Relativistic distortions
in large-scale structure
Camille Bonvin
CERN, Switzerland
University of Sussex
March 2015
Galaxy survey
Credit: M. Blanton, SDSS
The distribution of galaxies is determined by:
The initial conditions
The theory of gravity
The content of the universe
To interpret properly the information from large-scale
structure, we need to understand what we are measuring.
University of Sussex
Camille Bonvin
p. 2 /55
Galaxy survey
N − N̄
We count the number of galaxies per pixel: ∆ =
N̄
How is ∆ related to: the initial conditions, the theory of
Credit: M. Blanton, SDSS
gravity and dark energy?
University of Sussex
Camille Bonvin
p. 3 /55
Galaxy distribution
Simple picture:
dark matter is inhomogeneously distributed
it creates gravitational potential wells
baryons fall into them and form galaxies.
More dark matter
Less dark matter
δρ
∆=
≡δ
ρ
University of Sussex
Camille Bonvin
p. 4 /55
Complications
Bias: the distribution of galaxies does not trace directly
the distribution of dark matter ∆ = b · δ
We never observe directly the position of galaxies, we
observe the redshift and the direction of incoming
photons.
In a homogeneous universe:
(x1 , x2 , x3 )
we calculate r(z)
light propagates on straight lines
r(z)
Observer
n
University of Sussex
Camille Bonvin
p. 5 /55
Redshift
In an inhomogeneous universe: the redshift is affected by
fluctuations, e.g. Doppler effect due to peculiar velocities.
radial shift in the galaxy position
More dark matter
Less dark matter
Redshift distortions
Observer
University of Sussex
Camille Bonvin
p. 6 /55
Lensing
In an inhomogeneous universe: light is lensed by matter
between the galaxies and the observer
transverse shift in the galaxy position
More dark matter
Less dark matter
Lensing distortions
Observer
University of Sussex
Camille Bonvin
p. 7 /55
Galaxy distribution
Credit: M. Blanton, SDSS
The structures seen on a galaxy map do not reflect
directly the underlying dark matter structures. The observed
positions of galaxies are shifted radially and transversally.
To extract information from a galaxy map, we need to
understand exactly what the distortions are.
University of Sussex
Camille Bonvin
p. 8 /55
Outline
Calculate the distortions that affect the large-scale structure:
number density ∆ and convergence κ (galaxy’s size).
∆ = density + redshift distortions
+ lensing + relativistic distortions
κ = lensing + relativistic distortions
The relativistic distortions should not be considered as a
noise but rather as a new signal.
Impact of the different terms on the correlation function:
the relativistic distortions change the properties of the twopoint function
we can isolate them.
We can combine the relativistic distortions with standard
observables to test the theory of gravity.
University of Sussex
Camille Bonvin
p. 9 /55
The over-density of galaxies
We count N (z, n) galaxies in a pixel of volume V (z, n)
We want to calculate the fluctuations in N (z, n) with
respect to the average number.
