Inflaton dark matter from incomplete decay

0.62 MB·English

by  Mar Bastero-Gil

Inflaton dark matter from incomplete decay
Mar Bastero-Gil,1,∗ Rafael Cerezo,1,† and Jo˜ao G. Rosa2,‡
1Departamento de F´ısica Te´orica y del Cosmos, Universidad de Granada, Granada-18071, Spain
2Departamento de F´ısica da Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal
(Dated: January 23, 2015)
We show that the decay of the inflaton field may be incomplete, while nevertheless successfully
reheatingtheuniverseandleavingastableremnantthataccountsforthepresentdarkmatterabun-
dance. Wenote,inparticular,thatsincethemassoftheinflatondecayproductsisfield-dependent,
onecan constructmodels, endowedwith an appropriate discretesymmetry,where inflatondecay is
kinematically forbidden at late times and only occurs during the initial stages of field oscillations
after inflation. We show that this is sufficient to ensure the transition to a radiation-dominated
5 era and that inflaton particles typically thermalize in the process. They eventually decouple and
1 freezeout,yieldingathermaldarkmatterrelic. Wediscusspossibleimplementationsofthisgeneric
0 mechanism within consistent cosmological and particle physics scenarios, for both single-field and
2 hybridinflation.

n
a
J I. INTRODUCTION that pφ ρφ [8]. These two regimes will generically
be presen≪t in inflationary potentials, which further sug-
2
2 Inflatonanddarkmattercandidatesinparticlephysics gestsacommonframeworkforinflationanddarkmatter.
models share several common features, both being typi- The main difficulty in achieving such a unified descrip-
h] callyassumedtobeweaklyinteractingandneutralfields. tionlies,however,inthefactthatinflationmustendwith
p The inflaton scalar field must interact weakly with itself a transition to a radiation-dominated universe, in order
- and other degrees of freedom in order to ensure the re- to recover the standard “Big Bang” evolution at least
p quired flatness of the associated scalar potential, which before the freeze-out of light nuclear abundances takes
e
h could be spoiled by large radiative corrections [1]. Sim- place [9]. An efficient transfer of energy between the
[ ilarly, dark matter particles should form a stable non- inflaton field and radiation generically requires the for-
relativistic and non-luminous fluid at late times that ac- mer to decay into light degrees of freedom following the
1 counts for the observed galaxy rotation curves [2] and period of inflationary slow-roll [10], even though other
v
the large scale structure in the universe as inferred from non-perturbativeprocessessuchasparametricresonance
9
3 Cosmic Microwave Background [3, 4] and weak-lensing amplificationcouldcontributesignificantlytothereheat-
5 observations [5, 6] (for a recent review see [7]). Both ing process[11]. A plausible alternativeis alsothat non-
5 inflation and dark matter are essential features in the equilibrium dissipative processes continuously source a
0 moderncosmologicalparadigmandcannotbe accounted radiation bath during inflation that smoothly takes over
1. for within the present frameworkof the Standard Model asthe fieldexitstheslow-rollregime,apossibilitygener-
0 of particle physics. It is therefore interesting to consider ically known as warm inflation [12, 13].
5 the possibility that the same field accounts for both ac- Nevertheless, efficient reheating does not imply the
1 celeratedexpansionin the early universe and the hidden complete decay of the inflaton field and the possibility
:
v matter component at late times. thata stableremnantis leftafter inflationhasbeen con-
i Scalarfieldshavetheinterestingpropertyofmimicking sidered in a few studies in the literature [14–18]. In this
X
fluids with different equations of state depending on the work, we propose a concrete realization of this generic
ar kinematicalregimeconsidered. Forahomogeneousscalar ideainquantumfieldtheory,wherethedecayoftheinfla-
field φ with potential V(φ), we have: tonis truly incomplete,occurringonly forafinite period
after the end of the slow-roll regime.
1 1
ρ = φ˙2+V(φ) , p = φ˙2 V(φ) . (1.1) We describe a simple mechanism based on standard
φ φ
2 2 − Yukawa coupling between the inflatonand fermionfields
Hence, on the one hand a slowly-varying field, φ˙2/2 (and potentially their superpartners), endowed with an
V(φ) acts as an effective cosmologicalconstant, which≪is appropriatesymmetrythat protectsthe full decayofthe
the regime typically considered in canonical inflationary inflaton field, and discuss the parametric regimes where
models. On the other hand, a field oscillating about the inflaton decay into such fermions is incomplete. We also
minimumofits potentialwhereV(φ) m2φ2/2behaves discuss the embedding of this generic mechanism in con-
as non-relativistic matter, with φ˙2/2≃=φ V(φ) such crete inflationary models and possible scenarios for the
h i −h i interactionsbetweenthe fermionfieldsandthe Standard
Model degrees of freedom that allow for the presence of
the latter in the post-inflationary thermalbath. We also
∗ [email protected] show that the inflaton remnant is not necessarily in the
† [email protected] form of a coherent condensate of bosonic particles and
‡ [email protected] that, in particular, for not too small couplings the evap-
2
oration of this condensate is inevitable. In this case, the wherewenote that,asaresultofthe discrete symmetry,
thermalized inflaton particles eventually decouple from the two fermions have the same tree-level mass m but
f
theradiationbathandtheirabundancefreezesout,yield- opposite Yukawa coupling to the inflaton field. The ac-
ing a thermal inflaton relic with properties similar to tionofthediscretesymmetryisrestrictedtotheinflaton-
other WIMP candidates [19]. We suggestively denote fermion sector, such that all other fields, including the
this as the “WIMPlaton” scenario. Standard Model fields, are invariant under this symme-
Inthiswork,weconsidertwodynamicallydistinctsce- try.
narios. Firstly, we will consider a minimal model with a This symmetry implies that the inflaton potential is
single dynamical scalar field that simultaneously drives an even function of the field and we assume that it is
inflation, reheats the universe through incomplete decay unbroken at the minimum of the potential, which must
and leaves a stable non-relativistic remnant. Secondly, therefore lie at the origin, φ = 0. As a result, since no
we consider a (supersymmetric) hybrid inflation model othertermslinearinφexceptfortheaboveYukawaterms
with an additional dynamical waterfall sector, which is are allowed by the discrete symmetry, the only possible
responsible for ending inflation andreheating of the uni- decay channels of the inflaton at the minimum are φ
verse. In this case, we show that despite reheating be- ψ¯ ψ . If the inflaton mass at the minimum is given b→y
± ±
ing ensured by a different field, the inflaton field must m2 = V′′(0) > 0 and we require m > m /2, these
φ f φ
decay in order for radiation to fully come to dominate decays will be kinematically forbidden for φ=0 and the
the energy balance in the universe. As in the minimal inflaton will be stable when it settles at the minimum
model, this decay may nevertheless be incomplete and of its potential. Hence, the discrete symmetry and the
we discuss the parametric regimes in which the inflaton masshierarchyensurethe stability ofthe inflatonatlate
remnant constitutes a suitable dark matter candidate in times, such that it will contribute to the present dark
the hybrid framework. matter abundance.
