A Study of the Accelerating Universe in \(\boldsymbol{f(R)}\) Modified Gravity Using the Dynamical System Approach

The accelerated expansion of the Universe constitutes one of the biggest challenges in present-day cosmology. To understand and explain this phenomenon in the framework of general relativity, corrections and extensions to it are required, which make the so-called extended theories of gravity (ETGs). In these theories, the geometry of space-time that represents the gravitational sector at the left hand side of the Einstein field equation \({G_{\mu\nu}}=8\pi{T_{\mu\nu}}\) is necessarily modified. These theories have attracted much attention since the time the accelerated expansion was discovered. A class of these theories known as \(f(R)\) gravity, offers a potent candidacy for this purpose, in addition to matter content modifications. The gravitational sector depending on the Ricci scalar invariant \(R\) is basically replaced with some its general nonlinear function which consists of higher-order curvature terms. In this work, we attempt to realize the late-time accelerated expansion in the context of \(f(R)\) gravity using the dynamical system approach. Analyzing the dynamical system arising from a particular \(f(R)\) model, its stability is studied for the cosmological inferences. The particular model \(f(R)={R^{p}}\exp({qR})\) with \(m=\frac{{R{f_{,RR}}}}{{{f_{,R}}}}=\frac{{p(p-1)+2pqR+{q^{2}}{R^{2}}}}{{p+qR}}\) and \(r=-\frac{{R{f_{,R}}}}{f}=-(p+qR)\) and with the geometric curve \(m(r)=-\frac{{{r^{2}}-p}}{r}\), is studied in this paper. We use the geometric approach for the curve \(m(r)\) in the plane \((r,m)\) which provides some properties of the model. In the case of a matter-dominated era the viability conditions at \(r=-1\), \(m(r)=0\) and \(dm/dr>-1\) are investigated. On the other hand, for the late-time acceleration, however, at \(r=-2\), either of the two conditions \(m(r)=-r-1\) with \(dm/dr<-1\), \(1\geq m>(\sqrt{3}-1)/2\) and \(1\geq m\geq 0\) are sought to fulfill. In the first place, the cosmic content is assumed to comprise matter and radiation only in the absence of a cosmological constant \(\Lambda\). In this case, an interaction of any kind is disregarded. Afterwards, as the second consideration, an interaction term in the presence of a cosmological constant representing dark energy is taken into account. The effects of linear and nonlinear interactions between matter and dark energy are also taken into account orderly in this case. The results are presented for each case, along with a discussion of critical points, their eigenvalues, and the equation of state parameter.