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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.
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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
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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.
HKUST Camille Bonvin June 2011 p. /255
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.
HKUST Camille Bonvin June 2011 p. /25
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 ?
HKUST Camille Bonvin June 2011 p. /25
We look at correlations between the position and the shape of galaxies.
7
Weak lensingThe distortions created by weak lensing can be split in two parts: the convergence and the shear.
Convergence
We cannot look at a single galaxy, because we do not know its intrinsic shape.
How do we measure the convergence and the shear ?
HKUST Camille Bonvin June 2011 p. /25
We look at correlations between the position and the shape of galaxies.
8
Weak lensingThe distortions created by weak lensing can be split in two parts: the convergence and the shear.
Convergence 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 ?
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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
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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)
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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
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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
intrinsic corrections
CB 2008
HKUST Camille Bonvin June 2011 p. /2513
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
redshift corrections
CB 2008
HKUST Camille Bonvin June 2011 p. /2514
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)
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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
HKUST Camille Bonvin June 2011 p. /2516
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.
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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
HKUST Camille Bonvin June 2011 p. /2518
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
HKUST Camille Bonvin June 2011 p. /2519
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
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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
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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
Relativistic couplings
Newtonian non-linear evolution
Standard couplings
Vector and tensor modes
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
Relativistic couplings
Newtonian non-linear evolution
Standard couplings
Vector and tensor modes
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%
HKUST Camille Bonvin June 2011 p. /25
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
HKUST Camille Bonvin June 2011 p. /2525
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.
HKUST Camille Bonvin June 2011 p. /2526
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
