Scale dependence of f NL in N-flation
aa r X i v : . [ a s t r o - ph . C O ] D ec Preprint typeset in JHEP style - HYPER VERSION
Scale dependence of f N L in N-flation
Qing-Guo Huang ∗ Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China
Abstract:
Adopting the horizon-crossing approximation, we derive the spectral index of f NL in general N-flation model. Axion N-flation model is taken as a typical model forgenerating a large f NL which characterizes the size of local form bispectrum. We find thatits tilt n f NL is negligibly small when all inflatons have the same potential, but a negativedetectable n f NL can be achieved in the axion N-flation with different decay constants fordifferent inflatons. The measurement of n f NL can be used to support or falsify the axionN-flation in the near future. Keywords:
N-flation, non-Gaussianity. ∗ [email protected] ontents
1. Introduction 12. Scale dependence of f NL in general N-flation 23. Axion N-flation 5 N ∗ ≃ N tot
4. Discussions 14A. The trispectrum in N-flation 15
1. Introduction
In the single-component slow-roll inflation the different Fourier modes of the curvatureperturbation are roughly uncorrelated and their distribution is almost Gaussian [1]. Using δN formalism [2], the curvature perturbation can be expanded to the non-linear orders ζ = δN = X i N ,i δφ i + 12 X i,j N ,ij δφ i δφ j + 16 X i,j,k N ,ijk δφ i δφ j δφ k + ... , (1.1)where the subscript , i denotes the derivative with respect to φ i . The above formula indi-cates that the curvature perturbation is determined by the evolution of the homogeneousuniverse corresponding both to the classical inflation trajectory and to nearby trajectories.In principle, an important difference for the multi-field model from the single-componentcase is that the classical trajectory is not uniquely specified by the potential, but ratherhas to be given as a separate piece of information. It provides an opportunity to gen-erate a large non-Gaussianity in multi-field slow-roll inflation even at the inflationaryepoch [3, 4, 5], not at the end of inflation [6, 7, 8]. See some other relevant referencesin [9, 10, 11, 12, 13, 14, 15, 17, 16, 18].On the other hand, not only the size of bispectrum f NL is measurable, but also itsscale dependence characterized by n f NL is possibly measured by the accurate cosmologicalobservations, such as Planck and CMBPol. In [19] the authors concluded that Planck and– 1 –MBPol are able to provide a 1- σ uncertainty on the spectral index of f NL for local formbispectrum as follows ∆ n f NL ≃ . f NL p f sky for Planck , (1.2)and ∆ n f NL ≃ .
05 50 f NL p f sky for CMBPol . (1.3)Recently the scale dependence of f NL for inflation and curvaton model is discussed in[20, 21, 22, 23, 24] and a large parameter space for generating a detectable n f NL in curvatonmodel is illustrated in [22, 23]. The value of n f NL will be an important discriminator todistinguish different models.In this paper we focus on a class of multi-field inflation, so-called N-flation, in whichthere are no cross couplings among the inflaton fields or the cross couplings are negligiblysmall. For an example, we consider the axion N-flation with a detectably large f NL . Butwe find that n f NL is negligibly small when all inflatons have the same potential. However,a negative n f NL with detectably large absolute value can be obtained in the model withdifferent decay constants for different inflatons.Our paper is organized as follows. In Sec. 2 we derive the spectral index of f NL forgeneral N-flation. We explore the observables, including n f NL , in the axion N-flation inSec. 3. Some discussions are contained in Sec. 4.
