We study Dirac-Born-Infeld (DBI) inflation models with multiple scalar fields. We show that the adiabatic and entropy modes propagate with a common effective sound speed and are thus amplified at the sound horizon crossing. In the small sound speed limit, we find that the amplitude of the entropy modes is much higher than that of the adiabatic modes. We show that this could strongly affect the observable curvature power spectrum as well as the amplitude of non-Gaussianities, although their shape remains as in the single-field DBI case.
We study DBI spinflation models with angular potentials that are derived in string theory. We analyse the background dynamics with different parameter sets and study the impact of changing each parameter on inflationary dynamics. It is known that the conversion of the entropy perturbation into the curvature perturbation gives multi-field DBI inflation models a possibility of satisfying the observational constraints by relaxing the stringent microphysical constraint that disfavours single field DBI inflation models. we show that our model is excluded by the Planck satellite observations even wi…
We study cosmological perturbations in general inflation models with multiple scalar fields and arbitrary kinetic terms, with special emphasis on the multi-field extension of Dirac-Born-Infeld (DBI) inflation. We compute the second-order action governing the dynamics of linear perturbations in the most general case. Specializing to DBI, we show that the adiabatic and entropy modes propagate with a {\it common} effective sound speed and are thus amplified at sound horizon crossing. In the small sound speed limit, we find that the amplitude of the entropy modes is much higher than that of the ad…
We study the non-Gaussianity in the simplest infrared (IR) model of the DBI inflation. We show that the non-Gaussianity in such a model is compatible with the current observational bound, and is within the sensitivity of future experiments. We also discuss the scale dependence of the non-Gaussianity. In the DBI inflation, such a feature can be used as a probe to the properties of the background geometry of the extra dimensions or internal space.
We study multi-field DBI inflation models with a waterfall phase transition. This transition happens for a D3 brane moving in the warped conifold if there is an instability along angular directions. The transition converts the angular perturbations into the curvature perturbation. Thanks to this conversion, multi-field models can evade the stringent constraints that strongly disfavour single field ultra-violet DBI inflation models in string theory. We explicitly demonstrate that our model satisfies current observational constraints on the spectral index and equilateral non-Gaussianity as well …
We analyze the trajectories for multifield DBI inflation, which can arise in brane inflation models, and show that the trajectories are the same as in typical slow roll inflation. We calculate the power spectrum and find that the higher derivative terms of the DBI action lead to a suppression of the contribution from the isocurvature perturbations. We also calculate the bispectrum generated by the isocurvature perturbation, and find that it leads to distinctive features.
We study multifield aspects of Dirac-Born-Infeld (DBI) inflation. More specifically, we consider an inflationary phase driven by the radial motion of a D-brane in a conical throat and determine how the D-brane fluctuations in the angular directions can be converted into curvature perturbations when the tachyonic instability arises at the end of inflation. The simultaneous presence of multiple fields and non-standard kinetic terms gives both local and equilateral shapes for non-Gaussianities in the bispectrum. We also study the trispectrum, pointing out that it acquires a particular momentum de…
We study a cosmological scenario in which the DBI action governing the motion of a D3-brane in a higher-dimensional spacetime is supplemented with an induced gravity term. The latter reduces to the quartic Galileon Lagrangian when the motion of the brane is non-relativistic and we show that it tends to violate the null energy condition and to render cosmological fluctuations ghosts. There nonetheless exists an interesting parameter space in which a stable phase of quasi-exponential expansion can be achieved while the induced gravity leaves non trivial imprints. We derive the exact second-order…
Very few explicit inflationary scenarios are known to generate a large bispectrum of orthogonal shape. Dirac-Born-Infeld Galileon inflation, in which an induced gravity term is added to the DBI action, is one such model. We formulate it in the language of the effective field theory of inflation by identifying the unitary gauge operators that govern the behavior of its cosmological fluctuations. We show how to recover rather easily from this its power spectrum and bispectrum, which we calculated previously using standard cosmological perturbation theory. We push our calculations up to the deter…
In this letter we study adiabatic and isocurvature perturbations in the frame of inflation with multiple sound speeds involved. We suggest this scenario can be realized by a number of generalized scalar fields with arbitrary kinetic forms. These scalars have their own sound speeds respectively, so the propagations of field fluctuations are individual. Specifically, we study a model constructed by two DBI type actions. We find that the critical length scale for the freezing of perturbations corresponds to the maximum sound horizon. Moreover, if the mass term of one field is much lighter than th…
