The curvaton scenario in the MSSM and predictions for non-Gaussianity

0.11 MB·English

by  Anupam Mazumdar

The curvaton scenario within the MSSM and predictions for non-Gaussianity
Anupam Mazumdar1,2 and Seshadri Nadathur3
1Physics Department, Lancaster University, Lancaster LA1 4YB, UK
2Niels Bohr Institute, Copenhagen, Blegdamsvej-17, Denmark
3Rudolf Peierls Centre for Theoretical Physics,
University of Oxford, Oxford OX1 3NP, UK
We provide a model in which both the inflaton and the curvaton are obtained from within the
minimal supersymmetric Standard Model, with known gauge and Yukawa interactions. Since now
boththeinflaton andcurvatonfieldsaresuccessfully embeddedwithin thesamesector,theirdecay
productsthermalizeveryquicklybeforetheelectroweakscale. Thisresultsintwoimportantfeatures
2
of themodel: firstly, there will be no residual isocurvature perturbations, and secondly, observable
1 non-Gaussianities can be generated with the non-Gaussianity parameter fNL ∼O(5−1000) being
0
determined solely bythecombination of weak-scale physics and theStandard Model Yukawas.
2

n
a The curvaton scenario [1–4] is an alternative the curvatonis solelyresponsible forexciting allthe
J mechanismforthe generationofthe primordialper- SM dof so it must carry the SM charges [11, 12].

9 turbationswhosespectrumisobservedinthecosmic For the curvaton model to be observationallydis-
microwave background (CMB) [5]. In this scenario, tinguishable, we wish the model to be able to cre-
]
h the density perturbations are sourced by the quan- ate detectable non-Gaussianity. For this, r must
p tum fluctuations of a light scalar field φ, the cur- be small and both the inflaton and curvaton decay
- vaton, which makes a negligible contribution to the products must thermalize before the time of nucle-
p
energy density during inflation and decays after the osynthesis,as there are stringent constraintson any
e
h decay of the inflaton field σ. (For a review on infla- non-SMlikehiddenradiationafterBBN[9]. Inorder
[ tionincludingthecurvatonmechanism,see[6].) The to achieve this, we wish to place the entire inflaton-

advantage of the curvaton mechanism is that it can curvaton paradigm within a particle physics model
2
in principle generate measurable non-Gaussianity wherealltheinteractionsarewellconstrainedbythe
v
8 [1,7]intheprimordialdensityperturbationsandalso weak scale physics.
7 significant residual isocurvature perturbations, nei- Recently, the inflationary paradigm has been em-
0 therofwhicharepossibleintheusualsingle-fieldin- bedded within the minimal supersymmetric Stan-
4 flation models. Both signatures are detectable, and dard Model (MSSM) with known gauge interac-
7. if either were to be observed, this would strongly tions [13, 14]. The aim of this letter is to show, for
0 favour the curvaton hypothesis. the first time, that it is possible to embed both the
1 inflatonandcurvatonwithinMSSM,withoutinvolv-
If the curvaton does not completely dominate the
1 ing any hidden sector. We thus provide a solution
energydensityatthetimeofitsdecay,theprocessof
:
v conversion of initial isocurvature perturbations into to a general problem of the curvaton scenario, i.e.,
i how to generate measurable non-Gaussianity with-
X adiabatic curvature perturbations canenhance non-
out large residual isocurvature fluctuations.
Gaussian fluctuations to the level where they might
r Let us first consider the total potential to be the
a beconstrainedbythePlancksatellite. Theenhance-
sum of inflaton vacuum energy, denoted by V , and
ment in non-Gaussianity is given by f 5/(4r) 0
NL
∼ curvaton potential V(φ)
for r < 1, where r ρ /ρ at the time the cur-
φ rad
≡
vaton decays [1]. Planck is expected to be able to
V =V +V(φ). (1)
total 0
detect non-Gaussianity of the order f &5 [8]. To
NL
achieve detectable f thus requires small r. We assume V′′(φ) m2(φ ) H2 V /M2
NL ∼ φ I ≪ I ∼ 0 P
However,ifeitherthe curvatonortheinflatonbe- (MP 1018 GeV) where the subscript I indicates
∼
long to a hidden sector beyond the Standard Model the quantities are evaluated during inflation. This
(SM), they may decay into other fields beyond the condition is required for a successful curvaton sce-
SM degrees of freedom (dof). There is no guaran- nario. Thecurvatonacquiresvacuuminducedquan-
tee that the hidden and visible sector dof should tum fluctuations, which have amplitude
reach thermal equilibrium before Big Bang Nucle-
H
I
osynthesis(BBN)[9]takesplace. Inthiscase,resid- δ = . (2)
2πφ
ualisocurvatureperturbationsareexpectedtobe in I
conflictwithCMBdata,whichconstrainthemtobe These fluctuations are converted into the adiabatic
lessthan10%[5]. Ifthecurvatonbelongstothevis- densityperturbationswhenthecurvatondecaysdur-
iblesectorbuttheinflatondoesnot,avalueofr 1 ing its coherentoscillationsorrotations. In orderto
∼
would avoid this conflict [10] but would render any matchtheobservedamplitudeofthefluctuationson
non-Gaussianity undetectable. Note that if r 1 the CMB, rδ 10−5.
