Transformation problem Limber eqn. Accuracy check Conclusions
How accurate is Limber’s equation?
Patrick Simon
Argelander-Institut für Astronomie, AIfA
Rheinische Friedrich-Wilhelms-Universität Bonn, Germany
19th of Sept. 2006
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
1
Angular and real-space clustering
Defining structure statistically
Clustering of galaxies
Projecting random fields
2
Limber’s approximation and its breakdown
Derivation of Limber’s equation
Application of Limber’s equation
Where Limber’s approximation fails
3
Testing Limber’s equation against exact solution
Exact solution for angular clustering
Accuracy of Limber’s equation
Where else is Limber’s equation important?
4
Conclusions
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Definitions Clustering of galaxies 3D-2D projection
Distribution of galaxies in redshift space
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Definitions Clustering of galaxies 3D-2D projection
Defining a two-point correlation of galaxy clustering
Imagine volume being subdivided into small
cells dV , #galaxies inside each cell is random
variable N(~r )
All cells together make up a random field with
joint PDF
P(N(~r1 ), N(~r2 ), . . .)
The PDF is characterised by its moments
hN(~ri )i = hNi
hN(~ri ) N(~rj )i
hN(~ri ) N(~rj ) N(~rk )i
...
first order
second order
third order
...
Cosmological Principle →
All moments depend solely on |~ri − ~rj |, i.e.
homogeneity and isotropy of PDF!
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Definitions Clustering of galaxies 3D-2D projection
Defining a two-point correlation of galaxy clustering
Average number of pairs of galaxies in cells
with mutual distance ∆~r is:
hN(~r ) N(~r + ∆~r )i = hNi2 [1 + ξ(|∆~r |)] ,
ξ(|∆~r |): correlation function (2nd -order)
Alternative:
ξ(|∆~r |) = hδ(~r )δ(~r + ∆~r )i
where δ(~r ) is the fluctuation inside cell ~r :
δ(~r ) ≡
N(~r )
−1
hNi
also called: density contrast
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Definitions Clustering of galaxies 3D-2D projection
Clustering of galaxies
The real-space clustering of galaxies is
more or less a power law
“ r ”γ
0
ξ(r ) =
r
with the clustering length,
r0 ∼ 5.4 h−1 Mpc, and a slope, γ ∼ 1.8.
The angular clustering of galaxies is also
well described by a power law
ω(θ) = Aω θ−δ ,
Springel et al. (2005),
Nature, 435, 629
with the clustering amplitude Aω and slope
δ.
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Definitions Clustering of galaxies 3D-2D projection
Clustering of galaxies
The real-space clustering of galaxies is
more or less a power law
“ r ”γ
0
ξ(r ) =
r
with the clustering length,
r0 ∼ 5.4 h−1 Mpc, and a slope, γ ∼ 1.8.
The angular clustering of galaxies is also
well described by a power law
ω(θ) = Aω θ−δ ,
Scranton et al. (2002),
ApJ, 579, 49
with the clustering amplitude Aω and slope
δ.
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Definitions Clustering of galaxies 3D-2D projection
Relation between real-space and angular clustering
~ are projections of the
Fluctuations in angular number density, δ̂(ϑ),
~
~
real-space fluctuations, δ(r , r ), in direction ϑ:
Z ∞
Z ∞
~ =
~ r) ;
δ̂(ϑ)
dr p(r ) δ(r ϑ,
dr p(r ) = 1 ,
0
0
p(r ) is a filter selecting galaxies from some (comoving) distance r .
Two-point correlation of the projected random field is
ω(θ)
=
=
≈
where R ≡
~ 1 )δ̂(ϑ
~ 2 )i
hδˆ1 (ϑ
Z ∞ Z ∞
~ 1 , r1 )δ2 (r2 ϑ
~ 2 , r2 )i
dr1
dr2 p(r1 )p(r2 )hδ1 (r1 ϑ
0
Z ∞
Z0 ∞
“ r +r ”
1
2
,
dr1
dr2 p(r1 )p(r2 ) ξ R,
2
0
0
q
~1 , ϑ
~2 .
r12 + r22 − 2r1 r2 cos θ, and θ = ^ϑ
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Definitions Clustering of galaxies 3D-2D projection
Relation between real-space and angular clustering
~ are projections of the
Fluctuations in angular number density, δ̂(ϑ),
~
~
real-space fluctuations, δ(r , r ), in direction ϑ:
Z ∞
Z ∞
~ =
~ r) ;
δ̂(ϑ)
dr p(r ) δ(r ϑ,
dr p(r ) = 1 ,
0
0
p(r ) is a filter selecting galaxies from some (comoving) distance r .
