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Bigravity is one of the natural extensions of general relativity and contains an additional massive spin-2 field which can be a good candidate for dark matter. To discuss the production of spin-2 dark matter, we study fixed point solutions of the background equations for axisymmetric Bianchi type-I Universes in two bigravity theories without Boulware-Deser ghost, i.e., Hassan-Rosen bigravity and Minimal Theory of Bigravity. We investigate the local and global stability of the fixed points and classify them. Based on the general analysis, we propose a new scenario where spin-2 dark matter is produced by the transition from an anisotropic fixed point solution to isotropic one. The produced spin-2 dark matter can account for all or a part of dark matter and can be directly detected by laser interferometers in the same way as gravitational waves., Comment: 15 pages, 3 figures

We study self-gravitating bound states of a complex vector field, known as Proca stars, with a new type of quartic-order self-interaction which does not exist in the case of either a complex scalar field or a real vector field. Depending on the sign of the coupling constant, this quartic self-interaction can yield a distinct feature of Proca stars from the previously investigated self-interaction of the vector field. We find that self-gravitating solutions can be so compact that the photon sphere could form. However, we also show that the self-interaction gives rise to a ghost instability for the stars whose compactness is close that for the formation of a photon sphere, which might invalidate the formation of the photon sphere., 14 pages, 6 figures, the version to appear in PRD

It has been argued that a self-interacting massive vector field is pathological due to a dynamical formation of a singular effective metric of the vector field, which is the onset of a gradient or ghost instability. We discuss that this singularity formation is not necessarily a fundamental problem but a breakdown of the effective field theory (EFT) description of the massive vector field. By using a model of ultraviolet (UV) completion of the massive vector field, we demonstrate that a Proca star, a self-gravitating condensate of the vector field, continues to exist even after the EFT suffers from a gradient instability without any pathology at UV, in which the EFT description is still valid and the gradient instability in EFT may be interpreted as a standard dynamical instability of a high-density boson star from the UV perspective. On the other hand, we find that the EFT description is broken before the ghost instability appears. This suggests that a heavy degree of freedom may be spontaneously excited to cure the pathology of the EFT as the EFT dynamically tends to approach the onset of the ghost instability., Comment: 14 pages, 7 figures, to appear in PRD

Since the discovery of the accelerated expansion of the present Universe, significant theoretical developments have been made in the area of modified gravity. In the meantime, cosmological observations have been providing more high-quality data, allowing us to explore gravity on cosmological scales. To bridge the recent theoretical developments and observations, we present an overview of a variety of modified theories of gravity and the cosmological observables in the cosmic microwave background and large-scale structure, supplemented with a summary of predictions for cosmological observables derived from cosmological perturbations and sophisticated numerical studies. We specifically consider scalar-tensor theories in the Horndeski and DHOST family, massive gravity/bigravity, vector-tensor theories, metric-affine gravity, and cuscuton/minimally-modified gravity, and discuss the current status of those theories with emphasis on their physical motivations, validity, appealing features, the level of maturity, and calculability. We conclude that the Horndeski theory is one of the most well-developed theories of modified gravity, although several remaining issues are left for future observations. The paper aims to help to develop strategies for testing gravity with ongoing and forthcoming cosmological observations., Comment: 115 pages, 8 figures

We investigate the relativistic effective field theory (EFT) describing a non-dissipative gravitating continuum. In addition to ordinary continua, namely solids and fluids, we find an extraordinary more symmetric continuum, aether. In particular, the symmetry of the aether concludes that a homogeneous and isotropic state behaves like a cosmological constant. We formulate the EFT in the unitary/comoving gauge in which the dynamical degrees of freedom of the continuum (phonons) are eaten by the spacetime metric. This gauge choice, which is interpreted as the Lagrangian description in hydrodynamics, offers a neat geometrical understanding of continua. We examine a thread-based spacetime decomposition with respect to the four-velocity of the continuum which is different from the foliation-based Arnowitt-Deser-Misner one. Our thread-based decomposition respects the symmetries of the continua and, therefore, makes it possible to systematically find invariant building blocks of the EFT for each continuum even at higher orders in the derivative expansion. We also discuss the linear dynamics of the system and show that both gravitons and phonons acquire "masses" in a gravitating background., 41 pages + appendices, 1 figure, 3 tables, added explanations, published version

