Introduc)on to Perturba)ons. Ay 127 April 18, 2013

Transcription

1 Introduc)on to Perturba)ons Ay 127 April 18,

2 Outline 1. Hydrodynamics in an Expanding Universe 2. Linear Perturba)ons in Cold MaHer 3. The Growth Func)on 4. Jeans Length and the Development of Perturba)ons in the Baryons 5. Primordial Perturba)ons 6. Evolu)on of Perturba)ons, Transfer Func)on 7. Acous)c Oscilla)ons Ref.: Chapters 11, 12 of Longair 2

3 Summary: Evolu)on of Different Scales a IGM rehea)ng maher- Λ equality recombina)on maher- radia)on equality No causal communica,on possible Rela,vis,c analysis necessary kpc 10 kpc 100 kpc 1 Mpc 10 Mpc 100 Mpc 1 Gpc 10 Gpc [comoving scale] π/k 3

4 Hydrodynamics In a non- expanding Universe, we generally use the Navier- Stokes equa)ons: ρ + v ρ = ρ v t v p + (v )v = Φ t ρ Combine with Poisson equa)on: 2 Φ = 4πGρ These equa)ons form the basis for most of astrophysical hydrodynamics. 4

5 in an expanding Universe Three modifica)ons are needed in the case of an expanding Universe: 1. Scale factor a: we use comoving coordinates in cosmology, but Navier- Stokes equa)ons are in physical units. 2. Veloci)es: we use peculiar veloci)es in cosmology, not total veloci)es. 3. Poten)als: we consider only gravita)onal poten)als from the inhomogenei)es, since the mean density is already in the Friedmann equa)on. 5

6 in an expanding Universe Three modifica)ons are needed in the case of an expanding Universe: 1. Scale factor a: we use comoving coordinates in cosmology, but Navier- Stokes equa)ons are in physical units. [1/a in deriva)ves] 2. Veloci)es: we use peculiar veloci)es in cosmology, not total veloci)es. [complicated] 3. Poten)als: we consider only gravita)onal poten)als from the inhomogenei)es, since the mean density is already in the Friedmann equa)on. [ρ δρ in Poisson equa)on] 6

7 Peculiar Veloci)es Claim: In an expanding universe, peculiar velocity of nonrela)vis)c maher declines as ~1/a. This effec)vely leads to a new force: F H = Hp = mhv Quantum Mechanical Argument: De Broglie wavelength of par)cle: λ ~ a. Momentum p = h/λ ~ 1/a. Velocity v = p/m ~ 1/a. 7

8 Peculiar Veloci)es Part II Classical argument: this Hubble fric)on is a fic))ous force arising from the use of a posi)on- dependent reference frame. Decompose total velocity V into a peculiar velocity v and a velocity of the reference frame v H = ahx: V = v + v H = v + ahx Consider a par)cle at the origin: instantaneously x=0. In the absence of any forces on a par)cle, V remains constant but v changes according to: v = ah x = Hv 8

9 The Revised Equa)ons Old: ρ + v ρ = ρ v t v p + (v )v = Φ t ρ 2 Φ = 4πGρ New: ρ t + a 1 v ρ = a 1 ρ (v + ahx) v t + p a 1 (v )v = a 1 Φ a 1 ρ Hv a 2 2 Φ = 4πGδρ It s common to re- write the density equa)ons in terms of the density contrast: Δ δ m ρ m ρ m ρ m = δρ m ρ m 9

10 Conversions We know the background density evolu)on for maher: ρ m t = 3Hρ m The density contrast evolves according to: δ m t = ρ m 1 = ρ ρ + ρ (3Hρ ) m m m m 2 t ρ m ρ m = ρ m + 3Hρ m ρ m Subs)tu)on allows us to build perturba)on equa)ons in terms of δ m, v m. 10

11 The Revised Equa)ons Old: ρ + v ρ = ρ v t v p + (v )v = Φ t ρ 2 Φ = 4πGρ New: δ m t v m t + a 1 v m δ m = a 1 (1+ δ m ) v m + a 1 (v m )v m = Φ a p m aρ m Hv m a 2 2 Φ = 4πGρ m δ m 11

12 Linear perturba)on theory Assume perturba)ons are small. Ignore quan))es second- order in δ m, v m, Φ. Valid in early Universe (e.g. forma)on of CMB, pre- galac)c medium) Valid today averaged over sufficiently large scales (>>10 Mpc). We ll do this without pressure (dark maher!) first, and then come back and consider pressure. 12

