The Weak Lensing Bispectrum

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by  Scott Dodelson

The Weak Lensing Bispectrum
Scott Dodelson1,2,3 and Pengjie Zhang1
1NASA/Fermilab Astrophysics Center Fermi National Accelerator Laboratory, Batavia, IL 60510-0500
2Department of Astronomy & Astrophysics, The University of Chicago, Chicago, IL 60637-1433 and
3Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208
(Dated: February 2, 2008)
Weak gravitational lensing of background galaxies offers an excellent opportunity to study the
intervening distribution of matter. While much attention to date has focused on the two-point
function of the cosmic shear, the three-point function, the bispectrum, also contains very useful
cosmologicalinformation. Here,wecomputethreecorrectionstothebispectrumwhicharenominally
of the same order as the leading term. We show that the corrections are small, so they can be
ignored when analyzing present surveys. However, they will eventually have to be included for
accurate parameter estimates from futuresurveys.
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0
0 I. INTRODUCTION
2

n Weaklensingofferscosmologiststheopportunitytoprobethedistributionofmassintheuniverse[1]. Thisprospect
a is so alluring because theories make first-principles predictions about this distribution, so we can hope to extract
J
important constraints on fundamental cosmological parameters from weak lensing surveys [2, 3, 4, 5, 6, 7, 8, 9, 10].

4 In many senses, this promise is similar to that felt by those who studied the cosmic microwave background (CMB)

a decade ago: theoretical predictions are straightforward; experiments have detected the effect (anisotropies in that
1
case and cosmic shear in this); and there are grand plans for the future (which have been realized in the case of the
v
3 CMB).
6 Armed with this optimism, cosmologistsare quick to throw in warning labels: the signal is extremely small, so the
0 only way to measure cosmic shear is to average over many background galaxies. Each individual galaxy is observed
1 with its own set of systematics (seeing, elliptical point spread functions, calibration, unknown or at least uncertain
0
redshift, etc.) and these vary from one galaxy to another. Measurements of cosmic shear are unlikely to produce
5
smoothmaps since there inevitably will be brightstars whichmust be maskedout. Accounting for these masks leads
0
to complicated window functions. It is not clear then that weak lensing measurements will eventually pay off as did
/
h those of the CMB.
p
There is one area though in which weak lensing measurements have an advantage over the CMB: the higher point
-
o functions of the cosmic shear field are potentially simpler to interpret and more relevant than those in the CMB.
r Naively, this is what one would expect, for the cosmic shear field is sensitive to the matter density which has gone
t
s nonlinearandthereforewillhavelargecorrectionstothe Gaussianlimit. Temperaturefluctuations onthe otherhand
a arestillstuckatthe10−5 level,soareexpectedtobeveryclosetoGaussian(recallthattheinitialdistributionofboth
:
v temperatureandmatterinhomogeneitieswaslikelyGaussianandforaGaussiandistributionthehigherpointfunctions
i aretriviallyrelatedtothe two-pointfunction). We mightexpectthenthe 3-pointfunctionofCMBanisotropiesto be
X
very small, while that of the cosmic shear to be quite large, at least on small scales. Much work has been done over
r
a thepastfewyearsattemptingtodebunkthesenaiveideas. WehavefoundthatthehigherpointfunctionsoftheCMB
are quite interesting: lensing [11, 12, 13], hot gas [14], peculiar velocities [15], and reionization [16, 17] leave their
imprint in these higher point functions. Nonetheless, the fact remains that to date there has been no detection of a
non-zero3-pointfunction,forexample,intheCMB,whileseveralgroups[18,19]haveclaimedsuchadetectioninthe
cosmicshearfield. Further,a numberofauthors[2, 3, 4, 20, 21]haveshowedthat the bispectrum ofthe cosmicshear
field will be able to constrain important cosmologicalparameters, including properties of the dark energy. Certainly,
then, we need to obtain accurate predictions of the bispectrum of the shear.
