The critics argue that some of the assumptions made in the no dark matter paper are known to be counterfactual. The criticism is a valid one to raise, but weaker than it appears. Physicists make counterfactual assumptions all the time in models (e.g. modeling matter distributions in a galaxy as a continuous distribution fitting some function reasonably close to reality, rather than a collection of discrete, nearly point-like masses) in order to make computation more manageable. Even supercomputers don't have the power to solve these kinds of massively multibody problems exactly by analytical or fine grained numerical models without some simplying assumptions.
The hard to evaluate question is whether the counterfactual assumptions are ones that materially impair the accuracy of the conclusion, and if the alternative assumptions proposed remedy that problem if there is one.
Determining just how much room for error simplying assumptions introduce into a numerical approximation of a complex system's dynamics is more art than science. Many kinds of systemic error sources are frequently underestimated or ignored entirely, if present. On the other hand, measurement and sampling errors, while a serious concern in some kinds of scientific work, tend to be materially overestimated by physicists, particularly because physicists are prone to simply adding up linearly multiple error sources that are genuinely independent of each other and tend to mitigate each other due to the law of averages if one does a more sophisticated analysis of the sources and relationships of these kinds of errors.
In this case, critics have a fairly low hurdle to overcome. Since the expected contribution of dark matter effects in the general vicinity of the solar system is so slight, and since all of the data points and numerical approximation procedures used have some margin of error, a critic of a finding that purports to have an expected value of zero and error bars sufficient small to rule out the low expected dark matter density distribution in the region under a conventional wisdom spherical halo of cold dark matter hypothesis need only dispense with assumptions that slightly enlarge the magnitude of the estimated margin of error between the calculated dark matter value and zero. The critics don't need to actually affirmatively show an expected value for a dark matter density that is has a value different with zero with multiple sigma statistical significance.
There are hypothesis testing tools in statistics that can compare the relative likelihood of different dark matter hypotheses and a null hypothesis of no dark matter with either a frequentist or Baysian approach. But, making a genuine apples to apples comparison of the possibilities in a way that isn't really offering up a straw man extreme version of the dark matter hypothesis being tested is subtle stuff.
Often the secondary factual questions that validate or invalidate the assumptions relied upon in competing interpretations of the data are themselves based upon a fragile set of assumptions with their own warts and uncertainties. The overall enterprise is a bit like doing quality control in a massively complex engineering project like building a rocket that it is hard to test before you need to know the answer. And, while most rockets launch successfully, we know from experience that even large teams of highly talented and well funded scientific and engineers can frequently miss the one or two missteps (that often seem pedestrian and obvious in isolation and retrospect once you know which flaw you are looking for) that will make the end result disasterously wrong.
The particular claim made by the critics in this case is that:
An analysis of the kinematics of 412 stars at 1-4 kpc from the Galactic mid-plane by Moni Bidin et al. (2012) has claimed to derive a local density of dark matter that is an order of magnitude below standard expectations.The original study had concluded that: "We extrapolate a dark matter (DM) density in the solar neighborhood of 0+-1 mM_sun pc^-3, and all the current models of a spherical DM halo are excluded at a confidence level higher than 4sigma."
We show that this result is incorrect and that it arises from the invalid assumption that the mean azimuthal velocity of the stellar tracers is independent of Galactocentric radius at all heights; the correct assumption---that is, the one supported by data---is that the circular speed is independent of radius in the mid-plane. We demonstrate that the assumption of constant mean azimuthal velocity is physically implausible by showing that it requires the circular velocity to drop more steeply than allowed by any plausible mass model, with or without dark matter, at large heights above the mid-plane.
Using the correct approximation that the circular velocity curve is flat in the mid-plane, we find that the data imply a local dark-matter density of 0.008 +/- 0.002 Msun/pc^3= 0.3 +/- 0.1 Gev/cm^3, fully consistent with standard estimates of this quantity. This is the most robust direct measurement of the local dark-matter density to date.
Thus, the critics are arguing that the original paper is highly sensitive to the circular velocity of the small number of stars that are far above or below the galactic plane midway between the central black hole of the Milky Way galaxy and its fringe, where we live, while the original authors claimed that their method of approximation was "the most robust" method used to date, implying that the result is not very sensitive to parameters that aren't known very well right now. The relevant assumptions are discussed at pages 13-18 and pages 20-22 of the original pre-print.
More specifically, the body of the critical paper argues as it preliminarily lays out the gist of its argument that:
The main error is that they assume that the mean azimuthal (or rotational) velocity ¯ V of their tracer population is independent of Galactocentric cylindrical radius R at all heights (i.e., ¯ V (R, Z) = ¯ V (Z)).Put another way, the critics argue that it is inappropriate to extrapolate results that the original paper uses to support its assumptions about the relationship between circular velocity and distance from the galactic plane for out of plane objects from the galactric middle where the data was obtained to the galactric fringe where the solar system is located, and that there are good motivations for believing that out of plane stars in the galactric fringe (which have not been directly observed by anyone) are have circular velocities quite different from those in the data set and are common enough to materially impact the result.
This assumption is not supported by the data, which instead imply only that the circular speed Vc is independent of radius in the mid-plane (e.g., Gunn et al. 1979; Feast & Whitelock 1997). In the solar neighborhood, the circular speed is larger by & 35 km s−1 than the mean azimuthal velocity for the warm tracer population used by MB12, a phenomenon known as asymmetric drift. The asymmetric drift is expected to vary with R, although this variation cannot be measured for the sample of MB12 as the data do not span a large enough range in R.
In the absence of a measurement of the (R, Z) dependence of ¯ V for the tracers, we show that the assumption of an R-independent ¯ V at all heights Z is highly implausible. By using instead the data-driven assumption that the circular speed is independent of radius at Z = 0, we demonstrate that the measurements and analysis of MB12 are fully consistent with the standard estimate of the local dark matter density, approximately 0.01M⊙ pc−3, and indeed provide the best available direct measurement of this quantity.
Until someone observes a few dozen of these far out of the galactic plane stars in the galactric fringe and measures their dynamics with great precision, however, there is fundamentally just a difference in assumption between the two studies that isn't ultimately amenable to a definitive resolution without resort to some sort of supplementary empirical data set. Cold dark matter theory doesn't have a particular good track record of making accurate predictions about empirical behavior in astronomical contexts that weren't used to set the parameters of the theory in the first place, and the underlying spherical dark matter distribution assumption that goes into the theoretical expectation have very little justification, apart from mathematical simplification considerations associated with a sphere's many symmetries, in galactric dynamics or hypothetical non-baryonic dark matter dynamics.