The Noether symmetry approach is useful tool to restrict the arbitrariness in a gravity theory when the equations of motion are underdetermined due to the high number of functions to be determined in the ansatz. We consider two scalar-coupled theories of gravity, one motivated by induced gravity, the other more standard; in Bianchi I, Bianchi III and Kantowski-Sachs cosmological models. For these models, we present a full set of Noether gauge symmetries, which are more general than those obtained by the strict Noether symmetry approach in our recent work. Some exact solutions are derived using the first integrals corresponding to the obtained Noether gauge symmetries.

In this study, we consider a phantom cosmology in which a scalar field is minimally coupled to gravity. For anisotropic locally rotational symmetric (LRS) Bianchi type I space–time, we use the Noether symmetry approach to determine the potential of such a theory. It is shown that the potential must be in the trigonometric form as a function of the scalar field. We solved the field equations of the theory using the result obtained from the Noether symmetry. Our solution shows that the universe has an accelerating expanding phase.

The solutions for the field equations of f (R) gravity are investigated in static cylindrically symmetric space-time. Conserved quantities of the system, as well as unknown functions, can be determined with the help of the Noether symmetry method. In this article, some unknown values of the equations of state parameter (EoS) have emerged as a result of the constraints obtained by analyzing the Noether symmetry equations for the f (R) = f0R case. Consequently, several new exact solutions have been found for cases of General Relativity in static cylindrically symmetrical space-time for the non-dust matter.

The late time crossover from a power-law to an exponential expansion of the Universe evolution is the major problem in today’s physical cosmology. Unless this critical transition problem is solved, it is not possible to reach a holistic theory of cosmology. In this study, we propose a simple model in the FLRW framework, where dark matter and dark energy interact through a potential. We analytically solve this model and obtain scale factor a(t) from the presented model. Mainly, employing numerical solutions we show that the scale parameter has a hybrid form which includes power and exponential terms. The numerical results clearly show that there is a time crossover tc in the scale factor a(t) curve, which indicates the transition from the power-law to the exponential expansion of the Universe. We fit these unscaled curves and obtain that scale factor behaves as a(t) ∝ t 2/3 below t ≤ tc, and as a(t) ∝ exp(H0t) with H0 = 0.4 and H0 = 0.3 for the relatively weak and strong interactions above t > tc, respectively. It is the first time that we explicitly obtain a hybrid scale factor incorporating the power and exponential terms as a(t) ∝ t 2/3 e H0t . We conclude that the presented model can solve the late time transition problem of the Universe based on dark matter and dark energy interaction. Additionally, we numerically obtain other kinematic parameters depending upon the scale factor. We discuss the limit behaviors of all relevant cosmological parameters. Our results are completely in good agreement with observational data. Finally, we state that this work makes essential steps towards solving a critical outstanding problem of the cosmology, and has a potential to creates a paradigm for future studies in this field.

The f(R) theory is considered for static cylindrically symmetric and plane-symmetric spacetimes. In order to find solutions to the field equations of these models, the Noether symmetry method is used. First, we examine the GR case for cylindrically symmetrical space-time with the $$w=-1$$ w = - 1 dark energy state. Then, with the assumption of $$f(R) = f_0R^n$$ f ( R ) = f 0 R n , cases with matter and non-matter are examined and general solutions are determined for both space-times. Thus, it is shown that inclusive new solutions are obtained, considering the Noether symmetric conditions. In addition, the GR limit for each cases are examined.