de Sitter Exponential Decay and Spectral Gaps

A positive cosmological constant changes the endpoint of the radial scattering problem. In the static region of a subextremal Schwarzschild–de Sitter black hole, both ends are horizons, so the effective potential is exponentially short-range in the tortoise coordinate. For a scalar field, the late-time state at fixed radius is consequently controlled by isolated resonances rather than the asymptotically flat zero-frequency branch cut. The qualification “state” matters: a minimally coupled scalar has a constant zero resonance, whereas the field after subtracting that stationary contribution decays exponentially under the spectral and data hypotheses below.

Required background. PDEs and Green Functions for Gravitation supplies retarded resolvents, contour deformation, and local-energy estimates.

Helpful background. Black-Hole Geometry and Mechanics supplies the two-horizon static region; Formulations of General Relativity distinguishes field amplitudes from energy norms; Causality and Global Structure supplies horizon-regular coordinates; and Black-Hole Perturbation Theory supplies quasinormal boundary conditions.

Two horizons make the radial potential short-range

Write the four-dimensional Schwarzschild–de Sitter metric as

$$ds^2=-F(r)dt^2+F(r)^{-1}dr^2+r^2d\Omega^2, \qquad F(r)=1-\frac{2\mu}{r}-\frac{\Lambda r^2}{3}.$$

Here $\mu=GM$ is the geometric mass. For subextremal parameters, $F$ has a black-hole root $r_b$ and a cosmological root $r_c>r_b$. The static exterior is $r_b<r<r_c$. Define $dr_*/dr=F^{-1}$ and positive surface gravities $\kappa_i=\lvert F'(r_i)\rvert/2$. Then $r_*\to-\infty$ at $r_b$ and $r_*\to+\infty$ at $r_c$.

For a curvature-coupled scalar

$$(\Box_g-\xi R)\Phi=0, \qquad R=4\Lambda,$$

the decomposition $\Phi=r^{-1}\psi_{\ell m}(t,r)Y_{\ell m}(\theta,\phi)$ gives

$$\left(-\partial_t^2+\partial_{r_*}^2-V_\ell\right)\psi_{\ell m}=0, \qquad V_\ell=F\left[\frac{\ell(\ell+1)}{r^2} +\frac{F'}{r}+\xi R\right].$$

Because $F$ vanishes linearly at each nondegenerate horizon, $V_\ell=O(e^{2\kappa_b r_*})$ as $r_*\to-\infty$ and $V_\ell=O(e^{-2\kappa_c r_*})$ as $r_*\to+\infty$. This is the decisive contrast with Schwarzschild’s inverse-power far potential. With time dependence $e^{-i\omega t}$, a retarded mode is regular as $e^{-i\omega v}$ at the future event horizon, $v=t+r_*$, and as $e^{-i\omega u}$ at the future cosmological horizon, $u=t-r_*$. These conditions replace an outgoing condition at spatial infinity. At exceptional Frobenius frequencies, select the analytic continuation of the retarded branch: smoothness alone need not distinguish the two local solutions.

Meromorphic response and the spectral gap

Let

$$P_\ell(\omega)=\partial_{r_*}^2+\omega^2-V_\ell$$

with the two horizon-regular boundary conditions just stated. Its continued inverse has isolated poles $\omega_j$, the quasinormal resonances. In the regime where a resonance expansion and a resonance-free strip are available, deforming the retarded inverse-transform contour yields schematically

$$\psi_\ell(t,r) =\sum_{-\operatorname{Im}\omega_j<\nu} c_j(r)e^{-i\omega_j t}+O(e^{-\nu t}).$$

Thus the relevant gap is not simply the least damping rate in a published frequency table. For the chosen data and observable, define

$$\alpha =\inf\left\{-\operatorname{Im}\omega_j: \omega_j\ne0,\;c_j\ne0\right\}.$$

If a simple excited pole $\omega_*$ attains this gap, its real-field partner is $-\overline{\omega_*}$, not $\overline{\omega_*}$. Their combined contribution is proportional to $e^{-\alpha t}\cos(\operatorname{Re}\omega_*t+\delta)$ when both residues are nonzero. The two poles have the same negative imaginary part. A pole of higher order adds a polynomial in $t$; an infimum need not be attained. Under the stated contour and remainder bounds, an exponential estimate can instead use a rate $0\lt\alpha'\lt\alpha$ below the gap. Symmetry, initial data, or the measured field component can set $c_j=0$, so the observable gap can exceed the geometric gap. At fixed $r_b<r<r_c$ the time is the static $t$; on the event horizon it is advanced time $v$, and on the cosmological horizon it is retarded time $u$.

For slowly rotating Kerr–de Sitter, the scalar stationary resolvent is meromorphic and the local wave energy decays exponentially after its zero resonance is removed. The theorem assumes sufficiently small rotation and compactly supported, adequately regular initial data in a compact subregion; it is not a formula for the exact least-damped frequency (Dyatlov 2011, Theorems 2 and 4–6, pp. 2–4).

