arXiv subjects

Eemeli Tomberg

Publications and source records attributed to Eemeli Tomberg.

At least 19 recordsLinked to original sources

arXiv2026

Compaction function in stochastic inflation: a \texttt{FOREST} of type I and II primordial black holes

We show how to compute the compaction function within stochastic inflation, by solving the random field dynamics on stochastic binary trees. In this framework, the compaction function is directly related to the ratio of the volumes emerging from the sibling and child branches of a given node. This construction also determines whether or not the areal radius of a perturbation increases monotonically with the radial coordinate, thereby distinguishing between type-I and type-II fluctuations. As an application, we investigate primordial black hole (PBH) formation in a single-field toy model with a constant potential slope, using stochastic-tree realizations generated with the public code \texttt{FOREST}. In the classical regime, where quantum diffusion is subdominant, the PBH mass function is narrowly distributed and type-II fluctuations are strongly suppressed relative to type I. By contrast, in the quantum and near-critical (i.e. close to eternal inflation) regimes, the PBH mass distribution spans several orders of magnitude, the overall PBH abundance is enhanced, and type-II fluctuations outnumber type I. In that case cloud-in-cloud effects are also important, highlighting the need for a better understanding of the evolution and collapse of type-II fluctuations in order to obtain robust PBH predictions when stochastic effects are significant.

astro-ph.CO↗

arXiv2026

Stochastic constant-roll inflation beyond the hilltop with the spectral method

Stochastic inflation can be used to study large inflationary perturbations. This paper presents such a study for a quadratic hilltop potential, corresponding to constant-roll inflation. I solve the perturbation distribution using the spectral method, with detailed solutions of the eigenvalues and eigenfunctions of the Fokker-Planck operator. Contrary to previous studies of stochastic constant-roll inflation, the solution allows trajectories that cross the hilltop and get stuck near a reflecting boundary on the other side, tunneling out slowly in a way dictated by the lowest eigensolution. Despite their rarity, these trajectories turn out to dominate the mean first-passage time. For this reason, I argue the mean does not properly describe the inflationary background. Using the median instead, I compute the distribution of the coarse-grained $\Delta N$ distribution and show that its well-known exponential tail first flattens out and then forms a peak near a maximal $\Delta N$ value. I argue similar intricacies arise in primordial black hole models.

astro-ph.CO↗

arXiv2025

Evading the BBN bound with a soft stiff period

Cosmic inflation is the leading theory to explain early Universe history and structure formation. Non-oscillatory inflation is a class of models which can naturally introduce a post-inflationary stiff period of the Universe's evolution which boosts the signal of primordial gravitational waves (GWs), making it possible to observe them in forthcoming GW experiments. However, this pushes the GW energy density high enough to destabilise the process of Big Bang Nucleosynthesis (BBN). This problem can be overcome by "softening" the stiff period, so that the field is gradually tending towards freefall from a frozen start. Here, we consider a modified hybrid inflation model where the stiff period is driven by the waterfall field, allowing the barotropic parameter of the Universe to vary, so that it does not violate the BBN constraint but produces a characteristic gravitational wave spectrum soon to be observable.

hep-ph↗

arXiv2025

Effect of stochastic kicks on primordial black hole abundance and mass via the compaction function

We study stochastic effects in viable ultra-slow-roll inflation models that produce primordial black holes. We consider asteroid, solar, and supermassive black hole seed masses. In each case, we simulate $10^8$ patches of the universe that may collapse into PBHs. In every patch, we follow $4\times10^4$ momentum shells to construct its spherically symmetric profile from first principles, without introducing a window function. We include the effects of critical collapse and the radiation era transfer function. The resulting compaction function profiles are very spiky due to stochastic kicks. This can enhance the PBH abundance by up to 36 orders of magnitude, depending on the mass range and collapse criterion. The PBH mass function shifts to higher masses and widens significantly. These changes may have a large effect on observational constraints of PBHs and make it possible to generate PBHs with a smaller amplitude of the power spectrum. However, convergence issues for the mass function remain. The results call for redoing collapse simulations to determine the collapse criterion for spiky profiles.

astro-ph.CO↗

arXiv2025

Eternal inflation near inflection points: a challenge to primordial black hole models

