arXiv subjects

Pierre Auclair

Publications and source records attributed to Pierre Auclair.

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

Comment on: "Third-order corrections to the slow-roll expansion: Calculation and constraints with Planck, ACT, SPT, and BICEP/Keck [2025 PDU 47 101813]"

We point out that several terms in the third-order corrections to the slow-roll power spectra presented by Ballardini et al. [1] are incorrect. The authors of that work claim that their result differ from the ones originally presented by Auclair & Ringeval [2] due to some different approximation schemes. However, in our original work, all terms at all orders have been derived exactly and any difference between two expansions performed at the same pivot wavenumber signals a problem. As we show in this comment, Ballardini et al. [1] have misevaluated some definite three-dimensional integrals by integrating a truncated Taylor expansion instead of Taylor expanding an integral. Our claim is backed-up with a Monte-Carlo numerical integration of the incriminated three-dimensional integrals, which, unsurprisingly, matches the analytical value derived in Auclair & Ringeval [2].

astro-ph.CO↗

arXiv2026

Excursion-set for Primordial Black Holes I: white noise and moving barrier

In the excursion-set formalism, the mass distribution of primordial black holes (PBHs) is derived from the first-passage time of a random walk describing the density contrast as the coarse-graining scale varies. We address two recent concerns that have been raised about this approach. First, it was argued that the random walks are subject to colored (i.e. correlated over time) noise, making the first-passage-time problem cumbersome. We show that this arises from an incorrect separation of drift and noise when sampling on the Hubble-crossing surface: if Fourier modes are uncorrelated, the noise is strictly white. Moreover, sampling along the Hubble-crossing surface precludes using the density dispersion as a time variable, explaining some pathologies. Sampling instead on a synchronous surface removes both issues. This requires solving a first-passage-time problem with a moving barrier, for which we provide an efficient numerical framework. Second, it was suggested that cloud-in-cloud (i.e. that large black holes may engulf smaller ones) is irrelevant for PBHs and that the excursion set is therefore not needed. While valid for widely separated scales, this statement fails for broad power spectra with enhanced continua of modes. We further show that Press-Schechter estimates neglecting boundary evolution can break down even without cloud-in-cloud effects.

astro-ph.CO↗

arXiv2025

Breaking the Strings: the signatures of Cosmic String Loop Fragmentation

We study the impact of fragmentation on the cosmic string loop number density, using an approach inspired by the three-scale model and a Boltzmann equation. We build a new formulation designed to be more amenable to numerical resolution and present two complementary numerical methods to obtain the full loop distribution including the effect of fragmentation and gravitational radiation. We show that fragmentation generically predicts a decay of the loop number density on large scales and a deviation from a pure power-law. We expect fragmentation to be crucial for the calibration of loop distribution models.

astro-ph.CO↗

arXiv2025

The Science of the Einstein Telescope

Einstein Telescope (ET) is the European project for a gravitational-wave (GW) observatory of third-generation. In this paper we present a comprehensive discussion of its science objectives, providing state-of-the-art predictions for the capabilities of ET in both geometries currently under consideration, a single-site triangular configuration or two L-shaped detectors. We discuss the impact that ET will have on domains as broad and diverse as fundamental physics, cosmology, early Universe, astrophysics of compact objects, physics of matter in extreme conditions, and dynamics of stellar collapse. We discuss how the study of extreme astrophysical events will be enhanced by multi-messenger observations. We highlight the ET synergies with ground-based and space-borne GW observatories, including multi-band investigations of the same sources, improved parameter estimation, and complementary information on astrophysical or cosmological mechanisms obtained combining observations from different frequency bands. We present advancements in waveform modeling dedicated to third-generation observatories, along with open tools developed within the ET Collaboration for assessing the scientific potentials of different detector configurations. We finally discuss the data analysis challenges posed by third-generation observatories, which will enable access to large populations of sources and provide unprecedented precision.