At each redshift, we average over the direction: N̄ (z)
The observed over-density is:
N (z, n) − N̄ (z)
∆(z, n) =
N̄ (z)
Relation with the dark matter density:
N (z, n) = ρ(z, n) · V (z, n)
and N̄ (z) = ρ̄(z) · V̄ (z)
University of Sussex
Camille Bonvin
p. 10/55
Derivation
ρ̄ + δρ
V̄ + δV
ρ(z, n) · V (z, n) − ρ̄(z) · V̄ (z)
∆=
ρ̄(z) · V̄ (z)
We keep only linear terms:
δρ(z, n) δV (z, n)
∆=
+
ρ̄(z)
V̄ (z)
the background redshift is different
from the observed redshift
ρ(z, n) − ρ̄(z̄)
δ(z, n) ≡
ρ̄(z̄)
!=
ρ(z, n) − ρ̄(z)
δρ(z, n)
=
ρ̄(z)
ρ̄(z)
University of Sussex
Camille Bonvin
p. 11/55
Derivation
ρ̄ + δρ
V̄ + δV
ρ(z, n) · V (z, n) − ρ̄(z) · V̄ (z)
∆=
ρ̄(z) · V̄ (z)
We keep only linear terms:
δρ(z, n) δV (z, n)
∆=
+
ρ̄(z)
V̄ (z)
δ
the background redshift is different
from the observed redshift
ρ(z, n) − ρ̄(z̄)
δ(z, n) ≡
ρ̄(z̄)
!=
ρ(z, n) − ρ̄(z)
δρ(z, n)
=
ρ̄(z)
ρ̄(z)
University of Sussex
Camille Bonvin
p. 12/55
Derivation
z = z̄ + δz
Taylor:
ρ̄(z) = ρ̄(z̄ + δz) ≃ ρ̄(z̄) + ∂z ρ̄ · δz
ρ(z, n) − ρ̄(z)
ρ(z, n) − ρ̄(z̄) − ∂z ρ̄ · δz
δρ(z, n)
=
≃
ρ̄(z)
ρ̄(z)
ρ̄(z̄)
δz
δρ(z, n)
= δ(z, n) − 3
ρ̄(z)
1+z
δz
δV (z, n)
−3
∆(z, n) = b · δ(z, n) +
V
1+z
describes what we observe
in the linear regime
University of Sussex
Camille Bonvin
p. 13/55
Derivation
z = z̄ + δz
Taylor:
ρ̄(z) = ρ̄(z̄ + δz) ≃ ρ̄(z̄) + ∂z ρ̄ · δz
same redshift bin
ρ(z, n) − ρ̄(z)
ρ(z, n) − ρ̄(z̄) − ∂different
δρ(z, n)
z ρ̄ · δz physical volume
=
≃
ρ̄(z)
ρ̄(z)
ρ̄(z̄)
δz
δρ(z, n)
= δ(z, n) − 3
ρ̄(z)
1+z
Observer
δz
δV (z, n)
−3
∆(z, n) = b · δ(z, n) +
V
1+z
dzdΩ
describes what we observe
in the linear regime
University of Sussex
Camille Bonvin
p. 14/55
Derivation
z = z̄ + δz
Taylor:
ρ̄(z) = ρ̄(z̄ + δz) ≃ ρ̄(z̄) + ∂z ρ̄ · δz
same solid angle
ρ(z, n) − ρ̄(z)
ρ(z, n) − ρ̄(z̄) − ∂different
δρ(z, n)
z ρ̄ · δz physical volume
=
≃
ρ̄(z)
ρ̄(z)
ρ̄(z̄)
δz
δρ(z, n)
= δ(z, n) − 3
ρ̄(z)
1+z
Observer
δz
δV (z, n)
−3
∆(z, n) = b · δ(z, n) +
V
1+z
dzdΩ
describes what we observe
in the linear regime
University of Sussex
Camille Bonvin
p. 15/55
Derivation
z = z̄ + δz
Taylor:
ρ̄(z) = ρ̄(z̄ + δz) ≃ ρ̄(z̄) + ∂z ρ̄ · δz
same radial bin
ρ(z, n) − ρ̄(z)
ρ(z, n) − ρ̄(z̄) − ∂z ρ̄ different
δρ(z, n)
· δz distance
=
≃
ρ̄(z)
ρ̄(z)
ρ̄(z̄)
δz
δρ(z, n)
= δ(z, n) − 3
ρ̄(z)
1+z
Observer
δz
δV (z, n)
−3
∆(z, n) = b · δ(z, n) +
V
1+z
dzdΩ
describes what we observe
in the linear regime
University of Sussex
Camille Bonvin
p. 16/55
Redshift
!"
CB and Durrer (2011)
ds2 = −a2 1 + 2Ψ dη 2 + 1 − 2Φ δij dxi dxj
#
"
#
$
νS
ES
=
Effect of inhomogeneities on the redshift: 1 + z =
νO
EO
Photons travel on null geodesics.
" rS
!
#
aO
1+z =
1 + VS · n − VO · n + Ψ O − Ψ S −
dr(Φ̇ + Ψ̇)
aS
0
Doppler
Gravitational redshift:
Gravitational redshift
Integrated Sachs-Wolfe
Observer
University of Sussex
Camille Bonvin
p. 17/55
Redshift
!"
