Kazuyuki Furuuchi, Noel Jonathan Jobu, Suvedha Suresh Naik
Extra-natural inflation is (de)constructed. Explicit models are compared with cosmological observations. The models successfully achieve trans-Planckian inflaton field excursions.
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In this paper, we carefully calculated the tensor-to-scalar ratio, the running spectral index, and the running of running spectrum for (extra) natural inflation in order to compare with recent BICEP2 data, PLANCK satellite data and future 21 cm data. We discovered that the prediction for running spectral index and the running of running spectrum in natural inflation is different from that in the case of extra natural inflation. Near future observation for the running spectral index may achieve enough accuracy to allow us distinguishing between extra natural inflation from natural inflation. Di…
Inflation in Extra-Dimensions with one or two branes
本文研究了额外维度框架下的单胞和α-吸引子膨胀模型，发现α-吸引子膨胀在拟合观测数据时表现优异，而单胞膨胀则不被青睐。
We propose a new inflation scenario in flux compactification, where a zero mode scalar field of extra components of the higher dimensional gauge field is identified with an inflaton. The scalar field is a pseudo Nambu-Goldstone boson of spontaneously broken translational symmetry in compactified spaces. The inflaton potential is non-local and finite, which is protected against the higher dimensional non-derivative local operators by quantum gravity corrections thanks to the gauge symmetry in higher dimensions and the shift symmetry originated from the translation in extra spaces. We give an ex…
Analysis of inflationary models in higher-dimensional uniform inflation
本文分析高维均匀暴胀中额外维度与三维空间同速率膨胀的暴胀模型，计算微扰导出谱指数与张量标量比，结合普朗克2018约束，发现除一维额外维度外，该膨胀方式不可取。
At the beginning of inflation there could be extra dynamical scalar fields that will soon disappear (become static) before the end of inflation. In the light of multi-field inflation, those extra degrees of freedom may alter the time-dependence of the original spectrum of the curvature perturbation. It is possible to remove such fields introducing extra number of e-foldings prior to $N_e\sim 60$, however such extra e-foldings may make the trans-Planckian problem worse due to the Lyth bound. We show that such extra scalar fields can change the running of the spectral index to give correction of…
We propose a simple but effective mechanism to realize an inflationary early universe consistent with the observed WMAP, Planck and/or BICEP2 data, which would be incorporated in various supersymmetric models of elementary particles constructed in the (effective) five-dimensional spacetime. In our scenario, the inflaton field is identified with one of the moduli appearing when the fifth direction is compactified, and a successful cosmological inflation without the so-called eta problem can be achieved by a very simple moduli stabilization potential. We also discuss the related particle cosmolo…
Large extra dimensions from higher-dimensional inflation
本研究提出紧致额外维可通过高维暴胀获得大尺度，推导引力基本标度上限，指出仅单微米级额外维符合原初密度涨落观测约束，且五维暴胀后胀子模可稳定在暗能量标度的真空态。
Suppressing Extra-Dimensional Axion Isocurvature Dynamically
研究额外维度 QCD 轴子在暴胀时量子涨落产生等曲率微扰限制暴胀尺度的问题，提出在翘曲五维模型中的动态抑制机制，可满足 CMB 等曲率界限并允许更高暴胀尺度。
This paper considers Einstein-Brans-Dicke action being coupled with a Higgs sector. It is shown that classically graceful exit from inflation is not possible in principle, rejecting the claim of La and Steinhardt. Further, wormhole solutions with and without conserved charges are explored. Finally, the wavefunction of the universe in this model is constructed with the boundary condition, as proposed by Vilenkin. The interesting feature of the wavefunction is that it has been constructed for an arbitrary factor-ordering index.
