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Redshift Space Distortions

Dr. Mariana Vargas Magana Instituto de Fisica, UNAM.

Mariana Vargas Magana Latest Results from BOSS and their Implications for Cosmology28 Noviembre, Instituto de Fisica, UNAM

Observables


Perturbations Evolution


Review: Linear Perturbation Theory

• Small initial deviations from homogeneous FLRW model

• Inhomogenities grew by gravitacional inestability.

� =⇢� ⇢̄

⇢̄<< 1

linearity condition


Dynamics: Growth Factor

Evolution of perturbations in a expanding Universe is given by

which has the general solution

The growing solution D+(t) is called linear growth function and is normalized such that D+(t0) = 1.


Dynamics: Growth FactorStructure formation at large scales

Linear Growth D(a), D(a=1)=1 Snapshots de 2 N-body simulaciones a diferentes tiempos, muestra fluctuaciones de densidad mayores en un modelo LCDM comparado con un universo dominado por materia (EdS).


Dynamics: Growth Factor

Evolution of perturbations in a expanding Universe is given by

which has the general solution

The growing solution D+(t) is called linear growth function and is normalized such that D+(t0) = 1.

The growth factor is defined as: f(a) =d lnD

d ln a


Growth factor

The growth factor can be right with a good approximation as:

f(a) ⇡ ⌦�m(a)

γ is named « growth index »and can be calculated for different models (Linder 2005)

CDM ) � = 0.6

⇤CDM ) � = 0.55

DGP ) � ⇡ 0.68

!CDM ) � = 0.55 + 0.05[1 + !]


Growth Factor


Why: Testing General Relativity

• Growth function G(a): δ(k, a) = aG(a)δi(k)

• In General Relativity G(a) is determined once H(a) is specified/measured; generically this relation is different in modified gravity models

Beth Reid INPA Sept 23



• allowed growth in ΛCDM given 2009 CMB, SN, BAO, H0

• constrained by RSD


Mor

tons

on, H

u, H

uter

er 2

010,

PR

D, 8

1, 0

6300

7



• General Quintessence models, massive neutrinos suppress RSD signal


Mor

tons

on, H

u, H

uter

er 2

010,

PR

D, 8

1, 0

6300

7

Non-flat quintessence models with early dark energy


Redshift Space Distortions: What

Beth Reid 5

x x

real to redshift space separations

|vp| ~ d σ8/d ln a = σ8 * f

isotropic squashed along line of sight

z∇ ⋅ vp = -aHf δm

INPA Sept 23

f = d ln σ8 /d ln a


Redshift Space Distortions (RSD)

Hamilton 1997

Unreconstructed no RSD Unreconstructed with RSD

Reconstructed no RSD Reconstructed with RSD


Distorsiones de Corrimiento al rojo

distancia por corrimiento al rojo

Verdadera distancia velocidad peculiar

Estas distorsiones generan un incremento del agrupamiento a lo largo de la linea de visión en comparación con la dirección perpendicular.


Real space



Redshift space



Real Space Correlation Function


Redshift Space Correlation function


Redshift Space Distortions: How

• ξ(rσ, rπ): number of pairs in excess of random as a function of pair separation OR the avg density around a galaxy

Beth Reid 6

White et al. 2011 mock catalogs

ξ(rσ, rπ)

INPA Sept 23


Redshift Space Distortions: How

• Sidenote: Alcock-Paczynski effect is degenerate with RSD

Beth Reid 7

White et al. 2011 mock catalogs

ξ(rσ, rπ)

INPA Sept 23

α∥

α⊥


Background: Linear RSD (Kaiser 1987)

Beth Reid 13 INPA Sept 23


Background: Legendre Polynomial moments

Beth Reid 14

General Expansion

Linear theory prediction

INPA Sept 23


Resultados cosmológicos de RSD con BOSS

Alam, Ho & Vargas 2014


CLPT(Wang 2014, Carlson 2013)

pairwise!velocity

dispersion of pairwise velocity

depends on linear power spectrum

Real space Correlation function and velocity statistics is provided by CLPT


Gaussian streaming Model (GSM )

Mean LOS velocity

velocity dispersion

Redshift space Correlation function is provided by GSM



Alam, Ho & Vargas 2014


Test with simulations


Medotodologia



In linear theory, b and f are completely degenerate with 𝞼8, and observed clustering is only sensitive to their combination b𝞼8 and f𝞼8


Diversidad de Modelos para RSD en BOSS

Reference MODEL SPACE Range

Samushia 2013 LPT CS 24-152

Beuther 2013 eTNS(Tayura) RegPT FS 0.01-0.2

Chuang 2013 Dew +FOG CS 32-200

Sanchez 2013 “PT” CS(W)

Alam 2014 CLPT CS 30-130

Reid 2013 LPT+HOD CS 0.8-32


200 220 240 260 280 300Radial Index

200

220

240

260

280

300

Ind

ex T

ran

sver

se

Preliminary

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Fingers of God(Dedos de dios)

Efecto Kaiser

escalas pequeñasgrandes escalas

BAO



RSD small scales


Background: Fingers-of-God

Beth Reid 15

x

x

x x

x

REAL SPACE: r ~ 1 Mpc/h

REDSHIFT SPACE: r ~ 15 Mpc/hFinger-of-God features mix small and large scale power

Central galaxies

Satellite galaxies

INPA Sept 23


Fisher Matrix forecast for BOSS

Beth Reid 16

2% theoretical uncertainty “goal” for ξ0,2

INPA Sept 23

IF we can determine σ from small-scale clustering (e.g.,

HOD), gain factor of 2 on RSD


Outline

• What are redshift space distortions (RSD) and why are they the coolest cosmological probe in town?

• Projections for BOSS RSD constraints

• Step away from the Kaiser formula: building intuition in configuration space

• A model for the redshift space distortions of halos



Legendre Polynomial moments

Beth Reid 19

General Expansion

Linear theory prediction

INPA Sept 23


Legendre Polynomial moments

Beth Reid 20

General Expansion

Relation to Pℓ(k)

INPA Sept 23


Legendre Polynomial moments:scale dependence

Beth Reid 21

L0(μ) = 1

L2(μ) = 1.5μ2 - 0.5

L4(μ) = 4.375μ4 - 3.75μ2 + 0.375

INPA Sept 23


Configuration space quantities of interest:

Beth Reid 22

rσ

rπr vz

v̅r, σ2r

v̅t=0, σ2ty rπ

INPA Sept 23

Perp ||


Linear theory pairwise velocities (δg = bδm)


mean infall velocity

velocity dispersion


Fisher 1995: the Kaiser formula in configuration space

Beth Reid 24

• δ(x), v(x’) correlated Gaussian fields

• Expand around y = rπ

• Equivalent to Kaiser formula

INPA Sept 23


Pairwise velocity statistics in linear theory



Small scales RSD

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