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We extend our previous study of supersymmetric Higgs inflation in the context of rnal of rnal Article funded byunder SCOAP the terms of the Creative Commons Attribution 3.0 License . ou An IOP and SISSA journal An IOP and Theoretical Particle Physics andDepartment Cosmology of Group, Physics, King’sLondon College WC2R London, 2LS, U.K. Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland Institute of Modern PhysicsBeijing and 100084, Center China for HighCenter Energy for Physics, High Tsinghua University, EnergyBeijing Physics, 100871, Peking China University, Center of Mathematical SciencesDepartment and of Applications Physics, and Harvard University, Massachusetts 02138, U.S.A. E-mail: , [email protected] , [email protected] [email protected] b c e d a Any further distribution ofand this the work must title maintain of attribution the to work, the author(s) journal citation and DOI. independent of the reheatingscalar temperature. tilt We and derive thewell the tensor-to-scalar as corresponding ratio discussing predictions the in for gravitino cosmic the production microwave following background inflation. perturbations,Keywords: as ArXiv ePrint: Received June 21, 2016 Accepted August 18, 2016 Published August 30, 2016 Abstract. no-scale supergravity and grand unification, tothe include Pati-Salam models group. based on themodels Like flipped whose the inflation SU(5) previous predictions and interpolate SU(5) betweenand those GUT Starobinsky-like of model, the inflation, quadratic these while chaoticanalyse inflation yield avoiding the a tension class reheating with of proton process inflation decay in limits. these We models, further and derive the number of

John Ellis, J Higgs inflation, reheating and gravitino production in no-scale Supersymmetric GUTs JCAP08(2016)068 1 2 and since 1 5 – 1 – The resolution of this problem probably calls for some 3 , 2 such as supersymmetry (SUSY), which can help to stabilize 4 (125 GeV) holds a unique position in the Standard Model (SM) h One problem of minimal Higgs inflation in the SM is the instability or metastability of For a review seeFor ref. examples, [9] see and refs. referencesIn [10–18] therein. some and non-Higgs references inflation therein. models, vacuum stability could be restored byFor adding examples, a see non-minimal refs.See Higgs- [20–29] refs. and [32, references33] therein. and references therein. 3.1 The3.2 motion of the Particle inflaton3.3 production after inflation8 Gravitino production 11 15 2.1 Higgs2.2 inflation in the Higgs flipped inflation SU(5) in GUT3 the Pati-Salam GUT6 1 2 3 4 5 gravity coupling (withcoupled radiative to corrections) the and SM [10, assuming19]. inflation is driven by new physics not directly new physics beyond thethe Higgs SM, potential [30], whileas also the predicting 125 a GeV fairly scalarmagnitudes light of Higgs particle quantum boson discovered corrections that at to canthe the the be parameters energy LHC. identified of SUSY scale the could inflationary ofembed potential also Higgs inflation [31]. serve inflation is Since to into around certain controlframework. SUSY the that grand It of unified SUSY is theory (GUT), gauge particularly within unification, attractive a we supergravity to are choose tempted no-scale to supergravity (SUGRA), the Higgs potential at high scales. fermions (quarks and leptons), itsticles. vacuum expectation On the value other generates hand, thenear-exponential cosmological masses expansion inflation of [1–7] of SM postulates the par- a verythe scalar early inflaton observed field Universe, perturbations to whose in drive quantumscale the fluctuations structure. cosmic generate However, microwave the background identitytify (CMB), of the and the inflaton thereby inflaton with large- is theseek unknown SM possible so Higgs tests far. boson from as cosmological ItHiggs postulated observations is inflation and in would natural be collider models to a measurements. of iden- trulyflation, Higgs In economical and inflation this and provide [8], regard, predictive a mechanism welcome for direct the link cosmological in- between the SM and the early-Universe cosmology. it emerges naturally fromwhich are simple advantageous string for compactifications cosmological applications [34], [35]. and provides flat directions 1 Introduction The scalar Higgs boson 3 Reheating after Higgs inflation in no-scale GUTs7 4 Conclusions of particle physics. Via its interactions with the spin-1 weak gauge bosons and the spin- 16 Contents 1 Introduction1 2 Higgs inflation in no-scale supersymmetric GUTs3 JCAP08(2016)068 One special 8 -flat direction D In the current study, we show that the 7 GeV. and the Starobinsky-like no-scale supergravity 13 6 – 2 – -folds, which are sensitive to the mechanism for reheating. The on-going e We have given previously an explicit realization of a no-scale supersymmetric GUT After presenting the no-scale Higgs inflation models `ala flipped SU(5) and Pati-Salam, In this work, we first show how the structure of our previous no-scale SU(5) GUT See e.g. refs. [36–39]One and way references to therein. avoid this problem isFor to examples, see invoke refs. non-minimal [44, contributions45]. to the gauge kinetic function in 6 7 8 supergravity, which could modify the gauge unification condition and thus relax the proton decay bound [29]. scenario for inflation29], [ in in no-scale which supergravity. Higgs inflation In is this realized in model, an the SU(5) inflaton GUT is embedded identified as the we investigate the reheatinginflation, and process launches in the Universe theseof into the GUT the complete hot models. inflation Big-Bangalmost era, theory. Since determined it However, reheating should by unlike be the the happenscated treated inflationary inflationary after and as potential epoch a involves alone, when detailed part the theof dynamical physics reheating the properties is period reheating of process is theimprint is quite model. an of compli- While interesting the topic a post-inflationary inthe detailed evolution itself, description number of our the of analysis inflaton here on is motivated primordial by fluctuations the through of the twoThe Higgs predictions doublets of in thisbackground the (CMB) model interpolate minimal for between the supersymmetric scalarStarobinsky predictions and of model. extension quadratic tensor An of chaotic perturbations important inflation thecan and feature in be the of SM achieved the without this (MSSM). introducing cosmic model asymmetry is microwave non-minimal Higgs-gravity on that coupling a the or fairly imposing K¨ahlerpotential,provides a flat since the shift inflaton desired potential the flat no-scaleall direction. supergravity Higgs In structure bosons this couple of no-scalenon-minimal minimally this coupling SUSY to between model GUT the gravity approach Higgs viainflation fields to the and approach Higgs energy-momentum the differs inflation, tensor, Ricci in scalar. without anSUSY any essential Hence and way our GUT from no-scale extensions Higgs traditional in SM the Higgs literature, inflation [8,9], other constructions with the flipped SU(5)colored model Higgs or mass with from the the Pati-Salamaddition, model scale neither can of of disentangle inflation, the making therepresentation, model flipped so building SU(5) much they and more can flexible.constructions Pati-Salam of be GUTs In the require embedded flipped any morepredictions SU(5) field and as naturally in Pati-Salam the into GUT an SU(5) string HiggsStarobinsky adjoint inflation no-scale potentials. theory. models model make However,], [29 similar our also interpolating between the quadratic and measurements of CMB observablestrivial with constraints increasing on the precisionfeature reheating are of scenario beginning the within to GUT athe impose post-inflationary models given era, non- model we due of study toor inflation. GUT is exponentially symmetry that flat breaking. the inflationary Asinflation, inflaton potential a making result, changes potential the the into changes Universe original a dramatically effectively quadratic quartic radiation-dominated. in monomial The after moment the of end this of effective inflationary model can be generalized42] to other and GUT models, Pati-Salam specifically [43]SUSY the GUTs. flipped SU(5) SU(5) [41, One GUT motivationon is for proton this already decay. generalization tightlycolored is Higgs constrained These fields that to by the cause have masses simplest experiments, tension around particularly 10 with the the limits construction of [29], which requires the scenario proposed in [40]. JCAP08(2016)068 (2.2) (2.1) -folds e that may arise T , 1) representations ! c d − , e H H ,  10

