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Hamiltonian cosmology of bigravity
Abstract
This article is written as a review of the Hamiltonian formalism for the bigravity with de Rham–Gabadadze–Tolley (dRGT) potential, and also of applications of this formalism to the derivation of the background cosmological equations. It is demonstrated that the cosmological scenarios are close to the standard ΛCDM model, but they also uncover the dynamical behavior of the cosmological term. This term arises in bigravity regardless on the choice of the dRGT potential parameters, and its scale is given by the graviton mass. Various matter couplings are considered.
... In our previous work, we considered the Hamiltonian approach to bigravity in metric variables [15,16], in tetrad ones [17,18], and in the minisuperspace [19]. The peculiarities of the different formalisms were that in the metric variables we did not use any explicit expression of the potential, and worked with some equations this potential should fulfill, whereas in the tetrad variables calculations were limited to the minimal potential case β 1 = 0. ...
... Now we will see that they are independent of the formalism, and their explicit meaning is transparent from this work. We are sorry to acknowledge some misprints in the published [19] formula for the dRGT potential in tetrad variables. This work is to complete the analysis of the problem and to compare the results with the ones obtained by other authors. ...
... We prefer to use the same notations as in [17][18][19]. In particular, for spacetime coordinate indices running from 0 to 3, we use small Greek letters; for internal indices running from 1 to 3, we use small Latin letters from the beginning of the alphabet, for spatial indices small letters from the middle of the alphabet are used, for internal indices running from 0 to 3 the capital Latin letters are used. ...
The constraint algebra is derived in the second order tetrad Hamiltonian formalism of the bigravity. This is done by a straightforward calculation without involving any insights, implicit functions, and Dirac brackets. The tetrad approach is the only way to present the bigravity action as a linear functional of lapses and shifts and the Hassan–Rosen transform (characterized as ‘a complicated redefinition of the shift variable’ according to the authors) appears here not as an ansatz but as fixing of a Lagrange multiplier. A comparison of this approach with the other ones is provided.
... In our previous work, we considered the Hamiltonian approach to bigravity in metric variables [15,16], in tetrad ones [17,18], and in the minisuperspace [19]. The peculiarities of the different formalisms were that in the metric variables we did not use any explicit expression of the potential, and worked with some equations this potential should fulfill, whereas in the tetrad variables calculations were limited to the minimal potential case β 1 = 0. ...
... Now we will see that they are independent of the formalism, and their explicit meaning is transparent from this work. We are sorry to acknowledge some misprints in the published [19] formula for the dRGT potential in tetrad variables. This work is to complete the analysis of the problem and to compare the results with the ones obtained by other authors. ...
... We prefer to use the same notations as in Refs. [17,18,19]. In particular, for spacetime coordinate indices running from 0 to 3, we use small Greek letters; for internal indices running from 1 to 3, we use small Latin letters from the beginning of the alphabet, for spatial indices small letters from the middle of the alphabet are used, for internal indices running from 0 to 3 the capital Latin letters are used. ...
The constraint algebra is derived in the 2nd order tetrad Hamiltonian formalism of the bigravity. This is done by a straightforward calculation without involving any insights, implicit functions, and Dirac brackets. The tetrad approach is the only way to present the bigravity action as a linear functional of lapses and shifts, and the Hassan-Rosen transform (characterized as ``complicated redefinition of the shift variable'' according to the authors) appears here not as an ansatz but as a fixing of a Lagrange multiplier. A comparison of this approach with the others is provided.
... from one critical value to the next critical value. 12 This is due to the fact that the standard (observable) variables such as the energy density of matter or of vacuum become some explicit functions of unobservable variable r: ...
This article discusses cosmology, bimetric gravity and their possible interplay.
... A. A. Logunov and his group proposed a bi-metric theory of gravity with a massive graviton and investigated its cosmological consequences in papers [24][25][26][27][28][29] at the period of low popularity of such theories of massive gravity due to a presence of pathologies such as discontinuities and ghosts, see also recent publications of this group and references therein [30][31][32]. ...
