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In this work we shall address the ghost issue of $F\left( R,\mathcal{G} \right)$ gravity, which is known to be plagued with ghost degrees of freedom. These ghosts occur due to the presence of higher than two derivatives in the field equations, and can arise even when considering cosmological perturbations, where superluminal modes may arise in the theory. If we consider the quantum theory, the ghosts generate the negative norm states, which give the negative probabilities, and therefore the ghosts are physically inconsistent. Motivated by the importance of $F\left( R,\mathcal{G} \right)$ gravity for providing viable inflationary and dark energy phenomenologies, in this work we shall provide a technique that can render $F\left( R,\mathcal{G} \right)$ gravity theories free from ghost degrees of freedom. This will be done by introducing two auxiliary scalar fields, and by employing the Lagrange multiplier technique, the theory is ghost free in the Einstein frame. Also the framework can be viewed as a reconstruction technique and can be used as a method in order to realize several cosmological evolutions of interest. We demonstrate how we can realize several cosmologically interesting phenomenologies by using the reconstruction technique., Comment: NPB Accepted

Recent observation shows that general relativity (GR) is not valid in the strong regime. $\mathit{f(R)}$ gravity where $\mathit{R}$ is the Ricci scalar, is regarded to be one of good candidates able to cure the anomalies appeared in the conventional general relativity. In this realm, we apply the equation of motions of $\mathit{f(R)}$ gravity to a spherically symmetric spacetime with two unknown functions and derive original black hole (BH) solutions without any constrains on the Ricci scalar as well as on the form of $\mathit{f(R)}$ gravity. Those solutions depend on a convolution function and are deviating from the Schwarzschild solution of the Einstein GR. These solutions are characterized by the gravitational mass of the system and the convolution function that in the asymptotic form gives extra terms that are responsible to make such BHs different from GR. Also, we show that these extra terms make the singularities of the invariants much weaker than those of the GR BH. We analyze such BHs using the trend of thermodynamics and show their consistency with the well known quantities in thermodynamics like the Hawking radiation, entropy and quasi-local energy. We also show that our BH solutions satisfy the first law of thermodynamics. Moreover, we study the stability analysis using the odd-type mode and shows that all the derived BHs are stable and have radial speed equal to one. Finally, using the geodesic deviations we derive the stability conditions of these BHs., 21 pages, 21 figures, will appear in Phys. Rev. D. arXiv admin note: text overlap with arXiv:2010.04701

In this paper we investigate how to realize various quite well known cosmological bouncing models in the context of the recently developed unimodular $F(R)$ gravity. Particularly, we shall study the matter bounce scenario, the singular bounce, the superbounce and a symmetric bounce scenario. We present the behavior of the Hubble radius for each of the bouncing models we shall take into account and we investigate which era of the bouncing model is responsible for the cosmological perturbations. As we shall demonstrate, the various bouncing models do not behave in the same way, so the cosmological perturbations for each model may correspond to a different era, in comparison to other models. Also we present which unimodular $F(R)$ gravity realizes each model. We also show that Newton's law is not modified in the unimodular $F(R)$ gravity, which also is proven to be a ghost-free theory, and in addition we discuss the matter stability issue. Finally, we demonstrate how it is possible to solve a cosmological constant problem in the context of unimodular $F(R)$ gravity., PRD version

We review the accelerating (mainly, dark energy) cosmologies in modified gravity. Special attention is paid to cosmologies leading to finite-time future singularities in $F(R)$, $F(G)$ and $\mathcal{F}(R,G)$ modified gravities. The removal of the finite-time future singularities via addition of $R^2$-term which simultaneously unifies the early-time inflation with late-time acceleration is also briefly mentioned. Accelerating cosmology including the scenario unifying inflation with dark energy is considered in $F(R)$ gravity with Lagrange multipliers. In addition, we examine domain wall solutions in $F(R)$ gravity. Furthermore, covariant higher derivative gravity with scalar projectors is explored., 16 pages, no figure, to appear in: Proceedings of the 7th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics (Sept. 9--19, 2012, Belgrade, Serbia). Editors B. Dragovich and Z. Rakic. Published by Institute of Physics, Belgrade 2013, Serbia

