Tracking Dark Energy from Axion-Gauge Field Couplings

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by  Stephon Alexander

Tracking Dark Energy from Axion-Gauge Field Couplings
Stephon Alexander,1,∗ Robert Brandenberger,2,† and Ju¨rg Fr¨ohlich3,‡
1Department of Physics, Brown University, Providence, RI, 02912, USA
2Physics Department, McGill University, Montreal, QC, H3A 2T8, Canada, and
Institute for Theoretical Studies, ETH Zu¨rich, CH-8092 Zu¨rich, Switzerland
3Institute of Theoretical Physics, ETH Zu¨rich, CH-8093 Zu¨rich, Switzerland
(Dated: December 13, 2016)
We propose a toy model of Dark Energy in which the degrees of freedom currently dominating
theenergydensityoftheuniversearedescribedbyapseudo-scalar“axionfield”linearlycoupledto
thePontryagindensity,tr(F∧F),i.e.,theexteriorderivativeoftheChern-Simonsform,ofagauge
field. Weassumethattheaxionhasself-interactionscorrespondingtoanexponentialpotential. We
argue that a non-vanishing magnetic helicity of the gauge field leads to slow-rolling of the axion at
field values far below the Planck scale. Our proposal suggests a “Tracking Dark Energy Scenario”
6 in which the contribution of the axion energy density to the total energy density is constant (and
1 small),duringtheearlyradiationphase,untilaseculargrowthtermproportionaltothePontryagin
0 density of the gauge field becomes dominant. The initially small contribution of the axion field to
2
the total energy density is related to the observed small baryon-to-entropy ratio.

c
e
D I. INTRODUCTION self-interactions described by a potential term, V(φ), in
the action functional. As has previously been observed
2
in the context of inflationary models in [9, 10] (see also
1 Asiswellknown,theStrongCPProblemrelatedtothe
[11]), the coupling of the axion to the instanton den-
vacuum structure of QCD, as described by the vacuum
] angle θ, can be solved by promoting θ to a pseudo-scalar sity of Aµ can lead to slow-rolling of φ also for values
h of φ much smaller than the Planck mass. We show that
t field, the axion [1]. This field gives rise to a new species
- slow-rolling of φ leads to a “tracking solution” with the
of light particles. It can be interpreted as the phase,
p property that the energy density of φ tracks that of the
e relatedtoaU(1)-symmetry,ofacomplexscalarfield[2].
radiation-dominatedbackgroundoftheearlyUniverseup
h A non-vanishing vacuum expectation value of the scalar
to a time t when a secular growth term in the magnetic
[ fieldleadstothespontaneousbreakingofthissymmetry. c
helicity of the gauge field starts to dominate. From this
3 The field quanta of the axion are the Goldstone bosons
timeon,thecontributionofφtotheenergydensityofthe
v accompanyingthebreakingoftheU(1)-symmetry. They
Universe starts to grow until it might actually dominate
7 may acquire a mass through instanton effects and can
it at some late time. Choosing parameter values moti-
5 be made “invisible” by choosing the symmetry breaking
0 scale to be sufficiently high [3]. The axion may then be vated by the observed small baryon-to-entropy ratio, we
0 arrive at a scenario in which the currently observed dark
a candidate for dark matter; (see e.g. [4]). Some time
0 energy in the Universe may come from the axion field
ago, it has been suggested [5] that, besides the QCD
. φ. Thus, our mechanism might represent a realization of
1 axion, therecouldexistaneffectiveaxionfieldconjugate
0 the tracking dark energy scenario previously discussed in
to the anomalous axial vector current in QED. The time
6 [12]; (see also [13]).
derivative of this axion field would then play the role of
1
a space-time dependent chemical potential for the axial In the following section we describe some key features
:
v charge density in QED and, through the chiral anomaly, of our scenario. One such feature is a secular growth in
Xi would give rise to an instability triggering the growth of cosmological time of the electric component of the gauge
low-frequency magnetic fields with non-trivial helicity; field tensor. This is discussed in more detail in Section
r
a see also [6]. Possible applications of this observation to III, where we derive the gauge field equations of motion
earlyuniversecosmology,andinparticulartotheissueof inthepresenceofatermcouplingthePontryagindensity,
the generation of primordial magnetic fields, have been i.e., the exterior derivative of the Chern-Simons 3-form,
discussed in [7]; (see also [8], [6]). to the axion field. We then attempt to find solutions of
these equations that yield a homogeneous and isotropic
In this paper, we explore the possibility that an axion
Pontryagin density. It turns out that such solutions only
field, φ, linearly coupled to the Pontryagin- or “instan-
exist for non-abelian gauge fields. In Section IV we con-
ton” density, tr(F ∧ F), of a non-abelian gauge field,
sider an exponential potential for the self-interactions of
A , could contribute to the dark energy of the Uni-
µ φandtrytofindoutunderwhatconditionstrackingdark
verse. We assume that the axion field has non-trivial
energy arises. In Section V we discuss tentative particle
physicsconnectionsofourscenario. Someconclusionsare
presentedinSectionVI.Aninterestingvariantofoursce-
nario involving a complex scalar field whose phase plays
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] the role of the new axion field introduced in the present
‡Electronicaddress: [email protected] paper will be discussed in forthcoming work.
