Efficient calculation of cosmological neutrino clustering

Transcription

1 Efficient calculation of cosmological neutrino clustering MARIA ARCHIDIACONO RWTH AACHEN UNIVERSITY ARXIV: MA, STEEN HANNESTAD COSMOLOGY SEMINAR HELSINKI INSTITUTE OF PHYSICS

2 Cosmic history

3 Cosmic structure SDSS

4 The role of neutrinos Δm2 = ev Δm3 = ev Gonzalez-Garcia et al. (205) The Effect of Thermal Neutrino Motion on the Matter Power Spectrum CDM HDM Σmν = 0.3 ev Σmν = 0.6 ev 5 z=4 z=0 % accuracy! Euclid (2020) Brandbyge et al. (2008) Figure 2. Images of the CDM and neutrino density distributions in a slice of the simulation volume. The images span 52 h Mpc on a side and has a depth of

5 Neutrino free-streaming λ FS (t) 2π 2 3 v th p m 3T ν m v th H(t) 7.7 # ev & 50(+ z) % (km s $ m ' + z Ω Λ + Ω m (+ z) 3 % ' & ev m ( *Mpc h ) " z nr,i 890 m % ν,i $ ' # ev & " m k nr 0.0 Ω ν,i % m $ ' # ev & /2 hmpc k [h/mpc] 0 k fs (m ν = 0.05 ev) k fs (m ν = 0. ev) k fs (m ν = 0.5 ev) k FS (t) 2π a(t) λ FS (t) = " 4πGρ(t)a2 (t)% $ 2 ' # v th & / z

6 Matter power spectrum in the presence of ν P(k, z) = δ m (k, z) 2 δ m = δρ m ρ m ν = ν = k 2 a 2 φ = 4πG(δρ m ) H 2 = 8πG 3 ( ρ γ + ρ b + ρ cdm + ρ ν + ρ Λ ) / Λ = Castorina et al. (205) δ cdm a only cold dark matter [ - ] δ cdm a 3/5 f ν in the presence of ν cdm+hdm=mdm P(k) ΛMDM P(k) ΛCDM 8 f ν

7 Matter power spectrum in the presence of ν P(k, z) = δ m (k, z) 2 δ m = δρ m ρ m k 2 a 2 φ = 4πG(δρ m ) H 2 = 8πG 3 ( ρ γ + ρ b + ρ cdm + ρ ν + ρ Λ ) / Λ Linear perturbation theory = ν = ν = Castorina et al. (205) δ cdm a only cold dark matter [ - ] δ cdm a 3/5 f ν in the presence of ν cdm+hdm=mdm P(k) ΛMDM P(k) ΛCDM 8 f ν

8 Linear perturbation theory f (x i, P j,τ ) = f 0 (q)" # + Ψ(x i, q, n j,τ ) $ % f τ + f x i x i τ + f q q τ + f n i n i τ = 0 Ψ τ + i q ε ( k! ˆn)Ψ + d ln f " 0 d lnq " h η! + 6!η ( k! $ ) ˆn) 2 * = 0 # 2 % Phase-space distribution Collisionless Boltzmann equation Synchronous gauge, Fourier space T µν = δρ = a 4 dp dp 2 dp 3 ( g) P µp /2 ν f (x j, P P 0 j,τ ) εq 2 dq dωf 0 (q)ψ δ p = q 4 3 a 4 dq dωf 0 (q)ψ ε δt i 0 = a 4 i Σ j q 3 dq dωn i f 0 (q)ψ = a 4 q 4. dq dω n i n j ε 3 δ 0 ij 3 f 0 (q)ψ / 2 Energy-momentum tensor Perturbed components: Energy density Pressure Energy flux Shear stress ( ρ + p)θ ik i 0 δt i # ( ρ + p)σ n i n j 3 δ & i % ij (Σ j $ '

