the Universe
Gabriel Zsembinszki
GrupdeFísicaTeòricaandInstitutdeFísicad’AltesEnergies
7 UniversitatAutònomadeBarcelona
0 08193Bellaterra,Barcelona,Spain
0
2
Abstract.
n
We presenta modelwith a complexanda realscalarfieldsanda potentialwhosesymmetryis
a
J explicitlybrokenbyPlanck-scalephysics.Forexponentiallysmallbreaking,themodelaccountsfor
theperiodofinflationintheearlyuniverseandfortheperiodofaccelerationofthelateuniverseor
2
forthedarkmatter,dependingonthesmallnessoftheexplicitbreaking.
1
Keywords: inflation,darkenergy,darkmatter
1
PACS: 98.80.Cq,95.36.+x,95.35.+d
v
9
6
INTRODUCTION
3
1
0
The Standard Model (SM) of particle physics based on the gauge group SU(3)
7
×
0 SU(2) U(1)is considered to be a successful model, able to accommodate all existing
×
/ empirical data with high accuracy. Nevertheless, there are many deep questions for
h
p which the SM is unable to give the right answer, such that many physicists believe that
-
itisnot theultimatetheoryofnature. In anyextensionoftheSM, theideaofsupposing
o
r new additional symmetries is quite justified, taking into account that there are known
t
s symmetries that at low energies are broken, but at higher energies are restored. If we
a
: assume that global symmetries are valid at high energies, we should expect that they
v
are only approximate, since Planck-scale physics breaks them explicitly [1, 2]. Even
i
X
with an extremely small breaking, very interesting effects may appear. As discussed in
r
[3, 4], when aglobalsymmetryis spontaneouslybroken and inadditionthere isa small
a
explicit breaking, the corresponding pseudo-Golstone boson (PGB) can play a role in
cosmology.Thefocusin[3]wastoshowthatthePGBcouldbeadarkmatterconstituent
candidate, whereas in [4] it might play the role of a quintessence field responsible for
thepresent acceleration oftheuniverse.
In the present contribution we will relate the period of very early acceleration of the
universe (inflation) either with the present period of acceleration, or with the mysteri-
ous dark matter, depending on the smallness of the effects of Planck-scale physics in
breaking global symmetries. Direct or indirect observational evidence for the existence
ofdarkenergyanddarkmattertogetherwiththeneedforinflationcomemainlyfromsu-
pernovaeoftypeIaasstandardcandles[5],cosmicmicrowavebackgroundanisotropies
[6], galaxy counts [7] and others [8]. The physics behind inflation, dark matter or dark
energy may be completely unrelated, but it is an appealing possibility that they have a
commonorigin.Anideaforthiskindofunificationis"quintessentialinflation",thathas
beenforwardedbyFriemanandRosenfeld[9].Theirframeworkisanaxionfieldmodel
wherethereisaglobalU(1) symmetry,whichisspontaneouslybrokenatahighscale
PQ
and explicitlybroken byinstantoneffects at thelowenergy QCD scale. Thereal part of
thefield is able to inflate in the early universewhilethe axion boson could be responsi-
bleforthedark energy period.Theauthorsof[9]comparetheirmodelofquintessential
inflation with other models of inflation and/or dark energy. Here, in the framework of
a global symmetry with Planck-scale explicit breaking, we offer an explicit scenario of
quintessential inflation. As an alternative, we also consider the possibility that, in the
same framework, the axion boson is a dark matter constituent. We may have one alter-
nativeortheotherdependingon themagnitudeoftheexplicitsymmetrybreaking.
THE MODEL
In our model, we have a complex field Y that is charged under a certain global U(1)
symmetryand apotentialthat containsthefollowingU(1)-symmetricterm
1
V (Y )= l [ Y 2 v2]2 (1)
1
4 | | −
where l is a coupling constant and v is the energy scale of the spontaneous symmetry
breaking(SSB).
WithoutknowingthedetailsofhowPlanck-scalephysicsbreaksourU(1)symmetry,
weintroducethemostsimpleeffectiveU(1)-breakingterm
1
V (Y )= g Y n Y e id +Y ⋆eid (2)
non−sym − MPn−3| | (cid:16) − (cid:17)
with an integern>3. We base our model on the idea that the coupling g is expected to
be very small [10]. If g is of order 10 30 then we will see that the resulting PGB is a
−
darkmattercandidate, whileforg-valuesoforder10 119 itwillbeaquintessencefield.
−
Thecomplexscalarfield Y maybewrittenin theform
Y =f eiq /v. (3)
Ourbasicideaisthattheradialpartf ofthefieldY isresponsibleforinflation,whereas
the angular part q can play either the role of the present dominating dark energy of the
universe, or of the dark matter, depending on the values of g parameter that appears in
(2).