2
A word on our notation: Our space-time metric has where the prime denotes a derivative of V with respect
signature (−,+,+,+). We work in units in which the to φ. Following a hypothesis introduced in the context
speed of light, Planck’s constant and Boltzmann’s con- of inflationary models in [9] and in [10], we assume that
stant are all set to 1. The cosmological scale factor is the term proportional to the Pontryagin is responsible
denoted by a(t), where t is physical time. The Hubble for slow-rolling of φ, in the sense that the terms in (4)
expansion rate is H(t) = a˙(t), and t denotes the time proportional to first and second time derivatives of φ are
a eq
of equal matter and radiation. negligible as compared to the two remaining terms. If
this assumption can be justified the equation of motion
for φ reduces to
II. KEY FEATURES OF OUR SCENARIO
In this section we introduce our dark energy model, λ
V(cid:48)(φ) (cid:39) E(cid:126) ·B(cid:126) , (5)
postponing a discussion of its origins in particle physics 8f a a
to Section V.
A key element of our model is a pseudo-scalar axion
field, φ, that couples linearly to the Pontryagin density, an equation that determines the time-dependence of φ,
tr(F ∧ F), of a (massive) non-abelian gauge field, Aµ. onceoneknowsthetime-dependenceofE(cid:126) ·B(cid:126) . Wenote
a a
Thedynamicsoftheaxionfieldφ,thegaugefieldA and
µ that slow rolling can arise at sub-Planckian field values,
the space-time metric g is determined by the following
µν [9], in contrast to the usual slow-roll in large-field infla-
action functional:
tionaryscenarios,whichrequiressuper-Planckianvalues.
(cid:90) √ (cid:20) R (cid:21) In the context of inflation, a scenario based on the two
S = d4x −g +L , (1)
16πG m basic features of our model described so far is sometimes
called chromo-natural inflation [10].
where the matter Lagrangian is given by
Athirdkeyfeatureofourscenarioconcernsthesecular
1
L = ∂ φ∂µφ−V(φ) (2) growth of a spatially homogeneous configuration of the
m 2 µ electric field E , in excess of its usual dynamics. This
a
−1F Fµν − λφF F˜µν +mass terms. growthisinducedbythecouplingofthegaugefieldtothe
4 aµν a f aµν a axion field φ, as in (2). Under the assumption that the
couplingconstantλissufficientlylarge,wewillshowthat
The second but last term, henceforth called “ magnetic
seculargrowthofE ,whencombinedwithEq. (5),yields
helicity term”, can be understood as arising from cou- a
an axion field configuration that gives rise to tracking
pling thegradientof φ to an anomalousaxial vector cur-
dark energy.
rent and then invoking the chiral anomaly [14]. We will
discuss possible particle physics origins of the field φ, of The main point is that a non-vanishing magnetic he-
an anomalous axial vector current, and of a heavy gauge licity, which originates from the coupling of the gauge
field (with field strength denoted by F) in Section V. In field to the axion as expressed by the “magnetic helic-
Eq. (2), repeated indices are to be summed over, the ity term,” acts as an extra “friction term” that ensures
index a is a gauge group index, µ and ν are space-time that φ will slowly roll down its potential – even at sub-
indices, λ is a dimensionless coupling constant, and f is Planckian field values. Thus, the resulting equation of
a reference field value that also appears in the axion po- state for the energy density of the axion is dominated by
tential V(φ). In this paper we consider an exponential thepotentialenergyterm, whichyieldsacontributionto
potential: dark energy. Equations (3) and (5) then tell us that if
the Pontryagin density E(cid:126) ·B(cid:126) exhibits secular growth,
V(φ) = µ4eφ/f, (3) a a
the contribution of φ to the total energy density of the
whereµsetstheenergyscaleofthepotential. Thischoice Universe can become important at late times.
