Static, spherically symmetric objects in Type-II minimally modified gravity

Abstract

Static, spherically symmetric solutions representing stars made of barotropic perfect fluid are studied in the context of two theories of type-II minimally modified gravity, VCDM and VCCDM. Both of these theories share the property that no additional degree of freedom is introduced in the gravity sector, and propagate only two gravitational waves besides matter fields, as in General Relativity (GR). We find that, on imposing physical boundary conditions on the Misner-Sharp mass of the system, the solutions in V(C)CDM exactly coincide with the ones in GR, namely they also satisfy the Tolman-Oppenheimer-Volkoff equation. I. INTRODUCTIONWe live in a particular era in cosmology due to the remarkable precision achieved in various cosmological observations. As a matter of fact, since the discovery of gravitational waves [1], General Relativity (GR) has been confirmed by yet another independent experimental data. GR then appears more and more to be the theory of classical gravitational interactions. This picture is quite astonishing for a theory that was introduced more than one hundred years ago.To this beautiful picture and powerful result of theoretical physics, cosmological observations are adding to it several discrepant results. One of the most embarrassing among them is the tension among different measurements of the expansion rate of our universe today, which is called the Hubble parameter H 0 . Although this is one of the oldest and elementary measurements in cosmology, on assuming GR to hold at all times after the big bang, we find that different data sets find different values for this unique observable. The only ways out: experiments are wrong, statistical analysis is not correct, or, surprisingly, the assumed underlining theory does not hold.If GR is not suspicious in the context of cosmology, then either the cosmological data or the statistical analysis in determining H 0 cannot be trusted. Evidently, this statement (if correct) needs then to be completed by finding the reason why the data or/and the analysis are wrong. As long as we do not have a clear explanation for it, the other possibilities need to be explored. One could also conclude that the classical theory of gravity as described by GR is correct; what is missing is a proper description of the matter present in the universe. Then, we need to understand better what kind of matter can be responsible for the different values of H 0 without contradicting with all experiments and observations so far. On top of that, one would like to give possible predictions from the requirement that the extra matter responsible for the tension would be "visible" only in cosmology and have no (or negligible) effects otherwise, e.g. at solar system scales.Recently more cosmological data have been adding to this obscure picture another unsolved puzzle [2][3][4]. In the context of GR, these data sets tend to give a prediction for the growth rate of the structure which is lower than and in tension with the predictions coming from early-times data sets [5]. On top of that...

VCDM and Cuscuton

“…In summary this shows that all GR solutions, written in the constant-K slicing (whenever this choice of slicing is allowed), are also solutions of VCDM. An example of this case is given in [29], where the extrinsic curvature of the solutions is vanishing, finding indeed that the static profile of spherically symmetric stars solutions are also solutions of VCDM. 5 Here we can integrate out the Lagrange multiplier λ because its equation of motion is purely algebraic, getting a Lagrangian equivalent to the VCDM Lagrangian.…”

“…From the previous investigations of different spherically symmetric solutions of VCDM theory (see e.g. [27,29]), we know that there exist solutions inside VCDM which are the same as those of GR, provided that we set appropriate physical boundary conditions for the shadowy mode, e.g., the finiteness of the (generalized) Misner-Sharp mass for a spherically symmetric isolated compact gravitational body/system. Nevertheless, even though these GR/VCDM solutions do exist, VCDM theory, by construction, is different from GR.…”

“…The spherically symmetric static star solutions were also studied in the context of the VCDM theory 2 . It was shown that once we fix the physical boundary for the Misner-Sharp mass of the system the solution exactly matches those of GR [29]. The cosmology of the VCDM theory was also explored and it was shown that the H 0 tension can be reduced/addressed within this theory [30], since the theory allows for general dynamics for H(z) (with H(z) > 0) without introducing unstable/ghost degrees of freedom.…”

Static and spherically symmetric general relativity solutions in Minimal Theory of Bigravity

“…As a consequence, if we consider a static and spherically symmetric background, the solutions which satisfy the imposed ansatz for MTBG, having K = −3b 0 = constant, coincide with the general solutions given in [44] for a theory of minimally modified gravity, namely VCDM and VCCDM *3 [45][46][47][48], which can now be written as…”

“…We note that the constraint ( 12) is trivially satisfied for the spatially-flat coordinates (47). We also assume that the matter in the physical and fiducial sectors ( 43) is given by the perfect fluids with the energy-momentum tensors, Eq.…”