In this paper, we study a curvaton model described by a Dirac-Born-Infeld-like action. We calculate the spectrum of curvature perturbation induced by DBI-curvaton and estimate its non-Gaussianity. We find that in the limit of low sound speed the amplitude of curvature perturbation is enhanced more than that in DBI inflation. This result also means that the inflationary scale with DBI-curvaton may be lower than that in usual curvaton model. In addition, we also find that the local non-Gaussianity level in DBI-curvaton is generally about 10 times larger than that in usual curvaton model, which is interesting for current observations. This work in some sense explores a new application of brane-cosmology in inflation.
In curvaton-type models, observable non-gaussianity of the curvature perturbation would come from a contribution of the form $(δσ)^2$, where $δσ$ is gaussian. I analyse this situation allowing $δσ$ to be scale-dependent. The actual curvaton model is considered in more detail than before, including its cosmic uncertainty and anthropic status. The status of curvaton-type models after WMAP year three data is considered.
The generation of non-gaussianity is studied in a three fluid curvaton model. By utilizing second order perturbation theory we derive general formulae for the large scale temperature fluctuation and non-gaussianity parameter, $f_{NL}$, that includes the possibility of a non-adiabatic final state. In the adiabatic limit we recover previously known results. The results are applied to a three fluid curvaton model where the curvaton decays into radiation and matter. We find that the amount of non-gaussianity decreases as the final state of the system becomes more adiabatic and that the generated n…
We use a generalized delta N formalism to study the generation of the primordial curvature perturbation in the curvaton brane scenario inspired by stringy compactifications. We note that the non-Gaussian features, especially the trispectra, crucially depend on the decay mechanism in a general curvaton scenario. Specifically, we study the bispectra and trispectra of the curvaton brane model in detail to illustrate the importance of curvaton decay in generating nonlinear fluctuations. When the curvaton brane moves nonrelativistically during inflation, the shape of non-Gaussianity is local, but t…
We discuss non-Gaussianity and baryonic isocurvature fluctuations in the curvaton scenario, assuming that the baryon asymmetry of the universe originates only from the decay products of the inflaton. When large non-Gaussianity is realized in such a scenario, non-vanishing baryonic isocurvature fluctuations can also be generated unless the baryogenesis occurs after the decay of the curvaton. We calculate the non-linearity parameter f_NL and the baryonic isocurvature fluctuations, taking account of the primordial fluctuations of both the inflaton and the curvaton. We show that, although current …
We study the curvaton scenario using gauge-invariant second order perturbation theory and solving the governing equations numerically. Focusing on large scales we calculate the non-linearity parameter f_nl in the two-fluid curvaton model and compare our results with previous analytical studies employing the sudden decay approximation. We find good agreement of the two approaches for large curvaton energy densities at curvaton decay, Omega_dec, but significant differences of up to 10 percent for small Omega_dec.
Inflating curvaton can create curvature perturbation when the curvaton density is slowly varying. Using the delta-N formalism, we discuss the evolution of the curvature perturbation during curvaton inflation and find analytic formulation of the non-Gaussianity parameter. We first consider the inflating curvaton with sufficiently long inflationary expansion. Then we compare the result with short curvaton inflation.
