The Outtakes
• CMB Transfer Function
• Testing Inflation
• Weighing Neutrinos
• Decaying Neutrinos
• Testing Λ• Testing Quintessence
• Polarization Sensitivity
• SDSS Complementarity
• Secondary Anisotropies
• Doppler Effect
• Vishniac Effect
• Patchy Reionization
• Sunyaev-Zel'dovich Effect
• Rees-Sciama & Lensing
• Foregrounds
• Doppler Peaks?
• SNIa Complementarity
• Polarization Primer
• Gamma Approximation
• ISW Effect
Back to Talk
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 minima
Velocity maxima
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
• Zero point not shifted by baryon drag
• Increased baryon inertia decreases effectmeff V2 = const. V ∝ meff
–1/2 = (1+R)–1/2
V||
V||
η
∆T/T
η∆T
/T
−|Ψ|/3
−|Ψ|/3Velocity minima
Velocity maxima
No baryons
Baryons
Doppler Peaks?• Doppler effect has lower amplitude and weak features from projection
observer
jl(kd)Yl0 Y1
0
l
(2l+
1)j l'
(100
)no peak
observer
d d
jl(kd)Yl0 Y0
0
l
(2l+
1)j l(
100)
peakTemperature Doppler
Hu & Sugiyama (1995)
Relative Contributions
5
500 1000 1500 2000
10
Spat
ial P
ower
kd
totaltempdopp
Hu & Sugiyama (1995); Hu & White (1997)
Relative Contributions
5
10
5
500 1000 1500 2000
10
Ang
ular
Pow
erSp
atia
l Pow
er
l
kd
totaltempdopp
Hu & Sugiyama (1995); Hu & White (1997)
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 jl(kd)
observer
d
jl(kd)Yl0 Y0
0
l
(2l+
1)j l(
100)
peak
Bond & Efstathiou (1987) Hu & Sugiyama (1995); Hu & White (1997)
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 T2(k,l) ~ (2l+1)2 [∆T/T]2 jl
2(kd)
Hu & Sugiyama (1995)
0.5 1 1.5 2
-3.5
-3
-2.5
-2
-1.5
0.5 1 1.5 2
log(
k · M
pc)
log(
x)
SW
Acoustic
StreamingOscillations
0.5
1
1.5
2
2.5
log(l)
ProjectionOscillations
Main Projection
Transfer Function Bessel Functions
Measuring the Potential
• Remove smooth damping(independent of perturbations)
• Measure relative peak heights
• Relate to RΨ at last scattering
• Compare with large scale structure Ψ today
• Residual is smooth potentialenvelope and measuresmatter–radiation ratio
Hu & White (1996)
Uses of Acoustic Oscillations
• Distinct features
• Presence/absence unmistakenable
• Sensitive to background parametersthrough fluid parameters
• Sensitive to perturbations throughgravitational potential wellswhich later form structure
• Robust measures of
Angular diameter distance (curvature)Baryon–photon ratio
Uses of Baryon Drag
• Measures baryon–photon ratioat last scattering + zlast scattering + TCMB → Ωbh2
• Measures potential wells at lastscattering (compare with large–scalestructure today)
• Removes phase ambiguity by distinguishingcompression from rarefaction peaks(separates inflation from causal seed models)
Uses of Damping
• Sensitive to thermal history and baryon content
• Independent of (robust to changes in) perturbation spectrum
• Robust physical scale for angular diameter distance test(ΩK, ΩΛ)
Integrated Sachs–Wolfe Effect
• Potential redshift: g00=–(1+Ψ)2 δij
blueshift redshift
Kofman & Starobinskii (1985) Hu & Sugiyama (1994)
Integrated Sachs–Wolfe Effect
• Potential redshift: g00=–(1+Ψ)2 δij
• Perturbed cosmological redshiftgij=a2(1+Ψ)2 δij δT/T = –δa/a = Ψ
blueshift redshift
Kofman & Starobinskii (1985) Hu & Sugiyama (1994)
Integrated Sachs–Wolfe Effect
• Potential redshift: g00=–(1+Ψ)2 δij
• Perturbed cosmological redshiftgij=a2(1+Ψ)2 δij δT/T = –δa/a = Ψ
• Time–varying potentialRapid compared with λ/c
δT/T = –2∆ΨSlow compared with λ/c
redshift–blueshift cancel
• Imprint characteristic timescale of decay in angular spectrum
blueshift redshift
(2Ψ)2
Pow
er
l
lISW~ d/∆η
Kofman & Starobinskii (1985) Hu & Sugiyama (1994)
Testing Inflation / Initial Conditions• Superluminal expansion (inflation) required to generate superhorizon
curvature (density) perturbations
• Else perturbations are isocurvature initially with matter moving causally
• Curvature (potential) perturbations drive acoustic oscillations
• Ratio of peak locations
• Harmonic series:curvature 1:2:3...isocurvature 1:3:5...
