camille bonvin - blue-shift.ch · introduction to weak gravitational lensing. presentation of the...

Relativistic effects in weak gravitational lensing

Camille Bonvin KICC and DAMTP, Cambridge

Cosmology since Einstein, HKUST, June 2011

In collaboration with Francis Bernardeau, Filippo Vernizzi and Nicolas van de Rijt

HKUST Camille Bonvin June 2011 p. /252

Motivation

Gravitational lensing is a very powerful tool to map large-scale structure in our Universe.

At small scales lensing is theoretically well understood.

However future experiments will cover very large areas of the sky measure correlations at large scales.

We will probe regime where general relativity is relevant.

We need to compute gravitational lensing in a relativistic way.


Outline

Introduction to weak gravitational lensing.

Presentation of the standard small scales formula for the observable quantities: shear and convergence.

Computation of the relativistic corrections First order correctionSecond order corrections


Gravitational lensingLensing describes the deflection of light by matter between the source and the observer. It modifies the position and shape of the sources.

Light deflection does not depend on the nature of matter lensing is a powerful tool to map the dark matter distribution.

Measuring the distortions gives information on the distribution of matter.


Gravitational lensing

There are two different regimes in gravitational lensing, depending on the mass of the lens.

Strong lensing: multiple images of the same source

Weak lensing: small deflection of light smooth distortion of the shape of the galaxies. This can be used to measure the matter power spectrum.


We look at correlations between the position and the shape of galaxies.

6

Weak lensingThe distortions created by weak lensing can be split in two parts: the convergence and the shear.

We cannot look at a single galaxy, because we do not know its intrinsic shape.

How do we measure the convergence and the shear ?



7


Convergence





8


Convergence Shear




Observations

Convergence: The average intrinsic size of the galaxies is unknown. Schneider 1992, Bartelmann 1995, Broadhurst 1995 proposed to look at the number density of galaxies. Faint galaxies become observable through magnification bias. To separate this effect from intrinsic clustering one can either correlate objects at different redshifts or use the flux dependence of magnification bias. Zhang and Pen 2005. Recent measurements of the convergence agree with those of the shear. Scranton et al. 2005, Menard et al. 2009, Hildebrandt et al. 2009

Shear: Correlations between the ellipticity of galaxies are measured. First measurements in 2000: Bacon et al., Kaiser et al., Wittman et al., van Waerbeke et al.

Recently with CFHTLS and COSMOS

http://adsabs.harvard.edu/abs/1992A%2526A...254...14S

http://adsabs.harvard.edu/abs/1992A%2526A...254...14S

http://www.slac.stanford.edu/spires/find/hep/www?irn=3144631








http://www-library.desy.de/cgi-bin/spiface/find/hep/www?irn=8190267















The shear and the convergence at small scalesWe model the Universe as a homogeneous and isotropic expanding background + perturbations

χ = η − ηO

Measurements of the shear and the convergence provide measurement of the integrated potential.

At small scales

ds2 = −a2(1 + 2φ)dη2 + a2(1− 2ψ)δijdxidxj

κ(χS) =

� χS

0dχ

(χS − χ)χ

2χS(∂2

x1+ ∂2

x2)(φ+ ψ)

γ(χS) =

� χS

0dχ

(χS − χ)χ

2χS

�∂2x2

− ∂2x1

+ i∂x1∂x2

�(φ+ ψ)


Relativistic corrections

We can compute the relativistic corrections by solving Sachs equation

γ =

� χS

0dχ

χS − χ

2χχS

/∂2(φ+ ψ)

κ =

� χS

0dχ

χS − χ

2χχS

/∂ /∂(φ+ ψ) + ψS −� χS

0

dχ

χS(φ+ ψ)

+

�1

HSχS− 1

��φS + n · vS −

� χS

0dχ(φ̇+ ψ̇)

�

no relativistic corrections

CB 2008

Sachs 1961






γ =

� χS

0dχ

χS − χ

2χχS

/∂2(φ+ ψ)

κ =

� χS

0dχ

χS − χ

2χχS

/∂ /∂(φ+ ψ) + ψS −� χS

0

dχ

χS(φ+ ψ)

+

�1

HSχS− 1

��φS + n · vS −

� χS

0dχ(φ̇+ ψ̇)

�


intrinsic corrections

CB 2008






γ =

� χS

0dχ

χS − χ

2χχS

/∂2(φ+ ψ)

κ =

� χS

0dχ

χS − χ

2χχS

/∂ /∂(φ+ ψ) + ψS −� χS

0

dχ

χS(φ+ ψ)

+

�1

HSχS− 1

��φS + n · vS −

� χS

0dχ(φ̇+ ψ̇)

�


redshift corrections

CB 2008




Effect on the convergenceThe two contributions affect the magnification bias.

The angular power spectrum contains two contributions:

�δg(zS ,n)δg(zS ,n�)� =�

�

2� + 14π

C�(zS)P�(n · n�)

We expand in spherical harmonics, and we determine the angular power spectrum

δg

C� = Cst� + Cv

�

δg = 2(α− 1)(κst + κv)


Results

Standard contribution Velocity contribution

100 200 300 400 500

10!8

10!7

10!6

10!5

10!4

l

l!l"1"#2#

C lst

z!0.2z!0.5z!0.7z!1z!1.5

100 200 300 400 500

10!8

10!7

10!6

10!5

10!4

l

l!l"1"#2#

C lvel z!1.5

z!1z!0.7z!0.5z!0.2

With 10% accuracy we see the velocity term up to With 1% accuracy we see the velocity term up to

z = 0.6

z = 1

By combining shear and convergence measurements, one can extract peculiar velocities.

