Cosmological Structure Formation Dr. Asa Bluck

Transcription

1 Cosmological Structure Formation Dr. Asa Bluck Week 6 Structure Formation in the Linear Regime II CMB as Rosetta Stone for Structure Formation Week 7 Observed Scale of the Universe in Space & Time Week 8 Evidence for the Lambda CDM Cosmology Contact: asa.bluck@phys.ethz.ch 1

2 Week 6 Structure in the Universe Part II Basic growth modes Modification of the primordial power spectrum Fluctuations on the CMB LSS The inhomogeneous Universe 2

3 From W5 Jeans-Lifshitz Week 6 Newtonian fluid in physical coordinates: Continuity equation: Equation of motion (Euler equation): Poisson equation: ρ t = ρ v v t + (v )v = 1 p φ ρ 2 φ = 4πGρ Perturb the fluid (i.e. ρ = ρ + δρ, v = v + δv etc.) and treat it a first order Convert to comoving frame Perform Fourier transform Use c s2 = dp/dr (i.e. speed of sound) See Exercise 3

4 From W5 The growth of Δ through gravitational instability Week 6 1. Jeans-Lifshitz analysis for given Fourier component ρ = ρ ρ = ρ +δρ 2. Application of Friedmann solutions for homogeneous Universes to a local density excess d 2 Δ k dτ 2 + 2! R R dδ k dτ = # 4πGρ k 2 c $ s % R 2 2 & ' ( Δ k Damping term from expansion of Driving term from Pressure support Universe gravity 4

5 From W5 d 2 Δ k dτ & R R dδ k dτ = " 4πGρ k 2 c # s $ R 2 2 % & ' Δ k Week 6 Neglecting for the moment the second term (i.e. non-expanding medium) the solutions clearly depend on the sign of RHS: When negative (pressure dominates), we have oscillatory solutions with standing acoustic wave of constant amplitude and oscillation frequency { } ~ k 2 c s 2 / R 2 ω 2 = k 2 c s 2 / R 2 4πGρ When positive (gravity dominates), we have exponential increase in D k Δ = Δ 0 exp( ik r + Γτ ) with Γ = 4πGρ k 2 c 2 s / R ~ 4πGρ This is called gravitational instability. In a static medium it is very fast, with a timescale of (4pGr) -1/2 If there is pressure, then there is a critical wavelength (the Jeans length ) above which pressure cannot support the density perturbation: k J = R 4πGρ c s 2 or λ J = 2π c s 4πGρ ~ 2π c s Γ 1 5

6 What is the Jeans length for matter/radiation at different epochs? Coupling between matter and radiation prior to recombination allows pressure support from radiation to be applied to the matter giving rise to high sound speed and large Jeans mass (mass within the Jeans length). Note that non-interacting dark matter experiences no pressure and has zero sound speed and Jeans mass (see later). Week 6 Looking at just matter and radiation: p 2 lj = cs cs = Gr dp dr At R < R eq, radiation dominates both r and p: At R > R eq, baryonic matter dominates r but radiation provides all the p through scattering After recombination, coupling is lost and the pressure comes from thermal pressure of gas p = 1 3 ρc2 so c s 2 = c2 c s 2 = c2 3! 4ρ " rad # 3ρ b + 4ρ rad c 2 5 s = 3 $ % & 3 kt m p 6

7 What is the density at different epochs? Consider W = 1 Universes dominated by matter or radiation ρ = 3H 2 8πG with H = 2 3τ matter-dominated H = 1 2τ radiation-dominated In the radiation-dominated era, the Jeans length is essentially the horizon scale. λ J = 2π c s 4πGρ = 2π 2 cτ ~ d H In the matter-dominated era before recombination, it is easy to show that it is the horizon scale at R eq After recombination, it drops to low masses (of order 10 6 M Sun ) 7

8 log mass 15 GROWTH Horizon Mass We will see that the mass in Horizon at R eq is a key quantity M H (τ eq ) ~ (Ω m,0 h 2 ) 2 M sun >> M galaxy 6 OSCILLATION Jeans Mass Req log R Rrec Remember: this is just for the photo-baryon fluid because it reflects pressure effects 8

