Dark $$f(\mathcal{R},\varphi,\chi)$$ universe with Noether symmetry
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DARK f (R, ϕ, χ) UNIVERSE WITH NOETHER SYMMETRY M. F. Shamir,∗ A. Malik,∗ and M. Ahmad†
Using the Noether symmetry approach, we investigate f (R, ϕ, χ) theories of gravity, where R is the scalar curvature, ϕ is the scalar field, and χ is the kinetic term of ϕ. Based on the Lagrangian for f (R, ϕ, χ) gravity, we obtain the determining equations. We consider f (R, ϕ, χ) models of a flat Friedmann–Robertson– Walker universe. Using the obtained solutions, we find conserved quantities. In the framework of this scenario, the continuity equation is extremely important for analyzing the energy density and pressure. Using the first integral of motion, we present a graphical analysis of the energy density, pressure component, and parameter of the equation of state. The negativity of the pressure observed in the considered cases in fact suggests that this theory can describe a Noether universe with dark matter.
Keywords: conserved quantity, dark matter, Noether symmetry DOI: 10.1134/S0040577920120107
1. Introduction Data recently obtained from diverse sources such as type-Ia supernovas [1], [2] confirms that the Universe is expanding in the late epoch. Moreover, the acceleration of this expansion has been established by reliable evidence such as relict radiation, the large-scale structure [3], [4], baryon acoustic oscillations [5], and X-ray measurements [6]. It is now generally assumed that the cause of this accelerated expansion of the Universe is dark matter, which is an invisible unknown source of energy in the Universe, and this dark matter produces a large negative pressure. Observational data indicates that the Universe has a spatially flat structure and is composed of approximately 70% dark energy and 30% unknown matter source together with cold dark matter, baryons, and some negligibly small part of radiation. The astrophysical source of dark energy is currently enigmatic, and explaining the nature of dark energy is in fact a challenging problem in general relativity (GR). Moreover, there are two main approaches for investigating the phenomenon of the late-time accelerating expansion of the Universe. The first of them consists in analyzing the source of dark energy based on GR, and the second consists in introducing corrections to the theory of gravity as an alternative to GR (see [7]–[11] devoted to modified theories of gravity). Such modified theories are f (R), f (R, T ), f (R, G), f (G), f (G, T ), f (R, ϕ), and f (R, ϕ, χ) theories of gravity and others. The modified f (R) theory of gravity has been the most attractive topic of discussion for many researchers. Some viable f (R) gravity models have been proposed that show that early-time inflation and ∗
Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore, Pakistan, e-mail: [email protected] (corresponding author), adnanmalik [email protected]. †
National University of Computer and Emerging Sciences, Chiniot-Faisalabad Campus, Pakistan, e-mail: [email protected]. Prepared from an English manuscript submitted by
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