Horndeski theory meets the McVittie solution: A scalar field theory for accretion onto cosmological black holes

We show that the generalized McVittie spacetime, which represents a black hole with time-dependent mass in an expanding universe, is an exact solution of a subclass of the Horndeski family of actions. The heat-flow term responsible for the energy transfer between the black hole and the cosmological background is generated by the higher-order kinetic gravity braiding term, which generalizes the cuscuton action that yields McVittie with constant mass as a solution. Finally, we show that this generalization can be understood in terms of a duality realized by a disformal transformation, connecting the cuscuton field theory to an extension of the Horndeski action which does not propagate any scalar degrees of freedom. Our finding opens a novel window into studies of nontrivial interactions between dark energy/modified gravity theories and astrophysical black holes.

Figure 1

Examples of the causal structure of the McVittie and generalized McVittie metrics, depicting the apparent horizons and surfaces of constant time. A Schwarzschild–de Sitter extension is patched at the left of each diagram. In both examples, the expansion is given by Eq. (31). Note that, in this class of metrics, the null infinity ${\mathcal{I}}^{+}$ is spacelike. (a) McVittie metric (15) with constant mass. (b) Generalized McVittie with increasing mass. The mass function is of the form $m(t)=m_0[\frac{1}{2}+\text{tanh}(t-t_0)]$.