Θ+Ψ
−Ψ
−Ψ
η
∆Τ/Τ
(a) Adiabatic
Θ+Ψ
η
∆Τ/Τ
(b) Isocurvature
Hu & White (1996)
Testing Inflation / Initial Conditions• Superluminal expansion (inflation) required to generate superhorizon
curvature (density) perturbations
• Else perturbations are isocurvature initially with matter moving causally
• Curvature (potential) perturbations drive acoustic oscillations
• Ratio of peak locations
• Harmonic series:curvature 1:2:3...isocurvature 1:3:5...
Hu & White (1996)
Pow
er
l
2
500 1000 1500
4
6
CDM InflationAxion Isocurvature
Hidden 1st peak
Weighing Neutrinos• Massive neutrinos suppress power strongly on small scales
[∆P/P ≈–8Ων/Ωm]: well modeled by [ceff2=wg, cvis
2=wg, wg: 1/3→1]
• Degenerate with other effects [tilt n, Ωmh2...]
• CMB signal small but breaks degeneracies
• 2σ Detection: 0.3eV [Map (pol) + SDSS]
Power Suppression Complementarity
k (h Mpc-1)
mv = 0 eVmv = 1 eV
P(k
)
0.01 –0.01 0.0
0.6
0.8
1.0
1.2
1.4
0.01 0.02 0.030.1
0.1
1 SDSS
Ωνh2 = mν/94eV
n
SDSSonly
MAP only
Joint
Hu, Eisenstein, & Tegmark (1998); Eisenstein, Hu & Tegmark (1998)
Cosmology and the
Neutrino Anomalies
10-3
10-2
10-1
100
sin22θ
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
∆m2 (
eV2 )
Solar
Solarνe−νµ, τ
νe−νµ limit
νµ−ντ limit
BBNLimit
νµ−νs
νe−νs
νe- νµ,τ
LSNDνµ- νe
Atmosνµ−ντ
Solarνe- νµ,τ,s
Hata (1998)
CosmologicallyExcluded
Cosmology and the
Neutrino Anomalies
10-3
10-2
10-1
100
sin22θ
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
∆m2 (
eV2 )
Solar
Solarνe−νµ, τ
νe−νµ limit
νµ−ντ limit
BBNLimit
νµ−νs
νe−νs
νe- νµ,τ
LSNDνµ- νe
Atmosνµ−ντ
Solarνe- νµ,τ,s
Hata (1998)Hu, Eisenstein & Tegmark (1998)
Detectable inRedshift Surveys
CosmologicallyExcluded
Cosmology and the
Neutrino Anomalies
10-3
10-2
10-1
100
sin22θ
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
∆m2 (
eV2 )
Solar
Solarνe−νµ, τ
νe−νµ limit
νµ−ντ limit
BBNLimit
νµ−νs
νe−νs
νe- νµ,τ
LSNDνµ- νe
Atmosνµ−ντ
Solarνe- νµ,τ,s
Hata (1998)Hu, Eisenstein & Tegmark (1998)Hu & Tegmark (1998)
Detectable inWeak Lensing
Detectable inRedshift Surveys
CosmologicallyExcluded
10-8 10-6 10-4 10-2 1
0.2
0.4
0.01
0.1
1
a
aeq
wddm
ζ
Φ
∆T/T
Decaying Dark Matter• Example: relativistic matter goes non-
relativistic, decays back into radiation
• Model decay and decay products as a single component of dark matter
• Novel consequences: scale–invariant curvature perturbation from scale–invariant isocurvature perturbations
10 100 1000l
20
40
60
80
100
∆T (µ
K)
100
104
103P
(k)
0.01 0.1 1k (h Mpc-1)
Hu (1998)
τ=3yrsτ=5yrsτ=8yrs
m=5keV
Testing Λ• If wg<0, GDM has no effect on
acoustic dynamics → (kpeaks, heights)independent of wg, Ωg, ceff, cvis
• CMB sensitive to GDM/Λ mainly through angular diameter distance [dA =f(wg,Ωg...)]
Hu, Eisenstein, Tegmark & White (1998)
MAP(no pol with pol)
Ωg
wg
0
–0.5
–1.00.2 0.4 0.6 0.8 1.0
l (rescaled to dA)
Pow
er (
×10-1
0 )
ceff2=1=–1/6=–1/3=–2/3=–1
wg
10 100 1000
2
4
6
8
degeneracy
Testing Λ• If wg<0, GDM has no effect on
acoustic dynamics → (kpeaks, heights)independent of wg, Ωg, ceff, cvis
• CMB sensitive to GDM/Λ mainly through angular diameter distance [dA =f(wg,Ωg...)]