We study compatibility of the Standard Model of particle physics and General Relativity by means of gravitational positivity bounds, which provide a necessary condition for a low-energy gravitational theory to be UV completable within the weakly coupled regime of gravity. In particular, we identify the cutoff scale of the Standard Model coupled to gravity by studying consistency of light-by-light scattering. While the precise value depends on details of the Pomeron effects in QCD, the cutoff scale reads $10^{16}$GeV if the single-Pomeron exchange picture works well up to this scale. We also demonstrate that the cutoff scale is lowered to $10^{13}$GeV if we consider the electroweak theory without the QCD sector., 6 pages, 5 figures, updated to match the version to appear in PRL

We show how a class of multi-field scalar-field theories in a Lorentz-breaking background imposes consistency conditions on its effective theory of a single field and provides an example of order-unity violation of a naively applied positivity bound, assuming a large hierarchy between the masses of the lightest field and the others., Comment: 50 pages, 3 figures, 1 table, v2: improved explanations, new section VF added, references added, to appear in JCAP

We find unitary and local theories of higher curvature gravity in the vielbein formalism, known as the Poincar\'{e} gauge theory by utilizing the equivalence to the ghost-free massive bigravity. We especially focus on three and four dimensions but extensions into a higher dimensional spacetime are straightforward. In three dimensions, a quadratic gravity $\mathcal{L}=R+T^2+R^2$, where $R$ is the curvature and $T$ is the torsion with indices omitted, is shown to be equivalent to zwei-dreibein gravity and free from the ghost at fully non-linear orders. In a special limit, new massive gravity is recovered. When the model is applied to the AdS/CFT correspondence, unitarity both in the bulk theory and in the boundary theory implies that the torsion must not vanish. On the other hand, in four dimensions, the absence of ghost at non-linear orders requires an infinite number of higher curvature terms, and these terms can be given by a schematic form $R(1+R/\alpha m^2)^{-1}R$ where $m$ is the mass of the massive spin-2 mode originating from the higher curvature terms and $\alpha$ is an additional parameter that determines the amplitude of the torsion. We also provide another four-dimensional ghost-free higher curvature theory that contains a massive spin-0 mode as well as the massive spin-2 mode., Comment: 14 pages, no figure, published version

We study the validity of positivity bounds in the presence of a massless graviton, assuming the Regge behavior of the amplitude. Under this assumption, the problematic $t$-channel pole is canceled with the UV integral of the imaginary part of the amplitude in the dispersion relation, which gives rise to finite corrections to the positivity bounds. We find that low-energy effective field theories (EFT) with "wrong" sign are generically allowed. The allowed amount of the positivity violation is determined by the Regge behavior. This violation is suppressed by $M_{\rm pl}^{-2}\alpha'$ where $\alpha'$ is the scale of Reggeization. This implies that the positivity bounds can be applied only when the cutoff scale of EFT is much lower than the scale of Reggeization. We then obtain the positivity bounds on scalar-tensor EFT at one-loop level. Implications of our results on the degenerate higher-order scalar-tensor (DHOST) theory are also discussed., Comment: figures and references added, typos and eqs. (4.31) (4.33) corrected, improved explanations