= H + h.c. -folds and the spectral index in 1) and ( H e , , H 10 ! c u 3 ζ . We then study the reheating process H U(1) at the GUT scale [41, 42], and thus H + r

⊗ 2 | j = Φ | ) in the ( 1 3 G – 3 – − ∗ G, ,H T         2 3 1 + c G G G G 0 2) representations that are responsible for SM symmetry ν T , u u u  ¯ 5 3 2 1 0 G G G d d d 3 log 2 1 − c G GUT [43]. These GUT groups are both major alternatives to the 0 c G d 2) and ( d = R − − , 3 K 5 and the tensor-to-scalar ratio 0 c G d s . SU(2) n 0 G ⊗         L = ) in the ( U(1) that are responsible for the GUT symmetry breaking, and a pair of Higgs G H ⊗ SU(2) H, ⊗ As in [29], we introduce the following deformed no-scale K¨ahlerpotential: This paper is organized as follows. In section2, we construct models of Higgs inflation To realize Higgs inflation in the flipped SU(5) GUT, we need only the minimal field of SU(5) fields ( breaking, respectively. We express the components of the GUT Higgs multiplets as follows, and similarly for after Higgs inflation in these models, and compute their prediction for the numbers of in no-scale SUSY GUTsIn with section3, the we flipped firstfor SU(5) analyze the and the scalar Pati-Salam predictions tilt gauge from groups, our respectively. no-scale GUT Higgs inflation models radiation domination can be quiteof well determined the in reheating our models, process.our and models is are In well essentially before consequence, independentrelatively the of the precisely. onset the number reheating of temperature, and can be determined during inflation. We furthermodels. discuss the Finally, we issue draw of our gravitino conclusions production in after section4. inflation in2 these Higgs inflation inIn no-scale this supersymmetric section, GUTs generalizingconstruct new our models of previous Higgs no-scale inflationlar, in we SU(5) no-scale choose GUT GUTs as with two model other concreteSU(4) GUT realizations construction groups. the [29], flipped In SU(5) we particu- GUT [41, ]42 and the Pati-Salam minimal SU(5) GUT, andin can string readily be theory. embedded into the SO(10)2.1 GUT or accommodated Higgs inflationThe in flipped SU(5) the GUT flipped has the SU(5) gauge GUT group SU(5) content, namely a pair of Higgs fields ( is only a partialFirstly, it unification. can However, naturally thiselectroweak split Higgs framework the bosons. masses has Secondly, of a theproton super-heavy number decay problematic colored is of dimension-5 absent Higgs operator attractive in bosons thatHiggs this features. and causes theory. fields rapid TeV-scale Thirdly, in the construction anstring of theory. adjoint this model In representation, does addition, andfrom not it so require string is can compactification. easy to be incorporate embedded a easily singlet into modulus perturbative field JCAP08(2016)068 = × T 4 . i (2.4) (2.5) (2.6) (2.3) 2 GeV, 16 ' ( 10 P to be light; , × ) M d , and all the -term in (2.2) ) becomes is the inverse 2 ζ . Each of the H H α terms generate G 2 1 ) . Using SU(5) H ' H = 0 . Inspecting ¯ − λ term is important IJ α G, G )( and G G M/ K = 0 , the simple no- β (2 v G G u ζ p G . This is a significant H and ( M/ G γ = and , etc. The λ = + λv G H 2 -term scalar potential: terms collaborate to break 2 v † i as the inflaton, which has a ) to denote the chiral fields, ) F and other multiplets Φ G | H α = 2 H |W| 0 . d ··· G i ≡ . = 1 , the potential K¨ahler has a c , ≡ h 2 H T H | c 2 G ( | H c  ζ 2 ) H, ¯ ν | and β 3 h M + H G | + log H , and the resultant inflationary model | + | − 2 = G 0 M 2 u a ) v αG . Then, the ∗ J i G, H, 2 H G , and similarly for 2 | G Φ c 2 G G ¯ λ ∂ m ν ∂ − G G, h = γv ( ) and I H G ≡ K + 4 G α ˆ h Φ of the model, we first write down the following G M k` = ( ∂ = 2 † ( ∂ | + G j c 1 G G – 4 – ) via the following terms in the scalar potential: . As done previously, we assume the modulus W ij c m − IJ H H G | ∗ J G -term scalar potential involving ( K H Tr( G 2 G Φ , F  1 2 v ∂ G c 2 ) with constant G I such that, ¯ λ e a λ H to vary between 0 and 1 . When ijk`m ≡ ) = 2 Φ 4  + + 2 G 4 1 G ζ | during inflation. This provides a large positive contribution by some high scale physics (see [46, 47] for discussions of ( /∂ H ⊃ G P | K V 2 1 ≡ a, 2 V ) to denote the modulus M i (Φ) = and -flat component ∂ + = 1. We use Φ = Φ V D H λGGH = i P G ( , leading to minimization of the potential at ∗ T, + terms enable the electroweak Higgs doublets GGH G M T IJ → h γ , H K ) ij = ( = H G i and mH I ij T terms make the colored Higgs fields heavy; and (iv) the H, h G − ) collectively. In the above, m 2 1 ¯ λ G ··· ≡ and H, MG G ) light, we impose the condition λ − d G Next, we exploit the no-scale structure of the K¨ahlerpotential to analyse the inflation In order to discuss the superpotential To understand these points more clearly, we consider the = GeV) is unity, i.e., ,H G, H, u 18 W G, H where we use Φ potential. We identify the ( so the colored Higgsadvantage over fields the can minimal no-scale be SU(5) heavy, GUT with inflationary masses model studied in [29]. this point). Then, at low energies, the where the dots representadopt the fermions shorthand that notations, are irrelevant to inflation model building. Also, we of the metric, K¨ahler is stabilized at Hence, we have asymmetry, we GUT can symmetry-breaking rotate vacuumother components with vanish. At the( same time, in order to make the electroweak Higgs doublets Supersymmetric GUT unification of the gauge couplings implies that where for convenience we have used units in which the reduced Planck mass has a quadratic potential.parameter choices, which It is is quite notable different from that many the models shift that symmetry incorporate supergravity. is absent for most of our where terms in (2.3)SU(5); is (ii). the important for our model: (i) the masses for the colored Higgs fields ( most general terms up to dimension 4: (iii) the for obtaining a flat inflaton potential during inflation. value around the Planck scale to the effective mass of 10 is a natural andand slight in the deformation following of we will the allow standard no-scale supergravity K¨ahlerpotential, scale K¨ahlerpotential is recovered, and theas resultant inflation the model original has the Higgsshift same inflation. symmetry predictions ( On the other hand, when JCAP08(2016)068 , m ) ) at (2.8) (2.9) (2.7) = 0 , δ ˆ h (2.10) ( ζ + V ζ − . (1 as the original = 1 yields the 3 1 , provided that ! ˆ h ζ  ˆ h ) = ζ β − 6 , with a non-minimal = 16 (1 ˆ h r ζ . , which is always satisfied , and the round (square) r 2

