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We introduce a quasidilaton scalar field to the minimal theory of massive gravity with the Minkowski fiducial metric, in such a way that the quasidilaton global symmetry is maintained and that the theory admits a stable self-accelerating de Sitter solution. We start with a precursor theory that contains three propagating gravitational degrees of freedom without a quasidilaton scalar and introduce St\"uckelberg fields to covariantize its action. This makes it possible for us to formulate the quasidilaton global symmetry that mixes the St\"uckelberg fields and the quasidilaton scalar field. By the Hamiltonian analysis we confirm that the precursor theory with the quasidilaton scalar contains four degrees of freedom, three from the precursor massive gravity and one from the quasidilaton scalar. We further remove one propagating degree of freedom to construct the minimal quasidilaton theory with three propagating degrees of freedom, corresponding to two polarizations of gravitational waves from the minimal theory of massive gravity and one scalar from the quasidilaton field, by carefully introducing two additional constraints to the system in the Hamiltonian language. Switching to the Lagrangian language, we find self-accelerating de Sitter solutions in the minimal quasidilaton theory and analyze their stability. It is found that the self-accelerating de Sitter solution is stable in a wide range of parameters., Comment: 19 pages, 4 figures; title changed to adapt to published version, references added, minor typos corrected (equations (60) and (84)), minor changes to text

Variable-length codes are the bases of the free submonoids of a free monoid. There are some important longstanding open questions about the structure of finite maximal codes. In this paper we discuss this conjectures and their relations with factorizations of cyclic groups.

Gaia Data Release 1 (Gaia DR1) contains astrometric results for more than 1 billion stars brighter than magnitude 20.7 based on observations collected by the Gaia satellite during the first 14 months of its operational phase. We give a brief overview of the astrometric content of the data release and of the model assumptions, data processing, and validation of the results. For stars in common with the Hipparcos and Tycho-2 catalogues, complete astrometric single-star solutions are obtained by incorporating positional information from the earlier catalogues. For other stars only their positions are obtained by neglecting their proper motions and parallaxes. The results are validated by an analysis of the residuals, through special validation runs, and by comparison with external data. Results. For about two million of the brighter stars (down to magnitude ~11.5) we obtain positions, parallaxes, and proper motions to Hipparcos-type precision or better. For these stars, systematic errors depending e.g. on position and colour are at a level of 0.3 milliarcsecond (mas). For the remaining stars we obtain positions at epoch J2015.0 accurate to ~10 mas. Positions and proper motions are given in a reference frame that is aligned with the International Celestial Reference Frame (ICRF) to better than 0.1 mas at epoch J2015.0, and non-rotating with respect to ICRF to within 0.03 mas/yr. The Hipparcos reference frame is found to rotate with respect to the Gaia DR1 frame at a rate of 0.24 mas/yr. Based on less than a quarter of the nominal mission length and on very provisional and incomplete calibrations, the quality and completeness of the astrometric data in Gaia DR1 are far from what is expected for the final mission products. The results nevertheless represent a huge improvement in the available fundamental stellar data and practical definition of the optical reference frame., Comment: Accepted for publication in Astronomy & Astrophysics

We present a thorough stability analysis of modified gravity theories in the presence of matter fields. We use the Effective Field Theory framework for Dark Energy and Modified Gravity to retain a general approach for the gravity sector and a Sorkin-Schutz action for the matter one. Then, we work out the proper viability conditions to guarantee in the scalar sector the absence of ghosts, gradient and tachyonic instabilities. The absence of ghosts can be achieved by demanding a positive kinetic matrix, while the lack of a gradient instability is ensured by imposing a positive speed of propagation for all the scalar modes. In case of tachyonic instability, the mass eigenvalues have been studied and we work out the appropriate expressions. For the latter, an instability occurs only when the negative mass eigenvalue is much larger, in absolute value, than the Hubble parameter. We discuss the results for the minimally coupled quintessence model showing for a particular set of parameters two typical behaviours which in turn lead to a stable and an unstable configuration. Moreover, we find that the speeds of propagation of the scalar modes strongly depend on matter densities, for the beyond Horndeski theories. Our findings can be directly employed when testing modified gravity theories as they allow to identify the correct viability space., Comment: 21 pages, 1 figure; appendix B added, comments added in Secs IV A and B; Matches published version

