Relativistic Structure Formation in the Universe

Ruth Durrer

Departement´ de physique theorique,´ Universite´ de Geneve`

CosmoBack LAM, Mai 2018

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 1 / 35 Outline

1 Introduction

2 Linear perturbation theory Gauge transformations / gauge invariance Simple solutions What do we observe Measuring the lensing potential / relativistic effects

3 Relativistic N-body simulations

4 Conclusions

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 2 / 35 Introduction

The observed Universe is not homogeneous and isotropic. It contains galaxies, clusters of galaxies, voids, ﬁlaments...

M. Blanton and the Sloan Digital Sky Survey Team.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 3 / 35 Introduction

We assume that on large scales there is a well defined mean density and on intermediate scales, the density differs little from it. This is a highly non-trivial assumption, which is best justified by the isotropy of the cosmic microwave background, the CMB. Under the hypothesis that cosmic structure grew out of small initial fluctuations we can study their evolution using perturbation theory. At late times and sufficiently small scales fluctuations of the cosmic density are not small. The density inside a galaxy is about 105 times higher than the mean density of the Universe. Perturbation theory is then not adequate and we need N-body simulations to study structure formation on galaxy - cluster scales of a few Mpc and less. Since this is mainly relevant on scales much smaller than the Hubble scale (3000h−1Mpc), it has been studied in the past mainly with non-relativistic simulations. Since a couple years several groups have started to develop also relativistic simulations.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 4 / 35 Introduction

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 4 / 35 Linear cosmological perturbation theory

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 5 / 35 Linear perturbation theory: Gauge transformation

Splitting the true metric into a background and a perturbation,

µ ν 2 h 2 i i j i j i gµν dx dx = a (t) −(1 + 2A)dt − 2Bi dtdx + (1 + 2HL)γij dx dx + 2Hij dx dx

2 µ ν µ ν = a (t) [¯ηµν + δηµν ] dx dx = [g¯µν + δgµν ] dx dx is not unique. Under a linearized coordinate transformation (gauge transformation), x µ → x µ + X µ, the metric (and any other tensor ﬁeld) changes like

gµν 7→ gµν + LX gµν , δgµν 7→ δgµν + LX g¯µν .

The same is true for the energy momentum tensor,

Tµν = T¯µν + δTµν , δTµν 7→ δTµν + LX T¯µν .

A variable (tensor ﬁeld) V is called gauge invariant if it does not change under gauge transformations. As we see from the above equations, this happens if LX V¯ = 0 ∀ X, hence V¯ = constant. (Stewart Walker lemma). Furthermore, even if the total metric and energy momentum tensor satisfy Einstein’s equation, this is in general not true for arbitrary splits into a background and a perturbation. We call a split ’admissible’ if it is true. For an admissible split also the perturbations satisfy Einstein’s equations.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 6 / 35 Linear perturbation theory: Gauge transformation

Splitting the true metric into a background and a perturbation,

µ ν 2 h 2 i i j i j i gµν dx dx = a (t) −(1 + 2A)dt − 2Bi dtdx + (1 + 2HL)γij dx dx + 2Hij dx dx

gµν 7→ gµν + LX gµν , δgµν 7→ δgµν + LX g¯µν .

The same is true for the energy momentum tensor,

Tµν = T¯µν + δTµν , δTµν 7→ δTµν + LX T¯µν .

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 6 / 35 Linear perturbation theory: Gauge invariant variables

We typically split spatial tensor ﬁelds into helicity zero (scalar) helicity one (vector) and helicity two (tensor) parts. These do not mix within linear perturbation theory due to rotational symmetry of the background. The same is true for each Fourier component (translational symmetry)

1 B = ∇ B + B(V ) , H = ∇ ∇ − ∆γ H(S) + ∇ H(V ) + ∇ H(V ) + H(T ) i i i ij i j 3 ij i j j i ij

i (V ) i (V ) i (T ) j (T ) ∇ Bi = ∇ Hi = ∇ Hij = Hi = 0 . I shall not discuss vector perturbations any further. They seem not to be relevant in cosmology. µ In the background Friedmann-Lemaˆıtre universe the Weyl tensor C ναβ and the anisotropic stress tensor Πµν vanish ⇒ their perturbations are gauge-invariant,

Πµν = Tµν − (ρ + P)uµuν − Pgµν .