∼ ∼
2
Let us first discuss the origin of the curvaton, where u and d are squark scalars. Note that udd
which we take to be an R-parity conserving D-flat and LLe remain two independent directions for the
direction of the MSSM (for a review see [15]). Two entire reange ofeVEVs.
candidate flat directions are LLe (where L denotes This flat direction will also be lifted by the non-
the left-handed slepton superfield and e the right- renormalizable operators. However, at larger VEVs
handed superfield) and udd (where u and d de- the potential energy density stored in the udd di-
notetheright-handedsquarksuperfields),whichare rection will be larger than for the LLe, so it would
lifted by the non-renormalizable operator: be lifted by higher order terms:
λ Φn λ σ3m
W , (3) W = m . (8)
⊃ nMn−3 3mM3m−3
P mX≥2 P
where λ is a non-renormalizable coupling. For con-
The potential at lowest order would be:
creteness, we take the curvaton to be LLe so that
the scalar component of the Φ superfield is: σ5 σ8 σ11 2
V = λ +λ +λ +... (9)
(cid:12) 2M3 3M6 4M9 (cid:12)
φ=(L+L+e)/√3, (4) (cid:12) P P P (cid:12)
(cid:12) (cid:12)
where ... co(cid:12)ntainthe higher order terms. N(cid:12)ote that
where L and e are thee sleeptoen and selectron scalar the λ in Eq. (8) are all non-renormalizable cou-
m
fields. At the lowest order the potential along the φ plings induced by either gravity or by integrating
directioen is giveen by: outtheheavyfieldsattheintermediatescale. Aten-
ergies below the cut-off scale these coefficients need
V(φ)= m2φ2|φ|2+λ2|φM|22(nn−−31)+(cid:18)AλMφnn−3 +h.c.(cid:19) , noPtonteecnetsisaalrsillyikbeeEoqf.O(9()1)w. ere studied in Refs. [16,
P P
(5) 17]. For λ2 λ3 λ4 λn (1), they
≪ ≪ ≪ ≤ O
whereA m (100 1000)GeV,m isthesoft provide a unique solution for which first and sec-
φ φ
SUSY-br∼eaking∼mOass ter−m, and n=6 for LLe [15]. ond derivatives of the potential vanish along both
During inflation if m2 H2, the fluctuations radial and angular direction in the complex plane:
along this nearly massleφss≪directIion would create a ∂V/∂σ = ∂V/∂σ∗ = ∂2V/∂σ2 = ∂2V/∂σ∗2 = 0 (a
homogeneouscondensatewithavacuumexpectation saddlepointcondition)[18]. Forthefirstthreeterms
value (VEV) given by [15] in Eq. (9), it is possible to show that this happens
when
φI ∼ mφMPn−3 1/n−2 ∼1014 GeV, (6) λ2 = 55λ λ , (10)
(cid:0) (cid:1) 3 16 2 4
assuming λ (1). For m 100 1000 GeV,
φ
∼ O ∼ −
and n = 6, in order to match the amplitude of the at the VEVs: σ = σ exp[iπ/3, iπ, i5π/3], σ =
0 0
density perturbations δ, the Hubble expansion rate (2/11)(λ /λ )1/3M . Concentrating on the real di-
3 4 P
during inflation should be H 1010 GeV if r 1. rection, the potential energy density stored in the
I
∼ ∼
Thereis a distinctionbetweenapositiveandneg- inflaton sector is given by:
ativephaseoftheAterm. Thedifferenceindynam-
ics arises after the end of inflation. In the case of 153 2 σ10
V λ2 0 , (11)
positive A term the curvaton starts rolling towards 0 ∼(cid:18) 88 (cid:19) 2M6
P
the originimmediately, butin the caseofanegative
phase,for values of A≥√40mφ, it mayremainin a whereσ0 ≪MP. Notethatinflationoccursnearthe
falsevacuumwiththeVEVgivenbyEq. (6). Inthis saddle point σ0, where the effective mass vanishes.