ˆ h m , ). ) 2 = 59 , the value computed in 3 ζ 00 e m − h . V V 1 2 N − alone, and it takes the following arcsin > M/γ 10 , and hence the same predictions 2 2 P 1] (from left to right). In plot (b), i − × (1 ˆ h ˆ h h ζ 2 , 1 3 2 2 2 M ζ ) via H
. 9 ) − to remove the singularity of . 2 6 h m ˆ h 1 H ( -folds (1 ˆ h = = 2 [0 h µ e ζ
the canonically-normalized inflaton field. V ± m ∂ 2 6 − β ∈ s ) ( 1 ˆ h ζ h ζ 2 − m β 2
− − 2 2 ˆ h 1 , η ) ˆ h via (1 − ζ  – 5 – ζ 2 2 2 3 1 = 0 for the inflation analysis. In addition, we can − 6  6 1 − ˆ h ˆ h 0 1 , with h (1 ) G ) by using a modified condition ζ V = 2 V 1] (from right to left), while the upper strip attached ζ . − h ) to zero except for the inflaton field ˆ h 0  , r β and the tensor-to-scalar ratio − − 1 d ) s , H 2 1 ζ P ) =  (1 n 2 2 [0 V/ ζ h − M 2 ( 6 1 H, ∈ + 2 (1 V ζ − = 1) . The horizontal strip attached to the lower round dot = η = d ] = 1 6 ζ  ˆ h  00 [ 9 h 6 − V is connected to L r varies within the range of h = 0 ( δ = 1 and ζ s n h d ), which would otherwise lead to an exponentially steep potential. Under V/ 6 arctanh ζ √ − = d = are obtained from the inflation potential 0 (1 h / V h η In figure1, we compare the predictions of this model with the recent results of the It is instructive to compare the flipped SU(5) model with the minimal SU(5) GUT For another model interpolating between Starobinsky and quadratic chaotic inflation, see ref. [48]. = 6 9 and 2 ˆ section3. In plot (a), we impose the condition h kinetic term and a quadratic potential: models of Higgs inflationquadratic [8,9] chaotic and inflation. Starobinsky inflation. The other limit Planck Collaboration], [49 selecting the number of dot corresponds to describes the effect of varying this assumption, we derive the following Lagrangian for the inflaton model presented in [29]. Since the inflation potentials in the two models are identical, the In order to apply the standard slow-roll formalism, in which the first two slow-roll parameters where We find that the field There are then twowhich interesting gives limits an that exponentially-flatfor can potential the be in scalar studied terms tilt analytically. of One limit is to the square dotwe presents also the analyze effect the ofwhere predictions varying the of ( parameter the superpotential, we findduring that inflation. this happens Hence, when we can just set the potential is indeedthis minimized when stability these of components inflationwrite vanish the trajectory during inflation. scalar is potential Checking simple important, form: as and a was function done of in the [29]. inflaton We can therefore set all the other components of ( As in], [29 we impose the condition JCAP08(2016)068 = and β over = 0 , 1.00 s δ δ n . The e N 0.99 symmetry in 1] (from left to 95) and 2 0.98 . , is also a partial -folds ext +lensing+ 0 9 Z . e , ) representations, R [0 ¯ 2 98 , 0.97 . ∈ s 0 1] (from right to left), 1 . n ζ , , 0 ¯ 4 SU(2) , = 1) . The horizontal strip LowPlanckTT+ P LowPlanckTT+ P+BKP 0.96 ⊗ [0 ζ = (1 L ∈ ζ ζ GeV. In the SU(5) model [29], 0.95 ) and ( = 0 ( 2 13 ζ SU(2) , 1 ⊗ , 0.94 , and the colored Higgs boson mass 4 (b) 2 G = 59 from3.25), eq.( and compared with at tree-level, so the colored Higgs boson e γv 0.93

N in the inflationary potential is fixed by the ) in (

= m 0.05 0.00 0.25 0.20 0.15 0.10 r 5 9 G m m = G, -folds c – 6 – e H 1.00 M 0.99 0.98 ext +lensing+ 0.97 s . Hence, the colored Higgs boson is naturally heavy with a mass , and the strip attached to each dot presents the effect of varying n G ). 3 m , given the number of LowPlanckTT+ P LowPlanckTT+ P+BKP ) are the same when inputting the same number of λv 0.96 ) − r δ , r 10 s + = 2 × n ζ 2 0.95 c . H − (1 GeV. M ± (1 1 3 0.94 is imposed and the round (square) dot corresponds to 16 (a) . Predictions from our no-scale GUT models of Higgs inflation for the scalar tilt = are automatically absent due to the charge assignments, and there is no need to m ) β 0.93 ζ G Secondly, in the case of minimal SU(5), we need to impose a discrete Firstly, we recall that the mass parameter

