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: m22 - m12 ≈ (0.009 eV)2 , |m32 - m12| ≈(0.05 eV)2 Either “normal” or “inverted” hierarchies 1 1 2 2 3 3

Neutrinos masses

• Total mass not (yet) measured by particle

physics experiments, but must be at least ∑mν ≳ 0.06 eV (normal hierarchy) or ∑mν ≳ 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: ∑mν ≲ 0.2 - 0.3 eV

Cosmological neutrinos

• Decouple at T ~ 1 MeV, while ultra-relativistic.
• Keep a relativistic Fermi-Dirac distribution

f(p, z) =

g h3

⇣ exp[

p Tν(z)] + 1

⌘−1 , Tν(z) = (1 + z)Tν(0) Tν(0) = 1.95 K = 1.68 × 10−4 eV

• Become non-relativistic at
• Contribute a fraction of the total DM

znr ≈ 200 P mν 0.3 eV fν = 1 Ωmh2 P mν 94eV ≈ 0.02 P mν 0.3eV

• Affect the background expansion (in particular time
• f matter-radiation equality), hence CMB.

WMAP + H0 + BAO: ∑mν ≲ 0.6 eV Planck +WMAP +SPT+ACT+BAO: ∑mν ≲ 0.23 eV

Cosmological effects

6000 5000 4000 3000 2000 1000 1400 1200 1000 800 600 400 200 2 l(l+1)Cl / 2π (µK)2 l no ν's fν=0 fν=0.1

from Lesgourgues & Pastor (2006)

• Slow down the growth of structure on scales smaller

than the free-streaming scale. kfs ≈ 0.08 (1+z)1/2 (∑mν/0.3 eV)

Cosmological effects

0.2 0.4 0.6 0.8 1 1.2 1 10-1 10-2 10-3 10-4 P(k)fν / P(k)fν=0 k (h/Mpc) k knr

fν = 0.01 fν = 0.1

from Lesgourgues & Pastor (2006)

In linear regime:

• Most LSS probes are sensitive to mildly non-linear

modes (Ly α, galaxy distribution) or to full non- linear evolution (clusters).

• Current constraints: ∑mν ≲ 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.

Cosmological effects

Third Order Solutions For n = 3, the continuity and Euler equations are given by 3˙ a(τ)a2(τ)g3(k, τ)δ3,c(k) + a3(τ)˙ g3(k, τ)δ3,c(k) + ˙ a(τ)a2(τ)h3(k, τ)θ3,c(k) = ˙ a(τ)a2(τ) 1 (2π)6 dq1dq2dq3δD(q1 + q2 + q3 − k)δ1,c(q1)δ1,c(q2)δ1,c(q3) × k · q1 q2

1

g1(q1)g2(q23)F (s)

2

(q2, q3) + k · q12 q2

12

h2(q12)g1(q3)G(s)

2 (q1, q2)

• ≡ ˙

a(τ)a2(τ)A3(k), (B18)

• ¨

a(τ)a2(τ) + 2˙ a2(τ)a(τ)

• h3(k, τ)θ3,c(k) + ˙

a(τ)a2(τ)˙ h3(k, τ)θ3,c(k) + 2 τ ˙ a(τ)a2(τ)h3(k, τ)θ3,c(k) + 6 τ 2 a3(τ)δ3,c(k) − 6 τ 2 k2 k2

J

a3(τ)δ3,c(k) = ˙ a2(τ)a(τ) 1 (2π)6 dq1dq2dq3δD(q1 + q2 + q3 − k)δ1,c(q1)δ1,c(q2)δ1,c(q3) ×

• −k2(q1 · q23)

2q2

1q2 23

g1(q1)h2(q23)G(s)

2 (q2, q3) − k2(q12 · q3)

2q2

12q2 3

h2(q12)g1(q3)G(s)

2 (q1, q2)

− 3 4 k2 k2

J

g1(q1)g2(q23)F (s)

2

(q2, q3) − 3 4 k2 k2

J

g2(q12)g1(q3)F (s)

2

(q1, q2) + 1 2 k2 k2

J

g1(q1)g1(q2)g1(q3)

• ≡ ˙

a2(τ)a(τ)B3(k). (B19) In an EdS universe, a(τ) = τ 2 , we have

Shoji & Komatsu 2009

Nonlinear regime: I) higher-order perturbations

See also Lesgourgues et al 2009

Still, simplifying assumptions for neutrinos (either described with simple pressure term