of V(φ) leads to an explicit breaking of parity and time-
Finally, another key feature of our scenario is that the
reversal invariance. (To avoid this, one might consider
initial value of the energy density of φ is proportional
replacing exp(φ/f) by cosh(φ/f)−1 in Eq. (3).) A more
to a small number in cosmology, such as the baryon to
naturalchoiceofself-interactionsnotbreakingthesesym-
entropy ratio n /s, (n and s being the baryon and pho-
metriesexplicitlywillbeconsideredinforthcomingwork. b b
ton number densities, respectively). As far as relating
A basic feature of our model is related to the expecta-
a late-time cosmological observable to the small baryon
tion that φ is very slowly rolling at sub-Planckian field
to entropy ratio (via a term in the Lagrangian coupling
values, due to its coupling to the gauge field. Assuming
the axion to an anomalous current) is concerned there
spatialhomogeneity, thefieldequationofmotionforφis
are similarities of our work to the one in [15], where the
given by
tensor-to-scalar ratio, (i.e., the ratio of the strength of
λ gravitational waves to that of scalar cosmological fluctu-
φ¨+3Hφ˙+V(cid:48)(φ) = E(cid:126) ·B(cid:126) , (4)
8f a a ations), is related to nb/s.
3
III. GAUGE FIELD DYNAMICS IN THE and
PRESENCE OF THE “ANOMALY TERM”
λ
∇ Ei =− ∇ φBi, (15)
i a f i a
The equation of motion for the field strength tensor of
the gauge field in the presence of a Chern-Simons term with
(but neglecting mass terms) is given by
∂
εijk∇ E =− Bi (16)
DabFbαβ − 4λ(cid:15)µνβα∂ab(φFb )=0. (6) j ak ∂t a
α f α µν
In the following we will consider a spatially homoge-
In this equation D denotes the covariant derivative, neous gauge field configuration. Of course, in general
which is defined by non-vanishing electric and magnetic background fields
break rotational invariance; but spatial homogeneity can
Dab ≡δab∇ −gfabcAc (7)
α α α be preserved in gauge-invariant combinations of E(cid:126) and
where ∇ is the space-time covariant derivative, g is the B(cid:126). As a result of a non-vanishing instanton condensate,
α
gauge coupling constant, and the fabc are the structure CP is broken, which may have interesting consequences
constants of the gauge group. that we will return to elsewhere. The instanton conden-
We write the equation of motion for the gauge field sateE(cid:126) ·B(cid:126) canbenon-zero,homogeneousand isotropic,
a a
in terms of the “electric” and “magnetic” components of i.e., translation- and rotation invariant.
the field tensor, For a non-Abelian gauge group, such as SU(2), we
make the ansatz of a spatially homogeneous background
Ea = Fa uν, (8)
µ µν gauge field. Following [16] (see also [17]) we take
1
Bµa = −2(cid:15)µνρσFaρσuν, A0 = 0 (17)
A (t) = aψ(t)δaJ
where uµ = (1,0,0,0) is the four-velocity of a comoving i i a
observer in an FRW spacetime. In manifestly covariant
where J are the generators of the non-Abelian SU(2)
a
form, the equations of motion are gauge group and δi is a Kronecker delta symbol combin-
a
uαDabEbσ + 2HEaσ−u (cid:15)µσαβDabBb inganupperinternal(Lie-algebra)index,a,withalower
α µ α β spatial index, i. The field strength tensor elements are
= −8λ(cid:0)u (cid:15)µσαβ∂ φEa+uα∂ φBaσ(cid:1), then
f µ α β α
8λ Fa = a(a˙ψ)δa
DabEbα = ∂ φBaα. (9) 0i i
α f α Fa = −g(aψ)2(cid:15)a , (18)
ij ij
In standard three-vector form, the equations are which in particular implies that the electric field can be
written as
1
DabEb + 2HEa− Dab×Bb (10)
0 a Ea ∼ E(t)δa. (19)
(cid:18) (cid:19) i i
8λ 1
= − ∇φ×Ea+φ˙Ba
f a Beforethegaugefieldacquiresamass, theequationof
motion for ψ is (see [10])
8λ
Dab·Eb = ∇φ·B (11)
f ψ¨+3Hψ˙ +(cid:0)H˙ +2H2(cid:1)ψ+2g˜2ψ3 = g˜λφ˙a3/2ψ2. (20)
where Dab is the spatial part of Dab. f
α
Using the general definition of the electric and mag-
where the term on the right hand side of the equation
netic components of the field tensor,
comesfromthemagnetichelicity(instanton)terminthe
E = ∂ Aa−Dab(A)Ab (12) actionwhichleadstoacouplingoftheaxiontotheback-
ai 0 i i 0
ground gauge field. The coefficient of the term linear in
and ψ vanishes in the radiation epoch.
