Cosmological Structure Formation

Transcription

1 Cosmological Structure Formation Beyond linear theory Lagrangian PT: Zeldovich approximation (Eulerian PT: Spherical collapse) Redshift space distortions

2

3

4

5

6 Recall linear theory: When radiation dominated (H = 1/2t): (d 2 δ/dt 2 ) + 2H (dδ/dt) = (d 2 δ/dt 2 ) + (dδ/dt)/t = 0 δ(t) = C 1 + C 2 ln(t) (weak growth) In distant future (H = constant): (d 2 δ/dt 2 ) + 2H Λ (dδ/dt) = 0 δ(t) = C 1 + C 2 exp(-2h Λ t) If flat matter dominated (H = 2/3t): δ(t) = δ + t 2/3 + δ - t -1 a(t) at late times Linear growth just multiplicative factor, so if initial conditions Gaussian, linearly evolved field is too, with P(k,t) = D 2 (t)/d 2 (t init ) P(k,t init )

7 At small k, only amplitude grows At high k, shape also changes Bumps and wiggles = cosmic variance Ratio of Pk not sensitive to cosmic variance

8 Linear theory shape Nonlinear evolution changes shape

9 Initially Gaussian fluctuation field becomes very non- Gaussian Linear growth just multiplicative factor, so cannot explain non- Gaussianity at late times

10

11 Variance of δ R (δ smoothed on R) depends on R: σ 2 (R) = dk/k (k) W 2 (kr)

12 Variance of δ R (δ smoothed on R): σ 2 (R) = dk/k (k) W 2 (kr) Correlations in smoothed field (k) = k 3 P(k)/2π 2 R 1R2 (k) = (k) W(kR 1) W(kR 2 ) ξ R 1R2 (r) = dk/k R1R2 (k) j 0(kr)

13 E.g. Power-law P(k) ξ(r) = dk/k [k 3 Ak n /2π 2 ] j 0 (kr) r -3-n if n>-3 σ 2 (R) = (A/2π 2 ) dk/k k n+3 exp(-k 2 R 2 ) = (A/2π 2 ) Γ[(n+3)/2]/2 R -3-n ξ R (r) = (A/2π 2 ) dk k 2+n exp(-k 2 R 2 ) j 0 (kr) = (A/2π 2 ) (π/2r) erf(r/2r) if n=-2 ξ 0 (r) when r R (smoothing irrelevant on large scales? BAO )

14 Estimate of nonlinear scale <δ 2 (t)> = dk/k 4π k 3 P(k,t) W 2 (kr) If P(k) = Ak n then <δ 2 (t)> ~ R -(3+n) ~ M -(3+n)/3 converges only for n>-3. Convergence of potential fluctuations only if n=1. Note: P(k,t) = D + 2 (t) P(k), so <δ 2 (t)> ~ 1 means nonlinear structure on scales smaller than R nl ~ D + 2/(3+n) ~ t (4/3)/(3+n) Hierarchical structure formation for -3<n<1

15 In general 2 r φ = 4πG ρ δ (Poisson equation) ρ/ t = - r (ρv) (Continuity equation) Since ρ = ρ 0 (1 + δ)/a 3 we have ρ/ t = -3ρH + ρ ( δ/ t)/(1 + δ) = - ρ (Hr) + ρ ( δ/ t)/(1 + δ) (ρv) = ρ /(1 + δ) (1+δ)(v-Hr) + ρ/(1 + δ) (1+δ)(Hr) = ρ /(1 + δ) (1+δ)v pec +ρ (Hr) + ρ/(1 + δ) Hr δ δ/ t - r v pec = - x (v pec /a) = - x u (x = r/a)

16 Fourier transform δ(x,t) = k δ k (t) e -ik x and u(x,t) = k u k (t) e -ik x r 2 φ = 4πG ρδ = (3Ω 0 H 02 /2a 3 ) δ (matter domination) - k 2 φ k = (3Ω 0 H 02 /2a) δ k When Ω=1 then δ k a so the potential does not evolve! δ/ t = - x u δ k / t = -iku k δ k / t = lnd/ t δ k = ( lnd/ lna) H δ k = fh δ k So u k /(fh) = i (k/k) (δ k /k) Note that u/h has units of distance Velocities are more sensitive to small k (large scales), because of the factor of 1/k In practice, this means that once φ k is set, then u k and δ k are completely determined.

