Black-hole evaporation for cosmological observers

TL;DR

This paper addresses how primordial black hole evaporation is altered when the black hole resides in a de Sitter-like expanding universe. Using the Vaidya-de Sitter metric and Hayward’s dynamical thermodynamics, the authors model Hawking evaporation from the perspective of cosmological observers comoving with the expansion, deriving an observer-dependent evaporation law and analytic mass evolution in terms of the cosmological time $\tau'$. They find that evaporation timescales and end-states can differ dramatically from the standard Schwarzschild picture, including cases where complete evaporation is never observed for certain observers. The results challenge conventional PBH constraints and highlight the importance of the cosmological frame in interpreting PBH signals, motivating extensions to more realistic cosmological backgrounds and implications for PBH phenomenology.

Abstract

This work investigates the evaporation of black holes immersed in a de Sitter environment, using the Vaidya-de Sitter spacetime. The role of cosmological observers is highlighted in the development and Hayward thermodynamics for non-stationary geometries is employed in the description of the compact objects. The results of the proposed dynamical model are compared with the usual description based on stationary geometries, with specific results for primordial black holes (PBHs). The timescale of evaporation is shown to depend significantly on the choice of cosmological observer and can differ substantially from predictions based on stationary models at late times. Deviations are also shown with respect to the standard assertion that there is a fixed initial mass just below $10^{15} \, \text{g} \sim 10^{-18} M_\odot$ for the PBHs which are completing their evaporation process at the present epoch.

Figures (6)

• Figure 1: On the left, the Penrose diagram for the de Sitter spacetime is shown. On the right, it is highlighted the details of the two disconnected regions that are covered by the $(t, r, \theta, \phi)$ coordinates.
• Figure 2: Representation of a comoving observer (dash-dotted red line), $\mathcal{O}$, in the Penrose diagram of the de Sitter geometry. The dashed green lines indicate the coordinate $u$ at $\tau = 0$ and $\tau \rightarrow \infty$ for the observer.
• Figure 3: Dimensionless coordinate $u'$ as a function of the dimensionless cosmological time $\tau'$ for different cosmological observers (labeled by the dimensionless initial radial position). From top to bottom, each curve represents a cosmological observer that reaches the asymptotic regime more quickly according to their proper time.
• Figure 4: Black-hole mass as a function of dimensionless cosmological time for different values of initial masses. An observer characterized by ${r'_0 = 0.866}$ is considered. The vertical line indicates the present epoch (${\tau' \sim 0.75}$, while $\tau'$ is varied.) The curve where ${M_{0} = 1.25 \times 10^{-19} M_{\odot}}$ exemplifies a scenario in which a PBH would still be evaporating today. The other two curves exhibit behaviors similar to those of the traditional Schwarzschild-based model.
• Figure 5: Asymptotic mass of an evaporating black hole at the infinite future for distinct cosmological observers (labeled by their initial radial positions). From left to right, each curve corresponds to a cosmological observer for which an asymptotically vanishing black hole has a larger initial mass.