This work is organized as follows. In the next section Thetwofermionfieldsareindistinguishableapartfrom
we discuss the minimal model of inflaton dark matter theircouplingtotheinflaton,whichleadstoamasssplit-
from incomplete decay, starting with its basic properties ting away from the origin:
and dynamics and then discussing the possibility of con-
m = m hφ . (2.2)
densate evaporation, its embedding in a consistent infla- ± f
| ± |
tionary model and different phenomenological possibili-
This implies that, although inflaton decay into these
tiesforreheatingtheStandardModeldegreesoffreedom.
fermionsisforbiddenatlatetimesasthefieldapproaches
In Section III we discuss these aspects in the hybrid re-
the origin, that need not be the case if the amplitude of
alization of inflaton dark matter. Finally, in Section IV
field oscillations after inflation is sufficiently large. In
we summarize our main results and conclusions.
particular, decay will be allowed for field values satisfy-
ing:
II. MINIMAL MODEL m hφ <m /2 . (2.3)
f φ
| ± |
Thisimplies,inparticular,thatdecayiskinematicallyal-
A. Basic properties and dynamics
lowedforfieldamplitudes φ &m /h. Thepartialdecay
f
| |
widthsassociatedtothetwofermionicdecaychannelsare
The minimal model for inflaton dark matter considers
then given by:
asingledynamical(real)scalarfield,the inflatonφ,with
a potential energy V(φ) such that a period of slow-roll 3/2
h2 4m2
canoccurforsomefieldrange. Wetakethe inflatonfield Γ = m 1 ± , (2.4)
to be coupledto fermionfields ψ+ andψ− throughstan- ± 8π φ − m2φ !
dardYukawatermsandimposeadiscretesymmetry1[20]
withΓ =Γ +Γ . Notethatduetotheoppositesignof
C Z S on the Lagrangiansuch that the scalar in- φ + −
fla2to⊂nt2ra×nsfo2rmsundertheZ groupasφ φandthe the Yukawacouplings,the inflatonwillalternatelydecay
2 →− intoeachfermionspeciesasitoscillatesbetweennegative
fermions are simultaneously interchanged by the permu-
and positive values. The inflaton equation of motion as
tation symmetry ψ ψ . This yields for the resulting
+ ↔ − it oscillates about the minimum of its potential is then
Lagrangiandensity:
given by:
1
L= 2∂µφ∂µφ−V(|φ|) φ¨+3Hφ˙+Γφφ˙ +m2φφ=0 , (2.5)
+ψ¯+(iγµ∂µ−mf)ψ++ψ¯−(iγµ∂µ−mf)ψ− which, upon multiplying by φ˙ is equivalent to:
hφψ¯ ψ +hφψ¯ ψ , (2.1)
+ + − −
− ρ˙ +3H(ρ +p )= Γ φ˙2 . (2.6)
φ φ φ φ
−
The term on the right hand side gives the rate at which
1 Thissubgroupcontainsonlyelementsthattransformsimultane- energy density is transferred from the inflaton field into
ouslyunderZ2 andS2. thefermionsψ . Letusassumethatthefermionsquickly
±
3
thermalize,anassumptionthatwewillcheckaposteriori, HaL
in the process exciting g∗ relativistic degrees of freedom 10-5
and forming a radiation bath at temperature T, with
energy density ρR = (π2/30)g∗T4. Note that since the 10-8 ΡΦ
fermionmassesoscillateduetothevaryinginflatonfield,
they only contribute periodically to the number of rela- 10-11
tivistic degrees of freedom. These may also include the Ρ ΡR
inflaton for T & mφ and other species such as the Stan- 10-14
dard Model particles. Note that the latter must be ex-
cited before the cosmological synthesis of light nuclear 10-17
elements takes place, as we discuss in more detail in sec-
tion IID. For simplicity, we consider a fixed value of g ,
∗ 10-20
which is not a bad approximation since, as we will show 10 1000 105 107
later on, our results exhibit only a mild dependence on t
this parameter.
Energy conservation then implies that the energy lost HbL
bytheinflatonfieldinEq.(2.6)isgainedbytheradiation
106
bath, which then follows the dynamical equation:
ρ˙ +4Hρ =Γ φ˙2 . (2.7) 104
R R φ
Since both the inflaton and the radiation contribute to 100
the energy density in the universe, we may write the (cid:144)nsΦ
Friedmann equation as: 1
H2 = ρφ+ρR = 12φ˙2+ 21m2φφ2+ρR . (2.8) 0.01
3M2 3M2
P P
10-4
Eqs. (2.5), (2.7) and (2.8) then form a complete set of 10 1000 105 107
differentialequationsthatcanbesolvedforgivenchoices
t
oftheparameters(m ,m ,h)andinitialconditions. The
φ f
non-linearity of the equations precludes a complete an-
HcL
alytical description of the problem, although as we dis-
cuss below a few simple analytical estimates can be per-
formed, but it is straightforward to integrate them nu- 10-4
merically. Itisusefultowritethefermiontree-levelmass
as: 10-7
3H
m
φ
mf = (1+δ) , (2.9) GΦ
2 10-10
such that the decay is kinematically forbidden (allowed) GΦ
at late times for δ >0 (<0). 10-13
Figure 1 shows an example of the results obtained by
numerically solving the inflaton-radiation equations for
m = 10−3M , δ = 0.1 and h = 1. In this example we 10 1000 105 107
φ P
take as initial conditions at t = t = 10−2m−1, φ(t ) = t
i φ i
M , φ˙(t ) = 0 and ρ (t ) = 10−16ρ (t ), although the
P i R i φ i FIG. 1. Results of the numerical integration of the inflaton-
resultsareessentiallyunchangedaslongasthefieldvalue radiation dynamical equations for mφ = 10−3MP, δ = 0.1
issufficientlylargebeforeoscillationsbeginforthedecay and h = 1, showing the time evolution of (a) the inflaton
to be kinematically allowed as discussed above and the (solid bluecurve)andradiation (solid redcurve)energyden-
radiation energy density is initially negligible, as should sities; (b) the inflaton-to-entropy ratio; and (c) the inflaton
be expected after 50-60 e-folds of inflation. decay width (solid blue curve) compared to the Hubble pa-
TheplotinFigure1(a)illustratesthemaindynamical rameter (solid green curve). The blue dashed curves in (a)
and (c) give the evolution of the inflaton energy density in
features that are generically obtained. The inflaton field
the absence of decay and the maximum value of the decay
begins to oscillate about the origin with frequency m
φ width, respectively. All quantities are given in Planck units
for t m−1 and behaves initially as cold dark matter,
∼ φ such that MP =1.