In order for f to inflate, one has to introduce a new real field c that assists f to
inflate. The c field is supposed to be massive and neutral under U(1). In the process
of SSB at temperatures T v in the early universe, the scalar field f develops in time,
startingfromf =0andgoi∼ngtovaluesdifferentfromzero,asininvertedhybridinflation
[11,12]models.Weshallfollowref.[12]andcouplec toY witha Y Y c 2 term.More
∗
−
specificallyweintroducethefollowingcontributiontothepotential
1 a 2 Y 2c 2 2
V2(Y ,c )= 2m2c c 2+(cid:18)L 2− |4L |2 (cid:19) (4)
where a is a coupling and L and mc are mass scales. The interaction between the two
fieldswillgivetheneededbehavioroftherealpartofY togiveinflation.Suchmodelsof
inflationarerealizedinsupersymmetry,usingagloballysupersymmetricscalarpotential
[12].
Tosummarize,ourmodelhasacomplexfieldY andarealfieldc withatotalpotential
V(Y ,c )=V (Y ,c )+V (Y )+C (5)
sym non sym
−
whereC is a constant that sets the minimumof the effective potential to zero. The non-
symmetricpart isgivenby(2), whereas thesymmetricpart is thesumof(1)and (4),
V (Y ,c )=V (Y )+V (Y ,c ) (6)
sym 1 2
Inflation
Let us study, firstly, the conditions to be imposed on our model to describe the
inflationary stage of expansion of the primordial Universe. In order to do this, we will
only work with the symmetric part of the effective potential, which dominates over the
non-symmetricpartat early times,and aftermakingthereplacement(3) weobtain
1 a 4f 4c 4 1
Vsym(f ,c )=L 4+2 m2c −a 2f 2 c 2+ 16L 4 +4l (f 2−v2)2, (7)
(cid:16) (cid:17)
Here, f is the inflaton field and c is the field that plays the role of an auxiliary field,
which ends the inflationary regime through a "waterfall" mechanism. We note that the
f 4c 4 term in Eq.(7) does not play an important role during inflation, but only after it
ends,and itsetsthepositionoftheglobalminimumofV (f ,c ).
sym
From (7) we notice that the field c has an effectivemass givenby M2 =m2 a 2f 2,
c c
−
so that for f <f = mc , the only minimum ofV (f ,c ) is at c =0. The curvature of
c a sym
theeffective potentialin the c direction is positive,whilein the f direction is negative.
Because we expect that after theSSB, f is close to the origin and displaced from it due
to quantum fluctuations, it will roll down away from the origin, while c will stay at
its minimum c =0 until the curvature in c direction changes sign. That happens when
f >f and c becomesunstableand startstoroll downits potential.
c
Theconditionstobeimposedonourmodelare thefollowing:
• Thevacuumenergy termin (7)shoulddominateovertheothers:L 4 > 1l v4
4
• The absolute mass squared of the inflaton should be much less than the c -mass
squared, mf2 = l v2 m2c , which fixes the initial conditions for the fields: c is
| | ≪
initially constrained at the stable minimum c =0, and f may slowly roll from its
initialpositionf 0
≃
• Slow-rollconditionsinf -direction,whicharegivenbythefollowingrequirements:
e MP2 Vs′ym 2 1, h MP2Vs′y′m 1, where a prime means derivative with
res≡pe1c6tp to(cid:16)Vfsym(cid:17) ≪ | | ≡ (cid:12)(cid:12)8p Vsym(cid:12)(cid:12) ≪
(cid:12) (cid:12)
• Swuhfefirceiefnt numf b(etr o)f=e-ffolmdsarokfsitnhfleaetniodno:fNsl(ofw)-r=ollRtitennfldaHti(otn)dt = M8pP2 Rffend VVss′yymmdf
end end c
≡
• Fastrollofc fieldattheendofinflation: D Mc2 H2,where D Mc2 istheabsolute
variationofthec -masssquaredinaHubb|letim|e≫H,aroundth|epoin|twheref f
c
≃
• Fast roll of f after c settles down to the minimum. This is possible because the
potentialhasanon-vanishingfirstderivativeatthatpointwhichforcesf tooscillate
around the minimum of the potential, with a frequency w which we want to be
greaterthan theHubbleparameterH:w >H.
Fromthelastconditionweobtainan upperlimitfortheSSB scalev
v<M . (8)
P
Dark matter
As stated above, our idea is that the PGB q that appears after the SSB of U(1) can
play theroleofquintessenceorof dark matter, depending on thevaluesof g-parameter.
Let us start investigating the case where q describes dark matter. For a detailed study
we send the reader to our work [3]. Here, we will just highlight the main features and
conclusionsofourstudyin [3].
Duetothesmallexplicitbreaking oftheU(1)symmetry,q getsamass
n 1
v
m2 =2g − M2 (9)
q (cid:18)M (cid:19) P
P
which depends on the two free parameters v and g. In what follows, we fix the value of
n=4 exceptifexplicitlymentioned.
