tokens: may have a significant impact on dynamical processes that remain poorly known, such as star formation in molecular clouds, dissolution of open clusters in galactic tidal fields, tidal heating of thin discs, etc. Crucially, our results open up the possibility to test the existence of planet-sized microhaloes via experiments that are sensitive to tidal force fluctuations induced by dark matter substructures, e.g. by measuring the heating rate of weakly bound gravitational systems in different regions of the stellar halo, as well as in the dwarf spheroidal galaxies of the Milky Way (see also Peñarrubia et al. 2010 ). This type of experiments may also constrain a wide range of cosmological theories where dark matter is made of ‘warm’ particles or ULAs (see Section 4.2 ). The mathematical technique explored in this paper provides a computationally efficient tool to model the gravitational perturbations induced by an arbitrarily large number of CDM microhaloes. In follow-up contributions, we plan to run numerical experiments that follow the trajectories of tracer particles in smooth galactic potentials that incorporate the force and tidal distributions derived in Sections 2 and 3 as external stochastic fields. Such codes may provide a useful tool to model observational data (e.g. the separation function of wide binaries, see Section 4.3 ) at a low computational cost. Finally, it is worth stressing that our statistical method can also be used to describe stochastic fluctuations in the force and tidal fields generated by baryonic substructures covering a wide range of mass and sizes, such as GMCs, MACHOs, black holes, stellar clusters, individual stars, planetoids, etc., which may have several applications in areas other than cosmology. APPENDIX A: THE LOCAL APPROXIMATION In Sections 2 and 3 , we derive the distributions and , respectively, under the assumption that the number density of substructures remains constant across the entire system, and that corresponds to the value measured at the galactocentric location of the tracer particle, . Although the local approximation greatly simplifies analytical derivations 1 of the Fourier transform , in general it fails on scales r ≳ d , where d = |∇ n / n | −1 is the distance associated with the spatial variation of the number density profile ( 61 ) The fact that γ 1 in cosmological simulation of structure formation implies that the scalelength d decreases towards the centre of the potential, suggesting that the local approximation may not be valid at small galactocentric radii. The second scale of relevance corresponds to the average separation between substructures, D ( R ). In Section 2 , we found that the large-force fluctuations generated by non-overlapping subhalo populations are dominated by the nearest substructure, which is typically located at a distance D ( R ) ≈ [2π n ( R )] − 1/3 . Hence, the local approximation is justified for d ≫ D , that is when the number density profile varies on
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