Formalism for Primordial Black Hole Formation in Spherical Symmetry (1504.02071v1)

Abstract: 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.

• The paper introduces a formalism that combines Misner-Sharp and Hernandez-Misner approaches to model primordial black hole formation in a radiation-dominated FRW universe.
• It employs numerical simulations to derive critical density thresholds and ensure stable evolution through refined non-reflecting boundary conditions.
• The study highlights the pivotal role of spherical symmetry, providing insights that connect PBH formation to potential dark matter and supermassive black hole seed scenarios.

Formalism for Primordial Black Hole Formation in Spherical Symmetry

The paper "Formalism for Primordial Black Hole Formation in Spherical Symmetry" by Jolyon Bloomfield, Daniel Bulhosa, and Stephen Face presents a detailed theoretical framework for investigating the formation of primordial black holes (PBHs) in the early universe. This framework is centered on the utilization of the Misner-Sharp and Hernandez-Misner formalisms, allowing the authors to simulate the gravitational collapse of perturbations in a cosmological context. Below is an in-depth discussion of the paper's methodologies, results, and implications for the field of theoretical astrophysics.

Analytical Foundations

The authors begin by revisiting the derivation of the Misner-Sharp formalism, which is particularly adept at addressing spherically symmetric perturbations within a fluid medium. They extensively explore the equations of motion governing the dynamics of the system, considering the effects of a perfect fluid in spherical coordinates. The formalism is applied to a cosmological background, specifically a Friedmann-Robertson-Walker (FRW) universe, and assumes a fluid with a radiation-like equation of state, $w = 1/3$.

The paper carefully examines the threshold conditions necessary for PBH formation, primarily focusing on spherical symmetry. This approach aligns with the notion that the largest peaks resulting from cosmological perturbations tend to be nearly spherical, as suggested by peaks theory. The simulations derive critical parameters such as the critical mass overdensity $\delta_c$ required for black hole formation and assess their dependence on the initial density profiles.

Numerical Techniques

Numerical simulations play a pivotal role in this paper, facilitating the evaluation of how initial perturbations evolve into black holes. The authors implement a dimensionless framework for the equations of motion, emphasizing accuracy and stability during numerical execution. They explore the convergence of linear and nonlinear modes and propose a derivative expansion to account for higher-order effects. The boundary and initial conditions are rigorously examined to reflect realistic settings in the early universe.

Particularly noteworthy is the treatment of boundary conditions. The authors propose a non-reflecting condition for the outer boundary of the computational domain. This innovation minimizes the unphysical reflections typically encountered in simulations, thereby enhancing the reliability of the results.

Black Hole Detection and Mass Calculation

A critical section of the paper addresses the detection of black hole formation and the subsequent calculation of PBH masses. A black hole is deemed to form when the mass enclosed within a given radius satisfies the Schwarzschild condition, $2m/R \geq 1$. The Misner-Sharp formalism initially handles the evolution until a singularity is approached, at which point the Hernandez-Misner formalism offers a continuation of the evolution, circumventing the singularity by employing null slicing.

The authors provide a method for extracting the final mass of a PBH, employing a combination of the vanishing lapse and integrated mass within a defined radius. This approach is crucial for ensuring consistency and physical accuracy within the simulations.

Implications and Future Directions

The formulation established in this paper has substantial implications for the understanding of PBH formation. The method facilitates a clearer interpretation of the conditions and processes involved in the early universe's structure formation. The potential for PBHs to become seeds for supermassive black holes or to contribute to dark matter is explored, connecting cosmological phenomena to observational implications.

Future research may explore the interplay of non-Gaussianities or isocurvature perturbations and assess further the cosmological impacts of PBHs across different scales. Additionally, the integration of this formalism with advanced numerical relativity codes could open new avenues for probing PBH formation in a variety of inflationary scenarios.

In conclusion, the detailed formalism laid out in this paper is a significant step toward a comprehensive understanding of primordial black hole formation in the context of the early universe. By merging analytical rigor with numerical precision, the authors provide a robust tool for researchers to investigate this intriguing area of cosmology with heightened clarity and accuracy.