Cluster Formation in Lagrangian Perturbation Theory. Lavinia Heisenberg

Transcription

1 Cluster Formation in Lagrangian Perturbation Theory Zentrum fu r Astronomie (ITA/ZAH) Universita t Heidelberg , University of Canterbury, Christchurch

2 outline 1 Introduction 2 LPT 3 ELL 4 PINOCCHIO

3 PINOCCHIO standing for PINpointing Orbit-Crossing Collapsed HIerarchical Objects

4 Structure Formation Universe as made of a uniform smooth background with inhomogeneities study of structure formation=study of the growth of inhomogeneities if the inhomogeneities are small: linear perturbations theory (using Euler) otherwise, techniques for non-linear evolution analytical treatment not possible, numerical simulations (N-body simulation)

5 Why should we study the non-linear regime? to understand the formation of large structures as galaxy clusters Why are they interesting objects? because they are the most massive gravitationally bound systems in the Universe

6 PINOCCHIO instead of N-body sim.: PINOCCHIO (using Lagrangian Perturbation Theory) first developed by Monaco et al.(2001) is an algorithm to study the formation and evolution of dark matter haloes in a given initial linear density field uses Lagrangian perturbative approximation + ellipsoidal collapse density inhomogeneities replaced by the perturbed trajectories about the linear initial displacement

7 Lagrangian system of equations for a gravitating collisionless fluid Euler equation (1) d 2 dt 2 x( q, t) = [ f( q, t) 2H u( q, t)] a(t) (2) continuity equation (3) 1 + δ[ x( q, t), t] = 1 J( q, t) (4) Poisson equation (5) f = 4πGaρ b δ (6) irrotationality (7) x u = 0 (8)

8 Mapping from the Eulerian to the Lagrangian The path of each fluid element is followed during its evolution x( q, τ) q + D( q, τ). (9) D fully characterizes the map between the Eulerian and the Lagrangian coordinates Transformation of the Eulerian Fields in Lagrangian coordinates The Euler equation, the mass conservation relation, the Poisson equation and the Eulerian irrotationality condition expressed by the displacement field D v[ x( q, τ), τ] = d D( q, τ)/dτ (10) 1 + δ[ x( q, τ), τ] = det(i + D) 1 (11) x D = α(τ) δ[ x( q, τ), τ] (12) x D = 0 (13)

9 Lagrangian Perturbation Theory Transformation of the Eulerian Fields in Lagrangian coordinates Replacing the differentiation with respect to the Eulerian position x by the differentiation with respect to the Lagrangian q. the Poisson equation: [ (1 + D) δ αβ D αβ + D C αβ] Dβα = α(τ)[j( q, τ) 1], (14) the Eulerian irrotationality condition: ɛ αβγ [(1 + D) δ βσ D βσ + D C βσ] Ḋγσ = 0, (15)

10 Lagrangian Perturbation Theory Lagrangian Perturbative Approximation expand the trajectory D in a perturbative series D( q, τ) = g 1 (τ) D (1) ( q)+g 2 (τ) D (2) ( q)+g 3 (τ) D (3) ( q)+. (16) the Jacobian determinant as well J( q, τ) = 1 + D αα + 1/2 [D αα D ββ D αβ D βα ] + det(d αβ ) (17) First-Order Solution: The Zel dovich Approximation g 1 (τ) α(τ) g 1 (τ) = 0. (18) with the solution [ ] τ 1 g 1 (τ) = (τ 2 1) 1 + τ ln, (19) τ + 1

11 Lagrangian Perturbation Theory Lagrangian Perturbative Approximation expand the trajectory D in a perturbative series D( q, τ) = g 1 (τ) D (1) ( q)+g 2 (τ) D (2) ( q)+g 3 (τ) D (3) ( q)+. (16) the Jacobian determinant as well J( q, τ) = 1 + D αα + 1/2 [D αα D ββ D αβ D βα ] + det(d αβ ) (17) First-Order Solution: The Zel dovich Approximation g 1 (τ) α(τ) g 1 (τ) = 0. (18) with the solution [ ] τ 1 g 1 (τ) = (τ 2 1) 1 + τ ln, (19) τ + 1

12 Lagrangian Perturbation Theory Lagrangian Perturbative Approximation temporal solutions g 2 = 3 14 g2 1Ω a (20) g 3a = 1 9 g3 1Ω b (21) g 3b = 5 42 g3 1Ω c (22) g 3c = 1 14 g3 1Ω d (23)

13 Lagrangian Perturbations and ellipsoidal Collapse Ellipsoid the potential of a homogeneous ellipsoid in its principal reference frame the corresponding displacements ψ( q) = 1 2 (λ 1q λ 2 q λ 3 q 2 3) (24) D (1) a,b ψ,ab( q). (25) D (2) a,b = ψ,abψ,cc ψ,ac ψ,bc D (3a) a,b = ψ,ac ψ C,bc (26) D (3b) a,b D (3c) a,b = 0 = 1 2 [D(2) ab ψ,cc D (2) bc ψ,ac + ψ,ab D c,c (2) ψ,bc D a,c] (2)

14 Lagrangian Perturbations and ellipsoidal Collapse Ellipsoid collapse time J( q, g c ) = 0. (27) neglecting the Ω dependence of the time functions collapse time is given by the third order algebraic equation 1+λ i g c 3 ( 14 λ i(δ l λ i )gc 2 I ) 84 λ iδ l (δ l λ i ) gc 3 = 0 (28)

