In the present paper, we study the non-Gaussianity of multiple-field inflation model using the method of the field equation. We start from reviewing the background and the perturbation theory of multiple-field inflation, and then derive the Klein-Gorden equation for the perturbations at second order. Afterward, we calculate the tree-level bispectrum of the fields' perturbations and finally give the corresponding parameter $f_{NL}$ for the curvature perturbation $ζ$ in virtue of the $δN$ formalism. We also compare our result with the one already obtained from the Lagrangian formalism, and find they are consistent. This work may help us understand perturbation theory of inflation more deeply.
We study the level of primordial non-Gaussianity in slow-roll two-field inflation. Using an analytic formula for the nonlinear parameter f_nl in the case of a sum or product separable potential, we find that it is possible to generate significant non-Gaussianity even during slow-roll inflation with Gaussian perturbations at Hubble exit. In this paper we give the general conditions to obtain large non-Gaussianity and calculate the level of fine-tuning required to obtain this. We present explicit models in which the non-Gaussianity at the end of inflation can exceed the current observational bou…
A model of multi-component hybrid inflation, dubbed multi-brid inflation, in which various observable quantities including the non-Gaussianity parameter f_{NL} can be analytically calculated was proposed recently. In particular, for a two-brid inflation model with an exponential potential and the condition that the end of inflation is an ellipse in the field space, it was found that, while keeping the other observational quantities within the range consistent with observations, large non-Gaussianity is possible for certain inflationary trajectories, provided that the ratio of the two masses is…
Recent analysis of the WMAP three year data suggests $f_{NL}^{local}\simeq86.8$ in the WMAP convention. It is necessary to make sure whether general single field inflation can produce a large positive $f_{NL}$ before turning to other scenarios. We give some examples to generate a large positive $f_{NL}^{equil}$ in general single field inflation. Our models are different from ghost inflation. Due to the appearance of non-conventional kinetic terms, $f_{NL}^{equil}\gg1$ can be realized in single field inflation.
We study the generation of non-Gaussianity in models of hybrid inflation with two inflaton fields, (2-brid inflation). We analyse the region in the parameter and the initial condition space where a large non-Gaussianity may be generated during slow-roll inflation which is generally characterised by a large f_NL, tau_NL and a small g_NL. For certain parameter values we can satisfy tau_NL>>f_NL^2. The bispectrum is of the local type but may have a significant scale dependence. We show that the loop corrections to the power spectrum and bispectrum are suppressed during inflation, if one assume th…
We study the evolution of cosmological perturbations on large scales, up to second order, for a perfect fluid with generic equation of state. Taking advantage of super-horizon conservation laws, it is possible to follow the evolution of the non-Gaussianity of perturbations through the different stages after inflation. We find that a large non-linearity is generated by the gravitational dynamics from the original inflationary quantum fluctuations. This leads to a significant enhancement of the tiny intrinsic non-Gaussianity produced during inflation in single-field slow-roll models.
We calculate the bispectrum of single-field braneworld inflation, triggered by either an ordinary scalar field or a cosmological tachyon, by means of a gradient expansion of large-scale non-linear perturbations coupled to stochastic dynamics. The resulting effect is identical to that for single-field 4D standard inflation, the non-linearity parameter being proportional to the scalar spectral index in the limit of collapsing momentum. If the slow-roll approximation is assumed, braneworld and tachyon non-Gaussianities are subdominant with respect to the post-inflationary contribution. However, b…
In the literature there have been incompatible estimates for the amount of non-Gaussianity in hybrid inflation. In this note we point out the sources for the discrepancies and show that the results for the amount of non-Gaussianity in hybrid inflation obtained by two different methods, namely, perturbing Einstein equation to second order and the separate universe approach, indeed are compatible. This provides confidence in the methods themselves and in the actual computation of non-Gaussianities.
We study the non-gaussianity from the bispectrum in multi-field inflation models with a general kinetic term. The models include the multi-field K-inflation and the multi-field Dirac-Born-Infeld (DBI) inflation as special cases. We find that, in general, the sound speeds for the adiabatic and entropy perturbations are different and they can be smaller than 1. Then the non-gaussianity can be enhanced. The multi-field DBI-inflation is shown to be a special case where both sound speeds are the same due to a special form of the kinetic term. We derive the exact second and third order actions inclu…
This is a review of models of inflation and of their predictions for the primordial non-Gaussianity in the density perturbations which are thought to be at the origin of structures in the Universe. Non-Gaussianity emerges as a key observable to discriminate among competing scenarios for the generation of cosmological perturbations and is one of the primary targets of present and future Cosmic Microwave Background satellite missions. We give a detailed presentation of the state-of-the-art of the subject of non-Gaussianity, both from the theoretical and the observational point of view, and provi…
We calculate the tree-level bispectrum of the inflaton field perturbation directly from the field equations, and construct the corresponding f_NL parameter. Our results agree with previous ones derived from the Lagrangian. We argue that quantum theory should only be used to calculate the correlators when they first become classical a few Hubble times after horizon exit, the classical evolution taking over thereafter.