13 Perturba)on Equa)ons Exact: Approximate: δ m t + a 1 v m δ m = a 1 (1+ δ m ) v m δ m t = a 1 v m v m t + a 1 (v m )v m = Φ a Hv m a 2 2 Φ = 4πGρ m δ m v m t = Φ a Hv m a 2 2 Φ = 4πGρ m δ m 13

14 Solving the Perturba)on Equa)ons In the linearized equa)on, define: Then the equa)ons reduce to: θ a 1 v δ m = θ t θ t = 2 Φ a 2 Hθ a a θ = 4πGρ mδ m 2Hθ This becomes a single, 2 nd order ODE: δ m + 2H δ m 4πGρ m δ m = 0 14

15 Example: Einstein- de SiHer Model We will illustrate this equa)on for the EdS model: Simple Basic physics applies to other cases Actually describes Universe from z~1000 to z~1 In EdS, the density is related to )me via ρ m = ρ c = 3H 2 Growth equa)on is 8πG = 3( 2 3t )2 8πG = 1 6πGt 2 δ m + 4 δ m 2 3t 3t δ = 0 2 m 15

16 EdS model, Part III Real density evolu)on will be a superposi)on of the two modes: δ m = C 1 t 2 / 3 + C 2 t 1 In real Universe at late )mes, growing mode will dominate: δ m t 2 / 3 a. Veloci)es obey: θ v a = δ m = Hδ m Poten)als: 16 Φ a 2 ρ m 2 δ m a 2 a 3 a = constant

17 Defini)ons Growth func)on: δ m G(a) Defined for growing mode (δ m non- divergent at a=0). Normalized to G(a)=a during maher- dominated phase. f (a) d lng(a) d ln a = δ m = v Hδ m ahδ m Growth rate: During EdS phase: G(a)=a, f(a)=1. Measuring these func)ons is a major goal of dark energy projects. 17

18 Perturba)ons in ΛCDM Recall growth ODE: δ m + 2H δ m 4πGρ m δ m = 0 With Λ>0, and at fixed t: Hubble term is larger (d ln a/d ln t > ⅔) MaHer term is smaller (lower mean density) Therefore G(t) grows more slowly than t 2/3. Fixed scale factor a is reached at earlier )me, so G(a) grows more slowly than a. 18

19 Growth func)on Growth func,on, G(a) Scale factor, a EdS Lambda CDM Open CDM Λ, open models Assume Ω m0 =0.3 19

20 Perturba)ons during Radia)on Era Consider dark maher perturba)ons growing in a radia)on- dominated background. (Ignore baryons for the moment.) Radia)on is smoothly distributed so we can ignore its contribu)on to Φ and use the analysis of the previous pages. Background Hubble expansion: a~t 1/2, H=1/(2t). δ m + 1 δ m 4πGρ m δ m = 0 t 20

21 Radia)on Era, Part II MaHer density? ρ m = ρ m ρ r ρ r = a a eq ρ c From last week s lecture: a = ρ r,eq a eq ρ r 1/ 4 ρ r,eq ρ c 1/ 4 So: 1/ ρ m = ρ 4 3 / r,eq ρ 4 c = / 4 πg H 1/ 2 eq H 3 / 2 = / 4 πg 1/ 2 H eq t 3 / 2 Simplify growth ODE: δ m + t 1 δ m 3H 1/ 2 eq t 3 / 2 δ 2 11/ 4 m = 0 At t<<t eq, the third term is negligible (examine powers of t): δ m + t 1 δ m = 0 21

22 Solu)ons in Radia)on Era Dimensionally homogeneous equa)on: δ m + t 1 δ m = 0 Try power law solu)on, δ m ~t n : n(n 1)t n 2 + t 1 nt n 1 = 0 n=0 is double root Implies general solu)on is: δ m = C 1 + C 2 ln t Perturba)ons in maher grow logarithmically during radia)on era. 22

23 Pressure Effects and Jeans Length Basic Ques)on: Do the baryons (observable) fall into the poten)al wells (not directly observable, and o}en caused by dark maher)? Answer: Some)mes! 23

24 Perturba)ons with Pressure Linearized equa)ons for fluid i with δp=c s2 δρ: δ i t = a 1 v i v i t = Φ a Hv i a 1 c s 2 δ i When considering perturba)ons with pressure it s convenient to work in Fourier space: δ m (k) = δ m (r)e ik r d 3 r δ m (r) = 1 (2π) 3 δ m (k)e ik r d 3 k 24