gr-qc↗

arXiv2025

Harvesting primordial black holes from stochastic trees with $\texttt{FOREST}$

We introduce a novel framework to implement stochastic inflation on stochastic trees, modelling the inflationary expansion as a branching process. Combined with the $\delta N$ formalism, this allows us to generate real-space maps of the curvature perturbation that fully capture quantum diffusion and its non-perturbative backreaction during inflation. Unlike lattice methods, trees do not proceed on a fixed background since new spacetime units emerge dynamically as trees unfold, naturally incorporating metric fluctuations. The recursive structure of stochastic trees also offers remarkable numerical efficiency, and we develop the FOrtran Recursive Exploration of Stochastic Trees ($\texttt{FOREST}$) tool and demonstrate its performance. We show how primordial black holes blossom at unbalanced nodes of the trees, and how their mass distribution can be obtained while automatically accounting for the "cloud-in-cloud" effect. In the "quantum-well" toy model, we find broad mass distributions, with mild power laws terminated by exponential tails. We finally compare our results with existing approximations in the literature and discuss several prospects.

astro-ph.CO↗

arXiv2024

Clocking the End of Cosmic Inflation

Making observable predictions for cosmic inflation requires determining when the wavenumbers of astrophysical interest today exited the Hubble radius during the inflationary epoch. These instants are commonly evaluated using the slow-roll approximation and measured in e-folds $\Delta N=N-N_\mathrm{end}$, in reference to the e-fold $N_\mathrm{end}$ at which inflation ended. Slow roll being necessarily violated towards the end of inflation, both the approximated trajectory and $N_\mathrm{end}$ are determined at, typically, one or two e-folds precision. Up to now, such an uncertainty has been innocuous, but this will no longer be the case with the forthcoming cosmological measurements. In this work, we introduce a new and simple analytical method, on top of the usual slow-roll approximation, that reduces uncertainties on $\Delta N$ to less than a tenth of an e-fold.

astro-ph.CO↗

arXiv2024

LISA Definition Study Report

The Laser Interferometer Space Antenna (LISA) is the first scientific endeavour to detect and study gravitational waves from space. LISA will survey the sky for Gravitational Waves in the 0.1 mHz to 1 Hz frequency band which will enable the study of a vast number of objects ranging from Galactic binaries and stellar mass black holes in the Milky Way, to distant massive black-hole mergers and the expansion of the Universe. This definition study report, or Red Book, presents a summary of the very large body of work that has been undertaken on the LISA mission over the LISA definition phase.

astro-ph.CO↗

arXiv2024

Small-scale clustering of Primordial Black Holes: cloud-in-cloud and exclusion effects

Using an excursion-set approach, we revisit the initial spatial clustering of Primordial Black Holes (PBHs) originating from the Hubble re-entry of large Gaussian density fluctuations in the early universe. We derive the two-point correlation functions of PBHs, properly accounting for the "cloud-in-cloud" mechanism. Our expressions naturally and intrinsically correlate the formation of pairs of PBHs, which is a key difference with the Poisson model of clustering. Our approach effectively includes short-range exclusion effects and clarifies the clustering behaviors at small scale: PBHs are anti-correlated at short distances. Using a scale-independent collapse threshold, we derive explicit expressions for the excess probability to find pairs of PBHs separated by a distance $r$, as well as the excess probability to find pairs with asymmetric mass ratio. Our framework is model-independent by construction and we discuss possible other applications.

astro-ph.CO↗

arXiv2023

Practical approaches to analyzing PTA data: Cosmic strings with six pulsars

We search for a stochastic gravitational wave background (SGWB) generated by a network of cosmic strings using six millisecond pulsars from Data Release 2 (DR2) of the European Pulsar Timing Array (EPTA). We perform a Bayesian analysis considering two models for the network of cosmic string loops, and compare it to a simple power-law model which is expected from the population of supermassive black hole binaries. Our main strong assumption is that the previously reported common red noise process is a SGWB. We find that the one-parameter cosmic string model is slightly favored over a power-law model thanks to its simplicity. If we assume a two-component stochastic signal in the data (supermassive black hole binary population and the signal from cosmic strings), we get a $95\%$ upper limit on the string tension of $\log_{10}(G\mu) < -9.9$ ($-10.5$) for the two cosmic string models we consider. In extended two-parameter string models, we were unable to constrain the number of kinks. We test two approximate and fast Bayesian data analysis methods against the most rigorous analysis and find consistent results. These two fast and efficient methods are applicable to all SGWBs, independent of their source, and will be crucial for analysis of extended data sets.

