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JHEP04(2020)077 Springer April 14, 2020 March 26, 2020 : October 17, 2019 : February 20, 2020 : : Published Accepted Revised Received Published for SISSA by https://doi.org/10.1007/JHEP04(2020)077 [email protected] , . 3 1908.11378 The Authors. c Cosmology of Theories beyond the SM, Eﬀective Field Theories

Primordial non-Gaussianity signatures of extremely heavy particles are re- , [email protected] College Park, MD 20742, U.S.A. E-mail: Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, Open Access Article funded by SCOAP ArXiv ePrint: in standard inflation, consistentdynamics. with This explicit brings effective various motivated fieldgauge-charged particle physics theory scalars signatures, and control such fermions, as of loops within inflationary of future heavy observational reach. Keywords: examined within athe simple primordial alternative fluctuations to anddifferent the the fields. inflationary standard The spacetime inflationaryroles curvaton expansion paradigm, are scenario are played provides sourced in by an by theof which example two the curvaton of and curvaton to this the heavyshow in inflaton particles fields, that which with respectively. the they masses distinct We of can study order lead couplings the to inflationary non-Gaussian Hubble scale, signals and orders of magnitude larger than those Abstract: Soubhik Kumar and Raman Sundrum Cosmological collider physics and the curvaton JHEP04(2020)077 10 14 12 18 , which could lie well beyond the 5 H 11 17 – 1 – 20 9 ] for a review) in the very early universe is an attractive 1 . If such particles can decay into inflatons, they can give 19 13 H 6 ∼ 21 6 23 1 ]. Importantly, such properties can be tested with even better precision with 2 5.1 Higgs exchange5.2 in the broken Charged phase scalar5.3 exchange in the Charged symmetric Dirac phase fermion 4.1 Higgs exchange4.2 in the broken Charged phase scalar4.3 exchange in the Charged symmetric Dirac phase fermion 3.1 Cosmological history 3.2 Observational constraints era. In particular,insight any detection into of physics primordial atreach non-Gaussianity the of (NG) inflationary terrestrial can Hubble experiments. givenew scale us The particles crucial time with dependent masses rise inflationary spacetime to can a produce distinctive non-analytic momentum dependence and angular dependence of the that eventually seed the temperatureand fluctuations in the cosmic inhomogeneities microwave background in (CMB) inflation large predict scale that structureproximately the Gaussian (LSS). primordial — fluctuations all Theobservations three are simplest [ of adiabatic, models these scale of propertiesupcoming are cosmic invariant surveys verified and and to this ap- a would allow good us extent to get by a the more detailed picture of the inflationary 1 Introduction An era of cosmicparadigm inflation to (see explain [ the origin and the properties of the primordial density perturbations A Charged scalar loop B Fermion loop 6 Conclusions and future directions 5 Charged heavy particles in the curvaton paradigm 3 Curvaton paradigm 4 Charged heavy particles in the standard inflationary paradigm Contents 1 Introduction 2 Observables and cosmological collider physics JHEP04(2020)077 ]. NL f 34 (1.1) , 33 implies a larger φ ] give rise to small, ]. Thus we will con- 23 ], charged scalars and is a scale by which the 35 [ φ 23 3 ]. For various recent ideas , − 19 18 – , 3 . Current CMB constraints on –10 4 14 , and Λ 2 being a constant, which is broken − – O c , 10 12 ) during inflation. To ensure that it O n ∼ , the strength of NG is small, but can H ) φ physics at very high energy scales. This , with NL ∂φ c f ( σ 4 + described in terms of scalar fields. The pri- − ) φ , and a smaller value of Λ ], many of the prime targets of the cosmolog- φ O – 2 – 3 → on-shell 1 exist, with varying levels of conservatism and as- φ is schematically described within an effective field φ +dim( n O 2 φ explicitly Λ can be while our description still remains in theoretical φ ], Kaluza-Klein modes of the graviton [ ]. Upcoming LSS experiments will improve this by having a ]. 31 30 , . can be a reasonable choice, assuming no new physics comes in ] which will be useful to probe heavy particle-induced NG [ 30 – 1 29 pl 32 , 20 M . With such a high value of Λ 20 ) is a light field (with mass (5–50) [ , (1) [ pl φ ∼ as the ultimate limiting strength of NG for observability. Given this O M φ 18 4 , < − ∼ O to a generic operator | ) is the scaling dimension of the operator 14 10 – φ and NL O NL f ∼ f 12 σ | H NL f and thus Λ varies depending on the particular shape of NG under consideration and broadly, is Due to quantum gravity effects any EFT description is expected to break down at The inflaton ( To understand the prospects of detection of such heavy particle-induced NG, we can pl NL sumptions about the ultra-violet (UV)scales physics. is A shown schematic in representation figure of the relevant M between where dim( EFT description must breakthus down. characterized by Strengths inversecoupling, of powers leading such of to non-renormalizable Λ aimportant couplings larger to are strength ask of how NG.control. small Several So Λ possible from choices an of observational perspective, Λ it is coupling of theory (EFT) framework as, understand what suppresses such NG contributions in the standardremains paradigm. light in the presenceone of normally potentially imposes large a radiative shiftonly corrections symmetry weakly due by to its heavy potential. states, Under the restriction of such a shift symmetry, the dominant and sometimes even unobservable,where strength the of dynamics NG in ofmary the inflation goal standard is of inflationary the paradigm above present mentioned work NGs is are to naturallycan describe orders naturally of a be magnitude simple, larger, brought alternative and into paradigm, the the in associated scope which targets of the the cosmological collider program. Let us first 21-cm experiment which cansider roughly achieve lower bound, with someical rare collider exceptions program [ suchfermions as, [ massive gauge bosons [ characterize the strength ofin NG the by literature, a whose dimensionlessf definition parameter, will conventionally be called givengiven in by section precision The ultimate sensitivity in this regard will be provided by an only cosmic-variance-limited careful measurement of such dependencies,particles one — can a infer rare theis opportunity mass the to focus and probe of spin the ofin “cosmological such this collider heavy direction physics” see, program [ [ three (and higher)-point correlation function of the primordial fluctuations. Through a JHEP04(2020)077 . ]. 1 4 is inf 2 inf [ V 1 4 V inf 7 (1.2) & V − φ 10 & ≈ φ 4 2 0 ˙ φ H ], which describes 37 , during the reheating R on tensor-to-scalar ratio. T , a weaker restriction of 5 2 ) − 4 φ are respectively, the inflationary Λ ∂φ 10 ( pl × M 7 . . Thus an EFT with Λ 2 is the highest energy scale available 1 4 and inf < 1 4 H, V inf H pl ) denotes the homogeneous part of the in- V ) implies that the NG mediated by heavy t ∼ 250 ( 0 are respectively the potential and kinetic energy are respectively the potential and kinetic energy 1.2 > φ 0 H/M here. 0 ˙ σ ˙ ] 1 4 φ φ inf 2 – 3 – √ ). . To ensure control over the description of just V q 0 ˙ φ & and 3.22 ]. Stronger NG can be obtained by taking Λ φ ], where and q 4 / Λ 4 1 36 / 23 σ & during inflation, but not the dynamics driving inflation, , 1 inf V φ V 10 , 3 does not explicitly capture the scalar field source of inflation, fluctuations φ is still conservative because is the potential energy density that drives the inflationary expansion. φ 2 pl can be obtained [ M H 2 60 H ∼ ∼ 0 ˙ . Various energy scales discussed in this work. φ inf V q Even more agnostically, the Goldstone effective theory of inflation [ We note that if one only demands theoretical control of the series of higher dimen- & φ However, such low Λ We will not pursue this less conservative Λ only the cosmological can avoid even the restriction Λ sional terms in the EFTΛ which involve anflaton expansion field and in we have used the fact that the scalar power spectrum implies where we have used the PlanckWe will constraint [ show later thatparticles the in above restriction several ( scenarios of interest are quite small or even unobservable. Such a choice ofduring Λ and after inflation. Forstage example at the the reheat end temperature, capable of of inflation describing the canin universe be this both maximally work during we and will after take inflation. Keeping this in mind, scales of the curvatonbenchmark field. parameter point A given sample in set eq. of ( values of the above scalesbe can observable be in obtained a using fewwhere the cases [ Figure 1 Hubble scale and the Planck scale. scales of the inflaton field. Similarly, JHEP04(2020)077 , σ σ and (1.3) φ interactions is σ can arise in several φ since the latter scales, Λ 4 / 1 inf does not have to drive the V σ , such couplings need only be could represent a composite- . An example along this line σ σ φ . Then an argument similar to σ and inf 0 V ˙ φ q , one need only have the EFT cut off , dynamics. H 1 4 σ . φ φ V can be ∼ could represent the masses of new mediator Λ & σ σ – 4 – V σ respectively. Since Λ σ σ ]) which achieve compatibility between the two sets belong to two different sectors which are sequestered . Such a scenario with Λ σ 39 φ sector and/or heavy fields interacting with it. Via the , Λ − 38 and Λ -mass particles, but not to and σ , and thus one can then have orders of magnitude bigger NG < -mass particles now couple to does not lead to a break down of the EFT description for φ φ H φ , which automatically implies separate cutoffs for inflationary H 1 4 1 4 inf φ . Another possibility is that Λ inf V V 5 and and σ ), will only imply ], whose role is to predominantly give rise to primordial fluctuations σ σ σ 42 1.2 – , instead of Λ . One can then have stronger couplings between the inflationary fluctu- 40 σ ). H -mass particles, leading to significantly larger NG than what is obtained by 1.2 & are consistent with higher scales such as H Since the two sectors are decoupled (up to gravitational effects) and can undergo Concretely, one can have This Goldstone description is completely agnostic about whether such low values of are localized on two distinct “branes” and the extra-dimensional warping (redshifts) fluctuations fluctuations