Gravitational lensing system

Figure 2.4:Sketch of a gravitational lensing system. The distances𝐷_s, 𝐷_d, and𝐷_ds are the angular diameter distances to the source, to the lens, and from the lens to source, respectively. From Bartelmann and Schneider (2001).

2.4.1 Lens equation

We consider gravitational lensing in the weak-field limit of GR, with the lensing systems embedded in Minkowskian space-time. For this, we assume that the lenses gravitational potentialΦ, its typical scales 𝐿and intrinsic velocityvare all small, so,

|Φ| 𝑐² , 𝐿 𝑐

𝐻₀ , |v| 𝑐. (2.87)

Figure 2.4 shows a schematic sketch of a gravitational lens system. Light from a background object, thesource, is bent by the gravitational potential of a foreground object, thelens. This deflection shifts the apparent position of the source. Using the distances and angles defined in Fig. 2.4 and the assumption that the gravitational potential of the lens lies on a single plane,

𝜷 =𝜽− 𝐷_ds

𝐷_s 𝜶ˆ =: 𝜽−𝜶 , (2.88)

where we defined thereduced deflection angle𝜶. The reduced deflection angle depends on the surface mass densityΣof the lens,

Σ(𝜽) =

∫

d𝑟₃𝜌(𝒓) , (2.89)

with𝒓 =𝑟₃(𝜽, 1). With this surface mass density, we can define the lensing potentialΨ, Ψ(𝜽)= 4𝐺

𝑐2

𝐷_d𝐷_ds 𝐷_s

∫

d²𝜃⁰ Σ(𝜽⁰) ln(|𝜽−𝜽⁰|) . (2.90)

2.4 Gravitational lensing

The reduced deflection angle𝛼is the gradient of this potential, so𝜷is related to the lensing potential by

𝜷=𝜽− ∇Ψ. (2.91)

Taking the gradient of Eq. (2.91) and linearizing it, leads to

𝜕_𝑗𝛽_𝑖 =𝛿_{𝑖 𝑗} −𝜕_𝑖𝜕_𝑗Ψ =: 𝐴_{𝑖 𝑗}, (2.92) where𝐴is the Jacobian of the lensing potential. It is given by

𝐴(𝜽) =

1−𝜕²

𝑖Ψ −𝜕_𝑖𝜕_𝑗Ψ

−𝜕_𝑖𝜕_𝑗Ψ 1+𝜕²

𝑗Ψ

=:

1−𝜅−𝛾₁ −𝛾₂

−𝛾₂ 1−𝜅+𝛾₁

, (2.93)

with theconvergence 𝜅and thecomplex shear𝛾_c=𝛾₁+i𝛾₂.

The shear is often more conveniently expressed with respect to a given orientation𝜙. The rotated shear𝛾(𝝑;𝜙)is defined as

𝛾(𝝑;𝜙) =−e^−2i𝜙𝛾_c(𝝑)=: 𝛾_t(𝝑;𝜙) +i𝛾_×(𝝑;𝜙) , (2.94) where𝛾_tis thetangential shear and𝛾_× is thecross shear.

The convergence is a normalised version of the surface mass density, 𝜅(𝜽) = 4𝜋 𝐺

𝑐2

𝐷_d𝐷_ds 𝐷_s

Σ(𝜽) =:Σ⁻_crit¹ (𝑧_d,𝑧_s)Σ(𝜽) , (2.95) whereΣ_critis the critical surface mass density⁶and𝑧_dand𝑧_sare the redshifts of the lens and source respectively.

Shear and convergence are related to the reduced shear𝑔, 𝑔 = 𝛾_c

1+𝜅

= 𝛾₁+i𝛾₂

1+𝜅 . (2.96)

Since𝜅and𝛾_care both derivatives of the lensing potential, their Fourier transforms ˆ𝜅(ℓ)and ˆ

𝛾_c(ℓ) can be transformed into each other, using the Kaiser-Squires relation (Kaiser and Squires, 1993)

ˆ

𝛾_c(ℓ) =e^2i𝜙ℓ𝜅ˆ(ℓ) , (2.97)

where𝜙_ℓ is the polar angle ofℓ.

