Reanalyzing the upper limit on the tensor-to-scalar perturbation ratio r_T in a quartic potential inflationary model
RReanalyzing the upper limit on the tensor-to-scalar perturbation ratio r T ina quartic potential inflationary model R. Kabir a, ∗ , A. Mukherjee a, , D. Lohiya a, a Department of Physics and Astrophysics, University of Delhi, Delhi-110007, India
Abstract
We study the polynomial chaotic inflation model with a single scalar field in a double well quartic potentialwhich has recently been shown to be consistent with Planck data. In particular, we study the effects oflifting the degeneracy between the two vacua on the inflationary observables, i.e., spectral index n s andtensor-to-scalar perturbation ratio r T . We find that removing the degeneracy allows the model to satisfythe upper limit constraints on r T from Planck data, provided the field starts near the local maximum.We also calculate the scalar power spectrum and non-Gaussianity parameter f NL for the primordial scalarperturbations in this model.
1. Introduction
Inflation is regarded as the standard cosmolog-ical paradigm to describe the physics of the veryearly Universe. It leads to a causal mechanism togenerate almost scale invariant fluctuations on cos-mological scales, with small deviations that followfrom the precise microphysics of inflation. This pre-diction is consistent with the recently announcedmeasurements of the cosmic microwave background(CMB) anisotropies by the Planck satellite. Thelatest data allow us to constrain the inflationarymodel besides giving a slightly red tilted spectral in-dex n s = 0.9603 ± n s = 1 at over 5 σ [1]. The Planck data alsoprovide observational bounds on primordial non-Gaussianity parameter, i.e., local f NL = 2 . ± . V ( φ ) = φ n with n ≥
2, which were already dis-favored or marginally disfavored by WMAP. (Thisdoes not apply to alternative inflationary scenarios ∗ Corresponding author
Email addresses: [email protected] (R.Kabir), [email protected] (A. Mukherjee), [email protected] (D. Lohiya) e.g., warm inflation [3]). However Croon et al. [4]demonstrated that the double well degenerate po-tential V ( φ ) = Aφ ( v − vBφ + φ ) is consistentwith Planck data, although with severe constraintson the initial condition for φ and on the allowedrange of B . The polynomial quartic potential withdouble well is the most studied potential in a varietyof settings. It is also well motivated by the physicsbeyond the Standard Model, viz. supergravity andsuperstring theories [5]. The characteristic inter-esting features [4, 6] of the potential have been theprimary motivation for a lot of effort to explore itsconsistency with the Planck data [1].
2. Basic Formalism
We review the formalism in [4] to set up our no-tation. For an inflationary model described by thepotential V ( φ ) = Aφ ( v − vBφ + φ ), the slow-rollparameters are defined as (cid:15) ≡ M (cid:18) V φ V (cid:19) =2 M (cid:0) v − vBφ + 2 φ (cid:1) φ ( v − vBφ + φ ) , (1) η ≡ M (cid:18) V φφ V (cid:19) = 2 M (cid:0) v − vBφ + 6 φ (cid:1) φ ( v − vBφ + φ ) (2) Preprint submitted to Elsevier October 6, 2018 a r X i v : . [ a s t r o - ph . C O ] A ug nd ξ ≡ M (cid:18) V φ V φφφ V (cid:19) (3)= 24 M ( − vB + 2 φ ) (cid:0) v − vBφ + 2 φ (cid:1) φ ( v − vBφ + φ ) where V φ ≡ dVdφ , V φφ ≡ d Vdφ and V φφφ ≡ d Vdφ . (4)In terms of the slow-roll parameters, the scalarspectral index is expressed as n s = 1 − (cid:15) + 2 η (5)and the tensor-to-scalar ratio as r T = 16 (cid:15) . (6)Finally the number of e-folds is given by N = − M (cid:90) φ e φ i ( VV φ ) dφ (7)where φ e,i are the values of φ at the end and begin-ning of the inflationary epoch.The scalar power spectrum P s ( k ) is described interms of the spectral index n s ( k ) by P s ( k ) = A exp[( n s −
1) ln( k/k ) + 12 α s ln ( k/k )](8)where α s ≡ dn s d ln k (9)which in terms of slow-roll parameters is α s = 18 π [ − ξ η(cid:15) − (cid:15) ] . (10) B = 8 (cid:144) B = V (cid:72) x (cid:76) xB = B = (a)
