Revised f NL parameter in Curvaton Scenario
Lei-Hua Liu, Bin Liang, Ya-Chen Zhou, Xiao-Dan Liu, Wu-Long Xu, Ai-Chen Li
RRevised f NL of Curvaton Scenario Bin Liang , Ya-Chen Zhou , Xiao-Dan Liu , Lei-Hua Liu , ∗ Wu-Long Xu , and Ai-Chen Li ,
1. College of Civil Engineering, Hunan University of Technology, Zhuzhou, 412007, China2. Department of Physics, College of Physics,Mechanical and Electrical Engineering,Jishou University, Jishou 416000, China3. Institute of Theoretical Physics, Beijing University of Technology, Beijing 100124, China4. Departamento de matematica da Universidade de Aveiro and CIDMA,Campus de Santiago, 3810-183 Aveiro, Portugal
We revise the Non-Gaussianity of canonical curvaton scenario with a generalized δN formalism, in which it could handle the generic potentials. In various curvatonmodels, the energy density is dominant in different period including the secondary in-flation of curvaton, matter domination and adiation domination. Our method couldunify to deal with these periods since the non-linearity parameter f NL associatedwith Non-Gaussianity is a function of equation of state w . We firstly investigatethe most simple curvaton scenario, namely the chaotic curvaton with quadratic po-tential. Our study shows that most parameter space satisfies with observationalconstraints. And our formula will nicely recover the well-known value of f NL in theabsence of non-linear evolution. From the micro origin of curvaton, we also investi-gate the Pseudo-Nambu-Goldstone curvaton. Our result clearly indicates that thesecond short inflationary process for Pseudo-Nambu-Goldstone curvaton is ruled outin light of observations. Finally, our method sheds a new way for investigating theNon-Gaussianity of curvaton mechanism. I. INTRODUCTION
In traditional diagram of producing the curvature perturbation, it is sourced by thequantum fluctuations of inflationary field. In these broad class of single inflationary field ∗ Electronic address: [email protected] a r X i v : . [ a s t r o - ph . C O ] J u l theories, it experiences some initial condition problems associated with its correspondingpotential. In order to relax these restrictions of single field inflation, one nice alternativecalled curvaton mechanism was proposed [1–3], in which the energy density of curvaton issubdominant comparing with inflaton’s during inflationary period. After inflation decay, therole of curvaton will be more and more significant producing the isocurvature perturbation,which can be transferred into curvature perturbation seeding the temperature fluctuationon cosmological microwave background (CMB).Due to the appearance of CMB, there are huge data waiting for the investigations. In par-ticular, the most common method is calculating the power spectrum of scalar field (drivingthe curvature perturbation) characterizing by two point function, its corresponding spectralindex and tensor to scalar ratio. However, most data are still mysterious expecting a newtheoretical method for exploring these treasures. Under this background, the calculationof Non-Gaussianity (NG) identified with three point function was proposed [4]. Combiningwith curvaton scenario, NG, associated with its fraction of energy density among total en-ergy density, could also be produced as curvaton dominates over the energy density [5–7].Upon relaxing this condition (curvaton dominates over energy density), it could yield largeNG [8]. However, current observation constrains these models [9], namely characterizing bythe local non-linearity parameter f NL that cannot be large. This local f NL is suppressedby the quadratic potential plus quartic potential [10] and also in string axionic potential[11, 12]. Furthermore, the observable f NL also puts an enhanced constraint for the decayepoch of curvaton and its