DWARF GALAXY ANNIHILATION AND DECAY EMISSION PROFILES FOR DARK MATTER EXPERIMENTS

In order to accurately quantify uncertainties in the spatial distribution of dark matter it is necessary to use a suitably flexible functional form for the density profile (Bonnivard et al. 2015). Following Charbonnier et al. (2011), we adopt the functional form introduced by Zhao (1996) to generalize the Hernquist (1990) profile. In this spherically symmetric model, the density of dark matter at halo-centric radius r is

$$\begin{equation} \rho (r) = \frac{\rho _s}{\left(r / r_s \right)^\gamma \left[1 + \left(r / r_s \right)^\alpha \right] ^{(\beta -\gamma) / \alpha }}. \end{equation} \tag{ 7 }$$

This five-parameter profile, normalized by the scale density ρ_s, describes a split power law with inner logarithmic slope $d\log \rho /d\log r|_{r\ll r_s}=-\gamma$ and outer logarithmic slope $d\log \rho /d\log r|_{r\gg r_s}=-\beta$. The transition happens near the scale radius r_s, with α specifying its sharpness. For (α, β, γ) = (1, 3, 1) one recovers the two-parameter Navarro–Frenk–White (NFW) profile that characterizes CDM halos formed in dissipationless numerical simulations (Navarro et al. 1997). However, the profile can also describe halos with even steeper central "cusps" (γ > 1), or halos with "cores" of uniform central density (γ ∼ 0), as are usually inferred from observations of real galaxies (de Blok 2010, and references therein; Walker et al. 2011; Donato et al. 2009). This flexibility lets us explore a wide range of physically plausible dark matter profiles.

From Equation (3), the flux of annihilation by-products depends on the density of dark matter particles within the source, and thus on the source's gravitational potential. For collisionless stellar systems like dwarf galaxies, the gravitational potential is related fundamentally to the phase-space density of stars f(r, u), defined such that f(r, u) d³r d³u gives the expected number of stars lying within the phase-space volume d³r d³u centered on (r, u). However, dwarf galaxies are sufficiently far away that current instrumentation resolves only the projection of their internal phase-space distributions, effectively providing information in just three dimensions: position as projected onto the plane perpendicular to the line of sight, and velocity along the line of sight (from Doppler redshift). Given these limitations, it is common to infer the gravitational potential Φ by considering its relation to moments of the phase-space distribution: the stellar density profile,

$$\begin{equation} \nu (r)\equiv \int f({{\bf r}},{{\bf u}}) \,\, d^3{{\bf u}}, \end{equation} \tag{ 8 }$$

and the stellar velocity dispersion profile,

$$\begin{eqnarray} \overline{u^2}(r) &=& \overline{u_r^2}(r)+\overline{u_{\theta }^2}(r) + \overline{u_{\phi }^2}(r) \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} &=& \frac{1}{\nu (r)}\int u^2 f({{\bf r}},{{\bf u}}) \,\, d^3{{\bf u}}. \end{eqnarray} \tag{ 10 }$$

Assuming dynamic equilibrium and spherical symmetry, these quantities are related according to the spherical Jeans equation (Binney & Tremaine 2008),

$$\begin{eqnarray} &&\frac{1}{\nu (r)}\frac{d}{dr}\left[\nu (r) \overline{u_r^2}(r)\right]+2\frac{\beta _a(r)\overline{u_r^2}(r)}{r}=-\frac{d\Phi }{dr}=-\frac{GM(r)}{r^2},\nonumber\\ \end{eqnarray} \tag{ 11 }$$

where

$$\begin{equation} \beta _a(r)\equiv 1-\frac{\overline{2u_{\theta }^2}(r)}{\overline{u_r^2}(r)} \end{equation} \tag{ 12 }$$

characterizes the orbital anisotropy and the enclosed mass profile

$$\begin{equation} M(r)=4\pi \int _{0}^{r} s^2\rho (s)ds \end{equation} \tag{ 13 }$$

includes contributions from the dark matter halo.

Equation (11) has the general solution (van der Marel 1994; Mamon & Łokas 2005)

$$\begin{equation} \nu (r)\overline{u^2_r}(r)=\frac{1}{f(r)}\displaystyle \int _{r}^{\infty }f(s) \, \nu (s) \,\frac{GM(s)}{s^2} \,ds, \end{equation} \tag{ 14 }$$

where

$$\begin{equation} f(r)=2 \, f(r_1) \, \exp \left[ \int _{r_1}^{r}\beta _a(s)s^{-1} \, \, ds \right]. \end{equation} \tag{ 15 }$$

Projecting along the line of sight, the mass profile relates to observable profiles, the projected stellar density Σ(R), and line-of-sight velocity dispersion σ(R), according to (Binney & Tremaine 2008)

$$\begin{equation} \sigma ^2(R) \, \Sigma (R)=2 \, \displaystyle \int _{R}^{\infty }\left (1-\beta _a(r)\frac{R^2}{r^2}\right) \frac{\nu (r) \, \, \overline{u_r^2}(r) \, \, r}{\sqrt{r^2-R^2}} \, \, dr. \end{equation} \tag{ 16 }$$

We use Equation (16) to fit models for ρ(r) and β_a(r) to observed velocity dispersion and surface brightness profiles under the following assumptions:

• 1.  
  Dynamic equilibrium and spherical symmetry, both implicit in the use of Equation (11);
• 2.  
  The stars are distributed according to a Plummer (1911) profile,

  $$\begin{equation} \nu (r)=\frac{3L}{4\pi R_e^3}\frac{1}{\left(1+R^2/R_e^2\right)^{5/2}}, \end{equation} \tag{ 17 }$$

  implying surface brightness profiles of the form

  $$\begin{equation} \Sigma (R)=\frac{L}{\pi R_e^2}\frac{1}{\left(1+R^2/R_e^2\right)^2}, \end{equation} \tag{ 18 }$$

  where L is the total luminosity and R_e is the projected halflight radius;
• 3.  
  The stars contribute negligibly to the gravitational potential, such that R_e is the only meaningful parameter in ν(r) and Σ(R);
• 4.  
  β_a = constant;
• 5.  
  The distribution of stellar velocities is not significantly influenced by the presence of binary stars.