2. Scale dependence of f NL in general N-flation In this section we will derive the spectral index of f NL in general N-flation with N f un-coupled inflaton fields. The potential of inflatons is given by V = N f X i =1 V i ( φ i ) . (2.1)The dynamics of inflation is governed by H = 13 N f X i =1 (cid:18)
12 ˙ φ i + V i ( φ i ) (cid:19) , (2.2)¨ φ i + 3 H ˙ φ i = − V ′ i , (2.3)where V ′ i = ∂V i /∂φ i . In this paper we work on the unit of M p = 1. In the slow-roll limit,the above equations are simplified to be H ≃ V , (2.4)3 H ˙ φ i ≃ − V ′ i . (2.5)– 2 –imilarly, we introduce some ‘slow-roll’ parameters, such as ǫ i ≡ (cid:18) V ′ i V i (cid:19) . (2.6)From Eq.(2.4), we find ˙ H = − X i V ′ i /V, (2.7)and then ǫ H ≡ − ˙ HH = 12 P i V ′ i V , (2.8)or equivalently, ǫ H = X i ( V i V ) ǫ i . (2.9)Inflation happens when ǫ H <
1. The multi-field slow-roll inflation can be achieved evenwhen some inflaton fields have steep potentials because the friction term for each inflatonis contributed from all of fields.Following [25], the number of e-folds before the end of inflation is given by N = N f X i =1 Z φ i φ endi V i V ′ i dφ i . (2.10)Therefore we obtain N ,i = V i V ′ i , (2.11)and N ,ij = (cid:18) − V i V ′′ i V ′ i (cid:19) δ ij . (2.12)Using the δN formalism [2], the power spectrum of scalar perturbation is given by P ζ = (cid:18) H π (cid:19) X i N ,i . (2.13)A slow-roll inflation model predicts a near scale-invariant power spectrum of scalar per-turbation. The deviation from exact scale invariance is described by the spectral index n s which is defined by n s ≡ d ln P ζ d ln k = 1 − ǫ H − P i,j ǫ ij N ,i N ,j P k N ,k , (2.14)where ǫ ij ≡ V ′ i V V ′ j V − V ′′ i V δ ij . (2.15)– 3 –ee [25, 26] in detail. In this paper, we assume, without loss of generality, ˙ φ i <
0, so that V ′ i > ∀ i . On the other hand, the gravitational wave perturbation is also generatedduring inflation and the amplitude of its power spectrum is determined by the inflationscale P T = H π / , (2.16)and then the tensor-scalar ratio r T becomes r T ≡ P T /P ζ = 16 ,X i ǫ i . (2.17)The amplitude of gravitational perturbation is also near scale-invariant and its tilt is definedby n T ≡ d ln P T d ln k = − ǫ H . (2.18)In single-field inflation, r T = 16 ǫ H which leads to a consistency relation n T = − r T /
8. For N -field inflation, it is modified to be n T ≤ − r T / . (2.19)Here we used the inequality ǫ H ≥ ,X i ǫ i , (2.20)where the equality is satisfied when ǫ i = ǫ H V /V i for ∀ i .From Eq.(1.1), the non-Gaussianity parameter f NL can be written by f NL = 56 X i,j N ,i N ,j N ,ij ( P k N ,k ) . (2.21)Since N ,ij ∝ δ ij for N-flation, the above formula is simplified to be f NL = X i f iNL , (2.22)where f iNL = 56 N ,i N ,ii ( P k N ,k ) = 56 r T
128 1 ǫ i (cid:18) − V i V ′′ i V ′ i (cid:19) . (2.23)Following [21], the spectral index of f NL for N-flation is n f NL = 1 f NL X i f iNL (2 n multi,i + n f,i ) , (2.24)– 4 –here n multi,i = 2 X k,l ǫ kl N ,k N ,l P j N ,j − δ il N ,k N ,i ! , (2.25) n f,i = − X k N ,k F (2) kii N ,ii , (2.26)and F (2) kii = − V ′ i V ) V ′ k V + V ′′ i V V ′ k V + 2 V ′ i V V ′′ i V δ ki − V ′′′ i V δ ki . (2.27)Considering N ,i = V i /V ′ i , we obtain n multi,i = (1 − n s − ǫ H ) − V i V ǫ i (cid:18) − V i V ′′ i V ′ i (cid:19) , (2.28) n f,i = 4( V i V ) ǫ i − V i V (cid:18) V ′′ i V i − V ′′′ i V ′ i (cid:19) (cid:30)(cid:18) − V i V ′′ i V ′ i (cid:19) . (2.29)Using Eqs.(2.22), (2.23), (2.24), (2.28) and (2.29), we can easily calculate the spectral indexof f NL in general N-flation. The first term in (2.28) contribute 2(1 − n s − ǫ H ) to n f NL . If ǫ H ≪
1, 2(1 − n s − ǫ H ) ≃ .
