the cmb te cross correlation and primordial gravitational waves nathan miller cass journal club...
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The CMB TE Cross Correlation and Primordial Gravitational Waves
Nathan MillerCASS Journal Club 11/6/07
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Outline
Intro to CMB and Polarization Signature of PGWs in the TE Cross
Correlation Power Spectrum Tests used to look for PGWs Results
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References
Polnarev, Miller, Keating “The CMB TE Cross Correlation and PGWs I: The Zero Multipole Method” (2007) arXiv:0710.3649
Miller, Keating, Polnarev “The CMB TE Cross Correlation and PGWs II: Wiener Filtering and Tests Based on Monte Carlo Simulations” (2007) arXiv:0710.3651
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13.7 Gyr
380 kyr
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CMB
• Universe was much smaller, hotter
• Photons in equilibrium with the proton/electron plasma
• As universe expanded, wavelength expanded, eventually energy smaller than required to keep equilibrium in proton/electron plasma
• Photons free-streamed to us today
• Density perturbations before recombination give rise to photon anisotropies and polarization
• Primordial Gravitational Waves also contribute to temperature anisotropies and polarization Boomerang 03 Flight
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Gravitational Waves on the CMB
CMB B-mode or “Curl” Polarization Generated by Primordial GWB at large (1o)
angular scales Density perturbations do not create B-modes
Detection is limited by Lensing at small (5’) scales
Large Scale Structure Neutrinos
Foregrounds E/B Mixing for observations on small patches of
sky CMB TE Cross Correlation
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How a blackbody becomes polarized (Thomson scattering)
100% polarized
Plane of Polarization
unpolarized
Polarization ~ cos2Θ – Quadrupole Scattering
electron
Courtesy of Brian Keating
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How a blackbody becomes polarized II
Incoming unpolarized light with quadrupole anisotropy
Outgoing linear polarized light
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How is the CMB polarized by density perturbations?
Direction of wavevector
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How is the CMB polarized by DP?
Compressional Wavevector
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How is the CMB polarized by GW?
Direction of GW
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How is the CMB polarized by GW?
Gravitational Wavevector
e-
Courtesy of Brian Keating
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GW + CMB Plasma
This process leads to…. Courtesy of Brian Keating
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Gravitational Waves + CMB
Caldwell & Kamionkowski
Temperature and Polarization caused by single wave in +z direction.
Courtesy of Brian Keating
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Polarization Patterns
E-mode B-mode
• Density fluctuations give scalar perturbations => E-mode• Gravity Waves give tensor perturbation => B, E modes• Both give rise to TE cross correlation
• Polarization Generation by Thomson Scattering
Wayne Hu
Courtesy of Brian Keating
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Current WMAP Results
Anticorrelation on large scales
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Future CMB Experiments
Measurements of the B-mode power spectrum are the focus of future CMB grounds/balloon/space based experiments
How well can the experiments constrain r using
only the TE cross correlation?
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Why TE?
TE is much easier to measure Several orders of magnitude larger than BB
power spectrum Different systematics
T/B, E/B leakage (beam uncertainties) E/B mixing (partial sky)
Can be used as insurance against false detection from improper removal of systematics
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TE Correlation for Scalars
kShading is Φ, gravitational potential. Blue is more negative Φ
Blue is temperature trough (Photons flow from hot to cold)
Photons coming above and below are relatively blueshifted. Photons coming from the side are not.
Positive TE Correlation on large scales. (Cold – radial, hot-tangential)
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TE Correlation for Tensors
Blue indicates more negative hzz
(transverse-traceless gauge)
Temperature distortion is proportional to hzz (Sachs-Wolfe integral). Local quadrupole is different than what is seen on the sky. Blue is temperature trough
Expanded into picture, contracted horizontally in picture for temperature trough
Photons coming from side are relatively blueshifted. Photons coming from above/below are not
Negative TE Correlation on large scales (cold – tangential, hot - radial)
x
y
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Mathematically
Perturbation to the metric
1st Derivative
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Mathematically II
Difference in sign of 1st derivative of h leads to difference in sign of TE cross correlation PGWs modes are decreasing for n<100 The relevant modes for density perturbations
are the growing modes
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Scalars
Tensors, r=1
+
-
WMAP3 Parameters (lcdm model)
53
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Parameters and Errors
• Power spectra are evaluated when the wavelength in question leaves the horizon
• Tensor-to-scalar ratio, r, defined as ratio of power spectra at some wavenumber, k0
• No assumption about generation of power spectra. Every parameter assumed independent.