Real galaxies violate all of these assumptions at some level and it is important to consider that the error distributions that we derive for J values will not include the resulting systematic errors. For the present work, we are concerned primarily with quantifying statistical uncertainties that arise from finite sizes of stellar-kinematic samples. For a thorough study of systematic errors that can arise due to different stellar density profiles, non-spherical symmetry, and more complicated behaviors of the velocity anisotropy, we refer the reader to the recent study by Bonnivard et al. (2015).

For the Milky Way's eight most luminous "classical" dwarf galaxies, we adopt projected halflight radii listed in Table 1 of Walker et al. (2009b; original source is Irwin & Hatzidimitriou 1995). We use the stellar-kinematic data published by Mateo et al. (2008) for Leo I, and by Walker et al. (2009b) for Carina, Fornax, Sculptor, and Sextans.

For Draco, Leo II, and Ursa Minor we use stellar-kinematic data acquired with the Hectochelle spectrograph at the 6.5 m MMT. These data have previously been analyzed by Walker et al. (2009c) and Charbonnier et al. (2011), and will soon be made public (M. Walker et al., in preparation).

For the Milky Way's less luminous "ultra-faint" satellites discovered over the past seven years (e.g., Willman et al. 2005; Zucker et al. 2006b; Belokurov et al. 2007, 2008; Belokurov et al. 2009), we make use of published data from a variety of sources. We adopt projected halflight radii from the review of McConnachie (2012). Original sources are Martin et al. (2008) and/or discovery papers for satellites discovered after that work (Belokurov et al. 2008; Belokurov et al. 2009).

For Coma Berenices, Canes Venatici I, Canes Venatici II, Leo IV, Leo T, Ursa Major I, and Ursa Major II, we utilize the Keck/Deimos velocity data of Simon & Geha (2007), generously provided by Marla Geha (2011, private communication).

For Hercules we adopt the stellar-kinematic data published by Adén et al. (2009a). After using Stromgren photometry to identify and remove foreground contamination independently of velocity, Adén et al. (2009b) use these data to measure a global velocity dispersion of 3.7 ± 0.9 km s⁻¹, slightly smaller than the earlier estimate of 5.1 ± 0.9 km s⁻¹ by Simon & Geha (2007). In the interest of deriving conservative estimates on the dark matter density (and ultimately the exclusion limits for particle models), we adopt the Hercules data of Adén et al. (2009a).

For Boötes I, we adopt the stellar-kinematic data published by Koposov et al. (2011). The observing strategy of Koposov et al. (2011) provided ∼10 independent velocity measurements of each star over the course of one month, thereby enabling a direct examination of intrinsic velocity variability (e.g., due to unresolved binary stars) that can potentially inflate observed velocity dispersions (and hence the inferred dark matter density) above the values attributable to the galaxy's gravitational potential (McConnachie & Côté 2010). Koposov et al. (2011) directly resolve velocity variability—including near-constant accelerations of ∼10 km s⁻¹ month⁻¹—for ∼10% of the stars in their sample. In order to obtain conservative estimates of the dark matter density, we adopt the most restrictive sample of Boötes I members identified by Koposov et al. (2011; marked with a "B" in their Table 1), which includes only the 37 stars that show no evidence for velocity variability and have small (⩽2.5 km s⁻¹) velocity measurement errors.

For Leo V, we adopt the stellar-kinematic data published by Walker et al. (2009a). These include seven stars identified as likely members. However, while five of these stars lie within three times the projected half-light radius of Leo V (r_h ∼ 40 pc; Belokurov et al. 2008), the other two lie ≳ 10r_h ∼ 400 pc away from Leo V's center and in the direction toward Leo IV, which itself lies only ∼20 kpc from Leo V. This configuration is highly improbable for a dynamically relaxed system—even given the small number of stars—fueling speculation that Leo IV and Leo V are interacting gravitationally. In that case, the two outermost stars in the Leo V sample may trace a stellar "bridge" of low surface brightness. However, deep photometric studies have not yielded unambiguous evidence for such a structure (de Jong et al. 2010; Jin et al. 2012; Sand et al. 2010, 2012). In any case, once again in the interest of conservatism, we consider only the five innermost members in the analysis of Leo V.

For Segue 1, we adopt the stellar-kinematic data published by Simon et al. (2011). These data include repeat measurements for tens of stars with velocities originally measured by Geha et al. (2009), enabling an analysis of velocity variability. Martinez et al. (2011) perform a Bayesian analysis and conclude that the presence of unresolved binary stars is likely to have only mild (a ∼10% effect) influence on estimates of Segue 1's intrinsic velocity dispersion. Along with velocity measurements, we adopt the "Bayesian" membership probabilities listed for individual stars in Table 3 of Simon et al. (2011).

For Segue 2, we adopt the stellar-kinematic data published by Kirby et al. (2013) for 25 member stars. These measurements do not resolve Segue 2's internal velocity dispersion, instead placing a 95% upper limit of σ < 2.6 km s⁻¹. A previous study by Belokurov et al. (2009) reported a velocity dispersion of σ ∼ 3.5 km s⁻¹, based on a sample of ∼5 member stars that they cautioned might be contaminated by members of a dynamically hotter stream in the same vicinity. Again in the interest of placing conservative limits on the expected dark matter signal, we adopt the sample of Kirby et al. (2013)—not only because it implies a smaller velocity dispersion, but also because it provides a larger number of member stars. While the small velocity dispersion estimated by Kirby et al. (2013) does not, by itself, require that Segue 2 is embedded within a dark matter halo, Kirby et al. (2013) argue that the variance in metallicity among Segue 2's stars constitutes indirect evidence for a dark matter halo (whose deeper potential would help to retain chemically enriched gas despite strong winds generated by star formation).

