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This content was downloaded from IP address 136.152.209.140 on 30/07/2018 at 13:38 JCAP12(2016)014 iclehysics P t ar doi:10.1088/1475-7516/2016/12/014 strop b,c [email protected] and Keisuke Harigaya osmology and A and Cosmology a 1607.07058 inflation, particle physics - cosmology connection, non-gaussianity

Spectator field models such as the curvaton scenario and the modulated reheat- [email protected], rnal of rnal

ou An IOP and SISSA journal An IOP and 2016 IOP Publishing Ltd and Sissa Medialab srl Stanford Institute for Theoretical PhysicsStanford, and CA Department 94306, of U.S.A. Physics, StanfordDepartment University, of Physics, University ofBerkeley, CA California, 94720, U.S.A. Theoretical Physics Group, LawrenceBerkeley, Berkeley CA National 94720, Laboratory, U.S.A. E-mail: b c a c Tomohiro Fujita Hubble induced mass afterspectator inflation field in models Received August 16, 2016 Accepted November 21, 2016 Published December 7, 2016 Abstract. ing are attractive scenariosconstraints for on inflation the models generation areduced of relaxed. masses the on In the cosmic this dynamicsto curvature paper, of the perturbation, we spectator Hubble discuss as fields induced the after the cally effect mass inflation. unsuppressed of by We but Hubble the pay often in- kinetic particular overlooked.relaxes attention energy the In of constraint the an on curvaton the oscillating scenario,temperature property inflaton, the and of which Hubble the the is induced inflaton inflation generi- mass baryogenesis and in scale. the the curvaton, curvaton such We scenario. ase.g. comment the In the on reheating the non-gaussianity the modulated can reheating, implicationof be the the of considerably predictions modulated our altered. of reheating models discussion Furthermore,the utilizing we the for local propose Hubble non-gaussianity a induced parameter. mass new which model realizes a wideKeywords: range of ArXiv ePrint: J JCAP12(2016)014 2 6 4 5 6 6 7 8 9 2 1 7 11 12 12], – 1 – 2.3.1 Thermal2.3.2 leptogenesis Sneutrino2.3.3 curvaton The Affleck-Dine baryogenesis In general, any light scalar field obtains a fluctuation of its field value during inflation In this paper, we discuss the influence of a Hubble induced mass after inflation on 2.3 Implication to baryogenesis 3.1 Review3.2 of the modulated Field-dependent reheating 3.3 coupling Field-dependent mass of inflaton decay products 2.2 Hubble induced mass after inflation 2.1 Review of the curvaton scenario 3 Modulated reheating with Hubble induced mass 4 Summary A Hubble induced mass in supergravity and can be theduring inflation. origin of We refer thethe to cosmic fluctuation such turns perturbation, field into even as afield if fluctuation a dominates of their spectator the the field. energy energy energy density density In of is the density the negligible curvaton of universe scenario as the [6–10], the universe. spectator In the modulated reheating [11, the dynamics of thescenario. spectator A field. Hubble We induced mass focusor of on kinetic a the field energy is curvaton of given andunder by the the the any inflaton. coupling modulated symmetry, of reheating the the Ashas field Hubble the to an the induced potential approximate potential and mass shift the symmetry. is kinetic Actually, in the energy general Hubble terms unsuppressed, induced are unless mass neutral the is field a generic feature 1 Introduction The large scale structureground of (CMB) the are universe and naturallytion the explained [1–5]. fluctuation by of In thethe the the perturbation inflation cosmic simplest and generated microwave scenario, the back- during a generation cosmic of scalar infla- the field cosmic called perturbation. an inflaton is responsible both for the decay rate ofdepends the on inflaton (or theis any field modulated. fields value dominating Spectator of theresponsible field for energy the models the density cosmic are spectator perturbation, of attractive andFor field, the example, hence since constraints