Beyond-$\Lambda$CDM models, which were proposed to resolve the "Hubble tension", 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 propose a simple method to visualize the relation between the two parameters $H_0$ and $\sigma_8$: For a given extension of the $\Lambda$CDM model and data set, we plot the relation between $H_0$ and $\sigma_8$ for different amplitudes of the beyond-$\Lambda$CDM physics. We use this visualization method to illustrate the trend of selected cosmological models, including non-minimal Higgs-like inflation, early dark energy, a varying effective electron mass, an extra number of relativistic species and modified dark energy models. We envision that the proposed method could be a useful diagnostic tool to illustrate the behaviour of complex cosmological models with many parameters in the context of the $H_0$ and $\sigma_8$ tensions.
Comment: 19 pages, 3 figures, 5 tables