The theory of gravity with a quadratic contribution of scalar curvature is investigated using a dynamical systems approach. The simplest Friedmann–Robertson–Walker metric is employed to formulate the dynamics in both the Jordan frame and the conformally transformed Einstein frame. We show that, in both frames, there are stable de Sitter states where the expansion of the Hubble function naturally includes terms corresponding to an effective dark matter component. Using the invariant centre manifold, we demonstrate that, in the Einstein frame, there exists a zero-measure set of initial conditions that lead from an unstable to a stable de Sitter state. Additionally, the initial de Sitter state is associated with a parallelly propagated singularity. We show that the formulations of the theory in the Jordan frame and the Einstein frame are physically nonequivalent.

Poland

Orest Hrycyna

On the structural stability of a simple cosmological model in Theory of Gravity

Dynamical systems methods are used to investigate a cosmological model with non-minimally coupled scalar field and asymptotically quadratic potential function. We found that for values of the non-minimal coupling constant parameter <math><mfrac><mrow><mn>3</mn></mrow><mrow><mn>16</mn></mrow></mfrac><mo>&lt;</mo><mi>ξ</mi><mo>&lt;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math> there exists an unstable asymptotic de Sitter state, free from a parallelly propagated singularity for <math><mfrac><mrow><mn>5</mn></mrow><mrow><mn>24</mn></mrow></mfrac><mo>≤</mo><mi>ξ</mi><mo>&lt;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math>, giving rise to non-singular beginning of the universe. The energy density associated with this state depends on value of the non-minimal coupling constant and can be much smaller than the Planck energy density. For <math><mi>ξ</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math> we found that the initial state is in form of the static Einstein universe. Proposed evolutional model, contrary to the seminal Starobinsky's model, do not depend on the specific choice of initial conditions in phase space, moreover, a small change in the model parameters do not change the evolution thus the model is generic and structurally stable. The values of the non-minimal coupling constant can indicate for a new fundamental symmetry in the gravitational theory. We show that Jordan frame and Einstein frame formulation of the theory are physically nonequivalent.

A new generic and structurally stable cosmological model without singularity

Dynamical systems methods are used to investigate dynamics of a flat Friedmann–Robertson–Walker cosmological model with the non-minimally coupled scalar field and a potential function. Performed analysis distinguishes the value of non-minimal coupling constant parameter  $$\xi =\frac{3}{16}$$  <math> <mrow> <mi>ξ</mi> <mo>=</mo> <mfrac> <mn>3</mn> <mn>16</mn> </mfrac> </mrow> </math>  , which is the conformal coupling in five dimensional theory of gravity. It is shown that for a monomial potential functions at infinite values of the scalar field there exist generic de Sitter and Einstein–de Sitter states. The de Sitter state is unstable with respect to expansion of the Universe for potential functions which do not change faster than linearly. This leads to a generic cosmological evolution without the initial singularity.

The non-minimal coupling constant and the primordial de Sitter state

In dynamical system describing evolution of universe with the flat Friedmann–Robertson–Walker symmetry filled with barotropic dust matter and non-minimally coupled scalar field with a constant potential function an invariant manifold of the de Sitter state is used to obtain exact solutions of the reduced dynamics. Using observational data coming from distant supernovae type Ia, the Hubble function H(z) measurements and information coming from the Alcock–Paczyński test we find cosmological constraints on the non-minimal coupling constant ξ between the scalar curvature and the scalar field. For all investigated models we can exclude negative values of this parameter at the 68% confidence level. We obtain constraints on the non-minimal coupling constant consistent with condition for conformal coupling of the scalar field in higher dimensional theories of gravity.