6This critical surface mass density is not the comoving critical surface mass densityΣ_{crit, com}, defined by Σ⁻_crit,com¹ (𝑧_d,𝑧_s)= 4𝜋 𝐺

𝑐2

𝐷_A(𝑧_d,𝑧_s)𝐷_A(𝑧_d) (1+𝑧_d)𝐷_A(𝑧_s) ,

and used in some gravitational lensing studies. Appendix C in Dvornik et al. (2018) discusses the implications of different definitions of the critical surface mass density.

Aside from the distortion of the source shape, gravitational lensing also magnifies images. This magnification𝜇affects the observed flux𝑠of a source flux𝑠₀ as

𝑠 =𝜇 𝑠₀ , (2.98)

and is given as

𝜇= 1

(1−𝜅)²− |𝛾_c|² . (2.99)

Magnification by gravitational lenses can be used as a ‘natural telescope’ as it enables the observation of faint sources, which would be undetectable otherwise (see e.g. Richard et al., 2011; Schmidt et al., 2017). It also affects the observed number density of galaxies and therefore impacts measurements of GGL (see Sect. 2.4.4) and other weak gravitational lensing effects. We study its effect on G3L in Chapter 3.

2.4.2 Weak gravitational lensing

Equation (2.88) can be used in two different regimes. For𝜅 & 1, we are in thestrong lensing regime, where source galaxy shapes are strongly distorted. Strongly lensed sources appear as arcs, complete rings, or even multiple images. However, for this work, we are concerned with weak lensing, where𝜅 1 (see Bartelmann and Schneider, 2001, for a review). Weak lensing distorts galaxy shapes only slightly. Consequently, the images of weakly lensed galaxies look far less impressive for the casual observer than of strongly lensed galaxies. The shape distortion due to weak lensing is usually smaller than the intrinsic scatter of galaxy shapes, so it is not noticeable by observing individual galaxies. However, there are many more weakly lensed galaxies than strongly lensed sources. Therefore, weak lensing is ideal for measurements of the statistics of the matter distribution, while strong lensing gives mainly information on specific high-density objects such as galaxy clusters.

To first order, weak gravitational lensing only changes the apparent position and ellipticity of a source galaxy. The ellipticity of a galaxy with semi-major and semi-minor axes𝑎and𝑏is

𝜖 = 𝑎−𝑏

𝑎+𝑏e^2i𝜙 , (2.100)

where𝜙is the angle between the galaxies semi-major axis and the𝑥-axis of the coordinate frame. Due to weak lensing, the observed ellipticity is composed of the intrinsic ellipticity𝜖_int and the reduced shear𝑔,

𝜖 = 𝜖_int+𝑔

1+𝑔^∗𝜖_int , (2.101)

where the asterisk denotes complex conjugation. In the weak lensing regime𝑔' 𝛾_c, so we can in principle estimate the shear directly from the observed ellipticity.

However, for weak lensing, the shear is usually small compared to the intrinsic ellipticity of a galaxy. Moreover, as the intrinsic ellipticity is unknown, we cannot estimate the shear for

2.4 Gravitational lensing

any single galaxy. Therefore, instead of measuring the shear from a single galaxy, we average over the observed ellipticities of many sources. If the intrinsic ellipticities of the sources are uncorrelated to the shear, the average observed ellipticity is

h𝜖i=h𝜖_inti + h𝑔i ' h𝜖_inti + h𝛾_ci . (2.102) The first term vanishes, if galaxies are randomly orientated and have uncorrelated ellipticities, so we can find the shear with

h𝛾_ci ' h𝜖i . (2.103)

The ellipticities of source galaxies are accordingly unbiased estimators of𝛾_c.

2.4.3 Projected spectra and Limber equation

In lensing, all observables are projections on the sky. The deflection angle alone only constrains the surface mass density and cannot directly yield the three-dimensional matter distribution.

The three-dimensional density contrast𝛿is related to the lensing convergence by projecting it along the comoving distance𝑤with

𝜅(𝜽) = 3𝐻²

0Ω_m 2𝑐2

∫ ∞

0 d𝑤 𝑞(𝑤) 𝑓_𝐾(𝑤) 𝛿[®x(𝜽,𝑤),𝑡_𝑤]

𝑎(𝑤) , (2.104)

where 𝑓_𝐾 is defined by Eq. (2.3), 𝑡_𝑤 is the cosmic time at comoving distance 𝑤, 𝒙(𝜽,𝑤) = (𝑓_𝐾(𝑤)𝜽,𝑤)and