CB and Durrer (2011)
ds2 = −a2 1 + 2Ψ dη 2 + 1 − 2Φ δij dxi dxj
#
"
#
$
νS
ES
=
Effect of inhomogeneities on the redshift: 1 + z =
νO
EO
Photons travel on null geodesics.
" rS
!
#
aO
1+z =
1 + VS · n − VO · n + Ψ O − Ψ S −
dr(Φ̇ + Ψ̇)
aS
0
Doppler
Gravitational redshift
Integrated Sachs-Wolfe
ISW: integrated along the trajectory, sensitive to dark energy.
Observer
University of Sussex
Camille Bonvin
p. 18/55
Volume fluctuation
CB and Durrer (2011)
We want to calculate the relation between a physical volume
element at the position of the galaxies and an observed pixel.
dz
dx
dΩ
galaxy
nµ
The null geodesic equation tells us how directions
and distances are perturbed.
θS = θO + δθ
ϕS = ϕO + δϕ
r = r̄ + δr
Observer
University of Sussex
Camille Bonvin
p. 19/55
Result
density
Yoo et al (2010)
CB and Durrer (2011)
Challinor and Lewis (2011)
redshift space distortion
1
∆(z, n) = b · δ − ∂r (V · n)
H
lensing
! r
′
r
−
r
dr′
∆Ω (Φ + Ψ)
−
gravitational
′
rr
0
#
redshift
Doppler "
Ḣ
2
1
1
+ 1− 2 −
V · n + V̇ · n + ∂r Ψ
H
rH
H
H
! r
H
2
1
dr′ (Φ + Ψ)
+ Ψ − 2Φ + Φ̇ − 3 V +
H
k
r 0
"
#$
%
! r
potential
2
Ḣ
′
+
dr
(Φ̇ + Ψ̇)
+
Ψ
+
2
H
rH
0
University of Sussex
Camille Bonvin
p. 20/55
Result
Yoo et al (2010)
CB and Durrer (2011)
Challinor and Lewis (2011)
density
1
∆(z, n) = b · δ − ∂r (V · n)
H
lensing
! r
′
r
−
r
dr′
∆Ω (Φ + Ψ)
−
′
rr
0
"
#
Ḣ
2
1
1
+ 1− 2 −
V · n + V̇ · n + ∂r Ψ
H
rH
H
H
! r
H
2
1
dr′ (Φ + Ψ)
+ Ψ − 2Φ + Φ̇ − 3 V +
H
k
r 0
"
#$
%
! r
2
Ḣ
′
+
dr
(Φ̇ + Ψ̇)
+
Ψ
+
2
H
rH
0
University of Sussex
Camille Bonvin
p. 21/55
Result
redshift space distortion
Kaiser 1987, Hamilton 1992
ρ − ρ̄
Real space∆ = δ =
Redshift space
ρ̄
Two well known corrections:
1
∆(z, n) = b · δ − ∂r (V · n)
Bias between luminous and dark
H
lensing
! r
′
matter.