We present Feynman type diagrams for calculating the n-point function of the primordial curvature perturbation in terms of scalar field perturbations during inflation. The diagrams can be used to evaluate the corresponding terms in the n-point function at tree level or any required loop level. Rules are presented for drawing the diagrams and writing down the corresponding terms in real space and Fourier space. We show that vertices can be renormalised to automatically account for diagrams with dressed vertices. We apply these rules to calculate the primordial power spectrum up to two loops, th…
We investigate consistency relations for non-Gaussianity. We provide a model-independent dynamical proof for the consistency relation of 3-point correlation functions from the Hamiltonian and field redefinition. This relation can be applied to single field inflation, multi-field inflation and the curvaton scenario. This relation can also be generalized to $n$-point correlation functions up to arbitrary order in perturbation theory and with arbitrary number of loops.
We extend the δN formalism so that it gives all of the stochastic properties of the primordial curvature perturbation ζif the initial field perturbations are gaussian. The calculation requires only the knowledge of some family of unperturbed universes. A formula is given for the normalisation \fnl of the bispectrum of ζ, which is the main signal of non-gaussianity. Examples of the use of the formula are given, and its relation to cosmological perturbation theory is explained.
We study brane inflation scenarios in a warped throat geometry and show that there exists a consistency condition between the non-Gaussianity of the curvature perturbation and the amplitude and scale-dependence of the primordial gravitational waves. This condition is independent of the warping of the throat and the form of the inflaton potential. We find that such a relation could be tested by a future CMB polarization experiment if the Planck satellite is able to detect both a gravitational wave background and a non-Gaussian statistic. In models where the observable stage of inflation occurs …
We consider a class of multi-component hybrid inflation models whose evolution may be analytically solved under the slow-roll approximation. We call it multi-brid inflation (or $n$-brid inflation where $n$ stands for the number of inflaton fields). As an explicit example, we consider a two-brid inflation model, in which the inflaton potentials are of exponential type and a waterfall field that terminates inflation has the standard quartic potential with two minima. Using the $δN$ formalism, we derive an expression for the curvature perturbation valid to full nonlinear order. Then we give an ex…
Correlated adiabatic and isocurvature perturbation modes are produced during inflation through an oscillation mechanism when extra scalar degrees of freedom other than the inflaton field are present. We show that this correlation generically leads to sizeable non-Gaussian features both in the adiabatic and isocurvature perturbations. The non-Gaussianity is first generated by large non-linearities in some scalar sector and then efficiently transferred to the inflaton sector by the oscillation process. We compute the cosmic microwave background angular bispectrum, providing a characteristic feat…
Using the "$δN$-formalism", We obtain the expression of the non-Gaussianity of multiple-field inflationary models with the nontrivial field-space metric. Further, we rewritten the result by using the slow-rolling approximation.
We compute the fourth order action in perturbation theory for scalar and second order tensor perturbations for a minimally coupled single field inflationary model, where the inflaton's lagrangian is a general function of the field's value and its kinetic energy. We obtain the fourth order action in two gauges, the comoving gauge and the uniform curvature gauge. Using the comoving gauge action we calculate the trispectrum at leading order in slow-roll, finding agreement with a previously known result in the literature. We point out that in general to obtain the correct leading order trispectrum…
The scalar mode density perturbations in a the warm inflationary scenario are analysed with a view to predicting the amount of non-gaussianity produced by this scenario. The analysis assumes that the inflaton evolution is strongly damped by the radiation, with damping terms that are temperature independent. Entropy fluctuations during warm inflation play a crucial role in generating non-gaussianity and result in a distinctive signal which should be observable by the Planck satellite.
In this paper, we use the result in [7] to calculate the non-Gaussianity of the racetrack models in [3, 5]. The two models give different non- Gaussianities. Both of them are reasonable.
