An Acoustic Primer. Wayne Hu Astro 448. l (multipole) BOOMERanG MAXIMA Previous COBE. W. Hu Dec. 2000

Transcription

1 An Acoustic Primer 100 BOOMERanG MAXIMA Previous 80 T (µk) COBE W. Hu Dec l (multipole) Wayne Hu Astro 448

2 CMB Anisotropies COBE Maxima Hanany, et al. (2000) BOOMERanG de Bernardis, et al. (2000)

3 Ringing in the New Cosmology

4 Physical Landscape

5 Acoustic Oscillations

6 z > 1000; T γ > 3000K Hydrogen ionized Free electrons glue photons to baryons γ e p compton scattering coulomb interactions Photon baryon fluid Potential wells that later form structure Thermal History tightly coupled fluid cold hot mean free path << wavelength hot

7 Thermal History z > 1000; T γ > 3000K Hydrogen ionized Free electrons glue photons to baryons γ e p compton scattering coulomb interactions Photon baryon fluid Potential wells that later form structure z ~ 1000; T γ ~ 3000K Recombination Fluid breakdown z < 1000; T γ < 3000K Gravitational redshifts & lensing Reionization; rescattering last scattering λ k -1 surface z=1000 θ l -1 recombination observer

8 Initial Conditions

9 Inflation and the Initial Conditions Inflation: (nearly) scale-invariant curvature (potential) perturbations Superluminal expansion superhorizon scales "initial conditions" Accompanying temperture perturbations due to cosmological redshift cold Time Newtonian Comoving hot Space Potential perturbation Ψ = time-time metric perturbation δt/t = Ψ δt/t = δa/a = 2/3δt/t = 2/3Ψ Sachs & Wolfe (1967); White & Hu (1997)

10 Initial Conditions & the Sachs-Wolfe Effect Initial temperature perturbation Observed temperature perturbation Gravitational redshift: Ψ + Initial temperature: + Θ Time Newtonian Comoving Potential = time-time metric perturbations Ψ = δt/t Space

11 Initial Conditions & the Sachs-Wolfe Effect Initial temperature perturbation Observed temperature perturbation Gravitational redshift: Ψ + Initial temperature: + Θ Time Newtonian cold Comoving Potential = time-time metric perturbations Ψ = δt/t Matter dominated expansion: a t 2/3, δa/a = 2/3 δt/t Temperature falls as: T a 1 Temperature fluctuation: δt/t = δa/a hot Space

12 Initial Conditions & the Sachs-Wolfe Effect Initial temperature perturbation Observed temperature perturbation Gravitational redshift: Ψ + Initial temperature: + Θ Time Newtonian cold Comoving Potential = time-time metric perturbations Ψ = δt/t Matter dominated expansion: a t 2/3, δa/a = 2/3 δt/t Temperature falls as: T a 1 Temperature fluctuation: Result Initial temperature perturbation: δt/t = δa/a Θ δt/t = δa/a = 2/3 δt/t = 2/3 Ψ Observed temperature perturbation: (δt/t ) obs = Θ+Ψ = 1/3 Ψ hot Space Sachs & Wolfe (1967) White & Hu (1997)

13 Acoustic Oscillations

14 Gravitational Ringing Potential wells = inflationary seeds of structure Fluid falls into wells, pressure resists: acoustic oscillations

15 Photon pressure resists compression in potential wells Acoustic oscillations Acoustic Oscillations Photon Pressure Effective Mass Peebles & Yu (1970) Potential Well Ψ Hu & Sugyama (1995); Hu, Sugiyama & Silk (1997)

16 Acoustic Oscillations Photon pressure resists compression in potential wells Acoustic oscillations Gravity displaces zero point Θ δt/t = Ψ Oscillation amplitude = initial displacement from zero pt. Θ (-Ψ) = 1/3Ψ T/T Θ+Ψ η initial displ. zero point Ψ /3 Peebles & Yu (1970) Hu & Sugyama (1995); Hu, Sugiyama & Silk (1997)