separate reheating, Λ at the end of inflation.suppressed If by the Λ of primordial density fluctuations. Comparedthis to the scenario Goldstone can EFT of successfully inflation describe framework, inflationary and post-inflationary dynamics as will ness/confinement scale for the AdS/CFT duality, this latter scenarioσ is dual to anbetween extra-dimensional the set-up two where branes explains why Λ while still allowing Λ ways. One possibilityfields is which that couple the to will scale be Λ studied in section from each other (say, viaing having their different locations own in EFTbackground an cutoffs, expansion, extra its Λ dimension) energy with density the each one hav- leading to ( EFT cutoff scales for fluctuations and the homogeneous background.way, This lead can to then naturally, large but NG inparametrically a if smaller controlled the than EFT the scale EFT characterizing cutoff nonrenormalizable of is readily obtained. Thisof is freedom achieved responsible by for havingations. the an inflationary We explicit background assume separation and thatthe of for “curvaton” along seeding the [ the with field density degrees the(today), fluctu- inflaton whereas field, the role there ofsion exists the (with a inflaton is subdominant second reduced fluctuations). light to field just Then sourcing the it background is expan- completely consistent to have separate which control the homogeneous background,ics do of not inflationary even fluctuations. appearhomogeneous in The inflationary the Goldstone background Goldstone description is dynam- simply given.the assumes Goldstone There that description do a (e.g. exist suitable [ of subtle scales. mechanisms beyond In this work we explore a very simple alternative where such a compatibility Λ ations and demanding ( Λ the fluctuations which probe energy scales JHEP04(2020)077 . . , ]. ]. ζ 5 3 A 28 28 , (2.4) (2.1) (2.2) (2.3) 3 tuning k = 2 k = classical 1 . k 0 i +) sign convention ) , n 4, absent in ref. [ ] and also makes the k + ~ / ( , 5 28 ζ + − , ··· = − ) . 1 0 i loc k ~ NL ) ( f 3 ζ h k ~ ) ( 3 n ζ . k ) . k ~ ) 0 2 P i k ~ 1 ) + ( , and in the curvaton scenario, section k , we will set up the notation and discuss k ~ ζ P 4 2 ) − ··· 1 ( k, k, k ] appeared which discusses how Higgs fluc- k ( ~ ζ + ( ) 28 F ζ 1 k ~ h ( k ~ 5 – 5 – ( 18 ζ 3 h δ = ) = 3 = 3 ) k π NL , k P f 2 , k = (2 1 i k ) ( n F k ~ ( ζ , giving us an example of a controlled field theoretic scenario having ··· 3 ) . Throughout this paper we work in the ( 1 k ~ 6 ( ζ h defined above is in general momentum dependent and thus it is conventional F we will calculate the NG contributions explicitly to confirm those statements. We As this paper was being completed, ref. [ This paper is organized as follows. In section B The function in the literature to define an “amplitude” of NG in the equilateral limit The dimensionless three point function parametrizing the strength of NG is defined as, We denote the gauge invariantand curvature use perturbation, the which primed will notation be to defined denote below, its by momentum space correlation functions, The power spectrum is denoted by, robust prediction of the strengthNevertheless, there of are a some “local” structural type similarities of between NG, the present work2 and ref. [ Observables and cosmological collider physics for the spacetime metric. tuations, different from inflatonthrough fluctuations, reheating can via source Higgs-modulated primordialhere inflaton density allows perturbations decay. a The larger curvaton enhancement scenario of we NG present signals compared to ref. [ tuning, we willscenario show will, that on suchmagnitude the tunings larger other and NG hand, enhancements comparedand have are to no the limited. such standardconclude tunings inflationary in The but section scenario. curvaton still gives In rise appendices to orders of both in the standard inflationary scenario,Three section types of NGbroken contribution charged scalars of (Higgs heavyand bosons), particles charged and Dirac — loop-level fermions tree-level will effectsthat effects be of arises considered. of unbroken in spontaneously We charged will the scalars mediated discuss inflationary NG an paradigm issue are and of unobservably see small. that While if enhanced we NG forego signals such can tuning, be the obtained loop- with potentially larger NG than the standard paradigm. some essential aspects ofcurvaton the scenario and cosmological note colliderThe the physics detailed current program. analysis set of of the We observational EFT will constraints couplings review on of it the the in heavy particles section will then be carried out, be discussed in section JHEP04(2020)077 ) ]. M 42 and 0 or ( – (3.2) (3.1) (2.5) (2.6) s πM/H . → − 3 40 M e 4 − / , ∼ ) 9 so that the ) } ) for which –10 − 3 4 χ 2 k M { − 2.3 ( s σ, 10 /H f µ 2 ∂ ∼ ( M of the type, NL int σ in eq. ( p c.c. f F i L σ F , + ) + 2 2 θ / H . In particular, heavy fields σ 1 3 2 k k ) depend on the mass m (cos θ . ) = 3 s 1 2 m 1 ζ P M ) ( − (cos 0 2 M s ( ) 21-cm s 1 . and is given by, ,P 2 ∆ . N ) ∂σ | σ ( ) 2 √ 1 2 3 1 M ( k k m M s 21-cm ∼ ( − 1 2 s N ) 2 , f i f – 6 – ) ], one has very roughly i ) | } 3 = being some constant. Furthermore, for simplicity, becomes analytically dependent on , as a pseudo Nambu-Goldstone boson (pNGB) 2 χ ζ 43 M ζ σ σ c h M [ { ( h F ( V s s , f φ, 16 ∼ µ 1 H with k ∂ 10 one has a very rough estimate of the precision by which ( NL c = f ∼ 3 becomes a function of only int φ σ + k NL belong to two separate sectors sequestered from each other L M σ f to give us on-shell mass and spin information. F σ → 21-cm H ) + N φ σ ∼ ( and inf φ non-analytic V F . The functions ∆ − 3 2 ˆ k ) · . In that case, 1 ∂φ , can mediate non-analytic momentum dependence of 2 . Thus the cosmological collider program operates most efficiently in a window ( ˆ k k . Furthermore, for 1 2 H H = ≈ H − ∼ 1 ), production of heavier particles are “Boltzmann suppressed” with θ , forms the basis of cosmological collider physics. In the following, we will quantify of the heavy particle and can be calculated given its coupling to the inflaton. The = k s M M H s P We will model the curvaton field, While the time dependent inflationary spacetime readily produces particles with masses We will be interested in the so-called “squeezed limit” of can be measured in an only-cosmic-variance limited 21-cm experiment. Such a precision L ( M 3 NL respect a shift symmetry we will assume that (say, by different locationsbetween in them. an Thus extra-dimensional our geometry) model and is ignore specified any by interaction the lagrangian, whose shift symmetry is broken (softly) by a mass term Any significant interaction term involving the curvaton and another field will need to 3.1 Cosmological history We will now briefly review the cosmologicalsome history of in the the curvaton paradigm important and differencesmore emphasize details between on it the and curvaton the paradigm, the standard reader inflationary is paradigm. referred to For the original papers [ longer apparent. For example, for3 a for scalar particle, ∆ of heavy masses around 3 Curvaton paradigm the strength of NG by the absolute value ∼ O for distinctive non-analytic, on-shell information characterizing heavy-particle mediation is no where spin prospect of extracting theand mass and spin of the such heavy fields via measuring ∆ k with f is controlled only by the number of modes Thus using the estimate Using the above definition of JHEP04(2020)077 so , is 2 (3.7) (3.8) (3.9) (3.3) (3.4) (3.5) (3.6) , the σ ) and 5 2 exit − ζ m t, ~x 1 2 10 ( defined by, = 0, δφ ζ are decoupled ψ ) φ σ ( ) + t ( inf that we will specify 0 V } φ and χ { φ , ) = ··· + t, ~x ) into a homogeneous and a ( j φ . However, significant curvature , . 6 dx t, ~x i − ( . = 0 2 pl dx 10 δρ . ) = 0 ) 0 . t 0 . M 2 0 0 σ ∼ ( 0 ˙ 0 ˙ 2 φ 2 ρ δρ φ ˙ 2 ( φ σ δφ captures the shift-symmetric interactions 0 ) + a ˙ H 2 m t 2 φ H ) 0 ) acts as a spectator field. inf 2 ( H H π H σ 0 2 m + V ( 3 − σ ρ int φ ··· ≈ 0 + ˙ ψ L ∼ σ – 7 – ≈ − 2 + 0 ˙ − ≈ − ˙ H H φ H ij 0 exit δ = ) = exit ˙ H ) σ ζ ζ + 3 ζ ψ ≡ − 0 ), t, ~x 2 + 3 ( ¨ σ ρ 0 − ¨ φ 3.22 = ((1 2 1, the curvaton rolls very slowly along its potential, satisfying ). The equations of motion (EOM) for ds in a perturbative manner so that they do not affect the free EOMs at the lead- t, ~x to a good approximation and thus, in a gauge in which ( ) σ ( δσ int φ δφ and is a spatial metric fluctuation appearing as, L 2 ψ ) + t H ) is the inflaton potential and . During inflation, the potential energy is dominated by ( φ 0 ( 5 σ 1 2 inf drives the inflationary expansion and V m ) = φ To describe the fluctuations, we will split both the inflaton and the curvaton We will treat 1 t, ~x ( characteristic size of the observed primordialset fluctuations. of For parameters example, given forperturbations in the can benchmark eq. get ( generated after the end of inflation sinceing there order. is a second light field sourced only by One of thethe important inflaton are features subdominant of to the those curvaton of paradigm the curvaton, is and that in the particular fluctuations of and we havefluctuation split part. the For brevity,tensor density we fluctuations. have not Since explicitly thecurvature written inflaton perturbation dominates the when the other energy scalar, the density vector relevant during and momentum inflation, modes the exit the horizon, In the above, Since Curvature fluctuations can be characterized by the gauge invariant quantity Assuming that the kinetic energywe of get the the inflaton standard is relation, much bigger than that of the curvaton, σ (neglecting gravitational backreaction)given and by, in particular, the homogeneous EOMs are of the inflaton (curvaton) with ain collection section of the other heavythat fields fields into homogeneous and fluctuating components: where JHEP04(2020)077 2 (3.11) (3.12) (3.13) (3.10) matter − σ remains conserved on super- ζ , ) = 0 the curvature perturbation σ f ψ − , ∗ | ∗ ) dilutes in an identical way so as to | 0 (1 0 σ δσ , and matter due to the curvaton energy σ δσ , t, ~x rad ( rad 2 3 σ σ rad ρ = 2 δρ ρ ρ δσ δρ = 1 4 0 are respectively fluctuations corresponding to 1 3 σ σ δσ = 0 gauge), and they are conserved on super- ζ – 8 – + ≈ and rad starts oscillating around its minimum and dilutes rad σ keeps rolling very slowly along its potential until ψ ≈ ρ ζ 0 = 2 f δρ σ σ σ σ σ σ σ ρ final ρ δρ δρ ζ and 1 3 σ ) and