Multiple $��$CDM cosmology is studied in a way that is formally a classical analog of the Casimir effect. Such cosmology corresponds to a time-dependent dark fluid model or, alternatively, to its scalar field presentation, and it motivated by the string landscape picture. The future evolution of the several dark energy models constructed within the scheme is carefully investigated. It turns out to be almost always possible to choose the parameters in the models so that they match the most recent and accurate astronomical values. To this end, several universes are presented which mimick (multiple) $��$CDM cosmology but exhibit Little Rip, asymptotically de Sitter, or Type I, II, III, and IV finite-time singularity behavior in the far future, with disintegration of all bound objects in the cases of Big Rip, Little Rip and Pseudo-Rip cosmologies., LaTeX 11 pages, 10 figures

We study the $f(R)$ theory of gravity using metric approach. In particular we investigate the recently proposed model by Hu-Sawicki, Appleby $-$ Battye and Starobinsky. In this model, the cosmological constant is zero in flat space time. The model passes both the Solar system and the laboratory tests. But the model parameters need to be fine tuned to avoid the finite time singularity recently pointed in the literature. We check the concordance of this model with the $H(z)$ and baryon acoustic oscillation data. We find that the model resembles the $\Lambda$CDM at high redshift. However, for some parameter values there are variations in the expansion history of the universe at low redshift., Comment: 16 pages and 9 figures, typos corrected, few references and minor clarifications added, revised version to appera in PRD

A general approach to viable modified $f(R)$ gravity is developed in both the Jordan and the Einstein frames. A class of exponential, realistic modified gravities is introduced and investigated with care. Special focus is made on step-class models, most promising from the phenomenological viewpoint and which provide a natural way to classify all viable modified gravities. One- and two-steps models are explicitly considered, but the analysis is extensible to $N$-step models. Both inflation in the early universe and the onset of recent accelerated expansion arise in these models in a natural, unified way. Moreover, it is demonstrated that models in this category easily pass all local tests, including stability of spherical body solution, non-violation of Newton's law, and generation of a very heavy positive mass for the additional scalar degree of freedom., Comment: 15 pages, 2 figures, version to appear in Physical Review D

The functional formulation and one-loop effective action for scalar self-interacting theory non-linearly coupled with some power of the curvature are studied. After the explicit one-loop renormalization at weak curvature, we investigated numerically the phase structure for such unusual phi^4 theory. It is demonstrated the possibility of curvature-induced phase transitions for positive values of the curvature power, while for negative values the radiative symmetry breaking does not take place. The dynamical mechanism for the explanation of the current smallness of the cosmological constant is presented for several models from the class of theories under consideration., Comment: 25 pages, 20 eps figures, use iopart.cls; minor corrections, references added, to appear in JCAP

We consider thermal effects in the propagation of gravitational waves on a cosmological background. In particular, we consider scalar field cosmologies and study gravitational modes near cosmological singularities. We point out that the contribution of thermal radiation can heavily affect the dynamics of gravitational waves giving enhancement or dissipation effects both at quantum and classical level.These effects are considered both in General Relativity and in modified theories like $F(R)$ gravity which can be easily reduced to scalar-tensor cosmology. The possible detection and disentanglement of standard and scalar gravitational modes on the stochastic background are also discussed., Comment: 21 pages, accepted for publication in Physics of the Dark Universe

The detection of gravitational wave modes and polarizations could constitute an extremely important signature to discriminate among different theories of gravity. According to this statement, it is possible to prove that higher-order non-local gravity has formally the same gravitational spectrum of higher-order local gravity. In particular, we consider the cases of $f \left( R, \Box R, \Box^2 R, \cdots, \Box^n R \right) = R + \sum_{i=1}^n \alpha_i R \Box^i R$ gravity, linear with respect to both $R$ and $\Box^i R$ and $ f \left( R, \Box R \right) = R + \alpha \left(\Box R\right)^2 $ gravity, quadratic with respect to $\Box R$, where it is demonstrated the graviton amplitude changes if compared with General Relativity. We also obtain the gravitational spectrum of higher-order non-local gravity $ f \left( R, \Box^{-1} R, \Box^{-2} R, \cdots, \Box^{-n} R \right) = R + \sum_{i=1}^n \alpha_i R \Box^{-i} R$. In this case, we have three state of polarization and $n+3$ oscillation modes. More in detail, it is possible to derive two transverse tensor $(+)$ and $(\times)$ standard polarization modes of frequency $\omega_{1}$, massless and with 2-helicity; $n+1$ further scalar modes of frequency $\omega_{2},\dots,\omega_{n+2}$, massive and with 0-helicity, each of which has the same mixed polarization, partly longitudinal and partly transverse., Comment: 15 pages, to appear in Physics. Lett. B