• Galaxy surveys determines h
• CMB determines Ωmh2 → Ωm
• Flatness Ωg = 1 – Ωm
Hu, Eisenstein, Tegmark & White (1998)
SDSSOnly
MAP(no pol with pol)
MAP+ SDSS
Ωg
wg
0
–0.5
–1.00.2 0.4 0.6 0.8 1.0
l (rescaled to dA)
Pow
er (
×10-1
0 )
ceff2=1=–1/6=–1/3=–2/3=–1
wg
10 100 1000
2
4
6
8
degeneracy
Testing Λ• If wg<0, GDM has no effect on
acoustic dynamics → (kpeaks, heights)independent of wg, Ωg, ceff, cvis
• CMB sensitive to GDM/Λ mainly through angular diameter distance [dA =f(wg,Ωg...)]
• Galaxy surveys determines h
• CMB determines Ωmh2 → Ωm
• Flatness Ωg = 1 – Ωm
• SNIa determines luminosity distance [dL =f(wg,Ωg)]
Hu, Eisenstein, Tegmark & White (1998)
SDSSOnly
SNIaOnly
MAP(no pol with pol)
MAP+ SNIa
MAP+ SDSS
Con
sist
ency
Ωg
wg
0
–0.5
–1.00.2 0.4 0.6 0.8 1.0
Com
plem
etar
ity
l (rescaled to dA)
Pow
er (
×10-1
0 )
ceff2=1=–1/6=–1/3=–2/3=–1
wg
10 100 1000
2
4
6
8
degeneracy
SN data July 1998
Testing Λ• If wg<0, GDM has no effect on
acoustic dynamics → (kpeaks, heights)independent of wg, Ωg, ceff, cvis
• CMB sensitive to GDM/Λ mainly through angular diameter distance [dA =f(wg,Ωg...)]
• Galaxy surveys determines h
• CMB determines Ωmh2 → Ωm
• Flatness Ωg = 1 – Ωm
• SNIa determines luminosity distance [dL =f(wg,Ωg)]
Garnavich et al (1998); Riess et al (1998); Perlmutter et al (1998)Ωg
wg
0
–0.5
–1.00.2 0.4 0.6 0.8 1.0
l (rescaled to dA)
Pow
er (
×10-1
0 )
ceff2=1=–1/6=–1/3=–2/3=–1
wg
10 100 1000
2
4
6
8
degeneracy
Hu, Eisenstein, Tegmark & White (1998)
Is the Missing Energy a Scalar Field?
• Scalar Fields have maximal sound speed [ceff =1, speed of light]
• CMB+LSS → Lower limit on ceff>0.6 at wg=–1/6[2.7σ: MAP+SDSS; 7.7σ: Planck+SDSS]
[in 10d parameter space, including bias, tensors]
• Strong constraints for wg > –1/2
Large Scale Structure CMB Anisotropies
k (h Mpc–1) l
P(k
)
Pow
er (
×10-1
0 )
wg=–1/6 wg=–1/6ceff2=0 ceff2=0
1/6 1/6
11scalar
fields scalar fields
0.001 0.01 0.1 10 100 1000
104
103
102
4
6
2
Polarization from Thomson Scattering
• Thomson scattering of anisotropic radiation → linear polarization
• Polarization aligned with cold lobe of the quadrupole anisotropy
QuadrupoleAnisotropy
Thomson Scattering
e—
Linear Polarization
ε’
ε’
ε
Perturbations & Their Quadrupoles
m=0
v
Scalars:
hot
hot
cold
m=1
vv
Vectors:m=2
Tensors:
Hu & White (1997)
• Orientation of quadrupole relative to wave (k) determines pattern
• Scalars (density) m=0 • Vectors (vorticity) m=±1 • Tensors (gravity waves) m=±2
Polarization Patterns
π/20
0
π/2
ππ 3π/2 2π
θ
l=2, m=0
E, B
l=2, m=1 π/20
0
π/2
ππ 3π/2 2π
θ
π/20
0
π/2
ππ 3π/2 2πφ
θ
l=2, m=2
Scalars
Vectors
Tensors
Electric & Magnetic Patterns
Global ParityFlip
E
B
LocalAxes
Principal PolarizationKamionkowski, Kosowski, Stebbins (1997)Zaldarriaga & Seljak (1997)Hu & White (1997)
• Global view: behavior under parity
• Local view: alignment of principle vs. polarization axes
Patterns and Perturbation Types
Kamionkowski, Kosowski, Stebbins (1997); Zaldarriaga & Seljak (1997); Hu & White (1997)
• Amplitude modulated by plane wave → Principle axis
• Direction detemined by perturbation type → Polarization axis
Scal
ars
Vec
tors
