Axion Phantom Energy

0.15 MB·

by  Pedro F. Gonzalez-Diaz

IMAFF-RCA-03-03
Axion Phantom Energy
Pedro F. Gonz´alez-D´ıaz
Centro de F´ısica “Miguel A. Catal´an”, Instituto de Matema´ticas y F´ısica Fundamental,
Consejo Superior de Investigaciones Cient´ıficas, Serrano 121, 28006 Madrid (SPAIN).
(Dated: February 1, 2008)
Theexistenceofphantomenergyinauniversewhichevolvestoeventuallyshowabigripdoomsday
isapossibilitywhichisnotexcludedbypresentobservationalconstraints. Inthisletteritisargued
thatthefieldtheoryassociatedwithasimplequintessencemodeliscompatiblewithafielddefinition
which isinterpretablein termsof arank-threeaxionic tensorfield,wheneverweconsider aperfect-
fluidequation ofstatethatcorrespondstothephantomenergyregime. Explicit expressions forthe
axionic field and its potential, both in terms of an imaginary scalar field, are derived which show
that these quantitiesboth divergeat thebig rip, and that theonset of phantom-energy dominance
must takeplace just at present.
4
0 PACSnumbers: 14.80.Mz,98.80. Cq
0
2
I. INTRODUCTION clusters,galaxies,stars,planetsand,ultimately,nucleons
n
and leptons in it are all ripped apart. Even though such
a
a big rip does not take place in some phantom-energy
J Cosmology has become the current commonplace
models that use Chaplygin-like equations of state [16],
3 wherethegreatestproblemsofphysicsarebeingconcen-
it seems to be the simplest and most natural possibility
1 trated or clearest manifest themselves. Along with the
nature of dark matter, the origin of supermassive black stemming fromphantomenergy. Onthe other hand, the

1 big ripcosmic scenariointerestinglyadds anextra quali-
holes and several unexplained huge amounts of released
v tativefeaturetotheknownsetofcosmologicalmodelsas
astrophysicalenergyareonlysomeillustrativeexamples.
2 it introduces a curvature singularity, other than that at
However, perhaps the biggest problem of all of physics
8
thebigbang,atafinite,nonzerovalueofthecosmictime.
0 be the so-called dark energy problem [1] by which it is
However, in spite of the feature that the blowing up of
1 knownthatnearlytheseventypercentofthetotalenergy
thescalefactorappearstobeunavoidableinmodelswith
0 in the universe is in the form of an unobserved vacuum
4 energy which is responsible for the present accelerating equationofstatep=ωρandω < 1,acompleteaccount
−
0 ofthenatureandpropertiesofthefieldtheoryassociated
expansionoftheuniverse[2,3]andwhosenatureremains
/ withsuchmodelshasnotbeendoneyet. Thispaperaims
h being a mysterytoday [4]. There has been a rather huge
at investigating the characteristicsof the massless scalar
t influx of papers in recentyears trying to shedsome light
- field which allow emergence of a big rip singularity in
p onto the existence andthe kind ofthe possible stuff that
e may make up dark energy [5]. the simplest of such phantom energy models, assuming
h a perfect-fluid equation of state. It will be shown that
Four main candidates to represent dark energy have
: the stuff making up phantom energy can be interpreted
v been hitherto suggested: A positive cosmological con-
Xi stant [6], the quintessence fields (which may [7] or may to be a vacuum sea of cosmic axions which can be de-
scribed in terms of the kind of rank-three tensor field
not [8] be tracked), some generalizations of the Chaply-
r strengths predicted in supergravity and strings theories.
a gin gas [9], and the so-called tachyon model of Padman-
abhan et al. [10]. Even though such models could be This paper can be outlined as follows. In Sec. II we
argue in favor of the idea that superlight axions are the
all accommodated to present observational constraints,
sourceofcosmic phantomenergy. Asimple cosmological
these models are posing new or traditional problems in
model accounting for a big rip singularity in the case
such a way that none of them become completely sat-
that vacuum is filled with phantom energy is discussed
isfactory. In particular, cosmological data from current
inSec. III. Sec. IV containsthe solutionofthe phantom
observations do not exclude [11] but may be suggesting
fieldtheorywithinthe simple cosmologicalmodelofSec.