25 Pressure in Fourier Space In Fourier space, the perturba)on equa)ons are: δ i t v i t = ia 1 k v i = ikφ a With velocity divergence: δ i t θ i t = θ i = k 2 Φ a 2 Hv i ia 1 c 2 s kδ i 2Hθ i + c 2 s k 2 δ a 2 i θ i = ik v i a 25

26 Case I: Radia)on Era Consider baryon- photon fluid (single fluid because of Thomson scahering). This fluid dominates the poten)al so: δ i t θ i t = θ i = 4πGρ i δ i 2Hθ i + c 2 s k 2 δ a 2 i Perturba)on growth depends on Jeans wavenumber: k J a c s 4πGρ = 3 2 ah c s = 3 8 a c s t For long wavelengths (k<k J ), pressure term is neglgibile. For short wavelengths (k>k J ), self- gravity is negligible. 26

27 Case I: Radia)on Era In reality, there are complica)ons: Jeans wavelength λ J = 2π = 32 k π a 3 J c s t is the distance a sound wave can travel in the life)me of the Universe. But for radia)on- dominated plasma, c s =c/ 3. Thus not even light could travel more than ~1 λ J in the life)me of the Universe. Need rela)vis)c analysis. But it s correct that on scales <<λ J the photon- baryon fluid does not clump (ODE is a harmonic oscillator). 27

28 Case II: MaHer Era Now suppose poten)al is dominated by dark maher. Evolu)on equa)on for baryons is 2 nd order: δ b + 2H δ b c 2 s k 2 δ a 2 b k 2 Φ = 0 a 2 δ b + 2H δ b c 2 s k 2 δ a 2 b + Φ 2 c s Two limits depending on sound- crossing )me: = 0 If k>a/c s t (short wavelengths), 3 rd term is large and we reach pressure equilibrium: δ b = Φ/c s2. If k<a/c s t (long wavelengths), 3 rd term is irrelevant and baryons behave like DM. 28

29 Case II: MaHer Era Let s consider the short wavelengths in a bit more detail. Pressure equilibrium combined with Poisson equa)on gives: δ b = Φ c = 4πGa2 ρ dm δ dm = 2 a 2 s k 2 2 c s 3 kc s t So when k~a/c s t, the two solu)ons match up. At smaller scales the perturba)ons in the baryons are suppressed. 2 δ dm 29

30 Summary: Evolu)on of Different Scales a Nonlinear transi)on, Δ~1 Baryons smoothly distributed IGM rehea)ng DM switches to growth δ~a DM perturba,ons grow logarithmically Acous,c oscilla,ons in baryon- photon plasma Baryon and DM perturba,ons grow ~ G(a) maher- Λ equality recombina)on maher- radia)on equality No causal communica,on possible Rela,vis,c analysis necessary kpc 10 kpc 100 kpc 1 Mpc 10 Mpc 100 Mpc 1 Gpc 10 Gpc [comoving scale] π/k 30

31 Notes Without DM, perturba)ons on galac)c scales cannot grow before recombina)on. Would need ~1% density fluctua)ons at z=1000 to create galaxies. But observed perturba)ons in CMB are a few parts in The first baryonic objects form at minimum Jeans scale, ~ few kpc. Implied mass is M ~ 4 3 πr3 ρ m 0 ~ 10 3 M Sun 31

32 Primordial Perturba)ons 32

33 Superhorizon Regime Perturba)ons with very long wavelengths: k < k H ah c a ct are superhorizon light cannot travel ~1 perturba)on wavelength in the age of the Universe. Since during radia)on era, a~t 1/2, k H at early )mes. Therefore all perturba)ons start out superhorizon. 33

34 Behavior in Superhorizon Regime Since different regions in the superhorizon perturba)on are causally disconnected, they behave as separate universes. Any spa)al geometry is legal on large scales since locally (within any causally connected patch) it looks flat. These local patches of the Universe each obey the flat FRW solu)on. 34

35 Classifica)on Line element for the lumpy universe: ds 2 = dt 2 a2 (t) 3 3 [1+ 2ζ(x)] (dx i ) h c 2 ij (x)dx i dx j i=1 i, j=1 h ij is a traceless symmetric 3x3 matrix. Depends on 6 func)ons of x 1 in ζ, 5 in h By change of spa)al coordinates, we can always make h divergence- free ( transverse ): 3 h ij = 0 x j j=1 35

36 Classifica)on, Part II This leaves 3 free func)ons of x: ζ(x): the curvature perturba)on. Describes local deforma)ons of the scale factor. These will become density perturba)ons. h ij (x): 2 free func)ons. These cannot become density perturba)ons by symmetry (no way to construct a scalar out of them) and have no Newtonian analogue. They represent primordial gravita)onal waves. O}en called scalars and tensors. 36

