The CMB TE Cross Correlation and Primordial Gravitational Waves

Nathan MillerCASS Journal Club 11/6/07

Outline

Intro to CMB and Polarization Signature of PGWs in the TE Cross

Correlation Power Spectrum Tests used to look for PGWs Results

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

13.7 Gyr

380 kyr

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

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

How a blackbody becomes polarized (Thomson scattering)

100% polarized

Plane of Polarization

unpolarized

Polarization ~ cos2Θ – Quadrupole Scattering

electron

Courtesy of Brian Keating

How a blackbody becomes polarized II

Incoming unpolarized light with quadrupole anisotropy

Outgoing linear polarized light

How is the CMB polarized by density perturbations?

Direction of wavevector

How is the CMB polarized by DP?

Compressional Wavevector

How is the CMB polarized by GW?

Direction of GW

How is the CMB polarized by GW?

Gravitational Wavevector

e-

Courtesy of Brian Keating

GW + CMB Plasma

This process leads to…. Courtesy of Brian Keating

Gravitational Waves + CMB

Caldwell & Kamionkowski

Temperature and Polarization caused by single wave in +z direction.

Courtesy of Brian Keating

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

Current WMAP Results

Anticorrelation on large scales

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?

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

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)

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

Mathematically

Perturbation to the metric

1st Derivative

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

Scalars

Tensors, r=1

+

-

WMAP3 Parameters (lcdm model)

53

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.

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

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

Location of Zero Multipole I

-0.3 0

nt

-0.2 -0.1

Location of Zero Multipole II

l0

nt=0.5

nt=-0.5

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

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

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

S/N as a function of r

Uncertainty is independent of r

Mean is linear in r

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

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

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

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

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

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)

Ideal Experiment ErrorsNo Noise, full sky

Best sensitivity to TE possible

r=0.3

Realistic Experiment ErrorsSimilar to Current experiments

3% of sky, ~1 degree beam FWHM

r=0.3

WMAP Errors

Taken from published WMAP data

r=0.3

Planck Errors

Taken from Planck bluebook

r=0.3

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

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

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

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

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

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

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

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

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

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

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

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