This study explores the feasibility of an effective Friedmann equation in removing the classical Big Bang initial singularity and replacing it with a non-singular bounce occurring at a critical energy density value. In a spatially flat, homogeneous, and isotropic universe, the effective theory is obtained by introducing a function parametrically dependent on the critical energy density. This function measures the deviation from the benchmark theory, which is recovered as the critical energy density approaches infinity. Focusing on the covariant single-field formulation in viable Horndeski gravity, our analysis shows that both the effective and the benchmark theories belong to the same scalar–tensor theory, without any additional propagating degrees of freedom: the cuscuton and extended cuscuton models.

本研究探讨了有效的弗里德曼方程在消除经典大爆炸初始奇点并将其替换为在临界能量密度值处发生的非奇异反弹的可行性。在空间平坦、均匀和各向同性的宇宙中，有效理论是通过引入一个参数化依赖于临界能量密度的函数来获得的。此函数测量与基准理论的偏差，当临界能量密度接近无穷大时，该偏差会恢复。专注于可行的 Horndeski 引力中的协变单场公式，我们的分析表明，有效理论和基准理论都属于同一个标量-张量理论，没有任何额外的传播自由度：cuscuton 和扩展 cuscuton 模型。