I have investigated many kinds of modified gravity theories and considered the various applications based on different approaches. I briefly introduce my research accomplishments with simple reviews of topics related to each work.
The scalar-tensor theory is the gravitational theory coupling with scalar field. The simplest case is expressed as the GR with the minimally coupling canonical scalar field. The most general scalar-tensor theory with the single scalar field is known as the Horndeski theory, which produces the second-order equation of motion.
The sequestering mechanism was proposed as one possible way to solve the cosmological constant problem, which cancels a large amount of the quantum vacuum energy but leaves the tiny energy for the observed DE. The schematic idea is as follows: the energy-momentum tensor is divided into two parts, the classical and quantum parts. The new constraints are introduced so that the quantum contribution to DE is canceled in the equation of motion, and that the effective cosmological constant consists of the classical contribution averaged by the whole Universe.
It was shown that the effective cosmological constant could not explain the observed accelerated expansion of the Universe because it gives the negative contribution to the DE. The other candidate for DE is inevitable, and then, the scalar field in the scalar-tensor theory can be utilized to compensate for the sequestering mechanism. We made a theoretical prediction for the ratio of the DE to other components in the current Universe based on the scenario of the sequestering mechanism.
The parameter region where one can explain the observed DE-matter ratio was specified in the scalar-tensor theory. This work is the first to show that the sequestering mechanism works property in the modified gravity although the original sequestering mechanism was formulated in the GR.
The action of F(R) gravity is replaced by the function of the Ricci scalar instead of the Einstein-Hilbert action of GR. The functional degrees of freedom in F(R) gravity is written in terms of the scalar field, and then, it is known that the F(R) gravity is equivalent to the scalar-tensor theory at the classical level.
From the viewpoint of cosmology, the new field corresponding to the modification of gravity is responsible for the DE. In the F(R) gravity, the scalar field degree of freedom arises, and it is responsible for the DE at the classical level. Moreover, it was proposed that the new field corresponding to the modification can be a DM candidate when we treat this new field as a new particle. When the oscillation of this scalar field around the potential minimum is quantized in the F(R) gravity, we may get a particle picture of the new scalar field, called the scalaron.
The scalaron is a massive scalar and singlet to the SM, and its coupling to the SM particles are very weak of its gravitational origin. In this point, the scalaron can be a CDM candidate. Unlike the most of DM candidates in beyond-SM, however, the scalaron does not have extra symmetry and can decay to the SM particles. We calculated the decay width in the framework of the QFT and discussed the lifetime of scalaron.
In the standard sense, the mass of scalar field should be naively at the Hubble scale. Such a light scalar field is, however, easily detected an excluded by the observation because the Compton wavelength is very large. On the other hand, if the scalar field is massive enough for its propagation to be frozen out, the Compton wavelength is small, and the scalar field can avoid the observational constraints. However, such a heavy scalar field is irrelevant for the cosmology and gives the inconsequential effect for the DE problem. In order to solve this conflict between two different requirements in different scales, the screening mechanism was proposed.
In the chameleon mechanism which is one of the screening mechanisms, the scalar field potential depends on the energy-momentum tensor of matter fields. This results in that the mass of scalar field changes depending on the matter fields surrounding the scalar field. If the potential of scalar field could be chosen so that the mass increases in high-density region and decreases in the low-density region, the chameleonic scalar field makes the theory relevant in two different regions: the cosmological scale and Solar-System scale.
The viable models of F(R) gravity possess the chameleon mechanism for the consistency with the Solar-System constraints. And also, it is naively expected that the decay width depends on the mass of the scalaron. Therefore, taking into account the environment dependence of the chameleonic scalar field is of the great importance to evaluate the lifetime of scalaron. It was shown that the scalaron is very light in the current Universe although it is very heavy in the early Universe because the energy-momentum tensor decreases according to the expansion of the Universe. The coincidence problem was also addressed, to find the observed ratio of DE to DM with respect to energy density is relevant to this framework.
The mainstream of research on DM is to study new particles in the extended framework of SM. In my work, the new particle, scalaron, is induced from the modification of gravity, and it has a salient feature that the mass changes according to the chameleon mechanism. Such properties cannot be found in the ordinary DM candidates in the existing models, and thus, we proposed the totally new DM candidate as an application of modified gravity.
The Born-Infeld gravity is a non-linear extension of GR motivated by the QG. This theory is different from GR in the high energy scale although it restores GR in the low energy scale. If the Born-Infeld gravity is treated as an effective theory of QG, we can evaluate the quantum effect in this theory.
The inflation scenario predicts the singularity at the beginning of the Universe, so-called the initial singularity. In such an epoch, one expects the quantum effects become important and avoid the initial singularity. If the quantum effects induced by the QG create repulsive force, the universe would bounce before the singularity forms. As a result, the Universe would bounce, and there appears no singularity.