• r assumed linear)

II) Particle-based simulations

Simulation from Brandyge & Hannestad (2009). ∑mν = 0.6 eV, z=4

CDM Neutrinos

0.01 0.1 1.0 k [h Mpc-1] 10-4 10-3 10-2 10-1 100 101 102 Pν [h-1 Mpc]3

49 24 4

0.01 0.1 1.0 k [h Mpc-1]

49 24 4

0.01 0.1 1.0 k [h Mpc-1]

49 24 4

Particle-based simulations: shot noise

∑mν = 0.3 eV ∑mν = 0.6 eV ∑mν = 1.2 eV

Particle-based simulations

Neutrino power-spectrum from Brandyge & Hannestad (2009).

Shot noise P(k) = 1/n

Characteristic scales

1 2 3 4 5 z 0.01 0.10 1.00 10.00 k (h/Mpc) knl kfs(m = 0.2 eV) kfs(m = 0.1 eV) kfs(m = 0.05 eV)

CDM is linear Neutrinos free-stream

Echelles caractéristiques

k kfs knl Linear regime Free streaming

➡

Neutrinos should be nearly linear at all scales

Characteristic scales

Linear evolution of neutrino perturbations with non-linear CDM :

• Vlasov equation for neutrino distribution function f(⌧, ~

x, ~ q ⌘ am~ v) : @⌧f + ~ q ma · @~

xf ma @~ x · @~ qf = 0

• Linearize around f0(q) then Fourier transform :

@⌧(f) + i~ q · ~ k ma f = ima q (~ q · k)d f0 dq

• Write down explicit integral solution
• Integrate over momenta to get ⌫:

⌫(⌧,~ k) = F[f(⌧i,~ k)] + Z ⌧

⌧i

G(k, ⌧ 0, ⌧)(~ k, ⌧ 0)d⌧ 0, G(k, ⌧ 0 ! ⌧) ! 0, G(⌧ ⌧ 0 ⌧cross(k)) ! 0.

→ We have a prescription for δν given previous φ

• Given φ, update δc with N-body code
• Close the system with Poisson equation:

k2φ = −4πa2(ρcδc + ρνδν)

☛ We have replaced following 109 neutrinos by performing a simple integral

Results

Effect on the total matter power spectrum

fully linear theory particles (Bird et al 2012) this work

Agreement with particle method: at z = 0, 0.2% for ∑mν = 0.3 eV 1% for ∑mν = 0.6 eV 4% for ∑mν = 1.2 eV at z > 1, all agree to better than 1%

Results

Neutrino power spectrum at z = 1, ∑mν = 0.3 eV

ν’s all linear ν’s, this work ν’s all N-body

Limitation

This method does not account for the non-linear clustering of neutrinos in massive clusters at z = 0

CDM linear CDM N-body ν’s all linear ν’s, this work ν’s all N-body

Non-linear neutrino clustering

Non-linear neutrino clustering in massive haloes

• At z = 0, characteristic rhalo ≲ 1 Mpc << Lfs
• vν ≈500 km/s (0.1 eV/mν) << |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 Tν ⇠ > mν Tν,0 1 p H0∆tφ ⇣ 2H0 r0 p |2φ0| ⌘1/2 ⇡ (H0∆tφ)−1/2 mν 0.1 eV ✓ r0 0.5 h−1Mpc ◆1/2 p |φ0| 3000 km/s !1/2

About 94 % of neutrinos have p > T for Fermi- Dirac distribution. ☛ Most remain linear, a small fraction get captured and become very non-linear

10 100 1000 10000 r [ h-1 kpc ] 1 10 δν + 1

1015 1014 1013 1012

Non-linear neutrino clustering in massive haloes

Brandbyge et al. 2010 ∑mν = 0.3 eV

10 100 1000 r [ h-1 kpc ] 100 101 102 103 104 105 106 δm + 1

1015 1014 1013 1012

Still have < δν2 > << 1 on all scales, because haloes make a small fraction of the total volume.

ν’s all linear ν’s, this work ν’s all N-body

• Total power spectrum is not very sensitive to exact

clustering of neutrinos on small scales. In practice, δν(k>>kfs) << δCDM 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 ∑mν ≲ 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.

Conclusions