1 Flows on 100 h -1 Mpc scales Hume A. Feldman 43 rd Rencontres de Moriond Peculiar Velocity Moments for Estimating Flows on 100 h -1 Mpc Scales Hume A. Feldman Physics & Astronomy University of Kansas
Jan 29, 2016
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
20
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
29
Flows on 100 h-1 Mpc scalesHume A. Feldman 43rd Rencontres de Moriond
Window Function Design
30
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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