Recently, the LIGO-Virgo collaboration discovered gravitational waves and in their first publication on the subject the authors also presented a graviton mass constraint as eV Abbott et al. (2016). In the paper we analyze a potential to reduce upper bounds for graviton mass with future observational data on trajectories of bright stars near the Galactic Center.Since gravitational potentials are different for these two cases, expressions for relativistic advance for general relativity and Yukawa potential are different functions on eccentricity and semimajor axis, it gives an opportunity to improve current estimates of graviton mass with future observational facilities. In our considerations of an improvement potential for a graviton mass estimate we adopt a conservative strategy and assume trajectories of bright stars and their apocenter advance will be described with general relativity expressions and it gives opportunities to improve graviton mass constraints. In contrast with our previous studies, where we present current constraints on parameters of Yukawa gravity (Borka et al., 2013) and graviton mass (Zakharov et al. 2016) from observations of S2 star, in the paper we express expectations to improve current constraints for graviton mass, assuming the GR predictions about apocenter shifts will be confirmed with future observations. We concluded that if future observations of bright star orbits during around fifty years will confirm GR predictions about apocenter shifts of bright star orbits it give an opportunity to constrain a graviton mass at a level around eV or slightly better than current estimates obtained with LIGO observations.
The non-Euclidean geometry created by Bolyai, Lobachevsky and Gauss has led to a new physical theory—general relativity. In due turn, a correct mathematical treatment of the cosmological problem in general relativity has led Friedmann to a discovery of dynamical equations for the universe. And now, after almost a century of theoretical and experimental research, cosmology has a status of the most rapidly developing fundamental science. New challenges here are problems of dark energy and dark matter. As a result, a lot of modifications of general relativity appear recently. The bigravity is one of them, constructed with a couple of interacting space–time metrics accompanied by some coupling to matter. We discuss here this approach and different kinds of the coupling.
The status of experimental tests of general relativity and of theoretical frameworks for analyzing them is reviewed and updated. Einstein’s equivalence principle (EEP) is well supported by experiments such as the Eötvös experiment, tests of local Lorentz invariance and clock experiments. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, the Nordtvedt effect in lunar motion, and frame-dragging. Gravitational wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse-Taylor binary pulsar, and a growing family of other binary pulsar systems is yielding new tests, especially of strong-field effects. Current and future tests of relativity will center on strong gravity and gravitational waves.
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 perturbations. Previous works have investigated special branch of solutions. Here we perform a complete perturbation analysis for the general background equations of motion completing previous analysis.
We analyse the late time cosmology and the gravitational properties of doubly coupled bigravity in the vielbein formalism when the mass of the massive graviton is of the order of the present Hubble rate. We focus on one of the two branches of background cosmology where the ratio between the scale factors of the two metrics is algebraically determined. The Universe evolves from a matter dominated epoch to a dark energy dominated era where the equation of state of dark energy can always be made close to -1 now by appropriately tuning the graviton mass. We also analyse the perturbative spectrum of the theory in the quasi static approximation well below the strong coupling scale where no instability is present and we show that there are five scalar degrees of freedom, two vectors and two gravitons. In a cosmological FRW background for both metrics, four of the five scalars are Newtonian potentials which lead to a modification of gravity on large scales. In this scalar sector, gravity is modified with effects on both the growth of structure and the lensing potential. In particular, we find that the parameter governing the Poisson equation of the weak lensing potential can differ from one in the recent past of the Universe. Overall, the nature of the modification of gravity at low energy, which reveals itself in the growth of structure and the lensing potential, is intrinsically dependent on the couplings to matter and the potential term of the vielbeins. We also find that the time variation of Newton's constant in the Jordan frame can easily satisfy the bound from solar system tests of gravity. Finally we show that the two gravitons present in the spectrum have a non-trivial mass matrix whose origin follows from the potential term of bigravity. This mixing leads to gravitational birefringence.
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 Lorentz violation needs to occur below eV, which still agrees with observations. Finally, we discuss the relevance to other theories of modified gravity.
We show from theoretical considerations, that if the graviton is massive, its mass is constrained to be about 10−32eV/c2. This estimate is consistent with those obtained from experiments, including the recent gravitational wave detection in advanced LIGO.
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 choice of parameters in dRGT bimetric gravity. In addition, we confirm that the usual consistency relation holds and we have a steeper spectrum for the gravitational waves.