With this in mind, it is important to emphasize just how important “higher order” corrections to the bispectrum
might be. To understand this, recall that, to lowest order, the shear is a line-of-sight integral over the matter
overdensity:
γ(θ)= dχf(χ)δ(x(χ,θ)) (1)
Z
where θ is the angular position on the sky, f is a weighting function, and χ is the comoving distance along the line
of sight. Roughly then the bispectrum, which is proportionalto hγ3i, is proportionalto hδ3i. It vanishes therefore in
the large-scale limit where the density field is Gaussian and nonlinearities are irrelevant. The main contribution to
the bispectrumthencomes fromthe factthat, due togravity,δ evolvesnonlinearly. Inperturbationtheory,we would
2
write δ = δ(1)+δ(2)+... where δ(1) is the linear overdensity and δ(2) is proportional to δ2. This main contribution
L
to the bispectrum then comes from terms proportional to hδ(1)δ(1)δ(2)i and so is proportional to δ4.
L
It is clear then that any correction which alters the linear relation between shear and overdensity in Eq. (1) is of
the same order in δ as the “main contribution.” Here we study three such corrections first identified by Schneider
L
et al. [22] and compute their effect on the bispectrum:
• ReducedShearWeestimateshearbymeasuringellipticitiesofbackgroundgalaxies,invokingtherelation[1,23]
ǫ = 2γ , where the subscript i refers to the two components of ellipticity/shear. This relation though is only
i i
approximate; the full relation is
2γ
i
ǫ = (2)
i
1−κ
where κ is the convergence. Expanding the denominator, we see that the ellipticities used to estimate cosmic
shear have terms quadratic in the perturbations (2γ κ). This quadratic term contributes to the bispectrum at
i
the same order as the main term.
• Lens-Lens Coupling Eq. (1) does not account for the fact that lenses are correlated along the line of sight.
This lens-lens coupling induces another quadratic term in the relation between shear and overdensity.
• Born Approximation When computing the shear one integrates along the photon path back towards the
source. There is a complication inherent in this integration encoded in the argument of the overdensity in
Eq. (1): what is the position of the photon at radial distance χ if its observed angular position today is θ?
the naive answer x(0) = χ(θ,1) is the position corresponding to the path taken by an undeflected photon.
Expanding about this zero order position leads to a correction proportional to ∇δ·[x−x(0)]. Since x differs
fromthe undeflected position only if δ is nonzero,this correctionis secondorder in δ. It too contributes a term
of order δ4 to the bispectrum.
The next section reviews some basic lensing results including the standard computations of the power spectrum
and the bispectrum. §III computes the correction to the bispectrum from the three effects enumerated above. The
bispectrum cannot be simply plotted on a 2D graph since it depends on the three variables required to specify a
triangle. Therefore, §IV examines various ways of condensing the information contained in these corrections. The
goal is to see whether the corrections are important.
II. REVIEW OF BASIC RESULTS
Thedeformationtensorisdefinedasthedeviationfromunityofthe Jacobianrelatingtheundeflectedposition(θS)
to the actual position (θ):
∂θ
S,i
ψ ≡δ − . (3)
ij ij
∂θ
j
The elements of this 2×2 matrix are the two components of shear and the convergence:
κ+γ γ
ψ ≡ 1 2 . (4)
ij (cid:18) γ2 κ−γ1(cid:19)