using the gauge σ = ρ δρ need not necessarily be conserved on superhorizon scales. ζ 3.7 . ζ to the quantum fluctuations of the curvaton field which will which however is far subdominant in the curvaton scenario. σ rad σ ρ ρ δρ exit ζ , following which is related to the energy density in the curvaton field compared to the = m rad ]. . In the meantime, ρ . rad σ rad σ ρ ρ δρ 45 +4 3 , σ 1 4 ρ 44 3 ], . Thus using eq. ( = σ denotes the fact that fluctuations are evaluated at the epoch of horizon exit. This 46 ρ σ , ∗ f 1 when we can write, 41 We will now relate We assume that at the end of inflation, the inflaton reheats into a radiation bath largely This is to be contrasted with single-field inflation where quite generally ≈ 2 during inflation and thus σ which relates the final curvaturecurvaton perturbation field. in It terms is of in the this quantum limit fluctuations that ofhorizon the the scales adiabatic [ curvaton fluctuations can be identified give [ where then finally gives, adiabatic ones. later help us tocurvature write perturbation expressions is for negligibleonly NG at potential of the for primordial end the density of curvaton, perturbations. inflation both and Since we the are assuming a mass- which remains conserved on superhorizonfluids scales during subsequently. the We later assumematter stage all of all the evolution originate relevant i.e. fromisocurvature the the SM fluctuations photon, decay at neutrinos, of aand baryons the radiation later and has curvaton. dark stage. converted the This The initial way differential isocurvature we fluctuations evolution do in between the not curvaton generate field any into we will have Now, importantly since radiationtime dilutes between the faster start than off matter, curvaton oscillation assuming and there its decay, is we sufficient will reach a stage at which where radiation energy density the curvaton and thehorizon radiation scales (in since the thewhich two is fluids weak do on not these interact scales. with Since each the other, radiation other bath than originates via from gravity the inflaton decay, from the inflaton decay, having energydensity density after inflaton reheating can be written as, Let us now see how this happens. decoupled from the Hubble scale like matter. At such a point the content of the universe comprises of radiation coming σ JHEP04(2020)077 (3.20) (3.16) (3.17) (3.18) (3.19) (3.14) (3.15) is still , the tilt 2 pl 2 0 2 is a purely σ M 2 0 and simply ˙ 2 2 H φ -mode. The g π k H ζ 1 9 2 final ζ = ≈ 0 i ) 2 ˙ H k ~ H , where , 2 g − ( 3 ζ ζ for a given k 2 ) 2 0 loc NL ≡ − k ~ H σ f ( ] requires, 2 3 5 aH ζ 2 h 4 9 2 + = 3 π ] then implies, k g = 2 k ζ 0 31 . i , . ], = ) 2 σ = . k ~ η . ζ 04 49 ζ . 4 2 0 2 3 . − – 2 0 pl is a light spectator field during inflation, − σ ( σ pl δσ + 47 σ M 10 18 δσ M ] requires ) ≈ − 2 2 1 3 × k ~ = 06 σ ( . − 4 η . 2 2 0 − 2 0 2 pl – 9 – 4 2 3 δσ = σ H H Defining ∆ h 9 M < 4 8 σ ζ 2 δσ 0 ≈ + 0 k , is given by, 4 σ 2 3 0 σ = r 9 2 H ln σ r ln ∆ = d = − d ζ 0 i ) k ~ − Due to the fact that ( The ratio of the power spectrum of tensor fluctuations to that ζ ) k ~ 06 from Planck data [ ( . ζ 0 A very stringent constraint on the curvaton paradigm comes from the ], the inflaton fluctuations having become completely subdominant. h acquire an approximately scale invariant spectrum. Thus the scalar 37 r < is fixed by the mass of the curvaton and δσ 2 2 m H = σ η to denote the primordial density perturbations which act as “initial” conditions for ζ Gaussian field. Here theis NG non-Gaussian. arises In due theuniverse to during scenario the before when fact its the that decay, curvaton square one dominates can of the derive a [ energy Gaussian density fluctuation of the The upper bound Non-Gaussianity. upper bound on the “local” type of NG, defined as Tensor-to-scalar ratio. of the scalar fluctuations, denoted by where determined by the homogeneous inflaton field. Planck data [ Tilt of the scalarcan power be spectrum. derived as, where the r.h.s. isamplitude evaluated of at scalar the power time spectrum of from horizon Planck exit data [ 3.2 Observational constraints Scalar power spectrum. its fluctuations power spectrum is given by, theory of inflation [ Unless otherwise mentioned, in the following,use we will omit the subscriptthe in modes that subsequently re-enterto the note horizon the after present inflation. observational We constraints are on now this in paradigm. a position as the Goldstone mode for spontaneous time translation breaking in the Goldstone effective JHEP04(2020)077 σ is a ) also (3.21) (3.22) δσ 3.21 , ]. We expect 3 − 35 ) lies below the – is a tell-tale sign in eq. ( 33 3.21 loc = 10 NL loc NL f σ f η . σ in eq. ( 02; . ). As will be explained below, and loc NL f φ ). Multifield inflationary models = 0 3.22 2.6 ; which is broken only by its potential. pl c . M 5 4 is parametrically larger than the slow-roll + 6 can not give such non-analytic momentum − − φ differs by an overall sign. loc NL 10 = H f → NL – 10 – × 3 f loc NL φ 2 f . = 2 H ]. is parametrically suppressed in the latter. ; 32 pl in single-field inflationary models, as dictated by single-field ] in the squeezed limit. Thereby in this curvaton scenario, even loc NL M ] our definition of f 3 51 loc NL 47 , − f 50 . Although the robust prediction of 10 4 − × can be much smaller than the kinetic and the potential energy stored in 10 σ , and the case of a Dirac fermion to both = 5 σ × 0 5 . σ in light of non-renormalizabality and shift symmetry of the couplings. We will and = 4 is a light field, we can model it as a pNGB in the low energy EFT, just like the φ φ r φ The fact that in the curvaton paradigm the background inflationary expansion and the While it is true that the signal in the squeezed limit is dominated by the curvaton itself, The above constraints can easily be satisfied. For example, one can choose a benchmark Note that compared to [ 3 , as can be checked by using the benchmark point in eq. ( 4 Charged heavy particlesSince in the standard inflationarycurvaton, paradigm and impose a shift symmetry this feature can allow significantlythan stronger to coupling of someillustrate new this degrees by of considering the freedomto coupling to both of a charged scalar, with and without Higgsing, (eventual) primordial density perturbationsan are interesting sourced possibility. by In particular, twoenergy during different stored inflation in fields, both opens theφ up kinetic and the potential “oscillations” in the squeezed limithaving as additional explained in particles eq. with ( dependence. masses The observabilityliterature of in such the oscillatory contextthat of signals with single-field have some inflation, been adaptations for thedetailed investigated above example, investigation in studies lies in the continue beyond refs. to the [ be scope applicable of in the our present case, paper. but a to get Planck upper bound on NG,upcoming quite LSS excitingly, observations such [ a strength of NG will soonthe be distinctive tested by signaures of the heavy fields are imprinted in characteristic non-analytic set of values: consistency relations [ in the absence ofof heavy fields beyond that single-field will inflationaryserves be as considered dynamics. a below, crucial a The differencesingle-field large between above inflation the since value curvaton of paradigm and Goldstone description of Gaussian field, and in particular we have, It should be notedparameter that the suppressed above value of This has precisely the same form as the local type of NG defined above since JHEP04(2020)077 2 φ 2 0 ˙ φ Λ and (4.4) (4.5) (4.1) (4.2) (4.3) = χ α m where χ † αχ ⊃ − are respectively the VEV , χ 2 † ) χ χ χ 2 † ) ) in mind while considering the χ ( and ˜ ∂φ . χ ( , charged under some gauge/global 1.2 v λ )) in the unitary gauge, to read off 2 φ 2 χ 1 loop suppression, the associated NG Λ χ − 2 χ. H. H. + ˜ α. . π χ † 1 † v χ 16 60 eff + ( χ 2 250 ) 2 χ, ) 2 2 χ since, ∼ 1 ∼ α √ > m 0 m χ ∂φ , ˙ ( + φ 1 4 then reads as, ] and the summary is that one can have three inf = 2 φ 2 χ – 11 – 1 q χ Λ 18 is the EFT cutoff in the inflationary sector. As = (0 eff m 0, leading to a Higgsing of the symmetry. Due to ( > V > φ 2 χ, χ < φ φ − m L ⊃ Λ Λ 2 eff | 2 χ, ∂χ m is given by, where Λ can be placed if we demand that the EFT explicitly describes χ 2 ) 4 φ φ Λ ∂φ L ⊃ −| ( , the lagrangian for ), χ ), one can expand λ 1.2 4.3 The details have been discussed in [ The leading coupling of the inflaton to a scalar field coupling in eq. ( the vertices necessary forand tree fluctuations level of NG. the In Higgs field. the above, types of diagrams giving rise to NG as shown in figure We first discuss the casespontaneous symmetry when breaking, one canin now the have inflationary heavy couplings field.and which are since Consequently, such one linear processes can do haveare not tree more have readily the level observable. usual processes We that consider a can U(1) mediate symmetry NG group for simplicity. Given the Two scenarios arise depending on the sign of 4.1 Higgs exchange in the broken phase a quartic coupling where the effective mass for This term will also contribute tois the approximately mass constant of in slow-roll inflation. In the presence of a “bare” mass In the following we willstrengths keep of only NG. the restriction in eq. ( symmetry group, is given by a dimension-6 operator the scalar/gravity dynamics of inflationto and reheating. an inflaton All known potential descriptionsan as of EFT the this requires source refer ( of inflationary expansion. Thus the control of such derivative expansion in discussed in the introduction, theoretical control of such an expansion implies, A stronger restriction on Λ This implies that the interaction of the inflaton will be characterized predominantly by a JHEP04(2020)077 , 2 χ χ )), H m 2 and (4.7) (4.8) (4.6) ∼ φ /H 2 1 in a few ρ ( 2 O , H 4 α 2 1 ∼ H ρ (1 + 4 2 0 χ, ˙ φ † H tuning between | ∼ χ and consequently fine ∼ ! 2 k α P )) 3 . ), corresponding to each of k triple, tree δφ classical χ ( χ, 2.6 ∂ ( f | 2 0 and ˜ ˙ α , also ensuring a controlled EFT φ 0 . φ ˙ . φ + 2 φ 2 0 1 ˙ . φ ; Λ q 0 ˙ 4 α δφ 2 1 = . 0 > α H ρ ˙ . With the symmetry being unbroken the away from its tuned value | φ 2 . Since a quadratic mixing between α defined eq. ( χ 1 4 2 are given by, inf † eff , in which case all three diagrams will give , a larger strength of NG can be obtained. − f χ | ∼ 2 2 χ 2 χ, 2 /H α ) 1 2 H H > V m – 12 – − ) and ρ ∂φ