We show explicitly that massive, Abelian, vector, just like (properly defined) massive tensor, fields limit smoothly to their massless, gauge, versions: they emit only maximal helicity radiation and mediate Coulomb and (special relativistic) Newtonian, forces between their (conserved) sources. Our main motivation, though, is to show that the recent gravitational wave detection probably cannot directly rule out very long-range gravity: Even though the waves were emitted in a strong field regime, their being detected in the weak field wave zone means the above equivalences apply. There remains the, not unlikely, possibility that no strong field generation of radiation in massive models can reproduce the observed ring-down patterns. Separately, the smooth linear limiting behaviors show that the discontinuity lies not in the mass alone, but rather in Abelian versus non-Abelian, Yang-Mills and General Relativity, regimes, whose respective massive versions are known to be non-physical.
Recently LIGO collaboration discovered gravitational waves \cite{Abbott_16} predicted 100 years ago by A. Einstein. Moreover, in the key paper reporting about the discovery, the joint LIGO \& VIRGO team presented an upper limit on graviton mass such as (Abbott et al. (LIGO collaboration) PRL 116 (2016) 061102). Since the graviton mass limit is so small the authors concluded that their observational data do not show violations of classical general relativity. We consider another opportunity to evaluate a graviton mass from phenomenological consequences of massive gravity and show that an analysis of bright star trajectories could bound graviton mass with a comparable accuracy with accuracies reached with gravitational wave interferometers and expected with forthcoming pulsar timing observations for gravitational wave detection. It gives an opportunity to treat observations of bright stars near the Galactic Center as a wonderful tool not only for an evaluation specific parameters of the black hole but also to obtain constraints on the fundamental gravity law such as a modifications of Newton gravity law in a weak field approximation. In particular, we obtain bounds on a graviton mass based on a potential reconstruction at the Galactic Center.
Massive gravity in the presence of doubly coupled matter field via en effective composite metric yields an accelerated expansion of the universe. It has been recently shown that the model admits stable de Sitter attractor solutions and could be used as a dark energy model. In this work, we perform a first analysis of the constraints imposed by the SNIa, BAO and CMB data on the massive gravity model with the effective composite metric and show that all the background observations are mutually compatible at the one sigma level with the model.
This paper is dedicated to scrutinizing the cosmology in massive gravity. A matter field of the dark sector is coupled to an effective composite metric while a standard matter field couples to the dynamical metric in the usual way. For this purpose, we study the dynamical system of cosmological solutions by using phase analysis, which provides an overview of the class of cosmological solutions in this setup. This also permits us to study the critical points of the cosmological equations together with their stability. We show the presence of stable attractor de Sitter critical points relevant to the late-time cosmic acceleration. Furthermore, we study the tensor, vector and scalar perturbations in the presence of standard matter fields and obtain the conditions for the absence of ghost and gradient instabilities. Hence, massive gravity in the presence of the effective composite metric can accommodate interesting dark energy phenomenology, that can be observationally distinguished from the standard model according to the expansion history and cosmic growth.
We review recent progress in the construction of modified gravity models as alternatives to dark energy as well as the development of cosmological tests of gravity. Einstein's theory of general relativity (GR) has been tested accurately within the local universe i.e. the Solar System, but this leaves the possibility open that it is not a good description of gravity at the largest scales in the Universe. This being said, the standard model of cosmology assumes GR on all scales. In 1998, astronomers made the surprising discovery that the expansion of the Universe is accelerating, not slowing down. This late-time acceleration of the Universe has become the most challenging problem in theoretical physics. Within the framework of GR, the acceleration would originate from an unknown dark energy. Alternatively, it could be that there is no dark energy and GR itself is in error on cosmological scales. In this review, we first give an overview of recent developments in modified gravity theories including f(R) gravity, braneworld gravity, Horndeski theory and massive/bigravity theory. We then focus on common properties these models share, such as screening mechanisms they use to evade the stringent Solar System tests. Once armed with a theoretical knowledge of modified gravity models, we move on to discuss how we can test modifications of gravity on cosmological scales. We present tests of gravity using linear cosmological perturbations and review the latest constraints on deviations from the standard [Formula: see text]CDM model. Since screening mechanisms leave distinct signatures in the non-linear structure formation, we also review novel astrophysical tests of gravity using clusters, dwarf galaxies and stars. The last decade has seen a number of new constraints placed on gravity from astrophysical to cosmological scales. Thanks to on-going and future surveys, cosmological tests of gravity will enjoy another, possibly even more, exciting ten years.