− 0.00 0.15 0.10 0.05 0.25 0.20 r (1 1 3 unification, and can also bethis embedded embedding readily since into the SO(10). breakingFor However, of we the SO(10) will not is current elaborate irrelevant to study,It on the we contains realization a first of pair Higgs inspect inflation. of the GUT relevant Higgs field multiplets ( content of the Pati-Salam GUT. the 68% and 95% C.L. contours from cosmological observations]. [49 In plot (a), theright). condition In plot (b),where the three dots from top to bottom correspond to is rather light,However, and in hence the case in ofis tension flipped given with SU(5) by the we have non-trivial constraint from proton stability. order to remove anynecessary odd to power produce of the adjointtoo desired GUT light. inflation Higgs However, potential fields in andfield in the avoid the current the flipped superpotential, colored SU(5)impose which Higgs GUT is mass any model extra being odd discrete symmetry. powers of the GUT2.2 Higgs Higgs inflationLike in the the flipped Pati-Salam SU(5) GUT GUT, the Pati-Salam group SU(4) around 10 tensor-to-scalar ratio while the upper strip attached to the square dot depicts the effect ofthe varying range of this is related to the colored Higgs mass Figure 1 Planck normalization of the scalar spectrum to be around 10 predictions for ( attached to the lower round dot describes to the effect of varying same also holds for thesection predictions2.2. of Pati-Salam However, model,SU(5) it as model is will of worthwhile be Higgs to discussed inflation. note in the some following new features in the case of our flipped JCAP08(2016)068 ) = . G i c G reh (2.11) (2.13) (2.12) G, ν T h = i , 2 c G ) components . The ( ¯ ν c G H h H ), acquire heavy ) c G , d G . c G , ν G ! ¯ d ( c G terms are responsible c G c G γ ν e ν α . As a result, eight real + as the inflaton. During 3 3 L 2 c G c G | − ) d u fields can be parameterized 0 and ) multiplet 2 2 B ¯ 2 2 2 H H , H | c G c G M ( 2 d u β , + U(1) 1 1 1 | ) of the MSSM after the breaking + ⊗ c G c G 0 1 2 2 C d u ) direction, namely ) H , c ) . The | G

. Our next step is to estimate G ,H 2 1 ! , , ν G = = 0 and a ( 2 + ( . 2 2 reh H c G SU(3) , α T G ˆ ν h H H 1 M D terms ensure that the ( → + ) acquire large expectation values around the . The form of the superpotential is similar to representation of SO(10) after its breaking to 0 ∼ R 1 − ¯ λ 1 G k` G ) + ( H H – 7 – D 10 G 1 , G,

, ˜ D and 1 ! ijk` SU(2) ¯ λ = ,  c G c G λ ⊗ 6 ¯ ¯ e ν + ( H ≡ ) multiplet ) are eaten to give masses to gauge bosons corresponding 3 3 1 c G c G ij c G → , Higgs doublets ( -flat direction ¯ ¯ d u ˜ 1 , e D D L , 2 c G 2 10 6 e acquires a large background value around Planck scale, while λDGG c G c G ¯ ¯ d u + ˆ h field can be represented by an antisymmetric tensor of SU(4). 1 1 , and also receive masses 2 c G c G j` ¯ D ¯ d u ) and (¯ D H

mH c Gi ik = − H , u G k` G c Gi  u ij fields arise naturally from a  GeV. This breaks SU(4) MG 1 2 . At the same time, the H 16 − . Finally, the ≡ M 10 = 2 R ∼ and × H 2 W For the superpotential, we choose the following: As before, we choose the D ' G for the GUT breaking. They ensure that ( that in the previous flipped SU(5) model (2.3). As before, the multiplets can be parameterized as follows: The the Pati-Salam group, namely as where respectively, together with a ( which splits into theof two SU(2) SU(2) GUT scale, whichv can be chosencomponents to in (¯ lie into broken the symmetries, (¯ and themasses other eight components, together with (¯ and all components of inflation, only the inflaton all other fieldspotential remain and at superpotential zero. usingis (2.4) the Thus, is same again the as given scalar for by the potential (2.7), flipped derived and SU(5) the from3 model. rest the of above the K¨ahler analysis Reheating after HiggsAs we inflation have seen, in the no-scale inflatoncomponents in in GUTs all the our Higgs GUT doubletsfields models of the is rather MSSM. a strongly, As linear compared such, combinationinflation it of with couples the models. the to two neutral various gravitational types Consequently, couplings ofquickly the matter appearing the reheating in energy many processclassical originally typical can oscillation stored be of in rather theas the inflaton efficient, gauge inflaton field and bosons after potential and transfer theof sfermions) to inflation. their are other produced number In particles in densities, particular,particles this during when there due process to bosons the could Bose-Einstein with (such be statistics, a theproduced a significant phenomenon particles period accumulation known as in of stochastic this resonance. exponentiallytheir period The effective fast can masses, production which be ofthe depend either these radiation-dominated on relativistic era the or once inflatonativistic most non-relativistic, background. particles, of depending and the The on the inflaton Universe collisionsthermal potential of would energy equilibrium these enter was with relativistic released particles reheating into could temperature rel- then build up a quasi- JCAP08(2016)068 s ∼ h (3.1) (3.2) = 0 , h = 0 at = 0 due s s 1] : , [0 ∈ ζ ) fields. The red solid h, s . = 0 , 2 is a component of the adjoint GUT Higgs 0) , h = 0 . br 2 reaches unity, which happens around χ h h = 0 fails to be the local minimum in the s  M s )=( 1 2 – 8 – GeV . , as we defined in [29]. The interesting feature h,s ( ' χ 13

