Massive gravity may be viewed as a suitable limit of bimetric gravity. The limiting procedure can lead to an interesting interplay between the "background" and "foreground" metrics in a cosmological context. The fact that in bimetric theories one always has two sets of metric equations of motion continues to have an effect even in the massive gravity limit. Thus, solutions of bimetric gravity in the limit of vanishing kinetic term are also solutions of massive gravity, but the contrary statement is not necessarily true.
We perform cosmological perturbation theory in Hassan-Rosen bimetric gravity for general homogeneous and isotropic backgrounds. In the de Sitter approximation, we obtain decoupled sets of massless and massive scalar gravitational fluctuations. Matter perturbations then evolve like in Einstein gravity. We perturb the future de Sitter regime by the ratio of matter to dark energy, producing quasi-de Sitter space. In this more general setting the massive and massless fluctuations mix. We argue that in the quasi-de Sitter regime, the growth of structure in bimetric gravity differs from that of Eins…
Recently, Kenna-Allison et.al. claimed that bimetric gravity cannot give rise to a viable cosmological expansion history while at the same time being compatible with local gravity tests. In this note we review that claim and combine various results from the literature to provide several simple counter examples. We conclude that the results of Kenna-Allison et.al. cannot hold in general.
We discuss the subtle relationship between massive gravity and bimetric gravity, focusing particularly on the manner in which massive gravity may be viewed as a suitable limit of bimetric gravity. The limiting procedure is more delicate than currently appreciated. Specifically, this limiting procedure should not unnecessarily constrain the background metric, which must be externally specified by the theory of massive gravity itself. The fact that in bimetric theories one always has two sets of metric equations of motion continues to have an effect even in the massive gravity limit, leading to …
In this paper we establish the correspondence between ghost-free bimetric theory and a class of higher derivative gravity actions, including conformal gravity and New Massive Gravity. We also characterize the relation between the respective equations of motion and classical solutions. We illustrate that, in this framework, the spin-2 ghost of higher derivative gravity is an artifact of the truncation to a 4-derivative theory. The analysis also gives a relation between the proposed partially massless (PM) bimetric theory and conformal gravity, showing, in particular, the equivalence of their eq…
We study primordial tensor power-spectra generated during inflation in bimetric gravity. More precisely, we examine a homogeneous expanding spacetime in a minimal bimetric model with an inflaton and calculate tensor perturbations on the homogeneous background under slow-roll approximation. In terms of the mass eigenstates, only the power-spectrum of the massless state remains constant and both the power-spectrum of the massive state and the cross power-spectrum rapidly decay during inflation. The amplitude of the physical power-spectrum is suppressed due to the flavor mixing. All power-spectra…
In this letter, we consider cosmological solutions of bimetric theory without assuming that only one metric is coupled to gravity. We conclude that any cosmology can be described by fixing the matter content of the space that we are not inhabiting. On the other hand, we show that some conclusions can still be extracted independently of the matter content filling both spaces. In particular, we can conclude the occurrence of some extremality events in one universe if we know that they take place in the other space.
Bimetric gravity is a ghost-free and observationally viable extension of general relativity, exhibiting both a massless and a massive graviton. The observed abundances of light elements can be used to constrain the expansion history of the Universe at the period of Big Bang nucleosynthesis. Applied to bimetric gravity, we readily obtain constraints on the theory parameters which are complementary to other observational probes. For example, the mixing angle between the two gravitons must satisfy $θ\lesssim 18^\circ$ in the graviton mass range $m_\mathrm{FP} \gtrsim 10^{-16} \, \mathrm{eV}/c^2$,…
We investigate primordial gravitational waves and curvature perturbations in de Rham-Gabadadze-Tolley (dRGT) bimetric gravity. We evaluate the power-spectra in the leading order in slow roll. Taking into account the decay of massive graviton, we find that the action up to the second order reduces to the Einstein theory with a non-minimally coupled scalar field, which is simplified to a minimally coupled model by conformal transformation. We also find that the tensor to scalar ratio for large field inflation with power law potential is larger than the general relativity counterpart for any choi…
A ghost-free theory of gravity with two dynamical metrics both coupled to matter is shown to be consistent and viable. Its cosmological implications are studied, and the models, in particular in the context of partially massless gravity, are found to explain the cosmic acceleration without resorting to dark energy.