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 correspond to decoupling the massive or the massless spin-2 field from
the matter sector, respectively. More general classical solutions are reviewed
and the present status of bimetric cosmology is summarised. An interesting
corner in the bimetric parameter space which could potentially give rise to a
nonlinear theory for partially massless spin-2 fields is also discussed.
Relations to higher-curvature theories of gravity are explained and finally we
give an overview of possible extensions of the theory and review its
formulation in terms of vielbeins.
We study a previously largely unexplored branch of homogeneous and isotropic
background solutions in ghost-free massive bigravity with consistent double
matter coupling. For a certain family of parameters we find `self-inflated'
FLRW cosmologies, i.e. solutions with an accelerated early-time period during
the radiation-dominated era. In addition, these solutions also display an
accelerated late-time period closely mimicking GR with a cosmological constant.
Interestingly, within this family, the particular case of
gives bouncing cosmologies, where there is an infinite contracting past, a
non-zero minimum value of the scale factor at the bounce, and an infinite
expanding future.
There is a no-go theorem forbidding flat and closed FLRW solutions in massive gravity on a flat reference metric, while open solutions are unstable. Recently it was shown that this no-go theorem can be overcome if at least some matter couples to a hybrid metric composed of both the dynamical and the fixed reference metric. We show that this is not compatible with the standard description of cosmological sources in terms of effective perfect fluids, and the predictions of the theory become sensitive either to the detailed field-theoretical modelling of the matter content or to the presence of additional dark degrees of freedom. This is a serious practical complication. Furthermore, we demonstrate that viable cosmological background evolution with a perfect fluid appears to require the presence of fields with highly contrived properties. This could be improved if the equivalence principle is broken by coupling only some of the fields to the composite metric, but viable self-accelerating solutions due only to the massive graviton are difficult to obtain.
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 the
flavor eigenstates coincide with each other up to the first order of the
slow-roll parameter.
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, , 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 CDM, but with
a technically-natural value for the cosmological constant. We find should
be no larger than the electroweak scale in order for cosmological perturbations
to be stable by big-bang nucleosynthesis.
We study the cosmic expansion history of massive bigravity with a viable matter coupling which treats both metrics on equal footing. We derive the Friedmann equation for the effective metric through which matter couples to the two metrics, and study its solutions. For certain parameter choices, the background cosmology is identical to that of ACDM. More general parameters yield dynamical dark energy, which can still be in agreement with observations of the expansion history. We study specific parameter choices of interest, including minimal models, maximally-symmetric models, and a candidate partially-massless theory.
This paper is devoted to the Hamiltonian analysis of bimetric gravity in the vierbein formulation. We identify all constraints and determine their nature. We also show the existence of an additional constraint so that the scalar mode can be eliminated.
We study the cosmology of bimetric theory with a composite matter coupling.
We find two possible branches of background evolution. We investigate the
question of stability of cosmological perturbations. For the tensor and vector
perturbations, we derive conditions on the absence of ghost and gradient
instabilities. For the scalar modes, we obtain conditions for avoiding ghost
degrees. In the first branch, we find that one of the scalar modes becomes a
ghost at the late stages of the evolution. Conversely, this problem can be
avoided in the second branch. However, we also find that the constraint for the
second branch prevents the doubly coupled matter fields from being the standard
ingredients of cosmology. We thus conclude that a realistic and stable
cosmological model requires additional minimally coupled matter fields.
We study cosmological aspects of a bigravity dRGT model where matter couples
to both metrics. At linear order in perturbations two mass scales emerge: an
hard one from the dRGT potential, and an environmental dependent one from the
coupling of bigravity with matter. During early times the dynamics is dictated
by the second mass scale, of order of Hubble scale. Perturbations can be
classified according to two different combinations. The first is coupled to
matter and follows closely the behavior of GR. The second combination of
fluctuations shows no issues in the scalar sector, while problems arise in the
tensor and vector sectors. During radiation domination, the tensor mode grows
with a power law at super-horizon scales. More dangerously, the propagating
vector mode features an exponential instability on sub-horizon scales. We
discuss the consequences of such instabilities and speculate on possible ways
to deal with them.
