Abstract
Some cosmological models based on the gravitational theory f(R)=R+zeta R^2, and on fluids obeying to the equations of state of Redlich–Kwong, Berthelot, and Dieterici are proposed for describing smooth transitions between different cosmic epochs. A dynamical system analysis reveals that these models contain fixed points which correspond to an inflationary, a radiation dominated and a late-time accelerating epoch, and a nonsingular bouncing solution, the latter being an asymptotic fixed point of the compactified phase space. The infinity of the compactified phase space is interpreted as a region in which the non-ideal behaviors of the previously mentioned cosmic fluids are suppressed. Physical constraints on the adopted dimensionless variables are derived by demanding the theory to be free from ghost and tachyonic instabilities, and a novel cosmological interpretation of such variables is proposed through a cosmographic analysis. The different effects of the equation of state parameters on the number of equilibrium solutions and on their stability nature are clarified. Some generic properties of these models, which are not sensitive to the particular fluid considered, are identified, while differences are critically examined by showing that the Redlich–Kwong scenario admits a second radiation-dominated epoch and a Big Rip Singularity.
Highlights
On the other hand, extended gravity theories in which a certain curvature invariant is added to, or used to replace, the Ricci scalar inside the Einstein-Hilbert Lagrangian can provide as well an evolution between different cosmic epochs as a consequence of the modified field equations themselves [10,11,12,13,14,15,16]
We will merge the fluid and the modified gravity approaches and propose some cosmological models in which the gravity sector is accounted for by a Lagrangian of the type f (R) = R + ζ R2, while the matter content is assumed to obey to some non-ideal equations of state with a well-established thermodynamical foundation known under the names of Redlich–Kwong, Berthelot, and Dieterici fluid separately
We will obtain a cosmological dynamics with a rich variety of different behaviors like a nonsingular bounce, two de Sitter-like epochs (thanks to the non-linear equation of state of the cosmic fluid in which w(ρ) is not a constant), possibly two radiation-dominated epochs, and possibly a phantom regime
Summary
On the other hand, extended gravity theories in which a certain curvature invariant is added to, or used to replace, the Ricci scalar inside the Einstein-Hilbert Lagrangian can provide as well an evolution between different cosmic epochs as a consequence of the modified field equations themselves [10,11,12,13,14,15,16]. 3 constitutes the main part of our work: in 3.1 we will recast the equations governing the dynamics of our models as a system of autonomous first order equations in terms of a set of dimensionless variables on which we will derive appropriate physical restrictions; in 3.2 we will identify the cosmologically meaningful equilibrium solutions, explain for which ranges of the matter equation of state parameters they can arise pointing out possible bifurcations among them for particular types of matter contents, and report their stability showing that radiation-dominated, de Sitter-like and power law cosmologies can arise; in 3.3 we will compactify the phase space and perform the analysis at infinity showing that a nonsinglar bounce occurs; in 3.4, 3.5 and 3.6 we will investigate the dynamics in the invariant submanifolds both numerically by plotting the trajectories in the phase spaces, by deriving analytically their stability, and by finding analytical results for the phase orbits in some specific cases; in 3.7 we will relate the dimensionless variables we have adopted to the deceleration, jerk and snap cosmographic parameters which can be astrophysically measured. The analytical computations of the stability of the isolated fixed points and of the invariant submanifolds are reported in the “Appendices B, C, D, E” which make use of both the standard notion of linear stability and of a much more advanced technique like the “ center manifold analysis”
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