g The equation (20) displayed above holds at all times
B = (cid:15) (∂ Aa− (cid:15)abcAbAc) (13)
ai ijk j k 2 j k for an unbroken gauge theory. However, we are more in-
terested in a gauge theory that is spontaneously broken
thefieldequationsforthegaugefieldcoupledtotheaxion
atafairlylargemassscalem. Thegaugefieldofthethe-
field take the form
ory acquires its mass after the symmetry breaking phase
∂ g transition, a transition occurring when the temperature,
Ei − (cid:15)abcA Ei+2HEi −εijk∇ B (14)
∂t a 2 0b c a j ak T(t),oftheUniverseisoftheorderofm. Afterthephase
λ transition, at temperatures below the transition temper-
= − [φ˙Bi −εijk∇ φE ]
f a j ak ature, the energy density of the gauge field scales like
4
that of matter, i.e., ρ ∼ a(t)−3, for times greater where ψ (t) is the solution without the magnetic helic-
gauge 0
than t , where t is determined by ity term which satisfies the given initial conditions at an
m m
initial time t , and ψ (t) is the first order Born approxi-
i 1
T(tm) ≈ m, (21) mation calculated from
λ
where T denotes the temperature of radiation.This cor- ψ¨ +3Hψ˙ +m2ψ = g˜ φ˙a3/2ψ2 ≡ S(t), (27)
1 1 1 f 0
responds to the scaling
where S(t) stands for the “source” term in the Born ap-
E(t),B(t) ∼ a(t)−3/2. (22) proximation. The solution can be written as
(cid:90) t
Once the gauge field acquires a mass, there will be an
ψ (t) = dt(cid:48)G(t,t(cid:48))S(t(cid:48)), (28)
extramasstermintheequationofmotionforψ,yielding 1
ti
ψ¨ + 3Hψ˙ +(cid:0)H˙ +2H2(cid:1)ψ+m2ψ+2g˜2ψ3 where G(t,t(cid:48)) is the Green function which is given by
= g˜λφ˙a3/2ψ2. (23) G(t,t(cid:48)) = w(t(cid:48))−1(cid:0)u1(t)u2(t(cid:48))−u2(t)u1(t(cid:48))(cid:1), (29)
f whereu andu arethetwobasissolutionsofthehomo-
1 2
geneous equation of motion, and
Since we are assuming that the mass will be much larger
thanthevalueofH atthetimeofequalmatterandradi- w(t) = u˙ (t)u (t)−u˙ (t)u (t) (30)
1 2 2 1
ation,thethirdtermonthelefthandsideof(23)isnegli-
is the Wronskian.
gibleeveninthematterera. Thenonlineartermpropor-
In the radiation epoch, i.e. for t<t we have
tionaltoψ3 becomesincreasinglyunimportantcompared eq
to the ψ¨ ∼ m2ψ term as time goes on since the ampli- u (t) = (cid:0)ti(cid:1)3/4cos(mt)
tude of ψ decreases. The approximate form of (23) then 1 t
becomes u (t) = (cid:0)ti(cid:1)3/4sin(mt). (31)
2 t
λ
ψ¨+3Hψ˙ +m2ψ = g˜ φ˙a3/2ψ2. (24) Hence the leading term in the Wronskian gives
f
w(t(cid:48)) (cid:39) m(cid:0)ti(cid:1)3/2. (32)
In terms of electric and magnetic fields, this equation t(cid:48)
corresponds to (suppressing the gauge group index a)
As we will see later
1 1
∂ E + 3HE = −λ[φ˙B ] |φ˙| = n˜ , (33)
∂t i 2 i f i f t
3 wheren˜ isanumberoforderone. Computing(28),using
B˙ + HB = 0 (25)
i 2 i (29), (31), (32), and (33), yields a term ψ0(t), multiplied
by a certain integral denoted G(t),
The key point is that the coupling of the gauge field
to the axion field φ has the consequence that the elec- ψ1(t) = ψ0(t)G(t). (34)
tric fielddecaysless rapidlythan it would inthe absence Here G(t) is a “secular growth factor” given by
of φ. This effect entails that the magnetic helicity and
t
theenergydensity, 1(E(cid:126)2+B(cid:126)2)ofthegaugefield,which, G(t) (cid:39) g˜n˜λm−1ψ (t )log( ). (35)
2 0 i t
initially, scale like that of radiation, then like the one of i
matter, turn out to grow relative to the energy density Theseculargrowthofψ leadstoacorrespondingsecular
of matter, at late times. growth of the magnetic helicity tr(E(cid:126) ·B(cid:126)). The analysis
In the absence of the magnetic helicity term propor- for t > t is analogous, except that the mode functions
eq
tional to φ˙, or if the φ-field is time-independent, the u and u now scale as t−1.