ρ a−3, being the dominant energy component such
φ
∝
that a t2/3 and H =2/3t. While decay is blocked be-
∝
fore the onset of oscillations, since m hφ m , it
± φ
∼ | | ≫
4
becomes kinematically allowed as soon as the field goes then gives:
through the origin. Since in this example the ampli-
√6h
tude ofthefieldoscillationslargelyexceedsthe tree-level ρ m3M , (2.13)
masses, with hΦ m , decay occurs for two narrow R ≃ 32π2 φ P
f
≫
field ranges close to (and on both sides of) the origin. whichis inverygoodagreementwiththe numericalsim-
The decay width then corresponds to the series of peri- ulations. Noting that, from Eq. (2.7), ρ˙ 0 implies
R
≃
odic narrowpeaksshowninFigure1(c), withmaximum 4Hρ Γ φ˙2 Γ ρ , we see that that Γ 3H for
R φ φ φ φ
value Γmφax = (h2/8π)mφ. While initially Γmφax ≪ 3H ρR ∼ρ≃φ,sothat≃asobservednumericallythein∼flatonen-
as illustrated in Figure 1 (c), such that the inflaton’s ergydensityisonlyreducedsignificantlywhenitbecomes
energy density remains essentially unaffected by the de- comparable to the radiation energy density.
cayinto fermions,the sourceterm inthe radiationequa- FromEq.(2.13)wecaneasilydeterminetheassociated
tionquicklybecomessignificant,leadingtoajumpinthe temperature, which remains approximately constant up
value ofthe radiationenergydensity. The latterremains toinflaton-radiationequalityandthusconstitutesthere-
approximately constant until the inflaton’s energy den- heating temperature:
sity is sufficiently redshifted. When they become com-
parablein magnitude,the inflatoneffectively decays and 15√6 1/4 m 3/4
T g−1/4h1/4 φ M
rfiaedldiaatimonpltitaukdees odveecrreaassetsh,ethdeommianxainmtucmomdpeocnayenwt.idAths tbhee- R ≃ 16π4! ∗ (cid:18)MP(cid:19) P
comesprogressivelysmalleruntildecayisfinallyblocked. 2.7 106g−1/4h1/4 mφ 3/4 GeV.(2.14)
∗
The oscillating inflaton then becomes stable and once ≃ × 1 TeV
(cid:16) (cid:17)
morebehavesascolddarkmatter,eventuallytakingover From this we conclude, in particular, that the reheating
the radiationas the dominant component at later times. temperature is generically larger than the inflaton mass
A peculiar dynamical feature of the evolution is the for:
approximate constancy of the radiation energy density
h
achieved just after the first few oscillations. This is in- m <5.6 1016 GeV . (2.15)
φ
× g
herent to the fact that inflaton decay occurs in short ∗
burstsineachoscillation,whichdoesnotoccurifthe de- After inflaton-radiation equality, the field decays ex-
cay were allowed for all field values. Before the effect of ponentially fast until its amplitude drops sufficiently for
decay into fermions becomes significant,for t&m−1 the decay into fermions to become inefficient. Numerically,
φ
inflaton behaves as a damped harmonic oscillator with we observe that this occurs before the decay becomes
kinematically forbidden, for Φ . m /h, corresponding
f
8MP 1 roughly to when the (maximum) decay width becomes
φ(t) Φ(t)sin(m t+α ) , Φ(t)= ,
≃ φ φ 3 m t less thanthe Hubble rate. Since afterwardsthe field sta-
r φ
bilizesandbehavesasnon-relativisticmatter,itis useful
(2.10)
to compare its number density n = ρ /m a−3 with
φ φ φ
∝
where α is a phase depending on the initial field and the radiation entropy density:
φ
velocity values. For hΦ m , one can easily see that
decay into each fermion i≫s allofwed during a short period s= 2π2g T3 , (2.16)
∗
45
τ (hΦ)−1 2π/m as the field goes through the ori-
d φ
gin≃, which oc≪curs twice every oscillation period. Since which redshifts in a similar way up to changes in the
the average decay width in this interval can be taken as number of relativistic degrees of freedom, with T a−1.
∝
rΓimφodaxt/h2earanddiatthieonfieelndervgeylodceitnysiitsyφ˙in≃cremaφseΦs,deuveertyoihnafllfatpoen- tAesridlleucsatyrabteedcoimneFsiginuerffie1ci(ebn)t,,tahnedrantuimonerφic/asllsytawbeiliozbestaaifn-
decay by an amount: the following expression:
n
∆ρdecay Γmφax(m Φ)2(2τ ) h m3Φ , (2.11) sφ ≃0.5g∗−1/4h−3.6(mφ/MP)1.02f3(δ) , (2.17)
R ≃ 2 φ d ≃ 8π φ
where for 0<δ .1:
where we have taken into account the decays into both
ψ and ψ . When decay is forbidden, radiation simply f(δ)=1+4.8√δ+0.5δ . (2.18)
+ −
redshiftswithexpansion,whichcounteractstheenhance-
Ifnootherprocesseschangetheinflatonparticlenumber
ment due to decay by an amount:
density in the oscillating field, as we investigate below
in more detail, this ratio will remain constant until the
∆ρHubble 4Hρ (π/m ) . (2.12)
R ≃− R φ present day. Assuming the oscillating inflaton field ac-
countsforallthedarkmatterintheuniverse,thepresent
Since H = 2/3t = (m /√6M )Φ, it is easy to see that
φ P
inflaton-to-entropy ratio is given by:
the amount of radiation produced by inflaton decay can
be balanced exactly by Hubble expansionto yield a con- n 3 T Ω g M
φ0 = 0 c0 ∗0 10−28 P , (2.19)
stant energy density. Equating ∆ρdecay = ∆ρHubble s 4m Ω g ≃ m
R − R 0 φ R0 ∗s0 φ
5
where T = 2.73 K is the present CMB temperature, inEq.(2.1)andwhichmayplay animportantroleas we
0
while Ω h2 = 0.12 and Ω h2 4.17 10−5 are the discuss below.