Inflation with an inflection point potential is a popular model for producing primordial black holes. The potential near the inflection point is approximately flat, with a local maximum next to a local minimum, prone to eternal inflation. We show that a sufficient condition for eternal inflation is $\lambda_1 \leq 3$, where $\lambda_1$ is the index of the `exponential tail,' the lowest eigenvalue of the Fokker--Planck equation over a bounded region. We write $\lambda_1$ in terms of the model parameters for linear and quadratic regions. Wide quadratic regions inflate eternally if the second slow-roll parameter $\eta_V \geq -6$. We test example models from the literature and show this condition is satisfied; we argue eternal inflation is difficult to avoid in inflection point PBH models. Eternally inflating regions correspond to type II perturbations and form baby universes, hidden behind black hole horizons. These baby universes are inhomogeneous on large scales and dominate the multiverse's total volume. We argue that, if volume weighting is used, eternal inflation makes inflection point primordial black hole models incompatible with large-scale structure observations.

astro-ph.CO↗

arXiv2025

How deep is the dip and how tall are the wiggles in inflationary power spectra?

We study linear scalar perturbations in single-field models of inflation featuring a non-attractor phase. These models lead to a peak in the curvature power spectrum that may result in the formation of primordial black holes. We develop a transfer-matrix formalism, analogous to the S-matrix program in quantum-field theory, that maps perturbations throughout the transitory phase. At scales smaller than the peak, the power spectrum features damped oscillations, and the duration of the transition sets the scale at which power-law damping switches to exponential damping. At scales larger than the peak, we demonstrate that a dip appears in the power spectrum if and only if the inflaton's velocity does not flip sign. We show that the amplitude at the dip always scales as the inverse square-rooted amplitude of the peak, and comment on the physical consequences of this universal relationship. We also test the robustness of our results with a few toy models and interpret them with an intuitive mechanical analogy.

astro-ph.CO↗

arXiv2024

It\^{o}, Stratonovich, and zoom-in schemes in stochastic inflation

The It\^{o} and Stratonovich approaches are two ways to integrate stochastic differential equations. Detailed knowledge of the origin of the stochastic noise is needed to determine which approach suits a particular problem. I discuss this topic pedagogically in stochastic inflation, where the noise arises from a changing comoving coarse-graining scale or, equivalently, from `zooming in' into inflating space. I introduce a zoom-in scheme where deterministic evolution alternates with instantaneous zoom-in steps. I show that this alternating zoom-in scheme is equivalent to the It\^{o} approach in the Markovian limit, while the Stratonovich approach doesn't have a similar interpretation. In the full non-Markovian setup, the difference vanishes. The framework of zoom-in schemes clarifies the relationship between computations in stochastic inflation, linear perturbation theory, and the classical $\Delta N$ formalism. It informs the numerical implementation of stochastic inflation and is a building block for a first-principles derivation of the stochastic equations.

astro-ph.CO↗

arXiv2024

Stochastic Axion-like Curvaton: Non-Gaussianity and Primordial Black Holes Without Large Power Spectrum

We discuss a mechanism of primordial black hole (PBH) formation that does not require specific features in the inflationary potential, revisiting previous literature. In this mechanism, a light spectator field evolves stochastically during inflation and remains subdominant during the post-inflationary era. Even though the curvature power spectrum stays small at all scales, rare perturbations of the field probe a local maximum in its potential, leading to non-Gaussian tails in the distribution of curvature fluctuations, and to copious PBH production. For a concrete axion-like particle (ALP) scenario we analytically determine the distribution of the compaction function for perturbations, showing that it is characterized by a heavy tail, which produces an extended PBH mass distribution. We find the ALP mass and decay constant to be correlated with the PBH mass, for instance, an ALP with a mass $m_a = 5.4 \times 10^{14}$ eV and a decay constant $f_a = 4.6 \times 10^{-5} Mpl$ can lead to PBHs of mass $M_{\rm PBH} = 10^{21}$ g as the entire dark matter (DM) of the universe, and is testable in future PBH observations via lensing in the NGRST and mergers detectable in the LISA and ET Gravitational Waves (GW) detectors. We then extend our analysis to mixed ALP and PBH dark matter and Higgs-like spectator fields. We find that PBHs cluster strongly over all cosmological scales, clashing with CMB isocurvature bounds. We argue that this problem is shared by all PBH production from inflationary models that depend solely on large non-Gaussianity without a peak in the curvature power spectrum and discuss possible remedies.