We discuss a realization of a small field inflation based on string inspired supergravities. In theories accompanying extra dimensions, compactification of them with small radii is required for realistic situations. Since the extra dimension can have a periodicity, there will appear (quasi-)periodic functions under transformations of moduli of the extra dimensions in low energy scales. Such a periodic property can lead to a UV completion of so-called multi-natural inflation model where inflaton potential consists of a sum of multiple sinusoidal functions with a decay constant smaller than the …
We explore inflationary cosmology in a theory where there are two scalar fields which non-minimally couple to the Ricci scalar and an additional $R^2$ term, which breaks the conformal invariance. Particularly, we investigate the slow-roll inflation in the case of one dynamical scalar field and that of two dynamical scalar fields. It is explicitly demonstrated that the spectral index of scalar mode of the density perturbations and the tensor-to-scalar ratio can be consistent with the observations acquired by the recent Planck satellite. The graceful exit from the inflationary stage is achieved …
Bayesian inference techniques are used to investigate situations where an additional light scalar field is present during inflation and reheating. This includes (but is not limited to) curvaton-type models. We design a numerical pipeline where $\simeq 200$ inflaton setups $\times\, 10$ reheating scenarios $= 2000$ models are implemented and we present the results for a few prototypical potentials. We find that single-field models are remarkably robust under the introduction of light scalar degrees of freedom. Models that are ruled out at the single-field level are not improved in general, beca…
The interrelation between inflationary cosmology and new physics beyond the Standard Model (SM) is studied in a $U(1)_{B-L}$ extension of the SM embedded in a (4+1)-dimensional spacetime. In the scenario we study, the inflaton arises from the Wilson loop of the $U(1)_{B-L}$ gauge group winding an extra-dimensional cycle. Particular attention is paid to the coupling between the inflaton and SM particles that are confined on a brane localised in the extra dimension. We find that the inflaton decay channels are rather restricted in this scenario and the resulting reheating temperature is relative…
We consider a 4+N dimensional Einstein gravity coupled to a non-linear sigma model. This theory admits a solution in which the N extra dimensions contract exponentially while the ordinary space expand exponentially. Physically, the non-linear sigma fields induce the dynamical compactification of the extra dimensions, which in turn drives inflation. No inflatons are required.
We study a model of quintessential inflation constructed in $R^2$ modified gravity with a non-minimally coupled scalar field, in the Palatini formalism. Our non-minimal inflaton field is characterised by a simple exponential potential. We find that successful quintessential inflation can be achieved with no fine-tuning on the model parameters. Predictions on the characteristics of dark energy will be tested by observations in the near future, while contrast with existing observations provides insights on the modified gravity background, such as the value of the non-minimal coupling and its run…
While moving down the potential on its classical slow roll trajectory, the inflaton field is subject to quantum jumps, which take it up or down the potential at random. In "stochastic inflation", the impact of these quantum jumps is modeled by smoothing out the field over (at least) Hubble-patch sized domains and treating fluctuations on smaller scales as noise. The inflaton thus becomes a stochastic process whose values at a given time are calculated using its probability distribution. We generalize this approach for non-canonic kinetic terms of Dirac Born Infeld (DBI) type and investigate th…
We study inflationary solutions in the M-theory. Including the fourth-order curvature correction terms, we find three generalized de Sitter solutions, in which our 3-space expands exponentially. Taking one of the solutions, we propose an inflationary scenario of the early universe. This provides us a natural explanation for large extra dimensions in a brane world, and suggests some connection between the 60 e-folding expansion of inflation and TeV gravity based on the large extra dimensions.
Two simple photon gauges in inflation
在任意维度幂律暴胀中，通过两族单参数非协变规范构造了形式更简单的光子传播子，使涉及无质量矢量场的维度正规化圈计算变得可行。
We find the natural embedding of the (R+R^2)-inflationary model into the recently constructed N=1 F(\cal R)-supergravity. It gives a simple and viable realization of chaotic inflation in supergravity. The only requirement for a slow-roll inflation is the existence of the (\cal R)^3-term with an anomalously large coefficient in Taylor expansion of the F(\cal R) function, where \cal R is the covariantly-chiral scalar supercurvature superfield.
We present a new model of inflation in which the inflaton is the extra component of a gauge field in a 5d theory compactified on a circle. The chief merit of this model is that the potential comes only from non-local effects so that its flatness is not spoiled by higher dimensional operators or quantum gravity corrections. The model predicts a red spectrum (n ~ 0.96) and a significant production of gravitational waves (r ~ 0.11). We also comment on the relevance of this idea to quintessence.
We discuss inflation in models with large extra dimensions, driven by a bulk scalar field. The brane inflaton is then a single effective field, obtained from the bulk scalar field by scaling. The self interaction terms of the effective brane inflaton are then naturally suppressed. The picture is consistent with a fundamental string scale in the TeV range without the problem of a superlight inflaton. If hybrid inflation is considered, the right prediction for the density perturbations as observed by COBE can be obtained without any fine tunning. The bulk inflaton then decays preferentially into…
In this paper, a inflationary model from the rotation of D4-brane is constructed. We show that for a very wide rage of parameter, this model satisfies the observation and find that regarded as inflaton, the rotation of branes may be more nature than the distance between branes. Our model offers a new avenue for brane inflation.