r
−
r
′
dr
∆Ω (Φ + Ψ)
−
gravitational
′
rr
Observer
Observer
0
"
#
redshift
Ḣ
2
1
1
+ 1− 2 −
V · n + V̇ · n + ∂r Ψ
H
rH
H
H
! r
H
2
1
dr′ (Φ + Ψ)
+ Ψ − 2Φ + Φ̇ − 3 V +
H
k
r 0
"
#$
%
! r
2
Ḣ
′
+
dr
(Φ̇ + Ψ̇)
+
Ψ
+
2
H
rH
0
University of Sussex
Camille Bonvin
p. 22/55
Result
1
∆(z, n) = b · δ − ∂r (V · n)
H
lensing
! r
′
r
−
r
dr′
∆Ω (Φ + Ψ)
−
′
rr
Observer
0
"
#
Ḣ
2
1
1
+ 1− 2 −
V · n + V̇ · n + ∂r Ψ
H
rH
H
H
! r
H
2
1
dr′ (Φ + Ψ)
+ Ψ − 2Φ + Φ̇ − 3 V +
H
k
r 0
"
#$
%
! r
2
Ḣ
′
+
dr
(Φ̇ + Ψ̇)
+
Ψ
+
2
H
rH
0
University of Sussex
Camille Bonvin
p. 23/55
Result
Yoo et al (2010)
CB and Durrer (2011)
Challinor and Lewis (2011)
1
∆(z, n) = b · δ − ∂r (V · n)
H
! r
′
r
−
r
dr′
∆Ω (Φ + Ψ)
−
′
rr
0
#
Doppler "
Ḣ
2
1
1
+ 1− 2 −
V · n + V̇ · n + ∂r Ψ
H
rH
H
HObserver
! r
H
2
1
dr′ (Φ + Ψ)
+ Ψ − 2Φ + Φ̇ − 3 V +
H
k
r 0
"
#$
%
! r
2
Ḣ
′
+
dr
(Φ̇ + Ψ̇)
+
Ψ
+
2
H
rH
0
University of Sussex
Camille Bonvin
p. 24/55
Result
Yoo et al (2010)
CB and Durrer (2011)
Challinor and Lewis (2011)
1
∆(z, n) = b · δ − ∂r (V · n)
H
! r
′
r
−
r
dr′
∆Ω (Φ + Ψ)
−
gravitational
′
rr
0
"
#
redshift
Ḣ
2
1
1
Observer
+ 1− 2 −
V · n + V̇ · n + ∂r Ψ
H
rH
H
H
! r
H
2
1
dr′ (Φ + Ψ)
+ Ψ − 2Φ + Φ̇ − 3 V +
H
k
r 0
"
#$
%
! r
2
Ḣ
′
+
dr
(Φ̇ + Ψ̇)
+
Ψ
+
2
H
rH
0
University of Sussex
Camille Bonvin
p. 25/55
Result
Yoo et al (2010)
CB and Durrer (2011)
Challinor and Lewis (2011)
1
∆(z, n) = b · δ − ∂r (V · n)
H
! r
′
r
−
r
dr′
∆Ω (Φ + Ψ)
−
′
rr
0
"
#
Ḣ
2
1
1
+ 1− 2 −
V · n + V̇ · n + ∂r Ψ
H
rH
H
H
! r
H
2
1
dr′ (Φ + Ψ)
+ Ψ − 2Φ + Φ̇ − 3 V +
H
k
r 0
"
#$
%
! r
2
Ḣ
′
+
dr
(Φ̇ + Ψ̇)
+
Ψ
+
2
H
rH
0
University of Sussex
Observer
potential
Camille Bonvin
p. 26/55
Result
density
Yoo et al (2010)
CB and Durrer (2011)
Challinor and Lewis (2011)
redshift space distortion
1
∆(z, n) = b · δ − ∂r (V · n)
H
lensing
! r
′
r
−
r
dr′
∆Ω (Φ + Ψ)
−
gravitational
′
rr
0
#
redshift
Doppler "
Ḣ
2
1
1
+ 1− 2 −
V · n + V̇ · n + ∂r Ψ
H
rH
H
H
! r
H
2
1
dr′ (Φ + Ψ)
+ Ψ − 2Φ + Φ̇ − 3 V +
H
k
r 0
"
#$
%
! r
potential
2
Ḣ
′
+
dr
(Φ̇ + Ψ̇)
+
Ψ
+
2
H
rH
0
University of Sussex
Camille Bonvin
p. 27/55
Result
Yoo et al (2010)
CB and Durrer (2011)
Challinor and Lewis (2011)
standard expression
1
lensing: important at high z
∆(z, n) = b · δ − ∂r (V · n)
H
! r
′
relativistic distortions:
r
−
r
−
dr′
∆Ω (Φ + Ψ)
important at large scale
′
rr
0
#
"
1
1
2
Ḣ
H
V · n + V̇ · n + ∂r Ψ
+ 1− 2 −
δ
H
rH
H
H
k
! r
H
2
1
dr′ (Φ + Ψ)
+ Ψ − 2Φ + Φ̇ − 3 V +
! "2
H
k
r 0
H
"
#$
%
! r
δ
2
Ḣ
′
k
+
dr
(
Φ̇
+
Ψ̇)
+
Ψ
+
H2
rH
0
University of Sussex
Camille Bonvin
p. 28/55
Convergence
Galaxy surveys observe also the shape and the luminosity
et al (2012)
of galaxies
measure of the convergence. Schmidt
Alsing et al (2014)
The convergence κ measures distortions in the size.
