Primary black-hole scalar charges and kinetic screening in $K$-essence-Gauss-Bonnet gravity

TL;DR

This work assesses how a noncanonical kinetic term in a scalar-tensor theory with a scalar-Gauss-Bonnet coupling modifies black-hole hair and kinetic screening. By combining a K-essence action with a GB coupling and a conformal coupling to matter, the authors analyze static, asymptotically-flat BHs and BHs embedded in a self-accelerating cosmology, using an eikonal expansion to derive a quartic dispersion relation for the coupled scalar-gravity system. They introduce a coordinate-invariant characteristic invariant I ≡ Tr Q/8 to diagnose stability and quantify screening, and show that time dependence can render the BH scalar charge α_BH a primary parameter, not fixed by the mass, while kinetic screening remains active near the horizon. In an explicit toy model with K(X) = η X + c_2 X^2/M^2, they demonstrate regions where α_BH > 4α/r_s^2 and where the solution is real and proto-stable, with q_crit controlling the reality of the branches; screening strengths grow with the separation of scales between the BH and cosmological horizons. Overall, the paper offers a framework to study dark-energy motivated modifications to BH phenomenology, highlighting how cosmological dynamics and kinetic screening shape observable strong-field signatures and suggesting directions for future work on more general operators and multi-body systems.

Abstract

Black holes beyond General Relativity may carry non-standard charges that impact their phenomenology. We study how the scalar charge that is induced by the scalar-Gauss-Bonnet coupling is affected by the presence of a nontrivial kinetic term $K(X)$. We discuss the corresponding kinetic screening in the asymptotically flat, static solution first. We then turn to the case where self-accelerating cosmology is driven by $K(X)$, finding that the time-dependence of the scalar field opens up the parameter space, turning the black-hole scalar charge from secondary to primary. We provide a stability analysis and a measure of the intensity of the kinetic screening from the quartic dispersion relation of the mixed scalar and gravitational modes.

Figures (4)

• Figure 1: Screening for asymptotically-flat black holes. We plot (blue) the normalized scalar gradient for a solution featuring screening in a Schwarzschild BH background. We set $\alpha_\text{BH} = 4\alpha/r_s^2= 2/5$ and $c_2/r_s^2 M^2 = -10^{24}$ for illustrative purposes. The screening radius is located at $r_k/r_s \simeq 3 \times 10^5$ and corresponds to the knee in the blue line. In dashed lines, we show guiding slopes with the expected behaviour. For reference, we also plot (orange) a solution where there are no nonlinear operators to make the gradient suppression apparent. At the location of the BH horizon (vertical line), both solutions coincide.
• Figure 2: Three branches of solutions to the scalar field equation \ref{['scalar-eom-ex']}, for $H r_s = 10^{-4}$, $\alpha_\text{BH} = 2$, and $q = 10 q_c$. For these choices all three branches are real, but only the lower branch ($\varphi' < 0$, orange) satisfies the correct near-horizon form, Eq. \ref{['near-horizon-varphip']}. Inset: Zoom-in to the vicinity of the black-hole horizon.
• Figure 3: Region in the parameter space of the solutions to the scalar field equation \ref{['scalar-eom-ex']} for $H r_s = 10^{-8}$, indicating where the regular branch with correct cosmological asymptotics is real (in color). The plot extends symmetrically to $q<0$. The proto-stable ($\mathcal{I} > 0$) and unstable ($\mathcal{I} < 0$) subregions are colored green and red, respectively. Good solutions exist for continuous values of the scalar charge $\alpha_\text{BH} > 4\alpha/r_s^2$, making it a primary charge.
• Figure 4: Value of $\mathcal{I}$ for example test-field solutions to the scalar field equation on a Schwarzschild-de Sitter background metric, with $H r_s \sim 10^{-4}, 10^{-6}, 10^{-8}$, in red, green, and blue, respectively. The scalar charge is fixed at $\alpha_\text{BH} = 2$, while the time-gradient $q$ is adjusted in each case to be mildly larger than $q_\text{crit}$ [Eq. \ref{['qcrit']}], i.e. $q=10 q_c,50 q_c,200 q_c$. The quantity $\mathcal{I}$ is closely related to the conformal factor $\Omega_S$ of the scalar-like perturbations, serving as a measure of screening. The effect can be observed to be active, i.e. $\mathcal{I} \gg 1$, within a region of size $r_k$ surrounding the black hole, which grows with the separation of scales. The peak strength $\mathcal{I}_\text{max}$ also scales similarly, but it is always located in close proximity to the black-hole horizon. The value of $\mathcal{I}$ remains large at the horizon proper.