[12] values of the parameter ω in the perfect-fluid equa-
III. We check that both the scalar field and its potential
tion of state p = ωρ of the most popular quintessence
also have a singularity at the big rip. The results are
models which are smaller than -1. If this were the case,
summarized in Sec. V.
thendarkenergywouldbecomewhatisnownamedphan-
tomenergy[13],anentity violatingthe dominantenergy
conditionandtherebyallowingthe naturaloccurrenceof
wormholes and ringholes and even their corresponding II. AXIONS AS THE SOURCE OF PHANTOM
time machines in the universe [14]. ENERGY
Caldwell, Kamionkowski and Weinberg have recently
noticed[15]thatintheframeworkofquintessencemodels Thedefinitionofthemasslessscalarfieldφwhichisas-
phantomenergymayleadtoadoomsdayfortheuniverse sumedto makeupdarkenergyisusuallytakentobe the
- which would take place at a big rip singularity - once conventionalsimplest one; that is, in terms of pressurep
2
and energy density ρ, beingthescalefactor),isadded;i.e. inthe flatgeometry
case
1 1
ρ= φ˙2+V(φ), p= φ˙2 V(φ), (1)
2 2 − T
1 T a˙2 φ˙2 a3φ˙φ
S = dta3 +8πG , (6)
wsthaeterepV=(φω)ρisantdheEfiqe.ld(1p)oitteinmtimale.diFartoemlytfohleloewqsuathtiaotn of ′ 16πGZ0 "−N N #− N (cid:12)(cid:12)0
(cid:12)
(cid:12)
(cid:12)
φ˙2 whereN isthelapsefunction. Thekeypointthenisthat
ρ= . (2)
thesolutiontotheaboveequationsofmotionisevenalso
1+ω
preserved when we omit such an extra boundary term
Thus, if the weak energy condition ρ 0 is taken to be in the action for the scalar field φ, provided this scalar
≥
always satisfied [17], the requirement p+ρ <0, ω < 1 fieldissimultaneouslyrotatedtothepureimaginaryaxis,
−
from phantom energy in this kind of models necessarily φ iΦ. If the scalar field is equipped with a potential
implies thatthe field φ oughtto be pure imaginary;that V(→φ), then an extra term T dtNA3V(φ) should be
istosay,ifthevacuumenergydensityforaphantomvac- added to the action. In th−at0case, the axionic action
uum as referred to any timelike observer has to be posi- can be obtained by simply rotRating φ iΦ in an action
tive, then the massless scalar field making up the phan- containing the above potential term, w→ithout any extra
tom stuff should be pure imaginary. I will argue in what boundary term.
follows that, in the classicalframework, such a massless,
Such an property, which was first noticed in Eu-
pure imaginary scalar field actually represents an axion
clideanized solutions such as wormholes and other in-
describable as a rank-three antisymmetric tensor field,
stantons describing nucleation of baby universes [19], is
considering after the associatedcosmic theory for such a
actually independent of the metric signature and, there-
field. In fact, the Lorentzianactionthat couples such an
fore,appliesalsotoourLorentziancosmologicalcontext.
axion field to gravity can generally be written as
Thus, one can interpret that the stuff that makes up
phantom energy can be regarded to be a rank-three an-
R
S = d4x√ g A2+L , (3) tisymmetric tensor axionic field. It is usually thought
m
− 16πG −
Z (cid:18) (cid:19) thataxionscanexplaintheabsenceofanelectricaldipole
moment for the neutron and thereby solve the so-called
where L is the Lagrangian for observable matter and
m strong CP problem [20]. The axions are chargeless and
A2 A Aµνα if we choose the axion to be given by a
thre≡e-foµrmναA=dB field strength so that dA=0, that is spinlessparticleswithverytinymasswhichinteractwith
ordinarymatter only very weakly. Such particlesare be-
as a rank-three antisymmetric tensor field strength of a
lievedtohavebeenabundantlyproducedinthebigbang.
type arising in supergravity-theory motivated quantum-
Itisworthnoticingthatwhereasrelicaxionsareanexcel-
gravity solutions [18]. The equations of motion derived
lent candidate for the dark matter in the universe [21],
from action (3) are
their vacuum quantum background could make cosmic
phantom energy.