The shear γ measures distortions in the shape.
Relativistic distortions affect the convergence at linear order.
We solve the geodesic
deviation equation
D2 δxα (λ)
α
β µ
ν
=
R
k
k
δx
βµν
Dλ2
University of Sussex
Camille Bonvin
p. 29/55
Convergence
Gravitational lensing
1
κ=
2r
Doppler lensing
′
"
CB (2008)
Bolejko et al (2013)
Bacon et al (2014)
#
1
−r
dr
∆Ω (Φ + Ψ) +
−1 V·n
′
r
rH
0
#! r
"
! r
1
1
′
−
dr (Φ + Ψ) + 1 −
dr′ (Φ̇ + Ψ̇)
r 0
rH
0
#
"
Integrated
terms
1
Ψ+Φ
+ 1−
Sachs Wolfe
rH
!
r
′r
University of Sussex
Camille Bonvin
p. 30/55
Convergence
Gravitational lensing
1
κ=
2r
Doppler lensing
′
"
CB (2008)
Bolejko et al (2013)
Bacon et al (2014)
#
1
−r
dr
∆Ω (Φ + Ψ) +
−1 V·n
′
r
rH
0
#! r
"
! r
1
1
′
−
dr (Φ + Ψ) + 1 −
dr′ (Φ̇ + Ψ̇)
r 0
rH
0
#
"
Integrated
terms
1
Ψ+Φ
+ 1−
Sachs Wolfe
rH
!
r
′r
Gravitational lensing
Observer
University of Sussex
Camille Bonvin
p. 31/55
Convergence
Gravitational lensing
1
κ=
2r
Doppler lensing
′
"
CB (2008)
Bolejko et al (2013)
Bacon et al (2014)
#
1
−r
dr
∆Ω (Φ + Ψ) +
−1 V·n
′
r
rH
0
#! r
"
! r
1
1
′
−
dr (Φ + Ψ) + 1 −
dr′ (Φ̇ + Ψ̇)
r 0
rH
0
#
"
Integrated
terms
1
Ψ+Φ
+ 1−
Sachs Wolfe
rH
!
r
′r
Gravitational lensing
Doppler lensing
The moving galaxy
is further away it
looks smaller, i.e.
demagnified
Observer
Observer
University of Sussex
Camille Bonvin
p. 32/55
Observations
Due to relativistic effects, ∆ and κ contain additional information.
δ, V, Φ, Ψ
(Φ + Ψ), V, Φ, Ψ
This can help testing gravity by probing the relations
between density, velocity and gravitational potentials.
Two difficulties:
The relativistic effects are small: we need to go to large scales.
We always measure the sum of all the effects.
We need a way of isolating the relativistic effects.
Method: look for anti-symmetries in the correlation function.
University of Sussex
Camille Bonvin
p. 33/55
Correlation function
Credit: M. Blanton, SDSS
We do not compare ∆(n, z) with observations, because
we do not know the values of the density, velocity and
gravitational potentials at (n, z)
We compare
ensemble averages
ξ = !∆(x)∆(x′ )"
University of Sussex
Camille Bonvin
p. 34/55
Density
x′
The density contribution ∆ = b · δ ,
generates an isotropic correlation
function.
s
x
ξ(s) = !∆(x)∆(x′ )" depends only on
the separation s = |x − x′ |
2
ξ(s) =
Ab D12
2π 2
!
dk
k
"
n
Observer
#ns −1
k
Tδ2 (k) j0 (k · s)
H0
University of Sussex
Camille Bonvin
p. 35/55
Density
The density contribution ∆ = b · δ ,
generates an isotropic correlation
function.
s
x
ξ(s) = !∆(x)∆(x′ )" depends only on
the separation s = |x − x′ |
2
ξ(s) =
Ab D12
2π 2
!
dk
k
"
x′
n
Observer
#ns −1
k
Tδ2 (k) j0 (k · s)
H0
University of Sussex
Camille Bonvin
p. 36/55
Redshift distortions
Real space
Redshift space
Redshift distor tions break the
isotropy of the correlation function.
