Radiative falloff in Schwarzschild-de Sitter spacetime

TL;DR

The study investigates how a scalar field propagates in Schwarzschild–de Sitter spacetime, revealing a three-stage evolution: an early Schwarzschild-like regime, a transitional phase with a power-law tail that shifts to exponential decay, and a late-time regime governed by the de Sitter structure. Central to the result is a curvature-coupling constant $oldsymbol{\xi}$, which yields a critical value $oldsymbol{\xi_c=rac{3}{16}}$ separating monotonic exponential decay from oscillatory late-time behavior; the decay rate is captured by $oldsymbol{p = l+rac{3}{2}-rac{1}{2}igl(9-48oldsymbol{\xi}igr)^{1/2}+O(r_e/r_c)}$ with $oldsymbol{oldsymbol{\kappa_c}=1/r_c}$. The analysis combines numerical evolution of the wave equation with an analytical treatment of the Green’s function, showing that late-time dynamics are dictated by the cosmological horizon rather than near-horizon details, and extends to electromagnetic and gravitational perturbations. The results underscore a universal exponential tail dictated by de Sitter asymptotics and provide explicit conditions under which oscillations arise in the late-time signal. This has implications for radiative observables in black-hole spacetimes embedded in expanding universes and informs the interpretation of late-time decay in cosmological settings.

Abstract

We consider the time evolution of a scalar field propagating in Schwarzschild-de Sitter spacetime. At early times, the field behaves as if it were in pure Schwarzschild spacetime; the structure of spacetime far from the black hole has no influence on the evolution. In this early epoch, the field's initial outburst is followed by quasi-normal oscillations, and then by an inverse power-law decay. At intermediate times, the power-law behavior gives way to a faster, exponential decay. At late times, the field behaves as if it were in pure de Sitter spacetime; the structure of spacetime near the black hole no longer influences the evolution in a significant way. In this late epoch, the field's behavior depends on the value of the curvature-coupling constant xi. If xi is less than a critical value 3/16, the field decays exponentially, with a decay constant that increases with increasing xi. If xi > 3/16, the field oscillates with a frequency that increases with increasing xi; the amplitude of the field still decays exponentially, but the decay constant is independent of xi.

Figures (5)

• Figure 1: Absolute value of the wave function $\psi_l(t,r)$ as a function of time $t$, evaluated at $r^* = 10$ in Schwarzschild spacetime ($r_e = 1$) and SdS spacetime ($r_e = 1$ and $r_c = 2000$). The cases $l=0$ and $l=1$ are considered, and the wave functions are plotted on a log-log scale. In such a plot, a straight line indicates power-law behavior, and a change of sign in the wave function is represented by a deep trough. We see that the early portion of $\psi_1$ is oscillatory, and that for SdS spacetime, $\psi_0$ changes sign at $t \sim 260$.
• Figure 2: Absolute value of the wave function $\psi_l(t,r)$ as a function of time $t$, evaluated at $r^* = 10$ in SdS spacetime ($r_e = 1$ and $r_c = 100$). The cases $l=0, 1, 2$ are considered, and the wave functions are plotted on a semi-log scale. In such a plot, a straight line indicates exponential behavior. Notice that the final change of sign of the wave function occurs at $t \sim 50$ for $l=0$, $t \sim 190$ for $l=1$, and $t \sim 220$ for $l=2$. Notice also that the numerical integration becomes noisy when $|\psi_l|$ drops below $10^{-14}$.
• Figure 3: Absolute value of the wave function $\psi_0(t,r)$ as a function of time $t$, evaluated at $r^* = 10$ in SdS spacetime ($r_e = 1$ and $r_c = 100$). Several values of $\xi$ are considered, in the interval between $\xi = 0$ and $\xi = \frac{1}{2}$. The wave functions are plotted on a semi-log scale. The noteworthy features are these: (i) For $\xi < \xi_c$, the wave function decays exponentially, with a decay constant that increases with increasing $\xi$; (ii) for $\xi > \xi_c$, the wave function still decays exponentially, but with a decay constant that no longer varies with $\xi$; (iii) for $\xi > \xi_c$, the wave function oscillates, with a frequency that increases with $\xi$.
• Figure 4: Absolute value of the wave function $\psi_0(t,r)$ as a function of time $t$, evaluated at $r^* = 0.5$ in pure de Sitter spacetime (with parameter $r_c = 1$). Several values of $\xi$ are considered, in the interval between $\xi = 0$ and $\xi = \frac{1}{2}$. The wave functions are plotted on a semi-log scale. These plots show the same features as in the preceding figure.
• Figure 5: Absolute value of the wave function $\psi_1(t,r)$ as a function of time $t$, evaluated at $r^* = 0.5$ in pure de Sitter spacetime (with parameter $r_c = 1$). Several small values of $\xi$ are considered, together with the special case $\xi = 0$. The plots make it quite clear that the late-time behavior associated with $\xi \ll 1$ is qualitatively different from the behavior associated with $\xi = 0$. As explained in the text, this qualitative change of behavior is caused by pole cancellation in $1/W$.