We analyze seven different viable $f(R)$-gravities towards the Solar System tests and stochastic gravitational waves background. The aim is to achieve experimental bounds for the theory at local and cosmological scales in order to select models capable of addressing the accelerating cosmological expansion without cosmological constant but evading the weak field constraints. Beside large scale structure and galactic dynamics, these bounds can be considered complimentary in order to select self-consistent theories of gravity working at the infrared limit. It is demonstrated that seven viable $f(R)$-gravities under consideration not only satisfy the local tests, but additionally, pass the above PPN-and stochastic gravitational waves bounds for large classes of parameters., Comment: 23 pages, 8 figures

Phantom cosmology allows to account for dynamics and matter content of the universe tracing back the evolution to the inflationary epoch, considering the transition to the non-phantom standard cosmology (radiation/matter dominated eras) and recovering the today observed dark energy epoch. We develop the unified phantom cosmology where the same scalar plays the role of early time (phantom) inflaton and late-time Dark Energy. The recent transition from decelerating to accelerating phase is described too by the same scalar field. The (dark) matter may be embedded in this scheme, giving the natural solution of the coincidence problem. It is explained how the proposed unified phantom cosmology can be fitted against the observations which opens the way to define all the important parameters of the model., Comment: LaTeX file, 9 pages, no figure, version to appear in PLB

We discuss the PPN Solar-System constraints and the GW stochastic background considering some recently proposed $f(R)$ gravity models which satisfy both cosmological and stability conditions. Using the definition of PPN-parameters $\gamma$ and $\beta$ in terms of $f(R)$-models and the definition of scalar GWs, we compare and discuss if it is possible to search for parameter ranges of $f(R)$-models working at Solar System and GW stochastic background scale., Comment: 4 pages, Grassmann Meeting 2009, Annalen der Physik

The future finite-time singularities emerging in alternative gravity dark energy models are classified and studied in Jordan and Einstein frames. It is shown that such singularity may occur even in flat spacetime for the specific choice of the effective potential. The conditions for the avoidance of finite-time singularities are presented and discussed. The problem is reduced to the study of a scalar field evolving on an effective potential by using the conformal transformations. Some viable modified gravity models are analyzed in detail and the way to cure singularity is considered by introducing the higher-order curvature corrections. These results maybe relevant for the resolution of the conjectured problem in the relativistic star formation in such modified gravity where finite-time singularity is also manifested., 21 pages, 12 figures, published version in PRD

Dark energy dynamics of the universe can be achieved by equivalent mathematical descriptions taking into account generalized fluid equations of state in General Relativity, scalar-tensor theories or modified F(R) gravity in Einstein or Jordan frames. The corresponding technique transforming equation of state description to scalar-tensor or modified gravity is explicitly presented. We show that such equivalent pictures can be discriminated by matching solutions with data capable of selecting the true physical frame., Comment: LaTeX file, 10 pages, 3 figures, the fit of two equivalent models with observational data is added, new references

It is shown that phantom scalar models can be mapped into a mathematically equivalent, modified $F(R)$ gravity, which turns out to be complex, in general. Only for even scalar potentials is the ensuing modified gravity real. It is also demonstrated that, even in this case, modified gravity becomes complex at the region where the original phantom dark energy theory develops a Big Rip singularity. A number of explicit examples are presented which show that these two theories are not completely equivalent, from the physical viewpoint. This basically owes to the fact that the physical metric in both theories differ in a time-dependent conformal factor. As a result, an FRW accelerating solution, or FRW instanton, in the scalar-tensor theory may look as a deccelerating FRW solution, or a non-instantonic one, in the corresponding modified gravity theory., Comment: LaTeX file, 9 pages. Some references are added. The version to appear in Physcs Letters B