1
∆ = b · δ − ∂r (V · n)
H
n
n
Observer
Quadrupole
Hamilton (1992)
P2 (cos β) =
A
µℓ (s) =
2π 2
4f
4f
+
3
7
!
dk
k
"
2
"
1
3
cos2 β −
2
2
1.0
µ2 (s)P2 (cos β)
#ns −1
k
Tδ2 (k) jℓ (k · s)
H0
P2!cosΒ"
ξ2 =
−D12
!
Observer
0.5
0.0
!0.5
0
45
90
135
180
Β
University of Sussex
Camille Bonvin
p. 37/55
Redshift distortions
x′
β
Redshift distor tions break the
isotropy of the correlation function.
x
s
1
∆ = b · δ − ∂r (V · n)
H
Observer
Quadrupole
n
Hamilton (1992)
P2 (cos β) =
A
µℓ (s) =
2π 2
4f
4f
+
3
7
!
dk
k
"
2
"
1.0
µ2 (s)P2 (cos β)
#ns −1
k
Tδ2 (k) jℓ (k · s)
H0
P2!cosΒ"
ξ2 =
−D12
!
1
3
cos2 β −
2
2
0.5
0.0
!0.5
0
45
90
135
180
Β
University of Sussex
Camille Bonvin
p. 38/55
Redshift distortions
x′
Redshift distor tions break the
isotropy of the correlation function.
max distortion
x
1
∆ = b · δ − ∂r (V · n)
H
Observer
Quadrupole
n
Hamilton (1992)
P2 (cos β) =
A
µℓ (s) =
2π 2
4f
4f
+
3
7
!
dk
k
"
2
"
1.0
µ2 (s)P2 (cos β)
#ns −1
k
Tδ2 (k) jℓ (k · s)
H0
P2!cosΒ"
ξ2 =
−D12
!
1
3
cos2 β −
2
2
0.5
0.0
!0.5
0
45
90
135
180
Β
University of Sussex
Camille Bonvin
p. 39/55
Redshift distortions
Redshift distor tions break the
isotropy of the correlation function.
x′
x
min distortion
1
∆ = b · δ − ∂r (V · n)
H
Observer
Quadrupole
n
Hamilton (1992)
P2 (cos β) =
A
µℓ (s) =
2π 2
4f
4f
+
3
7
!
dk
k
"
2
"
1.0
µ2 (s)P2 (cos β)
#ns −1
k
Tδ2 (k) jℓ (k · s)
H0
P2!cosΒ"
ξ2 =
−D12
!
1
3
cos2 β −
2
2
0.5
0.0
!0.5
0
45
90
135
180
Β
University of Sussex
Camille Bonvin
p. 40/55
Redshift distortions
x′
β
Redshift distor tions break the
isotropy of the correlation function.
x
s
1
∆ = b · δ − ∂r (V · n)
H
Observer
Quadrupole
n
Hamilton (1992)
P2 (cos β) =
A
µℓ (s) =
2π 2
4f
4f
+
3
7
!
dk
k
"
2
"
1.0
µ2 (s)P2 (cos β)
#ns −1
k
Tδ2 (k) jℓ (k · s)
H0
P2!cosΒ"
ξ2 =
−D12
!
1
3
cos2 β −
2
2
0.5
0.0
!0.5
0
45
90
135
180
Β
University of Sussex
Camille Bonvin
p. 41/55
Redshift distortions
x′
β
Redshift distor tions break the
isotropy of the correlation function.
x
s
1
∆ = b · δ − ∂r (V · n)
H
Observer
Hamilton (1992)
2
8f
µ4 (s)P4 (cos β)
ξ4 = D12
35
A
µ4 (s) = 2
2π
!
dk
k
"
"
1!
P4 (cos β) =
35 cos4 β − 30 cos2 β + 3
8
1.0
P4!cosΒ"
Hexadecapole
n
#ns −1
k
Tδ2 (k) j4 (k · s)
H0
0.5
0.0
!0.5
0
45
90
135
180
Β
University of Sussex
Camille Bonvin
p. 42/55
Redshift distortions
1
ξ0 (s) =
2
!