∼ ∼ ( 4.5 natural, tree 2 φ | ∼ 2 χ 1 = Λ eff χ, m double, tree χ f 2 χ, | † χ, χ m ∼ f 2 implies Λ | ) α 2 ∂φ natural, tree H ( 0 so that there is no spontaneous symmetry breaking and 1 for perturbativity of such corrections. Then we see that NG . 2 φ ; χ, 1 0 ∼ 2 f Λ 2 1 > | ρ α . H to obtain ⊃ eff 2 will generically push ], ) then implies that in the absence of any 2 χ χ denotes conformal time which ends at the end of inflation. χ, 2 mass given by eq. ( 2 2 0 | ∼ − m 18 ˙ ) it is clear that by choosing a larger value of η /H φ φ Λ m 4.5 χ 2 1 denotes the quadratic mixing between L ρ 4.6 = 0 ˙ αv α φ 2 single, tree = . Massive Higgs mediated (in red) tree level “in-in” contributions to the inflaton (in black) χ, 1 we need to have both f ρ | α From eq. ( For observable strengths of NG, we need the masses of the heavy particles The rough strengths of NG, in terms of also gives rise to a correction to the scalar power spectrum ˜ relevant couplings between the inflaton and with an effective 4.2 Charged scalarHere exchange we in the assume symmetricmediated phase NG appear only viathe loop same operator diagrams. as The above, coupling namely to the inflaton is described by tuning it against However, one can notevolution do of this tuning toHubble more times. than a percent level since the (slow) time contributions are given by, The natural choice of description. suppressed. Eq. ( and similar strength of NG, χ we will require where otherwise the cosmological production of the massive particle will be severely Boltzmann labelled from left toexchange diagram. right: (a) single exchange diagram, (b) double exchange diagram, (c)the triple diagrams are [ Figure 2 three point function. Depending on the number of massive scalar propagators, these diagrams are JHEP04(2020)077 Ψ 5 γ µ (4.9) , the φ γ (4.10) λ Λ ¯ for the Ψ φ 4 µ ∂ − is somewhat 10 Ψ × . m 2 0 ˙ φ few 3 2 α . H 2 F π 1 16 , 9 . The associated NG can be − | ∼ 3 10 Ψ can be eliminated by integration ], we find ∼ µ γ 29 4 2 0 [ ¯ ˙ Ψ φ H triple, loop φ λ H 2 µ χ, π ∂ 1 f (1) “contamination” from the inflationary < | φ 1 16 O Λ ), there are two loop diagrams that can con- 3 1 . Such a strength of NG is unobservably small. k k 7 , such a contamination is small but the strength via NG would not have given us the underlying – 13 – 4.8 | ∼ − 2 ; ), as 2 2 0 eff H 10 ˙ φ α 2 χ, , and demand theoretical control of the calculation 2.6 ∼ 2 receives m H π 4 1 2 0 α ˙ φ eff H 15 16 for the fermion Ψ once the inflaton is set to its background 2 χ, natural, loop φ . m χ, | ∼ Λ . For f , λ / ). We will see in the next section that the curvaton scenario | 2 χ 2 0 ˙ φ m H defined eq. ( 2.3 = f ) gives, ∼ ) i.e. λ α 4.9 double, loop 1.2 χ, f without paying significant Boltzmann suppression. However, if we impose ]. It can help production of Ψ even when the fermion mass, | , eq. ( H 30 2 , denotes conformal time which ends at the end of inflation. H 29 η , . Massive charged particle mediated (in red) loop level “in-in” contributions to the inflaton ¯ ΨΨ. A dimension-5 operator of type ∼ 2 ) 20 α There could exist another dimension-5 operator involving the axial current, Based on the couplings given in eq. ( ∂φ ( 3 φ 1 Λ the restriction in ( of NG in thefunction squeezed defined limit, in which eq.will forces ( allow a larger strengthabove. of NG Furthermore, with in just the the analog regime of where the dimension-7 there operator is defined a substantial NG signal due to if it is not forbidden“chemical potential” by parity. Such avalue [ coupling is special sinceheavier it than gives rise to an effective 4.3 Charged DiracWe fermion will consider a charged Dirac fermion coupled to the inflaton via aby dimension-7 parts operator and current conservation if the Ψ couplings respect a U(1) symmetry. where we have used the fact that Note already with background and a measurementvalue of of the “pure” mass of NG becomes even smaller. The above discussion of thewhen “classical” tuning also applies here and in the natural case i.e. tribute to a threeestimated, in point terms function of which we list in figure (in black) three point function.these Depending diagrams on are the labelled numberdiagram. of from massive left charged particle to propagators, right: (a) double exchange diagram, (b) triple exchange Figure 3 JHEP04(2020)077 φ f , so (5.1) Ψ (4.14) (4.15) (4.11) (4.12) (4.13) m from the eff , Ψ . Because of Ψ . m m 2 0 H ˙ 3 further. φ , β Ψ 2 5 ¯ ΨΨ π γ , not belong to two different 1 µ λ φ ! 16 γ σ Λ 2 ¯ need not even be relevant Ψ φ )) 4 µ | ∼ / ∂ and δφ 1 inf ( φ V ∂ ( . 2 0 ˙ 9 β φ and − triple, loop + 2 , β. 10 / Ψ 1 0 ˙ sources the observed fluctuations and ˙ + . , f δφ φ that drives the inflationary expansion and | 0 1 4 | ∼ 0 σ ˙ σ β Ψ φ ˙ φ φ 2 m q − > V ; = – 14 – σ β 2 eff Λ β < 2 0 − , H ˙ φ 2 Ψ such NG are again unobservably small with

