Grey-Body Factors and Thermodynamics of Asymptotically de Sitter Black Holes in Generalized Proca Theory

Abstract: Generalized Proca theory supplements gravity with a massive vector field whose derivative self-interactions can support black holes carrying primary vector hair. In the asymptotically de Sitter branch considered here, the de Sitter scale is effective: it is generated by the vector sector rather than imposed through a bare cosmological-constant term. We compute grey-body factors and effective absorption cross-sections for massive scalar, electromagnetic, and massless Dirac test fields on this background. The transmission curves are obtained from a sixth-order WKB barrier calculation and are compared with the quasinormal-mode reconstruction; the two descriptions mostly agree, with small visible differences at lower mutlipoles. Increasing the scalar mass raises and broadens the scalar barrier, suppresses transmission at fixed frequency, and shifts efficient transmission to higher frequencies. The couplings considerably affect the grey-body factors and black hole thermodynamics.

• The paper computes grey-body factors for scalar, electromagnetic, and Dirac perturbations using sixth-order WKB methods and quasinormal-mode reconstruction, finding close agreement except near transmission thresholds and for low multipoles.
• The results show that scalar mass, the couplings α and β, and primary hair Q harden effective potential barriers and suppress transmission, while increasing c₁ softens the electromagnetic barrier and shifts transmission to lower frequencies.
• The paper links stronger Proca hair to roughly fourfold reductions in black-hole temperatures near extremality, explaining lower absorption and emission, while identifying limitations involving de Sitter cross-sections, WKB accuracy, and incomplete thermodynamic stability analysis.

Background and motivation

Grey-body factors quantify how efficiently test fields penetrate the effective potential barrier separating the event and cosmological horizons of an asymptotically de Sitter black hole, and therefore control both classical scattering and Hawking emission spectra. This paper computes them for massive scalar, electromagnetic, and massless Dirac perturbations on static, spherically symmetric black holes in generalized Proca theory (Heisenberg, 2014), in the branch where the de Sitter asymptotic is effective: the cosmological scale $\Lambda_{\rm eff} = \frac{3}{2\alpha}(\sqrt{B}-A)$ is generated by the vector-sector couplings rather than by a bare cosmological constant (Charmousis et al., 26 Mar 2026). The same Proca sector that supports primary vector hair thus sets the geometry of the static patch, tying transmission properties directly to the coupling parameters $(\alpha,\beta,\lambda,c_1,Q)$.

The work extends a recent quasinormal-mode analysis of the same geometry (Skvortsova, 12 May 2026) and complements studies of Proca-haired asymptotically flat black holes (Lütfüoğlu, 26 Aug 2025, Konoplya et al., 7 Oct 2025).

Geometry and master equations

The metric function is

$$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$

with $A=1-\beta\lambda/2$ and $B=1-\beta\lambda(1-c_1\lambda/4)$. The static patch lies between the two outer positive roots $r_h$ and $r_c$ of $f(r)=0$. All three test sectors reduce to a Schrödinger-type equation with spin-dependent potentials: the massive scalar potential $V_\mu = f(r)[\ell(\ell+1)/r^2 + f'(r)/r + \mu^2]$ contains both a curvature term and the mass scale; the electromagnetic barrier $V_1 = f(r)\ell(\ell+1)/r^2$ is purely centrifugal; and the Dirac problem involves supersymmetric partner potentials $(\alpha,\beta,\lambda,c_1,Q)$0 with superpotential $(\alpha,\beta,\lambda,c_1,Q)$1, which are isospectral under standard boundary conditions.

For the representative configuration $(\alpha,\beta,\lambda,c_1,Q)$2, the horizons sit at $(\alpha,\beta,\lambda,c_1,Q)$3 and $(\alpha,\beta,\lambda,c_1,Q)$4. The scalar barrier ($(\alpha,\beta,\lambda,c_1,Q)$5, $(\alpha,\beta,\lambda,c_1,Q)$6) peaks at $(\alpha,\beta,\lambda,c_1,Q)$7 near $(\alpha,\beta,\lambda,c_1,Q)$8, versus $(\alpha,\beta,\lambda,c_1,Q)$9 for electromagnetism and $$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$0 for Dirac — establishing the scalar barrier as the highest and broadest of the three.

Grey-body factors: WKB versus QNM reconstruction

Transmission probabilities are computed two ways: a sixth-order WKB treatment of the single barrier, $$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$1, and an independent reconstruction from fundamental quasinormal frequencies via the recently established QNM–grey-body-factor correspondence (Konoplya et al., 2024, Konoplya et al., 2024). The correspondence requires using only the Schwarzschild branch of modes; the nonoscillatory de Sitter branch cannot be reproduced by WKB and is excluded from the reconstruction. The authors also note explicitly that the correspondence assumes a single smooth dominant maximum — if the potential develops double-well or double-barrier topology, the reconstruction becomes unreliable.

The two methods agree over most of the frequency range, with visible differences concentrated near the transmission threshold where the grey-body factor rises from reflection to transmission — precisely where the local barrier-top approximation is most delicate. Disagreement is most pronounced for low multipoles and for the lightest scalar field near threshold, while for $$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$2 the two determinations are nearly indistinguishable.