𝑞(𝑤) =

∫ ∞

𝑤

d𝑤⁰ 𝑝_s(𝑤⁰) 𝑓_𝐾(𝑤⁰−𝑤)

𝑓_𝐾(𝑤⁰) , (2.105)

with the distribution 𝑝_sof sources with comoving distance. With this relation for𝜅, we can infer theprojected matter power spetrum𝑃_{𝜅 𝜅}(ℓ)which is defined by

(2𝜋)³𝛿_D(ℓ₁+ℓ₂)𝑃_{𝜅 𝜅}(ℓ₁) =h𝜅ˆ(ℓ₁)𝜅ˆ(ℓ₂)i , (2.106) where ˆ𝜅is the Fourier transform of 𝜅. This projected power spectrum can be derived from the three-dimensional power spectrum𝑃(𝑘,𝑡).

In general, this would require decomposing𝜅 into spherical harmonics and evaluating the correlation between density fluctuations at different cosmic times. However, under two as-sumptions, the Limber approximation (Kaiser, 1992) can be used. These assumptions are, first, that the sky can be approximated by a plane (flat-sky-approximation), and second, that𝑞(𝑤) varies little over the coherence length of the described structures. These assumptions hold if the angular scales on which𝑃_{𝜅 𝜅} is evaluated are small, and the 𝑞are not too narrow. For example, Simon (2007) found, that for broad𝑞, the Limber approximation is accurate at the 10% level for scales less than a few degrees. With these assumptions, the projected matter power spectrum is Universe is

𝑃_{𝜅 𝜅}(ℓ) = 9𝐻⁴

0Ω²_m 4𝑐4

∫ d𝑤

𝑞²(𝑤)

𝑎(𝑤) 𝑃(ℓ/𝑓_𝐾(𝑤),𝑡_𝑤) . (2.107)

We also define the projected galaxy number density 𝑁(𝜽), which is related to the three-dimensional number density𝑛[𝒙(𝜽,𝑤),𝑡_𝑤] at comoving distance𝑤 by the selection function 𝜈(𝑤),

𝑁(𝜽) =

∫

d𝑤 𝜈(𝑤)𝑛[𝒙(𝜽,𝑤),𝑡_𝑤] . (2.108) The selection function gives the fraction of galaxies at comoving distance𝑤included in the galaxy sample. For a flux-limited sample, this corresponds to the fraction of galaxies brighter than the magnitude limit. The selection function𝜈(𝑤) is related to the distribution𝑝(𝑤) of the galaxies with comoving distance, according to (Schneider, 2005)

𝜈(𝑤) = 𝑝(𝑤)

∫ d²𝜃 𝑁(𝜽)

∫ d²𝜃 𝑛[𝒙(𝜽,𝑤),𝑡_𝑤] . (2.109) With𝑁(𝜽), we can define the galaxy convergence𝜅_gas

𝜅_g(𝜽) = 𝑁(𝜽)

¯ 𝑁

−1 , (2.110)

where𝑁 is the mean projected galaxy number density. The galaxy convergence determines the projected galaxy-matter power spectrum𝑃_g𝜅, defined as

(2𝜋)³𝛿_D(ℓ₁+ℓ₂) 𝑃_g𝜅(ℓ₁) = ˆ

𝜅(ℓ₁)𝜅ˆ_g(ℓ₂)

, (2.111)

and theprojected galaxy-galaxy-matter bispectrum𝐵_gg𝜅, defined as (2𝜋)³𝛿_D(ℓ₁+ℓ₂+ℓ₃) 𝐵_gg𝜅(ℓ₁,ℓ₂,𝜙) =

ˆ

𝜅(ℓ₁)𝜅ˆ_g(ℓ₂)𝜅ˆ_g(ℓ₃)

. (2.112)

These can be derived from their three-dimensional counterparts in a similar way as the projected matter power spectrum. Under the same assumptions as for the Limber approximation, that is a flat sky and a slowly varying𝜈(𝑤), they are (Schneider and Watts, 2005)

𝑃_g𝜅(ℓ) = 3𝐻²

0Ω_m 2𝑐2

∫ d𝑤

𝑞(𝑤)𝑝(𝑤)

𝑤 𝑎(𝑤) 𝑃_g𝛿(ℓ/𝑓_𝐾(𝑤),𝑡_𝑤), (2.113) 𝐵_gg𝜅(ℓ₁,ℓ₂,𝜙) = 3𝐻²

0Ω_m 2𝑐2

∫ d𝑤

𝑞(𝑤)𝑝²(𝑤)