µ ν µ µ µ Here ρ and u are deﬁned by Tν u = −ρu , u uµ = −1 and Tµ + ρ = 3P.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 7 / 35 Linear perturbation theory: Gauge invariant variables

For scalar perturbations we can deﬁne the so called Bardeen potentials. In Fourier space (k = wavenumber) they are given by Ψ = A − Hk −1σ − k −1σ˙ , 1 Φ = −H − H(S) + Hk −1σ = −R + Hk −1σ L 3 1 σ = k −1H˙ (S) − B R = H + H(S) . L 3 µ The non-vanishing components of Cναβ and Πµν are then given by µ ν 0 1 1 E ≡ C uµu = −C = − (Ψ + Φ) − ∆(Ψ + Φ)γ ij iνj i0j 2 |ij 3 ij 1 8πGa2Π = (Ψ − Φ) − ∆(Ψ − Φ)γ . ij |ij 3 ij For the last eqn. we used Einstein’s equation. In longitudinal (Newtonian) gauge, B = H(S) = 0, the perturbed metric is given by

2 h 2 i j i δgµν = −2a Ψdt + Φγij dx dx .

Tensor perturbations are gauge invariant (there are no tensor-type gauge-transformations).

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 8 / 35 Linear perturbation theory: Gauge invariant variables

Apart from the anisotropic stress, the entropy perturbation

c2 Γ = (δP + s δρ)/ρ¯ w is gauge invariant. To obtain gauge invariant perturbations of the velocity and density perturbations, δ = (ρ − ρ¯)/ρ¯ one has to combine them with metric perturbations. The most common combinations are 1 V ≡ v − H˙ = v long k T −2 −1 long Ds ≡ δ + 3(1 + w)H(k H˙ T − k B) ≡ δ , H H D ≡ δlong + 3(1 + w) V = δ + 3(1 + w) (v − B) k k H = D + 3(1 + w) V , s k 1 D ≡ δ + 3(1 + w) H + H(S) = δlong − 3(1 + w)Φ g L 3

= Ds − 3(1 + w)Φ .

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 9 / 35 Density perturbation spectrum

The resulting density perturbation spectrum depends on the choice of the perturbation < −1 variable only on very large scales, k ∼ H0 .

104

1000 P(k) 100

0.001 0.01 0.1 1k Comparing density ﬂuctuations in different gauges: comoving gauge, D (blue), longitudinal gauge, Ds (red) and spatially ﬂat gauge, Dg (green)

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 10 / 35 Linear perturbation theory: Perturbation equations

Einstein’s equations( H = a˙ /a = Ha)

4πGa2ρD = −(k 2 − 3K )Φ (00) 4πGa2(ρ + P)V = k HΨ + Φ˙ (0i) 8πGa2PΠ(S) = k 2 (Φ − Ψ) (i =6 j) h 2 i 2 1 2 ¨ ˙ ˙ ˙ 2 k 4πGa ρ 3 D + cs Ds + wΓ = Φ + 2HΦ + HΨ + 2H + H − 3 Ψ(ii) 8πGa2PΠ(T ) = H¨ (T ) + 2HH˙ (T ) + 2K + k 2 H(T ) (tensor)

2 ˙ Conservation equations for scalar perturbations( w = P/ρ, cs = P/ρ˙) 3K D˙ − 3wHD = − 1 − [(1 + w)kV + 2HwΠ] , k 2 c2 w 2 3K w V˙ + HV = k Ψ + s D + Γ − 1 − Π . 1 + w 1 + w 3 k 2 1 + w Bardeen equation for scalar perturbations

2 h 2 2 2 2 2i Φ¨ + 3H(1 + cs )Φ˙ + 3(cs − w)H − (2 + 3w + 3cs )K + cs k Φ 8πGa2P 1 k 2 = HΠ˙ + [2H˙ + 3H2(1 − c2/w)]Π − k 2Π + Γ . k 2 s 3 2

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 11 / 35 Linear perturbation theory: Perturbation equations

Einstein’s equations( H = a˙ /a = Ha)

2 h 2 2 2 2 2i Φ¨ + 3H(1 + cs )Φ˙ + 3(cs − w)H − (2 + 3w + 3cs )K + cs k Φ 8πGa2P 1 k 2 = HΠ˙ + [2H˙ + 3H2(1 − c2/w)]Π − k 2Π + Γ . k 2 s 3 2

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 11 / 35 Linear perturbation theory: Perturbation equations

Einstein’s equations( H = a˙ /a = Ha)