We use the delta N -formalism to investigate the non-Gaussianity of the primordial curvature perturbation in the curvaton scenario for the origin of structure. We numerically calculate the full probability distribution function allowing for the non-instantaneous decay of the curvaton and compare this with analytic results derived in the sudden-decay approximation. We also present results for the leading-order contribution to the primordial bispectrum and trispectrum. In the sudden-decay approximation we derive a fully non-linear expression relating the primordial perturbation to the initial cu…
We discuss the signatures of non-Gaussianity in the curvaton model where the potential includes also a non-quadratic term. In such a case the non-linearity parameter f_NL can become very small, and we show that non-Gaussianity is then encoded in the non-reducible non-linearity parameter g_NL of the trispectrum, which can be very large. Thus the place to look for the non-Gaussianity in the curvaton model may be the trispectrum rather than the bispectrum. We also show that g_NL measures directly the deviation of the curvaton potential from the purely quadratic form. While g_NL depends on the str…
We study non-Gaussianity of density perturbations generated by an axionic curvaton, focusing on the case that the curvaton sits near the hilltop of the potential during inflation. Such hilltop curvatons can generate a red-tilted density perturbation spectrum without invoking large-field inflation. We show that, even when the curvaton dominates the Universe, the non-Gaussianity parameter fNL is positive and mildly increases towards the hilltop of the curvaton potential, and that fNL = O(10) is a general and robust prediction of such hilltop axionic curvatons. In particular, we find that the non…
We investigate density perturbations sourced by a curvaton with a generic energy potential. The key feature of a curvaton potential which deviates from a quadratic is that the curvaton experiences a non-uniform onset of its oscillation. This sources additional contributions to the resulting density perturbations, and we especially find that the non-Gaussianity parameter f_NL can become large with either sign no matter whether the curvaton dominates or subdominates the universe when it decays. Such non-quadratic curvaton potentials are required in order to produce a red-tilted density perturbat…
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…
We revise the Non-Gaussianity of canonical curvaton scenario with a generalized $δN$ formalism, in which it could handle the generic potentials. In various curvaton models, the energy density is dominant in different period including the secondary inflation of curvaton, matter domination and radiation domination. Our method could unify to deal with these periods since the non-linearity parameter $f_{\rm NL}$ associated with Non-Gaussianity is a function of equation of state $w$. We firstly investigate the most simple curvaton scenario, namely the chaotic curvaton with quadratic potential. Our …
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 the density perturbation by a curvaton with a double well potential and estimate the nonlinear parameters for non-Gaussianity and the amplitude of gravitational wave background generated during inflation. The predicted nonlinear parameters strongly depend on the size of a curvaton self-coupling constant as well as the reheating temperature after inflation for a given initial amplitude of the curvaton. The difference from usual massive self-interacting curvaton is also emphasized.
In this paper, we explore the parameter space for a Right-Handed (RH) sneutrino curvaton that can generate large non-Gaussianity without assuming any particular inflation sector. The mass of the RH sneutrino is suggested from a discussion on the initial condition of the curvaton field. It is shown that a small Yukawa coupling is generally required for a successful RH sneutrino curvaton. However, the Yukawa coupling can be larger if we consider the braneworld scenario. Some general discussion about the spectral index in curvaton scenario is also provided.
We derive the spectral indices and their runnings of local form $f_{NL}$ and $g_{NL}$ from a DBI isocurvature field and we find that the indices are suppressed by the sound speed $c_s$. This effect can be interpreted by the Lorentz boost from the viewpoint in the frame where brane is moving.
We study a model where two scalar fields, that are subdominant during inflation, decay into radiation some time after inflation has ended but before primordial nucleosynthesis. Perturbations of these two curvaton fields can be responsible for the primordial curvature perturbation. We write down the full non-linear equations that relate the primordial perturbation to the curvaton perturbations on large scales, calculate the power spectrum of the primordial perturbation, and finally go to second order to find the non-linearity parameter, fNL. We find large positive values of fNL if the energy de…
We show that the local type non-Gaussianity in a class of curvaton models is suppressed, i.e. the non-linearity parameters f_{NL} and those related with higher order statistics can be at most O(1), even if the curvaton energy density is subdominant at the decay. This situation is naturally realized in a very simple curvaton potential with quadratic term plus quartic term.
We propose a class of curvaton models which we call passive curvaton. In this paper, two kinds of passive curvaton is considered. The first one is a pseudoscalar curvaton couples to a gauge field. Different from the inflaton case, the constraint from formation of primordial black holes (PBHs) is much weaker and large non-gaussianity (of the equiliteral type) can be produced. The second model is a dilaton-like scalar curvaton couples to a gauge field. We investigate the scale dependence of non-gaussianity in this model. In both models, the spectrum and non-Gaussianity are enhanced by the slow-r…
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.
