Beyond-$\mathrm{\Lambda }\mathrm{CDM}$ models have been proposed to address various shortcomings of the standard cosmological model, such as the “Hubble tension.” These models often have an impact on the discrepancy in the amplitude of matter clustering, the “${\sigma }_8$ tension.” To explore the interplay between the two tensions, we suggest a simple method to visualize the relation between the two parameters: $H_0$ and ${\sigma }_8$. For a given extension of the $\mathrm{\Lambda }\mathrm{CDM}$ model and dataset, we plot the relation between $H_0$ and ${\sigma }_8$ for different amplitudes of the beyond-$\mathrm{\Lambda }\mathrm{CDM}$ physics. In this work, we use this visualization method to illustrate the trend of selected cosmological models, including nonminimal Higgs-like inflation, early dark energy, a varying effective electron mass, an extra number of relativistic species and modified dark energy models. Although already studied in the literature, some of these models have not been analyzed in view of the two joint tensions. We stress that the method used here could be a useful diagnostic tool to illustrate the behavior of complex cosmological models with many parameters in the context of the $H_0$ and ${\sigma }_8$ tensions.

中文翻译：

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用于评估 H0−σ8 平面中的张力解析模型的视觉工具

超过-$\mathrm{\Lambda }\mathrm{清洁发展机制}$人们已经提出模型来解决标准宇宙学模型的各种缺点，例如“哈勃张力”。这些模型通常会对物质聚类的幅度差异产生影响，即“${\sigma }_8$紧张。”为了探索两种张力之间的相互作用，我们建议使用一种简单的方法来可视化两个参数之间的关系：$H_0$和${\sigma }_8$。对于给定的扩展$\mathrm{\Lambda }\mathrm{清洁发展机制}$模型和数据集，我们绘制之间的关系$H_0$和${\sigma }_8$对于不同幅度的超越$\mathrm{\Lambda }\mathrm{清洁发展机制}$物理。在这项工作中，我们使用这种可视化方法来说明所选宇宙学模型的趋势，包括非最小希格斯暴胀、早期暗能量、变化的有效电子质量、额外数量的相对论物种和修改的暗能量模型。尽管已经在文献中进行了研究，但其中一些模型尚未针对两种关节张力进行分析。我们强调，这里使用的方法可能是一个有用的诊断工具，可以用来说明具有许多参数的复杂宇宙学模型在宇宙背景下的行为。$H_0$和${\sigma }_8$紧张局势。