1 2
equations (24) or (25) imply the behavior E ∼ a−3/2 Based on the above analysis we are able to determine
i
and B ∼a−3/2. Hence they imply that the energy den- the scaling of the magnetic helicity term. Taking into
i
sityofthegaugefield(i.e.,oftheE(cid:126)-andB(cid:126)-fields)scales account the fact that B(t) scales as a(t)−2, for t < tm,
like that of matter (For times t < tm, the energy den- and as a(t)−3/2, for t>tm, we find that
sity of the E(cid:126)- and B(cid:126)- fields decays like that of radiation. E(cid:126) ·B(cid:126) ∼ a(t)−4 (36)
a a
This corresponds to a scaling of these fields proportional
to a−2. Let us denote the ψ field in the absence of the for t<tm, as
“magnetic helicity term” by ψ0(t). E(cid:126) ·B(cid:126) ∼ a(t)−3 (37)
a a
Wemaythendeterminetheeffectscausedbythe“mag-
netic helicity term” using the Green function method. for tm <t<tc, and as
We write E(cid:126) ·B(cid:126) ∼ a(t)−3G(t), (38)
a a
ψ(t) = ψ (t)+ψ (t), (26) for t>t .
0 1 c
5
IV. LATE TIME ACCELERATION FOR AN Thus, the secular growth term grows logarithmically in
EXPONENTIAL POTENTIAL time.
We now search for conditions implying that φ is a vi-
Next, we analyze how the dynamics of the axion φ is able candidate for dark energy. First of all, φ has to be
affectedbygaugefieldconfigurationswithnon-vanishing slowly rolling as a function of time in order for the equa-
magnetic helicity. We recall that, for a spatially homo- tion of state of φ to be that of dark energy. Second, we
geneousconfigurationofaxions,thefieldequationofφis have to show that the energy density of φ has the po-
given by tential to dominate over the background energy density
shortly before the present time. To complete our analy-
φ¨+3Hφ˙+V(cid:48)(φ) = λ E(cid:126) ·B(cid:126) , (39) sis, we need to make sure that the energy density of the
8f a a gauge field which the axion field φ couples to remains
subdominant.
where the prime indicates a derivative of V with respect
Under the assumption that the slow-rolling conditions
to φ. As in the context of inflationary models in [9], we
aresatisfied,Eq. (42)immediatelyleadstoanexpression
assume that the term on the right side of (39) gener-
for the contribution, Ω , of the φ- field to the total en-
ates slow-rolling of φ, namely that the terms φ¨and 3Hφ˙ φ
ergy density of the universe. From the above discussion
in (39) are negligible as compared to the two remaining
it follows that, for t<t , the energy density of φ scales
m
terms. Wewillchecktheself-consistencyofthisassump-
like that of radiation and hence leads to a constant con-
tion below. The evolution of φ is then determined by
tribution to Ω . For t < t < t , the potential energy
φ m eq
of φ decreases less fast than the background radiation
λ
V(cid:48)(φ) = E(cid:126) ·B(cid:126) (40) density, leading to a contribution to Ω that grows lin-
8f a a φ
early in a(t). We should emphasize, however, that the
contributionofφtoΩscalesinthesamewayasthecon-
For an exponential potential,
tribution of dark matter to Ω. Once t > t , but before
eq
V(φ) =µ4eφ/f, (41) t=tc, both the background density and the energy den-
sity of φ scale as a(t)−3, and hence the contribution of
Eq. (40) yields φ to Ω is constant. Finally, once t > t , Ω increases in
c φ
time. Specifically, for late times t>t , we obtain that
eq
λ
V(φ) = E(cid:126) ·B(cid:126) (42)
8 a a V(φ(t))
Ω (t) (cid:39) (46)
The proportionality of V(φ) to tr (E(cid:126) · B(cid:126)) is a special φ ρ0(t)
feature of the exponential potential. For polynomial and = λ(E(cid:126) ·B(cid:126))(ti)(cid:0)a(teq)(cid:1)(cid:2)1+n˜λB(ti)log(t)(cid:3),
