High-Scale Axions without Isocurvature from Inflationary Dynamics

2.6 MB·

by  John Kearney

FERMILAB-PUB-16-005-T
MCTP-16-02
High-Scale Axions without Isocurvature from Inflationary Dynamics
John Kearney
Theoretical Physics Department,
Fermi National Accelerator Laboratory
Batavia, IL 60510 USA
Nicholas Orlofsky and Aaron Pierce
Michigan Center for Theoretical Physics (MCTP),
Department of Physics, University of Michigan
Ann Arbor, MI 48109 USA
(Dated: June 7, 2016)
Observableprimordialtensormodesinthecosmicmicrowavebackground(CMB)wouldpointto
6 ahighscaleofinflationH . IfthescaleofPeccei-Quinn(PQ)breakingf isgreaterthan HI,CMB
I a 2π
1 constraintsonisocurvaturena¨ıvelyruleoutQCDaxiondarkmatter. Thisassumesthepotentialof
0 theaxionisunmodifiedduringinflation. Werevisitmodelswhereinflationarydynamicsmodifythe
2 axionpotentialanddiscusshowisocurvatureboundscanberelaxed. Wefindthatmodelsthatrely
n solelyonalargerPQ-breakingscaleduringinflationfI requireeitherlate-timedilutionoftheaxion
u abundance or highly super-Planckian fI that somehow does not dominate the inflationary energy
J density. Models that have enhanced explicit breaking of the PQ symmetry during inflation may
allowf closetothePlanckscale. Avoidingdisruptionofinflationarydynamicsprovidesimportant
6 a
limits on the parameter space.

]
h
-p I. INTRODUCTION If the PQ symmetry breaks after inflation (fa < H2πI),
p
topological defects will form. These topological defects
e
h The axion [1, 2] is an elegant solution to the strong will lead to overclosure due to either domain wall sta-
[
CP problem [3]. It arises as a pseudo-Nambu-Goldstone bility if the domain wall number N > 1, or axion
DW
2
boson (pNGB) of a spontaneously broken Peccei-Quinn overproduction from cosmic strings and domain walls if
v
9 (PQ) symmetry at a scale fa. If its potential is dom- fa>∼ 1011 GeV [12]. Larger PQ-breaking scales and do-
4
inated by QCD instanton effects, the axion settles into main wall numbers other than one are allowed only if
0
3 an approximately CP-conserving minimum. In addition, the PQ symmetry is broken before the end of inflation
0
1. the axion is an attractive candidate for the dark matter (fa > H2πI) so that topological defects are inflated away.1
0 (DM) in our universe [4–6]. However, in this case the (massless) axion will obtain in-
6
Axion cosmology depends on whether the PQ symme- flationary fluctuations ∼ HI similar to the inflaton [15–
1 2π
: try is spontaneously broken before inflation ends. The 17]. Unlike the inflaton fluctuations, the fluctuations of
v
Xi Hubble parameter during inflation HI can be related the axion would today appear as isocurvature and are
r to the tensor perturbation amplitude in the cosmic mi- bounded by observations of the CMB [11].
a
crowave background (CMB) [7]. Current bounds from Narrowing to the case that the PQ symmetry is bro-
BICEP2/Keck/Planck constrain the tensor-to-scalar ra- ken before the end of inflation, once inflation ends the
tior <.07[8,9],whileplannednear-futuredetectorswill axial component of the PQ field remains at the value
probe the region r>∼ 2×10−3 [10]. If such a primordial taken during inflation until the temperature-dependent
gravity wave detection is made by these experiments, it QCD instanton axion mass m2 becomes comparable
QCD
will indicate to the Hubble expansion rate (typically near the QCD
(cid:18)π2 (cid:19)1/2 phase transition). At this point, the field will begin os-
HI = 2 MP2∆2Rr >∼1013 GeV, (1) cillating around the minimum of the low-energy poten-
wherethemeasuredscalarperturbationamplitude∆2 =
R
2.142×10−9 [11] and the reduced Planck mass M =
P 1 Alternatively,topologicaldefectsmaybedestabilizedviaavery
2.435×1018 GeV. delicatelychosentilttothepotential[13,14].
2
tial. These coherent oscillations correspond to an energy latedtoanexplicitPQsymmetrybreaking[31,34].2 We
density stored in the axion field, meaning that the axion explore the measures necessary to suppress isocurvature
can behave as cold DM. The QCD axion DM fraction and the constraints that must be taken into account as
generated in this manner is given by [18, 19] these measures are implemented.
In Section II, we discuss the viability of exclusively
Ω (cid:26)5.6×107(cid:10)θ2(cid:11)(f /M )7/6, f <fˆ,
R = a (cid:39) a P a a suppressingisocurvatureviaaninflationaryPQ-breaking
a Ω 1.6×108(cid:10)θ2(cid:11)(f /M )3/2, f >fˆ,
DM a P a a
scale that is larger than the present scale. While this
(2)
where (cid:10)θ2(cid:11) = θ2 + δθ2 is the axion misalignment an- has been previously considered as a viable mechanism
i
toreduceisocurvature, weshowthatitoftenfacesinsur-
gle including contributions from the initial misalignment
mountableconstraintsduetothelargehierarchyrequired
θ and the primordial fluctuations δθ (neglecting anhar-
i
monic factors [20]). The scale fˆ (cid:39) .991×1017 GeV (cid:39) between the two scales. In Section III, we explore the
a
MP correspondstothetransitionbetweenoscillationsbe- possibility that isocurvature can be suppressed with an
2.5
enhanced explicit breaking of the PQ symmetry, demon-
ginning before or after the QCD phase transition (see
stratingthemodel-buildingandexperimentalconstraints
[21, 22] for corrections to this approximation).