field value at the horizon exit [13]. The implications of NG featuresin curvaton scenario were also studied in Refs. [14–16]. On contrary, NG could be producedin various curvaton models [17–19].In most curvaton scenarios, curvaton usually is considered as an independent field. Iftaking the thermal effects into account, the large NG is the necessary product due to theobserved curvature perturbation [20]. From another perspective of independent curvaton, itnaturally embeds into two field inflationary theory, in which it could produce the sizable NGwithin observations [21]. Very recently, Ref. [22] rigorously realizes the curvaton mechanismunder the covariant framework of field space. Taking the curvaton and inflaton into accountfor the perturbation, NG could be generated by inflaton curvaton mixed model [18], even thecurvaton could drive the second inflationary process [23]. However, current observationalconstraints are not capable for distinguishing between the inflaton-curvaton mixed modeland single field inflation [24]. As curvaton explicitly couples to the super-heavy matter, itwill lead to observational signal including NG [25, 26]. From another aspect, curvaton isdubbed as some scalar fields, i.e. , Pseudo-Nambu-Goldstone Boson or right-handed sneutrinocurvaton etc [27–31]. Due to the unification of string theory, the curvaton scenario couldalso be applied into the string cosmology framework [32, 33], in which it yields considerableNG. Since the energy scale of inflation is far from Planck scale. Curvaton scenario could beembedded into the minimal supersymmetric Standard Model [34]. Another origin comes viathe inflaton decay [35].The NG is associated with three point function, in order to investigate the NG, δN formalism was proposed [6] depending on the surface of energy density slicing. Its huge meritis only needing the relation of the corresponding background field and e-folding number.Based on previous work, δN formalism was systematically developed by [36]. δN formalismhas become a standard procedure to evaluate the power spectrum and NG in the multi-field inflationary framework including the curvaton scenario (the canonical kinetic term offield space). Ref. [37] modified the δN formalism at the slice of curvaton energy density,their method could proceed the curvaton mechanism in various periods (matter domination,radiation domination, second inflationary period) explicitly associated with equation of state w (EoS). However, this traditional δN formalism cannot analytically evaluate the variouscurvaton models (distinct potentials). In order to compensate this flaw, Refs [38, 39] alsoproposed a modified δN formalism, in which this method could deal with various curvatonpotentials analytically in principle. However, they assumed that different periods havesimple attractor solution characterizing by a simple parameter c , in which the kinetic termis neglected and its contribution is enrolled into parameter. This estimation of their methodis too rough comparing to traditional calculation. The best way is including the contributionEoS w since it is model independent. In light of above theoretical motivation, we suggest ageneralized δN formalism unified to evaluate the Non linearity parameter f NL .This paper is organized as follows. In section II, we will revise the δN formalism basedon [36] and meanwhile we also give our central formula of Non linearity parameter f NL . Insection III, we study the most classical curvaton model whose potential is quadratic andPseudo-Nambu-Goldstone curvaton. Section IV gives our main conclusions.All of the calculations are adopted in the natural units which G = M P = c = 1, where G is the Newton constant, M P is the Planck mass and c is the speed of light. II. THE GENERALIZED δN FORMALISM OF CURVATON DECAY