natural, loop β , m 2 H Ψ π f 1 | ∼ 16 ), ¯ ΨΨ = β 2 , where the internal lines now represent Ψ, in terms of ) ¯ 1.3 ΨΨ, the VEV of the inflaton will give a contribution to the 3 and heavy fields. The only relation that is relevant for control . Furthermore, even if we choose to tune, the requirement of | ∼ 2 ) ∂φ σ H ( 3 φ ∂φ 1 ∼ ( Λ φ 3 1 β Λ double, loop inversely bound the strength of EFT couplings, making the associated L ⊃ , 4 Ψ / ), f 1 | inf . Thus the effective fermion mass becomes, V 3 φ 2 0 2.6 ˙ φ Λ already implies, 0 = ˙ φ and β 2 q / 1 0 Having discussed the case with inflationary couplings, let us see how things change In the coupling ˙ > φ φ sectors, sequestered from eachfor other, the the couplings between scales of the curvaton EFT is, eq. ( In the inflationary scenario, italso is the sources same the field observedas primordial fluctuations. ThatNG is small. why the However, in verydrives high the the inflationary scales curvaton expansion. scenario, such In particular, assuming when the heavy charged scalarinflaton and as fermion the fields dominant are source coupled of to fluctuations. a curvaton instead5 of the Charged heavy particles in the curvaton paradigm We see in the natural case i.e. As in the case ofinflationary the background charged to scalar, be if small, we the demand NG the becomes “contamination” even to smaller. loop process givendefined in eq. figure ( Naturalness requires Λ Focusing on the natural case we can estimate the strength of NG for a fermion-mediated where that the non-analytic signaturesthese reasons are we predominantly will sensitive not consider to the dimension-5 operator fermion mass as for charged scalars, and we can write the relevant couplings as, inflationary background also significantly contaminates the “pure” fermion mass JHEP04(2020)077 ) (5.4) (5.5) (5.2) (5.3) 3.15 tunings so With the above . This is analogous classical 1 4 Given the derivative σ . . However, even with remains small for the can be we write, H χ † σ ··· > V 10 ), χ V σ + 2 2 σ 0 ˙ & is actually smaller than the σ 2 Λ 1.2 , ) 4 σ H δσ H ). Thus one can ask whether it . i 10 ⊃ − σ ∂ η H ( χ ∼ 0 4 σ † 3.15 ˙ × σ 4 Λ δσ χ 0 / 6 2 2 ˙ 1 is more than sufficient to avoid current σ ) ) σ 2 σ t 10 1 V ( Λ ∂σ H ( a curvaton self int. reduces to Λ × 4 | ∼ & f , 4 σ 0 5 4 1 ˙ σ . σ & χ σ Λ . This will contribute to both bispectrum and 2 Λ 4 . We then see that for the benchmark choice of 4 σ ) 2 2 – 15 – ≈ > V − m H 2 ∂σ 0 3 , the trispectrum is also smaller than the current σ to satisfy eq. ( ( σ . As an example, the contribution of the curvaton σ H ˙ = 4 σ 2 1 2 σ σ δσ Λ σ m Λ 4 σ 0 Λ η ˙ 250 1 2 σ Λ curvaton, self-int. < allows us to have using eq. ( f & | = plays a central role in giving significantly larger NG in the 2 0 3 φ = 4 , a choice which will be justified below, and using eq. ( σ 4 ), σ ) and H Λ / 4 V 1 H ) and correspondingly there is no “contamination” to the “pure” σ 10. This is below the current upper bound on orthogonal and 4 2.6 . V σ ∂σ 3.15 ∼ 0 ∼ ( & σ 4 σ σ 1 σ Λ Λ Λ , and Λ , the requirement Λ 2 3 0 − σ H . φ 10 χ ∼ defined eq. ( ∼ 0 m f σ discussed in section , we do still need σ ]. is the energy density in the curvaton. To see how big η curvaton, self- int. 3 H f σ 31 − V 10 10 ∼ ∼ σ σ both the derivative expansion remainsto under the full problem of control super-Planckian andby inflaton Λ the field Lyth range bound. ineffective high Thus, field scale one ranges, inflation models, can suchnisms given borrow as illustrate the axion that mechanisms monodromy having that or field are multi-axion ranges used alignment. larger to than These create the mecha- large EFT cutoff can naturally emerge bound [ Cut-off of the effectivechoice theory of and field rangeΛ of the curvaton. is problematic to have curvaton field range much bigger than the EFT cut-off, although we get, equilateral type ofDoing NG a which more is carefulabove the crude calculation kind estimate, shows so of that thatconstraints. NG a choice induced With of by such Λ the a choice above of self-interaction. Λ in terms of Thus with ˙ expansion, we can also writetrispectrum. a term For bispectrum, the relevant couplings are, From such a coupling we can do a naive estimate of curvaton self-interaction mediated NG, to the mass termabove of choices the of charged ˙ scalarscalar field mass Constraints from curvaton self-interaction mediated NG. This relation Λ curvaton paradigm while also ensuringstandard theoretical inflationary control paradigm, of in thelong the set-up. curvaton as scenario Compared we there to make are the sure no ˙ where we have used eq.η ( where JHEP04(2020)077 2 ], f 57 still .A & – (5.8) (5.9) (5.6) (5.7) σ (5.10) σ Λ H 52 and we Λ 10 H > H < 10 2 4 σ ∼ ≈ Nf . eff σ, 1 1 that is parametrically . f σ 2 2 2 f 2 2 V Nf N cos ∼ = − 2 . 1 , one gets Λ m H 2 χ. 2 V † 2 and . χ ¯ σ + χ, ΨΨ (5.11) σ Nf σ † = 10 Σ Σ Λ χ Σ Σ y σ = 2 2 Σ µ The curvaton will couple to heavy particles µ ) can be brought down from 2 2 2 M cos σ f σ σ ∂σ Σ + and we can consistently have ( 2 − = Σ + + ) σ and Λ 2 – 16 – 2 1 ) f 1 Λ eff Σ 1 ∂σ 2 Σ H 2 σ, µ as in the inflationary scenario. As we have discussed ( f ) with 2 ∂σ Nσ ( Λ σ V M φ 1 1 σ Λ Λ 3.1 = 6 1, there is a heavier curvaton which gets stabilized at 1 = Λ has an effective field space Σ 2 eff cos σ V − N H, µ 1 1 = 3 , so that we can have stronger couplings leading to significant NG. and . This UV completion then serves as a proof-of-principle that it is 2 V 2 Σ N f = Nf M ) around its minima gives rise to the approximately mass-only potential ∼ : can still interact with heavy fields of interest with suppressions given by χ curv 1 2 5.7 f V σ , 2 0 so that the effective light curvaton potential is given by, V is consistent with EFT control. However, the restriction of Λ ≈ ∼ φ rather than 2 2 1 Λ f σ 2 V f , instead of multiple powers of + For example, one can have the following coupling between the curvaton, the mediator σ ∼ 1 1 Λ 1 σ σ f Nσ As an example, for similar procedure can be repeated for fermionic couplings by starting with, which implies an effective cutoff, Σ and the scalar Upon integrating out the mediator Σ, we can get an effective dimension-6 operator, above, the strength ofΛ NG in the curvatoncorresponds scenario to can somewhat suppressed therefore NG. be“mediator” We much will particles, now stronger the show since that effective incan scale the obtain of presence even of Λ stronger heavier NG. Expanding eq. ( for the curvaton considered in eq. ( Mediators and stronger EFT couplings. via higher dimensional operators andof hence the NG will be proportional to multiple powers bigger than the fundamentalfor field a space sufficiently large consistent to have field ranges thatlight are curvaton significantly bigger than theΛ EFT cutoff scales. The We see that the light curvaton one can imagine having two axionic curvatons with potential, With from controlled UV completions. In particular, following the bi-Axion mechanism [ JHEP04(2020)077 . eff ! σ, µ ··· . In (5.15) (5.16) (5.12) (5.13) (5.14) + H 4 → − 3 µ . ˜ χ & σ v + χ eff iµ , compared to λ σ, + 2 which gives the , we can get the 3 2 − H and Λ but for numerical | 4 ˜ χ 3 1 Σ 2 where the external ∼ eff ) k k tree 0 σ, 2 M χ, σ f ∂δσ | )) ( 2 σ πµ v ] using which we get, any classical tuning. Λ 10 + . Since ˙ v 2 sinh( 2 σ 0 )) as in section i ˜ , χ σ Λ v 0 2 2 2 ˙ ), we can have orders of magnitude ˙ ˙ ρ δσ σ without χ. + † (1+ 2 σ 0 = . 4.6 . χ ˙ × χ σ χ Λ σ (˜ 2 2 3 2 ) ρ Λ m 1 iµ − 0 √ 2 Σ 10 i , ∂σ ˜ + χ ) ( for the same choice of M 3 × 5 2 ˙ 2 σ k ~ δσ σ , 3 1 = Λ 0 v − we plot the function = (0 – 17 – , can be found in [ i ! 2 σ 0 ( ) eff ζ µ σ χ ⊃ 4 Λ | ∼ 2 3 ) iµ 3 σ, 2 ˙ k ~ 3 χ ( Λ k + ~ − − tree ζ ( φ → − 3 2 ) ζ χ, 2 h L µ 0 f k ~ | ⊃ i ) ( + ) . For brevity, in the rest of the paper we will denote Λ ζ 2 1 , ) iµ ) iµ k ~ 1 H 2 σ χ + k ~ † − 3 2 − ( 4. In figure ( χ ) as to ζ / ζ = 4 ( h Γ( 9 ) 3 1 χ χ 1 2 2.6 k k λ eff − k ~ ( σ, 2 − ζ ) iµ h χ µ ( † + /H 1 = χ 2 χ 1 2 iδ 2 in the presence of somewhat heavy mediators, gives us Λ e ) m 3 1