that ultimately limit the range that f may take in such
Using this expression for the axion DM fraction and a
models. Numerous previous works have observed that
assuming a detection of primordial tensor modes—
explicit PQ-breaking modifies the axion potential, and
indicating a large H and thus large isocurvature fluc-
I
so in general could be a suitable strategy for suppress-
tuations δθ—current constraints on isocurvature would
ing isocurvature. Here, however, we focus on developing
ruleoutthesimplestaxionmodelsforfa > H2πI, see, e.g.,
concrete models, which allows us to study the various
[19, 23]. For high-scale axion models to be viable in the
interrelated field dynamics and constraints, and we are
presenceofsuchadetection, theaxionfluctuationsmust
thus able to more fully address the strengths and short-
be suppressed.
comings of this approach.
Itisimportanttounderstandtherobustnessofisocur-
As we are interested in modifying the behavior of the
vature constraints for two reasons. First, string theories
axion during inflation, our primary focus here will be on
genericallypredictaPQ-breakingscalearoundthegrand
howthesolutionsimplementedaffectthedynamicsofthe
unifiedtheory(GUT)scale,f ∼1016 GeV[24,25]. Sec-
a
axion and inflaton fields. However, the solutions under
ond, several experiments have recently been proposed to
considerationintheabovesectionscanalsoaffectthebe-
lookforlarge-f axionDM[26–29]. Ifprimordialgravity
a
havior of the radial component of the PQ-breaking field.
wavesareobserved,itwillbeimportanttoknowwhether
This leads to model-dependent, but potentially severe,
such theories are permissible and thus which regions of
constraints related to ensuring a consistent cosmologi-
parameter space these experiments should target. More
cal history. We review these issues in Section IV. Many
optimistically, if signals are seen both in tensor modes
of these constraints could be evaded if the PQ field re-
and in a hunt for large-f axions, we should know what
a
laxed adiabatically to its minimum today without oscil-
types of new physics would be required to reconcile the
lation. However, in Section V, we discuss how models
two signals.
thataredesignedtopermitsuchadiabaticrelaxationwill
We examine two classes of solutions that have been
generally only partially mitigate constraints. Moreover,
discussedtocircumventisocurvatureconstraintsandres-
we highlight some previously unappreciated obstacles to
urrectmodelswithlargef . Bothinvolveaninflationary
a constructingsuchmodels,whichmakeitunclearwhether
shift of the PQ sector away from its zero temperature
minimum and/or potential. In the first, the PQ scale
2 Othermechanismstosuppressisocurvaturehavealsobeenpro-
is large during inflation f (cid:29) f [17, 30–33]. In the
I a posed [35–44], though notably many of these do not work for
second, the axion has a large mass during inflation re- GUT-scaleaxionDMwithhigh-scaleinflation.
3
adiabaticrelaxationcaninfactbeimplemented. Wecon- and PQ sectors, making such a mechanism unfavorable;
clude in Section VI. further details are provided in footnote 13. As such, the
likely source of the inflationary radial field displacement
II. SUPPRESSING ISOCURVATURE VIA WAVE is Hubble friction, i.e., the situation in which the radial
FUNCTION RENORMALIZATION
field is displaced from the minimum today at the onset
of inflation and remains light (compared to H ) during
I
The complex PQ field S can be written in terms of
inflation.
its radial and axial components, σ and a, and the PQ
In fact, regardless of the source of the radial field dis-
breaking scale, f ,
PQ placement, such super-Planckian field values could still
(cid:20) (cid:21)
1 ia disrupt inflation. Depending on the form of the poten-
S = √ (σ+f)exp , (3)
2 f
tial for S, at large field values it may dominate the Uni-
wheref = fPQ ,withf =f andf =f denotingtheef- verse’s energy density, superseding the supposed infla-
NDW a I
fective breaking scale today and during inflation, respec- tionary sector. This effect is exacerbated as the required
tively. Duringinflation,fluctuations∼ H2πI areinducedin fI increases. Moreover, such large values for f have re-
thecanonicallynormalizedaxionfield, whichcorrespond sisted embedding in string theory [45], perhaps related
to fluctuations in the field today given by [17, 30], to the weak-gravity conjecture [46–49].
H f This concern could be avoided if the PQ sector were
δa= I a. (4)
2πf responsibleforinflation, assuggestedin[33]. Wewillre-
I
Thus, f > f will reduce axion fluctuations. However, turntothisexampleattheendofthissection,but,aswe
I a
duetothestringentboundsonisocurvature,α <.0019 shall see, the other constraints outlined below generally
iso
at the 95% confidence level [11], f (cid:29) f is necessary require additional late-time dilution of axions for such
I a
to achieve the required suppression. The isocurvature “PQ sector inflation” to be a viable solution.
