Topical Lectures on DM@Nikhef 13 December 2017 Dark matter distribution Large and small scale structure Shin’ichiro Ando GRAPPA, University of Amsterdam
Evidence for dark matter: Rotation curves v 2 = GM ( < r ) r • Inferring enclosed mass for luminous matter (stars assuming reasonable mass-to-light ratio) significantly under- predicts rotation-curve data • Implication : “Dark” matter exists (but it doesn’t exclude not-so-bright stars or black holes) Bergström, Rep. Prog. Phys. 63 , 793 (2000)
Evidence for dark matter: Bullet clusters • Red: X-ray image Bullet cluster (1E0657-56) (baryonic gas) • Blue: Weak gravitational lensing (dark matter) • Gas is collisional (Coulomb force) so feels drag from each other; dark matter goes through • Implication : Dark matter is collionless σ m . 1 cm 2 / g = 2 barn / GeV
Evidence for dark matter: Baryon acoustic oscillation (BAO) Couples to baryons 6000 Planck 2015 5000 A ff ected by both baryons 4000 [ µ K 2 ] and dark matter 3000 D TT � 2000 1000 • Measured both in CMB and 0 galaxy power spectrum 600 60 300 30 ∆ D TT � 0 0 • Implication : Dark matter can -30 -300 -60 -600 not be made of baryons 2 10 30 500 1000 1500 2000 2500 �
Result from all the cosmology data • CMB, galaxy power spectrum, Planck 2015 weak lensing, supernova Ia, etc. • 27% of the total energy / 85% of the total matter is made of dark matter • Properties of dark matter • Collisionless • Non-baryonic • Doesn’t interact with photons • Cold (or warm; hot dark matter erases too many structures)
Dark matter: Origin of all the structures Not yet observed Observed clusters of micro-halos dwarf galaxies galaxies -10 -6 -2 2 6 8 12 16 Free-streaming scale of WIMP galaxies log( M/M � )
Dark matter: Origin of all the structures • How do dark matter structures form? — Spherical collapse model Today • What is abundance, mass distribution, etc.? — Halo mass Today function • Impact on dark matter annihilation in cosmological halos — Tomorrow Indirect dark matter searches • Implications for properties of dark matter particles — Cold, warm, self-interacting? Tomorrow
Spherical collapse model • Deriving two magic numbers analytically • Over-density of virialized halos compared with background Useful for simulations ∆ vir = 18 π 2 to find halos • Linear extrapolation of over-density for halos that just collapsed Useful for analytic δ c = 1 . 686 calculations to estimate number of halos
Spherical collapse model ρ b θ = π ρ h R θ = 2 π θ = 0 A 3 Parameterized solution t = ( θ − sin θ ) (cf., expanding closed Universe) √ GM R = A 2 (1 − cos θ )
Spherical collapse model When do halos virialize? Virialization! Virial theorem: 2 K vir + U vir = 0 (for 1/ R potential) Total energy conservation: K vir + U vir = U ta U vir = 2 U ta R vir = R ta 2
Spherical collapse model A 3 t = ( θ − sin θ ) √ R vir = R ta GM 2 R = A 2 (1 − cos θ ) How dense is a virialized halo compared with background? 3 M = 6 M 3 π ( t col = 2 t ta ) ρ vir = = 4 π R 3 π R 3 Gt 2 ta vir col 1 ρ b ( t col ) = 6 π Gt 2 ρ vir col ρ b ( t col ) = 18 π 2
Spherical collapse model Halos are defined as regions with density larger than 18 π 2 compared with average The Millennium Simulation@MPA Garching
Spherical collapse model • Deriving two magic numbers analytically ✔ • Over-density of virialized halos compared with background Useful for simulations ∆ vir = 18 π 2 to find halos • Linear extrapolation of over-density for halos that just collapsed Useful for analytic δ c = 1 . 686 calculations to estimate number of halos
Spherical collapse model • Deriving two magic numbers analytically ✔ • Over-density of virialized halos compared with background ∆ vir = 18 π 2 + 82[ Ω m ( z ) − 1] − 39[ Ω m ( z ) − 1] 2 Λ CDM: Bryan & Norman (1998) • Linear extrapolation of over-density for halos that just collapsed Useful for analytic δ c = 1 . 686 calculations to estimate number of halos