Finally, we note that while high-quality stellar-kinematic data sets are available for the Milky-Way satellites Sagittarius and Willman 1, these objects show strong evidence for tidal disruption and/or non-equilibrium kinematics (Ibata et al. 1997; Willman et al. 2011). Since any mass inference based on the Jeans equation relies fundamentally on the assumption of dynamic equilibrium, we do not consider these objects here.

Table 1 lists central coordinates, distances (from the Sun), absolute V-band magnitudes, and projected halflight radii for the Milky Way satellites we consider here. The last column gives the number of member stars with velocity measurements available for the kinematic analysis. Figures 1 and 2 display projected velocity dispersion profiles for each dwarf. These binned profiles are included only for the purpose of display; the fitting that we describe below uses the unbinned data directly.

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Table 1. Properties of Milky Way Satellites^a and Stellar-kinematic Samples

Object	R.A. (J2000)	Decl. (J2000)	Distance	MV	Rhalf	Nsample	rmax
	(hh:mm:ss)	(dd:mm:ss)	(kpc)	(mag)	(pc)		(pc)
Carina	06:41:36.7	−50:57:58	105 ± 6	−9.1 ± 0.5	250 ± 39	774	$2224^{+885}_{-441}$
Draco	17:20:12.4	+57:54:55	76 ± 6	−8.8 ± 0.3	221 ± 19	292	$1866^{+715}_{-317}$
Fornax	02:39:59.3	−34:26:57	147 ± 12	−13.4 ± 0.3	710 ± 77	2483	$6272^{+2616}_{-1366}$
Leo I	10:08:28.1	+12:18:23	254 ± 15	−12.0 ± 0.3	251 ± 27	267	$1948^{+794}_{-407}$
Leo II	11:13:28.8	+22:09:06	233 ± 14	−9.8 ± 0.3	176 ± 42	126	$824^{+345}_{-178}$
Sculptor	01:00:09.4	−33:42:33	86 ± 6	−11.1 ± 0.5	283 ± 45	1365	$2673^{+1099}_{-569}$
Sextans	10:13:03.0	−01:36:53	86 ± 4	−9.3 ± 0.5	695 ± 44	441	$2544^{+1109}_{-587}$
Ursa Minor	15:09:08.5	+67:13:21	76 ± 3	−8.8 ± 0.5	181 ± 27	313	$1580^{+626}_{-312}$
Bootes I	14:00:06.0	+14:30:00	66 ± 2	−6.3 ± 0.2	242 ± 21	37	$544^{+252}_{-135}$
Canes Venatici I	13:28:03.5	+33:33:21	218 ± 10	−8.6 ± 0.2	564 ± 36	214	$2030^{+884}_{-468}$
Canes Venatici II	12:57:10.0	+34:19:15	160 ± 4	−4.9 ± 0.5	74 ± 14	25	$352^{+105}_{-28}$
Coma Berenices	12:26:59.0	+23:54:15	44 ± 4	−4.1 ± 0.5	77 ± 10	59	$238^{+103}_{-53}$
Hercules	16:31:02.0	+12:47:30	132 ± 12	−6.6 ± 0.4	$330_{-52}^{+75}$	30	$638^{+295}_{-147}$
Leo IV	11:32:57.0	−00:32:00	154 ± 6	−5.8 ± 0.4	206 ± 37	18	$443^{+197}_{-95}$
Leo V	11:31:09.6	+02:13:12	178 ± 10	−5.2 ± 0.4	135 ± 32	5	$201^{+95}_{-43}$
Leo T	09:34:53.4	+17:03:05	417 ± 19	−8.0 ± 0.5	120 ± 9	19	$534^{+183}_{-60}$
Segue 1	10:07:04.0	+16:04:55	23 ± 2	−1.5 ± 0.8	$29_{-5}^{+8}$	70	$139^{+56}_{-28}$
Segue 2	02:19:16.0	+20:10:31	35 ± 2	−2.5 ± 0.3	35 ± 3	25	$119^{+45}_{-18}$
Ursa Major I	10:34:52.8	+51:55:12	97 ± 4	−5.5 ± 0.3	319 ± 50	39	$732^{+338}_{-181}$
Ursa Major II	08:51:30.0	+63:07:48	32 ± 4	−4.2 ± 0.6	149 ± 21	20	$294^{+139}_{-74}$

Note. ^aCentral coordinates, distances, absolute magnitudes and projected half-light radii are adopted from the review of McConnachie (2012, see references to original sources therein).

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Given the form of Equation (7), the annihilation rate will drop rapidly at galactocentric distances r ≫ r_s where the density profile is steeply falling (β > 3). However, within r_s the radial distribution of the emission is determined by the slope of the inner density profile γ. For sufficiently cuspy profiles (γ > 1) the emission is dominated by annihilation near the halo center. For γ ∼ 1 the annihilation rate receives approximately equal contributions from all radii and for γ < 1 the emission comes primarily from the largest radii within r_s.

Unfortunately, the current data sets—even for the classical dwarfs—do not place strong upper bounds on r_s, thereby allowing emission that extends to an arbitrarily large radius.⁴ Therefore, the question of where the halo ends has important consequences for the expected dark matter signal from a dwarf galaxy. The data for nearly all dwarf galaxies are consistent with density profiles described by single power laws with logarithmic slopes dlog ρ/dlog r > −3—indeed, despite its unphysically infinite mass, the "isothermal sphere," characterized by ρ(r)∝r⁻², has long been used to model kinematics of spheroidal galaxies (Binney & Tremaine 2008). It is therefore important to define some means for preventing the outer parts of a halo—i.e., regions outside the orbits of the observed stellar populations—from dominating the integral used to calculate the J-profile (Equation (5)).