parameter is Following inflation, once H decreases below the radial
α 1 (cid:18) (H /(2π))2(cid:19) field’s mass, σ will start to oscillate with a large initial
iso (cid:39) 4R2 I . (5)
1−α ∆2 a f2(cid:104)θ2(cid:105) amplitude corresponding approximately to its displace-
iso R I
mentduringinflation.3 AswediscussinSec.IV,theam-
As such, employing Eq. (2),
plitudeanddecayoftheseoscillationsareconstrainedby
(cid:115)
f (cid:16) r (cid:17)(cid:18) .0019 (cid:19)(cid:18)10−1M (cid:19)5/12
fI >∼1.4×104 Ra .002 α f P cosmological observations—the energy density must be
a iso,max a dissipated efficiently and appropriately (Sec. IVC) and
(6)
fluctuations induced in a light, Hubble-trapped σ may
(valid for f < fˆ, while a similar expression can be de-
a a
also be bounded (Sec. IVA). Both of these constraints
rived for the other case). Thus, for a theory with de-
can be evaded via appropriate model building that is
tectable gravitational waves r>∼ 2×10−3, a large sepa-
largely independent of the axion. Alternately, if σ is the
ration is required between f and f to evade bounds on
I a
inflaton, these constraints are superseded by requiring
isocurvature.
generation of the observed curvature perturbations and
This displacement must be generated in a way con-
appropriate reheating.
sistent with inflation. For instance, (cid:104)|S|(cid:105) could be dis-
Our main focus here, however, is that the axion field
placed via a model that leverages inflationary dynamics
itselfcanexperiencenonperturbativeeffectsfromlargeσ
to give an explicit inflationary minimum for |S| larger
than its present one. However, as seen from Eq. (6), f
I
must be super-Planckian for fa>∼ 4×10−7MP. Such a
3 AdiabaticrelaxationasdescribedinSec.V,whichrequiresanin-
large displacement is likely to seriously disrupt the infla-
flationaryminimumsetbyinflationarydynamics,isnotpossible
ton potential via the couplings between the inflationary inthismodelwhichreliespurelyonHubblefriction.
4
oscillations, along the lines of the parametric resonance behaves as radiation. Thus, avoiding symmetry restora-
effectsthatcanbeinducedbyinflatonoscillationsduring tionrequiresthattheinitialamplitudeofoscillations(or,
preheating [50, 51]. Constraints from these effects turn at least, the initial amplitude when the potential starts
out to be most relevant for smaller f , and thus com- behaving quartically [56]) is somewhat small.7
a
plementary to the concerns about the disruption of the Comparing Eqs. (6) and (7), for fa <∼ 10−1MP,
inflation potential. Specifically, radial oscillations can one cannot simultaneously suppress isocurvature using
excite fluctuations in the axial field via parametric reso- wavefunctionrenormalizationaloneandavoidsymmetry
nance. If the fluctuations in the axial field grow to order restoration. Notef inEq.(6)isthefieldvaluewhenthe
I
f so that θ =a/f takes all possible values throughout modes relevant to the CMB exit the horizon, as opposed
a a
the universe with roughly equal probability, topological to when oscillations commence. However, we expect |S|
i
defects form and symmetry can be considered restored to differ from √fI only by O(10%) due to the flatness of
2
[52].4 Since topological defects overclose the universe for the PQ potential required to avoid dominating the infla-
the models with larger f (or N ) that are of interest tionary energy density. Thus, suppressing isocurvature
a DW
here,PQsymmetryrestorationmustbeavoided.5 More- via this mechanism is challenging and likely only viable
over, if the symmetry is restored, this clearly defeats the for a limited range of f and potentials.
a
purpose of suppressing the axion fluctuations in the first Theconstraintfromnonthermalsymmetryrestoration
place. could be relaxed somewhat if the PQ field began oscil-
Axion fluctuations are more strongly enhanced for lating in a potential dominated by |S|2M with M ≥ 3
larger initial σ oscillation amplitude, which results in an [56]. However, constraints still apply once quartic or
increased duration of oscillations and oscillation speed. quadratic terms come to dominate. Alternatively, the
Basedonlatticesimulations[53],topologicaldefectsform required hierarchy in f and f could be reduced in the
I a
when the initial amplitude case of a late-decaying scalar that dilutes the axion relic
(cid:18)f (cid:19) abundance [23, 57, 58]. In this case,
|S| >∼104 √a (7)
i 2 R (cid:39)1.7×107(cid:18) TRH (cid:19)(cid:10)θ2(cid:11)(cid:18) fa (cid:19)2, (9)
in a quartic potential and matter-dominated back- a 6 MeV M
P
ground.6 The bound is even stronger in the case where
where T is the reheat temperature at which the scalar
RH
the PQ field dominates the energy density [52, 54], in
decays, with the requirements TRH>∼6 MeV to not spoil
which case topological defects form when
Big Bang nucleosynthesis (BBN) [59] and TRH<∼ ΛQCD
(cid:18) (cid:19)
f
|S| >∼2×102 √a . (8) for the dilution to occur. Then, the isocurvature bound
i 2
requires
A similar bound should apply in the case of radiation
(cid:115)
f (cid:16) r (cid:17)(cid:18) .0019 (cid:19)(cid:18) T (cid:19)
dominationasaPQfieldoscillatinginaquarticpotential fI >∼3.0×103 Ra .002 α 6 MRHeV , (10)
a iso,max
whichcouldstillbepermittedbyEq.(7)dependingonr
4 The fluctuations can also be viewed as inducing a large, posi-
tiveeffectivemasssquaredfortheradialdirectionsuchthatthe andα . However,futurenonobservationofisocurvature
iso
potentialnolongerexhibitsasymmetry-breakingform[52].