We discussed the FRW Universe with the flat spatial part filled with pressure-less dust in the Born-Infeld gravity. It is well-known that the Born-Infeld gravity with the standard metric formalism leads to the appearance of ghost mode, and thus, we considered the Palatini formalism in which the metric and connection are independent of each other. In the Palatini formalism, the shrinking FRW Universe was investigated, and the condition for the bounce was analyzed.
The bounce of the shrinking Universe was confirmed in the certain parameter region of the Born-Infeld gravity. It was found that the modified equation of motion is very similar to that in the loop quantum cosmology, and we concluded that the singularity does not show up because of the effective repulsive force by modification of gravity.
Based on the analysis of bouncing Universe, the black hole formation was also addressed; that is, shrinking FRW Universe filled with dust can be interpreted as the collapse of a spherically symmetric and uniform ball made of dust. It is interesting to compare the condition for bounce with the size of black hole horizon. If the bounce occurs outside the Schwarzschild radius, it may imply that the black hole does not form.
We made quantitative speculation of the gravitational collapse because we found it difficult to discuss the qualitative behavior of the gravitational collapse, due to unknown junction condition at the boundary. One of the salient features of the Born-Infeld gravity is that the presence of matter fields derives the modification of gravity: the absence of matter makes the equation of motion to be the Einstein's equation. Thus, one needs to consider the connection between two different theories at the boundary. However, we used the junction condition in GR as an approximation.
We observed that the bouncing of the dust sphere would occur after the black hole horizon appears in a specific parameter. If bounce occurs inside the Schwarzschild radius, the black hole horizon forms but there would be a remnant inside the black hole without singularity. Such a phenomenon was proposed in the context of the quantum gravity, which is called the Planck star.
The de Rham-Gabadaze-Tolley (dRGT) massive gravity describes non-linear interacting massive spin-2 field, which is referred to the massive graviton. The interaction terms are constructed to avoid the ghost mode and realize the Vainshtein mechanism.
Various attempts have been made to explain the massive neutron stars in the framework of the modified gravity. It was known that the F(R) gravity could explain the massive neutron star, however, non-perturbative effects in the strong-gravity regime depend on details of the theory. Therefore, we need to study the neutron stars or compact stars in other modified gravity. It is a significant test of the modified gravity in the strong-gravity regime.
We examined the neutron star and quark star in the massive gravity. The modified Tolman-Oppenheimer-Volkoff (TOV) equation was derived, and its mathematical structure was analyzed, comparing with the original TOV equation in GR. We assumed the minimal combination of the parameters and the standard equation of state, and obtained the M-R relations of the compact stars. The M-R relation suggests that the maximal mass in the massive gravity becomes smaller than that in GR.
We concluded that the minimal model of the dRGT massive gravity could not explain the massive neutron stars, and successfully constrained the massive gravity from the viewpoint of astrophysics. It was found that the decrease of the maximal mass is related to the absence of the Vainshtein mechanism in the minimal model.
The bigravity theory includes two independent metric tensors. This theory contains a massive spin-2 field called massive graviton in addition to the ordinary massless spin-2 gravity, and it has been successfully constructed as the generalization of dRGT massive gravity.
We calculated the black hole entropy based on the holographic principle and discussed the effect of the massive spin-2 field around the black hole. It was confirmed that the bigravity theory has the static and spherically symmetric black hole, so-called the Schwarzschild black hole. We found that the massive graviton certainly contributes to the black hole entropy, and thus, the entropy takes different value from that in GR.
The massive graviton is categorized as one of the higher spin fields predicted in the string theory. So, we studied the black hole entropy with the higher spin field in the framework of modified gravity. This work is the first to evaluate the black hole entropy explicitly in the bigravity theory.
Because of the new massive spin-2 field, the number of degrees of freedom in bigravity is larger than those in GR. We studied the perturbation corresponding to these new degrees of freedom and evaluated the stability of the black holes under the perturbation. In previous researches, the stability of the Schwarzschild black hole and the Kerr black hole had been investigated, to show that these black holes are unstable. We examined the stability of Nariai solution in the bigravity.
The Nariai space-time is a special case of the Schwarzschild-de Sitter black hole which is the static and spherically symmetric space-time with the positive cosmological constant. The Schwarzschild-de Sitter black hole has two horizons: black hole horizon and cosmological horizon. These two horizons are degenerate in the Nariai space-time. We showed the equations of perturbations in the bigravity are same as those in GR. We found that the Nariai space-time is stable under the perturbation which keeps the space-time symmetry invariant, which stems the extra symmetry in the Nariai space-time.
The Nariai solution is considered to be realized in the early Universe, thus, we studied the stability of the primordial black hole in the bigravity theory. It is known that the Nariai solution is unstable in the F(R) gravity, therefore, the difference between F(R) gravity and the bigravity was examined in this work.