Polynomial Chaotic Inflation in the Planck Era
aa r X i v : . [ h e p - ph ] M a y UT-13-18 TU-932 IPMU13-0070
Polynomial Chaotic Inflation in the Planck Era
Kazunori Nakayama a,c , Fuminobu Takahashi b,c and Tsutomu T. Yanagida c a Department of Physics, University of Tokyo, Tokyo 113-0033, Japan b Department of Physics, Tohoku University, Sendai 980-8578, Japan c Kavli IPMU, TODIAS, University of Tokyo, Kashiwa 277-8583, Japan
We propose a chaotic inflation model in supergravity based on polynomial interactions of the inflaton.Specifically we study the chaotic inflation model with quadratic, cubic and quartic couplings in thescalar potential and show that the predicted scalar spectral index and tensor-to-scalar ratio can liewithin the 1 σ region allowed by the Planck results. There is overwhelming observational evidence for infla-tion in the early Universe [1]. Not only does the inflationprovide beautiful explanations for the observed homo-geneity and isotropy of the Universe, but it also predictstiny density fluctuations with distinct properties; namely,they are nearly scale-invariant and Gaussian. Both prop-erties have been confirmed with unprecedented accuracyby the Planck satellite [2]. The last missing piece towardthe observational confirmation of the inflation theory willbe detection of the primordial gravitational waves gen-erated during inflation. Although the strength of thegravitational waves depends on inflation models, a sim-ple class of models, called chaotic inflation [3], predictsgravitational waves with a large amplitude within thereach of the Planck satellite and future CMB observa-tion experiments.The recent Planck observations have tightly con-strained possible inflation models [2]. Specifically, achaotic inflation model based on a quartic potential ishighly disfavored by the observation, and that based ona quadratic potential is marginally consistent with theobservation at 2 σ level. Those with a linear or fractionalpower potential lie outside 1 σ but within 2 σ allowed re-gion. When combined with the polarization data, thePlanck data will be able to discover or put even severeconstraints on those inflation models.The inflaton field slowly rolls down to the minimumof its potential during inflation, and it travels more thanthe Planck scale in the chaotic inflation. In order to con-trol the inflaton dynamics over such large variation, it isworthwhile to build a successful chaotic inflation modelin a framework of supergravity or superstring theory. Inthis letter we extend the chaotic inflation model in su-pergravity proposed by Kawasaki, Yamaguchi, and oneof the present authors (Yanagida) [4], by allowing higherorder interactions of the inflaton in the superpotential.We start with the following K¨ahler and super- potentials, a K = c ( φ + φ † ) + 12 ( φ + φ † ) + | X | + · · · , (1) W = X (cid:0) d + d φ + d φ + d φ + · · · (cid:1) , (2)where φ and X are chiral superfields, and c , and d i are real and complex numerical coefficients, and the dotsrepresent higher-order interactions. In the following, thefirst term in the superpotential, d , is taken to be zeroby an appropriate shift of φ . Here and in what followswe adopt the Planck unit where M P ≃ . × GeVis set to be unity. The K¨ahler potential respects a shiftsymmetry φ → φ + iC with a real constant C . Thanksto the shift symmetry, the imaginary component of φ , ϕ ≡ Im[ √ φ ], does not appear in the K¨ahler poten-tial, which enables us to identify ϕ with the inflaton.The real component, σ ≡ Re[ √ φ ], on the other hand,receives supergravity corrections and cannot go beyondthe Planck scale. We also assume that X and φ havean R -charge +2 and 0, respectively. The superpotentialterms involving φ explicitly break the shift symmetry,and hence | d | , | d | , · · · are much smaller than unity, andcan be viewed as the order parameters of the shift sym-metry breaking.In our analysis we focus on the d and d terms inthe superpotential, and drop the higher-order terms, as-suming they are sufficiently suppressed. b Let us define m ≡ | d | , λ ≡ | d | and θ ≡ arg[ d ∗ d ] for later use. Forthe moment we take c = 0, and the effect of non-zero c will be discussed later.Substituting (1) and (2) into the general expression forthe scalar potential in supergravity V = e K h K i ¯ j ( D i W )( D ¯ j ¯ W ) − | W | i , (3) a This type of generalization was considered in Refs. [5, 6]. InRef. [5], it was pointed out that the form of the superpotential, W = Xφ n , leads to a chaotic inflation based on ϕ n potential.General inflation models based on W = Xf ( φ ) were studied inRefs. [6, 7]. See Refs. [8–10] for the realization of inflation modelsbased on a linear or fractional power potential in supergravityand also Refs. [11–13] in string theory. b This will be the case if the effective cut-off scale in the super-potential is more than one order of magnitude larger than thePlanck scale. we find V ≃ ϕ (cid:18) m − √ mλ sin θ ϕ + λ ϕ (cid:19) . (4)Here we have assumed that both σ and X are stabilized at | σ | ≪ h X i ∼ h W i is suppressed dueto h X i ∼
