We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

We update the constraints on the fraction of the Universe going into primordial black holes in the mass range ${10}^{9}--{10}^{17}\text{ }\text{ }\mathrm{g}$ associated with the effects of their evaporations on big bang nucleosynthesis and the extragalactic photon background. We include for the first time all the effects of quark and gluon emission by black holes on these constraints and account for the latest observational developments. We then discuss the other constraints in this mass range and show that these are weaker than the nucleosynthesis and photon background limits, apart from a small range ${10}^{13}--{10}^{14}\text{ }\text{ }\mathrm{g}$, where the damping of cosmic microwave background anisotropies dominates. Finally we review the gravitational and astrophysical effects of nonevaporating primordial black holes, updating constraints over the broader mass range $1--{10}^{50}\text{ }\text{ }\mathrm{g}$.

New cosmological constraints on primordial black holes

We study the effect of primordial non-Gaussianity on large-scale structure, focusing upon the most massive virialized objects. Using analytic arguments and N-body simulations, we calculate the mass function and clustering of dark matter halos across a range of redshifts and levels of non-Gaussianity. We propose a simple fitting function for the mass function valid across the entire range of our simulations. We find pronounced effects of non-Gaussianity on the clustering of dark matter halos, leading to strongly scale-dependent bias. This suggests that the large-scale clustering of rare objects may provide a sensitive probe of primordial non-Gaussianity. We very roughly estimate that upcoming surveys can constrain non-Gaussianity at the level of |f{sub NL}| < or approx. 10, which is competitive with forecasted constraints from the microwave background.

Imprints of primordial non-Gaussianities on large-scale structure: Scale-dependent bias and abundance of virialized objects

This article reviews current understanding of primordial black holes (PBHs), with particular focus on those massive examples ( ) which remain at the present epoch, not having evaporated through Hawking radiation. With the detection of gravitational waves by LIGO, we have gained a completely novel observational tool to search for PBHs, complementary to those using electromagnetic waves. Taking the perspective that gravitational-wave astronomy will make significant progress in the coming decades, the purpose of this article is to give a comprehensive review covering a wide range of topics on PBHs. After discussing PBH formation, as well as several inflation models leading to PBH production, we summarize various existing and future observational constraints. We then present topics on formation of PBH binaries, gravitational waves from PBH binaries, and various observational tests of PBHs using gravitational waves.

https://iopscience.iop.org/article/10.1088/1361-6382/aaa7b4/pdf

Primordial Black Holes - Perspectives in Gravitational Wave Astronomy -

Gravitational waves (GWs) have a great potential to probe cosmology. We review early universe sources that can lead to cosmological backgrounds of GWs. We begin by presenting proper definitions of GWs in flat space-time and in a cosmological setting (section 2). Following, we discuss the reasons why early universe GW backgrounds are of a stochastic nature, and describe the general properties of a stochastic background (section 3). We recap current observational constraints on stochastic backgrounds, and discuss the basic characteristics of present and future GW detectors, including advanced LIGO, advanced Virgo, the Einstein telescope, KAGRA, and LISA (section 4). We then review in detail early universe GW generation mechanisms, as well as the properties of the GW backgrounds they give rise to. We classify the backgrounds in five categories: GWs from quantum vacuum fluctuations during standard slow-roll inflation (section 5), GWs from processes that operate within extensions of the standard inflationary paradigm (section 6), GWs from post-inflationary preheating and related non-perturbative phenomena (section 7), GWs from first order phase transitions related or not to the electroweak symmetry breaking (section 8), and GWs from general topological defects, and from cosmic strings in particular (section 9). The phenomenology of these early universe processes is extremely rich, and some of the GW backgrounds they generate can be within the reach of near-future GW detectors. A future detection of any of these backgrounds will provide crucial information on the underlying high energy theory describing the early universe, probing energy scales well beyond the reach of particle accelerators.

http://iopscience.iop.org/article/10.1088/1361-6382/aac608/pdf

Cosmological Backgrounds of Gravitational Waves

We calculate the full probability density function (PDF) of inflationary curvature perturbations, even in the presence of large quantum backreaction. Making use of the stochastic-δ N formalism, two complementary methods are developed, one based on solving an ordinary differential equation for the characteristic function of the PDF, and the other based on solving a heat equation for the PDF directly. In the classical limit where quantum diffusion is small, we develop an expansion scheme that not only recovers the standard Gaussian PDF at leading order, but also allows us to calculate the first non-Gaussian corrections to the usual result. In the opposite limit where quantum diffusion is large, we find that the PDF is given by an elliptic theta function, which is fully characterised by the ratio between the squared width and height (in Planck mass units) of the region where stochastic effects dominate. We then apply these results to the calculation of the mass fraction of primordial black holes from inflation, and show that no more than ~ 1 e-fold can be spent in regions of the potential dominated by quantum diffusion. We explain how this requirement constrains inflationary potentials with two examples.

