1. 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

2. 6dF Workshop - 26-27 April 2005 - Sydney Cosmological Parameters From 2MRS: Anaïs Rassat, Ofer Lahav From 6dFv: Alexandra Abate, Sarah Bridle Cosmology Group, University College, London, UK.

3. 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, K s <11.25  Current data: 25K galaxies (nearly complete)

4. 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

5. 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

6. 6dF Workshop - 26-27 April 2005 - Sydney Why do we need  m and  8 ? Bridle, Lahav, Ostriker & Steinhardt (2003) Science Need more measurements to improve precision

7. 6dF Workshop - 26-27 April 2005 - Sydney UCL Cosmology: 2MRS: Anais Rassat & Ofer Lahav 6dF: Alexandra Abate & Sarah Bridle

8. 6dF Workshop - 26-27 April 2005 - Sydney 2MRS data

9. 6dF Workshop - 26-27 April 2005 - Sydney 2MRS data

10. 6dF Workshop - 26-27 April 2005 - Sydney Redshift Space vs. Real Space V pec = v. r is the component along the line of sight of the peculiar velocity z = λ obs /λ emit -1

11. 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.

12. 6dF Workshop - 26-27 April 2005 - Sydney Spherical Harmonics The Spherical Harmonics are defined for m≥0 by : where m = -l, -l+1,..., 0, …, l-1, l

13. 6dF Workshop - 26-27 April 2005 - Sydney Spherical Harmonics Any function can be expanded as a function of the Ylm’s, as :

14. 6dF Workshop - 26-27 April 2005 - Sydney Spherical Harmonics The coefficients of expansion for the density field can be obtained by : Spherical Harmonics and Likelihood formalism developed and applied to IRAS data by: Fisher, Scharf and Lahav (1994) (Similar method in Heavens and Taylor (1995) ).

15. 6dF Workshop - 26-27 April 2005 - Sydney Spherical Harmonics: Theory 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:

16. 6dF Workshop - 26-27 April 2005 - Sydney Spherical Harmonics: Theory So that one can write : Sum of Real-Space and Redshift-Space contributions to the harmonics.

17. 6dF Workshop - 26-27 April 2005 - Sydney Spherical Harmonics: Theory Where

18. 6dF Workshop - 26-27 April 2005 - Sydney Spherical Harmonics: Theory Calculate the mean weighted harmonic power spectrum: Predicted harmonics = Theoretical Harmonics + Poisson Shot Noise component

19. 6dF Workshop - 26-27 April 2005 - Sydney Spherical Harmonics: Real and Redshift Space Dashed line is alm_s Solid line is alm_r

20. 6dF Workshop - 26-27 April 2005 - Sydney Spherical Harmonics: Real and Redshift Space

21. 6dF Workshop - 26-27 April 2005 - Sydney Spherical Harmonics: Data Decompose the density field in redshift space, using: Calculate the mean weighted harmonic power spectrum:

22. 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

23. 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, σ 8, Ω m are.

24. 6dF Workshop - 26-27 April 2005 - Sydney σ 8 from 6dFv: Alexandra Abate, Sarah Bridle Error bar on σ 8 to be determined. Galaxies simulated, and each assigned a velocity. Via likelihood analysis a constraint on σ 8 was obtained. Further issues  dealing with non-linear effects  using N (z) expected from 6dF  vary other cosmological parameters including Ω m Assumptions  # galaxies randomly distributed in 100 h -1 Mpc  velocities drawn from linear theory correlation matrix:

25. 6dF Workshop - 26-27 April 2005 - Sydney σ 8 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 σ 8 =1.69 ± 0.25 6df expects to get 15000 velocities Back of the envelope calculation:  error we expect = (1300/15000) ½ · Freudling’s error  σ 8 ± 0.07

26. 6dF Workshop - 26-27 April 2005 - Sydney How? Velocities correlation function ξ 12 Basic definition where S 1 and S 2 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

27. 6dF Workshop - 26-27 April 2005 - Sydney Full correlation function Final form It can be split up into its parallel and perpendicular parts

28. 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)

29. 6dF Workshop - 26-27 April 2005 - Sydney Current work Aim: to constrain σ 8 Use: ξ 12 and Maximum Likelihood technique

30. 6dF Workshop - 26-27 April 2005 - Sydney Maximum Likelihood technique Bayes’ Theorem P(d|M) is the likelihood function, this is used to constrain σ 8

31. 6dF Workshop - 26-27 April 2005 - Sydney Likelihood function Form used To constrain σ 8, plot likelihood function against range of σ 8 values Peak corresponds to σ 8 Width of peak corresponds to error on σ 8

32. 6dF Workshop - 26-27 April 2005 - Sydney Progress so far…. Simulating galaxies Attaching a velocity to each via ξ 12, assuming concordance with a fixed σ 8. Calculating likelihood for each ξ 12 for a range of σ 8 Plot likelihood to find 1 σ constraint on σ 8 To be repeated with actual data when released

33. 6dF Workshop - 26-27 April 2005 - Sydney 6dF and 2MRS : Summary Constraining Cosmological Parameters 2MRS: Spherical Harmonic Analysis 6dFv: Velocity Correlations