9 Neutrino perturbations Ψ( k,! ˆn, q,τ ) = ( i) l (2l +)Ψ l ( k,! q,τ )P l ( k! ˆn) δρ = 4πa 4 l=0 εq 2 dqf 0 (q)ψ 0 δ p = 4π q 4 3 a 4 ε dq f 0 (q)ψ 0 ( ρ + p)θ = 4πka 4 q 3 dqf 0 (q)ψ ( ρ + p)σ = 8π q 4 3 a 4 ε dq f 0 (q)ψ 2!Ψ 0 = k q ε Ψ + 6!Ψ = k q 3ε (Ψ 0 2Ψ 2 )!h d ln f 0 d lnq!ψ 2 = k q 5ε (2Ψ ' 3Ψ 3 )!h ! * ) η, d ln f 0 ( + d lnq!ψ l = k q (2l +)ε (lψ l (l +)Ψ l+ ), l 3 Legendre expansion Energy-momentum conservation Note: Ψ 0 and Ψ are gauge dependent Neutrino free-streaming

10 The moment hierarchy truncation!ψ l = k q (2l +)ε (lψ l (l +)Ψ l+ ), l 3 In the absence of any gravitational source term, or for very high l Ψ l (kτ ) j l (kτ ) m ν =0.05 ev k =0.0 Mpc q CLASS = 2l + Ψ l+ = kτ Ψ Ψ l l l = 2!Ψ 2 = k q 5ε (2Ψ # 3Ψ 3 )!h ! & % η( d ln f 0 $ ' d lnq Ψ 3 5ε kτ Ψ 2 Ψ Note: gauge dependent, while the real one is gauge invariant Ψ Ψ 2 Ψ 3,exact Ψ 3,trunc a Archidiacono & Hannestad, arxiv:

11 A new moment hierarchy truncation In a non-expanding Universe and in the absence of gravity, the Boltzmann hierarchy: α!ψ l = (2l +) (lψ (l +)Ψ ) l l+ with solutions Ψ l j l (ατ ) 0!Φ l = Ansatz: α (2l +) (lφ l (l +)Φ l+ )+ f (τ )(δ l0 +δ l2 ) Φ l = g(τ ) 2l + ατ >> Φ Φ 2 Φ l = ατ 2l + Φ l ατ Φ 3 ατ 7 Φ 2 Ψ 3 # % % % % $ x β 7 x β x + x β x & ( β ( Ψ ( 2, ( ' x = qkτ ε Φ3/τ α = Φ 0 (0) = Φ l 0 (0) = τ # % Ψ 3 % % $ x x + x & () ( k, +. * h / Mpc- ( ' 0.2 Ψ 2 Archidiacono & Hannestad, arxiv:

12 Comparison 00 m ν =0.05 ev k =0.0 Mpc q CLASS =0. 00 m ν =0.05 ev k =Mpc q CLASS = Ψ Ψ 2 Ψ 3,exact Ψ 3,trunc Ψ 3,new a a 00 m ν =0.5 ev k =0.0 Mpc q CLASS =0. 00 m ν =0.5 ev k =Mpc q CLASS = a a Archidiacono & Hannestad, arxiv:

13 Comparison 00 m ν =0.05 ev k =0.0 Mpc q CLASS = 00 m ν =0.05 ev k =Mpc q CLASS = Ψ Ψ 2 Ψ 3,exact Ψ 3,trunc Ψ 3,new a a 00 m ν =0.5 ev k =0.0 Mpc q CLASS = 00 m ν =0.5 ev k =Mpc q CLASS = a a Archidiacono & Hannestad, arxiv:

14 The fluid approximation Lesgourgues & Tram (20) Writing the hierarchy in terms of fluid quantities (δ, θ, σ) Differentiate Same ansatz ( ρ + p)σ = 8π q 4 3 a 4 ε dq f 0 (q)ψ 2 Ψ 3 5ε kτ Ψ 2 Ψ ( σ! = 3 τ +!a " 2 * $ a 3 c 2 g ) # 3 2 c vis = 3wc g 2 c 2 g =!p!ρ = w ( 3(+ w) 5!p + * - ) p,!p = 4π q 6 3 a 4 ε dq f 3 0 (q) 0 "p % + '-σ c vis " p&, 3 + w # 2θ + h! % & CLASS built-in approximation Pros: no momentum integration!