9
ξ4 (s) =
2
1
x
A
µ4 (s) = 2
2π
s
dµ ξ(s, µ)
−1
Measure separately:
b Observer
· σ8 and nf · σ8
1Hamilton (1992)
dµ ξ(s, µ)P4 (µ)
d ln D1
f=
d ln a
1.0
−1
P4!cosΒ"
µ = cos β
!
x′
β
1
!
∆ = b · δ 1− ∂r (V · n)
5
H
ξ2 (s) =
dµ ξ(s, µ)P2 (µ)
2 −1
!
Samushia et al (2014)
dk
k
"
#ns −1
k
Tδ2 (k) j4 (k · s)
H0
0.5
0.0
!0.5
0
45
90
135
180
Β
University of Sussex
Camille Bonvin
p. 43/55
Relativistic distortions
The relativistic distortions break
the symmetry of the correlation
function.
The correlation function differs
for galaxies behind or in front of
the central one.
CB, Hui and Gaztanaga (2013)
McDonald (2009)
x′
x
Observer
n
This differs from the breaking of isotropy, which is
symmetric: redshift distortions have even powers of cos β .
The amplitude is the same for β = 0 and β = π
To measure the asymmetry, we need two populations
of galaxies: faint and bright.
University of Sussex
Camille Bonvin
p. 44/55
Relativistic distortions
The relativistic distortions break
the symmetry of the correlation
function.
The correlation function differs
for galaxies behind or in front of
the central one.
CB, Hui and Gaztanaga (2013)
McDonald (2009)
x
x′
Observer
n
This differs from the breaking of isotropy, which is
symmetric: redshift distortions have even powers of cos β .
The amplitude is the same for β = 0 and β = π
To measure the asymmetry, we need two populations
of galaxies: faint and bright.
University of Sussex
Camille Bonvin
p. 45/55
Relativistic distortions
CB, Hui and Gaztanaga (2013)
McDonald (2009)
F
The relativistic distortions break
the symmetry of the correlation
function.
The correlation function differs
for galaxies behind or in front of
the central one.
B
F
Observer
n
This differs from the breaking of isotropy, which is
symmetric: redshift distortions have even powers of cos β .
The amplitude is the same for β = 0 and β = π
To measure the asymmetry, we need two populations
of galaxies: faint and bright.
University of Sussex
Camille Bonvin
p. 46/55
Anti-symmetries
density
redshift space distortion
1
∆(z, n) = b · δ − ∂r (V · n)
H
lensing
! r
′
r
−
r
∆Ω (Φ + Ψ)
−
dr′
gravitational
′
rr
0
#
redshift
Doppler "
Ḣ
1
2
1
+ 1− 2 −
V · n + V̇ · n + ∂r Ψ
H
rH
H
H
! r
1
H
2
+ Ψ − 2Φ + Φ̇ − 3 V +
dr′ (Φ + Ψ)
H
k
r 0
#$
"
%
! r
potential
2
Ḣ
′
dr
(Φ̇ + Ψ̇)
+
Ψ
+
+
2
H
rH
0
University of Sussex
Camille Bonvin
p. 47/55
Cross-correlation
The following terms break the symmetry:
∆rel =
!
Ḣ
2
1− 2 −
H
rH
"
1
1
V · n + V̇ · n + ∂r Ψ
H
H
University of Sussex
Camille Bonvin
p. 48/55
Dipole in the correlation function
F
CB, Hui and Gaztanaga (2013)
H
2
ξ(s, β) = D1 f
H0
A
ν1 (s) = 2
2π
!
dk
k
!
β
Ḣ
2
+
2
H
rH
"
"
B
(bB − bF )ν1 (s) · cos(β)
#ns −1
k
Tδ (k)TΨ (k) j1 (k · s)
H0
Observer
n
By fitting for a dipole in the correlation function, we can
measure relativistic distortions, and separate them
from the density and redshift space distortions.
3
ξ1 (s) =
2
!
