Reading the Milky Way satellite limits on warm and fuzzy dark matter

Read How warm and fuzzy dark matter erase small ripples first; the transfer function and half-mode mass come from there. Recompute the warm cutoff, the fuzzy cutoff and the soliton checks the warm half-mode formula this paper builds on. You should be comfortable with a likelihood and a 95 per cent upper limit. A subhalo is a small halo orbiting inside a larger one, and a satellite galaxy is the stars that a subhalo holds.

Which version this guide reads

The paper is Nadler et al., Milky Way Satellite Census III, Constraints on Dark Matter Properties from Observations of Milky Way Satellite Galaxies, read here as arXiv:2008.00022v2, posted on 1 March 2021. Its arXiv record links the published version, Physical Review Letters 126, 091101 (2021). We read the full arXiv text, including the Supplemental Material, but did not compare it line by line with the typeset journal article. The authors' code and subhalo catalogs are linked from the paper; we did not run them.

The paper uses $h=0.7$ , $\Omega_{\mathrm m}=0.286$ , $\sigma_8=0.82$ and $n_{\mathrm s}=0.96$ , and the Bryan–Norman virial overdensity $\Delta_{\mathrm{vir}}\simeq99.2$ . Our course uses the Planck 2018 values, which give $102.5$ . Keep the two apart when you compare masses.

Figure 1 is the whole argument

Figure 1 has three panels, and reading them from left to right gives the logic of the paper.

The left panel shows $T^2(k)$ for the warm, interacting and fuzzy models that the analysis rules out: 6.5 keV, a dark matter–proton cross section of $8.8\times10^{-29}\,\mathrm{cm^2}$ at 100 MeV, and $2.9\times10^{-21}\,\mathrm{eV}$ . This is linear theory, the same quantity as in chapter 3.

The middle panel turns each cutoff into $f_{\mathrm{DM}}(M)$ , the fraction of cold-dark-matter subhalos of mass $M$ that survive. A vertical dashed line marks the 95 per cent upper limit on the lowest halo mass known to host a Milky Way satellite, taken from the authors' earlier census paper. Subhalos below that line hold no galaxy we can see, so their loss cannot be tested.

The right panel shows the predicted number of satellites brighter than each luminosity, against the DES and PS1 data. The caption warns that this panel is a one-dimensional summary. The actual fit uses the luminosity, size and sky position of each satellite in each survey footprint.

Equations 1 to 7, with numbers

Equation 1 defines $T^2=P_{\mathrm{DM}}/P_{\mathrm{CDM}}$ , and the half-mode is where $T^2=0.25$ . Equation 2 gives the half-mode mass, as in chapter 3. Equation 3 says the only thing the dark matter model changes is the subhalo mass function:

\left(\frac{dN_{\mathrm{sub}}}{dM}\right)_{\mathrm{DM}}=f_{\mathrm{DM}}(M,\theta_{\mathrm{DM}})\left(\frac{dN_{\mathrm{sub}}}{dM}\right)_{\mathrm{CDM}}.

For warm dark matter, equations 4 and 5 are

f_{\mathrm{WDM}}=\left[1+\left(\frac{\alpha M_{\mathrm{hm}}}{M}\right)^{\beta}\right]^{\gamma},\qquad M_{\mathrm{hm}}=5\times10^8\,\mathrm M_\odot\left(\frac{m_{\mathrm{WDM}}}{3\,\mathrm{keV}}\right)^{-10/3},

with $\alpha=2.7$ , $\beta=1.0$ and $\gamma=-0.99$ , taken from simulations of Milky Way-mass hosts by Lovell and collaborators. For fuzzy dark matter, equation 6 has the same form with mass scale $M_0$ and mass-dependent exponents listed in the Supplemental Material, and equation 7 is

M_0=1.6\times10^{10}\,\mathrm M_\odot\left(\frac{m_\phi}{10^{-22}\,\mathrm{eV}}\right)^{-4/3}.

Nadler et al. 2021, equations 1 to 7

Example 6.1. What does the warm limit say about subhalo numbers? Put the fiducial limit $M_{\mathrm{hm}}=3.8\times10^7\,\mathrm M_\odot$ into equation 4. At $M=10^8\,\mathrm M_\odot$ , $f=[1+2.7\times0.38]^{-0.99}=2.026^{-0.99}=0.497$ . At $10^9\,\mathrm M_\odot$ , $f=1.103^{-0.99}=0.908$ . So the excluded model removes about half the $10^8\,\mathrm M_\odot$ subhalos and about a tenth of those ten times heavier. The data allow no more loss than that.

