AbstractThe present article investigates the existence of Noether and Noether gauge symmetries of flat Friedman–Robertson–Walker universe model with perfect fluid matter ingredients in a generalized scalar field formulation namely $$f(R,Y,\phi )$$
f
(
R
,
Y
,
ϕ
)
gravity, where R is the Ricci scalar and Y denotes the curvature invariant term defined by $$Y=R_{\alpha \beta }R^{\alpha \beta }$$
Y
=
R
α
β
R
α
β
, while $$\phi $$
ϕ
represents scalar field. For this purpose, we assume different general cases of generic $$f(R,Y,\phi )$$
f
(
R
,
Y
,
ϕ
)
function and explore its possible forms along with field potential $$V(\phi )$$
V
(
ϕ
)
by taking constant and variable coupling function of scalar field $$\omega (\phi )$$
ω
(
ϕ
)
. In each case, we find non-trivial symmetry generator and its related first integrals of motion (conserved quantities). It is seen that due to complexity of the resulting system of Lagrange dynamical equations, it is difficult to find exact cosmological solutions except for few simple cases. It is found that in each case, the existence of Noether symmetries leads to power law form of scalar field potential and different new types of generic function. For the acquired exact solutions, we discuss the cosmology generated by these solutions graphically and discuss their physical significance which favors the accelerated expanding eras of cosmic evolution.
A Riccati equation based approach to isotropic scalar field cosmologies