1. Density and Velocity Fields from the 2MASS Redshift Survey
Pirin Erdoğdu
Nottingham University
MNRAS, 373, 45 (2006)

2. The 2MRS Team Pirin Erdoğdu Ofer Lahav John Huchra Matthew Colless
Roc Cutri
Emilio Falco
Teddy George
Thomas Jarrett
Heath Jones
Lucas Macri
Jeff Mader
Nathalie Martimbeau
Michael Pahre
Quentin Parker
Anaïs Rassat
Will Saunders

3. 2MASS Redshift Survey Multiple telescopes to get all-sky coverage.
Data from ZCAT, SDSS (although a lot of our gals are too bright), FLWO, McDonald, Arecibo, Nancay and from CTIO, 6dF, Parkes, South Africa
2MASS XSC is the primary sample for 6dF
Started ~ 1998 as 2MASS produced objects

4. 2MASS Redshift Survey
K_s < ~ 24,832 galaxies all sky , currently 22,750 of 22,966 |b| > 5 degrees
11.25 is now essentially complete.
IR galaxy colors have incredibly small dispersion, < 0.08 mag J-H, 0.09 mag H-K
~ 40% Early types in a K selected sample
versus 30% Early types in a B selected sample
K_s < ,957 |b|>10; 28,671 w/z’s 99%
K_s < , “ ,578 w/z’s 94%

5. The 2MASS Redshift Survey
The 2MRS survey is the densest all sky survey to date. This plot has about galaxies. In northern hemisphere observations by fred lawrance wipple observatory’s 1.5 m telescope. In southern hem. As a part of 6dF galaxy survey. And at low latitudes by cerro tololo interamerican observatory. The magnitude limit is K_s=11.25 after extinction correction corresponding to a median z=0.02 (6000 km/sec, 60 Mpc) I’ll show you the z dist. in a bit. Mag K_20: magnitude inside a circular isophote corresp.surf brightness: 20 mag/arcsec^2.
Stellar contam. is low and reduced by inspecting gals below cz=200 by eye.
can see most of the well known structure; so why bother with all the reconstruction?
z ≤ < z ≤ < z < ≤ z

6. Motivation Statistical Uncertainties due to Shot Noise
Incomplete Galaxy Sampling
Incomplete Sky Coverage
Recovering one field from another (real-space mass and velocity fields from galaxies in redshift-space)

7. The 2MRS Redshift Distribution
The 2MRS is magnitude limited, at large distances we cannot observe the faint galaxies and this can be seen in the z distribution histogram in black. We need to model this and incorporate it into the analysis. We do that by calculating the selection function by fitting to the histogram (black line). Selection function is the probability of observing a galaxy at a given distance, so by weighing each galaxy by the inverse of selection function we can account for the unobserved galaxies. Also plotted is the PSCz survey z distribution. 2mrs samples better out to km/sec (z=0.05) but pscz has a longer tail. Although, analysis goes up to 200 Mpc, only show results up to 140 where shot noise exceeds 10%

8. Notice the Zone of Avodiance
incomplete sky; sky divided in l of 10. and and dist 10 mpc/h. gals are sample from adjacent strips plus a poisson deviate. Method is robust for 2MRS (bzoa<15)
z ≤ < z ≤ < z < ≤ z

9. Expansion of the Density Field in Spherical Coordinates
Spherical Bessel Function
Spherical Harmonic
Normalization Coefficient
will expand the field in spherical harmonics. They are great coz careful choice reduce statistical uncertainties and non-linear effects. Separating den field into angular and radial modes concentrates z-distortion to 1d, secondly spher. Har and Bessel func. Orthogonal and together they form the eigen function of the laplacian operator: simple rel.ship between den. Vel and pot. Fields.exp. in the fol way. Lmax nmax tend to inf but in practice gal. surveys are finite so we need to truncate them. Flux limit so coefficients in z space is defined as.

10. Boundary Conditions 1 ) For r>R we set 2) We require
is continuous at r=R
3) We require
is continuous at r=R
AND
2 others: den=0 vel=0, den=0 give high pec vel., vel=0 ok til 100 Mpc then goes to zero.

11. Resolution (Peebles, 1980) have an effective resolution of so and
too low: struc lost too high funny features.

12. (For details see Zaroubi et al 1995, ApJ, 449, 446)
Wiener Filter
(For details see Zaroubi et al 1995, ApJ, 449, 446)
The Wiener Filter is then the optimal estimator of the underlying density field and which is chosen so that the variance of the difference between the estimator and the true field is minimum. If we minimize this we end up with the so called wiener filter.
which is roughly prior/(prior+noise). so if the signal is high compared to the noise, the wiener filter is close to one and it goes to zero when signal/noise is too low, as in the absence of no data zero field is the best estimator. At this point that so far the only requirement for WF is a model for signal and noise.
Underlying Signal (ΛCDM)
Noise

13. signal/(signal + noise)
where the matrix WF is chosen such that the residual:
is minimised.
Hence:
The Wiener Filter is then the optimal estimator of the underlying density field and which is chosen so that the variance of the difference between the estimator and the true field is minimum.
If we minimize this we end up with the so called wiener filter. which is roughly prior/(prior+noise). so if the signal is high compared to the noise, the wiener filter is close to one and it goes to zero when signal/noise is too low, as in the absence of no data zero field is the best estimator. At this point that so far the only requirement for WF is a model for signal and noise.
signal/(signal + noise)

14. Noise matrix Wiener Filter and
Fisher, Lahav, Hoffman, Lynden-Bell, Zaroubi, 1995, MNRAS,
Wiener Filter
Noise matrix
and
Since we expanded in fourier bessel series the redshift distortion will only effect the radial components and can be modeled by a radial coupling matrix. Z depends on sel, beta and H and not on P(k)

15. Scatter in the field is 0.3 then rises at 100 Mpc/h

17. Poisson’s equation:
is the eigenfunction
of Laplacian operator
In linear theory
where

19. Acceleration on the Local Group
Linear theory:
Linear theory:
From CMB: vLG = 627 km/sec towards l=273o and b=29o
From galaxy cat.

21. Summary
Recovered density and velocity fields from the 2MRS survey using WF technique:
Expanded the observed density field into spherical harmonics and spherical Bessel Functions
The redshift distortions were corrected for using a coupling matrix, Z
The real-space density field was estimated using WF.
Velocity field is reconstructed from the WFed density field
The reconstructed density field resolves most of the known structures as well as new clusters and voids
There is a backside infall towards the GA
There is a strong outflow towards Shapley. Shapley plays an inportant role in the local dynamics.
The reconstructed LG dipole suggests most of the acceleration is generated within 50 Mpc/h