) 2) ) ) , where 10 / h χ 3 ( × 2 h, s 4 ( − V . that is the order parameter of GUT symmetry breaking. = 0 , the path of the inflaton deviates from ∂s , 1 V 2 s 2 h / ' = Re( ∂ can be found by solving the condition, 3 s h − br , M 1 h , 1 ) of the flipped SU(5) model as a function of ( , , where the trajectory h, s ( br V h 15 diag(1 / 2 p . Post inflationary trajectory of no-scale Higgs inflation. This 3-dimensional plot presents ⊃ We introduce a scalar field The analyses of the reheating process are rather similar for the minimal SU(5) model [29] . The inflaton then starts to oscillate around the minimum of the potential at P Σ:Σ direction. The position of where the mass parameter In the minimal SU(5) model, here is that,to in GUT the symmetry vicinity breaking.the of branch Quantitatively, point the scalar trajectory deviates from M with decreasing amplitude dueoscillation, to the potential the is cosmic well expansion. described by At a the quadratic first function stage for of all this damped and the flipped SU(5)sider the and minimal Pati-Salam SU(5) GUTdifferences case models from in the introduced other the in two following section2. models as whenever an We needed. explicit will example, con- 3.1 and comment on The the motion ofInflation the ends inflaton when after the inflation slow-roll parameter curve depicts the trajectory ofcurve the denotes inflaton an before and (imagined) after continued passing path the under branch point. The blue dashed Figure 2 the scalar potential JCAP08(2016)068 ) h ( = 0 (3.4) (3.5) (3.6) (3.3) s V = 0 , so 4 in this h ) . Hence, / 2 . In the 1 P / is small and 1 M 6 the local min- ( h 6 γ α O br λ α GeV at 0 in this model, the 16 , 4 10 h < h × 2 λ χ 1 4 = 1 to be α, γ > ' 6= 0 , due to the GUT sym-  + s G = 0 does not have a quadratic 2 v ) is obtained from the following χ h , , , = 2 G h, s s λ α ( γ γ α α λv is basically a free parameter at this 2 4 2 1 V λ − − − 1 1+ 4 1 2 ). This is much smaller than the value of ) around ) r P r r 0 h d ( M + + H 2 + 2 V 0 u γ γ α α − ) at this stage will be different from (3.1). λ α H 2 h – 9 – ( ( 2 (10 β − V − 1+ 2 1 + 2 s s 0 d ∼ O G G s H , the inflaton trajectory stays at the local minimum v v br G ) 2 h v br χ = 2 = 2 , the inflaton continues to oscillate around its local minimum = − 06 is fixed by the amplitude of the curvature perturbation, so br . br br br 0 . The existence of a branch point requires 2 G h h > h direction is given by eq. (3.1), whereas for 0 0 , and the existence of a branch point requires, h h h are free parameters, so they can easily satisfy this condition. v ( h 6 ' 0 u γ . The potential h is around α G αH v α, λ > br α, γ > and h = = 03 , which is easily satisfied since . s α 0 W 6 λ shifts away from 0 , and gradually increases to . The absence of a mass term can be readily understood. At the global minimum, 2 s , the motion of the inflaton resembles oscillation in the quadratic potential (3.1). h br . In particular, the Taylor expansion of h ∝ G The important point here is that the potential becomes very flat when In all three models, for Similarly, we find that the branch point in the Pati-Salam model is given by In the flipped SU(5) model presented in section2 the location of the branch point is v = 0 , but with ' the mass of themuch smaller inflaton than the (namely scales the of inflaton MSSM and Higgs reheating under boson) consideration. lies In at general, the weak scale, which is s term the corresponding scalar potential deviates fromthe eq. post-inflationary (3.1). oscillation when As the athan oscillation result, amplitude during of the first the stage inflatonHowever, of is as much the greater amplitudemetry damps, breaking. the When inflaton trajectoryh has the stability of the inflaton trajectory is notand affected the by scalar the potential in appearanceimum of of the branch point. the inflaton field at the end of inflation, as set by the condition where the couplings minimal SU(5) model, we require where the couplings which differs from the correspondingchoice of expression couplings (3.5) in in thebranch the superpotentials point flipped (3.6) (2.3) SU(5) is and model, always (2.13).of due present. the to Since From branch our eqs. point (3.4)–(3.6), we further note that the position In the minimal SU(5) model ofsuperpotential: [29], the scalar potential stage. It is importantmodel to [29], note and that differ the from parameters the above couplings are defined couplings for in the the new minimal models SU(5) in our previous section. case. Here both and the branch point position derived from the condition (3.2) is JCAP08(2016)068 , ∝ P ) M t from (3.7) (3.8) (3.9) ( (3.10) h 04 h . A ) in the ω = 0 and = 0 . The , h 2 s h, s / ( 1 and t V h ∝ A ) . 33). The red solid , and then via an t . ( h 0 a br , A h br ) reduces to 0 h t h ( 06 . M h A 4) 4) / . / ) ) = (0 br , Γ(3  π ) α, γ 2Γ(5 0 t , √ . ( ) h < h h h ( 3 , which implies ω = ( 0 t ρ/ V d ) = 0 t )] h + = h , ( Z 0 2 ( 4 p  ˙ h h V fit 1 2 V . The motion of the inflaton is governed by the – 10 – 2 h 2 br + ) that vary slowly with time. Ignoring the cosmo- ) sin t for typical parameter choices. Hence, soon after h − ˙ br t = h ( M 2 ( ) h h ˙ a a 2 h h d br ω  A < h h A ˙ a a is larger than the branch point ( + 3 h  ) = fit h ¨ h 3 h A V ) = ( A t ( 2[ . But for our purpose, it is a very good approximation to fit fit h V h p h h A A . − 2 Z / 1 1 π ) and frequency t − t ( = h ∝ 1 A decreases rapidly during the first few oscillations. In fact, the amplitude ) − h t ω ( h h ω is no longer harmonic. For our purpose, it is a good enough approximation to undergoes a period of oscillations with decreasing amplitude, initially in a quadratic G and Hence, the expansion of the Universe during this period is the same as in a conventional During this stage, the oscillation of the inflaton around the local minimum Before considering particle production and reheating, we first study the motion of the v h 2 / 1 = − t radiation-dominated universe. As far as thepotential cosmological have an expansion effective is equation concerned, of we state note that oscillations in the quartic In figure2, weflipped present SU(5) a model, three-dimensional where picture we have of the the sample scalar inputs ( potential following equations: During the firstsolution stage for governed bydecreases a so quadratic fast potential,already that the reaching after amplitude the of justinflation branch the one the point oscillating Universe oscillation enters the the amplitude second stage ofs oscillation governed by adescribe quartic the potential. motion of the inflaton as energy conservation: potential when the amplitude inflaton after inflationary epoch,ton switching off all interactions. As discussed above, the infla- is a complicated functionthis of potential by a quartic monomial: curve describes the trajectorya of reference, the the inflaton bluepotentials before dashed in and curve the after depicts Pati-Salam passing anflipped model the (imagined) SU(5) branch model, and continued point. path as thepoint long under minimal As as exists. SU(5) the are couplings are quite chosen similar such to that the that corresponding of branch the effective quartic potential when with amplitude logical expansion for the moment, it is easy to find the relation between JCAP08(2016)068 , = 2 ϕ u,d 2 ) = ϕ may H M t ( (3.11) (3.12) (3.13) (3.14) (3.15) M + h h 2 ) /a k = ( k  ) as follows: x through an interaction ). The behaviour of a , , 2 h t,  can be written as ( h ω x · ϕ k . by dividing the total energy (4 ϕ i / − ) = 0 , k 2 h e x ) ϕ A , which depends on the value of = 0 t . As shown in figure3, the plane t, 2 . ( ( ϕ g k q ∗ k would oscillate simply as
ϕ 2 ϕ ϕ ) = 0 |  = † k  h ξ k ) ) ( a q ϕ and | ϕ + ) 2 k h, a A h, a  ξ in the absence of the background inflaton ( x ( · 2 ϕ 2 ϕ k + ϕ reduces to the well-known Mathieu equation: 2 M +i , and M | cos 2 e k k 2 h ) + q ˙ + ϕ t ϕ 2 i | ( couples to the inflaton /ω – 11 – 2 2 k ) ) by the energy of each particle, a − k , the effective mass of 2 2 h ∂x ϕ k 2 ϕ i created by the mode  |  k ∂ 1 g A A k 2 a 2 k + ∂x ϕ  g as follows: | n k + ( 3 2 2 1 2 k 1 = ˙ ) k a  ϕ ) ϕ is the mass of 3 π ˙ ξ + k a a + 2 d ( − 0 n (2 0 2 ξ | ϕ ϕ 2 ϕ d 2 k ∂ Z + 3 ∂t ˙ d M ϕ M ˙ k | a a ( ¨ ϕ + 1 2 ) = 2 + 3 x = k 2 GUT Higgs bosons Σ, and finally the two MSSM Higgs doublets t, 2 k ) satisfies the following equation: ( t ∂ ∂t , where E ϕ ( = ( t 24 = 1 , and the background inflaton  k h ϕ A a ω with coupling constant , 2 ) is the effective mass of the quantum field t 2 h sin ϕ ω h, a 2 2 h ( . Then, if the quantum field h 2 ϕ = t A 2 2 h g ξ M g ω 2 1 ) can be divided into stable and unstable regions, where the stable regions (unshaded) + 0 sin 2 ϕ h A, q stored in this mode where the background inflaton fieldis and negligible the in metric. most The cases, contribution so from we the make background a metric mode decomposition of where the mode We can infer the number of particles A term M field. In consequence, the mode equation for To obtain an intuitive picturean how idealised the case resonancewe would in happen, would which it have the is instructive expansion to of consider the Universe could be ignored. In this case, where solution to this equation depends on the parameters 3.2 Particle production We now consider interactions and particle production. The oscillating inflaton field themselves. We can formulatethe the production inflaton of and these particles theevolution in Friedman-Robertson-Walker of the metric the standard as way, quantum treating motion backgrounds, fluctuations for and of a studying all given the quantum decay field products. We can write the equation of ( correspond to ordinary oscillatingto exponentially-amplified solutions, solutions. and It theparticles. is unstable the regions latter (shaded) that correspond give rise to resonant production of decay perturbatively toallowed. all However, it particles turns it outthan that couples non-perturbative perturbative resonant to, decays decays canconsider so in be all possible more long certain channels important for as casesbosons, inflaton decays this sfermions, ]. [50–54 into bosonic decay final is states, To which kinematically see include gauge this point more explicitly, we yielding JCAP08(2016)068 ) h 2 h ω and (4 (3.17) (3.18) (3.16) h / 2 h gA . =      q µ ) , 24 µ 2 40 V 2 h 3 2 10 W g i 3 q 1 4 / . 1 − − j µ − = µ µ 1 3 H Y i 24 W 2 W V ( ψ 2 2 ij 30 1 1 M √ √ T 5 3 d − y = µ ) + µ 2 24 W 2 m V W 3 H 10 3 k` / q q 4 20 + i T − µ µ ,M − ij 1 2 µ T X h 3 W 2 – 12 – 5 matrix: ( g W : 2 × 1 1 4 2 2 ijk`m 1 | √  √ 2 , + u 1 y µ 2 G H 10 1 8 24 v µ 2 V . The shaded regions represent instability bands, where the = g D 2 | 2 h 15 3 5 q /ω = 3 ) 3 / 0 + / 2 2 Y ϕ Yukawa a µ ) scan a large number of instability bands within a few oscillations. +4 +1 µ µ M M G W 0 Y X + = A, q 5 0

