Probing the nature of dark matter particles with stellar streams
Nilanjan Banik, Gianfranco Bertone, Jo Bovy, Nassim Bozorgnia
PPrepared for submission to JCAP
Probing the nature of dark matterparticles with stellar streams
Nilanjan Banik, a,b
Gianfranco Bertone, a Jo Bovy c,d and NassimBozorgnia a,e a GRAPPA Institute, Institute for Theoretical Physics Amsterdamand Delta Institute for Theoretical Physics, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, The Netherlands b Lorentz Institute, Leiden University, Niels Bohrweg 2,Leiden, NL-2333 CA, The Netherlands c Department of Astronomy and Astrophysics, University of Toronto,50 St. George Street, Toronto, ON, M5S 3H4, Canada d Alfred P. Sloan Fellow e Institute for Particle Physics Phenomenology, Department of Physics,Durham University, Durham, DH1 3LE, United KingdomE-mail: [email protected]
Abstract.
A key prediction of the standard cosmological model – which relies on the as-sumption that dark matter is cold, i.e. non-relativistic at the epoch of structure formation –is the existence of a large number of dark matter substructures on sub-galactic scales. Thisassumption can be tested by studying the perturbations induced by dark matter substruc-tures on cold stellar streams. Here, we study the prospects for discriminating cold from warmdark matter by generating mock data for upcoming astronomical surveys such as the LargeSynoptic Survey Telescope (LSST), and reconstructing the properties of the dark matterparticle from the perturbations induced on the stellar density profile of a stream. We discussthe statistical and systematic uncertainties, and show that the method should allow to setstringent constraints on the mass of thermal dark matter relics, and possibly to yield anactual measurement of the dark matter particle mass if it is in the O (1) keV range. Keywords:
Dark matter theory, dark matter substructures
Preprint numbers:
IPPP/18/24 a r X i v : . [ a s t r o - ph . C O ] J un ontents Understanding the nature of dark matter is one of the most pressing problems in cosmologyand particle physics [1–4]. A key prediction of the standard cosmological model, based on theassumption that dark matter is cold, i.e. non-relativistic at the epoch of structure formation,is that a large number of dark matter substructures should exist in Milky Way-like galaxies[5, 6]. Detecting these subhalos will not only be a strong indicator for the existence of darkmatter but also will give valuable information about its particle nature. Depending on theircouplings and production mechanism, in fact, dark matter particles can achieve non-negligiblevelocity dispersions – thus act as warm dark matter (WDM) – leading to a suppression ofthe primordial density fluctuations on small scales, and thus a cutoff at small halo masses,as already realised long ago [7].We discuss below the case of a thermal WDM relic, for which the cutoff in the powerspectrum depends only on the WDM particle mass, but the discussion can be generalisedto sterile neutrinos [8–13] by taking into account the lepton asymmetry parameter (see nextsection). The subhalo mass function of WDM and cold dark matter (CDM) differs formasses smaller than those of dwarf galaxies. On these scales, such substructures would becompletely dark matter dominated, and contain too few stars to be observed directly. Anumber of strategies have been proposed to indirectly probe low mass dark matter halos,including analyses of the impact on Ly α forest observations [14–20] and perturbations ofstrong gravitational lenses [21–29].Here, we study the prospects for identifying the nature of dark matter particles bystudying the perturbations induced by sub-dwarf galaxy clumps on cold stellar streams. Astellar stream, created as a result of tidal disruption of a globular cluster or dwarf galaxy bythe Milky Way potential, has more or less uniform stellar density along its length. A flyby– 1 –ubhalo gravitationally perturbs the stars in the stream resulting in a region of low stellardensity or gap whose size increases with time. There has been a lot of work recently [30–35]focusing on the study of gaps in stellar streams as a result of subhalo encounters. Specifically,ref. [34] showed that with accurate measurements of the density and phase-space structureof stellar streams, which is possible with the near future galaxy surveys like LSST and theGaia mission currently taking data, we will be able to measure subhalo impacts with massas low as 10 M (cid:12) . In ref. [36], a novel framework was presented for inferring properties ofthe impacting subhalos by analyzing the power spectrum of the fluctuations in the densityand mean track of the perturbed stream. This method was applied on Pal 5 density dataand was shown to be sensitive to subhalos with mass as low as 10 . M (cid:12) . The authors alsopredicted that with better data this method will be sensitive to even 10 M (cid:12) subhalos. Inorder to discriminate between WDM and CDM, we will in particular exploit the differencein the density power spectrum arising from the different subhalo populations.This paper is structured as follows. In section 2, we describe the WDM model thatwe used to simulate the stream-subhalo encounters in the WDM scenario. In section 3, wediscuss our method of generating fast stream density simulations of the GD-1 stream and howwe model subhalo impacts on it. In section 4, we present our results of the stream densityperturbations and their power spectra, and apply the Approximate Bayesian Computation(ABC) technique to