My research interests are focused on the theoretical investigation of the gravitational theory beyond the general relativity (GR), so-called modified gravity. Modified gravity is extension or generalization of the GR in which one can introduce the new terms, new fields, or new principles. Many theories of modified gravity have been proposed and investigated so far.

There are a lot of possible ways to modify the GR, however, what they have in common is that the modified gravity predicts the deviations from the GR. The GR is constructed on several assumptions: four-dimensional space-time, general coordinate transformation invariance, metric theory, the second-order equation of motion. Technically speaking, the GR is a unique theory of the Lorentz-invariant massless spin-2 field, and one requires to break, at least, one of the assumptions to introduce the modification of gravity. Thus, we are to find specific features of modified gravity, which leads to the deviation from the GR.

If we observe deviations from the GR, it would be an evidence of the modified gravity theories. If not, the modified gravity would be constrained or excluded. So, it is necessary to investigate the modified gravity theories, comparing with the results in the GR. This is literally the test of gravitational theory, and it would give us a hint of understanding what the gravitational theory should be. Furthermore, the modified gravity could predict new physics; that is, modifications of gravitational theory would provide us the predictions of new gravitational phenomena which are unexplainable in the GR. This is one of the motivations to study the modified gravity from the viewpoint of theoretical physics. If such new physics are particular in each theory, we could specify the gravitational theory with observational data.

I am interested in the application of the modified gravity to particular situations in the framework of not only the cosmology but also the astrophysics and particle physics. I am studying the theoretical predictions in the modified gravity to know the deviation from the predictions in the GR. I am also seeking the new theoretical tools that enable us to distinguish the gravitational theories with the existing and forthcoming observations and experiments.

There are four conventionally accepted fundamental forces in our Universe. The electromagnetic, weak, strong forces are described in the standard model (SM) of particle physics based on the quantum field theory (QFT), and the gravity is described by the GR. Despite plenty of success in the SM and GR, several phenomena are still unrevealed.

Late-time accelerated expansion of the Universe has been confirmed by several independent observations. In order to explain the accelerated expansion, it could be inevitable to include the dark energy (DE) as the new energy source. In addition to the existence of the DE, the rotation curve of galaxies, the gravitational lensing, and several other observations indicate the presence of new matters, called the dark matter (DM), which do not have the electromagnetic interaction but the gravitational one. The origin of dark sector involving the DE and DM is still mysterious among fields of the particle physics and astrophysics.

The Λ-CDM model provides us the simplest way to account for the DE and DM, in which the cosmological constant Λ and the cold dark matter (CDM) are introduced in the framework of the GR. The DE is interpreted as vacuum energy induced from the fluctuation of matter fields. This model successfully describes the almost all of the cosmic history. However, it suffers from two theoretical problems about the cosmological constant:

- Fine-tuning problem: Why the cosmological constant is so small? The observed value of the cosmological constant is larger than is the theoretical value by about 120 orders of magnitude. Even if it can be canceled by unknown contributions, the cancellation has to be accurate to 120 decimal places.
- Coincidence problem: Why the cosmological constant is the observed value? DE density is not only small but also comparable to the current energy density of the Universe. We have no explanation for the ratio of the DE to the CDM and baryon with respect to the current energy density.

The Λ-CDM does not give us any answer to the above problems, which would imply the necessity of a new paradigm in the study of gravitational theory. It would be the modified gravity theory that can solve the problems, which has been intensively investigated so far. The modified gravity theories introduce literally the modified or generalized gravitational action instead of the Einstein-Hilbert action in the GR. The modification of gravity leads to the emergence of new degrees of freedom, which is expressed by introducing new fields. Thus, if the new dynamical field mimics the role of the cosmological constant, one can explain the late-time cosmic acceleration without the cosmological constant. The fine-tuning problem is, therefore, initially not present, but translated into the tuning of the parameter in the modified gravity. If the new field is responsible for the DE, the observed small DE density would be explained without fine-tuning by the new dynamics or mechanism of the field.

This feature of the alternative solution to the DE problem is well known, and it motivates to study the cosmological application of modified gravity theory. However, there are many other motivations for the study of the modified gravity. In the following, I introduce two major topics in which modifications of gravity play essential roles.