c0 R0
≃ ×
present abundances of cold dark matter and radiation, The classical inflaton field corresponds to a collective
respectively [4]. We have also taken into account the state of zero-momentum scalar bosons, assuming that
present difference between the number of relativistic de- no large field inhomogeneities are formed at the end of
grees of freedom contributing to the radiation and en- the slow-roll inflationary regime. Inflaton particles in
tropy densities, g 3.36, g 3.91. Equating (2.17) this condensate can interact with the fermions that re-
∗0 ∗s0
≃ ≃
and (2.19), we obtain for the inflaton mass: sultfromits decayand,in particular,these fermions can
scatter some of the bosons out of the condensate and
m 72g0.12h1.78f(δ)1.49 TeV . (2.20)
φ ≃ ∗ promote them to higher-momentum states that become
For fermion masses above the kinematical limit and of part of the thermal bath. These correspond to scatter-
the order of the inflaton mass, 0 < δ . 1, and taking ing processes ψ± φ ψ±φ, where we denote by φ
h i → h i
g = 10 100, we conclude from Eq. (2.14) that the re- andφ scalarparticlesin the zero-momentumcondensate
∗
heating t−emperature is above 100 MeV for Yukawa cou- andinhigher-momentummodes,respectively,andwhich
plingsh&10−6 10−5,correspondingtoinflatonmasses are mediated through both s- and t-channel fermion ex-
m & 10 keV. F−or larger (and arguably more natural) change. Moreover,these processes may occur as soon as
φ
couplings h & 10−3 10−2, the inflaton may account the field begins oscillating and decay into ψ± becomes
for the dark matter −in the universe for masses in the kinematically allowed, potentially leading to the evapo-
GeV TeV range, similarly to the mass range obtained ration of the condensate and the transfer of the inflaton
for W−IMP-like thermal relics. Qualitatively, it is easy to particle number into the thermal bath.
understand the parametric dependence of the required As we have seen earlier, soon after the onset of infla-
inflaton mass. The (incomplete) decay into fermions is tonoscillations,thetemperatureofthethermalbathrises
more efficient for larger couplings, which affect both the sharply to a value that remains approximately constant
overallvalueofthedecaywidthandtheeffectivefermion until inflaton-radiation equality and that corresponds to
masses, and lighter fermions, yielding a smaller inflaton- the reheating temperature in Eq. (2.14). In particular,
to-entropyratioatlatetimesandthusallowingforlarger this temperature exceeds the inflaton and fermion (tree-
inflaton masses to match the present dark matter abun- level) masses in the parameter space region that yields
dance. Ontheotherhand,smallercouplingsandheavier the present-daydark matter abundance. Further assum-
fermions lead to a largerinflaton abundance, which may ingthatlocalthermalequilibriumis quicklyachieved,as
overclose the universe unless the inflaton is sufficiently we check below, we may then take the phase-space dis-
light. tributionsforinflatonandfermionspeciesinthethermal
Wenotealsothatthereisnogaininconsideringfinely- bathtobetherelativisticBose-EinsteinandFermi-Dirac
tuned fermion masses, i.e. δ 0, since even though distributions,respectively. Takingintoaccounttheabove
decay is kinematically allowed f≪or longer as δ decreases, scatteringsandthe inverseprocesses,the netcondensate
decaybecomesinefficientwhenΓ H,aconditionthat evaporation rate is given by [21]:
φ
∼
becomes independent of the tree-level fermion mass in
this limit. Although we have restricted the numerical Γ = 1 4 d3pi (2π)4δ4(p +p p p )
analysis to values of δ . 1, we expect incomplete de- evap n (2π)32E 1 2− 3− 4
φ Z i=1 i
cay to be efficient as long as Γ > H for inflaton field Y
φ M2[f f (1+f )(1 f ) f f (1+f )(1 f )] ,
values satisfying the kinematical condition in Eq. (2.3). 1 2 3 4 3 4 1 2
×| | − − −
In particular, since the decay width takes its maximum (2.22)
value Γmax = (h2/8π)m for hφ = m , i.e. when the
φ φ | | f whereMisthescatteringamplitudefor φ (p )ψ (p )
fdeurrminiognisnflaaretoneff-deoctmivinelaytiomna,swsleescso,nacnluddeHth∼atmdeφc|φay|/MwiPll φ(p3)ψ±(p4)andfi thecorrespondingphhaise-s1pac±ed2istr↔i-
bution factors. Since the condensate is inherently char-
be efficient for:
acterizedbylargeoccupationnumbersf 1,weobtain
1
mf .h3MP , (2.21) to leading order for T ≫mφ,m±: ≫
whichallowsforquitelargefermionmassesiftheYukawa h4 T
coupling is not too suppressed. Γevap ≃ 12π3 1+log m T , (2.23)
(cid:18) (cid:18) φ(cid:19)(cid:19)
where we have taken into account the contribution of
B. Condensate evaporation: the WIMPlaton both fermion species. Note that this is only valid when
scenario the fermions are relativistic, while for non-relativistic
fermions in local thermal equilibrium the evaporation
Although the imposed discrete symmetry protects the rateisexponentiallysuppressed. Thismeansthatduring
inflatonfromdecayingintoanyotherparticlesexceptfor the first few oscillations before the inflaton amplitude is
the fermions ψ , the above analysis neglects the effects significantlyreduced,condensateevaporationonlyoccurs
±
of additional interactions induced by the Yukawa terms duringtheshortperiodswheredecayisalsokinematically
6
allowed. As we have concluded above, for the parameter between the value obtained in Eq. (2.14) and this max-
values yielding the present dark matter abundance we imum value if condensate evaporation occurs before it
obtain T & m ,m , and since m m after reheat- decays significantly. The reheating temperature is given
R φ f ± f
∼
ingtheaboveexpressionholdsuntileithertheinflatonor byEq.(2.14)ifcondensateevaporationonlyoccursinthe
the fermions become non-relativistic. Since in the radi- radiation era, which corresponds to Yukawa couplings
ation era H (π/√90)g1/2T2/M , condensate evapora- h . 10−4 for δ . 1 and g = 10 100. Note that in
∗ P ∗
≃ −
tion becomes progressively more efficient as the temper- this parametric regime thermalization of the fermions in
ature drops. We may then determine a lower bound on the plasmacanonly be efficientif their interactionswith
the Yukawa coupling such that condensate evaporation otherdegreesoffreedomarestrongerthanthoseinduced
isinefficientforT &m ,m . Forcomparablemasses,we bytheYukawaterms,whileforh&10−4 thelatteroccur
φ f
obtain: sufficientlyfasttomaintainlocalthermalequilibrium,as
we analyze in more detail in section IID.