5 Inadditiontothepossibilityofsymmetryrestoration,largelocal would ultimately restrict even this case.
fluctuations could contribute to the axion relic density (though Finally, even if axion fluctuations are made negligible,
not isocurvature, as the relevant modes are very small wave-
length) or even result in the field locally relaxing to different the initial displacement of the axion field is also con-
minimaafterinflation,producingdomainwalls. Asfluctuations
strainedtoavoidtheoverproductionofDM,particularly
growexponentially,though,theselatterconcernsareonlylikely
toberelevantveryclosetotheregionwherethereisriskoffully
restoringthesymmetry.
6 Refs. [52–54] use the linear basis, S = √1 (X+iY), to explore
2
this effect, but it can also be understood in the nonlinear basis 7 Another more model-dependent nonthermal effect that can re-
ofEq.(3)viaderivativeinteractionsbetweenσ anda,see[55]. storethePQsymmetryisdescribedinSec.IVB.
5
for larger values of f . Eq. (2) implies data. The measured value of ∆2 specifies the relation-
a R
(cid:26)1.3×10−4(f /M )−7/12, f <fˆ, ship between ξ and λ. With the potential so fixed, the
θi<∼ 7.9×10−5(fa/MP)−3/4, fa >fˆa. (11) slow-roll parameters and inflaton field value throughout
a P a a
inflation (and thus f ) can be calculated. One finds the
In other words, the inflationary value for the PQ phase I
must not differ significantly from the value today. Here, requirement ξ>∼(few)×10−3 to satisfy bounds on r and
n . ThedetailsareworkedoutinRef.[33](seealso[64]),
the initial misalignment at the onset of inflation corre- s
sponds to that in the Hubble patch that gave rise to which identifies a window 1012 GeV<∼ fa <∼ 1015 GeV
where isocurvature constraints can be avoided.
our Universe. Such a small misalignment angle could
However, this model suffers from the previously de-
be justified by an anthropic argument—the axion takes
scribed parametric resonance constraints, wherein oscil-
ondifferentinitialmisalignmentsindifferentinflationary
lations in σ can restore the symmetry. Because the PQ
patches,andweresideinonesuchpatchthatgivesriseto
field dominates the energy density, a stronger bound
a not-too-large axion abundance [60, 61]. Alternatively,
more analogous to Eq. (8) applies. However, due to
dilution of the axion abundance by a late-decaying par-
the nonminimal coupling, we cannot directly apply the
ticle would permit larger initial misalignment angles for
bound of Eq. (8). First, the potential is modified: in the
a given f .
a
Einstein frame,
Overall,itseemschallengingtoimplementasolutionin
which axion isocurvature is suppressed by wave function λ(|S|2−f2/2)2
V = a , (14)
renormalization alone. E (1+2ξ|S|2/M2)2
P
which for large values of |S| and ξ >0 is flatter than the
PQ Sector Inflation
wine bottle. Second, σ has a rolling mass parametrized
by,
Someofthesechallenges,namelythosearisingfromthe
interplay between the PQ and inflationary sectors, could (1+ξσ2)2
Z(σ)= , (15)
be surmounted if these sectors were identified. How- 1+ξσ2+6ξ2σ2
ever, this approach is complicated by nonperturbative
where a heavier rolling mass, Z(σ) < 1, corresponds to
effects that result in the PQ symmetry being restored in
slower motion.
much of the same parameter space where the isocurva-
Ratherthanattemptingafullreanalysissimilartothat
ture fluctuations are sufficiently suppressed. As we shall
of [52, 54] including these contributions, we present two
see, this leads to the additional requirement of late-time
bounds that we believe bracket the true one. They both
dilution of the axion abundance to yield a viable infla-
comefromtheruleofthumbofEq.(8), butusedifferent
tionary model.
choices of |S| . In the first, |S| (cid:39) |S| , where |S|
i i end end
If the PQ field is in a “wine-bottle” potential,
is the value for |S| at the end of inflation, defined to be
V =λ(cid:18)|S|2− fa2(cid:19)2, (12) when the slow-roll parameter (cid:15) (cid:39) 1. This requirement
2
is likely a bit too restrictive, especially for larger values
it will have a |S|4-type inflationary potential [62], which of ξ, since the radial field will begin rolling more slowly
isinconsistentwithBICEP2/Keck/Planckdataonr and due both to its rolling mass and the shallower potential
the scalar tilt n [8, 9, 11]. The introduction of a non- compared to the case of ξ = 0. The second possibility
s
minimal coupling of S to gravity [63] is to take |S| to be the point after inflation has ended
i
where both Z(σ) (cid:39) 1 and dlogVE (cid:39) 4, i.e., where the
V ⊃ξ|S|2R, (13) dlogσ
rolling mass is close to its standard value of 1 and the
whereRistheRicciscalarandξ isaconstant,allowsfor potential is similar to Eq. (12). This second bound is
aninflationarymodelpotentiallyconsistentwithexisting likely not quite constraining enough, as the radial field
6
to the scale when modes relevant to the CMB exit the
10-1
horizon. Also shown are current BICEP2/Keck/Planck
bounds on r [8, 9]. Note that future experiments will be
10-2
able to probe all values of r for this model [10].
