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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., Comment: 32 pages, 4 figures. v2: Corrected typos, updated notation

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., Comment: 30 pages + Appendices, 7 figures; comments are welcome !

We present here results of numerical simulations of the formation and early evolution of rocky planets through pebble accretion, with an with an emphasis on hydrogen envelope longevity and the composition of the outgassed atmosphere. We model planets with a range in mass from 0.1 to 5 Earth masses that orbit between 0.7 and 1.7 AU. The composition of the outgassed atmosphere is calculated with the partial pressure of free oxygen fit to geophysical models of magma ocean self-oxidation. XUV radiation powered photoevaporation is considered as the main driver of atmospheric escape. We model planets that remain below the pebble isolation mass and hence accrete tenuous envelopes only. We consider slow, medium or fast initial stellar rotation for the temporal evolution of the XUV flux. The loss of the envelope is a key event that allows the magma ocean to crystallise and outgas its bulk volatiles. The atmospheric composition of the majority of our simulated planets is dominated by CO$_2$. Our planets accrete a total of 11.6 Earth oceans of water, the majority of which enters the core. The hydrospheres of planets lighter than the Earth reach several times the mass of the Earth's modern oceans, while the hydrospheres of planets ranging from 1 to 3.5 Earth masses are comparable to those of our planet. However, planets of 4-5 Earth masses have smaller hydrospheres due to trapping of volatiles in their massive mantles. Overall, our simulations demonstrate that hydrogen envelopes are easily lost from rocky planets and that this envelope loss triggers the most primordial partitioning of volatiles between the solid mantle and the atmosphere., Comment: Accepted for Astronomy & Astrophysics

This brief note presents standard computations of primordial black hole mass $M$ given perturbations of scale $k$, and their late-time abundance $\Omega_\text{PBH}$ given their initial density fraction $\beta$. I recap the assumptions made in these computations and present formulas and reference charts useful for a working cosmologist., Comment: Brief note. 7 pages, 1 table, 2 figures

We study the formation of primordial black holes (PBH) with ultra-slow-roll inflation when stochastic effects are important. We use the $\Delta N$ 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., Comment: 5+3 pages, 3 figures. v2: Added an appendix with a plot, fixed a typo. Published version

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., Comment: 23 pages, 5 figures, matches the published version

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., Comment: 23 pages, 7 figures, published version, license updated

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 $\sigma_k^2$ and the second slow-roll parameter $\epsilon_2$. With a large $\epsilon_2$, 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 $\Delta N$ formalism., Comment: 11 pages, 3 figures, 1 table. v2: Corrected typos, expanded Section II. Matches published version

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., Comment: matches version published in International Journal of Geometric Methods in Modern Physics

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., Comment: 22 pages, 7 figures, revised version: added a section on $F(R,X)$ models, title, abstract and conclusions revised

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., Comment: 34 pages, 8 figures, 1 table. v2: Minor revisions in text. Published version

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 $\Lambda$CDM., Comment: 45 pages, 11 figures

We construct an analytically solvable simplified model that captures the essential features for primordial black hole (PBH) production in most models of single-field inflation. The construction makes use of the Wands duality between the constant-roll (or slow-roll) and the preceding ultra-slow-roll phases and can be realized by a simple inflaton potential of two joined parabolas. Within this framework, it is possible to formulate explicit inflationary scenarios consistent with the CMB observations and copious production of PBHs of arbitrary mass. We quantify the variability of the shape of the peak in the curvature power spectrum in different inflationary scenarios and discuss its implications for probing PBHs with scalar-induced gravitational wave backgrounds. We find that the COBE/Firas $\mu$-distortion constraints exclude the production of PBHs heavier than $10^4 M_\odot$ in single-field inflation., Comment: 47 pages, 10 figures, extended discussion, added references

We study a model of quintessential inflation constructed in $R^2$ modified gravity with a non-minimally coupled scalar field, in the Palatini formalism. Our non-minimal inflaton field is characterised by a simple exponential potential. We find that successful quintessential inflation can be achieved with no fine-tuning on the model parameters. Predictions on the characteristics of dark energy will be tested by observations in the near future, while contrast with existing observations provides insights on the modified gravity background, such as the value of the non-minimal coupling and its running., Comment: 20 pages, 7 figures, 1 table; published version

