Multi-Scale Initial Conditions Oliver Hahn (KIPAC/Stanford) MULTI - PowerPoint PPT Presentation

Multi-Scale Initial Conditions Oliver Hahn (KIPAC/Stanford) MULTI SCALE Hahn & Abel (2011) INITIAL CONDITIONS

Some pre-to-post-CMB physics: Inflation leads to near scale-invariant primordial density spectrum δ ¯ ⌦ ↵ P prim ( k ) = n s . 1 δ ∝ k n s sub-horizon matter-dom. z=200000 Gets processed by growth on sub- z=20000 6 10 z=2000 and super-horizon scales (GR): z=200 z=20 4 10 P late ( k ) ∝ T 2 ( k ) P prim ( k ) 2 10 T(k) radiation-dom. 0 10 Multi-species fluid of CDM+baryon+photon+neutrino → linear Boltzmann solver − 2 10 (e.g. Ma & Bertschinger 1995) super-horizon − 4 10 − 4 − 2 0 2 10 10 10 10 comoving k [h/Mpc] Multi-scale ICs Oliver Hahn SC comparison workshop 08/2012

Peaks vs. halos Identify the peak (or region) from which an object forms 1:1 mapping corresponding peak patch e.g. cluster halo at z=0 in white noise field We want to increase the resolution locally in this patch... Multi-scale ICs Oliver Hahn SC comparison workshop 08/2012

Disentangling scales...adaptive meshes Gaussian density perturbation field: Region of interest at high resolution galaxy, cluster, first star... Large-scale modes at low resolution environment, sample variance Need to find an algorithm to generate such multi-scale density perturbation fields hard in Fourier space! (cf. Bertschinger 2001, GRAFIC-2) Multi-scale ICs Oliver Hahn SC comparison workshop 08/2012

Thinking in real space... because that’s where the peak patch lives... Remember the generation of a density field with given power spectrum: r ) = F − 1 n o k n s / 2 T ( k ) G (0 , 1) � ( ⇥ These are products in k-space, and thus become convolutions (cf. also Salmon 1996) r ) = F − 1 n o ⇥ F − 1 { G (0 , 1) } k n s / 2 T ( k ) � ( ⇤ = T ( r ) � G (0 , 1) real space TF Gaussian white noise What does it mean? Multi-scale ICs Oliver Hahn SC comparison workshop 08/2012

Real space : the baryon acoustic wave The T(r) kernel for baryons over cosmic time: Stalled wave for z<1000 Propagating wave for z>1000 sound speed drops after recomb sound speed ~ c/3 perturbations grow − 6 − 6 3 x 10 3 x 10 z=2000 z=1000 z=1400 z=700 z=1000 z=500 2.5 2.5 2 2 1.5 1.5 δ (r) 1 δ (r) 1 0.5 0.5 0 0 − 0.5 − 0.5 − 1 − 1 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 comoving radius [Mpc/h] comoving radius [Mpc/h] Convolution superimposes waves and growing modes on noise. Linear regime: no interaction between waves. Multi-scale ICs Oliver Hahn SC comparison workshop 08/2012

Multi-scale convolution picture � ( ~ r ) = T ( r ) ? G (0 , 1) Advantages: •Operating in real space •No inherent periodicity (Sirko 2005) •Easy to deal with finite support •No problems with sharp boundaries Multi-scale convolutions relatively easy to deal with: sample “propagator” at different resolutions a) top grid b) subgrid Ω � Ω 1 p Ω � Ω 2 ,p Ω 2 N 2 N important: need to be locally-mass conserving Multi-scale ICs Oliver Hahn SC comparison workshop 08/2012

DM (N-body) initial conditions Lagrangian perturbation theory relates density perturbations to displacements and velocities x ( t ) = d ˙ x ( t ) = q + L ( q , t ) , d t L ( q , t ) ( at 1st order, displacement field is proportional to gravitational force (Zel’dovich 1970) L ( q ) ∝ r q Φ ( q , t ) need to solve Poisson’s equation ∆ q Φ ∝ δ adaptive multi-grid (Fedorenko 1961, Brandt 1973,1977) can achieve this on nested grids. But uses finite differences ! straightforward to generalize to 2LPT Multi-scale ICs Oliver Hahn SC comparison workshop 08/2012

� ��� ���� � ����� ���� ������������� ������������ ������������������� ������� �� ������ ������� �� ������������� �������������������� �� �� ���� ��� � ��������� �� Fourier space properties of finite differences Order n Laplacian L Gradient G ∂ 2 exact: ∂ x x ⇥ ⇤ ⇥ ⇤ 1 2: 1 � 2 1 � 1 0 1 ⇥ ⇤ ⇥ ⇤ 2 1 1 4: � 1 16 � 30 16 � 1 1 � 8 0 8 � 1 ⇥ ⇤ ⇥ ⇤ 12 12 1 1 6: 2 � 27 270 � 490 270 � 27 2 � 1 9 � 45 0 45 � 9 1 180 60 � k 2 exact: � i k 2: � 2 [ � cos( k ) + 1] � i sin( k ) � i � 1 4: 6 [cos(2 k ) � 16 cos( k ) + 15] 6 [ � sin(2 k ) + 8 sin( k )] � i � 1 6: 90 [ � 2 cos(3 k ) + 27 cos(2 k ) � 270 cos( k ) + 245] 30 [sin(3 k ) � 9 sin(2 k ) + 45 sin( k )] Attenuation of power Need a hybrid Poisson solver. on small scales! ⇧ ⌃ k 2 − G ( n ) i k j � j v ⇥ � j ( k ) = f ( k ) ��� L ( n ) � ������������� ��� Correct displacements/velocities ��� on finest grid. � �� ������ ��� � �� ������ Keep long-range, inter-grid interaction ��� � �� ������ from multi-grid � ��� � ����� �� Bad with CDM Multi-scale ICs Oliver Hahn SC comparison workshop 08/2012

Multi-scale initial conditions (IC errors) 1 level, error in std. dev of the field Grafic-2 2 level, error in std. dev of the field Multi-scale ICs Oliver Hahn SC comparison workshop 08/2012

Resimulating a galaxy cluster... 1 level 2 level degraded, hybrid T(r), hybrid multi-scale, hybrid To test, refine region just around the cluster peak patch See José Oñorbe’s talk for details about errors related to the choice of Lagrangian region and resolution... Multi-scale ICs Oliver Hahn SC comparison workshop 08/2012