Γ h4 g −1/2 M
evap ∗ P .1 .(2.24) After reheating, inflaton and fermion particles will be
H ≃ 4π3 10 m
(cid:12)(cid:12)T=mφ (cid:16) (cid:17) (cid:18) φ(cid:19) kept in local thermal equilibrium by annihilation and
(cid:12) elastic scattering processes. Once these become inef-
UsingEq,(2.(cid:12)20)fortheinflatonmass,wethenobtainfor
ficient, the abundance of inflaton particles will freeze
δ =1:
out, as for other conventionalWIMP darkmatter candi-
h.10−5g0.28 . (2.25) dates. Assuming this occurs when both the inflaton and
∗ the fermions are non-relativistic, the relevant (fermion
As we had seen above, this is in tension with the lower t-channel) annihilation cross section is given by:
bound onthe coupling requiredfor a reheating tempera-
ture above 100 MeV, such that initial conditions for Big h4
Bang Nucleosynthesis (BBN) are already in place after σφφ ≃ 8πm2 , (2.27)
inflaton decay [22]. This conclusion is essentially com- φ
mon to all fermion mass values m > m /2. Although
f φ
a more detailed analysis of the Boltzmann equation de- which is independent of the fermion mass in this limit.
termining the evolution of the inflaton condensate may Following the standard calculation for the thermal relic
berequired,thisestimateindicatesthatinthephysically abundanceofadecouplednon-relativisticspecies,weob-
interestingparameterrange,where the condensate could tain for the inflaton mass:
accountforthepresentdarkmatterabundancewhilesat-
isfying the BBN constraint, condensate evaporation is Ω h2 1/2 g 1/4 x −3/4
most likely inevitable. mφ 1.4h2 φ0 0 ∗F F TeV ,
≃ 0.1 10 25
Sinceevaporationsimplytransfersthezero-momentum (cid:18) (cid:19) (cid:16) (cid:17) (cid:16) (cid:17)
(2.28)
condensate particles into excited states, the conclusion
above does not imply that inflaton particles cannot ac-
count for dark matter. One can easily check that all 4- where g∗F denotes the number of relativistic degrees of
bodyprocessesinvolvingrelativisticinflatonandfermion freedom at freeze-out and xF =mφ/TF, with TF denot-
particles, including scatterings and annihilations, occur ingthefreeze-outtemperature. Thisissomewhatsmaller
ata rate comparableto the evaporationrate obtainedin than the mass values obtained assuming the oscillating
Eq. (2.23). Hence, evaporation of the condensate should inflaton condensate survives until the present day, al-
lead to a bath of fermions and inflaton particles (as well though in a comparable range and exhibiting a similar
as other species) in local thermal equilibrium. In fact, dependence on the Yukawa coupling.
this may even occur just after the onset of inflaton os- This then gives us a more realistic dynamical picture
cillations, for H . mφ, when the temperature reaches ofwhatwesuggestivelydenoteasthe“WIMPlaton”sce-
an approximately constant value T & mφ,mf as seen nario. After inflation, the scalar inflaton begins oscillat-
above, for h . O(1) couplings. The condensate’s energy ing about the minimum of its potential, decaying into
is in this case quickly transferred to the thermal bath, fermions in short bursts every oscillation. These may
increasing its temperature to a maximum value: thermalize and excite other degrees of freedom in the
plasma, and scatter off the inflaton particles in the con-
1/4
Tmax = 90 g−1/4 M m densate,leadingtoitsevaporation. Bothdecayandevap-
R π2 ∗ P φ oration increase the relative abundance of radiation and
(cid:18) (cid:19)
p m 1/2 decreasetheamplitudeofoscillations,untileventuallyra-
8.5 1010g∗−1/4 φ GeV .(2.26) diationbecomesdominantandinflatondecayisnolonger
≃ × 1 TeV
kinematically allowed. The stable inflaton particles re-
(cid:16) (cid:17)
Note that this was obtained for H = m , which con- maininthermalequilibriumuntilthetemperaturedrops
φ
stitutes only an upper bound since fermion particles are below their mass and they decouple from the plasma,
only produced for H . m as shown by our numerical their frozenabundance yielding the inferreddarkmatter
φ
simulations. Thetruereheatingtemperaturewillthenlie component of our present universe.
7
C. Embedding in a consistent inflationary model thepost-inflationaryevolutionwillnecessarilydifferfrom
the dynamical picture discussed in this work. It would
As we have concludedfrom the analysis above,the in- neverthelessbe interesting to investigatewhether the in-
flaton field can account for dark matter in the universe flaton can act as a stable dark matter candidate at late
atlatetimesformassesbeloworaroundtheTeVscalein timesinthiscontext,althoughthisliesoutsidethe scope
the WIMPlaton scenario, with a similar mass range ob- of the present discussion.
tained assuming the inflaton condensate does not evap- Anotherinterestingpossibilityistheinclusionofanon-
orate. This implies that the inflaton potential cannot minimalcouplingtothegravitationalsector,inparticular
be given solely by a quadratic mass term, since the am- a coupling between the inflaton and Ricci scalar of the
plitude of CMB temperature anisotropies would yield in formξφ2R,whichiscompatiblewiththediscreteZ2sym-
this case: metry. The resulting inflationary scenario yields a per-
turbation spectrum that smoothly interpolates between
m 6π2∆2RM 1.4 1013 60 GeV , the minimal quartic model and the Starobinsky model
φ P
≃ N ≃ × N as the non-minimal coupling constant increases. On the
p e (cid:18) e(cid:19)
(2.29) one hand, the latter is characterized by a low tensor-
to-scalar ratio and a spectral index n = 0.96 0.97 in
s
which for 50 60 e-folds of slow-rollinflation largely ex- agreement with the Planck results; on the oth−er hand,
−
ceedstheTeVscale. However,ouranalysisassumedonly a small non-minimal coupling constant is preferred to
that mφ is the inflaton mass as it oscillates about the obtain a non-negligible tensor-to-scalar ratio . We refer
minimum of the potential at the origin, while the effec- the reader to [26, 27] for a more detailed discussion of
tive inflaton mass can be much larger if slow-roll occurs these scenarios, since here we are mainly interested in
forsignificantlylargerfieldvalues whereself-interactions the post-inflationary dynamics. The effect of the non-
becomeimportant. Forexample,thediscreteZ2 symme- minimal coupling on the effective scalar potential in the
try allows for quartic self-interactions such that: Einstein frame becomes negligible for ξφ2/M2 .1, such
P
that consistency of our analysis implies
λ 1
V(φ)= φ4+ m2φ2 , (2.30) m
4! 2 h& ξ f , (2.32)
M
P
with the quartic term dominating for |φ| > 12/λmφ. which is generically less pstringent than Eq. (2.31) for
Since it increases the inflaton mass, the quartic self-
p masses in the TeV range and ξ <1016.