An obvious choice for a conservative truncation radius is that of the outermost member star used to estimate the velocity dispersion profile. For stars well beyond the luminous scale radius R_e, the projected distances that we observe are likely to be similar to the de-projected distances r. However, if we observe enough stars close to the center it becomes likely that some of these stars lie at large galactocentric distances. Therefore, we use the entire distribution of projected radii of the kinematic sample to estimate the maximum galactocentric distance r_max among those stars.

We estimate r_max in the following way. Given spherical symmetry, it is straightforward to find the probability distribution for the unprojected distance to the outermost observed star given the projected distances to the observed stars. We start by considering an individual star. Given its projected distance R we take the probability of its line of sight distance z (relative to the halo center) to be proportional to the (deprojected) Plummer density profile:

$$\begin{equation} {P}(z | R) \propto \left(1 + \frac{z^2 + R^2}{R_e^2}\right)^{-5/2}, \end{equation} \tag{ 20 }$$

where R_e is the projected halflight radius (Section 3). Once the above probability has been normalized (by integrating over z) we can construct the probability distribution for the unprojected distance r given the projected distance R:

$$\begin{equation} {P}(r | R) = \int _z {P}(r|z,R) \,\, {P}(z|R) \,dz. \end{equation} \tag{ 21 }$$

In the above P(r|z, R) is simply the Dirac delta function $\delta (r - \sqrt{z^2 + R^2})$. Note that the above integral is zero unless r > R, in which case the delta function picks out two values of z. The result we will need is the cumulative distribution function (CDF) of r given R:

$$\begin{eqnarray} {\rm CDF}(r|R) &=& \int ^r_0 {P}(r^{\prime } | R) dr^{\prime }\nonumber\\ & =& \frac{\left(r^2 - R^2\right)^{1/2} \left(r^2 + \frac{1}{2}\left(3R_h^2 + R^2\right)\right)}{\left(r^2 + R_h^2\right)^{3/2}}, \end{eqnarray} \tag{ 22 }$$

for r > R and CDF(r|R) = 0 for r < R.

To find the CDF for the distance to the outermost of n observed stars we simply multiply the CDFs for each of the n stars:

$$\begin{equation} {\rm CDF}_{\rm max}(r | R_1, \dots, R_n) = {\rm CDF}(r|R_1) \cdots {\rm CDF}(r|R_n), \end{equation} \tag{ 23 }$$

where each term on the right-hand side is given by Equation (22) and R_i is the measured projected distance to the ith member star.

For each dwarf, Equation (23) can be used to estimate the distance to the outermost star used in the Jeans analysis. The median estimate and ±1σ confidence intervals for the distance to the outermost member star for each dwarf are shown in the last column of Table 1. When computing J-profiles we truncate all halo profiles obtained from the Jeans/MultiNest sampling analysis at the median estimate to the outermost member star.

Note that this truncation is not imposed on the mass profile in Equation (14) when calculating the integral in Equation (16) during the Jeans/MultiNest sampling. However, the integral in Equation (16) is dominated by the contribution from radii r < R_e, such that as long as the truncation radius is larger than the luminous effective radius (as it is for every dwarf galaxy we consider), the result from the Jeans/MultiNest analysis is insensitive to whether or not we truncate the halo density profile at the outermost star in the kinematic sample. We have verified this argument by a controlled experiment and found that the significant effect of truncation is on the subsequent integration over the density profile that enters the calculation of the annihilation signal (see Equation (5)). For the purpose of this work (quantifying the expected dark matter flux from a dwarf) this particular choice of truncation is a conservative one. A spherical Jeans analysis cannot, in principle, constrain the mass distribution far beyond the outermost member stars and we therefore set the density to zero at these distances.

For some allowed models, however, the halo density at the galactocentric radius of the outermost star is far smaller than that expected for the Milky Way halo at the same location. This situation would be inconsistent, as the outermost star (and dark matter particles) would likely be lost due to tides. Therefore we impose an additional, physically motivated filter by requiring that the tidal radius of any acceptable halo be larger than the distance to the outermost star.

The magnitude of the tidal radius r_t depends on the internal and external potentials (i.e., those of the dwarf and Milky Way, respectively), the orbit of the dwarf, and on the orbital configuration within the dwarf (orbits that are prograde with respect to the dwarf's orbit are more easily stripped than those that are retrograde; Read et al. 2006).

For each kinematically allowed halo profile we follow von Hoerner (1957) and King (1962) and estimate a tidal radius r_t by solving

$$\begin{eqnarray} &&\left.r_t^3 - D^3 \frac{M(r_t)}{ M_{\mathrm{MW}}(D)} \left[ 2 + \frac{\omega ^2 D^3}{G M_{\mathrm{MW}}(D)} - \frac{d \ln M_{\mathrm{MW}}}{d \ln r} \right \vert _{r=D} \right]^{-1} = 0. \nonumber\\ \end{eqnarray} \tag{ 24 }$$

Here, D is the distance between the Milky Way center and the dwarf, M(r_t) is the mass within the tidal radius of the dwarf galaxy, M_MW(r) is the mass enclosed within radius r in the Milky Way, and ω is the angular speed of the dwarf about the Galactic center (which we take to be a linear speed of 200 km s⁻¹ divided by the distance from the dwarf to the Galactic center). The Milky Way mass model is taken to be an NFW profile with virial mass M_MW = 10¹²M_☉, a scale radius r_{s, MW} = 21.5 kpc, and a concentration c_MW = 12 (Klypin et al. 2002; for a more recent treatment of the subject see, e.g., Nesti & Salucci 2013; the results are not sensitive to the uncertainties in the Milky Way mass model).

