Higher-order Gravity Theories Research Articles

The recollapse problem of closed Friedmann–Robertson–Walker models in higher-order gravity theories

We study the closed universe recollapse conjecture for positively curved Friedmann–Robertson–Walker models with a perfect fluid matter source and a scalar field which arises in the conformal frame of the R+αR2 theory. By including ordinary matter, we extend the analysis of a previous work. We analyze the structure of the resulted four-dimensional dynamical system with the methods of the center manifold theory and the normal form theory. It is shown that an initially expanding closed FRW universe, starting close to the Minkowski space-time, cannot avoid recollapse. We discuss the posibility that potentials with a positive minimum may prevent the recollapse of closed universes.

Noether symmetry in the higher order gravity theory

Noether symmetry in the higher order gravity theory

Reconciling dark energy models withf(R)theories

Higher-order theories of gravity have recently attracted a lot of interest as alternative candidates to explain the observed cosmic acceleration without the need of introducing any scalar field. A critical ingredient is the choice of the function $f(R)$ of the Ricci scalar curvature entering the gravity Lagrangian and determining the dynamics of the Universe. We describe an efficient procedure to reconstruct $f(R)$ from the Hubble parameter $H$ depending on the redshift $z$. Using the metric formulation of $f(R)$ theories, we derive a third order linear differential equation for $f[R(z)]$ which can be numerically solved after setting the boundary conditions on the basis of physical considerations. Since $H(z)$ can be reconstructed from the astrophysical data, the method we present makes it possible to determine, in principle, what is the $f(R)$ theory which best reproduces the observed cosmological dynamics. Moreover, the method allows to reconcile dark energy models with $f(R)$ theories finding out what is the expression of $f(R)$ which leads to the same $H(z)$ of the given quintessence model. As interesting examples, we consider ``quiessence'' (dark energy with constant equation of state) and the Chaplygin gas.

Higher-order gravity theories and scalar–tensor theories

We generalize the known equivalence between higher-order gravity theories and scalar–tensor theories to a new class of theories. Specifically, in the context of a first-order or Palatini variational principle where the metric and connection are treated as independent variables, we consider theories for which the Lagrangian density is a function f of (i) the Ricci scalar computed from the metric, and (ii) a second Ricci scalar computed from the connection. We show that such theories can be written as tensor–multi-scalar theories with two scalar fields with the following features: (i) the two-dimensional σ-model metric that defines the kinetic energy terms for the scalar fields has constant, negative curvature; (ii) the coupling function determining the coupling to matter of the scalar fields is universal, independent of the choice of function f; and (iii) if both mass eigenstates are long range, then the Eddington post-Newtonian parameter γ has value 1/2. Therefore, in order to be compatible with solar system experiments at least one of the mass eigenstates must be short range.

Noether symmetry in the higher order gravity theory

We explore the conditions for the existence of Noether symmetries for higher order gravity theory, after introducing an auxiliary variable, which gives the correct quantum description of the theory. It turns out that the application of Noether theorem in higher order theory of gravity is a powerful tool to find the solution of the field equations. A few such physically reasonable solutions like power law inflation are presented.

Dynamical system approach to FRW models in higher-order gravity theories

We study the late time evolution of positively curved FRW models with a scalar field which arises in the conformal frame of the R+αR2 theory. The resulted three-dimensional dynamical system has two equilibrium solutions corresponding to a de Sitter space and an ever expanding closed universe. We analyze the structure of the first equilibrium with the methods of the center manifold theory and, for the second equilibrium, we apply the normal form theory to obtain a simplified system, which we analyze with special phase plane methods. It is shown that an initially expanding closed FRW space–time avoids recollapse.

CURVATURE QUINTESSENCE

The issues of quintessence and cosmic acceleration can be discussed in the framework of higher order theories of gravity. We can define effective pressure and energy density directly connected to the Ricci scalar of curvature of a generic fourth order theory and then ask for the conditions to get an accelerated expansion. Exact accelerated expanding solutions can be achieved for several fourth order theories so that we get an alternative scheme to the standard quintessence scalar field, minimally coupled to gravity, usually adopted. We discuss also conformal transformations in order to see the links of quintessence between the Jordan and Einstein frames.

The effect of higher-order curvature terms on string quantum cosmology

Several new results regarding the quantum cosmology of higher-order gravity theories derived from superstring effective actions are presented. After describing techniques for solving the Wheeler–DeWitt equation with appropriate boundary conditions, it is shown that this quantum cosmological model may be compared with semiclassical theories of inflationary cosmology. In particular, it should be possible to compute corrections to the standard inflationary model perturbatively about a stable exponentially expanding classical background.