15 Comparison: observables kτ 3 l =7 kτ > 3 approximation or truncation (Ptot,approx Ptot,exact) /Ptot,exact (Pν,approx Pν,exact) /Pν,exact CLASS m ν =0.05 ev -0.8 m ν =0. ev m ν =0.5 ev CLASS (Ptot,approx Ptot,exact) /Ptot,exact (Pν,approx Pν,exact) /Pν,exact Ψ 3,CLASS Ψ 3,CLASS (Pν,approx Pν,exact) /Pν,exact (Ptot,approx Ptot,exact) /Ptot,exact Ψ 3,new Ψ 3,new z=0 z=0 (< % Euclid) (Cl,approx Cl,exact) /Cl,exact CLASS l (Cl,approx Cl,exact) /Cl,exact Ψ 3,CLASS l (Cl,approx Cl,exact) /Cl,exact Ψ 3,new l Archidiacono & Hannestad, arxiv:

16 Comparison: runtime Only one thread, no OpenMPI CLASS (kτ > 3) New Ψ 3 (kτ > 3) New Ψ 3 (kτ > 2) CPUt/ CPUt(l=7) (Ptot,approx Ptot,exact) /Ptot,exact m ν =0.05 ev,kτ =3 m ν =0.05 ev,kτ =6 4 m ν =0.05 ev,kτ =2 m ν =0.5eV,kτ =3 m 3 ν =0.5eV,kτ =6 m ν =0.5eV,kτ = Archidiacono & Hannestad, arxiv:

17 The non-linear regime Ly-α / Λ Linear perturbation theory ν = ν = = Bird, Viel, Haehnelt (204) Castorina et al.(205) [ - ]

18 Non-linear methods Beyond linear order perturbation theory Ø Fuhrer & Wong (204) Ø Blas, Garny, Konstandin, Lesgourgues (204) Ø Dupuy & Bernardeau (204) N-body simulations Ø Hybrid methods: Brandbyge & Hannestad (2009 & 200) Ø Semi-linear methods: Ali-Haimoud & Bird (202) Our approach: using HALOFIT, we account for the non-linear growth of cold dark matter overdensities and gravitational potential, then we evolve linear neutrino perturbations in the non-linear gravitational potential. The entire computation is in Fourier k space.

19 Free-streaming scale vs non-linear scale 0 k nl k fs (m ν = 0.05 ev) k fs (m ν = 0. ev) k fs (m ν = 0.5 ev) k [h/mpc] z Archidiacono & Hannestad, arxiv:

20 Neutrino power spectrum ν m ν =0.05 ev m ν =0. ev m ν =0.5 ev Archidiacono & Hannestad, arxiv:

21 Neutrino power spectrum 4 mν =0.05 ev 0 z = mν =0.05 ev 0 z = 0 2 Pν(k)[(Mpc/h) 3 ] Pν(k)[(Mpc/h) 3 ] mν =0. ev 0 z = mν =0. ev 0 z = 0 2 Pν(k)[(Mpc/h) 3 ] Pν(k)[(Mpc/h) 3 ] mν =0.3 ev 0 z = mν =0.3 ev 0 z = 0 2 Pν(k)[(Mpc/h) 3 ] Pν(k)[(Mpc/h) 3 ] Archidiacono & Hannestad, arxiv:

22 Neutrino power spectrum 0 4 m ν =0. ev z = m ν =0. ev z = 0 2 Pν(k)[(Mpc/h) 3 ] Pν(k)[(Mpc/h) 3 ] Ali-Haimoud & Bird (202) Figure 8. Neutrino component of the matter power spectrum. Solid lines represent the linear predictions, dashed lines show the results of semi-linear approximation. Archidiacono & Hannestad, arxiv:

23 Conclusions We have demonstrated that the neutrino evolution hierarchy can be solved very accurately even if truncated at l = 2. Our approximation for the l = 3 term allowed us to reliably calculate the neutrino power spectrum to better than 5% precision for masses up to.5 ev. The matter power spectrum has a precision of better than 0.5% because of the relatively small direct contribution of neutrinos to this quantity. The new approximation to Ψ 3 is significantly more precise than previously used once. We showed how the neutrino power spectrum can be calculated using the full non-linear gravitational potential, but keeping the entire computation in k- space. The results obtained using this technique are completely consistent with those from N-body simulations implementing neutrinos in Fourier-space. However, in our case the neutrino power spectrum can be obtained in a few seconds whereas the N-body technique requires far bigger computational resources.

24 Backup 0-2 Pν(k)/P tot(k) nl m ν =0.05 ev m ν =0. ev m ν =0.5 ev