The idea of gauge theories of gravity predicts that there should exist not only the massless graviton but also massive particles carrying the gravitational force. We study the cosmology in a quadratic gravity with dynamical torsion where gravity may be interpreted as a gauge force associated with the Poincar\'{e} group. In addition to the massless spin-2 graviton, the model contains four non-ghost massive particle species: a couple of spin-0, a spin-1 and a spin-2. Supposing the restoration of the local Weyl invariance in the UV limit and the parity invariance, we find the most general minisuperspace action describing a homogeneous and isotropic universe with a flat spatial geometry. We then transform the minisuperspace action to a quasi-Einstein frame in which the field space is a hyperboloid and the field potential is a combination of those of a Starobinsky-like inflation and a natural inflation. Remarkably, thanks to the multi-field dynamics, the Starobinsky-like inflationary trajectory can be realized even if the initial condition is away from the top of the Starobinsky-like potential. We also study linear tensor perturbations and find qualitatively different features than the Starobinsky inflation, spontaneous parity violation and mixing of the massless and massive spin-2 modes, which might reveal the underlying nature of gravity through inflationary observables., Comment: 35 pages, 5 figures, published version

Ostrogradsky theorem states that Hamiltonian is unbounded when Euler-Lagrange equations are higher than second-order differential equations under the nondegeneracy assumption. Since higher-order nondegenerate Lagrangian can be always recast into an equivalent system with at most first-order derivatives by introducing auxiliary variables and constraints, it is conceivable that the link between ghost and higher derivatives may be reinterpreted as a link between ghost and constraints and/or auxiliary variables. We find that the latter point of view actually provides more general perspective than the former, by exploring the un/boundedness of the Hamiltonian for general theories containing auxiliary variables, for which Euler-Lagrange equations can be essentially second order or lower than that. For Lagrangians including auxiliary variables nonlinearly, we derive the degeneracy condition to evade the Ostrogradsky ghost that can apply even if auxiliary variables can be solved only locally. For theories with constraints with Lagrange multipliers, we establish criteria for inclusion of nonholonomic (velocity-dependent) constraints leading to the absence of local minimum of Hamiltonian. Our criteria include the Ostrogradsky theorem as a special case, and can detect not only ghost associated with higher-order derivatives, but also ghost coming from lower-order derivatives in system with constraints. We discuss how to evade such a ghost. We also provide various specific examples to highlight application and limitation of our general arguments., Comment: 28 pages, 1 figure, improved discussions, accepted for publication in JCAP

We consider generic derivative corrections to the Einstein gravity and find new classes of theories without ghost around the Minkowski background by means of an extension of the spacetime geometry. We assume the Riemann-Cartan geometry, i.e. a geometry with a non-vanishing torsion, and consider all possible terms in the Lagrangian up to scaling dimension four. We first clarify the number, spins, and parities of all particle species around the Minkowski background and find that some of those particle species are ghosts for generic choices of parameters. For special choices of the parameters, on the other hand, those would-be ghosts become infinitely heavy and thus can be removed from the physical content of particle species. Imposing the conditions on the coupling constants to eliminate the ghosts, we find new quadratic curvature theories which are ghost-free around the Minkowski background for a range of parameters. A key feature of these theories is that there exist a non-ghost massive spin-2 particle and a non-ghost massive spin-0 particle in the graviton propagator, as well as the massless spin-2 graviton. In the limit of the infinite mass of the torsion, the Riemann-Cartan geometry reduces to the Riemannian geometry and thus the physical content of particle species coincides with that of the well-known quadratic curvature theory in the metric formalism, i.e. a massive spin-2 ghost, a massive spin-0 particle and the massless spin-2 graviton. Ghost-freedom therefore sets, besides other constraints, an upper bound on the mass of the torsion. In addition to the above mentioned particle species, the ghost-free theory contains either the set of a massive spin-1 and a massive spin-0 (Class I) or a couple of spin-1 (Class II). These additional particle species mediate gravity sourced by the spin of matter fields., Comment: 17 pages, published version

