Dwarf spheroidal galaxies as degenerate gas of free fermions arxiv[hep-ph] 1409.3167 Valerie Domcke SISSA, INFN Trieste in collaboration with ~ Alfredo Urbano Hunting the invisible WIMPs Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 2 Hunting the invisible This talk: non-interacting fermion degenerate quantum gas WIMPs $ properties of dwarf galaxies Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 2 Outline quantum mechanics and dwarf galaxies a first estimate quantitative analysis testing against data velocity dispersion escape velocity free-streaming length finite temperature and larger galaxies conclusion and outlook Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 3 N 1=3h=R [Pauli] h2ρ5=3 m8=3 GM 2 R4 h2 R ∼ GM 1=3m8=3 m ∼ 20 eV fermion w. m ∼ eV − keV: DM halo stabilized by quantum pressure? Quantum pressure versus gravity A first estimate: degenerate spherical configuration of free particles bosons: fermions: p ∼ h=R [Heisenberg] h2ρ P ∼ nvp ∼ Q m2R2 GM 2 P ∼ G R4 h2 P = P ) R ∼ Q G GMm2 ) m ∼ 10−25 eV |||||||| see also [Sin '94, Harko et al '07/'11, Chavanis '12, Rindler-Daller and Shapiro '12/'14, Davidson '13/'14, . ] (bosons) and [Vega et al '13/'14] (fermions) Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 4 Quantum pressure versus gravity A first estimate: degenerate spherical configuration of free particles bosons: fermions: 1=3 p ∼ h=R [Heisenberg] N h=R [Pauli] h2ρ h2ρ5=3 P ∼ nvp ∼ Q m2R2 m8=3 GM 2 GM 2 P ∼ G R4 R4 h2 h2 P = P ) R ∼ R ∼ Q G GMm2 GM 1=3m8=3 ) m ∼ 10−25 eV m ∼ 20 eV fermion w. m ∼ eV − keV: DM halo stabilized by quantum pressure? |||||||| see also [Sin '94, Harko et al '07/'11, Chavanis '12, Rindler-Daller and Shapiro '12/'14, Davidson '13/'14, . ] (bosons) and [Vega et al '13/'14] (fermions) Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 4 In a bit more detail Assume degenerate configuration (consistency check later): Z pF 8πp3 ( F n = 2 dΠp = 3 1 p ≤ pF 0 3h fFD = ; Z 0 p > pF PQ = p v(p) n(p) dp h2 3 2=3 ) P = ρ5=3 Q 8=3 8π 5mf hydrostatic equilibrium: dP (r) GM Q = − ρ(r) dr r2 density profile ρ(r) depends on two parameters: mf and ρ0 Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 5 Density profile 10-26 for D 3 - 10-27 mf = 200 eV cm kg -28 ρ(r) ! @ 10 M(r); R; σ(r)::: L r H test vs data r 10-29 `cored' profile 10-30 0.0 0.2 0.4 0.6 0.8 1.0 r @kpcD universal cored profile ! can address cusp/core problem Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 6 Testing the against data: velocity dispersion velocity dispersion along the line of sight for the eight bright dwarf galaxies of the Milky Way: 14 m f = 150 eV HdashedL D 12 Carina 1 - m f = 200 eV HsolidL Draco s 10 Fornax æ km 8 æ @ æ æ æ æ æ æ Leo I æ æ æ æ æ æ æ 6 æ æ æ æ Leo II æ æ æ æ æ Sculptor LOS 4 ææ æ Sextans s 2 Ursa Minor Carina 0 0 200 400 600 800 1000 r @pcD data from [Walker et al '10]. Theory input: M(r) + assumptions on distribution of visible matter: Plummer profile for stellar densities, marginalized over the orbital anisotropy of the stellar component and ρ0. mf = 150 − 200 eV provides a good fit to the data Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 7 Escape velocity: a lower bound on the DM mass Fermi velocity is bounded by escape velocity v1: [Tremaine, Gunn '79] " 1=4#1=3 12 ρ0 eV obs vF ∼ 8:2 × 10 3 km/s ≤ v1 (σ) kg/cm mf ) mf ≥ 200 eV (Leo II) q 2GM −4=3 obs Note: v1 = R / mf ! for degenerate configuration use v1 , weaker bound on mf than `usual' Tremaine-Gunn bound! ) DM mass fixed to mf ' 200 eV Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 8 Free-streaming length: cosmological history simplest setup: inflaton decay φ ! f f 1013 structure formation: l < 0.1 0.3 0.5 1 free-streaming length FS 1012 R v ! λFS = a dt < 0:5 Mpc D (suppressed vs case of GeV 11 thermal relic) @ 10 RH T 2 WDMh = 0.12 total abundance: ! 