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 Ρ 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 Σ 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 v∞: [Tremaine, Gunn ’79]
" 1/4#1/3 12 ρ0 eV obs vF ∼ 8.2 × 10 3 km/s ≤ v∞ (σ) kg/cm mf
⇒ mf ≥ 200 eV (Leo II)
q 2GM −4/3 obs Note: v∞ = R ∝ mf → for degenerate configuration use v∞ , 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: Λ < 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
mΦ 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 Ρ 0 -27.5 10 æ k → 0: degenerate FD 27 D - 10 26.5 k → ∞: 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 ∞ 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 √ 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 ∞ dI dR 3L 1 ν(r) = − √ = . 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 Ρ 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
Ρ 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