1. Redshift-Space Distortions of the Galaxy Distribution Alex Szalay, JHU

2. The Growth of Fluctuations Jeans (1902): gravitational instability in a P=0 fluid When the pressure is large, sound waves At around recombination: ◦ Phase interference => modulation of fluctuations ◦ Behaves like a resonant cavity time observer P=/3 p,e, zero mean free path Oscillating sound wavesGravitational growth/decay P=0 H, no scattering Surface of last scattering Nonlinear growth

3. Changing Equations of State Sharp changes in the equations of state: ◦ Equality ◦ Recombination All happens between z=1300 and z=1000 Radiation drag acting on the baryons until z=200

4. The Effect of Sound Waves The remnants of early sound waves show up as small bumps in the LSS P(k) They fully affect the CMB Wavelength given by the sound horizon (Sakharov) Amplitude of LSS bumps depends on The position of the bumps determined from first principles: the speed of sound and the age of the universe at recombination An perfect ruler to measure cosmological parameters How can we detect these? W. Hu (1992)

5. Observables We can measure: Galaxy 2D positions Galaxy 3D positions Fluxes Distances ◦ angular diameter distance ◦ luminosity distance Estimate galaxy velocities Estimate correlation functions, power specra

6. Correlation Function Defined through the ensemble average Can be expressed through the Fourier transform The correlation function only depends on the distance The correlation and the power spectrum form a Fourier transform pair

7. Distance from Redshift Redshift measured from Doppler shift Hubble’s Law v = H 0 r Gives distance to zero th order But, galaxies are not at rest in the comoving frame: ◦ Distortions along the radial directions ◦ Originally homogeneous isotropic random field, now anisotropic!

8. Redshift Space Distortions Three different distortions Linear infall (large scales) ◦ Flattening of the redshift space correlations ◦ Kaiser 86 Thermal motion (small scales) ◦ ‘Fingers of God’ ◦ Cuspy exponential Nonlinear infall (intermediate scales) ◦ Caustics

9. The 2dF Correlation Function

10. Redshift Space Power Spectrum Expressing the power spectrum where  is the angle between the line of sight and the wave vector P(k) and  (r) are duals: ◦ P gets elongated ◦ gets compressed along the line of sight This is due to linear infall Fingers of God on the axis

11.  (r) from Linear Theory Kaiser (1987) ◦ Linear infall, plane parallel approximation Lilje and Efstathiou (1989) Hamilton (1992) ◦ Working in correlation function space rather than in P(k) Cole, Fischer and Weinberg (1994) Heavens and Taylor (1995) ◦ Multipole expansion Zaroubi and Hoffman (1996) …. Szalay, Matsubara, Landy (1998) ◦ Using Bipolar spherical harmonics to expand wide angles Szapudi and Papai (2009) ◦ Computing neglected term for wide angles, improved coordinate system

12. BEKS 1990 – peaks at 128 h -1 Mpc Broadhurst, Ellis, Koo and Szalay, Nature 1990

13. BEKS Power Spectrum

14. AS, In Proc 18 th Texas Symposium, 1996

15. Las Campanas Redshift Survey Landy et al 1995

16. SDSS Redshift Samples Main Galaxies ◦ 800K galaxies, high sampling density, but not too deep Luminous Red Galaxies ◦ 100K galaxies, color and flux selected ◦ m r < 19.5, 0.15 < z < 0.45, close to volume-limited Quasars ◦ 20K QSOs, cover huge volume, but too sparse

17. Finding the BAO in DR4 Eisenstein et al (2005) – LRG sample Correlation function

18. Primordial Sound Waves in SDSS Power Spectrum (Percival et al 2006, 2007) SDSS DR6+2dF SDSS DR5 800K galaxies

19. Redshift Space Correlation Fn Azimuthal average wipes out the P 2 and P 4 terms, only the real space correlation function remains  terms contain a second derivative (Hamilton ‘92) Interesting effect on power spectra with sharp features: redshift-space distortions make features even sharper! Baryon Acoustic Oscillations !!!

20.  (r) from linear theory + BAO Mixing of  0,  2 and  4 ◦ Along the line of sight   r r 2 (r)

22. 2D Symmetry There is a planar symmetry: ◦ Observer + 2 galaxies Thus 2D correlation of a slice is the same We usually average over dcos  Very little weight along the  axis: Sharp features at the line-of-sight go away with averaging!