These are simply definitions. The physics comes from solving the geodesic equation and expressing the distortion
tensor in terms of the gravitational potential Φ:
χs ∂2Φ(x(θ,χ);χ)
ψ (θ,χ )= dχW(χ,χ ) [δ −ψ (θ,χ)]. (5)
ij s s kj kj
Z ∂x ∂x
0 i k
Here,weareassumingaflatuniverse;sourcesareassumedtolieatχ ;thefirstargumentoftheΦisthe3Dcomoving
s
position (in the small angle limit)
χ χ
′ ′ ′
x(θ,χ)=[χθ ,χθ ,χ]− dχW(χ,χ) ∇Φ(x(θ,χ);χ) (6)
x y Z χ′
0
while the second (χ) refers to the cosmic time at which the photon path passed by this position. The weighting
function is
W(χ,χ )=2χ(1−χ/χ )Θ(χ −χ). (7)
s s s
3
We observe ellipticities of backgroundgalaxies ǫ , which are related to the elements of the distortion tensor via
i
2γ
i
ǫ = . (8)
i
1−κ
One way to estimate the convergence,which is the projected density, is to work in Fourier space; then,
1
κˆ˜(l)= T (l)ǫ˜(l). (9)
l2 i i
Here the variable conjugate to θ is l. As usual, large l corresponds to small scales. The trigonometric functions in
Eq. (9) are defined with respect to an arbitraryx- axis as
l2−l2 l2
T (l) ≡ x y = cos(2φ )
1 l
2 2
l2
T (l) ≡ l l = sin(2φ ). (10)
2 x y l
2
Here φ is the angle between l and the x- axis. In the limit in which ǫ →2γ , the estimator in Eq. (9) reduces to
l i i
reduced shear
κˆ˜(l) −→ cos(2φ )γ˜ (l)+sin(2φ )γ˜ (l)
l 1 l 2
z}=|{ cos(2φ )(ψ˜ (l)−ψ˜ (l))/2+sin(2φ )ψ˜ (l)
l 11 22 l 12
lens−lens −l2
−→ ψ˜(l). (11)
2
z}|{
The approximation on the first line neglects the fact that ellipticities are sensitive to the reduced shear; the approx-
imation on the third line neglects the second order term in the brackets in Eq. (5), the term which accounts for the
fact that the lens distribution is correlated. The projected potential in Eq. (11) is defined as
χs W(χ,χ )
ψ(θ,χ )= dχ s Φ(x(θ,χ);χ). (12)
s Z χ2
0
In the Born approximation the gravitational potential in the integral along the line of sight is evaluated at the
unperturbed path. In that case, its Fourier transform reduces to
Born χs W(χ,χ )
ψ˜(l,χ )−→ψ˜(0)(l,χ )= dχ s φ˜(l;χ) (13)
s s Z χ2
0
z}|{
with
1 dk
φ˜(l,χ)≡ 3Φ˜(l/χ,k ;χ)eik3χ. (14)
χ2 Z 2π 3
Note that φ˜ is dimensionless unlike Φ˜ which has dimensions of (length)3.
The statistics of ψ˜(0) follow directly from those of φ˜, which are simple if we include only modes with k small, i.e.
3
the Limber approximation [23, 24],
hφ˜(l,χ)φ˜(l′,χ′)i=(2π)2δ2(l+l′)δ(χ−χ′)P (l/χ;χ)/χ2. (15)
Φ
Here P is the power spectrum of the gravitational potential. Similarly, the three point-function is related to the
Φ
spatial bispectrum [16]:
hφ˜(l ,χ )φ˜(l ,χ )φ˜(l ,χ )i=(2π)2δ2(l +l +l )δ(χ−χ′)δ(χ−χ′′)B (l /χ,l /χ,l /χ;χ)/χ4. (16)
1 1 2 2 3 3 1 2 3 Φ 1 2 3
Then, we have
hψ˜(0)(l)ψ˜(0)(l′)i=(2π)2δ2(l+l′)P (l), (17)
2
with
∞ W2(χ,χ )
s
P (l)= dχ P (l/χ;χ). (18)
2 Z χ6 Φ
0
4
Similarly, the three-point function is
hψ˜(0)(l )ψ˜(0)(l )ψ˜(0)(l )i=(2π)2δ2(l +l +l )P (l ,l ,l ), (19)
1 2 3 1 2 3 3 1 2 3
where now the projected power is a line-of-sight integral over the bispectrum:
∞ W3(χ)
P (l ,l ,l )= dχ B (l /χ,l /χ,l /χ;χ). (20)
3 1 2 3 Z χ10 Φ 1 2 3
0
The power spectrum, or the C’s, are defined as the coefficient of (2π)2δ(l+l′) when computing the variance of κˆ˜.
l
Since κˆ˜ =−l2ψ(0)/2 in the standard computation, we have
Cκ =l4P (l)/4. (21)
l 2
Similarly, the bispectrum of the κ estimator is
Bκ(l ,l ,l )=−l6P (l ,l ,l )/8, (22)
1 2 3 3 1 2 3
in agreement1 with previous results [4]. The bispectrum with all l’s equal, the equilateral configuration, is shown in
Fig. 1.