Tachyonic preheating is realized when the inflaton repeatedly returns to a convex region of the potential during the post-inflationary oscillating phase. This will induce a strong tachyonic instability and lead to a rapid fragmentation of the coherent field that can complete within a fraction of an $e$-fold. In this paper, we study the linear regime of this process in a model-independent way. To this purpose, we construct simplified models that provide an analytic Floquet theoretic description of mode growth. This approach captures the essential features of well-motivated tachyonic preheating scenarios, including scenarios in which the inflaton is part of a larger scalar multiplet. We show that tachyonic preheating is efficient if the field excursions are sub-Planckian, can produce gravitational waves in the frequency range of current and future gravitational wave interferometers, and can be consistent with any experimentally allowed tensor-to-scalar ratio., Comment: 28 pages, 7 figures

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 $F(R) = R + \alpha R^n$ and find that, as in the previously known $n=2$ case, a large $\alpha$ 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., Comment: 20 pages, 5 figures, 1 table, matches the published version

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 $\Delta N$ formalism, we find the probability distribution $P(\mathcal{R})$ of the comoving curvature perturbation $\mathcal{R}$. 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 $\mathcal{O}(10)-\mathcal{O}(10^8)$, 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., Comment: 31+16 pages, 16 figures. v2. Added references and clarifications. Published version

This note collects together useful unit conversions and numerical values from early universe cosmology. It is a quick reference that can be used to make easy order-of-magnitude estimates. Included are tables for unit conversions, the thermal history of the universe, and collected properties of astronomical objects. The note also introduces a modifiable Mathematica package NaturalUnits (newest version 2.0), which makes it easy to convert between natural and physical units., Comment: 7 pages, 7 tables. Corrected typos, updated for v2.0 of the Mathematica package

Certain models of cosmic inflation produce strong cosmological perturbations at short length scales, which may later collapse into primordial black holes. To find the statistics of these strong perturbations and the ensuing black holes, it is necessary to go beyond linear perturbation theory. Stochastic inflation provides a way to take the leading non-linear effects into account. In this contribution, I discuss recent progress in numerical computations of stochastic inflation. A numerical approach can include more of the non-linearities than an analytical calculation, and can be applied to single-field inflationary models with any potential., Comment: 5 pages, 2 figures. Based on Phys.Rev.Lett. 127 (2021) 10, 101302, arXiv:2012.06551. Proceedings of the 17th International Conference on Topics in Astroparticle and Underground Physics (TAUP 2021)

Plateau inflation is an experimentally consistent framework in which the scale of inflation can be kept relatively low. Close to the edge of the plateau, scalar perturbations are subject to a strong tachyonic instability. Tachyonic preheating is realized when, after inflation, the oscillating inflaton repeatedly re-enters the plateau. We develop the analytic theory of this process and expand the linear approach by including backreaction between the coherent background and growing perturbations. For a family of plateau models, the analytic predictions are confronted with numerical estimates. Our analysis shows that the inflaton fragments in a fraction of an $e$-fold in all examples supporting tachyonic preheating, generalizing the results of previous similar studies. In these scenarios, the scalar-to-tensor ratio is tiny, $r<10^{-7}$., Comment: 32 pages, 7 figures. v2: Updated to match published version. Added references

We study preheating in the Palatini formalism with a quadratic inflaton potential and an added $\alpha R^2$ term. In such models, the oscillating inflaton field repeatedly returns to the plateau of the Einstein frame potential, on which the tachyonic instability fragments the inflaton condensate within less than an e-fold. We find that tachyonic preheating takes place when $\alpha \gtrsim 10^{13}$ and that the energy density of the fragmented field grows with the rate $\Gamma/H \approx 0.011 \times \alpha^{0.31}$. The model extends the family of plateau models with similar preheating behaviour. Although it contains non-canonical quartic kinetic terms in the Einstein frame, we show that, in the first approximation, these can be neglected during both preheating and inflation., Comment: Matches published version