constrain the mass of the dark matter particle from the stream powerspectra. We show that for typical cases we are able to distinguish between WDM and CDMmodels. In section 5, we demonstrate the risks of using only one stream to constrain theparticle mass of dark matter. With these examples we make a case for the need of applyingour method on multiple streams to make robust predictions on the mass of the dark matterparticle. Finally we conclude in section 6. In appendix A we discuss how the scale radiusof subhalos in the WDM scenario differ from the CDM scenario and why this variation haslittle effect on our method. We discuss below a thermal WDM relic, for which the cutoff in the power spectrum dependsonly on the WDM particle mass, but the discussion can be generalised to sterile neutrinos bytaking into account the lepton asymmetry parameter . For example, the power spectrum ofa 3.3 keV thermal WDM matches to a high degree with that of a 7 keV sterile neutrino withlepton asymmetry parameter equal to 8.66 (see figure 1 in ref. [37]), with both power spectradeviating from the CDM case at around log( k ) (cid:38) . h Mpc − , where k is the comovingwave number and h = H /
100 km / s / Mpc is the dimensionless Hubble parameter. The 7keV sterile neutrino is especially interesting since its decay products have been attributedto be the source of the 3.5 keV X-ray line [38, 39] detected in stacked spectrum of clustersas well as in the spectra of Andromeda and Perseus cluster. Notice that the case of the 7keV sterile neutrino with lepton asymmetry of 8.66 corresponds to the coldest in the 7 keV The lepton asymmetry is defined as L ≡ ( n ν e − n ¯ ν e ) /s , where n ν e and n ¯ ν e are the number densitiesin electron neutrinos and anti-neutrinos respectively, and s is the entropy density of the Universe. For veryhigh and low values of lepton asymmetry the power spectrum of sterile neutrinos can be well approximatedby that of a thermal WDM. – 2 –terile neutrino family, so any constraints on it can be extended to all the other members ofthe family.Various studies have set a lower bound on the mass of thermal WDM based on theobserved clumpiness in the Ly- α forest. Ref. [17] found this lower bound to be 3.3 keV (2 σ ),while ref. [18] found it to be 4.09 keV (95% CL) from Ly- α data alone, and 2.96 keV (95% CL)when combining Ly- α data with CMB data from Planck 2016. Very recently, ref. [40] foundthis lower limit to be 4.17 keV (95% CL) using two different Ly- α measurements, and 4.65keV (95% CL) when adding a third set of data. Notice that the constraints on the thermalWDM mass depend on the thermal history assumed for the intergalactic medium [19].In this work we model WDM as a thermal relic and emphasize on the case of 3.3 keVparticle mass. We follow the same framework that was adopted in WDM simulation works[37, 41, 42]. The WDM and CDM power spectra are related by a transfer function T ( k ): P WDM ( k ) = T ( k ) P CDM ( k ) , (2.1)where T ( k ) is approximated by the fitting formula [43]: T ( k ) = (1 + ( αk ) ν ) − /ν , (2.2)with ν and α constants. Ref. [15] found that for k < h Mpc − , the best fit of the transferfunction is obtained with ν = 1 . α is the cutoff scale as a result of free streaming in theWDM power spectrum relative to CDM, which following ref. [15], is given by α = 0 . (cid:16) m WDM keV (cid:17) − . (cid:18) Ω WDM . (cid:19) . (cid:18) h . (cid:19) . h − Mpc . (2.3)Here m WDM is the WDM particle mass and Ω
WDM is the WDM contribution to the densityparameter.The length scale at which the transfer function drops by a factor of 2 is the half mode wavelength, λ hm , and the mean mass enclosed within a sphere of diameter equal to λ hm isthe half mode mass, M hm . The half mode mass quantifies the threshold mass below whichthe WDM subhalo mass function is strongly suppressed. For example, for a 3.3 keV thermalWDM, M hm ∼ × h − M (cid:12) [17, 44, 45].Based on this WDM model, ref. [46] performed a series of high-resolution N-body sim-ulations and obtained a functional fit for the differential mass function of WDM relative toCDM as (cid:18) dndM (cid:19) WDM = (cid:18) M hm M (cid:19) − β (cid:18) dndM (cid:19) CDM , (2.4)where M is the subhalo mass and β is a free parameter found to be equal to 1.16. Ref. [41]studied the abundance and structure of WDM subhalos within a Milky Way-like host halousing high resolution N-body simulations based on the Aquarius Project [6]. They foundthat the above functional fit is improved by introducing another parameter γ , such that (cid:18) dndM (cid:19) WDM = (cid:18) γ M hm M (cid:19) − β (cid:18) dndM (cid:19) CDM , (2.5)with γ = 2 . β = 0 .
99. – 3 –ince the number of subhalo encounters experienced by a particular stellar stream de-pends on the number of subhalos within the average radius of the stream from the galacticcenter, we need a model for the radial dependence of the number density of WDM subhalos.In ref. [6], it was shown that for a Milky Way like host galaxy, the radial distribution ofCDM subhalos in the mass range [10 − ] M (cid:12) is described by an Einasto profile. Fur-thermore, the CDM subhalo mass function for those halos were shown to be well describedby dn/dM ∝ M − . . These results were combined in ref. [47] to estimate the normalizedsubhalo profile at a given mass and at a certain galactrocentric radius: (cid:18) dndM (cid:19) CDM = c (cid:18) Mm (cid:19) − . exp (cid:26) − α (cid:20)(cid:18) rr − (cid:19) α − (cid:21)(cid:27) (2.6)with c = 2 . × − M − (cid:12) kpc − , m = 2 . × M (cid:12) , α = 0 .