Recently, the primordial non-Gaussianity in the curvaton model has been predicted assuming sudden decay of the curvaton. We extend the calculation to non-instantaneous decay by employing delta N -formalism. The difference between the sudden-decay approximation and our numerical result is larger than 1% only if the non-linearity parameter is small, -1.16 < f_NL < 60. Thus it is safe to use the sudden-decay approximation when deriving constraints for the curvaton model from WMAP3 (f_NL < 114), but with the Planck forecast |f_NL| <5 one should employ the fully numerical result. Often, the curvato…
In this paper, we consider large non-Gaussianity generated after D-term inflation in the case that the adiabatic curvature perturbation produced is dominated by a right-handed (RH) sneutrino curvaton. The cosmic string problem can also be evaded with around 10% contribution to the CMB power spectrum and the spectral index $n_s \simeq 1$. The non-Gaussianity produced can be as large as $10<f_{NL}<100$ for the right-handed sneutrino mass being around $m_Φ=10 {GeV}$ with the Yukawa coupling $λ_ν\sim 10^{-10}$. In this case, the lightest neutrino mass is $10^{-10}{eV}$ which favors RH sneutrino le…
We propose a curvaton model in which the initial condition of the curvaton oscillation is determined by its attractor behavior during inflation. Assuming a chaotic inflation model, we find that the initial condition determined by the attractor behavior is appropriate to generate a sizable non-Gaussianity contribution to the curvature perturbation, which will be tested in the foreseeable future. Implications on the thermal history of the universe and on particle physics models are also discussed.
We study observational implications of the stochastic gravitational wave background and a non-Gaussian feature of scalar perturbations on the curvaton mechanism of the generation of density/curvature fluctuations, and show that they can determine the properties of the curvaton in a complementary manner to each other. Therefore even if Planck could not detect any non-Gaussianity, future space-based laser interferometers such as DECIGO or BBO could practically exhaust its parameter space.
We calculate curvature perturbations in the scenario in which the curvaton field decays into another scalar field via parametric resonance. As a result of a nonlinear stage at the end of the resonance, standard perturbative calculation techniques fail in this case. Instead, we use lattice field theory simulations and the separate universe approximation to calculate the curvature perturbation as a nonlinear function of the curvaton field. For the parameters tested, the generated perturbations are highly non-Gaussian and not well approximated by the usual fNL parameterisation. Resonant decay pla…
We investigate the scale-dependence of f_NL in the self-interacting curvaton model. We show that the scale-dependence, encoded in the spectral index n_{f_NL}, can be observable by future cosmic microwave background observations, such as CMBpol, in a significant part of the parameter space of the model. We point out that together with information about the trispectrum g_NL, the self-interacting curvaton model parameters could be completely fixed by observations. We also discuss the scale-dependence of g_NL and its implications for the curvaton model, arguing that it could provide a complementar…
In this paper we consider how non-Gaussianity of the primordial density perturbation and the amplitude of gravitational waves from inflation can be used to determine parameters of the curvaton scenario for the origin of structure. We show that in the simplest quadratic model, where the curvaton evolves as a free scalar field, measurement of the bispectrum relative to the power spectrum, fNL, and the tensor-to-scalar ratio can determine both the expectation value of the curvaton field during inflation and its dimensionless decay rate relative to the curvaton mass. We show how these predictions …
A red tilted primordial power spectrum is preferred by WMAP five-year data and a large positive local-type non-Gaussianity $f_{NL}$ might be observed as well. In this short note we find that a red tilted and large non-Gaussian primordial power spectrum cannot be naturally obtained in curvaton model, because $f_{NL}$ is related to the initial condition of inflation.
