Spherical scalar collapse in a type-II minimally modified gravity

We investigate the spherically symmetric gravitational collapse of a massless scalar field in the framework of a type-II minimally modified gravity theory called VCDM (where V replaces $\Lambda$ in the $\Lambda$CDM abbreviation). This theory propagates only two local physical degrees of freedom (DoF) supplemented by the so-called instantaneous (or shadowy) mode. Imposing asymptotically flat spacetime in the standard Minkowski time slicing, one can integrate out the instantaneous mode. Consequently, the equations of motion reduce to those in general relativity (GR) with the maximal slicing. Unlike GR, however, VCDM lacks 4D diffeomorphism invariance, and thus one cannot change the time slicing that is preferred by the theory. We then numerically evolve the system to see if and how a black hole forms. For small amplitudes of the initial scalar profile, we find that its collapse does not generate any black hole, singularity or breakdown of the time slicing. For sufficiently large amplitudes, however, the collapse does indeed result in the formation of an apparent horizon in a finite time. After that, the solution outside the horizon is described by a static configuration, i.e., the Schwarzschild geometry with a finite and time-independent lapse function. Inside the horizon, on the other hand, the numerical results indicate that the lapse function keeps decreasing toward zero so that the central singularity is never reached. This implies the necessity for a UV completion of the theory to describe physics inside the horizon. Still, we can conclude that VCDM is able to fully describe the entire time evolution of the Universe outside the black hole horizon without knowledge about such a UV completion.