Weak Field Newtonian Motion Gauges in collaboration with C Rampf, R - PowerPoint PPT Presentation

Weak Field Newtonian Motion Gauges in collaboration with C Rampf, R Crittenden, K Koyama, T Tram and D Wands Institut für theoretische Teilchenphysik und Kosmologie Christian Fidler Today

The Large Scale Structure Christian Fidler Institut für theoretische Teilchenphysik und Kosmologie N-body 1/ 9

The Newtonian Motion Gauge Idea Gauge Freedom of General Relativity The gauge defines the coordinates The gauge specifies the dynamical equations Can we find a gauge that has a Newtonian dynamics? N -body gauge Newtonian motion gauge N -body simulation τ 3 τ 3 τ 3 τ 2 τ 2 τ 2 τ 1 τ 1 τ 1 τ ini τ ini τ ini Christian Fidler Institut für theoretische Teilchenphysik und Kosmologie N-body 2/ 9

The Newtonian Motion Gauge Idea The post Newtonian forces in the N-body gauge act only on large scales Instead of separating pairs of particles, relativistic corrections move them together. This may be used to define a novel gauge, the Newtonian motion gauge. Christian Fidler Institut für theoretische Teilchenphysik und Kosmologie N-body 3/ 9

The Newtonian Motion Gauge − a 2 (1 + 2 A ) d η 2 − 2 a 2 ˆ d s 2 ∇ i B d η d x i = � � ∇ j + δ ij � � ∇ i ˆ ˆ + a 2 d x i d x j δ ij (1 + 2 H L ) + 2 H T 3 Gauge Condition We want Newtonian trajectories: v cdm = v N ➔ A + ( ∂ τ + H ) K − 2 ˙ H T = − Φ N The relativistic density is related to the coordinate density via the volume perturbation: ρ = (1 − 3 H L ) ρ N ➔ 4 π Ga 2 δρ N = K 2 Φ N Combined the gauge condition becomes ( ∂ τ + H ) ˙ H T = 4 π Ga 2 ( δρ γ + 3 H ( ρ γ + p γ ) K − 1 ( v − K − 1 ˙ H T ) − ρ cdm (3 ζ − H T )) + 8 π Ga 2 Σ Christian Fidler Institut für theoretische Teilchenphysik und Kosmologie N-body 4/ 9

The Newtonian Motion Gauge The scheme is self-consistent: All metric perturbations remain small in the weak field sense The evolution of H T decouples from the non-linear matter perturbations and may be solved in SPT The Newtonian motion gauge decouples the full relativistic evolution Into the non-linear but Newtonian collapse of matter ➔ Can be simulated by existing N-body codes And the relativistic but linear analysis of the underlying space-time ➔ Can be implemented in existing linear Boltzmann codes Christian Fidler Institut für theoretische Teilchenphysik und Kosmologie N-body 5/ 9

The Metric 10 0 A 0 . 4 A (1) 10 − 2 Φ N 0 . 3 ( ∂τ + H ) ˙ H T 10 − 4 0 . 2 A 10 − 6 A (1) 0 . 1 Φ N 10 − 8 | ( ∂τ + H ) ˙ H T | 0 . 0 10 − 5 10 − 3 10 − 1 10 1 10 − 5 10 − 3 10 − 1 10 1 [Mpc − 1 ] [Mpc − 1 ] k k Christian Fidler Institut für theoretische Teilchenphysik und Kosmologie N-body 6/ 9

Comparison to gevolution 1.00 0.99 relative power before Nm → Nb 0.98 linear prediction after Nm → Nb 0.97 z = 0.0 0.96 0.001 0.01 0.1 k [ h/ Mpc] Christian Fidler Institut für theoretische Teilchenphysik und Kosmologie N-body 7/ 9

The ICS Effect Light Transport on a Non-Trivial Metric The simulation potential Φ N bends light rays: Lensing Corrections from H T introduce a rotation in the photon direction ➔ The effect is integrated along a trajectory comparable to the ISW ➔ ICS = Integrated coordinate shift Nm trajectory Poisson trajectory L N m P Christian Fidler Institut für theoretische Teilchenphysik und Kosmologie N-body 8/ 9

Conclusions Newtonian motion gauges allow a consistent embedding of Newtonian simulations in general relativity, from the large to the small scales Numerically efficient and simple to use Caution is needed in the interpretation of the data, a Newtonian simulation lives on a NM gauge Thank You For Your Attention Christian Fidler Institut für theoretische Teilchenphysik und Kosmologie N-body 9/ 9