The CMB TE Cross Correlation and Primordial Gravitational Waves Nathan Miller CASS 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: Miller, Keating, Polnarev “The CMB TE Cross Correlation and PGWs II: Wiener Filtering and Tests Based on Monte Carlo Simulations” (2007) arXiv:

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 (1 o ) 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 ~ cos 2 Θ – 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-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-modeB-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 k Shading 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 h zz (transverse- traceless gauge)‏ Temperature distortion is proportional to h zz (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 1 st Derivative

Mathematically II Difference in sign of 1 st 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, k 0 No assumption about generation of power spectra. Every parameter assumed independent.

Zero Multipole Method C l TE (density) > 0 (up to l=53)‏ C l TE (PGW) 90)‏ Calculation of where C l TE first changes sign, denoted as l 0, is a measure of PGWs Going from l 0 to r depends on other cosmological parameters

Calculation of Zero Multipole Can approximate (l+1)C l TE /2π as a line Fit measured values to a line and solve Monte Carlo simulations to determine distribution of l 0 for a given experiment l 0 in absence of PGWs l 0 for r=2.0 and n t =0

Location of Zero Multipole I ntnt

Location of Zero Multipole II l0l0 n t =0.5 n t =-0.5

Other Cosmological Parameters Effect of n s is opposite as that of n t  Value can be gotten from other measurements (TT,EE)‏ Values for energy densities gotten from TT and other measurements  Assumed to be WMAP3 values A s 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 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 C l TE (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 (R i )‏ U i = R i – n i (n i +1)/2

Interpreting U Know distribution of U i 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: (lowest to highest)‏ R 1 = = 16 U 1 = 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  Ideal Experiment – Full sky, no instrumental noise (best possible experiment)‏  Realistic Experiment – 3% of sky, instrumental noise consistent with current experiments  WMAP – noise consistent with 3 years  Planck 4 Experiments were used to test the PGW detection techniques (2 toy, 2 real)‏

Ideal Experiment Errors No Noise, full sky Best sensitivity to TE possible r=0.3

Realistic Experiment Errors Similar 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 C l and ΔC l  Either 1 experiment with r=0.3 or 2 experiments (r=0.3 and r=0.0)‏ Randomly generate measure C l 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: Δl 0 =1.3  Realistic Experiment: Δl 0 =10  WMAP: Δl 0 =15  Using real data, l 0 =48, no evidence of PGWs  Planck: Δl 0 =3.8 r=0.3, n t =0.0 ⇉ l 0 =49

r, n t Contours Ideal experiment Realistic experiment Values of r and n t that fall within 1σ of l 0 (with r=0.3 and l 0 =49)‏ Inflationary consistency relation, n t =-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: U avg – m U = -1.23σ U Realistic: U avg – m U = -0.2σ U

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: U avg – m U = -0.66σ U WMAP: U avg – m U = -0.01σ U

Which Test is Best?  Zero Multipole Method: 3σ  S/N Test: 2.3σ  Sign Test: 1.8σ  Wilcoxon Rank Sum Test: 1.2σ r=0.3, Ideal Experiment

Future Possibilities See how n t affects the separated power spectra results (only looked for its effect in zero multipole method)‏ Look at effects of n s 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