We consider the production of gravitational KK modes via cosmological photon-photon and electron-positron annihilation in models with large factorisable extra dimensions. We place constraints on this production using recent results from a joint analysis of the power spectra of the 2dF Galaxy Redshift Survey (2dFGS) and the cosmic microwave background (CMB) anisotropies. We obtain a more accurate upper limit for the temperature corresponding to matter-radiation equality and show that, even for the case of 6 extra dimensions and a fundamental scale of 1 TeV, a period of inflation is required tha…
We have constructed a large field N-flation model in the supergravity framework. In this simple set-up, N fields collectively drive inflation where each field traverses sub-Planckian field values. This has been realised with a generalisation of the single field chaotic inflation in Supergravity. Interestingly, despite of the presence of the field interactions, the dynamics can be described in terms of an effective single field. The observable predictions of our model i.e. tensor to scalar ratio r and scalar spectral index n_s are akin to the chaotic inflation.
We propose to address the fine tuning problem of inflection point inflation by the addition of extra vacuum energy that is present during inflation but disappears afterwards. We show that in such a case, the required amount of fine tuning is greatly reduced. We suggest that the extra vacuum energy can be associated with an earlier phase transition and provide a simple model, based on extending the SM gauge group to SU(3)_C \times SU(2)_L\times U(1)_Y\times U(1)_{B-L}, where the Higgs field of U(1)_{B-L} is in a false vacuum during inflation. In this case, there is virtually no fine tuning of t…
Many theories beyond the Standard Model include extra dimensions, though these have yet to be directly observed. In this work we consider the possibility of a compactification mechanism which both allows extra dimensions and is compatible with current observations. This compactification is predicted to leave a signature on the CMB by altering the amplitude of the low l multipoles, dependent on the amount of inflation. Recently discovered CMB anomalies at low multipoles may be evidence for this. In our model we assume the spacetime is the product of a four-dimensional spacetime and flat extra d…
In this note, we refine the racetrack inflation model constructed in arXiv:hep-th/0406230 by including the open string modulus. This modulus encodes the embedding of our braneworld inside some Calabi-Yau throat. We argue that in generic this open string modulus dynamically runs with the inflaton field thanks to its nonlinear coupling. A full analysis becomes difficult because the scalar potential changes progressively during the inflation epoch. Nevertheless, by explicit construction we are still able to build a realistic model through appropriate choices of the initial conditions.
Recently a new approach to inflation proposal has been constructed via the smeared coherent state picture of spacetime noncommutativity. Here we generalize this viewpoint to a Randall-Sundrum II braneworld scenario. This model realizes an inflationary, bouncing solution without recourse to any axillary scalar or vector fields. There is no initial singularity and the model has the potential to produce scale invariant spectrum of scalar perturbations.