0, which enables the inflation for ϕ ≫
1. Theschematic picture of the scalar potential is shown in Fig. 1for a few different values of θ . The quadratic chaotic in-flation model is reproduced in the limit of λ →
0. Forsin θ >
0, the second term in (4) gives a negative contri-bution to the scalar potential, making the potential flat-ter or negatively curved at large ϕ . It is worth stressingthat a variety of the inflation models can be realized sim-ply by taking the d and d terms in (2) with a differentrelative phase. Note that this is effectively a single-fieldinflation, since the other degrees of freedom can be safelystabilized during inflation.Before proceeding further, let us mention the worksin the past. In Ref. [14], the inflation model basedon the scalar potential equivalent to (4) was studied ina non-supersymmetric framework, where the inflaton isa real scalar field. Recently, the model was revisitedand its global supersymmetric extension was proposedin Ref. [15], where it was shown that the predicted spec-tral index and the tensor-to-scalar ratio can be consistentwith the Planck data for θ ≈ π/
2. However, the infla-ton is a complex scalar field and it is not clear how tostabilize the inflationary trajectory for θ = π/
2. We alsonote that, in supergravity, the large expectation of their h W i would lead to a negative scalar potential for the in-flaton field value greater than the Planck scale, spoilingthe inflation. This is known as one of the difficulties toimplement chaotic inflation in supergravity with a sin-gle superfield. The latter problem can be avoided in theno-scale supergravity [16–18].Now let us continue with our discussion of the inflationmodel (4). The global shape of the potential depends onthe relative phase θ as shown in Fig. 1. In a case of θ = π/
2, there appear one local maximum at ϕ = m/ ( √ λ ) ≡ ϕ t and two degenerate minima at ϕ = 0 and 2 ϕ t . c Forsmaller values of θ , the minimum at ϕ = 0 is lifted. Aslong as there are such local minimum and maximum, theinitial value of the inflaton field should be below the localmaximum since otherwise the inflaton would be trappedin the false vacuum. The false vacuum disappears for | sin θ | < √ /
3. In this case successful chaotic inflationtakes place for an arbitrary large initial field value [3].Interestingly, if the relative phase θ marginally satisfies c In the case of θ = π/
2, the inflaton dynamics is equivalent to thatin the spontaneous symmetry breaking model first considered in[19]. See also Ref. [20].
FIG. 1: The schematic picture of the scalar potential (4). the inequality, there appears a flat plateau at around ϕ = ϕ t . As we shall see shortly, the predicted spectral indexas well as the tensor-to-scalar ratio are then significantlymodified and they can lie within the 1 σ allowed region.The chaotic inflation with quadratic potential, V = m ϕ /
2, is reproduced in the limit ϕ t ≫ O (10). Asmentioned before, the model is at odds with the recentPlanck result at the 2 σ level. An interesting situationappears when ϕ t ∼ O (10). In this case, the last 50 or60 e-foldings of the inflation occurs around ϕ ∼ ϕ t wherethe potential is flatter due to the contribution from the ϕ and ϕ terms. As a result, the inflation energy scalecan be significantly lowered compared with the quadraticchaotic inflation model, when the Planck normalizationof the density perturbations is imposed. Thus it will beable to relax the tension between the prediction of thechaotic inflation model and observations.We have numerically solved the equation of motion of ϕ with the scalar potential (4) and calculated the scalarspectral index, n s = 1 − ǫ + 2 η , and the tensor-to-scalarratio, r = 16 ǫ , where 2 ǫ = ( V ′ /V ) and η = V ′′ /V evaluated at ϕ = ϕ ( N e ) [21]. Here ϕ ( N e ) is calculatedfrom N e = Z ϕ ( N e ) ϕ end VV ′ dϕ, (5)where ϕ end denotes the field value at the end of inflation,at which max[ ǫ, | η | ] = 1. The results are shown in the toppanel of Fig. 2 N e = 60 (red, solid) and 50 (red, dashed)together with observational constraints from the Plancksatellite [2]. Here we have taken θ = 23 π/