678 and r − = 162 . (cid:18) dndM (cid:19) WDM = c (cid:18) Mm (cid:19) − . exp (cid:26) − α (cid:20)(cid:18) rr − (cid:19) α − (cid:21)(cid:27) (cid:18) γ M hm M (cid:19) − β . (2.7)Notice that we have ignored the evolution of the subhalo number density over the ageof the stream, since we do not expect it to cause any significant change in the results. Thisis because gaps fill up over time due to the internal velocity dispersion in the stream andtherefore very old gaps will not be visible today.Baryonic effects have been shown to tidally disrupt dark matter subhalos thereby reduc-ing their number. Ref. [48] used APOSTLE simulations in the ΛCDM framework to estimatethat the substructure abundance in the mass range [10 . − . ] M (cid:12) inside a Milky Waymass halo is greatly affected over a lookback time of up to 5 Gyr. Within the radius of theGD-1, they predicted a reduction of dark matter substructures by ∼ − − ] M (cid:12) by 47%. For WDM subhalos this factor may be even greater asthey are more easily tidally disrupted owing to their lower concentration at the time of theirinfall. However, we have ignored this in the present work.Figure 1 shows the cumulative number of subhalos for different WDM particle massesobtained by integrating eq. (2.7) up to a subhalo mass of 10 M (cid:12) , shown by the solid lines.The dashed lines represent the cumulative number of subhalos taking the 47% reduction intoaccount. In order to analyze the gap power spectrum of a stellar stream due to impacts from thermalWDM subhalos, we need a mock stream that had minimal gap inducing perturbations frombaryonic effects. In refs. [49, 50], it was shown that the Milky Way bar can induce gaps instellar streams such as Pal 5 [51], which is in prograde motion with respect to the pattern– 4 – M sub (M (cid:12) ) − − N ( > M s ub ) Dashed: × N r < kpc Figure 1 . Cumulative number of subhalos within a Galactocentric radius of 23 kpc of a Milky Waysized host halo for different WDM particle mass scenarios. Dashed lines indicate the cumulativenumbers taking into account the 47% reduction of the number of subhalos due to baryonic effects. speed of the bar. Such gaps are not induced in streams like the GD-1 [52] which is in retro-grade motion with respect to the bar [49, 53, 54]. Furthermore, ref. [55] studied the effectsof giant molecular clouds on the GD-1 stream and found that because of the giant molecularclouds’ larger pericentre and steeper mass function, and also due to GD-1’s retrograde motionthere is no strong density perturbations induced on the GD-1 stream. Taking all these pointsinto consideration, the density gaps found in GD-1 stream are expected to be solely due todark matter subhalo impacts. Therefore, we generate a mock GD-1 stream for our presentanalysis.To generate a mock GD-1 stream and simulate the impacts due to dark matter subhalos,we make use of the simple model of stream evolution and subhalo impacts in the space oforbital frequency Ω , and orbital angle θ , that was developed in ref. [56] and is includedin the galpy code [57]. For a detailed step-by-step explanation of this entire method seerefs. [36, 56]. We briefly summarize the method here. Given a model host potential, currentprogenitor phase-space information, velocity-dispersion parameter σ v of the progenitor, and adisruption time t d at which disruption started, a leading or trailing tail model of the streamis generated in ( Ω , θ ) space. In this work we use the well-tested three component MilkyWay potential MWPotential2014 from ref. [57] as the host potential. The GD-1 progenitor’sphase space coordinate was taken from ref. [54] : ( φ , φ , D, µ φ , µ φ , V los ) = (0 ◦ , − ◦ . ± ◦ . , . ± . , − . ± . − , − . ± .
10 mas yr − , − ± − ), where φ and φ are custom sky coordinates as used in ref. [53], D is the distance to the progenitor, µ φ and µ φ are the proper motion along φ and φ , and V los is the line of sight velocity.It should be noted that the progenitor’s location φ is set to 0 ◦ , σ v = 0 .
365 km s − , andwe model the whole GD-1 stream as a leading arm. The true extent of the GD1 stream is– 5 –imited by the edge of current surveys as well as by the galactic disk. Hence, constrainingthe disruption time t d is difficult with the present data. We therefore have assumed that thedisruption occurred 9 Gyr ago which made our stream very old and long.From the progenitor orbit, an approximate Gaussian action ( J ) distribution for thetidally stripped stars is constructed [58], which is then transformed into frequency space usingthe Hessian matrix evaluated at the progenitor’s actions ( ∂ Ω /∂ J ) J P . The resulting variancetensor in frequency space has the principal eigenvector along the direction in frequency spacein which the stream spreads, Ω (cid:107) . The eigenvalues of the eigenvectors perpendicular to Ω (cid:107) are less than the largest eigenvalue by a factor of 30 or more [59]. Tidally disrupted starsare generated with a frequency distribution that is modeled as a Gaussian. This Gaussianhas a mean equal to the frequency offset from the progenitor and its variance tensor isobtained by transforming the Gaussian action distribution to frequency space and multiplyingby the magnitude of the parallel frequency, which is done for the purpose of simplifyinganalytic calculations. The dispersion of the stellar debris in frequency space is scaled by thevelocity dispersion, σ v , and the relative eigenvalues. Once stripped, the gravitational effectson the stellar debris from its progenitor is neglected and they are evolved solely in their hostpotential. Their future locations are computed based on their linear evolution in angle spacewith their frequency remaining constant. For a given disruption time t d , stellar debris isgenerated with a distribution of constant stripping time with a maximum time t d . Followingrefs. [54, 56], we can transform the stellar debris from ( Ω , θ ) space to configuration spaceusing linearized transformation near the track of the stream.Based on this approach, we generate a smooth GD-1 stream whose properties, namelypath of the smooth stream in Galactic latitude and longitude, parallel angle variation ∆ θ (cid:107) with respect to the Galactic longitude, and density variation are similar to those shown infigure 1 of ref. [36]. The perigalacticon of the stream is ∼
14 kpc, the apogalacticon is ∼ ∼
23 kpc. As shown in figure 2, in the customsky coordinates ( φ , φ ), the full stream is aligned along φ and stretches over ∼ ◦ , whilealong φ the stream only extends over ∼ ◦ . The gray points are samples drawn from thesmooth stream model after applying the angular cut − ◦ < φ < − ◦ to match the observeddata from ref. [53] which are shown as red points. The data agree well with the mock stream’sangular position in the sky. The minor offset between the data and the simulated track of thestream is because the phase space coordinate of the progenitor from ref. [54] was obtainedfrom a fit in which the Milky Way potential was left to vary freely, whereas we have evolvedthe stream in a fixed MWPotential2014 . This offset however does not affect how well we candistinguish between WDM and CDM.
A close encounter of a dark matter subhalo with a stellar stream imparts perturbations to theorbits of the stars in the stream which can be computed by the impulse approximation [30, 32,33, 35]. In this approximation, the subhalo-stream encounter is modeled as an instantenousvelocity kick imparted to the stars in the stream at the point of closest approach. In ref. [35]it was shown that subhalo-stream interactions can be efficiently modeled in frequency-anglespace by transforming the velocity kicks, δ v g , in frequency-angle space using the Jacobians ∂ Ω /∂ v and ∂ θ /∂ v . The equations of motion for the stars in frequency-angle space before thesubhalo impact are Ω = Ω = constant and θ = Ω t + θ , where ( Ω , θ ) is the frequency-– 6 – − − − −
20 0 φ (degree) − − − − − − φ ( d e g r ee ) Figure 2 . GD-1 stream model generated using the framework developed in ref. [56] and using galpy ’s MWPotential2014 for a stream age of 9 Gyr, velocity-dispersion parameter σ v = 0 .