V(x) x (b)
Fig. 1: (color online). (a) An illustration of thepotential for different values of B ; (b) Inset in (a)zoomed to highlight the behavior as x → + . On parameterizing the field as φ → xv , where x is dimensionless, for the chosen potential in [4], theinflationary observables reduce to : (cid:15) = 2 M (cid:0) x − xB (cid:1) v x (1 + x − xB ) , (11) η = 2 M (1 + 6 x ( x − B )) v x (1 + x − xB ) , (12) ξ = 24 M (2 x − B ) (cid:0) x − xB (cid:1) v x (1 + x − xB ) (13) In [4], a minus sign is missing in the expression for N .Further, the power of (1 − x ) in their Eq.(10), generalisedin our Eq.(13), should be 4 and not 2. This error creepsinto the numerical calculation (in table) also. A summary ofstandard formulae can be found in [7]. N = − v M (cid:90) x e x i ( VV x ) dx . (14)We point out that the range of possible initialfield values is severely restricted if B > (cid:112) / N will not converge on choos-ing an initial field value beyond the local max-imum (see Fig. 6). This rules out the possi-bility of investigation of deviation from slow-rollor multiphase-inflation. Punctuated inflation [6]is the simplest and interesting example of multi-phase-inflation which occurs in our model when B ∼ (cid:112) / P s ( k ) and f NL as B is variedfrom 1 to 0 within the slow-roll regime. Our analy-sis suggests a broad range of B for which predictionin the ( n s , r T ) plane lies within the 1 σ level. The global shape of the potential depends on thecoefficient B as shown in Fig. 1. For B = 1, thereis one local maximum at x = 0 . x = 0 and 1.0. For B < .
0, theminimum at x = 1 . B < (cid:112) /
9. In-terestingly, if B marginally satisfies the inequality,there appears a flat plateau at around the point ofinflection. If one starts at a suitable value of thefield beyond the point of inflection in the above po-tential, it is found that one can achieve two epochsof slow roll inflation sandwiching a brief period ofdeparture from inflation (lasting for a little less thanone e-fold ), a scenario which has been dubbed aspunctuated inflation [6]. In fact, it is the pointof inflection, around which the potential exhibits aplateau with an extremely small curvature, whichpermits such an evolution to be possible.It needs to be kept in mind that this potentialdoes not provide a single inflation model, but rathera class of the inflationary models as we vary B from0 to 1. Fig. 1 describes the potential profile for fourvalues of B . Interestingly, there are two ways tomimic the quadratic potential case: (a) by chang-ing the class of potential by decreasing B from 1to 0 and (b) by taking the initial field value very near to zero (near the first minimum) [4]. However,as shown in Fig. 1b, each member of the class isasymptotic to the quadratic case if x i < .
12 irre-spective of the value of B . N (cid:61) N (cid:61) (cid:72) n s (cid:76) T e n s o r (cid:45) t o (cid:45) S ca l a r R a ti o (cid:72) r . (cid:76) (a) (cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199) (cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247)(cid:247) (cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228)(cid:228) (cid:228) (cid:228) (cid:228) (cid:228) (cid:228)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170)(cid:170) (cid:170) (cid:170) (cid:170) (cid:170) (cid:170) (cid:170) (cid:170) (cid:170) (cid:170)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169)(cid:169) (cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199)(cid:199) (cid:72) n s (cid:76) T e n s o r (cid:45) t o (cid:45) S ca l a r R a ti o (cid:72) r . (cid:76) (b) Fig. 2: (color online). The reproduction of the workby Nakayama et al. [5]. (a) The two curves corre-spond to N = 50 (green) and 60 (black). Each curveis generated by varying x i from 0.005 to 1.0 in stepsof 0.012. In this plot we have fixed B = 0 .