60, whichmarginally satisfies the condition for the disappearanceof the false vacuum leading to a flat plateau in the infla-ton potential. In the bottom panel of Fig. 2, results forvarious values of θ are shown for N e = 60. We can clearlysee that almost entire region allowed by the Planck datacan be covered by our model, and importantly, the pre-dicted value of r is testable in future/on-going B-mode FIG. 2: (Top) The prediction of polynomial chaotic inflationmodel is shown in ( n s , r ) plane for N e = 60 (red, solid) and50 (red, dashed). In this plot we have taken θ = 23 π/ σ (dark) and 2 σ (light) con-straints from the Planck satellite [2]: Planck + WMAP po-larization (gray), Planck + WMAP polarization + high- ℓ CMB measurement (red), Planck + WMAP polarization +baryon acoustic oscillation (blue). Filled circles connectedby line segments show the predictions from chaotic inflationwith V ∝ ϕ (green), ϕ (black), ϕ (yellow), ϕ / (red) and R inflation (orange), for N e = 50 (small circle)–60 (big cir-cle). Purple band shows the prediction of natural inflation [2].(Bottom) Same as top panel, but for various values of θ . Herewe have taken N e = 60. polarization search experiments. Note that the predicted( n s , r ) lies within the 1 σ region for θ = π/ ∼ π/
2. Inthe following we take θ = 23 π/
60 unless otherwise stated.In Fig. 3, the scalar spectral index as a function of ϕ t isshown. It is seen that in the large ϕ t limit, the predictionapproaches to that of the chaotic inflation with quadraticpotential, as expected. By choosing ϕ t = O (10), the pre-dicted n s and r can lie within the 1 σ region allowed by thePlanck data. In particular, the predicted r is testable in n s φ t / M P N e =60N e =50 FIG. 3: The scalar spectral index as a function of ϕ t /M P for θ = 23 π/ -7 -7 -7 -7 -6 -6 -6 -6 -6 -6 -6 λ mN e =60N e =50 FIG. 4: The parameters λ and m (in Planck unit) whichreproduce the Planck normalization of the CMB anisotropyfor θ = 23 π/ future/on-going B-mode polarization search experiments.While the predicted n s and r depends only on m/λ ,the Planck normalization of the CMB anisotropy fixes arelation between m and λ . We have confirmed that theyare approximately given by m ≃ (1 − × GeV and λ < ∼ × − for the spectral index allowed by the Planckdata as shown in Fig. 4. In terms of d and d , they areroughly related to each other as | d | = O (0 . | d | forthe parameters of our interest.The reheating can be induced by introducing the fol-lowing coupling to the Higgs doublets in the superpoten-tial: W ⊃ κXH u H d , (6)with a numerical constant κ . This allows the φ ( X ) decayinto the Higgs boson and higgsino pair. Note here that φ and X are maximally mixed with each other to formthe mass eigenstate around the vacuum. The reheatingtemperature is estimated as T R ∼ × GeV (cid:16) κ − (cid:17) (cid:16) m GeV (cid:17) / . (7)Therefore, thermal leptogenesis [22] works, and the WinoLSP produced by the decay of thermally produced grav-itinos can account for the observed dark matter abun-dance in a heavy gravitino scenario.Finally we mention the gravitino production from theinflaton decay [23–26]. Assuming that the dominantchannel of the gravitino production is that through theinflaton decay into the hidden hadrons [25], the gravitinoabundance is given by n / s ≃ × − (cid:18) GeV T R (cid:19) (cid:18) h φ i GeV (cid:19) (cid:16) m GeV (cid:17) , (8)where n / and s are the gravitino number density andthe entropy density, respectively. Thus the gravitinoabundance crucially depends on the VEV of the inflaton,and its impact on cosmology depends on the gravitinomass. In general, c in (1) is non-zero and then the VEVis effectively given by h φ i ∼ c . To be concrete, we as-sume a heavy gravitino, m / = O (10 − c < ∼ − to avoid the overproduction of theLSPs produced by the gravitino decay. Alternatively, ifthe R -parity is violated by a small amount, the LSPs thusproduced soon disappear before the big-bang nucleosyn-thesis begins. There is no cosmological gravitino problemin such a case. d In summary, we have proposed a polynomial chaoticinflation model defined in (1) and (2). Focusing on the d and d terms in the superpotential, we have shownthat it is possible to realize a wide variety of inflationmodels due to the different relative phase θ . Importantly,the inflation dynamics is described by single-field infla-tion, since the other degrees of freedom can be safelystabilized. We studied the inflaton dynamics for variousvalues of θ , and showed that the predicted spectral indexand tensor-to-scalar ratio can lie within the 1 σ regionallowed by the Planck results for π/ < ∼ θ < ∼ π/
2. Inter-estingly, the tensor-to-scalar ratio is relatively large, andwill be testable in future/on-going B-mode polarizationsearches.
Acknowledgments
We are grateful to Andrei Linde for letting us knowabout Ref. [14] and his works [19, 20]. TTY thanks d See also Ref. [27] to suppress the gravitino overproduction fromthe inflaton decay.