The standard $\Lambda$-CDM predicts a growth of structures which tends to be higher than the values of redshift space distortion (RSD) measurements, if the cosmological parameters are fixed by the CMB data. In this paper we point out that this discrepancy can be resolved/understood if we assume that the graviton has a small but non-zero mass. In the context of the Minimal Theory of Massive Gravity (MTMG), due to infrared Lorentz violations measurable only at present cosmological scales, the graviton acquires a mass without being haunted by unwanted extra degrees of freedom. While the so-called self-accelerating branch of cosmological solutions in MTMG has the same phenomenology for the background as well as the scalar- and vector-type linear perturbations as the $\Lambda$-CDM in General Relativity (GR), it is possible to choose another branch so that the background is the same as that in GR but the evolution of matter perturbations gets modified by the graviton mass. On studying the fit of such modified dynamics to the above-mentioned RSD measurements, we find that the $\Lambda$-CDM model is less probable than MTMG by two orders of magnitude. With the help of the cross correlation between the integrated Sachs-Wolfe (ISW) effect and the large scale structure (LSS), the data also pin-down the graviton mass squared around $\mu^2\approx - (3\times 10^{-33} \rm{eV})^2$, which is consistent with the latest bound $|\mu^2|<(1.2\times 10^{-22} \rm{eV})^2$ set by the recent LIGO observation., Comment: 4 pages, 4 figures. Revised to match the PRL version

We consider the finite interactions of the generalized Proca theory including the sixth-order Lagrangian and derive the full linear perturbation equations of motion on the flat Friedmann-Lema\^{i}tre-Robertson-Walker background in the presence of a matter perfect fluid. By construction, the propagating degrees of freedom (besides the matter perfect fluid) are two transverse vector perturbations, one longitudinal scalar, and two tensor polarizations. The Lagrangians associated with intrinsic vector modes neither affect the background equations of motion nor the second-order action of tensor perturbations, but they do give rise to non-trivial modifications to the no-ghost condition of vector perturbations and to the propagation speeds of vector and scalar perturbations. We derive the effective gravitational coupling $G_{\rm eff}$ with matter density perturbations under a quasi-static approximation on scales deep inside the sound horizon. We find that the existence of intrinsic vector modes allows a possibility for reducing $G_{\rm eff}$. In fact, within the parameter space, $G_{\rm eff}$ can be even smaller than the Newton gravitational constant $G$ at the late cosmological epoch, with a peculiar phantom dark energy equation of state (without ghosts). The modifications to the slip parameter $\eta$ and the evolution of growth rate $f\sigma_8$ are discussed as well. Thus, dark energy models in the framework of generalized Proca theories can be observationally distinguished from the $\Lambda$CDM model according to both cosmic growth and expansion history. Furthermore, we study the evolution of vector perturbations and show that outside the vector sound horizon the perturbations are nearly frozen and start to decay with oscillations after the horizon entry., Comment: 23 pages, 3 figures

We develop an exceptional generalised geometry formalism for massive type IIA supergravity. In particular, we construct a deformation of the generalised Lie derivative, which generates the type IIA gauge transformations as modified by the Romans mass. We apply this new framework to consistent Kaluza-Klein reductions preserving maximal supersymmetry. We find a generalised parallelisation of the exceptional tangent bundle on S^6, and from this reproduce the consistent truncation ansatz and embedding tensor leading to dyonically gauged ISO(7) supergravity in four dimensions. We also discuss closely related hyperboloid reductions, yielding a dyonic ISO(p,7-p) gauging. Finally, while for vanishing Romans mass we find a generalised parallelisation on S^d, d=4,3,2, leading to a maximally supersymmetric reduction with gauge group SO(d+1) (or larger), we provide evidence that an analogous reduction does not exist in the massive theory., Comment: 69 pages; v2: version published in JHEP

We consider a massive vector field with derivative interactions that propagates only the 3 desired polarizations (besides two tensor polarizations from gravity) with second-order equations of motion in curved space-time. The cosmological implications of such generalized Proca theories are investigated for both the background and the linear perturbation by taking into account the Lagrangian up to quintic order. In the presence of a matter fluid with a temporal component of the vector field, we derive the background equations of motion and show the existence of de Sitter solutions relevant to the late-time cosmic acceleration. We also obtain conditions for the absence of ghosts and Laplacian instabilities of tensor, vector, and scalar perturbations in the small-scale limit. Our results are applied to concrete examples of the general functions in the theory, which encompass vector Galileons as a specific case. In such examples, we show that the de Sitter fixed point is always a stable attractor and study viable parameter spaces in which the no-ghost and stability conditions are satisfied during the cosmic expansion history., Comment: 23 pages, 7 figures, revised version