1 1
R g R=16πG 3A2 g A2+T(m) (4)
µν − 2 µν µν − 2 µν µν The coincidence and fine tuning problems could be
(cid:18) (cid:19) thought to become exacerbated in the present scenario
where one sets ω constant and < 1. However, in dark
−
energy models such as the generalized Chaplygin gas [9]
d A=0, (5)
and tachyon models [10], dark energy and dark matter
∗
are described as separate limiting cases from an exist-
in which A2 = A Aαβ, the asterisk denotes Hodge
µν µαβ ν ing unique field. Partly inspired by these models one
dual, and T(m) is the momentum-energy tensor for ordi- couldnaivelyassumetheexistenceofauniqueaxionfield
µν
nary observable matter. One now can check that any which,whenexcited,wouldmakedarkmatter,andwhen
explicit solution to these equations of motion subject atits vacuumgroundstate wouldbe the sourceof phan-
to the usual Friedmann-Robertson-Walker (FRW) sym- tom energy. Coincidence time could then be interpreted
metry for the metric and the corresponding spherically asthetimewhenboththevacuumandtheexcitedstates
symmetric ansatz for the axion antisymmetric tensor are approximately equally populated. Of course, this
A = f(r)ǫ (with ǫ the volume form which, when inte- wouldnotsolvethecoincidenceandfinetuningproblems
grated over a surface of constant radius, yields the area but providedsome explanationto these problems andto
of the unit three-sphere A = 2π2/Γ(2)), if we set e.g. thepredictionofgeneratingsuchstrangelysmallamounts
L = 0 and f(r) = Const., is the same as the solution of homogeneously distributed axions. It remains never-
m
obtained from the equations of motion derived from the theless an intriguing possibility that such a small value
FRW action integral describing coupling of a real mass- ofthecosmologicalconstantbesuppliedbythepotential
less scalar field φ to gravity where an extra boundary energy density of an ”ultrainvisible” axion field which
term accounting for the axion properties, a3φ˙φT (a can be dubbed quintessence axion [22].
− |0
3
III. THE COSMOLOGICAL MODEL IV. THE FIELD THEORY
We shall consider in what follows a simple model Now we shall consider the field theory which is asso-
where the massless scalar field φ is a quintessential field, ciated with phantom energy, starting with the general
equipped with a potential V(φ), whichis minimally cou- formalism for any value of ω. From the equation of mo-
pled to Hilbert-Einstein gravity. The action for this tion for the scalar field φ,
model is
dV(φ)
φ¨+3Hφ˙ = , (13)
R 1 − dφ
S = d4x√ g + ∂iφ∂ φ V(φ)+L ,
c i m
− 16πG 2 −
Z (cid:18) (cid:19) where H a˙/a and V(φ) again is the potential for the
(7) ≡
field φ, and the expression for kinetic energy of the field
where L again is the Lagrangianfor observable matter
m [Eq. (2)], we can obtain a relation between dV/dφ and
fields. If we again restrict ourselves to the case where
the scale factor which can be expressed in the form
(i)L =0,(ii)boththescalarfieldandHilbert-Einstein
m
gravitysatisfytheFriedmann-Robertson-Walkersymme- 3 dV
try, and (iii) we assume a perfect fluid equation of state, A(1+ω)(1 ω)a−3(1+ω) = . (14)
2 − −dφ
then by integrating the conservation law for cosmic en-
p
ergy, dρ = 3(p+ρ)da/a, it turns out that the energy On the other hand, integrating the equation of motion
−
density for the quintessence scalar field will be given by for the field φ we can derive the dependence of this field
with time t.
ρ=Ra 3(1+ω), (8)
− 2 a3(1+ω)/2+ 3(1+ω)√A(t t )
φ(t)= ln 0 2 − 0 ,
in which R is an integration constant. From the Fried- 3√1+ωℓp " B0 #
mann equation derived from action (7) subject to the
(15)
FRW symmetry and L =0,
m where ℓ = R/A is the Planck length and B is an
p 0
integration constant.
a˙ 2 8πGR p
=Aa−3(1+ω), A= , (9) From Eqs. (10) and (15) an expression for the scale
a 3 factor in terms of the scalar field can now be derived. It
(cid:18) (cid:19)
is:
we can obtain the solution
2 ℓ φ
a(t)= a3(1+ω)/2+ 3(1+ω)√A(t t ) 3(12+ω) , (10) a(φ)=B03(1+ω) exp(cid:18)√1p+ω(cid:19). (16)
" 0 2 − 0 # UsingEqs. (14)and(16)wefinallyobtaintheexpression
of the field potential for any ω < 1/3 as a function of
−
where a0 and t0 are the initial radius and time, respec- the field φ
tively. Note that for ω > 1 this solution describes
− 1
awnaradcsceinlefirnaittiyngasuntiverse.wWheosaereschaelerefamctoosrt iinntcerreeassteesdtoin- V(φ)=V0+2(1−ω)√RB0−2exp −3√1+ωℓpφ , (17)
the observationally→no∞t excluded yet case where ω < 1 in which V is a constant. Potentia(cid:0)ls with this for(cid:1)m nat-
− 0
which correspondsto the so-calledphantomdark energy urally arise in supergravity models [23] and have been
for which the dominant energy condition is generally vi- used in a variety of context, ranging from accelerating
olated, i.e. [15] expansion models [24] to cosmological scaling solutions
[25].