FIG.1: Equilateralbispectrum. Solidcurveisthestandardresult;short-dashed[red]curveisorderofmagnitudeofcorrections
considered here; long-dashed [blue] curve is cosmic-variance error. The signal to noise from a single configuration is therefore
extremely small.
1 Onesubtletywhencomparingwithotherresultsisthesign. ThesignhereisnegativebecauseΦ˜ ∝−δ˜.
5
A few qualitative comments are in order here. The standard measure of the amplitude of fluctuations is l2Cκ =
l
l6P /4. Let’sdoanorderofmagnitudeestimateforthisquantityl6P intermsoftheamplitudeofdensityfluctuations,
2 2
∆2 ≡ k3P /2π2. Since l ∼ (k/H ) and since Φ ∼ (H /k)2δ, we have P ∼ P /l4. Now, Eq. (18) suggests that
δ 0 0 Φ δ
P ∼ P /χ3; since χ ∼ l/k, P ∼ (k/l)3P ∼ (k/l)3P /l4 ∼ ∆2/l7. We expect then that l6P should be of order
2 Φ 2 Φ δ 2
∆2/l. Whatthis meansphysicallyisthatprojectioneffectssuppressthe 2Dpowerspectrumbyafactorof1/l. There
is even a nice explanation of this in terms of Fourier modes [23, 25]: only modes with small k contribute; these are
3
a fraction of 1/l of the total number of modes. The bottom line then is that the power spectrum of the convergence
is smaller than the power spectrum of the 3D density field.
Similar order of magnitude estimates relate the angular bispectrum, −l6P /8, to the 3D bispectrum of the density
3
field:
l6P ∼l6 (k/l)6B ∼H6B . (23)
3 Φ 0 δ
(cid:2) (cid:3)
The corrections we consider below are all of order l8P2, which by the arguments of the preceding paragraph are
2
of order H6P2. The 3D bispectrum B is nominally of the same order as the square of the power spectrum, P2.
0 δ δ δ
However,numericallyit is abit larger[26],as indicatedin Fig.1, so the correctionswecompute arenotas important
as we would have hoped.
III. HIGHER ORDER TERMS
WenowcomputethecorrectionstothebispectrumfromgoingbeyondtheapproximationsinEq.(11)andEq.(13).
A. Reduced Shear
The first-order correction to the ǫ−γ relation is
ǫrs =2γ κ. (24)
i i
When weswitchto Fourierspace,the relationbetweenellipticity andshearis aconvolutionintegral(products inreal
space correspond to convolutions in Fourier space):
d2l′
˜ǫrs(l)= l′2T (l−l′)ψ˜(0)(l′)ψ˜(0)(l−l′) (25)
i Z (2π)2 i
When we form the bispectrum of the κ estimator (Eq. (9)), the first order correctionemerges by replacing one of the
three ellipticities with the higher order Eq. (25). So this correction to the bispectrum estimator becomes:
T (l )l2l2 d2l′
hκˆ˜(l )κˆ˜(l )κˆ˜(l )irs = i 1 2 3 l′2T (l −l′)hψ˜(0)(l′)ψ˜(0)(l −l′)ψ˜(0)(l )ψ˜(0)(l )i
1 2 3 4l2 Z (2π)2 i 1 1 2 3
1
+(l ↔l )+(l ↔l ). (26)
1 2 1 3
The four-point function for the potential gets contributions from the connected part – the trispectrum – and the
disconnected part: the product of power spectra. Here we consider only the latter set of terms as these are expected
to dominate. That is, let
hψ˜(0)(l′)ψ˜(0)(l −l′)ψ˜(0)(l )ψ˜(0)(l )i → (2π)4P (l )P (l )
1 2 3 2 2 2 3
× δ2(l′+l )δ2(l −l′+l )+δ2(l′+l )δ2(l −l′+l ) .
