Kiselev black hole and the ultra-slow evaporating behavior

Abstract: Kiselev solution is a metric that describes black holes immersed in a quintessence-like dark energy background. By introducing a dynamic state parameter $w_q$, the Kiselev solution is supposed to help comprehend the effect of quintessential matter on black holes. In this work, we study the evaporation behaviors of Kiselev black holes. By varying the state parameter $w_q$, we find that the decreasing state parameter lowers the non-final stage temperature and markedly prolongs the evaporation lifetime. We also find that the ultra-slow evaporation mechanism of Kiselev black holes differs vastly from the perfect fluid dark matter (PFDM) black holes and Horndeski black holes, which share the analogous ultra-long lifetime. These results illuminate the effects of dynamic dark energy background on black hole evaporation, provide a potential laboratory to constrain the value of $w_q$, and may complement cosmological and astrophysical observations, e.g., the DESI's preference for thawing dark energy and the observation of exploding black holes based on ultra-slow evaporation.

• The paper demonstrates that the quintessential term in the Kiselev metric significantly suppresses Hawking temperature, leading to markedly prolonged black hole lifetimes.
• It employs numerical simulations and empirical scaling laws to compare evaporation dynamics with PFDM and Horndeski black hole models.
• The findings imply that dynamic dark energy could extend primordial black hole lifespans and serve as a probe for cosmological parameters.

Evaporation Dynamics of Kiselev Black Holes in Quintessence-Like Backgrounds

Introduction

The Kiselev solution provides a spherically symmetric black hole metric embedded in a dark energy environment characterized by a quintessence-like equation of state parameter $w_q$. This model has attracted interest as both cosmological and astrophysical data, including recent results from DESI, challenge the $\Lambda$CDM paradigm and suggest possible dynamical evolution of dark energy. The current work investigates the thermodynamic and evaporation behavior of Kiselev black holes (KBHs), focusing on how varying $w_q$ impacts their thermal properties and evaporation lifetimes, and comparing these results to other extended black hole models such as those in perfect fluid dark matter (PFDM) and Horndeski backgrounds.

Thermodynamic Framework

The Kiselev solution alters the Schwarzschild metric by introducing a quintessential term via the metric function

$f(r) = 1 - \frac{2M}{r} - \frac{a}{r^{3w_q+1}}$

where $a$ is a parameter proportional to the quintessence energy density and $w_q$ obeys $-1 < w_q < -1/3$. When $a=0$, one recovers Schwarzschild; for $w_q \rightarrow -1$, the metric approaches Schwarzschild-de Sitter. Event horizons are modified, and for analytic tractability, the case $w_q = -2/3$ admits closed-form radii.

The Hawking temperature and Bekenstein-Hawking entropy follow:

$\Lambda$0

where $\Lambda$1 is the event horizon radius. For negative $\Lambda$2 approaching $\Lambda$3, the temperature is significantly suppressed at fixed $\Lambda$4, due to the dominance of the quintessential correction.

Evaporation Behavior and Asymptotics

Black hole evaporation is described using the Stefan–Boltzmann law, with the emission rate highly sensitive to $\Lambda$5. For KBHs, in the relevant range of $\Lambda$6, the metric modification leads to a substantial reduction in temperature for moderate and large event horizon radii. Numerical calculation of the evaporation process demonstrates that as $\Lambda$7 decreases (approaching $\Lambda$8), the characteristic lifetime of a KBH increases dramatically, exhibiting ultra-slow evaporation behavior. This effect is not a minor correction; for sufficiently negative $\Lambda$9, the lifetime can become orders of magnitude longer than the Schwarzschild case:

$w_q$0

The exponent $w_q$1 becomes significant in the quintessential regime, producing a power-law cancellation that amplifies the suppression of $w_q$2 and, thus, the evaporation rate.

Comparison with Other Ultra-Slow Evaporation Models

The ultra-slow evaporation in KBHs is distinct from that in PFDM black holes and Horndeski theories:

• PFDM Black Holes: Extended lifetimes arise from a reduced surface gravity and additional energy loss through dark matter Schwinger pair production.
• Horndeski Black Holes: Non-minimal matter couplings produce a complex three-stage evaporation process, with changes in both emission temperature and channels.
• Kiselev Black Holes: The extreme suppression results purely from the thermal correction induced by the dark energy background, as encoded in the metric exponent $w_q$3; no extra particle creation is involved.

Numerical Results and Scaling Laws

Quantitative simulations (with realistic $w_q$4) confirm that for $w_q$5, the evaporation lifetime remains comparable to the Schwarzschild case, $w_q$6 Gyr for benchmark primordial black hole masses. As $w_q$7 approaches $w_q$8, the lifetime increases exponentially and can be described by an empirical formula:

$w_q$9

A small change in $f(r) = 1 - \frac{2M}{r} - \frac{a}{r^{3w_q+1}}$0 near $f(r) = 1 - \frac{2M}{r} - \frac{a}{r^{3w_q+1}}$1 leads to an enormous increase in lifetime, highlighting the sensitivity of the ultra-slow evaporation mechanism to even modest deviations from the cosmological constant limit.

Implications and Prospects

The results have practical and theoretical consequences. Practically, if quintessence-like dark energy backgrounds pervade the universe, primordial black holes (PBHs) formed in such environments would have extended lifespans, potentially altering predictions for PBH remnants and constraints from Hawking evaporation. The dynamic nature of $f(r) = 1 - \frac{2M}{r} - \frac{a}{r^{3w_q+1}}$2 can, in principle, be constrained by observations of PBH candidates and their current evaporation status.

Theoretically, the work clarifies that ultra-slow black hole evaporation is not a universal consequence of all dark sector backgrounds. Instead, the effect is realized through specific interplay between the equation of state parameter and the black hole metric. This mechanism is independent of additional particle production channels and relies purely on the generalized Hawking temperature.

Assessment of exploding black holes, as recently discussed in the literature, could also be informed by this mechanism. If dark energy evolves towards more negative $f(r) = 1 - \frac{2M}{r} - \frac{a}{r^{3w_q+1}}$3, the probability of witnessing black hole explosions may decrease, contrary to predictions from models with rapid late-time evaporation.

These results support the use of black hole evaporation properties as probes for fundamental cosmological parameters and their evolution, providing a complementary approach to standard cosmology.

Conclusion

Kiselev black holes in quintessence-like dark energy backgrounds demonstrate a unique ultra-slow evaporation mechanism, driven by the suppression of Hawking temperature due to the metric modification parameterized by $f(r) = 1 - \frac{2M}{r} - \frac{a}{r^{3w_q+1}}$4. This behavior is distinct from other extended black hole solutions and provides a potential avenue to constrain dynamic dark energy behavior via black hole thermodynamics. The framework established here invites further investigation into PBH abundance, quantum gravity phenomenology, and the interplay between dynamical cosmological parameters and compact object physics.