interactionwouldmakethedecayintofermionsmoreeffi-
Finally, an important aspect in embedding the in-
cientandspeedupthereheatingprocess. Ouranalysisis
teractions in Eq. (2.1) within a consistent inflationary
neverthelessvalidifthe quartictermissub-dominantfor
model is the fact that the discrete Z S symmetry
the field values |φ|∼mf/h at which decay into fermions does not protect the scalar potential fr2om× ra2diative cor-
occurs, which requires:
rections. In particular, the Yukawa interactions induce
λ m m loop-corrections of the Coleman-Weinberg form, which
h& f 3 10−8 f , (2.31) for large inflaton field values take the leading form:
12 m ≃ × m
r (cid:18) φ(cid:19) (cid:18) φ(cid:19)
h4φ4 h2φ2 3
where we haveused λ≃10−14 as imposedby the COBE ∆Vf ≈−16π2 log µ2 − 2 (2.33)
normalizationforinflationwithaquarticpotential. This (cid:18) (cid:18) (cid:19) (cid:19)
is easily satisfied if the fermions are not much heavier and therefore induce an effective quartic term in the
than the inflaton given the more stringent bounds on potential. The effect of this term does not necessarily
the Yukawa coupling discussed earlier. The addition of spoil the predictions of the non-minimally coupled quar-
a quartic term is, however, not sufficient to produce a tic model, as discussed in [28], although one must en-
consistent model of inflation, since it predicts a too red- surethatthe observednormalizationofthe perturbation
tiltedspectrumforcurvatureperturbationsandatensor- spectrumisobtained. Whileforξ 1theeffectivequar-
≪
to-scalar ratio already outside the bounds obtained by tic coupling must have approximately the same value as
Planck [23] and BICEP2 [24]. in the minimally coupled case, which requires h.10−3,
A consistentspectrum maybe achievedinwarminfla- significantlylargervaluescanbeaccommodatedforlarge
tionscenarios[12],wherethe couplingbetweenthe infla- non-minimal couplings.
ton and other sectors plays a significant role during in- Radiativecorrectionscan,however,besignificantlyre-
flation. For example, renormalizable couplings between duced in supersymmetric scenarios,and a supersymmet-
theinflatonandadditionalfermionicandbosonicspecies ricversionofthe modelinEq.(2.1) witha C2 Z2 S2
⊂ ×
may lead to dissipation of the inflaton’s energy into ra- symmetrycanbe easilyobtainedbyconsideringasuper-
diation during the slow-roll period, leading to an obser- potential of the form:
vationally consistent spectrum if inflaton particle states h m
W = Φ Y2 Y2 + f Y2+Y2 +
thermalize with the other relativistic degrees of freedom 2 +− − 2 + −
[d2o5m].inSainnctectohmispnoanteunrtalaltytlehaedesntdoorafdtihaetisolnowb-ercoolml rienggimthee, + mφΦ(cid:0)2+ λΦ2(cid:1)Z , (cid:0) (cid:1) (2.34)
2 2
8
wheretheinflatonandfermionsψ areembeddedwithin ton s channel exchange occurs at a rate:
±
−
thechiralsuperfieldsΦandY ,respectively,andtheaux-
± 9ζ(3)
iliary superfield Z induces the quartic term in the infla- Γ h4T . (2.36)
ψψ ≃ 16π3
ton potential (noting that the discrete symmetry forbids
cubic inflaton terms in the superpotential). Supersym- Since after the onset of oscillations H . m and tak-
φ
metry then cancels the leading contributions of scalars ing the temperature value obtained in Eq. (2.14) from
and bosons to the 1-loop Coleman-Weinberg potential, inflaton decay, we have:
which becomes:
Γ Γ m −1/4
ψψ > ψψ 60g−1/4h17/4 φ ,(2.37)
∗
h2 h2φ2 H mφ ≃ 1 TeV
∆V log V(φ) . (2.35) (cid:16) (cid:17)
SUSY ≈ 16π2 µ2 suchthatforO(1)couplingsYukawainteractionscanlead
(cid:18) (cid:19)
to thermalization from the start of oscillations. Recall
This contribution is thus necessarily smaller than the that, as seen above, evaporation of the condensate oc-
tree-level potential V(φ) λ2 φ4/4 for h . 1, therefore curs at a comparable rate, so that as the temperature
≃ | |
avoidingthegenerationoflargeeffectiveself-interactions rises due to evaporation both processes become progres-
during inflation. sively more efficient. If the Yukawa couplings are more
BesidestheYukawatermsconsideredsofar,thesuper- suppressed, one may also envisage scenarios where for
symmetric model also yields scalar interactions between example the fermions are charged under a gauge sym-
the inflaton and the scalar partners y of the fermions metrythatis unbrokenatthe relevanttemperaturesand
±
ψ , which apart from SUSY splittings that vanish at thermalizationoccurs throughgauge bosonexchange for
±
the origin have the same mass m = m +hφ. Trilin- sufficiently strong couplings.
± f
| |
ear terms in the scalar potential also lead to the decay Outside the inflatonfieldrangefor whichthe fermions
φ y y , with analogous kinematics and comparable are effectively light, their number density will become
± ±
→
widths to the fermionic decay channels, therefore yield- Boltzmann-suppressed, and any interactions will neces-
ingasimilarincompletedecayoftheinflatonasanalyzed sarily become inefficient in keeping the fermions in local
above. We note that the incomplete decaydynamics can thermal equilibrium. We thus expect them to decouple
be fully described in terms of scalar fields and is there- for most of the oscillation period, transferring their en-
forenotexclusiveoffermionYukawacouplings,although tropyintolightdegreesoffreedomsuchastheinflatonit-
the required form of the scalar mass terms is more nat- selfore.g.gaugebosons. Ifinteractionsoccurfasterthan
urally motivated within a supersymmetric context. We theinflatonoscillationratem ,asgiveninEq.(2.37)for
φ
also note the existence of quartic terms in the scalar po- the Yukawa scattering processes, fermions will drop in
tentialwhichinducethe3-bodydecayφ zy y ,where andoutoflocalthermalequilibriumastheyoscillatebe-
± ±
→
z is the scalar component of the Z chiral multiplet. The tween the relativistic and non-relativistic regimes. This
associated couplings have opposite signs for y and pro- will lead to an oscillating g , but as mentioned at the
± ∗
portional to λ, so that they are typically sub-dominant start of our discussion this is of little consequence since
withrespecttothe2-bodydecaysandtheyarealsokine- our results exhibit only a mild dependence on this pa-
matically forbidden at late times, so that they do not rameter.
affect our earlier conclusions. Since the inflaton is typically takenas a gaugesinglet,
the structureofthe Yukawainteractionsimplies thatψ
±
arenon-chiralfermions(eitherDiracorMajorana),asop-
posedtotheknownSMfermions,beingthusunlikelythat
D. Reheating the Standard Model
they are explicitly charged under the SM gauge group.
