Evaporation of Primordial Black Holes in a Thermal Universe: A Thermofield Dynamics Approach

TL;DR

This work addresses how a finite-temperature environment, such as the early Universe radiation bath, alters Hawking radiation from black holes. By employing Thermofield Dynamics, the authors derive finite-temperature occupation numbers for scalar and fermionic fields in Schwarzschild and Kerr backgrounds, incorporating the rotation via the frequency shift $ ilde{ u}= u-m\Omega_h$. They show that the ambient bath induces stimulated emission, modifying the standard spectra to expressions like $n_ u=rac{\nabla_ u}{e^{2\pi u/κ}-1}[1+(e^{2\pi u/κ}+1)/(e^{β u}-1)]$ for scalars in Schwarzschild and analogous Kerr generalizations for both bosons and fermions. Applying this to reheating-era PBHs, the finite-temperature corrections accelerate mass and angular momentum loss, shortening PBH lifetimes and affecting cosmological constraints. The results demonstrate that thermal environments must be incorporated in PBH evaporation analyses, with potential implications for gravitational-wave backgrounds and dark matter considerations; the framework provides a consistent operator-level treatment of thermal effects in curved spacetime Hawking radiation, especially for Kerr PBHs during reheating.

Abstract

We investigate the impact of a finite temperature environment on the Hawking radiation from black holes (BHs), with particular focus on Kerr BHs immersed in a cosmological thermal bath. The emitted particles from BHs interact with the thermal background and thermalize, leading to a modification in the Hawking radiation spectrum. By employing the methods of Thermofield Dynamics (TFD), a real time formalism of thermal quantum field theory, we derive the modified occupation numbers of the Hawking spectrum for asymptotically flat spacetimes like the Schwarzschild and the Kerr geometries. These corrections depend on the interplay between the BH temperature and the ambient bath temperature. We apply this formalism in the early universe reheating background scenario arising after inflation and demonstrate that the thermal correction to Hawking spectrum enhances the evaporation rate of primordial black holes (PBHs). As a result, the lifetime of PBH shortens compared to the zero temperature vacuum and leads to interesting cosmological consequences.

Figures (6)

• Figure 1: Penrose diagram of a collapsing star. The light ray at $v=v_{0}$ is the last null ray scattered from the black hole geometry.
• Figure 2: Left Panel: Variation of $\epsilon_i$ (Eq. \ref{['epsilon_i']}) with bath temperature $T_b$ for Schwarzschild (dashed lines) and Kerr (solid lines) BHs, plotted for three particle spins: scalar ($s = 0$, red), fermion ($s = 1/2$, blue), and vector boson ($s = 1$, green). Right Panel: Dependence of $\gamma_i$ (Eq. \ref{['gamma_i']}) on $T_b$ for particles with spin $s = 1/2$ (blue) and $s = 1$ (green), shown for both Schwarzschild (dashed) and Kerr (solid) BHs.
• Figure 3: Lifetime of BHs vs. bath temperature $T_b$ (Eq. \ref{['lifetime']}) for two initial masses, $1\,\mathrm{g}$ (red) and $10\,\mathrm{g}$ (yellow), shown for Schwarzschild (dashed) and Kerr (solid) BHs.
• Figure 4: Left Panel: Evolution of BH mass as a function of the scale factor for an initial mass of $5\, \mathrm{g}$ and initial bath temperature $10^{15}\, \mathrm{GeV}$. The solid purple line corresponds to the Kerr BH with thermal corrections, the dashed purple line to the Schwarzschild BH with thermal corrections, and the red dashed line to the Schwarzschild BH without thermal corrections. Right Panel: Evolution of the spin parameter $a_*$ for the same initial conditions as the left panel.
• Figure 5: Schematic evolution of background radiation temperature.