casethecurvatonrotatesinsteadofoscillatesaround However,the third derivative of the potential is not
its globalminimum atφ=0. In either scenario,the negligible, V′′′ λ2σ7/M6 =0, which leads to slow
∼ 2 0 P 6
curvaton mass is negligible compared to the Hubble rollinflation. Thepotentialfortheinflatonbecomes
expansion rate. In fact, for A= √40m and a neg- flat enough to sustain a large number of e-foldings.
φ
ative phase the curvaton is actually massless along As written, the condition Eq. (10) represents a
the realdirection, andobtains inflaton-inducedran- completefine-tuning. Somedeviationfromthiscon-
dom fluctuations of order δφ H /2π. dition will be possible, changing the saddle point to
I
Wenowturntotheorigino≈fV withintheMSSM. a point of inflection, so long as V′ remains small
0
Letusconsideraflat-directionorthogonaltothecur- enoughforsufficiente-foldingsofinflation. Detailed
vaton. IfthecurvatonisLLe,thiscouldbetheudd discussiononfine-tuningininflectionpointinflation
direction. We take the inflaton direction to be: can be found in Ref. [13].
The amplitude of perturbations of the inflaton
σ =(u+d+d)/√3, (7) is given by δ V′′′(σ )N2 /30πH [13]. The
H ∼ 0 COBE I
e e e
3
corresponding Hubble expansion rate is given by The kinematical blocking due to the curvaton VEV
H (153/88)λ σ5/M4. For σ 1017.5 GeV and enhances the efficiency factor, r, therefore the cur-
λI ∼10−6, it is2po0ssiblPe to obta0in∼H 1010 GeV, vatonrotations prolong the mater-dominatedepoch
2 I
∼ ∼
required for a successful curvaton scenario. For the till it decays completely. For soft SUSY-breaking
above values, the inflaton perturbations are negligi- mass m . 1 TeV, the inefficiency parameter is
φ
ble, i.e. δH < 10−5, therefore all the observed per- r (1)h2/3. AlthoughtheLHChasalreadyplaced
∼O
turbations are created mainly by the decay of the severe constraints on the parameter space for low-
curvaton. scaleSUSY,the currentlimits donotexcludeheavy
Now let us consider the aftermath of inflation. squark and slepton masses & 500 GeV [22]. Since
The inflaton would decay primarily into the MSSM our flat directions are all made up of squarks and
dof. The coherent oscillations of the inflaton would sleptons, there is a large parameter space available
giveriseto instantpreheatingandthermalizationof in which this condition may be satisfied if SUSY is
the light MSSM dof as discussed in Ref. [19], with a discoveredat the LHC.
reheat temperature Since the curvaton decay is delayed due to the
kinematical blocking, r 1 is different for each de-
TR ∼[HIMP]1/2 ∼1013 GeV. (12) caychannel. Whatrang≤eoffNL we expect fromthe
However, not all of the MSSM dof will be in ther- variousdominantdecaychannelsofthecurvatonde-
mal equilibrium in our case. For the given choice pendsonthedifferentvaluesofh. Ifweconsiderthe
of flat-direction fields, if both inflaton and curvaton SM gaugecouplings, then h 0.1 and we would ex-
simultaneously take large VEVs, the SU(2)W dof pect the largest fNL ∼(5/4r∼)∼O(1)h−2/3 ∼O(5).
would not reach in thermal equilibrium, since the However,the curvaton also has the Yukawa interac-
LLe VEV would induce large masses to those dof. tions, especially when the curvaton decays into lep-
This will play a crucialrole in determining the non- tons and sleptons, for which:
Gaussianityparameterf , aswe shallshow below.