astro-ph.CO↗

arXiv2024

Primordial black hole compaction function from stochastic fluctuations in ultra-slow-roll inflation

We study the formation of primordial black holes (PBH) with ultra-slow-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 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 three orders of magnitude in mass. The results point to a need to redo numerical simulations of PBH formation with spiky profiles.

astro-ph.CO↗

arXiv2024

Primordial black hole numbers: standard formulas and charts

This brief note presents standard computations of primordial black hole mass $M$ given perturbations of scale $k$, and their late-time abundance $Ω_\text{PBH}$ given their initial density fraction $β$. I recap the assumptions made in these computations and present formulas and reference charts useful for a working cosmologist.

astro-ph.CO↗

arXiv2024

Beyond (and back to) Palatini quadratic gravity and inflation

We study single-field slow-roll inflation embedded in Palatini $F(R)$ gravity where $F(R)$ grows faster than $R^2$. Surprisingly, the consistency of the theory requires the Jordan frame inflaton potential to be unbounded from below. Even more surprisingly, this corresponds to an Einstein frame inflaton potential bounded from below and positive definite. We prove that for all such Palatini $F(R)$'s, there exists a universal strong coupling limit corresponding to a quadratic $F(R)$ with the wrong sign for the linear term and a cosmological constant in the Jordan frame. In such a limit, the tensor-to-scalar ratio $r$ does not depend on the original inflaton potential, while the scalar spectral index $n_s$ does. Unfortunately, the system is ill-defined out of the slow-roll regime. A possible way out is to upgrade to a $F(R,X)$ model, with $X$ the Jordan frame inflaton kinetic term. Such a modification essentially leaves the inflationary predictions unaffected.

gr-qc↗

arXiv2023

Observable Gravitational Waves from Hyperkination in Palatini Gravity and Beyond

We consider cosmology with an inflaton scalar field with an additional quartic kinetic term. Such a theory can be motivated by Palatini $R+R^2$ modified gravity. Assuming a runaway inflaton potential, we take the Universe to become dominated by the kinetic energy density of the scalar field after inflation. Initially, the leading kinetic term is quartic and we call the corresponding period hyperkination. Subsequently, the usual quadratic kinetic term takes over and we have regular kination, until reheating. We study, both analytically and numerically, the spectrum of primordial gravitational waves generated during inflation and re-entering the horizon during the subsequent eras. We demonstrate that the spectrum is flat for modes re-entering during radiation domination and hyperkination and linear in frequency for modes re-entering during kination: kinetic domination boosts the spectrum, but hyperkination truncates its peak. As a result, the effects of the kinetic period can be extended to observable frequencies without generating excessive gravitational waves, which could otherwise destabilise the process of Big Bang Nucleosynthesis. We show that there is ample parameter space for the primordial gravitational waves to be observable in the near future. If observed, the amplitude and `knee' of the spectrum will provide valuable insights into the background theory.

gr-qc↗

arXiv2023

Primordial black holes and inflation from double-well potentials

We investigate the formation of large peaks in the inflationary curvature power spectrum from double-well potentials. In such scenarios, the initial CMB spectrum is created at large field values. Subsequently, the inflaton will cross one of the minima and will decelerate rapidly as it reaches the local maximum at the origin, either falling back or crossing it. During this final phase, a significant peak in the curvature power spectrum can be generated. Our analysis reveals that this class of models produces more pronounced peaks than the more commonly studied quasi-inflection point scenarios with less tuning for the model parameters. Finally, we construct an explicit theoretically motivated inflationary scenario that is consistent with the latest CMB observations and capable of generating sufficiently large curvature perturbations for primordial black holes.

