Non-linear structure formation with massive neutrinos Yacine - PowerPoint PPT Presentation

Non-linear structure formation with massive neutrinos Yacine Ali-Haïmoud and Simeon Bird (IAS) MNRAS, 2013 Séminaire IHES, 10 octobre 2013

Neutrinos masses From oscillation experiments: m 22 - m 12 ≈ (0.009 eV) 2 , |m 32 - m 12 | ≈ (0.05 eV) 2 Either “normal” or “inverted” hierarchies 3 2 1 2 1 3

Neutrinos masses • Total mass not (yet) measured by particle physics experiments, but must be at least ∑ m ν ≳ 0.06 eV (normal hierarchy) or ∑ m ν ≳ 0.1 eV (inverted hierarchy) • Cosmological observations mostly probe the total mass. If sensitive enough can eventually lead to the absolute neutrino masses. Current constraint: ∑ m ν ≲ 0.2 - 0.3 eV

Cosmological neutrinos • Decouple at T ~ 1 MeV, while ultra-relativistic. • Keep a relativistic Fermi-Dirac distribution ⌘ − 1 ⇣ g p f ( p, z ) = exp[ T ν ( z ) ] + 1 T ν ( z ) = (1 + z ) T ν (0) , h 3 T ν (0) = 1 . 95 K = 1 . 68 × 10 − 4 eV P m ν • Become non-relativistic at z nr ≈ 200 0 . 3 eV • Contribute a fraction of the total DM P m ν P m ν 1 f ν = 94eV ≈ 0 . 02 Ω m h 2 0 . 3eV

Cosmological effects • Affect the background expansion (in particular time of matter-radiation equality), hence CMB. WMAP + H0 + BAO: ∑ m ν ≲ 0.6 eV Planck +WMAP +SPT+ACT+BAO: ∑ m ν ≲ 0.23 eV no ν 's 6000 f ν =0 f ν =0.1 5000 l(l+1)C l / 2 π ( µ K) 2 4000 3000 2000 1000 from Lesgourgues & Pastor (2006) 0 2 200 400 600 800 1000 1200 1400 l

Cosmological effects • Slow down the growth of structure on scales smaller than the free-streaming scale. k fs ≈ 0.08 (1+z) 1/2 ( ∑ m ν /0.3 eV) 1.2 k nr k 1 f ν = 0.01 In linear 0.8 P(k) f ν / P(k) f ν =0 regime: 0.6 0.4 f ν = 0.1 0.2 from Lesgourgues & Pastor (2006) 0 1 10 -4 10 -3 10 -2 10 -1 k (h/Mpc)

Cosmological effects • Most LSS probes are sensitive to mildly non-linear modes (Ly α , galaxy distribution) or to full non- linear evolution (clusters). • Current constraints: ∑ m ν ≲ 0.2-0.3 eV. Could get much better in future, provided we model their effect accurately enough. • Neutrinos are “simple” (gravity only!), so we should be able to model their effect very precisely.

Nonlinear regime: I) higher-order perturbations Third Order Solutions Shoji & Komatsu 2009 For n = 3, the continuity and Euler equations are given by a ( τ ) a 2 ( τ ) g 3 ( k , τ ) δ 3 ,c ( k ) + a 3 ( τ )˙ a ( τ ) a 2 ( τ ) h 3 ( k , τ ) θ 3 ,c ( k ) 3˙ g 3 ( k , τ ) δ 3 ,c ( k ) + ˙ 1 � � � a ( τ ) a 2 ( τ ) = ˙ d q 1 d q 2 d q 3 δ D ( q 1 + q 2 + q 3 − k ) δ 1 ,c ( q 1 ) δ 1 ,c ( q 2 ) δ 1 ,c ( q 3 ) (2 π ) 6 � k · q 1 ( q 2 , q 3 ) + k · q 12 � g 1 ( q 1 ) g 2 ( q 23 ) F ( s ) h 2 ( q 12 ) g 1 ( q 3 ) G ( s ) 2 ( q 1 , q 2 ) × 2 q 2 q 2 1 12 a ( τ ) a 2 ( τ ) A 3 ( k ) , ≡ ˙ (B18) h 3 ( k , τ ) θ 3 ,c ( k ) + 2 a ( τ ) a 2 ( τ )˙ a ( τ ) a 2 ( τ ) + 2˙ a 2 ( τ ) a ( τ ) a ( τ ) a 2 ( τ ) h 3 ( k , τ ) θ 3 ,c ( k ) � � ¨ h 3 ( k , τ ) θ 3 ,c ( k ) + ˙ τ ˙ k 2 + 6 τ 2 a 3 ( τ ) δ 3 ,c ( k ) − 6 a 3 ( τ ) δ 3 ,c ( k ) τ 2 k 2 J 1 � � � a 2 ( τ ) a ( τ ) = ˙ d q 1 d q 2 d q 3 δ D ( q 1 + q 2 + q 3 − k ) δ 1 ,c ( q 1 ) δ 1 ,c ( q 2 ) δ 1 ,c ( q 3 ) (2 π ) 6 − k 2 ( q 1 · q 23 ) 2 ( q 2 , q 3 ) − k 2 ( q 12 · q 3 ) � g 1 ( q 1 ) h 2 ( q 23 ) G ( s ) h 2 ( q 12 ) g 1 ( q 3 ) G ( s ) 2 ( q 1 , q 2 ) × 2 q 2 1 q 2 2 q 2 12 q 2 23 3 k 2 k 2 k 2 � − 3 ( q 2 , q 3 ) − 3 ( q 1 , q 2 ) + 1 g 1 ( q 1 ) g 2 ( q 23 ) F ( s ) g 2 ( q 12 ) g 1 ( q 3 ) F ( s ) g 1 ( q 1 ) g 1 ( q 2 ) g 1 ( q 3 ) 2 2 k 2 k 2 k 2 4 4 2 J J J a 2 ( τ ) a ( τ ) B 3 ( k ) . ≡ ˙ (B19) In an EdS universe, a ( τ ) = τ 2 , we have See also Lesgourgues et al 2009 Still, simplifying assumptions for neutrinos (either described with simple pressure term or assumed linear)