2

20 15 10

2

X a SU(3) k - 2q A λ M 2 1 = ( √ q      2 = − is the SU(5) gauge coupling at the GUT scale. . Stability-instability chart of the Mathieu equation in the plane of µ A g A We consider next the sfermions in the supersymmetric SU(5) model, which couple to With this in mind, we now consider the couplings between the inflaton field In a more realistic case, the large amplitude of the first few oscillations and the cosmic the background inflaton via Yukawa terms in the superpotential: directly from the kinetic terms where As a result, the resonantreason, production this of period particles of can particle behave production in is a stochastic termed manner. stochastic resonance. For this expansion of the Universethat the make parameters the ( situation more complicated. It is possible, in general, versus exponentially-amplifying solutions are located,the whereas unshaded the regions. conventional perturbative solutions lie in various bosons into whichcan it be may parameterized via decay. the We following consider 5 first the SU(5) gauge bosons, which We can deduce the effective masses for various gauge bosons in the inflaton background Figure 3 JCAP08(2016)068 q h × 2 24 ω = 0 , (1) − (3.21) (3.19) (3.20) 0 O A ϕ is a large M can safely ) = d 2 q h ω H . 2 with (4 of particles in a h / 2 d 2 h ϕ k and -term couplings to y . In consequence, n A 1 4 parameter increases h u F 2 g A H q = , 2 =         5) components of the is a left-handed multiplet 2 νR c c c − q , L − L 1 L 2 L 3 L i ν e A d d d M ψ = 4         = 1) that can be determined by GeV, we see that . = 2 `L i, j (0 13 ( ψ O M , 10 ij 4 = h , so that for a field ∼ field), the 2 , 2 h s 2 h dR α         -flat direction, so its 4 1 /ω 2 L 3 L 1 L M + L 0 ) D d d d e 0 + 2 ϕ 3 L 2 L 1 L 0 4 G ,M u u u M v 2 . On the other hand, for heavy particles such as 2 1, as is clear from eq. (3.3). Finally, the coupling representation, and – 13 – h 2 L + ) and λ L ) 2 u − k ) P c 2 2 y c 1 0 10 u k = 1 4 is a coefficient of M u ( 2 2 Σ − = k − = ( µ L M 10 q ) 2 uR 2 (10 c 3 0 ( u field is proportional to the amplitude − ). Hence a broad resonance can readily appear when A 0 ( 5 ∼ O h ,M 2         are all of higher order and thus are highly suppressed. , where br h (10 t ) h h = 2 d O ω u,d y k T H µ also gives effective masses to the Σ 2 + have mass dimension e 2 u h ) to α y 4 ∝ ( 1 4 ) and for sfermions. k (10 is easily achieved for small and n 24 = O λ q . Recalling GeV. Hence, at this stage the perturbative decay of inflaton is kinematically for- 2 2 is antisymmetric, and for simplicity we suppress all the flavor indices. The effective 2 uL ) is a left-handed multiplet in a 13 − h W, V during the era of inflaton oscillation, and hence are much larger than inflaton mass ij M ij representation, which can be parameterized as follows: parameter in the Mathieu equation remains constant with T 10 A ) gauge bosons and Σ bosons (including the h T /M An interesting feature of resonant production here is that the oscillation frequency The above analysis shows that the effective masses of various bosons are generally of A broad resonance leads to an exponential increase of the number The inflaton 5 q ∼ For example, see also ref.]. [55 br h 10 gh X,Y of the inflaton to the other components of the MSSM Higgs doublets order M bidden, and the leadingFermion channel production for through energy perturbative transfer decaynant. is may through also present, non-perturbative but resonances. is generally subdomi- where both be neglected, since theother inflaton components is in moving in the of the background inflaton the ( number, of is small. Also, wesmall note that ( linearly with time, sosee the that broad particle resonance productionbosons is via ( suppressed broad for resonance these could species. be In efficientgiven summary, only mode, we for the light gauge where In the realistic casebe with obtained three by diagonalization generationsdepend in of on flavor fermions, space model the and assumptions.details sfermion will of However, mass be for the more spectrum the flavor complicated would structure. sake in of general, illustration and weGUT do Higgs not multiplet elaborate Σ , on as follows: masses extracted from the scalar potential are Here in a JCAP08(2016)068 , , i of br 1), . W h , we when (0 reh h M (3.24) (3.22) (3.23) = h T A 2 h rad g . Hence, ∼ O ρ h ∼ |i ∝ g A h W h| , which is given h decay can disrupt bosons is given by ) and 4 W − W is small, the scattering . in our models does not .
of e (10 ! W 2 W is of order Γ N n n λ α W ± , taking values between 0.15 n 2 W ∼ O q σ W − br 1 − , /h 3 , and the fact that r W h g |i n 2 1 + h M 0 W h| λ α . µ Γ 3 ). g t given by eqs. (3.4),3.5), ( and (3.6) for the − . So, we should compare the production − 2 1 h h to determine whether 1 ω 2 W br W M 0

