Limits on Dark Matter annihilation in dwarf galaxies with prior-free astrophysical factors Andrea Chiappo andrea.chiappo@fysik.su.se Co-authors: Jan Conrad, Nils Håkansson, Johann Cohen-Tanugi, Louis E. Strigari Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 1
Premises & Motivation Dwarf spheroidal satellite galaxies (dSphs) of the Milky Way: - nearby ideal targets for Dark Matter (DM) indirect detection - DM dominated - low γ -ray contamination - low Galactic foregrounds Searching for DM decay or annihilation products (e + , e - , γ …) Photon flux from DM annihilation: ⌥ E max ⌥ ⌥ Φ γ ( ∆Ω ) = ⇥ σ v ⇤ d N 1 ρ 2 ( r ( l ))d l d Ω d E γ 2 m 2 d E γ 4 π E min ∆Ω l.o.s ⌦ � ↵ ⌦ � ↵ (Bergström et al. 1998) J factor particle physics factor • one of main uncertainties in the statistical analysis of the γ -ray data • previously obtained with Bayesian methods • γ -ray analyses are performed in a frequentist manner priors influence on DM particle properties inference Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 2
Objectives I. BUILD PRIOR-FREE D S PHS J -FACTORS LIKELIHOODS • JEANS ANALYSIS ASSUMING SPHERICAL SYMMETRY • MAXIMUM LIKELIHOOD TECHNIQUE TO FIT PARAMETERS • VALIDATION ON SIMULATIONS BY GAIA CHALLENGE • APPLIED ON STELLAR KINEMATIC DATA FROM 9 D S PHS II. UPDATE UPPER LIMITS � σ v ⇥ 95% • COMBINE NEW J -LIKELIHOODS WITH PUBLISHED γ -LIKELIHOODS ( from arxiv:1503.02641) Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 3
Jeans Equation For dSphs as spherically symmetric, collision-less steady state systems � ∞ los ( R ) = 2 G K ( r, R ) ν ( r ) M ( r )d r σ 2 (Mamon & Ł okas 2004) I ( R ) r R : line-of-sight velocity dispersion los ( R ) : surface brightness I ( R ) : stellar density profile ⌃ ( r ) : velocity anisotropy kernel function K ( r, R ) ⌥ s : DM Halo mass ρ ( r ) r 2 d r M ( s ) = 0 CAVEAT: only valid for DM-dominated systems (like dSphs) Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 4
Jeans Equation ρ 0 DM profile: (generalised NFW) ρ DM ( r ; r 0 , ρ 0 , a, b, c ) = ⇥ a ⇥ b � c � ⇥ c � � a r r (Zhao 1996) 1 + r 0 r 0 � � Stellar profile: (Hernquist 1990) ⌃ Plummer ( r ; r � , � � ) = ⇥ 2 ⌅ 5 ⇤ � ⇥ 2 � r 1 + r ⇥ � � ⌃ Plummer-like ( r ; r � , � � ) = ⇤ ⇥ ⌅ ⇥ 2 ⌅ 4 . 9 � ⇥ � ⇥ 0 . 1 ⇤ 2 � � r r 1 + r ⇥ r ⇥ � � ⌃ non-Plummer ( r ; r � , � � ) = ⇥ 2 ⌅ 2 ⇥ ⇤ � � r r 1 + r ⇥ r ⇥ ⇣ ⌘ (construction of J likelihood is Velocity profile: β iso ( r ) = 0 unreliable when the stellar anistropy is allowed as free parameter) Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 5
Likelihood Unbinned Gaussian Likelihood on stars’ velocity (Walker et al. 2006, Strigari et al. 2008) − ( vi − u )2 N F N F 2 � 2 i ) + ( v i − u ) 2 g |D ) = − ln L = 1 ⇧ ⌃ e i ⌥ ln(2 ⌥ 2 L ( ⌦ g |D ) = L ( ⌦ 2 ⌦ 2 2 ⌥ 2 i i =1 i =1 i CAVEAT: Gaussian Likelihood is an approximation to the true underlying velocity distribution function where • : with the measurement uncertainty of � i v i 2 i = ⇧ 2 i + 2 los ( R i ; ⌦ g ) • : the data vectors ⇧ , ⌦ D = ( ⌦ R ) v, ⌦ CAVEAT: only valid for √ From and 2 2 velocity-independent J ∝ � 2 los ( R ) ∝ los ( R ) ∝ M ( r ) ∝ � 0 J 0 annihilation cross-section We fit via two possible schemes: J = log 10 ( J ) manual-profiling J-sampling • vary over likely range J = • sample over with MCMC L g = J , a, b, c, r 0 ~ • sample over with MCMC L g = a, b, c, r 0 ~ • retain the likelihood evaluations • retaining the likelihood evaluations • envelope the likelihood in direction J = • intepolate between pairs ( J , L J ) profile likelihood L ( J ) Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 6
Profile likelihood: example log-normal J likelihood profile J likelihood (manual-profiling) profile J likelihood (J-sampling) Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 7
Validation GAIA CHALLENGE: one-component, spherical models used to validate the method: velocity DM profile stellar profile CAVEAT: distribution not generated with Cusped Isotropic Plummer-like Gaussian sampling Cusped Isotropic non-Plummer distribution Cored Isotropic Plummer-like Cored Isotropic non-Plummer Examples: Isotropic N ★ = 100 N ★ = 1000 Cored Plummer-like Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 8
Validation: coverage and bias Partition the full dataset (N ★ = 10 4 ) into sets of N ★ = 10,100,1000 BIAS 1 - σ COVERAGE Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 9
Validation: bias extended Explore the statistical properties by considering the cases: N ★ N PE 10 1000 we fit a power law to the bias estimates ➝ index ~ 0.5 20 500 50 200 using the best-fit power law we get an estimate of the 100 100 minimum N ★ to achieve a bias < 10% in J 200 50 ↓ 500 20 (on average) N ★ ≳ 200 1000 10 Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 10
Results for real data: examples Data utilised: θ MAX = 0.5° • kinematic data from 9 dSphs DM profile: generalised NFW (galaxies with N ★ ≳ 200) surface brightness: Plummer • consisting of ( R , v , ε ) velocity anisotropy: Isotropic (projected radius, velocity, velocity uncertainty) Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 11
Updated 〈 σ v 〉 upper limits optimise � ⇥ ⌅ v ⇤ J N dwarfs ⇧ ⇤ N bins ⌅⌃ ⇥ normalisation where: ⌥ ⌥ Φ exp ⇤ v ⌅ 0 , J 0 L d L ( ⇥ ⌅ v ⇤ | m χ ) = min ( m χ ) L d ( J ) b b ⇥ ⌅ v ⇤ 0 J 0 J values of Φ exp ( m χ ) d =1 b =1 b example: Draco 10 − 4 0 Draco 1 Energy Flux (MeV cm − 2 s − 1 ) 2 10 − 5 3 4 − ∆ log L 10 − 6 5 6 7 10 − 7 8 9 10 − 8 10 3 10 4 10 5 (Ackermann et al. 2014) Energy (MeV) preliminary Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 12
Future work •Determine frequentist J -factor for recently discovered (DES) dSphs •Derive new upper limits using more γ -ray data from more dSphs •Perform coverage test on •Study systematics arising from different model assumptions or Likelihood •Improvements to the code (public release is planned) - implement non-Gaussian likelihoods - explore different optimisation algorithms •Include probability of interloper Milky Way stars Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 13
Thank you Halo Substructure and Dark Matter searches, Andrea Chiappo, IFT June 2018 14
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