the universe) and on the chaotic inflation inflation inflaton hence modelsthe [13] are field initial is the relaxed. itself condition one timing problem of is [14]. thetensor no of most fraction The more attractive the and chaotic scenarios inflation, the as reheating however, simplestspectator it generically model is field predicts free is models, a from disfavored large this by constraint the is observations evaded of the CMB [15]. In Contents 1 Introduction 2 Curvaton dynamics with Hubble induced mass JCAP12(2016)014 (2.1) 21] in , is generated δσ , the curvaton begins σ m is much smaller than the σ m 2, we discuss the influence of the , 2 σ 3, we investigate the modulated reheating 2 σ m 2 1 is devoted to summary and discussion. 4 – 2 – ) = (σ V . In the following, we assume that the inflaton decays is its mass. We assume that dec t σ . Then the curvaton is expected to have a non-negligible field = I m t H 1 . Furthermore, the perturbation of the field value, i σ , which is the origin of the cosmic perturbation. i is a curvaton field and σ σ After inflation, the Hubble scale decreases. As it drops below In the curvaton scenario, the magnitude of the cosmic perturbation strongly depends on The organization of this paper is as follows. In section See ref. [22] for a derivation of the curvature perturbation for a generic potential. 1 around its oscillation around theradiation-dominant origin. at Eventually, a the time curvaton decays and the universe becomes value during inflation, where Hubble scale during inflation, the evolutions of theinduced inflaton mass and of the the spectatorthe curvaton after field curvaton inflation field (curvaton) can after can enhanceconstraints inflation. more the on easily The field dominate inflation Hubble value the of modelsthe energy the reheating and curvaton, density temperature, and of curvaton are the models, considerably universe. relaxed. e.g. As This those was a on pointed result, out the in inflation refs. [20, scale and in supersymmetric models–19]. [16 inflation, namely The the Hubbleto coupling induced to mass explain the of the thethe potential almost spectator energy coupling scale field of of during invariantunsuppressed. the the spectrum inflaton, Such spectator coupling of must field generates be the a to suppressed sizable curvature the Hubble perturbation. kinetic induced mass energy after However, of inflation. the inflaton is expected to be the context of thegeneric leptogenesis curvaton by models, the and comment sneutrino onIn curvaton. an the implication Here modulated to we the reheatingdependence extend scenarios scenario, of the of the as discussion the baryogenesis. the to timingthat spectator of the field the magnitudes moves reheating of duringalso the on the propose curvature the reheating, perturbation a spectator the new and field modelof the value of non-gaussianity the is the are spectator modulated modified. affected. field reheatingof We due We which the find is to local enabled the non-gaussianity by Hubble parameter. the induced time mass. evolution TheHubble induced model mass can on realizefindings the a to curvaton the wide scenario. scenarios range We of alsoscenario baryogenesis. comment and In on propose section the a implications new of model. our Section 2 Curvaton dynamics withIn Hubble this induced section, mass wenamics discuss of the the effect curvaton.induced of We mass. first the discuss Hubble Wethat the then induced the dynamics take mass Hubble of into after induced the account massinflation inflation curvaton can the models. on ignoring considerably effect the relax the of dy- the Hubble the constraint on Hubble curvaton induced models2.1 mass. and We Review find ofBefore the discussing curvaton scenario thecurvaton effect ignoring of the the Hubblepotential Hubble induced is induced mass. a quadratic mass, To one, let be concrete, us we review assume the that dynamics the of curvaton a JCAP12(2016)014 (2.6) (2.2) (2.4) (2.5) (2.3) , where Pl , is constant /M ) ,I (k 1 ζ 2 RH δσ/σ P 2 Pl horizon exit > T 2