17 Acoustic Oscillations Photon pressure resists compression in potential wells Acoustic oscillations Gravity displaces zero point Θ δt/t = Ψ Oscillation amplitude = initial displacement from zero pt. Θ (-Ψ) = 1/3Ψ Gravitational redshift: observed (δt/t) obs = Θ +Ψ oscillates around zero T/T Θ+Ψ First Extrema η Ψ /3 Peebles & Yu (1970) Hu & Sugyama (1995); Hu, Sugiyama & Silk (1997)

18 Acoustic Oscillations Photon pressure resists compression in potential wells Acoustic oscillations Gravity displaces zero point Θ δt/t = Ψ Oscillation amplitude = initial displacement from zero pt. Θ (-Ψ) = 1/3Ψ Gravitational redshift: observed (δt/t) obs = Θ +Ψ oscillates around zero T/T Θ+Ψ Second Extrema η Ψ /3 Peebles & Yu (1970) Hu & Sugyama (1995); Hu, Sugiyama & Silk (1997)

19 Plane Waves Potential wells: part of a fluctuation spectrum Plane wave decomposition

20 Harmonic Modes Frequency proportional to wavenumber: ω=kc s Twice the wavenumber = twice the frequency of oscillation

21 Harmonic Peaks Oscillations frozen at last scattering Wavenumbers at extrema = peaks Sound speed c s T/T Θ+Ψ Last Scattering First Peak Ψ /3 k 1 =π/ sound horizon η Hu & Sugyama (1995); Hu, Sugiyama & Silk (1997)

22 Harmonic Peaks Oscillations frozen at last scattering Wavenumbers at extrema = peaks Sound speed c s T/T Θ+Ψ Last Scattering Frequency ω = kc s ; conformal time η last scattering Phase k ; φ = 0 dη ω = k Harmonic series in sound horizon sound φn = nπ kn = nπ/ First Peak T/T Θ+Ψ horizon Last Scattering sound horizon Ψ /3 k 1 =π/ sound horizon Ψ /3 Hu & Sugyama (1995); Hu, Sugiyama & Silk (1997) η k 2 =2k 1 Temporal Coherence in IC η Second Peak

23 Seeing Sound Oscillations frozen at recombination Compression=hot spots, Rarefaction=cold spots

24 Harmonic Modes Frequency proportional to wavelength: ω=kc s Modes reaching extrema are integral multiples harmonics 1:2:3

25 Peaks in Angular Power Oscillations frozen at recombination Distant hot and cold spots appear as temperature anisotropies

26 Projection into Angular Peaks Peaks in spatial power spectrum Projection on sphere Spherical harmonic decomposition Maximum power at l = kd Extended tail to l << kd Described by spherical bessel function j l (kd) peak d observer j l (kd)y l 0 Y 0 0 (2l+1)j l (100) l Bond & Efstathiou (1987) Hu & Sugiyama (1995); Hu & White (1997)

27 Projection into Angular Peaks Peaks in spatial power spectrum Projection on sphere Spherical harmonic decomposition Maximum power at l = kd Extended tail to l << kd 2D Transfer Function T 2 (k,l) ~ (2l+1) 2 [ T/T] 2 j l 2 (kd) log(k Mpc) Transfer Function Streaming Oscillations SW Acoustic log(x) log(l) Bessel Functions Projection Oscillations Main Projection Hu & Sugiyama (1995)

28 Doppler Effect Relative velocity of fluid and observer Extrema of oscillations are turning points or velocity zero points Velocity π/2 out of phase with temperature Velocity maxima Velocity minima

29 Doppler Effect Relative velocity of fluid and observer Extrema of oscillations are turning points or velocity zero points Velocity π/2 out of phase with temperature No baryons Zero point not shifted by baryon drag Increased baryon inertia decreases effect m eff V 2 1/2 = const. V m eff = (1+R) 1/2 T/T η Ψ /3 V Velocity maxima Baryons V T/T η Velocity minima Ψ /3

30 Doppler Peaks? Doppler effect has lower amplitude and weak features from projection d observer j l (kd)y l 0 Y 0 0 d observer j l (kd)y l 0 Y 1 0 Temperature peak Doppler (2l+1)j l (100) (2l+1)j l '(100) no peak l l Hu & Sugiyama (1995)

31 Relative Contributions Spatial Power 10 5 total temp dopp Hu & Sugiyama (1995); Hu & White (1997) kd

32 Relative Contributions l Angular Power 10 5 Spatial Power 10 5 total temp dopp Hu & Sugiyama (1995); Hu & White (1997) kd