II) Particle-based simulations CDM Neutrinos Simulation from Brandyge & Hannestad (2009). ∑ m ν = 0.6 eV, z=4

Particle-based simulations: shot noise Particle-based simulations Neutrino power-spectrum from Brandyge & Hannestad (2009). 10 2 49 49 49 24 24 24 10 1 4 4 4 0 0 0 10 0 ∑ m ν = 0.3 eV ∑ m ν = 0.6 eV ∑ m ν = 1.2 eV P ν [ h -1 Mpc] 3 10 -1 10 -2 10 -3 10 -4 0.01 0.1 1.0 0.01 0.1 1.0 0.01 0.1 1.0 k [ h Mpc -1 ] k [ h Mpc -1 ] k [ h Mpc -1 ] Shot noise P(k) = 1/n

Characteristic scales 10.00 k nl k fs (m � = 0.2 eV) k fs (m � = 0.1 eV) k fs (m � = 0.05 eV) CDM is linear 1.00 k (h/Mpc) Neutrinos free-stream 0.10 0.01 0 1 2 3 4 5 z

Echelles caractéristiques Characteristic scales Linear regime k nl k k fs Free streaming ➡ Neutrinos should be nearly linear at all scales

Linear evolution of neutrino perturbations with non-linear CDM : • Vlasov equation for neutrino distribution function f ( ⌧ , ~ v ) : x, ~ q ⌘ am ~ ~ q @ ⌧ f + q f = 0 ma · @ ~ x f � ma @ ~ x � · @ ~ • Linearize around f 0 ( q ) then Fourier transform : q · ~ @ ⌧ ( � f ) + i ~ ma � f = ima k q · k ) d f 0 q ( ~ dq � • Write down explicit integral solution • Integrate over momenta to get � ⌫ : Z ⌧ � ⌫ ( ⌧ , ~ k ) = F [ � f ( ⌧ i , ~ G ( k, ⌧ 0 , ⌧ ) � ( ~ k, ⌧ 0 ) d ⌧ 0 , k )] + ⌧ i G ( k, ⌧ 0 ! ⌧ ) ! 0 , G ( ⌧ � ⌧ 0 � ⌧ cross ( k )) ! 0 .

→ We have a prescription for δ ν given previous φ • Given φ , update δ c with N-body code • Close the system with Poisson equation: k 2 φ = − 4 π a 2 ( ρ c δ c + ρ ν δ ν ) ☛ We have replaced following 10 9 neutrinos by performing a simple integral

Results Effect on the total matter power spectrum Agreement with particle method: at z = 0, 0.2% for ∑ m ν = 0.3 eV 1% for ∑ m ν = 0.6 eV 4% for ∑ m ν = 1.2 eV fully linear theory particles (Bird et al 2012) at z > 1, all agree to this work better than 1%

Results Neutrino power spectrum at z = 1, ∑ m ν = 0.3 eV ν ’s all N-body ν ’s, this work ν ’s all linear

Limitation This method does not account for the non-linear clustering of neutrinos in massive clusters at z = 0 CDM N-body CDM linear Non-linear neutrino ν ’s all N-body clustering ν ’s, this work ν ’s all linear

Non-linear neutrino clustering in massive haloes • At z = 0, characteristic r halo ≲ 1 Mpc << L fs • v ν ≈ 500 km/s (0.1 eV/m ν ) << | ϕ | 1/2 ~800-3000 km/s

� � � Non-linear neutrino clustering in massive haloes If halo grows on ~ Hubble timescale, neutrinos may be captured. Escape condition: ⇡ p > m ν 1 ⌘ 1 / 2 ⇣ p 2 H 0 r 0 | 2 φ 0 | T ν ⇠ T ν , 0 p H 0 ∆ t φ ! 1 / 2 ◆ 1 / 2 p ✓ | φ 0 | m ν r 0 ⇡ ( H 0 ∆ t φ ) − 1 / 2 0 . 1 eV 0 . 5 h − 1 Mpc 3000 km / s About 94 % of neutrinos have p > T for Fermi- Dirac distribution. ☛ Most remain linear, a small fraction get captured and become very non-linear

Non-linear neutrino clustering in massive haloes ∑ m ν = 0.3 eV 10 6 10 15 10 15 10 5 10 14 10 14 10 13 10 13 10 4 10 12 10 12 10 δ m + 1 δ ν + 1 10 3 10 2 10 1 10 0 1 10 100 1000 10 100 1000 10000 r [ h -1 kpc ] r [ h -1 kpc ] Brandbyge et al. 2010

Still have < δ ν 2 > << 1 on all scales, because haloes make a small fraction of the total volume. ν ’s all N-body ν ’s, this work ν ’s all linear

Conclusions • Total power spectrum is not very sensitive to exact clustering of neutrinos on small scales. In practice, δ ν (k>>k fs ) << δ CDM is what really counts. May as well use a simple method! • It is accurate to better than 0.2% for the matter power spectrum at z = 0 for ∑ m ν ≲ 0.3 eV and nearly exact at z > 1. • Our method is about 20% faster than particle-based method. Patch for GADGET publicly available (S. Bird webpage) • Future work: accurately modeling the clustering of neutrinos themselves / use hybrid methods.