1 4 Γ k k = 1), σ | q ∂σ , ) (

defined eq. ( y χ . µ 2 σ 2 2 = ( 1 σ f m Λ H > V ρ 0 0 µ tree σ σ σ ⊃ 8 ˙ 3 χ, While the above is a rough estimate, the necessary ingredients for a precise calculation To summarize, demanding theoretical control of the curvaton derivative expansion χ f tree − − χ, σ ≡ | = F where strength of NG as a function of the scalar mass, where we have denotedthe the result quadratic in mixing the as inflationarylarger paradigm NG given in in the eq. curvaton ( scenario, and importantly, of the single exchange diagram in figure comes from the singleterms exchange of diagram for which one can roughly estimate the NG, in L The above couplings givelegs NG represent mediated fluctuations of by the diagrams curvaton given instead of in the figure inflaton. The leading contribution Expanding around the correctrelevant vacuum, terms coupling ˜ results we will take Λ by just Λ 5.1 Higgs exchange inThe coupling the of broken the phase curvaton with the charged scalar is given by the dimension-6 operator, This effective cutoff is again smaller than Λ and Λ the following, we will give parametric estimates of NG in terms of Λ to get (for JHEP04(2020)077 ), ! for µ ), χ 4.10 (5.20) (5.19) (5.18) (5.17) m whose without → − 3 A.10 µ + 3 1 k k iµ, �� interaction terms are ). 3+2 χ 5.16 �� F ) , ), due to the double exchange iµ 2 ! χ. ) 2.3 . µ † 0 χ iµ �� H σ → − 2 × 2+ . The result is given by eq. ( / χ. )Γ(3+2 µ 2 )) † ), is given by, A 4 σ + �� iµ χ /� H δσ 2 2 Γ(3 0 0 Λ χ ( iµ is defined in eq. ( ) 2.6 i 2 ˙ σ ∂ ) ) − | � ( ) 3 2 2 ∂σ 3+2 µ k iµ 2 σ ~ π / ( ( 1 1 �� 3 0 Λ − − 2 σ – 18 – i 16 1 3 1 ( − ) Λ tree k k ζ Γ( + 3 ) χ, 5 k Γ( ~ 3 f ˙ π ( | ∼ 2 1 ) | k ~ δσ / ζ �� ( µ 16 5 L ⊃ 1 0 ) ( 2 σ ζ defined eq. ( π 2 2 σ h loop 4 0 Λ k ~ iδ 2 ˙ i f χ, ( e )

f ζ − 1 | )