New Ekpyrotic Cosmology is an alternative scenario of the early universe which relies on a phase of slow contraction before the big bang. We calculate the 3-point and 4-point correlation functions of primordial density perturbations and find a generically large non-Gaussian signal, just below the current sensitivity level of CMB experiments. This is in contrast with slow-roll inflation, which predicts negligible non-Gaussianity. The model is also distinguishable from alternative inflationary scenarios that yield large non-Gaussianity, such as DBI inflation and the simplest curvaton-like models…
We compute the leading order contribution to the four-point function of the primordial curvature perturbation in a class of single field models where the inflationary Lagrangian is a general function of the inflaton and its first derivative. This class of models includes string motivated inflationary models such as DBI inflation. We find that the trispectrum for some range of parameters could potentially be observed in future experiments. Moreover, the trispectrum can distinguish DBI inflation from other inflation models with large non-Gaussianities which typically have a similar bispectrum. W…
We study multi-field Dirac-Born-Infeld (DBI) inflation models, taking into account the NS-NS and R-R bulk fields present in generic flux compactifications. We compute the second-order action, which governs the behaviour of linear cosmological perturbations, as well as the third-order action, which can be used to calculate non-Gaussianities in these models. Remarkably, for scalar-type perturbations, we show that the contributions due to the various form fields exactly cancel in both the second- and third-order actions. Primordial perturbations and their non-Gaussianities are therefore unaffecte…
This contribution gives an overview on primordial non-Gaussianities from a theoretical perspective. After presenting a general formalism to describe nonlinear cosmological perturbations, several classes of models, illustrated with examples, are discussed: multi-field inflation with non-standard Lagrangians, modulaton fields, curvaton fields. In the latter case, a special emphasis is put on the isocurvature perturbations, which could leave a specific signature in non-Gaussianities.
We investigate the noncommutative effect on the non-Gaussianities of primordial cosmological perturbation. In the lowest order of string length and slow-roll parameter, we find that in the models with small speed of sound the noncommutative modifications could be observable if assuming a relatively low string scale. In particular, the dominant modification of non-Gaussianity estimator f_{NL} could reach O(1) in DBI inflation and K-inflation. The corrections are sensitive to the speed of sound and the choice of string length scale. Moreover the shapes of the corrected non-Gaussianities are dist…
We analyze primordial non-gaussianities in presence of an ultra-slow phase during the inflationary dynamics, focusing on scenarios relevant for the production of primordial black holes. We compute the three-point correlation function of comoving curvature perturbations finding that non-gaussianities are sizable, and predominantly local. In the context of threshold statistics, we analyze their impact for the abundance of primordial black holes, and their interplay with the non-gaussianities arising from the non-linear relation between density and curvature perturbations. We find that non-gaussi…
This is a pedagogical review on primordial non-Gaussianities from inflation models. We introduce formalisms and techniques that are used to compute such quantities. We review different mechanisms which can generate observable large non-Gaussianities during inflation, and distinctive signatures they leave on the non-Gaussian profiles. They are potentially powerful probes to the dynamics of inflation. We also provide a non-technical and qualitative summary of the main results and underlying physics.
We investigate primordial perturbations and non-gaussianities in the Hořava-Lifshitz theory of gravitation. In the UV limit, the scalar perturbation in the Hořava theory is naturally scale-invariant, ignoring the details of the expansion of the Universe. One may thus relax the exponential inflation and the slow-roll conditions for the inflaton field. As a result, it is possible that the primordial non-gaussianities, which are ``slow-roll suppressed'' in the standard scenarios, become large. We calculate the non-gaussianities from the bispectrum of the perturbation and find that the equilateral…
We re-analyse current single-field inflationary models related to primordial black holes formation. We do so by taking into account recent developments on the estimations of their abundances and the influence of non-gaussianities. We show that, for all of them, the gaussian approximation, which is typically used to estimate the primordial black holes abundances, fails. However, in the case in which the inflaton potential has an inflection point, the contribution of non-gaussianities is only perturbative. Finally, we infer that only models featuring an inflection point in the inflationary poten…
We utilize the Canny edge detection algorithm as an estimator for primordial non-Gaussianities. In preliminary tests on simulated sky patches with a window size of 57 degrees and multipole moments $l$ up to 1024, we find a $3σ$ distinction between maps with local non-Gaussianity $f_{NL}=350$ (or $f_{NL}=-700$) and Gaussian maps. We present evidence that high resolution CMB studies will strongly enhance the sensitivity of the Canny algorithm to non-Gaussianity, making it a promising technique to estimate primordial non-Gaussianity.