37 Classifica)on, Part III It is also possible for there to be perturba)ons that are not represented in the geometry of the Universe but rather the distribu)on of different types of maher. These are called isocurvature perturba)ons. Two main examples: Baryon isocurvature: baryon/photon ra)o η is spa)ally variable. Dark maher isocurvature: DM par)cle/photon ra)o is spa)ally variable. Neither has an effect on geometry at early )mes since baryons, DM don t contribute significantly to the cosmic pie. 37

38 Classifica)on of Primordial Perturba)ons Adiaba,c Isocurvature Scalar (curvature perturba)on, ζ) Tensor (gravita)onal waves, h ij ) Baryon Dark maher (any conserved quan)ty can have an isocurvature perturba)on) Adiaba'c scalars are the only kind of perturba'ons observed so far, so we ll focus on them. 38

39 Evolu)on on Large Scales The largest- scale perturba)ons in the universe can be understood by considering a slightly curved universe as a perturba)on to a flat universe. This is valid on scales larger than the maximum sound- crossing length (~150 Mpc comoving). Birkhoff s theorem: a local overdensity doesn t know what s outside of it as long as it s spherically symmetric. All curvature perturba)ons are superposi)ons of spherically symmetric perturba)ons with different centers. Subtlety: in full GR theory, must do a change of coordinates ( gauge transforma)on ) to recover Newtonian picture at late )mes. 39

40 Evolu)on on Large Scales The largest- scale perturba)ons in the universe can be understood by considering a slightly curved universe as a perturba)on to a flat universe. Recall that under stereographic projec)on, the length element of a hypersphere is: N=O d 2 = dx 2 + dy 2 + dz 2 [ R 2 (x 2 + y 2 + z 2 )] 2 S 40

41 Correspondence to Curvature Perturba)on Since R is related to Ω K, this gives us an alternate line element for the curved universe: ds 2 = dt 2 a 2 curved (t) dx 2 + dy 2 + dz 2 c Ω 4 Kc 2 H 2 0 (x 2 + y 2 + z 2 ) [ ] 2 To linear order (Ω K small), this is equivalent to a curvature perturba)on: ζ = 1 4 Ω K c 2 H 0 2 (x 2 + y 2 + z 2 ) Alternate (and more general) form: 2 ζ = 3 2 Ω Kc 2 H

42 Density Contrast Evolu)on Ignore Λ for the moment assume we re perturbing around EdS. Compare density evolu)on in the flat universe: to curved universe: ρ curved = 1 6πGt 2 ρ EdS = 1 6πGt Ω K The density contrast between the curved solu)on and the background is: δ m = 3 5 Ω K t t 0 2 / 3 t t 0 2 / 3 = 3 5 Ω K a + O(Ω 2 K ) 42

43 Density Contrast Evolu)on, Part II Let s express this in terms of the curvature perturba)on: But the Newtonian poten)al is also related to the density: Combine to get: δ m = 3 5 Ω K a = 2 5 c 2 a H ζ 4πGa 2 ρ m δ m = 2 Φ 2c 2 a Φ = 4πGa 2 ρ m 5H ζ = 8πG c 2 ρ m a 3 ζ = H 0 5 c 2 ζ This relates poten)als in maher era to primordial ζ. 43

44 Inclusion of Λ Since the universe passed through an EdS phase, and we have solved for perturba)on dynamics including Λ, we can simply write down the answer: δ m (x) = 2 5 Φ(x) = 3 5 c 2 G(a) a c 2 Ω m H 0 2 G(a) 2 ζ(x) ζ(x) 44

45 Smaller Scales At scales <150 Mpc, perturba)ons went through a phase where the sound- crossing )me was less than the age of the universe. This results in more complicated equa)ons for the growth of structure, as described earlier. Usually packaged into a transfer func)on, T(k): δ m (k) = 2 5 Φ(k) = 3 5 c 2 G(a) a c 2 Ω m H 0 2 G(a)k 2 T(k)ζ(k) T(k)ζ(x) 45

46 a Small vs. Large Scale Evolu)on Nonlinear transi)on, Δ~1 Baryons smoothly distributed IGM rehea)ng DM switches to growth δ~a DM perturba,ons grow logarithmically Acous,c oscilla,ons in baryon- photon plasma small scale Baryon and DM perturba,ons grow ~ G(a) large scale maher- Λ equality recombina)on maher- radia)on equality No causal communica,on possible Rela,vis,c analysis necessary kpc 10 kpc 100 kpc 1 Mpc 10 Mpc 100 Mpc 1 Gpc 10 Gpc [comoving scale] π/k 46