93 forboth curves. (b) The seven curves correspond to B:0.96, 0.95, 0.94, 0.92, 0.90, 0.80, 0.70 for N = 60.Each curve is again generated by varying x i as in(a).3 (cid:61) N (cid:61) N (cid:61) r T (a) N (cid:61) N (cid:61) N (cid:61) n s (b) N (cid:61) N (cid:61) N (cid:61) (cid:72) n s (cid:76) T e n s o r (cid:45) t o (cid:45) S ca l a r R a ti o (cid:72) r . (cid:76) (c) Fig. 3: (color online). (a) The tensor-to-scalar per-turbation ratio r T as a function of B for N = 50(red), 60 (green) and 70 (blue). In this plot we havetaken x i = 0 . r T = 0 .
11. (b) spec-tral index n s as a function of B with same colorcoding as in Fig. (a). Two horizontal lines showthe constraint on n s , i.e., n s = 0 . ± . n s , r T ) plane correspondingto (a) and (b). For example, each red point in the( n s , r T ) plane corresponds to a value of r T ( redpoint in (a)) and a value of n s (red point in (b)),both calculated for a particular value of B . N (cid:61) N (cid:61) N (cid:61) r T (a) N (cid:61) N (cid:61) N (cid:61) n s (b) N (cid:61) N (cid:61) N (cid:61) (cid:72) n s (cid:76) T e n s o r (cid:45) t o (cid:45) S ca l a r R a ti o (cid:72) r . (cid:76) (c) Fig. 4: (color online). Same as in Fig. 3, but herewe have taken x i = 0 . . Numerical Results We set out to explore the model for the entirerange of B values from 0 to 1 and for an arbitraryinitial field value φ i (or equivalently x i ). We needto keep in mind that an arbitrary large initial fieldvalue can be taken only for those members of theclass for which B < (cid:112) /
9, otherwise either devia-tion from slow roll will occur (when B = (cid:112) / B > (cid:112) / -5 -4 -3 -2 k1.21.31.41.51.61.71.81.92.0 P S ( k )
1e 9
Scalar Power Spectrum P S ( k ) as function of k N=70N=65 (a) -5 -4 -3 -2 k0.01440.01420.01400.01380.01360.01340.01320.01300.0128 f N L Behaviour of f NL as function of k N=70N=65 (b) Fig. 5: (color online). (a) The scalar power spec-trum as a function of k plotted on a logarithmicscale for B = 0 .
96 and x i = 0 . f NL as a function of k on a logarithmicscale. The values of B and x i are the same as in(a). ( n s , r T ) plane Fig. 3 exhibits how our analysis differs from thepast work of Nakayama et al. [5] (see Fig. 2), whereeffects mainly due to varying initial field value areconsidered. In contrast, we have varied B continu-ously for each fixed initial field value—from the onenear the minimum at x = 0 to the other near thelocal maximum—and for three e-fold values: N =50, 60 and 70. However, we have shown the analysisonly for the two extreme initial field values becauseof clarity and reasons of space (see Figs. 3 and 4).Plots only on the ( n s , r T ) plane, as done in [5],hide valuable information of initial conditions forwhich the model lies within 1 σ . Notice that verynear to x = 0 . n s , r T ) plane) start from the quadratic potential(the black line), found concordant with the Planckdata, and then graze the contour as x is increased;see Figs. 2 and 7.From Fig. 3a, it is clear that for N = 50 (red)and 60 (green), this model is completely above thebound imposed by Planck ( r T < .
11) for the entirerange of B —here the field starts near the origin;whereas for N = 70 (blue), the model is at themargin of this bound.When we contrast the above situation with Fig.4a, we find that the upper limit on r T is satisfied forsufficiently large values of B (red and green curvesfall below the horizontal line for B (cid:38) .
75 and B (cid:38) .
65 respectively). It is emphasized that this is thecase when the field starts near the local maximum( x ∼ . N = 50) is within 1 σ ofthe Planck value. Although there is no appreciabledisagreement between Planck and BICEP2 resultsas clarified in [8], a slightly larger tensor-to-scalarratio ( r T ∼ . B ∼ .