We provide a model in which both the inflaton and the curvaton are obtained from within the minimal supersymmetric Standard Model, with known gauge and Yukawa interactions. Since now both the inflaton and curvaton fields are successfully embedded within the same sector, their decay products thermalize very quickly before the electroweak scale. This results in two important features of the model: firstly, there will be no residual isocurvature perturbations, and secondly, observable non-Gaussianities can be generated with the non-Gaussianity parameter $f_\mathrm{NL}\sim {\cal O}(5-1000)$ being …
In this paper, we consider the effect of two right-handed sneutrino curvaton decays and investigate the parameter space. We compare the difference of the result between single- and two-curvaton cases. We find one Yukawa coupling of the right-handed snetrinos can be as large as $λ\sim 0.1$ while the other one is much smaller which is consistent with the idea that only one generation of the neutrino mass may be much smaller than others. When the curvatons decay, we assume both of them subdominate the energy density of the universe. We find that, unlike a single curvaton case, here a small or neg…
Since a positive future detection of non-linearity in the cosmic microwave background anisotropy pattern might allow to descriminate among different mechanisms giving rise to cosmological adiabatic perturbations, we study the evolution of the second-order cosmological curvature perturbation on super-horizon scales in the curvaton scenario. We provide the exact expression for the non-Gaussianity in the primordial perturbations including gravitational second-order corrections which are particularly relevant in the case in which the curvaton dominates the energy density before it decays. As a byp…
We consider the early Universe scenario which allows for production of non-Gaussian curvature perturbations at small scales. We study the peculiarities of a formation of primordial black holes (PBHs) connected with the non-Gaussianity. In particular, we show that PBH constraints on the values of curvature perturbation power spectrum amplitude are strongly dependent on the shape of perturbations and can significantly (by two orders of magnitude) deviate from the usual Gaussian limit ${\cal P}_ζ\lesssim 10^{-2}$. We give examples of PBH mass spectra calculations and PBH constraints for the parti…
We investigate how the background evolution affects the curvature perturbations generated by the curvaton, assuming a curvaton potential that may deviate slightly from the quadratic one, and parameterizing the background fluid density as ρ\propto a^{-α}, where a is the scale factor, and αdepends on the background fluid. It turns out that the more there is deviation from the quadratic case, the more pronounced is the dependence of the curvature perturbation on α. We also show that the background can have a significant effect on the nonlinearity parameters f_NL and g_NL. As an example, if at the…
We study the thermalization process in the self-interacting curvaton preheating scenario. We solve the evolution of the system with classical lattice simulations with a recently released symplectic PyCOOL program during the resonance and the early thermalization periods and compare the results to the inflaton preheating. After this we calculate the generated non-gaussianity with the $ΔN$ formalism and the separate universe approximation by running a large number of simulations with slightly different initial values. The results indicate a high level of non-gaussianity. We also use this paper t…
In this paper we study a curvaton model obtained by considering a probe anti-D3-brane with angular motion at the bottom of a KS throat with approximate isometries. We calculate the spectrum of curvature perturbations and the non-Gaussianities of this model. Specifically, we consider the limit of relativistic rotation of the curvaton brane which leads to a small sound speed, and thus it can be viewed as an implementation of the DBI-curvaton mechanism. We find that the primordial power spectrum is nearly scale-invariant while the non-Gaussianity of local type is sizable and that of equilateral t…
In this paper, we study a curvaton model where the curvaton is acted by Galileon field. We calculate the power spectrum of fluctuation of G-curvaton during inflation and discuss how it converts to the curvature perturbation after the end of inflation. We estimate the bispectrum of curvature perturbation induced, and show the dependence of non-Gaussianity on the parameters of model. It is found that our model can have sizable local and equilateral non-Gaussianities to up to ${\cal O}(10^2)$, which is illustrated by an explicit example.
