tokens: with haloes, and the slight difference is because the two tracer classes do not correspond to exactly the same halo population, since central galaxies are populated in haloes in a random way. Third, for the two cases in which we cross-correlate the dark matter field from the F5 simulation with tracers from the GR simulation (cyan dashed and red dot-dot-dot–dashed), ξ gm decreases towards very small separations ( r ≲ 0.3 h −1 Mpc), and this is because GR tracers usually do not coincide with the highest density peaks in the dark matter field of the F5 simulation so that the highest values of ξ gm appear to be away from the GR tracers themselves. All in all, the test highlights the importance of using carefully constructed mock galaxy catalogues when we compare the predictions of GGL in different models. As an example, the 3D real-space two-point correlation functions and the projected 2D correlation functions in our GR and F5 HOD catalogues agree with each other within 2 per cent–3 per cent in the range of separations considered here, but such agreements are in a statistical, rather than an object-by-object, sense. When cross-correlated with the same (F5) matter field, the ξ gm results by using F5 and GR galaxies are indeed different (cf. blue dotted and cyan dashed curves in Fig. 8 ), in particular at small separations. This indicates that high-resolution simulations enable to accurately resolve the internal structures of haloes and to allow realistic galaxy mocks to be constructed, are necessary in order to make GGL more useful for distinguishing f ( R ) gravity and GR. 3.3 Galaxy bias Finally, we compare the galaxy biases in GR and f ( R ) gravity. We are interested in this comparison this for two reasons. First, in standard ΛCDM, the linear bias factor, defined as is scale-independent on large scales. On the other hand, it is known that f ( R ) gravity, and in general chameleon models, predict a scale-dependent linear growth rate of matter density perturbations, which may result in scale-dependent b lin , because we have tuned the HOD parameters to match ξ gg ( r ) in the two models, while their ξ mm ( r ) can have a scale-dependent difference. We want to check whether, at least for F5, such a scale dependence of b lin is significant enough to make it of interest in observations. Secondly, assuming that the linear bias factor b lin gives a correct quantification of galaxy bias at some given scale r , the galaxy–matter cross-correlation coefficient R gm , defined as is equal to 1. This offers an opportunity to derive the matter correlation function ξ mm from observational determinations of ξ gg and ξ gm . We want to see whether the minimum length-scale R min above which this can be performed without worrying about the non-linear effects in galaxy bias is significantly different in GR and in F5. Fig. 9 shows the results of our tests to answer these questions. The top bunch of curves in each panel (left for GR and right for F5) show the
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