23. Lower Dimensional Subsets For a homogeneous isotropic process, the correlation function in a lower dimensional subset is identical Statement even true if the field is not isotropic, as long as subsets obey the same symmetry Projection-Slicing theorem

24. Projection and Slicing Theorem The basis of CAT-SCAN / Radon transform

25. Effect of Projections Projection in r-space Equivalent to slicing in k -space 2D projection onto a plane (like angular corr) Corresponding power spectrum is slicing at k z =0 Reverse transform for the 2D correlations

26. Tomography of SDSS SDSS DR7 Main Galaxy Sample ◦ Limit distances to 100<r<750 h -1 Mpc Compute 3D correlation function Also cut 3D data into thin angular slices ◦ Project down to plane (only 2D info) ◦ Different widths (2.5, 5, 10 deg) ◦ Rotate slicing direction by 15 degrees ◦ Ensemble of 661 slices to work with Analyze 2D correlation functions  ( ,  ) (Tian, Budavari, Neyrinck and AS 2011)

27. SDSS DR7 MGS, Stripes 9 through 37, Northern Cap only 0.001 0.9, zErr<0.1 Remove all the objects in the incomplete areas 527,362 objects 17M random galaxies Slices: From 0 to 165°, 15° increments, 12 angular orientations, 2.5° thickness, 20°<width<80°, 661 slices total

28. Computations on GPUs Generated 16M randoms with correct radial and angular selection for SDSS-N Done on an NVIDIA GeForce 260 card 600 trillion galaxy/random pairs Brute force massively parallel code, much faster than tree-code for hi-res  All done inside the JHU SDSS database Correlation function is now DB utility

29. Full 3D correlation function

30. Radial Distribution

31. 300-750 Mpc/h Cut Lose 50% of galaxies, only 1/27 th of volume

32. 300-750 Mpc/h only

33. Wavelet Transform Mexican hat wavelet transform ◦ Compensated filter ◦ Enhances localized “bump” Zero signal for constant background Decreases correlations among bins

34. Wavelet Transform

35. S/N of the Wavelet Transform Flat thetaLOS(6 deg) Noise estimated from slices, divided by 16

36. Velocity effects Velocity correlations have longer range Nonlinear contributions in velocity correlations will extend much farther than in the density

37. Far Side Infall (140Mpc) Center: 140 h -1 Mpc, width: 25 h -1 Mpc Still shows some skewness  LOS (140)

38. Strong Non-Linear Infall (55Mpc) Distribution of LOS Mexican Hat wavelet coefficients over the 660 slices, ◦ Center at 55 h -1 Mpc, width 25 h -1 Mpc  LOS (55)

39. Perturbative Approach Zeldovich approximation 2 nd order in Eulerian space Correlation function is Easily generalized to redshift space (McCullagh and Szalay)

40. The Indra Simulations o Quantifying cosmic variance o Suite of dark matter N-body simulations o Gadget2 o 512 different 1 Gpc/h box, 1024 3 particles per simulation o Data loaded into SQL database, will be available to the public o Random initial conditions, WMAP7 cosmology o Particle data: o Ids, positions, velocities for 64 snapshots of each simulation o 35T particles total o Halo catalogs o Fourier modes: o Course density grid for 512 time steps of each run Bridget Falck (ICG Portsmouth), Tamás Budavári (JHU), Shaun Cole (Durham), Daniel Crankshaw (JHU), László Dobos (Eötvös), Adrian Jenkins (Durham), Gerard Lemson (MPA), Mark Neyrinck (JHU), Alex Szalay (JHU), and Jie Wang (Durham/Beijing)

41. Current N-body Simulations

42. Covariance matrix Real spaceLine of sight

43. Pairwise Velocity Distribution Thermal velocities (“finger of God”) neither Maxwellian or Exponential Distribution also anisotropic ◦ Longitudinal more elongated than transverse σ r

44. Pairwise Velocity Distribution R=1 Mpc/hR=15 Mpc/h

45. Redshift space distortions amplify and sharpen BAO features along the line of sight Near and far side infall onto the BAO bump Angular averaging wipes out most of this effect Evidence for BAO in SDSS DR7 MGS at ~110 h -1 Mpc, potentially constraining the equation of state at low z Trough at 55 h -1 Mpc indicates effects of strong nonlinear infall on these scales Nonlinearities important even on BAO scales!