365 km s − andusing the phase-space coordinates from ref. [54]. The blue line is the mean stream track of the fullstream in angular coordinates. The gray points show a sampling of mock stream data from the modelafter applying the angular cut − ◦ < φ < − ◦ in order to match the observed stream. The redpoints are the stream data positions from ref. [53]. angle coordinate when the star was stripped from its progenitor. If the subhalo impactsat t g , then the equations of motion of the star becomes Ω = Ω + δ Ω g = constant and θ = Ω t + δ Ω g ( t − t g ) + δ θ g + θ , where δ Ω g and δ θ g are the frequency and angle kickscorresponding to δ v g . Following refs. [35, 36], instead of computing δ v g over the full 6-dimensional phase space volume, we only compute along the 1-dimensional mean track ofthe stream at the time of impact. Moreover, the angle kicks, δ θ g were shown to be smallcompared to the frequency kicks, δ Ω g after one orbital period [35], therefore in the context ofthe GD-1 stream, which is much older than its orbital period, we can neglect the angle kicks.Extensive tests of these approximations were performed in ref. [54] and they concluded thatthe approximations work well at least in the subhalo mass range of interest here.In order to simulate the effects of multiple subhalo impacts of different masses, atdifferent times in the orbit and locations along the stream, we need a prescription for samplingmultiple subhalo impacts. For this we follow the sampling procedure described in detail insection 2.3 of ref. [36]. The only difference in the WDM case is the Poisson sampling of thenumber of impacts of different masses.We consider the subhalos to follow a Hernquist profile, and use the relation for the scaleradius, r s , from ref. [47], r s = 1 .
05 kpc (cid:18) M sub M (cid:12) (cid:19) . , (3.1)which was obtained for CDM subhalos with Hernquist profile by fitting the circular velocity -mass relation from the publicly available Via Lactea II catalogs [5]. We elaborate in appendixA on why we are justified in using the same r s fitting formula for WDM subhalos. Lessmassive and smaller dark matter halos need to pass closer to a stellar stream compared tomore massive and larger subhalos to result in an observable effect on the stream. To capturethis effect in our simulations, we set the maximum impact parameter equal to five times the– 7 –
10 20 30 40 50 no of impacts . . . . . . P D F Figure 3 . PDF of the number of impacts that a GD-1 like stream had over 9 Gyr in different casesof dark matter particle mass. Each PDF was constructed out of 2100 simulations. scale radius of the dark matter subhalo following ref. [36]. The maximum impact parameteris therefore smaller for lower mass subhalos and larger for more massive subhalos. To samplethe velocity distribution of the DM subhalos, we use a Gaussian with a velocity dispersion of120 km s − which is the measured radial velocity dispersion within a galactocentric distanceof 30 kpc for a collection of halo objects as found in ref. [60].For a particular mass of thermal WDM, one can find the number of subhalos in a specificmass range within a spherical radius by integrating eq. (2.7) over the subhalo mass rangeand the chosen radial range. For the GD-1 stream generated in this work, the mean sphericalradius of the stream is ∼
23 kpc. If dark matter constitutes of a thermal WDM of particlemass 3.3 keV (or a 7 keV sterile neutrino with lepton asymmetry parameter equal to 8.66),we find 0.47 subhalos in the mass range [10 − ] M (cid:12) within 23 kpc from the galactic centerif subhalo disruption due to baryonic effects are not taken into account. By taking the 47%reduction of number of subhalos due to baryonic effects, this number is reduced to ∼ . ∼ .
78 subhalos when no baryonic effect is included, and ∼ . N over the entire mass range of subhalos in the WDM scenario by sampling the number of– 8 –mpacts from a Poisson distribution for the expected number of impacts. We then sample N masses from the impact-mass distribution. Note that this distribution differs from eq. (2.7),because it needs to account for the shrinking maximum impact parameter as the subhalomass decreases and vice versa. All other parameters are then sampled exactly as in theCDM case considered in ref. [36]. The total rate of impact is 0.72 for a 3.3 keV thermalWDM, 1.95 for a 5 keV thermal WDM, and 24.8 for a 1 GeV CDM particle. These ratesare calculated by assuming that the rate of impact for each mass is that corresponding toits mass decade and is computed at the center of the logarithmic mass bin. In the subhalomass bins [10 − ] M (cid:12) , [10 − ] M (cid:12) , and [10 − ] M (cid:12) , the impact rates for the 3.3keV WDM scenario are ∼ . ∼ .
17 and ∼ .
49, for the 5 keV WDM are ∼ . ∼ . ∼ .
18 and for the 1 GeV CDM scenario they are ∼ . ∼ .
34, and ∼ .