gr-qc↗

arXiv2023

Repeated Gravitational Wave Bursts from Cosmic Strings

A characteristic observational signature of cosmic strings are short duration gravitational wave (GW) bursts. These have been searched for by the LIGO-Virgo-KAGRA (LVK) collaboration, and will be searched for with LISA. We point out that these burst signals are repeated, since cosmic string loops evolve quasi-periodically in time, and will always appear from essentially the same position in the sky. We estimate the number of GW repeaters for LVK and LISA, and show that the string tension that can be probed scales as detector sensitivity to the sixth power, which raises hope for detection in future GW detectors. The observation of repeated GW bursts from the same cosmic string loop helps distinguish between the GW waveform parameters and the sky-localization.

gr-qc↗

arXiv2023

Cosmic string bursts in LISA

Cosmic string cusps are sources of short-lived, linearly polarised gravitational wave bursts which can be searched for in gravitational wave detectors. We assess the capability of LISA to detect these bursts using the latest LISA configuration and operational assumptions. For such short bursts, we verify that LISA can be considered as ``frozen", namely that one can neglect LISA's orbital motion. We consider two models for the network of cosmic string loops, and estimate that LISA should be able to detect 4-30 bursts per year assuming a string tension $G\mu \approx 10^{-10.6} - 10^{-10.1}$ and detection threshold $\rm{SNR} \ge 20$. Non-detection of these bursts would constrain the string tension to $G\mu\lesssim 10^{-11}$ for both models.

gr-qc↗

arXiv2023

Spatial Curvature from Super-Hubble Cosmological Fluctuations

We revisit how super-Hubble cosmological fluctuations induce, at any time in the cosmic history, a non-vanishing spatial curvature of the local background metric. The random nature of these fluctuations promotes the curvature density parameter to a stochastic quantity for which we derive novel non-perturbative expressions for its mean, variance, higher moments and full probability distribution. For scale-invariant Gaussian perturbations, such as those favored by cosmological observations, we find that the most probable value for the curvature density parameter $\Omega_\mathrm{K}$ today is $-10^{-9}$, that its mean is $+10^{-9}$, both being overwhelmed by a standard deviation of order $10^{-5}$. We then discuss how these numbers would be affected by the presence of large super-Hubble non-Gaussianities, or, if inflation lasted for a very long time. In particular, we find that substantial values of $\Omega_\mathrm{K}$ are obtained if inflation lasts for more than a billion e-folds.

astro-ph.CO↗

arXiv2022

Mean-field approach to Random Apollonian Packing

We revisit the scaling properties of growing spheres randomly seeded in d=2,3 and 4 dimensions using a mean-field approach. We model the insertion probability without assuming a priori a functional form for the radius distribution. The functional form of the insertion probability shows an unprecedented agreement with numerical simulations in d=2, 3 and 4 dimensions. We infer from the insertion probability the scaling behavior of the Random Apollonian Packing and its fractal dimensions. The validity of our model is assessed with sets of 256 simulations each containing 20 million spheres in 2, 3 and 4 dimensions.