9 Now look at the case of an expanding medium when we can neglect pressure, either because we are looking at scales l >> l J or because we are dealing with non-interacting dark matter d 2 Δ k dτ 2 + 2H dδ k dτ = 4πGρ Δ k Look at matter dominated case: H = 2 3 τ 1 and 8πGρ = 3H 2 4πGρ = 2 3 τ 2 d 2 Δ k dτ + 4 dδ k 2 3τ dτ 2 3τ Δ = 0 2 k Clearly there will be power-law solutions, with D k proportional to τ n 4 2 n( n -1) + n - = 0 Two solutions: n = 2/3 and n = Growing mode solution: Δ k τ 2/3 R Not exponential (!) Independent of k 9

10 Look at radiation-dominated case. Previous equation becomes d 2 Δ k dτ 2 + 2H dδ k dτ = 32 3 πgρ Δ k d 2 Δ k dτ dδ k τ dτ 1 τ Δ = 0 2 k Power-law Ansatz: n(n 1)+ n 1= 0 Two solutions: n = +1 and -1 Growing mode solution: Δ k τ 1 R 2 Look at low density Universe r ~ 0. Now have H = t -1 d 2 Δ k dt 2 + 2H dδ k dt = 0 n(n 1)+ 2n = 0 Two solutions: n = 0 and -1 There is no growing mode solution. Fluctuations are at best frozen in at constant amplitude 27

11 Growth outside of the horizon Week 6 It can be shown using General Relativity that outside of the horizon perturbations effectively grow like Δ k 1 H 2 R 2 R 2 (radiation dominated) R (matter dominated) This behavior is caused by the fact that perturbations outside the horizon are essentially frozen. 11

12 Entering the horizon Week 6 Interesting calculation: For a given initial spectrum, defined at some fixed epoch with some primordial value of n, what is the amplitude of a given mass scale when it equals the horizon scale? We have: d H ~ (cτ ) R 2 (radiation dominated) M H ~ ρ matter d H 3 R 3/2 R 3 Δ k 3/2 R 3/2 Δ k 3/2 (matter dominated) (radiation dominated) (matter dominated) Therefore the amplitude of a mode Δ k (relative to its initial amplitude at a fixed starting time in the past) when it enters the horizon is proportional to M H 2/3 in both the radiation and matter dominated case. Therefore: Δ MH (3+n) 6 M initial M 2 (1 n) 3 6 M H (for all τ) at t when k = k H (t) (i.e. when mode k enters the horizon) The Harrison-Peebles-Zeldovich n = 1 spectrum has D m independent of scale (mass or time) as measured on the horizon scale often called scale-free spectrum It can be shown by using quantum field theory and GR that the simplest models of Inflation produces a nearly scale-free spectrum n 1. 12

13 Should you be worried by the idea of growth of D scales larger than the horizon? on Nowhere in our derivation of Friedmann R(t) etc. did we consider horizons. ρ = ρ You can derive exact development of density inhomogeneities using Friedmann solutions for each part ρ = ρ +δρ 30

14 Consider fluctuations in the curvature The average flat Universe will have H 2 = 8π 3 Gρ The perturbed region will have some curvature Rearranging H 2 = 8π 3 G ρ! k k A δ 2 R 2 = Δ (c H ) 2 c2 A δ 2 R 2 So the radius of curvature due to the density perturbation is A δ = ω H kδ Radiation dominated: w H µt 1/2 D µ t Matter dominated: w H µt 1/3 D µ t 2/3 A d stays constant during expansion and linear development of structure (as it must) 31

15 Growth within the horizon Need to worry about causal physical effects Density fluctuations may undergo any of the following: oscillate as sound-waves (baryons + photons) be damped out through diffusion effects undergo growth but at a reduced rate Dark matter, matter and radiation may behave differently because of different causal physics (e.g. pressure effects on baryons but not on dark matter) 15

16 On scales smaller than the (baryon+photon) Jeans Mass, baryon/photon fluctuations will oscillate. Week 6 Dark matter fluctuations are free to grow because DM does not interact (with itself or with baryons/photons). But if Universe is radiation dominated, growth of DM fluctuations is suppressed ( stagnation effect, Meszaros effect ) log mass GROWTH Horizon Mass 15 Here baryons and photons oscillate due to pressure 6 Here DM growth suppressed because r DM << r rad OSCILLATION Here DM grows normally because it dominates the density Jeans Mass Req log R Rrec 16