The least pole and the exceptional constant mode

For a small but nonzero Schwarzschild–de Sitter black hole, $0\lt r_b/r_c\ll1$, the leading late-time scalar exponent is

$$p=\ell+\frac32-\frac12\sqrt{9-48\xi} +O\!\left(\frac{r_b}{r_c}\right), \qquad \Phi_\ell\sim e^{-p\kappa_c t}.$$

For $0\le\xi<3/16$, $p$ is real and the dominant contribution is nonoscillatory. For $\xi>3/16$, the square root is imaginary and the same pair of poles gives, at leading small-$r_b/r_c$ order,

$$\Phi_\ell\sim e^{-(\ell+3/2)\kappa_c t} \cos\!\left[\frac{\kappa_c t}{2}\sqrt{48\xi-9}+\delta\right].$$

Direct characteristic evolutions at fixed radius and on both horizons recover this transition, with $t$ replaced by $v$ or $u$ on the corresponding horizon (Brady et al. 1999, §§ II–IV, pp. 2–8). The displayed formula is an analytic small-black-hole approximation, not a universal spectrum for arbitrary Schwarzschild–de Sitter parameters.

The exact pure-de Sitter limit requires care. At $\xi=0$, all nonzero candidate poles in this Green function cancel for every $\ell$; only the $\ell=0$, $\omega=0$ pole survives. At $\xi=1/6$, all candidate poles cancel for every $\ell$. Thus the exact-limit late-time field cannot be inferred by simply setting $r_b=0$ in the small-black-hole exponential. The cancellation is absent for the small nonzero black holes considered here Brady et al. 1999, §§IV D and IV G, pp. 7 and 9, Eq. (4.28), Open PDF.

At minimal coupling, $\xi=0$, the leading small-black-hole formula gives $p=\ell$. Hence the monopole has $p=0$: generic scalar data can approach a nonzero constant. Higher multipoles decay, and the monopole after subtraction of its constant part also decays. This is precisely why rigorous statements are phrased as decay toward a constant or require zero projection onto the zero resonance. The constant field has zero derivative energy, so exponential local-energy decay does not contradict a nonzero limiting field amplitude.

What the exponential statement does and does not say

The result concerns linear fields on a fixed, nondegenerate de Sitter black hole. Its rate depends on the field equation, angular sector, boundary regularity, background parameters, data projection, and observation region. It should not be extrapolated through an extremal limit in which surface gravities or resonance gaps may close. Nor does a scalar theorem by itself fix the decay of every electromagnetic or gravitational gauge-invariant component.

Within this chapter, the point is the analytic mechanism: changing the far end from asymptotically flat infinity to a cosmological horizon replaces the zero-frequency continuum responsible for Price powers by a meromorphic resonance problem. Full nonlinear decay and stability theorems belong with mathematical relativity, and fitting late-time observations remains a downstream problem.

An independent numerical check should solve the same radial boundary-value problem and characteristic initial-value problem. First locate the least-damped nonzero zero of the Wronskian while increasing radial precision. Then evolve two smooth compact pulses, subtract the fitted constant monopole when $\xi=0$, and compare the remaining logarithmic slope with $-\operatorname{Im}\omega_*$. Extract separately at fixed radius, the event horizon in $v$, and the cosmological horizon in $u$. Agreement of the three damping rates after the coordinate replacement tests the resonance interpretation; agreement only before subtraction of a drifting constant does not. Varying the initial pulse also tests whether an accidentally small residue has exposed the next pole rather than the true gap.

Exercise: isolate the zero resonance

For $\xi=0$ and $\ell=0$, verify directly that $\Phi=C$ is a solution. Explain why a theorem can simultaneously say that $\Phi$ approaches $C$ and that its local derivative energy decays exponentially.

Solution

For any constant $C$, $\nabla_a C=0$, and therefore $\Box_g C=0$. It is the $\ell=0$, $\omega=0$ resonant state. Write a general solution as $\Phi=C+\widetilde\Phi$, where the coefficient $C$ is the data’s projection onto that state. The nonzero-resonance part obeys $\widetilde\Phi=O(e^{-\alpha' t})$ locally for an admissible rate $0\lt\alpha'\lt\alpha$ when the positive gap and the required resonance-expansion remainder bounds are available. Using a rate strictly below the gap also absorbs possible polynomial factors from higher-order poles. Since every derivative of $C$ vanishes, an energy constructed from $\nabla\Phi$ equals the energy of $\nabla\widetilde\Phi$ and decays even though the undifferentiated field tends to $C\ne0$. Subtracting $C$, or imposing zero projection onto the resonance, gives exponential decay of the field itself.

References

• Brady, Patrick R., Chris M. Chambers, William G. Laarakkers, and Eric Poisson. “Radiative Falloff in Schwarzschild–de Sitter Spacetime.” Physical Review D 60 (1999): 064003. DOI. Open PDF
• Dyatlov, Semyon. “Quasi-Normal Modes and Exponential Energy Decay for the Kerr–de Sitter Black Hole.” Communications in Mathematical Physics 306 (2011): 119–163. DOI. Open PDF