/pdf/quantum-diffusion-during-inflation-and-primordial-black-2qc2cvqqtm.pdf

Quantum diffusion during inflation and primordial black holes

We investigate the generation of gravitational waves due to the gravitational instability of primordial density perturbations in an early matter-dominated era which could be detectable by experiments such as laser interferometer gravitational wave observatory (LIGO) and laser interferometer space antenna (LISA). We use relativistic perturbation theory to give analytic estimates of the tensor perturbations generated at second order by linear density perturbations. We find that large enhancement factors with respect to the naive second-order estimate are possible due to the growth of density perturbations on sub-Hubble scales. However very large enhancement factors coincide with a breakdown of linear theory for density perturbations on small scales. To produce a primordial gravitational-wave background that would be detectable with LIGO or LISA from density perturbations in the linear regime requires primordial comoving curvature perturbations on small scales of order 0.02 for advanced LIGO or 0.005 for LISA; otherwise numerical calculations of the nonlinear evolution on sub-Hubble scales are required.

/pdf/gravitational-waves-from-an-early-matter-era-c27zaz22tr.pdf

Gravitational waves from an early matter era

We calculate the full probability density function (PDF) of inflationary curvature perturbations, even in the presence of large quantum backreaction. Making use of the stochastic-$\delta N$ formalism, two complementary methods are developed, one based on solving an ordinary differential equation for the characteristic function of the PDF, and the other based on solving a heat equation for the PDF directly. In the classical limit where quantum diffusion is small, we develop an expansion scheme that not only recovers the standard Gaussian PDF at leading order, but also allows us to calculate the first non-Gaussian corrections to the usual result. In the opposite limit where quantum diffusion is large, we find that the PDF is given by an elliptic theta function, which is fully characterised by the ratio between the squared width and height (in Planck mass units) of the region where stochastic effects dominate. We then apply these results to the calculation of the mass fraction of primordial black holes from inflation, and show that no more than $\sim 1$ $e$-fold can be spent in regions of the potential dominated by quantum diffusion. We explain how this requirement constrains inflationary potentials with two examples.

We consider the stochastic background of gravitational waves produced during the radiation-dominated hot big bang as a constraint on the primordial density perturbation on comoving length scales much smaller than those directly probed by the cosmic microwave background or large-scale structure. We place weak upper bounds on the primordial density perturbation from current data. Future detectors such as BBO and DECIGO will place much stronger constraints on the primordial density perturbation on small scales.

Constraints on primordial density perturbations from induced gravitational waves

We study the stochastic formalism of inflation beyond the usual slow-roll approximation. We verify that the assumptions on which the stochastic formalism relies still hold even far from the slow-roll attractor. This includes demonstrating the validity of the separate universe approach to evolving long-wavelength scalar field perturbations beyond slow roll. We also explain that, in general, there is a gauge correction to the amplitude of the stochastic noise. This is because the amplitude is usually calculated in the spatially-flat gauge, while the number of e-folds is used as the time variable (hence one works in the uniform-$N$ gauge) in the Langevin equations. We show that these corrections vanish in the slow-roll limit, but we also explain how to calculate them in general. We compute them in difference cases, including ultra-slow roll and the Starobinsky model that interpolates between slow roll and ultra-slow roll, and find the corrections to be negligible in practice. This confirms the validity of the stochastic formalism for studying quantum backreaction effects in the very early universe beyond slow roll.

Hooshyar Assadullahi

Papers

Quantum diffusion during inflation and primordial black holes

Gravitational waves from an early matter era

Quantum diffusion during inflation and primordial black holes

Constraints on primordial density perturbations from induced gravitational waves

Stochastic inflation beyond slow roll