In this section, we will generalize the δN formalism. In light of Ref. [37] and Ref. [38, 39],our extending framework contains their merits. The main advantage of [38, 39] for f NL isthat they build the explicit relation of the onset of oscillation of curvaton and the curvatonvalue as inflation ends, which is not included in a traditional δN formalism. As for Ref.[37], they construct f NL associated with equation of state w (EoS) except the fraction ofcurvaton energy density to total energy density denoted by r decay . Firstly, we will reviewthe δN formalism. A. Recap of δN formalism for curvaton decay In a traditional curvaton scenario, it will generate the Non-Gaussianity essentially char-acterizing by non-local Nonp-Gaussianity parameter f NL . In order to obtain its explicitformula, the most common method for copying with is so-called δN formalism [36] since itonly requires the relation between the background field and e-folding number.The curvatureperturbation can be expanded as order by order, ζ ( x ) = ζ ( x ) + 12! ζ ( x ) + 13! ζ ( x ) , (1)with ζ = f NL ζ and ζ = g NL ζ , where ζ is explicitly proportional to Gaussian field, ζ and ζ are related to Non-Gaussian field associated with Gaussian field with non-LocalNon-Gaussianity parameter f NL and g NL . Here, we only concern with f NL since g NL ∝ f and it will be suppressed at higher order. f NL originates from the three point correlationfunctions, (cid:104) ζ ( k ) ζ ( k ) ζ ( k ) (cid:105) = (2 π ) B ( k , k , k ) δ ( (cid:88) n =1 k n ) , (2)where B ( k , k , k ) = f NL ( P ( k ) P ( k ) + 2 perm . ) with P ( k i ) is the power spectrum of ζ k i field.In order to relate to e-folding number N , once adopting uniform density hypersurfaces ofcurvaton, then the curvature perturbation can be denoted in terms of non-linear curvatureperturbation, ζ ( x ) = δN ( x ) + 13 (cid:90) ρ ( x )¯ ρ ( t ) d ˜ ρ ˜ ρ + ˜ P , (3)where δN is the perturbed expansion, ˜ ρ is the local energy density and ˜ P is the localpressure. Given that the curvaton decay occurs in matter domination period (MD), thenone naturally neglects the contribution of pressure. Subsequently, integrating both sides ofEq. (3) and choosing flat slice, one obtains ρ χ = ¯ ρ χ exp(3 ζ χ ) . (4)For curvaton field, its perturbation can be defined by, χ ∗ = ¯ χ + δ χ ∗ , (5)where δ χ ∗ denotes the vacuum fluctuations of curvaton field. For depicting the curvatureperturbation of curvaton, we need relating the Hubble crossing value to the initial amplitudeof curvaton oscillation. In order to achieve this goal, one could use the Taylor expansion tobuild their relation, g ( χ ∗ ) = g ( ¯ χ + δ χ ∗ ) = ¯ g + ∞ (cid:88) n =1 g ( n ) n ! (cid:18) δ χ osc g (cid:48) (cid:19) n , (6)where g (cid:48) = dgdχ ∗ and χ osc denotes the value of curvaton field that begins to oscillate. Appar-ently, g ( χ ∗ ) depends on the model. Until present, the discussion for curvature perturbation ofcurvaton is generic which means that the curvaton potential is general. In order to relate tosome specific curvaton models, Ref. [36] assumes that the simplest potential (quadratic po-tential) for curvaton. Apparently, it shows that g ( χ ∗ ) ∝ χ ∗ . Subsequently, one can considerthis potential as energy density and then expand them to the second order of perturbationof curvaton field for comparing, finally we find that ζ χ = 23 δ χ ¯ χ , (7) ζ χ = − (cid:0) − gg ” g (cid:48) (cid:1) . (8)Nextly, we need to find the relation between ζ χ and ζ . Following the sudden decay ap-proximation, this relation can be analytically obtained, which is realized on a uniform totaldensity hypersurface as H = Γ χ (the decay rate of curvaton). On this curvaton decayhypersurface, one accordingly have ρ r ( t decay ) + ρ χ ( t decay ) = ¯ ρ ( t decay ) , (9)where ¯ ρ denotes the background field