Stable black hole solutions with cosmological hair

Abstract

Dynamical dark energy theories generically introduce a time-dependent field that causes the accelerated expansion of the Universe on large scales. When embedding black hole solutions in such a cosmological space-time, this time dependence naturally gives rise to cosmological hair, i.e. the local black hole physics is no longer controlled by just the mass and spin of the black hole, but also impacted by the dark energy field. However, known such solutions are unstable. Focusing on the cubic Galileon as a concrete and illustrative example, we discuss the restrictions imposed on physical solutions by their regularity and stability in detail. We explicitly derive regular and stable solutions, that both recover the desired cosmological long-range behaviour and give rise to well-behaved short-range dynamics around black holes. We show how the nature of the scalar hair around these local black hole solutions encodes cosmological information, highlighting novel and tantalising prospects of directly probing cosmological dynamics with black hole observations.

Figures (6)

• Figure 1: Here we show numerical solutions to \ref{['eq:EoMsDimensionless1']}-\ref{['eq:EoMsDimensionless3']} for a specific example case, where we have chosen $\alpha_1=10^7$ and $\alpha_2=8\times10^{-15}$\ref{['eq:DimensionlessParameters']} (see main text for a discussion of the chosen values). Going from left to right, we compare the full numerical solutions for $h(x),f(x)$ and $\Xi(x)$ together with their analytical short- and long-range approximations (\ref{['eq:ShortRangeSolutions']},\ref{['eq:LongRangeSolutions']}), where $x \equiv r/r_s$. $x = 1$ and $x = r_c/r_s$ denote black hole and cosmological horizons, respectively. $x=(r_c/r_s)^{2/3}$ is the scale where the short-range solution is generically expected to become inaccurate (see appendix \ref{['sec:AppendixShortRangeLimitAccuracy']}). Top row: Here we start the evolution in the $\Xi_+$ branch \ref{['eq:BranchDefinitions']} at short scales and integrate out to large $x$. The one parameter freedom in this solution, captured by the parameter $\beta$ in \ref{['eq:ShortRangeSolutions']}, is then fixed via the shooting method: scanning over initial conditions of $h'$ to match the solution to the desired (cosmological) long-range solution. We find an excellent fit for $\beta=1.53$. Bottom row: Here we instead start in the $\Xi_-$ branch on short scales. No value of $\beta$ could be found here to connect to the desired cosmological limit. Instead the solution on large scales generically diverges away from that desired limit, as is illustrated here with a solution for $\beta = 1.53$ in this branch. In section \ref{['sec:BranchingStructure']} we discuss how this can be understood in terms of the overall branching structure of solutions across all $x$ values.
• Figure 2: Here we show the numerical solution for $\Xi(x)$ from the bottom row of figure \ref{['fig:NumericalSolutions']}, integrated up to large $x$, as well as the $\Xi_\pm(x)$ branches calculated from the solutions for $f(x)$ and $h(x)$. The solution fully follows the $\Xi_-$ branch as expected and we find no branch points anywhere in this range.
• Figure 3: Here we show a numerical solution for the specific case highlighted at the end of section \ref{['sec:NumericalSolutions']}, namely a choice of $\alpha_i$ such that $q = q_0$ for which $\beta = 1$ is a solution and hence the respective limits of a Schwarzschild-de Sitter spacetime are recovered at short and long distances. We show such a numerical solution, as well as the corresponding small and large scale analytic approximations for the example values $\alpha_1=10^7$, $\alpha_2=\frac{2}{3}\times10^{-14}$ here. Left plot: Given the asymptotic analytic behaviour, one may expect that Schwarzschild-de Sitter is a good fit throughout all $x$ values. Here we see that this is indeed appears to be the case when looking at the overall form of the full numerical solution. However, while the solution for the metric superficially resembles Schwarzschild-deSitter, there are important differences in the branching structure that can be understood when looking at the discriminant $\Delta$\ref{['eq:BranchDefinitions']}. These will be crucial in light of the stability constraints related to the different branches of solutions in section \ref{['sec:BlackHolePerturbations']}. Right plot: Here we illustrate the branching structure of the solution by plotting the discriminant $\Delta$ for the numerical solution shown in the left plot, $\Delta_{\rm num}$, as well as for a true Schwarzschild-deSitter solution, $\Delta_{\rm SdS}$. At small and large scales, the numerical solution closely follows the Schwarzschild-deSitter solution. In particular we recover a common branch point close to $r_c$. However, on intermediate scales the two solutions deviate from each other, with the crucial difference that the numerical solution has an additional (odd multiplicity) branch point. This induces an additional switch of branch, which means that in the numerical solution the (unstable) short-range $\Xi_+$ branch connects to the cosmological limit instead of the desired $\Xi_-$ branch, rendering this solution unstable altogether (see section \ref{['sec:BlackHolePerturbations']}).
• Figure 4: The coefficients $a_0$ and $a_1$ in the kinetic part of the Hamiltonian density \ref{['eq:ExplicitlyStableHamiltonianQuasiStationary']} in explicitly stable coordinates for a specific value of the accretion rate ($\alpha_4\sqrt{\beta}/\alpha_1=-1+10^{-16}$). For boundedness from below these must be positive. This is the case up to some maximum radius, $r_\text{max}=r_s x_\text{max}$ with $x_\text{max}$ defined in \ref{['eq:xMax']}. Therefore, for this choice of $\alpha_4$, $r_\text{max}\gg(r_c^2r_s)^{1/3}$, the radius up to which our stability analysis with approximate short-range solutions is accurate, which means the approximate solutions in the $\Xi_+$ branch are stable in the region that we have studied. Note that fixing $\alpha_4\sqrt{\beta}/\alpha_1=-1$ takes $r_\text{max}\rightarrow\infty$.
• Figure 5: The quasi-stationary numerical solution obtained using the shooting method described in section \ref{['sec:NumericalSolutions']} for $\alpha_1=10^7$, $\alpha_2=8\times10^{-15}$, $\alpha_4=-8084527.4$ and $\Xi$ starting off in the $\Xi_+$ branch. We find that the solution connects to the correct cosmological asymptotes at large scales for $\beta=1.5299975$. For these values, the solution is stable against perturbations in the short-range regime.