In this work we study the partially constrained vielbein formulation of the new quasidilaton theory of massive gravity which couples to both physical and fiducial metrics simultaneously via a composite effective metric. 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., Comment: 13 pages

For a massive vector field with derivative self-interactions, the breaking of the gauge invariance allows the propagation of a longitudinal mode in addition to the two transverse modes. We consider generalized Proca theories with second-order equations of motion in a curved space-time and study how the longitudinal scalar mode of the vector field gravitates on a spherically symmetric background. We show explicitly that cubic-order self-interactions lead to the suppression of the longitudinal mode through the Vainshtein mechanism. Provided that the dimensionless coupling of the interaction is not negligible, this screening mechanism is sufficiently efficient to give rise to tiny corrections to gravitational potentials consistent with solar-system tests of gravity. We also study the quartic interactions with the presence of non-minimal derivative coupling with the Ricci scalar and find the existence of solutions where the longitudinal mode completely vanishes. Finally, we discuss the case in which the effect of the quartic interactions dominates over the cubic one and show that local gravity constraints can be satisfied under a mild bound on the parameters of the theory., Comment: 20 pages, 3 figures, revised version

In Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, it is known that the conical singularity arises at the center of a spherically symmetric body ($r=0$) in the case where the parameter $\alpha_{{\rm H}4}$ characterizing the deviation from the Horndeski Lagrangian $L_4$ approaches a non-zero constant as $r \to 0$. We derive spherically symmetric solutions around the center in full GLPV theories and show that the GLPV Lagrangian $L_5$ does not modify the divergent property of the Ricci scalar $R$ induced by the non-zero $\alpha_{{\rm H}4}$. Provided that $\alpha_{{\rm H}4}=0$, curvature scalar quantities can remain finite at $r=0$ even in the presence of $L_5$ beyond the Horndeski domain. For the theories in which the scalar field $\phi$ is directly coupled to $R$, we also obtain spherically symmetric solutions inside/outside the body to study whether the fifth force mediated by $\phi$ can be screened by non-linear field self-interactions. We find that there is one specific model of GLPV theories in which the effect of $L_5$ vanishes in the equations of motion. We also show that, depending on the sign of a $L_5$-dependent term in the field equation, the model can be compatible with solar-system constraints under the Vainshtein mechanism or it is plagued by the problem of a divergence of the field derivative in high-density regions., Comment: 25 pages, 1 figure

We investigate the minimal theory of massive gravity (MTMG) recently introduced. After reviewing the original construction based on its Hamiltonian in the vielbein formalism, we reformulate it in terms of its Lagrangian in both the vielbein and the metric formalisms. It then becomes obvious that, unlike previous attempts in the literature of Lorentz-violating massive gravity, not only the potential but also the kinetic structure of the action is modified from the de Rham-Gabadadze-Tolley (dRGT) massive gravity theory. We confirm that the number of physical degrees of freedom in MTMG is two at fully nonlinear level. This proves the absence of various possible pathologies such as superluminality, acausality and strong coupling. Afterwards, we discuss the phenomenology of MTMG in the presence of a dust fluid. We find that on a flat homogeneous and isotropic background we have two branches. One of them (self-accelerating branch) naturally leads to acceleration without the genuine cosmological constant or dark energy. For this branch both the scalar and the vector modes behave exactly as in general relativity (GR). The phenomenology of this branch differs from GR in the tensor modes sector, as the tensor modes acquire a non-zero mass. Hence, MTMG serves as a stable nonlinear completion of the self-accelerating cosmological solution found originally in dRGT theory. The other branch (normal branch) has a dynamics which depends on the time-dependent fiducial metric. For the normal branch, the scalar mode sector, even though as in GR only one scalar mode is present (due to the dust fluid), differs from the one in GR, and, in general, structure formation will follow a different phenomenology. The tensor modes will be massive, whereas the vector modes, for both branches, will have the same phenomenology as in GR., Comment: 21 pages