The expression in Equation (24) should be taken as a crude approximation for several reasons (see, for example, discussions in Binney & Tremaine 2008; Mo et al. 2010). First, it makes the assumption that the dwarf galaxy is on a circular orbit (tidal radii for systems with eccentric orbits are not well defined). Second, the three dimensional tidal surface is not of constant radius and thus does not correspond to one unique value for r_t. Third, Equation (24) does not include the effect of orbital dynamics of the particles within the dwarf galaxy itself (manifested as some variance about the angular velocity ω). Nevertheless, the utility of this prescription has been thoroughly explored in studies of globular clusters, dark matter substructure, and dwarf galaxies (Johnston 1998; Johnston et al. 2001; Taylor & Babul 2001, 2004; Zentner & Bullock 2002, 2003), and gives a reasonably intuitive definition of "tidal radius."

Therefore, we reject any halo profiles for which this estimate of the tidal radius is smaller than the radius we estimate for the outermost member in the kinematic samples.⁵ This consistency condition turns out to affect only two dwarfs, the ultra-faints Segue 2 and Leo IV, removing profiles with small values of both ρ_s and r_s (i.e., profiles that do not lie on the typical ρ_s versus r_s constraint seen in Figure 3).

As a final filter, we reject halo profiles that would be cosmologically "implausible" in the sense that their formation would have required extremely rare peaks in the primordial matter density field. We estimate the rareness of a candidate halo in the following way. Cosmological simulations show that the density profile near the center of a dark matter halo is more or less set at the time of its formation (modulo feedback effects from baryon-physical processes). The outer profile then evolves as mass is accumulated by accretion in a hierarchical scenario (see, e.g., Navarro et al. 1997; Wechsler et al. 2006; Gao et al. 2008). To a first approximation, then, the primordial mass of a halo (the virial mass of the collapsing overdensity) is greater (or on the order of) the mass M within r_s today. In addition, the density of the inner parts of a halo is also roughly set at the time of formation, following a distribution in concentration, c ≡ R_vir/r_s, where R_vir is the radius of the virialized region (Navarro et al. 1996).

For example, the scale density of an NFW profile is ρ_s = δ_cρ_critf(c_v), where δ_c is the characteristic overdensity of a collapsed object and ρ_crit (scaling as (1 + z)³ in the matter-dominated era) is the critical density of the universe at the time of collapse (e.g., Gao et al. 2008). The function f(c_v) = ln (1 + c_v) + c_v/(1 + c_v) is a function of order unity. There is thus an approximately one-to-one mapping between the collapse redshift and the scale density ρ_s of a halo (ignoring the intrinsic variation in concentration; Wechsler et al. 2006; Gao et al. 2008). For simplicity, and due to a lack of knowledge of the initial conditions that give rise to the intrinsic properties of the dwarf galaxies at their formation time, we use the approximation (Navarro et al. 1996)

$$\begin{equation} \frac{1 + z_{\rm col}}{20} \gtrsim \left(\frac{\rho _s}{M_\odot \,{\rm pc}^{-3}}\right)^{1/3} \end{equation} \tag{ 25 }$$

as an upper limit on the value of the characteristic density of each halo in the chains.

The collapse redshift, along with an estimate of a halo's mass at collapse M, can be used to quantify the rarity of the density perturbation that collapsed to form the halo. In order to estimate the mass of the collapsed perturbation we use the conventional theory of spherical collapse. A region of space has collapsed and virialized when its average overdensity (with respect to the background as computed in linear perturbation theory) is δ_collapse ∼ 1.686. The overdensity at redshift z, averaged over a region of mass M, is a Gaussian random variable with mean 0 and standard deviation $\sigma (M) {\cal {D}}(z)$, where $\sigma (M) = \langle \delta _M^2 \rangle ^{1/2}$ is the standard deviation of the overdensity today when smoothed over a region of mass M and ${\cal {D}}(z)$, the growth function, quantifies the linear growth of perturbations such that ${\cal {D}}(z=0)=1$. Therefore, the rarity (in standard deviations) of a region of mass M collapsing at redshift z is given by (see, e.g., Bond et al. 1991; Bower 1991; Lacey & Cole 1993),

$$\begin{equation} \nu (M,z) = \frac{\delta _{\rm collapse}}{\sigma (M) {\cal {D}}(z)}. \end{equation} \tag{ 26 }$$

The probability that such a region has collapsed by redshift z is the tail probability of a standard normal distribution to the right of ν. We can make a rough estimate of the probability that the Milky Way contains any objects of mass M that formed at (or before) a redshift z by incorporating a trials factor which counts the number of "independent" regions of the universe of mass M that eventually make up the Milky Way today. We estimate this quantity simply as M_MW/M, where M_MW = 10¹²M_☉ is the mass of the Milky Way. We take the mass M of a halo to be the lesser of the mass within r_s and the mass within the outermost star. For the halos generated by the Monte Carlo sampling the masses range from about 10⁵ to 10⁷M_☉. This leads to trials factors between 10⁵ and 10⁷. The probability that the Milky Way contains a halo that collapsed with mass M by redshift z is then

$$\begin{equation} {P}(M,z) = 1 - \Phi \left[\nu (M,z)\right]^{M_{\mathrm{MW}}/M}, \end{equation} \tag{ 27 }$$

where Φ(x) is the CDF of a standard normal distribution and ν(M, z) is given by Equation (26). We apply the constraint by demanding that the Milky Way is "typical" at the 3σ level, i.e., we reject a halo if P(M, z) < 0.003. For a trials factor of 10⁶ this constraint is equivalent to demanding that ν ≲ 6. The results are insensitive to the threshold value for P(M, z)—the probability of the Milky Way containing a halo goes to zero extremely rapidly as the halo's ν value increases beyond about ν ≈ 6.