15 Collapse equation to solve Collapse Times collapse time g (1) λ 3 = 0 g (1) c = 1 λ λ i g (2) c 3 14 λ i(δ l λ i )(g (2) c ) 2 = 0 g (2) c = 7λ 3+ 7λ 3 (λ 3 +6δ l ) 3λ 3 ( λ 3 +δ l ) λ i g c 3 14 λ i(δ l λ i )gc 2 (3) g c 1 = 2 q cos(θ/3) (δ l /λ i 1)/14c + I λ iδ l (δ l λ i ) gc 3 = 0 gc(3) 2 = 2 q cos((θ + 2π)/3) (δ l /λ i 1)/ (3) g c 3 = 2 q cos((θ + 4π)/3) (δ l /λ i 1)/

16 Introduction LPT PINOCCHIO: Collapse ELL PINOCCHIO (first part) Collapse Time Compute the power spectrum of the initial density fluctuations numerically and therefrom generate the density contrast δl, afterwards the initial displacements. Smoothe the field with a Gaussian window function Calculate the deformation tensor and its eigenvalues Define the collapse time as J(~q, bc ) = 0 using ellipsoidal collapse as a truncation of the Lagrangian series

17 Fragmentation (second part) first case If none of the 6 Lagrangian neighbours have collapsed, then the particle is a local maximum of the inverse collapse time This particle is a seed for a new halo having the unit mass of the particle and been created at the particle s position. Obviously the particle with the first collapse time is the first halo.

18 Fragmentation (second part) first case In the case that the collapsing particle touches only one halo, then the accretion condition, if the halo is close enough, is checked. When the accretion condition is satisfied, then the particle is added to the halo, otherwise it is marked as belonging to a filament. The particles that only touch filaments are marked as filaments as well.

19 Fragmentation (second part) first case In the case of more than one touching, the merging condition has to be checked for all the halo pairs. The pairs that satisfy the conditions are merged together. The accretion condition for the particle is checked for all the touching halos. It accretes onto that halo for which d/r N is the smaller.

20 Fragmentation (second part) first case If the collapsing particle does not accrete onto the candidate halos in tha case they are too far, it becomes a filament. But later for this filament particle there is still the posibility to accrete when its neighbour particle accretes onto a halo. This is done in order to mimic the accretion of filaments onto the halos.

21 The Code

22 Fragmentation Accretion Condition The collapsing particle accretes onto the touching halo, if it is close enough, i.e. the Eulerian comoving distance d between particle and halo is smaller than a fraction of the halo s size R N, d < f a R N + f ra + δd (29) The size R of a halo of N particles is assumed to be R N = N 1/3. Merging Condition Two haloes merge if the distance d between them is smaller than a fraction of the Lagrangian radius of the larger halo: d < f m max(r N1, R N2 ) + f rm (30) This condition reflects the fact that the centre of mass of the smaller halo is within a distance f m R N1 of the centre of mass of the larger halo.

23 Fragmentation Accretion Condition The collapsing particle accretes onto the touching halo, if it is close enough, i.e. the Eulerian comoving distance d between particle and halo is smaller than a fraction of the halo s size R N, d < f a R N + f ra + δd (29) The size R of a halo of N particles is assumed to be R N = N 1/3. Merging Condition Two haloes merge if the distance d between them is smaller than a fraction of the Lagrangian radius of the larger halo: d < f m max(r N1, R N2 ) + f rm (30) This condition reflects the fact that the centre of mass of the smaller halo is within a distance f m R N1 of the centre of mass of the larger halo.

24 Mass function and positions of haloes (first order) mass function (first order) numerical analytical Lagrange positions of haloes 3-dimensional (first order) masses: log dn/dmdv log M [M 0 ] z [Mpc] y [Mpc] x [Mpc]

25 Mass function and positions of haloes (third order) mass function (third order) third order analytical Lagrange positions of haloes 3-dimensional (third order) masses: log dn/dmdv log M [M 0 ] z [Mpc] y [Mpc] x [Mpc]

26 Parameter space mass function (Monaco parameters) mass function (three parameters changed) -16 numerical numerical analytical -16 analytical log dn/dmdv -18 log dn/dmdv log M [M 0 ] log M [M 0 ]

27 Parameter space mass function (parameter f a changing) mass function (parameter f m changing) f a f m f a f a f m f m analytical f a -17 analytical f m analytical f a analytical f m analytical f a analytical f m log dn/dmdv log dn/dmdv log M [M 0 ] log M [M 0 ] mass function (parameter f ra changing) mass function (parameter f rm changing) f ra f rm f ra f ra f rm f rm analytical f ra analytical f ra analytical f ra analytical f rm analytical f rm analytical f rm log dn/dmdv log dn/dmdv log M [M 0 ] log M [M 0 ]

28 please be careful: any correlation between the mass function and the parameter space should be studied very carefully and physically well justified (PS:don t extrapolate to 2010)

29 Merger Tree merging at each merger of two haloes a new halo is created with the identification number (ID) the mass of each halo involved in the merging process is recorded together with the redshift the progenitor haloes are not deleted, instead, for each of them one has to keep track at all times of the parent halo they are presently incorporated within

30 Merger Tree