2 1 3 3 1 2
h i
(27)
The integral over l′ then leaves the coefficient of (2π)2δ2(l +l +l ) as
1 2 3
T (l )l2P (l )l2P (l )
i 1 2 2 2 3 2 3 l2T (l )+l2T (l ) (28)
4l2 2 i 3 3 i 2
1 (cid:2) (cid:3)
plus permutations. The contraction over the geometric factors involves
l2l2
T (l )T (l )= 1 2 cos(2φ ) (29)
i 1 i 2 12
4
6
where φ is the angle between l and l .
12 1 2
Defining B ≡B(l ,l ,l ) as the coefficient multiplying (2π)2δ2(l +l +l ), we therefore have
123 1 2 3 1 2 3
l4l4P (l )P (l )
Brs = 2 3 2 2 2 3 [cos(2φ )+cos(2φ )] (30)
123 16 13 12
plus permutations.
B. Lens-Lens Coupling
Lens-lens coupling in encapsulated by the second term in square brackets in Eq. (4). This contribution to the
distortion tensor is then
ψll (θ,χ )=− dχW(χ,χ )Φ ψ (θ,χ). (31)
ab s Z s ,ac cb
At second order in Φ, this reduces to
ψll (θ,χ )=− dχW(χ,χ )Φ (θ,χ) dχ′W(χ′,χ)Φ (θ,χ′). (32)
ab s Z s ,ac Z ,cb
Using Eq. (9), we can compute how this term contributes to the estimator of convergence. The second-order
contribution is
2T (l)γll(l)
κˆll(l) = i i
l2
−T (l) d2l′ dχ dχ′
= i E (l′,l−l′) W(χ,χ ) W(χ′,χ)φ˜(l′,χ)φ˜(l−l′,χ′). (33)
l2 Z (2π)2 i Z χ2 s Z χ′2
Here the geometrical factors are defined as
E (l ,l ) ≡ l ·l [l l −l l ]
1 1 2 1 2 1,x 2,x 1,y 2,y
E (l ,l ) ≡ l ·l [l l +l l ]. (34)
2 1 2 1 2 1,x 2,y 1,y 2,x
The estimator for the bispectrum of the convergence then gets a Gaussian contribution from the lens-lens term.
One such term is
−l2l2T (l ) dχ dχ′ dχ dχ
hκˆll(l )κˆ(0)(l )κˆ(0)(l )i = 2 3 i 1 W(χ,χ ) W(χ′,χ) 2W(χ ,χ ) 3W(χ ,χ )
1 2 3 4l2 Z χ2 s Z χ′2 Z χ2 2 s Z χ2 3 s
1 2 3
d2l′
× E (l′,l −l′)hφ˜(l′,χ)φ˜(l −l′,χ′)φ˜(l ,χ )φ˜(l ,χ )i. (35)
Z (2π)2 i 1 1 2 2 3 3
Two other terms exist with l ↔ l and l ↔ l . In this case, the trispectrum does not contribute in the Lim-
1 2 1 3
ber approximation. Physically, the Limber approximation sets all lenses close to each other; mathematically, this
corresponds to enforcing the constraint that the line of sight distances are all equal [16]. Here this constraint sets
′ ′
χ = χ = χ = χ , so that W(χ,χ) vanishes. The only relevant terms therefore are the two products of two-point
2 3
functions. Momentum conservation from the first such pair enforces l′ = −l and l −l′ = −l . Each two point
2 1 3
function is evaluated using the Limber approximationas in Eq. (15).