3) as shown in Fig. 4a.Another way to realize a large r T is by assumingthe non-monotonicity of the slow-roll parameter (cid:15) ,as considered in a model somewhat similar to ours[9]. Non-Gaussianity is now a standard cosmologicalobservable, comparable to the spectral index ( n s )and tensor-to-scalar ratio r T , and is a powerful dis-criminant between competing models. It is there-fore desirable to study aspects of non-Gaussianityfor this model too. We have numerically calculatedthe equilateral limit of the local f NL parameter [10].5n Fig. 5a, the power spectrum is plotted as a func-tion of k , from which it is clear that the spectrum isnot scale invariant on any scale (otherwise it wouldhave been a horizontal straight line). In Fig. 5b,the non-Gaussianity parameter f NL are plotted asa function of k , from which we observe that thevery low values of | f NL | ( ∼ .
01) lie in the rangepredicted by Planck data for local f NL = 2.7 ± r T [11](see also [12]), is compatible with our results also[13]. This observation is also consistent with thefact that single-field slow roll models predict a verysmall non-Gaussianity [14].
4. Summary and Conclusion
In this Letter, we have explored in detail the(
B, φ i ) space within the slow-roll regime for thequartic potential. We have explicitly shown thaton varying B , interesting patterns on the variationof n s and r T arise only when field dynamics cap-tures the deviation from the quadratic case, i.e.,the field should start near the local maximum, notnear the origin. Even though this parameter spacehas been moderately studied in [5], the authors re-stricted their analysis to a few discrete values of B .Similarly Ellis et al. [15] have carried out a statisti-cal study with just four explicit values of numericalcoefficient in a two field model to explore concor-dance with the preferred observational values of n s and r T .One of the main well-known problems—constraints from Planck on the upper limit on r T —has been reanalyzed here and we find that thereis no respite for the model solely due to remov-ing degeneracy between the vacua. On the otherhand, merging variations of B with the field starting near the local maximum, the model’s predictionsfor ( n s , r T ) lie well within 1 σ for a large part of the( B, φ i ) space (see Fig. 4). Further, our Fig. 4aclearly shows that large r T can be generated whenthe potential is non-degenerate and the field startsnear the local maximum. On non-Gaussianity as-pects ( which were not considered in previous works[4],[5]), we have explicitly calculated the local f NL parameter for each mode of our interest and verifiedthat this model indeed satisfies the constraint im-posed by Planck data on non-Gaussianity [2], andis compatible with BICEP2 measurements. Acknowledgments
RK thanks Dheeraj Kumar Hazra for useful com-munication and Tarun Choudhari for assistancewith Latex. We are grateful to Rudnei O. Ramosand Sayantan Choudhury for pointing out theirworks [3] and [9, 12] respectively. We also thankan anonymous reviewer whose comments helped ushighlight the subtle points of our study. The workof RK was partially supported by CSIR, New Delhiwith the award of JRF (No. 09/045(0930)/2010-EMR-I).
Appendix (a)(b)(c)
Fig. 6: Schematic plot of integrand for N for (a) B = 0 .
92, (b) B = 0 .
94 and (c) B = 0 .
97; see Eq.14.6n Fig. 6, we plot the integrand for N for 3 valuesof B . (As already mentioned, one minimum disap-pears if B < (cid:112) / . VV x develops as B increasesabove (cid:112) /
9. One cannot take any point on the x axis beyond the local maximum, which is near x = 0 . B (seeFig. 1).Fig. 7 is from Planck Data on which we super-impose our predictions—following [4] and [5]. n s ) . . . . . . T e n s o r - t o - S c a l a r R a t i o ( r . ) C o n v e x C o n c a v e Planck +WP
