δN Formalism Research Articles

Issues on generating primordial anisotropies at the end of inflation

We revisit the idea of generating primordial anisotropies at the end of inflation in models of inflation with gauge fields. To be specific we consider the charged hybrid inflation model where the waterfall field is charged under a U(1) gauge field so the surface of end of inflation is controlled both by inflaton and the gauge fields. Using δN formalism properly we find that the anisotropies generated at the end of inflation from the gauge field fluctuations are exponentially suppressed on cosmological scales. This is because the gauge field evolves exponentially during inflation while in order to generate appreciable anisotropies at the end of inflation the spectator gauge field has to be frozen. We argue that this is a generic feature, that is, one can not generate observable anisotropies at the end of inflation within an FRW background.

Conformal invariance of curvature perturbation

We show that in the single component situation all perturbationvariables in the comoving gauge are conformally invariant to allperturbation orders. Generally we identify a special time slicing, theuniform-conformal transformation slicing, where all perturbations areagain conformally invariant to all perturbation orders. We applythis result to the δN formalism, and show its conformalinvariance.

Waterfall field in hybrid inflation and curvature perturbation

We study carefully the contribution of the waterfall field to the curvature perturbation at the end of hybrid inflation. In particular we clarify the parameter dependence analytically under reasonable assumptions on the model parameters. After calculating the mode function of the waterfall field, we use the δN formalism and confirm the previously obtained result that the power spectrum is very blue with the index 4 and is absolutely negligible on large scales. However, we also find that the resulting curvature perturbation is highly non-Gaussian and hence we calculate the bispectrum. We find that the bispectrum is at leading order independent of momentum and exhibits its peak at the equilateral limit, though it is unobservably small on large scales. We also present the one-point probability distribution function of the curvature perturbation.

Large-scale perturbations from the waterfall field in hybrid inflation

We estimate large-scale curvature perturbations from isocurvature fluctuations in the waterfall field during hybrid inflation, in addition to the usual inflaton field perturbations. The tachyonic instability at the end of inflation leads to an explosive growth of super-Hubble scale perturbations, but they retain the steep blue spectrum characteristic of vacuum fluctuations in a massive field during inflation. The power spectrum thus peaks around the Hubble-horizon scale at the end of inflation. We extend the usual δN formalism to include the essential role of these small fluctuations when estimating the large-scale curvature perturbation. The resulting curvature perturbation due to fluctuations in the waterfall field is second-order and the spectrum is expected to be of order 10−54 on cosmological scales.

Inflationary infrared divergences: geometry of the reheating surface vs. δN formalism

We describe a simple way of incorporating fluctuations of the Hubble scale during the horizon exit of scalar perturbations into the δN formalism. The dominant effect comes from the dependence of the Hubble scale on low-frequency modes of the inflaton. This modifies the coefficient of the log-enhanced term appearing in the curvature spectrum at second order in field fluctuations. With this modification, the relevant coefficient turns out to be proportional to the second derivative of the tree-level spectrum with respect to the inflaton ϕ at horizonexit. A logarithm with precisely the same coefficient appears in acalculation of the log-enhancement of the curvature spectrum basedpurely on the geometry of the reheating surface. We take thisagreement as strong support for the proposed implementation of theδN formalism. Moreover, our analysis makes it apparent that the log-enhancement of the inflationary power-spectrum is indeed physical if this quantity isdefined using a global coordinate system on the reheating surface (or anyother post-inflationary surface of constant energy density). However, it can be avoided by defining the spectrum using invariant distances on this surface.

Use of δN formalism—difficulties in generating large local-type non-Gaussianity during inflation

We discuss the generation of non-Gaussianity in density perturbation through the super-horizon evolution during inflation by using the so-called δN formalism. We first provide a general formula for the nonlinearity parameter generated during inflation. We find that it is proportional to the slow-roll parameters, multiplied by the model-dependent factors that may enhance non-Gaussianity to the observable ranges. Then we discuss three typical examples to illustrate how difficult it is to generate sizable non-Gaussianity through the super-horizon evolution during inflation. The first example is the double inflation model, which shows that temporal violation of slow-roll conditions is not enough for the generation of non-Gaussianity. The second example is the ordinary hybrid inflation model, which illustrates the importance of taking into account perturbations on small scales. Finally, we discuss the Kadota–Stewart model. This model gives an example in which we have to choose rather unnatural initial conditions even if large non-Gaussianity can be generated.