53, respectively.In order to efficiently calculate the stream-subhalo impacts, we use the line-of-parallel-angle approach as explained in ref. [36]. In this approach, multiple impacts occurring at thesame time do not increase the computational cost. Therefore, the impacts are allowed tohappen at a set of equally spaced discrete times along the past orbit of the stream. In orderfor the structure of the stream not to be affected by the discrete time sampling, the samplingneeds to be high enough. Furthermore, sampling impacts more frequently than the radialperiod of the stream allows us to explore stream-subhalo encounters at different epochs ofthe streams orbit. In Appendix C of ref. [36] it was shown that the statistical properties ofa perturbed stream converge for greater than 16 different impact times. We considered 64different impact times for the 9 Gyr stream. This would amount to a time interval of ∼ ∼
400 Myr for the GD-1 stream.
Using the above mentioned machinery for generating perturbed stellar streams, we carry out2100 simulations each for the 3.3 keV and 5 keV thermal WDM and 1 GeV CDM scenariosfor a GD-1 like stream, considering subhalo impacts in the mass range of [10 − ] M (cid:12) .For each simulation, we transform the density of the stream from frequency-angle space toconfiguration space ( φ , φ ) and apply angular cuts to match the observed extent of the GD-1stream. As evident from figure 2, the angular extent of the stream along φ is very small,therefore for the rest of this work, we will only analyze the stream as a function of φ .In figure 4 we show the density contrast (density of the perturbed stream divided bythat of the unperturbed stream) of four different cases from the 2100 simulations for the5 keV and 1 GeV scenarios. Black curves denote randomly chosen density contrasts, whilegreen curves denote density contrasts for cases whose power is close to the median power ofthe 2100 simulations as shown in figure 5. We treat the cases shown in green as fiducial caseswhich are used to constrain the mass of dark matter in section 4.2. The red curve in the 5keV WDM scenario in the left panel of figure 4 corresponds to a case whose power is close tothe upper 2 σ bound of the power spectrum dispersion, while the blue curves in both panelsof the same figure correspond to cases whose power is close to the lower 2 σ bound as shownin figure 5. For the 5 keV WDM, the case close to the lower 2 σ bound occurs for a streamthat had no subhalo impacts, and as a result the density contrast is equal to one. We do notconsider a case in the 1 GeV CDM scenario whose power is close to the upper 2 σ bound of– 9 – . . . fiducial upper σ . . . random − − − − − φ (degree) . . . lower σ p e rt u r b e d / s m oo t hd e n s i t y fiducial random random − − − − − φ (degree) . . . lower σ p e rt u r b e d / s m oo t hd e n s i t y Figure 4 . Density contrast of four different realizations of the perturbed GD-1 stream. The leftpanel shows the case in which dark matter is composed of thermal WDM with particle mass 5 keVand the right panel shows the case for a 1 GeV CDM candidate. The black curves show randomlychosen density contrasts. The green curves denote density contrast for the fiducial cases which wetreat as mock observed data and are analyzed for inferring the mass of dark matter in section 4.2. Thered curve indicates a realization whose power is close to the upper 2 σ bound of the power spectrumdispersion, while the blue curves indicate the realizations that are close to the lower 2 σ bound. Inthe 5 keV case, this happens when the stream had no subhalo impact. See section 5 for discussion onthese extreme cases. the power spectrum dispersion because as we will discuss in section 4.2, our method can notdistinguish between dark matter models if the mass of dark matter is greater than ∼
15 keV.In section 5, we investigate these cases of extreme power and how they can bias our analysis.In our stream simulation, the progenitor emits new stars constantly. In scenarios wherethe stream had many impacts, the bulk of the stream gets disrupted and stars are pushedtowards the extremities. In such cases, the stars which are pushed towards the progenitoradd up to the new stars which have recently been emitted from the progenitor, causing alarge overdensity near the location of the progenitor ( φ = 0 ◦ ). Such large overdensities aresomewhat unphysical since in a real stream some stars pass the progenitor, moving from onearm of the stream to the other, and our simulations ignore this effect. In order to remove thispotentially artificial peak, we cut the stream at φ > − ◦ . This choice of the 5 ◦ cut removesthe large peaks close to the progenitor, while leaving the smaller physical peaks which areexpected due to the disruption of the subhalo.For each case we binned the density in 0 . ◦ φ bins. If we assume that LSST can gobelow the main-sequence turn-off point and accurately resolve almost all the member starsof the GD-1 stream which has ≈ ,
000 stars deg − [53], then this would correspond to a shotnoise of 10% for a bin width equal to 0 . ◦ . Therefore, we have considered a constant noiseof 10% of the density contrast in each φ bin.We compute the 1-dimensional power spectrum of the density contrast of the stream P δδ ( k φ ) = (cid:104) δ ( k φ ) (cid:105) using the csd routine in scipy [61] and do not divide by the samplingfrequency. In the power spectrum plots, we have plotted the square root of the power along y -axis and inverse of the frequency (1/ k φ ) along x -axis. In this form, the power spectrum plotsrepresent the power at a particular angular separation which is indicative of the correlation– 10 –
10 100 /k φ (deg) . . . (cid:112) P δδ ( k φ )
10 % upper σ /k φ (deg)
10 % upper σ fiducial /k φ (deg)
10 % lower σ fiducial Figure 5 . Power spectrum of the density contrast of the mock GD-1 stream in the 3.3 keV WDM(left panel), 5 keV WDM (middle panel) and 1 GeV CDM (right panel) scenarios. The black solidline is the median power of the 2100 simulations. For the 3.3 keV case, the median power is close to 0and hence not shown. The error bars represent the 2 σ (95%) spread of the power spectra of the 2100stream density simulations. The green solid curves (wherever shown) represent the power spectrumof the fiducial cases whose density contrasts are shown in figure 4. The red and blue curves (wherevershown) represent the power for cases close to the upper and lower 2 σ bound of the dispersion inpower spectrum, respectively. The dashed curves indicate the median power corresponding to 10%noise in the density and their color represents the case they correspond to. The colored dots specifythe angular scales at which the power is above the noise floor. of the density contrast at that angular scale. In figure 5, we show the power spectrumof the density contrast of the GD-1 stream for the 3.3 keV and 5 keV WDM, and the 1GeV CDM scenarios. The black solid curves show the median power spectrum of the 2100simulations. For the 3.3 keV case, owing to very few impacts in majority of the simulationsthe median power is close to 0. The error bars show their 2 σ dispersion around the median.The green solid curves show the power spectrum of our fiducial cases whose density contrastsare shown in figure 4. The red and blue solid curves show a case close to the upper and lower2 σ bound of the power spectrum dispersion, respectively. In both the 3.3 keV and 5 keVWDM scenarios the lower bounds are close to zero. This is