CB 2008




Second-order effects

The propagation of light and the theory of gravity are non-linear we expect corrections in the convergence and in the shear that are quadratic in the metric potentials.

Bernardeau, CB and Vernizzi 2010

We solved Sachs equation up to second-order and we extracted the shear.

These corrections are small, but they contain information on the geometrical and dynamical couplings.

It is possible to isolate them by looking at the three-point correlation functions.






Second-order shear

g = −� χs

0dχ

χS − χ

χSχ/∂

�− /∂Ψ+

1

χ

�/∂2Ψ

� χ

0dχ�χ− χ�

χ�/∂Ψ+ /∂ /∂Ψ

� χ

0dχ�χ− χ�

χ�/∂Ψ

��

− 2

� χs

0dχ

χS − χ

χχS

�1

2/∂2Ψ2 +

1

2ψ(χS) /∂

2Ψ−Ψ1

χ

� χ

0dχ� /∂2Ψ+ /∂2

�Ψ̇

� χ

0dχ�Ψ

�

+ /∂Ψ1

χ

� χ

0dχ�χ− χ�

χ�/∂Ψ

�+

2

χS

� χS

0dχ

�Ψ

� χ

0dχ� 1

χ�/∂2Ψ+

1

χ/∂2

�Ψ

� χ

0dχ�Ψ

��

− 2h(χS)−1

2χS

� χS

0dχ

�χS − χ

χ/∂2(ωr +

1

2hrr) +

χS

χ/∂(1ω + 1hr)

�

+

� χS

0dχ

χS − χ

χχS

/∂ /∂Ψ

� χS

0dχ�χS − χ�

χ�χS

/∂2Ψ

+ ψ(χS)

� χS

0dχ

χS − χ

χχS

/∂2Ψ− 2

� χS

0dχΨ

� χS

0dχ�χS − χ�

χ�χ2S

/∂2Ψ

+1 + zS

χ2SHS

�φ(χS) + vS · n− 2

� χS

0dχΨ̇

�� χS

0

/∂2Ψ

Bernardeau, CB and Vernizzi 2010






Relativistic corrections: non-linearities in the Riemann tensorcouplings between longitudinal perturbations and lensesredshift perturbations

Non-linear evolution of the gravitational potential:second-order scalar, vector and tensor modes

Standard second-order newtonian couplings: lens-lens couplings corrections to Born approximationreduced shear

Second-order shear

Bernardeau et al. 1997Cooray and Hu 2002Dodelson et al. 2005Shapiro and Cooray 2006


Three-point correlations

The two-points correlation function at second-order are extremely small we look at the three-points correlation functions.

Since the primordial potential is gaussian

At first order

At second order

We computed the bispectrum

�φ3� = 0

�φ4�B�1�2�3(zS)

�γ(1)(zS ,n1)γ(1)(zS ,n2)γ

(1)(zS ,n3)� = 0

�γ(2)(zS ,n1)γ(1)(zS ,n2)γ

(1)(zS ,n3)� �= 0


Three-point correlations

The two-points correlation function at second-order are extremely small we look at the three-points correlation functions.

Since the primordial potential is gaussian

At first order

At second order

We computed the bispectrum

�φ3� = 0

�φ4�B�1�2�3(zS)

�1

�2

�3

�γ(1)(zS ,n1)γ(1)(zS ,n2)γ

(1)(zS ,n3)� = 0

�γ(2)(zS ,n1)γ(1)(zS ,n2)γ

(1)(zS ,n3)� �= 0


Three-points correlations

�

Relativistic couplings

Bernardeau, CB,Van de Rijt and Vernizzi in preparation

Newtonian non-linear evolution

Standard couplings

Vector and tensor modes

�1 = 10

�

�

B10��

2C10C� + C2� zS = 1

200 400 600 800 100010!4

0.001

0.01

0.1

1

10







HKUST Camille Bonvin June 2011 p. /2522�

Varying the shape



Standard couplings


B�1�2�3

C�1C�2 + C�2C�3 + C�1C�3

200 400 600 800 100010!4

0.001

0.01

0.1

1

10

HKUST Camille Bonvin June 2011 p. /2523�

Varying the shape



Standard couplings


B�1�2�3

C�1C�2 + C�2C�3 + C�1C�3

200 400 600 800 100010!4

0.001

0.01

0.1

1

10

relativistic v. standard 8% 68%


The amplitude of the bispectrum depends on the amount of non-gaussianities that can be parameterised by

24

Primordial non-gaussianities

From WMAP7 : Local

A non-gaussian primordial potential would also generate a non-zero shear bispectrum.

Equilateral

In order to use large-scale structure to probe non-gaussianities, we need to know the contamination from non-linearities.

We computed the effective for local type: fNL = 8

fNL

cos = 0.66fNL

−10 < fNL < 74

−214 < fNL < 266


Conclusion

At linear order, relativistic corrections have an impact on the convergence only.

The dominant contribution comes from peculiar velocities of galaxies. It is large enough to be observed by future weak lensing experiments.

At second order, relativistic corrections affect also the shear.

We computed the three-point correlations functions associated with the non-linearities and we found that the importance of the relativistic corrections depends strongly on the configuration.


Consistency relationThe standard contribution of the convergence and the shear are related through the consistency relation:

What we measure is the sum of the standard term and the velocity term.

The observed angular power spectrum obey therefore the new consistency relation:

This can be used to measure galaxies peculiar velocities

Cκst� =

�(�+ 1)

(�+ 2)(�− 1)Cγ

�

Cκv� = Cκobs

� − �(�+ 1)

(�+ 2)(�− 1)Cγobs

�

camille bonvin - blue-shift.ch · introduction to weak gravitational lensing. presentation of the...

Documents