Two-point correlation of the projected random field is
ω(θ)
=
=
≈
where R ≡
~ 1 )δ̂(ϑ
~ 2 )i
hδˆ1 (ϑ
Z ∞ Z ∞
~ 1 , r1 )δ2 (r2 ϑ
~ 2 , r2 )i
dr1
dr2 p(r1 )p(r2 )hδ1 (r1 ϑ
0
Z ∞
Z0 ∞
“ r +r ”
1
2
,
dr1
dr2 p(r1 )p(r2 ) ξ R,
2
0
0
q
~1 , ϑ
~2 .
r12 + r22 − 2r1 r2 cos θ, and θ = ^ϑ
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Derivation Application Breakdown
The Limber equation (Limber 1953): Derivation
2
We use different variables: 1. r̄ = r1 +r
, 2. ∆r = r2 − r1 , and:
2
Z ∞
Z ∞
“ r +r ”
1
2
ω(θ) =
dr1
dr2 p(r1 )p(r2 ) ξ R,
2
0
0
„
« „
«
Z ∞ Z +2r̄
∆r
∆r
=
dr̄
d∆r p r̄ −
p r̄ +
ξ(R, r̄ )
2
2
0
−2r̄
„
« „
«
Z ∞ Z +∞
∆r
∆r
(APPROX.1) ≈
dr̄
d∆r p r̄ −
p r̄ +
ξ(R, r̄ )
2
2
0
−∞
Z ∞ Z +∞
dr̄
d∆r p(r̄ ) p(r̄ ) ξ(R, r̄ )
(APPROX.2) ≈
−∞
0
Z
=
∞
dr̄ [p(r̄ )]2
0
where R =
√
Z
+∞
d∆r ξ(R, r̄ ) ,
−∞
r̄ 2 θ2 + ∆r 2 , small θ . 30◦ assumed (not necessary).
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Derivation Application Breakdown
The Limber equation: Application
Limber’s equation (in flat-sky approximation)
Z ∞
Z +∞
“p
”
ω(θ) =
dr̄ [p(r̄ )]2
d∆r ξ
r̄ 2 θ2 + ∆r 2 , r̄
−∞
0
explains why a power-law ξ(r ) is a power-law ω(θ):
„ «−γ
«1−γ
„
r
θ
; ξ(r ) =
ω(θ) = Aω
1 RAD
r0
Z
√ γ Γ(γ/2 − 1/2) ∞
Aω =
π r0
dr̄ [p(r̄ )]2 r̄ 1−γ
Γ(γ/2)
0
Approximation is often used to relate angular clustering of Ly-break
galaxies, Lyα emitters, AGNs, etc. to real-space clustering in cases
where only available 3D-information is p(r )
BUT: what is solution for ω(θ) if p(r ) is peaked, i.e. p(r ) = δD (r − rm )?
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Derivation Application Breakdown
The Limber equation: Breakdown
Assume p(r ) is narrow top-hat with centre rm and width 2∆r . Limber
says:
Z rm +∆r
1 1−γ
1
dr̄ r̄ 1−γ ≈
rm ,
Aω ∝
4∆r 2 rm −∆r
2∆r
which diverges for ∆r → 0.
⇒ Limber’s equation is bound to fail for narrow p(r ), and systematically
over predicts the amplitude of ω(θ).
Correct solution for peaked p(r ), go back to original equation:
“ √ √
” „ r θ «−γ
m
ω(θ) = ξ rm 2 1 − cos θ, rm ≈
.
r0
It can be shown that this solution is asymptotically approached for
(almost) all p(r ) (see paper), for
„
«
√ σ
θ
σ
& 2 tan−1 √
≈ 2 ,
1 RAD
rm
2rm
where σ is the 1σ-variance and rm the mean of p(r ).
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Derivation Application Breakdown
The Limber equation: Breakdown
Assume p(r ) is narrow top-hat with centre rm and width 2∆r . Limber
says:
Z rm +∆r
1 1−γ
1
dr̄ r̄ 1−γ ≈
rm ,
Aω ∝
4∆r 2 rm −∆r
2∆r
which diverges for ∆r → 0.
⇒ Limber’s equation is bound to fail for narrow p(r ), and systematically
over predicts the amplitude of ω(θ).
Correct solution for peaked p(r ), go back to original equation:
“ √ √
” „ r θ «−γ
m
ω(θ) = ξ rm 2 1 − cos θ, rm ≈
.
r0
It can be shown that this solution is asymptotically approached for
(almost) all p(r ) (see paper), for
„
«
√ σ
θ
σ
& 2 tan−1 √
≈ 2 ,
1 RAD
rm
2rm
where σ is the 1σ-variance and rm the mean of p(r ).
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Derivation Application Breakdown
The Limber equation: Breakdown
The full solution is a broken power-law, ω(θ) ∝ θ1−γ for small θ (Limber)
and ω(θ) ∝ θ−γ for large θ.
Define break position as intersection point between both asymptotes, for
top-hat p(r ):
√
Γ(γ/2)
θbreak
2 3
σ
≈ √
.