The bispectrum from lens-lens coupling is then
−l2l2T (l )E (l ,l ) dχ dχ′
Bll = 2 3 i 1 i 2 3 W2(χ,χ ) W(χ′,χ)W(χ′,χ )
123 4l2 Z χ6 s Z χ′6 s
1
′ ′ ′ ′
×[P (l /χ;χ)P (l /χ;χ)+P (l /χ;χ)P (l /χ;χ)]+(l ↔l )+(l ↔l ). (36)
Φ 2 Φ 3 Φ 3 Φ 2 1 2 1 3
Here we have used the fact that E (l ,−l )=E (l ,l )=E(l ,l ).
i 1 2 i 1 2 2 1
The geometrical factor in front reduces to
l2l2l2cos(2φ )
T (l )E (l ,l )= 1 2 3 23 cos(φ +φ ). (37)
i 1 i 2 3 12 13
2
7
Therefore,
−l4l4cos(2φ )cos(φ +φ ) dχ dχ′
Bll = 2 3 23 12 13 W2(χ,χ ) W(χ′,χ)W(χ′,χ )
123 8 Z χ6 s Z χ′6 s
′ ′ ′ ′
×[P (l /χ;χ)P (l /χ;χ)+P (l /χ;χ)P (l /χ;χ)]+(l ↔l )+(l ↔l ). (38)
Φ 2 Φ 3 Φ 3 Φ 2 1 2 1 3
C. Born Approximation
The distortion tensor in Eq. (5) evaluates the potential everywhere along the unperturbed path of the light. To go
beyond the Born approximation, we need to evaluate the potential at x=x +δx where
0
χ
′ ′ ′
δx (θ,χ)=− dχW(χ,χ) Φ (x ;χ). (39)
a Z χ′ ,a 0
This leads to a new contribution to the distortion tensor, which in Fourier space, reads
dχ dχ′ d2l′
ψ˜Born(l,χ )=− W(χ,χ ) W(χ′,χ) l′l′l′(l−l′) φ˜(l′;χ)φ˜(l−l′;χ′). (40)
ab s Z χ2 s Z χ′2 Z (2π)2 a b c c
This extra term in the distortion tensor contributes to the estimator for the convergence
κˆ˜Born = −Ti(l) dχW(χ,χ ) dχ′W(χ′,χ) d2l′ F (l′,l−l′)φ˜(l′;χ)φ˜(l−l′;χ′). (41)
l2 Z χ2 s Z χ′2 Z (2π)2 i
Here
F (l ,l )≡2l ·l T (l ). (42)
i 1 2 1 2 i 1
Thisexpressionisidenticalinformtothatfromlens-lenscoupling,withthesubstitutionE →−F . Wecantherefore
i i
copy the result from Eq. (36) to get
−l2l2T (l ) dχ dχ′
BBorn = 2 3 i 1 W2(χ,χ ) W(χ′,χ)W(χ′,χ )
123 4l2 Z χ6 s Z χ′6 s
1
′ ′ ′ ′
×[P (l /χ;χ)P (l /χ;χ)F (l ,l )+P (l /χ;χ)P (l /χ;χ)F (l ,l )]+(l ↔l )+(l ↔l ). (43)
Φ 2 Φ 3 i 2 3 Φ 3 Φ 2 i 3 2 1 2 1 3
But, T (l )F (l ,l )=l2l3l cos(2φ )cos(φ ), so
i 1 i 2 3 1 2 3 12 23
−l3l3cos(φ ) dχ dχ′
BBorn = 2 3 23 W2(χ,χ ) W(χ′,χ)W(χ′,χ )
123 8 Z χ6 s Z χ′6 s
× l2P (l /χ;χ)P (l /χ′;χ′)cos(2φ )+l2P (l /χ;χ)P (l /χ′;χ′)cos(2φ ) +(l ↔l )+(l ↔l )(.44)
2 Φ 2 Φ 3 12 3 Φ 3 Φ 2 13 1 2 1 3
(cid:2) (cid:3)
D. Summary
Here we collect the results from the previous three subsection. The reduced shear correction can be expressed in
terms of the 2-point function P :
2
l4l4P (l )P (l )
Brs = 2 3 2 2 2 3 [cos(2φ )+cos(2φ )]
123 16 13 12
l4l4P (l )P (l )
+ 1 3 2 1 2 3 [cos(2φ )+cos(2φ )]
23 12
16
l4l4P (l )P (l )
+ 1 2 2 2 2 1 [cos(2φ )+cos(2φ )] (45)
13 23
16
The other two corrections are best expressed in terms of
′
dχ dχ
I(l ,l )≡l3l3 W2(χ,χ ) W(χ′,χ)W(χ′,χ )P (l /χ;χ)P (l /χ′;χ′). (46)
1 2 1 2Z χ6 s Z χ′6 s Φ 1 Φ 2
8
Then, the lens-lens term is
−l l cos(2φ )cos(φ +φ )
Bll = 2 3 23 12 13 [I(l ,l )+I(l ,l )]
123 8 2 3 3 2
l l cos(2φ )cos(φ +φ )
