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Hojman Symmetry Approach for Scalar-Tensor Cosmology
Abstract
Scalar-tensor Cosmologies can be dealt under the standard of the Hojman conservation theorem
that allows to fix the form of the coupling F( \phi), of the potential V (\phi ) and to find out exact
solutions for related cosmological models. Specifically, the existence of a symmetry transformation
vector for the equations of motion gives rise to a Hojman conserved quantity on the corresponding
minisuperpace and exact solutions for the cosmic scale factor a and the scalar field \phi can be achieved.
In particular, we take advantage of the fact that minimally coupled solutions, previously obtained
in the Einstein frame, can be conformally transformed in non-minimally coupled solutions in the
Jordan frame. Some physically relevant examples are worked out.
... Motivated by the above-mentioned features of f (R) gravity, a Noether symmetry approach has already been developed by the authors [21] to obtain (2 + 1)-dimensional black hole solutions in the framework of f (R) gravity. An alternative approach, so-called Hojman symmetry approach, has recently been received attention by which one can find new exact solutions [22][23][24][25][26][27][28][29]. Unlike the Noether symmetry approach which needs Lagrangian and Hamiltonian functions, in the Hojman symmetry approach, we just need the symmetry vectors and the corresponding conserved charges which are easily obtained by using the equations of motion. ...
... By using Eqs. (24) and (73) we have ...
... By this choice and using Eqs. (24), (80) and (81), we obtain ...
In this paper, we use the Hojman symmetry approach to find new [Formula: see text]-dimensional [Formula: see text] gravity solutions, in comparison to Noether symmetry approach. In the special case of Hojman symmetry vector [Formula: see text], we recover [Formula: see text]-dimensional Bañados–Teitelboim–Zanelli (BTZ) black hole and generalized [Formula: see text]-dimensional BTZ black hole solutions, obtained by Noether symmetry approach, and the interesting point is that the cosmological constant is appeared as the direct manifestation of Hojman symmetry.
... In general, its conserved quantities and the exact solutions can be quite different from the ones via Noether symmetry. In fact, recently Hojman symmetry has been used in cosmology and gravity theory [33][34][35]. It is found that Hojman symmetry exists for a wide range of the potential V (φ) of quintessence [33] and scalar-tensor theory [34], and the corresponding exact cosmological solutions have been obtained. ...
... In fact, recently Hojman symmetry has been used in cosmology and gravity theory [33][34][35]. It is found that Hojman symmetry exists for a wide range of the potential V (φ) of quintessence [33] and scalar-tensor theory [34], and the corresponding exact cosmological solutions have been obtained. While Noether symmetry exists only for exponential potential V (φ) [19,26,27], Hojman symmetry can exist for a wide range of potentials V (φ), including not only exponential but also power-law, hyperbolic, logarithmic and other complicated potentials [33,34]. ...
... It is found that Hojman symmetry exists for a wide range of the potential V (φ) of quintessence [33] and scalar-tensor theory [34], and the corresponding exact cosmological solutions have been obtained. While Noether symmetry exists only for exponential potential V (φ) [19,26,27], Hojman symmetry can exist for a wide range of potentials V (φ), including not only exponential but also power-law, hyperbolic, logarithmic and other complicated potentials [33,34]. On the other hand, it is also found that Hojman symmetry exists in f (T ) theory and the corresponding exact cosmological solutions are obtained [35]. ...
Nowadays, $f(R)$ theory has been one of the leading modified gravity theories
to explain the current accelerated expansion of the universe, without invoking
dark energy. It is of interest to find the exact cosmological solutions of
$f(R)$ theories. In fact, symmetry has been proved as a powerful tool to find
exact solutions in physics. As is well known, Noether symmetry has been
extensively used in cosmology and gravity theories. Recently, the so-called
Hojman symmetry was also considered in the literature. Hojman symmetry directly
deals with the equations of motion, rather than Lagrangian or Hamiltonian,
unlike Noether symmetry. In this work, we consider Hojman symmetry in $f(R)$
theories in both the metric and Palatini formalisms, and find the corresponding
exact cosmological solutions of $f(R)$ theories via Hojman symmetry. We show
that the results are different from the ones obtained by using Noether symmetry
in $f(R)$ theories. The present work confirms that Hojman symmetry can bring
new features to cosmology and gravity theories.