el Comparing even the conservative estimate of the sym-
10-3 d
o
P m
M n metry restoration bounds to the isocurvature bounds as-
i
/ e
fa 10-4 sibl sumingtheaxionsareproducedinthemannerofEq.(2)
s
e
cc (i.e., without a late-decaying scalar), it is clear that
a
n
I
10-5 some additional cosmological model building is neces-
sarytomakethisaviableinflationarymodel. Withlate-
10-6 time dilution, some parameter space may not yet be ex-
10-2 10-1
cluded depending on where the true symmetry restora-
r
tion bounds fall. Nevertheless, this parameter space is
FIG. 1. Constraints on the radial part of the PQ field act- narrow and only potentially allows PQ-breaking scales
ing as the inflaton. The red curves denote isocurvature con-
straints after taking into account fI > fa assuming Ra = 1 fa<∼ 2×10−3MP. Additionally, a detection of r>∼ 10−2
withanabundancedeterminedbyeitherEq.(2)(solid,shaded
would definitively exclude this model. Of course, larger
upwards) or Eq. (9) with T = 6 MeV (dot-dashed). The
RH
twobluecurvesshowtwoprescriptionsfordeterminingwhere values of fa and r could be allowed if the requirement
the PQ symmetry may be restored by parametric resonance
that R = 1 is relaxed, though this would require an
a
(seetextfordetails);thesolidshadedboundshouldbeviewed
asdefinitivelyexcluded. Theblackverticalregionontheright even smaller initial misalignment angle without any par-
shows the current BICEP2/Keck/Planck bounds on the ten-
ticular motivation for such a tuning.
sor to scalar ratio, and the gray vertical region on the left is
inaccessible in this model. This model illustrates an important tension. Suffi-
cient suppression of isocurvature requires a very large
will already have begun rolling from a larger field value hierarchy between f and f . For just such a hierar-
I a
and will have attained some “velocity” from doing so. chy, nonperturbative dynamics have the potential to re-
These symmetry restoration bounds are plotted in storethePQsymmetry. Consequently,additionalcosmo-
Fig. 1, with the solid blue curve and shaded region cor- logical mechanisms are necessary to permit models that
responding to constraints with |S| determined by con- suppressisocurvatureviawavefunctionrenormalization,
i
ditions on Z(σ) and V ,8 and the dashed blue curve and the allowed values for f are still restricted. More-
E a
corresponding to taking |S| = |S| . Also shown are over, if isocurvature perturbations continue unobserved,
i end
isocurvature bounds if the axion makes up all of the eventheseremainingvaluesforf mayeventuallybeex-
a
DM abundance (i.e., R = 1) assuming either Eq. (2) cluded.
a
(solid red region)9 or Eq. (9) with maximal possible di- Having demonstrated the difficulties inherent in mod-
lution (corresponding to TRH =6 MeV) (dot-dashed red els that seek to suppress isocurvature via axion wave
line). For the purposes of calculating the isocurvature functionrenormalizationalone,wenowturntomodelsin
constraints we take fI to be the field value 60 e-folds which field displacement enhances explicit PQ breaking
before the end of inflation, approximately corresponding in the early Universe.
III. AXION MASS FROM ENHANCED
8 In our study, we require Z ≥0.9 and dlogVE ≥3.9. The solid
dlogσ EXPLICIT U(1)PQ BREAKING DURING
blue curve is mildly sensitive to this choice, but the overlap of INFLATION
theregionsexcludedbythesolidblueandredcurvesisnot.
9 Our bounds on isocurvature are stronger by roughly a factor of
8π comparedtoRef.[33]becausetheirEq.(15)usesthePlanck
Scalar field fluctuations are very efficiently suppressed
masswhiletheirotherexpressionsusethereducedPlanckmass.
WealsousetheupdatedPlanckboundsonαiso [11]. forfieldswithlargemassesm>∼HI (see,e.g.,[65]). With
7
this motivation, in this section we will consider models additional contributions (such that θ (cid:39) |θ −θ | is the
i N 0
in which U(1) is only an approximate symmetry that initial displacement). Then, assuming θ is tuned to be
PQ i
is explicitly broken to a discrete subgroup Z . Then, as small as required by cosmology but not more so,
N
higher dimension operators allowed by the discrete sym- θ¯m2
metry but forbidden by the continuous U(1)PQ generate m2eff,0<∼ |θN −QCθD0|. (17)
contributionstotheaxionmassinadditiontothosefrom
Note, owing to the necessarily small denominator, this
QCD. If these operators were enhanced during inflation,
constraint is somewhat weaker than the usual con-
thelargeinflationarymasscouldsufficientlysuppressax-
straint for O(1) displacement between the operators—
ion isocurvature while still yielding a model consistent
constraints would be more stringent for larger displace-
with strong CP constraints.
mentsasallowedby,e.g.,dilution. Herewehaveapprox-
Supposing the axial component of the field is heavy
imated the QCD potential as quadratic in the vicinity of
during inflation, it will evolve to the minimum favored
the minimum,
by the operator(s) responsible for generating its large
1
V (cid:39) m2 (a−f θ )2, (18)
mass. This minimum need not coincide with those of 2 QCD a 0
theQCDpotentialandingeneralwouldnotbeexpected and m (cid:39)6 µeV(cid:0)1012 GeV/f (cid:1) [67].