1 3 13 21 23
− [I(l ,l )+I(l ,l )]
1 3 3 1
8
l l cos(2φ )cos(φ +φ )
2 1 21 32 31
− [I(l ,l )+I(l ,l )]. (47)
2 1 1 2
8
And the Born term is
−cos(φ )
BBorn = 23 l2I(l ,l )cos(2φ )+l2I(l ,l )cos(2φ )
123 8 2 2 3 12 3 3 2 13
(cid:2) (cid:3)
cos(φ )
− 13 l2I(l ,l )cos(2φ )+l2I(l ,l )cos(2φ )
8 1 1 3 12 3 3 1 23
(cid:2) (cid:3)
cos(φ )
− 21 l2I(l ,l )cos(2φ )+l2I(l ,l )cos(2φ ) . (48)
8 2 2 1 32 1 1 2 13
(cid:2) (cid:3)
One word of caution: all of the above assume that our estimator for κ is as given in Eq. (9). One could also
imagine defining the bispectrum as the three-point function of one-half of the trace of the distortion tensor. These
two expressions agree in the zeroth order case, but they disagree when these higher order corrections are included,
becausethedistortiontensorisnolongerthesecondderivativeofapotentialψ. Werewetobeinterestedinthelatter
definition, then the cos(2φ) terms inside the square brackets in Eq. (48) would be replaced by 1. For the lens-lens
term, the productofcosinesonthe firstline ofEq.(47)wouldbe replacedby cos2φ ; onthe secondline by cos2φ ;
23 13
and on the third by cos2φ . Practically, we think that the estimator of Eq. (9) is more relevant, since it is the
12
different components of ellipticity that are measured, not the distortion tensor.
IV. HOW IMPORTANT ARE THE CORRECTIONS?
There are a number of ways of assessing the importance of the corrections considered here. First, we compute the
skewnessasafunctionofsmoothingangle. The smoothnessatanygivenangleis anintegraloverallconfigurationsof
the bispectrum with a particular weighting scheme. Thus it reduces all elements to a single number. Second, we can
computetheequilateralconfigurationasafunctionofmultipolemomentl. Comparingthesignalwiththeanticipated
noise allowsus to see whether this one configuration,in whichalll’s areequal,is sensitive to the corrections. Finally,
we compute the anticipated bias on a cosmological parameter from future measurements of the bispectrum if these
correctionsareneglected. Ifthisbiasisverysmall,smallerthantheanticipatedstatisticalerror,thenthereisnoneed
to worry about the corrections.
A. Skewness
The convergence skewness is defined as
S ≡hκ¯3i/hκ¯2i2 (49)
3
where κ¯ is the convergence smoothed over certain window function. In the weakly nonlinear regime where second
order perturbation theory applies, Bernardeau et al. [27] showed that the skewness does not depend on the density
fluctuationamplitude σ but is verysensitiveto the meanmatter density Ω . This behaviorholdseveninthe highly
8 m
nonlinearregime[21]. SoS isparticularlyusefultobreakthedegeneracyofΩ andσ inthelensingpowerspectrum.