... In [12], Hojman showed that this conservation theorem can drastically restrict the functional form of point symmetry transformations, and a harmonic oscillator system was considered as a concrete example. Recently, Hojman conservation theorem has been used in cosmology and gravity theory [15,16]. It is found that Hojman conserved quantities exist for a wide range of the potential V (φ) of quintessence [15] and scalar-tensor theory [16], and the corresponding exact cosmological solutions have been obtained. ...
... Recently, Hojman conservation theorem has been used in cosmology and gravity theory [15,16]. It is found that Hojman conserved quantities exist for a wide range of the potential V (φ) of quintessence [15] and scalar-tensor theory [16], and the corresponding exact cosmological solutions have been obtained. As mentioned above, Hojman conserved quantities and other related quantities can be different from the ones of Noether. ...
... As mentioned above, Hojman conserved quantities and other related quantities can be different from the ones of Noether. In fact, Noether symmetry exists only for exponential potential V (φ) [1,6,7], while Hojman symmetry exists for a wide range of potentials V (φ), including not only exponential but also power-law, hyperbolic, logarithmic and other complicated potentials [15,16]. Therefore, Hojman symmetry might give rise to new features in cosmology and gravity theory. ...
In this work, we consider Hojman symmetry in $f(T)$ theory. Unlike Noether
conservation theorem, the symmetry vectors and the corresponding conserved
quantities in Hojman conservation theorem can be obtained by using directly the
equations of motion, rather than Lagrangian or Hamiltonian. We find that Hojman
symmetry can exist in $f(T)$ theory, and the corresponding exact cosmological
solutions are obtained. We find that the functional form of $f(T)$ is
restricted to be the power-law or hypergeometric type, while the universe
experiences a power-law or hyperbolic expansion. These results are different
from the ones obtained by using Noether symmetry in $f(T)$ theory.
... Hojman [34] and González-Gascón [35] explored a new approach in the search for first integrals, which has become more and more important over the last several years for its applications to f (R)-gravity and FRW cosmology [36][37][38][39][40][41][42]. ...
A geometric approach to the integrability and reduction of dynamical systems, both when dealing with systems of differential equations and in classical physics, is developed from a modern perspective. The main ingredients of this analysis are infinitesimal symmetries and tensor fields that are invariant under the given dynamics. A particular emphasis is placed on the existence of alternative invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to the Noether theorem and non-Noether constants of motion. We also recall the geometric approach to Sundman infinitesimal time-reparametrisation for autonomous systems of first-order differential equations and some of its applications to integrability, and an analysis of how to define Sundman transformations for autonomous systems of second-order differential equations is proposed, which shows the necessity of considering alternative tangent bundle structures. A short description of alternative tangent structures is provided, and an application to integrability, namely, the linearisability of scalar second-order differential equations under generalised Sundman transformations, is developed.
... In the preceding sections 4 and 5 we have first summarised the usual way of searching firstintegrals by means of infinitesimal symmetries via the first Noether theorem and presented then a second procedure for finding non-Noether constants of motion, which is not based on symmetries but on the existence of alternative geometric structures for the description of the vector field, what leads to the existence of a recursion operator. We mention next a third approach started by Hojman [36] and González-Gascón in [37] and that it is becoming more and more important during the last years for its applications in f (R)-gravity and FRW cosmology [38,39,40,41,42,43,44]. ...
A geometric approach to integrability and reduction of dynamical system is developed from a modern perspective. The main ingredients in such analysis are the infinitesimal symmetries and the tensor fields that are invariant under the given dynamics. Particular emphasis is given to the existence of invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to Noether theorem and non-Noether constants of motion. The geometric approach to Hamilton-Jacobi equation is shown to be a particular example of the search for related field in a lower dimensional manifold.
