Mariana Vargas Magana - Berkeley Center for Cosmological Physics

Redshift
Distortions
Mariana Vargas Magana
Laboratorie Astroparticule & Cosmologie- Universite Paris 7-Denis Diderot
ssential Cosmology for the next generation 2011 BCCP-IAC Puerto Vallarta , Mexico, January, 2011

INTRODUCTION
A FIRST EXPLORATION
OUTLINE
LOGNORMAL SIMULATIONS IN LINEAR REGIME
MONTE CARLO REALIZATIONS OF A N-BODY
SIMULATION (HORIZON PROJECT)
PERSPECTIVES.

INTRODUCTION TO
REDSHIFT DISTORTIONS
IN REDSHIFT SURVEYS WE MAP THE UNIVERSE IN 3 DIMENSIONS
USING REDSHIFT TO MESURE DISTANCE.
REDSHIFT DISTANCES S ( ) DIFFERS FROM TRUE DISTANCES
R ( )BY PECULIAR VELOCITIES.
r = H0d
s = r + v
s ≡ cz
v ≡ �r · v
PECULIAR VELOCITIES CAUSE THAT DISTANCES TO APPEAR
DISPLACED ALONG THE LINE OF SIGHT IN REDSHIFT SPACE.

ILUSTRATION OF REDSHIFT
DISTORTIONS
LARGE
SCALES
SMALL
SCALES
GALAXIES UNDERGOING INFALL VELOCITIES
TOWARS A SPHERICAL OVERDENSIY
LARGE SCALES,
PECULIAR
VELOCITY OF
INFALL SHELL IS
SMALL
COMPARED WITH
ITS RADIUS.
SMALL SCALES,
RADIUS OF
SPHERE IS
SMALLER,
ALSO PECULIAR
VELOCITY IS
LARGER

WHY STUDING REDSHIFT
DISTORTIONS?
REDSHIFT DISTORTIONS GIVE ACCESS TO THE LINEAR GROWTH
RATE FACTOR OF DENSITY PERTURBATION “THAT IS A POWERFUL
OBSERVABLE QUANTITY OF FUTURE LARGE REDSHIFT SURVEYS TO PROBE PHYSICAL
PROPERTIES OF DARK ENERGY AND TO DISTINGUISH AMONG VARIOUS GRAVITY
THEORIES.”[ TEPPEI OKUMURA AND Y. P. JING arXiv:1004.3548v2 ]

LINEAR REGIME APPROXIMATION
IN LINEAR REGIME , KAISER’S FORMULA’S [1987] APPLIES, THE EFFECT OF
REDSHIFT DISTORTIONS IS THE AMPLIFICATION OF THE UNREDSHIFTED MODE BY
A FACTOR
AMPLITUDE REDSHIFT
DISTORTIONS
β = f(ΩM )= dLnD
dLna
P s (k) = (1 + βµ 2 k) 2 P (k)
MASS DENSITY
PARAMETER AT Z
ANGLE BETWEEN WAVE
VECTOR & LOS DIRECTION
LINEAR GROWTH RATE OF DENSITY
PERTURBATIONS
GROWTH INDEX
f(ΩM )= γ =0.55 FOR GR
(LINDER 2005)
ΩM
γ
b
LIGHT TO MASS BIAS
δ = bδM
REDSHIFT DISTORTIONS ALLOWS TO PROBE DEVIATIONS FROM GR.
DEGENERATE PARAMETER COMBINATION (DENSITY OF MATTER & BIAS)

DECOMPOSITION IN HARMONICS
IN THE LINEAR REGIME
FOR PLANE PARALLEL APROXIMATION, POWER SPECTRUM CAN BE WRITTEN
AS A SUM OF EVEN HARMONICS.
P s (k) =ΣPl(µk)P s
l (k)
(KAISER’S FORMULA )IN LINEAR THEORY REDSHIFTED POWER SPECTRUM IS
COMPLETLY CHARACTERZED BY L=0,2,4, THE HEXADECAPOLE (L=4) IS
SMALL COMPARED WITH THE OTHERS & NOISY
HARMONICS OF REDSHIFT CORRELATION ARE DEFINED SIMILARLY TO
THOSE OF POWER SPECTRUM [HAMILTON (1992)]
ξ s (s, µ) =P0(µs)ξ s 0(s)+P2(µs)ξ s 2(s)+P4(µs)ξ s 4(s)
ξ s 0(s) = (1 + 2/3β +1/5β 2 )ξ(r)
ξ s 2(s) = (4/3β +4/7β 2 )(ξ(r) − ¯ ξ(r))
ξ0(s)
ξ(r)
2 1
= 1 + β +
3 5 β2
¯ ξ(r) = 3r −3
ξ2(s)
ξ s 0 (s) + (3/s3 ) � s
0 ξs 0 (s′ )s ′ 2 ds ′ =
THESE RATIOS YIELD AN ESTIMATE OF AT EACH DISTANCE.
β
� r
0
4 4
3β + 7β2 1+ 2
3
ξ(s)s 2 ds
1 = G(β)
β + 5β2