∼ n h n br µ 6 W G pairs can annihilate through scattering, with a 4) h h v W / 2 ω ± – 14 – 0 4) α , but is determined by the energy density µ / 2 W Γ(3 exp(2 2 , with π reh ' 3 ∼ T (1) uncertainty may alter this picture, and a period of √ 2 br a 2Γ(5 0 n h 2 br O µ 2 h h = 2 2 mass during an oscillation of the inflaton h M
boson is so quick that resonant production cannot take place M W W ' . As a result, the number density W n ' 2 3 − rad a i = 0 is a rapidly varying function of ρ TeV [56–58], due to the quartic shape of the effective potential (3.7). rad t q W have the same dependence on the oscillation amplitude ρ d 14 d 2 boson as an example. The decay rate of M − . At the same time, W W |i ∼ h A h h| W is the averaged g σ for and Γ 1 2 in (3.22) with the decay width Γ i k can be determined without ambiguity. This is because h (1) uncertainty in the coefficient. Since = W h µ ω e i O ω M 0 N h W µ Without getting involved in these details, we see that the reheating temperature Although a more precise estimate of the reheating temperature depends whether the The above analysis ignores processes that can decrease the number of produced particles, , it is reasonable to estimate the number of produced particles by considering zero M h =0 k -folds process is rare and its rate is suppressed by rate 2 resonant production. From eq. (3.10) and Γ the decay process is a more rapid process than resonant production when see that three models, respectively. For instance, in the minimal SU(5) GUT model, we have we see that the decay of the we immediately deduce, cross section efficiently. This conclusionnot is parametrically small, certainly so not some definitive, since the left-hand-side of (3.23) is decays of gauge bosons anddepends sfermions would on disrupt numerical their resonante details, production, it which further is important and interesting to note that the number of solving equation (3.13) or theresultant Mathieu equation3.15) ( by ignoring theand cosmic 0.35. expansion. The Thethe total particle number numbers of ofµ produced all modes. particles can Since then themodes be zero-mode only, in inferred particles which by have case the integrating we largest over have coefficient In the first stage of reheating during which the number density of with an not-very-efficient resonant production may happen. this model may bemated significantly to lower be than around in 10 We conventional SM recall Higgs that inflation, a which lowover-production is reheating of esti- temperature gravitinos (as in we supersymmetric discuss GUTs below) may as help well to as avoid the unwanted topological defects. depend on the reheating temperature radiation begins to dominatepotential the changes from Universe. a Since quadratic the shape above to analysis a shows quartic that one the at scalar the branching point including decays and scatteringimportant, with and may other destroy particles.illustrate the using the resonant These production processes of turn gauge out bosons to and be sfermions, very as we where by JCAP08(2016)068 , . r 4 G e 2 2 , v . / 2 /ρ N 3 2 α Y / (3.26) (3.25) 2 1 = 1 3 ρ c ' ≡ , only differs 2 / end 3 59 is not affected ρ Y rad e ρ ' ' N ∗ g , is generally subject to 2 begin 59 , and is independent log / V 3 r Γ ' and tensor-to-scalar ratio 1 ρ t 12 e − s e − N n !  2 2 suppressed by a small coefficient / / (1) and contributes little to rad end 2 3 2 1 ρ ρ O m m depends only logarithmically on the  rad ρ e is roughly the GUT symmetry-breaking log N 558 and radiation . -folds to be is the effective number of degrees before 1 59 by inputting 12 e 2 01 . For the flipped SU(5) model and the end . / ∗ ρ 3 0 ' + g ρ 1 + 0 e 