p+ρ<0, (11)
Together with the expression for the field φ in terms
of time t, this expression solves the field-theory problem
even though the energy density is surprisingly ever in-
for any value of ω < 1/3. According to our discussion
creasing. Notice furthermore that in this case the scale −
above, for axionic phantom energy we have to perform
factor blows up at a finite time,
the continuation φ iΦ in addition to taking ω < 1.
→ −
In such a case, the solution of the field theory problem
2
t =t + , (12) would read
∗ 0 3(ω 1)a3(|ω|−1)/2
| |− 0
Φ(t)=
giving rise to a ”big rip” [15]. One would expect that
in the vecinity of time t the size of all local geometrical
othbejelcatrsgeinrcar0eaasnedd iωn athrea∗tnheearredrrtahmeadtoicomwasyd.ayW,aensdeehethnacet 2 ln a−03(|ω|−1)/2− 3(|ω|−21)√A(t−t0)
the time expected| t|o have macroscopically increased ge- −3 ω 1ℓp " B0 #
| |−
ometrical objects in our universe. (18)
p
4
for phantomenergy dominationand catastrophicbig rip
with ω < 1 and
−
V(Φ)V(Φ)
V ′ ′′
Γ= =1,
[V(Φ)]2
′
at a critical point defined by
V(Φ)
′
8πGλ = = 3 ω 1ℓ ,
c p
−V(Φ) − | |−
p
forwhichω = 1 λ2/3(See alsophantommodels with
− − c
Born-Infeld type Lagrangians [27].)
Note that the potential V(Φ) takes on a value
1
V V =V0+ 2(1+|ω|)√Ra30(|ω|−1) (20)
0
attheinitialtimet=t tosteadilyincreasethereafterup
0
toinfinitywhent=t ,atthebigrip. Fromthatmoment
onthe potential woul∗dcontinuously decrease,tending to
reachitsconstantminimumvalueV ast (seesolid
0
→∞
curves in Fig. 1). That behaviour is in sharp contrast
withthatofpotentialV(φ)fordarkenergywithω > 1,
−
which starts with a value
t t
0 * 1
time V =V0+ 2(1−ω)√Ra−03(1+ω)
FIG. 1: The potential V for imaginary (solid lines) and real tomonotonouslydecreasedowntothevalueV0ast ,
→∞
(dashedline)scalarfield. Thescalesforthepotentialandtime such as it is also shown by the dashed curve in Fig. 1.
are arbitrary. t is the time at which the big-rip singularity We have solved in this way the field theory which is as-
∗
wouldoccurinthefuture. Itisworthnoticingthatthesingu- sociated with dark and phantom energy if the latter is
larityinthespacetimecurvatureatt∗ coincideswithasingu- interpreted as originating from the existence of an ax-
larityintheimaginaryfieldΦandthepotentialV(Φ). Incon- ionicfieldwhichisexpressibleasamassless,purelyimag-
structingthisfigureithasbeenassumedthattheinitialscale
inaryscalarfield. Theexplicitemergenceofaxionsinthe
factorandtheparameterenteringtherespectiveequationsof
cosmologicalcontextof phantomdarkenergy appearsto
state are such that a0 >> [(1−ωd)/(1+|ωp|)]1/[3(ωd+|ωp|)], be just another example where such rather elusive par-
where ω >−1 and ω <−1, with a sufficiently large.
d p 0
ticles are invoked as candidates for inexorably needed
cosmic constituents which are themselves defined to be
unobservable. Actually, apart of being a key ingredi-
ent to solve the strong-CP problem [20] or to represent
1
V(Φ)=V0+ 2(1+|ω|)√RB0−2exp 3 |ω|−1ℓpΦ . qreuaadnytubmee-ngrwaviditeylytuopseodlotgoysochlvaenagevsar[1ie9t]y, oafxaiosntrsohpahvyesicaal-l
(cid:16) p ((cid:17)19)
and cosmologicalshortcomings [28].