We study primordial fluctuations generated during inflation in a class of models motivated by the DBI Galileons, which are extensions of the DBI action that yield second order field equations. This class of models generalises the DBI Galileons in a similar way with K-inflation. We calculate the primordial non-Gaussianity from the bispectrum of the curvature perturbations at leading order in the slow-varying approximations. We show that the estimator for the equilateral-type non-Gaussianity, $f_{\rm NL} ^{equil}$, can be applied to measure the amplitude of the primordial bispectrum even in the …
We propose a novel inflationary scenario in string theory in which the inflaton field is a 'Wilson line' degree of freedom in the worldvolume of a probe Dp-brane, in a warped flux compactification. Kinetic terms for Wilson line fields on the world volume of a D-brane take a nonstandard Dirac-Born-Infeld (DBI) form. Thus, we work in the framework of DBI inflation. This extends the original slow roll Wilson line inflationary scenario, where only the quadratic piece was considered. Warped DBI Wilson line inflation offers an attractive alternative to ordinary (position field) DBI inflation, inasmu…
We investigate DBI inflation using $N$ multiple branes and show how the configuration is equivalent to a single wrapped $D5$-brane with flux. We then show how 1/N corrections can be implemented, and we examine the sound speed and levels of non-Gaussianities in two distinct cases. For models with constant warping we find that the non-Gaussian amplitude is bounded from above (as a function of $γ$). For $AdS$ backgrounds we find that the signature is generally large and positive, although is no longer globally defined over the full phase space. We then investigate an inflationary mechanism using …
We compute the non-Gaussianities that arise in single field, slow roll inflationary models arising from arbitrary homogeneous initial states, as well as subleading contributions to the power spectrum. Non Bunch-Davies vacuum initial states can arise if the transition to the single field, slow roll inflation phase occurs only shortly before observable modes left the horizon. They can also arise from new physics at high energies that has been integrated out. Our general result for the bispectrum exhibits several features that were previously seen in special cases.
We set up cosmological perturbation theory and study the cosmological implications of the so-called ``generalized Galileon'' developed in \cite{Deffayet:2011gz,horndeski}. This is the most general scalar field theory whose Lagrangian contains derivatives up to second order while keeping second order equations of motion, and contains as sub-cases $k$-inflation, $G$-inflation and many other models. We calculate the power spectrum of the primordial curvature perturbation, finding a modification of the usual consistency relation of the tensor-to-scalar ratio in $k$-inflation or perfect fluid model…
We perform a general study of the primordial scalar non-Gaussianities in multi-field inflationary models in Einstein gravity. We consider models governed by a Lagrangian which is a general function of the scalar fields and their first spacetime derivatives. We use $δN$ formalism to relate scalar fields and curvature perturbations. We calculate the explicit cubic order perturbation action and the three-point function of curvature perturbation evaluated at horizon-crossing. Under reasonable assumptions, in the limit of small slow-varying parameters and a sound speed $c_s$ close to one, we find t…
We investigate the primordial trispectra of the general multifield DBI inflationary model. In contrast with the single field model, the entropic modes can source the curvature perturbations on the super horizon scales, so we calculate the contributions from the interaction of four entropic modes mediating one adiabatic mode to the trispectra, at the large transfer limit ($T_{RS}\gg1$). We obtained the general form of the 4-point correlation functions, plotted the shape diagrams in two specific momenta configurations, "equilateral configuration" and "specialized configuration". Our figures show…
It was pointed out recently that in some inflationary models quantum loops containing a scalar of mass $m$ that couples to the inflaton can be the dominant source of primordial non-Gaussianities. We explore this phenomenon in the simplest such model focusing on the behavior of the primordial curvature fluctuations for small $m/H$. Explicit calculations are done for the three and four point curvature fluctuation correlations. Constraints on the parameters of the model from the CMB limits on primordial non-Gaussianity are discussed. The bi-spectrum in the squeezed limit and the tri-spectrum in t…
We investigate the relation between the non-Gaussianities of the primordial perturbations and the sound speed of the tensor perturbations, that is, the propagation speed of the gravitational waves. We find that the sound speed of the tensor perturbations is directly related not to the auto-bispectrum of the tensor perturbations but to the cross-bispectrum of the primordial perturbations, especially, the scalar-tensor-tensor bispectrum. This result is in sharp contrast with the case of the scalar (curvature) perturbations, where their reduced sound speed enhances their auto-bispectrum. Our find…
In this paper, we study the generation of a large scale magnetic field with amplitude of order $μ$G in an inflationary model which has been introduced in hep-th/0310221. This inflationary model based on existence of a speed limit for inflaton field. Generating a mass for inflaton at scale above the $ϕ_{IR}$, breaks the conformal triviality of the Maxwell equation and causes to originate a magnetic field during the inflation. The amplitude strongly depends on the details of reheating stage and also depends on the e-foldings parameter N. We find the amplitude of the primordial magnetic field at …