𝑤3𝑎(𝑤) 𝐵_gg𝛿(ℓ₁/𝑓_𝐾(𝑤),ℓ₂/𝑓_𝐾(𝑤),𝜙,𝑡_𝑤) . (2.114)

2.4.4 Galaxy-galaxy-lensing

Weak gravitational lensing is an excellent tool to measure the galaxy-matter correlations. To estimate the galaxy-matter power spectrum𝑃_g𝛿(𝑘), the method-of-choice is GGL. For GGL, we measure the ellipticity of background source galaxies and their angular separation from foreground lens galaxies. Then, we average the ellipticities of all sources with separation𝝑to a lens to find an estimate of

h𝛾_ti (𝝑)= 1

¯ 𝑁

h𝑁(𝜽)𝛾_t(𝜽+𝝑;𝜙)i , (2.115)

We can study the galaxy-galaxy-matter bispectrum with G3L (Schneider and Watts, 2005).

This effect includes the lensing of source galaxies by lens galaxy pairs, which determines the lens-lens-shear correlation. Unlike GGL or galaxy clustering, G3L depends on the galaxy-matter three-point correlation and the HOD of galaxy pairs. In principle, it also depends on the ellipticity of dark matter halos as well as misalignments between the galaxy and matter distribution because the galaxy pair orientation introduces a preferred direction.

The lens-lens-shear correlation was measured for lens pairs separated by several Mpc to detect inter-cluster filaments (Mead et al., 2010; Clampitt et al., 2016; Epps and Hudson, 2017; Xia et al., 2020). However, for the assessment of SAMs, it is more suitable to study the correlation at smaller, sub-Mpc scales. At these scales, the G3L signal is more sensitive to the small-scale physics that vary between different SAMs, because it depends primarily on galaxy pairs with

Figure 2.5:Geometry of a G3L system with one source and two lens galaxies. Adapted from Schneider and Watts (2005).

galaxies in the same dark matter halo. For lens pairs with galaxies of similar stellar mass or colour, the small-scale lens-lens-shear correlation was determined by Simon et al. (2008) in the Red-sequence Cluster Survey (RCS; Gladders and Yee, 2005) and Simon et al. (2013) in the CFHTLenS. The G3L measured in CFHTLenS was compared to predictions by multiple SAMs implemented in the MR by Saghiha et al. (2017) and Simon et al. (2019). They demonstrated that G3L is more effective in evaluating SAMs than GGL and that the H15 SAM agreed with the observations in CFHTLenS, while the L12 SAM predicts too large G3L signals.

For this work, we are concerned with the lensing of single sources around pairs of lens galaxies.

Figure 2.5 depicts the corresponding geometric configuration of lens and source galaxies. The main observable of this type of G3L is the correlation function ˜G, defined as

G (𝝑˜ ₁,𝝑₂) = 1 𝑁²

D

𝑁(𝜽+𝝑₁)𝑁(𝜽+𝝑₂)𝛾(𝜽; 𝜑₁+𝜑₂ 2 )E

, (2.119)

orG, which is

G (𝝑₁,𝝑₂) = D

𝜅_g(𝜽+𝝑₁)𝜅_g(𝜽+𝝑₂)𝛾(𝜽;𝜑₁+𝜑₂ 2 )E

. (2.120)

The two correlation functions are linked by G (𝝑₁,𝝑₂) =G (𝝑˜ ₁,𝝑₂) − 1

𝑁

h𝑁(𝜽+𝝑₁)𝛾_t(𝜽;𝜑₁)ie^−i𝜙− 1 𝑁

h𝑁(𝜽+𝝑₁)𝛾_t(𝜽;𝜑₂)ie^i𝜙 (2.121)

=G (𝝑˜ ₁,𝝑₂) − h𝛾_ti (𝝑₁)e^−i𝜙− h𝛾_ti (𝝑₂)e^i𝜙 , (2.122)

2.4 Gravitational lensing

so ˜Gcontains terms arising from GGL, whileGis only the third-order correlation due to G3L.