1
dµ ξ(s, µ) · µ
µ = cos β
−1
University of Sussex
Camille Bonvin
p. 49/55
Multipoles
z = 0.25
Monopole
Quadrupole
0
120
100
!20
80
s 2 Ξ2
s 2 Ξ0
60
40
!40
!60
20
0
!80
!20
50
100
150
200
50
s !Mpc"h#
Hexadecapole
200
150
200
Dipole
2.0
6
1.5
s 2 Ξ1
s 2 Ξ4
150
s !Mpc"h#
8
4
1.0
0.5
2
0
100
50
100
s !Mpc"h#
150
200
0.0
50
100
s !Mpc"h#
University of Sussex
Camille Bonvin
p. 50/55
1
κg =
2r
!
Convergence
r
dr
′r
0
′
−r
∆Ω (Φ + Ψ)
′
r
κv =
!
"
1
−1 V·n
rH
We can isolate the Doppler lensing by looking for
anti-symmetries in !∆κ"
redshift space
real space
demagnified
κ
κ
∆
∆
κ
magnified
κ
Observer
n
Observer
n
University of Sussex
Camille Bonvin
p. 51/55
κ
Dipole
β
∆
CB, Bacon, Andrianomena, Clarkson and Maartens (in preparation)
2A
2 H
ξ(s, β) = 2 2 D1
f
9π Ωm
H0
!
"!
"
3f
1
b+
ν1 (s)cos(β)
1−
Hr
5
Observer
z = 0.1
n
z = 0.5
0.5
CB
0
Gravitational lensing
0.0
Gravitational lensing
!0.5
!4
s 2 Ξ1
s 2 Ξ1
!2
Doppler lensing
!6
!1.0
!1.5
!8
!10
50
100
s !Mpc"h#
150
200
!2.0
Doppler lensing
50
100
150
200
s !Mpc"h#
The dipole due to gravitational lensing is completely subdominant.
University of Sussex
Camille Bonvin
p. 52/55
Testing Euler equation
The monopole and quadrupole in ∆ allow to measure V
The dipole allows to measure:
∆rel
1
= ∂r Ψ +
H
!
Ḣ
2
1− 2 −
H
rH
"
1
V · n + V̇ · n
H
If Euler equation is valid: V̇ · n + HV · n + ∂r Ψ = 0
∆rel = −
!
Ḣ
2
+
2
H
rH
"
V·n
With the dipole, we can test the Euler equation.
University of Sussex
Camille Bonvin
p. 53/55
Measuring the anisotropic stress
The dipole in the convergence is sensitive to:
κv =
!
"
1
−1 V·n
rH
1
The standard part κg =
2r
!
r
0
′
r
−
r
dr′ ′ ∆Ω (Φ + Ψ)
r
can be measure through !κκ" and !γγ"
Assuming Euler equation, we can test the relation between
the two metric potentials Φ and Ψ .
University of Sussex
Camille Bonvin
p. 54/55
Conclusion
Our observables are affected by relativistic effects.
These effects have a different signature in the
correlation function: they induce anti-symmetries.
By measuring these anti-symmetries we can isolate the
relativistic effects and use them to test the relations
between the density, velocity and gravitational potentials.
University of Sussex
Camille Bonvin
p. 55/55
Contamination
The density and velocity evolve with time: the density of the
faint galaxies in front of the bright is larger than the density
behind. This also induces a dipole in the correlation function.
Larger redshift
smaller density
F
2
relativistic
1
s 2 Ξ1
B
!1
F
Observer
!2
n
total
0
evolution
50
100
150
200
s !Mpc"h#
University of Sussex
Camille Bonvin
p. 56/55
Dipole in the correlation function
ξ(s, β) =
D12
A
ν1 (s) = 2
2π
H
f
H0
!
!
"
dk
k
2
Ḣ
+
2
H
rH
"
F
B
#ns −1
k
Tδ (k)TΨ (k) j1 (k · s)
H0
Observer
2.0
1.5
s 2 Ξ1
β
(bB − bF )ν1 (s) · cos(β)
n
z = 0.25
1.0
z = 0.5
0.5
0.0
z=1
50
100
s !Mpc"h#
150
200
bB − bF ≃ 0.5
University of Sussex
Camille Bonvin
p. 57/55