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
6dF Workshop - 26-27 April 2005 - Sydney
Cosmological Parameters
From 2MRS: Anaïs Rassat, Ofer LahavFrom 6dFv: Alexandra Abate, Sarah Bridle
Cosmology Group, University College, London, UK.
6dF Workshop - 26-27 April 2005 - Sydney
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 Workshop - 26-27 April 2005 - Sydney
6dF Velocity Survey (6dFv):
6dF survey of southern hemisphere:Velocity Survey (15K galaxies)
Distances determined by diameter/velocity dispersion → Peculiar Velocities
6dF Workshop - 26-27 April 2005 - Sydney
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
6dF Workshop - 26-27 April 2005 - Sydney
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
6dF Workshop - 26-27 April 2005 - Sydney
2MRS data
6dF Workshop - 26-27 April 2005 - Sydney
2MRS data
6dF Workshop - 26-27 April 2005 - Sydney
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
6dF Workshop - 26-27 April 2005 - Sydney
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.
6dF Workshop - 26-27 April 2005 - Sydney
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
6dF Workshop - 26-27 April 2005 - Sydney
Spherical Harmonics
( ) ( )∑∑∞
=
+
−=
=0
,,l
l
lmlmlmYag φθφθ
Any function can be expanded as a function of the Ylm’s, as :
6dF Workshop - 26-27 April 2005 - Sydney
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) ).
6dF Workshop - 26-27 April 2005 - Sydney
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
6dF Workshop - 26-27 April 2005 - Sydney
Spherical Harmonics: Theory
( ) ( ) ( )∫∞
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.
6dF Workshop - 26-27 April 2005 - Sydney
Spherical Harmonics: Theory
( )dr
rdfrondependskC
l
)()(φΨ
( ) )()( rfrondependskRl φΨWhere
bm
6.0=β
6dF Workshop - 26-27 April 2005 - Sydney
Spherical Harmonics: Theory
12
2
2 +=SNl
theorylm
s
l a
aC
Calculate the mean weighted harmonic power spectrum:
Predicted harmonics = Theoretical Harmonics + Poisson Shot Noise component
6dF Workshop - 26-27 April 2005 - Sydney
Spherical Harmonics: Real and Redshift Space
Dashed line is alm_sSolid line is alm_r
6dF Workshop - 26-27 April 2005 - Sydney
Spherical Harmonics: Real and Redshift Space
6dF Workshop - 26-27 April 2005 - Sydney
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
6dF Workshop - 26-27 April 2005 - Sydney
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
6dF Workshop - 26-27 April 2005 - Sydney
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.
6dF Workshop - 26-27 April 2005 - Sydney
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
6dF Workshop - 26-27 April 2005 - Sydney
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
6dF Workshop - 26-27 April 2005 - Sydney
Full correlation function Final form
It can be split up into its parallel and perpendicular parts
6dF Workshop - 26-27 April 2005 - Sydney
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)
6dF Workshop - 26-27 April 2005 - Sydney
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 Workshop - 26-27 April 2005 - Sydney
6dF and 2MRS : Summary
Constraining Cosmological Parameters
2MRS: Spherical Harmonic Analysis
6dFv: Velocity Correlations
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