A unified model of inflation and dark energy based on the holographic spacetime foam
本文通过将哈勃视界内时空泡沫的经典能量设为宇宙学方程的精确解来构造暴胀子势，从而提出一个能统一描述早期暴胀与晚期暗能量的模型。
We construct an inflationary model in 6D supergravity that is based on explicit time-dependent solutions to the full higher-dimensional field equations, back-reacting to the presence of a 4D inflaton rolling on a space-filling codimension-2 source brane. Fluxes in the bulk stabilize all moduli except the `breathing' modulus (that is generically present in higher-dimensional supergravities). Back-reaction to the inflaton roll causes the 4D Einstein-frame on-brane geometry to expand, a(t) ~ t^p, as well as exciting the breathing mode and causing the two off-brane dimensions to expand, r(t) ~ t^q…
Extra-natural production of superheavy Kaluza-Klein particles
该研究在五维QED框架下构建额外自然暴胀模型，证明紧致维度中的轻规范场可通过Schwinger效应产生超重KK粒子，这些粒子可能成为暗物质或主导宇宙。
In Aligned Natural Inflation, an alignment between different potential terms produces an inflaton excursion greater than the axion scales in the potential. We show that, starting from a general potential of two axions with two aligned potential terms, the effective theory for the resulting light direction is characterized by four parameters: an effective potential scale, an effective axion constant, and two extra parameters (related to ratios of the axion scales and the potential scales in the $2-$field theory). For all choices of these extra parameters, the model can support inflation along v…
Cosmic inflation driven by branes wrapping the extra dimensions involves Kaluza-Klein (KK) degrees of freedom in addition to the zero-mode position of the brane which plays the role of the inflaton. As the wrapped brane passes by localized sources or features along its inflationary trajectory in the extra dimensional space, the KK modes along the wrapped direction are excited and start to oscillate during inflation. We show that the oscillating KK modes induce parametric resonance for the curvature perturbations, generating sharp signals in the perturbation spectrum. The effective four dimensi…
Inflation models are compared with observation on the assumption that the curvature perturbation is generated from the vacuum fluctuation of the inflaton field. The focus is on single-field models with canonical kinetic terms, classified as small- medium- and large-field according to the variation of the inflaton field while cosmological scales leave the horizon. Small-field models are constructed according to the usual paradigm for beyond Standard Model physics
A model of inflation is proposed in which compact extra dimensions allow a graceful exit without recourse to flat potentials or super-Planckian field values. Though bubbles of true vacuum are too sparse to uniformly reheat the Universe by colliding with each other, a compact dimension enables a single bubble to uniformly reheat by colliding with itself. This mechanism, which generates an approximately scale invariant perturbation spectrum, requires that inflation be driven by a bulk field, that vacuum decay be slow, and that the extra dimension be at least a hundred times larger than the false…
We examine whether F-term supersymmetric hybrid inflation can be embedded within the minimal SO(10) model, in a natural way. We show that none of the singlets of the Standard Model symmetries in the minimal set of SO(10) representations can satisfy the conditions which are necessary for a scalar field to play the role of the inflaton. As a consequence, one has to introduce an extra scalar field, which however may spoil the naturalness of inflation within the context of SO(10). Nevertheless, if we add an extra scalar field, we are then able to construct a model that can accommodate flat directi…
We embed the supersymmetric standard model of hybrid inflation based on the next-to-minimal superpotential term $λNH_uH_d$ supplemented by an inflaton term $κϕN^2$, into an extra-dimensional framework, in which all the Higgs fields and singlets live in the bulk, while all the matter fields live on the brane. All the parameters of the effective 4d model can then be naturally understood in terms of a fundamental (``string'') scale $M_*\sim 10^{13}$ GeV and a brane supersymmetry breaking scale $10^{8}$ GeV, of the same order as the height of the inflaton potential during inflation. In particular …
Within the framework of a five-dimensional brane world with a stabilized radion, we compute the cosmological perturbations generated during inflation and show that the perturbations are a powerful tool to probe the physics of extra dimensions. While we find that the power spectrum of scalar perturbations is unchanged, we show that the existence of the fifth dimension is imprinted on the spectrum of gravitational waves generated during inflation. In particular, we find that the tensor perturbations receive a correction proportional to $(HR)^2$, where $H$ is the Hubble expansion rate during infl…
Cosmological history after higher dimensional inflation
本文提出一种宇宙学历史，连接高维暴胀结束到再加热温度，避免早期宇宙中体引力子过度产生，并解释额外维度在暴胀后的稳定化。
A semiclassical gravitation model is outlined which makes use of the Casimir energy density of vacuum fluctuations in extra compactified dimensions to produce the present-day cosmological constant as rho_LAMBDA ~ M^8/M_P^4, where M_P is the Planck scale and M is the weak interaction scale. The model is based on (4+D)-dimensional gravity, with D = 2 extra dimensions with radius b(t) curled up at the ADD length scale b_0 = M_P/M^2 ~ 0.1 mm. Vacuum fluctuations in the compactified space perturb b_0 very slightly, generating a small present-day cosmological constant. The radius of the compactified…
We propose a natural scenario for the cosmological inflation with the non-minimal coupling term invoking compact hyperbolic extra dimensions. Thanks to the unique mathematical properties of compact hyperbolic space, the large volume of extra dimensions, which provides a natural understanding of the proper size of couplings, does not necessarily accompany with the low Kaluza-Klein scale so that the model allows a single field inflation with a scale around $10^{13}$ GeV. The model fulfills all the observed data and predicts a sizable gravitational perturbation, r~0.003.