The current effort to test General Relativity employs multiple disparate
formalisms for different observables, obscuring the relations between
laboratory, astrophysical and cosmological constraints. To remedy this
situation, we develop a parameter space for comparing tests of gravity on all
scales in the universe. In particular, we present new methods for linking
cosmological large-scale structure, the Cosmic Microwave Background and
gravitational waves with classic PPN tests of gravity. Diagrams of this
gravitational parameter space reveal a noticeable untested regime. The untested
window, which separates small-scale systems from the troubled cosmological
regime, could potentially hide the onset of corrections to General Relativity.
We compute the one-loop quantum corrections to the interactions between the
two metrics of the ghost-free massive bigravity. When considering gravitons
running in the loops, we show how the structure of the interactions gets
destabilized at the quantum level, exactly in the same way as in its massive
gravity limit. A priori one might have expected a better quantum behavior,
however the broken diffeomorphism invariance out of the two initial
diffeomorphisms in bigravity has similar consequences at the quantum level as
the broken diffeomorphism in massive gravity. From lessons of the generated
quantum corrections through matter loops we propose yet other types of
effective composite metrics to which the matter fields can couple. Among these
new effective metrics there might be one or more that could provide interesting
phenomenology and important cosmological implications.
Recently, several works have investigated the coupling to matter in
ghost-free massive (bi-&multi-)gravity and a new effective coupling to matter
has been proposed. In this note we clarify some confusion on the existence and
the implications of a ghost above the strong coupling scale. We confirm that
the standard constraint which is otherwise typically present in this type of
theories disappears on generic backgrounds as soon as this new coupling is
considered. This implies the re-emergence of the Boulware-Deser ghost.
Nevertheless the absence of ghost in the decoupling limit implies that the
cut-off scale (if identified with the scale at which the ghost enters) is
higher than the strong coupling scale. Therefore there is a valid interesting
region of applicability for these couplings at scales below the cut-off.
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.
In this paper we construct a family of consistent ways in which matter can
couple to one or more `metrics'/spin-2 fields in the vielbein formulation. We
do so subject to requiring the weak equivalence principle and the absence of
ghosts from pure spin-2 interactions generated by the matter action. Results
are presented for Massive, Bi- and Multi-Gravity theories and we give explicit
expressions for the effective matter metric in all of these cases.
This document briefly reviews recent short-range gravity experiments that
were performed at below laboratory scales to test the Newtonian inverse square
law of gravity. To compare sensitivities of these measurements, estimates using
the conventional Yukawa parametrization are introduced. Since these experiments
were triggered by the prediction of the large extra-dimension model,
experiments performed at different length scales are compared with this
prediction. In this paper, a direct comparison between laboratory-scale
experiments and the LHC results is presented for the first time. A laboratory
experiment is shown to determine the best limit at and
. In addition, new analysis results are described for
atomic systems used as gravitational microlaboratories.
We investigate the coupling to matter in ghost-free massive (bi-)gravity.
When species in the matter sector couple covariantly to only one metric, we
show that at one-loop these couplings do not spoil the special structure of the
graviton potential. When the same species couples directly to both metrics we
show that a ghost is present at the classical level and that loops destroy the
special structure of the potential at an unacceptably low scale. We then
propose a new `composite' effective metric built out of both metrics. When
matter fields couple covariantly to this effective metric, the would be
Boulware--Deser ghost is absent in different representative limits. At one-loop
such couplings do not detune the special structure of the potential. We
conjecture that matter can couple covariantly to that effective metric in all
generality without introducing any Boulware-Deser ghost below a cut-off scale
parametrically larger than the strong coupling scale. We also discuss
alternative couplings to matter where the kinetic and potential terms of the
matter field couple to different metrics. In both cases we discuss preliminary
implications for cosmology.
We perform a covariant constraint analysis of massive gravity valid for its
entire parameter space, demonstrating that the model generically propagates
five degrees of freedom; this is also verified by a new and streamlined
Hamiltonian description. The constraint's covariant expression permits
computation of the model's caustics. Although new features such as the
dynamical Riemann tensor appear in the characteristic matrix, the model still
exhibits the pathologies uncovered in earlier work: superluminality and likely
acausalities.