�� k ~ 1 | H ) 4 σ k ~ 0 − ( µ ( Λ σ ( ζ ζ 2 ˙ h ) � 0 1 �� . The function loop �� k 2 σ ~ 2 (

χ, ��

ζ H �� χ f � | | h − will be calculated in appendix − = = ≡ | 3 , compared to the result in the inflationary paradigm given in eq. ( = 0 0 ˙ ) we can estimate the strengths of NG. The relevant diagrams are still given φ 3 1 σ k k q , χ 5.18 except the external legs now represent curvaton fluctuations instead of inflaton and ˙ . The strength of NG for tree level Higgs exchange as a function of Higgs mass m 3 σ H 3 . The dimensionless three point function, defined in eq. ( loop = 0 χ, 2 F any classical tuning. diagram in figure Since Λ we can have orders of magnitude bigger NG in the curvaton scenario, and again, Using eq. ( by figure fluctuations. The leading NGparametric is strength, given in by terms of the double exchange diagram in figure In the absencegiven by, of symmetry breaking, the relevant curvaton- Figure 4 ρ 5.2 Charged scalarThe exchange leading in coupling the is symmetric again phase given by a similar dimension-6 operator, JHEP04(2020)077 ), 3 for χ 4.14 (5.21) (5.22) (5.23) m without and Ψ if it is σ �� we plot the function 5 ). between Ψ 5.20 5 �� γ µ . σ γ . Λ ¯ Ψ ∗ ), is given by, | ¯ σ ΨΨ 0 4. In figure µ �� ∂ / H σ 2.6 9 2 × )) − . 4 2 /� δσ 6 σ χ ( H ¯ . �� ΨΨ is defined in eq. ( 0 Λ ∂ /H χ 2 � ˙ ( | σ ) 2 χ ) 3 σ m 2 µ 1 m ( Λ π ∂σ 1 – 19 – defined eq. ( ( q 16 3 σ + �� f loop 1 Λ = ˙ χ, ∼ δσ f | µ 0 3 σ σ Λ loop 2 ˙ , �� Ψ − f ) and A.4 �� . The function

�� 2

, except the external legs now represent curvaton fluctuations instead �� χ � | | H 3 − , compared to the result in the inflationary paradigm given in eq. ( 0 = ˙ ) we can estimate the strengths of NG for which the relevant diagrams are φ 0 σ q as a function of the scalar mass, 5.22 | is defined in eq. ( ) and ˙ . The strength of NG for loop level scalar exchange as a function of scalar mass µ σ ( F H Through a reasoning identical to the case of scalars, one can check that the mass loop = 4 χ, σ f Since Λ once again we canany have classical orders tuning. of magnitude bigger NG in the curvaton scenario Using eq. ( still given by figure of inflaton fluctuations. The leadingwhose NG is parametric given strength, by the in double terms exchange diagram of in figure correction to the fermionparameters due as to above. the The curvaton coupling relevant curvaton-fermion is interaction negligible terms with are similar given choice by, of As before, there cannot be forbidden a by parity. dimension-5 However, axialical such coupling a potential coupling as does in notenhancement the give of rise inflationary NG to scenario, signals. afocusing and significant only chem- correspondingly We on leave there the a is dimension-7 detailed no operator study substantial in of the present this work. coupling for future work, 5.3 Charged DiracWe fermion again consider a chargedsimilar Dirac dimension-7 fermion, operator, and its coupling to the curvaton is given by a Figure 5 Λ where | JHEP04(2020)077 |   H ˜ µ loop , 250 ), (5.24) Ψ . Such f → − | > different inf B.5 µ 1 4 V inf ). + ˜ V , the heavy φ is the energy & Λ 3 1 5.24 φ σ k k V ˜ µ, for the benchmark i σ 4 ) �� Λ H 4+2 . , where (10 1 4 1 4 F σ σ ) �� V V ∼ , ), due to the double exchange ˜ µ we plot the function i is defined in eq. ( ! & 2 6 2.3 | ˜ µ ) ) σ ˜ µ µ i (˜ �� ) → − ˜ )Γ(4+2 µ . The result is given by eq. ( , which has to satisfy Λ loop Γ(2+ i µ , ˜ µ 2 φ ) i Ψ B + ˜ ˜ /� µ f 2 i | ˜ µ Ψ 0 (1+2 . In figure i �� − − i 2 ) 2 � / 3 / /H 4+2 k ~ 5 Ψ Γ(1 0 − − – 20 – 3 1 i 5 ( m 3 ) k k �� π ζ 3 Γ( ) = 2 k − ~ 3 / ( ) k ~ 5 1 µ ζ µ ( π (˜ ) ζ 4 3 . The function 2 . h 2 0 iδ k �� ~ Ψ i   ( e H ) ζ 3 m 1

) ) and ˜ − | k ~ 1 6 σ H ) k ~ = 0 − µ Λ ( ( A.4 σ (˜ 0 ζ �� ζ 2 ˙ σ h ) 0 1 loop k 2 σ , ~

(

3 ��

Ψ and ˙

ζ �� Ψ � | | f h − will be calculated in appendix H leads to small or even unobservable strengths of NG in various otherwise = = ≡ | 3 φ = 4 3 1 σ k k , Ψ is defined in eq. ( for Λ . The strength of NG for loop level charged fermion exchange as a function of fermion m Ψ F m We have studied a non-standard inflationary paradigm in which the space-time expan- The dimensionless three point function, defined in eq. ( loop , Ψ F fields. In particular, weare have played focused by on the the inflaton curvatonto and scenario, the the where curvaton curvaton the respectively. suppressed abovedensity Heavy two only contained particles roles in by can the inverse curvaton then field powers couple whichparameter of can choice be Λ as considered low in as this work. With the choice if we demand that the EFTa explicitly high describes value the of scalar Λ fieldwell-motivated source scenarios of of inflation cosmological collider physics. sion and the predominant production of primordial fluctuations are due to two 6 Conclusions and futureIn directions the standard inflationary scenario,by the (at inflaton-heavy least) inverse particle powers of couplings an are EFT suppressed cutoff scale Λ where as a function of fermion mass, Figure 6 mass diagram in figure JHEP04(2020)077 2 σ H (A.2) (A.1) could ∼ 0 O σ ], bosons, in the , , where 10 W O 3 1 2 k k ]. This was done by ) , are not essential, and 10 ∆ ∂φ 0 ( σ g . F ∆ | 0 (∆) ∆ ~x 0 ). Considering trispectra (four- c free − ηη c . This will then allow smaller Λ ~x 3.15 | σ ∆ V X 3 3 ∆ c k 1 3 1 ) and ( k ∆ X 0 ˙ 3.6 φ 2 in the “in-in” diagrams. Demanding ˙ = 2 – 21 – g i 0 ), ( ) ]. σ 0 − 23 5.2 , ~x = 0 0 η i ( ) 3 O ) k ~ ( and the scaling dimensions ∆ which appear in the late-time δφ η, ~x via eqs. ( ∆ ) ( c 2 4 k ~ ) hO ( H δφ ) 1 (10 k ~ ], especially the signal due to the loops of massive ( ] along the lines of [ 0) of the position space two point correlation function: ∼ 60 δφ σ → 18 h – , V 0 58 14 – η, η ’s for the set of ∆’s, the three point function can be written as [ 12 ∆ c We have restricted our attention to bispectra of primordial scalar fluctuations which Several future directions remain open. We have considered a scenario in which the Given be either an elementary orevaluating a the composite coefficients operator, waslimit calculated (i.e. in [ Institute for Theoretical Physics,Energy UC and Electroweak Santa Scales” workshop, Barbara, and the during support the by the “Origin NSF grant of PHY-174958. A the Vacuum Charged scalar loop The three point function induced by an interaction of the type Acknowledgments The authors would likesupported to in part thank by the Anson NSF grantsCenter Hook PHY-1620074 for and for PHY-1914731, Fundamental helpful and Physics by discussions. the (MCFP). Maryland This RS research acknowledges was the hospitality of the Kavli consequently one can haveand even larger smaller NG. valuessignals of It [ would alsocurvaton scenario be that very was interesting presented in to this analyze work. all the “heavy-lifted” SM unification [ required us to usethen a implied non-zero valuepoint of correlation ˙ functions) on the other hand, such insertions of ˙ curvaton and the inflatonwhich belong can to naturally twothe be decoupled two achieved, sectors fields for (up are(KK) example, to localized gravitons by gravitational is on a effects), having two robustesting an different feature to branes. extra of see such dimension whether Since anscenario. appreciable in extra-dimensional the KK-graviton In which set-up, presence mediated particular, it of NG this would Kaluza-Klein can be may be inter- be applicable generated to using the our strongly motivated case of orbifold of magnitude larger NGary while and still reheating dynamics. ensuringcharged controlled To scalars, EFT illustrate both this description in fact,Dirac of fermions. the we the In have Higgs inflation- particular, considered phase we NG have and shown mediated that in even by loop-level the NG effects unbroken are phase, observable! as well as charged particles can couple to the primordial fluctuations much more strongly and lead to orders JHEP04(2020)077 . , = 7 . ! i ) ∆ µ 0 (A.9) (A.5) (A.6) (A.7) (A.8) (A.3) (A.4) , ~x 3 1 0 . k k η → − ( ! 1 µ µ χ ) + ∆) 2 ) − → − η, ~x ( 2 ). We can quickly iµ µ 1 = 1. / χ gives rise to scaling ) to get, + h H , A.2 2 + , ) χ 2 / † 3 1 i iµ 5.18 χ ) k k 0 , Γ(3 2 2 , ~x ) , 0 ) iµ, η iµ ∆) ( iµ 1 − )Γ(3 + 2 − χ 3 + 2 ) 2 + Γ( iµ / 2)))∆(∆ + 1)Γ(3 2 ∆)Γ(∆) Γ(2 / iµ η, ~x F 3 ( − − ) Γ(3 1 2 2 2 3+2 − χ ) iµ ]. This is schematically shown in figure / / 2 h ) 3 iµ 2 10 (∆ | = − 0 iµ − Γ(3 π 0 i ~x 2 Γ( ) / 0 Γ( ) and used the fact that – 22 – 2 ηη 5 − 2 5 1 2 + / π )Γ(3 + 2 , ~x 5 / π ~x 1 0 1 4 iχ | π η iµ 16 4 ( ), we will have, 2 + Γ(3 χ (1 + sin( 2