17 The Zeldovich Approximation I. The physical displacement of a particle in time dt is dr = v dt so comoving displacement is dx = dr/a = (dr/dt)/a dt = (v/a) dt. Hence, in linear theory, dx/dd = (v/a) (dt/dd) ~ δ i (v/a) (dt/dδ) ~ δ i (v/a) (r/v) ~ δ i x = constant Hence, if initial comoving position was q, then comoving position x at a later time, when the growth factor is D, is x = q + D(t) u(q)/(fh) (note that u/h is a distance)

18 Structure grows because of perturbations in the initial velocity field

19 Because of these motions, the fluctuation field can become very non-gaussian (even though the displacements themselves are Gaussian)

20 The Zeldovich Approximation II. x = q + D(t) u(q)/(fh) = q + D(t) S(q) How are Zeldovich displacements S (for shift) related to density? dx i /dq j = δ ij + D(t) ds i /dq j = δ ij - D(t) d[dφ/dq i ]/dq j Evidently, displacements are related to one derivative of potential so Jacobian of x-q transformation involves second derivatives of potential: a 3x3 matrix. The 3 eigenvalues of Φ ij, say λ 1, λ 2, λ 3, describe the principal axes of an ellipsoid (not a sphere!): in this respect, Zeldovich is more general than spherical.

21 Zeldovich approximation III. In principal axis frame: dx i /dq i = 1 - D(t) λ i Thus D(t) λ describes how the axis shrinks (or expands). Hence, the density is 1 + δ(t) = Π i=13 (1 D(t)λ i ) 1 To lowest order this is 1 + δ(t) = 1 + D(t) λ i + D 2 (t) (λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 ) + = 1 + D(t) δ initial + D 2 (t) (λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 ) + Evidently, δ Linear is just the trace of Φ ij. This is why it can be arbitrarily negative, and even when it is, the true overdensity is still sensible. Second order is combination of δ 2 Linear and tidal effects.

22 N-body simulations of gravitational clustering in an expanding universe

23 It s a capitalist s life Most of the action is in the big cities Newcomers to the city are rapidly stripped of (almost!) all they have Encounters generally too high-speed to lead to long-lasting mergers Repeated harassment can lead to change Real interactions take place in the outskirts A network exists to channel resources from the fields to feed the cities

24 Nonlinear evolution Assume a spherical cow. (Gunn & Gott 1972)

25 Zeldovich sphere Expansion/contraction same in all directions means λ 1 =λ 2 =λ δ(t) = [1 D(t) λ] 3 = [1 D(t) δ/3] 3 There is a critical density: δ = 3 Is this generic? Compare with exact sphere

26 Spherical evolution model d 2 R/dt 2 = GM/R 2 + ΛR = ρ (4πG/3H 2 ) H 2 R + ΛR = ½ Ω(t)H(t) 2 R + ΛR Note: currently fashionable to modify gravity. Should we care that only 1/R 2 or R give stable circular orbits?

27 decaying mode growing mode Excise a sphere and replace with smaller one of same mass Perturbing time of big bang decaying mode Perturbing energy of sphere growing mode

28 Spherical evolution model Initially, E i = GM/R i + (H i R i ) 2 /2 (Λ = 0) Shells remain concentric as object evolves; if denser than background, object pulls itself together as background expands around it At turnaround : E = GM/r max = E i So GM/r max = GM/R i + (H i R i ) 2 /2 Hence (R i /r) = 1 H i2 R i3 /2GM = 1 (3H 2 i /8πG) (4πR i3 /3)/M = 1 1/(1+ i ) = i /(1+ i ) i

29 Virialization Final object virializes: W = 2K E vir = W+K = W/2 = GM/2r vir = GM/r max so r vir = r max /2: Ratio of initial to final size = (density) ⅓ final density determined by initial overdensity To form an object at present time, must have had a critical over-density initially Critical density same for all objects! To form objects at high redshift, must have been even more over-dense initially

30 Spherical evolution model Collapse depends on initial over-density i ; same for all initial sizes Critical density depends on cosmology Final objects all have same density, whatever their initial sizes Collapsed objects called halos are ~ 200 denser than critical (background?!), whatever their mass Tormen 1997 (Figure shows particles at z~2 which, at z~0, are in a cluster)

31 Nonlinear evolution: Spherical collapse Turnaround: E = -GM/r max R(t) R init Virialize: -W=2K E = W+K = W/2 r vir = r max /2 t/t init Modify gravity modify collapse

32 Exact Parametric Solution (R i /R) vs. θ and (t/t i ) vs. θ very well approximated by (R initial /R) 3 = Mass/(ρ com Volume) = 1 + δ (1 D Linear (t) δ i /δ sc ) δsc Dependence on cosmology from δ sc (Ω,Λ), but this is rather weak

33 1 + δ (1 δ Linear /δ sc ) δsc As δ Linear δ sc ( 1.686), δ infinity This is virialization limit Zeldovich (approximation) has δ sc = 3 Standard perturbation theory has δ sc = 21/13 = 1.61 As δ Linear 0, δ δ Linear If δ Linear = 0 then δ = 0 This does not happen in modified gravity models where D(t) D(k,t) Related to loss of Birkhoff s theorem when r 2 lost? Note 1+δ 0 as δ Linear - Why is δ Linear < -1 sensible?

