Linear perturbation theory for tidal streams and the small-scale CDM power spectrum

Tidal streams in the Milky Way are sensitive probes of the population of low-mass dark matter subhaloes predicted in cold dark matter (CDM) simulations. We present a new calculus for computing the effect of subhalo fly-bys on cold streams based on the action-angle representation of streams. The heart of this calculus is a line-of-parallel-angle approach that calculates the perturbed distribution function of a stream segment by undoing the effect of all relevant impacts. This approach allows one to compute the perturbed stream density and track in any coordinate system in minutes for realizations of the subhalo distribution down to 10(5) M-circle dot, accounting for the stream's internal dispersion and overlapping impacts. We study the statistical properties of density and track fluctuations with large suites of simulations of the effect of subhalo fly-bys. The one-dimensional density and track power spectra along the stream trace the subhalo mass function, with higher mass subhaloes producing power only on large scales, while lower mass subhaloes cause structure on smaller scales. We also find significant density and track bispectra that are observationally accessible. We further demonstrate that different projections of the track all reflect the same pattern of perturbations, facilitating their observational measurement. We apply this formalism to data for the Pal 5 stream and make a first rigorous determination of 10(-6)(+11) dark matter subhaloes with masses between 10(6.5) and 10(9) M-circle dot within 20 kpc from the Galactic centre [corresponding to 1.4(-0.9)(+1.6) times the number predicted by CDM-only simulations or to f(sub)(r < 20 kpc) approximate to 0.2 per cent] assuming that the Pal 5 stream is 5Gyr old. Improved data will allow measurements of the subhalo mass function down to 10(5) M-circle dot, thus definitively testing whether dark matter is clumpy on the smallest scales relevant for galaxy formation.