Local non-Gaussianity from inflation

The non-Gaussian distribution of primordial perturbations has the potential to reveal the physical processes at work in the very early universe. Local models provide a well-defined class of non-Gaussian distributions that arise naturally from the nonlinear evolution of density perturbations on super-Hubble scales starting from Gaussian field fluctuations during inflation. I describe the δN formalism used to calculate the primordial density perturbation on large scales and then review several models for the origin of local primordial non-Gaussianity, including the curvaton, modulated reheating and ekpyrotic scenarios. I include an appendix with a table of sign conventions used in specific papers.

On the generation of a non-gaussian curvature perturbation during preheating

The perturbation of a light field might affect preheatingand hence generate a contribution to the spectrum and non-gaussianity ofthe curvature perturbation ζ. The field might appear directly in the preheatingmodel (curvaton-type preheating) or indirectly through its effect on amass or coupling (modulated preheating).We give general expressions for ζ based on the δN formula, and apply them to the cases of quadratic and quarticchaotic inflation. For the quadratic case, curvaton-type preheating is ineffective in contributing to ζ, but modulated preheating can be effective. For quartic inflation, curvaton-typepreheating may be effective but the usual δN formalism has to be modified. We see under what circumstances the recent numerical simulation of Bond et al. [0903.3407] may be enough to provide a rough estimate for this case.

Non-gaussianity from the trispectrum and vector field perturbations

We use the δN formalism to study the trispectrum Tζ of the primordial curvature perturbation ζ when the latter is generated by vector field perturbations, considering the tree-level and one-loop contributions. The order of magnitude of the level of non-gaussianity in the trispectrum, τNL, is calculated in this scenario and related to the order of magnitude of the level of non-gaussianity in the bispectrum, fNL, and the level of statistical anisotropy in the power spectrum, gζ. Such consistency relations will put under test this scenario against future observations. Comparison with the expected observational bound on τNL from WMAP, for generic inflationary models, is done.

Delta-N formalism for the evolution of the curvature perturbations in generalized multi-field inflation

The δN formalism is considered to calculate the evolution of the curvature perturbation in generalized multi-field inflation models. The result is consistent with the usual calculation of the standard kinetic term. For the calculation of the generalized kinetic term, we improved the definition of the adiabatic field. Our calculation improves the usual calculation of R˙ based on the field equations and the perturbations, giving a very simple and intuitive argument for the evolution equations in terms of the perturbations of the inflaton velocity. Significance of non-equilibrium corrections are also discussed, which is caused by the small-scale (decaying) inhomogeneities. This formalism based on the modulated inflation scenario (i.e., calculation based on the perturbations related to the inflaton velocity) provides a powerful tool for investigating the signature of moduli that may appear in string theory.

Evolution of curvature perturbation in generalized gravity theories

Using the cosmological perturbation theory in terms of the δN formalism, we find the simple formulation of the evolution of the curvature perturbation in generalized gravity theories. Compared with the standard gravity theory, a crucial difference appears in the end-boundary of the inflationary stage, which is due to the non-ideal form of the energy–momentum tensor that depends explicitly on the curvature scalar. Recent study shows that ultraviolet-complete quantum theory of gravity (Hořava–Lifshitz gravity) can be approximated by using a generalized gravity action. Our paper may give an important step in understanding the evolution of the curvature perturbation during inflation, where the energy–momentum tensor may not be given by the ideal form due to the corrections from the fundamental theory.

The curvature perturbation in the axion-type curvaton model

We study the axion-type curvaton model, with emphasis on the largefield regime where analytic results are very difficult to obtain. Weevaluate the tensor-scalar ratio r using WMAP normalization andthe non-linearity parameters fNL and gNL by solving theequations numerically using the δN formalism. We comparethem with results for the curvaton with quadratic potential. We findthat r is much smaller for the axion-type case compared to theresult from the quadratic potential, with the differenceincreasingly more pronounced at larger field values. gNL isfound to be positive, unlike the quadratic case where it isnegative, and the amplitude of gNL is much larger. Moreover,there is a nearly linear scaling between gNL and fNL, withsmall deviation from linearity at large field values. The slopebetween gNL and fNL depends on the parameterscharacterizing the axion-type curvaton model. We further consider amixed scenario where both the inflaton and the curvaton contributeto the primordial power spectrum and the non-Gaussianity parametersare found to be much larger than those in the case with quadraticpotential.