2 h 2 2 2 2 2i Φ¨ + 3H(1 + cs )Φ˙ + 3(cs − w)H − (2 + 3w + 3cs )K + cs k Φ 8πGa2P 1 k 2 = HΠ˙ + [2H˙ + 3H2(1 − c2/w)]Π − k 2Π + Γ . k 2 s 3 2

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 11 / 35 Linear perturbation theory: Simple solutions

We consider adiabatic fluctuations of a perfect fluid (Γ = Π = 0) with K = 0 and 2 w = cs =constant. In this case the Bardeen eqn. simplifies to 2 q Φ¨ + 3H(1 + w)Φ˙ + 3wk 2Φ = 0 , H = t −1 = 1 + 3w t with solution 1 Φ = [Aj (c kt) + By (c kt)] a q s q s

On large scales, cskt < 1 the growing mode ∝ A is constant, on small scales, cskt > 1 it oscillates and decays. 2 For radiation, w = cs = 1/3, q = 1. 2 The case of dust, cs = w = 0 has been treated apart. In this case the solution is B Φ = A + . (kt)5 Again, the growing mode is constant.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 12 / 35 Bardeen potential

2 3 n−1 Starting from scale invariant initial conditions, h|Ψ| ik = AS(k/H0) , n = 1 one ﬁnds today a Bardeen potential of roughly the following form:

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 13 / 35 Linear perturbation theory: Simple solutions

From the Bardeen potential we can now compute the density and velocity perturbations. For dust we obtain for the growing modes

V ∝ t D ∝ t 2 ∝ a . √ For radiation we ﬁnd, x = kt/ 3

2 D = 2A cos(x) − sin(x) , g x √ √ 3 3A 2 2 V = − D0 = sin(x) + cos(x) − sin(x) . 4 g 2 x x 2

These are the acoustic oscillations which we see in the CMB temperature anisotropy spectrum.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 14 / 35 CMB spectrum

Planck Collaboration: Cosmological parameters

6000

5000

] 4000 2 K µ [ 3000 TT

D 2000

1000

0 600 60 300 30 TT 0 0 D

-300 -30 -600 -60 2 10 30 500 1000 1500 2000 2500 The Planck Collaboration, 2015 Fig. 1. The Planck 2015 temperature power spectrum. At multipoles ` 30 we show the maximum likelihood frequency averaged temperature spectrum computed from the Plik cross-half-mission likelihood with foreground and other nuisance parameters determined from the MCMC analysis of the base ⇤CDM cosmology. In the multipole range 2 ` 29, we plot the power spectrum estimates from the Commander component-separation algorithm computed over 94% of the sky. The best-ﬁt base ⇤CDM theoretical spectrum ﬁtted to the Planck TT+lowP likelihood is plotted in the upper panel. Residuals with respect to this model are shown in Ruth Durrerthe (Universit lower panel.e´ de The Gen erroreve)` bars show 1 uncertainties.Structure Formation in the Universe Marseille, Mai 2018 15 / 35 ±

sults to the likelihood methodology by developing several in- the full “TT,TE,EE” likelihoods) would di↵er in any substantive dependent analysis pipelines. Some of these are described in way had we chosen to use the CamSpec likelihood in place of Planck Collaboration XI (2015). The most highly developed of Plik. The overall shifts of parameters between the Plik 2015 these are the CamSpec and revised Plik pipelines. For the likelihood and the published 2013 nominal mission parameters 2015 Planck papers, the Plik pipeline was chosen as the base- are summarized in column 7 of Table 1. These shifts are within 2⌧ line. Column 6 of Table 1 lists the cosmological parameters for 0.71 except for the parameters ⌧ and Ase which are sen- base ⇤CDM determined from the Plik cross-half-mission like- sitive to the low multipole polarization likelihood and absolute lihood, together with the lowP likelihood, applied to the 2015 calibration. full-mission data. The sky coverage used in this likelihood is In summary, the Planck 2013 cosmological parameters were identical to that used for the CamSpec 2015F(CHM) likelihood. pulled slightly towards lower H0 and ns by the ` 1800 4-K line However, the two likelihoods di↵er in the modelling of instru- systematic in the 217 217 cross-spectrum, but⇡ the net e↵ect of mental noise, Galactic dust, treatment of relative calibrations and this systematic is relatively⇥ small, leading to shifts of 0.5 or multipole limits applied to each spectrum. less in cosmological parameters. Changes to the low level data As summarized in column 8 of Table 1, the Plik and processing, beams, sky coverage, etc. and likelihood code also produce shifts of typically 0.5 or less. The combined e↵ect of CamSpec parameters agree to within 0.2 , except for ns, which these changes is to introduce parameter shifts relative to PCP13 di↵ers by nearly 0.5 . The di↵erence in ns is perhaps not sur- 2⌧ prising, since this parameter is sensitive to small di↵erences in of less than 0.71 , with the exception of ⌧ and Ase . The main scientiﬁc conclusions of PCP13 are therefore consistent with the the foreground modelling. Di↵erences in ns between Plik and CamSpec are systematic and persist throughout the grid of ex- 2015 Planck analysis. tended ⇤CDM models discussed in Sect. 6. We emphasise that Parameters for the base ⇤CDM cosmology derived from the CamSpec and Plik likelihoods have been written indepen- full-mission DetSet, cross-year, or cross-half-mission spectra are dently, though they are based on the same theoretical framework. in extremely good agreement, demonstrating that residual (i.e. None of the conclusions in this paper (including those based on uncorrected) cotemporal systematics are at low levels. This is

