Session K6: Primordial Black Holes, Spacetime Structure and Thermodynamics

Primordial Black Holes from First Principles (Overview)

Given a power spectrum from inflation, our goal is to calculate, from first principles, the number density and mass spectrum of primordial black holes that form in the early universe. Previously, these have been calculated using the Press- Schechter formalism and some demonstrably dubious rules of thumb regarding predictions of black hole collapse. Instead, we use Monte Carlo integration methods to sample field configurations from a power spectrum combined with numerical relativity simulations to obtain a more accurate picture of primordial black hole formation. We demonstrate how this can be applied for both Gaussian perturbations and the more interesting (for primordial black holes) theory of hybrid inflation. One of the tools that we employ is a variant of the BBKS formalism for computing the statistics of density peaks in the early universe. We discuss the issue of overcounting due to subpeaks that can arise from this approach (the “cloud-in-cloud” problem).   

Primordial Black Holes from First Principles (Statistics)

To compute estimates for the number density of candidates for black hole formation, we will examine the statistics governing peaks in the density perturbation field arising from inflation. The number density of peaks was calculated for gaussian random density perturbations by BBKS (1984). However, we are interested in hybrid inflation, where the perturbation spectrum is governed by ``chi-squared'' random fields. We will review the formalism of BBKS and extend it to the chi-squared case. The chi-squared field statistics present mathematical challenges due to the participation of multiple inflaton fields in the generation of density perturbations. We exploit a symmetry of these fields to reduce the density calculation to a numerically tractable integration. Surprisingly, the result for an arbitrarily large number of inflaton fields is simpler than the two and three field cases. We will relate these exceptional cases to the dimensionality of space and resulting topological defects. The final number density estimate depends on a single parameter derived from the power spectrum of the gaussian fields that comprise the chi-squared perturbation field.   

Primordial Black Holes from First Principles (numerics)

In order to compute accurate number densities and mass spectra for primordial black holes from an inflationary power spectrum, one needs to perform Monte Carlo integration over field configurations. This requires a method of determining whether a black hole will form, and if so, what its mass will be, for each sampled configuration. In order for such an integral to converge within any reasonable time, this requires a highly efficient process for making these determinations. We present a numerical pipeline that is capable of making reasonably accurate predictions for black holes and masses at the rate of a few seconds per sample (including the sampling process), utilizing a fully-nonlinear numerical relativity code in 1+1 dimensions.   

Fermi LAT Limits on Primordial Black Hole Evaporation

Primordial black holes (PBHs) of sufficiently small mass emit gamma rays in the Fermi Large Area Telescope (LAT) energy range. PBHs with lifetimes shorter than the Fermi observation time will appear as moving point sources with gamma-ray emission that becomes harder and brighter with time until the PBH completely evaporates. Previous searches for gamma rays from PBHs have focused on either short time scale bursts or the contribution of PBH bursts to the isotropic diffuse emission. Here we use Fermi LAT point source catalogs to search for PBH candidates that evaporate on a time scale of several years. In addition to looking for the spectral signatures of a PBH, we also develop an algorithm to detect proper motion. There are a few unassociated point sources with spectra consistent with PBH evaporation; however, none of these sources show significant proper motion. We derive a conservative limit on PBH evaporation rate in the vicinity of the Earth by using a threshold on the gamma-ray flux above 10 GeV such that there are no sources above this threshold with spectra consistent with Hawking radiation from PBHs. The derived limit is more stringent than the limits obtained with ground-based gamma-ray observatories.   

Inhomogeneities in an expanding universe: the nonlinear and relativistic regimes

I will discuss the dynamics, and observational consequences of inhomogeneities in an expanding universe. In particular, I will concentrate on how the tools of numerical relativity can be used to study this problem in a fully general-relativistic setting, where traditionally employed approximations may break down. I will show how this can be used to explore and quantify the cosmological regime where the evolution of the inhomogeneities becomes nonlinear, and where relativistic effects may become important. This includes applications to primordial black hole formation, as well as other settings in the early universe where strong-field gravity plays a role.   

Perihelion advance of a test particle in the Kerr field.

Here I represent a Perihelion advance of a test particle in the Kerr field. I assume that the spin of the central body to be very small and planar motion occurs only in the equatorial plane. I find some physical picture which is different from the case of Schwarzschild field and can recover the picture for Schwarzschild field. We use perturbation method to solve the equation of motion.   

Invariant Laws of Thermodynamics and Validity of Hasen\"{o}hrl Mass-Energy Equivalence Formula m $=$ (4/3) $E$/c$^{\mathrm{2}}$ at Photonic, Electrodynamic, and Cosmic Scales

According to a scale-invariant statistical theory of fields$^{\mathrm{1}}$ electromagnetic photon mass is given as $m_{em,k} =\sqrt {hk/c^{3}} $. Since electromagnetic energy of photon is identified as $amu=\sqrt {hkc} $, all baryonic matter is composed of light (photons) $E_{em} =Nm_{em,k} c^{2}=M_{em,k} c^{2}\thinspace [Joule]$ or equivalently $M_{em,k} c^{2}/8338\thinspace [kcal]=Namu=M_{a} [kg]$ where 8338 is De Pretto number$^{\mathrm{1}}$. Besides particle \textit{electromagnetic} energy one requires \textit{potential energy} associated with Poincar\'{e} $^{\mathrm{2\thinspace }}$ stress for particle stability leading to rest enthalpy$^{\mathrm{1}} \quad \hat{{h}}_{o} =\hat{{u}}_{o} +p_{o} \hat{{v}}=\hat{{u}}_{o} +\hat{{u}}_{o} /3=(4/3)m_{em,k} c^{2}$ in accordance with Hasen\"{o}hrl$^{\mathrm{3}}$. The 4/3 problem of electrodynamics (Boyer, T. H., Phys. Rev. Lett.\textbf{ 25}, 1982) is also related to Poincar\'{e} $^{\mathrm{2}}$ stress thus the potential energy$p_{o} \hat{{v}}=\hat{{u}}_{o} /3$. Hence, the factor 4/3 is identified as Poisson polytropic index $b=c_{p} /c_{v} $ and total particle rest mass will be composed of \textit{electromagnetic} and \textit{gravitational} parts $m_{o} =m_{em} +m_{gr} =(3/4)E_{o} /c^{2}+(1/4)E_{o} /c^{2}$. At cosmological scale, respectively 3/4 and 1/4 of the total mass of closed universe will be electromagnetic (\textit{dark energy}) and gravitational (\textit{dark matter})$^{\mathrm{1}}$ in nature as was emphasized by Pauli (\textit{Theory of Relativity}, Dover, 1958). Also, Poincar\'{e}-Lorentz \textit{dynamic} versus Einstein \textit{kinematic} theory of relativity will be discussed.\\ $^{\mathrm{1}}$ Sohrab, S. H.,\textit{ ASME J. Energy Resources and Technology} \textbf{138}: 1-12 (2016). $^{\mathrm{2}}$ Poincar\'{e}, H., \textit{Rend. del Circ. Mat. Palermo }\textbf{21}: 129-176 (1906). $^{\mathrm{3}}$ Hasen\"{o}hrl, F., \textit{Annalen der Physik }\textbf{321}: 589-592 (1905).