QCD a
to as the two contributions to the axion potential arise
This severe constraint makes it impossible to arrange
fromdifferentsources. However,itwilldeterminetheini-
for a large inflationary mass via explicit PQ symmetry
tial misalignment angle θ . So, for axions with large f ,
i a breaking under the assumption that f = f . However,
I a
barringamechanismfortheaxiontoadiabaticallyevolve
displacement of the PQ field during inflation from its
to the vicinity of the minimum today, cosmological con-
minimumtodaycanenhancethenon-QCDcontributions
siderations do require a rather dissatisfying coincidence
to m2. In fact, these models can achieve an effective
a
betweenoperatorstoavoidoverproductionofaxionDM, inflationary axion mass m2eff,inf>∼ HI2 even for fI<∼ MP,
seeEq.(11).10 Thisincrediblecoincidencecouldbemiti-
so the required field displacement can be substantially
gatedif, forinstance, alate-decayingparticledilutedthe
less than the (super-)Planckian values of (cid:104)|S|(cid:105) needed to
axion abundance, allowing larger initial misalignment.
suppress isocurvature by changing the normalization of
Moreover, if non-QCD contributions to the axion po-
the axion field alone (Sec. II).
tentialareincompletelyturnedoff,measurementsofCP-
However, the symmetry breaking responsible for giv-
violating observables today tightly constrain the size of
ing the axion a large inflationary mass will also give the
such contributions [66]. For the axion to still provide a
radial field a mass of the same order, m ∼ H . Thus,
σ I
solution to the strong CP problem, the minimum today
in this case, f > f cannot simply arise from Hubble
I a
mustcorrespondtotheQCDminimumθ toaveryhigh
0 trapping, but instead requires a modification of the PQ
degree,
potential during inflation. As such, the PQ field must
(cid:28)(cid:12)(cid:12)(cid:12)(cid:12)fa −θ0(cid:12)(cid:12)(cid:12)(cid:12)(cid:29)≤θ¯(cid:39)10−11, (16) cpoliunpglsettootihneflaintifloantaiorynadryynsaemctiocsr.cIamnpdoirstraunpttlyt,hseuflchatcnoeuss-
a
of the fragile inflaton potential. To be concrete, the con-
whereθ¯representsthecurrentconstraintsontheeffective
tributions of couplings between the PQ and inflationary
θ angle. Let m2 represent additional contributions to
eff,0
sectors, denoted ∆V, will result in additional contribu-
m2 today and θ represent the minimum favored by the
a N
tions to the slow-roll parameters,
M2 (cid:18)V(cid:48)(cid:19)2 V(cid:48)(cid:48)
(cid:15)≡ P , η ≡M2 , (19)
2 V P V
10 The anthropic argument that we simply reside in an inflation-
ary patch where the initial misalignment is small is no longer
whereprimesdenotederivativeswithrespecttoI. There
valid. Anyattemptedanthropicargumentmustinsteadconsider
amultiverseselectionamongstappropriatepotentialparameters. are no hard and fast constraints on such contributions,
8
as the part of the potential that depends only on the in- the axion mass today,
flaton V could always be tuned to give values consistent
I m2
eff,0 sin(N|θ −θ |)
with CMB observations. However, large contributions N N 0 <∼θ¯, (22)
m2 +m2 cos(N|θ −θ |)
from ∆V would require particularly severe (and poten- QCD eff,0 N 0
tially dynamical, if f changes over the course of infla- whereasabovewehaveapproximatedtheQCDpotential
I
tion)tunings. Assuch,V wouldhavetobeverycarefully asquadraticinthevicinityoftheminimum,seeEq.(18).
I
arranged to appropriately balance the contribution from For small m2 (i.e., supposing sin(N|θ −θ |) not in-
eff,0 N 0
∆V throughout inflation and maintain an appropriate credibly small and so neglecting the subdominant term
inflationary trajectory. in the denominator), we write the constraint as
We study the restrictions arising from these contribu- Nθ¯m2
tions to the inflationary potential as well as from strong m2eff,0<∼ sin(N|θQC−Dθ |). (23)
N 0
CP constraints in the context of specific models below.
Taking θ = |θ −θ | small as required by cosmology,
We will consider three models. The first will consist of i N 0
see Eq. (11), and expanding this equation reproduces
only a single PQ field S charged under Z , and we will
N
Eq. (17). Unless sin(N|θ −θ |) is extremely close to
see that the combination of strong CP constraints and N 0
zero—in particular, much closer even than required by
avoiding excessive disruption of the inflaton potential
cosmology—thisplacesastringentlowerboundonN for
limits the values of f that can be reasonably allowed.
a
a given (|k|,f ) that generally requires N to be large,
Consequently, we will consider other models in which a
additional fields charged under U(1) (or the discrete N>∼O(10). Inotherwords,asonemightexpect,U(1)PQ
PQ
needs to be a good symmetry to a high degree in order
symmetry)—eithertheinflatonoradditionalPQfields—
to solve the strong CP problem.
acquirelargevaluesduringinflation, potentiallyrelaxing
Clearly, this precludes the SN operator from giving a
constraints on models with larger f .
