SLIDE 1
The Large Scale Structure
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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 - - PowerPoint PPT Presentation
Weak Field Newtonian Motion Gauges in collaboration with C Rampf, R Crittenden, K Koyama, T Tram and D Wands Institut fr theoretische Teilchenphysik und Kosmologie Christian Fidler Today The Large Scale Structure Christian Fidler Institut
SLIDE 2
SLIDE 3
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?
τini τ1 τ2 τ3
N-body gauge
τini τ1 τ2 τ3
Newtonian motion gauge
τini τ1 τ2 τ3
N-body simulation
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SLIDE 4
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.
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SLIDE 5
The Newtonian Motion Gauge
ds2 = −a2 (1 + 2A) dη2 − 2a2 ˆ ∇iBdηdx i +a2
- δij (1 + 2HL) + 2
- ˆ
∇i ˆ ∇j + δij 3
- HT
- dx idx j
Gauge Condition
We want Newtonian trajectories: vcdm = vN
➔ A + (∂τ + H) K−2 ˙ HT = −ΦN
The relativistic density is related to the coordinate density via the volume perturbation: ρ = (1 − 3HL)ρN
➔ 4πGa2δρN = K2ΦN
Combined the gauge condition becomes (∂τ + H) ˙ HT = 4πGa2(δργ + 3H(ργ + pγ)K−1(v − K−1 ˙ HT) − ρcdm(3ζ − HT)) + 8πGa2Σ
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SLIDE 6
The Newtonian Motion Gauge
The scheme is self-consistent: All metric perturbations remain small in the weak field sense The evolution of HT 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
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SLIDE 7
The Metric
10−5 10−3 10−1 101
k [Mpc−1]
0.0 0.1 0.2 0.3 0.4
A A(1) ΦN (∂τ + H) ˙ HT
10−5 10−3 10−1 101
k [Mpc−1]
10−8 10−6 10−4 10−2 100
A A(1) ΦN |(∂τ + H) ˙ HT|
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SLIDE 8
Comparison to gevolution
0.001 0.01 0.1 0.96 0.97 0.98 0.99 1.00 z = 0.0
before Nm → Nb
linear prediction
after Nm → Nb
relative power k [h/Mpc]
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SLIDE 9
The ICS Effect
Light Transport on a Non-Trivial Metric
The simulation potential ΦN bends light rays: Lensing Corrections from HT introduce a rotation in the photon direction
➔ The effect is integrated along a trajectory comparable to the ISW ➔ ICS = Integrated coordinate shift Poisson trajectory L
N m P
Nm trajectory
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SLIDE 10
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
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