Emergence of Dark Phases in Scalar Particles within the Schwarzschild-Kiselev-Letelier Spacetime

TL;DR

The paper investigates how a quintessence fluid, modeled within the Schwarzschild-Kiselev-Letelier spacetime that also includes a cloud of strings, influences the radial dynamics of a spin-0 particle through the Klein-Gordon equation. By solving the Einstein equations, it derives the metric f(r) with parameters r_s, dc{a}, N_Q, and α_Q, and analyzes event horizon formation for α_Q in {0, 1/2, 1}. It then presents exact and near-horizon solutions to the Klein-Gordon equation in terms of confluent Heun functions, revealing quintessence-induced dark phases as explicit phase shifts in the radial wave function that depend on N_Q and α_Q. The results show that these dark phases are more subtle in spherically symmetric spacetimes than in cylindrical cases, yet they provide a potential observable handle on dark energy through quantum-level effects and may connect to phenomena like Hawking radiation and HBT interferometry in future work.

Abstract

This work focuses on the emergence of dark phases (dark energy-induced phases) in the radial wave function of scalar particles. We achieve this by presenting novel solutions to the Klein-Gordon equation in a spherically symmetric spacetime, which encompasses a black hole, a quintessential fluid, and a cloud of strings. We determine the exact solution for the spacetime metric, analyze the admissible ranges for its physical parameters, and discuss the formation of the event horizon. Subsequently, we detail the solution of the Klein-Gordon equation and explore three distinct cases of dark phases, corresponding to the quintessence state parameter $α_{Q}$ taking the values $0$, $1/2$, and $1$. Notably, the case where $α_{Q} = 1$ holds particular significance due to current observational constraints on dark energy.

Figures (20)

• Figure 1: Plot of the quintessence parameter, $N_{ \raisebox{-\srblobdepth}{$Q$} \raisebox{-\srblobdepth}{$Q$} }$, as a function of the state parameter $\alpha_{ \raisebox{-\srblobdepth}{$Q$} \raisebox{-\srblobdepth}{$Q$} }$. As shown in \ref{['N_Q_Estimate']}, $N_{ \raisebox{-\srblobdepth}{$Q$} \raisebox{-\srblobdepth}{$Q$} }$ exponentially decreases as $\alpha_{ \raisebox{-\srblobdepth}{$Q$} \raisebox{-\srblobdepth}{$Q$} }$ increases.
• Figure 2: Contribution of the quintessential term on the metric function $f(r)$. As represented in \ref{['Contribution_Numeric_Estimate']}, the influence of the quintessence term is most significant at smaller values of $\alpha_{ \raisebox{-\srblobdepth}{$Q$} \raisebox{-\srblobdepth}{$Q$} }$. In contrast, as the state parameter approaches its upper bound, significantly larger radii are required for the quintessence effect to become substantial.
• Figure 3: Behavior of the metric function for the state parameter $\alpha_{ \raisebox{-\srblobdepth}{$Q$} \raisebox{-\srblobdepth}{$Q$} }=0,\, 1/2,\, 1$. The cloud contribution and the Schwarzschild radius are set as $\overline{a}=10^{-2}$ and $r_{s}=2 \times 10^{3}$ m, respectively, while the quintessential term is represented according to \ref{['Contribution_Numeric_Estimate']}, where $r_{\text{obs}} = 4.4 \times 10^{26}$ m. As highlighted in the inset plot, the curves concerning the two higher state parameters will substantially distinguish themselves near $r \propto r_{\text{obs}}$.
• Figure 4: $f(r)$ for different fractions of quintessence. We set the cloud parameter $\overline{a}=10^{-2}$, the Schwarzschild radius $r_{s}=2 \times 10^{3}$ m, and the dark term according to \ref{['fraction_NQ_term']}. The left panel displays the metric function for $\alpha_{ \raisebox{-\srblobdepth}{$Q$} \raisebox{-\srblobdepth}{$Q$} } = 0$, while the right panel illustrates $f(r)$ for $\alpha_{ \raisebox{-\srblobdepth}{$Q$} \raisebox{-\srblobdepth}{$Q$} } = 0.02$.
• Figure 5: Behavior of the metric function for different quintessence fractions. The cloud parameter was set to $\overline{a}=10^{-2}$, the Schwarzschild radius is $r_{s}=2 \times 10^{3}$ m, and the dark term was considered according to \ref{['fraction_NQ_term']}. The left panel displays the metric function for $\alpha_{ \raisebox{-\srblobdepth}{$Q$} \raisebox{-\srblobdepth}{$Q$} } = 1/2$, while the right panel depicts $f(r)$ for $\alpha_{ \raisebox{-\srblobdepth}{$Q$} \raisebox{-\srblobdepth}{$Q$} } = 1$.