08 for n s = 0 .
3. Axion N-flation
In this section we focus on axion N-flation model [5] in which there are N f inflaton fieldsand the potential of φ i is given by V i = Λ i (1 − cos α i ) , (3.1)where α i = 2 πφ i /f i and f i is the i-th axion decay constant. The ‘slow-roll’ parameters ǫ i ’sare given by ǫ i = 2 π f i α i − cos α i , (3.2)and N ,ii = 1 − V i V ′′ i V ′ i = 11 + cos α i . (3.3)Now the number of e-folds before the end of inflation becomes N ≃ N f X i =1 (cid:18) f i π (cid:19) ln 21 + cos α i . (3.4)– 5 –uoting the results in [5], the observables of axion N-flation are P ζ = H π X i ǫ i , (3.5) n s = 1 − ǫ H − π H X i Λ i f i ǫ i ,X j ǫ j , (3.6) r T = 16 ,X i ǫ i , (3.7) f NL = 56 r T X i ǫ i
11 + cos α i . (3.8)Using the results in the previous section, we can calculate the spectral index of f NL for theaxion N-flation, namely n multi,i = (1 − n s − ǫ H ) − π H Λ i f i , (3.9) n f,i = 4( V i V ) ǫ i − V i V ǫ i , (3.10)and then n f NL = 2(1 − n s − ǫ H ) − π H X i Λ i f i ǫ i
11 + cos α i ,X j ǫ j
11 + cos α j + 2 X i (cid:18) V i V ) − V i V (cid:19)
11 + cos α i ,X j ǫ j
11 + cos α j . (3.11)The first and second lines in Eq.(3.11) are contributed by n multi,i and n f,i respectively.Since the second term in n multi,i is large compared to n f,i for α i ≃ π which corresponds toa large f NL , the spectral index of f NL is mainly determined by n multi,i . In this subsection we discuss the model with Λ i = Λ and f i = f for ∀ i in detail. Now thetensor-scalar ratio and f NL become r T = 32 π f ,X i − cos α i α i , (3.12) f NL = 10 π f X i − cos α i (1 + cos α j ) ,X j − cos α j α j , (3.13)– 6 –nd the spectral indices of scalar power spectrum and f NL are simplified to be n s = 1 − ǫ H − π Λ H f , (3.14) n f NL = f π (1 − n s − ǫ H ) X i ǫ i (1 − cos α i ) ,X j ǫ j
11 + cos α j − r T
96 1 f NL (1 − n s − ǫ H ) , (3.15)where Λ H = 1 ,X i (1 − cos α i ) , (3.16)and ǫ H = 2 π f X i − cos 2 α i ,X j (1 − cos α j ) . (3.17)We see that n multi,i = 0 and n f NL is expected to be small in this simple setup. We will showthe smallness of n f NL and extend the discussions in [5] in the following two subsubsections. N ∗ ≃ N tot Assuming the initial conditions for α i satisfy a uniform distribution, the number of e-foldsbefore the end of inflation is roughly given by N tot ≃ ln 22 π f N f . (3.18)A large enough total number of e-folds, for example not less than 60, can be achieved aslong as the number of inflatons is large enough, namely N f & . × /f .Similar to [5], we consider the case with N ∗ ≃ N tot in this subsubsection, where N ∗ denotes the number of e-folds corresponding to the CMB scale. So we have N f f ≃ π N ∗ ln 2 , (3.19)and Λ / (3 H ) ≃ /N f , ǫ H ≃ π / ( N f f ). The spectral index of the scale power spectrumis related to N ∗ by n s ≃ − N ∗ , (3.20)which is the same as that in [5]. For N ∗ ≃ n s ≃ . N ∗ ≃ N tot ≃
60 and f = 1 / O (10 − ∼ − ). As long as N f is large enough, from Eqs. (3.12) and (3.13), both r T and– 7 – .941 0.942 0.943 n s
20 40 60 f NL - - - - - r T - - - - f NL - n f N L Figure 1:
The observables of axion N-flation with the same potential for different inflatons. Here f = 1 / α i is assumed to be the uniform distribution. f NL are proportional to 1 /f . From the numerical results, we find that f NL is boundedfrom above, namely f NL . π f . (3.21)For f .