We consider quantum diffusion in ultra-slow-roll (USR) inflation. Using the $\Delta N$ formalism, we present the first stochastic calculation of the probability distribution $P(\mathcal{R})$ of the curvature perturbation during USR. We capture the non-linearity 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 $P(\mathcal{R})$. Studying the production of primordial black holes in a viable model, we find that stochastic effects during USR increase their abundance by a factor $\sim 10^5$ compared to the Gaussian approximation., Comment: 6 pages, 2 figures. v2: 5+3 pages, 2 figures. Improved analysis and text, gathered 400 times more data. Small change in results. Published version

We study Higgs inflation in the Palatini formulation with the renormalisation group improved potential in the case when loop corrections generate a feature similar to an inflection point. Assuming that there is a threshold correction for the Higgs quartic coupling $\lambda$ and the top Yukawa coupling $y_t$, we scan the three-dimensional parameter space formed by the two jumps and the non-minimal coupling $\xi$. The spectral index $n_s$ can take any value in the observationally allowed range. The lower limit for the running is $\alpha_s>-3.5\times10^{-3}$, and $\alpha_s$ can be as large as the observational upper limit. Running of the running is small. The tensor-to-scalar ratio is $2.2\times 10^{-17}

A hyperk\"ahler manifold $M$ has a family of induced complex structures indexed by a two-dimensional sphere $S^2 \cong \mathbb{CP}^1$. The twistor space of $M$ is a complex manifold $Tw(M)$ together with a natural holomorphic projection $Tw(M) \to \mathbb{CP}^1$, whose fiber over each point of $\mathbb{CP}^1$ is a copy of $M$ with the corresponding induced complex structure. We remove one point from this sphere (corresponding to one fiber in the twistor space),and for the case of $M = \mathbb{R}^{4n}$, $n\in\mathbb{N}$, equipped with the standard hyperk\"ahler structure, we construct one quantization that replaces the family of Berezin-Toeplitz quantizations parametrized by $S^2-\{ pt\}$. We provide semiclassical asymptotics for this quantization., Comment: 23 pages; minor changes

We study preheating in plateau inflation in the Palatini formulation of general relativity, in a special case that resembles Higgs inflation. It was previously shown that the oscillating inflaton field returns to the plateau repeatedly in this model, and this leads to tachyonic production of inflaton particles. We show that a minimally coupled spectator scalar field can be produced even more efficiently by a similar mechanism. The mechanism is purely gravitational, and the scalar field mass can be of order $10^{13}$ GeV, larger than the Hubble scale by many orders of magnitude, making this a candidate for superheavy dark matter., Comment: Section 4 extended, references added, typos fixed, matches published version

We present the first high-resolution measurements and rotational analysis of the fundamental asymmetric stretching vibrational band ${\nu_3}(F_2)$ of radiocarbon methane. A spectrometer consisting of a mid-infrared continuous-wave optical parametric oscillator and a cantilever-enhanced photoacoustic detector was employed to determine the wavenumbers of 43 rovibrational lines. A spectroscopic model could reproduce all the observed transition wavenumbers within the accuracy of the experiment. Our work contributes to the development of radiocarbon methane detectors based on laser spectroscopy.

We study initial conditions for inflation in scenarios where the inflaton potential has a plateau shape. Such models are those most favored by Planck data and can be obtained in a large number of model classes. As a representative example, we consider Higgs inflation with and without an $R^2$ term in the context of Palatini gravity. We show that inflation with a large number of e-folds generically occurs in a large part of the parameter space without any fine-tuning of parameters even when the scale of inflation and the inflaton field value during inflation are much smaller than the Planck scale. We discuss consequences for detection of primordial gravitational waves and spectral tilt of curvature perturbations, as well as the recently proposed "Trans-Planckian Censorship" conjecture., Comment: 16 pages, 6 figures. v2: A slight change of title, discussion and references added. Matches the version published in JCAP