33 Mechanics of the Calculation Radiation distribution: f(x=position,t=time;n=direction,ν=frequency) Expand in basis functions: (local angular dependence spatial dependence) m e ik x sy l Sources (radiation & metric: monopole, dipole, quadrupole) hierarchy Addition of Angular Momentum integral plane wave ~ gradient plane wave j 0 ly l m 0 sy l' Y 1 m 0 sy l' Y l'' original codes Bond & Efstathiou (1984) Vittorio & Silk (1983) l'±1 l±l' l'+l'' Σ Observables CMBFast Seljak & Zaldarriaga (1996) Hu & White (1997) (clebsch-gordan) j l l' l'' semi-analytic Hu & Sugiyama (1995) Seljak (1994)

34 Semi-Analytic Calculation Treat Sources in the Tight Coupling Approximation Expand the Boltzmann hierarchy equations in 1/(optical depth per λ) Also the trick to numerically integrating the stiff hierarchy equations Closed Euler equation + Continuity equation = Oscillator equation Project Sources at Last Scattering Integral equations Visibility function of recombination + =

35 The First Peak

36 Shape of the First Peak Consistent with potential wells in place on superhorizon scale (inflation) Sharp fall from first peak indicates no continuous generation (defects) BOOMERanG MAXIMA Previous T (µk) l~ l (multipole) 1000

37 Angular Diameter Distance A Classical Test Standard(ized) comoving ruler Measure angular extent Absolute scale drops out Infer curvature Upper limit 1st Peak Scale (Horizon) Upper limit on Curvature Calibrate 2 Physical Scales Sound horizon (peak spacing) Diffusion scale (damping tail) Ω m h 2 Ω b h 2 IC s Flat Kamionkowski, Spergel & Sugiyama (1994) Hu & White (1996) Closed Power Ω m curvature 0.1 flat 1.0 critical 0.1 open Ω m h 2 =0.25; Ω b h 2 = l

38 Curvature in the Power Spectrum Features scale with angular diameter distance Angular location of the first peak

39 A Flat Universe!...? ΩΛ h<0.8 Ω b h 2 <0.025 Perlmutter et al. (1998) Riess et al. (1998) BOOMERanG 0.2 MAXIMA closed open Priors! Ω m

40 First Peak Location BOOM's parabolic peak fit 185<l 1 <209 (2σ) MAX's value l 1 ~ BOOMERanG MAXIMA Previous T (µk) l (multipole) l

41 Are They Consistent? Using ΛCDM models: 184<l 1 <216 (2σ; BOOM) Joint analysis: 194<l 1 <218 (2σ; BOOM+MAX) 20 χ upper limits consistent! lower limits tightened Hu, Fukugita, Zaldarriaga, Tegmark (2000) 2.5σ BOOM BOOM+MAX l 1

42 Is the Scale too Large? Fiducial flat Ω m =0.35, h=0.65 model: l 1 =221 (excluded at ~2.5σ) (a) positive curvature (b) high Hubble constant / matter (c) dark energy h l 1 /l 1 = 1.4 Ω tot /Ω tot 0.48 h/h 0.15 Ω m /Ω m 0.11 w/w Ω b h 2 /Ω b h Hu, Fukugita, Zaldarriaga, Tegmark (2000) Ω m Ω b h 2 > 0.019

43 Age and Dark Energy Flat solutions involve decreasing age of universe through h, Ω m or dark energy equation of state w=p/ρ (< Gyr) h Gyr Gyr 12Gyr Hu, Fukugita, Zaldarriaga, Tegmark (2000) Ω m

44 Age and Dark Energy Region of consistency shrinking headed to crisis? New physics? w, m ν, α f b f b 0.9 σ 8 σ 8 h 0.8 h 0.7 f b BBN t l Hu, Fukugita, Zaldarriaga, Tegmark (2000) Ω m

45 The Second Peak

46 Baryon & Inertia Baryons add inertia to the fluid Equivalent to adding mass on a spring Same initial conditions Same null in fluctuations Unequal amplitudes of extrema

47 Baryons provide inertia Relative momentum density R = (ρ b + p b )V b / (ρ γ + p γ )V γ Ω b h2 Effective mass m eff = (1 + R) Baryon Drag T/T zero pt Low Baryon Content Ψ /3 η Hu & Sugiyama (1995)