1, the value of f NL can be of order O (10) which is large enough to be detected inthe near future, but its tilt n f NL ( −O (10 − ) ∼ −O (10 − )) is too small to be detected inthis case. – 8 – .1.2 A few inflatons stays around the hilltop In [5], the authors proposed an alternative way to achieve a large f NL . Roughly speaking,the summation of the function which is proportional to 1 /ǫ i is dominated by those fieldswith the smallest ǫ i , or equivalently staying around the hilltop. Suppose some number ¯ N of fields have roughly comparable ǫ i , of order ¯ ǫ which is related to the angle ¯ α by¯ ǫ ≃ π f (1 + cos ¯ α ) ≃ π f ( π − ¯ α ) , (3.22)with ( π − ¯ α ) ≪
1. In this case, P ζ , r T , and f NL becomes P ζ ≃ H π ¯ N ¯ ǫ ≃ H ¯ N f π ( π − ¯ α ) , (3.23) r T ≃ ǫ ¯ N ≃ π ¯ N f ( π − ¯ α ) , (3.24) f NL ≃ π N f . (3.25)Considering ¯ N ≥ f NL . π f , (3.26)which is the same as that in Sec. 3.1.1. Now the spectral index of f NL is given by n f NL ≃ − α − n s − ǫ H ) , (3.27)which is much smaller than (1 − n s − ǫ H ). So the spectral index of f NL is undetectablein this case as well.Because the potential become very flat around the hilltop, one may worry that theclassical motions of those inflatons staying around the hilltop are not reliable. Requiringthat the classical motion of inflaton ¯ φ = f ¯ α/ π per Hubble time be not less than itsquantum fluctuation H/ π yields( π − ¯ α ) & f H π Λ = 2 ξ − Hf , (3.28)where ξ = 8 π Λ / (3 H f ) = (1 − n s − ǫ H ) ≪ On the other hand, from (3.23), ( π − ¯ α )can be estimated by ( π − ¯ α ) = √ ¯ N Hf π ∆ R , (3.29)where ∆ R = p P ζ . Therefore p ¯ N f & π ∆ R ξ . (3.30) In [5], the authors consider | φ hilltop − ¯ φ | & H/ π , namely ( π − ¯ α ) & H/f , which is much looser thanwhat we require, where φ hilltop ≡ f/ – 9 –aking Eq.(3.25) into account, f NL is bounded from above by f NL . ξ ∆ R .p ¯ N . (3.31)WMAP normalization [27] implies P ζ = ∆ R = 2 . × − , and then f NL . × ξ/ √ ¯ N .Since ξ . O (10 − ), f NL . / √ ¯ N . The value of f NL cannot be very large in this case. In this subsection, we consider a class of axion N-flation in which Λ i = Λ for ∀ i , but thedecay constants can be different for different inflatons. For simplicity, we only focus on thecases with only two different decay constants. In this