Planck +WP+highL
Planck +WP+BAONatural InflationPower law inflationLow Scale SSB SUSY R Inflation V ∝ φ / V ∝ φ V ∝ φ V ∝ φ N ∗ =50 N ∗ =60 Fig. 7: (color online). Marginalized joint 68% and95% CL regions for n s and r . . Also shown areobservational 1 σ (dark) and 2 σ (light) constraintsfrom the Planck satellite [1]: Planck + WMAP po-larization (gray), Planck + WMAP polarization +high- (cid:96) CMB measurement (red), Planck + WMAPpolarization + baryon acoustic oscillation (blue).Filled circles connected by line segments show thepredictions from chaotic inflation with V ∝ φ (green), φ (black), φ (yellow), φ / (red) and R inflation (orange), for N = 50 (small circle)–60 (bigcircle). The purple band shows the prediction ofnatural inflation [1]. References [1] P. Collaboration, P. A. R. Ade, Planck 2013 results.XXII. Constraints on inflation arXiv:1303.5082 .[2] P. Collaboration, P. A. R. Ade, Planck 2013 re-sults. XXIV. constraints on primordial non-gaussianity,arXiv:1303.5084 [astro-ph].URL http://arxiv.org/abs/1303.5084 [3] R. O. Ramos, L. A. da Silva, Power spectrum for in-flation models with quantum and thermal noises, Jour-nal of Cosmology and Astroparticle Physics 2013 (03)(2013) 032–032, arXiv: 1302.3544. doi:10.1088/1475-7516/2013/03/032 .URL http://arxiv.org/abs/1302.3544 [4] D. Croon, J. Ellis, N. E. Mavromatos, Wess-zuminoinflation in light of planck, arXiv e-print 1303.6253,physics Letters B 724 (2013), 165 (Mar. 2013).URL http://arxiv.org/abs/1303.6253 [5] K. Nakayama, F. Takahashi, T. T. Yanagida, Polyno-mial chaotic inflation in the planck era, arXiv e-print1303.7315, Phys.Lett.B725, 111 (2013) (Mar. 2013).URL http://arxiv.org/abs/1303.7315 [6] R. K. Jain, P. Chingangbam, J.-O. Gong, L. Sri-ramkumar, T. Souradeep, Punctuated inflationand the low CMB multipoles, J. Cosmol. As-tropart. Phys. 2009 (01) (2009) 009, 00049. doi:10.1088/1475-7516/2009/01/009 .URL http://iopscience.iop.org/1475-7516/2009/01/009 [7] E. W. Kolb, M. S. Turner, The early universe, Addison-Wesley, Reading, Mass., 1990.[8] B. Audren, D. G. Figueroa, T. Tram, A note ofclarification: BICEP2 and planck are not in tension,arXiv:1405.1390 [astro-ph]arXiv: 1405.1390.URL http://arxiv.org/abs/1405.1390 [9] S. Choudhury, A. Mazumdar, An accurate boundon tensor-to-scalar ratio and the scale of infla-tion, Nuclear Physics B 882 (2014) 386–396, 00000. doi:10.1016/j.nuclphysb.2014.03.005 .URL [10] D. K. Hazra, L. Sriramkumar, J. Martin, BINGO:a code for the efficient computation of the scalarbi-spectrum, Journal of Cosmology and Astroparti-cle Physics 2013 (05) (2013) 026–026, arXiv:1201.0926[astro-ph, physics:gr-qc, physics:hep-ph, physics:hep-th]. doi:10.1088/1475-7516/2013/05/026 .URL http://arxiv.org/abs/1201.0926 [11] G. D’Amico, M. Kleban, Non-gaussianity afterBICEP2, arXiv:1404.6478 [astro-ph, physics:hep-th]arXiv: 1404.6478.URL http://arxiv.org/abs/1404.6478 [12] S. Choudhury, A. Mazumdar, Reconstructing infla-tionary potential from BICEP2 and running of ten-sor modes, arXiv:1403.5549 [astro-ph, physics:gr-qc,physics:hep-ph, physics:hep-th]00002.URL http://arxiv.org/abs/1403.5549 [13] D. K. Hazra, A. Shafieloo, G. F. Smoot, A. A.Starobinsky, Whipped inflation, arXiv:1404.0360 [astro-ph, physics:gr-qc, physics:hep-ph, physics:hep-th]arXiv:1404.0360.URL http://arxiv.org/abs/1404.0360 [14] J. Maldacena, Non-gaussian features of primordialfluctuations in single field inflationary models, Jour-nal of High Energy Physics 2003 (05) (2003) 013–013, 01332 arXiv:astro-ph/0210603. doi:10.1088/1126-6708/2003/05/013 .URL http://arxiv.org/abs/astro-ph/0210603 [15] J. Ellis, N. E. Mavromatos, D. J. Mulryne, Ex-ploring two-field inflation in the wess-zumino model,arXiv:1401.6078 [astro-ph, physics:hep-ph, physics:hep-th]arXiv: 1401.6078.URL http://arxiv.org/abs/1401.6078http://arxiv.org/abs/1401.6078