The correlation functions are related to the projected galaxy-galaxy-matter bispectrum by G (𝝑₁,𝝑₂) =

∫ d²ℓ₁ (2𝜋)²

∫ d²ℓ₂

(2𝜋)² e^{−i(ℓ1·𝝑1+ℓ2·𝝑2)} 1

|ℓ₁+ℓ₂|²

ℓ₁e^i𝜙1+ℓ₂e^i𝜙2₂

𝐵_gg𝜅(ℓ₁,ℓ₂,𝜙_ℓ) , (2.123) where𝜙_ℓ is the angle betweenℓ₁ andℓ₂.

Due to statistical homogeneity and isotropy, ˜GandGonly depend on the lens-source separations 𝜗₁and𝜗₂and the angle𝜙between𝝑₁and𝝑₂. Consequently, we write

G (𝝑˜ ₁,𝝑₂) =: ˜G (𝜗₁,𝜗₂,𝜙), (2.124) G (𝝑₁,𝝑₂) =: G (𝜗₁,𝜗₂,𝜙). (2.125) Simon et al. (2008) showed how to estimate ˜G (𝜗₁,𝜗₂,𝜙) by averaging the ellipticities of source galaxies over all lens-lens-source triplets, where𝜗₁ (𝜗₂) is the separation between the first (second) lens and the source. Their estimator of ˜Gin a bin𝐵of𝜗₁,𝜗₂and𝜙for 𝑁_ssource, and 𝑁_dlens galaxies is

G˜_est(𝐵) =− Í^𝑁_d

𝑖,𝑗=1Í^𝑁_s

𝑘=1𝑤_𝑘𝜖_𝑘e^{−i(𝜑𝑖 𝑘+𝜑𝑗 𝑘)}

1+𝜔(|𝜽𝑖−𝜽𝑗|)

Δ𝑖 𝑗 𝑘(𝐵) Í^𝑁_d

𝑖,𝑗=1Í^𝑁_s

𝑘 𝑤_𝑘Δ𝑖 𝑗 𝑘(𝐵) (2.126)

=:− Í

𝑖,𝑗,𝑘𝑤_𝑘𝜖_𝑘e^{−i(𝜑𝑖 𝑘+𝜑𝑗 𝑘)}

1+𝜔(|𝜽𝑖−𝜽𝑗|)

Δ𝑖 𝑗 𝑘(𝐵) Í

𝑖,𝑗,𝑘 𝑤_𝑘Δ𝑖 𝑗 𝑘(𝐵) , (2.127)

with

Δ𝑖 𝑗 𝑘(𝐵) =

(1 for |𝜽𝑘 −𝜽𝑖|,|𝜽𝑘−𝜽𝑗|,𝜙_{𝑖 𝑗 𝑘}

∈𝐵

0 otherwise . (2.128)

The angle𝜑_{𝑖 𝑘}(𝜑_{𝑗 𝑘}) is the polar angle of𝜽𝑘−𝜽𝑖(𝜽𝑘−𝜽𝑗) and𝜙_{𝑖 𝑗 𝑘} is the opening angle between 𝜽𝑘 −𝜽𝑖and𝜽𝑘 −𝜽𝑗. The𝑤_𝑘 are the weights of the measured ellipticities. Source galaxies with more precise shape measurements receive a higher ellipticity weight𝑤_𝑘. The weight, therefore, increases the contribution of source galaxies with more precise shapes to the estimator. For the simulated shear data in Chapters 3 and 4, we set the weights to𝑤_𝑘 =1 for all sources. This estimator also includes the angular two-point correlation function𝜔, which takes account of the clustering of lens galaxies. The two-point correlation can be estimated with the Landy-Szalay estimator (Landy and Szalay, 1993),

𝜔(𝜃) = 𝑁_r²𝐷 𝐷(𝜃) 𝑁²

d𝑅 𝑅(𝜃) −2 𝑁_r𝐷 𝑅(𝜃)

𝑁_d𝑅 𝑅(𝜃) +1 . (2.129)

Here,𝐷 𝐷(𝜃) is the paircount of lens galaxies,𝑅 𝑅(𝜃)is the paircount of randoms, which are unclustered galaxies, and𝐷 𝑅(𝜃)is the cross paircount of lenses and randoms at separation𝜃. The total numbers of lenses and randoms are denoted by𝑁_dand𝑁_r.

This estimator does not take into account redshift information on the galaxies. Therefore, lens-lens-source triplets with lenses separated along the line-of-sight have the same weight in the estimator as triplets whose lens galaxies are physically close. These separated lens pairs decrease the signal-to-noise ratio (S/N), as shown by Simon et al. (2019). We, therefore, propose a new estimator that uses lens redshift information to improve the S/N in Chapter 3.