3 m 8
Detections of skewness have been reported by several groups [18, 19]; future surveys such as Canada-France-Hawaii
Telescope Legacy Survey [28] could determine Ω to 10% in this fashion.
m
Correctionstothelensingbispectrumaffectthepredictionofskewnessandthusbiastheconstraintsofcosmological
parameters. We quantify corrections of reduced shear, lens-lens coupling and deviation from Born approximation to
hκ¯3i. hκ¯3i is related to the lensing bispectrum by
d2l d2l
hκ¯3i= B(l ,l ,l )W(l )W(l )W(l ) 1 2 . (50)
Z 1 2 3 1 2 3 (2π)2(2π)2
9
Here, W(l)is the Fourier transformof the window function W(θ). We study twowindow functions, the compensated
Gaussian and the aperture:
W (θ) = (1−θ2/2θ2)exp(−θ2/2θ2)
CG f f
W (θ) = (1−θ2/θ2)(1/3−θ2/θ2)Θ(θ −θ) (51)
aperture f f f
where θ is the characteristic scale in both cases. For both, hκ¯3i is then a function of θ . The corrections from the
f f
three effects considered in §III and the total are shown in Fig. 2; on interesting scales, corrections are smaller than
about 2%. Since S ∝Ω−α where α∼0.8 [27, 28], these corrections could bias the determination of Ω by less than
3 m m
about 2%. So they can be safely neglected in the near future.
FIG. 2: Corrections tolensing skewness. The window functionswe adopt are compensated Gaussian (top panel) and aperture
function (bottom panel). Individualand combined corrections are less than about 2% on all scales of interest.
10
B. Equilateral Configuration
One configuration which is often used as a standard is the equilateral configuration, wherein l =l =l ≡l. The
1 2 3
bispectrum can then be plotted as a function of l. Let’s consider the corrections to the equilateral bispectrum.
For the reduced shear correction, all the cosines in Eq. (30) are −0.5, so adding up all the permutations leads to
−3l8P2(l)
Brs = 2 . (52)
lll 16
For the lens-lenscorrection,inthe firstline ofEq.(47) φ +φ =2π; cos(2φ )=−1/2;andall three permutations
12 13 23
contribute equally, so
3l2
Bll = I(l,l). (53)
lll 8
In the equilateral case of Eq. (48), all the cosines are equal to −1/2, so
BBorn =−Bll /2. (54)
lll lll
The resulting corrections are shown in Fig. 3
Thenewtermsareonlyabouttenpercentofthefirstordertermusuallyconsideredinthisequilateralconfiguration.
Nonetheless, they may still be important for precision cosmology where we sum over many different configurations.
C. Cosmological Parameter Bias
There is a simple formula relating the error in a cosmological parameter to a mis-estimate in the theoretical
prediction.
∂B
∆p=F−1 w(l ,l ,l ) l1l2l3∆B . (55)
1 2 3 ∂p l1,l2,l3
l1X,l2,l3
Here the bias in the parameter is ∆p; F is the Fisher matrix (here just one number since we treat the simple case
of only one parameter); w is the weight, or the inverse variance, from the experiment of interest, and ∆B is the
mis-estimate in the bispectrum, here taken to be the full set of corrections computed above. The Fisher matrix too
depends on the survey. It is
2
∂B
F = w(l ,l ,l ) l1l2l3 . (56)
1 2 3
(cid:18) ∂p (cid:19)
l1X,l2,l3
The weights for a particular configuration depend on the survey in question. We are interested in the question of
whether these corrections can ever be important, so we take the minimum possible errors: cosmic variance due to
simple Gaussian fluctuations. Following Takada and Jain [4], the weights are
w(l ,l ,l )−1 =∆ C C C (57)
1 2 3 123 l1 l2 l3
where the C ’s are l4P (l)/4, and ∆ = 1 when all l’s are different, ∆ = 2 when two l’s are equal, and 6 when all l’s
l 2
are equal. Under this weighting, the sum extends over l ≤l ≤l .
1 2 3
AsimpleapplicationofthisformulaistoconsidertheparametertobetheamplitudeAofthebispectrumassuming
the shape is known. Then, Eq. (55) reduces to
∆A = l1,l2,l3w(l1,l2,l3)Bl1l2l3∆Bl1,l2,l3. (58)
(cid:18) A (cid:19) P w(l ,l ,l )B2
bias l1,l2,l3 1 2 3 l1l2l3
P
This is to be compared with the fractional statistical error,
−1/2
∆A
(cid:18) A (cid:19) = w(l1,l2,l3)Bl21l2l3 . (59)
statistical l1X,l2,l3
 