Boulware-Deser Ghost Research Articles

On the absence of ghosts in quadratic bigravity

An extension of the bimetric theory of gravity is considered that includes quadratic Ricci curvature terms associated with each metric. The issue of the Boulware-Deser ghost is analyzed. The Hamiltonian constraint is derived and the existence of a secondary constraint is shown, proving that the theory is ghost-free.

Bimetric-affine quadratic gravity

Bimetric gravity is a theory of gravity that posits the existence of two interacting and dynamical metric tensors. The spectrum of bimetric gravity consists of a massless and a massive spin-2 particle. The form of the interactions between the two metrics ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$ and ${f}_{\ensuremath{\mu}\ensuremath{\nu}}$ is constrained by requiring absence of the so-called Boulware-Deser ghost. In this work we extend the original bimetric theory to its bimetric-affine counterpart, in which the two connections, associated with the Ricci scalars, are treated as independent variables. We examine in detail the case of an additional quadratic in the Ricci scalar curvature term ${\mathcal{R}}^{2}(g,\mathrm{\ensuremath{\Gamma}})$ and we find that this theory is free of ghosts for a wide range of the interaction parameters, not excluding the possibility of a dark matter interpretation of the massive spin-2 particle.

Spin-2 dark matter from an anisotropic universe in bigravity

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.

Linear growth of structure in projected massive gravity

In the present paper we investigate the linear growth of matter fluctuations based on a concrete model of the projected massive gravity, which is free of the Boulware-Deser ghost and preserves the global Lorentz symmetry. We found that at subhorizon scales, the modification to the linear growth is strongly suppressed even without nonlinear screening of an additional force. In addition, we obtain observational constraints from distance and redshift space distortion measurements and find that there is a parameter region that is consistent both observationally and theoretically.

Bi-gravity with a single graviton

We analyze a bi-gravity model based on the first order formalism, having as fundamental variables two tetrads but only one Lorentz connection. We show that on a large class of backgrounds its linearization agrees with general relativity. At the non-linear level, additional degrees of freedom appear, and we reveal the mechanism hiding them around the special backgrounds. We further argue that they do not contain a massive graviton, nor the Boulware-Deser ghost. The model thus propagates only one graviton, whereas the nature of the additional degrees of freedom remains to be investigated. We also present a foliation-preserving deformation of the model, which keeps all symmetries except time diffeomorphisms and has three degrees of freedom.

Ghost free mimetic massive gravity

The mass of the graviton can be generated using a Brout-Englert-Higgs mechanism with four scalar fields. We show that when one of these fields is costrained as in mimetic gravity, the massive gravity obtained is ghost free and consistent. The mass term is not of the Fierz-Pauli type. There are only five degrees of freedom and the sixth degree of freedom associated with the Boulware-Deser ghost is constrained and replaced by mimetic matter to all orders. The van Dam-Veltman-Zakharov discontinuity is also absent.

Doubly coupled matter fields in massive bigravity**Supported by the Chinese National Youth Thousand Talents Program (71000-41180003), JSPS Grant-in-Aid for Scientific Research (15H02082, 25287054, 26610062), L. H. Acknowledges Financial Support from Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation

In the context of massive (bi-)gravity, non-minimal matter couplings have been proposed. These couplings are special in the sense that they are free of the Boulware-Deser ghost below the strong coupling scale and can be used consistently as an effective field theory. Furthermore, they enrich the phenomenology of massive gravity. We consider these couplings in the framework of bimetric gravity and study the cosmological implications for background and linear tensor, vector, and scalar Previous works have investigated special branches of solutions. Here we perform a complete perturbation analysis for the general background equations of motion, completing previous analyses.

Mimetic massive gravity: beyond linear approximation

We present a theory of ghost-free massive gravity where the mass of the graviton is generated through the Brout-Englert-Higgs (BEH) mechanism and one of the four scalar fields used is that of mimetic gravity. The mass term is not of the Fierz-Pauli type and the constraint eliminates the Boulware-Deser ghost which is absent to all orders. We perform a detailed analysis using the methods of cosmological perturbation theory and consider quantum fluctuations of the degrees of freedom of massive graviton and mimetic matter. It is shown that for three of the degrees of freedom of the graviton of mass m the nonlinear corrections become comparable to the linear terms already at a length scale of order {m}^{-frac{1}{2}} . Thus, at smaller scales they become strongly coupled and the graviton remains with two transverse degrees of freedom which get strongly coupled only at Planck scale. The mimetic field behaves as cold particles of half of the graviton mass and could well explain the source of dark matter in our universe. In the weakly coupled domain mimetic matter is completely decoupled from the massive graviton.