periodicpotentials,V(φ)endsupbeingproportionaltoa 8 ρ (t ) a(t ) E(t ) t
r i m i i
poweroftr (E(cid:126)·B(cid:126))greaterthan1,andhencewoulddecay
faster in time, in an expanding universe. This makes it where ρ (t) is the background energy density at time t,
0
more difficult to interpret φ as a dark-energy candidate. andρ (t )istheenergydensityofradiationattheinitial
r i
We will study such types of potentials in the context of timet ,(whichisapproximatelyequaltothetotalenergy
i
a different (possibly more natural) model in a follow-up density at that time, since we have assumed that t is
i
paper. chosen to belong to the radiation period). Equivalently,
Combining (41) and (42) we find the following expres- we can express Ω in terms of the background matter
φ
sion for the axion field φ: density, ρ , at the initial time
m
fφ = log(cid:0)λ8µ−4E(cid:126)a·B(cid:126)a(cid:1), (43) Ωφ(t) (cid:39) Vρ(φ((tt))) (47)
0
which leads to = λ(E(cid:126) ·B(cid:126))(ti)(cid:0) a(ti) (cid:1)(cid:2)1+n˜λB(ti)log(t)(cid:3),
8 ρ (t ) a(t ) E(t ) t
1 1 m i m i i
|φ˙| = n˜ , (44)
f t
Figure 1 presents a sketch of the time evolution of Ω .
φ
Sincetheenergydensityofthenewgaugefieldislarger
where the number n˜ is n˜ =2, for t<t and for t>t ,
and equals n˜ = 3/2, for t < t < t .mThis expressieoqn than tr(E(cid:126) ·B(cid:126)), (by the Schwarz inequality), a necessary
m eq
is important for the evaluation of the magnitude of the condition for φ to be a dark energy candidate is that
seculargrowthterm(35). Infact,inserting(44)into(25)
and taking into account that B and E scale the same V(φ) (cid:29) tr (E(cid:126) ·B(cid:126)), (48)
0
way as a function of time we find that
which, by (42), can only hold provided
E (t) = E (t)(cid:2)1+n˜λBa(ti)log(t)(cid:3). (45)
a a0 E (t ) t λ (cid:29) 1. (49)
a i i
6
FIG. 1: Sketch of the time evolution of the fractional contribution Ω of the φ field to energy density of the Universe. The
φ
horizontalaxisistime,theverticalaxisisthevalueofΩ . Thecontributionisconstantuntilthetimet whenthegaugefield
φ m
massbecomesimportant. Itthenrisesasthescalefactor,tobecomeconstantagainfort>t . Oncetheseculargrowthofthe
eq
E field becomes dominant at the time t the contribution of φ to Ω once again begins to rise as given by the secular growth
c
factor G(t). Note that t is the present time.
0
Forφtobeagoodtracking quintessencecandidate,we ment that B(t )/E(t ) needs to be slightly smaller than
i i
needtheseculargrowthtermin(35)tobecomedominant λ−1, which is a tuning condition we need to impose.
at a time t , with Above we have assumed that the slow-roll conditions
c
t <t <t , (50) φ¨(cid:28)V(cid:48)(φ) and 3Hφ˙ (cid:28)V(cid:48)(φ) (51)
eq c 0
where t is the present time. This leads to the require- are satisfied, and that the equation of state of φ leads
0
7
to acceleration. It is easy to check that the slow-roll candidatescalarfieldfordarkenergyneedstocouplevery
conditions are satisfied, provided weakly to standard model matter [18].
The idea underlying our proposal is that the field φ
f (cid:28) m . (52)
pl responsible for dark energy could be a new axion field
conjugate to an anomalous matter current [14]. A pos-
It is not hard to check that the equation of state for φ
sible example would be an anomalous lepton current, in
is dominated by the potential energy if condition (52)
which case the gauge field would be a weak SU(2) field
is satisfied. Thus, the field φ is indeed a candidate for
(see, e.g., [19] ). The gradient of φ can then be linearly
tracking dark energy.