energy density. Meanwhile, we have δN = ζ on thecurvaton decay hypersurface. Observing that the production of curvaton decay is relativisticand total pressure P = ρ , consequently one easily obtains that ρ r = ¯ ρ r exp[4( ζ r − ζ )] , (10) ρ χ = ¯ ρ χ exp[3( ζ χ − ζ )] . (11)Using these two formulas into Eq. (9) and defining a dimensionless quantity Ω χ = ¯ ρ χ / ( ¯ ρ χ +¯ ρ r ), after some algebra, one obtains that(1 − Ω χ ) exp[4( ζ r − ζ )] + Ω χ exp[3( ζ χ − ζ )] = 1 . (12)Once deriving this central formula of δN formalism, we can set the relations between the ζ χ and ζ . Expanding up to the second order of Eq. (12), we collect these relations, ζ = r decay ζ χ , (13) ζ = (cid:20) r decay (cid:0) gg (cid:48)(cid:48) g (cid:48) (cid:1) − − r decay (cid:21) ζ χ , (14)where we have defined r decay = 3Ω χ, decay − Ω χ, decay = 3 ¯ ρ χ ρ χ + 4 ¯ ρ r . (15)It naturally yields non-linearity parameter using the sudden decay approximation [6, 7], f NL = 54 r decay (cid:0) gg (cid:48)(cid:48) g (cid:48) (cid:1) − − r decay . (16)Observing that this non-linearity parameter highly depends on the r decay , meanwhile mildlydepends on the structure of model showing in g and g (cid:48) . Although we adopted the simplestpotential for curvaton, the final result is almost quadratic potential independent. Actually,one can roughly estimate this result since when expanding the energy density of curvatonup to the second order. Subsequently, one can discover via Eq. (4, 6) that the backgroundof curvaton will be cancelled as comparing them through their equation.Furthermore, the generic potential of curvaton should be taken into account. The timeof occurrence of curvaton mechanism (various decays of curvaton models will happen in RDor MD) is also different. In order to compensate these two missing places into curvatonmechanism, some distinct generalized δN formalisms are proposed. B. Generalized δN formalism In this section, we will construct a generalized δN formalism with a generic potentialand EoS w . Consequently, it is valid for broad kinds of curvaton models. In Ref. [37], theyinnovatively assumed that the curvaton decay occurs on a uniform curvaton density slice.Being different with definition of total energy density in section II A, they found that ζ = ζ χ + 14 ln (cid:0) ρ r + 3( ¯ ρ χ + ¯ P χ )4 ρ r + 3( ¯ ρ χ + ¯ P χ ) (cid:1) . (17)By inserting Eq. (11) into Eq. (17), they obtained (cid:18) − − w χ (cid:19) exp[4( χ − χ r )] = (1 − Ω χ ) exp[4( ζ r − ζ χ )] + 3(1 + w )4 Ω χ , (18)where they defined w = ¯ P χ ¯ ρ χ . Following the standard procedure, the relation between the ζ and ζ χ can be derived order by order, ζ = ˜ r decay ζ χ , (19) ζ ζ χ = 3(1 + w )2˜ r decay (cid:18) gg (cid:48)(cid:48) g (cid:48) (cid:19) + 1 − w ˜ r decay − , (20)where ˜ r decay = w )Ω χ w − χ is introduced. Apparently, the non-linearity parameter associatedwith non-Gaussianity can be explicitly derived by, f NL = 54 1 + w ˜ r decay (cid:18) gg (cid:48)(cid:48) g (cid:48) (cid:19) + 56 1 − w ˜ r decay − . (21)Observing that the value of f NL will be enhanced in the limit of w → r decay →
0. Ref. [37] has noticed that this case will be appeared in the secondary inflation.The similar process was also discussed in various curvaton models [40, 41]. Consequently,one can conclude that w is a possible criteria for assessing the occurrence of secondaryinflationary process.This non-linearity parameter is tiny different comparing to (16). This difference comesfrom the slice of energy density. In inflationary period, there are at least two components ifrequiring the existence of curvaton field. In order to remove the influence of other field tothe non-Gaussianity, this method is necessary and more precise comparing to traditional δN formalism. However, one cannot manage it analytically with generic potential besides thequadratic potential. Ref. [38, 39] accordingly proposed another generalized δN formalism fordealing with the generic potential analytically. In their method, the non-linearity parameteris written by f NL = − r decay −