We investigate thermal inflation in double-screen entropic cosmology. We find that its realization is general, resulting from the system evolution from non-equilibrium to equilibrium. Furthermore, going beyond the background evolution, we study the primordial curvature perturbations arising from the universe interior, as well as from the thermal fluctuations generated on the holographic screens. We show that the power spectrum is nearly scale-invariant with a red tilt, while the tensor-to-scalar ratio is in agreement with observations. Finally, we examine the non-Gaussianities of primordial cu…
The origin of the seeds which develop into the observed super-massive black holes at high redshifts may be hard to interpret in the context of the standard $ΛCDM$ of early universe cosmology based on Gaussian primordial perturbations. Here we consider the modification of the halo mass function obtained by introducing skewness and kurtosis of the primordial fluctuations. We show that such primordial non-Gaussianities constrained by the current observational bounds on the nonlinearity parameters of $f_{NL}$ and $g_{NL}$ are not effective at greatly increasing the number density of seeds which co…
Deviations from Gaussian statistics of the cosmological density fluctuations, so-called primordial non-Gaussianities (NG), are one of the most informative fingerprints of the origin of structures in the universe. Indeed, they can probe physics at energy scales inaccessible to laboratory experiments, and are sensitive to the interactions of the field(s) that generated the primordial fluctuations, contrary to the Gaussian linear theory. As a result, they can discriminate between inflationary models that are otherwise almost indistinguishable. In this short review, we explain how to compute the n…
We study the cosmological collider phenomenology of neutrinos in an effective field theory. The mass spectrum of neutrinos and their characteristic oscillatory signatures in the squeezed limit bispectrum are computed. Both dS-covariant and slow-roll corrections are considered, so is the scenario of electroweak symmetry breaking during inflation. Interestingly, we show that the slow-roll background of the inflaton provides a chemical potential for the neutrino production. The chemical potential greatly amplifies the oscillatory signal and makes the signal observably large for heavy neutrinos wi…
We calculate the three-point correlation function evaluated at horizon crossing for a set of interacting scalar fields coupled to gravity during inflation. This provides the initial condition for the three-point function of the curvature perturbation in the Sasaki--Stewart δN formulation. We find that the effect is small, of the order of a slow-roll parameter, and that the non-gaussianity can be determined on large scales once the unperturbed background evolution is known. As an example of the use of our formalism, we calculate the primordial non-gaussianity arising in a model of assisted infl…
We have studied primordial non-Gaussian features from a model of potential driven single field DBI Galileon inflation. We have computed the bispectrum from the three point correlation function considering all possible cross correlation between scalar and tensor modes from the proposed setup. Further, we have computed the trispectrum from four point correlation function considering the contribution from contact interaction, scalar and graviton exchange diagrams in the in-in picture. Finally we have obtained the non-Gaussian consistency conditions from the four point correlator, which results in…
Within the so-called cosmological collider program, imprints of new particles on primordial non-Gaussianities have been studied intensively. In particular, their non-analytic features in the soft limit provide a smoking gun for new particles at the inflation scale. While this approach is very powerful to probe particles of the mass near the Hubble scale, the signal is exponentially suppressed for heavy particles. In this paper, to enlarge the scope of the cosmological collider, we explore a new approach to probing spins of heavy particles from signs of Wilson coefficients of the inflaton effec…
We calculate the three-point function for primordial scalar fluctuations in a single field inflationary scenario where the scalar field Lagrangian is a completely general function of the field and its first derivative. We obtain an explicit expression for the three-point correlation function in a self-consistent approximation scheme where the expansion rate varies slowly, analogous to the slow-roll limit in standard, single-field inflation. The three-point function can be written in terms of the familiar slow-roll parameters and three new parameters which measure the non-trivial kinetic struct…
It has been pointed out that matter bounce cosmology driven by a k-essence field cannot satisfy simultaneously the observational bounds on the tensor-to-scalar ratio and non-Gaussianity of the curvature perturbation. In this paper, we show that this is not the case in more general scalar-tensor theories. To do so, we evaluate the power spectra and the bispectra of scalar and tensor perturbations on a general contracting background in the Horndeski theory. We then discuss how one can discriminate contracting models from inflation based on non-Gaussian signatures of tensor perturbations.
