6dF Workshop - 26-27 April 2005 - Sydney Cosmological Parameters from 6dF and 2MRS Anaïs Rassat (University College London) 6dF workshop, AAO/Sydney, 26-27.

6dF Workshop - 26-27 April 2005 - Sydney

Cosmological Parameters from 6dF and 2MRS

Anaïs Rassat (University College London)

6dF workshop, AAO/Sydney, 26-27 April 2005


Cosmological Parameters

From 2MRS: Anaïs Rassat, Ofer LahavFrom 6dFv: Alexandra Abate, Sarah Bridle

Cosmology Group, University College, London, UK.


6dF and 2MRS

6dF survey of southern hemisphere:Spectroscopic Redshift Survey (150K galaxies)

2MRS: Whole Sky Survey (Huchra et al.) Southern Sky: 6dFRS Northern Sky: FLWO, Arizona, USA Low Latitutes: CTIO, Chile

Median redshift: z=0.02 Flux limited survey, Ks<11.25 Current data: 25K galaxies (nearly complete)


6dF Velocity Survey (6dFv):

6dF survey of southern hemisphere:Velocity Survey (15K galaxies)

Distances determined by diameter/velocity dispersion → Peculiar Velocities


The two structure formation parameters:

m = matter density of the universe

dark + baryonic matter

in units of the critical density

8 = clumpiness of the universe

rms fluctuation in 8 Mpc spheres

present day

Velocities and clustering probe these parameters as they trace the underlying mass


Why do we need m

and 8 ?

Bri

dle

, La

hav,

Ost

rike

r &

Ste

inh

ard

t (2

003)

Sci

ence

Need more measurements to improve precision


2MRS data


2MRS data


Redshift Space vs. Real Space

rv ˆ0 ⋅+== rHczvrecession

Vpec = v.r is the component along the line of sight of the peculiar velocity

z = λobs/λemit-1


Redshift Space vs. Real Space

Observed Redshift is not only due to Hubble flow – it is also due to the Peculiar Velocity along the line of sight.

Peculiar Velocities: are due to large-scale streaming motions or local velocities within clusters

These are not always negligible in the local universe – unfortunately they are not always easy to measure – especially at large scales.


Spherical Harmonics

The Spherical Harmonics ( )φθ ,mlY

are defined for m≥0 by :

( ) ( ) ( )( ) ( ) 0cos

!

!

4

121,

21

≥⎥⎦

⎤⎢⎣

⎡+

−+−= mforeP

ml

mllY imm

lmm

lφθ

πφθ

( ) ( ) ( )[ ] 0,1,*

<−=− mforYY ml

mml φθφθ

where m = -l, -l+1, ..., 0, …, l-1, l


Spherical Harmonics

( ) ( )∑∑∞

=

+

−=

=0

,,l

l

lmlmlmYag φθφθ

Any function can be expanded as a function of the Ylm’s, as :


Spherical Harmonics

The coefficients of expansion for the density field can be obtained by :

( ) ( )ilm

N

iilm sYfa

g

ˆ1

∑=

= s

Spherical Harmonics and Likelihood formalism developed and applied to IRAS data by:Fisher, Scharf and Lahav (1994)(Similar method in Heavens and Taylor (1995) ).


Spherical Harmonics: Theory

( )ρρδ Δ

=ss

How is the harmonic decomposition in redshift space related to that in real space? Assume the density fluctuations:

If the perturbations induced by peculiar motions are small, then can expand redshift quantities to first order:

( ) ( ) ( )( )rV ˆ)( ⋅−+= obsrUdr

rdfrfsf



( ) ( ) ( )∫∞

Ψ+Ψ=0

222 2kkkPdkka C

lR

lRslm β

π

( ) ( ) ( )[ ] ( )k̂2

32

*

lmCl

Rl

Rk

lslm Ykkdk

ia ∫ Ψ+Ψ= βδ

π

So that one can write :

Sum of Real-Space and Redshift-Space contributions to the harmonics.



( )dr

rdfrondependskC

l

)()(φΨ

( ) )()( rfrondependskRl φΨWhere

bm

6.0=β



12

2

2 +=SNl

theorylm

s

l a

aC

Calculate the mean weighted harmonic power spectrum:

Predicted harmonics = Theoretical Harmonics + Poisson Shot Noise component


Spherical Harmonics: Real and Redshift Space

Dashed line is alm_sSolid line is alm_r


Spherical Harmonics: Real and Redshift Space


Spherical Harmonics: Data

SNl

l

lmlm

la

al

C2

2

2 121 ∑

−=+=

Decompose the density field in redshift space, using:

Calculate the mean weighted harmonic power spectrum:

( ) ( )ilm

N

iilm sYfa

g

ˆ1

∑=

= s


Spherical Harmonics: Data

The method of Sphercial Harmonics can only be used on Whole-Sky data. In our analysis, we must mask the region of the Zone of Avoidance and fill it in order to obtain a whole-sky map.

This masking can be done in several ways:

• The ZoA is filled with a random distribution of galaxies, with the same density as the mean density for the rest of the sky

• The distribution of galaxies in the ZoA is interpolated from the neighbouring distribution


Spherical Harmonics: Data vs. Theory

Next steps :

Test different methods of including the masked region

Use Selection Function obtained from 2MRS, Pirin Erdoğdu et al. in Preparation.

Compare results using different Power Spectra

Simulate universes with different cosmological parameters and apply the same method to them. This will permit us to quantify how accurate our measurements of β, σ8, Ωm are.


σ8 from 6dFv: Alexandra Abate, Sarah Bridle

Previous work on velocities: Freudling et al 1999 calculated velocity correlations,

similar to our present work 1300 velocities from SFI catalogue Found σ8=1.69 ± 0.25 6df expects to get 15000 velocities Back of the envelope calculation:

error we expect = (1300/15000)½ · Freudling’s error σ8 ± 0.07


How? Velocities correlation function ξ12

Basic definition

where S1 and S2 are peculiar velocities Full version derived from:

Continuity equation Linear theory

Giving velocities in terms of densities

Power spectrum definition Only can measure radial velocity component


Full correlation function Final form

It can be split up into its parallel and perpendicular parts


Physical meaning of Ψ═ and Ψ┬

Ψ═ tells you how correlated galaxy velocities are in the line of sight direction (shown in red)

Ψ┬ tells you how correlated galaxy velocities are perpendicular to the line of sight direction (shown in blue)


Progress so far….

Simulating galaxies Attaching a velocity to each via ξ12,

assuming concordance with a fixed σ8.

Calculating likelihood for each ξ12 for a range of σ8

Plot likelihood to find 1 σ constraint on σ8

To be repeated with actual data when released


6dF and 2MRS : Summary

Constraining Cosmological Parameters

2MRS: Spherical Harmonic Analysis

6dFv: Velocity Correlations

6dF Workshop - 26-27 April 2005 - Sydney Cosmological Parameters from 6dF and 2MRS Anaïs Rassat (University College London) 6dF workshop, AAO/Sydney, 26-27.

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