53 + 52 r decay (1 + A ) , (22)where A is given by A = (cid:20) V (cid:48) ( χ osc ) V ( χ osc ) − X ( χ osc ) χ osc (cid:21) − (cid:20) X (cid:48) ( χ osc )1 − X ( χ osc ) + V (cid:48)(cid:48) ( χ osc ) V (cid:48) ( χ osc ) − (1 − X ( χ osc )) V (cid:48)(cid:48) ( χ ∗ ) V (cid:48) ( χ osc ) (cid:21) + (cid:20) V (cid:48) ( χ osc ) V ( χ osc ) − X ( χ osc ) χ osc (cid:21) − (cid:34) V (cid:48)(cid:48) ( χ osc ) V ( χ osc ) − (cid:18) V (cid:48) ( χ osc ) V ( χ osc ) (cid:19) − X (cid:48) ( χ osc ) χ osc + 3 X ( χ osc ) χ (cid:35) . (23)Here A is characterized by a curvaton with a generic energy potential, in which it experi-ences a non-uniform onset of its oscillation. Its validity only requires starting a sinusoidaloscillation as satisfying with H = V (cid:48) ( χ osc ) cχ osc (24)where c is given by 9 / c characterizing by the attractor solution.In order to relate the method of Ref. [37], we need to find their correspondence betweenEq. (22) and Eq. (21). Before finding the correspondence, the relation between Eq. (22)and Eq. (16) is necessary since these two methods are adopted in the total energy densityslice. Maybe this slice for [38, 39] is not explicit. However, one can easily check thatthe whole calculation is depending on the total energy density in the curvaton dominantperiod after inflation. Furthermore, the total energy slice is approximately equalled tothe curvaton energy density slice after inflation, since the curvaton is dominant which isalso an assumption for original curvaton scenario. In light of this logic, we should find thecorrespondence between Eq. (22) and Eq. (16) and then explicitly adopt this correspondencefor Eq. (22). Comparing with Eq. (22) and Eq. (16), an explicit correspondence can befound by 1 + 2 A = gg (cid:48)(cid:48) g (cid:48) . (25)Using this correspondence into Eq. (21), we obtain f NL = 52 1 + w ˜ r decay (cid:18) A (cid:19) + 56 1 − w ˜ r decay − . (26)In this formula, we observe that ˜ r decay is also the function of w . Following the traditionallogic, we will work with f NL in terms of r decay and w . In order to achieve this goal, therelation between r decay and ˜ r decay is mandatory. In light of their relation, the non-linearityparameter can be rewritten by f NL = 5(3 Aw + 3 A + 4)6 r decay ( w + 1) + 5 (3 Aw + 3 Aw − w + 1) . (27)Thus, we obtain the central result of this paper, in which it could tackle the genericpotential analytically and it could assess the existence of second inflationary process forcurvaton field. In the next section, we will investigate the non-linearity parameter f NL invarious curvaton models under the observational constraints. III. CASE STUDY
The realization of curvaton mechanism depends on the models, particularly it dependson the potential of curvaton. The shape of potential for curvaton will lead to the differencein various curvaton models, i . g . chaotic curvaton model, axionic curvaton, e . t . c . Before discussing the Non-Gaussianity identified with non-linearity parameter f NL . Theconsideration of power spectrum of curvaton must be taken into account. Recalling that ourderivation of f NL is mainly according to the framework of [38, 39], they found that the powerspectrum of curvaton is nearly scale invariant in different values of k for various models ofcurvaton (exactly speaking for the various potentials of curvaton). Furthermore, Ref. [42]also studied that power spectrum is only depending on r decay and χ explicitly. Thus, thepower spectrum of curvaton is the same for various models of curvaton. This issue can beeasily checked in [36, 38, 39]. A. Chaotic curvaton