again due to either no subhaloimpacts or impacts that did not result in significant density fluctuations. The dashed linesshow the 10% noise power and their color represents the case they correspond to. The noisepower spectrum is the median of 10,000 Gaussian noise realizations of the stream density andtherefore independent of the density bin width. For the fiducial 1 GeV case, the noise poweris much lower than the other cases since the bulk of the stream is pushed away due to subhaloencounters leaving the density along most of the stream close to 0 (top right panel figure 4).The density noise being 10% of the density has therefore very low power. The colored dotsrepresent the angular scales at which the power is above the corresponding noise floor andhence are used in the ABC analysis to infer the dark matter particle mass (see section 4.2).As evident from figure 1, there are many more subhalos in the mass range [10 − ] M (cid:12) in the 1 GeV case compared to the 5 keV and 3.3 keV cases, which leads to a higher chance of– 11 –tream-subhalos encounters as suggested in figure 3, resulting in higher density fluctuationsalong the stream in the 1 GeV case compared to the 5 keV and 3.3 keV cases. This is easilyseen from the density contrasts plotted in figure 4.The difference in the density fluctuations translates to the power spectrum and henceexplains the more power at the largest scales in the 1 GeV case compared to the 5 keVcase. The wide dispersion in the power spectra in either case is due to the range of possibleways the stream-subhalo impacts can occur resulting in different levels of density fluctuationsalong the stream. Comparing the 5 keV and 1 GeV cases in figure 5, it is clear that thereis more power at smaller scales in the 1 GeV case. This is because if dark matter is coldthen there are more low mass subhalos whose impacts with the stream give rise to densityfluctuations at smaller scales. On the contrary, if dark matter is warm then as discussedin section 2, substructures less massive than the half-mode mass are strongly suppressed.As a result, majority of the impacts are by the more massive subhalos, giving rise to largescale density fluctuations. The noise power appears flat since we considered a constant noiselevel throughout the stream. Power at scales below the noise floor do not convey any usefulinformation. In this section we use the statistical properties of gaps in a stellar stream to distinguishbetween CDM and WDM. To achieve this, we use the power spectrum of the density contrastof the perturbed stream to constrain the mass of the dark matter particle. We employ theABC method to construct an approximate posterior probability distribution function (PDF)of the mass of the dark matter particle. The ABC method is a likelihood-free approach ofBayesian parameter inference in which an approximate posterior PDF of the parameters in theproblem is constructed using simulator outputs and by comparing the outcome with observeddata. We run the simulations by randomly drawing the mass of the dark matter particle froma prior distribution of the mass, which we have assumed to be a uniform distribution between[0.1,16] keV. The upper limit of the prior was picked based on the result presented in figure6, which shows that the median power spectrum for cases with mass of WDM greater than15 keV converge and are indistinguishable from each other. This is consistent with figure 1,which shows that increasing the mass of the WDM particle shifts the subhalo mass functionclose to the CDM case in the subhalo mass range [10 − ] M (cid:12) .The ABC works by accepting those simulations which are within some pre-defined tol-erance of the data summaries. As our data summaries we have taken the power at the threelargest observed scales in the 5 keV case and the five largest observed scales in the 1 GeVcase, since those scales are above the noise floor of the fiducial cases (see figure 5). We donot consider the 3.3 keV case here, since as seen in figure 5 the median power is close to zerofor this case. Additionally, we do not consider power at scales below the noise floor becausethey are very noisy and thus are incapable of discriminating between different WDM models.The level of noise in the data is therefore crucial for our method. The tolerances are made assmall as it is allowed by the noise in the data. In order to sample the entire range of the priordistribution properly, we ran ∼ ,
000 simulations. Since the most time consuming part ofthe ABC approach is running these simulations, we follow the same strategy as adopted inref. [36], i.e. for each WDM mass we produce 100 simulations by adding 100 realizations ofthe noise. Therefore, effectively, we ran ∼ , ,
000 simulations.– 12 –
10 100 /k φ (deg) . . . p P δδ ( k φ ) Figure 6 . Power spectrum of the median of 2100 simulations of the GD-1 stream in different WDMparticle mass scenarios. The median power spectra for WDM particle mass larger than ∼
10 keV tendto overlap, and hence it is no longer possible to distinguish at high significance between WDM modelsthat have particle mass greater than 10 keV. The power spectrum of cases with WDM particle massgreater than 15 keV is very close to the 1 GeV CDM case and therefore WDM models with particlemass greater than ∼
15 keV can not be distinguished from the CDM model.
Figure 7 shows the posterior PDF of the dark matter particle mass obtained by runningsimulations of the GD-1 stream and using the ABC method as explained above in the 5 keVWDM and the 1 GeV CDM scenarios. For the 5 keV case, the PDF peaks at 5 . +5 . − . keV(68% confidence) and the 95% upper limit is 15.0 keV. In the 1 GeV CDM case, the PDFplateaus indicating the true dark matter mass is beyond the upper limit of the prior. The95% lower limit is 4.3 keV.Next, we investigate the case in which the intrinsic power of the stream is below thenoise floor. As shown in the power spectrum dispersion plots in figure 5, such cases mayarise when dark matter is warm and the stream suffers few or no subhalo impacts (e.g., thedensity contrast for the 5 keV case corresponding to the lower 2 σ bound). In such cases themeasured power is dominated by noise. The left panel of figure 8 shows one randomly pickedrealization of the noise in the 5 keV case which we use as the measured power. We considerthe power at the three largest scales to be data summaries. Since the power at all these threescales are below the median noise floor, the ABC accepts any simulation whose power atthose scales are below the noise floor. The resulting posterior PDF of the dark matter massis shown in the right panel of figure 8, which demonstrates that the 95% upper limit on themass of WDM is 5.3 keV. This is consistent with the fact that for a lower mass WDM, there– 13 – m WDM (keV) . . . . . . . . P D F m WDM = 5 . +5 . − . keV (68 %)m WDM < . m WDM (keV) . . . . . . . . P D F m WDM > . Figure 7 . Posterior PDFs of the mass of dark matter obtained by running simulations of the GD-1stream and using the ABC method to match the power at the largest scales above the noise floor(shown by red dots in figure 5) to the fiducial cases. The left panel shows the posterior PDF in the5 keV WDM scenario and the right panel shows the posterior PDF in the 1 GeV CDM scenario. Forthese typical cases the ABC is able to distinguish between WDM and CDM and also constrain themass of WDM. are fewer subhalos and hence lower chances of stream-subhalo encounters, compared to thecase of a higher mass WDM.