k M σ  2 5H i m 6π σ 2δσ and/or large enough infla- 1 9 is then given by ,  ,σ . 2 × k . Note that ,  dec . We assume that the curvaton RH ,I dec , 2 f f T . −1 dec   dec RH = 1, we obtain the similar bound i f 1 decay − f 2 Pl 2 T i k , )| 4 σ M H dec σ ,σ ≤ GeV 2πσ is the energy density of the radiation f ρ 5 6 = )  RH,I RH I 12 + [22]. The energy fraction of the curvaton T 2 ρ (k T ' 1 =0.002Mpc I ζ 10 k  2 Pl 2 i P decay 2 osc 2

H osc σ ) /(ρ 1. For M dec i 2 H Pl σ  H f 2 k dec σ 5 6 ρ – 3 – (σ 2 M Pl 2δσ ≡ → from eqs. (2.3) and (2.5), we obtain − 2  2f V 5H i = σ 4 /5 σ GeV 3M 6π dec 2 σ  RH,I 2 f 9 min T ) σ ' dec m + 3ρ ρ f 10 ' ) = dec I ' osc f × dec f (k 2 4ρ dec f ζ − f H P = (4 24] GeV 2 × in the following discussion. The curvaton start its oscillation when decay 10

Pl σ 1, by eliminating × σ RH,I ρ M T δρ < σ σ = 7 = m dec ,σ f √ σ is the reheating temperature by the curvaton. + 3ρ ρ RH → I T is the Hubble scale at the horizon exit of the scale is the energy density of the curvaton, 4ρ . The upper bound can be understood as the condition under which the curvaton k σ RH,σ ,σ RH,I ρ T H = is the reheating temperature by the inflaton. For the other case, one needs the replace- T As we will see, the upper bound on the reheating temperature restricts the possibility The successful curvaton scenario requires low enough The curvature perturbation in the constant energy slice is fixed after the curvaton Let us evaluate the energy fraction of the curvaton ζ RH ,I T RH of baryogenesis. on dominates the energy density of the universe. where the inequality is saturated for the Hubble scale drops below where originating from thebetween inflaton, the horizon and exit andvalue the is decay linear. of the The curvaton, power as spectrum the of evolution of the the curvature curvaton perturbation field is given by starts its oscillation before the inflaton decays. This is the case for at that time is given by The ratio isproportion conserved to until the the scale inflaton factor decays of the into universe. radiation, The and fraction increases afterward in where decays, and is given by [23, ment before the curvaton decays, otherwise theuniverse. curvaton cannot To dominate generate the energy largethe density fluctuation enough of the of cosmic the perturbationsgaussianity. curvaton from energy the density sub-dominant must curvaton, be large, which leads to too large non- where T tion scale. For JCAP12(2016)014 0), ) to (2.9) /2 = (2.8) (2.7) I (2.10) (2.11) (2.12) −2 ρ 0(< 10 = . c > | ave 0 2 21]. For , h∂φ∂φ/2i  1 2 , − 2 σ > , 2 i  σ 1 cH × 2 3 , − 2 , γ , c 2 σ i = = 0. 3 σ σ 2 1 16 ≡ 2 2 σ 1 2 26].) σ α 2 × − 2 1  4c 1 3t (φ) 25, 5 V ave q + √ / 2 ∂φ∂φ Pl 0 osc,I σ A.) = σ c 2 – 4 – Pl H M d c dt m M t 2  h∂φ∂φi , α ) = = + α × 2 Pl i c (σ  σ σ M 2 2 V (φ) is its potential, should be suppressed (|c − L d t ' dt osc,I V t ) = , becomes comparable to the Hubble scale. The equation of osc  σ (σ σ i σ m eff V −1/2. (See appendix (t) = c, the curvaton begins its oscillation. As a result, the field value of the = σ c denotes the time-average, and we have used the relation is the time at which the inflaton begins its oscillation. Here we assume that (1). We note that in single field small inflation models in supergravity, the value ave O is an inflaton field and /2. osc,I = t 2 φ h· · ·i Pl c The Hubble induced mass after inflation, on the other hand, is not necessarily sup- The Hubble induced mass affects the dynamics of the curvaton [20, During inflation, the kinetic energy of the inflaton is negligible and hence the interaction is determined, M 3/16. For larger In multi-field inflaton models such as hybrid inflation, the potential and the kinetic energy after inflation 2 2 c can explain the red-tilt spectrum without invoking a large first slow-roll parameter nor the 0 pressed. Even if thethe coupling coupling between with the the curvaton kinetic and term the inflaton of potential the is inflaton suppressed, is generically un-suppressed, can be dominated by aHubble field induced other than masses. the inflaton.inflation If is the The also coupling curvaton with suppressed. has those an energy approximate also induce shift un-suppressed symmetry, the Hubble induced mass after where 3H the curvaton field value decreasesthe (increases) during mass the of oscillation phase the of curvaton, the inflaton until motion of the curvaton is given by curvaton when it start its oscillation is different from with of The solution to this equation is where c < in eq. (2.8) doesbegins its not oscillation. generate Then a the sizable curvaton Hubble effectively induced obtains its mass. mass, After inflation, the inflaton hiltop-type curvaton potential (see e.g. refs. [20, 2.2 Hubble inducedThe mass Hubble after induced inflation mass of the curvaton during inflation, where explain the almost scale invariantc power spectrum of the curvature perturbation. A negative JCAP12(2016)014 Pl as a /M γ (2.13) . As a i σ 3 2 . RH,I × T Pl . σ γ → /m σ < M . m 2 ,I  osc,I RH H 27]. Otherwise, large γ 100 /T is modified. Eqs. (2.2)  24, ,σ and 2 c 0.0 2.1  RH 0, T 1, we show the value of −1 p c < Pl GeV M 0.5 for given - 12 γ ' =0.002Mpc 10 k osc H σ  c 1.0 10 2 - 10 ,I 3 GeV – 5 – RH 10 9 4 can be considerable. For the case where the inflaton T 10 10 γ 1.5 = - γ × is considerably relaxed if is eqs. (2.4) and (2.5) should be replaced with GeV i 2.0 6 σ - 4 3 2 6 5 RH,σ 1 10 10