In this paper, we discuss the ghost freeness in the case when we add matter coupled to two metrics to the ghost-free bigravity. In this paper we show that the Boulware–Deser ghost generally revives in the presence of doubly coupled matter and that ghost freeness strongly restricts the model of kinetically doubly coupled matter. This result may anticipate difficulties in the attempt to derive the ghost-free bigravity as a low-energy effective theory, starting with a model applicable at high energies.
Matter can be consistently coupled to two metrics at once. This is allowed in
the most general ghost-free, bimetric theory of gravity, and it unlocks an
additional symmetry with respect to the exchange of the metrics. This double
coupling, however, raises the problem of identifying the observables of the
theory. It is shown that there is no physical metric to which matter would
universally couple, and that moreover such an effective metric generically does
not exist even for an individual matter species. A resolution is suggested in
the context of Finsler geometry.
One of the driving aims of modern cosmology is to turn the Universe into a laboratory. By studying cosmic history at both early and late times, we have access to a range of energy scales far exceeding that which we can probe on Earth. It falls to us only to construct the experimental tools for gathering data and the theoretical tools for connecting them to fundamental physics.
Various kinematical relations, holding between hypersurface projections of spacetime tensor fields in an arbitrary Riemannian spacetime, are studied in terms of differential geometry in hyperspace. A criterion is given that a collection of hypertensor fields is generated by the projections of a single spacetime tensor field intersected by the embeddings. From here, it is shown that the super-Hamiltonian of an arbitrary tensor field splits into two parts, Hφ↑ and Hφ+, H φ↑ being local in the field momenta and H φ+ containing their first derivatives. The form of Hφ+ for an arbitrary tensor field is determined from the field behavior under hypersurface tilts. The kinematical equations for the intrinsic metric and the extrinsic curvature are written in a quasicanocical form, and their connection with the closing relations for the gravitational super-Hamiltonian is exhibited. The conservation laws of charge, energy and momentum, and the contracted Bianchi identities, are written as hypertensor equations.
Hyperspace is heuristically defined as an (infinitely dimensional) manifold of all spacelike hypersurfaces embedded in a given Riemannian spacetime. The Riemannian structure (M,g) of spacetime induces a rich geometrical structure in hyperspace. Part of that structure, especially the moving normal frames in hyperspace, Lie derivatives, and symmetrical ▽ and asymmetrical ▽*covariant hyperderivatives, are studied in detail. The formalism introduced in this paper sets the stage for the geometrical study of hypersurface kinematics and dynamics of general tensor fields with derivative gravitational coupling, and of the Dirac-ADM geometrodynamics with such tensor sources, in the following papers.
We develop the Hamiltonian hypersurface dynamics of the gravitational field derivatively coupled to general tensor sources. The closing of constraints follows from the independence of the hypersurface action on the path in the space of embeddings. The derivative coupling breaks the DeWitt supennetric in Riem (m).
Recently, aLIGO has announced the first direct detections of gravitational waves, a direct manifestation of the propagating degrees of freedom of gravity. The detected signals GW150914 and GW151226 have been used to examine the basic properties of these gravitational degrees of freedom, particularly setting an upper bound on their mass. It is timely to review what the mass of these gravitational degrees of freedom means from the theoretical point of view, particularly taking into account the recent developments in constructing consistent massive gravity theories. Apart from the GW150914 mass bound, a few other observational bounds have been established from the effects of the Yukawa potential, modified dispersion relation and fifth force that are all induced when the fundamental gravitational degrees of freedom are massive. We review these different mass bounds and examine how they stand in the wake of recent theoretical developments and how they compare to the bound from GW150914.
The gravitational-wave signal GW150914 was first identified on September 14, 2015, by searches for short-duration gravitational-wave transients. These searches identify time-correlated transients in multiple detectors with minimal assumptions about the signal morphology, allowing them to be sensitive to gravitational waves emitted by a wide range of sources including binary black hole mergers. Over the observational period from September 12 to October 20, 2015, these transient searches were sensitive to binary black hole mergers similar to GW150914 to an average distance of similar to 600 Mpc. In this paper, we describe the analyses that first detected GW150914 as well as the parameter estimation and waveform reconstruction techniques that initially identified GW150914 as the merger of two black holes. We find that the reconstructed waveform is consistent with the signal from a binary black hole merger with a chirp mass of similar to 30 M-circle dot and a total mass before merger of similar to 70 M-circle dot in the detector frame.