17 Meszaros effect Can approach this in two ways: Compare collapse timescale of DM and expansion timescale of the Universe τ = 1 Γ 1 = ( 4πGρ DM ) 1/2 2H =! 32πGρ rad # " 3 $ & % 1/2 ρ rad >> ρ rad Γ 1 >> τ So it doesn t happen Replace driving density on RHS with r DM D k, effectively zero when r DM << r rad d 2 Δ k dτ 2 d 2 Δk dτ 2 + 2H dδ k dτ = 32 3 πgρ Δ k + 2H dδ k dτ = 0 à Perturbations are frozen 17

18 Meszaros effect flattens the shape of D M log mass GROWTH Horizon Mass k 3/2 [P(k)] 1/2 ( = k 3/2 A k ) Primordial slope n = +1 Δ M M 3+n M 15 6 OSCILLATION Jeans Mass Must be flat on scales << M H,eq i.e. n mod = -3 Req log R Rrec k 3 M 18

19 Meszaros effect Introduce transfer function T(k) to describe modified spectrum: P(k) = k n T(k) 2! T(k) = 1+ ak + bk "# { ( ) 3/2 + ( ck) 2 } ν with a = 6.4( Ω m,0 h 2 ) 1 Mpc b = 3.0( Ω m,0 h 2 ) 1 Mpc c =1.7( Ω m,0 h 2 ) 1 Mpc ν=1.13 $ %& 1/ν 19

20 Damping processes The fluid dynamical approach used so far does not consider dissipative effects. Are there processes that will damp out density fluctuations? Recall that Key idea: If individual particles travel a distance (over the age of the Universe) that is large compared with the wavelength of the fluctuation, then the fluctuation will be damped out simply by diffusion and averaging. Two cases of interest Non-interacting dark matter that is either relativistic or non-relativistic (free-streaming damping) Photons interacting with matter (Silk damping) [As we discuss these, remember: we are considering Fourier modes with amplitudes << unity.] 20

21 Free-streaming damping Non-interacting particles move in a straight line. (If they are relativistic v = c. If they are non-relativistic v << c.) Relativistic particles will move a distance equal to the (particle) horizon scale. Non-relativistic particles move a much smaller (negligible) distance. For as long a non-interacting DM particles are relativistic then we would expect DM density fluctuations to be erased on all scales below the horizon. D M =k 3/2 [P (k)] 1/2 M H (t NR ) (=k 3/2 A k ) If the particles later become nonrelativistic, then there will be a maximum in DM fluctuations on the scale corresponding to M fs ~ M H (t NR ) M fs ~ m 30eV 1 d fs ~ 40 m 30eV M sun comoving Mpc M 21 k 3

22 Cold, Hot and Warm Dark Matter M fs ~ m 30eV M sun Week 6 M fs defines the nature of DM, and (probably) the mass of the DM particle. CDM M fs << M galaxy (10 12 M sun ) i.e. free-streaming damping is negligible WDM HDM M fs ~ M galaxy M fs >> M galaxy Note: If particles are born completely non-interacting, they may never be thermalised to high temperatures (e.g. axion) In this case CDM particles could be of low mass. Relative to CDM, WDM has less power on small scales, resulting in fewer low mass objects, fewer early forming haloes (see later for why). Best observational constraint comes from P(k) in Lyman a forest at high redshift: d fs < 150 kpc, M fs < M Sun m DM > 8 kev HDM-dominated is strongly ruled out. Warm-ish DM (or mixture) could solve some problems 22