We reveal the existence of a certain hidden symmetry in general ghost-free scalar-tensor theories which can only be seen when generalizing the geometry of the spacetime from Riemannian. For this purpose, we study scalar-tensor theories in the metric-affine (Palatini) formalism of gravity, which we call scalar-metric-affine theories for short, where the metric and the connection are independent. We show that the projective symmetry, a local symmetry under a shift of the connection, can provide a ghost-free structure of scalar-metric-affine theories. The ghostly sector of the second-order derivative of the scalar is absorbed into the projective gauge mode when the unitary gauge can be imposed. Incidentally, the connection does not have the kinetic term in these theories and then it is just an auxiliary field. We can thus (at least in principle) integrate the connection out and obtain a form of scalar-tensor theories in the Riemannian geometry. The projective symmetry then hides in the ghost-free scalar-tensor theories. As an explicit example, we show the relationship between the quadratic order scalar-metric-affine theory and the quadratic U-degenerate theory. The explicit correspondence between the metric-affine (Palatini) formalism and the metric one could be also useful for analyzing phenomenology such as inflation., Comment: 15 pages, no figure, published version

We study cosmology in a class of minimally modified gravity (MMG) with two local gravitational degrees of freedom. We classify modified gravity theories into type-I and type-II: theories of type-I have an Einstein frame and can be recast by change of variables as general relativity (GR) with a non-minimal matter coupling, while theories of type-II have no Einstein frame. Considering a canonical transformation of the lapse, the 3-dimensional induced metric and their conjugate momenta we generate type-I MMG. We then show that phenomenological deviations from GR, such as the speed of gravitational waves $c_T$ and the effective gravitational constant for scalar perturbations $G_{\rm eff}$, are characterized by two functions of an auxiliary variable. We study the phenomenology of several models all having $c_T=1$. We obtain a scenario with $c_T=1$ in which the effective equation-of-state parameter of dark energy is different from $-1$ even though the cosmic acceleration is caused by a bare cosmological constant, and we find that it is possible to reconstruct the theory on choosing a selected time-evolution for the effective dark energy component., 14 pages, 1 figure, updated to match the version to appear in JCAP

We study a theory of minimally modified gravity called cuscuton/VCDM that propagates only two gravitational degrees of freedom. Despite being apparently different from general relativity (GR), it is in principle possible that this theory might be obtained via a field redefinition starting from the GR action. This would make the vacuum theory equivalent to GR and the theory would differ from GR only in the presence of matter. In this paper, studying the dispersion relation of gravitational waves and the dynamics of the Bianchi-I universe, we prove that such a field redefinition does not exist and that the theory differs from GR already in vacuum., 19 pages, 1 figure

We study the slow-roll single field inflation in the context of the consistent $D\to4$ Einstein-Gauss-Bonnet gravity that was recently proposed in \cite{Aoki:2020lig}. In addition to the standard attractor regime, we find a new attractor regime which we call the Gauss-Bonnet attractor as the dominant contribution comes from the Gauss-Bonnet term. Around this attractor solution, we find power spectra and spectral tilts for the curvature perturbations and gravitational waves (GWs) and also a model-independent consistency relation among observable quantities. The Gauss-Bonnet term provides a nonlinear $k^4$ term to the GWs dispersion relation which has the same order as the standard linear $k^2$ term at the time of horizon crossing around the Gauss-Bonnet attractor. The Gauss-Bonnet attractor regime thus provides a new scenario for the primordial GWs which can be tested by observations. Finally, we study non-Gaussianity of GWs in this model and estimate the nonlinear parameters $f^{s_1s_2s_3}_{\rm NL,\;sq}$ and $f^{s_1s_2s_3}_{\rm NL,\;eq}$ by fitting the computed GWs bispectra with the local-type and equilateral-type templates respectively at the squeezed limit and at the equilateral shape. For helicities $(+++)$ and $( -- )$, $f^{s_1s_2s_3}_{\rm NL,\;sq}$ is larger while $f^{s_1s_2s_3}_{\rm NL,\;eq}$ is larger for helicities $(++-)$ and $(--+)$., 30 pages, 4 figures, observational bound is corrected

