We present a comprehensive formalism for the description of primordial black hole formation in spherical symmetry based on the formalisms of Misner, Sharp, and Hernandez, which can be used to predict whether or not a black hole will form, and extract the resulting black hole mass when formation does occur. Rigorous derivations of all aspects of the formalism are provided, including a thorough investigation of appropriate initial and boundary conditions. We connect our formalism with numerous other approaches in the literature. Some implementation details for numerical code are provided. We include animations of simulated primordial black hole formation as supplemental material.
A $δN$ formalism is used to study the non-Gaussianity of the primordial curvature perturbation on an uniform density hypersurfaces generated by the warm inflation for the first time. After introducing the framework of the warm inflation and the $δN$ formalism, we obtain an analytic expression for the nonlinear parameter $f_{NL}$ that describes the non-Gaussianity in slow roll approximation, and find that the $δN$ formalism gives a very good result. We analyse the magnitude of $f_{NL}$ and compare our result with those of the standard inflation. Then we discuss two concrete examples: the quarti…
Primordial black holes, which could have been formed in the very early Universe due to the collapse of large curvature fluctuations, are nowadays one of the most attractive and fascinating research areas in cosmology for their possible theoretical and observational implications. This review article presents the current results and developments on the conditions for primordial black hole formation from the collapse of curvature fluctuations in spherical symmetry on a Friedman-Lemaitre-Robertson-Walker background and its numerical simulation. We review the appropriate formalism for the condition…
We explore a plausible mechanism that the hemispherical power asymmetry in the CMB is produced by the spatial variation of the primordial sound speed parameter. We suggest that in a generalized approach of the $δN$ formalism the local e-folding number may depend on some other primordial parameters besides the initial values of inflaton. Here the $δN$ formalism is extended by considering the effects of a spatially varying sound speed parameter caused by a super-Hubble perturbation of a light field. Using this generalized $δN$ formalism, we systematically calculate the asymmetric primordial spec…
Using the concept of apparent horizon for dynamical black holes, we revisit the formation of primordial black holes (PBH) in the early universe for both linear and non-linear regimes. First, we develop the perturbation theory for spherically symmetric spacetimes to study the formation of spherical PBHs in linear regime and we fix two gauges. We also introduce a well defined gauge invariant quantity for the expansion. Using this quantity, we argue that PBHs do not form in the linear regime. Finally, we study the non-linear regime. We adopt the spherical collapse picture by taking a closed FRW m…
We have studied the merger rate of primordial black holes (PBHs) in the ellipsoidal-collapse model of halo to explain the dark matter abundance by the PBH merger estimated from the gravitational waves detections via the Advanced LIGO (aLIGO) detectors. We have indicated that the PBH merger rate within each halo for the ellipsoidal models is more significant than for the spherical models. We have specified that the PBH merger rate per unit time and per unit volume for the ellipsoidal-collapse halo models is about one order of magnitude higher than the corresponding spherical models. Moreover, w…
We construct spacetimes which provide spherical and nonspherical models of black hole formation in the flat Friedmann--Lemaitre--Robertson--Walker (FLRW) universe with the Lemaitre--Tolman--Bondi solution and the Szekeres quasispherical solution, respectively. These dust solutions may contain both shell-crossing and shell-focusing naked singularities. These singularities can be physically regarded as the breakdown of dust description, where strong pressure gradient force plays a role. We adopt the simultaneous big bang condition to extract a growing mode of adiabatic perturbation in the flat F…
The existence (and abundance) of primordial black holes (PBHs) is governed by the power spectrum of primordial perturbations generated during inflation. So far no PBHs have been observed, and instead, increasingly stringent bounds on their existence at different scales have been obtained. Up until recently, this has been exploited in attempts to constrain parts of the inflationary power spectrum that are unconstrained by cosmological observations. We first point out that the simple translation of the PBH non-observation bounds into constraints on the primordial power spectrum is inaccurate as …
Primordial Black Holes in a Radiation-Dominated Universe
Bowdoin College（鲍登学院）；Queen Mary University of London（伦敦大学玛丽女王学院）；University of Pennsylvania（宾夕法尼亚大学）；Kenyon College（肯扬学院）；Case Western Reserve University（凯斯西储大学）；The Ohio State University（俄亥俄州立大学）
We develop a formalism to calculate the merger rate density of primordial black hole binaries with a general mass function, by taking into account the merger history of primordial black holes. We apply the formalism to three specific mass functions, monochromatic, power-law and log-normal cases. In the former case, the merger rate density is dominated by the single-merger events, while in the latter two cases, the contribution of the multiple-merger events on the merger rate density can not be ignored. The effects of the merger history on the merger rate density depend on the mass function.