Variational and conformal structure of nonlinear metric-connection gravitational Lagrangians

We examine the variational and conformal structures of higher-order theories of gravity that are derived from a metric-connection Lagrangian that is an arbitrary function of the curvature invariants. We show that the constrained first-order formalism when applied to these theories may lead consistently to a new method of reduction of order of the associated field equations. We show that the similarity of the field equations that are derived from appropriate actions via this formalism to those produced by Hilbert varying purely metric Lagrangians is not merely formal but is implied by the diffeomorphism covariant property of the associated Lagrangians. We prove that the conformal equivalence theorem of these theories with general relativity plus a scalar field, holds in the extended framework of Weyl geometry with the same forms of field and self-interacting potential but, in addition, there is a new “source term” that plays the role of a stress. We point out how these results may be further exploited and address a number of new issues that arise from this analysis.

The Quantum Evolution of the Closed Friedman Universe with “Bouncing Off”

The homogeneous and isotropic closed Friedmanuniverse evolution in higher-order gravity theories isconsidered. The model takes into account vacuumpolarisation of conformal and nonconformal fields. That leads to the following addition in the Einsteinlagrangian: R2 ln |R/R0|. Near theregular minimum of the scale factor the model has ananalytical solution depending on an integration constantC. If |R/R0| > 1, the solution passes through the regularminimum, experiences inflation with a decreasing valueof R and approaches to the critical value R =R0. In the interval |R/R0| < 1,the solutions have non-linear oscillations (i.e. the scalaronstage). On this stage of the evolution the universe isfilled with relativistic plasma. The continuoustransition through the critical point R = R0is possible in only one type of solution, the separatrix.Though other solutions have no features in this point,they experience the discontinuity in derivatives of R.It is unsuitable since higher-order gravity theories are considered. Thus the measure of continuoussolutions giving a hot universe is negligible. Howeversolutions of the model can be continued in the imaginarytime. In such a case the Euclidean action will have a non-zero value because of the spaceclosed boundedness of the universe and the finiteness ofthe imaginary time interval (instanton). The last allowsus to calculate the probability of the quantum tunnelling of the Friedman universe from theinflation region into the scalaron region.

Conservation Laws and Symmetry Properties of a Class of Higher Order Theories of Gravity

We consider a class of fourth order theories of gravity with arbitrary matter fields arising from a diffeomorphism invariant Lagrangian density \(\mathcal{L}_T = \mathcal{L}_G + \mathcal{L}_M\), with \(\mathcal{L}_G = \sqrt { - g} \left[ {R + h\left( R \right)} \right]\)and \(\mathcal{L}_M\)the phenomenological representation of the nongravitational fields. We derive first the generalization of the Einstein pseudotensor and the von Freud superpotential. We then show, using the arbitrariness that is always present in the choice of pseudotensor and superpotential, that we can choose these superpotentials to have the same form as those for the Hilbert Lagrangian of general relativity (GR). In particular we may introduce the Moller superpotential of GR as associated with a double-index differential conservation law. Similarly, using the Moller superpotential we prove that we can choose the Komar vector of GR to construct a conserved quantity for isolated asymptotically flat systems. For the example R + R2theory we prove then, that the active mass is equal to the total energy (or inertial mass) of the system.

Comment: No wormholes with real minimally coupled scalar fields

Cotsakis et al. [Phys. Rev. D 49, 6489 (1994)] claimed that wormhole solutions can be obtained with real scalar fields which could in turn be simulated with higher order gravity theories. Because of various sign errors this conclusion is not justified, and instead a complex scalar field is necessary. Higher derivative theories can have wormholes by choosing the ``wrong signs'' for the coefficients in front of the functions of scalar curvature. This ensures that there is a minus sign in front of the kinetic term in the conformally related scalar field theory, thus ensuring that the Ricci tensor has negative eigenvalues. A real scalar field with a negative definite potential $V(\ensuremath{\varphi})&lt;0$, although satisfying the Ricci tensor condition, does not support a wormhole, at least when using a FRW metric.

Nonlinear lagrangians and the isotropy of the universe

We analyse the question of whether, in higher derivative gravity theories, the present isotropic state of the universe could have been obtained by starting from “arbitrary” initial conditions. We show that the Collins-Hawking conclusion holds in this extended case, that is, the set of spatially homogeneous cosmologies which can approach isotropy at late times is of measure zero in the space of all spatially homogeneous universe models. This involves an examination of a generalized form of the Raychaudhuri equation valid in a general higher-order theory of gravity in four spacetime dimensions. We show that this equation is conformally equivalent to what we call a Raychaudhuri system, that is, to the usual Raychaudhuri equation in general relativity coupled to a scalar-wave equation satisfied by the scalar field generated by the conformal transformation with the usual effective potential. This result is also used to prove directly in higher-order gravity the known forms of cosmic no-hair theorems in the Sitter and power-law inflation in general relativity. Generalizations to arbitrary dimensionality and also the addition of other curvature invariants are discussed.

