Noether symmetry approach in non-minimal derivative coupling gravity

Eur. Phys. J. C (2022) 82:556 https://doi.org/10.1140/epjc/s10052-022-10408-8 Regular Article - Theoretical Physics Noether symmetry approach in non-minimal derivative coupling gravity Muhammadsorfee Dolohtahe1,a , Watcharakorn Srikom2,b , Phongpichit Channuie3,4,c , Narakorn Kaewkhao1,d 1 Department of Physics, Faculty of Science, Prince of Songkla University, Hatyai 90112, Thailand 2 Department of Innovative Technology for Renewable Energy, Faculty of Science and Technology, Suratthani Rajabhat University, Surat Thani 84100, Thailand 3 School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand 4 College of Graduate Studies, Walailak University, Nakhon Si Thammarat 80160, Thailand Received: 17 October 2021 / Accepted: 6 May 2022 © The Author(s) 2022 Abstract In this work, we examine solutions of the system of equations obtained by applying the Noether gauge symmetry (NGS) and its conserved quantity for the standard general relativity (GR) and the non-minimal derivative coupling (NMDC) cosmological model. We discover two salient features of the solutions. The first one is a(t) ∝ t 1/3 for a kineticdominant phase which may emerge before inflationary period at very early time for GR case. The second one is a new form of scalar field φ(t) govern by the exponential cosmological solution for NMDC case, φ(t) = (c1 + c2 t)e−λt + c3 . Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Non-minimal-derivative coupling gravity . . . . . . . 3 Hessian matrix, EL equations and NMDC universe . . 4 Noether gauge symmetries . . . . . . . . . . . . . . . 4.1 Standard GR cosmology . . . . . . . . . . . . . 4.2 NMDC universe . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction Astrophysical observations including Type Ia Supernovae [1,2], cosmic microwave background (CMB) radiation [3–9], large scale structure [10], baryon acoustic oscillations (BAO) a e-mail: b e-mail: c e-mail: d e-mail: (corresponding author) 0123456789().: V,-vol [11] as well as weak lensing [12] provide a strong evidence that the expansion of the universe is presently accelerating. In spite of its successes, the so-called Lambda cold dark matter (CDM) [13] is plagued by the cosmological problem [14] and the coincident problem [15]. The phase of late-time cosmic acceleration receives much attention. However, the introduction of the so-called “dark energy (DE)” in the context of conventional general relativity is one of the promising explanations. Additionally, another possible scenario is to engineer Einstein gravity on the large-scale methodology. There were some reviewed articles published so far regarding the mentioned issues, see for example [16–19] and references therein. However, very little is known about the DE sector of the universe and it poses one of the unsolved problems in physics. Alternative paradigms by engineering the Einstein field equations either in the geometric part or in the stress-energy tensor are widely accepted to explain effects of dark ingredients [20]. The f (R) theories of gravity serves as one of the simplest modifications to the standard general relativity. Here the Lagrangian density of f is an arbitrary function of the scalar curvature R [21,22]. It is worth noting that there were rigorous reviews on f (R) theories [23,24] as well as on Born–Infeld inspired modifications of gravity [25]. In Ref. [26], the authors investigated the cosmological implications of the modified theories of gravity on inflation, bounce and late-time evolution. Apart from these modified theories of gravity, theories of non-minimal derivative coupling to gravity attract much attention from theoretical and phenomenological points of view, see, e.g., [27–38]. More specifically, their applications on inflation and its consequences were proposed by a number of authors [39–47]. The important role of the Noether symmetry in cosmology has received increasing attention within decades in order to 123 556 Page 2 of 9 Eur. Phys. J. C select the viable models [48]. Conserved quantities of the system, as well as unknown functions, can be determined with the help of the Noether symmetry approach. More specifically, by using the Noether symmetry, we can obtain the exact solutions. The Noether symmetry approach has been applied to study various cosmological scenarios so far including nonlocal f (T ) gravity [49–51], viable mimetic f (R) and f (R, T ) theories [52], f (R) cosmology [53], the cosmological alpha-attractors [54],and f (G) theory [55]. Moreover, the exact solutions for potential functions, scalar field and the scale factors in the Bianchi models have been investigated in Refs. [56–58] and the solutions of the field equations of f (R) gravity are investigated in static cylindrically symmetric space-time using the Noether symmetry technique [59]. The second kind of Noether symmetry approach for cosmological studies in the literature is the so-called Noether gauge symmetry (NGS) approach [60–62]. It is a generalization of the conventional one. Very recently, the authors of Ref. [63] have discussed the NGS approach for the Eddingtoninspired Born–Infeld theory. In the present work, we study a formal framework of the non-minimal derivative coupling (NMDC) gravity scenario through the NGS approach and present a detailed calculation of the point-like Lagrangian. The point-like Lagrangian of the Einstein-Hilbert action including the non-minimal derivative coupling (NMDC) sector are examined with spatially flat FLRW spacetime in which matter in such a universe only has a scalar field and a matter field. The latter model is expected to quantify to what extend the field kinetic term affects the evolution of the universe. This paper is organized as follows: We will start by making a short recap of a formal framework of NMDC gravity and derive the point-like Lagrangian for underlying theory in Sect. 2. In Sect. 3, we study a Hessian matrix and quantify the Euler–Lagrange equations and Hamiltonian equations of the Einstein (GR) and the NMDC universes. In Sect. 4, the NGS approach for the GR and NMDC is discussed. We discuss exact cosmological solutions of both theories with the help of the Noether symmetries of point-like Lagrangian. Finally, we conclude our findings in the last section. = (2022) 82:556 1 μν G ∂μ φ∂ν φ . M2 (2.2) The appearance of minus sign in front of M12 G μν is due to the avoidance of ghosts in the scalar field sector, i.e., φ̇ 2 + 2V (φ) > M12 G 00 g 00 g 00 φ̇ 2 = M32 ( ȧa )2 φ̇ 2 . Hence, the potential term dominates over the NMDC term [42]. The NMDC Lagrangian was first proposed in [27] and the action is given by SNMDC (g)    4 √ ,μ ,ν = d x −g R − (gμν + κG μν )φ φ − 2V (φ) +Sm (gμν , ), (2.3) where κ ≡ M −2 is a NMDC free parameter that has dimension of [M P−2 ],and Sm (gμν , ) denotes the matter field action. It is (...truncated)