Chaotic curvaton indicates that the potential of curvaton is quadratic. These kinds ofcurvaton have been investigated broadly, in particular, for the non-Gaussianity characteriz-ing by non-linearity parameter f NL [6, 7]. In light of quadratic potential, Ref. [36] proposeda generalized δN formalism to investigate the non-Gaussianity, in which curvature perturba-tion can be derived up to any order. We accordingly concern the second order of curvatureperturbation associated with f NL .0We will give a analysis of f NL for chaotic curvaton based on our central result (26). In ourprevious work [42], we clearly show that A = − as the potential of curvaton proportional to χ where χ denotes the value of curvaton field, in which it is explicitly consistent with simpleanalysis of Ref. [39] (only adopting the different notation for the fraction of curvation energydensity among the total energy density). Accordingly, the central result for f NL becomes f NL = − w − r decay ( w + 1) − w + 3 w + 8)12( w + 1) . (28)We will use this formula for investigating the non-Gaussianity comparing to previous relevantwork. This f NL is a generic formula for curvaton associated with Non-Gaussianity.Case a : w → − In various models, the EoS w could have different values. In Ref. [37], they constructed acurvaton scenario under the framework of brane world, in which the corresponding w → − f NL will be divergent exceeding the range of currentobservational constraints [43].Case b : w → In this case, the curvaton behaves as the pressureless matter. f NL simplifies into f NL = 2512 r decay − . (29)In limit of r decay → f NL = − which nicely recovers with Eq. (26) in Ref. [36] in theabsence of non-linear evolution for the curvature perturbation of curvaton (also emphasizedin Ref. [6]), in which the curvaton scenario is the simplest curvaton model whose potentialis m χ χ ( χ denotes the curvaton field) and behaves as pressureless matter according to ouranalysis. Meanwhile, curvaton dominates the energy density. For large Non-Gaussiantiy, itrequires that r decay →
0. In order to better understand the possible range of r decay , we willplot Eq. (29).Case c : w → In this case, curvaton decay is a relativistic process. Then, f NL becomes f NL = 54 r decay − . (30)A similar analysis will be given as in case b. In limit of r decay → f NL → − . Thevalue is almost the same with case b, in which one cannot distinguish the tiny differencebetween case b and case c . Frankly speaking, curvaton is an independent and extra field1during inflationary process (even including the preheating process), however curvaton couldbe induced by the inflaton decay whose realization occurs from the transferring of entropyperturbation to curvature perturbation [42].We have discussed the non-linearity parameter with various cases of chaotic curvaton,whose potential is proportional to χ . Although we cannot distinguish the difference forcase b and case c via observational contraints, it is expecting for obtaining the distinctvalues for its corresponding cases. Ref [9] tells that f NL = 2 . ± .
7, afterwards, combiningwith Eq. (29,30), we could plot for comparing them. In figure 1, it explicitly depicts thatthe constraints of r decay for case b and case c , respectively. The corresponding values are0 .
18 for left panel (case b ) and 0 .
11 for right panel (case c ). This trend is logical since case c illustrates curvaton behaves as relativistic matter meaning the curvaton will last longer-timeoccurrence of its decay. r decay f NL r decay f NL FIG. 1: Left panel shows the non-linearity parameter f NL for case b and right panel illustrates thecase case c . The brown and blue line denote the upper and lower bound for f NL . The correspondingvalue of r decay is 0 .
18 and 0 .
