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In spite of the reasonable agreement of the $\Lambda$CDM model with various data sets, pressing historical issues like the cosmological constant problem and more recent issues like the H0 tension and the $\sigma$8 tension make relevant investigating the following question: can a modified gravity based late-time model exactly mimic a $\Lambda$CDM-like cosmological evolution? Indeed, this question is frequently addressed in the context of various modified gravity models, mostly via the reconstruction method. In my talk, I will revisit $\Lambda$CDM-mimicking $f(R)$-gravity models. An important question in this context is, even if the $\Lambda$CDM model and $\Lambda$CDM-mimicking f(R) models are cosmographically identical at the background level, can we obtain distinctive signatures at the perturbation level? The usual consensus would be that to answer this question we need to first reconstruct the form of the $\Lambda$CDM-mimicking f(R). This has been done previously and the answer, unfortunately, is too complicated (Hypergeometric functions) for any practical purpose. The main point of my talk is that such an explicit reconstruction of $\Lambda$CDM-mimicking f(R) is not necessary. There is a way around it. Consequently, we have been able to find analytically what kind of distinctive signatures can such models give rise to at the level of density perturbations. The talk is based on 2103.02274 and 2408.03998. |