Matrix Template Matching Reloaded Third ATNF Gravitational Wave Workshop 2009 Willem van Straten

Synopsis “The Matrix”: polarisation and pulsar timing invariant interval & matrix template matching “The One”: PSR J0437-4715 use bright source to calibrate itself “The Reload”: PSR J1022+1001 use bright source to calibrate weak sources “The Future”

The Matrix Statistical definition of polarisation in terms of the second moments of the electric field, variance on diagonal, covariance off diagonal Rho = coherency matrix E = column vector Outer product is implied, Hermitian transpose, ensemble average … Rho is Hermitian -> 4 degrees of freedom

Stokes Parameters S_k = four Stokes parameters (Einstein notation is used) Sigma_0 = 2x2 identity matrix; Sigma_1-3 are the Pauli matrices, Tr is the matrix trace operator. Coherency matrix is a linear combination of hermitian basis matrices; Stokes parameters are projections onto those basis matrices

Stokes 4-vector Total intensity (time-like) S0 = I Polarization vector (space-like) S = (S1, S2, S3)

Transformations Jones matrix, J Mueller matrix, M Linear transformations of electric field Congruence transformation of coherency matrix Lorentz transformation of Stokes parameters

Rotations and Boosts Rotations: Boosts: rotate S about an axis by an angle leave S0 unchanged Boosts: boost along an axis by a Lorentz factor modify both S0 and S

Boosts: Instrumental Non-orthonormality of receptors: differential gain (along S1 axis) cross-coupling (along S2 & S3 axes) Mixing of S0 and S depends on S• Produce pulse phase dependent distortions of total intensity profile

Template Matching P() = aS(+ ) + b  P() = aS()e-i2 + ()b FFT P() = aS()e-i2 + ()b

Frequency Domain High frequency power yields greatest constraint on slope,     

The One: PSR J0437-4715 The good: The ugly: closest and brightest MSP known narrow pulse peak The ugly: OPM sweep near pulse peak TOA strongly distorted by calibration errors

PSR J0437-4715 profile

PSR J0437-4715 spectrum Fluctuation power spectra (power in Fourier transform of pulse profile) Black = total intensity, Red = polarized intensity At high frequencies (power on shortest time scales), polarized flux provides more power

Parallactic Variation of Arrival Time Polarisation also impacts on high-precision timing 6.5 hours of time-of-arrival ~9 microsecond peak-to-peak Due to parallactic rotation of receiver

Even After Calibration Ideal feed assumption 2 us peak-to-peak

Enter the Invariant Form the invariant profile, Sinv(), where Sinv2 = S02 - |S|2 Dramatically reduced systematic error in PSR J0437-4715 data (Britton 2000; van Straten et al 2001)

Caveats Sinv  0 as |S|/S0  1 Limited by S/N: depolarization due to time/frequency variations of instrumental response baseline estimation error/bias no longer normally distributed error Limited by degree of polarization: Sinv  0 as |S|/S0  1

Matrix Template Matching (MTM) Replace scalars with matrices 6 new degrees of freedom 4 times the number of constraints

Relative TOA Errors Pulsar MTM invariant J0437-4715 0.8318(1) 1.4314(3) J1022+1001 0.669(4) 1.80(2) J1713+0747 0.877(2) 1.543(5) B1855+09 0.9314(9) 1.430(2) J1909-3744 0.959(4) 1.464(8) B1937+21 0.862(4) 1.451(9) Relative uncertainty with respect to timing total intensity profile Does not consider systematic error

Limitations Also limited by S/N: Calibration before integration … depolarization due to time/frequency variations of instrumental response Calibration before integration …

Calibration Options (2007) Measurement equation modeling (MEM; van Straten 2004) unknown source and calibrators observed over a range of parallactic angles Matrix template matching (MTM; van Straten 2006) known source matched to a single (short) observation; no calibrator input

Parallactic variation of Stokes parameters How is J determined? Variation of Stokes parameters as a function of parallactic angle (pulsars provide many constraints, one page for each pulse phase bin)

MTM Reloaded Merge MEM and MTM methods use template to also constrain calibrator model time-variations and multiple observations Derive 20 MEM solutions (~8hr each) Integrate into standard (~150hr) Derive 150 MEM/MTM solutions (~3.5hr) both receiver and calibrator

Receiver solution 2003

Receiver solution 2008

Calibrator solution 2003

Calibrator solution 2008

Ellipticity time variation 54

Ellipticity time variation 74

Calibrator time variation 54

Calibrator time variation 74

PSR J1022+1001 profile

PSR J1022+1001 spectrum

Arrival Time Comparison Intensity TM Matrix TM Postfit r.m.s. 1.453 s 0.942 s Reduced 2 2.03 1.01 two CPSR2 bands at 20cm 234 TOAS (Nfree=217) r.m.s reduced by 39% (predicted 33%)

Conclusions Long-term stability of PSR J0437-4715 can be exploited to calibrate instrumental polarization Matrix Template Matching can do better than doubling integration length

The Future Turn MEM/MTM solutions back on PSR J0437-4715 Quantify long-term stability of MSP polarization Fit time- & frequency-varying receiver & calibrator model to MEM/MTM solutions * MTM-only data from 2006 have underestimated errors * Variations in MSP polarization would appear as phase-dependent residuals * Well-fit model with fewer parameters than data increases instantaneous precision