In this work, we consider four $f(R)$ gravity models—the Hu-Sawicki, Starobinsky, Exponential and Tsujikawa models—and use a range of cosmological data, together with Markov Chain Monte Carlo sampling techniques, to constrain the associated model parameters. Our main aim is to compare the results we get when ${\mathrm{\Omega }}_{k,0}$ is treated as a free parameter with their counterparts in a spatially flat scenario. The bounds we obtain for ${\mathrm{\Omega }}_{k,0}$ in the former case are compatible with a flat geometry. It appears, however, that a higher value of the Hubble constant $H_0$ allows for more curvature. Indeed, upon including in our analysis a Gaussian likelihood constructed from the local measurement of $H_0$, we find that the results favor an open universe at a little over $1\sigma$. This is perhaps not statistically significant, but it underlines the important implications of the Hubble tension for the assumptions commonly made about spatial curvature. We note that the late-time deviation of the Hubble parameter from its $\mathrm{\Lambda }\mathrm{CDM}$ equivalent is comparable across all four models, especially in the nonflat case. When ${\mathrm{\Omega }}_{k,0}=0$, the Hu-Sawicki model admits a smaller mean value for ${\mathrm{\Omega }}_{\mathrm{cdm},0}{h}^2$, which increases the said deviation at redshifts higher than unity. We also study the effect of a change in scale by evaluating the growth rate at two different wave numbers $k_{†}$. Any changes are, on the whole, negligible, although a smaller $k_{†}$ does result in a slightly larger average value for the deviation parameter $b$.

中文翻译：

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空间曲率 inf(R)gravity

在这项工作中，我们考虑了四个 $F(\mathrm{电阻})$引力模型——Hu-Sawicki、Starobinsky、指数和辻川模型——并使用一系列宇宙学数据以及马尔可夫链蒙特卡罗采样技术来约束相关的模型参数。我们的主要目的是比较我们在以下情况下得到的结果${\mathrm{\Omega }}_{到,0}$在空间平坦的场景中被视为自由参数。我们获得的界限${\mathrm{\Omega }}_{到,0}$在前一种情况下，与平面几何兼容。然而，看起来更高的哈勃常数值$H_0$允许更多的曲率。事实上，在我们的分析中包含一个由局部测量构建的高斯似然$H_0$，我们发现结果有利于开放宇宙 $1\sigma$. 这可能在统计上并不显着，但它强调了哈勃张力对通常关于空间曲率的假设的重要影响。我们注意到，哈勃参数与其的后期偏差$\mathrm{\Lambda }\mathrm{清洁发展机制}$等效在所有四个模型中具有可比性，尤其是在非平坦情况下。什么时候${\mathrm{\Omega }}_{到,0}=0$，Hu-Sawicki 模型承认较小的平均值 ${\mathrm{\Omega }}_{\mathrm{清洁剂},0}{H}^2$，这增加了红移高于统一的所述偏差。我们还通过评估两个不同波数的增长率来研究规模变化的影响${到}_{†}$. 总体而言，任何变化都可以忽略不计，尽管较小${到}_{†}$ 确实导致偏差参数的平均值稍大 $乙$.