coupled to the anomalous axial vector current Jµ, intro-
Finally, we study the magnitude of the contribution 5
ducing a term proportional to
of φ to the dark energy budget. Evaluating (46) at the
presenttimet andassumingt >t weobtain(dropping
0 0 c ∂ φ·Jµ (55)
Lie-algebra indices on E(cid:126) and B(cid:126)) µ 5
in the Lagrangian of the theory. Apparently, the time
Ω (t ) (cid:39) λ(E(cid:126) ·B(cid:126))(ti)(cid:0)a(teq)(cid:1)λB(ti)log(t0). (53) derivativeofφthenplaystheroleofaspace-timedepen-
φ 0 8 ρ (t ) a(t ) E(t ) t
r i m i i dent axial chemical potential for an axial charge density
If we do not want to introduce a new mass hierarchy [5], (e.g., the 0-component of the left-handed lepton cur-
into our model, it is natural to assume that t ∼ t . In rent). This may furnish an ingredient in a mechanism
m i
this case, an initial value of tr (E(cid:126) ·B(cid:126)) comparable to the responsible for the observed matter-antimatter asymme-
try. Thanks to the anomaly equation [14], the term (55)
initial matter density is required in order for the order
is equivalent to a term proportional to
of magnitude of (53) today to be close to unity. This
is ensured if, initially, at time t , the ratio between the
instanton density and the energiy density of radiation is φ(F ∧F + 1 (cid:88)m ψ¯ γ ψ ), (56)
f j j 5 j
proportional to the ratio between baryon- and entropy j
density, i.e.,
withj labelingfermionspecies,wherespeciesj hasmass
m and is described by a spinor field ψ .
j j
tr (E(cid:126) ·B(cid:126))(ti) ∼ nB(t ), (54) Instanton effects are usually expected to lead to a po-
ρ (t ) s i tential for φ that is periodic in φ, and this possibility is
r i
studied in forthcoming work. One may imagine, how-
wheren isthebaryonnumberdensityandstheentropy
B ever, that axion shift-symmetry breaking effects might
density. Hence, the smallness of the initial contribution
generate an exponential potential. The value of the pa-
of φ to dark energy is guaranteed by the observed small
rameterf isrelatedtothesymmetrybreakingscale,and
baryon to entropy ratio. This factor is believed to be of
theenergy-scaleparameterµissetbythestrengthofthe
the order 10−10.
instanton effects.
B) Universal axion of string theory:
V. PARTICLE PHYSICS CONNECTIONS
Axions arise naturally in superstring theory [20].
Specifically, string compactifications generate Peccei-
A) An axion coupling to an anomalous matter
Quinn type symmetries often broken at the string scale
current:
[21]. For example [22], there is an axion field a that is
Thestandardaxionfield,a(x,t),alongwiththePeccei-
in the same chiral superfield S as the four dimensional
Quinnsymmetryhasbeenintroducedtosolvethestrong
dilaton ϕ
CP problem of QCD; see [2]. The mechanism leading to
the spontaneous breaking of the Peccei-Quinn symmetry S = e−ϕ + ia. (57)
involves a complex scalar field with a standard symme-
try breaking potential whose angular variable is the ax- In addition, there is an axion field a˜ in the superfield S˜
ion field a [1]. The coefficient of the Pontryagin density, of the volume scalar ρ:
tr(F ∧F), in the QCD Lagrangian then becomes a dy-
namical variable. At the perturbative level, the axion S˜ = eρ + ia˜. (58)
has a flat potential. Non-perturbative instanton effects
create however a non-trivial potential, V(a), for the ax- The Peccei-Quinn symmetries of string theory are al-
ion. This potential is periodic in a, which is an “angular ways broken by stringy instanton effects, leading to a
variable.” Theperiodicityofthepotentialisnotunprob- coupling of the axion to some tr(F ∧F)- term. This can
lematic, since it could give rise to an axion domain-wall be shown explicitly by reducing the ten-dimensional su-
problem. pergravityactiontofourspace-timedimensionsviacom-
The axion of QCD is a candidate for dark matter [4], pactification on some internal Calabi-Yau manifold; (see
but cannot be a candidate for dark energy, since it in- e.g. [22]). Such a compactification also generates po-
teracts too strongly with electromagnetism. Any viable tentials for the superfields to which the axions belong.
8
These potentials are typically exponential in the radial close to the present time t . This is only the case if, at
0
direction,butaremnantoftheexponentialpotentialmay the initial time, the φ- field energy is a small contribu-
alsoaffectthepotentialintheaxiondirection; especially tiontothetotalenergydensity. Ourproposalisthatthis
ifstringyeffectsleadtoabreakingoftheshiftsymmetry smallinitialvalueofΩ islinkedtothesmallvalueofthe
φ
in the axion direction, as happens in axion-monodromy lepton to entropy ratio. This would imply that the sec-
models [23, 24]. For some explicit constructions of expo- ular growth term becomes important only at rather late
nential potentials see [25]. times. Thus, our model represents an implementation
of the “tracking quintessence” scenario of [12]. We have
C) Axion monodromy: shown that, in order to obtain the currently observed
Indeed,ithasrecentlybeenrealizedthatstringyeffects value of dark energy in our model, it suffices to require a
break the shift symmetry of the axion. The axion ceases fairly mild tuning of dimensionless coupling constants.
