Cosmology Adamek

1. General Relativistic N-body Simulations
of Cosmic Large-Scale Structure
Julian Adamek
General Relativistic effects in cosmological
large-scale structure, Sexten, 19. July 2018

2. Gravity
The Newtonian limit
• conceptually simple
• describes the dynamics of nonrelativistic
matter under its own gravity
• works well in ΛCDM
Isaac Newton
Julian Adamek Queen Mary University of London 1 / 10

3. Gravity
The Newtonian limit
• conceptually simple
• describes the dynamics of nonrelativistic
matter under its own gravity
• works well in ΛCDM
Isaac Newton
Albert Einstein
The limit of Newtonian
• effects on light propagation (lensing,
Shapiro delay. . . )
• distortion of geometry (e.g. volume
deformation)
• gravitational fields of relativistic sources
Julian Adamek Queen Mary University of London 1 / 10

4. A Brief Overview of gevolution
gevolution, a general relativistic N-body code
Adamek, Daverio, Durrer & Kunz, Nature Phys. 12 (2016) 346–349
spin-1 metric perturbation
with gevolution
• based on weak-field expansion (in
Poisson gauge)
• for any given Tµ
ν computes the six
metric d.o.f. (Φ, Ψ, Bi, hij)
• N-body particle ensemble evolved using
relativistic geodesic equation
https://github.com/gevolution-code/gevolution-1.1.git
Julian Adamek Queen Mary University of London 2 / 10

5. Strategy
• choose ansatz for the metric (perturbed FLRW)
ds2 =a2(τ)
−e2Ψdτ2+ e−2Φδijdxidxj + hijdxidxj − 2Bidxidτ
Julian Adamek Queen Mary University of London 3 / 10

6. Strategy
• choose ansatz for the metric (perturbed FLRW)
ds2 =a2(τ)
−e2Ψdτ2+ e−2Φδijdxidxj + hijdxidxj − 2Bidxidτ
• metric components are evolved with Einstein’s equations
Gµ
ν = 8πGTµ
ν
Julian Adamek Queen Mary University of London 3 / 10

7. Strategy
• choose ansatz for the metric (perturbed FLRW)
ds2 =a2(τ)
−e2Ψdτ2+ e−2Φδijdxidxj + hijdxidxj − 2Bidxidτ
• metric components are evolved with Einstein’s equations
Gµ
ν = 8πGTµ
ν
• stress-energy tensor is determined by solving the EOM’s of
all sources of stress-energy
Tµν
m =
P
n
m(n)
δ(3)(x−x(n))
√
−g
−gαβ
dxα
(n)
dτ
dxβ
(n)
dτ
− 1
2 dxµ
(n)
dτ
dxν
(n)
dτ
Julian Adamek Queen Mary University of London 3 / 10

8. Canonical Momentum
One-particle action ⇒ canonical momentum
S = −m
R q
−gµν
dxµ
dτ
dxν
dτ dτ ⇒ q = ∂L
∂v
Julian Adamek Queen Mary University of London 4 / 10

9. Canonical Momentum
One-particle action ⇒ canonical momentum
S = −m
R q
−gµν
dxµ
dτ
dxν
dτ dτ ⇒ q = ∂L
∂v
Geodesic equation
dqi
dτ = − ∂
∂xi
eΨ
p
q2e2Φ − qjqkhjk + m2a2 + qjBj
dxi
dτ = ∂
∂qi
eΨ
p
q2e2Φ − qjqkhjk + m2a2 + qjBj
Stress-energy tensor
T0
0 = −δ(3)(x−x(n))e3Φ
a4
p
q2e2Φ − qiqjhij + m2a2 + qiBi
Julian Adamek Queen Mary University of London 4 / 10

10. Einstein’s Equations
−a2
2 G0
0 = 3
2e−2Ψ (H − Φ′)2
+ e2Φ
h
∆Φ − 1
2 (∇Φ)2
i
a2
2 G0
i = e−Ψ∇i
e−Ψ (H − Φ′)
− 1
4∆Bi
a2
Gi
j − 1
3δi
jGk
k
=
δikδl
j − 1
3δi
jδkl
h
eΦ+Ψ∇k∇leΦ−Ψ − 2e2Φ (∇kΨ) (∇lΨ) +
B′
(k,l) + 2HB(k,l) + 1
2h′′
kl + Hh′
kl − 1
2 ∆hkl
i
Here I dropped quadratic and higher-order terms only with Bi
or hij.
For computational efficiency the exponentials can be expanded
(weak-field expansion).
Julian Adamek Queen Mary University of London 5 / 10

11. Power Spectra
k [h/Mpc] k [h/Mpc] k [h/Mpc]
∆(k)
Φ
Φ-Ψ
hij
B
z = 3 z = 1 z = 0
10
10
10
10
10
10
10
10
1 1 1
0.1 0.1 0.1
0.01 0.01 0.01
-10
-12
-14
-16
-18
-20
-22
-24
Julian Adamek Queen Mary University of London 6 / 10

12. Features
Version 1.1 (public)
• multiple particle species (CDM, baryons, neutrinos)
• initial condition generation “on the fly”
• auto- and cross-power spectra
• linear perturbations in the radiation field
• Newtonian mode compatible with radiation perturbations
(using N-body gauge)
• massive neutrinos can be treated as linear perturbations
and/or as particles
Version 1.2 (upcoming)
• particle metric light cones for ray tracing and
post-processing
• linear dark energy fluids (w-cs-parametrization)
Julian Adamek Queen Mary University of London 7 / 10

13. Ray Tracing
Instead of keeping snapshots = {data | τ = τsnap}, we store a
thick light cone = {data | τ − τo + r ∈ [−∆τ, ∆τ]}, where ∆τ is
chosen such that the perturbed light cone ⊂ thick light cone.
In a post-processing step, we integrate backwards in time
(without approximation):
null geodesic
equation
⇓
observed angles
redshifts
This allows us to construct the statistics of observed sources.
Julian Adamek Queen Mary University of London 8 / 10

14. Other Approaches?
Comparison to Numerical Relativity fluid simulations
• generally good agreement (but further studies warranted)
• comparison needs to be done based on observables
Adamek, Di Dio, Durrer Kunz, Phys. Rev. D89 (2014) 063543
Adamek, Gosenca Hotchkiss, Phys. Rev. D93 (2016) 023526
• fluid simulations have no access to the clustering /
multistream regime
• fluid simulations often
use coordinates in
which the light cone is
heavily distorted
Julian Adamek Queen Mary University of London 9 / 10

15. Summary
• GR framework for N-body simulations has been fully
implemented and tested
• follows first principles approach wherever possible
• requires minimal assumptions that are internally verified
• provides unified relativistic treatment to predict large-scale
structure observables
• v1.1 of the code is available on a public Git repository:
https://github.com/gevolution-code/gevolution-1.1.git
Julian Adamek Queen Mary University of London 10 / 10