astro-ph.CO↗

arXiv2023

Stochastic constant-roll inflation and primordial black holes

Stochastic inflation resolves primordial perturbations non-linearly, probing their probability distribution deep into its non-Gaussian tail. The strongest perturbations collapse into primordial black holes. In typical black-hole-producing single-field inflation, the strongest stochastic kicks occur during a period of constant roll. In this paper, I solve the stochastic constant-roll system, drawing the stochastic kicks from a numerically computed power spectrum, beyond the usual de Sitter approximation. The perturbation probability distribution is an analytical function of the integrated curvature power spectrum $σ_k^2$ and the second slow-roll parameter $ε_2$. With a large $ε_2$, stochastic effects can reduce the height of the curvature power spectrum required to form asteroid mass black holes from $10^{-2}$ to $10^{-3}$. I compare these results to studies with the non-stochastic $ΔN$ formalism.

astro-ph.CO↗

arXiv2023

Numerical stochastic inflation constrained by frozen noise

Stochastic inflation can resolve strong inflationary perturbations, which seed primordial black holes. I present a fast and accurate way to compute these perturbations in typical black hole producing single-field models, treating the short-wavelength Fourier modes beyond the de Sitter approximation. The squeezing and freezing of the modes reduces the problem to one dimension, and the resulting new form of the stochastic equations, dubbed `constrained stochastic inflation,' can be solved efficiently with semi-analytical techniques and numerical importance sampling. In an example case, the perturbation distribution is resolved in seconds deep into its non-Gaussian tail, a speed-up of factor $10^9$ compared to a previous study. Along the way, I comment on the role of the momentum constraint in stochastic inflation.

astro-ph.CO↗

arXiv2023

Implications of Palatini gravity for inflation and beyond

We present an introduction to cosmic inflation in the framework of Palatini gravity, which provides an intriguing alternative to the conventional metric formulation of gravity. In the latter, only the metric specifies the spacetime geometry, whereas in the former, the metric and the spacetime connection are independent variables-an option that can result in a gravity theory distinct from the metric one. In scenarios where the field(s) responsible for cosmic inflation are non-minimally coupled to gravity or the gravitational sector is extended, assumptions about the underlying gravitational degrees of freedom can have substantial implications for the observational effects of inflation. We examine this explicitly by discussing various compelling scenarios, such as Higgs inflation with non-minimal coupling to gravity, Higgs inflation with non-minimal derivative coupling, $\mathcal{R}^2$ inflation, and beyond. We also comment on reheating in these models. Finally, as an application of the general results of Palatini $\mathcal{R}^2$ inflation, we review a model of successful quintessential inflation, where a single scalar field acts initially as the inflaton and then becomes dynamical dark energy, in agreement will all experimental constraints.

gr-qc↗

arXiv2022

Palatini $R^2$ Quintessential Inflation

We construct a model of quintessential inflation in Palatini $R^2$ gravity employing a scalar field with a simple exponential potential and coupled to gravity with a running non-minimal coupling. At early times, the field acts as the inflaton, while later on it becomes the current dark energy. Combining the scalar sector with an ideal fluid, we study the cosmological evolution of the model from inflation all the way to dark energy domination. We interpret the results in the Einstein frame, where a coupling emerges between the fluid and the field, feeding energy from the former to the latter during the matter-dominated era. We perform a numerical scan over the parameter space and find points that align with observations for both the inflationary CMB data and the late-time behaviour. The final dark energy density emerges from an interplay between the model parameters, without requiring the extreme fine-tuning of the cosmological constant in $Λ$CDM.

gr-qc↗

arXiv2022

Slow-roll inflation in Palatini $F(R)$ gravity

We study single field slow-roll inflation in the presence of $F(R)$ gravity in the Palatini formulation. In contrast to metric $F(R)$, when rewritten in terms of an auxiliary field and moved to the Einstein frame, Palatini $F(R)$ does not develop a new dynamical degree of freedom. However, it is not possible to solve analytically the constraint equation of the auxiliary field for a general $F(R)$. We propose a method that allows us to circumvent this issue and compute the inflationary observables. We apply this method to test scenarios of the form $F(R) = R + αR^n$ and find that, as in the previously known $n=2$ case, a large $α$ suppresses the tensor-to-scalar ratio $r$. We also find that models with $F(R)$ increasing faster than $R^2$ for large $R$ suffer from numerous problems.

gr-qc↗