We investigate the observational consequences of scalar instabilities in bimetric theory, under the assumption that the Vainshtein mechanism restores general relativity within a certain distance from gravitational sources. We argue that early time instabilities have a negligible impact on observed structures. Assuming that the instabilities affect sub-horizon density fluctuations, we constrain the redshift, z_i, below which instabilities are ruled out. For the "minimal" beta_1-model, observational constraints are close to the theoretical expectations of z_i = 0.5, potentially allowing the mode…
The gravitational effective average action is studied in a bimetric truncation with a nontrivial background field dependence, and its renormalization group flow due to a scalar multiplet coupled to gravity is derived. Neglecting the metric contributions to the corresponding beta functions, the analysis of its fixed points reveals that, even on the new enlarged theory space which includes bimetric action functionals, the theory is asymptotically safe in the large $N$ expansion.
The problem of motion for different test particles, charged and spinning objects of constant spinning tensor in different versions of bimetric theory of gravity is obtained by deriving their corresponding path and path deviation equations, using a modified Bazanski in presence of Riemannian geometry. This method enables us to find path and path deviation equations of different objects orbiting very strong gravitational fields.
Ghost-free bimetric gravity is an extension of general relativity, featuring a massive spin-2 field coupled to gravity. We parameterize the theory with a set of observables having specific physical interpretations. For the background cosmology and the static, spherically symmetric solutions (for example approximating the gravitational potential of the solar system), there are four directions in the parameter space in which general relativity is approached. Requiring that there is a working screening mechanism and a nonsingular evolution of the Universe, we place analytical constraints on the p…
Bimetric gravity theories allow for many different types of cosmological solutions, but not all of them are theoretically allowed. In this work we discuss the conditions to satisfy the Higuchi bound and to avoid gradient instabilities in the scalar sector at the linear level. We find that in expanding universes the ratio of the scale factors of the reference and observable metric has to increase at all times. This automatically implies a ghost-free helicity-2 and helicity-0 sector and enforces a phantom dark energy. Furthermore, the condition for the absence of gradient instabilities in the sc…
We provide a novel model of gravity by using adjoint frame fields in four dimensions. It has a natural interpretation as a gravitational theory of a complex metric field, which describes interactions between two real metrics. The classical solutions establish three appealing features. The spherical symmetric black hole solution has an additional hair, which includes the Schwarzschild solution as a special case. The de Sitter solution is realized without introducing a cosmological constant. The constant flat background breaks the Lorentz invariance spontaneously, although the Lorentz breaking e…
We derive dynamical and gravitational lensing properties of local sources in the Hassan-Rosen bimetric gravity theory. Observations of elliptical galaxies rule out values of the effective length-scale of the theory, in units of the Hubble radius, in the interval 10^-6 < lambda_g/r_H < 10^-3, unless the proportionality constant between the metrics at the background level is far from unity, in which case general relativity is effectively restored for local sources. In order to have background solutions resembling the concordance cosmological model, without fine-tuning of the parameters of the mo…
We discuss the prospects for a consistent, nonlinear, partially massless (PM), gauge symmetry of bimetric gravity (BMG). Just as for single metric massive gravity, we show that consistency of BMG relies on it having a PM extension; we then argue that it cannot.
We study the cosmic no-hair in the presence of spin-2 matter, i.e. in bimetric gravity. We obtain stable de Sitter solutions with the cosmological constant in the physical sector and find an evidence that the cosmic no-hair is correct. In the presence of the other cosmological constant, there are two branches of de Sitter solutions. Under anisotropic perturbations, one of them is always stable and there is no violation of the cosmic no-hair at the linear level. The stability of the other branch depends on parameters and the cosmic no-hair can be violated in general. Remarkably, the bifurcation…
We perform the complete canonical analysis of the tetrad formulation of bimetric gravity and confirm that it is ghost-free describing the seven degrees of freedom of a massless and a massive gravitons. In particular, we find explicit expressions for secondary constraints, one of which is responsible for removing the ghost, whereas the other ensures the equivalence with the metric formulation. Both of them have a remarkably simple form and, being combined with conditions on Lagrange multipliers, can be written in a covariant way.
