In this work by using a numerical analysis, we investigate in a quantitative way the late-time dynamics of scalar coupled gravity. Particularly, we consider a Gauss–Bonnet term coupled to the scalar field coupling function ξ(ϕ), and we study three types of models, one with f(R) terms that are known to provide a viable late-time phenomenology, and 2 Einstein–Gauss–Bonnet types of models. Our aim is to write the Friedmann equation in terms of appropriate statefinder quantities frequently used in the literature, and we numerically solve it by using physically motivated initial conditions. In the case that f(R) gravity terms are present, the contribution of the Gauss–Bonnet related terms is minor, as we actually expected. This result is robust against changes in the initial conditions of the scalar field, and the reason is the dominating parts of the f(R) gravity sector at late times. In the Einstein–Gauss–Bonnet type of models, we examine two distinct scenarios, firstly by choosing freely the scalar potential and the scalar Gauss–Bonnet coupling ξ(ϕ), in which case the resulting phenomenology is compatible with the latest Planck data and mimics the Λ-cold-dark-matter model. In the second case, since there is no fundamental particle physics reason for the graviton to change its mass, we assume that primordially the tensor perturbations propagate with the speed equal to that of light’s, and thus this constraint restricts the functional form of the scalar coupling function ξ(ϕ), which must satisfy the differential equation . The latter equation is greatly simplified when late times are considered and can be integrated analytically to yield a relation for , which depends solely on the Hubble rate, in a model independent way. This leads eventually to an elegant simplification of the Friedmann equation, which when solved numerically, yields a viable late-time phenomenology. A common characteristic of the Einstein–Gauss–Bonnet models we considered is that the dark energy era they produce is free from dark energy oscillations.

在这项工作中，通过使用数值分析，我们以定量的方式研究了标量耦合重力的后期动力学。特别是，我们认为耦合到标量场耦合功能的高斯-博内长期ξ（φ），我们研究了三种类型的模型，一个用˚F（[R已知提供一个可行的后期时间现象）方面，2爱因斯坦-高斯-帽子模型的类型。我们的目的是根据文献中经常使用的适当的状态搜索器量来编写弗里德曼方程，并通过使用物理动机的初始条件对它进行数值求解。在f（R）存在引力项，正如我们实际预期的那样，高斯-邦纳特相关项的贡献很小。该结果对于标量场初始条件的变化具有鲁棒性，其原因是f（R）重力扇形在后期占主导地位。在爱因斯坦-高斯-邦尼特类型的模型中，我们研究了两种不同的情况，首先通过自由选择标量势和标量高斯-邦尼特耦合ξ（ϕ），在这种情况下，产生的现象学与最新的普朗克数据兼容，并且模仿了Λ-冷-暗-物质模型。在第二种情况下，由于引力子没有改变其质量的基本粒子物理学原因，我们假设张量扰动原本以等于光速的速度传播，因此该约束条件限制了标量耦合的功能形式。函数ξ（ϕ），必须满足微分方程。当考虑到较晚时间时，可以大大简化后一个方程式，并且可以通过分析积分得出以下方程式的关系：，它以独立于模型的方式完全取决于哈勃速率。最终，这导致了Friedmann方程的简洁化，当对其进行数值求解时，得出了可行的后期现象学。我们考虑过的爱因斯坦-高斯-邦内模型的一个共同特征是，它们产生的暗能量时代没有暗能量振荡。