T Here we focus on the thermal leptogenesis [31], the non-thermal

10 10 10 10 10

. The factor

σ I osc, / m H σ 4 × 7 36]. /m < osc,I H RH,σ T is the Hubble scale when the inflaton begins its oscillation. Typically, it is of the and c Figure 1. The value of the enhancement factor osc,I H Due to the change of the field value, discussion in section In the above analysis we have implicitly assumed that the curvaton field value is smaller The Hubble induced mass after theThe inflaton constraint decays from the can CMB be can be non-negligible evaded if if the the baryon couplings and the between dark the matter isocurvature perturbation 3 4 curvaton and the kinetic terms of the particlescancel in with the each thermal other bath [28–30]. is large. decays before the curvaton startsin its oscillation, eq. one (2.12), needs as the the replacement Hubble induced mass almost vanishes after the inflaton decays. the Planck scale. IfHubble the induced curvaton field mass, value Hubblecurvaton, becomes and induced close the higher to above the powereq. Planck analysis terms (2.13). scale is also Actually, by the invalid. affect a bound negative the This is dynamics assumption saturated of does when the not affect the bound in where function of and (2.3) are unchanged, but 2.3 Implication toHere baryogenesis we discuss the implicationsgenesis. of our findings In inafter the the the previous curvaton curvaton section dominates to scenario, the scenarios energy the of density baryo- of baryon the or universe [23, lepton asymmetry should be produced same order as the Hubble scale at the end of inflation. In figure result, the upper bound on baryon isocurvature perturbations are produced,observations which is of excluded the by the CMB. constraintleptogenesis from from the the decay of(AD) right-handed baryogenesis sneutrino [35, curvaton [32–34] and the Affleck-Dine JCAP12(2016)014 ,σ 10 ' & 10 RH T osc,I RH,σ H T = 0. The 9 c 10 GeV 8 GeV 10 / GeV GeV GeV I 2, we show the con- 9 8 7 10 10 , and the Hubble scale = 21]. 10 10 10 RH, σ T m RH,I T 7 10 2, the upper bound on 2 / −1/2 and assume that 1 = =- c c , and the Hubble scale during infla- 6 −1/2. For the dashed line, we take ,I 10 12 11 14 13 = RH GeV]. [37 In figure T

c

10 10 10 10

9

= .0 Mpc 0.002 k

/ GeV H - 1 −1/2. For the dashed line, we take 10 = & . Below the solid lines of the left panel, the upper = 0. It can be seen that the effect of the Hubble c – 6 – c −1 13 RH,σ 10 T GeV. GeV. Here, we take 6 .002Mpc 9 . Below the solid lines of the left panel, the upper bound 10 =0 GeV, and the thermal leptogenesis cannot produce sufficient k 12 −1 > 9 H 10 GeV. Here, we take 6 RH,σ T GeV GeV 11 12 GeV GeV / =0.002Mpc GeV I 9 k 10 11 10 10 = 10 RH, H 10 10 σ T m 10 is smaller than 10 10 2 . For the dashed line, we take / 1 −1 RH,σ =- T c in eq.2.13) ( is lower than 10 9 . The constraint on the reheating temperature of the inflaton, ,σ 10 13 12 14 RH