Universality of Primordial Anisotropies in Gravitational Wave Background
School of Physics Science and Engineering, Tongji University（同济大学物理科学与工程学院）；Graduate School of Science and Engineering, Yamaguchi University（山口大学理工学研究科）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo（东京大学宇宙线研究所 Kavli宇宙物理数学研究所）；Institute for Advanced Study of Tongji University（同济大学高等研究院）；Center for Gravitation and Cosmology, Yangzhou University（扬州大学引力与宇宙学中心）
Primordial black holes (PBHs) within the mass range $10^{17} - 10^{22}$ g are a favorable candidate for describing the all of the dark matter content. Towards the lower end of this mass range, the Hawking temperature, $T_{\rm H}$, of these PBHs is $T_{\rm H} \gtrsim 100$ keV, allowing for the creation of electron -- positron pairs; thus making their Hawking radiation a useful constraint for most current and future MeV surveys. This motivates the need for realistic and rigorous accounts of the distribution and dynamics of emitted particles from Hawking radiation in order to properly model detec…
We use the spherical collapse method to investigate the non-linear density perturbations of pressureless matter in the cosmological models with the extended quintessence as dark energy in the metric and Palatini formalisms. We find that for both formalisms, when the coupling constant is negative, the deviation from the $Λ$CDM model is the least according to the evolutionary curves of the linear density contrast $δ_{c}$ and virial overdensity $Δ_{v}$, and it is less than one percent. And this indicates that, in the extended quintessence cosmological models in which the coupling constant is nega…
Primordial black holes through preheating instabilities in $α$-attractor models
Universidade da Beira Interior（贝拉内陆大学）；Centro de Matemática e Aplicações da Universidade da Beira Interior（贝拉内陆大学数学与应用中心）；Jagiellonian University（雅盖隆大学）；University of Utah（犹他大学）；Institute of Science Tokyo（东京科学大学）；University of Portsmouth（朴茨茅斯大学）
We exploit a new numerical technique for evaluating the tree order contributions to the primordial scalar and tensor power spectra for scalar potential models of inflation. Among other things we use the formalism to develop a good analytic approximation which goes beyond generalized slow roll expansions in that (1) it is not contaminated by the physically irrelevant phase, (2) its 0th order term is exact for constant first slow roll parameter, and (3) the correction is multiplicative rather than additive. These features allow our formalism to capture at first order, effects which are higher or…
We develop an analytic formalism for computing the merger rate of primordial black hole binaries with a general mass function by taking into account the torques by the surrounding primordial black holes and linear density perturbations. We find that $α=-(m_{i}+m_{j})^2\partial^{2} \ln {\mathcal R}(m_{i},m_{j})/\partial m_{i} \partial m_{j}=36/37$ is independent of the mass function. Moreover, the ratio of the merger rate density of primordial black hole binaries by taking into account the torques by the surrounding primordial black holes to by the nearest primordial black hole is independent o…
We examine the implementation of Buchert's and Green & Wald's averaging formalisms in exact spherically symmetric and plane symmetric dust-filled cosmological models. We find that, given a cosmological space-time, Buchert's averaging scheme gives a faithful way of interpreting the large-scale expansion of space, and explicit terms that precisely quantify deviations from the behaviour expected from the Friedmann equations of homogeneous and isotropic cosmological models. The Green & Wald formalism, on the other hand, does not appear to yield any information about the large-scale properties of a…
We consider gravitational radiation and electromagnetic radiation from point mass binary with electric charges in a Keplerian orbit, and calculate the merger rate distribution of primordial black hole binaries with charges and a general mass function by taking into account gravitational torque and electromagnetic torque by the nearest primordial black hole. We apply the formalism to the extremal charged case and find that $α=-(m_i+m_j)^2\partial^2 \ln {\cal R}(m_i,m_j)/\partial m_i \partial m_j=12/11$, which is independent of the mass function.