We propose a dark energy model where a scalar field $\phi$ has nonlinear self-interactions in the presence of a dilatonic coupling with the Ricci scalar. This belongs to a sub-class of theories beyond Horndeski, which accommodates covariant Galileons and Brans-Dicke theories as specific cases. We derive conditions under which the scalar sound speed squared $c_{\rm s}^2$ is positive from the radiation era to today. Since $c_{\rm s}^2$ remains to be smaller than the order of 1, the deviation from Horndeski theories does not cause heavy oscillations of gauge-invariant gravitational potentials. In this case, the evolution of matter perturbations at low redshifts is similar to that in the coupled dark energy scenario with an enhanced gravitational interaction. On the spherically symmetric background with a matter source, the existence of field self-interactions suppresses the propagation of fifth force inside a Vainshtein radius. We estimate an allowed parameter space in which the model can be compatible with solar system constraints while satisfying conditions for the cosmological viability of background and perturbations., Comment: 15 pages, 8 figures

A splicing system is a formal model of a recombinant behaviour of sets of double stranded DNA molecules when acted on by restriction enzymes and ligase. In this survey we will concentrate on a specific behaviour of a type of splicing systems, introduced by P\u{a}un and subsequently developed by many researchers in both linear and circular case of splicing definition. In particular, we will present recent results on this topic and how they stimulate new challenging investigations., Comment: Appeared in: Discrete Mathematics and Computer Science. Papers in Memoriam Alexandru Mateescu (1952-2005). The Publishing House of the Romanian Academy, 2014. arXiv admin note: text overlap with arXiv:1112.4897 by other authors

We consider a consistent linear effective vielbein matter coupling without introducing the Boulware-Deser ghost in ghost-free massive gravity. This is achieved in the partially constrained vielbein formulation. We first introduce the formalism and prove the absence of ghost at all scales. As next we investigate the cosmological application of this coupling in this new formulation. We show that even if the background evolution accords with the metric formulation, the perturbations display important different features in the partially constrained vielbein formulation. We study the cosmological perturbations of the two branches of solutions separately. The tensor perturbations coincide with those in the metric formulation. Concerning the vector and scalar perturbations, the requirement of absence of ghost and gradient instabilities yields slightly different allowed parameter space., Comment: 25 pages

In a class of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, we derive both vacuum and interior Schwarzschild solutions under the condition that the derivatives of a scalar field $\phi$ with respect to the radius $r$ vanish. If the parameter $\alpha_{\rm H}$ characterizing the deviation from Horndeski theories approaches a non-zero constant at the center of a spherically symmetric body, we find that the conical singularity arises at $r=0$ with the Ricci scalar given by $R=-2\alpha_{\rm H}/r^2$. This originates from violation of the geometrical structure of four-dimensional curvature quantities. The conical singularity can disappear for the models in which the parameter $\alpha_{\rm H}$ vanishes in the limit that $r \to 0$. We propose explicit models without the conical singularity by properly designing the classical Lagrangian in such a way that the main contribution to $\alpha_{\rm H}$ comes from the field derivative $\phi'(r)$ around $r=0$. We show that the extension of covariant Galileons with a diatonic coupling allows for the recovery of general relativistic behavior inside a so-called Vainshtein radius. In this case, both the propagation of a fifth force and the deviation from Horndeski theories are suppressed outside a compact body in such a way that the model is compatible with local gravity experiments inside the solar system., Comment: 10 pages

The GERDA experiment at the Laboratori Nazionali del Gran Sasso (LNGS) searches for the neutrinoless double beta decay of 76-Ge. In view of the GERDA Phase II data collection, four new 228-Th radioactive sources for the calibration of the germanium detectors enriched in 76-Ge have been produced with a new technique, leading to a reduced neutron flux from ( alpha; n ) reactions. The gamma activities of the sources were determined with a total uncertainty of 4 percent using an ultra-low background HPGe detector operated underground at LNGS. The emitted neutron flux was determined using a low background LiI(Eu) detector and a 3-He counter at LNGS. In both cases, a reduction of about one order of magnitude with respect to commercially available 228-Th sources was obtained. Additionally, a specific leak test with a sensitivity to leaks down to 10 mBq was developed to investigate the tightness of the stainless steel capsules housing the sources after their use in cryogenic environment., Comment: 14 pages, 5 figures, 3 tables

We propose a new theory of massive gravity with only two propagating degrees of freedom. After defining the theory in the unitary gauge in the vielbein language, we shall perform a Hamiltonian analysis to count the number of physical degrees of freedom, and then study some phenomenologies. While the homogeneous and isotropic background cosmology and the tensor linear perturbations around it are described by exactly the same equations as those in the de Rham-Gabadadze-Tolley (dRGT) massive gravity, the scalar and vector gravitational degrees of freedom are absent in the new theory at the fully nonlinear level. Hence the new theory provides a stable nonlinear completion of the self-accelerating cosmological solution that was originally found in the dRGT theory., Comment: 5 pages. Matched with the published version