There are nevertheless several possibilities for exciting
Wehavesofarassumedthatthefermions(orscalarsas theSMdegreesoffreedomintheplasmaeitherbeforeor
discussedabove)resultingfrominflatondecaythermalize after radiation comes to dominate.
and excite other degrees of freedom, and in particular it The simplest case is perhaps that of unstable ψ±
is crucialthatStandardModelparticlesaregeneratedin fermions decayinginto a light scalaranda lightfermion,
the thermal bath at temperatures above 100 MeV so for which the decay width is:
∼
that BBN may occur following the standard freeze-out
h2
dynamics of light nuclear abundances. Γ = f m , (2.38)
ψ± 16π ±
Aswehavebrieflydiscussedabove,thefermionsthem-
selves cannot be treated as fully relativistic degrees of whereh denotestheassociatedcouplingconstant. Note
f
freedombeforetheinflatondecayssufficiently,sincetheir thatthisiscomputedinthefermions’restframe,whereas
mass varies between small and large values as the infla- in the plasma’s frame an additional time dilation factor
ton oscillates about the origin. For the short periods m /T suppressesthe decayforrelativisticfermions. For
±
when they are light and decay is allowed, thermaliza- asignificantpartofthe energydensityinradiationto be
tion through Yukawa interactions can be quite efficient. transferred into these degrees of freedom, the fermions
For example, fermion-fermion scatterings through infla- must decay before they become non-relativistic. Thus,
9
requiring Γ & H for T & m in the radiation era fromdirectcollidersearches(includingtheLHC)yielding
ψ± f
(wherem m ),weobtainthefollowingboundonthe ǫ . 0.1 for 1 GeV . m . few 100 GeV and indirect
± f f
≃ ×
coupling: bounds from the CMB anisotropyspectrum, which yield
ǫ . 10−4 for few 100 GeV . m . few TeV based on
f
m 1/2 × ×
h &8 10−8g1/4 f , (2.39) the effect of mili-chargedparticles on the baryon-photon
f × ∗f 1 TeV oscillations (for a gauge coupling g = 0.1). Note that
X
(cid:16) (cid:17)
where g is the number of relativistic degrees of free- for masses above the TeV range, mili-charged particles
∗f
dom at T = m . In addition, for this to happen be- may give a too large contribution to the dark matter
f
fore BBN we require m & 100 MeV. If the inflaton abundance in the universe, and more stringent bounds
f
and fermion masses are comparable, this corresponds to on ǫ apply in this case (see [30] and references therein).
h & 0.01 according to Eq. (2.28). These light degrees Thisthusconstitutesapromisingscenariowithpotential
of freedom may correspond to SM particles if, for ex- for experimental probing in the near future.
ample, the fermions coupled to the inflaton field corre- The discrete C2 Z2 S2 symmetry protects the in-
⊂ ×
spondtoapairofdegeneratesterileneutrinos,whichare flaton from decaying at late times, thus constituting a
singlets under the SM gauge group and may decay into viable dark matter candidate. One can consider, how-
a Higgs-lepton pair through Yukawa terms of the form ever, scenarios where this symmetry is broken and the
h H¯lψ . Note that this requires m > m = 125 GeV inflaton is only meta-stable, with a lifetime larger than
f ± f H
[29] and hence mφ <250 GeV, which is compatible with the age of the universe, t0 ∼ 14 Gyrs. Note that inter-
thepresentdarkmatterabundanceforcouplingsh.0.7 actions between the fermions ψ± and other light degrees
from Eq. (2.28). of freedom as in the scenarios outlined above can induce
If the fermions are stable, another possibility for re- thedecayofthe inflatonthroughradiativeeffectsorpro-
heating the SM degrees of freedom is through efficient cesses mediated by off-shell fermions. A few examples
annihilation. Apossiblescenarioisforthefermionstobe of these processes are illustrated in Figure 2 and in all
charged under a hidden U(1)X gauge group, which may cases the contribution of ψ+ and ψ− cancels if the dis-
mediate fermion scatterings and thus improve the ther- crete symmetry is exact.
malization efficiency. This U(1) hidden photon may
X
be kinetically mixedwiththe SMphotonorhypercharge (a) (b) (c)
gauge boson Yµ through a term of the form FµνFY ,
X µν
which may be generated radiatively if there are fields
charged under both gauge groups or simply via gravita-
tional interactions, as happens e.g. in string theory. Di-
agonalization of the gauge kinetic terms then induces a
small electric chargefor the fermions ψ , such that they
± (d)
may annihilate into SM charged particles via s-channel
photon exchange, ψ ψ γ qq,ll. At high tempera-
± ±
→ →
ture the annihilationcrosssectionis givenby the Thom-
sonscatteringformulaandthecorrespondinginteraction
rate for relativistic species is then given by:
4ζ(3) FIG. 2. Feynman diagrams for the 2-body decay of the in-
Γ ǫ2α2N T , (2.40)
th ≃ π ch flaton into (a) gauge bosons, (b) light scalars and (c) light
fermions, induced at the 1-loop level by gauge and Yukawa
where ǫ is the “mili-charge”of the fermions ψ±, α is the interactions of the ψ± fermions. In (d) we also show the 4-
fine-structure constant and Nch is the effective number bodydecayoftheinflatoninducedbytheexchangeofvirtual
of charged species in the final state, which for the full ψ± modes with Yukawainteractions with other light species.