GeV [38]. Below the solid lines of the right panel of figure 10 10 10

T

=0.002Mpc

= .0 Mpc 0.002

k 6 / H GeV - 1 = 0. It can be seen that the effect of the Hubble induced mass extends the parameter k during inflation at thebound pivot on scale, Figure 2 amount of the baryon asymmetry. Here we take 2.3.1 Thermal leptogenesis The thermal leptogenesis requires that light panel shows thecurvaton, similar which constraint requires for that the successful leptogenesis by the decay of the sneutrino straint on the reheating temperaturetion of at the the inflaton, pivot scale on H induced mass extends the parameter region consistent with the thermal2.3.2 leptogenesis. Sneutrino curvaton Next, we consider theorder leptogenesis to by produce the a decay of sufficient the amount right-handed of sneutrino the curvaton. lepton in asymmetry, it is required that in eq. (2.13) isc lower than 10 region consistent with the leptogenesis by the sneutrino curvaton [20, 2.3.3 The Affleck-Dine baryogenesis Finally, we consider thesymmetric AD standard model. baryogenesis scenario Asproperty the by of efficiency flat the of flat the directions direction ADwe in as baryogenesis comment well the depends as only on minimal the on the mediation super-its the detailed scheme oscillation constraint of the applicable after supersymmetry to the breaking, generic curvaton models. dominates the The energy AD density field of should the start universe. The Hubble 10 JCAP12(2016)014 (3.3) (3.1) (3.2) ) and (2.14) acquires . Under osc,I σ σ H , 2 = ) Pl φ φ 2 RH,I H M T ln Γ ln Γ . 4 2 σ 4 σ  ∂  does not significantly (∂ k , these observables are Pl are . σ γ H σ 5 M n σ 100 −5 . For example, in gravity   = = ∗ 4 , g 2  dom 2 ) NL  90 π H f −1 φ Γ H ∆N r ∆N 2 , f σ 2 σ  ∂ ) GeV σ (∂ δσ , the transition time from the inflaton ln (k 5 6 13 σ 6 ζ n 1 2 4 =0.002Mpc P . Here we have assumed that 10 − k = 4 + σ 25γ H ) is given by =  f NL  ζ H 2 φ 2916π – 7 – osc,I  Γ = is a power-law function of H = H φ  ,I ⇒ GeV Pl δσ, f = ln RH 2 ) RH,I 10 M T 3 2 φ T ∗ 10 = g ln Γ  2 n σ 90 π × gσ ∆N (∂ r during inflation and the field value of 4 1 6 i = . Contrary to curvaton models, the energy density of the spectator 4 Pl σ − σ φ 4 δσ depends on the field value of Γ = φ (1) TeV, the reheating temperature of the inflaton as well as the Hubble 36M 25γ O formalism [39–41], one can show that the contribution to the curvature )δσ = = 100 GeV δN ∆N dom σ H ∂ denotes the derivative with respect to σ = ( , the e-folding number between the onset of the inflaton oscillation ( ∂ φ ζ = Γ perturbation and the local non-gaussianity from the fluctuation of Since the AD fieldAD starts field, its the oscillation when mass the of Hubble the scale AD drops field below the must mass be of smaller the than oscillating era to thethe radiation sudden decay dominated approximation eraH which is assumes perturbed that by the the inflaton decays fluctuation instantaneously of at Based on the where a gaussian fluctuation evolve after inflation. Forcomputed instance, as if Γ scale when the curvaton dominates the energy density of the universe is given by mediated models, where thearound mass the of origin, the AD filed is as large as the masses of superparticles a sufficiently later time after its decay ( scale must be large.induced mass The relaxes those enhancement constraints. of the curvaton field3 value by the Modulated negative reheating Hubble withIn this Hubble section, induced we discuss mass scalar the field effect of whose the field Hubblethe value induced case determines without mass the the after Hubble inflation timinganalytic induced on of result mass a under the and spectator the then inflaton sudden takeimation decay. into decay by account approximation We numerical its as briefly