Analytic model of halo mass function • It is not possible to describe non-linear evolution of density fully analytically • However, one can extrapolate behavior in linear regime (that can be solved analytically), as if it continues • What does this 18 π 2 collapsed region correspond to, in terms of linear over-density, δ L ? • One can estimate the number of halos of given mass (i.e., halo mass function), by using this threshold δ c and by assuming density distribution is Gaussian (excellent approximation for CMB, hence must be true with linear extrapolation)
Over-density: Linear extrapolation A 3 t = ( θ − sin θ ) √ GM R = A 2 (1 − cos θ ) Exact solution: ρ h = 3 M 4 π R 3 = 3 M 1 4 π A 6 (1 − cos θ ) 3 1 1 M ρ b = 6 π Gt 2 = 6 π A 6 ( θ − sin θ ) 2 − 1 = 9( θ − sin θ ) 2 δ = ρ h 2(1 − cos θ ) 3 − 1 ρ b
Over-density: Linear extrapolation A 3 t = ( θ − sin θ ) A 3 √ GM θ 3 − → t ≈ √ GM 6 θ ⌧ 1 R = A 2 (1 − cos θ ) Linear extrapolation: ! 2 / 3 √ δ = 9( θ − sin θ ) 2 2(1 − cos θ ) 3 − 1 ≈ 3 GM δ L ≈ 3 6 20 θ 2 t A 3 20 At collapse: ( θ = 2 π ) t col = 2 π A 3 δ L = 3 20(12 π ) 2 / 3 ≈ 1 . 686 √ GM
Over-density: Linear extrapolation Exact A 3 t = ( θ − sin θ ) √ Exact GM δ = 9( θ − sin θ ) 2 Linear 2(1 − cos θ ) 3 − 1 Linear ! 2 / 3 √ GM δ L ≈ 3 6 t A 3 20
Gaussian random field Density field smeared rms over-density: over R , given by d 3 k Z σ 2 ( M ) = h δ 2 (2 π ) 3 P lin ( k ) W 2 R i = R ( k ) M = 4 π ρ R 3 3 ¯ Redshift evolution: σ ( M, z ) = σ ( M ) D ( z )
Gaussian random field: simple example Mean = 0 SD = 5 Pixel size = (0.23 deg) 2
Gaussian random field: simple example Mean = 0 Gaussian smoothing: σ = 0.5 deg SD = 5 Pixel size = (0.23 deg) 2
Gaussian random field: simple example Mean = 0 Gaussian smoothing: σ = 2 deg SD = 5 Pixel size = (0.23 deg) 2
Gaussian random field: simple example Raw map 0.5 deg smoothing 2 deg smoothing δ = 1 . 686 Smaller structures form first and then merge and accrete to form larger structures
Press-Schechter mass function Fraction of collapsed halos Z ∞ d δ P ( δ | M, z ) δ c Press-Schechter mass function [ ν ≡ δ c / σ ( M, z )] ◆ d ln σ − 1 r − ν 2 ✓ dn 2 ¯ ρ d ln M = M ν exp 2 d ln M π
Comparison with numerical simulations Springel et al., Nature 435 , 629 (2005) • Reasonable agreement 10 -1 with the Millennium simulations ( Red points) z = 0.00 z = 1.50 • Blue : Press- 10 -2 z = 3.06 Schechter mass function M 2 / ρ d n /d M z = 5.72 10 -3 • Black : Jenkins et al. (2001) mass function z = 10.07 10 -4 • Other representative models include Sheth & 10 -5 Tormen (2001), Tinker et al. (2008), many of which are based on ellipsoidal 10 10 10 11 10 12 10 13 10 14 10 15 10 16 M [ h -1 M O collapse model • ]
Spherical collapse model • Deriving two magic numbers analytically ✔ • Over-density of virialized halos compared with background Useful for simulations ∆ vir = 18 π 2 to find halos ✔ • Linear extrapolation of over-density for halos that just collapsed Useful for analytic δ c = 1 . 686 calculations to estimate number of halos
What is the smallest structure? E.g., 1d random walk followed • In the WIMP scenario: by free-streaming • After chemical decoupling , WIMPs can still interact with baryons and leptons through scattering • When this gets slower than Hubble expansion ( kinetic decoupling ), WIMPs start free-streaming • All the structures below this kd+free-streaming scale will be washed away • Finding small halos is key to distinguish di ff erent dark matter models
What is the smallest structure? Diamanti, Cabrera-Catalan, Ando, Phys. Rev. D 92 , 065029 (2015) • MCMC parameter scan for 9- parameter MSSM • Typical kinetic decoupling temperature: a few MeV • Typical smallest halo mass: 10 � 12 − 10 � 4 M �
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