' 1 2 N – 15 –  G , and ) is the decay rate of the inflaton (gravitino), and end P v h begin ρ 2 Γ V / M end 3  ρ  59 also holds well. This is because -folds is determined to be -folds e (Γ e log ' GeV, but is probably much lower, where the uncertainty is h 1 4 c e 06 and 14 . √ 0 N 00398 + . 0  ' 4 (1) , Γ P α ' O begin M denote the energy density of the inflaton at the beginning and the 2 V / = 3  Y c with end ρ log 6 G 4 1 v 2 derived, we can predict the values of scalar tilt and α e 2 ) are the gaugino mass at the GUT scale and the gravitino mass, respectively. It 62 + N 2 / ' 3 ' begin 08 as in eq. (3.25). In passing, we also note that the estimate of V . e rad , m 0 With In the above, we have mainly presented the explicit analysis for the minimal SU(5) (1) factors among the three models, and thus has negligible difference when computing ρ N 2 / O ' 1 , due to the very mild logarithmic dependence log more precisely, as represented by the yellow strips in1 figure (section 2.1), where the e m 1 12 scale of effective radiation dominance scale, which is the same for all threer models. predictions of our models are compared with3.3 the latest Planck results Gravitino in production 2015It [49, is also60]. desirable tobetween study gravitino the production energy in our densities models, of since the the ratio gravitino is also assumed thatfields. the It effective was degrees shown of recently freedom [66] during that the reheating above consist expression of is the a MSSM good estimate with and by N of details of the reheating process. This is because nontrivial constraints from gravitinoassuming production instant [61–65]. decay of The the standard inflaton calculation and of thermalization, yields ( even when we take intoHowever, account of as perturbative gravitino was productionmodels, shown prior and to above, gravitino thermalization. production non-perturbativeabove in standard resonance the estimation. non-perturbative would regime probably may or occur may in not affect our the Pati-Salam model, the result mainly due to the highlyon non-perturbative model dynamics details; of (ii). the the reheating number of process, which depends model29], [ butthe it flipped is clear SU(5)temperature that and can be the Pati-Salam as two high models as principal 10 given conclusions in in section2. this section Namely, apply (i). also the to reheating by the accumulated decaybecause products these particles even are wheneffectively also they dominated light by start and radiation highly to once relativistic, dominate the and the branch thus point energy the is Universe density, reached. is always where the constant end of observable inflation, respectively, and Hence, using [59, 60], we compute the number of where the moment of effective radiation dominance, which is Finally, we deduce the number of JCAP08(2016)068 4 − a (3.27) (3.28) ) can be 14 − fermion, with 2 1 (10 O < ) . This contribution 2 / 3 reh Y . /T 2 3 reh / states are essentially empty. 3 T f m ( ∼ O 4 100) TeV would have decayed into / . Since the radiation energy density 3 reh − k > k ρ . 3 reh states only, because they are produced 4 T . / (10 1 2 2 3 rad , we divide the gravitino energy density / ρ O 3 4 reh rad 1 GeV, the bound m ) > T . 2 2 ∼ / / ∼ 3  2 3 is diluted to states is similar to that of a spin- / , while all rad n – 16 – r an amount of 3 ) ( f ρ m 1 2 2 4 ρ k 3/2 / / r reh 3 3 ) m ) the scale factor, and the energy density scales as n 2 /ρ ( / 2 states (which couple to other fields only through grav- 3 a / rad 3 n 3 2 ρ /ρ of gravitinos produced during a non-perturbative resonance, reh = with ρ 2 cannot exceed the scale of the energy density during reheating, ( 2 / in eq. (3.24). Therefore, we derive the following upper limit on 3 / 3 3 f − ) GeV. n 4 ∼ k a Y is more flexible in these models, and can be heavier in both the / 14 rad 1 2 ρ c / 3 (10 M n O . reh T production through non-perturbative effects: The non-perturbative production of gravitinos can be important in our models. To esti- In order to compare this result with the thermal production (3.26), we note that the In this work, we have extended our previous study on no-scale inflation in the minimal 2 / 3 it is a good approximation to consider the helicity- low-momentum states dominating, sinceas the seen instability in region figure3.mentum is states denser In up at the low to most momentum, a efficient “fermi scenario, surface” the produced gravitinos occupy all mo- more efficiently than the helicity- n mate the number density number density scales as ity). The resonant production of helicity- The physical momentum which is always below The subscript indicates that thedominance. number density is evaluated at the time of effective radiation due to effective radiationof dominance. reheating, Hence, the at number the density time of ( thermalization, namely the time Thus, the energy density of gravitinos is by the radiation energycontributes density, and to find the that the ratio non-perturbative production of gravitinos radiation before Big-Bangof nucleosynthesis an (BBN), ultra-light and gravitino so with would mass be harmless. In the case 4 Conclusions Higgs inflation identifies theeconomical inflaton field approach as to the realizeexpansion observed the of Higgs cosmic the boson, inflation very anddesired provides early that a energy Universe could truly and scale have generatedunification, driven of the providing the a successful observed exponential strong large inflation motivatationsupersymmetric scale to lies GUTs. embed structure. around Higgs The inflation the into attractive scale no-scale ofSU(5) supersymmetric] [29 to gauge a class ofgroups, Higgs inflation namely models the in no-scalecolored flipped supersymmetric Higgs GUTs SU(5) with mass different andflipped Pati-Salam SU(5) group, and as Pati-Salam models, presented compared in to section2. the minimal The SU(5) model. This helps at the time of reheating is of the order could be madevery cosmologically light. acceptable A by heavy requiring gravitino the with mass gravitino to be either heavysatisfied or for JCAP08(2016)068 , 241 Phys. , Phys. Lett. Mon. Not. , , , we note that e (1982) 389 N Astrophys. J. , ]. B 108 (2013) 214001 30 ]. SPIRE 100 TeV) or ultra-light (much IN ]. − SPIRE in these models is generally lower Phys. Lett. in supergravity inflation models is , IN SPIRE ][ reh IN reh ]. T T (1983) 177[ in the near future. Unlike SPIRE s IN Class. Quant. Grav. Cosmological implications of the Higgs mass n , ]. ][ B 129 – 17 – (1982) 1220[ The Standard Model Higgs boson as the inflaton ]. SPIRE arXiv:0710.2484 48 -folds can be determined without ambiguity, due to the e IN Cosmology for grand unified theories with radiatively induced SPIRE Phys. Lett. at which particles were thermalized depends on more details IN , ]. reh ]. (2008) 002[ arXiv:0710.3755 T (1981) 467[ SPIRE 05 IN ]. Phys. Rev. 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Riotto, of the reheatingestimate process, has in shown that particular the the reheating efficiency temperature of resonant production. Our simple the reheating temperature than that in thetemperature conventional more SM precisely Higgs ingenerally inflation. these subject It models, to is since order an desirable to prevent important to over-production of estimate constraintrequire gravitinos, the from the a reheating relatively gravitino the high mass reheating to gravitinobelow temperature 1 be production GeV) would either [67], rate very which would heavy [65]. have (above important 10 implications for In supergravity phenomenology. Acknowledgments The work of JEHJH was and supported ZZX in was partand supported by 11135003) in the and part STFC by by Grant the Tsinghua ST/J002798/1, National University (under NSF and grant of the 20141081211). 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