34 1 + δ (1 δ Linear /δ sc ) δsc 1 + δ Linear + (1+1/δ sc ) δ Linear2 /2 + j a j δ Linear j SPT has δ sc = 21/13 so δ δ Linear + (17/21) δ Linear2 + Terms like δ Linear2 being products in real space are convolutions in k-space Therefore k-modes of nonlinear δ are coupled, so evolved density field is non-gaussian Spherical evolution not the full story

35 Only very fat cows are spherical. (Lin, Mestel & Shu 1963; Icke 1973; White & Silk 1978; Bond & Myers 1996; Sheth, Mo & Tormen 2001; Ludlow, Boryazinski, Porciani 2014)

36 size Triaxial collapse: initial sphere evolves because of triaxial shear time Evolution of 2 nd axis very similar to spherical model of same initial density Collapse of 1 st axis sooner than in spherical model; collapse of all 3 axes takes longer

37 Tri-axial (ellipsoidal) collapse Evolution determined by properties of initial deformation field, described by 3 3 matrix at each point (Doroshkevich 1970) Tri-axial because 3 eigenvalues/invariants; Trace = initial density δ in = quantity which determines spherical model; other two (e,p) describe anisotropic evolution of patch Critical density for collapse no longer constant: On average, δ ec (δ in,e,p) larger for smaller patches low mass objects

38 Convenient Approximations Zeldovich Approximation (1970): (1 + δ) Zel = Π i=13 (1 D(t)λ i ) 1 Zeldovich Sphere (λ 1 = λ 2 = λ 3 = δ Linear /3): (1 + δ) ZelSph = (1 δ Linear /3) 3 (1 + δ) EllColl (1 + δ) SphColl (1 + δ) Zel /(1 + δ) ZelSph

39 Redshift space distortions

40 Redshift space distortions: peculiar velocities driven by gravity cz obs = Hd + v pec

41 Linear redshift space distortions The same velocities which lead to Zeldovich displacements make redshift space position different from real space position. x s = x + [v(x).d los / d los ]/H = q + v(q)/(afh) + [v(q) d los / d los ]/H Hence, same Jacobian trick yields δ (Kaiser 1987) δ s = (1 + fµ 2 ) δ where µ is angle wrt los, so P s (k,µ) = (1 + fµ 2 ) 2 P(k) and averaging over µ (1 + 2f/3 + f 2 /5) P(k)

42 Virial Motions (within halos ) (R i /r vir ) ~ f( i ): ratio of initial and final sizes depends on initial overdensity Mass M ~ R i3 (since initial overdensity «1) So final virial density ~ M/r vir3 ~ (R i /r vir ) 3 ~ function of critical density: Hence, all virialized objects have the same density, vir ρ crit (z), whatever their mass V 2 ~ GM/r vir ~ (Hr vir ) 2 vir ~ (HGM/V 2 ) 2 vir ~ (HM) 2/3 : massive objects have larger internal velocities or temperatures; H decreases with time, so, for a given mass, virial motions (or temperature) higher at high z

43 Nonlinear Fingers-of-God Virial equilibrium: V 2 = GM/r = GM/(3M/4π200ρ) 1/3 Since halos have same density, massive halos have larger random internal velocities: V 2 ~ M 2/3 V 2 = GM/r = (G/H 2 ) (M/r 3 ) (Hr) 2 = (8πG/3H 2 ) (3M/4πr 3 ) (Hr) 2 /2 = 200 ρ/ρ c (Hr) 2 /2 = Ω (10 Hr) 2 Halos should appear ~ten times longer along line of sight than perpendicular to it: Fingers-of-God Think of V 2 as Temperature; then Pressure ~ V 2 ρ

44 Two redshift space distortions: Linear + nonlinear

45 Redshift space distortions On large scales, use Gaussian statistics to compute (Fisher 1995)

46 Alcock-Paczynski