Late-time-accelerated expansion esteemed from minisuperspace deformation

Eur. Phys. J. C (2022) 82:965 https://doi.org/10.1140/epjc/s10052-022-10941-6 Regular Article - Theoretical Physics Late-time-accelerated expansion esteemed from minisuperspace deformation Behzad Tajahmad1,2,a 1 Faculty of Physics, University of Tabriz, Tabriz, Iran 2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM)-Maragha, P.O. Box: 55134-441, Maragha, Iran Received: 28 September 2022 / Accepted: 21 October 2022 © The Author(s) 2022 Abstract The effects of minisuperspace deformation on Einstein–Hilbert action along with ordinary and phantom scalar fields as the matter contents are investigated. It is demonstrated that late-time-accelerated expansion and phase transition (from decelerated to accelerated) are obtained as a consequence of minisuperspace deformation. Finally, a mathematical theorem for distinguishing valid descriptions of the noncommutative frames is suggested. a e-mail: (corresponding author) Several approaches to noncommutative gravity [18–21] were developed from the initial interest in noncommutative field theory [22,23]. Noncommutative theories of gravity exhibit a highly nonlinear end result according to all of these formulations. Several aspects of the universe are studied in noncommutative cosmology to determine how noncommutativity affects them [24]. There have been observations in the literature that noncommutative deformations modify noncommutative fields and a full noncommutative theory of gravity would be expected to affect the minisuperspace variables. By introducing the Moyal product of functions into the Wheeler–DeWitt equation, similar to noncommutative quantum mechanics, this is accomplished. Historically, the noncommutative deformations of the minisuperspace have been analyzed at the quantum level in [24] where a Kantowski–Sachs universe was studied. Nonetheless, classical noncommutative formulations have been suggested utilizing an effective noncommutativity on the minisuperspace. Noncommutativity in classical theory is primarily founded on the assumption that modifying Poisson brackets yields noncommutative equations of motion [24– 27]. As a result, two generally different interpretations are given by phase space deformations, called the “C frame” and the “NC frame”, which, in general, are not physically equivalent [28]. In order to determine the valid range of the deformation parameters, a principle should be adopted [29]: “Deformed phase space models are only valid when the descriptions of C and NC frames are physically equivalent”. Physicists face a major challenge in explaining the nature and mechanism of our universe’s acceleration. Accelerated expansion of the universe has been confirmed by several astrophysical observations including supernova type Ia [30,31], CMB studies [32] weak lensing [33], large-scale structure [34], and baryon acoustic oscillations [35]. It contradicts Einstein’s theory of general relativity. The late-time- 0123456789().: V,-vol 123 1 Introduction Several researchers have studied the nature of gravitational theory at the quantum level in relation to the early evolution of the universe. There is evidence that quantum gravity plays a key role in understanding the very early evolution of the universe, according to the existing literature [1–4]. The method of revealing quantum gravity is not unique. A variety of approaches have been proposed over the years, including string theory [5], black hole physics [6–8], doubly special relativity [9–11], and etc. There was a consensus among all of them that there was a minimum length scale close to Planck length. In quantum gravity, this led to what is known as a generalized uncertainty principle (GUP) based on Heisenberg’s uncertainty principle [12]. GUP has many cosmological and astrophysical implications, for example, black hole thermodynamics [13], the origin of the magnetic fields in the Universe sector [14], and etc [15,16]. It is essential to investigate the effects of GUP on late regime of the universe physics because it plays such a crucial role in early universe physics. Therefore, in order to understand the universe’s complete dynamical picture, GUP should be investigated at both early and late regimes. Several attempts has been performed in the literature, for example see [16,17]. 965 Page 2 of 8 Eur. Phys. J. C accelerated expansion has generally been explained by two distinct classes of ideas (solutions). The acceleration that occurs in this process is a consequence of negative pressure; therefore, one way to explain it is that it is caused by an exotic liquid, so-called dark energy, which makes up about 70% of the universe. Einstein’s cosmological constant was thought to be the most likely solution to dark energy, but it failed to solve the ’fine tuning’ and ’cosmic coincidence’ problems [36]. Hence, other theoretical models such as the phantom field [37,38], quintessence [39,40], quintom [41,42], and tachyon field [43] have been proposed. Another option is that Einstein’s general relativity can be modified so its action is governed by a function of the curvature scalar ( f (R)-gravity) [44,45]. The approach is not limited to this type of change; various novel gravitational modification theories like scalar-tensor theories, f (T )-gravity, f (T )-gravity with an unusual term [46] and etcetera have recently been proposed. In the current paper, to introduce the deformation we will follow the approach in [47]. Indeed we revisit papers [47] and some parts of [48] to show that both decelerated (matterdominated era) and accelerated (dark-energy dominated era) epochs of the universe evolution can be obtained by minisuperspace deformation. One can easily compare to find that our solutions are different than these papers. or [49]  √   λϕ ; x = λ−1 ( a)3/2 sinh  √ y = λ−1 a 3/2 cosh  λϕ , √ where λ−1 = 8/3. Both transformations lead to a conserved equation:  ỹ 2 −  x̃ 2 = 1; We investigate a flat Friedmann–Robertson–Walker universe with scale factor a(t) and a homogeneous and isotropic scalar field ϕ(t). Assuming the signature of metric as (−, +, +, +) and dominating the scalar field over other matters degrees of freedom, the action takes the form S= 3 κ2  dt N −a ȧ N2  2   1 ϕ̇ 3 + dt N a  2 − 2V (ϕ) , 2 N (1)  Hc. = N    1 2 ω2 2 1 2 ω2 2 Px + x − N Py + y , 2 2 2 2  √  sin − λϕ ; x = λ−1 (− a)3/2√  y = λ−1 a 3/2 cos − λϕ , 123 (3) in which ω2 = −3/4. As is observed, for a phantom scalar field, the Hamiltonian is as a sum of two harmonic oscillators while for an ordinary scalar field, the Hamiltonian appeared as a ghost oscillator namely as a difference of two harmonic oscillators. The elements of our new configuration space, (x , y), and their conjugate momentums fulfill the following commutations based on the Poisson bracket: (4) where k and j can take 1 and 2, i.e. (x1 , x2 ) = (x, y) and δk j is the usual Kronecker delta. The equations of motion declaring the dynamics of our (...truncated)