We consider the effective stress-energy tensors for the foreground and background sectors in ghost-free bimetric gravity. By considering the symmetries of the theory, we show that the foreground and background null energy conditions (NECs) are strongly anti-correlated. In particular, the NECs can only be simultaneously fulfilled when they saturate, corresponding to foreground and background cosmological constants. In all other situations, either the foreground or the background is subject to a NEC-violating contribution to the total stress-energy.
We discuss the linear perturbation equations with the synchronous gauge in a minimal scenario of the bimetric massive gravity theory. We find that the matter density perturbation and matter power spectrum are suppressed. We also examine the ghost and stability problems and show that the allowed deviation of this gravitational theory from the cosmological constant is constrained to be smaller than $\mathcal{O}(10^{-2})$ by the large scale structure observational data.
Ghost-free bimetric gravity is a theory of two interacting spin-2 fields, one massless and one massive, in addition to the standard matter particles and fields, thereby generalizing Einstein's theory of general relativity. To parameterize the theory, we use five observables with specific physical interpretations. We present, for the first time, observational constraints on these parameters that: (i) apply to the full theory, (ii) are consistent with a working screening mechanism (i.e., restoring general relativity locally), (iii) exhibit a continuous, real-valued background cosmology (without …
In general relativity, the endpoint of spherically symmetric gravitational collapse is a Schwarzschild--[(A)dS] black hole. In bimetric gravity, it has been speculated that a static end state must also be Schwarzschild--[(A)dS]. To this end, we present a set of exact solutions, including collapsing massless dust particles. For these, the speculation is confirmed.
In the average action approach to the quantization of gravity the fundamental requirement of "background independence" is met by actually introducing a background metric but leaving it completely arbitrary. The associated Wilsonian renormalization group defines a coarse graining flow on a theory space of functionals which, besides the dynamical metric, depend explicitly on the background metric. All solutions to the truncated flow equations known to date have a trivial background field dependence only, namely via the classical gauge fixing term. In this paper we analyze a number of conceptual …
We consider modified gravity which may describe the early-time inflation and/or late-time cosmic acceleration of the universe. In particular, we discuss the properties of $F(R)$, $F(G)$, string-inspired and scalar-Einstein-Gauss-Bonnet gravities, including their FRW equations and fluid or scalar-tensor description. Simplest accelerating cosmologies are investigated and possibility of unified description of the inflation with dark energy is described. The cosmological reconstruction program which permits to get the requested universe evolution from modified gravity is developed. As some extensi…
Authors Hohmann and Wohlfarth have put forward a no-go theorem for bimetric gravity with positive and negative mass in arXiv:0908.3384v1 [gr-qc]. This comment shows that their no-go theorem does not apply to arXiv:0807.2838v1 [gr-qc].
Bimetric variational formalism was recently employed to construct novel bimetric gravity models. In these models an affine connection is generated by an additional tensor field which is independent of the physical metric. In this work we demonstrate how the ADM decomposition can be applied to study such models and provide some technical intermediate details. Using ADM decomposition we are able to prove that a linear model is unstable as has previously been indicated by perturbative analysis. Moreover, we show that it is also very difficult if not impossible to construct a non-linear model whic…
Bimetric theory describes gravitational interactions in the presence of an extra spin-2 field. Previous work has suggested that its cosmological solutions are generically plagued by instabilities. We show that by taking the Planck mass for the second metric, $M_f$, to be small, these instabilities can be pushed back to unobservably early times. In this limit, the theory approaches general relativity with an effective cosmological constant which is, remarkably, determined by the spin-2 interaction scale. This provides a late-time expansion history which is extremely close to $Λ$CDM, but with a …
We obtain the general cosmological evolution equations for a classically consistent theory of bimetric gravity. Their analytic solutions are demonstrated to generically allow for a cosmic evolution starting out from a matter dominated FLRW universe while relaxing towards a de Sitter (anti-de Sitter) phase at late cosmic time. In particular, we examine a subclass of models which contain solutions that are able to reproduce the expansion history of the cosmic concordance model inspite of the nonlinear couplings of the two metrics. This is demonstrated explicitly by fitting these models to observ…