8 What do we observe?

Measured quantities must be gauge invariant. They do not depend on the coordinate system in which we compute them. We have long ago (1990-95) computed what we measure (within linear perturbation theory) when we masure CMB anisotropies. A simpliﬁed formula neglecting Silk damping is (RD 1990)

∆T (n) 1 Z t0 = D(r) + V (b)nj + Ψ + Φ (t , x ) + (Ψ˙ + Φ)(˙ t, x(t)) dt . T 4 g j dec dec tdec This result is obtained by studying lightlike geodesics. To go beyond this and take into account polarisation and Silk damping, we have to study the propagation of photons with a Boltzmann equation approach.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 16 / 35 The observed number count

For density perturbations (even neglecting bias and non-linearities) it is more complicated. We consider the following measurement: we count the galaxies in a small redshift element dz around redshift z and solid angle dΩn around a direction n. To obtain the ﬂuctuation of this number count we subtract from it the mean number of galaxies inside a redshift bin dz and a solid angle dΩn and we also divide by this mean,

N(z, n) − N¯ (z) ∆(z, n) = . N¯ (z) Assuming that the number of galaxies is proportional to the matter density times the volume we have dV (z, n) dN(z, n) = ρ(z, n)dV (z, n) = ρ(z, n)ν(z, n)dzdΩn ν(z, n) = . dzdΩn To continue we have also to take into account that also the redshift and the direction from which we see the galaxy are perturbed. Taking this all into account we ﬁnd the perturbed expression for the observed number counts,

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 17 / 35 The observed number count

1 h i ∆ (n, z, m∗) = bD + Φ˙ + ∂ (V · n) − (3 − b )HV + g H r e H˙ 2 − 5s Z rs − · ˙ ˙ 2 + + 5s be Ψ + V n + dr(Φ + Ψ H rsH 0 Z rs 2 − 5s h rs − r i −(2 − 5s)Φ + Ψ + dr 2(Φ + Ψ) − ∆Ω(Φ + Ψ) 2rs 0 r (Bonvin & RD 2011, Challinor & Lewis 2011)

The ﬁrst term is simply the biased density contrast in comoving gauge. The second term is the redshift space distortion (the radial volume distortion) the last term is the lensing term (the transversal volume distortion). The other terms (Doppler, Shapiro time delay, integrated Sachs Wolfe term) are relevant mainly on very large scales. m∗ is the magnitude limit of the survey and L∗ is the limiting luminosity.

be is the evolution bias,

5 ∂ log ng (z, L) s = is the magniﬁcation bias. 2 ∂ log L L=L∗

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 18 / 35 The observed number count

The observed number count ﬂuctuation is not a function of position but a function of (observed) direction n and redshift z; (n, z) are observable coordinates on the background lightcone. We cannot observe distances. To infer a distance from observations in cosmology, we always use a model. Hence the real space correlation function and its Fourier transform, the power spectrum are model dependent. We can directly observe the angular-redshift correlation function,

ξ(n · n0, z, z0) = h∆(n, z)∆(n0, z0)i

0 0 or its angular power spectrum, C`(z, z ), n · n = cos θ

0 1 X 0 ξ(θ, z, z ) = (2` + 1)C`(z, z )P`(cos θ) 4π `

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 19 / 35 The observed number count