We classify the metric-affine theories of gravitation, in which the metric and the connections are treated as independent variables, by use of several constraints on the connections. Assuming the Einstein-Hilbert action, we find that the equations for the distortion tensor (torsion and non-metricity) become algebraic, which means that those variables are not dynamical. As a result, we can rewrite the basic equations in the form of Riemannian geometry. Although all classified models recover the Einstein gravity in the Palatini formalism (in which we assume there is no coupling between matter and the connections), but when matter field couples to the connections, the effective Einstein equations include an additional hyper energy-momentum tensor obtained from the distortion tensor. Assuming a simple extension of a minimally coupled scalar field in metric-affine gravity, we analyze an inflationary scenario. Even if we adopt a chaotic inflation potential, certain parameters could satisfy observational constraints. Furthermore, we find that a simple form of Galileon scalar field in metric-affine could cause G-inflation., 18 pages, 6 figures, comments welcome. v2: references added

We study scalar-tensor theories respecting the projective invariance in the metric-affine formalism. The metric-affine formalism is a formulation of gravitational theories such that the metric and the connection are independent variables in the first place. In this formalism, the Einstein-Hilbert action has an additional invariance, called the projective invariance, under a shift of the connection. Respecting this invariance for the construction of the scalar-tensor theories, we find that the Galileon terms in curved spacetime are uniquely specified at least up to quartic order which does not coincide with either the covariant Galileon or the covariantized Galileon. We also find an action in the metric-affine formalism which is equivalent to class ${}^2$N-I/Ia of the quadratic degenerated higher order scalar-tensor (DHOST) theory. The structure of DHOST would become clear in the metric-affine formalism since the equivalent action is just linear in the generalized Galileon terms and non-minimal couplings to the Ricci scalar and the Einstein tensor with independent coefficients. The fine-tuned structure of DHOST is obtained by integrating out the connection. In these theories, non-minimal couplings between fermionic fields and the scalar field may be predicted. We discuss possible extensions which could involve theories beyond DHOST., 13 pages, no figure, published version

We study canonical transformations of general relativity (GR) to provide a novel matter coupling to gravity. Although the transformed theory is equivalent to GR in vacuum, the equivalence no longer holds if a matter field minimally couples to the canonically transformed gravitational field. We find that a naive matter coupling to the transformed field leads to the appearance of an extra mode in the phase space, rendering the theory inconsistent. We then find a consistent and novel way of matter coupling: after imposing a gauge fixing condition, a matter field can minimally couple to gravity without generating an unwanted extra mode. As a result, the way matter field couples to the gravitational field determines the preferred time direction and the resultant theory has only two gravitational degrees of freedom. We also discuss the cosmological solution and linear perturbations around it, and confirm that their dynamics indeed differ from those in GR. The novel matter coupling can be used for a new framework of modified gravity theories., 11 pages, no figure, version accepted by Phys. Rev. D

We propose a scenario that can naturally explain the observed dark matter-baryon ratio in the context of bimetric theory with a chameleon field. We introduce two additional gravitational degrees of freedom, the massive graviton and the chameleon field, corresponding to dark matter and dark energy, respectively. The chameleon field is assumed to be non-minimally coupled to dark matter, i.e., the massive graviton, through the graviton mass terms. We find that the dark matter-baryon ratio is dynamically adjusted to the observed value due to the energy transfer by the chameleon field. As a result, the model can explain the observed dark matter-baryon ratio independently from the initial abundance of them., Comment: 13 pages, 6 figures; v2: references added, published version