Ten
sors
π/2
0 π/4 π/2
π/2
φ
θ
10 100l
0.5
1.0
0.5
1.0
0.5
1.0
Polarization Pattern Multipole Power
B/E=0
B/E=6
B/E=8/13
Polarization Raw Sensitivity
10
2
0.1
0.2
0.3
0.4
4
6
100 1000
10
l
∆T (
µK)
SDSS: Improving Parameter Estimation
Eisenstein, Hu & Tegmark (1998)
h 1.3Ωm 1.4ΩΛ 1.1ΩK 0.31
0.0120.0160.0240.011
0.23
0.25
0.20
0.057
100% 75% 50% 25% 0%
Relative Errors
MAP(no pol)
MAP(pol)
MAP+SDSS(pol, 0.2hMpc-1)
Classical Cosmology
Supernovae Type Ia
Ωg
wg
0
–0.5
–1.00.2 0.4 0.6 0.8
68%95
%99%
July 1998
Garnavich et al. (1998); Riess et al. (1998); Perlmutter et al. (1998)Figure: Hu, Eisenstein, Tegmark, White (1998)
MAP(P)
SDSS
SN
Ωg0.2 0.4 0.6 0.8 1.0
ProjectionSupernovae Type Ia, CMB & LSS
Hu, Eisenstein, Tegmark, White (1998)
wg
0
–0.5
–1.0
SDSS: Improving Parameter Estimation
Eisenstein, Hu & Tegmark (1998)
h 1.3Ωm 1.4ΩΛ 1.1ΩK 0.31
Ωmh2 0.029
Ωbh2 0.0027
Ωνh2 0.0094
ns 0.14α 0.30
T/S 0.48
log(A) 1.3
τ 0.69
0.0120.0160.0240.011
0.00820.00080.0019
0.0510.013
0.15
0.28
0.024
0.23
0.25
0.20
0.057
0.015
0.0013
0.0063
0.094
0.019
0.19
0.36
100% 75% 50% 25% 0%
Relative Errors
MAP(no pol)
MAP(pol)
MAP+SDSS(pol, 0.2hMpc-1)
Secondary Anisotropies
• Temperature and polarization anisotropies imprinted in the CMB after z=1000
• Rescattering Effects
• Linear Doppler Effect (cancelled)
• Modulated Doppler Effects (non–linear)
• by linear density perturbations → Ostriker–Vishniac Effect
• by ionization fraction → Inhomogeneous Reionization
• by clusters → thermal & kinetic Sunyaev–Zel'dovich Effects
• Gravitational Effects
• Gravitational Redshifts
• by cessation of linear growth → Integrated Sachs–Wolfe Effect
• by non–linear growth → Rees–Sciama Effect
• Gravitational Lensing
Cancellation of the Linear Effect
overdensity
e— velocity redshifted γ
blueshifted γ
Observer
Cancellation
Last Scattering Surface
Modulated Doppler Effect
overdensity,ionization patch,cluster...
e— velocity unscattered γ
blueshifted γ
Observer
Last Scattering Surface
Ostriker–Vishniac Effect
Ostriker–Vishniac
Primary
Doppler
Hu & White (1996)
Patchy ReionizationAghanim et al (1996)
Gruzinov & Hu (1998)
Knox, Scocciomarro & Dodelson (1998)
Thermal SZ Effect
Persi et al. (1995) Atrio–Barandela & Muecket (1998)
Rees–Sciama Effect Gravitational Lensing
Seljak (1996a,b)
Residual Foreground Effects
10
1
0.1
100
10
1
100
10
10 100 1000
1
0.01
100
total
totalincreased noisesynchrotron
totalincreased noisesynchrotron
x2
x2
x2
Temperature
E-polarization
B-polarization
dustsynchrotronfree-free
increased noisepoint sources
% d
egra
datio
n
l
1
1
10 100 1000
totaldustincreased noisepoint sources
totaldustincreased noisesynchrotron
totaldustincreased noisesynchrotron
x2
x2
x2
Temperature
E-polarization
B-polarization
l
MAP Planck
Foregrounds & Parameter Estimation
Tegmark, Eisenstein, Hu, de Oliviera Costa (1999)
Features in the Transfer Function• Features in the linear transfer function
• Break at sound horizon
• Oscillations at small scales; washed out by nonlinearities
T(k)
T(k
) / T
BB
KS8
6(k)
EH98
k (h Mpc-1)
Ωm=0.3, h=0.5, Ωb/Ω0=0.3
0.01
1.0
0.9
0.7
0.6
0.5
0.1
Eisenstein & Hu (1998)
PD94
S95
BBKS86