The parameter scans reveal consistent monotonic trends:

Variation	Effect on barrier	Effect on $$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$3
Scalar mass $$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$4: 0 → 2	Raised, broadened	Suppressed at fixed $$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$5; onset shifted to higher $$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$6
Coupling $$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$7: 0.1 → 1.9 (EM)	Hardened	Transmission delayed
Coupling $$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$8: 0.1 → 1.9 (EM)	Hardened	Strongest monotonic suppression among EM scans
Coupling $$f(r)=1-\frac{2(M-Q)}{r}+\frac{r^2}{2\alpha}\left(A-\sqrt{B+\frac{8\alpha}{r^3}\left[Q+\frac{\lambda(M-Q)}{2\beta}\right]}\right),$$9: 1.1 → 3.5 (EM)	Softened	Onset shifted to lower frequencies
Hair $A=1-\beta\lambda/2$0: 0 → 1.03 (Dirac)	Hardened	Neutral case most transmissive

The opposite behavior of $A=1-\beta\lambda/2$1 versus $A=1-\beta\lambda/2$2 is notable: increasing $A=1-\beta\lambda/2$3 enlarges the effective de Sitter contribution yet softens the electromagnetic barrier, reversing the ordering seen for the other couplings. Since the electromagnetic and Dirac barriers depend only on geometry and angular structure, these sectors isolate genuine background effects from field-mass effects.

Absorption cross-sections

Because no plane-wave region exists at spatial infinity in de Sitter, the paper defines a generalized static-patch absorption cross-section measuring how efficiently radiation incident from the cosmological horizon is captured by the event horizon, built from partial cross-sections $A=1-\beta\lambda/2$4 summed over propagating multipoles. These inherit the barrier physics directly. Comparing the neutral configuration against the near-extreme charged case $A=1-\beta\lambda/2$5, the larger hair value reduces both the dominant low-$A=1-\beta\lambda/2$6 partial-wave peaks and the high-frequency oscillation level of the total cross-section. Thus primary hair does not merely relocate horizons; it lowers the effective capture efficiency of radiation entering from the cosmological side.

Thermodynamics

Three temperatures are distinguished following the Schwarzschild–de Sitter literature [gr-qc/9606052]: the bare event-horizon temperature $A=1-\beta\lambda/2$7, the cosmological-horizon temperature $A=1-\beta\lambda/2$8, and the Bousso–Hawking normalized temperature $A=1-\beta\lambda/2$9 evaluated at the geodesic static observer where $B=1-\beta\lambda(1-c_1\lambda/4)$0. For the neutral reference point, $B=1-\beta\lambda(1-c_1\lambda/4)$1.

Increasing the hair $B=1-\beta\lambda(1-c_1\lambda/4)$2 drives the event horizon inward and drops the black-hole temperatures sharply — at $B=1-\beta\lambda(1-c_1\lambda/4)$3, $B=1-\beta\lambda(1-c_1\lambda/4)$4, roughly a fourfold reduction relative to the neutral case — while $B=1-\beta\lambda(1-c_1\lambda/4)$5 and $B=1-\beta\lambda(1-c_1\lambda/4)$6 remain nearly unchanged. This temperature collapse directly explains the suppressed absorption and emission channels at high charge. Among the couplings, $B=1-\beta\lambda(1-c_1\lambda/4)$7 has a strong nonmonotonic influence on $B=1-\beta\lambda(1-c_1\lambda/4)$8 because it alters both the near-horizon slope and the normalization point; increasing $B=1-\beta\lambda(1-c_1\lambda/4)$9 or $r_h$0 raises $r_h$1 while lowering $r_h$2. Throughout the scanned range, $r_h$3, as required by the normalization factor.

Limitations and open questions

Several caveats are stated plainly in the paper. The sixth-order WKB method is least reliable at very low multipoles, and the QNM-based reconstruction inherits the assumptions of a single smooth dominant maximum and the restriction to the Schwarzschild mode branch. The absorption cross-section lacks a standard plane-wave interpretation in de Sitter and should be read as a static-patch quantity. The thermodynamic analysis reports temperatures but does not compute entropy, phase structure, or stability bounds from the kinetic/sound-speed conditions known to govern viable generalized Proca branches (Felice et al., 2016, Felice et al., 2016). Finally, the Hawking-radiation program is incomplete: full Bose–Einstein and Fermi–Dirac emission sums across the parameter space, and direct time-domain scattering checks of the WKB curves, remain open questions raised but not answered here.

Conclusion

This paper provides a systematic grey-body-factor and absorption analysis for three test-field spins on effectively de Sitter generalized Proca black holes, cross-validated by sixth-order WKB and QNM-based reconstruction. The central quantitative findings are that scalar mass, the couplings $r_h$4 and $r_h$5, and the primary hair $r_h$6 all harden the barrier and suppress transmission, whereas $r_h$7 softens it; that hair-driven cooling of the black-hole horizon by roughly a factor of four near extremality strongly suppresses capture; and that the two semiclassical determinations of $r_h$8 agree except near transmission thresholds and at low multipoles. The results position grey-body factors as a sensitive diagnostic of Proca-hair parameters complementary to quasinormal spectra.