11 with respect to case b and case c , respectively. For the careful reader, they may find that there is still some losing information for thetransition from w → w → , since the curvaton will become the relativistic matteras the longtime occurrence of curvaton decay (from MD to RD). If considering this case, r decay will be a small number, but what the precise value is. We need the more detailedinvestigation of f NL varying w . In order to achieve this goal, we show the density plot ofnon-linearity parameter f NL depending on the parameter r decay and w in figure 2. It clearly2 - - - - - r w FIG. 2:
Contour plot of non-linearity parameter (28):
The horizontal line corresponds to r decay whose range is 0 (cid:54) r decay (cid:54) w locating from − , in which it includes that dark energy epoch,radiation domination period, matter domination period and it could indicate the transition fromone era to another era. The right panel shows that the value of f NL matching its correspondingcolor. indicates that the Non-Gaussianity will be dramatically enhanced as w → r decay → f NL is still within theobservational constraints [43], in which w is approaching − r decay < . B. Pseudo-Nambu-Goldstone curvaton
In this case, we will further consider the curvaton could origin from microscopic physics,namely, pseudo-Nambu-Goldstone boson with a broken U (1) symmetry. The curvaton masswill be suppressed by the approximating symmetry. Since curvaton has the periodicity of U (1) leading to minima and maxima along the potential. Therefore it will generate the3blue and red tiled curvature perturbation of curvaton. What we concern is the potential ofPseudo-Nambu-Goldstone curvaton, it reads as V ( χ ) = Λ (cid:20) − cos (cid:18) χf (cid:19) (cid:21) , (31)where f and Λ denote the energy scale. In order to obtain its corresponding f NL , the relationbetween the χ ∗ and χ osc is mandatory. For achieving this goal, we need the modified KGequation (24), one can deriveln (cid:20) tan( χ osc / f )tan( χ ∗ / f ) (cid:21) = − N ∗ H Λ f − c − χ osc /f sin( χ osc /f ) , (32)where N ∗ denotes the e-folding number at the horizon exit, H inf represents the Hubbleparameter during inflation. After some algebras, we can represent χ ∗ in terms of χ osc , χ ∗ = 1 f (cid:20) arccot (cid:18) exp (cid:18) − fχ osc csc ( χ osc f ) c − + N ∗ H f (cid:19) cot (cid:18) f χ osc (cid:19) (cid:19)(cid:21) + constant . (33)In this calculation, the constant can be set to zero and the maxima of χ osc is around 0 . . A corresponding to Pseudo-Nambu-Goldstone curvaton. Due to complicationof formula of A , all of these formulas will be tackled by Mathematica. Being armed withthese formulas, we will plot the non-linearity parameter in various epoches including secondinflationary process, RD and MD. Being different with investigating chaotic curvaton, A isalso a function of c whose various values corresponding to different periods. Due to thisparameter, we cannot vary with w to analyze nonlinearity parameter f NL . Finally we onlystudy the individual case referring to specific w and c .Case a : w = − c = 3The explicit of f NL is too complicated to express due to the complication of A . Actually,most curvaton models being with various potentials cannot find express A explicitly since therelation between χ osc and χ ∗ is almost not possible, taking placing by the numerical methodsas showing in Ref. [38, 39]. Once knowing these knowledge and meanwhile observing that4 χ osc χ * FIG. 3: The relation of χ oscc and χ ∗ according to their explicit relation (33). During the wholerange of χ osc , its corresponding maximal value of χ ∗ is 0 .
7. The parameters are setting as N ∗ = 50, f = 3 . × − , c = 9 / . × − and H inf = 10 − as adopting inRef. [38]. w = − c = 3 will lead to the divergence of f NL from Eq. (27). For better understandingthis case, the plot will be given. In figure 4, we could clearly see that the f NL varying with χ osc and r decay . The observational constraint gives the upper limit whose value is less than 10.From figure 4, it is almost impossible find this value, in particular, as r < . f NL alreadyexceeds the upper limit of observational constraint. Additionally, there is also divergenceas χ osc is between from 0 .
018 to 0 .
03. The varying trend of f NL will flip as crossing thesedivergent areas. To sum up, the secondary inflation for curvaton will not happen in light ofour discussion.Case a : w = 0 and c = 9 / r decay > .
2. The value of f NL will become negative as 0 . ≤ χ osc ≤ . f NL , it will give a strongconstraints of our mechanism for curvaton. Comparing with Ref. [38], our formula is notnot so highly depending on the field value of χ , in which we use replace χ ∗ with χ osc toinvestigate. In this case, the upper limit r decay is smaller comparing to chaotic curvaton,5 χ osc r FIG. 4: The horizontal line corresponds to r decay whose range is 0 (cid:54) r decay (cid:54) χ osc locating from 0 to 0 . f NL matching its corresponding color. The parameters areset the same as figure 3. which means that fraction of curvaton among the total energy could be less even in MD.case c : w = , c = 5In case, we will study the nonlinearity parameter in RD. Generically, the trend of figure6 is similar with figure 5. It contains lots of parameter spaces satisfied with observationalconstraints. The difference comes for the upper limit of r decay , its value is even smaller whoserange could reach 0 .