This review is dedicated to recent progress in the field of classical, interacting, massive spin-2 theories, with a focus on ghost-free bimetric theory. We will outline its history and its development as a nontrivial extension and generalisation of nonlinear massive gravity. We present a detailed discussion of the consistency proofs of both theories, before we review Einstein solutions to the bimetric equations of motion in vacuum as well as the resulting mass spectrum. We introduce couplings to matter and then discuss the general relativity and massive gravity limits of bimetric theory, which…
The bimetric generalization of general relativity has been proven to be able to give an accelerated background expansion consistent with observations. Apart from the energy densities coupling to one or both of the metrics, the expansion will depend on the cosmological constant contribution to each of them, as well as the three parameters describing the interaction between the two metrics. Even for fixed values of these parameters can several possible solutions, so called branches, exist. Different branches can give similar background expansion histories for the observable metric, but may have …
Ghost-free bimetric theory can describe gravity in the presence of an extra spin-2 field. We study certain aspects of dynamics in this theory: (1) It is shown that if either of the metrics is an Einstein solution then the other is always forced to be Einstein, too. For a class of bimetric models this constraint is stronger and as soon as one metric is Einstein, the other metric is forced to be proportional to it. As a consequence, the models in this class avoid a branch of pathological solutions that exhibit determinant singularities or nonlinear ghosts. These constraints persists in a general…
We find the general conditions for viable cosmological solution at the background level in bigravity models. Furthermore, we constrain the parameters by comparing to the Union 2.1 supernovae catalog and identify, in some cases analytically, the best fit parameter or the degeneracy curve among pairs of parameters. We point out that a bimetric model with a single free parameter predicts a simple relation between the equation of state and the density parameter, fits well the supernovae data and is a valid and testable alternative to $Λ$CDM. Additionally, we identify the conditions for a phantom b…
A formulation of linearized gravity which is manifestly invariant under electric-magnetic duality rotations in the internal space of the metric and its dual, and which contains both metrics as basic variables (rather than the corresponding prepotentials), is derived. In this bimetric formulation, the variables have a more immediate geometrical significance, but the action is non-local in space, contrary to what occurs in the prepotential formulation. More specifically, one finds that: (i) the kinetic term is non-local in space (but local in time); (ii) the Hamiltonian is local in space and in …
In this paper we investigate a particular ghost-free bimetric theory that exhibits the partially massless (PM) symmetry at quadratic order. At this order the global SO(1,4) symmetry of the theory is enhanced to SO(1,5). We show that this global symmetry becomes inconsistent at cubic order, in agreement with a previous calculation. Furthermore, we find that the PM symmetry of this theory cannot be extended beyond cubic order in the PM field. More importantly, it is shown that the PM symmetry cannot be extended to quartic order in any theory with one massless and one massive spin-2 fields.
In this paper we study the generation of primordial perturbations in a cosmological setting of bigravity during inflation. We consider a model of bigravity which can reproduce the $Λ$CDM background and large scale structure and a simple model of inflation with a single scalar field and a quadratic potential. Reheating is implemented with a toy-model in which the energy density of the inflaton is entirely dissipated into radiation. We present analytic and numerical results for the evolution of primordial perturbations in this cosmological setting. We find that the amplitude of tensor perturbati…
We consider a manifold endowed with two different vielbeins $E^{A}{}_μ$ and $L^{A}{}_μ$ corresponding to two different metrics $g_{μν}$ and $f_{μν}$. Such a situation arises generically in bimetric or massive gravity (including the recently discussed version of de Rham, Gabadadze and Tolley), as well as in perturbative quantum gravity where one vielbein parametrizes the background space-time and the other the dynamical degrees of freedom. We determine the conditions under which the relation $g^{μν} E^{A}{}_μ L^{B}{}_ν = g^{μν} E^{B}{}_μ L^{A}{}_ν$ can be imposed (or the "Deser-van Nieuwenhuize…
We develop the "generalized Gordon ansatz" for the ghost-free versions of both massive and bimetric gravity, an ansatz which is general enough to include almost all spacetimes commonly considered to be physically interesting, and restricted enough to greatly simplify calculations. The ansatz allows explicit calculation of the matrix square root gamma = sqrt{g^{-1} f} appearing as a central feature of the ghost-free analysis. In particular, this ansatz automatically allows us to write the effective stress-energy tensor as that corresponding to a perfect fluid. A qualitatively similar "generaliz…