In this section we discuss the perils of using only one stream for constraining the mass ofWDM using the stream density power spectrum method presented above. As evident fromfigure 5, a stream-subhalo interaction can take place in a wide range of possible ways givingrise to the dispersion in the power spectrum. Therefore, it is not unexpected that we couldend up with a stream that had either too few or too many subhalo hits resulting in itspower spectrum to be > σ away from the median. Such cases would result in a posteriorthat would rule out the true dark matter particle mass by > σ . We demonstrate this withthree examples here. In the 1 GeV CDM scenario we pick a density simulation whose powerspectrum is close to the lower bound of the 2 σ dispersion, and for each 3.3 keV and 5 keVWDM scenarios we pick one realization with a power close to the upper 2 σ bound. Thesecases are shown by the blue and red curves in figure 5. Treating them as mock data andconsidering a 10% density noise, we carry out the ABC steps with power at the two largestscales for the 1 GeV case, five largest scales in the 3.3 keV case and four largest scales in the 5keV case as data summaries to construct the posterior PDF for the mass of dark matter. Theresults are shown in figure 9. The posterior PDF predicts, in the 3.3 keV case (left panel), a– 14 –
10 100 /k φ (deg) . . . p P δδ ( k φ )
10 % m WDM (keV) . . . . . . . . P D F m WDM < . Figure 8 . Figure demonstrating the case in which the intrinsic power of the stream density is belowthe noise floor, as a result of either no subhalo impact or impacts that did not contribute to anysignificant density fluctuations. Left panel: One realization of the noise power (solid black line) whichwe treat as our mock measurement of the stream density power spectrum, and the 10% noise floor(dashed black line). The black dots represent the data summaries which are the power at the threelargest scales. Right panel: Posterior PDF of the dark matter mass obtained using the ABC method. m WDM (keV) . . . . . . . . P D F m WDM > . m WDM (keV) . . . . . . . . P D F m WDM > . m WDM (keV) . . . . . . . . P D F m WDM = 5 . +5 . − . keV (68%)m WDM < . Figure 9 . Posterior PDFs for the extreme cases. Left panel: upper 2 σ case in the 3.3 keV scenario;middle panel: upper 2 σ case in the 5 keV scenario; right panel: lower 2 σ case in the 1 GeV scenario. lower 95% limit on the mass of dark matter to be 4 . . +5 . − . keV at 68% and sets a95% upper limit at 14 . σ results of one stream. This shortcoming can be improved by using multiple stellar streamsto constrain the mass of dark matter. This will not only remove the effects of the outliercases but will also make the constraints more robust. We leave this analysis for a futurepublication. In this paper we presented a methodology of investigating the particle nature of dark matterby analyzing the statistical properties of density fluctuations, in the form of power spectrum,induced in a stellar stream as a result of its gravitational encounter with dark matter sub-halos. We accomplished this by exploiting the fact that if dark matter is warm then therewill be fewer subhalos in the Milky Way host halo compared to the CDM case. This differ-ence will result in a different rate of stream-subhalo encounters and will be apparent in theintrinsic power spectrum of the stream density. By using fast simulations of stream densityin frequency-angle space we generated mock GD-1 stream data and exposed it to subhaloencounters in a 3.3 keV WDM, 5 keV WDM and 1 GeV CDM scenario. Due to the higherrate of impacts in the CDM scenario, we found that the stream density power spectrum hasmore intrinsic power in the CDM scenario compared to the WDM case.Assuming that LSST will be able to resolve most of the member stars of the GD-1stream without contamination, we applied LSST-like noise to our mock data. With
Gaia ’smagnitude threshold of 20, it will not provide much direct data on GD-1 for the purpose ofour study. Nevertheless, Gaia will improve our knowledge of the Milky Way’s gravitationalpotential in general and of the orbits of streams, which will be pivotal for our method.We used an ABC technique to perform rigorous inference on the dark matter massusing mock streams for the 5 keV WDM and the 1 GeV CDM fiducial cases. In the WDMcase when the intrinsic power of the stream density is greater than the noise, we constrainedthe dark matter mass to 5 . +5 . − . keV (68% confidence) and <
15 keV at 95%. In the CDMscenario we found the lower limit on the mass of dark matter to be 4 . Acknowledgments
We thank Mark Lovell for many useful discussions and for providing us with the data fromAquarius warm dark matter simulations. We also thank Tameem Adel for discussions at anearly stage of this project. We acknowledge the support of the D-ITP consortium, a pro-gramme of the Netherlands Organization for Scientific Research (NWO) that is funded bythe Dutch Ministry of Education, Culture and Science (OCW). N. Bozorgnia is grateful to– 16 –he Institute for Research in Fundamental Sciences in Tehran for their hospitality during hervisit. G.B. (PI) and N. Bozorgnia acknowledge support from the European Research Councilthrough the ERC starting grant WIMPs Kairos. J. Bovy acknowledges the support of theNatural Sciences and Engineering Research Council of Canada (NSERC), funding referencenumber RGPIN-2015-05235, and from an Alfred P. Sloan Fellowship. N. Bozorgnia has re-ceived support from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sklodowska-Curie grant agreement No 690575.