Graviton creation and conformal noninvariance in higher order gravity theories.

The relation between graviton creation and conformal noninvariance is explored in the context of higher order gravity theories and some consequences for the amount of the created gravitons in the universe are given and compared to that in general relativity.

Past-instability conjecture and cosmological attractors in generalized isotropic universes.

We conjecture that all homogeneous and isotropic physically reasonable cosmological solutions of general relativity theory are in general past unstable in the generalized framework of higher order gravity theories. In this respect we provide a detailed perturbation analysis of the most interesting of these solutions and find that the results support our conjecture. We show that, in general, radiation solutions of higher order gravity are nonperturbative (bouncing or singular) as we approach the singularity in the past. A well-known flat radiation solution of quadratic gravity is shown not to be an attractor as t\ensuremath{\rightarrow}0. We prove that the quasiexponential phase in higher order gravity theories cannot be an attractor to solutions which may describe a preinflationary stage in these theories. However, this last conclusion may be altered if additional conditions are imposed and this situation is similar to the stability properties of the Starobinski inflationary solution. Other examples of a nongeneric-type in acordance with our conjecture are discussed and these include the Milne universe.

Escape rates for the quiescent self-regenerating universe

A detailed, probabilistic treatment based on the Fokker-Planck equation is given for the self-regenerating mechanism in higher order gravity theories. Nonstationary solutions to the Fokker-Planck equation exist and are obtained for certain forms of the drift, diffusion coefficients and spacetime dimensionality which correspond exactly to the well-known Wiener and Ornstein-Uhlenbeck processes respectively. We calculate the Kramers escape rates for certain choices of the gravitational lagrangian and find that in the resulting inflationary model, which we call the quiescent self-regenerating universe, particles stay near the global minimum of the effective potential with probability one. We conclude that the true vacuum state is stable against the quantum formation of bubbles of the inflationary phase. This provides a new proof, via a stochastic argument, of the important transient attractor property in quiescent inflation.

Power-law inflation and conformal transformations

We present a discussion of power-law inflation in higher-order gravity theories. Using the conformal equivalence theorem, we find power-law solutions in the context of pure higher-order gravity theories in arbitrary dimension. We also find a family of polar inflationary solutions. We give a perturbation analysis which establishes that the power-law solutions are stable against homogeneous and isotropic perturbations but that the polar solutions are not.

The fragility and rigidity of cosmological theories

The concepts of fragility and rigidity in theories containing a minimally coupled, self-interacting scalar field are introduced and defined for an arbitrary space-time dimension,D, by using the field as the dynamical variable. It is proved that inflation is rigid and the case of an exponential self-interaction is studied. The fragility of higher-order, scalar-tensor and Kaluza-Klein gravity theories is investigated by using their conformal equivalence to general relativity plus a scalar field. Whilst most higher-order theories are rigid, the conditions for Kaluza-Klein theories to become fragile depend strongly onD.

Conformal transformations single out a unique measure of distance

The conformal equivalence theorem states that higher-order gravity theories derived from a Lagrangian which is an analytic function of the scalar curvature are conformally equivalent to general relativity plus a scalar-field matter source with a particular self-interaction effective potential. We discuss the question of physical reality of the two metrics associated with the conformal transformation in the presence of matter fields. We find that of the two conformally related metrics only one can be chosen as the real, physical one. In particular, if the scalar field generated by the conformal transformation satisfies ${\ensuremath{\nabla}}_{a}{\ensuremath{\nabla}}_{b}\ensuremath{\varphi}=0$ then the original metric of the higher-order gravity theory cannot represent a real physical measure of distance and the true metric is that of the system: general relativity plus scalar field. This situation can be realized for a wide class of manifolds.

Coupling first-order phase transitions to curvature-squared inflation.

A new way to couple first-order phase transitions to inflation is proposed. The mechanism, dubbed getaway inflation, bypasses the problem of the "graceful exit" by letting the inflationary phase of the background have a classical end. At the same time, a stage of bubble production via semiclassical tunneling occurs, allowing speculations on the role of bubbles on the early structure formation. A realization of the proposed process is found in a nonminimally coupled higher-order gravity theory which by itself deserves further investigation.