The distance between two galaxies in directions n and n0 at redshifts z and z0, for K = 0 p d = r(z)2 + r(z0)2 − 2r(z)r(z0) cos θ depends on the cosmological model via

Z z dz0 1 Z z dz0 r(z) = = 0 p 0 3 0 0 H(z ) H0 0 Ωm(1 + z ) + Ωdef (z ) One therefore has to be very careful when estimating cosmological parameters via the real space correlation function or the power spectrum.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 20 / 35 The observed number count

z1 =z2 =0.7 0.035

0.030 Observed

0.025 2

Θ 0.020 L

Θ 0.015 H Ξ 0.010

0.005

0.000 Parameter dependence of the real 0 2 4 6 8 10 Θ space correlation function. deg z1 =z 2 =0.7 (Figure by F. Montanari) 0.035 @ D

2 0.030 True Wm = 0.24 Θ

D 0.025 Wrong W = 0.3

L m 2

z 0.020 , Wrong Wm = 0.5 1

z 0.015 , Θ

H 0.010 r @

Ξ 0.005

0.000 0 50 100 150 200 250 300 r Mpc h

@ D

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 21 / 35 Density and RSD

Most present analyses take into account only the density and rsd terms and consider s small survey in direction n in the sky. −1 0.57 Using kV /H = H D˙ = f (z)D, f (z) = d log D/d log a ∼ Ωm(z) one ﬁnds 2 ξobs(d, z) = ξ(d)TD(z)[β0(z)P0(µ) − β2(z)P2(µ) + β4(z)P4(µ)] ,

2 Pobs(k, z) = P(k)TD(z)[β0(z)P0(µ) + β2(z)P2(µ) + β4(z)P4(µ)] , with 2bf f 2 4bf 4f 2 8f 2 β = b2 + + , β = + , β = . 0 3 5 2 3 7 4 35

↵2 ↵2

µ = dˆ · n = cos γ, or µ = kˆ · n. s t

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 22 / 35 Lensing

The lensing term can become very important at high redshift and for different redshifts, z =6 z0. Contributions to the power spectrum at redshift z0 = z = 3, ∆z = 0.3 (from Bonvin & RD ’11)

1!10"4 5!10"5 l C Π

2 "5

"# 1!10

1 "6

# 5!10 l ! l

1!10"6 5!10"7 20 40 60 80 100 l DD zz Dz lensing C` (red), C` (green), 2C` (blue), C` (magenta).

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 23 / 35 Measuring the lensing potential

Well separated redshift bins measure mainly the lensing-density correlation:

h∆(n, z)∆(n0, z0)i ' h∆L(n, z)δ(n0, z0)i z > z0

∆L(n, z) = (2 − 5s(z))κ(n, z)

Bins 1-1 0.05 Bins 1-4 full 0.0020 0.04 only local 0.0015 ℓ

ℓ 0.0010 0.03 local-lensing 0.0005 0.02 0.0000 2π C ℓ(+1 )/ 0.01 ℓ(+1)/2πC -0.0005 -0.0010 0.00 -0.0015 1 10 100 1000 1 10 100 1000 ℓ ℓ (Montanari & RD) Bins 1-10 Bins 1-4, SKA [1506.01369] 0.0000 0.004 ℓ

ℓ -0.0005 0.003

-0.0010 0.002 ℓ(+1)/2πC ℓ(+1 )/2πC 0.001 -0.0015

0.000 1 10 100 1000 1 10 100 1000 ℓ ℓ

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 24 / 35 The radial power spectrum

0.001 0.1 5!10"4 l l C

0.01 C "4 Π 1!10 Π 2

2 "5 "# 5!10 "# 1 1

" 0.001 l # ! l l ! l 1!10"5 10!4 5!10"6

0.105 0.110 0.115 0.120 0.125 0.130 3.0 3.2 3.4 3.6 3.8 z’ z’ 0.01

0.001 l C

Π !4

2 10 "#

1 0 "

l !5 The radial power spectrum C`(z, z ) ! 10 l

!6 for 20 10 ` =

! Left, top to bottom: z = 0.1, 0.5, 1, 10 7 0.50 0.55 0.60 0.65 0.70 0.75 top right: z = 3 z’

0.001 lensing

l Standard terms (blue), C` (magenta), C

Π 10!4 Doppler grav 2 C (cyan), C (black), "# ` ` 1 " l

! !5 l 10 (from Bonvin & RD ’11)