We propose a new mechanism to generate primordial curvature perturbations, based on the resonant production of particles during inflation. It is known that this phenomenon can trap the inflaton for a fraction of e-fold. This effect is governed by the mass of the produced particles and by their coupling to the inflaton, parameters which can depend on the expectation value of other fields. If one of such additional fields - a 'modulaton' - is light, then its fluctuations, acquired during the earlier stages of inflation, will induce a spatial modulation of the trapping, and thus of the end of inf…
We propose a new method to numerically calculate higher-order correlation functions of primordial fluctuations generated from any early-universe scenario. Our key-starting point is the realization that the tree-level In-In formalism is intrinsically separable. This enables us to use modal techniques to efficiently calculate and represent non-Gaussian shapes in a separable form well suited to data analysis. We prove the feasibility and the accuracy of our method by applying it to simple single-field inflationary models in which analytical results are available, and we perform non-trivial consis…
It has been found that the primordial non-Gaussianity of the curvature perturbation in the case of non-Bunch-Davies initial states can be enhanced compared with those in the case of the Bunch-Davies one due to the interactions among the perturbations on subhorizon scales. The purpose of the present paper is to investigate whether tensor non-Gaussianities can also be enhanced or not by the same mechanism. We consider general gravity theory in the presence of an inflaton, and evaluate the tensor auto-bispectrum and the cross-bispectrum involving one tensor and two scalar modes with the non-Bunch…
We investigate the primordial non-Gaussianities from the trispectra in multi-field inflation models, which can be seen as generalization of multi-field $k$-inflation and multi-DBI inflation. We derive the full fourth-order perturbation action for the inflaton fields and evaluate the four-point correlation functions for the perturbations in the limit $\ca \ll 1$ and $\ce \ll1$. There are three types of momentum-dependent shape functions which arise from three types of four-point interaction vertices. The final trispectrum of the curvature perturbation can be expressed in terms of the deformatio…
The methods of effective field theory are used to study generic theories of inflation with a single inflaton field and to perform a general analysis of the associated non-Gaussianities. We investigate the amplitudes and shapes of the various generic three-point correlators, the bispectra, which may be generated by different classes of single-field inflationary models. Besides the well-known results for the DBI-like models and the ghost inflationary theories, we point out that curvature-related interactions may give rise to large non-Gaussianities in the form of bispectra characterized by a fla…
In the Horndeski's most general scalar-tensor theories, we derive the three-point correlation function of scalar non-Gaussianities generated during single-field inflation in the presence of slow-variation corrections to the leading-order term. Unlike previous works, the resulting bispectrum is valid for any shape of non-Gaussianities. In the squeezed limit, for example, this gives rise to the same consistency relation as that derived by Maldacena in standard single-field slow-roll inflation. We estimate the shape close to the squeezed one at which the effect of the term inversely proportional …
We investigate the features generated by a sharp turn along the inflationary trajectory in a two-field model of Dirac-Born-Infeld inflation, where one of the fields is heavy. Distinct features are generated by two different effects: the mixing of the light and heavy modes during the turn, on the one hand, and the resonance between the oscillations along the heavy direction after the turn, on the other hand. Contrary to models with standard kinetic terms, the resonance effect is not strongly suppressed because the action contains derivative interactions. Working in the potential basis, we study…
We completely clarify the feature of primordial non-Gaussianities of tensor perturbations in generalized G-inflation, i.e., the most general single-field inflation model with second order field equations. It is shown that the most general cubic action for the tensor perturbation (gravitational wave) $h_{ij}$ is composed only of two contributions, one with two spacial derivatives and the other with one time derivative on each $h_{ij}$. The former is essentially identical to the cubic term that appears in Einstein gravity and predicts a squeezed shape, while the latter newly appears in the prese…