... Apart from the usual way of finding first-integrals from infinitesimal symmetries via the Noether theorem, and a second procedure based on the existence of alternative geometric structures for the description of the vector field providing us a recursion operator, there is a third approach started by Hojman [75] and González-Gascón in [76], which is becoming more and more important because of its applications in f (R)-gravity and FRW cosmology [77][78][79][80][81][82][83]. It was introduced first for divergence-free vector fields in an oriented manifold (M, Ω), in the particular case of a SODE and then generalised to arbitrary SODE vector fields. ...
We review the general theory of the Jacobi last multipliers in geometric terms and then apply the theory to different problems in integrability and the inverse problem for one-dimensional mechanical systems. Within this unified framework, we derive the explicit form of a Lagrangian obtained by several authors for a given dynamical system in terms of known constants of the motion via a Jacobi multiplier for both autonomous and nonautonomous systems, and some examples are used to illustrate the general theory. Finally, some geometric results on Jacobi multipliers and their use in the study of Hojman symmetry are given.
... The third approach we want to study here is the one started in a coordinate dependent way by Hojman [4] for the case of a divergence-free non-autonomous system of second-order ordinary differential equations and later on extended by González-Gascón in [5] for a non divergence-free case, and that it is becoming more and more important during the last years for its applications in f (R)-gravity and FRW cosmology [6,7,8,9,10,11,12]. The geometric approach to autonomous systems provides a method to incorporate holonomic constraints by replacing the systems by vector fields in a differentiable manifold and allows to develop coordinate free formulations of the problem. ...
The geometric intrinsic approach to Hojman symmetry is developed and use is made of the theory of the Jacobi last multipliers to find the corresponding conserved quantity for non divergence-free vector fields. The particular cases of autonomous Lagrangian and Hamiltonian systems are studied as well as the generalization of these results to normalizer vector fields of the dynamics. The nonautonomous cases, where normalizer vector fields play a relevant role, are also developed.
... The third approach we want to study here is the one started in a coordinate dependent way by Hojman [4] for the case of a divergence-free non-autonomous system of second-order ordinary differential equations and later on extended by González-Gascón in [5] for a non divergence-free case, and that it is becoming more and more important during the last years for its applications in f (R)-gravity and FRW cosmology [6,7,8,9,10,11,12]. The geometric approach to autonomous systems provides a method to incorporate holonomic constraints by replacing the systems by vector fields in a differentiable manifold and allows to develop coordinate free formulations of the problem. ...
The geometric intrinsic approach to Hojman symmetry is developed and use is made of the theory of the Jacobi last multipliers to find the corresponding conserved quantity for non divergence-free vector fields. The particular cases of autonomous Lagrangian and Hamiltonian systems are studied as well as the generalization of these results to normalizer vector fields of the dynamics. The nonautonomous cases, where normalizer vector fields play a relevant role, are also developed.
... As a final comment, it is worth noticing that, apart from the Lie point symmetries a , which are the simplest kind of symmetries, there exist several other types of transformations for searching for symmetries in differential equations. In particular, in [19], S. Hojman proposed a conservation theorem where one uses directly the equations of motion, rather than the Lagrangian or the Hamiltonian of a system and, in general, the conserved quantities can be different from those derived from the Noether Symmetry Approach [20,21]. In addition, there exist higher order symmetries, such as contact symmetries; that is, when the equation of motion are invariant under contact transformations, which are defined as one parameter transformations, in the tangent bundle of the associated dynamical system [22]. ...
We review the Noether Symmetry Approach as a geometric criterion to select theories of gravity. Specifically, we deal with Noether Symmetries to solve the field equations of given gravity theories. The method allows to find out exact solutions, but also to constrain arbitrary functions in the action. Specific cosmological models are taken into account.