We find vacuum solutions such that massive gravitons are confined in a local spacetime region by their gravitational energy in asymptotically flat spacetimes in the context of the bigravity theory. We call such self-gravitating objects massive graviton geons. The basic equations can be reduced to the Schr\"odinger-Poisson equations with the tensor "wavefunction" in the Newtonian limit. We obtain a non-spherically symmetric solution with $j=2,\ell=0$ as well as a spherically symmetric solution with $j=0,\ell=2$ in this system where $j$ is the total angular momentum quantum number and $\ell$ is the orbital angular momentum quantum number, respectively. The energy eigenvalue of the Schr\"odinger equation in the non-spherical solution is smaller than that in the spherical solution. We then study the perturbative stability of the spherical solution and find that there is an unstable mode in the quadrupole mode perturbations which may be interpreted as the transition mode to the non-spherical solution. The results suggest that the non-spherically symmetric solution is the ground state of the massive graviton geon. The massive graviton geons may decay in time due to emissions of gravitational waves but this timescale can be quite long when the massive gravitons are non-relativistic and then the geons can be long-lived. We also argue possible prospects of the massive graviton geons: applications to the ultralight dark matter scenario, nonlinear (in)stability of the Minkowski spacetime, and a quantum transition of the spacetime., Comment: 16 pages, 7 tables, 3 figures; v2: references added, improved discussion, published version

We study the stability of the Vainshtein screening solution of the massive/bi-gravity based on the massive nonlinear sigma model as the effective action inside the Vainshtein radius. The effective action is obtained by taking the $\Lambda_2$ decoupling limit around a curved spacetime. First we derive a general consequence that any Ricci flat Vainshtein screening solution is unstable when we take into account the excitation of the scalar graviton only. This instability suggests that the nonlinear excitation of the scalar graviton is not sufficient to obtain a successful Vainshtein screening in massive/bi-gravity. Then to see the role of the excitation of the vector graviton, we study perturbations around the static and spherically symmetric solution obtained in bigravity explicitly. As a result, we find that linear excitations of the vector graviton cannot be helpful and the solution still suffers from a ghost and/or a gradient instability for any parameters of the theory for this background., Comment: 12 pages, no figure; v2: minor improvements, references added, published version

We study the stability of a spherically symmetric perturbation around the flat Friedmann-Lema$\hat{\i}$tre-Robertson-Walker spacetime in the ghost-free bigravity theory, retaining nonlinearities of the helicity-$0$ mode of the massive graviton. It has been known that, when the graviton mass is smaller than the Hubble parameter, homogeneous and isotropic spacetimes suffer from the Higuchi-type ghost or the gradient instability against the linear perturbation in the bigravity. Hence, the bigravity theory has no healthy massless limit for cosmological solutions at linear level. In this paper we show that the instabilities can be resolved by taking into account nonlinear effects of the scalar graviton mode for an appropriate parameter space of coupling constants. The growth history in the bigravity can be restored to the result in general relativity in the early stage of the Universe, in which the St\"uckelberg fields are nonlinear and there is neither ghost nor gradient instability. Therefore, the bigravity theory has the healthy massless limit, and cosmology based on it is viable even when the graviton mass is smaller than the Hubble parameter., Comment: 24 pages, 8 figures, references added

We study the origin of dark matter based on the ghost-free bigravity theory with twin matter fluids. The present cosmic acceleration can be explained by the existence of graviton mass, while dark matter is required in several cosmological situations [the galactic missing mass, the cosmic structure formation and the standard big-bang scenario (the cosmological nucleosynthesis vs the CMB observation)]. Assuming that the Compton wavelength of the massive graviton is shorter than a galactic scale, we show the bigravity theory can explain dark matter by twin matter fluid as well as the cosmic acceleration by tuning appropriate coupling constants., Comment: 15 pages, 7 figures, minor changes, references added

We study dynamics of Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime based on the ghost-free bigravity theory. Assuming the coupling parameters guaranteeing the existence of de Sitter space as well as Minkowski spacetime, we find two stable attractors for spacetime with "twin" dust matter fields: One is de Sitter accelerating universe and the other is matter dominated universe. Although a considerable number of initial data leads to de Sitter universe, we also find matter dominated universe or spacetime with a future singularity for some initial data. The cosmic no-hair conjecture does not exactly hold, but the accelerating expansion can be found naturally. The $\Lambda$-CDM model is obtained as an attractor. We also show that the dark matter component in the Friedmann equation, which originates from another twin matter, can be about 5 times larger than the baryonic matter, by choosing the appropriate coupling constants., Comment: 23 pages, 24 figures