A spherical planetary nebula is described as a geometric model. The nebula itself is considered as a thin-shell which visualized as a boundary of two spacetimes. The inner and outer curvature tensors of the thin-shell are found in order to get an expression of the energy-momentum tensor on the thin-shell. The energy density and pressure expressions are derived using the energy-momentum tensor. The time evolution of the radius of the thin-shell is obtained in terms of the energy density. The model is tested by using a simple power function for decreasing energy density and the evolution pattern…
Primordial black holes (PBHs) cannot be produced abundantly enough to be the dark matter in canonical single-field inflation under slow roll. This conclusion is robust to local non-Gaussian correlations between long- and short-wavelength curvature modes, which we show have no effect in slow roll on local primordial black hole abundances. For the prototypical model which evades this no go, ultra-slow roll (USR), these squeezed non-Gaussian correlations have at most an order unity effect on the variance of PBH-producing curvature fluctuations for models that would otherwise fail to form sufficie…
We investigate the formation of primordial black holes (PBHs) from the collapse of spherically symmetric domain wall bubbles, which spontaneously nucleate via quantum tunneling during inflation. Since the tension of domain walls changes with time and so domain walls nucleate in a short time interval, the mass function of PBHs in general has a spike-like structure. In contrast to models in which PBHs produced from overdense regions, our model avoids the uncertainties of PBHs production mechanism. PBHs from domain walls with mass around $10^{20}\mathrm{g}$ may constitute all dark matter, those w…
We measure statistically anisotropic signatures imprinted in three-dimensional galaxy clustering using bipolar spherical harmonics (BipoSHs) in both Fourier space and configuration space. We then constrain a well-known quadrupolar anisotropy parameter $g_{2M}$ in the primordial power spectrum, parametrized by $P(\vec{k}) = \bar{P}(k) [ 1 + \sum_{M} g_{2M} Y_{2M}(\hat{k}) ]$, with $M$ determining the direction of the anisotropy. Such an anisotropic signal is easily contaminated by artificial asymmetries due to specific survey geometry. We precisely estimate the contaminated signal and finally s…
Primordial bubbles that possibly nucleate through quantum tunneling during inflation in a multidimensional potential might have left some relic detectable at the present time. These bubbles turn into black holes during the radiation era, which may account for the LIGO black holes, supermassive black holes, and may play an important role in dark matter. Typically, these black holes are surrounded by an energy deficit in the form of a spherical sound wave packet propagating outwards. In this work we study how this perturbation of the cosmic plasma dissipates before the time of recombination, lea…
Long-term 3+1 simulations of primordial black hole formation during radiation domination
University of Chinese Academy of Sciences (UCAS)（中国科学院大学）；Hangzhou Institute for Advanced Study (HIAS)（杭州高等研究院）；Institute of Theoretical Physics, Chinese Academy of Sciences (CAS)（中国科学院理论物理研究所）；Ningbo University（宁波大学）；Asia Pacific Center for Theoretical Physics (APCTP)（亚太理论物理中心）；Nagoya University（名古屋大学）；Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI)（小林-益川粒子宇宙起源研究所）
The curvature perturbations produced during an early era of inflation are known to have quasi-Gaussian distribution functions close to their maximum, where they are well constrained by measurements of the cosmic microwave background anisotropies and of the large-scale structures. In contrast, the tails of these distributions are poorly known, although this part is the relevant one for rare, extreme objects such as primordial black holes. We show that these tails are highly non-Gaussian, and cannot be described with standard non-Gaussian expansions, that are designed to approximate the distribu…
We consider a member of the family of scalar-tensor theories of Inflation - the so-called Horndeski model for Inflation, for which we calculate the non-Gaussianity (NG) and the power spectrum of curvature perturbations. We concentrate on the determination of the power spectrum, and the bi-spectrum along with the associated $f_\mathrm{NL}$ parameter and the Spectral Index. We follow the methodology outlined within the Quantum field theory framework (S-matrix approach and Weinberg's 'in-in' formalism) and compare - whenever necessary - our findings with those obtained via the traditional $δ$N fo…
