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

2. 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.

3. 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-).

4. 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

5. 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):

6. 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.

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

8. 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.

9. Linear Redshift Distortion operator

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

11. 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.

12. Non-linear power spectra (1105.1194)

13. 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.

14. 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 = , f=
at z=0.15

15. Guzzo et al 2008,
Nature Letter,
using 10,000, galaxies
VIMOS VLT
Deep Survey
b = ,
f= ,
at z=0.8

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

17. Examples (w0 free)

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

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

22. RSD in f(R), Taruya et al

23. arXiv:

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

27. 1.Eulerian Dispersion Model (EDM)
, 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.

29. The end

30. 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.

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

32. 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?

33. Local reference system (the Local Group)

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

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

36. 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.

37. Maximum likehood

38. Beyond linear regime
The non-linear Kaiser model:

39. Quasi-linear regime

40. Non-linear regime
“Streaming” models:

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