47 The Program 1. Obtain range of scale factors a min a max for evolu)on within sound- crossing length. 2. Determine ini)al condi)ons at a min using curved- universe argument. 3. Propagate to a max and match to maher- dominated solu)on to find T(k). 47

48 1. Range of Scales Difference between large and small scales is that small scales pass through a regime where the DM perturba)ons grow only as ln a. The range of scale factors for which this occurs is from a min a max, where a max = a eq and ct(a min ) 3a min = π k Solve: k = 3πa min ct(a min ) = 2 3πa minh(a min ) c = 2 3πa min c 2 H eq a eq 2 2a min a min = 6π k eq k a eq where k eq = a eqh eq c = 0.01 Mpc 1 48

49 2. Ini)al Condi)ons In general at horizon entry (k~ah/c) we expect perturba)ons δ~ζ. (No dimensionless constants in problem.) e.g. for entry in maher era, δ m = 2 ck 5 ah Similar calcula)on for radia)on- dominated universe (homework exercise) gives: 2 ζ δ r = 1 ck 3 ah 2 ζ; δ m = 3 4 δ r = 1 4 ck ah [This is an exact solu)on for k<<ah/c in Lagrangian coordinate system; see Ma & Bertschinger 1995.] 2 ζ 49

50 2. Ini)al Condi)ons When sound speed becomes important: ck ah amin δ m = 1 ck 4 ah Expand from a min a eq : = 2ctk a amin = 2π 3 Match to defini)on of transfer func)on: 2 ζ = 3π 2 ζ δ m (a eq ) ~ 3π 2 ln a eq a min ζ = 3π 2 ln δ m (k,a eq ) = 2 5 k 6π k eq ζ c 2 Ω m H 0 2 a eq k 2 T(k)ζ(k) 50

51 3. Matching to MaHer Era Solu)on Use equa)on for equality scale factor: we find: T(k) ~ k eq = 3π 2 ln 2 c Ω m H a k 2 eq 0 c a eq H eq = T(k) ~ 15π 2 4 a eq 2 k eq k ln 2 Real answer is somewhat smaller due to matching condi)ons (e.g. at a eq, some of the perturba)on ends up in the decaying mode). 15π 2 / c k 6π k eq 2Ω m H a eq k 6π k eq

52 Transfer func)on: results 1 Eisenstein & Hu (1998), no baryon func,on T(k) 0.1 CDM k (Mpc 1 ) 52

53 Baryons Baryons are ~17% of the maher in the Universe. Baryons have lihle effect on the maher distribu)on at >>150 Mpc scales. But at smaller scales the perturba)ons in the DM could grow (logarithmically) during the radia)on era, whereas the perturba)ons in the baryons did not. Recall evolu)on equa)on for baryons when pressure dominates: δ i + 2H A harmonic oscillator! δ i + c 2 s k 2 a 2 δ i = 0 53

54 Baryons, Part II Baryon density oscillates through N cycles, where: N = 1 t rec ωdt = 1 ck t rec 2π 0 2π 3a dt = kη rec 0 2π 3 dt Quan)ty η = is called the conformal )me. a At )me of recombina)on, baryons are either going into DM wells or out of depending on phase of oscilla)on. This changes transfer func)on by ~17% - - if the oscilla)on survives un)l recombina)on. 54

55 Baryons, Part III Sound waves in baryon- photon fluid depend on photons providing pressure support. This won t happen if the photons can diffuse out of overdense regions. Photon diffusion length is shorter than light- crossing length by a factor of τ. From past lectures: r light = ct a τ = n e σct r diff = r light τ = 1 a ct ~ 10 recombination n e σ 55

56 Baryons, Part IV This diffusion length is called the Silk damping length and represents the scale below which photon diffusion can dissipate sound waves. 56

57 a Baryons smoothly distributed Evolu)on Including Baryons Nonlinear transi)on, Δ~1 IGM rehea)ng DM switches to growth δ~a Baryon and DM perturba,ons grow ~ G(a) maher- Λ equality recombina)on maher- radia)on equality DM perturba,ons grow logarithmically Acous,c oscilla,ons in baryon- photon plasma No causal communica,on possible Rela,vis,c analysis necessary kpc 10 kpc 100 kpc 1 Mpc 10 Mpc 100 Mpc 1 Gpc 10 Gpc [comoving scale] π/k 57

58 Transfer func)on: results 1 Eisenstein & Hu (1998) formulae T(k) CDM all k (Mpc 1 ) 58