Ultraviolet Sensitivity in Higgs-Starobinsky Inflation
本研究针对希格斯-斯塔罗宾斯基暴胀模型，将六维标量场与里奇标量的非导数算符作为微扰，分析其对成功暴胀预言的影响，发现标量六次项需普朗克标度压制，其余项所需压制标度更小。
It has been proposed that, without invoking supersymmetry, it is possible to solve the hierarchy problem provided the fundamental scale in the higher dimensional theory is at a much lower scale than the Planck scale. In this paper we consider a toy model where we allow $4+d$ dimensions to evolve from a region determined by the new fundamental scale, which is in our case is electro-weak scale. We further investigate whether it is possible to inflate not only the 3 uncompactified dimensions but also the extra dimensions. We require around 70 e-foldings to stabilize the size of two extra dimensio…
We develop an inflationary model without small parameters on the basis of multidimensional $f(R)$ gravity with a minimally coupled scalar field. The model is described by two stages of space expansion. The first one begins at energy scales about the D-dimensional Planck mass and ends with the de Sitter metric of our space and the maximally symmetric extra dimensions. In the following, the quantum fluctuations produce a wide set of inhomogeneous extra metrics in causally disconnected regions quickly generated in the de Sitter space. We find a specific extra space metric that leads to the effect…
In low scale quantum gravity scenarios the fundamental scale of nature can be as low as TeV, in order to address the naturalness of the electroweak scale. A number of difficulties arise in constructing specific models; stabilisation of the radius of the extra dimensions, avoidance of overproduction of Kaluza Klein modes, achieving successful baryogenesis and production of a close to scale-invariant spectrum of density perturbations with the correct amplitude. We examine in detail the dynamics, including radion stabilisation, of a hybrid inflation model that has been proposed in order to addres…
The possibility to construct an inflationary scenario for renormalization-group improved potentials corresponding to the Higgs sector of finite gauge models is investigated. Taking into account quantum corrections to the renormalization-group potential which sums all leading logs of perturbation theory is essential for a successful realization of the inflationary scenario, with very reasonable parameter values. The inflationary models thus obtained are seen to be in good agreement with the most recent and accurate observational data. More specifically, the values of the relevant inflationary p…
We consider how accelerated expansion, whether due to inflation or dark energy, imposes strong constraints on fundamental theories obtained by compactification from higher dimensions. For theories that obey the null energy condition (NEC), we find that inflationary cosmology is impossible for a wide range of compactifications; and a dark energy phase consistent with observations is only possible if both Newton's gravitational constant and the dark energy equation-of-state vary with time. If the theory violates the NEC, inflation and dark energy are only possible if the NEC-violating elements a…
In the context of (4+d)-dimensional general relativity, we propose an inflationary scenario wherein 3 spatial dimensions grow large, while d extra dimensions remain small. Our model requires that a self-interacting d-form acquire a vacuum expectation value along the extra dimensions. This causes 3 spatial dimensions to inflate, whilst keeping the size of the extra dimensions nearly constant. We do not require an additional stabilization mechanism for the radion, as stable solutions exist for flat, and for negatively curved compact extra dimensions. From a four-dimensional perspective, the radi…
What new physics can we extract from inflation using the ACT DR6 and DESI DR2 Observations?
本文利用ACT DR6和DESI DR2数据，分析暴胀模型的预测，探讨未来CMB和引力波实验对暴胀参数空间的约束，并展示多种理论模型的排除情况。
We consider trapped inflation in a higher dimensional field space: particle production at a dense distribution of extra species points leads to a terminal velocity at which inflation can be driven in steep potentials. We compute an additional, nearly scale invariant contribution to the power-spectrum, caused by back-scattering of the continuously produced particles. Since this contribution has a blue tilt, it has to be sub-dominant, leading to an upper bound on the coupling constant between the inflatons and the extra species particles. The remaining allowed parameter space is narrow.
