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The next generation of cosmological surveys will have unprecedented measurement precision, hence they hold the power to put theoretical ideas to the most stringent tests yet. However, in order to realize the full potential of these measurements, we need to ensure that we apply the most effective analytical tools. We need to identify which cosmological observables are the best cosmological probes. Two commonly used cosmological observables are galaxy redshift number counts and cosmic magnification. Both of these observables have been investigated extensively in cosmological analyses, but only separately. In the light of interacting dark energy (IDE) emerging as a plausible means of alleviating current cosmological tensions, we investigate both observables on large scales in a universe with IDE, using the angular power spectrum, taking into account all known terms, including relativistic corrections, in the observed overdensity. Our results suggest that (given multitracer analysis) measuring relativistic effects with cosmic magnification will be relatively better than with galaxy redshift number counts, at all redshifts z. Conversely, without relativistic effects, galaxy redshift number counts will be relatively better in probing the imprint of IDE, at all z. At low z (up to around |$z \, {=}\, 0.1$|), relativistic effects enable cosmic magnification to be a relatively better probe of the IDE imprint, while at higher z (up to |$z \, {\lt }\, 3$|) galaxy redshift number counts become the better probe of IDE imprint. However, at |$z \, {=}\, 3$| and higher, our results suggest that either of the observables will suffice. [ABSTRACT FROM AUTHOR]

f (R) gravity is one of the simplest viable modifications to General Relativity: it passes local astrophysical tests, predicts both the early-time cosmic inflation and the late-time cosmic acceleration, and describes dark matter. In this paper, we probe cosmic magnification on large scales in f (R) gravity, using the well-known Hu-Sawicki model as an example. Our results indicate that at redshifts |$z \, {\lt }\, 3$|⁠ , values of the model exponent |$n \, {\gt }\, 1$| lead to inconsistent behaviour in the evolution of the scalar perturbations. Moreover, when relativistic effects are taken into account in the large-scale analysis, our results show that as z increases, large-scale changes in the cosmic magnification angular power spectrum owing to integral values of n tend to share a similar pattern, while those of decimal values tend to share another. This feature could be searched for in the experimental data, as a potential 'smoking gun' for the given class of gravity models. Furthermore, we found that at |$z \, {=}\, 1$| and lower, relativistic effects lead to a suppression of the cosmic magnification on large scales in f (R) gravity, relative to the concordance model; whereas, at |$z \, {\gt }\, 1$|⁠ , relativistic effects lead to a relative boost of the cosmic magnification. In general, relativistic effects enhance the potential of the cosmic magnification as a cosmological probe. [ABSTRACT FROM AUTHOR]