to be an angular variable and, instead, has an infinite In this paper we have neglected the coupling of
range of values. Monodromy induces an axion potential the axion field φ in (56) to pseudo-scalar mass terms,
rising without bound, as φ increases to ∞; see, e.g., [24]. m ψ¯ γ ψ , of fermionic matter fields. Taking such cou-
j j 5 j
At large field values, the axion potential may be linear. plingsintoaccountwouldleadtoextratermsontheright
To make contact with our scenario we need to assume side of the axion equation of motion (39). For H >
that the potential is exponential at small field values. max m , these terms will decay as radiation, and, for
j j
We are not the first to connect an axion with a po- H < min m , they decay as matter. If m¯ ≡ max m <
j j j j
tential induced by stringy monodromy effects with dark m, there is a time interval t < t < t when the con-
m m¯
energy. In [26] it was in fact suggested that a stringy tribution due to the mass terms on the right side of (55)
axion may play the role of a quintessence field. The con- decays rapidly, relative to the one of the F ∧ F term.
structionin[26]makesuseofstandardslow-rollinflation Hence, as long as m > m¯, the extra terms in (55) will
and thus requires super-Planckian field values. It must not change our conclusions.
still be shown that such field values are consistent from
Ithasbeenpointedoutthatifthefieldresponsiblefor
the point of view of string theory, since for other axion
dark energy is a pseudo-scalar field then it could cou-
models they are not [27]. In our construction, the axion
ple to visible matter, and this leads to rather stringent
field values are sub-Planckian, because slow-rolling is in-
constraints. A discussion of the coupling of an axion to
duced by the coupling of the axion to the Chern-Simons
visible matter has been given in [18], where it has been
term of a gauge field.
assumed that the axion couples to the E(cid:126) · B(cid:126)- term of
electromagnetism. This would lead to a rotation of the
direction of polarization of light emitted by distant ra-
VI. CONCLUSIONS AND DISCUSSION
dio sources. The constraints resulting from this effect
are quite restrictive and would potentially rule out our
We have studied a model of tracking dark energy in
model if our axion were to couple to the electromagnetic
which dark energy arises from an axion field φ linearly
field. However,wehaveassumedthatouraxiondoesnot
coupledtothePontryagindensityofagaugefield,i.e.,to
interactwiththephotonandthusevadestheboundspre-
a term tr(F ∧F). Thanks to this coupling, the axion is
sented in [18] and in related work. In a future paper, we
rolling slowly even for sub-Planckian field values, It thus
willinvestigatecollidersignalsduetoapossiblecoupling
has the right equation of state to account for dark en-
of the axion field φ to W- and Z- bosons.
ergy. We have considered the example of an exponential
potential for the axion. The coupling between the axion
and the gauge fields leads to a secular growth term in
the electric field. At early times, the energy density in
Acknowledgement
φ tracks that of the background matter; but when the
seculargrowthtermbecomesimportantthecontribution
of φ to the density parameter Ω starts to increase. We One of us (RB) wishes to thank the Institute for The-
havestudiedtheevolutionofΩ (thefractionofthetotal oretical Studies of the ETH Zu¨rich for kind hospital-
φ
energy density required for a spatially flat universe due ity. RB acknowledges financial support from Dr. Max
totheaxionfieldφ)asafunctionoftimeandfoundthat R¨ossler, the Walter Haefner and ETH Zurich Founda-
itisconstantforearlytimest<t ,wheret isthetime tions, and from a Simons Foundation fellowship. The
m m
when the mass of the gauge field becomes important. It research of RB is also supported in part by funds from
grows linearly in the scale factor between time t and NSERC and the Canada Research Chair program. JF
m
the time, t , of equal matter and radiation. After time thanks A. H. Chamseddine and D. Wyler for very help-
eq
t , the value of Ω ceases to grow until the time when ful discussions on related ideas. SA thanks JiJi Fan and
eq φ
the secular growth term becomes dominant, after which Sam Cormack for helpful discussions, and acknowledges
it will start to grow again. support from the US Department of Energy under grant
In order for φ to be a successful candidate for dark DE-SC0010386. We wish to thank Tom Rudelius and
energy, the time when Ω approaches Ω = 1 has to be Gary Shiu for insightful communications.
φ
9
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