2.4 Gravitational lensing

while for G3L the relevant aperture statistics ishN N Mi,

hN N Mi (𝜃₁,𝜃₂,𝜃₃) (2.138)

=

N𝜃₁(𝝑₁) N𝜃₂(𝝑₂) M𝜃₃(𝝑₃)

= 1 𝑁²

∫ d²𝜗₁

∫ d²𝜗₂

∫

d²𝜗₃𝑈_𝜃

1(𝝑₁)𝑈_𝜃

2(𝝑₂)𝑈_𝜃

3(𝝑₃) h𝑁(𝝑₁)𝑁(𝝑₂)𝜅(𝝑₃)i . (2.139) The aperture statistics can be easier related to the projected galaxy-matter polyspectra than the direct GGL and G3L correlation functions. For GGL,

hN Mi (𝜃₁,𝜃₂) =

∫ d²ℓ (2𝜋)²𝑈ˆ_𝜃

1(ℓ)𝑈ˆ_𝜃

2(ℓ)𝑃_g𝜅(ℓ) , (2.140) and for G3L

hN N Mi (𝜃₁,𝜃₂) =

∫ d²ℓ₁ (2𝜋)²

∫ d²ℓ₂ (2𝜋)²𝑈ˆ_𝜃

1(ℓ₁)𝑈ˆ_𝜃

2(ℓ₂)𝑈ˆ_𝜃

3(|ℓ₁+ℓ₂|)𝐵_gg𝜅(ℓ₁,ℓ₂,𝜙) , (2.141) where ˆ𝑈_𝜃 is the Fourier transform of𝑈_𝜃.

Throughout this work, we use an exponential filter function, 𝑢(𝑥) = 1

2𝜋

1− 𝑥² 2

exp

−𝑥² 2

. (2.142)

For this choice, the correlation function ˜Gcan be connected to hN N Miwith an analytical expression,

hN N Mi (𝜃₁,𝜃₂,𝜃₃) +ihN N M_⊥i (𝜃₁,𝜃₂,𝜃₃) (2.143)

=

∫ ∞

0 d𝜗₁ 𝜗₁

∫ ∞

0 d𝜗₂ 𝜗₂

∫ ₂𝜋

0 d𝜙 G (˜ 𝜗₁,𝜗₂,𝜙) A_{N N M}(𝜗₁,𝜗₂,𝜙 | 𝜃₁,𝜃₂,𝜃₃) ,

with the kernel functionA_{N N M}(𝜗₁,𝜗₂,𝜙 |𝜃₁,𝜃₂,𝜃₃) given in the appendix of Schneider and Watts (2005). We measure aperture statistics only for equal scale radii𝜃₁ =𝜃₂ =𝜃₃. Therefore, we use the abbreviations

hN N Mi (𝜃) :=hN N Mi (𝜃,𝜃,𝜃) , (2.144) and

A_{N N M}(𝜗₁,𝜗₂,𝜙 |𝜃) :=A_{N N M}(𝜗₁,𝜗₂,𝜙| 𝜃₁,𝜃₂,𝜃₃) . (2.145) The aperture statistics can be used to constrain the galaxy bias factor, discussed in Sect. 2.1.3 (e.g Schneider and Watts, 2005). For two galaxy populations with linear deterministic bias factors 𝑏₁and𝑏₂ and aperture number countsN₁ andN₂, this simple model predicts for the aperture statistics

hN₁N₂Mi ∝ 𝑏₁𝑏₂ . (2.146)

From this follows that

𝑅(𝜃) := hN₁N₂Mi (𝜃)

√︁hN₁N₁Mi (𝜃) hN₂N₂Mi (𝜃)

= 𝑏₁𝑏₂

√︃

𝑏²

1𝑏²

2

=1 . (2.147)

In Chapter 4, we measure𝑅in the observation and simulation to assess the assumption of linear deterministic bias.

2.4.7 Gravitational Lensing in 𝑵-body simulations

To study gravitational lensing with𝑁-body simulations, the three-dimensional density con-trast𝛿, which is given by these simulations, needs to be converted to the two-dimensional convergence𝜅. To obtain𝜅, one usesray-tracing algorithms. There exist a variety of different ray-tracing methods, some of which were reviewed and compared by Hilbert et al. (2020).