The curvaton scenario for the generation of the cosmological curvature perturbation on large scales represents an alternative to the standard slow-roll scenario of inflation in which the observed density perturbations are due to fluctuations of the inflaton field itself. Its basic assumption is that the initial curvature perturbation due to the inflaton field is negligible. This is attained by lowering the energy scale of inflation, thereby highly suppressing the amount of gravitational waves produced during inflation. We compute the power-spectrum of the gravitational waves generated at secon…
The primordial black hole (PBH) formation is studied in light of the inflating curvaton. The typical scale of the PBH formation is determined by curvaton inflation, which may generate PBH with $10^{14}\mathrm{g}\le M_\mathrm{PBH}\le 10^{38}\mathrm{g}$ when curvaton inflation gives the number of e-foldings $5\leq N_2\leq 38$. The non-Gaussianity of the inflating curvaton does not prevent the PBH formation.
The evolution of the curvature perturbation after multi-field inflation is studied in the light of the curvaton mechanism. Past numerical studies show that many-field inflation causes significant evolution of the curvature perturbation after inflation, which generates significant non-Gaussianity at the same time. We reveal the underlying mechanism of the evolution and show that the evolution is possible in a typical two-field inflation model.
Massless scalar fields originating in a quantum vacuum state acquire a scale-invariant spectrum of fluctuations in a matter-dominated contracting universe. We show that these isocurvature fluctuations transfer to a scale-invariant spectrum of curvature fluctuations during a non-singular bounce phase. This provides a mechanism for enhancing the primordial adiabatic fluctuations and suppressing the ratio of tensor to scalar perturbations. Moreover, this mechanism leads to new sources of non-Gaussianity of curvature perturbations.
We investigate primordial non-Gaussianity and dark matter isocurvature fluctuations in the modulated reheating and the curvaton scenarios. In these scenarios, large non-Gaussianity can be generated, on the other hand, depending on how dark matter is produced, too large isocurvature fluctuations can also arise, which is inconsistent with current observations. In this paper, we study this issue in a mixed scenario where the curvature fluctuations can also be produced from the inflaton fluctuations as well as those from a light scalar field such as the modulus and the curvaton. We show that primo…
We study density perturbations, including their non-Gaussianity, in models in which the decay rate of the curvaton depends on another light scalar field, denoted the modulaton. Although this model shares some similarities with the standard curvaton and modulated reheating scenarios, it exhibits interesting predictions for f_NL and g_NL that are specific to this model. We also discuss the possibility that both modulaton and curvaton fluctuations contribute to the final curvature perturbation. Our results naturally include the standard curvaton and modulated reheating scenarios as specific limit…
A pseudo Nambu-Goldstone boson as curvaton avoids the eta-problem of inflation which plagues most curvaton candidates. We point out that a concrete realization of the curvaton mechanism with a pseudo Nambu-Goldstone boson can be found in the supersymmetric Peccei-Quinn mechanism resolving the strong CP problem. In the flaton models of Peccei-Quinn symmetry breaking, the angular degree of freedom associated with the QCD axion can naturally be a flat direction during inflation and provides successful curvature perturbations. In this scheme, the preferred values of the axion scale and the Hubble …
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…
In general relativity predictions for observable quantities can be expressed in a coordinate independent way. Nonetheless it may be inconvenient to do so. Using a particular frame may be the easiest way to connect theoretical predictions to measurable quantities. For the cosmological curvature bispectrum such frame is described by the Conformal Fermi Coordinates. In single field inflation it was shown that going to this frame cancels the squeezed limit of the density perturbation bispectrum calculated in Global Coordinates. We explore this issue in quasi single field inflation when the curvato…
We consider an inflationary curvaton scenario, where the curvaton decays into two non-interacting relativistic fluids and later during the cosmological evolution one of them becomes non-relativistic, forming dark matter component of the universe. We study the thermic properties and the generation of non-gaussianity in this three fluid curvaton model. By solving the evolution of the system and using several cosmological conditions we find that the allowed parameter space is strongly constrained. The naturalness of this curvaton scenario is also discussed.
