Abstract

We study the early inflationary phase of the universe driven by noncanonical scalar field models using an exponential potential. The noncanonical scalar field models are represented by Lagrangian densities containing square and square-root kinetic corrections to the canonical Lagrangian density. We investigate the Lie symmetry of the homogeneous scalar field equations obtained from noncanonical Lagrangian densities and find only a one-parameter Lie point symmetry for both canonical and noncanonical scalar field equations. We use the Lie symmetry generator to obtain an exact analytical group-invariant solution of the homogeneous scalar field equations from an invariant curve condition. The solutions obtained are consistent and satisfy the Friedmann equations subject to constraint conditions on the potential parameter \(\lambda\) for the canonical case and on the parameter \(\mu\) for the noncanonical case. In this scenario, we obtain the values of various inflationary parameters and make useful checks on the observational constraints on the parameters from Planck data by imposing a set of bounds on the parameters \(\lambda\) and \(\mu\). The results for the scalar spectral index (\(n_{S}\)) and the tensor-to-scalar ratio (\(r\)) are presented in the \((n_{S},r)\) plane in the background of Planck-2015 and Planck-2018 data for noncanonical cases and are in good agreement with cosmological observations. For theoretical completeness of the noncanonical models, we verify that the noncanonical models under consideration are free from ghosts and Laplacian instabilities. We also treat the noncanonical scalar field model equations for two power-law (kinetic) corrections by the dynamical system theory. We provide useful checks on the stability of the critical points for both cases and show that the group-invariant analytical noncanonical inflation solutions are stable attractors in phase space.

中文翻译：

---

具有规范和非规范标量场的早期膨胀阶段：基于对称的方法

摘要

我们使用指数势研究由非规范标量场模型驱动的宇宙早期膨胀阶段。非规范标量场模型由包含正则拉格朗日密度平方和平方根动力学校正的拉格朗日密度表示。我们研究了从非规范拉格朗日密度获得的齐次标量场方程的Lie对称性，并且对于正则和非规范标量场方程都只发现了一个参数的Lie点对称性。我们使用Lie对称生成器从不变曲线条件中获取齐次标量场方程的精确解析组不变解。所获得的解是一致的，并且满足Friedmann方程，并且受势参数约束的条件对于规范情况，为\（\ lambda \）；对于非规范情况，为参数\（\ mu \）。在这种情况下，我们通过在参数\（\ lambda \）和\（\ mu \）上施加一组边界，来获取各种膨胀参数的值，并对普朗克数据中的参数观测约束进行有用的检查。标量谱索引（\（n_ {S} \））和张量标量比（\（r \））的结果显示在\（（n_ {S}，r）\）中在非常规情况下的Planck-2015和Planck-2018数据的背景下飞行，并且与宇宙学观测非常吻合。对于非规范模型的理论完整性，我们验证了所考虑的非规范模型没有重影和拉普拉斯不稳定性。我们还通过动力学系统理论来处理两个幂律（动力学）校正的非规范标量场模型方程。我们对这两种情况下的临界点稳定性进行了有用的检查，并证明了组不变分析非规范膨胀解是相空间中的稳定吸引子。