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

There are a lot of possible ways to modify GR. The GR is constructed on several assumptions: (a) four-dimensional space-time, (b) general coordinate transformation invariance, (c) metric theory, (d) the second-order equation of motion. Technically, GR is a unique theory of the Lorentz-invariant massless spin-2 field. Thus, one requires to break, at least, one of the assumptions to introduce the modification of gravitation.

As a consequence of modification of GR, the modified gravity theory predicts the deviations from GR, which reflects specific features of each modified gravity theory. If we could observe such deviations, it would be an evidence of the modified gravity theories. Otherwise, the modified gravity would be constrained or excluded. Thus, it is necessary to investigate the physical observables and measurements of the modified gravity theories, comparing them with the known results in GR. From the phenomenological viewpoint, this is literally the test of gravitational theory, and it would give us a hint to understand what the gravitational theory should be in our Universe.

Furthermore, the modified gravity could predict new physics; that is, modifications of gravitational theory would provide us with new gravitational phenomena which are unexplainable in GR. Such a new phenomenon can solve existing problems in GR, and this is another motivation to study the modified gravity from the theoretical viewpoint. If such new physics are particular in each theory, we could specify the gravitational theory with observational data.

I am interested in applications of the modified gravity to particular situations, in the context 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 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 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 that has negative pressure. 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 side of the Universe (DSU) involving the DE and DM is still mysterious among the particle physics and astrophysics.

Recent cosmological observation further suggests that the ordinary matters, which compose atoms and stars, is only about 5% of our Universe, although DM and DE are about 27% and 68%, respectively. Thus, DSU occupies about 95% of the Universe, and our current understanding of the Universe is far from perfect.

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 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 smaller than the theoretically expected value by about 120 orders of magnitude. Even if the large discrepancy 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 GR. The modification of gravity leads to the emergence of new degrees of freedom, which is expressed by new dynamical fields.

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 nontrivial dynamics of the new fields 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 dynamical DE field. The alternative solution to the DE problem motivates to study the cosmological application of modified gravity theory.

There are many other motivations for the study of the modified gravity. In the following, I introduce the applications of modified gravity theories of my research interest.

(Last update: 2024/3/1)

Building approriate models for DE, one could explain the accelerated expansion of the late-time Universe. However, such models also need to explain whole cosmic history if the modified gravity is the true gravitational theory in our Universe. In this sense, it is inevitable to check the consistency with observational data inferred from the early Universe. Several models of modified gravity theoreies predicts an approach to GR at scales smaller than the cosmological scale or at high energy scales.

The Hubble tension problem has received widespread attention in recent years. The values of the Hubble constant measured by using the late-time-universe data are larger than those inferred from the early-universe data based on the Λ-CDM model. The tension between those persists at about 5σ confidence level, even though measurements in the late-time and early Universe have undergone extensive and rigorous scrutiny. The Hubble tension may imply modifications of the standard cosmic history, and cosmological models that go beyond the Λ-CDM model predicted by modified gravity could resolve this tension. At least, modifications to the physics in the early Universe would be required.

Although the DE problem is related to the accelerated expansion of the Universe at cosmological scale, there are also mysteries in the astrophysics at smaller scale. Recently, massive and compact neutron stars whose mass is comparable to two Solar mass were found. There could be two points of view to explain such 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 and compact object is realized in the hydrostatic equilibrium between repulsive force by matters and attractive force by gravity; therefore, the nature of the compact star is determined by the balance between the internal matter and gravity.

In order to explain the massive neutron star, three approaches seem to be reasonable: (i) we upgrade the equation of state of internal matter, (ii) we modify the gravitational theory, (iii) we accept cases (i) and (ii) simultaneously and assume the new physics in both matter and gravity sectors. From (i) based on the particle physics, new equations of state with the new particle contents and interactions have been proposed. From (ii) based on the gravitational physics, it has been suggested that some models of modified gravity can explain the massive 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 assuming the specific equations of state for matters.

In addition to compact star, on length scales smaller than 1 Mpc and mass scales smaller than 10^11 solar mass, the Λ-CDM model faces a number of challenges. For example, the observed cores of many dark-matter dominated galaxies are both less dense and less cuspy than predicted in Λ-CDM model. The number of small galaxies and dwarf satellites is also far below the predicted count of low-mass dark matter halos and sub-halos within similar volumes. These issues underlie the well-documented problems with Λ-CDM model: Cusp/Core, Missing Satellites, and Too-Big-to-Fail problems.

The new dynamical fields predicted in the modified gravity theory are not necessarily handled as the classical field theory, but can be as QFT. Quantization of new fields gives us the new elementary particles which go beyond SM of particle physics. This is similar to the conventional semi-classical description in QFT on curved spacetime; that is, quantum description for particles with classical description for spacetime and gravity. Because such a new particle originates from the gravity sector, interaction to the SM particles would be of order of the gravitational coupling, and thus, suppressed by the Planck scale. Even though it is very hard to directly detect the modified-gravity-inspired new particles, this approach can open a new window to constrain the modified gravity theory by ground-based observatories and particle colliders.

An interesting possibility is to identify the new particles as dark matter candidate. In this case, one can read constraints on DM, for instance, from direct search experiments as constraints on DE. Furthermore, this scenario can give us a fascinating scneario. By separating the dynamical DE field into two parts, the fluctuation and background, quantized fluctuation can be considered as particle DM, while background solution explains DE for late-time cosmic acceleration. In this scenario, DSU is naturally unified in the framework of modified gravity theories, and one can naturally address the coincidence problem, since both DM and DE stems from one theory.

At the beginning of the Universe or center of the black holes, 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 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 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 applications, the key is that modifications of gravity bring deviations from GR, and the deviations are unique in each theory of modified gravity. One has to pay particular attention to the theoretical predictions of modified gravity, which suggests the importance to study the modified gravity in concrete situations. As the ordinary sense of science, if one constructs 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 effects of modification should be observed in the many physics. Therefore, we need to investigate the application of modified gravity in multiple fields of physics, and interdisciplinary research is inevitable under the name of gravitation. We need to focus on the test of gravitationa theory not only in cosmology but also in astrophysics, particle physics, and even ground-based experiments.

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 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 different are the predictions in the modified gravity from those in 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 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.