We study a model of two-field ultra-slow-roll (USR) inflation bounded by a curve in the field space. Curvature perturbations and non-Gaussianities can be enhanced both during the USR phase and from the inhomogeneities at the boundary. We employ the full non-linear $δN$ formalism to calculate the probability distribution function (PDF) for curvature perturbation non-perturbatively and show that the non-linear effects can significantly enhance the abundance of the primordial black holes (PBHs). For large curvature perturbations, the PDF has a universal exponential tail, but for the intermediate …
In this work we have used for the first time pseudo-spectral methods to perform numerical simulations of spherically symmetric black hole formations on a Friedman-Robertson-Walker universe. With these methods, the differential equations describing the gravitational collapse are partially solved algebraically. With our publicly available code we then independently check, and confirm, previous numerical estimations of the thresholds to form primordial black holes. By using an excision technique and analytical estimations of accretion rates, we were also able to estimate the black holes mass even…
During inflation, vacuum quantum fluctuations are amplified and stretched to astrophysical distances. They give rise to fluctuations in the cosmic microwave background (CMB) temperature and polarisation, and to large-scale structures in our universe. They can also trigger the formation of primordial black holes (PBHs). Such objects could provide the progenitors of the recently detected black-hole mergers, and constitute part or all of the dark matter. Their observation would give invaluable access to parts of the inflationary sector that are unconstrained by the CMB. Since PBHs require large i…
It is well known that a primordial black hole (PBH) can be generated in the inflation process of the early universe, especially when the inflation field has a number of non-trivial features that could break the slow-roll condition. In this study, we investigate a toy model of inflation with bumpy potential, which has one or several bumps. We determined that the potential with multi-bump can generate power spectra with multi-peaks in small-scale region, which can in turn predict the generation of primordial black holes in various mass ranges. We also consider the two possibilities of PBH format…
In a Bohmian quantum cosmology scenario, we investigate some quantum effects on the evolution of the primordial universe arising from the adoption of an alternative non-trivial ordering to the quantization of the constrained Hamiltonian of a minimally coupled scalar field. The Wheeler-DeWitt equation has a contribution from the change in factor ordering, hence there are new quantum effects. We compare the results between the non-trivial and the trivial ordering cases, showing that the classical limit is valid for both orderings, but new bouncing and cyclic solutions are present in the non-triv…
We present a fully relativistic numerical method for the study of cosmological problems using the Baumgarte-Shapiro-Shibata-Nakamura formalism on a dynamical Friedmann-Lemaître-Robertson-Walker background. This has many potential applications including the study of the growth of structures beyond the linear regime. We present one such application by reproducing the Lemaître-Tolman-Bondi solution for the collapse of pressureless matter with arbitrary lapse function. The regular and smooth numerical solution at the center of coordinates proceeds in a natural way by relying on the Partially Impli…
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,…
Hierarchical formulation of the self-gravitating, n-dimensional, charged scalar field in spherical symmetry in affine null formalism
Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación（科尔多瓦国立大学数学、天文学、物理和计算学院）；Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET, IFEG（阿根廷国家科学研究委员会，CONICET，IFEG）；Facultad de Ingeniería, Universidad San Sebastián（圣塞巴斯蒂安大学工程学院）；Facultad de Ingeniería y Ciencias, Universidad Diego Portales（迭戈帕拉塔斯大学工程与科学学院）