10!6 1.0 1.1 1.2 1.3 1.4 1.5 Ruth Durrer (Universite´ dez’ Geneve)` Structure Formation in the Universe Marseille, Mai 2018 25 / 35 Relativistic N-body simulations

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 26 / 35 Going beyond linear perturbation theory

On intermediate to small scales, d < 10Mpc, density perturbations can become large. Inside a galaxy we have δ ∼ 105 and even inside a galaxy cluster the motion of galaxies is decoupled from the Hubble ﬂow, clusters due not expand. ⇒ We have to treat clustering non-linearly . But the gravitational potential of a galaxy remains small,

12 2 × 10 km −5 Φ ∼ Rs/D ∼ ' 10 10kpc Therefore, in Newtonian (longitudinal) gauge, metric perturbations remain small. In the past, this together with the smallness of peculiar velocities has been used to argue that Newtonian N-body simulations are sufﬁcient. Caveats: We start simulations at z ∼ 100 when the box size is comparable to the horizon and neglecting radiation is not a good approximation (aim: 1% accuracy). Neutrino velocities, even for massive neutrinos are not very small at z = 100. The relativistic gravitational ﬁeld has 6 degrees of freedom (2 helicity 0, 2 helicity 1 and 2 helicity 2) and Newtonian simulations only consider 1 of them. Observations are made on the relativistic, perturbed lightcone. We want to correctly simulate this.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 27 / 35 Going beyond linear perturbation theory

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 27 / 35 Relativistic N-body simulations

A full relativistic N-body simulation might, starting from the initial metric and particle positions at time t1

1 Move dark matter particles (and baryons) along geodesics of the metric at t1 to obtain their position at t1 + ∆t.

µ µ α β mp˙ + Γαβ p p = 0

2 Compute the energy-momentum tensor with a particle to mesh projection.

µν −1 X µ ν T (x, t) = m p (t)p (t)δ(x − xn(t)) n

3 Solve Einstein’s equations to determine the metric at t1 + ∆t.

4 Return to step 1 replacing t1 by t1 + ∆t1 In practice the time-stepping is replaced by a staggered leap-frog to render his procedure stable.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 28 / 35 Relativistic N-body simulations

A full relativistic N-body simulation might, starting from the initial metric and particle positions at time t1

1 Move dark matter particles (and baryons) along geodesics of the metric at t1 to obtain their position at t1 + ∆t.

µ µ α β mp˙ + Γαβ p p = 0

2 Compute the energy-momentum tensor with a particle to mesh projection.

µν −1 X µ ν T (x, t) = m p (t)p (t)δ(x − xn(t)) n

3 Solve Einstein’s equations to determine the metric at t1 + ∆t.

4 Return to step 1 replacing t1 by t1 + ∆t1 In practice the time-stepping is replaced by a staggered leap-frog to render his procedure stable.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 28 / 35 Relativistic N-body simulations

A full relativistic N-body simulation might, starting from the initial metric and particle positions at time t1

1 Move dark matter particles (and baryons) along geodesics of the metric at t1 to obtain their position at t1 + ∆t.

µ µ α β mp˙ + Γαβ p p = 0

2 Compute the energy-momentum tensor with a particle to mesh projection.

µν −1 X µ ν T (x, t) = m p (t)p (t)δ(x − xn(t)) n

3 Solve Einstein’s equations to determine the metric at t1 + ∆t.

4 Return to step 1 replacing t1 by t1 + ∆t1 In practice the time-stepping is replaced by a staggered leap-frog to render his procedure stable.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 28 / 35 Relativistic N-body simulations

A full relativistic N-body simulation might, starting from the initial metric and particle positions at time t1

1 Move dark matter particles (and baryons) along geodesics of the metric at t1 to obtain their position at t1 + ∆t.

µ µ α β mp˙ + Γαβ p p = 0

2 Compute the energy-momentum tensor with a particle to mesh projection.

µν −1 X µ ν T (x, t) = m p (t)p (t)δ(x − xn(t)) n

3 Solve Einstein’s equations to determine the metric at t1 + ∆t.

4 Return to step 1 replacing t1 by t1 + ∆t1 In practice the time-stepping is replaced by a staggered leap-frog to render his procedure stable.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 28 / 35 Relativistic N-body simulations

A full relativistic N-body simulation might, starting from the initial metric and particle positions at time t1