NL 5
The curvaton φ starts to rotate about the ori- fNL (1)h−2/3 10 103, (15)
∼ 4r ∼O ∼ −
gin when H = H m . The field value at
osc φ
this time is φ (m∼Mn−3)1/n−2. During this for h 10−2 10−5. This range of h covers all the
epoch the un|ivosecr|se∼is alφreaPdy radiation-dominated SMYu∼kawase−xceptthetopYukawawhichisoforder
following the decay of the inflaton. However, the h 0.1. Duetothesmallervaluesofh,thesedecays
∼
curvaton cannot decay immediately, due to the fact arekinematicallyallowedathigherVEVs. Anexact
thatthecurvatonVEVinduceslargemassesh φ(t) prediction for net effect on fNL requires a complete
for gauge bosons, gauginos and (s)leptons, whherei analysis of the decay modes for the LLe curvaton
h is the gauge or Yukawa coupling. The curva- which is beyond the scope of the current letter, but
ton’s decay at leading order is kinematically forbid- it can be seen that this model of the curvaton can
den if h φ mφ/2 (100 1000) GeV. Decays providefNL inarangewhichwillbe observationally
do not ohcciu≥r until t∼heOHubbl−e expansion has red- relevant in the near future.
shifted φ(t) down to m /2h. Note that the SM The temperature at which the curvaton decay
φ
Yukawahcoupilings are smaller than the gauge cou- productsreachthermalequilibriumisdeterminedby
plings. Therefore the decays via SM Yukawas be- Eq. (13). The final thermal bath filled with MSSM
come kinematically allowed at higher VEVs. dof would be obtained by the reheat temperature
Duringtherotations,thecurvatonVEVwillscale T (H M )1/2 104.5 106.5 GeV (16)
as φ(t) a−3/2, as a H−1/2 during the radiation- R,f ∼ dec P ∼ −
domina∝tedepoch. Th∝erefore,eachdecaychannelbe- for h 10−2 10−5. Such a temperature is suffi-
∼ −
comes allowed when [20] cient to excite weakly interacting massive particles
and for baryogenesis [21]. Note that both the tem-
H =Hdec mφ(mφ/hφ(t))4/3 , (13) peratures from Eqs. (12) and (16) are sufficiently
∼
high to excite thermal/non-thermal gravitinos and
For large φ(t) , the decay time is naturally longer
h i axinos. If the gravitinos or axinos are the lightest
than the normal decay rate into the massless dof.
SUSY particle, this causes two problems for this
The radiation energy density stored in the inflaton
scenario: over-production of gravitinos with both
decay products scales as ρ H2, where the sub-
vis ∝ helicities would be bad for BBN, and the graviti-
scriptdenotesthevisibledof. Theratiooftheenergy
nos and axinos would thermally decouple even be-
densitiesatthetimethecurvatondecaysisgivenby
fore the curvaton has started decaying. This would
ρ ρ H −1/2 generate large residual isocurvature perturbations,
φ φ dec
r , because gravitinos and axinos can never come into
≡ ρ ∼ ρ (cid:12) (cid:18)H (cid:19)
vis vis(cid:12)osc osc thermal equilibrium. Instead the ideal dark matter
2/(n−(cid:12)2) −2/3
mφ (cid:12) mφ candidate would be the neutralino, which decouples
1. (14)
∼ (cid:18)M (cid:19) (cid:18)hφ(cid:19) ≤ from the thermal plasma at T 40 50 GeV.
P ∼ −
4
Our discussion so far has been based on treating inflaton and curvaton belongs to the visible sector,
udd as the inflaton and LLe as the curvaton flat avoidingtheproblemofresidualisocurvaturefluctu-
direction. In principle, we could have swapped the ations,whilethecurvatonmechanismcancreateob-
roles of inflaton and curvaton, i.e. LLe as an infla- servable non-Gaussianity. The non-Gaussianity pa-
ton and udd to be the curvaton. The main aspects rameterf depends cruciallyonthe SMgaugeand
NL
of the analysis would not differ at all. Although Yukawacouplings,andrangesfrom (5)to (1000)
treatingudd asacurvatonwouldalsomakeSU(3) in the different decay channels (forOYukawaOs in the
c
dof heavy during the curvaton oscillations and this range h 10−2 10−5, which is the case for all
∼ −
wouldalterthedetaileddiscussionofthermalization, the Yukawas except the top). The model favours
nevertheless the range of f quoted above for the a visible-sector dark matter candidate such as the
NL
SM Yukawas in Eq. (15) would remain the same. lightest neutralino but will not work if the lightest
To summarize, we have discussed the possibility SUSY particle is a gravitino or axino type.
of constructing a model in which both the infla- Acknowledgements: The authors would like to
ton and curvaton are flat direction fields within the thank R. Allahverdi, P. Dayal, A. Liddle, and D.
MSSM. The radiationcreatedfromthe decayof the Wands for helpful discussions.
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