calculations. effect. well review It astion We is a derive and an found result the that without non-gaussianity the theWe can prediction approx- also for be propose the considerably a curvature new changed perturba- model due of to the the3.1 modulated Hubble reheating induced utilizing mass. the Review Hubble of inducedIn the mass. the modulated modulated reheating reheating scenario,tion the of curvature a perturbation spectator originatesfield field from may be the always fluctua- subdominant inrate this of scenario, the which we inflaton assume Γ in this paper. If the decay JCAP12(2016)014 . φ ), n 3. gσ (3.6) (3.7) (3.8) (3.5) (3.4) osc,I = 5 . )[42–44]. φ −3 osc,I , 10 are the energy ασ (namely Γ 2 3 − r is negligible and ∆N formally diverges ρ φ Γ ζ −2 σ ∂ 2 ) = ) and (10 , osc,I φ φ O n−1 osc,I ρ ρ ∆N nΓ . For Planck-suppressed (t Γ φ 2 . σ Pl H goes down to Γ + (∂ N , we obtain d ∂ = 0, = H + 1 n 1. Note that /2M H r ∆N σ 2 is the coefficient of the coupling , 2 Γ r nα σ ∂ can be computed. To compare the 2 c r is typically cρ |  6 5 when + 4ρ H c osc,I ˜ cρ + 1, respectively. Therefore the effect = 5 NL without the sudden decay approxima- r + ˜ 2 Pl H = − f H ρ |nα φ ) = 0 and M N nα NL NL cρ = 1 NL f ) decrease at the same rate and the inflaton and nα+1 osc,I . + L −1 ζ  t (t equals σ and r , f , ∂ N φ ρ osc,I – 8 – ζ φ (∝ ∂ osc,I osc,I σ ρ δσ ) H in eq. (3.1) by this Γ φ H H H ∆N, f Γ φ . Then  Γ + 1 ln − n ∂ has a Hubble induced mass, eq. (2.9), and Γ = ) and N osc,I = d nα σ ∂ osc,I nα φ 6 1 H t 3, the analytic expressions derived under the sudden decay − ensures that eq. (2.11) is correct and another linearly independent, more (∝ + 3ρ = φ + 1)Γ σ + (3 + φ ζ (see eq. (2.11)), Γ N ρ σ ∂ ). Replacing Γ α N t 2 N ∂ ∂ ∝ and particles in the thermal bath, ˜ are divided and multiplied by osc,I −6(nα hereafter. It is assumed that the energy density of σ σ (t . The initial condition is r = n NL as a function of Γ is constant between the horizon exit and the decay of the curvaton, as the ρ f −2 gσ + ana num = ζ φ ζ and and the radiation energy density, ρ δσ/σ = 10 denotes the derivative with respect to e-folding number, is the derivative with respect to Γ ζ σ = c osc,I N Γ 2 ∂ ∂ −1/n. It is because Γ H As one can see in figure In order to obtain the predictions for Running a numerical calculation with changing an initial value of Γ = . 2 This initial condition of Pl 5 σ α tion, we numerically solve the following equations: rapidly decreasing solutioncondition. vanishes. It is straightforward to extend the analysis to a more general initial between couplings between 3M one obtains ∆N approximation become less accurate as the Hubble induced mass or the initial value of Γ never decays in that case. where density of the inflaton and radiation, respectively, and ˜ We use ˜ numerical results and the analytic expressions, we plot the following quantities in figure where This is a typicalprevious result section, of however, the the conventionalof Hubble modulated induced reheating scenario. mass can As cause we a saw3.2 in non-trivial the time evolution Field-dependent coupling Now we consider the case where Note that if evolution of theeq. curvaton (3.3), field value is linear. Compared with the conventional result in Since it evolves as where Γ of the Hubble induced mass cannot be ignored unless JCAP12(2016)014 . . σ 1/n osc,I ) (3.9) , and 0.100 /2y ψψ φ approaches → and radiation, (m φ α ψ ≡ is ignored and I 0.010 d . The decay rate is σ σ φ osc, , the time evolution H / I σ > m particles, osc, ), however, the analytic ψ Γ α ψ 0.001 increases and 6 decreases after inflation with . increases in this case and the c ζ σ φ depends on (positive ψ 4 - c=0.05 c=0.1 c=0.15 c=0.18 c , and the deviation from the inflaton m φ 10

