Redshift Space Distortions & MG models Jorge Cervantes, ININ III Taller Cuernavaca July 30th, 2015

Theory The redshift distance (s) differs from the true distance (r) by its total velocity along the line of sight: In terms of local, peculiar velocities, one has: The peculiar velocities of galaxies appear displaced along the line of sight in redshift space.

RSD refers to distortions in the power spectrum due to large scale peculiar velocities falling down to a potential well (there is also a random component to take out -FoG-).

One computes quantities in real (where matter distribution is) and redshift space (from which galaxies are measured). It is also convenient to define Fourier modes (with a hat ^) both in real and redshift space. One defines the 2-point correlation function (or covariance function) which is the expectation value of the products of overdensites at a pair of randomly positioned points separated by r12 . The Fourier transforms is The Power spectrum

The peculiar velocity of a galaxy is its proper velocity in the comoving frame is the conformal time The continuity, Euler, and Poisson eqs are (for CDM):

Linear regime -> A modification of gravity implies a different growth rate. when only gravity operates, the peculiar velocity in the linear regime is purely gradient.

The continuity eq. becomes: and one defines, the linear growth rate: CDM in GR gives: LCDM in GR gives:

Bias DM dominates over galaxies (g) in mass, but what we see is a galaxy field. There are evidences from the different galaxy catalogs of the same objects that poses different correlation amplitudes. Galaxy overdensity Matter overdensity One expects galaxy velocities follow the velocity of matter: The continuity eq. implies: The bias factor evolves closer to unity as time passes. One defines: The continuity eq. is now and b is in fact measured! that means we measure the ratio of the peculiar velocity to the galaxy overdensity.

Linear Redshift Distortion operator

Linear plane-parallel (distance observer limit) RSD operator In Fourier space: with

For Galaxies, Kaiser (1987): Redshift space Fourier space But for DM It implies: In s-space In Fourier space There is a degeneracy between f and b Since Pm ~ s82, then the measurement depends on b s8 and f s8 . The model is well grounded in large scales (k -> 0), but it breaks down on small, non-linear scales. Detections of evolving dark energy or modified gravity proceed as follows: Every alternative model should provide a background dynamics (H,a) and a kinematical description of the perturbations (d, f). Thus, fits are performed to the LSS data. Also, one makes comparisons with other probes (CMB, BAO) and with estimates from cosmological simulations.

Non-linear power spectra (1105.1194)

The correlation function computed in redshift-space from 2dF data is shown, as a function of the line-of-sight π and transverse σ (Peacock et al. 2001). To emphasize deviations from circular symmetry, the first quadrant has been mirrored along both axes. The contours represent model predictions, with ξ = 10,5,2,1,0.5,0.2 and 0.1. The Fingers-of-God effect is clearly seen, and the Kaiser flattening can just be seen at large radii.

b =0.49+-0.09, f=0.91+-0.14 at z=0.15 Peacock et al 2001, Nature Letter, using 220,000 galaxies from 2dFGRS b =0.49+-0.09, f=0.91+-0.14 at z=0.15

Guzzo et al 2008, Nature Letter, using 10,000, galaxies VIMOS VLT Deep Survey b =0.70+-0.26, f=0.91+-0.36, at z=0.8

Song and Percival 2009 They compare LCDM with DGP, clumpling dark energy, interacting dark energy

Examples (w0 free) 1105.1194

C-H Chuang, F. Prada, et al, using DR9 CMASS BOSS galaxies,1303.4486 One measures bs8 and b f s8 (0.57)=0.126+0.008-0.010 b = 0.428 ± 0.066

Modified Gravity? Macaulay et al 2013 Data from L. Samushia, W. J. Percival & A. Raccanelli, (2012).

RSD in f(R), Taruya et al 1309.6783

arXiv:1407.1865

Shadab Alam, Shirley Ho, Mariana Vargas-Magaña, Donald P. Schneider, 1504.02100 They use the CLPT-GSRSD model

1.Eulerian Dispersion Model (EDM) 1408.5435, M. White et al: Tests of redshift-space distortions models in configuration space for the analysis of the BOSS final data release. Idea: to fit the N-body data using a set of theoretical models 1.Eulerian Dispersion Model (EDM) 2. Non-linear Dispersion Model (NDM) considers Pr from non-linear theory 3. Perturbation theory inspired model (PIM) 4. The Gaussian streaming model (GSM) 5. The Lagrangian streaming model (LSM) is a hybrid model that uses the Zeldovich approx. with a model for local Lagrangian bias.

The end

PPF formalism and g are the functions that parametrize modified gravity. It follows to solve these equations, to computed the PP and to fit to data.

The bias implies in the correlation function: Dekel & Lahav (1999): r is the dimensionless correlation coefficient between the distribution of mass and galaxies.

For Galaxies, Kaiser (1987): Redshift space Real space But for DM It implies: In s-space In real space There is a degeneracy between f and b One adds a Poisson distributed noise: The model is well grounded in large scales (k -> 0), but it breaks down on small, non-linear scales. The power spectrum is boosted in redshift space, since the velocity field is directed toward mass concentrations. Therefore, galaxies appear to be more concentrated. Detections of non-gaussianities or modified gravity?

Local reference system (the Local Group)

Methods to measure b Angle-average the power spectra This quantity is actually measured and from it one infers b

Ratio of the quadrupole to monopole harmonics of the power spectrum Monopole, l=0 Quadrupole, l=2 Hexadecapole, l=4

This ratio can be measured and thus b can be determine. The hexadecapole is in fact small < 0.1 of the monopole and quadrupole that determine: This ratio can be measured and thus b can be determine. Surveys measure at finite resolutions and therefore in practice introduce window functions, W(k) that imply And again the same power ratio holds, as above.

Maximum likehood

Beyond linear regime The non-linear Kaiser model:

Quasi-linear regime

Non-linear regime “Streaming” models:

Using the following expression Percival and White (2009) showed that one can in fact measure by using likehood fits.