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Zero Multipole Method
ClTE(density) > 0 (up to l=53)
ClTE(PGW) < 0 (up to l>90)
Calculation of where ClTE first changes
sign, denoted as l0, is a measure of PGWs
Going from l0 to r depends on other cosmological parameters
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Calculation of Zero Multipole
Can approximate (l+1)Cl
TE/2π as a line
Fit measured values to a line and solve
Monte Carlo simulations to determine distribution of l0 for a given experiment
l0 in absence of PGWs
l0 for r=2.0 and nt=0
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Location of Zero Multipole I
-0.3 0
nt
-0.2 -0.1
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Location of Zero Multipole II
l0
nt=0.5
nt=-0.5
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Other Cosmological Parameters
Effect of ns is opposite as that of nt
Value can be gotten from other measurements (TT,EE)
Values for energy densities gotten from TT and other measurements Assumed to be WMAP3 values
As and τ have no effect on results
rescaling of both power spectra by the same value will not change the results
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Scalar/Tensor Separation
Proposed using Wiener filtering to separate scalars and tensors
Requires some prior knowledge of power spectrum for filtering
Assume perfect filter and no reduction in errors
Resulting power spectrum is just due to tensors and errors is errors due to scalars+tensors
Similar to subtracting off predicted scalar power spectrum (if we know it perfectly)
3 Statistical Tests used to check for PGWs
Signal-to-Noise (S/N) Test
Sign Test
Wilcoxon Rank Sum Test
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Signal-to-Noise (S/N) Test
Calculate the variable Sign and amplitude can help determine r
S/N = 0 if r=0 S/N < 0 if r>0
Monte Carlo simulations used to determine distribution of S/N Not exact signal-to-noise ratio
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S/N as a function of r
Uncertainty is independent of r
Mean is linear in r
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Sign Test
If r=0, then ClTE(tensors)=0
Number of positive multipoles = Number of negative multipoles (on average)
Calculate number of positive and number of negative multipoles All we care about is sign of the multipole
Assuming the null hypothesis, the number of positive (negative) multipoles follows the binomial distribution
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Wilcoxon Rank Sum Test Non-parametric test based on ranks
Comparison between two data sets Null hypothesis is that both data sets have the same
value of r Needs a simulated data set
Can only make 1 real measurement Rank multipole values
Don't care about specific values, just the ranks
Sum ranks for each data set (Ri)
Ui = Ri – ni(ni+1)/2
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Interpreting U
Know distribution of Ui under the null hypothesis Look up in tables for small numbers of
multipoles Approximate as a Gaussian for large numbers
of multipoles
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Example of Wilcoxon Rank Sum Test
Data 1: 1.8, 2.7, 4.3, 8.7 Data 2: 1.2, 6.3, 6.5, 10.3 Rank: 21112212 (lowest to highest)
R1 = 2 + 3 + 4 + 7 = 16
U1 = 16 – 4*5/2 = 6
U < 1 is required to reject null hypothesis at 95% confidence
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Comparison of Tests
S/N: outliers affect results greatly One outlier could lead to a false detection
Sign: Amplitudes of measured values have no affect on results Only sign is important
WRS: Comparison between two sets of data Must always combine real data with Monte
Carlo simulated data
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Experiments
1)Ideal Experiment – Full sky, no instrumental noise (best possible experiment)
2)Realistic Experiment – 3% of sky, instrumental noise consistent with current experiments
3)WMAP – noise consistent with 3 years
4)Planck
4 Experiments were used to test the PGW detection techniques (2 toy, 2 real)
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Ideal Experiment ErrorsNo Noise, full sky
Best sensitivity to TE possible
r=0.3
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Realistic Experiment ErrorsSimilar to Current experiments
3% of sky, ~1 degree beam FWHM
r=0.3
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WMAP Errors
Taken from published WMAP data
r=0.3
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Planck Errors
Taken from Planck bluebook
r=0.3
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Monte Carlo Simulations
Assume some underlying Cl and ΔCl
Either 1 experiment with r=0.3 or 2 experiments (r=0.3 and r=0.0)
Randomly generate measure Cl from mean and uncertainty Bin together measurements according to
fraction of sky observed Run 1,000,000 times to determine
distributions of random variables used in statistical tests
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Zero Multipole Results
Ideal Experiment: Δl0=1.3
Realistic Experiment: Δl0=10
WMAP: Δl0=15
Using real data, l0=48, no evidence of PGWs
Planck: Δl0=3.8
r=0.3, nt=0.0 ⇉ l0=49
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r, nt Contours
Ideal experiment
Realistic experiment
Values of r and nt that fall within 1σ of l0 (with r=0.3 and l0=49)
Inflationary consistency relation, nt=-r/8
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S/N Results
Black is distribution of S/N. Red line S/N=0 (average if r=0)
Ideal: Mean = -17.1, SD = 7.2
Realistic: Mean = -1, SD= 2.6
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S/N Results
Black is distribution of S/N. Red line S/N=0 (average if r=0)
Planck: Mean = -6.2, SD = 5.1
WMAP: Mean = -0.2, SD= 5.1
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Sign Test Results
Black line is distribution of N+. Red line is average if r=0
Ideal: Mean=18
Realistic: Only 7 different uncorrelated multipoles. Always a decent chance to get any number of positive values
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Sign Test Results
Black line is distribution of N+. Red line is average if r=0
Planck: Mean = 10, 50% chance of 1σ detection
WMAP: Looks same as distribution if r=0
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Wilcoxon Rank Sum Test Results
Black line is distribution of U. Red is average if r=0 and blue is 1σ confidence region for r=0
Ideal: Uavg – mU = -1.23σU
Realistic: Uavg – mU = -0.2σU
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Wilcoxon Rank Sum Test Results
Black line is distribution of U. Red is average if r=0 and blue is 1σ confidence region for r=0
Planck: Uavg – mU = -0.66σU
WMAP: Uavg – mU = -0.01σU
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Which Test is Best?
1)Zero Multipole Method: 3σ
2)S/N Test: 2.3σ
3)Sign Test: 1.8σ
4)Wilcoxon Rank Sum Test: 1.2σ
r=0.3, Ideal Experiment
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Future Possibilities
See how nt affects the separated power spectra results (only looked for its effect in zero multipole method)
Look at effects of ns on zero multipole method results
Foregrounds Possible effect on TE correlation
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Conclusion
Used as an auxiliary method of constraining PGWs Provides insurance against false detection
due to improper foreground removal and other systematic effects
Cosmic variance limited experiment would have similar sensitivity to r in TE as current experiments have to r in BB