1 RAD
π Γ(γ/2 − 1/2) rm
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Derivation Application Breakdown
The Limber equation: Breakdown
The full solution is a broken power-law, ω(θ) ∝ θ1−γ for small θ (Limber)
and ω(θ) ∝ θ−γ for large θ.
Define break position as intersection point between both asymptotes, for
top-hat p(r ):
√
Γ(γ/2)
θbreak
2 3
σ
≈ √
.
1 RAD
π Γ(γ/2 − 1/2) rm
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Handy coordinates Accuracy Impacts
1
Angular and real-space clustering
Defining structure statistically
Clustering of galaxies
Projecting random fields
2
Limber’s approximation and its breakdown
Derivation of Limber’s equation
Application of Limber’s equation
Where Limber’s approximation fails
3
Testing Limber’s equation against exact solution
Exact solution for angular clustering
Accuracy of Limber’s equation
Where else is Limber’s equation important?
4
Conclusions
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Handy coordinates Accuracy Impacts
Exact solution in convenient coordinates
The exact solution for ω(θ)
„
« „
«
Z ∞ Z +2r̄
∆r
∆r
ω(θ) =
dr̄
d∆r p r̄ −
p r̄ +
ξ(R, r̄ )
2
2
0
−2r̄
is somewhat impractical for numerical integrations...
...but after some transformations one can obtain
ω(θ) =
2
1 + cos θ
Z
0
Z2r̄
∞
dr̄
√
r̄
where
1
∆≡ √
2
dR R
p(r̄ − ∆)p(r̄ + ∆)
ξ(R, r̄ ) ,
∆
2(1−cos θ)
r
R 2 − 2r̄ 2 (1 − cos θ)
.
1 + cos θ
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Handy coordinates Accuracy Impacts
Exact solution in convenient coordinates
The exact solution for ω(θ)
„
« „
«
Z ∞ Z +2r̄
∆r
∆r
ω(θ) =
dr̄
d∆r p r̄ −
p r̄ +
ξ(R, r̄ )
2
2
0
−2r̄
is somewhat impractical for numerical integrations...
...but after some transformations one can obtain
ω(θ) =
2
1 + cos θ
Z
0
Z2r̄
∞
dr̄
√
r̄
where
1
∆≡ √
2
dR R
p(r̄ − ∆)p(r̄ + ∆)
ξ(R, r̄ ) ,
∆
2(1−cos θ)
r
R 2 − 2r̄ 2 (1 − cos θ)
.
1 + cos θ
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Handy coordinates Accuracy Impacts
Accuracy of Limber’s equation
left: θ (line index) at which Limber’s eqn. becomes inaccurate by 10%
(arcmin),
right: systematic error in inferred r0 (solid) and γ (dotted, line index)
Accuracy depends essentially only on 1. ratio σ/rm , and 2. on slope γ.
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Handy coordinates Accuracy Impacts
Where else is Limber’s equation important
Other incarnations of Limber’s equation involving power spectra
Z ∞
sin (kr )
P(k, r̄ ) = 2π 2
dr r 2 ξ(r , r̄ )
kr
0
Z ∞
P(`) = 2π
dθ θ ω(θ) J0 (`θ)
0
suffer from same problem, inaccuracy beyond some ` . `10% depending
on σ/rm and γ.
Also used for predictions of higher-order correlations (bispectrum,
trispectrum etc.) and for cross-correlations.
Weak gravitational lensing statistics:
Z rs
2
~ =
~ r ) ; pκ (r ) = 3H0 Ωm (rs − r )r .
κ(θ)
dr pκ (r )δm (r θ,
2c 2
a(r )rs
0
σ/rm ≈ 0.22 for zs = 1.0, γ ∼ 1.8 on large scales.
Integral constrain corrections usually assume power-law ω(θ), motivated
by Limber’s equation.
Patrick Simon
Transformation problem Limber eqn. Accuracy check Conclusions
Conclusions
For a power-law real-space correlation ξ(r ) ∝ (r /r0 )−γ the angular
correlation is a broken power-law, the break is at
θbreak
σ
for γ ∈ [1.2, 2.1] .
≈ 0.8
1 RAD
rm
Limber’s equation is an approximation for small θ, ω(θ) = Aω θ−γ+1 .
The “thin layer solution” approximates ω for large θ, ω(θ) = (rm θ/r0 )−γ .
The accuracy of Limber’s equation depends only on 1. σ/rm and 2. γ.
Blindly applying Limber’s equation to narrow p(r )’s biases inferences
about ξ(r ): r0 is too small, γ is too large.
For σ/rm ∼ 0.22 and γ ∼ 1.8, as roughly in gravitational lensing,
θ10% ∼ 1 deg (cosmic shear correlations).
Remedy: perform numerical integration of exact equation!
for more details: see Simon, P., astro-ph/0609165
Patrick Simon