δN Formalism Research Articles

Non-Gaussian tails without stochastic inflation

We show, both analytically andnumerically,thatnon-Gaussian tails in the probability density function of curvature perturbations arise in ultra-slow-roll inflation from the δN formalism, without invoking stochastic inflation. Previously reported discrepancies between both approaches are a consequence of not correctly accounting for momentum perturbations. Once they are taken into account, both approaches agree to an excellent degree. The shape of the tail depends strongly on the phase space of inflation.

Primordial Black Hole Compaction Function from Stochastic Fluctuations in Ultraslow-Roll Inflation.

We study the formation of primordial black holes (PBHs) with ultraslow-roll inflation when stochastic effects are important. We use the ΔN formalism and simplify the stochastic equations with an analytical constant-roll approximation. Considering a viable inflation model, we find the spatial profile of the PBH compaction function numerically for each stochastic patch, without assumptions about Gaussianity or the radial profile. The stochastic effects that lead to an exponential tail for the density distribution also make the compaction function very spiky, unlike as assumed in the literature. Naively using collapse thresholds found for smooth profiles, the PBH abundance is enhanced by up to a factor of 10^{9}, and the PBH mass distribution is spread over 3 orders of magnitude in mass. The results point to a need to redo numerical simulations of PBH formation with spiky profiles.

Statistical Anisotropy of Primordial Gravitational Waves from Generalized δN Formalism.

In this Letter, we demonstrate how to use the generalized δN formalism, which enables us to compute the evolution of all the large-scale fluctuations, including gravitational waves, solely by solving the evolution of the background homogeneous Universe. Using the Noether charge density, we derive an analytic formula which describes the mapping between the fluctuations at the horizon crossing and the sourced gravitational waves at the end of inflation. This formula can apply also to an inflation model with an anisotropic background. Using this formula, we discuss the condition for the nonvanishing linear polarization and the qualitative difference between single- and multigauge field models.

The separate-universe approach and sudden transitions during inflation

The separate-universe approach gives an intuitive way to understand the evolution of cosmological perturbations in the long-wavelength limit. It uses solutions of the spatially-homogeneous equations of motion to model the evolution of the inhomogeneous universe on large scales. We show that the separate-universe approach fails on a finite range of super-Hubble scales at a sudden transition from slow roll to ultra-slow roll during inflation in the very early universe. Such transitions are a feature of inflation models giving a large enhancement in the primordial power spectrum on small scales, necessary to produce primordial black holes after inflation. We show that the separate-universe approach still works in a piece-wise fashion, before and after the transition, but spatial gradients on finite scales require a discontinuity in the homogeneous solution at the transition. We discuss the implications for the δN formalism and stochastic inflation, which employ the separate-universe approximation.

Non-Gaussianity in rapid-turn multi-field inflation

We show that theories of inflation with multiple, rapidly turning fields can generate large amounts of non-Gaussianity. We consider a general theory with two fields, an arbitrary field-space metric, and a potential that supports sustained, rapidly turning field trajectories. Our analysis accounts for non-zero field cross-correlation and does not fix the power spectra of curvature and isocurvature perturbations to be equal at horizon crossing. Using the δN formalism, we derive a novel, analytical formula for bispectrum generated from multi-field mixing on super-horizon scales. Rapid-turn inflation can produce a bispectrum with several potentially large contributions that are not necessarily of the local shape. We exemplify the applicability of our formula with a fully explicit model and show that the new contributions indeed can generate a large amplitude of local non-Gaussianity, f NL loc ∼ 𝒪(1). These results will be important when interpreting the outcomes of future observations.

An exact model for enhancing/suppressing primordial fluctuations

Abstract Enhancements of primordial curvature fluctuations in single field inflation often involve departures from attractor trajectories in the phase space. We study enhancement/suppression of primordial fluctuations in one of the simplest models with exact background solutions for arbitrary initial conditions: a single field inflationary model with a piecewise exponential potential. We then present close to exact analytical solutions for primordial fluctuations in a general transition between two slow-roll attractors, valid whether the first slow parameter increases or decreases. The main features in the primordial spectrum are determined by the ratio of exponents of the potential. We also discuss the imprint of such features in the induced GW spectrum. Lastly, we apply the δN formalism to discuss non-Gaussianities and the tail of the probability distribution. We find that while non-Gaussianities are at most 𝒪(1) in the case of enhancement, they can be very large in the case of suppression. Our work can be easily generalized to multiple piecewise exponential potentials.

