A novel determination of the local dark matter density. Riccardo Catena. Institut für Theoretische Physik, Heidelberg

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1 A novel determination of the local dark matter density Riccardo Catena Institut für Theoretische Physik, Heidelberg R. Catena and P. Ullio, arxiv: [astro-ph.co]. Riccardo Catena (ITP) Firenze 28/04/ / 31

2 Overview/Motivations - Direct detection signals depend from dark halo properties. - Example : Spin-independent dark matter-nucleus scattering. - The expected event rate reads dr = σp ρ DM(R 0 ) de r 2µ 2 p,dm m DM Z f DM(v, t) v min v dv A 2 F 2 (E r) - It crucially depends on ρ DM (R 0 ) (this talk) and f DM ( v, t). Riccardo Catena (ITP) Firenze 28/04/ / 31

3 Outline 1 The underlying Galactic Model 2 The experimental constraints 3 The method:bayesian inference with Markov Chain Monte Carlo 4 Results and Conclusions Riccardo Catena (ITP) Firenze 28/04/ / 31

4 Outline 1 The underlying Galactic Model 2 The experimental constraints 3 The method:bayesian inference with Markov Chain Monte Carlo 4 Results and Conclusions Riccardo Catena (ITP) Firenze 28/04/ / 31

5 Outline 1 The underlying Galactic Model 2 The experimental constraints 3 The method:bayesian inference with Markov Chain Monte Carlo 4 Results and Conclusions Riccardo Catena (ITP) Firenze 28/04/ / 31

6 Outline 1 The underlying Galactic Model 2 The experimental constraints 3 The method:bayesian inference with Markov Chain Monte Carlo 4 Results and Conclusions Riccardo Catena (ITP) Firenze 28/04/ / 31

7 The underlying Galactic Model Dark Halo Thick Disk Gas + Fluctuations Bulge/Bar Thin Disk Figure: Schematic representation of the Galaxy Riccardo Catena (ITP) Firenze 28/04/ / 31

8 The underlying Galactic Model Dark Halo Thick Disk Gas + Fluctuations Bulge/Bar Thin Disk Figure: Schematic representation of the assumed Galactic model Riccardo Catena (ITP) Firenze 28/04/ / 31

9 The underlying Galactic Model rotation curve halo disc balge/bar HI H2 200 Velocity (km/s) Radius (Kpc) Figure: Rotation curve of the Galaxy Riccardo Catena (ITP) Firenze 28/04/ / 31

10 The underlying Galactic Model - The stellar disk: ρ d (R, z) = Σ ( ) d e R R z d sech 2 2z d z d with R < R dm H. T. Freudenreich, Astrophys. J. 492, 495 (1998) - The dust layer: The distribution of the Interstellar Medium is assumed axisymmetric as well. T. M. Dame, AIP Conference Proceedings 278 (1993) 267. Riccardo Catena (ITP) Firenze 28/04/ / 31

11 The underlying Galactic Model - The stellar bulge/bar: [ ( ) ] ρ bb (x, y, z) = ρ bb (0) exp s2 b + sa 1.85 exp( s a ) 2 where s 2 a = q2 a (x 2 + y 2 ) + z 2 z 2 b [ ( ) 2 ( ) ] 2 2 ( ) 4 x y z sb 2 = + +. x b y b z b H. Zhao, arxiv:astro-ph/ Riccardo Catena (ITP) Firenze 28/04/ / 31

12 The underlying Galactic Model - The Dark Matter halo: where f is the Dark Matter profile. - M vir, and c vir as halo parameters: ( R ρ h (R) = ρ f a h ), ρ = ρ (M vir, c vir ) a h = a h (M vir, c vir ) Riccardo Catena (ITP) Firenze 28/04/ / 31

13 The underlying Galactic Model - The Dark Matter profile: [ ] f E (x) = exp 2 α E (x α E 1) J.F. Navarro et al., MNRAS 349 (2004) A.W. Graham, D. Merritt, B. Moore, J. Diemand and B. Terzic, Astron. J. 132 (2006) f NFW (x) = 1 x(1+x) 2 J.F. Navarro, C.S. Frenk and S.D.M. White, Astrophys. J. 462, 563 (1996); Astrophys. J. 490, 493 (1997). f B (x) = 1 (1+x) (1+x 2 ). A. Burkert, Astrophys. J. 447 (1995) L25. Riccardo Catena (ITP) Firenze 28/04/ / 31

14 The underlying Galactic Model 1e Dark matter profiles Einasto NFW Burkert Profiles Kpc Riccardo Catena (ITP) Firenze 28/04/ / 31

