Flows on 100 h -1 Mpc scales

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

Peculiar Velocity Moments

for Estimating Flows

on 100 h-1 Mpc Scales

Hume A. FeldmanHume A. FeldmanPhysics & AstronomyUniversity of Kansas

2Flows on 100 h-1 Mpc scalesHume A. Feldman 43rd Rencontres de Moriond

Local Group Velocity (20th Century Version)

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

Survey l b |VLG|

3Flows on 100 h-1 Mpc scalesHume A. Feldman 43rd Rencontres de Moriond

4Flows on 100 h-1 Mpc scalesHume A. Feldman 43rd Rencontres de Moriond

In large scale observations we look for

Estimators

We try to estimate an underlying quantity

Estimator = True quantity ⊗ Window function

e.g.

˜ p =N

d3k2π( )3 p

r k ( )W

r k ( )∫

¿¿¿Why ???

5Flows on 100 h-1 Mpc scalesHume A. Feldman 43rd Rencontres de Moriond

˜ p =N

d3k2π( )3 p

r k ( )W

r k ( )∫

6Flows on 100 h-1 Mpc scalesHume A. Feldman 43rd Rencontres de Moriond

Velocity FieldsThe 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

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

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.

Likelihood Methods for Peculiar Velocities

Likelihood Methods for Peculiar Velocities

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*

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

Likelihood function for the bulk flow components

Likelihood Methods for Peculiar Velocities

Likelihood Methods for Peculiar Velocities

Maximum likelihood solution for bulk flow

where

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

Likelihood Methods for Peculiar Velocities

Likelihood Methods for Peculiar Velocities

The measured peculiar velocity of galaxy n

A Gaussian with zero mean and variance

Rij = vi vj =Rij(v) +δij σ i

2 +σ *2

( )

Rij(v) =

12π( )3 P(v)(k)Wij

2(k)d3k∫

=H2 f 2 Ω0( )

2π2 P(k)Wij2(k)dk∫

Theoretical covariance matrix for the bulk flow components

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

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Comparing Velocity Field Surveys

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

Can we do better?

Get rid of small scale aliasing

Improve window function design

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

Window Function Design

depends on the spatial distribution and the errors.

The BF Maximum Likelihood Estimates of the weights (MLE)

Goal: Study motions on largest scales Require WF that

have narrow peaks small amplitude outside peak

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Consider an ideal survey Very large number of points Isotropic distribution Gaussian falloff

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

Window Function Design

Depth of the survey

The moments are specified by the weights

that minimize the variance

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

Expand out the variance

since the measurement error included in is uncorrelated with the bulk flow .

Minimize this expression with respect to

Window Function Design

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

For bulk flow moments:

where

Window Function Design

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

Enforce this constraint using Lagrange multiplier

Minimize with respect to

Window Function Design

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

Matrix form

individual velocity covariance matrix

Window Function Design

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

Solving to get the optimal weights

Minimum Variance (MV) weights

Window Function Design

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

Peculiar Velocity Surveys

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

Window Function Design

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

Window Function Design

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

Window Function Design

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

Comparing Surveys

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

Comparing Surveys

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

Power Spectrum Parameter Estimation

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

ConclusionsGiven 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

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

Velocity

Thank you

Peculiar

Field

Lagrange

Covariance

MatrixMultiplierThe End

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