... As a final comment, it is worth noticing that, apart from the Lie point symmetries a , which are the simplest kind of symmetries, there exist several other types of transformations for searching for symmetries in differential equations. In particular, in [19], S. Hojman proposed a conservation theorem where one uses directly the equations of motion, rather than the Lagrangian or the Hamiltonian of a system and, in general, the conserved quantities can be different from those derived from the Noether Symmetry Approach [20,21]. In addition, there exist higher order symmetries, such as contact symmetries; that is, when the equation of motion are invariant under contact transformations, which are defined as one parameter transformations, in the tangent bundle of the associated dynamical system [22]. ...
We review the Noether Symmetry Approach to select viable theories of gravity. Specifically, we deal with Noether Symmetries to solve field equations of given gravity theories. The method allows to find exact solutions, but also to constrain arbitrary functions in the action. Specific cosmological models are taken into account.
In this paper, we find exact string cosmological solutions for FRW cosmology, by using the Hojman symmetry approach. The string cosmology under consideration includes a scalar field [Formula: see text] with the potential [Formula: see text], and a totally antisymmetric field strength [Formula: see text] which is specifically defined in terms of the scale factor [Formula: see text]. We show that for this string cosmology, Hojman conserved quantities exist using which new exact solutions for the scale factor and the scalar field are obtained for specific potentials [Formula: see text] with some free parameters. The presence of these parameters, together with those of arising from Hojman symmetry, is an important advantage using which one can construct the solutions [Formula: see text] and [Formula: see text] with variety of cosmological behaviors.
Scalar-tensor gravitational theories are important extensions of standard general relativity, which can explain both the initial inflationary evolution, as well as the late accelerating expansion of the Universe. In the present paper we investigate the cosmological solution of a scalar-tensor gravitational theory, in which the scalar field $\phi $ couples to the geometry via an arbitrary function $F(\phi $). The kinetic energy of the scalar field as well as its self-interaction potential $V(\phi )$ are also included in the gravitational action. By using a standard mathematical procedure, the Lie group approach, and Noether symmetry techniques, we obtain several exact solutions of the gravitational field equations describing the time evolutions of a flat Friedman-Robertson-Walker Universe in the framework of the scalar-tensor gravity. The obtained solutions can describe both accelerating and decelerating phases during the cosmological expansion of the Universe.
We discuss the Noether Symmetry Approach in the framework of Gauss-Bonnet
cosmology showing that the functional form of the $F(R, {\cal G})$ function,
where $R$ is the Ricci scalar and ${\cal G}$ is the Gauss-Bonnet topological
invariant, can be determined by the presence of symmetries. Besides, the method
allows to find out exact solutions due to the reduction of cosmological
dynamical system and the presence of conserved quantities. Some specific
cosmological models are worked out
We investigate the main features of the flat
Friedmann-Lema{\i}tre-Robertson-Walker cosmological models in the f(T)
teleparallel gravity. In particular, a general approach to find out exact
cosmological solutions in f (T) gravity is discussed. Instead of taking into
account phenomenological models, we consider as a selection criterion, the
existence of Noether symmetries in the cosmological f(T) point-like Lagrangian.
We find that only power-law models admit extra Noether symmetries. A complete
analysis of such cosmological models is developed.
We present a new approach to find exact solutions for cosmological models. By
requiring the existence of a symmetry transformation vector for the equations
of motion of the given cosmological model (without using either Lagrangian or
Hamiltonian), one can find corresponding Hojman conserved quantities. With the
help of these conserved quantities, the analysis of the cosmological model can
be simplified. In the case of quintessence scalar-tensor models, we show that
the Hojman conserved quantities exist for a wide range of V(\phi)-potentials
and allow to find exact solutions for the cosmic scale factor and the scalar
field. Finally, we investigate the general cosmological behavior of solutions
by adopting a phase-space view.