23 Silk Damping: Damping processes in matter-radiation oscillations If photons can diffuse out of the fluctuation during the lifetime of the Universe, then density fluctuation will be severely damped (i.e. fluctuations will disappear ). Mean free path of photon: d T ~ 1 n e σ T ~ n 1 e cm Mean time between scattering: t T ~ d T c Distance travelled in age of Universe t in random walk is the geometric mean of mean free path and the horizon scale (both of which increase with time) 1/2 d Silk ~ d T n scat " c ~ d T $ τ H # d T ~ d T d H % ' & 1/2 Fluctuations on shorter scales than d Silk at t rec (when scattering ceases) will be damped out # Ω M Silk ~ b,0 % $ Ω m,0 & ( ' 3/2 ( Ω m,0 h 2 ) 5/4 M Sun 23

24 log mass 15 GROWTH Horizon Mass 6 OSCILLATION DAMPED Silk Mass No baryonic fluctuations should survive on mass scales < 3x10 13 M sun Jeans Mass Req log R Rrec This is a potentially very serious problem for baryon-only scenarios for galaxy formation since purely baryonic fluctuations on galactic scales do not survive to the end of recombination 24

25 Baryonic acoustic oscillations Week 6 Below the Jeans mass, photon-baryon fluctuations oscillate as standing sound waves with frequency w(k) = (k 2 -k J2 ) 1/2 c s /R They all start oscillating at fixed phase (i.e. maximum amplitude). The frequency of oscillation and the duration of oscillation both depend (only) on k. Therefore the phase of a given mode at a particular epoch depends on k. log mass 15 6 GROWTH OSCILLATION f = [ 0,2p ] depending on k Horizon Mass Jeans Mass Req log R Rrec 25

26 Baryonic acoustic oscillations SDSS Percival et al. (2007) Week 6 Figure 3. BAO recovered from the data for each of the redshift slices (solid circles with 1σ errors). These are compared with BAO in our default CDM model (solid lines). 26

27 Post-recombination growth By recombination, dark matter fluctuations dr/r will be larger than the matter/radiation fluctuations which will have been held at constant amplitude, or even damped out, while the DM continues to grow. This is a benefit as it means underlying dr/r is generally larger than is directly visible on last scattering surface at small scales What then? Regular matter quickly falls into DM potential wells, rapidly catching up in density contrast. # Δ b = Δ DM 1 R rec % $ R & ( ' 27

28 Post-recombination growth [ k 3 P / (2 π 2 ) ] 1/2 [ k 3 P / (2 π 2 ) ] 1/ z = z = k (Mpc -1 ) z = 800 z = k (Mpc -1 ) Figure 3.1. Power spectra (in dimensionless form k 3 P) of the density fluctuations of DM and baryons at di erent redshifts. The solid curve corresponds to DM perturbations and the dotted curve to baryon perturbations. The horizon scale at z = 1000 is about k Inside the horizon the baryonic acoustic oscillations in the power spectrum of the baryons are clearly visible and it is shown how over the timespan of z = 1200 to z = 200 the baryon power spectrum slowly catches up with that of DM, as the baryons fall into the gravitational potential wells of previously formed DM structures. (Adapted from Naoz & Barkana 2005) 28 from growing due to the photon pressure, while the DM perturbations grow unimpeded. After

29 Primordial spectrum P(k), probably from inflation Baryon-photon fluctuations oscillate on all scales below M H (t eq ) with phasing effects at given epoch Baryon-photon fluctuations damped out on scales below Silk Mass DM fluctuations have growth suppressed within horizon until t eq ( Meszaros effect ) DM fluctuations damped out on scales below M H (t NR ) Recombination and Last Scattering of the CMB Baryons fall into surviving DM fluctuations, so D B tends to D DM 29

30 Angular Scales on the Last Scattering Surface 1 degree = 260 comoving Mpc with concordance cosmological parameters Horizon at recombination ~ 350 comoving Mpc ~ 1.4 deg Horizon at R eq ~ 250 comoving Mpc ~ 1 deg Silk damping scale ~ 0.2 deg Finite thickness of LSS in w is equivalent to 0.2 deg 30

31 Imprinting dr/r as dt/t Week 6 There are three mechanisms to imprint density fluctuations dr/r on the LSS on the brightness of the CMB. All three produce pure temperature fluctuations dt/t (c.f. other processes along the line of sight, see later) 1. Adiabatic compression of baryon-photon fluids Consider a region of size l, compressed by dl Just like expanding Universe Yielding dr dl = -3 r l dt dl T = - l dt dr T ~ 1 3 r 31

