General Relativistic N-body Simulations of Cosmic Large-Scale Structure. Julian Adamek

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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 /

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 /

4 A Brief Overview of gevolution gevolution, a general relativistic N-body code Adamek, Daverio, Durrer & Kunz, Nature Phys. 12 (2016) spin-1 metric perturbation with gevolution based on weak-field expansion (in Poisson gauge) for any given T µ ν computes the six metric d.o.f. (Φ, Ψ, B i, h ij ) N-body particle ensemble evolved using relativistic geodesic equation Julian Adamek Queen Mary University of London 2 /

5 Strategy choose ansatz for the metric (perturbed FLRW) ds 2 =a 2 (τ) [ e 2Ψ dτ 2 +e 2Φ δ ij dx i dx j +h ij dx i dx j 2B i dx i dτ ] Julian Adamek Queen Mary University of London 3 /

6 Strategy choose ansatz for the metric (perturbed FLRW) ds 2 =a 2 (τ) [ e 2Ψ dτ 2 +e 2Φ δ ij dx i dx j +h ij dx i dx j 2B i dx i dτ ] metric components are evolved with Einstein s equations G µ ν = 8πGT µ ν Julian Adamek Queen Mary University of London 3 /

7 Strategy choose ansatz for the metric (perturbed FLRW) ds 2 =a 2 (τ) [ e 2Ψ dτ 2 +e 2Φ δ ij dx i dx j +h ij dx i dx j 2B i dx i dτ ] metric components are evolved with Einstein s equations G µ ν = 8πGT µ ν stress-energy tensor is determined by solving the EOM s of all sources of stress-energy T m µν = δ m (3) (x x (n) ) dx (n) g ( g α (n) αβ dτ n dx β (n) dτ ) 1 2 dx µ dx ν (n) (n) dτ dτ Julian Adamek Queen Mary University of London 3 /

8 Canonical Momentum One-particle action canonical momentum S = m g µν dx µ dτ dx ν L dτ dτ q = v Julian Adamek Queen Mary University of London 4 /

9 Canonical Momentum One-particle action canonical momentum S = m g µν dx µ dτ dx ν L dτ dτ q = v Geodesic equation ( ) dq i dτ = e Ψ q x 2 e 2Φ q j q k h i jk +m 2 a 2 +q j B j ( ) dx i dτ = q i e Ψ q 2 e 2Φ q j q k h jk +m 2 a 2 +q j B j Stress-energy tensor T 0 0 = δ(3) (x x (n) ) e3φ a 4 ( q 2 e 2Φ q i q j h ij +m 2 a 2 +q i B i ) Julian Adamek Queen Mary University of London 4 /

10 Einstein s Equations a2 2 G0 0 = 3 2 e 2Ψ (H Φ ) 2 +e 2Φ [ Φ 1 2 ( Φ)2] a 2 2 G0 i = e Ψ i [ e Ψ (H Φ ) ] 1 4 B i ) a (G 2 i j 1 3 δi j Gk k = )[ (δ ik δj l 1 3 δi j δkl e Φ+Ψ k l e Φ Ψ 2e 2Φ ( k Ψ)( l Ψ)+ ] B (k,l) +2HB (k,l) h kl +Hh kl 1 2 h kl Here I dropped quadratic and higher-order terms only with B i or h ij. For computational efficiency the exponentials can be expanded (weak-field expansion). Julian Adamek Queen Mary University of London 5 /

11 Power Spectra z = 3 z = 1 z = (k) Φ B Φ-Ψ -24 h ij k [h/mpc] k [h/mpc] k [h/mpc] Julian Adamek Queen Mary University of London 6 /

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-c s -parametrization) Julian Adamek Queen Mary University of London 7 /

13 Ray Tracing Instead of keeping snapshots = {data τ = τ snap }, we store a thick light cone = {data τ τ o +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 /

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) Adamek, Gosenca & Hotchkiss, Phys. Rev. D93 (2016) 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 /

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: Julian Adamek Queen Mary University of London /