Example 6.2. Turn the mass limits into particle masses. Inverting equation 5, $m_{\mathrm{WDM}}=3\,\mathrm{keV}\,(M_{\mathrm{hm}}/5\times10^8)^{-3/10}$ . The fiducial $3.8\times10^7$ gives $6.50\,\mathrm{keV}$ , and the unscaled $3.0\times10^7$ gives $6.98\,\mathrm{keV}$ , matching the printed 6.5 and 7.0 keV. Inverting equation 7, $m_\phi=10^{-22}\,\mathrm{eV}\,(M_0/1.6\times10^{10})^{-3/4}$ . The fiducial $M_0=1.8\times10^8$ gives $2.89\times10^{-21}\,\mathrm{eV}$ , matching the printed $2.9\times10^{-21}$ . The paper prints only the scaled particle mass; by our own calculation the unscaled $1.4\times10^8$ gives $3.50\times10^{-21}\,\mathrm{eV}$ .

How the fit is built

The data are the satellites found in DES and PS1, each with an observational selection function that says how likely a satellite of given brightness, size, distance and sky position is to be detected. The model starts from dark-matter-only zoom-in simulations of two Milky Way-like hosts with virial masses $1.57\times10^{12}$ and $1.26\times10^{12}\,\mathrm M_\odot$ . Both include an analog of the Large Magellanic Cloud, which the authors' earlier census paper showed is needed to reproduce how satellites are spread over the sky.

An empirical galaxy–halo model then places galaxies in subhalos. Its eight parameters cover the luminosity relation and its scatter, the size relation, the fraction of halos that host a visible galaxy and how efficiently the Milky Way disk destroys subhalos. The dark matter enters as one more parameter, $\log_{10}M_{\mathrm{hm}}$ or $\log_{10}M_0$ , with a flat prior. Each mock satellite's detection probability is multiplied by $f_{\mathrm{DM}}$ . The likelihood is Poisson in bins of surface brightness. The sampler emcee ran $10^5$ iterations with 36 walkers, discarding the first $2\times10^4$ steps. In the Supplemental table of 95 per cent intervals, the lower ends for $M_{\mathrm{hm}}$ and $M_0$ are marked as set by the prior, as expected when the data give only an upper limit.

The last step is a scaling. A heavier host has heavier subhalos, so observed satellites would sit in heavier subhalos, and the limit would loosen. The authors multiply each upper limit by the ratio of the largest Milky Way mass allowed by Gaia satellite kinematics to the simulated average, $1.4\times10^{12}\,\mathrm M_\odot$ . The printed limits move from $3.0$ to $3.8\times10^7$ and from $1.4$ to $1.8\times10^8$ , both factors near $1.3$ . The ratio itself is not printed in the parts we read. Our inference from these rounded numbers is a largest allowed mass near $1.8\times10^{12}\,\mathrm M_\odot$ .

Where the limits are soft

The fuzzy subhalo model is the largest theoretical uncertainty, and the paper says so. The fiducial $f_{\mathrm{FDM}}$ comes from a semi-analytic model that includes tidal stripping of subhalos with soliton cores. Using instead the mass function from Schive and collaborators, which has $\tilde\beta=1.1$ and $\tilde\gamma=-2.2$ , gives $M_0<3.4\times10^7\,\mathrm M_\odot$ and $m_\phi>9.1\times10^{-21}\,\mathrm{eV}$ , about three times stronger. The Supplemental Material notes that the Schive runs used ordinary particles with fuzzy initial ripples and did not solve the Schrödinger–Poisson equations, while the semi-analytic model does not treat the quantum-pressure term explicitly. Neither is a full wave calculation of subhalos.

The warm limit applies to thermal relics only. The authors note that a mass function built for resonantly produced sterile neutrinos is significantly less suppressed. They handle sterile neutrinos separately in the Supplemental Material.

Only two hosts were simulated. The authors list this as a limit on both accuracy and precision.

The model assumes the dark matter changes only how many subhalos there are, not where they orbit. For fuzzy dark matter they say this is less certain, because dynamical friction differs for waves, though they estimate a negligible effect for the $10^8\,\mathrm M_\odot$ subhalos that drive the limit.

Cores are not part of the warm fit. Read, Agertz and Collins 2016 state in their abstract that cored dwarfs are more easily destroyed by tides, which would remove subhalos in cold dark matter too. The disk-disruption parameter in Nadler's model is fitted, but it is not a model of feedback-made cores, and we have not checked how much cores could move the limit. Bullock and Boylan-Kolchin 2017, section 3.1 summarize simulations in which feedback cores are negligible at the lowest stellar masses, so for ultra-faint satellites the effect may be small.

A unit difference worth noticing

The abstract and Table I give the fuzzy de Broglie wavelength as $\lambda_{\mathrm{dB}}\lesssim0.5\,\mathrm{kpc}$ . The results paragraph gives $0.5\,h^{-1}\,\mathrm{kpc}$ . With $h=0.7$ these differ by a factor of $1.4$ . We have not worked out which the authors intended and do not correct either. When quoting the result, use the particle mass, $m_\phi>2.9\times10^{-21}\,\mathrm{eV}$ , which the abstract, Table I and the results text all agree on.