We give a basic introduction to ghost-free nonlinear theories involving massive spin-2 fields, focussing on bimetric theory. After motivating the construction of such models from field theoretical considerations, we review the linear theories for massive and massless spin-2 fluctuations propagating on maximally symmetric backgrounds. The structure of general nonlinear spin-2 interactions is explained before we specialise to the ghost-free case. We review the maximally symmetric solutions of bimetric theory, its mass spectrum and the parameter limit which brings the theory close to general rela…
The formulation of an exact functional renormalization group equation for Quantum Einstein Gravity necessitates that the underlying effective average action depends on two metrics, a dynamical metric giving the vacuum expectation value of the quantum field, and a background metric supplying the coarse graining scale. The central requirement of "background independence" is met by leaving the background metric completely arbitrary. This bimetric structure entails that the effective average action may contain three classes of interactions: those built from the dynamical metric only, terms which a…
The ghost-free bimetric theory describes interactions of gravity with another spin-2 field in terms of two Lorentzian metrics. However, if the two metrics do not admit compatible notions of space and time, the formulation of the initial value problem becomes problematic. Furthermore, the interaction potential is given in terms of the square root of a matrix which is in general nonunique and possibly nonreal. In this paper we prove that the reality of the square root matrix leads to a classification of the allowed metrics in terms of the intersections of their null cones. Then, the requirement …
We consider spatially homogeneous and isotropic Friedmann-Robertson-Walker (FRW) solutions of Milgrom's recently proposed class of bimetric theories of gravity. These theories have two different regimes, corresponding to high and low acceleration. We find simple power-law matter dominated solutions in both, as well as solutions with spatial curvature, and exponentially expanding solutions. In the high acceleration limit these solutions behave like the FRW solutions of General Relativity, with a cosmological constant term that is of the correct order of magnitude to explain the observed acceler…
We consider the parametrized post-Newtonian (PPN) limit of ghost-free massive bimetric gravity with two mutually non-interacting matter sectors coupled to the two metrics. Making use of a gauge-invariant differential decomposition of the metric perturbations, we solve the field equations up to the linear PPN order for a static, point-like mass source. From the result we derive the PPN parameter $γ$ for spherically symmetric systems, which describes the gravitational deflection of light by visible matter. By a comparison to its value measured in the solar system we obtain bounds on the paramete…
A bimetric gravity model with a variable speed of light is shown to be in agreement with the results reported from the Planck satellite in 2013. The predicted scalar mode spectral index is $n_s\approx 0.96$ and its running is $α_s\approx 8\times 10^{-4}$ when the fundamental length scale $\ell_0$ in the model is fixed to be $\ell_0\approx 10^5\ell_P$, where $\ell_P$ is the Planck length $\ell_P=1.62\times 10^{-33}\,{\rm cm}$, giving the observed CMB fluctuations: $δ_H\approx 10^{-5}$. The enlarged lightcone ensures that horizon and flatness problems are solved. The model is free from many of t…
An approximately scale invariant spectrum generating the seeds of structure formation is derived from a bimetric gravity theory. By requiring that the amplitude of the CMB fluctuations from the model matches the observed value, we determine the fundamental length scale in the model to be a factor of 10^5 times larger than the Planck length, which results in a scalar mode spectral index: n_s\approx 0.97, and its running: α_s\approx -5\times 10^{-4}. This is accomplished in the variable speed of light (VSL) metric frame, in which the dynamics of perturbations of the bimetric scalar field are det…
We argue that the most conservative geometric extension of Einstein gravity describing both positive and negative mass sources and observers is bimetric gravity and contains two copies of standard model matter which interact only gravitationally. Matter fields related to one of the metrics then appear dark from the point of view of an observer defined by the other metric, and so may provide a potential explanation for the dark universe. In this framework we consider the most general form of linearized field equations compatible with physically and mathematically well-motivated assumptions. Usi…
Bimetric gravity can reproduce the accelerated expansion of the Universe, without a cosmological constant. However, the stability of these solutions to linear perturbations has been questioned, suggesting exponential growth of structure in this approximation. We present a simple model of structure formation, for which an analytical solution is derived. The solution is well-behaved, showing that there is no physical instability with respect to these perturbations. The model can yield a growth of structure exhibiting measurable differences from $Λ$CDM.
