This paper investigates the Noether symmetry approach in modified [Formula: see text] theory of gravity, where [Formula: see text] is the Ricci scalar, [Formula: see text] is a scalar field and [Formula: see text] is the kinetic energy term. For this purpose, we consider Locally Rotationally Symmetric (LRS) Bianchi Type-I spacetimes. The cosmological solutions are developed through Noether symmetry approach. The determining equations are computed in the context of a point-like Lagrangian for [Formula: see text] gravity. In particular, three different cases are taken into account for the LRS Bianchi Type-I spacetimes. Some important conserved quantities for the Lagrangian in this modified theory of gravity are worked out through the determining equations. In this scenario, the graphical behavior of energy density, pressure component and equation of state parameter are reported and analyzed with the help of first integrals of motion. The negative trends of the strong energy conditions actually suggest that the said theory supports expanding Universe with dark energy.

中文翻译：

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f(ℛ,φ,χ) 引力下 LRS Bianchi I 型时空的 Noether 对称性

本文研究了修改后的[公式：见文]重力理论中的诺特对称方法，其中[公式：见文]是里奇标量，[公式：见文]是标量场，[公式：见文]是动能项。为此，我们考虑局部旋转对称 (LRS) Bianchi Type-I 时空。宇宙学解决方案是通过诺特对称方法开发的。确定方程是在 [公式：见文本] 重力的点状拉格朗日上下文中计算的。特别是，LRS Bianchi Type-I 时空考虑了三种不同的情况。在这个修正的引力理论中，拉格朗日量的一些重要守恒量是通过确定方程计算出来的。在这种情况下，能量密度的图形行为，借助运动的一阶积分报告和分析压力分量和状态方程参数。强能量条件的负面趋势实际上表明上述理论支持用暗能量膨胀宇宙。