DRGT theory of massive gravity from spontaneous symmetry breaking

In this note we propose a topological action for a Poincare times diffeomorphism invariant gauge theory. We show that there is Higgs phase where the gauge symmetry is spontaneous broken to a diagonal Lorentz subgroup and gives the Einstein–Hilbert action plus the dRGT potential terms. In this vacuum, there are five (three from Goldstone modes) propagating degrees of freedom which form polarizations of a massive spin 2 particle, an extra healthy heavy scalar (Higgs) mode and no Boulware–Deser ghost mode. We further show that the action can be derived in a limit from a topological de Sitter invariant gauge theory in 4 dimensions.

Ghosts in extended quasidilaton theories

We report on our independent investigations of the puzzle of cosmological perturbations in extended quasidilaton. We confirm the claims of presence of the Boulware-Deser ghost. We use both the language of cosmological perturbations with broken diffeomorphisms and the Stuckelberg approach.

Failures of homogeneous and isotropic cosmologies in extended quasidilaton massive gravity

We analyze the Extended Quasi-Dilaton Massive Gravity model around a Friedmann-Lema\^itre-Robertson-Walker cosmological background. We present a careful stability analysis of asymptotic fixed points. We find that the traditional fixed point cannot be approached dynamically, except from a perfectly fine-tuned initial condition involving both the quasi-dilaton and the Hubble parameter. A less-well examined fixed-point solution, where the time derivative of the 0-th St\"uckelberg field vanishes $\dot\phi^0=0$, encounters no such difficulty, and the fixed point is an attractor in some finite region of initial conditions. We examine the question of the presence of a Boulware-Deser ghost in the theory. We show that the additional constraint which generically allows for the elimination of the Boulware-Deser mode is only present under special initial conditions. We find that the only possibility corresponds to the traditional fixed point and the initial conditions are the same fine-tuned conditions that allow the fixed point to be approached dynamically.

Boulware-Deser ghost in extended quasidilaton massive gravity

In the extended quasidilaton massive gravity we perform a nonlinear transformation of the shift vector and then calculate the second derivatives of the Hamiltonian density with respect to the lapse function and the (nonlinearly transformed) shift vector. It is then shown that the $4\times 4$ Hessian matrix is invertible, meaning that the equations of motion for the lapse function and the shift vector simply determine themselves. Therefore, there is no primary constraint that removes the Boulware-Deser ghost.

On scale-free extensions of massive (bi-)gravity

We discuss a scale-free model of bigravity, in which the mass parameter of the standard bigravity potential is promoted to a dynamical scalar field. This modification retains the ghost-free bigravity structure, in particular it remains free of the Boulware-Deser ghost. We investigate the theory’s interaction structure, focusing on its consistent scaling limits and strong coupling scales. Furthermore we explore the model’s quadratic action, both around generic background configurations and paying special attention to cosmological backgrounds and to the associated background evolution. Finally we consider the possibility of realising a phase of late-time acceleration as well as a quasi-de Sitter inflationary stage at early times, when the promoted “mass scalar” becomes the inflaton.

A spectre is haunting the cosmos: quantum stability of massive gravity with ghosts

Many theories of modified gravity with higher order derivatives are usually ignored because of serious problems that appear due to an additional ghost degree of freedom. Most dangerously, it causes an immediate decay of the vacuum. However, breaking Lorentz invariance can cure such abominable behavior. By analyzing a model that describes a massive graviton together with a remaining Boulware-Deser ghost mode we show that even ghostly theories of modified gravity can yield models that are viable at both classical and quantum levels and, therefore, they should not generally be ruled out. Furthermore, we identify the most dangerous quantum scattering process that has the main impact on the decay time and find differences to simple theories that only describe an ordinary scalar field and a ghost. Additionally, constraints on the parameters of the theory including some upper bounds on the Lorentz-breaking cutoff scale are presented. In particular, for a simple theory of massive gravity we find that a breaking of Lorentz invariance is allowed to happen even at scales above the Planck mass. Finally, we discuss the relevance to other theories of modified gravity.

Absence of the Boulware-Deser ghost in novel graviton kinetic terms

Using novel nonlinear kinetic terms for gravitons, a large class of bi-gravity models were constructed, which are potentially free of the Boulware-Deser (BD) ghost. In this work, we derive their Hamiltonians using the ADM formalism, and verify that the BD ghost is eliminated by additional constraints. The general Hamiltonian structure is analogous to that of the other class of bi-gravity models free of the BD ghost.