Primordial Black Holes and loops in single-field inflation

Using the δN formalism we calculate the one-loop correction to the large-scale power spectrum of the curvature perturbation in the standard scenario where primordial black holes are formed in the early universe thanks to a phase of ultra-slow-roll in single-field inflation. We explicitly show that one-loop corrections are negligible when the transition from the ultra-slow-roll to the slow-roll phase is smooth. We conclude that the PBH formation scenario through a ultra-slow-roll phase is viable.

Highly asymmetric probability distribution from a finite-width upward step during inflation

We study a single-field inflation model in which the inflaton potential has an upward step between two slow-roll regimes by taking into account the finite width of the step. We calculate the probability distribution function (PDF) of the curvature perturbation P[ℛ] using the δN formalism. The PDF has an exponential-tail only for positive ℛ whose slope depends on the step width. We find that the tail may have a significant impact on the estimation of the primordial black hole abundance. We also show that the PDF P[ℛ] becomes highly asymmetric on a particular scale exiting the horizon before the step, at which the curvature power spectrum has a dip. This asymmetric PDF may leave an interesting signature in the large scale structure such as voids.

Multi-field inflation with large scalar fluctuations: non-Gaussianity and perturbativity

Recently multi-field inflation models that can produce large scalar fluctuations on small scales have drawn a lot of attention, primarily because they could lead to primordial black hole production and generation of large second-order gravitational waves. In this work, we focus on models where the scalar fields responsible for inflation live on a hyperbolic field space. In this case, geometrical destabilisation and non-geodesic motion are responsible for the peak in the scalar power spectrum. We present new results for scalar non-Gaussianity and discuss its dependence on the model's parameters. On scales around the peak, we typically find that the non-Gaussianity is large and close to local in form. We validate our results by employing two different numerical techniques, utilising the transport approach, based on full cosmological perturbation theory, and the δN formalism, based on the separate universe approximation. We discuss implications of our results for the perturbativity of the underlying theory, focusing in particular on versions of these models with potentially relevant phenomenology at interferometer scales.

Tail diversity from inflation

The tail of the distribution of primordial fluctuations (corresponding to the likelihood of realization of large fluctuations) is of interest, from both theoretical and observational perspectives. In particular, it is relevant for the accurate evaluation of the primordial black hole (PBH) abundance. In this paper, we first analyze the non-perturbative δN formalism as a method to non-perturbatively estimate the probability distribution function (PDF) of primordial fluctuations, discuss its underlying assumptions and deal with several subtleties that may arise as a result of considering large fluctuations. Next, we employ the method to study several non-attractor single-field inflationary models as the simplest examples that may lead to the abundant production of PBHs. We conclude that the Gaussian extrapolation from linear perturbation theory may fail drastically to predict the likelihood of large fluctuations. Specifically, we show that a truncation of the tail, a power-law tail, a double-exponential tail, and a doubly peaked distribution can all be realized for the curvature perturbation in the single-field non-attractor models of inflation. We thus show that there is a diverse zoo of possible tails from inflation so that a model-dependent, non-perturbative study of the distribution of the primordial fluctuations seems inevitable concerning PBH abundance.

Growth of curvature perturbations for PBH formation & detectable GWs in non-minimal curvaton scenario revisited

We revisit the growth of curvature perturbations in non-minimal curvaton scenario with a non-trivial field metric λ(ϕ) where ϕ is an inflaton field, and incorporate the effect from the non-uniform onset of curvaton's oscillation in terms of an axion-like potential. The field metric λ(ϕ) plays a central role in the enhancement of curvaton field perturbation δχ, serving as an effective friction term which can be either positive or negative, depending on the first derivative λ ,ϕ . Our analysis reveals that δχ undergoes the superhorizon growth when the condition η eff ≡ -2 √2ϵ M Pl λ ,ϕ /λ < -3 is satisfied. This is analogous to the mechanism responsible for the amplification of curvature perturbations in the context of ultra-slow-roll inflation, namely the growing modes dominate curvature perturbations. As a case study, we examine the impact of a Gaussian dip in λ(ϕ) and conduct a thorough investigation of both the analytical and numerical aspects of the inflationary dynamics. Our findings indicate that the enhancement of curvaton perturbations during inflation is not solely determined by the depth of the dip in λ(ϕ). Rather, the first derivative λ ,ϕ also plays a significant role, a feature that has not been previously highlighted in the literature. Utilizing the δ𝒩 formalism, we derive analytical expressions for both the final curvature power spectrum and the non-linear parameter f NL in terms of an axion-like curvaton's potential leading to the non-uniform curvaton's oscillation. Additionally, the resulting primordial black hole abundance and scalar-induced gravitational waves are calculated, which provide observational windows for PBHs.