math-ph↗

arXiv2022

Gravitational Waves from Current-Carrying Cosmic Strings

Cosmic strings are predicted by many Standard Model extensions involving the cosmological breaking of a symmetry with nontrivial first homotopy group and represent a potential source of primordial gravitational waves (GWs). Present efforts to model the GW signal from cosmic strings are often based on minimal models, such as, eg, the Nambu--Goto action that describes cosmic strings as exactly one-dimensional objects without any internal structure. In order to arrive at more realistic predictions, it is therefore necessary to consider nonminimal models that make an attempt at accounting for the microscopic properties of cosmic strings. With this goal in mind, we derive in this paper the GW spectrum emitted by current-carrying cosmic strings (CCCSs), which may form in a variety of cosmological scenarios. Our analysis is based on a generalized version of the velocity-dependent one-scale (VOS) model, which, in addition to the mean velocity and correlation length of the string network, also describes the evolution of a chiral (light-like) current. As we are able to show, the solutions of the VOS equations imply a temporarily growing fractional cosmic-string energy density, $\Omega_{\rm cs}$. This results in an enhanced GW signal across a broad frequency interval, whose boundaries are determined by the times of generation and decay of cosmic-string currents. Our findings have important implications for GW experiments in the Hz to MHz band and motivate the construction of realistic particle physics models that give rise to large currents on cosmic strings.

astro-ph.CO↗

arXiv2022

Slow-Roll Inflation at N3LO

The next generation of cosmological observations will be sensitive to small deviations from a pure power law in the primordial power spectrum of the curvature perturbations. In the context of slow-roll inflation, these deviations are expected and correspond to the so-called running of the spectral index. Their measurement would bring as much information as the discovery of deviations from scale invariance. However, robust parameter inference requires to marginalize over any possible higher order uncertainties, which have been, up to now, not fully determined. We tackle this issue by deriving the inflationary scalar and tensor slow-roll power spectra at next-to-next-to-next to leading order (N3LO), fully expanded around an observable pivot wavenumber, for all single field inflationary models having minimal and non-minimal kinetic terms. Our result therefore encompasses string-inspired inflationary models having a varying speed of sound.

astro-ph.CO↗

arXiv2022

Generation of gravitational waves from freely decaying turbulence

We study the stochastic gravitational wave background (SGWB) produced by freely decaying vortical turbulence in the early Universe. We thoroughly investigate the time correlation of the velocity field, and hence of the anisotropic stresses producing the gravitational waves. With hydrodynamical simulations, we show that the unequal time correlation function (UETC) of the Fourier components of the velocity field is Gaussian in the time difference, as predicted by the "sweeping" decorrelation model. We introduce a decorrelation model that can be extended to wavelengths around the integral scale of the flow. Supplemented with the evolution laws of the kinetic energy and of the integral scale, this provides a new model UETC of the turbulent velocity field consistent with the simulations. We discuss the UETC as a positive definite kernel, and propose to use the Gibbs kernel for the velocity UETC as a natural way to ensure positive definiteness of the SGWB. The SGWB is given by a 4-dimensional integration of the resulting anisotropic stress UETC with the gravitational wave Green's function. We perform this integration using a Monte Carlo algorithm based on importance sampling, and find that the result matches that of the simulations. Furthermore, the SGWB obtained from the numerical integration and from the simulations show close agreement with a model in which the source is constant in time and abruptly turns off after a few eddy turnover times. Based on this assumption, we provide an approximate analytical form for the SGWB spectrum and its scaling with the initial kinetic energy and integral scale. Finally, we use our model and numerical integration algorithm to show that including an initial growth phase for the turbulent flow heavily influences the spectral shape of the SGWB. This highlights the importance of a complete understanding of the turbulence generation mechanism.

astro-ph.CO↗

arXiv2022

Cosmology with the Laser Interferometer Space Antenna

The Laser Interferometer Space Antenna (LISA) has two scientific objectives of cosmological focus: to probe the expansion rate of the universe, and to understand stochastic gravitational-wave backgrounds and their implications for early universe and particle physics, from the MeV to the Planck scale. However, the range of potential cosmological applications of gravitational wave observations extends well beyond these two objectives. This publication presents a summary of the state of the art in LISA cosmology, theory and methods, and identifies new opportunities to use gravitational wave observations by LISA to probe the universe.

astro-ph.CO↗