Peculiar Velocity Moments for Estimating Flows on 100 h-1 Mpc Scales Hume A. Feldman Physics & Astronomy University of Kansas Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Local Group Velocity (20th Century Version) Survey l b |VLG| VCMB 271o +29o 620 km / s VLP 220o –28o 561 ± 284 km / s VRPK 260o +54o 600 ± 350 km / s VSMAC 195o 0o 700 ± 250 km / s VLP10k 173o +63o 1000 ± 500 km / s VSC 180o 0o 100 ± 150 km / s Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

¿¿¿ Why ??? In large scale observations we look for Estimators We try to estimate an underlying quantity Estimator = True quantity ⊗ Window function e.g. Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Velocity Fields The Modern Version Sarkar, HAF & Watkins, MNRAS 375 691-697 (2007) Watkins & HAF, MNRAS 379, 343-348 (2007) HAF & Watkins, arXiv:0802.2961 (2008) HAF, Watkins & Hudson, in Preparation Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Likelihood Methods for Peculiar Velocities A catalog of peculiar velocities galaxies, labeled by an index n Positions rn Estimates of the line-of-sight peculiar velocities Sn Uncertainties σn Assume that observational errors are Gaussian distributed. Model the velocity field as a uniform streaming motion, or bulk flow, denoted by U, about which are random motions drawn from a Gaussian distribution with a 1-D velocity dispersion σ* Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Likelihood Methods for Peculiar Velocities Likelihood function for the bulk flow components Maximum likelihood solution for bulk flow where Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Likelihood Methods for Peculiar Velocities The measured peculiar velocity of galaxy n A Gaussian with zero mean and variance Theoretical covariance matrix for the bulk flow components Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Comparing Velocity Field Surveys Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Improve window function design Can we do better? Get rid of small scale aliasing Improve window function design Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Window Function Design The BF Maximum Likelihood Estimates of the weights (MLE) depends on the spatial distribution and the errors. Goal: Study motions on largest scales Require WF that have narrow peaks small amplitude outside peak Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Window Function Design Consider an ideal survey Very large number of points Isotropic distribution Gaussian falloff Depth of the survey The moments are specified by the weights that minimize the variance Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Window Function Design Expand out the variance since the measurement error included in is uncorrelated with the bulk flow . Minimize this expression with respect to Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Window Function Design For bulk flow moments: where Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Window Function Design Enforce this constraint using Lagrange multiplier Minimize with respect to Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Window Function Design Matrix form individual velocity covariance matrix Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Window Function Design Solving to get the optimal weights Minimum Variance (MV) weights Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Peculiar Velocity Surveys Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Window Function Design Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Window Function Design Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Window Function Design Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Comparing Surveys Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Comparing Surveys Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Power Spectrum Parameter Estimation Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Conclusions Given appropriate window functions, velocity field surveys are consistent with each other. Bulk Flow Measurements agree. Maximum Likelihood parameter estimation are robust and mostly agree with other methods. Seems to be systematic bias towards large or small scale flow Optimization of window functions removes the bias and shows the flow Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond

Thank you Lagrange Peculiar Velocity Likelihood The End Multiplier Field Multiplier weight Covariance Matrix Hume A. Feldman Flows on 100 h-1 Mpc scales 43rd Rencontres de Moriond