subsubsection we consider a model with potentials V ( φ i ) = Λ (1 − cos α i ) , (3.32) V ( ¯ φ ) = Λ (1 − cos ¯ α ) , (3.33)where α i = 2 πφ i /f for i = 1 , , ..., N f and ¯ α = 2 π ¯ φ/ ¯ f . Here f is assumed to be differentfrom ¯ f . We consider N f ≫ α i satisfy a uniform distribution. The total number of e-folds before the end of inflation is given by N tot ≃ ln 22 π f N f + ¯ f π ln 21 + cos ¯ α . (3.34)We assume ¯ φ stays around the hilltop of its potential. Requiring that the classical motionof inflaton ¯ φ is reliable leads to ( π − ¯ α ini ) & f H π Λ , (3.35)where ¯ α ini denotes the initial value of ¯ α .In this model, the Hubble parameter is roughly related to Λ by H ≃ N f Λ /
3. Becausethe inflaton staying around the hilltop almost does not move, we also have ǫ H ≃ ln 22 (cid:30)(cid:18) N tot − ¯ f π ln 2 π − ¯ α (cid:19) . (3.36)The spectral index of power spectrum and f NL become n s ≃ − ln 2 N tot − ¯ f π ln π − ¯ α " c r ( N f ) N f + π − ¯ α ) c r ( N f ) N f + ( ¯ f /f ) π − ¯ α ) , (3.37) n f NL ≃ N tot − ¯ f π ln π − ¯ α " c r ( N f ) N f + π − ¯ α ) c r ( N f ) N f + ( ¯ f /f ) π − ¯ α ) − d r ( N f ) N f + π − ¯ α ) d r ( N f ) N f + ( ¯ f /f ) π − ¯ α ) , (3.38) Thank C. T. Byrnes for the helpful discussion. – 10 –here c r ( N ) = 1 N N X i =1 − cos α i α i , and d r ( N ) = 1 N N X i =1 − cos α i (1 + cos α i ) , (3.39)which depend on the total number of points and the random choice of the series ‘ r ’ for α i ∈ [0 , π ].Similar to Sec. 3.1.1, we consider that the total number of e-folds corresponds to theCMB scale, namely N ∗ ≃ N tot ≃
60. Considering ¯ f .
1, the total number of e-fold ismainly contributed by φ i for i = 1 , , ..., N f . If f = ¯ f , n s ≃ − /N ∗ and n f NL ≃ f < f , the second term in the bracket of Eq. (3.37) is larger than 4 and then n s . − /N ∗ . For ¯ f > f , we find n s & − /N ∗ . In general, it is difficultto get the analytical estimation for the model with f = ¯ f , but we can expect that theresults should be quite different the model with the same decay constants. The numericalresults are showed in Fig. 2. Here we keep f = 1 / π − ¯ α ini ) /π = 2 × − fixed andexplore the results for two choices of ¯ f . From Fig. 2, we see that a detectable negative n f NL is possibly generated for ¯ f = 1 /