Novel nonlinear kinetic terms for gravitons

A set of novel derivative terms for spin-2 fields are proposed. They are the wedge products of curvature two-forms and vielbeins. In this work, we investigate the properties of novel two-derivative terms in the context of bi-gravity. Based on a minisuperspace analysis, we identify a large class of bi-gravity models where the Boulware-Deser ghost could be absent. We give a new perspective that Weyl Gravity and New Massive Gravity belong to this class of bi-gravity models involving novel derivative terms. In addition, we discuss the UV cut-off scales, dynamical symmetric conditions and novel higher-derivative terms.

Cosmology with three interacting spin-2 fields

Theories of massive gravity with one or two dynamical metrics generically lack stable and observationally-viable cosmological solutions that are distinguishable from $\Lambda$CDM. We consider an extension to trimetric gravity, with three interacting spin-2 fields which are not plagued by the Boulware-Deser ghost. We systematically explore every combination with two free parameters in search of background cosmologies that are competitive with $\Lambda$CDM. For each case we determine whether the expansion history satisfies viability criteria, and whether or not it contains beyond-$\Lambda$CDM phenomenology. Among the many models we consider, there are only three cases that seem to be both viable and distinguishable from standard cosmology. One of the models has only one free parameter and displays a crossing from above to below the phantom divide. The other two provide scaling behavior, although they contain future singularities that need to be studied in more detail. These models possess interesting features that make them compelling targets for a full comparison to observations of both cosmological expansion history and structure formation.

New quasidilaton theory in partially constrained vielbein formalism

In this work we study the partially constrained vielbein formulation of the new quasidilaton theory of massive gravity, where the quasidilaton field couples to both physical and fiducial metrics simultaneously via a composite effective metric and Lorentz violation is introduced by a constraint on the vielbein. This formalism improves the new quasidilaton model since the Boulware-Deser ghost is removed fully non-linearly at all scales. This also yields crucial implications in the cosmological applications. We derive the governing cosmological background evolution and study the stability of the attractor solution.

Linear spin-2 fields in most general backgrounds

We derive the full perturbative equations of motion for the most general background solutions in ghost-free bimetric theory in its metric formulation. Clever field redefinitions at the level of fluctuations enable us to circumvent the problem of varying a square-root matrix appearing in the theory. This greatly simplifies the expressions for the linear variation of the bimetric interaction terms. We show that these field redefinitions exist and are uniquely invertible if and only if the variation of the square-root matrix itself has a unique solution, which is a requirement for the linearized theory to be well defined. As an application of our results we examine the constraint structure of ghost-free bimetric theory at the level of linear equations of motion for the first time. We identify a scalar combination of equations which is responsible for the absence of the Boulware-Deser ghost mode in the theory. The bimetric scalar constraint is in general not manifestly covariant in its nature. However, in the massive gravity limit the constraint assumes a covariant form when one of the interaction parameters is set to zero. For that case our analysis provides an alternative and almost trivial proof of the absence of the Boulware-Deser ghost. Our findings generalize previous results in the metric formulation of massive gravity and also agree with studies of its vielbein version.

Cutoff for extensions of massive gravity and bi-gravity

Recently there has been interest in extending ghost-free massive gravity, bi-gravity, and multi-gravity by including non-standard kinetic terms and matter couplings. We first review recent proposals for this class of extensions, emphasizing how modifications of the kinetic and potential structure of the graviton and modifications of the coupling to matter are related. We then generalize existing no-go arguments in the metric language to the vielbein language in second-order form. We give an ADM argument to show that the most promising extensions to the kinetic term and matter coupling contain a Boulware–Deser ghost. However, as recently emphasized, we may still be able to view these extensions as effective field theories below some cutoff scale. To address this possibility, we show that there is a decoupling limit where a ghost appears for a wide class of matter couplings and kinetic terms. In particular, we show that there is a decoupling limit where the linear effective vielbein matter coupling contains a ghost. Using the insight we gain from this decoupling limit analysis, we place an upper bound on the cutoff for the linear effective vielbein coupling. This result can be generalized to new kinetic interactions in the vielbein language in second-order form. Combined with recent results, this provides a strong uniqueness argument on the form of ghost-free massive gravity, bi-gravity, and multi-gravity.