= Γ 1.0 0.5 0.0 3.0 2.5 2.0 1.5 .

NL n f H is possible even for 2m yσ ψ .15 (green) and 0.18 (red). The solid lines are the = is not negligible at the beginning, the radiation – 9 – decreases to the critical value, and its mass ψ through the inflaton decay rate depending on σ ψ m /H ζ φ 0.100 can be significant in the case with the Hubble induced decays into radiation immediately after the inflaton decays into σ , respectively. The sudden decay approximation becomes less 46]. at the end of inflation and ψ NL φ f by the multi body factor as well as the coupling between I as 0.010 are comparable for a longer time. In addition, a large Γ φ > m σ osc, n H H / I ) becomes relevant. For negative .05 (blue), 0.1 (yellow), 0 is converted into osc, 2/3 and = 2yσ Γ t σ = 0 ψ φ ∝ c 0.001 while it is allowed once m 7 depends on ψ 4 - . The comparison between the analytic expressions derived under the sudden decay approx- = 2 and n 10 If the inflaton decays into a particle given in eq.) (2.11 can open the on-shell decay channel of the inflaton which is initially 0, the decay of the inflaton is kinematically prohibited at the beginning of the inflaton and the local non-gaussianity

In the following we assume that A conceptually similar mechanism to produce curvature perturbations through the evaporation of primor-

0.6 1.0 0.9 0.8 0.7

σ 7 6 ana num ζ ζ ζ / s. Then the decay of the inflaton through off-shell suppressed in comparison withand Γ can be neglected. dial black holes is discussed in refs. [45, ψ Figure 3 imation and the numerical resultsWe in fix the modulated reheatingnumerical scenario results with while the the Hubble dashedof induced lines mass. are the analytic ones. The left and right panel show the ratio accurate for larger decay rate and larger Hubble induced mass. increases. One expects that the approximation gets worse, as the fluctuation of −1/n, because Γ also invalidates the approximation;component if is Γ significant most ofoscillation the era time (a before and numerical results aresudden in decay good approximation agreement. is It more is accurate. because Γ 3.3 Field-dependent mass ofIn inflaton the decay conventional products modulated reheating scenario,Nevertheless, the the time time evolution evolution of of mass and we find the following new mechanism of generating of closed. In this paper, we consider that the inflaton decays into two the mass of Provided that 2 α < oscillating era, JCAP12(2016)014 d d 4. NL 10 f /H (3.10) (3.12) (3.11) 0.01 = I , osc , respectively. 5 H / −2 Γ d σ / with the numerical I osc, is close to unity, Γ NL σ stops evolving and this f Contrary to the analytic is boosted as seen in figure d = 10 and 10 α 8 and the on-shell decay channel t /σ NL 2 f d σ ∝ osc,I , osc,I 2 ψ σ σ 2 φ . significantly decreases and Γ/H m 4m reaches H − 1 = 5α. σ −1/α , 1 d  5 0 σ NL

v u u t 15 10

NL shows that an arbitrary value of the local d f is suppressed and 1. σ osc,I ζ 4 is small and σ (t))  σ  d , f in the case with Γ/H – 10 – − osc,I σ δσ d 10 osc,I =.05 0 (blue), 0 .1 (yellow), 0.15 (green) and.18 0 (red). NL /H 1 f c H 6α 10 − goes down to = = I becomes weak, d = σ σ = Γ Θ (σ H osc, ζ H 5 φ on / = 2 and −5/2, the numerical computation indicates that positive Γ Γ n d σ / I , one finds formally diverges. This is because osc, σ c=0.05 c=0.1 c=0.15 c=0.18 ζ = 5α > 0, is large, while ana 2 NL d → 4, we show the comparison of the analytic result of /σ > f α 1. osc,I . The comparison between the analytic expressions derived under the sudden decay ap- σ  1 d In figure /H In this case, the inflaton decay is not effective even when 0.5 0.0 0.5 1.0 1.5 2.0

8

- - - - NL f cannot be large enough toprediction, justify 0 the sudden decay approximation. can be produced in such cases. Indeed, figure mechanism cease to work. Itif should Γ be noted that the sudden decay approximation is justified one. If becomes large. On the other hand, if Γ where the Heaviside function Θ andand the the square-root factor kinematic represent suppression theof kinematic effect, prohibition the respectively. decay Here process wepossible is assume not factor that suppressed is by the straightforward. theHubble martix momentum parameter element Assuming of at that the the final other onset state. decay of channels Inclusion the of are inflaton such suppressed, decay can the be obtained as In the limit opens. Since the dependence of ∆N Figure 4 proximation and the numericalHubble results induced in the mass. new We type fix of the modulated reheating scenario with the The solid lines areright panel the show numerical the results localThe while non-gaussianity sudden the decay approximation dashed is lines valid are if the Γ/H analytic ones. The left and Substituting it into ∆N In this case, the inflaton decay rate is written as JCAP12(2016)014 9 (3.13) (3.14) with ne- is close to σ σ is accidentally [47]. σ NL f 3. The upper bound 2, as a function of . 1.0 = 1, ! NL n f 2n .  n for 0.9 3 i d = σ σ σ yσ  = 2 n ψ 0.8 = n 1) + d − , m σ by few factor. Still, it is necessary that / 2 ψ 0.7 2 φ σ NL (2n m f 4m – 11 –

as a function of 10 can be realized in this new mechanism of the is given by − 2n 0.6 ) 1 . NL  NL f i d v u u t f 1 σ 2015 σ NL = f n  0.5 = Γ 5 n . φ Planck( Γ = 5 0 0.4