We discuss the Hamiltonian dynamics for cosmologies coming from Extended
Theories of Gravity. In particular, minisuperspace models are taken into
account searching for Noether symmetries. The existence of conserved quantities
gives selection rule to recover classical behaviors in cosmic evolution
according to the so called Hartle criterion, that allows to select correlated
regions in the configuration space of dynamical variables. We show that such a
statement works for general classes of Extended Theories of Gravity and is
conformally preserved. Furthermore, the presence of Noether symmetries allows a
straightforward classification of singularities that represent the points where
the symmetry is broken. Examples of nonminimally coupled and higher-order
models are discussed.
From the above discussions, it is clear that the cosmological problem can be approached in several ways. The issue is essentially
connected with the fact that no «ultimate» theory still exists. However, some definitive results seem to have been acquired
in the past decades both from the theoretical and the observational point of view. First of all, the standard cosmological
model, which describes the observed Universe sufficiently well, fails in the very early stages of evolution since it cannot
coherently solve some shortcomings. They essentially are the flatness, the horizon, the isotropy problems and, mainly, the
singularity problem, that is the failure of GR at primordial eras. Furthermore, in that framework, it is not possible to explain
the large-scale structure starting from some theory of primordial perturbations.
We discuss scalar–tensor cosmology with an extra R−1 correction by the Noether symmetry approach. The existence of such a symmetry selects the forms of the coupling ω(φ), of the potential V(φ) and allows to obtain physically interesting exact cosmological solutions.
We quantize a flat FRW cosmology in the context of the f(R) gravity by Noether symmetry approach. We explicitly calculate the form of f(R) for which such symmetries exist. It is shown that the existence of a Noether symmetry yields a general solution of the Wheeler–DeWitt equation where can be expressed as a superposition of states of the form eiS. In terms of Hartle criterion, this type of wave function exhibits classical correlations, i.e. the emergent of classical universe is expected due to the oscillating behavior of the solutions of Wheeler–DeWitt equation. According to this interpretation we also provide the Noether symmetric classical solutions of our f(R) cosmological model.
In this study, we consider a flat Friedmann-Robertson-Walker (FRW) universe
in the context of Palatini $f(R)$ theory of gravity. Using the dynamical
equivalence between $f(R)$ gravity and scalar-tensor theories, we construct a
point Lagrangian in the flat FRW spacetime. Applying {\em Noether gauge
symmetry approach} for this $f(R)$ Lagrangian we find out the form of $f(R)$
and the exact solution for cosmic scale factor. It is shown that the resulting
form of $f(R)$ yield a power-law expansion for the scale factor of the
universe.
In this Letter by utilizing the Noether symmetry approach in cosmology, we
attempt to find the tachyon potential via the application of this kind of
symmetry to a flat Friedmann-Robertson-Walker (FRW) metric. We reduce the
system of equations to simpler ones and obtain the general class of the
tachyon's potential function and $f(R)$ functions. We have found that the
Noether symmetric model results in a power law $f(R)$ and an inverse fourth
power potential for the tachyonic field. Further we investigate numerically the
cosmological evolution of our model and show explicitly the behavior of the
equation of state crossing the cosmological constant boundary.
We discuss the f(R) gravity model in which the origin of dark energy is
identified as a modification of gravity. The Noether symmetry with gauge term
is investigated for the f(R) cosmological model. By utilization of the Noether
Gauge Symmetry (NGS) approach, we obtain two exact forms f(R) for which such
symmetries exist. Further it is shown that these forms of f(R) are stable.