4. We also check the requirement in (3.35) which issatisfied as long as ( π − ¯ α ) /π & × − for ¯ f = 1 /
4. Unfortunately, the spectral index n s ( . .
91) in the case with detectable n f NL is small compared to WMAP 7yr data [27]( n s = 0 . ± . f = 1, the spectral index can fitthe WMAP 7yr data nicely, but the scale dependence of f NL becomes negligibly small.See the red dots in Fig. 2. This setup indicates that a large n f NL can be achieved in axionN-flation. In this subsubsection we consider a more general axion N-flation model in which there aretwo kinds of inflatons who have different decay constants, V ( φ ,i ) = Λ (1 − cos α ,i ) , (3.40) V ( φ ,j ) = Λ (1 − cos α ,j ) , (3.41)where α ,i = 2 πφ ,i /f for i = 1 , , ..., N , and α ,j = 2 πφ ,j /f for j = 1 , , ..., N . Here f is assumed to be different from f . We also consider N , N ≫ α ,i and α ,j satisfy a uniform distribution respectively. In this model the total number ofe-folds before the end of inflation is given by N tot ≃ ln 22 π ( f N + f N ) . (3.42)In this model, the Hubble parameter is related to Λ by H = ( N + N )Λ / ǫ H = π ( N + N ) (cid:0) N /f + N /f (cid:1) . (3.43)– 11 – .88 0.90 0.92 0.94 0.96 0.98 n s f NL - - - - r T detectable region @ CMBPol D
10 20 30 40 50 60 - - - - f NL n f N L Figure 2:
The observables of axion N-flation with different decay constants. Here we adopt f = 1 / π − ¯ α ini ) /π = 2 × − . The blue and green dots correspond to ¯ f = 1 / f = 1. The green dots illustrate the case with detectable scale dependence of f NL . Similarly, we can also derive the spectral index of power spectrum and f NL , namely n s ≃ − π ( N + N ) (cid:18) N f + N f (cid:19) − π N + N c r ( N ) N + c r ( N ) N c r ( N ) f N + c r ( N ) f N , (3.44) n f NL ≃ π N + N (cid:20) c r ( N ) N + c r ( N ) N c r ( N ) f N + c r ( N ) f N − d r ( N ) N + d r ( N ) N d r ( N ) f N + d r ( N ) f N (cid:21) . (3.45)– 12 –gain, if f = f , the scale dependence of f NL is negligibly small and our results recoverthose in Sec. 3.1.1.For the case with f = f , we need to adopt numerical method to explore it. Con-sidering N ∗ ≃ N tot ≃
60, our numerical results are illustrated in Fig. 3. Here we adopt n s f NL - - - - - r T detectable region @ CMBPol D
50 100 150 200 250 - - - - f NL n f N L Figure 3:
The observables of a general axion N-flation with different decay constants. Here weadopt f = 1 / f = 1 /
4. The green dots illustrate the case with detectable scale dependenceof f NL . – 13 – = 1 / N = 4800, f = 1 / N = 8000. We see that a detectable scale dependenceof f NL can be obtained.
4. Discussions
In this paper we work out the spectral index of f NL in general N-flation model by adoptingthe horizon-crossing approximation. For an instance, we focus on the axion N-flationproposed in [5] and we find that f NL can be large enough to be detected in the near future. • All inflatons have the same potential. In the case with the total number of e-foldscorresponding to the CMB scale, f NL is roughly of order O (10) for a uniform distributionof initial angles if the decay constant f is around the Planck scale. Our numerical resultsindicate that f NL is bounded from above by 5 π / f , and τ NL ≃ (2 π/f ) / f / NL . Anotherway to obtain a large f NL is that a few inflatons stay around the hilltop and the summationsin the formula of f NL is mainly contributed by these inflatons. Typically f NL is less thanone hundred in the second case. In both cases the tilt of f NL is negligibly small. Thesmallness of n f NL in these two case comes from the accident cancellation in n multi,i becauseall inflatons have the same potential. • Different inflatons have different decay constants. In this kind of model, the accidentcancellation in n multi,i does not happen and a negative detectable n f NL can be obtained. Itimplies that a scale independent f NL is not generic prediction of N-flation. A more generalaxion N-flation is left to be studied in the future.In addition, we also study the trispectrum for N-flation in the Appendix. A. We findthat there is a universal relation between g NL and τ NL , namely g NL ≃ τ NL /
27, for thecases with large f NL . But the relation between f NL and τ NL depends on detail of themodel.If there is only one inflaton field, the axion N-flation becomes natural inflation. Theslow-roll parameters are roughly given by M p /f and then the slow-roll conditions aresatisfied if f > M p . However such a large decay constant cannot be achieved in somefundamental theories, such as string theory [28, 29]. In [30] the N-flation is supposed torealize a slow-roll inflation even when f < M p by calling multi inflaton fields. On theother hand, in [31] we proposed a conjecture that the effective gravity scale should beΛ G = M p / p N f for the case with multi scalar fields, not M p . If the decay constant ofaxion field is required to be smaller than Λ G , namely N f f <
1, the total number of e-foldsis less than one and the axion N-flation fails.
