Seminars of the Gravitational Lensing Working Group at IPM
Limber's Equation
Farbod Kamiab
Sharif University of Technology
June 2008
Overview
• The so-called Limber equation is widely used in the
literature to relate the projected angular clustering of
galaxies to the spatial clustering of galaxies in an
approximate way.
• In the study of cosmic shear in weak gravitational lensing,
the Limber equation relates the power spectrum of the
convergence to the power spectrum of the large scale
density fluctuations of the universe.
• The accuracy of the Limber's equation has been studied in
some cases, but still needs to be studied for weak lensing.
Outline
• Introduction
• Correlation functions and power spectra
• Relation between spatial and angular correlation function:
1- Exact relation with only some restrictions
2- Limber’s approximation
• A review of basic weak lensing
• Cosmic shear and the Limber's equation
• Accuracy of the Limber’s equation
Introduction
• One can see since the 1930s a tendency to think that the
general distribution of matter in the universe is so
complicated, and the data we can hope to have so
schematic that a full reduction to genera and species of
clustering might not be profitable or even possible.
• The alternative is to resort to statistical measures, such as
the correlation functions and power spectra.
• Separated parts of the universe manifest the same physical
process, but can be taken as independent realizations.
Independent samples of the visible universe, therefore, can
be taken to represent a statistical ensemble.
Correlation functions and power spectra
δP = nδV
N = MnδV
δP = n δV1δV2 × [1 + ξ (r12 )]
2
δP(2 1) = nδV2 × [1 + ξ (r12 )]
N
p
r
4
3
= π × r n + n ∫ ξ (r )dV
0
3
N = nV
Correlation functions and power spectra
ρ (r ) = n
ρ ( x) − ρ
δ ( x) =
ρ
ξ (r ) = δ ( x + r ) × δ ( x)
The power spectrum is the fourier transform of the
correlation function.
Relation between spatial and angular correlation
function: Exact relation with only some restrictions
• Two number density fields of galaxies: n1(r) and n2(r)
• Spacial correlation function:
,
• If we have n1 = n2, then
is a autocorrelation
function of a galaxy number density field.
Relation between spatial and angular correlation
function: Exact relation with only some restrictions
• Angular correlation function:
• Filter or weight function:
• Look-back time:
• Scale factor:
Relation between spatial and angular correlation
function: Exact relation with only some restrictions
• The definition of the density contrast of the projected
(number) density:
Relation between spatial and angular correlation
function: Exact relation with only some restrictions
• According to the definition of
we have for sight lines
Spanning an angle
where:
Relation between spatial and angular correlation
function: Exact relation with only some restrictions
• Two assumptions had to be made to arrive at (11): a) the
random fields
, and hence also their projections,
,
are statistically isotropic and homogeneous, and b) the
time-evolution of is small within the region where the
product p1(r1)p2(r2) is non-vanishing. Due to assumption
a) depends only on θ and is independent of the directions
.
. Owing to assumption b) we can approximate the
spatial correlation of fluctuations at different cosmic times
(radial distances) t(r1) and t(r2) by a representative _ at time
(r1 + r2)/2.
• Note that Eq. (11) is, under the previously stated
assumptions, valid even for large .
Limber’s approximation
• In order to find an approximation of Eq. (11), one
introduces for convenience new coordinates,
which are the mean radial comoving distance and difference
of radial distances, respectively, of a pair of galaxies.
where:
Limber’s approximation
• Especially the first approximation is characteristic for
Limber’s equation. It is justified if the weight functions p1,2
do not “vary appreciably” (Bartelmann & Schneider 2001,
p.43) over the coherence length of structures described by
,
typically a few hundred Mpc in the context of
cosmological large-scale structure –, which means we
consider cases in which the coherence length is small
compared to the width of the weight functions p1,2.
Limber’s approximation
• In total, these assumptions lead to the (relativistic) Limber
equation (Limber 1953; Peebles 1980):
• For historical reasons, as a further approximation it is
assumed in the above equations that we are dealing with
small angles of separation, , by which we can introduce:
Limber’s approximation
• These two approximations are accurate to about 10% for
angles smaller than
which covers the typical range
of investigated separations. Usually, when employing
Limber’s equation this approximation is automatically
used.
• Example:
⇒
A review of basic weak lensing
Cosmic shear and the Limber's equation
• Equation of geodesic deviation:
Cosmic shear and the Limber's equation
Cosmic shear and the Limber's equation
• Born approximation:
Cosmic shear and the Limber's equation
____________________________________________________
Cosmic shear and the Limber's equation
Cosmic shear and the Limber's equation
Accuracy of the Limber’s equation
Accuracy of the Limber’s equation
In the case of weak lensing, the filter function relevant for
lensing is the lensing efficiency that, quite naturally, has a wide
distribution; one can roughly estimate
if the
source galaxies have a typical value of zs = 1. The centre of
this filter is at about
. These values will depend on the
distribution of source galaxies in redshift and the fiducial
cosmological model. Taking these values as a rough estimate
of the relative lensing filter width and centre, and assuming
that the dark matter clustering is somewhere between the
clustering of red and blue galaxies, one can infer from Figure 3
that a two-point auto-correlation function of the convergence,
based on Limber’s equation, is accurate to about 10% for
separations less than several degrees; beyond that, an
alternative description should be used.
References:
The large scale structure of
the universe; Jim Peebles
How accurate is the Limber
equation? Patrick Simon
(astro-ph/0609165v2)
Weak gravitational lensing;
Peter Schneider
D. N. Limber