In this Letter, we find suitable potentials in the multiple scalar fields scenario by using the Noether symmetry approach. We discussed three models with multiple scalar fields: N-quintessence with positive kinetic terms, N-phantom with negative kinetic terms and N-quintom with both positive and negative kinetic terms. In the N-quintessence case, the exponential potential which could be derived from several theoretic models is obtained from the Noether conditions. In the N-phantom case, the potential $\frac{V_{0}}{2}(1-\cos(\sqrt{\frac{3N}{2}}\frac{\phi}{m_{pl}}))$, which could be derived from the Pseudo Nambu-Goldstone boson model, is chosen as the Noether conditions required. In the N-quintom case, we derive a relation $DV'_{\phi q}=-\tilde{D}V'_{\phi p}$ between the potential forms for the quintessence-like fields and the phantom-like fields by using the Noether symmetry. Comment: 22 pages, 3 figures
We study the cosmological models whose ordinary differential equations can be given a Lagrangian description on a two-dimensional configuration space.'' By requiring the existence of a Noether symmetry for such a Lagrangian, we are able to show that the potential must have an exponential form. With the help of the constants of motion we can get from that Lagrangian, we integrate the models and analyze the behavior for all the possible varying free parameters and plot some of the solutions. We also compare our results with others available in the literature.
We study Palatini f(R) cosmology using Noether symmetry approach for the
matter dominated universe. In order to construct a point-like Lagrangian in the
flat FRW space time, we use the dynamical equivalence between f(R) gravity and
scalar-tensor theories. The existence of Noether symmetry of the cosmological
f(R) Lagrangian helps us to find out the form of f(R) and the exact solutions
for cosmic scale factor. We show that this symmetry always exist for f(R)~R^n
and the Noether constant is a function of the Newton's gravitational constant
and the current matter content of the universe.
The debate on the physical relevance of conformal transformations can be faced by taking the Palatini approach into account to gravitational theories. We show that conformal transformations are not only a mathematical tool to disentangle gravitational and matter degrees of freedom (passing from the Jordan frame to the Einstein frame) but they acquire a physical meaning considering the bi-metric structure of Palatini approach which allows to distinguish between spacetime structure and geodesic structure. Examples of higher-order and non-minimally coupled theories are worked out and relevant cosmological solutions in Einstein frame and Jordan frames are discussed showing that also the interpretation of cosmological observations can drastically change depending on the adopted frame.
We search for spherically symmetric solutions of f(R) theories of gravity via the Noether Symmetry Approach. A general formalism in the metric framework is developed considering a point-like f(R)-Lagrangian where spherical symmetry is required. Examples of exact solutions are given.
Noether symmetry for higher order gravity theory has been explored, with the introduction of an auxiliary variable which gives the only correct quantum desccription of the theory, as shown in a series of earlier papers. The application of Noether theorem in higher order theory of gravity turned out to be a powerful tool to find the solution of the field equations. A few such physically reasonable solutions like power law inflation are presented.
As is well known, symmetry plays an important role in the theoretical physics. In particular, the well-known Noether symmetry is an useful tool to select models motivated at a fundamental level, and find the exact solution to the given Lagrangian. In the present work, we try to consider Noether symmetry in f(T)f(T) theory. At first, we briefly discuss the Lagrangian formalism of f(T)f(T) theory. In particular, the point-like Lagrangian is explicitly constructed. Based on this Lagrangian, the explicit form of f(T)f(T) theory and the corresponding exact solution are found by requiring Noether symmetry. In the resulting f(T)=μTnf(T)=μTn theory, the universe experiences a power-law expansion a(t)∼t2n/3a(t)∼t2n/3. Furthermore, we consider the physical quantities corresponding to the exact solution, and find that if n>3/2n>3/2 the expansion of our universe can be accelerated without invoking dark energy. Also, we test the exact solution of this f(T)f(T) theory with the latest Union2 Type Ia Supernovae (SNIa) dataset which consists of 557 SNIa, and find that it can be well consistent with the observational data in fact.
As one of the non-food crops, cassava is widely cultivated in some of the southern provinces of China as a source of starchy material for food and fermentation industry. In this study, cassava starch was used as the substrate for Saccharomycopsis fibuligera A11, a trehalose poor-assimilating mutant isolated from S. fibuligera sdu through chemical mutagenesis, to accumulate trehalose. The results indicated that trehalose accumulation by S. fibuligera A11 was optimal in the medium comprised of 20.0g/l cassava starch, 20.0g/l hydrolysate of soybean cake at pH 5.5 and 30°C for 48h. At the flask level, trehalose yield was 24.8g per 100g of cell dry weight. At the end of 2-l fermentation, trehalose yield was 25.8g per 100g of cell dry weight. This is the highest trehalose yield accumulated in the yeast cells reported so far. At the same time, 0.12g/100ml of reducing sugar and 0.21g/100ml of total sugar were observed in the fermented medium. After isolation and purification, the crystal trehalose was obtained from the culture extract.