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…
An analysis is given of inflation based on a supersymmetric Dirac-Born-Infeld (DBI) action in an axionic landscape. The DBI model we discuss involves a landscape of chiral superfields with one $U(1)$ shift symmetry which is broken by instanton type non-perturbative terms in the superpotential. Breaking of the shift symmetry leads to one pseudo-Nambu-Goldstone-boson which acts as the inflaton while the remaining normalized phases of the chiral fields generically labeled axions are invariant under the $U(1)$ shift symmetry. The analysis is carried out in the vacuum with stabilized saxions, which…
We suppose that a vector field perturbation causes part of the primordial curvature perturbation. The non-Gaussianity parameter fNL is then, in general, statistically anisotropic. We calculate its form and magnitude in the curvaton scenario and in the end-of-inflation scenario. We show that this anisotropy could easily be observable.
The perturbation of a light field might affect preheating and hence generate a contribution to the spectrum and non-gaussianity of the curvature perturbation ζ. The field might appear directly in the preheating model (curvaton-type preheating) or indirectly through its effect on a mass or coupling (modulated preheating). We give general expressions for ζbased on the δN formula, and apply them to the cases of quadratic and quartic chaotic inflation. For the quadratic case, curvaton-type preheating is ineffective in contributing to ζ, but modulated preheating can be effective. For quartic inflat…
We construct a model of cosmological inflation and perturbation based on the higher-dimensional gauge theory. The inflaton and curvaton are the scalar fields arising from the extra space components of the gauge field living in more than four dimensions. We take the six-dimensional (6D) Yang-Mills theory compactified on $T^2$ as a toy model, and apply the one-loop effective potential of the inflaton and the curvaton to the curvaton scenario. We have found that the curvaton is subdominant for the linear curvature perturbation, but that a significant non-Gaussianity and a sizable tensor to scalar…
We study the intermediate inflation in the mimetic Dirac-Born-Infeld model. By considering the scale factor as $a=a_{0}\exp(bt^β)$, we show that in some ranges of the intermediate parameters $b$ and $β$, the model is free of the ghost and gradient instabilities. We study the scalar spectral index, tensor spectral index, and the tensor-to-scalar ratio in this model and compare the results with Planck2018 TT, TE, EE+lowE+lensing +BAO +BK14 data at $68\%$ and $95\%$ CL. In this regard, we find some constraints on the intermediate parameters that lead to the observationally viable values of the pe…
We analyze the spectrum of density perturbations generated in models of the recently discovered "D-cceleration" mechanism of inflation. In this scenario, strong coupling quantum field theoretic effects sum to provide a DBI-like action for the inflaton. We show that the model has a strict lower bound on the non-Gaussianity of the CMBR power spectrum at an observable level, and is thus falsifiable. This in particular observationally distinguishes this mechanism from traditional slow roll inflation generated by weakly interacting scalar fields. The model also favors a large observable tensor comp…
We investigate density fluctuations in a scenario with gravitino dark matter in the framework of modulated reheating, which is known to generate large non-Gaussianity. We show that gravitino dark matter is disfavored if primordial fluctuations have large local-type non-Gaussianity in this framework. We also briefly discuss the case with the curvaton mechanism and some other possible dark matter scenarios.
The scale--independence of the primordial curvature perturbation suggests that it comes from the vacuum fluctuation during inflation of a light scalar field. This field may be the inflaton, or a different `curvaton' field. The observation of primordial non--gaussianity would be a smoking gun for the curvaton model, while the observation of gravitational waves originating during slow--roll inflation would rule out the model.
We consider the mixed inflaton-curvaton scenario in which quantum fluctuations of the curvaton field during inflation lead to a relatively large curvature perturbation spectrum at small scales. We use the model of chaotic inflation with quadratic potential including supergravity corrections leading to a large positive tilt in the power spectrum of the curvaton field. The model is characterized by the strongly inhomogeneous curvaton field in the Universe and large non-Gaussianity of curvature perturbations at small scales. We obtained the constraints on the model parameters considering the proc…