In applying this cosmological argument a posteriori on the chains we find that in the case of classical dwarfs (for which we have a large number of gravitational stellar tracers) stellar kinematics alone does not allow for rare peaks and this filter essentially has no effect. However, for the ultra-faint dwarfs the stellar kinematic data allow dark matter halos that appear to be extremely rare, suggesting that the origin of such rare peaks is due to the small size of kinematic samples available compared with the classical dwarfs. In practice, the halos that are eliminated are those with ρ_s ≳ 1 M_☉ pc³.

The gamma-ray flux from dark matter annihilation is fully described by the function dJ(θ)/dΩ and any dark matter search is properly conducted using this function (e.g., Geringer-Sameth et al. 2014). However, it is useful to explore the dark matter distributions in dwarfs using summary quantities based on the J-profiles.

Concerning indirect detection, the most important properties of the dwarf galaxies are the amplitude and spatial extent of the J-profile. The amplitude is often given as the integral of dJ(θ)/dΩ over some solid angle; i.e., we define

$$\begin{equation} J(\theta) \equiv \int _0^\theta \frac{dJ(\theta ^{\prime })}{d\Omega } 2\pi \sin (\theta ^{\prime })d\theta ^{\prime }, \end{equation} \tag{ 28 }$$

with an analogous definition for J_decay(θ). A detector-independent amplitude is the J-profile integrated out to the truncation radius of the halo. This corresponds to integrating dJ(θ)/dΩ out to an angle θ_max = arcsin(r_max/D), where r_max is the distance from the center of the dwarf to the outermost member star (Section 6.2) and D is the distance from Earth to the dwarf. Integrating the J-profile within θ_max gives the total dark matter flux expected from a halo. We use J or J_decay to denote this scalar quantity when there is no confusion.

Using the kinematic data discussed in Section 4, and applying the formalism described in Sections 3, 5, and 6, we apply Equations (5) and (6) to estimate the kinematically allowed values of J and J_decay. Figures 4 and 5 along with Table 2 show the main results of this work.

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Table 2. J Values for Annihilation and Decay in Dwarf Galaxies^a

Dwarf	θmax	θ0.5	θ0.5 decay	log10J(θmax)	log10J(05)	log10Jdecay(θmax)	log10Jdecay(05)
	(deg)	(deg)	(deg)	(GeV2 cm−5)	(GeV2 cm−5)	(GeV cm−2)	(GeVcm−2)
Carina	1.26	$0.15^{+0.15}_{-0.07}$	$0.46^{+0.16}_{-0.12}$	$17.92^{+0.19}_{-0.11}$	$17.87^{+0.10}_{-0.09}$	$18.15^{+0.34}_{-0.25}$	$17.90^{+0.17}_{-0.16}$
Draco	1.30	$0.40^{+0.16}_{-0.15}$	$0.64^{+0.06}_{-0.14}$	$19.05^{+0.22}_{-0.21}$	$18.84^{+0.12}_{-0.13}$	$18.97^{+0.17}_{-0.24}$	$18.53^{+0.10}_{-0.12}$
Fornax	2.61	$0.13^{+0.04}_{-0.05}$	$0.31^{+0.08}_{-0.05}$	$17.84^{+0.11}_{-0.06}$	$17.83^{+0.12}_{-0.06}$	$17.99^{+0.11}_{-0.08}$	$17.86^{+0.04}_{-0.05}$
Leo I	0.45	$0.13^{+0.05}_{-0.05}$	$0.22^{+0.02}_{-0.04}$	$17.84^{+0.20}_{-0.16}$	$17.84^{+0.20}_{-0.16}$	$17.91^{+0.15}_{-0.20}$	$17.91^{+0.15}_{-0.20}$
Leo II	0.23	$0.04^{+0.05}_{-0.02}$	$0.09^{+0.03}_{-0.05}$	$17.97^{+0.20}_{-0.18}$	$17.97^{+0.20}_{-0.18}$	$17.24^{+0.35}_{-0.48}$	$17.24^{+0.35}_{-0.48}$
Sculptor	1.94	$0.15^{+0.05}_{-0.05}$	$0.48^{+0.14}_{-0.11}$	$18.57^{+0.07}_{-0.05}$	$18.54^{+0.06}_{-0.05}$	$18.47^{+0.16}_{-0.14}$	$18.19^{+0.07}_{-0.06}$
Sextans	1.70	$0.58^{+0.32}_{-0.47}$	$0.87^{+0.10}_{-0.53}$	$17.92^{+0.35}_{-0.29}$	$17.52^{+0.28}_{-0.18}$	$18.56^{+0.25}_{-0.73}$	$17.89^{+0.13}_{-0.23}$
Ursa Minor	1.37	$0.06^{+0.07}_{-0.03}$	$0.25^{+0.14}_{-0.09}$	$18.95^{+0.26}_{-0.18}$	$18.93^{+0.27}_{-0.19}$	$18.13^{+0.26}_{-0.18}$	$18.03^{+0.16}_{-0.13}$
Boötes I	0.47	$0.22^{+0.05}_{-0.10}$	$0.26^{+0.02}_{-0.04}$	$18.24^{+0.40}_{-0.37}$	$18.24^{+0.40}_{-0.37}$	$17.90^{+0.23}_{-0.26}$	$17.90^{+0.23}_{-0.26}$
Coma	0.31	$0.16^{+0.02}_{-0.05}$	$0.17^{+0.01}_{-0.02}$	$19.02^{+0.37}_{-0.41}$	$19.02^{+0.37}_{-0.41}$	$17.96^{+0.20}_{-0.25}$	$17.96^{+0.20}_{-0.25}$
CVnI	0.53	$0.11^{+0.15}_{-0.09}$	$0.23^{+0.07}_{-0.17}$	$17.44^{+0.37}_{-0.28}$	$17.43^{+0.37}_{-0.28}$	$17.57^{+0.37}_{-0.73}$	$17.57^{+0.36}_{-0.72}$
CVnII	0.13	$0.07^{+0.01}_{-0.02}$	$0.07^{+0.00}_{-0.01}$	$17.65^{+0.45}_{-0.43}$	$17.65^{+0.45}_{-0.43}$	$16.97^{+0.24}_{-0.23}$	$16.97^{+0.24}_{-0.23}$
Hercules	0.28	$0.07^{+0.08}_{-0.06}$	$0.12^{+0.03}_{-0.09}$	$16.86^{+0.74}_{-0.68}$	$16.86^{+0.74}_{-0.68}$	$16.66^{+0.42}_{-0.40}$	$16.66^{+0.42}_{-0.40}$
Leo IV	0.16	$0.05^{+0.03}_{-0.04}$	$0.08^{+0.01}_{-0.06}$	$16.32^{+1.06}_{-1.69}$	$16.32^{+1.06}_{-1.69}$	$16.12^{+0.71}_{-1.14}$	$16.12^{+0.71}_{-1.14}$
Leo V	0.07	$0.03^{+0.01}_{-0.02}$	$0.04^{+0.00}_{-0.01}$	$16.37^{+0.94}_{-0.87}$	$16.37^{+0.94}_{-0.87}$	$15.86^{+0.46}_{-0.47}$	$15.86^{+0.46}_{-0.47}$
Leo T	0.08	$0.03^{+0.01}_{-0.02}$	$0.04^{+0.00}_{-0.01}$	$17.11^{+0.44}_{-0.39}$	$17.11^{+0.44}_{-0.39}$	$16.48^{+0.22}_{-0.25}$	$16.48^{+0.22}_{-0.25}$
Segue 1	0.35	$0.13^{+0.05}_{-0.07}$	$0.18^{+0.01}_{-0.05}$	$19.36^{+0.32}_{-0.35}$	$19.36^{+0.32}_{-0.35}$	$17.99^{+0.20}_{-0.31}$	$17.99^{+0.20}_{-0.31}$
Segue 2	0.19	$0.07^{+0.03}_{-0.05}$	$0.10^{+0.01}_{-0.05}$	$16.21^{+1.06}_{-0.98}$	$16.21^{+1.06}_{-0.98}$	$15.89^{+0.56}_{-0.37}$	$15.89^{+0.56}_{-0.37}$
Ursa Major I	0.43	$0.15^{+0.08}_{-0.12}$	$0.22^{+0.03}_{-0.14}$	$17.87^{+0.56}_{-0.33}$	$17.87^{+0.56}_{-0.33}$	$17.61^{+0.20}_{-0.38}$	$17.61^{+0.20}_{-0.38}$
Ursa Major II	0.53	$0.24^{+0.06}_{-0.11}$	$0.29^{+0.02}_{-0.04}$	$19.42^{+0.44}_{-0.42}$	$19.42^{+0.44}_{-0.42}$	$18.39^{+0.25}_{-0.27}$	$18.38^{+0.25}_{-0.27}$