SM is N = 20/3. In the radiation era, since H T2 For clarity, all light fields are represented bygreen lines.
ch
∝
annihilation becomes more efficient at smaller tempera-
tures. Once T . mf, however, the fermion abundance As anexample, we have consideredthe case where ψ±
becomesBoltzmann-suppressedandannihilationscanno are unstable, decaying into a light fermion and scalar,
longer be efficient. Thus, requiring that SM species are which induce diagrams (b)-(d) in Figure 2. Since these
excitedbefore ψ± become non-relativistic,we obtainthe processes have comparable magnitudes, we have com-
following bound on the mili-charge: puted the inflaton decay width for the 1-loop process
(b), where it decays into two light scalars. For concrete-
N −1/2 α−1 m 1/2
ǫ&5 10−7g1/4 ch f . ness we consider the case where ψ± are slightly non-
× ∗f 20/3 128 1 TeV degenerate with m = m and m = m (1+∆), for
(cid:18) (cid:19) (cid:18) (cid:19)(cid:16) (cid:17) − f + f
(2.41) ∆ 1. This gives for mφ =2mf (δ =0):
≪
This bound is not very stringent for fermion masses in h2h4
the GeV-TeV range, where the main constraints come Γφ(b) = 64πf5mφ∆2 , (2.42)
10
which is more suppressed for larger values of m . terfallfields. Onepossibilitytosatisfythisconditionand
f
Form yieldingthecorrectrelicabundancetoaccount simultaneously generate a constant vacuum energy is to
φ
fordarkmatterandusingthelowerboundonh obtained introduce an additional “driving” superfield, Z, which is
f
inEq.(2.39),weobtainthefollowingupperboundonthe not chargedunder the discrete symmetry and is coupled
inflaton lifetime: to the waterfall sector, along the lines proposed in [36].
We thus consider a superpotential of the form:
m −4 ∆ −2
τ <15.7 g−1 φ Gyrs,(2.43)
φ ∗f (cid:16)10 GeV(cid:17) (cid:18)0.01(cid:19) W = g2Φ(X+2 −X−2)+ h2Φ(Y+2−Y−2)+ m2f(Y+2+Y−2)
where we have taken the reference values for the present κ
+ Z(X2 +X2 M2)
dark matter abundance and freeze-out parameters in 2 + −−
Eq.(2.28). Hence,weconcludethatsignificantviolations h
ofthediscretesymmetrycanstillyieldasufficientlylong- + χ(X++X−)Q2+... , (3.1)
2
lived inflaton for 100 MeV . m . 10 GeV, such that
φ
whereM isaconstantmassscaleandwehaveincludeda
the fermions ψ decay before BBN and while they are
±
couplingbetweenthewaterfallsuperfieldsandadditional
still relativistic. For heavier inflaton particles, values of
chiralsuperfieldsQwhichgivetheirdecayproducts. The
∆belowthepercentlevelarerequired,signalingthatthe
dotsindicateadditionaltermsthatmaybeadded,involv-
discretesymmetrymostholdtoahighdegreeofaccuracy
ing the inflaton and the superfield Z. In particular, if
in this regime.
thescalarcomponentofthelatterhasasufficientlylarge
It is thus clearfromthe examples abovethat if the in-
mass,either fromsuperpotentialterms,softmassesfrom
flatonfieldcanonlydecaytothe ψ fermionsforafinite
±
SUSY breaking in other sectorsor non-minimal terms in
period following the end of the slow-roll regime, becom-
the K¨ahler potential, its expectation value will be set to
ing (meta-)stable at late times, it may account for the
zero both during and after inflation. The global mini-
dark matter in the universe while allowing for successful
mum of the scalar potential will then lie along the real
reheating of the SM particles and setting the necessary
direction X = X χ/√2,whichpreservesthedis-
conditions for BBN. + +
h i h i≡
crete symmetry, and the scalarpotential relevantfor the
inflationaryandpost-inflationarydynamicshastheusual
III. HYBRID MODEL hybrid form:
κ2 g2
A. Basic properties and dynamics V(φ,χ)= (χ2 M2)2+ φ2χ2+... , (3.2)
4 − 2
An alternativeframeworkto the one consideredin the where φ = √2 Φ is the real inflaton scalar field. We
h i
previoussectionisthesupersymmetrichybridmodel[31]. recover the usual SUSY hybrid case for κ = g/√2 and
In this scenario, the inflaton decay products need not for simplicity we will consider this parametric regime,
include or interact with the Standard Model degrees of althoughouranalysiscanbeextendedtothegenericcase.
freedom, since the additional waterfall sector can be re- Inflationtakesplaceforamplitudesoftheinflatonfield
sponsible for reheating after inflation [32, 33]. larger than a critical value, φ > φ = M/√2, such that
c
For the same reasons exposed in the minimal model, the waterfall field is held at the origin χ=0. As the in-
the symmetry C Z S is imposed on the super- flaton rolls towards its minimum at φ=0, its amplitude
2 2 2
⊂ ×
field containing the inflaton and all the superfields that falls below the critical value and the waterfall field can
it couples directly to. Hence, for the superpotential to roll to its true vacuum at χ = M, thus ending inflation.
be invariant under the action of this group, the inflaton After that point, both fields start to oscillate around its
is, as before, coupled to a pair of superfields Y which respectiveminima,triggeringtheprocessofreheatingthe
±
containthefermionsψ withmassesm ,andtoawater- universe into a radiation era.
± f
fall sector with a pair of superfields X . The group C In this scenario, the inflaton cannot decay into either
± 2
simultaneously changes Φ Φ and interchanges the thebosonicorfermioniccomponentsinthewaterfallsec-
→ −
superfields Y Y and X X . torduetokinematicalblocking. Thisiseasytoseeatthe
+ − + −
↔ ↔
The discrete symmetry forbids the linear term in the globalminimum, where m =m =gM, but extends to
φ χ
superpotential that is typically considered in SUSY hy- allfieldvalues. Asintheminimalmodel,theinflatoncan
brid inflation to generate the constant vacuum energy decayintotheY fieldsandthedecaywillbeincomplete
±
driving accelerated expansion. This may nevertheless for m >m /2=gM/2. For simplicity, we assume that
f φ
be generated either by a D-term contribution [34], or the scalar components of the Y fields acquire large soft
±
through a non-vanishing F-term coming from a SUSY masses from SUSY breaking and focus on the fermionic
breaking sector [35]. In addition, in order to ensure the decay channels, noting that the inclusion of both chan-
stability of the inflaton at late times, so that it may ac- nels will not change our conclusions significantly.
count for dark matter, the C symmetry must be pre- The waterfall fields will decay into the Q sector fields
2
served in the ground state, implying equal vacuum ex- and,forsimilarreasons,weincludeonlythefermionicde-
pectation values for the scalar components of both wa- cay channels in this case as well. We assume that these