These algorithms are usually applied to a simulation after it was fully calculated, that is in ‘post-processing’. This approach makes it easier to change source redshifts or observer orientations, as the simulation only needs to run once. However, the number of simulational snapshots limits the accuracy of the resulting convergence maps. If only a small number of snapshots are available, the convergence is averaged over a larger time interval, so the resulting maps can be biased. There are also approaches to compute the convergence ‘on the fly’ that is together with the full particle distribution, (e.g. Barreira et al., 2016), but these are computationally more expensive.

Post-processing ray-tracing algorithms all operate similarly. First, these algorithms project the matter in each snapshot, either on lens planes perpendicular to the line-of-sight (Hilbert et al., 2009; Giocoli et al., 2016) or on spheres centred on the observer (Fosalba et al., 2008; Fabbian et al., 2018). Then, light rays are traced backwards from the observer to the source plane. At each lens plane/sphere, the deflection angle of the rays is calculated from the lensing potential of all matter at the plane/sphere. Adding up the deflections of all planes up to the source plane gives the total deflection angle due to the matter distribution. The gradient of this deflection angle corresponds to the lensing Jacobian in Eq. (2.92). The Jacobian can be converted to maps of the shear𝛾and convergence𝜅.

While different ray-tracing algorithms differ in details, such as the choice of the projection method, Hilbert et al. (2020) found that the predicted convergence maps agree in general very well. Differences exist mainly in the predicted mean convergence. However, this quantity is not directly observable in weak lensing measurements, as the shear does not depend on it. After normalising the convergence maps of different ray-tracing algorithms, differences are only of the order of a few percents. Consequently, the choice of the ray-tracing algorithm is not particularly critical for the comparison of simulations to observations. We use the ray-tracing algorithm by Hilbert et al. (2009) on the MR to study G3L in the following chapters.

Improving the precision and

accuracy of galaxy-galaxy-galaxy lensing

3

This chapter is based on Linke et al. (2020a), published in Astronomy & Astrophys-ics.

In this chapter, we discuss how to improve measurements of G3L by increasing both the accuracy and precision of the measured aperture statistics. These improvements use precise redshift estimates for the lens galaxies, as well as an adaptive binning scheme for the estimation of the three-point correlation function ˜G. We motivate our improvements in Sect. 3.1, and explain their application in Sect. 3.2. We apply the improved and original measurement schemes to two different types of catalogues described in Sect. 3.3, one based on simple, but unrealistic assumptions on the galaxy distribution and one based on the SAM by H15, implemented in the MR. The resulting aperture statistics are presented in Sect. 3.4 and discussed in Sect. 3.5.

3.1 Motivation

As explained in Sect. 2.4.4, G3L is a sensitive probe of galaxy formation. However, previous measurements of this effect used only photometric data without precise redshift estimates.

Consequently, pairs of physically close lens galaxies, which are highly correlated, had the same weight as galaxy pairs separated along the line of sight with little to no correlation. These separated galaxies decrease the signal and lower the S/N.

Additionally, G3L is affected by the magnification of lens galaxies caused by the LSS in front of the lenses. This magnification affects the number density of lens galaxies in a survey.

Because source galaxies are also lensed by the LSS, the shear of sources correlates with the lens magnification, and an additional correlation signal arises. This signal has not yet been quantified for G3L, but was found to affect GGL by up to 5 % in CFHTLenS (Simon and Hilbert, 2018).

Consequently, we introduce three improvements to the G3L estimator used by Simon et al.

(2008, 2013) and given in Eq. (2.126). These are (i) weighting the lens galaxy pairs according to

3.2 Methods

estimate ˜G𝑍 with G˜𝑍,est(𝐵)=−

Í

𝑖 𝑗 𝑘 𝑤_𝑘𝜖_𝑘e^{−i(𝜑𝑖 𝑘+𝜑𝑗 𝑘)}

1+𝜔_𝑍 |𝜽𝑖−𝜽𝑗| 𝑍(Δ𝑧_{𝑖 𝑗})Δ𝑖 𝑗 𝑘(𝐵) Í

𝑖 𝑗 𝑘𝑤_𝑘𝑍(Δ𝑧_{𝑖 𝑗})Δ𝑖 𝑗 𝑘(𝐵) . (3.4) To estimate the redshift-weighted two-point correlation𝜔_𝑍, we use the𝑁_rrandoms, located at 𝜽⁰_𝑖, the𝑁_dlenses at the positions𝜽𝑖, and the estimator