This is my contribution to Proceedings of II Russian-Spanish Congress on Particle and Nuclear Physics at all Scales and Cosmology, Saint Petersburg, Russia, October 1 - 4, 2013. We give a brief overview of the ghost-free massive and bimetric gravity emphasising its non-perturbative aspects and ADM approach to calculating the precise number of degrees of freedom. To the standard material, we add some thoughts concerning existence and uniqueness problems of matrix square roots in non-perturbative metric formulations of massive gravity.
We put forward a two-dimensional nonlinear sigma model that couples (bosonic) matter fields to topological Horava gravity on a nonrelativistic worldsheet. In the target space, this sigma model describes classical strings propagating in a curved spacetime background, whose geometry is described by two distinct metric fields. We evaluate the renormalization group flows of this sigma model on a flat worldsheet and derive a set of beta-functionals for the bimetric fields. Imposing worldsheet Weyl invariance at the quantum level, we uncover a set of gravitational field equations that dictate the dy…
This article extends bimetric formulations of massive gravity to make the mass of the graviton to depend on its environment. This minimal extension offers a novel way to reconcile massive gravity with local tests of general relativity without invoking the Vainshtein mechanism. On cosmological scales, it is argued that the model is stable and that it circumvents the Higuchi bound, hence relaxing the constraints on the parameter space. Moreover, with this extension the strong coupling scale is also environmentally dependent in such a way that it is kept sufficiently higher than the expansion rat…
In this dissertation, we explore models based on the idea that there are two metrics in spacetime: One describes the standard gravity, and the other provides a geometry in which matter fields propagate. In order to do that, we provide the essentials of Finsler geometry and the rules to induce a metric for the propagation of matter. Such a description will cover some of the most critical features related to the field necessary to do the induction, these will arise in an attempt to build an action for this field. And finally, we provide an example to study the homogeneous limit of background FLR…
We study bimetric gravity through the context of the AdS/CFT correspondence, especially, in the first order hdrodynamic limit. If we put pure general relativity as a bulk field, the boundary field theory is interpreted as fluid of the N = 4 supersymmetric Yang-Mills plasma. The transport coefficients of this plasma are computed via the AdS/CFT correspondence. Then, we prepare a pair of gravitational fields on the bulk side and let them interact. We expect that two-component fluid emerge on the CFT boundary side because the number of metrics becomes double. However, the situation is rather comp…
Recently, some papers in the literature have shown that, from a bimetric theory of gravity, it is possible to produce massive gravitational waves which generate a longitudinal component in a particular polarization of the wave. After a review of previous works, in this paper the longitudinal response function of interferometers for this particular polarization of the wave is computed in two different gauges, showing the gauge invariance, and in its full frequency dependence, with specific application to the Virgo and LIGO interferometers.
In this paper, we use Statefinder hierarchy method to distinguish between bimetric theory of massive gravity, galileon modified gravity and DGP models applied to late time expansion of the universe. We also carry out comparison between bimetric and DGP models using Statefinder pairs {r, s} and {r, q}. We show that statefinder diagnostic can differentiate between ΛCDM and above mentioned cosmological models of dark energy, and finally show that Statefinder S2 is an excellent discriminant of ΛCDM and modified gravity models.
We discuss in detail a particularly simple example of a bimetric massive gravity model which seems to offer an alternative to the standard cosmological model at background level. For small redshifts, its equation of state is $w(z)\approx-1.22_{-0.02}^{+0.02}-0.64_{-0.04}^{+0.05}z/(1+z)$. Just like $Λ$CDM, it depends on a single parameter, has an analytical background expansion law and fits the expansion cosmological data well. However, confirming previous results, we find that the model is unstable at early times at small scales and speculate over possible ways to cure the instability. In the …