Notes. ^aErrors correspond to the 1σ range of the kinematic analysis (i.e., the 16th and 84th percentiles). J(θ) is the J-profile (Equation (5)) integrated over a cone with radius θ (Equation (28)); θ_0.5 is the "half-light radius" for dark matter emission (i.e., the angle containing 50% of the total emission: J(θ_0.5) = 0.5 × J(θ_max), see Section 7.2). Analogous definitions apply for dark matter decay. Halos are truncated at a radius corresponding to θ_max (Section 6.2).

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Figure 4 shows the derived J values for all of the dwarf galaxies considered in this analysis. Error bars represent the 1σ uncertainty in the value of J (e.g., the 16th and 84th percentiles of the posterior distribution). There are many interesting features of this distribution of J values among the dwarf galaxies. Concerning the overall amplitude of an annihilation signal, we find that among the classical dwarfs Draco and Ursa Minor have the largest expected flux. However, Sculptor and, to a lesser extent, Fornax and Carina have the most constraining kinematic sample, giving a very small range of allowed values for J and thus the smallest uncertainties (in agreement with the analysis of Charbonnier et al. 2011). Among the ultra-faint dwarf galaxies, Segue 1 and Ursa Major II have central J values that are higher then those of any of the classical dwarfs. However, the uncertainties in these values are also larger than those of all the classical dwarfs. Of all the dwarf galaxies in this sample, Sculptor has the smallest uncertainty in its total J value (about 15%). On the other hand, all ultra-faint dwarf galaxies exhibit 1σ errors that are about an order of magnitude (in some cases several orders of magnitude—e.g., Leo IV). This is an outcome of the limited size of the kinematic samples available to date. From the entire ultra-faint sample, Segue 1, Coma Berenices, and Ursa Major II exhibit well-constrained (less than an order of magnitude) J values, and due to their overall high amplitude they should be considered for current and future annihilation searches.

Figure 5 shows the corresponding J_decay values for the twenty dwarf galaxies analyzed. As in Figure 4, error bars represent the 1σ range in the value of J_decay. In the case of dark matter decay scenarios, the emission profile is set by the amount of mass along the line of sight (Equation (4)) and is less sensitive to the inner slope of the density profile. As a result, the preferential ordering of dwarfs according to their emission amplitude is different than when searching for annihilation products. In the case of decay, the kinematic analysis shows that Draco, and to a lesser extent, Sextans and Sculptor are the sources with the largest expected emission. As in the annihilation case, Fornax and Sculptor have the least systematic uncertainty in their emission amplitudes compared with the rest of the dwarfs.