For q to be a dark matter candidate, it should satisfy the following astrophysical and
cosmologicalconstraints:
• It shouldbestable,witha lifetimet q >t0, wheret0 isthelifetimeoftheuniverse
• Itsdensityshouldbecomparableto thedark matterdensityW q W DM 0.25
∼ ∼
• Because it can be produced in stars, it should not allow for too much energy loss
and rapidcoolingofstars
• Even if it is stable, q can be decaying in the present and thus contribute to the
diffusephotonbackgroundoftheuniverse,whichisboundedexperimentally.
In order to calculate the density of produced q -particles we took into account the dif-
ferent production mechanisms: thermal production in the hot plasma, and non-thermal
productionby q -field oscillationsand from the decay of cosmicstringsproduced in the
SSB. A detailed study [3] showed that for v < 7.2 1012 GeV, there is thermal pro-
duction of q particles, and the number density produ×ced is given by n 0.12T3. The
th
numberdensity produced by themisalignmentmechanism is nosc ≃ 21mq≃v2 and by cos-
mic strings decay is n v2/t . Also, we have to take into account that non-thermal
str str
produced q may finally t≈hermalize, depending on the values of g and v. Astrophysical
constraintsplacealimitonv, butnot ong
v>3.3 109GeV. (10)
×
The combinations of astrophysical and cosmological constraints lead to the following
valuesfor vandg forq tobeadark mattercandidate
v 1011GeV, g 10 30. (11)
−
∼ ∼
Asafinalcomment,wementionthatonecouldobtainvaluesofordertheelectriccharge
forg,ifoneputsn=7, withalln<7 prohibitedforsomeunknownreason.
Dark energy
Let us find now the values for v and g in order for q to be a quintessence field
responsible for the present acceleration of the universe. There are two conditions it
shouldsatisfy:
• The field q should be displaced from the minimum of the potential Vnon sym(q ),
and we suppose that its value is of order v; it will only start to fall tow−ards the
minimumin thefuture
mq <3H0 (12)
• The energy density of the q field, r 0, should be comparable to the present critical
densityr , ifwewant q to explainall ofthedark energy contentoftheuniverse.
c
0
r q r c (13)
∼ 0
Intheaboveequation(12),H istheHubbleconstant.Takingintoaccounttheexpression
0
n 1
forthemassofq , Eq. (9), mq =√2g(cid:16)MvP(cid:17) −2 MP,condition(12)becomes
v n 1 9H2
g − < 0 . (14)
(cid:18)M (cid:19) 2M2
P P
The energy density of the q field is givenby the valueof the non-symmetricpart of the
effective potential, V (f ,q ), with the assumption that the present values of both
non sym
fieldsare oforderv −
n 1
v
r q ≃Vnon−sym(v,v)=g(cid:18)MP(cid:19) − MP2v2. (15)
Introducing (15) into (13) and remembering that the present critical energy density
r = 3H02MP2, wehavethat
c0 8p
v n 1 3H2
g − 0 . (16)
(cid:18)M (cid:19) ≃ 8p v2
P
Combining(14)and (16)weobtaina constrainton v
1
v> M . (17)
P
6
This is the restriction to be imposed on v in order for q to be the field describing dark
energy. Notice that it is independent of n. It is also interesting to obtain the restriction
onthecouplingg, whichcan bedoneifweintroduce(17)into(16)giving
3 6n+1 H2
g< × 0 . (18)
8p M2
P
ReplacingthevalueforH 10 42 GeVandtakingthesmallestvaluen=4,weobtain
0 −
∼
thelimit
g<10 119. (19)
−
CONCLUSIONS
We havepresented a model that is able to explain inflation and dark energy, or inflation
and dark matter. Although it is possible that there is no connection between them,
the idea of unifying such important ingredients of cosmology into the same model is
exciting.
Our model contains two scalar field: one, Y , which is complex and charged under
a certain global U(1) symmetry, and another one, c , which is real and neutral under
U(1). The real part of Y is supposed to give inflation by coupling to the real field
c . The imaginary part of Y can be either a dark matter candidate, or a quintessence
field responsible for the recent acceleration of the universe. We suppose that we have
a U(1)-symmetric potential to which we add a small term which explicitly breaks the
symmetry due to Planck-scale physics. Our conclusion is that the explicit breaking has
to be exponentially suppressed. In fact, this is suggested by quantitative studies on the
breakingofglobalsymmetriesbygravitationaleffects[10].Ifthesuppressionparameter
g is of order 10 30 and v 1011 GeV, the PGB that appears after the SSB ofU(1) is a
−
dark matter candidate. Fo∼r a much stronger suppression g 10 119 and a higher SSB
−
∼
scalev M , thePGB isacandidateto thedark energy oftheuniverse.