We give a concise formula for the non-Gaussianity of the primordial curvature perturbation generated on super-horizon scales in multi-scalar inflation model without assuming slow-roll conditions. This is an extension of our previous work. Using this formula, we study the generation of non-Gaussianity for the double inflation models in which the slow-roll conditions are temporarily violated after horizon exit, and we show that the non-linear parameter $f_{NL}$ for such models is suppressed by the slow-roll parameters evaluated at the time of horizon exit.
We present an estimate for the non-linear parameter τ_NL, which measures the non-gaussianity imprinted in the trispectrum of the comoving curvature perturbation, ζ. Our estimate is valid throughout the inflationary era, until the slow-roll approximation breaks down, and takes into account the evolution of perturbations on superhorizon scales. We find that the non-gaussianity is always small if the field values at the end of inflation are negligible when compared to their values at horizon crossing. Under the same assumption, we show that in Nflation-type scenarios, where the potential is a sum…
We show a text-book potential for single-field inflation, namely, the Coleman-Weinberg model can induce double inflation and formation of primordial black holes (PBHs), because fluctuations that leave the horizon near the end of first inflation are anomalously enhanced at the onset of second inflation when the time-dependent mode turns to a growing mode rather than a decaying mode. The mass of PBHs produced in this mechanism lies in several discrete ranges depending on the model parameters. We also calculate the effects of non-Gaussian statistics due to higher-order interactions on the abundan…
We calculate the general solutions for a warm inflationary scenario with weak dissipation, reviewing the dissipative dynamics of the two-fluid system, and calculate the bispectrum of the gravitational field fluctuations generated in the case where dissipation of the vacuum potential during inflation is the mechanism for structure formation, but is the sub-dominant effect in the dynamics of the scalar field during slow-roll. The bispectrum is non-zero because of the self-interaction of the scalar field. We compare the predictions with both those of standard, or `supercooled', inflationary model…
Theories beyond the standard model such as string theory motivate low energy effective field theories with several scalar fields which are not only coupled through a potential but also through their kinetic terms. For such theories we derive the general formulae for the running of the spectral indices for the adiabatic, isocurvature and correlation spectra in the case of two field inflation. We also compute the expected non-Gaussianity in such models for specific forms of the potentials. We find that the coupling has little impact on the level of non-Gaussianity during inflation.
We analyze the non-Gaussianity for primordial curvature perturbations generated in multi-scalar slow-roll inflation model including the model with non-separable potential by making use of $δN$ formalism. Many authors have investigated the possibility of large non-Gaussianity for the models with separable potential, and they have found that the non-linear parameter, $f_{NL}$, is suppressed by the slow-roll parameters. We show that for the non-separable models $f_{NL}$ is given by the product of a factor which is suppressed by the slow-roll parameters and a possible enhancement factor which is g…
We consider reheating after inflation with a non-standard kinetic term. We show that the difference in the kinetic term in different Hubble patches inherited from the long-wavelength moduli inhomogeneities may generate a significant level of non-Gaussianity after inflation.
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.
The statistics of multi-field inflation are investigated using the stochastic approach. We analytically obtain the probability distribution function of fields with the scaling approximation by extending the previous work by Amendola. The non-Gaussian nature of the probability distribution function is investigated decomposing the fields into the adiabatic and isocurvature components. We find that the non-Gaussianity of the isocurvature component can be large compared with that of the adiabatic component. The adiabatic and isocurvature components may be correlated at nonlinear order in the skewn…
We argue that the typical energy density of a light scalar field should not be less than $H^4$ in the inflationary Universe. This requirement implies that the non-Gaussianity parameter $f_{NL}$ is typically bounded by the tensor-scalar ratio $r$ from above, namely $f_{NL}\lesssim 518\cdot r^{1\over 4}$. If $f_{NL}=10^2$, inflation occurred around the GUT scale.
We present a new approximation scheme that allows us to increase the accuracy of analytical predictions of the power spectra of inflationary perturbations for two specific classes of inflationary models. Among these models are chaotic inflation with a monomial potential, power-law inflation and natural inflation (inflation at a maximum). After reviewing the established first order results we calculate the amplitudes and spectral indices for these classes of models at higher orders in the slow-roll parameters for scalar and tensorial perturbations.