Many of the ultra-faint dwarfs do not appear to be as promising targets as several of the classical dwarfs. This is due to the conservative way we chose to truncate the halos when computing J and J_decay. The outermost observed member stars of the ultra-faints are much closer in than those in the classical dwarfs (see the θ_max values along the bottom of Figures 4 and 5 and the last column of Table 1). Therefore, we simply do not allow the ultra-faint halos to be as extended as those of the classical dwarfs. Since the emission profile from decay is more sensitive to the total mass of the halo, whereas the annihilation profile is more sensitive to the inner slope, the effect of the truncation has different effects when considering annihilation and decay. Whether this truncation is conservative or if it truly reflects the physical sizes of the ultra-faints' dark matter halos is unknown.

It is important to note that the ranking of the various J values implies a consistency check for any detection claim. First, it is likely that if a signal is seen, it will be seen in multiple dwarfs. Consider the pairs Draco/Ursa Minor (classical) and Segue 1/Ursa Major II (ultra-faint). The similar J values among the members of a pair imply that if an annihilation signal is detected in any of these dwarfs there should be a detection of similar amplitude in the other member of the pair (modulo differences in the diffuse gamma-ray background between the dwarfs).

In the farther future, this argument may be turned around to provide an example of "dark matter particle astronomy": the relative annihilation fluxes measured in multiple dwarfs can be used to constrain the dark matter distributions in their halos. For example, the detection of a signal in a highly constrained dwarf like Sculptor, Draco, or Ursa Minor would immediately tell us something about the dark matter distribution in Segue 1, an object less luminous by three orders of magnitude.

The question of whether a dark matter halo will appear as an extended source for a gamma-ray instrument is important as the additional information available from the angular distribution of emission can be used to increase the signal-to-noise ratio. More point-like sources of emission are more straightforward to detect. However, the detection of a spatially varying annihilation signal immediately reveals properties of the dark matter halo.

Figure 6 shows the constraints on the emission profiles for two benchmark dwarf galaxies: the classical dwarf Draco and the ultra-faint dwarf Segue 1. The envelopes show the ±1σ and median values of the emission profiles as a function of the angular separation from the center of the dwarf. It is important to emphasize that none of the curves in the figure correspond to an individual dark matter profile. Rather, the envelope should be thought of as constraining the value of dJ/dΩ or dJ_decay/dΩ at a given angular separation. In the lower panels we show the emission profiles integrated over solid angle out to an angular separation θ (Equation (28)). For Draco we see a familiar result: there is a particular radius at which the differential flux profile is most tightly constrained, and another (slightly larger) angle within which the total annihilation flux is best constrained (Walker et al. 2011; Charbonnier et al. 2011; Bonnivard et al. 2015). The uncertainty in the flux within 001 is about a factor of five and decreases to about 20% when integrating within about 03, an angle corresponding to twice the projected half-light radius. For Segue 1, however, the situation is somewhat different. While the integrated J value within 001 can be inferred to within a factor of six, similar to the case of Draco, and the minimum uncertainty again occurs when integrating within about twice the half-light radius (θ ≈ 015), even there J can only be determined to within a factor of 3.5. We do not see the drastic decrease in the uncertainty of Segue 1's expected emission that we see with most of the classical dwarfs. The larger uncertainty for Segue 1 is a direct consequence of the relatively small size of its available kinematic sample.

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We can quantify the extent to which halos can be spatially resolved in gamma-ray telescopes by comparing the derived emission profiles for either annihilation or decay with the point spread function (PSF) of specific instruments.

Figures 7 and 8 show the angular distribution of dark matter annihilation and decay. The bands show constraints on the "containment fraction" curves for the different dwarfs. The containment fraction, at angle θ, is defined simply as J(θ)/J(θ_max), where J(θ) is given by Equation (28). Each halo profile gives rise to a containment fraction curve and the dotted line corresponds to the median value of the containment fraction among all the allowed halos, computed at each θ. The shaded band corresponds to the 16th and 84th percentiles. For example, the constraint on the "half-light radius" of the dark matter emission profile is the intersection of the horizontal line y = 0.5 with the shaded band. We use θ_0.5 and θ_0.5 decay to denote the half-light radii for J- and J_decay-profiles and tabulate them in Table 2.

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The curves in Figures 7 and 8 illustrate the PSFs of two gamma-ray experiments. The containment fraction of a PSF is simply the probability that a gamma-ray will be reconstructed within an angle θ of its true origin. The solid blue, magenta, red, and green lines correspond to the PSF of the Fermi-LAT at photon energies of 0.5, 1, 2, and 10 GeV (computed using gtpsf—see software and documentation at the Fermi Science Support Center⁶). The dashed orange line corresponds to a two-dimensional Gaussian PSF with a 68% containment angle of 01 (e.g., a Rayleigh distribution with a mean of 0083). This corresponds to the benchmark PSF of current-generation Atmospheric Čerenkov Telescopes (ACTs). Figure 8 is identical to Figure 7 but shows the containment fractions for J_decay.

We find that for many of the classical dwarfs (Carina, Draco, Fornax, Leo I, Sculptor, Sextans) ACTs should be able to detect extended emission from dark matter annihilation (if the emission can be detected at all) and similarly for some of the ultra-faint dwarfs (Boötes I, Coma Berenices, and Ursa Major II). Regarding Fermi-LAT, at the highest energies (>10 GeV) only Draco and, perhaps, Ursa Major II appear to be extended enough to be detected, and therefore any limits derived using Fermi-LAT data will not be affected significantly by the assumption of point sources when it comes to dwarf galaxies (in agreement with Ackermann et al. 2014).