We introduce the notion of specular sets which are subsets of groups called here specular and which form a natural generalization of free groups. These sets are an abstract generalization of the natural codings of linear involutions. We prove several results concerning the subgroups generated by return words and by maximal bifix codes in these sets., Comment: arXiv admin note: substantial text overlap with arXiv:1405.3529

In the approach of the effective field theory of modified gravity, we derive the equations of motion for linear perturbations in the presence of a barotropic perfect fluid on the flat isotropic cosmological background. In a simple version of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, which is the minimum extension of Horndeski theories, we show that a slight deviation of the tensor propagation speed squared $c_{\rm t}^2$ from 1 generally leads to the large modification to the propagation speed squared $c_{\rm s}^2$ of a scalar degree of freedom $\phi$. This problem persists whenever the kinetic energy $\rho_X$ of the field $\phi$ is much smaller than the background energy density $\rho_m$, which is the case for most of dark energy models in the asymptotic past. Since the scaling solution characterized by the constant ratio $\rho_X/\rho_m$ is one way out for avoiding such a problem, we study the evolution of perturbations for a scaling dark energy model in the framework of GLPV theories in the Jordan frame. Provided the oscillating mode of scalar perturbations is fine-tuned so that it is initially suppressed, the anisotropic parameter $\eta=-\Phi/\Psi$ between the two gravitational potentials $\Psi$ and $\Phi$ significantly deviates from 1 for $c_{\rm t}^2$ away from 1. For other general initial conditions, the deviation of $c_{\rm t}^2$ from 1 gives rise to the large oscillation of $\Psi$ with the frequency related to $c_{\rm s}^2$. In both cases, the model can leave distinct imprints for the observations of CMB and weak lensing., Comment: 20 pages, 4 figures, published in JCAP

One century after its formulation, Einstein's general relativity has made remarkable predictions and turned out to be compatible with all experimental tests. Most of these tests probe the theory in the weak-field regime, and there are theoretical and experimental reasons to believe that general relativity should be modified when gravitational fields are strong and spacetime curvature is large. The best astrophysical laboratories to probe strong-field gravity are black holes and neutron stars, whether isolated or in binary systems. We review the motivations to consider extensions of general relativity. We present a (necessarily incomplete) catalog of modified theories of gravity for which strong-field predictions have been computed and contrasted to Einstein's theory, and we summarize our current understanding of the structure and dynamics of compact objects in these theories. We discuss current bounds on modified gravity from binary pulsar and cosmological observations, and we highlight the potential of future gravitational wave measurements to inform us on the behavior of gravity in the strong-field regime., Comment: 188 pages, 46 figures, 6 tables, 903 references. Matches version published in Classical and Quantum Gravity. Supplementary data files available at http://www.phy.olemiss.edu/~berti/research/ and http://centra.tecnico.ulisboa.pt/network/grit/files/

This paper presents the analytical solution of the inverse ray tracing problem for photons emitted by a star and collected by an observer located in the gravitational field of the Solar System. This solution has been conceived to suit the accuracy achievable by the ESA Gaia satellite (launched on December 19, 2013) consistently with the measurement protocol in General relativity adopted within the RAMOD framework. Aim of this study is to provide a general relativistic tool for the science exploitation of such a revolutionary mission, whose main goal is to trace back star directions from within our local curved space-time, therefore providing a three-dimensional map of our Galaxy. The results are useful for a thorough comparison and cross-checking validation of what already exists in the field of Relativistic Astrometry. Moreover, the analytical solutions presented here can be extended to model other measurements that require the same order of accuracy expected for Gaia., Comment: 29 pages, 1 figure

We consider a class of non-local gravity theories where the Lagrangian is a function of powers of the inverse d'Alembertian operator acting on the Ricci scalar. We take an approach in which the non-local Lagrangian is made local by introducing auxiliary scalar fields, and study the degrees of freedom of the localized Lagrangian. We find that among the auxiliary scalar fields introduced, some of them are always ghost-like. That is, in the Einstein frame they develop a negative kinetic term. Because of this, except for a particular case already known in the literature, in general, it is not clear how to quantize these models and how to interpret this theory in the light of standard field theory., Comment: 13 pages