Which is a better cosmological probe: Number counts or cosmic
  magnification?
  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 realise 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 the
galaxy number counts, and the cosmic magnification. The galaxy number counts
are a diagnostic of the growth of structure, hence are able to probe the nature
of dark energy (DE). On the other hand, the cosmic magnification is a
diagnostic of cosmic distances and sizes -- consequently, a diagnostic of the
geometry of the large scale structure -- hence is also able to probe the nature
of DE. Both of these observables have been investigated extensively in
cosmological analyses, but only separately. In this paper, by using the angular
power spectrum, we investigate both the galaxy number counts and the cosmic
magnification, on ultra-large scales. Our results suggest that measuring
relativistic effects in the cosmic magnification angular power spectrum will be
relatively better than in the number-count angular power spectrum, at all
redshifts z. On the other hand, we found that without relativistic effects, the
number-count angular power spectrum will be relatively better in probing the
imprint of interacting DE (IDE), at all z. At low z (up to around z = 0.1),
relativistic effects enables the cosmic magnification angular power spectrum to
be a relatively better probe of the IDE imprint; while at higher z (up to z <
3), the number-count angular power spectrum becomes the better probe of IDE
imprint. However, at z = 3 and higher, our results suggest that either the
number-count or the magnification angular power spectrum, will suffice to probe
IDE.