1 Move dark matter particles (and baryons) along geodesics of the metric at t1 to obtain their position at t1 + ∆t.

µ µ α β mp˙ + Γαβ p p = 0

2 Compute the energy-momentum tensor with a particle to mesh projection.

µν −1 X µ ν T (x, t) = m p (t)p (t)δ(x − xn(t)) n

3 Solve Einstein’s equations to determine the metric at t1 + ∆t.

4 Return to step 1 replacing t1 by t1 + ∆t1 In practice the time-stepping is replaced by a staggered leap-frog to render his procedure stable.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 28 / 35 Relativistic N-body simulations

At present, we do not solve the full Einstein equations. We work in longitudinal gauge (K = 0),

2 2 2Ψ 2 2 i 2 −2Φ i j ds = −a e dt − 2a Bi dtdx + a [e δij + hij ]dx dx and use the fact that Ψ ' Φ is very small. However, spatial derivatives, ∇Ψ/H ∼ V and second derivatives, ∆Ψ/H2 ' D are not small. Therefore, when computing the Einstein tensor go only to first order in the gravitational potentials and their time derivatives. We include quadratic terms of first spatial derivatives and all orders for second spatial derivatives. Since the Einstein tensor is linear in its second derivatives of the metric, this simplifies the equations significantly. We include also vector and tensor perturbations only at linear order. They are induced from scalar perturbations at second order and always remains small. When solving the geodesic equation we also neglect tensor perturbations as they only enter together with v 2 and are very suppressed.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 29 / 35 Relativistic N-body simulations

At present, we do not solve the full Einstein equations. We work in longitudinal gauge (K = 0),

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 29 / 35 Relativistic N-body simulations

At present, we do not solve the full Einstein equations. We work in longitudinal gauge (K = 0),

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 29 / 35 Gevolution

The code doing this is ’gevolution’. It is written by Julian Adamek and extensively uses packages of ’LATﬁeld2’ written for gevolution by David Daverio. The code is publicly available at github.com/gevolution-code/gevolution-1.0

(Figure by Julian Adamek)

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 30 / 35 Gevolution

(Figure by Julian Adamek)

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 31 / 35 Spectra from gevolution

z=10 z=1 z=0

-8 -8 10 10

-10 -10 10 10

10-12 10-12

10-14 10-14 ) k ( -16 -16 P 10 10 2 π 2

-18 -18 10 10

-20 -20 10 10 Φ B -22 -22 10 Φ-Ψ 10

hi j

10-24 10-24 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 k [h/Mpc] k [h/Mpc] k [h/Mpc] (From Adamek et al., 2016)

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 32 / 35 Spectra from gevolution

z = 0

m = 0.06 eV 1.0 ν P mν = 0.1 eV (normal) P mν = 0.1 eV (inverted) 0.9 Pmν = 0.2 eV P mν = 0.3 eV

h massless P ∆

/ 0.8 h massive ∆ 0.7

0.6

0.01 0.1 1 k [h/Mpc]

Difference of tensor spectra in presence of massive neutrinos. (From Adamek et al., 2017)

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 33 / 35 Performance of gevolution

Villaescusa-Navarro et al. 2013 1012 • H Inman et al. 2015 N Castorina et al. 2015 1011 Emberson et al. 2016 ⋆ ⋆ ⋆ ⋆ this work

10 cdm 10 N N H

109

108 • ••

0.001 0.01 0.1 1 10 100 1000 3 3 3 Lbox [Gpc /h ]

Gevolution has been run on the Swiss Supercomputer ’Piz Daint’. These simulations are among the largest worldwide. (From Adamek et al., 2017)

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 34 / 35 Conclusions

At sufficiently large scales and at early times structure formation can be treated within linear cosmological perturbation theory. Perturbations variables are usually gauge dependent, but observations are not. In cosmology we can observe directions, redshifts, fluxes, ··· , but not distances. Therefore, the real space correlation function and the power spectrum are model dependent. Galaxy number counts contain apart from density and rsd an important lensing term which is most relevant for z =6 z0 correlations and at high redshift. At intermediate scales we need simulations as density fluctuations become large. Relativistic weak field simulations are not much more costly than Newtonian simulations and they capture all the relativistic effects relevant in cosmology.

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 35 / 35 Conclusions

Ruth Durrer (Universite´ de Geneve)` Structure Formation in the Universe Marseille, Mai 2018 35 / 35