The local dark matter halo density Riccardo Catena Institut für Theoretische Physik, Heidelberg 17.05.2010 R. Catena and P . Ullio, arXiv:0907.0018 [astro-ph.CO]. To be published in JCAP Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 1 / 25
Overview/Motivations - Direct detection signals depend from dark halo properties. - Example : Spin-independent dark matter-nucleus scattering. - The expected event rate reads Z ∞ dE r = σ p ρ DM ( R 0 ) dR f DM ( v , t ) dv � A 2 F 2 ( E r ) � p , DM m DM v 2 µ 2 v min - It crucially depends on ρ DM ( R 0 ) (this talk) and f DM ( � v , t ) . Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 2 / 25
Outline The underlying Galactic Model 1 The experimental constraints 2 The method:Bayesian inference with Markov Chain Monte Carlo 3 Results and Conclusions 4 Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 3 / 25
Outline The underlying Galactic Model 1 The experimental constraints 2 The method:Bayesian inference with Markov Chain Monte Carlo 3 Results and Conclusions 4 Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 3 / 25
Outline The underlying Galactic Model 1 The experimental constraints 2 The method:Bayesian inference with Markov Chain Monte Carlo 3 Results and Conclusions 4 Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 3 / 25
Outline The underlying Galactic Model 1 The experimental constraints 2 The method:Bayesian inference with Markov Chain Monte Carlo 3 Results and Conclusions 4 Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 3 / 25
The underlying Galactic Model Dark Halo Thick Disk Gas + Fluctuations Thin Disk Bulge/Bar Figure: Schematic representation of the Galaxy Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 4 / 25
The underlying Galactic Model Dark Halo Thick Disk Gas + Fluctuations Thin Disk Bulge/Bar Figure: Schematic representation of the assumed Galactic model Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 5 / 25
The underlying Galactic Model - The stellar disk: � z ρ d ( R , z ) = Σ d � e − R with R < R dm Rd sech 2 2 z d z d 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 (GGI) 17/05/2010 6 / 25
The underlying Galactic Model - The stellar bulge/bar: − s 2 � � � � ρ bb ( x , y , z ) = ρ bb ( 0 ) b + s − 1 . 85 exp ( − s a ) exp a 2 where �� x � y � z � 2 � 2 a = q 2 a ( x 2 + y 2 ) + z 2 � 2 � 4 s 2 s 2 b = + + . z 2 x b y b z b b H. Zhao, arXiv:astro-ph/9512064. Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 7 / 25
The underlying Galactic Model - The Dark Matter halo: � R � ρ h ( R ) = ρ ′ f , a h where f is the Dark Matter profile. - M vir , and c vir as halo parameters: ρ ′ = ρ ′ ( M vir , c vir ) a h = a h ( M vir , c vir ) Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 8 / 25
The underlying Galactic Model - The Dark Matter profile: f E ( x ) = exp � α E ( x α E − 1 ) � − 2 J.F. Navarro et al., MNRAS 349 (2004) 1039. A.W. Graham, D. Merritt, B. Moore, J. Diemand and B. Terzic, Astron. J. 132 (2006) 2701. 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 (GGI) 17/05/2010 9 / 25
The underlying Galactic Model Dark matter profiles 1e+07 Einasto NFW 1e+06 Burkert 100000 10000 Profiles [GeV/cm 3 ] 1000 100 10 1 0.1 0.01 0.001 0.0001 0.001 0.01 0.1 1 10 Galactocentric distance [Kpc] Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 10 / 25
Parameter space Galactic components Parameters Disk Σ d R d Disk Bulge/bar ρ bb ( 0 ) Halo α E M vir Halo c vir Halo R 0 All components All components β ⋆ Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 11 / 25