A new conservation theorem is derived. The conserved quantity is constructed in terms of a symmetry transformation vector of the equations of motion only, without using either Lagrangian or Hamiltonian structures (which may even fail to exist for the equations at hand). One example and implications of the theorem on the structure of point symmetry transformations are presented.
Adopting the method described in previous papers, we search for Nöther's symmetries in point-like Friedman--Robertson--Walker (FRW) Lagrangians derived for general non-minimally coupled gravitational theories. We obtain exact solutions for flat models capable of producing inflation and recovering the Einstein regime at the present time (i.e. the scalar field and the coupling become constants directly related to the Newton constant ).
We consider Noether symmetry approach to find out exact cosmological
solutions in $f(T)$-gravity. Instead of taking into account phenomenological
models, we apply the Noether symmetry to the $f(T)$ gravity. As a result, the
presence of such symmetries selects viable models and allow to solve the
equations of motion. We show that the generated $f(T)$ leads to a power law
expansion for the cosmological scale factor.
Extended Theories of Gravity can be considered a new paradigm to cure
shortcomings of General Relativity at infrared and ultraviolet scales. They are
an approach that, by preserving the undoubtedly positive results of Einstein's
Theory, is aimed to address conceptual and experimental problems recently
emerged in Astrophysics, Cosmology and High Energy Physics. In particular, the
goal is to encompass, in a self-consistent scheme, problems like Inflation,
Dark Energy, Dark Matter, Large Scale Structure and, first of all, to give at
least an effective description of Quantum Gravity. We review the basic
principles that any gravitational theory has to follow. The geometrical
interpretation is discussed in a broad perspective in order to highlight the
basic assumptions of General Relativity and its possible extensions in the
general framework of gauge theories. Principles of such modifications are
presented, focusing on specific classes of theories like f (R)-gravity and
scalar-tensor gravity in the metric and Palatini approaches. The special role
of torsion is also discussed. The conceptual features of these theories are
fully explored and attention is payed to the issues of dynamical and conformal
equivalence between them considering also the initial value problem. A number
of viability criteria are presented considering the post-Newtonian and the
post-Minkowskian limits. In particular, we discuss the problems of neutrino
oscillations and gravitational waves in Extended Gravity. Finally, future
perspectives of Extended Gravity are considered with possibility to go beyond a
trial and error approach.
Classical generalization of general relativity is considered as gravitational
alternative for unified description of the early-time inflation with late-time
cosmic acceleration. The structure and cosmological properties of number of
modified theories, including traditional $F(R)$ and Ho\v{r}ava-Lifshitz $F(R)$
gravity, scalar-tensor theory, string-inspired and Gauss-Bonnet theory,
non-local gravity, non-minimally coupled models, and power-counting
renormalizable covariant gravity are discussed. Different representations and
relations between such theories are investigated. It is shown that some
versions of above theories may be consistent with local tests and may provide
qualitatively reasonable unified description of inflation with dark energy
epoch. The cosmological reconstruction of different modified gravities is made
in great detail. It is demonstrated that eventually any given universe
evolution may be reconstructed for the theories under consideration: the
explicit reconstruction is applied to accelerating spatially-flat FRW universe.
Special attention is paid to Lagrange multiplier constrained and conventional
$F(R)$ gravities, for last theory the effective $\Lambda$CDM era and
phantom-divide crossing acceleration are obtained. The occurrence of Big Rip
and other finite-time future singularities in modified gravity is reviewed as
well as its curing via the addition of higher-derivative gravitational
invariants.
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