a
large mass to the axion during inflation if f = f , even
I a
for|θ −θ |(cid:28)1. WhenI takesonlargevalues,though,
N 0
A. A Simple Model with a Single PQ Field
the I2|S|2 coupling can drive (cid:104)|S|(cid:105) = √fI > √fa. Then,
2 2
a large effective mass for the axial direction due to the
Perhaps the simplest example of an operator that ex-
SN term may stabilize the phase of the PQ field at θ
plicitly breaks U(1) to a discrete symmetry (in this N
PQ
and suppress fluctuations. After inflation, the effect of
case, Z ) is SN. We consider a basic model containing
N
the inflaton-PQ field cross-coupling will disappear, and
this operator as well as U(1) invariant terms,
PQ
the field will evolve to a minimum with (cid:104)|S|(cid:105) = √fa and
(cid:18) f2(cid:19)2 δ (cid:32) kSN (cid:33) 2
V ⊃λ |S|2− a − I2|S|2+ + h.c. . arg(S) (cid:39) θN after a period of rolling and oscillation,
2 2 MN−4
P where it will remain until m2 turns on.11
QCD
(20)
However, this coupling also influences the inflaton po-
The inclusion of the δ coupling is motivated by the fact
tential. Theinflationarytrajectoryisconstrainedbylim-
that it is not forbidden by any symmetries. In addition,
its on the slow-roll parameters given in Eq. (19),
for δ >0, this coupling can be responsible for generating
the necessary radial displacement fI >fa. r ≈16(cid:15)<.07, (24)
Theoperatorinthelasttermofthisequationwillgen-
n −1≈−6(cid:15)+2η ∈(−0.0413,−0.0253), (25)
s
erate an additional contribution to the axion mass
|k|N2(cid:104)|S|(cid:105)N−2
m2 = . (21)
eff MN−4
P 11 While the potential barriers in the axial direction are small at
small(cid:104)|S|(cid:105),thelargerbarrieratlarge(cid:104)|S|(cid:105)willproducea“fun-
If θ represents the minimum favored by the SN oper-
N nel” directing the phase towards θN or, if oscillations are suffi-
ator, strong CP constraints require the contribution to cientlylargetotakeS through(cid:104)|S|(cid:105)=0,towardsθN +π.
9
In Fig. 2, we present an example of the constraints on
50
such a model. Blue regions are excluded by strong CP
constraintswithcosmologicalboundsontheaxionabun-
40 dance taken into account, i.e., |θ −θ | set by Eq. (11).
N 0
Theconstraintsareindiscerniblymorestringentifoneas-
30 5×10-1 P sumes|θN −θ0|(cid:39)O(1)asmightbeexpectedinthecase
N M of late-time dilution. The red region denotes where the
/
fI inflatonpotentialissignificantlymodified,inviolationof
20 3×10-1 Eq. (26). If these constraints are weakened by a factor
of 3, this exclusion region moves up to N>∼ 130, poten-
10-1
tially allowing f (cid:39) 3×10−1. We take H (cid:39) 10−5M ,
10 a I P
10-2 consistent with 10−3<∼ r <∼ 10−2 that will be probed in
10-6 10-5 10-4 10-3 10-2 10-1 1 near future experiments.14 The Lyth bound [68] implies
fa/MP that, for such values of r, the excursion of the inflaton
FIG. 2. Bounds on the SN model in the parameter space field over the course of inflation is M∆PI >∼ 1. So, we set
of f and N for |k| = 1. The right axis shows the re- I = 5M at CMB mode horizon crossing (comparable
a ∗ P
quired value for f to give the axion a large enough infla-
I to the value for a Starobinsky-like model at the edge of
tionary mass to suppress isocurvature. Blue corresponds to
the strong CP constraint. Red indicates where the contri- observability [69]).
butions to the slow-roll parameters from the S potential are
The constraints shown are conservative in the sense
greater than present bounds. The hatched region indicates
where (cid:104)|S|(cid:105) >104√fa so PQ symmetry restoration via para- that,foreachvalueofN,wechooseλ,δtobethesmallest
e 2
metric resonance may occur (see text for caveats). Vertical
possible values such that the radial and axial directions
lines indicate the CASPEr Phase 2 (right) and ideal (left)
reach (prospective bounds extend to the right). bothhavemasses>∼HI—largercouplingswouldresultin
larger contributions to the slow-roll parameters.15 The
where we have given the 95% confidence bounds from axion mass condition is equivalent to requiring
[8, 11]. A reasonable constraint to avoid an overly tuned (cid:32) (cid:33)N−2
H2MN−4
inflationary model is to require that the additional con- fI (cid:39) |Ik|NP2 . (27)
tributions to (cid:15),η do not exceed the maximal values con-
The CASPEr experiment [26]—with its Phase 2 reach of
sistent with present CMB observations.12 For instance,
requiring that the contribution be <∼ 1 of the maximal fa>∼1.3×1016GeVandidealreachoffa>∼4×1013GeV—
may eventually be able to probe much of the allowed
value corresponds to
region as indicated by the vertical lines.
(cid:12)(cid:12)∆V(cid:48)(cid:12)(cid:12) (cid:12)(cid:12)∆V(cid:48)(cid:48)(cid:12)(cid:12)
(cid:12) (cid:12)≤0.094, (cid:12) (cid:12)≤0.021. (26) Thecross-hatchedregiondenoteswherethefieldvalue
(cid:12) V (cid:12) (cid:12) V (cid:12)
attheendofinflation(cid:104)|S|(cid:105) >104√fa,supposingthatthe
e 2
We stress though that, while it represents a severe, ad
valueoftheinflatonfieldattheendofinflationI isafac-
e
hoc, dynamical tuning and so should be taken seriously,
torof5smallerthanatCMBhorizoncrossing(suchthat
this constraint is aesthetic rather than experimental.13 (cid:104)|S|√(cid:105)e ≈ 1). According to the analysis of [53], for such
fI/ 2 5
values radial PQ field oscillations may excite large oscil-
lations in the axial field, potentially leading to nonther-
12 An analogous constraint on ξ or |∆V(cid:48)(cid:48)(cid:48)/V| from the running
spectralindexαsissubdominant. Inaddition,wehaveconfirmed
that∆V (cid:28)VI inthesemodels.