15 Parameter space Galactic components Parameters Disk Disk Σ d R d Bulge/bar ρ bb (0) Halo Halo Halo α E M vir c vir All components R 0 All components β Riccardo Catena (ITP) Firenze 28/04/ / 31

16 The experimental constraints Constraints: - Oort s constants: A B = Θ 0 R 0 ; A + B = Θ(R 0) R - terminal velocities - total mean surface density within z < 1.1kpc - local disk surface mass density - total mass inside 50 kpc and 100 kpc - l.s.r. velocity, proper motion and parallaxes distance of high mass star forming regions in the outer Galaxy - radial velocity dispersion of tracers from the SDSS - stellar motions around the massive black hole in the GC - peculiar motion of SgrA Riccardo Catena (ITP) Firenze 28/04/ / 31

17 The experimental constraints Constraints: - Oort s constants: A B = Θ 0 R 0 ; A + B = Θ(R 0) R - terminal velocities - total mean surface density within z < 1.1kpc - local disk surface mass density - total mass inside 50 kpc and 100 kpc - l.s.r. velocity, proper motion and parallaxes distance of high mass star forming regions in the outer Galaxy - radial velocity dispersion of tracers from the SDSS - stellar motions around the massive black hole in the GC - peculiar motion of SgrA Riccardo Catena (ITP) Firenze 28/04/ / 31

18 The experimental constraints: Terminal velocities Orbitating Material Line of Sight GC Tangential Point Sun The terminal velocity is the l.o.s. velocity of the material at the tangential point Riccardo Catena (ITP) Firenze 28/04/ / 31

19 The experimental constraints: Terminal velocities - How to measure a terminal velocity? From the Doppler shift of a reference absorption line. - How to calculate a terminal velocity? v term = Θ(R) Θ(R 0 ) R R 0 - It is a strong constraint in the range 3 kpc R < R 0 Riccardo Catena (ITP) Firenze 28/04/ / 31

20 The experimental constraints: Radial velocity dispersions -The dataset: population of stars with distances up to 60kpc from the Galactic center. The distances are accurate to 10% and the radial velocity errors are less than 30 km s 1. -It is a strong constraint in the range 10 kpc R 60 kpc -To compare the data to the predictions: Jeans Equation σ 2 r (r) = 1 r 2β ρ (r) r d r r 2β 1 ρ ( r)θ 2 ( r) - where β is the anisotropy parameter: β 1 σ 2 t /σ2 r. Riccardo Catena (ITP) Firenze 28/04/ / 31

21 The method: Bayesian approach Parametric model of the Galaxy Frequentist approach = Maximum Likelihood Bayesian approach = Posterior probability density - This work Bayesian approach Riccardo Catena (ITP) Firenze 28/04/ / 31

22 The method: Bayesian approach - Target: posterior pdf (Bayes theorem): p(η d) = L(d η)π(η) p(d) ; d = data; η = parameters - Output: the mean and the variance with respect to p(η d) of functions f(η). - We will focus on f = η and f = ρ DM (R 0 ). Riccardo Catena (ITP) Firenze 28/04/ / 31

23 The method: Markov Chain Monte Carlo - Monte Carlo expectation values: f(η) = dη f(η)p(η d) 1 N 1 f(η (t) ), N where η (t) was sampled from p(η d). t=0 - Monte Carlo technics require a method to sample η (t) = Markov chains. - Markov chains : p(η (0) ) T(η (t), η (t+1) ) = η(t) distributed according to p(η d). Riccardo Catena (ITP) Firenze 28/04/ / 31

24 Convergence of the Markov chains R (Scale reduction factor). Convergence: R < 1.1 and roughly constant. 1-R as a function of the iteration number: disc central surface density bulge-bar central mass density disc radial scale Sun s Galactocentric distance halo virial mass concentration parameter anisotropy beta parameter Multivariate Scale Reduction Fatcor Iteration number Riccardo Catena (ITP) Firenze 28/04/ / 31

25 Figure: Marginal posterior pdf of the Galactic model parameters (NFW profile). Riccardo Catena (ITP) Firenze 28/04/ / 31

26 Figure: Marginal posterior pdf of the Galactic model parameters (Einasto profile). Riccardo Catena (ITP) Firenze 28/04/ / 31

27 Figure: Marginal posterior pdf of a set of derived quantities (Einasto profile). Riccardo Catena (ITP) Firenze 28/04/ / 31

28 R 0 [Kpc] R 0 [kpc] c vir M [ M ] vir Θ c vir M [ M ] vir Θ Figure: Two dimensional marginal posterior pdf in the planes spanned by combinations of the Galactic model parameters (NFW profile). Riccardo Catena (ITP) Firenze 28/04/ / 31