We consider the possibility that the massive graviton is a viable candidate of dark matter in the context of bimetric gravity. We first derive the energy-momentum tensor of the massive graviton and show that it indeed behaves as that of dark matter fluid. We then discuss a production mechanism and the present abundance of massive gravitons as dark matter. Since the metric to which ordinary matter fields couple is a linear combination of the two mass eigenstates of bigravity, production of massive gravitons, i.e. the dark matter particles, is inevitably accompanied by generation of massless gravitons, i.e. the gravitational waves. Therefore, in this scenario some information about dark matter in our universe is encoded in gravitational waves. For instance, if LIGO detects gravitational waves generated by the preheating after inflation then the massive graviton with the mass of GeV is a candidate of the dark matter.
Observational evidence for the existence of Dark Matter is limited to its gravitational effects. The extensive program for dedicated searches has yielded null results so far, challenging the most popular models. Here we propose that this is the case because the very existence of cold Dark Matter is a manifestation of gravity itself. The consistent bimetric theory of gravity, the only known ghost-free extension of General Relativity involving a massless and a massive spin-2 field, automatically contains a perfect Dark Matter candidate. We demonstrate that the massive spin-2 particle can be heavy, stable on cosmological scales, and that it interacts with matter only through a gravitational type of coupling. Remarkably, these features persist in the same region of parameter space where bimetric theory satisfies the current gravity tests. We show that the observed Dark Matter abundance can be generated via freeze-in and suggest possible particle physics and gravitational signatures of our bimetric Dark Matter model.
On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO) simultaneously observed the binary black hole merger GW150914. We report the results of a matched-filter search using relativistic models of compact-object binaries that recovered GW150914 as the most significant event during the coincident observations between the two LIGO detectors from September 12 to October 20, 2015. GW150914 was observed with a matched filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203 000 years, equivalent to a significance greater than 5.1 {\sigma}.
Despite its continued observational successes, there is a persistent (and
growing) interest in extending cosmology beyond the standard model,
CDM. This is motivated by a range of apparently serious theoretical
issues, involving such questions as the cosmological constant problem, the
particle nature of dark matter, the validity of general relativity on large
scales, the existence of anomalies in the CMB and on small scales, and the
predictivity and testability of the inflationary paradigm. In this paper, we
summarize the current status of CDM as a physical theory, and review
investigations into possible alternatives along a number of different lines,
with a particular focus on highlighting the most promising directions. While
the fundamental problems are proving reluctant to yield, the study of
alternative cosmologies has led to considerable progress, with much more to
come if hopes about forthcoming high-precision observations and new theoretical
ideas are fulfilled.
We review some recent proposals for relativistic models of dark matter in the
context of bimetric gravity. The aim is to solve the problems of cold dark
matter (CDM) at galactic scales, and to reproduce the phenomenology of the
modified Newtonian dynamics (MOND), while still being in agreement with the
standard cosmological model -CDM at large scales. In this context a
promising alternative is dipolar dark matter (DDM) in which two different
species of dark matter particles are separately coupled to the two metrics of
bigravity and are linked together by an internal vector field. The
phenomenology of MOND then results from a mechanism of gravitational
polarization. Probably the best formulation of the model is within the
framework of recently developed massive bigravity theories. Then the
gravitational sector of the model is safe by construction, but a ghostly degree
of freedom in the decoupling limit is still present in the dark matter sector.
Future work should analyse the cosmological solutions of the model and check
the post-Newtonian parameters in the solar system.
Hyperspace is heuristically defined as an (infinitely dimensional) manifold of all spacelike hypersurfaces embedded in a given Riemannian spacetime. The Riemannian structure (M,g) of spacetime induces a rich geometrical structure in hyperspace. Part of that structure, especially the moving normal frames in hyperspace, Lie derivatives, and symmetrical ∇ and asymmetrical ∇* covariant hyperderivatives, are studied in detail. The formalism introduced in this paper sets the stage for the geometrical study of hypersurface kinematics and dynamics of general tensor fields with derivative gravitational coupling, and of the Dirac–ADM geometrodynamics with such tensor sources, in the following papers.