13 TSehci.sIcIonfrsotrmainatrigsienngerianllythpisrecmluadnensert.heF(cid:104)o|Sr|(cid:105)in>staMncPe,resquupiproedsining 14 Note that fI > H2πI throughout the parameter space, such that
∆V =−2δI2|S|2weexpectδI2>∼HI2suchthatm2σ>∼HI2and|S| thePQsymmetryisbrokenduringinflationevenforfa< H2πI.
evolvestothelargevev. Thisimplies|∆V(cid:48)/V|∼fI2/(IMP)>∼1 15 The required values of δ are sufficiently small that parametric
and|∆V(cid:48)(cid:48)/V|∼fI2/I2>∼1,exceedingthestatedbounds. excitationofS duetoinflatonoscillationsisnotaconcern.
10
mal symmetry restoration (see Eq. (7) and surrounding threatening to destabilize the fragile inflaton potential
discussion). In this model, the situation is complicated and at best constituting a very severe (field-dependent)
by the presence of additional operators and the lack of tuning.
initialfluctuations. Forinstance,thecurvatureofthepo- One reason that it is difficult to achieve a sufficiently
tentialintheaxialdirectionduetotheSN operatormay large mass whilst satisfying strong CP constraints is
suppressfluctuations. Alternatively,self-couplingsofthe that the same field is responsible for both the enhance-
axial field resulting from this operator may mitigate the ment of the explicit breaking and solving the strong CP
growthoffluctuations[70]. Thus,whilethisregionisnot problem—thelatterrequiresthepotentialofthisfieldbe
necessarily excluded, a further analysis of field dynamics largely U(1) invariant today, such that drastic mod-
PQ
post-inflation would be required to ensure a consistent ifications of its potential are required during inflation,
cosmology. readily disrupting inflationary dynamics. Consequently,
inthenextsubsections,wediscussthepossibilityofreal-
Thisanalysissuggeststhatitisdifficulttoreachlarger
izingalargeaxionmassduringinflationbycouplingS to
values of fa >∼ 10−2MP. While the conclusion that a
additional fields that acquire large vevs during inflation
significant modification of the PQ potential can disrupt
but are not subject to the same strong CP constraints.
inflation is robust, the exact limits do depend on the in-
We will see that such approaches do indeed extend the
flationary model. For larger values of I , the same f
∗ I
reach in f relative to this simple model.
can be achieved for smaller values of δ and λ, resulting a
in smaller contributions to slow-roll parameters. In ad-
dition, as long as I (cid:39) M , (cid:104)|S|(cid:105) would be reduced in B. Inflaton-Sourced Axion Mass
e P e
the case of large I , shrinking the nonthermal symmetry
∗
At first glance, coupling to the inflaton would ap-
restorationregion. Assuch, perhapsparadoxically, high-
pearanidealmethodforboostingtheaxionmassduring
scaleinflationmayreconcilemorereadilywithasolution
inflation—the large (super-Planckian) value of I could
of this type—the steeper potential could be less suscep- ∗
giveasignificantcontributiontom2 evenviaPlanck-
tible to disruption. It should be noted, however, that eff,inf
suppressed operators. Meanwhile, for small (cid:104)|I|(cid:105) today,
larger I would also likely correspond to larger values of
∗
thiscontributionwouldbesuppressed, alleviatingstrong
H (cid:39) 10−4M (r (cid:39) 10−2), implying imminent tensor
I P
CP constraints. However, as we discuss, the difficulty
mode observation. Meanwhile, in a specific inflationary
in this approach lies in generating the desired operators
model, violation of Eq. (26) may result in a worse than
withoutgeneratingeitheradditionalunwantedoperators
O(1) tuning. Recall, these equations were derived us-
orsymmetries(thelatterofwhich,forinstance,maypre-
ing current experimental constraints. For a Starobinsky-
vent fields from acquiring a necessary mass).
like model with r (cid:39) 2×10−3, the red region in Fig. 2
One straightforward approach is to consider a term
would more approximately correspond to tuning>∼ few,
that explicitly breaks U(1) such as
owing to the smaller denominators in Eq. (26). Requir- PQ
ing tuning ≤ 1 in this model would exclude N >∼ 31, kISN
V ⊃ + h.c. (28)
fa>∼2×10−3MP. MPN−3
In sum, strong CP constraints require large values of Such an operator was considered in, e.g., [31].16 If
N to ensure that the contribution to the axion mass due the rest of the potential maintains the symmetry I →
to the explicit breaking is small today. As such, fI must −I,SN → −SN, this explicitly breaks U(1)PQ → Z2N.
be somewhat larger than f to ensure the axion is suf-
a
ficiently heavy during inflation. In other words, large
16 Refs.[71,72]considerthepotentialforsimilaroperatorstogen-
modifications to the PQ potential are required. These
erateacross-correlationspectruminthecasewhereisocurvature
in turn “backreact” on the inflaton potential, at worst isunsuppressed.