29 R 0 [kpc] R 0 [kpc] M vir [10 12 M Θ ] M vir [ M Θ ] c vir α E c vir 15 c vir 15 α E M vir [ M Θ ] α E R 0 [kpc] Figure: Two dimensional marginal posterior pdf in the planes spanned by combinations of the Galactic model parameters (Einasto profile). Riccardo Catena (ITP) Firenze 28/04/ / 31

30 Figure: Marginal posterior pdf for the local Dark Matter density. Top left panel: Einasto profile, applying different subsets of constraints. Top right panel: Einasto profile. Bottom left panel: NFW profile. Bottom right panel: Burkert profile. Riccardo Catena (ITP) Firenze 28/04/ / 31

31 Numerical values - Numerically we find: ρ DM (R 0 ) = (0.385 ± 0.027) GeV cm 3 (Einasto) ρ DM (R 0 ) = (0.389 ± 0.025) GeV cm 3 (NFW) ρ DM (R 0 ) = (0.409 ± 0.029) GeV cm 3 (Burkert) - No strong dependences from the assumed halo profile. Riccardo Catena (ITP) Firenze 28/04/ / 31

32 2000 β 3 4 Σ d [M Θ pc 2 ] ρ DM (R 0 ) [GeV cm 3 ] ρ bb (0) [M Θ pc 3 ] ρ (R ) [GeV cm 3 ] DM 0 R d [kpc] ρ (R ) [GeV cm 3 ] DM 0 R 0 [kpc] ρ (R ) [GeV cm 3 ] DM 0 M vir [10 12 M Θ ] ρ (R ) [GeV cm 3 ] DM 0 c vir ρ (R ) [GeV cm 3 ] DM α E ρ DM (R 0 ) [GeV cm 3 ] ρ DM (R 0 ) [GeV cm 3 ] Figure: Two dimensional marginal posterior pdf in the planes spanned by the local Dark Matter energy density and the Galactic model parameters for the Einasto case. Riccardo Catena (ITP) Firenze 28/04/ / 31

33 A B [km s 1 kpc 1 ] A+B [km s 1 kpc 1 ] v c (R 0 ) [km s 1 ] ρ DM (R 0 ) [GeV cm 3 ] ρ (R ) [GeV cm 3 ] DM ρ (R ) [GeV cm 3 ] DM Σ * [M Θ pc 2 ] Σ z <1.1 kpc [M Θ pc 2 ] M(<50 kpc) [10 11 M Θ ] ρ (R ) [GeV cm 3 ] DM ρ (R ) [GeV cm 3 ] DM ρ (R ) [GeV cm 3 ] DM 0 12 M(<100 kpc) [10 11 M Θ ] ρ (R ) [GeV cm 3 ] DM Figure: Two dimensional marginal posterior pdf in the planes spanned by the local Dark Matter energy density and the derived quantities for the Einasto case. Riccardo Catena (ITP) Firenze 28/04/ / 31

34 Comparison with other recent related works - Maximum Likelihood approach: M. Weber and W. de Boer, arxiv: [astro-ph.co]. * Only three free parameters (many fixed a priori) * For some choice of the fixed parameters (with reasonable M vir ): ρ DM (R 0 ) = (0.39 ± 0.05) GeV cm 3 - Poisson equation approach: P. Salucci, F. Nesti, G. Gentile and C. F. Martins, arxiv: [astro-ph.ga]. RΘ 2 K, R=R 0 * Strategy: ρ DM (R 0 ) = 1 4πGR 2 0 R ρ DM (R 0 ) = (0.43 ± 0.11 ± 0.10) GeV cm 3 - Fisher matrix forecasts: L. E. Strigari and R. Trotta, JCAP 0911 (2009) 019 [arxiv: [astro-ph.he]]. * Assumed a reference point in parameter space it tests the reconstruction capabilities of a future direct detection experiment accounting for astrophysical uncertainties. Riccardo Catena (ITP) Firenze 28/04/ / 31

35 Conclusions We proved that Bayesian probabilistic inference is a good method to constrain the local dark matter density. Assuming a Einasto Dark Matter profile, we find ρ DM (R 0 ) = (0.385 ± 0.027) GeV cm 3. Assuming an NFW Dark Matter profile, we find ρ DM (R 0 ) = (0.389 ± 0.025) GeV cm 3. Assuming an Burkert Dark Matter profile, we find ρ DM (R 0 ) = (0.409 ± 0.029) GeV cm 3. For a given dark matter profile, we can therefore estimate the local value of the Dark Matter density with an accuracy of roughly the 7%. This result alleviate the astrophysical uncertainties on the theoretical predictions for the direct detection dark matter signals. Riccardo Catena (ITP) Firenze 28/04/ / 31