NL 20 15 10 30 25

NL −5/2 f f initially. This shows the importance of the Hubble induced mass in models 10.8 [47] (indicated by a dashed line), is satisfied only when d < , which makes the modulated reheating with a modulated mass less interesting. d NL 5, we show the prediction on σ f σ > σ , Let us comment on an implication of our result in the models of the modulated reheating. As we have shown, with the Hubble induced mass, the non-gaussianity is suppressed as NL Multi-body decay of the inflaton and/or the momentum suppressed matrix element of the decay can f 9 change the kinetic factor in eq. (3.13) and suppress glecting the Hubble induced mass. The dashed line shows thenon-gaussianity upper between bound on some specific value. It is a commonmass feature of in a the particlea standard model depends model as of on well modulated amodulated. as reheating value When the the of where beyond Hubble induced some theoriginates standard mass field. only mass of model the from of that spectator It the field a the is kinetic would daughter neglected, suppression be the particle factor, modulation thus of natural the to inflaton consider is modulated reheating scenario, depending on the parameters. Then the non-gaussianity parameter Figure 5. The prediction on the local non-gaussianity parameter long as of the modulated reheating. 4 Summary In this paper, wedynamics have of discussed spectator the fieldthe effect models Hubble of of induced a the mass Hubble generation of induced of the mass the spectator after cosmic field inflation perturbation. must on be Although the suppressed during inflation, it can In figure close to on JCAP12(2016)014 (A.2) (A.3) (A.1) ), higher Pl (M O is much smaller (1) free paramter. σ O term during inflation. .  . Comparing eqs.) (A.3 F Pl M is again an . N. σφ∂σ∂φ 2 c † 1 2 σ |φ|  1 2 + ) ΦN † d must be close to unity. On the other Φ d − 2 Pl d ∂σ∂σ term during inflation is in general different 2 1 M is given by 2 (φ)(1 F + φ σ V 1 2 N 2 Pl † – 12 – + 1 and N M φ (1) free parameter. In inflation models with multiple ) = ∂φ∂φ Φ + † 1 2 O 2 σ (φ, σ = Φ 1 2 V  is an K × c 1/2. d be the chiral multiplet whose lowest component include an inflaton , respectively. Assuming that the field value of σ = N ' − L c In a large field inflation models where the inflaton field value changes by We have found that the Hubble induced mass can enhance the field value of the curvaton, For the modulated reheating scenario, we find that the magnitudes of the curvature and a spectator field chiral field, a chiral fieldfrom a which chiral has field a containing an non-zero inflaton. Thus, the constant and (2.8), we obtain order Kahler potentialspectator terms field, are and also hence relevant for the Hubble induced mass term of the than the Planck scale, theof following the Kahler spectator potential is field, relevant for the Hubble induced mass To suppress the Hubblehand, induced the mass, derivative the couplings constant between The second and the third terms are negligible as long as In single field inflationThus, models, the the Hubble chiral field induced Φ mass obtains of a the non-zero spectator field during inflation is given by φ which helps the curvatonconstraints to on inflation dominate models thediscussed and energy the curvaton density models implications of of arethe the considerably our baryogenesis relaxed. finding universe. is to We extended. As have the also a scenarios of result, baryogenesis. the perturbation The possibility and of thepropose non-gaussianity a are new affected modelcan by of produce the the an modulated arbitrary Hubble reheating value induced of utilizing the mass. the local Hubble non-gaussianityAcknowledgments We induced parameter. also mass, which This work was supporteddoctoral in Fellowships part for by Researchof the Abroad, Energy, Japan Office Grant Society of No.05CH11231 for Science, (K. the H.), Office 27-154 and Promotion of (T. byPHY-1521446 of High the F.), (K. National Science Energy Science H.) by Post- Physics, Foundation the under under grants Department contract PHY-1316783 and No. DE-AC02- A Hubble induced massIn in this supergravity appendix, wean discuss inflaton the in coupling supergravity.chiral between field. Let a Let us spectator Φ first field and consider and a the small kinetic field term of inflation model with a single be sizable afterunsuppressed. inflation. 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