† 1 = ) − 2 A.6 5 χ χ ) = 2 ( · iµ iµ π (∆) = 1 / 2)) 2 2 iµ ) 1 − 3 / / 16 √ 3 3+2 3 free − Γ( c c π η, ~x = 5 2 − = ) and ( ( Γ( / π 1 0 3. More generally, in the squeezed limit, a loop diagram can be 2 χ (3 + 2 / χ † 16 → 5 0 (∆ 1 0 A.3 4, we see that the composite operator χ i π ∆+3 free π iµ, / h ) 4 c − 2 9 3 η,η | ), ( 4 k ~

2 ( − . Using the fact that the non-analytic pieces of the two point function i 3 3 i ) 2 χ 0 4 cos( ) k A.1 δσ 0 1 3 1 ) m , ~x 2 k = 0 , ~x ’s we first write, k ~ 0 η 4 σ q ( 0 η ( ∆ ˙ ( 1 c σ 1 3 2 2Λ δσ = χ k k ) . Reduction of a loop diagram into a linear combination of tree diagrams in the squeezed χ ) 1 , − ) µ k ~ ∆ ( = η, ~x Now we are ready to give the three point function induced by a Higgs loop. To get the η, ~x ( δσ ( 1 h 2 F χ χ h generalize this to the case of curvaton fluctuations by using eq. ( which can be used to get the inflaton three point function in eq. ( where dimensions ∆ = 3 decomposed into a setcorresponding of to such effective a “tree” scalingThus diagrams, dimension from ∆ each eqs. [ ( involving a dS mass eigenstate h are given by, associated where we have written and In this and the following appendix, we will work in the units where limit. where Figure 7 JHEP04(2020)077 )   ˜ µ, ! ˜ µ A.1 µ (B.5) (B.1) (B.2) (B.3) (B.4) ]. The (A.10) → − . 61 → − → − µ µ   µ ˜ µ + ˜ + ˜ + 2 ) 3 1 3 1 ˜ µ k k → − k k i µ ˜ µ, derived in [ i iµ, ) + ˜ i . ˜ µ ) ) i 0 Γ(2 + ˜ µ 2 4+2 3 1 , 3+2 ), we need the late time i ) , ~x k k 2 0 ˜ µ ) η . i . ˜ µ ˜ F µ, F 5.22 i (1 + 2 i ) ! ) ¯ − ! ]. We thus get using eqs. ( Ψ( ˜ ˜ µ µ ) µ 2 i iµ 2 28 / ) ) 2 )Γ(4 + 2 ) ˜ µ 4 + 2 η, ~x ˜ µ ˜ iµ i µ → − Γ(2 + i → − i Γ(1 2 2 . This can be calculated by squaring µ Ψ( ) µ i h 2+ ) F ˜ iµ ) µ + ˜ − ˜ )Γ(4+2 0 µ / + i ) )Γ(3+2 Γ(2+ i ˜ µ ˜ µ 2 0 ˜ 2 0 µ i (1 + 2 iµ ) i , ~x i 4+2 i iµ i − / 0 ˜ µ ) 2 ) 2 Γ(3 5 i η 2 3 3 (1+2 2 4+2 2 − − 3+2 / 2 k ) − k ~ − ) ~ | 2 2 ˜ 0 µ 2 / 0 0 − i − 0 / iµ 3 1 Γ( i / ~x i ( 3 1 ( ¯ ΨΨ( Γ(1 5 ) ) k k – 23 – 3 2 ζ k k ζ − Γ(1 · 3 ) 3 / ηη ) − ) 5 − ˜ )Γ(4 + 2 − µ 5 k 3 5 k ~ ) 3 ~ 3 Γ(2+ 1 i 3 π Γ( ( ˜ ( ) ~x µ π 2 π ) k k ~ ~ | 5 Γ( ) i ζ µ ζ Γ( ( ( µ 4 − (˜ ˜ ) π µ 2 ) 2 ( − 2 1 ζ ζ i 3 η, ~x / 2 2 2 / h (1+2 h 5 16 1 5 0 − 0 iδ 1 k = k 5 ~ ~ − iδ π i 2 i π 3 ( e ( e 4 π ) = ) / ) ˜ 4 µ 2 ζ ζ i 1 1 ¯ ˜

µ ) ΨΨ( / )

− | i k k ~ h   ), we derive the dimensionless three point function of the 1 ~

1 | 5 Γ(1 ) ) ), the dimensionless bispectrum is given by, k k ~ ~ 0 0 − 4+2 − 5 = µ − 6 σ ( 4 σ ( 3 µ ( ( c σ π (˜ σ ( ζ ζ Λ 0 ζ Λ ζ 2 ˙ 2 ˙ 3.13 h h Γ( ) ) − 3.13 → 0 2 0 1 1 (4 + 2 0 / loop loop k 2 σ , k 2 σ ~ ~ 5 1 0 ( ( 3 3 π i Ψ χ, η,η free ζ ζ 4 ) f f c h h i| − − 3 ) k ~ 0   = = ≡ | ( = ≡ | = 3 3 , ~x 0 k δσ 1 3 3 1 1 1 3 η ) ) we can get the three point function of the curvaton fluctuations, 2 . This matches with the answer obtained in [ k k k k k k , ~ , Ψ 6 σ ( 0 χ Ψ ¯ ˙ 5.22 ΨΨ( m σ ) m 2Λ δσ m ) ), = 1 − k ~ η, ~x µ ( = A.3 loop loop , δσ χ, h Ψ ¯ ΨΨ( F h F Using the above and eq. ( Using eq. ( and ( where ˜ To calculate the NGtwo induced point function by of the theand coupling type taking given a in trace eq.result of ( is, the spinor two point function B Fermion loop curvature perturbation, Using the above and eq. ( JHEP04(2020)077 , , ]. 12 11 (2010) SPIRE Phys. 04 IN , JHEP ][ , (2012) 021 JCAP ]. , 09 ]. 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