P
∼
Previous work on explicit breaking of global symmetries can also be found in [13],
and related to Planck-scale breaking, in [14]. Cosmological consequences of some
classesofPGBs are discussedin[15].
ACKNOWLEDGMENTS
I thank Eduard Massó for all the advices and the good things he taught me during last
years ofcollaboration.Thiswork was supportedby DURSI undergrant 2003FI00138.
REFERENCES
1. SeeforinstanceT.Banks,Physicalia12,19(1990),andreferencestherein.
2. S.B.GiddingsandA.Strominger,Nucl.Phys.B307,854(1988).
S.R.Coleman,Nucl.Phys.B310,643(1988).
G.Gilbert,Nucl.Phys.B328,159(1989).
3. E.Masso,F.RotaandG.Zsembinszki,Phys.Rev.D70,115009(2004)[arXiv:hep-ph/0404289].
4. E.MassoandG.Zsembinszki,“UnifiedmodelforinflationanddarkenergywithPlanck-scaleJCAP
0602,012(2006)[arXiv:astro-ph/0602166].
5. S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517, 565 (1999)
[arXiv:astro-ph/9812133].
A.G.Riessetal.[SupernovaSearchTeamCollaboration],Astron.J.116,1009(1998)[arXiv:astro-
ph/9805201].
P.Astieretal.,Astron.Astrophys.447,31(2006)[arXiv:astro-ph/0510447].
6. D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 148, 175 (2003) [arXiv:astro-
ph/0302209].
A. Balbi et al., Astrophys. J. 545, L1 (2000) [Erratum-ibid. 558, L145 (2001)] [arXiv:astro-
ph/0005124].
C.Pryke,N.W.Halverson,E.M.Leitch,J.Kovac,J.E.Carlstrom,W.L.HolzapfelandM.Dragovan,
Astrophys.J.568,46(2002)[arXiv:astro-ph/0104490].
C. B. Netterfield et al. [Boomerang Collaboration], Astrophys. J. 571 (2002) 604 [arXiv:astro-
ph/0104460].
J.L.Sieversetal.,Astrophys.J.591,599(2003)[arXiv:astro-ph/0205387].
A. Benoit et al. [the Archeops Collaboration], Astron. Astrophys. 399, L25 (2003) [arXiv:astro-
ph/0210306].
C.J.MacTavishetal.,arXiv:astro-ph/0507503.
A.G.Sanchezetal.,Mon.Not.Roy.Astron.Soc.366,189(2006)[arXiv:astro-ph/0507583].
M.Tegmarketal.[SDSSCollaboration],Phys.Rev.D69,103501(2004)[arXiv:astro-ph/0310723].
7. L.Verdeetal.,Mon.Not.Roy.Astron.Soc.335(2002)432[arXiv:astro-ph/0112161].
M.S.Turner,TheAstrophysicalJournal,576:L101-L104,2002.
A.LewisandS.Bridle,Phys.Rev.D66,103511(2002)[arXiv:astro-ph/0205436].
X. m. Wang, M. Tegmark and M. Zaldarriaga, Phys. Rev. D 65, 123001 (2002) [arXiv:astro-
ph/0105091].
8. J.Garcia-Bellido,arXiv:astro-ph/0502139.
9. R.RosenfeldandJ.A.Frieman,JCAP0509,003(2005)[arXiv:astro-ph/0504191].
10. R. Kallosh, A. D. Linde, D. A. Linde and L. Susskind, Phys. Rev. D 52, 912 (1995) [arXiv:hep-
th/9502069].
11. B.A.OvrutandP.J.Steinhardt,Phys.Rev.Lett.53,732(1984).
12. D.H.LythandE.D.Stewart,Phys.Rev.D54,7186(1996)[arXiv:hep-ph/9606412].
13. C.T.HillandG.G.Ross,Phys.Lett.B203,125(1988).
C.T.HillandG.G.Ross,Nucl.Phys.B311,253(1988).
14. M.Lusignoli,A.MasieroandM.Roncadelli,Phys.Lett.B252,247(1990).
S.Ghigna,M.LusignoliandM.Roncadelli,Phys.Lett.B283,278(1992).
D.Grasso,M.LusignoliandM.Roncadelli,Phys.Lett.B288,140(1992).
15. C.T.Hill,D.N.SchrammandJ.N.Fry,CommentsNucl.Part.Phys.19,25(1989).
A.K.Gupta,C.T.Hill,R.HolmanandE.W.Kolb,Phys.Rev.D45,441(1992).
J.A.Frieman,C.T.HillandR.Watkins,Phys.Rev.D46,1226(1992).
J. A. Frieman, C. T. Hill, A. Stebbins and I. Waga, Phys. Rev. Lett. 75, 2077 (1995) [arXiv:astro-
ph/9505060].