The generic prediction for the perturbations generated during slow--roll, single--field inflation, as they appear in the cosmic microwave background (CMB), is a flat, close--to--Gaussian spectrum. We calculate the general solution for a warm inflationary scenario with weak dissipation, reviewing the dissipative dynamics of the two--fluid system, and calculate the bispectrum of the gravitational field fluctuations generated in the case where dissipation of the vacuum potential during inflation is the mechanism for structure formation, but is the sub--dominant effect in the dynamics of the scala…
We analyse models of inflation in which isocurvature perturbations present during inflation are converted into the primordial curvature perturbation during instant preheating. This can be due to an asymmetry between the fields present either during inflation or during preheating. We consider all the constraints that the model must satisfy in order to be theoretically valid and to satisfy observations. We show that the constraints are very tight in all of the models proposed and special initial conditions are required for the models to work. In the case where the symmetry is strongly broken dur…
In this paper we calculate additional contributions to that part of the non-Gaussianity of the curvature perturbation, which come from the three-point correlator of the field perturbations. We estimate this contribution in the following models for its origin: single-component inflation, multi-component chaotic inflation, a two-component "hybrid" inflationary model, and the curvaton scenario. In all of these models, the additional contributions to the primordial non-gaussianity considered here, are too small to be ever detected.
We consider a two component hybrid inflation model, in which two fields drive inflation. Our results show that this model generates an observable non-gaussian contribution to the curvature spectrum, within the limits allowed by the recent WMAP year 3 data. We show that if one field has a mass less than zero, and an initial field value less than 0.06Mpl while the other field has a mass greater than zero, and initial field value ranging between 0.5Mpl and Mpl then the non-gaussianity is observable with 1<fnl<1.5, but that fnl becomes much less than the observable limit should we take both masses…
We propose a non-Gaussianity test for gravitational wave backgrounds by combining data streams of multiple detectors. This simple method allows us to check whether a detected background is "smooth" enough to be consistent with an inflation-type background, or is contaminated by individually undetectable weak burst signals. The proposed test would be quite useful for the Big Bang Observer or DECIGO whose primary target is a background from inflation at 0.1-1Hz where gravitational wave bursts from supernovae of population III stars might become a troublesome foreground.
This article investigates the generation of non-Gaussianity during inflation. In the context of multi-field inflation, we detail a mechanism that can create significant primordial non-Gaussianities in the adiabatic mode while preserving the scale invariance of the power spectrum. This mechanism is based on the generation of non-Gaussian isocurvature fluctuations which are then transfered to the adiabatic modes through a bend in the classical inflaton trajectory. Natural realizations involve quartic self-interaction terms for which a full computation can be performed. The expected statistical p…
We present a simple way to calculate non-Gaussianity in inflation using fully non-linear equations on long wavelengths with stochastic sources to take into account the short-wavelength quantum fluctuations. Our formalism includes both scalar metric and matter perturbations, combining them into variables which are invariant under changes of time slicing in the long-wavelength limit. We illustrate this method with a perturbative calculation in the single-field slow-roll case. We also introduce a convenient choice of variables to graphically present the full momentum dependence of the three-point…
We calculate the trispectrum T_ζof the primordial curvature perturbation ζ, generated during a {\it slow-roll} inflationary epoch by considering a two-field quadratic model of inflation with {\it canonical} kinetic terms. We consider loop contributions as well as tree level terms, and show that it is possible to attain very high, {\it including observable}, values for the level of non-gaussianity τ_{NL} if T_ζis dominated by the one-loop contribution. Special attention is paid to the claim in JCAP {\bf 0902}, 017 (2009) [arXiv:0812.0807 [astro-ph]] that, in the model studied in this paper and …
Recently we studied inflation models in which the inflaton potential is characterized by an underlying approximate global symmetry. In the first work we pointed out that in such a model curvature perturbations are generated after the end of the slow-roll phase of inflation. In this work we develop further the observational implications of the model and compute the degree of non-Gaussianity predicted in the scenario. We find that the corresponding nonlinearity parameter, $f_{NL}$, can be as large as 10^2.
