Gravitational LensingUsing the lensing magnification

Gravitational telescope Galactic and extragalactic microlensing

Weak lensing

OverviewLensing cross section and optical depth

Magnification bias

Lens searches

Gravitational telescope

Microlensing magnification

Galactic microlensing

Extragalactic microlensing

Weak lensing and cosmology

The highly magnified z=2.7 Ly alpha galaxy behind the z=0.37 galaxy cluster MS1512+16 (Credit: NASA)

Lensing probability

Lensing cross section Lensing optical depth Magnification bias

Illustration of the magnification bias (Negrello et al. 2010, Science, 330, 800)

Lensing cross-section

The lensing cross-section is a solid angle which angular radius corresponds to some minimum magnification μp. This is simply

If we express this in units of the Einstein radius by introducing y = β/θE we can write

where y can be computed using the expression for the magnification of any lens profile. If we choose the point-mass model y is easily found

Note that for a magnification of 1.34 we find, not surprisingly, that y2 = 1, so the source has to be within or on the Einstein radius of the lens

�(µp) = ⇡✓2(µp)

�(µp) = ⇡✓2Ey2(µp)

y2 = 2

0

@ µpqµ2

p � 1� 1

1

A

Lensing cross section

Note that there is not one single definition of the lensing cross section.

One can see it as the solid angle where a source must be in order for its images to have some given property.

The «property» can be a total magnification, a magnification ratio, or whatever be your wish of the moment.

In practice the lensing cross section is used to compute the lensing probability of sources.

This lensing probability is simply the total area of the sky where a source must be to obtain some desired lensing property, i.e., the number of lenses or their surface density multiplied by the cross section of individual lenses.

Lensing optical depth

Of course one is often interested in estimating the probability of lensing to occur.

An object can be considered as lensed if it is located within the Einstein radius of a lens.

Then the probability that a source within a solid angle Ω is lensed is the fraction of this solid angle «covered by Einstein radii of foreground lenses».

Ω

Ds

Dd

Lensing optical depth

The lensing probability of a source being lensed is also called the lensing optical depth.

To compute the lensing optical depth lets consider the number density n(Dd) of the lenses at distance Dd (number per volume element).

Each lens has a cross section indicating that the source is magnified by at least 1.34 and lies at a maximum distance from each lens equal to the Einstein radius.

The lensing probability is then

where the volume element dV is related to the solid angle element dω by

� = ⇡✓2E

⌧ =1⌦

Zn(Dd)⇡✓2

EdV

dV = d! D2d dDd

Lensing optical depth

The optical depth becomes

If we now replace the Einstein radius by its expression and if we use the mass density ϱ of lenses, we obtain

⌧ =1⌦

Zd!

Zn(Dd) ⇡✓2

E D2d dDd

⌧ =Z

n(Dd) ⇡✓2E D2

d dDd

⌧ =Z Ds

0

4⇡G⇢

c2

Dds

DdDsdDd

Lensing optical depth

The integral is easily solved if space is assumed Euclidean, i.e., Dds = Ds - Dd and if the mass density towards the source is constant.

The above assumptions are realistic in the Milky Way, hence the equation for the optical depth written here is the one used in Galactic microlensing studies.

Note that the optical depth does NOT depend on the masses of the lenses but ONLY on their mass density.

This is quite intuitive: what counts is the total surface covered on the sky by Einstein radii, whatever be the exact distribution of these Einstein radii.

⌧ =2⇡G

3c2⇢ D2

s

Magnification bias

The magnification bias affects flux limited samples of sources in two ways:

New sources are seen because they are «brightened-up» above the flux limit.

The number density of sources decreases as the magnification is due to a «stretch» of the light beam in the plane of the sky (remember that surface brightness is conserved).

The net effect is called the magnification bias.

If a number of unlensed sources N0(>S) is seen above a limiting flux S in some solid angle Ω, the lensed number counts, for some magnification μ is

N(> S) =1µ

N0(>S

µ)

Magnification bias

The lensed source counts are «diluted» with respect to the original counts and the limiting flux is smaller by a factor μ, i.e., one sees fainter sources.

Of course real source counts are affected by real lenses, so that the magnification follows a probability distribution function p(μ) normalized to unity. Then we obtain

N(> S) =1µ

N0(>S

µ)

N(> S) =1

hµi

Zp(µ)N0(>

S

µ) dµ

Magnification bias

Real sources counts can in most cases be approximated by a power law

so that the lensed counts can be re-written

N0(> S) = K S��

N(> S) = N0(> S)1

hµi

Zp(µ) µ� dµ

N

N0(> S) =

1hµi

Zp(µ) µ� dµ

N(> S) =1

hµi

Zp(µ) K

⇣µ

S

⌘�dµ

Magnification bias

The ratio of the lensed (observed) counts over the unlensed counts depends on the slope of of the source luminosity function

If β=1 there is no observed effect of the magnification bias

If β > 1 (steep slope) the source counts are increased

If β < 1 (shallow slope) the source counts are depleted

N

N0(> S) =

1hµi

Zp(µ) µ� dµ