32 2. Doppler effects Moving scatterers produce temperature shift dt/t ~ v/c What will the typical velocities associated with the density fluctuation be? dt T Well above Jeans mass v ~ dl t H v dl l ~ ~ ~ c ct ct H H 1 dr 3 r Well below Jeans mass ω 2 = c 2 s k 2 R 2 2! 2π $ = c s # & " λ % 2π v ~ ω δλ ~ c s λ δλ ~ c δρ s ρ 2 δt T ~ λ δρ d H ρ dt T ~ cs c dr r Increasing with wavelength, dominates Comparable to adiabatic 32 over adiabatic above d H

33 3. Gravitational effects (Sachs-Wolfe effect) Note: this is the only one that applies for DM fluctuations Gravitational redshift (to prove this we would need GR): δt T ~ 1 δφ 3 c 2 δφ ~ G δm λ Gρ = 3H 2 8π ~ 1 τ 2! ~ G δρλ 2 ~ G δρ $ # &ρλ 2 " ρ % df ~ c 2 dr r 2 l 2 D H dt T æ l ~ ö ç è D H ø 2 dr r 33

34 dt dr T ~ 1 3 r dt T ~ l dr r D H dt T æ l ~ ö ç è D H ø 2 dr r Different (l/d H ) dependencies mean that different physical processes will dominate on different scales Note that (1) and (2) produced only by baryon-photon dr/r, but (3) is produced by all dr/r as it is purely gravitational. Sachs-Wolfe dominates on large scales above the horizon, where the P(k) is still the primordial power-law with n ~ 1. We had for D M : D M n µ M θ λ M 1 3 Δ T θ 3+n 2 so and amplified by unity, q or q 2 adiabatic Doppler Sachs-Wolfe 34

35 A schematic representation: Week 6 primordial Δ T θ 2 for n =1 log D T DM multiplied by SW effect only DM modified by Meszaros effect Baryons modified by SW and Doppler/Adiabatic Baryons modified by oscillations and Silk damping Sachs-Wolfe regime q H (R eq ) Δ T θ 3+n 1 n 2 θ 2 2 θ -log q Temperature fluctuations should be of constant amplitude above horizon scale for n = 1 (another interesting consequence of the HPZ spectrum) 35

36 δt T θ,ϕ ( ) = a l,m Y l,m θ,ϕ l,m 2 ( ) C l = a l,m l(l+1)c l2 is the equivalent of k 3 P(k) and is the contribution to the variance in dt on a given multipole scale l from logarithmic interval of l 36

37 Typical CMB LSS spectrum for CDM Week 6 >> q H, Sachs- Wolfe effect, independent of q Increasing power q 2 leading up to peak near M H,eq = never having oscillated baryonic fluctuations q H Decline at high l due to (a) roll-over in CDM P k ; (b) suppression of growth of baryonic fluctuations below M J ; then (c) Silk damping; and eventually (d) LSS smearing; with acoustic phasing effects below horizon scale This is a Rosetta Stone for Cosmology: Why? 37

38 Planck fluctuation spectrum (March 2013) Week 6 38

39 Planck fluctuation spectrum (March 2013) Week 6 39

40 Key points amongst many Week 6 Gravitational instability causes D k to increase on scales above the Jeans length, below which pressure causes fluctuations to oscillate as sound waves. In an expanding Universe, the growth is not exponential but at best proportional to R or R 2. DM fluctuations are not affected by pressure effects, but they also cannot grow below the Jean s length for as long as the Universe is radiation dominated. ( stagnation effect, Meszaros effect ) Transport (diffusion) of particles can damp out the density fluctuations below certain scales. Free-streaming damping defines the nature of DM (e.g. CDM). Baryonic fluctuations are erased on scales below the Silk Mass. There are different ways to imprint dr/r on the LSS as a dt/t signature, which dominate on different scales. - Adiabatic compression of the gas - Doppler effects - Gravitational redshifts (Sachs-Wolfe) These all depend on dr/r but have different dependencies on length scale, and therefore dominate on different scales. These produce a complex spectrum of temperature fluctuations with angular scales, which matches well what is observed. 40