As to the physical motivation for the exponential ax-
It appears of some interest to briefly comment next
ionpotential(19),Iwillbrieflycommentontheinfluence
on a potentially fruitful consequence from the connec-
that the matter-field sources may have on the stability
tion of axions with our cosmological model. If we are
of the present axion model of phantom energy. If, corre-
actually living in a universe with phantom energy where
sponding to the physically interesting case that V =0,
m 6 there will occur a big rip singularity which can never be
wewouldlettheinfluenceofmattersourcesontheexpan-
circumventedbycausally-violatingconnectionstothefu-
sionfactorimmediatelybeforeofphantomdominationto
ture, then the contracting branch of solution (9) can be
have the form of a perturbation, and analyze the result-
physically disregardedas the big-rip singularityis a true
ing model via phase space [26], it can be seen that if we
curvature singularity. In such a case, only the branch
choose V =0 then potential (19) and the solution given
0 of potential V(Φ) before the singularity can have phys-
by Eq. (10) (with ω < 1) and Eq. (18), that is for
− ical significance, and hence the global minimum of that
2 1 a0 3(|ω|−1)/2 potential would become placed at t = t0 with the value
− a given by Eq. (20). It corresponds to a value of the field
t=t + ,
correspond (see0e.g.3Ra03eh(f|.ω|[−216(cid:0))]/)2t(cid:1)(o|ωa|−dy1n)a√miAical attractor Φm = 3ℓ 2ω 1ln B0a03(|ω|−1)/2 . (21)
p
| |− (cid:16) (cid:17)
p
5
Interpretingpotential(19)asthataxionpotentialresult- breaking scale, f 1019(ω 1) 1 GeV, would imply
A −
∼ | |−
ing from the QCD nonperturbative effects, we can take a mass of the phantom axions
[29] Φ = f θ , with f a constant and θ the ef-
m A eff A eff
fective θ parameter resulting after the diagonalization 107 GeV
m 0.62eV 10 12 ω 1eV,
of the quark masses in the QCD Lagrangian. Setting A ∼ f ∼ − | |−
A
thenfA =2/[3ℓp |ω|−1],θeff =ln B0a03(|ω|−1)/2 and i.e. just at the extremal minimum ofpthe allowed values
B0 = 1 it followspthat, since θeff < 1(cid:16)0−9 [29], the i(cid:17)nitial for the axion mass, and clearly well below the cut off
radiusofthe phantomenergyuniversea shouldbe very
0 required by Carroll et al. [30] for phantom energy to be
closetounity,thatistheonsetofsuchaphantomenergy
stable.
regime, if it ever at all occurred, must always be placed
at nearly just the present epoch. Actually, if θ is can-
eff
celled to completely solve the strong CP problem [29],
V. SUMMARY
then the choice B = 1 implied that a would exactly
0 0
satisfy a = 1. In such a case, the observational setting
0
Imposing the weak energy condition, the negative ki-
ofthevalueofthestateequationparameterω wouldalso
netic energy for a phantom field is interpreted as being
set the value of a , and hence of time t .
0
A potential problem with the propo∗sal in this paper originated from super-light axions. We have built up a
simple cosmological model encompassing constant equa-
could be that whereas current axion theories have a cut
tions of state for both ω > 1 andω < 1. In the latter
offonthescaleofinflationaryenergy,typicallyataround
− −
the GUT characteristicenergyof1016 GeV[29],inorder case the universe evolves toward a singularity at finite
time. The field theory is then solved for all cases, deriv-
to avoid phantom energy decaying into gravitons, a cut
ing an increasing exponential potential for the phantom
off of < 100 MeV should be introduced in the effective
field which also diverges at the singularity and shows a
phantom theory [30]. Not with standing, while the for-
dynamical attractor also when matter fields are present
mercutoffappearsrelevantfordarkmatteraxionswhich
that inexorably leads to a catastrophic big rip.
can initially be highly excited, the axion field proposed
in this paper as the source of phantom energy can never Acknowledgments
be excitedoutside the vacuum. Onthe otherhand, if we
keep the current values for color anomaly of the Peccei- The author thanks Carmen L. Sigu¨enza for useful dis-
Quinnsymmetry,thepionandquarkmassesandthepion cussions. This work was supported by DGICYT under
decay constant, then the above Peccei-Quinnsymmetry- Research Project BMF2002-03758.
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