John Ellis for useful communication. This work wassupported by the Grant-in-Aid for Scientific Researchon Innovative Areas (No. 21111006 [KN and FT],No.23104008 [FT], No.24111702 [FT]), Scientific Re-search (A) (No. 22244030 [KN and FT], 21244033 [FT],22244021 [TTY]), and JSPS Grant-in-Aid for Young Sci-entists (B) (No.24740135) [FT]. This work was also sup-ported by World Premier International Center Initiative(WPI Program), MEXT, Japan. [1] A. H. Guth, Phys. Rev. D , 347 (1981); A. A. Starobin-sky, Phys. Lett. B , 99 (1980); K. Sato, Mon. Not. Roy.Astron. Soc. , 467 (1981).[2] P. A. R. Ade et al. [ Planck Collaboration],arXiv:1303.5082 [astro-ph.CO].[3] A. D. Linde, Phys. Lett. B , 177 (1983).[4] M. Kawasaki, M. Yamaguchi, T. Yanagida, Phys. Rev.Lett. , 3572 (2000) [hep-ph/0004243].[5] M. Kawasaki, M. Yamaguchi, T. Yanagida, Phys. Rev.D , 103514 (2001) [hep-ph/0011104].[6] R. Kallosh, A. Linde, JCAP , 011 (2010)[arXiv:1008.3375 [hep-th]].[7] R. Kallosh, A. Linde, T. Rube, Phys. Rev. D , 043507(2011) [arXiv:1011.5945 [hep-th]].[8] F. Takahashi, Phys. Lett. B , 140 (2010)[arXiv:1006.2801 [hep-ph]].[9] K. Nakayama, F. Takahashi, JCAP , 009 (2010)[arXiv:1008.2956 [hep-ph]]; JCAP , 010 (2011)[arXiv:1008.4457 [hep-ph]]; JCAP , 039 (2010)[arXiv:1009.3399 [hep-ph]].[10] K. Harigaya, M. Ibe, K. Schmitz, T. T. Yanagida,arXiv:1211.6241 [hep-ph].[11] E. Silverstein, A. Westphal, Phys. Rev. D , 106003(2008) [arXiv:0803.3085 [hep-th]].[12] L. McAllister, E. Silverstein, A. Westphal, Phys. Rev. D , 046003 (2010) [arXiv:0808.0706 [hep-th]].[13] H. Peiris, R. Easther, R. Flauger, arXiv:1303.2616 [astro-ph.CO].[14] C. Destri, H. J. de Vega, N. G. Sanchez, Phys. Rev. D , 043509 (2008) [astro-ph/0703417].[15] D. Croon, J. Ellis, N. E. Mavromatos, arXiv:1303.6253[astro-ph.CO].[16] E. Cremmer, S. Ferrara, C. Kounnas and D. V. Nanopou-los, Phys. Lett. B , 61 (1983).[17] J. R. Ellis, A. B. Lahanas, D. V. Nanopoulos and K. Tam-vakis, Phys. Lett. B , 429 (1984); J. R. Ellis, C. Koun-nas and D. V. Nanopoulos, Nucl. Phys. B , 373(1984); J. R. Ellis, C. Kounnas and D. V. Nanopoulos,Nucl. Phys. B , 406 (1984).[18] H. Murayama, H. Suzuki, T. Yanagida, J. ’i. Yokoyama,Phys. Rev. D , 2356 (1994) [hep-ph/9311326].[19] A. D. Linde, Pisma Zh. Eksp. Teor. Fiz. , 606 (1983)[JETP Lett. , 724 (1983)]; Phys. Lett. B , 317(1983).[20] R. Kallosh, A. D. Linde, JCAP , 017 (2007)[arXiv:0704.0647 [hep-th]].[21] A. R. Liddle and D. H. Lyth, “Cosmological inflation andlarge scale structure,” Cambridge, UK: Univ. Pr. (2000).[22] M. Fukugita, T. Yanagida, Phys. Lett. B174 , 45 (1986). [23] M. Kawasaki, F. Takahashi and T. T. Yanagida, Phys.Lett. B , 8 (2006) [hep-ph/0603265]; Phys. Rev. D , 043519 (2006) [hep-ph/0605297]; T. Asaka, S. Naka-mura and M. Yamaguchi, Phys. Rev. D , 023520(2006) [hep-ph/0604132]; M. Dine, R. Kitano, A. Morisseand Y. Shirman, Phys. Rev. D , 123518 (2006)[hep-ph/0604140]; M. Endo, K. Hamaguchi and F. Taka-hashi, Phys. Rev. D , 023531 (2006) [hep-ph/0605091].[24] M. Endo, M. Kawasaki, F. Takahashi, T. T. Yanagida, Phys. Lett. B , 518 (2006) [hep-ph/0607170].[25] M. Endo, F. Takahashi and T. T. Yanagida, Phys. Lett.B , 236 (2008) [hep-ph/0701042].[26] M. Endo, F. Takahashi and T. T. Yanagida, Phys. Rev.D , 083509 (2007) [arXiv:0706.0986 [hep-ph]].[27] K. Nakayama, F. Takahashi and T. T. Yanagida, Phys.Lett. B718