1, the discussion is the same since one could consider that curvaton isthe production in MD as showing in our previous work [42]. Another distinct place is thatthe sign of f NL flips around 0 . ≤ χ osc ≤ . f NL to different curvaton models. Firstly, in lightof framework [38], it has already known that power spectrum is not varying dramaticallywith energy scale. According to this point, we only concern the nonlinearity parameter f NL .Our findings are the generic curvaton mechanism that will not experience the second infla-tionary process, although there is tiny choice of parameter space for chaotic curvaton. Asfor a Pseudo-Nambu-Goldstone curvaton, our findings show that no matter what curvatonbehaves as pressure or pressureless matter, most of parameter spaces satisfy with observa-6 χ osc r - - FIG. 5: The horizontal line corresponds to r decay whose range is 0 (cid:54) r decay (cid:54) χ osc locating from 0 to 0 . f NL matching its corresponding color. The parameters areset the same as figure 3. tional constraints [9]. The only differences are determined by their decay process, this pointis illustrated in Ref. [39] identified with comparison between t decay and t reheating . IV. CONCLUSION
In this paper, we have constructed a generalized δN formalism consisting of merits of Ref.[37, 38]. Our method could deal with curvaton models with generic potentials only requiringsinusoidal oscillation, meanwhile it can also handle curvaton mechanism in various periodsexplicitly showing by EoS w (secondary inflation, MD, RD) with corresponding parameter c in Section II B. Ref. [38] analyzes the non-Gaussianity associated with f NL . Althoughtheir method could work with different period (MD, RD, e.t.c ), they simply assumed thatthe different epoch corresponds to the various values of c by neglecting the contribution ofkinetic term. It is unavoidable for wrongly estimating the precise contribution of kineticterms. In order to compensate this flaw, we adopted the advantage of Ref. [37], directlyassociated with EoS w , for investigation.7 χ osc r - - - FIG. 6: The horizontal line corresponds to r decay whose range is 0 (cid:54) r decay (cid:54) χ osc locating from 0 to 0 . f NL matching its corresponding color. The parameters areset the same as figure 3. Once obtaining the key result for non-linearity parameter f NL (27), we implement it intotwo curvaton models. One is the chaotic curvaton, the other one is the Pseudo-Nambu-Goldstone curvaton. In light of framework of [39], we only concern the non-Gaussianityidentified with f NL since the power power spectrum is nearly scale invariant in variousmodels.For the chaotic curvaton, we investigate the f NL . In the limit of r decay → f → − nicelyrecovers the analysis of Ref. [36] in case a of chaotic curvaton and f NL will be divergentin the limit of r decay →
0. For case a , it indicates that the secondary inflationary processis ruled out by observational constraints. However, the occurrence of second inflationaryprocess will alive if there is a transition from DE era to MD showing in figure 2.The original curvaton mechanism assumed that it was an extra and independent fieldcomparing to inflaton field. One possibility for accounting for its origin is Pseudo-Nambu-Goldstone curvaton. In this model, the value of f NL shows the similar varying trend withchaotic curvaton as showing in figure 6, 5 and 4. Due to the complication of A written byEq. (23), we cannot transit w from one era to another era taking place by parameter c .8From these figures, it explicitly shows that most parameter spaces satisfy with observationalconstraints which determines the upper limit of r decay > .
1. And the case of a will be ruledout by the observations. Acknowledgements
LH is funded by Hunan Natural Science Foundation NO. 2020JJ5452 and Hunan Provin-cial Department of Education, NO. 19B464. WL is funded by NSFC 1175012.
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