We use the tetrad formalism to calculate a matrix square root occurring in the de Rham-Gabadadze-Tolley potential. In the minimal case, we obtain the constraints and their algebra. We show that the number of gravitational degrees of freedom corresponds to the massless and massive gravity fields. The Boulware-Deser ghost is eliminated because of two second-class constraints.
Non-minimal matter couplings have recently been considered in the context of
massive gravity and multi-gravity. These couplings are free of the
Boulware-Deser ghost in the decoupling limit and can thus be considered within
an Effective Field Theory setup. Beyond the decoupling limit the ghost was
shown to reemerge in the metric formulation of the theory. Recently it was
argued that this pathology is absent when formulated in terms of unconstrained
vielbeins. We investigate this possibility and show that the Boulware-Deser
ghost is always present beyond the decoupling limit in any dimension larger
than two. We also show that the metric and vielbein formulations have an
identical ghost-free decoupling limit. Finally we extend these arguments to
more generic multi-gravity theories and argue that for any dimension larger
than two a ghost is also present in the vielbein formulation whenever the
symmetric vielbein condition is spoiled and the equivalence with the metric
formulation is lost.
In this work we investigate the existence of relativistic models for dark
matter in the context of bimetric gravity, used here to reproduce the modified
Newtonian dynamics (MOND) at galactic scales. For this purpose we consider two
different species of dark matter particles that separately couple to the two
metrics of bigravity. These two sectors are linked together \textit{via} an
internal U(1) vector field, and some effective composite metric built out of
the two metrics. Among possible models only certain class of kinetic and
interaction terms are allowed without invoking ghost degrees of freedom. Along
these lines we explore the number of allowed kinetic terms in the theory and
point out the presence of ghosts in a previous model. Finally, we propose a
promising class of ghost-free candidate theories that could provide the MOND
phenomenology at galactic scales while reproducing the standard cold dark
matter (CDM) model at cosmological scales.
We consider a recently proposed non-minimal matter coupling in massive
gravity and multi-gravity. We argue that, when formulated in terms of
unconstrained vielbeins, this matter coupling is free of the Boulware-Deser
ghost to all orders away from the decoupling limit.
The tetrad approach is used to resolve the matrix square root appearing in
the dRGT potential. Constraints and their algebra are derived for the minimal
case. It is shown that the number of gravitational degrees of freedom
corresponds to one massless and one massive gravitational fields when two sorts
of matter separately interact with two metric tensors. The Boulware-Deser ghost
is then excluded by two second class constraints. In other case when the matter
couples to a linear combination of two tetrads this ghost re-appears.
After a decade and a half of research motivated by the accelerating universe,
theory and experiment have a reached a certain level of maturity. The
development of theoretical models beyond \Lambda, or smooth dark energy, often
called modified gravity, has led to broader insights into a path forward, and a
host of observational and experimental tests have been developed. In this
review we present the current state of the field and describe a framework for
anticipating developments in the next decade. We identify the guiding
principles for rigorous and consistent modifications of the standard model, and
discuss the prospects for empirical tests. We begin by reviewing attempts to
consistently modify Einstein gravity in the infrared, focusing on the notion
that additional degrees of freedom introduced by the modification must screen
themselves from local tests of gravity. We categorize screening mechanisms into
three broad classes: mechanisms which become active in regions of high
Newtonian potential, those in which first derivatives become important, and
those for which second derivatives are important. Examples of the first class,
such as f(R) gravity, employ the familiar chameleon or symmetron mechanisms,
whereas examples of the last class are galileon and massive gravity theories,
employing the Vainshtein mechanism. In each case, we describe the theories as
effective theories. We describe experimental tests, summarizing laboratory and
solar system tests and describing in some detail astrophysical and cosmological
tests. We discuss future tests which will be sensitive to different signatures
of new physics in the gravitational sector. Parts that are more relevant to
theorists vs. observers/experimentalists are clearly indicated, in the hope
that this will serve as a useful reference for both audiences, as well as
helping those interested in bridging the gap between them.