Logarithmic Duality of the Curvature Perturbation.

We study the comoving curvature perturbation R in the single-field inflation models whose potential can be approximated by a piecewise quadratic potential V(φ) by using the δN formalism. We find a general formula for R(δφ,δπ), consisting of a sum of logarithmic functions of the field perturbation δφ and the velocity perturbation δπ at the point of interest, as well as of δπ_{*} at the boundaries of each quadratic piece, which are functions of (δφ,δπ) through the equation of motion. Each logarithmic expression has an equivalent dual expression, due to the second-order nature of the equation of motion for φ. We also clarify the condition under which R(δφ,δπ) reduces to a single logarithm, which yields either the renowned "exponential tail" of the probability distribution function of R or a Gumbel-distribution-like tail.

An update on adiabatic modes in cosmology and δN formalism

In this paper, we generalize the Weinberg's procedure to determine the comoving curvature perturbation ℛ to non-attractor inflationary regimes. We show that both modes of ℛ are related to a symmetry of the perturbative equations in the Newtonian gauge. As a byproduct, we clarify that adiabaticity does not generally imply constancy of ℛ, not even in the k ⟶ 0 limit. We then show that there exist non-equivalent definitions of δN that would reproduce ℛ or the uniform density curvature perturbation ζ at linear order. We have then shown that the perturbative δN definition in terms of difference between the number of e-foldings of different gauges, can be extended non-perturbatively at leading order in gradient expansion. Nevertheless, the computer friendly definition in terms of the difference of e-foldings obtained from the evolution of a local FRW Universe, respectively with perturbed and un-perturbed initial conditions, might only give information about the linear order curvature perturbations, contrary to what is stated in the literature.

Bootstrapping multi-field inflation: non-Gaussianities from light scalars revisited

Primordial non-Gaussianities from multi-field inflation are a leading target for cosmological observations, because of the possible large correlations generated between long and short distances.These signatures are captured by the local shape of the scalar bispectrum. In this paper, we revisit the nonlinearities of the conversion process from additional light scalars into curvature perturbations during inflation.We provide analytic templates for correlation functions valid at any kinematical configuration, using the cosmological bootstrap as a main computational tool. Our results include the possibility of large breaking of boost symmetry, in the form of small speeds of sound for both the inflaton and the mediators.We consider correlators coming from the tree-level exchange of a massless scalar field.By introducing a late-time cutoff, we identify that the symmetry constraints on the correlators are modified. This leads to anomalous conformal Ward identities, and consequently the bootstrap differential equations acquire a source term that depends on this cutoff.The solutions to the differential equations are scalar seed functions that incorporate these late-time growth effects. Applying weight-shifting operators to auxiliary “seed” functions, we obtain a systematic classification of shapes of non-Gaussianity coming from massless exchange.For theories with de Sitter symmetry, we compare the resulting shapes with the ones obtained via the δN formalism, identifying missing contributions away from the squeezed limit. For boost-breaking scenarios, we derive a novel class of shape functions with phenomenologically distinct features in scale-invariant theories.Specifically, the new shape provides a simple extension of equilateral non-Gaussianity: the signal peaks at a geometric configuration controlled by the ratio of the sound speeds of the mediator and the inflaton.

Rare events are nonperturbative: Primordial black holes from heavy-tailed distributions

In recent years it has been noted that the perturbative treatment of the statistics of fluctuations may fail to make correct predictions for the abundance of primordial black holes (PBHs). Moreover, it has been shown in some explicit single-field examples that the nonperturbative effects may lead to an exponential tail for the probability distribution function (PDF) of fluctuations responsible for PBH formation—in contrast to the PDF being Gaussian, as suggested by perturbation theory. In this paper, we advocate that the so-called δN formalism can be considered as a simple, yet effective, tool for the nonperturbative estimate of the tail of the PDF. We discuss the criteria a model needs to satisfy so that the results of the classical δN formalism can be trusted and most possible complications due to the quantum nature of fluctuations can be avoided. As a proof of concept, we then apply this method to a simple example and show that the tail of the PDF can be even heavier than exponential, leading to a significant enhancement of the PBH formation probability, compared with the predictions of the perturbation theory. Our results, along with other related findings, motivate the invention of new, nonperturbative methods for the problem and open up new ideas on generating PBHs with notable abundance.

Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation

We study the impact of stochastic noise on the generation of primordial black hole (PBH) seeds in ultra-slow-roll (USR) inflation with numerical simulations. We consider the non-linearity of the system by consistently taking into account the noise dependence on the inflaton perturbations, while evolving the perturbations on the coarse-grained background affected by the noise. We capture in this way the non-Markovian nature of the dynamics, and demonstrate that non-Markovian effects are subleading. Using the ΔN formalism, we find the probability distribution P(ℛ) of the comoving curvature perturbation ℛ. We consider inflationary potentials that fit the CMB and lead to PBH dark matter with i) asteroid, ii) solar, or iii) Planck mass, as well as iv) PBHs that form the seeds of supermassive black holes. We find that stochastic effects enhance the PBH abundance by a factor of \U0001d4aa(10)–\U0001d4aa(108), depending on the PBH mass. We also show that the usual approximation, where stochastic kicks depend only on the Hubble rate, either underestimates or overestimates the abundance by orders of magnitude, depending on the potential. We evaluate the gauge dependence of the results, discuss the quantum-to-classical transition, and highlight open issues of the application of the stochastic formalism to USR inflation.

Generalized local ansatz for scale-dependent primordial non-Gaussianities and future galaxy surveys

Abstract We revisit a possible scale-dependence of local-type primordial non-Gaussianities induced by super-horizon evolution of scalar field perturbations. We develop the formulation based on δN formalism and derive the generalized form of the local-type bispectrum and also trispectrum which allows us to implement the scale-dependence and suitably compare model prediction with observational data. We propose simple but phenomenologically meaningful expressions, which encompass the information of a wide range of physically motivated models. We also formulate a large-scale power spectrum and bispectrum of biased objects in the presence of the scale-dependent primordial non-Gaussianities. We perform the Fisher analysis for future galaxy surveys and give the projected constraints on the parameters of the generalized local-form of primordial non-Gaussianities.

Non-Gaussian Tail of the Curvature Perturbation in Stochastic Ultraslow-Roll Inflation: Implications for Primordial Black Hole Production.

We consider quantum diffusion in ultraslow-roll (USR) inflation. Using the ΔN formalism, we present the first stochastic calculation of the probability distribution P(R) of the curvature perturbation during USR. We capture the nonlinearity of the system, solving the coupled evolution of the coarse-grained background with random kicks from the short wavelength modes, simultaneously with the mode evolution around the stochastic background. This leads to a non-Markovian process from which we determine the highly non-Gaussian tail of P(R). Studying the production of primordial black holes in a viable model, we find that stochastic effects during USR increase their abundance by a factor of ∼10^{5} compared with the Gaussian approximation.

Anisotropic separate universe and Weinberg's adiabatic mode

In the separate universe approach, an inhomogeneous universe is rephrased as a set of glued numerous homogeneous local patches. This is the essence of the gradient expansion and the δN formalism, which have been widely used in solving a long wavelength evolution of the universe. In this paper, we show that the separate universe approach can be generically used, as long as a theory under consideration is local and preserves the spatial diffeomorphism invariance. Focusing on these two conditions, we also clarify the condition for the existence of the so-called Weinberg's adiabatic mode. Remarkably, the separate universe approach and the δN formalism turn out to be applicable also to models with shear on large scales and also to modified theories of gravity, accepting violation of four-dimensional diffeomorphism invariance. The generalized δN formalism enables us to calculate all the large scale fluctuations, including gravitational waves. We also argue several implications on anisotropic inflation and ultra slow-roll inflation.

Failure of the stochastic approach to inflation beyond slow-roll

After giving a pedagogical review we clarify that the stochastic approach to inflation is generically reliable only at zeroth order in the (geometrical) slow-roll parameter ϵ1 if and only if ϵ22 ≪ 6/ϵ1, with the notable exception of slow-roll. This is due to the failure of the stochastic ΔN formalism in its standard formulation. However, by keeping the formalism in its regime of validity, we showed that, in ultra-slow-roll, the stochastic approach to inflation reproduces the power spectrum calculated from the linear theory approach.