Other motivation is for the physics in astrophysical scale. Recently, massive and compact neutron stars whose mass is comparable to two Solar mass were found. It could be hardly understood in the framework of the GR and hadron physics so far if one uses the stellar matter equations of state which are comfortable in astrophysics and hadron physics. Thus, there could be two points of view to explain the massive neutron stars. One is from particle physics side which requires any change of equation of state in high-density and high-temperature matter, and another is the gravitational physics side in which one namely consider the modified gravity. The relativistic compact object is realized in the hydrostatic equilibrium between repulsive force by matters and attractive force by gravity; therefore, the size of the compact star is determined by the balance between degeneracy force and gravitational force.

In order to explain the massive neutron star, three approaches seem to be reasonable: (i) the repulsive force is stronger than that realized with the standard equations of state with assuming the GR, (ii) the attractive force is weaker than that predicted in the GR while using the convenient equation of state for stellar matter, (iii) we accept cases (i) and (ii) simultaneously and assume the new physics in both matter and gravity sectors. From (i) based on the hadron physics, it was suggested that equations of state could be modiﬁed by introducing the new interactions. From (ii) based on the gravitational physics, it has been suggested that some models of modified gravity can explain the massive and compact neutron stars. In the study of modified gravity theories, it is often to investigate the mass-radius (M-R) relation of compact stars with changing the gravitational theory, but providing the specific equations of state for matters.

The last one is for the physics in high-energy scale. At the beginning of the Universe or center of the black holes, the GR predicts irremovable singularities in the space-time, where the physical observables diverge. The GR, therefore, loses the predictability at a certain high energy scale, which implies that GR may be not the complete theory but an effective theory of an unknown but complete gravitational theory. The typical scale where the GR becomes invalid is called the Planck scale. In order to describe the gravitational interaction at the Planck scale, it is inevitable to construct a new theory, the quantum gravity (QG). Although the QG is usually considered to be the quantization of GR, it is troublesome to quantize the GR in the framework of QFT due to the infinite number of uncontrollable divergence. Thus, many theories and models based on other quantization of gravity have been proposed and studied so far.

Instead of aiming at the complete theory of QG, the bottom-up approach to QG has also been discussed. At low energy scale, the quantum correction to GR can be mimicked in terms of the existing theories, GR and QFT. Conversely, we can obtain the implications to QG if the modification to GR is related to the quantum correction to GR. Several modified gravity theories can be interpreted as low energy effective theories of QG, and utilized to understand the physics in QG regime. Compared with the previous two motivations, this motivation is theoretical and hardly confirmed by the experiments or observation. Therefore, the mainstream in this research is to study the consistency check or qualitative understanding of the theories, rather than the prediction of physical observables.

In any cases, the modifications of gravity bring deviations from the GR, and the deviations are unique in each theory of modified gravity. Thus, one has to pay particular attention to the prediction of modified gravity, which suggests the importance to study the application of the modified gravity in concrete situations. As the ordinary sense of science, if one construct the new theory or model of modified gravity, it should be tested by observations or experiments. It should be properly understood that the cosmological observation is not only one way to distinguish the modified gravities. Because the gravity is considered to couple with all matters universally, the effect of modification should be observed in the many physics. Therefore, we need to investigate the application of modified gravity in many fields of physics, and focus on the test not only in cosmology but also in astrophysics, table-top experiments, or even particle physics.

From the phenomenological points of view in modified gravity, the deviations from GR are potentially utilized to test the modified gravity. Here, such deviations are characterized by the new degrees of freedom induced from the new fields or terms, and the new field degrees of freedom predict the new gravitational phenomena beyond the GR. From the theoretical points of view, it is attractive to study the role of new degrees of freedom for new physics. We may discover the new physics in the modified gravity theories, and thus, it is worth exploring the theories beyond the Einstein's general relativity. These two viewpoints raise two questions:

- How are the predictions in the modified gravity different from those in the GR? Provided that the matter fields be described in the SM of particles physics, the modified gravity should be responsible for all deviation from the "standard" physics based on the GR and SM.
- How can we use the new degree of freedom for unsolved problems? One way to utilize a new field induced from the modified gravity is a solution to DE problem. Moreover, such new field can be interpreted even as "new particle from beyond-GR", just like beyond-SM particles.