Shaft inflation is a model in which the inflaton potential approaches a plateau far from the origin, while it resembles chaotic inflation near the origin. Meanwhile, the Randall-Sundrum type II model (RSII) is an interesting extra-dimensional model to study cosmological phenomenology. In this paper, we study shaft inflation in the RSII model. We find that the predictions are in excellent agreement with observation. The fundamental five-dimensional Planck scale is found to be $M_5\simeq 10^{16}$ GeV, which is consistent with the lower bound $M_5\gtrsim 10^{9}$ GeV obtained from experimental New…
We explore the mechanics of inflation in simplified extra-dimensional models involving an inflaton interacting with the Einstein-Maxwell system in two extra dimensions. The models are Goldilocks-like in that they are just complicated enough to include a mechanism to stabilize the extra-dimensional size, yet simple enough to solve the full 6D field equations using basic tools. The solutions are not limited to the effective 4D regime with H << m_KK (the latter referring to the mass splitting of the Kaluza-Klein excitations) because the full 6D Einstein equations are solved. This allows an explor…
We constructed a model of natural inflation in the context of $α$-attractor supergravity, in which both the dilaton field and the axion field are light during inflation, and the inflaton may be a combination of the two. The T-model version of this theory is defined on the Poincare disk with radius |Z| = 1. It describes a Mexican hat potential with the flat axion direction corresponding to a circle of radius |Z| < 1. The axion decay constant $f_{a}$ in this theory can be exponentially large because of the hyperbolic geometry of the Poincare disk. Depending on initial conditions, this model may …
BICEP2 observations, interpreted most simply, suggest an era of inflation with energy densities of order ($10^{16}\, {\rm GeV})^4$, not far below the Planck density. However, models of TeV gravity with large extra dimensions might allow a very different interpretation involving much more modest energy scales. We discuss the viability of inflation in such models, and conclude that existing scenarios do not provide attractive alternatives to single field inflation in four dimensions. Because the detection of tensor modes strengthens our confidence that inflation occurred, it disfavors models of …
We investigate the longitudinal ghost issue in Abelian vector inflation. It turns out that, within the class of Lorentz-invariant vector field theories with three degrees of freedom and without any extra (scalar) fields, the possibilities are essentially exhausted by the classical solution due to Larry Ford with an extremely flat potential which doesn't feel the fast roll of its argument. And, moreover, one needs to fulfil an extra condition on that potential in order to avoid severe gradient instability. At the same time, some Lorentz-violating modifications are worth to be explored.
We consider conditions necessary for a successful implementation of so-called assisted inflation. We generalize the applicability of assisted inflation beyond exponential potentials as originally proposed to include standard chaotic (λϕ^4 or m^2 ϕ^2) models as well. We also demonstrate that in a purely 4-dimensional theory, unless the assisted sector is in fact decoupled, the additional fields of the assisted sector actually impede inflation. As a specific example of an assisted sector, we consider a 5-dimensional KK model for which the extra dimension may be somewhat or much larger than the i…
We discuss cosmology of models with universal extra dimensions, where the Standard Model degrees of freedom live in a $4+n$ dimensional brane, with $n$ compact and small extra spatial dimensions. In these models, the simplest way to obtain the conventional 4-dimensional Planck scale starting with a low string scale is to have also some larger extra dimensions, where only gravity propagates. In such theories, dimensional reduction generically leads to at least two radion fields, one associated with the total volume of the extra spatial dimensions, and the other with the ratio of the sizes of sm…
Brane inflation in superstring theory predicts that cosmic strings (but not domain walls or monopoles) are produced towards the end of the inflationary epoch. Here, we discuss the production, the spectrum and the evolution of such cosmic strings, properties that differentiate them from those coming from an abelian Higgs model. As D-branes in extra dimensions, some type of cosmic strings will dissolve rapidly in spacetime, while the stable ones appear with a spectrum of cosmic string tensions. Moreover, the presence of the extra dimensions reduces the interaction rate of the cosmic strings in s…
We extend some previous attempts to explain the origin and evolution of primordial magnetic fields during inflation induced from a 5D vacuum. We show that the usual quantum fluctuations of a generalized 5D electromagnetic field cannot provide us with the desired magnetic seeds. We show that special fields without propagation on the extra non-compact dimension are needed to arrive to appreciable magnetic strengths. We also identify a new magnetic tensor field $B_{ij}$ in this kind of extra dimensional theories. Our results are in very good agreement with observational requirements, in particula…
We consider the simplest extension to the Starobinsky model, by allowing an extra scalar field to help drive inflation. We perform our analysis in the Einstein frame and calculate the power spectra at the end of inflation to second order in the slow--roll parameters. We find that the model gives predictions in great agreement with the current Planck data without the need for fine-tuning. Our results encourage current efforts to embed the model in a supergravity setting.
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