What kind of claim each one is

**Table 6.1** Claims in and around the paper, sorted by status.
Claim	Status	Where
Satellite counts agree with cold dark matter down to $k\sim40\,h\,\mathrm{Mpc^{-1}}$	Published result, within the satellite model	Introduction
Thermal-relic warm dark matter above 6.5 keV	Published 95 per cent limit, thermal relics only	Results, Table I
Fuzzy dark matter above $2.9\times10^{-21}\,\mathrm{eV}$	Published 95 per cent limit, depends on the subhalo model	Results, Table I
Fuzzy limit about three times stronger with the Schive mass function	Published alternative, not the fiducial result	Supplemental Material
Fuzzy dark matter at $10^{-22}\,\mathrm{eV}$ strongly disfavored	Published interpretation	Results
The shapes $f_{\mathrm{WDM}}$ and $f_{\mathrm{FDM}}$	Predictions from simulations and a semi-analytic model	Equations 4 and 6
A wave-based subhalo mass function for fuzzy dark matter	Open	Supplemental Material
Later satellite surveys and limits since 2021	Not read for this guide	None

Connecting the paper to this course

In chapter 3, a 3 keV particle had a half-mode mass of $4.9\times10^8\,\mathrm M_\odot$ , about 13 times above Nadler's limit. Fuzzy dark matter at $m_{22}=1$ , the mass the paper says was invoked to explain dwarf cores, sits about 29 times below the fuzzy limit.

The companion Lab draws warm halo numbers with Schneider's field-halo formula, $(1+M_{\mathrm{hm}}/M)^{-1.16}$ , not with Nadler's subhalo formula. At $M_{\mathrm{hm}}=3.8\times10^7$ and $M=10^8\,\mathrm M_\odot$ the Schneider form gives $0.688$ and the Nadler form gives $0.497$ . They describe different objects, halos in the field against subhalos inside a host, and they come from different simulations. Read the Lab's abundance readout as a field-halo ratio, separate from the satellite-count limit.

A checklist for any satellite-count limit

Which survey footprints, which selection functions, and which satellites are in the sample?
Which simulated hosts, how many, and do they include an LMC analog?
Which subhalo suppression formula, and from which simulations?
Is the dark matter parameter given a flat prior in the logarithm, and does the limit sit near a prior edge?
How is the Milky Way mass handled: marginalized, fixed or scaled?
What does the model leave out: cores, wave interference, changes in subhalo orbits?
Which numbers are in the abstract, which in the tables, and do they agree?

Lab tasks

Predict. In the halo Lab, choose warm dark matter and read the half-mode mass at 6.5 keV. Is it close to the paper's $3.8\times10^7\,\mathrm M_\odot$ ? Remember the cosmologies differ.
Change one condition. Slide the fuzzy particle mass from $m_{22}=1$ up to $m_{22}=29$ in a $10^9\,\mathrm M_\odot$ halo. Watch the halo-number readout and the soliton core.
Explain. Set warm dark matter to 6.5 keV and the halo mass to $10^9\,\mathrm M_\odot$ . The Lab's halo-number readout is about $0.96$ , while equation 4 at the paper's limit gives $0.908$ . Give two reasons.

Exercises

Using equation 4, at what subhalo mass does the fiducial warm limit remove exactly 20 per cent of subhalos?
The Schive mass function gives $M_0<3.4\times10^7\,\mathrm M_\odot$ . Use equation 7 to check the printed $9.1\times10^{-21}\,\mathrm{eV}$ , and give the ratio to the fiducial limit.
Suppose a later analysis found the Milky Way's halo mass is at most $1.2\times10^{12}\,\mathrm M_\odot$ . Using the authors' linear scaling, would the warm limit get stronger or weaker, and roughly to what particle mass?

Answers

Solve $[1+2.7\times3.8\times10^7/M]^{-0.99}=0.8$ . The bracket must equal $0.8^{-1/0.99}=1.2526$ , so $2.7\times3.8\times10^7/M=0.2526$ and $M\simeq4.1\times10^8\,\mathrm M_\odot$ .
$(3.4\times10^7/1.6\times10^{10})^{-3/4}=(2.125\times10^{-3})^{-0.75}\simeq101$ , so $m_\phi\simeq1.01\times10^{-20}\,\mathrm{eV}$ , about 11 per cent above the printed $9.1\times10^{-21}$ . The same inversion reproduces the fiducial $2.9\times10^{-21}$ , so we cannot explain this gap from the text we read and leave it as an open question for the authors. The ratio of the printed limits is $9.1/2.9\simeq3.1$ .
A lighter host means satellites sit in lighter subhalos, so the limit gets stronger. The scale factor becomes $1.2/1.4=0.86$ instead of about $1.3$ , so $M_{\mathrm{hm}}\lesssim3.0\times10^7\times0.86=2.6\times10^7\,\mathrm M_\odot$ . Equation 5 then gives about $7.3\,\mathrm{keV}$ . The authors validated linear scaling only with their two hosts, so treat this as a rough estimate.