MODELING REDSHIFT DISTORTIONS
IN NONLINEAR REGIME
EXTRACT INFORMATION WHERE THE POWER SPECTRUM DOES NOT MATCH TO THE
LINEAR REGIME.
NON-LINEARITY IN THE VELOCITY AND DENSITY FIELDS PRODUCES DISTORTIONS IN AN
OPPOSITE SENSE TO THE LINEAR THEORY PREDICTIONS, LEADING TO LOWER ESTIMATES
OF
β
β
IN ORDER TO ESTIMATE USING CURRENTLY AVAILABLE REDSHIFT SURVEYS IS
ESSENTIAL TO MODEL THE NON-LINEARITIES ACCURATELY
STANDARD PROCEDURE CONSIST ON THE MODIFICATION OF KAISER'S FORMULA
INCLUDING A TERM TAKING IN ACCOUNT THE UNCORRELATED VELOCITY
DISPERSION .
P s (k) = (1 + βµ 2 k) 2 P (k) G(k, µ 2 )
limk→0G(k, µ 2 ) = 1
DISTRIBUTION OF RANDOM
PAIR VELOCITIES ,
(EXPONENCIAL. GAUSSIAN)

BIAS PARAMETER OVERSIMPLIFIED?
β = Ω0.6
m
b
RATIO OF RMS GALAXY FLUCTUATIONS TO
RMS MASS FLUCTUATIONS ON SCALE R.
RELATION BETWEEN THE GALAXY AND MASS DENSITY FIELDS DEPENDS ON THE COMPLEX
PHYSICS OF GALAXY FORMATION, AND IT CAN BE IN GENERAL NON-LINEAR, STOCHASTIC,
AND PERHAPS NON-LOCAL..
WAS ESTIMED USING DIFFERENT TECHNIQUES AND USING COMPLEX BIAS MODELS.
IT WAS FOUND THAT β ESTIMATED FROM DIFFERENT METHODS WAS SIMILAR TO THE
ASYMPTOTIC VALUE OF ( β = ) FOR R LARGE. [ ANDREAS A. BERLIND , VIJAY K.
NARAYANAN ,DAVID H. WEINBERG arXiv:astro-ph/0008305v1]
Ω0.6
m
bσ(r)
β
b(r) =σg(r)/σm(r)
β = Ω0.6
m
bσ(r)
CONVENTIONAL INTERPRETATION OF CONTINUES
TO HOLD EVEN FOR COMPLEX BIAS MODELS.
β

FIRST EXPLORATION....
FIRST GOAL INVESTIGATE HOW WELL I CAN RECONSTRUCT
AMPLIFICATION REDSHIFT PARAMETER WITH SIMULATIONS.
GENERATE LOG NORMAL RANDOM FIELD
FOLLOWS ΛCDM
.COSMOLOGY TO SIMULATE
DENSITY FIELD, AND USE THIS FIELD AS PDF TO
POPULATE WITH GALAXIES.
GENERATE VELOCITIES USING EXPRESSION FROM
LINEAR REGIME .

SIMULATION LOGNORMAL GALAXIES
REAL CONFIGURATION SPACE

SIMULATION LOGNORMAL GALAXIES
REDSHIFT CONFIGURATION SPACE

π [Mpc]
CORRELATION FUNCTION IN 2D
100
0
-100
C(r p,π)
-100 0 100
r p [Mpc]
NO VELOCITIES
-1.6
-1.8
-2.0
-2.2
s [Mpc]
BAO PEAK
ξ 0(s)× s 2
100
0
ξ 2(s)× s 2
-100
400
300
200
100
0
C(s,θ)
0 50 100 150 200
s [Mpc.h -1 -1.0
]
0 50 100 150 200
s [Mpc.h -1 -20
-40
-60
]
0
20
60
40
-2.0
-2.5
-50 0 50
θ [Degrees]
0.0
-0.5
-1.5
ξ 0(s)× s 2
ξ 2(s)× s 2

π [Mpc]
CORRELATION FUNCTION IN 2D
100
0
-100
C(r p,π)
-100 0 100
r p [Mpc]
WITH VELOCITIES
-1.6
-1.8
-2.0
ξ 0(s)× s 2
s [Mpc]
ξ 2(s)× s 2
100
0
-100
0 50 100
s [Mpc.h
150 200
-1 500
400
300
200
100
0
C(s,θ)
0.0
-0.5
]
-1.0
0 50 100 150 200
s [Mpc.h -1 -100
-200
-300
]
0
300
200
100
-50 0 50
-2.0
-2.5
θ [Degrees]
-1.5
2
ξ (s)× s 2

s(Mpc)
CORRELATION FUNCTION 2D &
MONOPOLE AND QUADRUPOLE ESTIMATION
ξ(s, θ)
θ( ◦ )
r �
ξ(r⊥,r �)
r⊥
FOR EACH BIN IN R , A
POLINOMIAL FIT IS DONE TO
CORRELATION FUNCTION
ξ s (s, µ) =P0(µs)ξ s 0(s)+P2(µs)ξ s 2(s)+P4(µs)ξ s 4(s)
ξ(ri, cos 2 θ)
cos 2 (θ)