1010 ΩDM = 0:12 −5 mφ −3 ) Br < 10 , T < 10 RH 109 106 107 108 109 mf GeV correct small scale structure for suitable mφ=TRH @ D Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 9 Outlook: finite temperature and larger galaxies estimate galaxy temperature with Virial theorem ) TVir=Tcrit increases for larger galaxies −1 " p2 !# +mf Φ−µ 2mf ) redo analysis for finite T : fFD = 1 + exp kB T 102 LSG æ degeneracy parameter -3 D cm @kg mf Φ−µ -28 k = exp : 10 k T = MSG B r 0 -27.5 10 æ k ! 0: degenerate FD 27 D - 10 26.5 k ! 1: Maxwell-Boltzmann kpc - 7 @ 1 k = 10 10 10 H SSG R æ Larger galaxies might corre- k = 1 spond to non-generate con- 100 figurations 109 1010 1011 1012 1013 MH @MD data from [Destri et al '13] Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 10 Conclusion and outlook A degenerate gas of fermions with mf ' 200 eV yields a good fit to the dwarf galaxy data velocity dispersion escape velocity free-streaming length Larger galaxies might be obtained in finite temperature limit Can address cusp/core and missing satellite problem Open questions: larger galaxies? structure formation? Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 11 backup slides Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 12 Mass-radius relation 1012 m f = 11 25 ρ0 eliminated 10 galaxies m eV f = m Large D 10 dwarf galaxies: f 50 10 = eV M 100 DM dominated @ eV compact objects 109 m M f m = 150 f = larger galaxies: 8 200 10 galaxies eV finite T (later) eV Dwarf see also [Vega, Sanchez '13] 107 10-1 100 101 102 R @kpcD focus on dwarf galaxies and mf ∼ 150 − 200 eV Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 13 +81 2 Observations: surface density Σ0 ≡ RH ρ0 = 141−52M =pc independent of galaxy luminosity [Donato et al '09] Predictions from non-degenerate Fermi model: 2 Σ0 ' 238 M =pc (SSG), 2 Σ0 ' 280 M =pc (MSG), 2 Σ0 ' 275 M =pc (LSG). ) given the precision of RH , this looks promising Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 14 Fit to the velocity dispersion projected velocity dispersion along the line-of-sight: see e.g. [Walker et al '10] Z 1 2 2G 0 0 0 2β−2 0 0 σLOS(R) = ν(r )M(r )(r ) F (β; R; r )dr ; I(R) R where 0 Z r R2 r−2β+1 F (β; R; r0) ≡ 1 − β p dr : 2 2 2 R r r − R We adopt the Plummer profile for the projected stellar density L 1 I(R) = 2 2 2 ; πrhalf [1 + (R=rhalf ) ] where L is the total luminosity and rhalf the half-light radius. ! 3-dimensional stellar density 1 Z 1 dI dR 3L 1 ν(r) = − p = : 2 2 3 2 5=2 π r dr R − r 4πrhalf [1 + (r=rhalf ) ] Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 15 A criterion for degeneracy A rough estimate: dwarf galaxies larger galaxies 2 pF −3 −4 critical temperature: Tc ∼ TF = ∼ 10 K ∼ 10 K 2mf kB GMmf −3 −1 Virial theorem: TVir ∼ ∼ 10 K ∼ 10 K RkB −1 " p2 !# +mf Φ−µ 2mf ) redo analysis for finite T : fFD = 1 + exp kB T Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 16 Mean density within half-light radius 100 Data from arXiv:0906.0341 ø Carina ç ç ÷ Draco D 3 ÷ ôç - ìçç ò Fornax pc § 10-1 ç ô Leo I M @ Leo II \ ø r X òç ì Sculptor è Sextans ç è § Ursa Minor 10-2 102 103 rhalf @pcD Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 17 Escape velocity obs D -26 3 10 vF > v¥ - cm ) DM mass fixed kg @ to mf ' 200 eV 0 -27 r 10 Carina 130 140 150 160 170 180 190 200 m f @eVD Dwarf spheroidal galaxies as degenerate gas of free fermions| Valerie Domcke | SISSA | 22.06.2015 | Page 18.