Appendix A Scale radius of warm dark matter subhalos
In this appendix we discuss the scale radius of WDM subhalos and how it compares to thatof CDM subhalos. As it was explained in section 3.2, we use eq. (3.1) for the scale radiusof CDM subhalos, which is obtained assuming Hernquist density profiles for the subhalos inthe Via Lactea II catalog.In order to compare the properties of WDM subhalos to their CDM counterparts, weuse the density profiles of subhalos available from the Virgo consortium Aquarius project [41]where in addition to a CDM simulation, four haloes with WDM particle mass of 1.5, 1.6, 2.0,and 2.3 keV are simulated. These include subhalos which are not spurious (see ref. [41] for adetailed discussion of removing spurious subhalos from their halo catalogs) and are within 2Mpc of the host halo center. We fit a Hernquist profile to the density profiles of CDM andWDM subhalos and check how the best fit scale radii change as a function of subhalo mass.To find the best fit Hernquist scale radius, for each individual simulated subhalo, weminimize the following χ function: χ ( r s ) = N (cid:88) i ( ρ i − ρ Hern ( r i , r s )) σ i , (A.1)where ρ i is the value of the DM density of the simulated subhalo at the radial bin i with r i denoting the bin center, σ i is the corresponding 1 σ Poisson error, N is the number of radialbins we consider to perform the fit, and ρ Hern ( r i , r s ) is the Hernquist density profile evaluatedat radius r i as a function of the scale radius r s .The density profiles are computed in spherical shells spaced equally in log( r ). We findthe inner and outer radii for performing the fit by using the criteria discussed in Springel etal. [6]. Namely, the inner radius for the fit, r min , is set to the radius in which convergenceis achieved according to the Power et al. criterion [62]. The outer radius, r max , is set to thelargest radius where the density of bound mass is more than 80% of the total mass density.After specifying the radial range for the fit, we introduce the following two criteria tofind good subhalos for performing the fit: (i) r max − r min > (ii) the number of binsis greater or equal to 5. These additional criteria ensure that we are not fitting over a smallradial range and we have enough bins for performing the fit. Finally, we need to consider theeffect of tidal stripping on the goodness of fit. With the criteria mentioned, we don’t obtain agood fit for some subhalos since we are probing a radial range where tidal stripping becomesimportant and there is a sharp decrease in the DM density from one bin to the next. Toavoid this, we set the last bin we consider, i f , as the bin where the DM density in the nextbin decreases by more than an order of magnitude, i.e. where ρ i f +1 < . ρ i f . With these– 17 – ■■ ■■■■■ ■ ■■■■■■■■ ■▲ ▲▲▲▲▲ ▲▲▲▲ ▲ ▲ ▲ ▲▲▲▲▲▲ ▲◆◆◆◆ ◆◆◆◆◆ ◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆ ◆▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ▼ ▼ CDM2.3 keV2.0 keV1.6 keV1.5 keV �� � �� � �� � �� � �� �� �� �� ���������������� � ��� ( ���� ) � � ( �� � ) Figure 10 . The mean values of the best fit Hernquist scale radii of subhalos in the CDM and WDMAquarius simulations as a function of the subhalo mass. The halos in the WDM simulations haveWDM particle masses in the range 1.5 – 2.3 keV. criteria we retain 4482 CDM subhalos over the mass range of [1 . × , ] M (cid:12) . Noticethat the lower boundary of the subhalo mass range is set by the requirement that the subhalohas at least 100 particles.The result is shown in figure 10 where we have plotted the mean values of the best fitHernquist scale radius at different masses, considering equal size bins in log ( M sub ). As it canbe seen in the figure, the overall deviation between the best fit scale radii of CDM halos andWDM halos with different WDM particle mass is small. Although for M sub (cid:38) M (cid:12) , thescale radii for WDM halos are in general larger than those of CDM halos, for M sub ≤ M (cid:12) which is the subhalo mass range probed in our study there is no clear trend. Moreover,because the impact parameters considered in this work are much larger than the scale radiusof the interacting subhalos with the streams, we use the same scale radius relation to describeboth WDM and CDM subhalos.To capture the overall variation of the scale radius with the subhalo mass, we findthe best fit r s ( M sub ) relation from the mean scale radii of CDM subhalos in the Aquariussimulations shown in figure 10, r s | fit = 1 .
24 kpc (cid:18) M sub M (cid:12) (cid:19) . . (A.2)This equation is slightly different from eq. (3.1) which is our fiducial scale radius relation.In figure 11 we show how this difference in the scale radius relation will affect our analysis.The black solid line is the median power of 2500 simulations with the fiducial scale radiusrelation given by eq. (3.1). The error bars represent the 1 σ dispersion of these simulations.The red curve is the median power of the simulations with the new scale radius fit given byeq. (A.2). The green and blue curves are the median power spectrum with 0.4 and 2.5 timesscale radius but keeping the maximum impact parameter b max fixed for a particular masssubhalo. All the power spectra are within 1 σ dispersion of the fiducial case, suggesting thatvarying the scale radius within what we discussed will not affect our analysis.– 18 –
10 100 /k φ (deg) . . . p P δδ ( k φ ) r s ( M ) | fid r s ( M ) | fit r s ( M ) | fid × . r s ( M ) | fid × . Figure 11 . Comparison of the median density power spectrum in the 1 GeV CDM scenario withdifferent choices for the scale radius. The black curve is for the fiducial scale radius relation given byeq. (3.1), the red curve is with the scale radius relation found in eq. (A.2) from a fit to the scale radiiof subalos in the Aquarius simulations. In both of these cases the maximum impact parameter b max is fixed to 5 times the scale radius for a particular mass subhalo. The error bars represent the 1 σ dispersion of the power spectrum around the median for the fiducial case. The blue and green curveshows the cases when the fiducial scale radii of the subhalos are increased and decreased 2.5 timesrespectively while keeping the maximum impact parameter the same as the fiducial case. References [1] G. Bertone (ed.),
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