We clarify the features of primordial non-Gaussianities of tensor perturbations in Gao's unifying framework of scalar-tensor theories. The general Lagrangian is given in terms of the ADM variables so that the framework maintains spatial covariance and includes the Horndeski theory and Gleyzes-Langlois-Piazza-Vernizzi (GLPV) generalization as specific cases. It is shown that the GLPV generalization does not give rise to any new terms in the cubic action compared to the case of the Horndeski theory, but four new terms appear in more general theories beyond GLPV. We compute the tensor 3-point cor…
We discuss a supersymmetric version of DBI (Dirac-Born-Infeld) inflation, which is a typical inflation model in string cosmology. The supersymmetric DBI action together with a superpotential always leads to correction terms associated with the potential into the kinetic term, which drastically change the dynamics of DBI inflation. We find two significant features of supersymmetric DBI inflation. The first one is that ultra-relativistic motion is prohibited to cause inflation, which leads to order of unity sound velocity squared and hence small non-Gaussianities of primordial curvature perturba…
We apply the time-renormalization group approach to study the effect of primordial non-Gaussianities in the non-linear evolution of cosmological dark matter density perturbations. This method improves the standard perturbation approach by solving renormalization group-like equations governing the dynamics of gravitational instability. The primordial bispectra constructed from the dark matter density contrast and the velocity fields represent initial conditions for the renormalization group flow. We consider local, equilateral and folded shapes for the initial non-Gaussianity and analyze as wel…
This talk briefly explains how the breaking of a Lorentz-invariant description of nature at tiny space-time intervals might affect the non-Gaussian character of the primordial fluctuations left by inflation. For example, a model that contains irrelevant operators that only preserve the spatial symmetries along constant-time surfaces can generate a larger non-Gaussian component in the pattern of primordial fluctuations than is ordinarily predicted by inflation. This property can be useful for constraining models that allow some Lorentz violation at short distances, beyond the constraints possib…
In single-field inflationary models with a low sound speed, the orthogonal shape of the primordial bispectrum arises due to partial cancellations between equilateral-type shapes. This fact allows for a speed of sound c_s as low as about 0.01, which is actually weakly preferred by WMAP data. For such values, the trispectrum, scaling like 1/c_s^4, is of order 10^8 and is therefore comparable to, and greater than, the 1 sigma observational bound t_NL^eq=(-3.11 +- 7.5)*10^6. Hence, the trispectrum is already constraining inflationary mechanisms candidates for generating an orthogonal bispectrum at…
We address the possibility of having an enhanced signal for tensor non-Gaussianities in presence of a source, as a signature of Primordial Gravitational Waves. We employ a nearly model-independent framework based on Effective Field Theory of inflation and compute tensor non-Gaussianities therefrom sourced by particle production during (p)reheating to arrive at an enhanced signal strength. We obtain the model-independent non-linearity parameters and also find that squeezed limit bispectra are more enhanced than equilateral limit.
The spatial curvature of the universe is not yet known. Even though at present the Universe is very close to being essentially flat and most signatures of curvature appear to have been diluted by inflation, if the number of e-foldings during inflation is close to the minimum necessary to explain the horizon problem, the curvature of the universe may have left imprints in the cosmic microwave background (CMB) that may be observable, especially at large angles. Motivated by general results on quantum cosmology and using effective field theory techniques, we develop a general approach for analyti…
We study non-linear primordial adiabatic and isocurvature perturbations and their non-Gaussianity. After giving a general formulation in the context of an extended delta N-formalism, we analyse in detail two illustrative examples. The first is a mixed curvaton-inflaton scenario in which fluctuations of both the inflaton and a curvaton (a light isocurvature field during inflation) contribute to the primordial density perturbation. The second example is that of double inflation involving two decoupled massive scalar fields during inflation. In the mixed curvaton-inflaton scenario we find that th…