The experimental constraints Constraints: A + B = − ∂ Θ( R 0 ) - Oort’s constants: A − B = Θ 0 R 0 ; ∂ R - terminal velocities - total mean surface density within | z | < 1 . 1 kpc - 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 Sgr A ∗ Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 12 / 25
The experimental constraints Constraints: A + B = − ∂ Θ( R 0 ) - Oort’s constants: A − B = Θ 0 R 0 ; ∂ R - terminal velocities - total mean surface density within | z | < 1 . 1 kpc - 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 Sgr A ∗ Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 12 / 25
The experimental constraints: Radial velocity dispersions -The dataset: population of stars with distances up to ∼ 60 kpc 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 � ∞ 1 r ( r ) = d ˜ r ˜ r 2 β ⋆ − 1 ρ ⋆ (˜ r )Θ 2 (˜ r ) σ 2 r 2 β ⋆ ρ ⋆ ( r ) r - where β ⋆ is the anisotropy parameter: β ⋆ ≡ 1 − σ 2 t /σ 2 r . Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 13 / 25
The method: Bayesian approach Frequentist approach = ⇒ Maximum Likelihood Parametric model of the Galaxy Bayesian approach = ⇒ Posterior probability density - This work → Bayesian approach Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 14 / 25
The method: Bayesian approach - Target: posterior pdf (Bayes’ theorem): p ( η | d ) = L ( d | η ) π ( η ) d = data ; η = parameters ; p ( d ) - 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 (GGI) 17/05/2010 15 / 25
The method: Markov Chain Monte Carlo - Monte Carlo expectation values: N − 1 � d η f ( η ) p ( η | d ) ≈ 1 f ( η ( t ) ) , � f ( η ) � = � N t = 0 where η ( t ) was sampled from p ( η | d ) . - Monte Carlo technics require a method to sample η ( t ) = ⇒ Markov chains. - Markov chains : p ( η ( 0 ) ) ⇒ η ( t ) distributed according to p ( η | d ) . = T ( η ( t ) , η ( t + 1 ) ) Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 16 / 25
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: 0.14 disc central surface density bulge-bar central mass density disc radial scale Sun’s Galactocentric distance 0.12 halo virial mass concentration parameter anisotropy beta parameter Multivariate Scale Reduction Fatcor 0.1 0.08 0.06 0.04 0.02 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Iteration number Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 17 / 25
Figure: Marginal posterior pdf of the Galactic model parameters (NFW profile). Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 18 / 25
Figure: Marginal posterior pdf of the Galactic model parameters (Einasto profile). Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 19 / 25
9.5 9.5 25 9 9 20 8.5 8.5 R 0 [Kpc] R 0 [kpc] c vir 8 8 15 7.5 7.5 10 7 7 6.5 6.5 5 1 2 3 4 5 10 15 20 25 1 2 3 4 M vir [10 12 M Θ ] M vir [10 12 M Θ ] c vir 0 0.2 0.4 0.6 0.8 1 Figure: Two dimensional marginal posterior pdf in the planes spanned by combinations of the Galactic model parameters (NFW profile). Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 20 / 25
9.5 9.5 4 9 9 M vir [10 12 M Θ ] 3 8.5 8.5 R 0 [kpc] R 0 [kpc] 8 8 2 7.5 7.5 1 7 7 6.5 6.5 1 2 3 4 5 10 15 20 25 0.1 0.2 0.3 0.4 M vir [10 12 M Θ ] α E c vir 0.4 25 25 20 20 0.3 c vir c vir α E 15 15 0.2 10 10 5 5 0.1 1 2 3 4 0.1 0.2 0.3 0.4 7 8 9 M vir [10 12 M Θ ] α E R 0 [kpc] 0 0.2 0.4 0.6 0.8 1 Figure: Two dimensional marginal posterior pdf in the planes spanned by combinations of the Galactic model parameters (Einasto profile). Riccardo Catena (ITP) Firenze (GGI) 17/05/2010 21 / 25
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 (GGI) 17/05/2010 22 / 25
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 (GGI) 17/05/2010 23 / 25
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