48 Baryons provide inertia Relative momentum density R = (ρ b + p b )V b / (ρ γ + p γ )V γ Ω b h2 Effective mass m eff = (1 + R) Baryon Drag Baryons drag photons into potential wells zero point Amplitude Frequency (ω m eff 1/2 ).. Constant R, Ψ: (1+ R) Θ + (k 2 /3)Θ = (1+ R) (k 2 /3)Ψ Θ + Ψ = [Θ(0) + (1+ R) Ψ(0)] cos [ kη/ 3 (1+ R)] RΨ Θ+Ψ T/T η zero pt High Baryon Content Ψ /3 Hu & Sugiyama (1995)

49 Baryons provide inertia Relative momentum density R = (ρ b + p b )V b / (ρ γ + p γ )V γ Ω b h2 Effective mass m eff = (1 + R) Baryon Drag Baryons drag photons into potential wells zero point Amplitude Frequency (ω m eff 1/2 ).. Constant R, Ψ: (1+ R) Θ + (k 2 /3)Θ = (1+ R) (k 2 /3)Ψ Θ + Ψ = [Θ(0) + (1+ R) Ψ(0)] cos [ kη/ 3 (1+ R)] RΨ T/T Θ+Ψ zero pt Alternating Peak Heights Ψ /3 η Hu & Sugiyama (1995)

50 Baryons in the CMB Power Baryon Drag Power/Damping Power High odd peaks 500 l 1000 Ω b h Additional Effects Time varying potential Dissipation/Fluid imperfections 0.1 Power W. Hu 2/ l 0.1

51 Baryons in the Power Spectrum

52 Height of the Second Peak BOOM and MAX both show a low power at l 2 = 2.4 l 1 H 2 = power at 1st/2nd = ( T l 2 / T l1 ) BOOMERanG MAXIMA Previous T (µk) H l (multipole) 1000

53 Height of the Second Peak BOOM : BOOM+MAX: H 2 =0.37 ±0.044 (1σ) H 2 =0.38 ±0.04 (1σ) χ < l < < l < 600 (Ω Λ, Ω b h 2, n) 5 BOOM BOOM+MAX H 2 2.5σ H 2 /H 2 = 0.65 Ω b h 2 /Ω b h n/n Ω m h 2 /Ω m h 2

54 Height of the Second Peak Excludes fiducial LCDM (n=1, Ω b h 2 =0.02, H 2 =0.51) at ~3.3σ Requires Ω b h 2 > 0.022n (if Ω m h 2 >0.16, from l 1, flat, h<0.8) Ω m h 2 = Ω b h n

55 Height of the Second Peak Excludes fiducial LCDM (n=1, Ω b h 2 =0.02, H 2 =0.51) at ~3.3σ Requires Ω b h 2 > 0.022n (if Ω m h 2 >0.16, from l 1, flat, h<0.8) Ω m h 2 = Ω b h ISM D Conservative BBN upper limit QSOa D Tytler et al. ~5% n

56 Score Card

57 Higher Peaks

58 Radiation and Dark Matter Radiation domination: potential wells created by CMB itself Pressure support potential decay driving Heights measures when dark matter dominates

59 Driving Effects and Matter/Radiation Potential perturbation: Radiation Potential: k 2 Ψ = 4πGa 2 δρ generated by radiation inside sound horizon δρ/ρ pressure supported δρ hence Ψ decays with expansion Ψ T/T η Θ+Ψ Hu & Sugiyama (1995)

60 Driving Effects and Matter/Radiation Potential perturbation: Radiation Potential: Potential Radiation: Feedback stops at matter domination k 2 Ψ = 4πGa 2 δρ generated by radiation inside sound horizon δρ/ρ pressure supported δρ hence Ψ decays with expansion Ψ decay timed to drive oscillation 2Ψ + (1/3)Ψ = (5/3)Ψ 5x boost Ψ T/T Θ+Ψ η Hu & Sugiyama (1995)