Acknowledgments
We would like to thank S. A. Kim and A. R. Liddle for useful discussions. QGH issupported by the project of Knowledge Innovation Program of Chinese Academy of Scienceand a grant from NSFC. – 14 – . The trispectrum in N-flation
From Eq. (1.1), the size of trispectrum is characterized by two parameters τ NL and g NL which are defined by τ NL = X i,j,k N ,ij N ,ik N ,j N ,k , ( X l N ,l ) , (A.1) g NL = 2554 X i,j,k N ,ijk N ,i N ,j N ,k , ( X l N ,l ) . (A.2)For N-flation, the above two equations are simplified to be τ NL = X i N ,ii N ,i , ( X l N ,l ) , (A.3) g NL = 2554 X i N ,iii N ,i , ( X l N ,l ) , (A.4)where N ,iii = 2 V i V ′′ i V ′ i − V ′′ i V ′ i − V i V ′′′ i V ′ i . (A.5)We remind the readers that the value of τ NL is bounded from below by ( f NL ) . For theaxion N-flation, we have N ,iii = 2 πf i sin α i (1 + cos α i ) . (A.6)Therefore τ NL = X i f i π − cos α i (1 + cos α i ) ,X j f j π − cos α j α j , (A.7) g NL = 2554 X i f i π (1 − cos α i ) (1 + cos α i ) ,X j f j π − cos α j α j . (A.8)For the case with f i = f for ∀ i , both τ NL and g NL are proportional to 1 /f . Similarto Sec. 3.1.1, we consider N ∗ ≃ N tot ≃
60 and the numerical results for f = 1 / τ NL and g NL are roughly related to f NL and f by τ NL ≃ (cid:18) πf (cid:19) / f / NL , g NL ≃ τ NL . (A.9)Roughly speaking, f NL ∼ /f and then g NL ≃ τ NL ∼ /f which is consistent withour estimation. On the other hand, combining (A.9) and τ NL ≥ ( f NL ) , we get f NL . π / f which is roughly the same as our results in Sec. 3.1.1.– 15 – f NL Τ NL g NL Figure 4:
The non-Gaussianity parameters of axion N-flation for the uniform distribution of α i when f = 1 /
2. Here we consider the case with N ∗ ≃ N tot ≃ The non-Gaussianity parameters of trispectrum for the case in which a few inflatonsstay around the hilltop were worked out in [5]: τ NL = ( f NL ) and g NL = τ NL , whichare different from the previous case unless f NL ≃ π / f .The non-Gaussianity parameters for the model in Sec. 3.2.1 are showed in Fig. 5.Fitting the green dots corresponding to a detectable scale dependence of f NL , we find τ NL ≃ . f . NL , g NL ≃ τ NL . (A.10)Fitting the blue and red dots respectively, we reach the same results τ NL ≃ . f . NL , g NL ≃ τ NL . (A.11)Here the relations between τ NL and f NL depend on the choice of the parameters, such as f , ¯ f and ¯ α , in the model.Similarly we also investigate the non-Gaussianity parameters for the model in Sec. 3.2.2.See the numerical results in Fig. 6. Fitting the blue and green dots respectively, we find τ NL ≃ . f . NL , g NL ≃ τ NL . (A.12)To summarize, there is a universal relation between g NL and τ NL , namely g NL ≃ τ NL . The value of g NL is the same order of τ NL . But the relations between τ NL and f NL are different for different cases. – 16 – f NL Τ NL g NL Figure 5:
The non-Gaussianity parameters in the axion N-flation where N f inflatons have thesame decay constant f , and the decay constant of another inflaton is ¯ f . Here we consider the casewith N ∗ ≃ N tot ≃
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