Evolution of the curvature perturbation during and after multi-field inflation

We study the evolution of the curvature perturbation on the super-horizon scalesstarting from the inflationary epoch until there remains only a single dynamicaldegree of freedom, presureless matter, in the universe. We consider the cosmicinflation driven by a multiple number of the inflaton fields, which decay into bothradiation and pressureless matter components. We present a complete set of the exactbackground and perturbation equations which describe the evolution of the universethroughout its history. By applying these equations to the simple but reasonablemodel of multi-field chaotic inflation, we explicitly show that the total curvatureperturbation is continuously varying because of the non-adiabatic components of thecurvature perturbation generated by the multiple inflaton fields throughout thewhole evolution of the universe. We also provide an useful analytic estimation ofthe total as well as matter and radiation curvature perturbations, assuming thatmatter is completely decoupled from radiation from the beginning. The resultingisocurvature perturbation between matter and radiation is at most sub-percent levelwhen the masses of the inflaton fields are distributed between 10−6mPl and 10−5mPl. We find that this result is robust unless we use non-trivial decayrates, and that thus, in general, it is hard to obtain large matter-radiationisocurvature perturbation. Also, by using the δN formalism, we point outthat the inflationary calculation, especially when involving multiple inflatonfields, is likely to lose the potentially important post-inflationary evolutionwhich can modify the resulting curvature perturbation.

Different mechanism to explain the 1∕f noise in n- and p-SOI-MOS transistors fabricated on (110) and (100) silicon-oriented wafers

The investigation of the low frequency noise in p- and n-channel semiconductor-on-insulator–metal oxide semiconductor transistors fabricated on (100) and (110) silicon-oriented wafers using high advanced processes has been carried out. While for both orientations the 1∕f noise in the n-channel transistors can be explained through the ΔN formalism, the p-channel ones seem to follow the Hooge model. In addition, the new orientation presents a noise level higher than the conventional one. Furthermore, the authors showed that when there is a gap of one decade between the Si(100) and Si(110) p-channel transistors, this one has been reduced to a factor of 2 in the case of the n-channel ones. Finally, by combining the highly advanced microwave-excited high-density plasma oxidation process with the alkali-free five-step cleaning process, it has been possible to fabricate on both surfaces a very high quality oxide with a very low density of traps.

Modulated inflation

We have studied modulated inflation that generates curvature perturbation from light-field fluctuation. As discussed in previous works, even if the fluctuation of the inflaton itself does not generate the curvature perturbation, fluctuation of a light field may induce fluctuation for the end-line of inflation and this may lead to generation of cosmological perturbation “at the end of the inflation”. Our scenario is different from this kind of modulated scenario, as clearly explained in this Letter by using δN formalism. We also explain the crucial difference from the standard multi-field inflation model. We show concrete examples of the modulated inflation scenario in which large non-gaussianity can be generated. We also discuss the running of the non-gaussianity parameter.

A note on nonlinear curvature perturbations in an exactly soluble model of multi-component slow-roll inflation

Using the nonlinear δN formalism, we consider a simple exactly soluble model of multi-component slow-roll inflation in which the nonlinear curvature perturbation can be evaluated analytically.

Analysis of the evolution of curvature perturbations in preheating by using δN formalism

We study the evolution of the curvature perturbation during preheating for a simple preheating model whose potential is given by with g2/λ = 2. In order to evaluate the final amplitude of the curvature perturbation after preheating, a method beyond linear perturbation theory is required. We make use of the δN formalism and compare our results with the previous studies which make use of the Hartree approximation. We find that the final amplitude of the curvature perturbation after preheating obtained by the δN formalism is smaller than that obtained in the previous studies.

A new δN formalism for multi-component inflation

The δN formula that relates the final curvature perturbation on comoving slices to the inflatonperturbation on flat slices after horizon crossing is a powerful and intuitive tool forcomputing the curvature perturbation spectrum from inflation. However, it iscustomarily assumed further that the conventional slow-roll condition is satisfied, andsatisfied by all components, during horizon crossing. In this paper, we develop a newδN formalism for multi-component inflation that can be applied in the most general situations.This allows us to generalize the idea of general slow-roll inflation to the multi-componentcase, in particular only applying the general slow-roll condition to the relevant component.We compute the power spectrum of the curvature perturbation in multi-component generalslow-roll inflation, and find that under quite general conditions it is invertible.