61 Driving Effects and Matter/Radiation Potential perturbation: Radiation Potential: Potential Radiation: Feedback stops at matter domination k 2 Ψ = 4πGa 2 δρ generated by radiation inside sound horizon δρ/ρ pressure supported δρ hence Ψ decays with expansion Ψ decay timed to drive oscillation 2Ψ + (1/3)Ψ = (5/3)Ψ 5x boost Ψ T/T Θ+Ψ η Hu & Sugiyama (1995)

62 Potential Envelope Similar to matter transfer function, acoustic oscillations, as a function of k, enveloped by a function that depends only on k/ω m h 2 Unlike matter transfer function, the potential envelope increases with k Combination distiguishes between dynamical effects and initial conditions: complementarity 4 2 k 3/2 (Θ 0 +Ψ) 0 2 Hu & White (1997) 4 0 Ω m h 2 =0.25 Ω m h 2 = k/ω m h 2 (Mpc 1 )

63 Matter Density in the CMB Power Ω m h 2 Power/Damping Amplitude ramp across matter radiation equality Radiation density fixed by CMB temperature & thermal history 5 Power Ω m h Measure Ω m h 2 from peak heights 1 Power W. Hu 2/ l

64 Dark Matter in the Power Spectrum

65 Damping Tail

66 Diffusion Damping Diffusion inhibited by baryons Random walk length scale depends on time to diffuse: horizon scale at recombination

67 Diffusion Damping Random walk during recombination Dissipation as hot meets cold Physical scale for standard ruler or calibration

68 Dissipation / Diffusion Damping Imperfections in the coupled fluid mean free path λ C in the baryons Random walk over diffusion scale: geometric mean of mfp & horizon λ D ~ λ C N ~ λ C η >> λ C Overtake wavelength: λ D ~ λ ; second order in λ C /λ Viscous damping for R<1; heat conduction damping for R>1 N=η / λ C 1.0 λ D ~ λ C N λ Power 0.1 perfect fluid instant decoupling Silk (1968); Hu & Sugiyama (1995); Hu & White (1996) l

69 Dissipation / Diffusion Damping Rapid increase at recombination as mfp Independent of (robust to changes in) perturbation spectrum Robust physical scale for angular diameter distance test (Ω K, Ω Λ ) Recombination 1.0 Power 0.1 perfect fluid instant decoupling recombination Silk (1968); Hu & Sugiyama (1995); Hu & White (1996) l

70 Time Evolution Time evolution of an acoustic mode (k=1/mpc, scdm) Acoustic Envelope Θ 0 +Ψ RΨ Hu & Sugiyama (1996)

71 Time Evolution Diffusion length compared with horizon at recombination baryons CMB Ω b h 2 = Ω m h 2 =0.25 Hu & Sugiyama (1996)

72 Acoustic Visibility Effective visibility for CMB and baryons accouting for damping Ω b h 2 = Ω m h 2 =0.25 CMB baryons Hu & Sugiyama (1996)

73 Damping Length as Ruler Damping scale presents another physical scale for curvature test Independent of phase of oscillation (Lasenby's demon: curvature vs. novel isocurvature perturbations)

74 Damping Tail in the Power Spectrum Calibrate the standard rulers in curvature test Depends on baryon-photon ratio and horizon scale at recombination (matter-radiation ratio)

75 The Peaks

76 Physical Decomposition & Information 4 combined Power 3 2 temperature doppler l

77 Physical Decomposition & Information Fluid + Gravity alternating peaks photon-baryon ratio Ω b h 2 4 baryon drag Power l

78 Physical Decomposition & Information Fluid + Gravity alternating peaks photon-baryon ratio Ω b h 2 driven oscillations matter radiation ratio Ω m h 2 Power matter radiation equality l

79 Physical Decomposition & Information Fluid + Gravity alternating peaks photon-baryon ratio Ω b h 2 driven oscillations matter radiation ratio Ω m h 2 Fluid Rulers sound horizon damping scale Power πk A k D diffusion damping perfect fluid 500 l

80 Physical Decomposition & Information Fluid + Gravity alternating peaks photon-baryon ratio Ω b h 2 driven oscillations matter radiation ratio Ω m h 2 Fluid Rulers sound horizon damping scale Geometry angular diameter distance f(ω Λ,Ω K ) + flatness or no Ω Λ, Ω Λ or Ω K Power l A =d A πk A ISW 500 l D =d A k D l