In shift-symmetric Horndeski theories, a static and spherically symmetric black hole can support linearly time-dependent scalar hair. However, it was shown that such a solution generically suffers from ghost or gradient instability in the vicinity of the horizon. In the present paper, we explore the possibility to avoid the instability, and present a new example of theory and its black hole solution with a linearly time-dependent scalar configuration. We also discuss the stability of solutions with static scalar hair for a special case where nonminimal derivative coupling to the Einstein tensor appears.

We perform an analytical and numerical study of static spherically symmetric solutions in the context of the Brans–Dicke-like cosmological model by Elizalde et al (2004 Phys. Rev. D 70 043539 ) with an exponential potential. In this model the phantom regime arises without the appearance of any ghost degree of freedom due to the specific form of coupling. For certain parameter ranges the model contains a regular solution that we interpret as a wormhole in an otherwise de Sitter Universe. We put several bounds on the parameter values: . The numerical solution could mimic the Schwarzschild one, so the original model is consistent with astrophysical and cosmological observational data. However, differences between our solution and the Schwarzschild one can be quite large, so black hole candidate observations could probably place further limits on the value.

We discuss a Brans-Dicke model with a cosmological constant, a negative value of the $w$ parameter, and an arbitrary (in general nonvanishing) scale factor at the big bang. The Friedmann equations for a flat universe are considered. The current observational values for Hubble constant $H_0$ and deceleration parameter $q_0$ play the role of initial conditions. We follow the approach of Uehara and Kim in order to solve field equations analytically. In K. Uehara and C. W. Kim [ Phys. Rev. D 26, 2575 (1982)] only positive values of $w$ were considered; we extend the study to a complete set of possible $w$ values. Our main result is that the scale factor (during its evolution back in time direction) may not vanish, unlike in the standard $\Lambda \mathrm{CDM}$ case. In other words, the considered model demonstrates a cosmological bounce instead of the initial singularity. The famous formula (24), that leads to the bounce, is valid only for the dust-filled universe with $p=0$ and, therefore, is not adequate for the early universe hot stage when the bounce happens. So, our results are qualitative in nature and must be used to obtain initial values for the hot stage of the Universe.

The geodesic equations for black hole solutions in the scalar–tensor Horndeski gravity model with non-minimal kinetic coupling have been investigated. The ranges of model parameters admitting the existence of bounded orbits have been determined. Constraints on the model parameters providing agreement of the model with the observational data on the accretion and motion of bodies in the Solar system have been obtained.

In this review, we focus our attention on scalar-tensor gravity models and their empirical verification in terms of black hole and wormhole physics. We focus on black holes, embedded in an expanding universe, describing both cosmological and astrophysical scales. We show that in scalar-tensor gravity it is quite common that the local geometry is isolated from the cosmological expansion, so that it does not backreact on the black hole metric. We try to extract common features of scalar-tensor black holes in an expanding universe and point out the issues that are not fully investigated.