In (Hansen et al. 2002) we presented a new approach for measuring non-Gaussianity of the Cosmic Microwave Background (CMB) anisotropy pattern, based on the multivariate empirical distribution function of the spherical harmonics a_lm of a CMB map. The present paper builds upon the same ideas and proposes several improvements and extensions. More precisely, we exploit the additional information on the random phases of the a_lm to provide further tests based on the empirical distribution function. Also we take advantage of the effect of rotations in improving the power of our procedures. The sugg…
We show in this paper that it is possible to attain very high, {\it including observable}, values for the level of non-gaussianity f_{NL} associated with the bispectrum B_ζof the primordial curvature perturbation ζ, in a subclass of small-field {\it slow-roll} models of inflation with canonical kinetic terms. Such a result is obtained by taking care of loop corrections both in the spectrum P_ζand the bispectrum B_ζ. Sizeable values for f_{NL} arise even if ζis generated during inflation. Five issues are considered when constraining the available parameter space: 1. we must ensure that we are i…
We investigate non-Gaussianity in general multiple-field inflation using the formalism we developed in earlier papers. We use a perturbative expansion of the non-linear equations to calculate the three-point correlator of the curvature perturbation analytically. We derive a general expression that involves only a time integral over background and linear perturbation quantities. We work out this expression explicitly for the two-field slow-roll case, and find that non-Gaussianity can be orders of magnitude larger than in the single-field case. In particular, the bispectrum divided by the square…
We study non-Gaussianity, the spectral index of primordial scalar fluctuations and tensor modes in models where fluctuations from the inflaton and the curvaton can both contribute to the present cosmic density fluctuations. Even though simple single-field inflation models generate only tiny non-Gaussianity, if we consider such a mixed scenario, large non-Gaussianity can be produced. Furthermore, we study the inflationary parameters such as the spectral index and the tensor-to-scalar ratio in this kind of models and discuss in what cases models predict the spectral index and tensor modes allowe…
Two different predictions for the primordial curvature fluctuation bispectrum are compared through their effects on the Cosmic Microwave Background temperature fluctuations. The first has a local form described by a single parameter f_{NL}. The second is based on a prediction from the warm inflationary scenario, with a different dependence on wavenumber and a parameter f_{WI}. New expressions are obtained for the angular bispectra of the temperature fluctuations and for the estimators used to determine $f_{NL}$ and f_{WI}. The standard deviation of the estimators in an ideal experiment is roug…
In the curvaton model of inflation, where a second scalar field, the "curvaton", is responsible for the observed inhomogeneity, a non-zero neutrino degeneracy may lead to a characteristic pattern of isocurvature perturbations in the neutrino, cold dark matter and baryon components. We find the current data can only place upper limits on the level of isocurvature perturbations. These can be translated into upper limits on the neutrino degeneracy parameter. In the case that lepton number is created before curvaton decay, we find that the limit on the neutrino degeneracy parameter is comparable w…
We study non-Gaussian signatures on the cosmic microwave background (CMB) radiation predicted within inflationary models with non-vacuum initial states for cosmological perturbations. The model incorporates a privileged scale, which implies the existence of a feature in the primordial power spectrum. This broken-scale-invariant model predicts a vanishing three-point correlation function for the CMB temperature anisotropies (or any other odd-numbered-point correlation function) whilst an intrinsic non-Gaussian signature arises for any even-numbered-point correlation function. We thus focus on t…
After simplifying and improving the non-Gaussian formalism we developed in previous work, we derive a quantitative expression for the three-point correlator (bispectrum) of the curvature perturbation in general multiple-field inflation models. Our result describes the evolution of non-Gaussianity on superhorizon scales caused by the nonlinear influence of isocurvature perturbations on the adiabatic perturbation during inflation. We then study a simple quadratic two-field potential and find that when slow roll breaks down and the field trajectory changes direction in field space, the non-Gaussi…
This paper explores single field inflation models with a constant, but arbitrary speed of sound c_s, obtained by deforming the kinetic energy terms to a Dirac-Born-Infeld form. Allowing c_s<1 provides a simple parameterization of non-gaussianity. The dependence of inflationary observables on the parameter c_s is considered in the leading order slow roll approximation. The results show that in most cases the dependence is actually rather weak for the range of c_s allowed by existing bounds on non-gaussianity.
We investigate the relative sensitivities of several tests for deviations from Gaussianity in the primordial distribution of density perturbations. We consider models for non-Gaussianity that mimic that which comes from inflation as well as that which comes from topological defects. The tests we consider involve the cosmic microwave background (CMB), large-scale structure (LSS), high-redshift galaxies, and the abundances and properties of clusters. We find that the CMB is superior at finding non-Gaussianity in the primordial gravitational potential (as inflation would produce), while observati…
If light scalar fields are present at the end of inflation, their non-equilibrium dynamics such as parametric resonance or a phase transition can produce non-Gaussian density perturbations. We show how these perturbations can be calculated using non-linear lattice field theory simulations and the separate universe approximation. In the massless preheating model, we find that some parameter values are excluded while others lead to acceptable but observable levels of non-Gaussianity. This shows that preheating can be an important factor in assessing the viability of inflationary models.