LOG-NORMAL SIMULATION
REAL SPACE

s [Mpc]
100
0
-100
LOG-NORMAL SIMULATION
C(s,θ)
-50 0 50
θ [Degrees]
-0.4
-0.6
-0.8
-1.0
-1.2
ξ 0(s)× s 2
ξ 2(s)× s 2
ξ 4(s)× s 2
150
100
50
0
Input without RS dist.
Linear RS dist.
Simulation
0 50 100
s [Mpc.h
150 200
-1 -50
]
400
200
0
-200
0 50 100
s [Mpc.h
150 200
-1 -400
]
150
100
50
0
0 50 100
s [Mpc.h
150 200
-1 -50
]
REDSHIFT SPACE

BETA STIMATION USING REDSHIFTED
AND REAL MONOPOLE RATIO
ξ 0(s) / ξ 0(r)
4
3
2
1
0
ξ0(s)
ξ(r)
β =1
Bias = 2.00
2 1
= 1 + β +
3 5 β2
Horizon rpi
50 100
s [Mpc.h
150 200
-1 ]

HORIZON SIMULATION
HALOS OF DARK MATTER OF N-BODY SIMULATION, 2 GPC SIDE
s [Mpc]
100
0
-100
C(s,θ)
-50 0 50
θ [Degrees]
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
ξ 0(s)× s 2
ξ 2(s)× s 2
400
300
200
100
0
20
60
40
0 50 100 150 200
s [Mpc.h -1 ]
0 50 100
s [Mpc.h
150 200
-1 -20
-40
-60
]
0

s [Mpc]
100
0
-100
HORIZON SIMULATION
C(s,θ)
-50 0 50
θ [Degrees]
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
ξ 0(s)× s 2
ξ 2(s)× s 2
0 50 100 150 200
s [Mpc.h -1 400
300
200
100
0
]
300
200
100
0 50 100
s [Mpc.h
150 200
-1 -100
-200
-300
]
0

ξ 0(s) / ξ 0(r)
BETA STIMATION IN HORIZON
BIAS USING RATE BETWEEN REDSHIFTED /REAL MONOPOLE
4
3
2
1
0
b =1
β = ΩM0
b
ξ0(s)
ξ(r)
Bias = 1.40
γ ∼ 0.46
2 1
= 1 + β +
3 5 β2
Horizon rtheta
50 100 150 200
s [Mpc.h -1 ]

PERSPECTIVES...
FINISH BIAS ESTIMATON USING QUADRUPOLE.
ADD ANGULAR AND RADIAL SELECTION FONCTIONS TO
SIMULATE REAL DATA.
APPLIED TO OTHERS MOCK CATALOGUES OF GALAXIES
(LAS DAMAS) TO SEE SMALL SCALES (FINGERS OF GOD).
GENERATE MOCK CATALOGUES OF GALAXIES USING FOF
OF HORIZON SIMULATION WITH DIFERENT PRESCRIPTIONS
FOR BIAS ?
APPLIED TO SDSS PUBLIC DATA (DR7) AND COMPARED
WITH PREVIOUS WORK
APPLIED TO SDSS-III BOSS DATA.

FIN

BIAS USING QUADRUPOLE
NORMALIZED
ξ2(s)
ξ s 0 (s) + (3/s3 ) � s
0 ξs 0 (s′ )s ′ 2 ds ′ =
4 4
3β + 7β2 1+ 2
3
1 = G(β)
β + 5β2

QUADRUPOLE ET BIAS IN HORIZON
GENERE PLOT
ξ2(s)
ξ s 0 (s) + (3/s3 ) � s
0 ξs 0 (s′ )s ′ 2 ds ′ =
4 4
3β + 7β2 1+ 2
3
1 = G(β)
β + 5β2

GENERATION DE SIMULATION
LOG NORMAL
GENERATION OF A LOG NORMAL RANDOM FIELD,FOR
EACH MODE K WITH VARIANCE PROPORCIONAL TO PK
INPUT.
LES GALAXIES ARE PLACED USING DENSITY FIELD AS A
PROBABILITY DENSITY FUNCTION.
VELOCITIES ARE GENERATED USING DENSITY FIELD IN
THE LINEAR APROXIMATION :
v = −β∇∇ −2 δ