𝜔_𝑍(𝜃) =

𝑁_r²𝐷 𝐷_𝑍(𝜃) 𝑁²

d𝑅 𝑅_𝑍(𝜃) −2𝑁_r𝐷 𝑅_𝑍(𝜃)

𝑁_d𝑅 𝑅_𝑍(𝜃) +1 , (3.5) with the modified pair-counts

𝐷 𝐷_𝑍(𝜃) =

𝑁_d

∑︁

𝑖=1 𝑁_d

∑︁

𝑗=1

Θ_H 𝜃+Δ𝜃/2− |𝜽𝑖−𝜽𝑗|

Θ_H −𝜃+Δ𝜃/2+ |𝜽𝑖−𝜽𝑗|

𝑍(Δ𝑧_{𝑖 𝑗}) , (3.6)

𝑅 𝑅_𝑍(𝜃) =

𝑁_r

∑︁

𝑖=1 𝑁_r

∑︁

𝑗=1

Θ_H 𝜃+Δ𝜃/2− |𝜽𝑖−𝜽𝑗| Θ_H

−𝜃+Δ𝜃/2+ |𝜽⁰_𝑖−𝜽⁰_𝑗|

𝑍(Δ𝑧_{𝑖 𝑗}), (3.7) and

𝐷 𝑅_𝑍(𝜃) =

𝑁_d

∑︁

𝑖=1 𝑁_r

∑︁

𝑗=1

Θ_H

𝜃+Δ𝜃/2− |𝜽⁰_𝑖−𝜽⁰_𝑗| Θ_H

−𝜃+Δ𝜃/2+ |𝜽⁰_𝑖−𝜽⁰_𝑗|

𝑍(Δ𝑧_{𝑖 𝑗}) . (3.8) Here,Θ_His the Heaviside step function andΔ𝜃is the bin size for which𝜔_𝑍 is estimated. For 𝑍 ≡1, this estimator reduces to the standard Landy-Szalay estimator in Eq. (2.129).

The aperture statistics from the redshift-weighted correlation function ˜G𝑍 are expected to have a higher S/N than the aperture statistics from the original ˜G. This expected improvement can be estimated with simplified assumptions. For this, we assume that the𝑁_totlens-lens-source triplets can be split into 𝑁_true physical triplets, each carrying the signal 𝑠, and 𝑁_tot − 𝑁_true triplets carrying no signal. We further assume that all triplets carry the same uncorrelated noise 𝑛. Then, the measured total signal𝑆, noise𝑁 and S/N are

𝑆= 𝑁_true 𝑁_tot

𝑠, 𝑁 = 1

√ 𝑁_tot

𝑛, and𝑆/𝑁 = 𝑁_true

√ 𝑁_tot

𝑠

𝑛 . (3.9)

With redshift weighting we decrease the effective number of triplets from𝑁_totto ˜𝑁_tot, while retaining the same number of physical triplets𝑁_true. The signal ˜𝑆, the noise ˜𝑁 and the new S/N

˜

𝑆/𝑁˜ are then

˜

𝑆 = 𝑁_true

˜ 𝑁_tot

𝑠, 𝑁˜ = 1

√︁𝑁˜_tot

𝑛, and ˜𝑆/𝑁˜ = 𝑁_true

√︁𝑁˜_tot 𝑠

𝑛 . (3.10) Consequently, redshift weighting increases the noise by a factor of(𝑁_tot/𝑁˜_tot)^1/2. Nonetheless, the S/N improves by(𝑁_tot/𝑁˜_tot)^1/2 because the signal increases by𝑁_tot/𝑁˜_tot. Accordingly, we expect the S/N to increase approximately by the square root of the signal increase.

The critical parameter for the redshift weighting is the width 𝜎_𝑧 of the weighting function.

For our application on the observational and simulated data described in Sect. 3.3, we choose 𝜎_𝑧 = 0.01. Because there is no clear division between lens pairs that carry signal and those that do not, the choice of this parameter needs to remain somewhat arbitrary. However, three arguments motivate our choice.

The first argument considers the galaxy correlation length. Farrow et al. (2015) measured the two-point correlation function of galaxies in GAMA and found correlation lengths between 3.28±0.42ℎ⁻¹Mpc and 38.17±0.47ℎ⁻¹Mpc, depending on the stellar masses of the galaxies.

Zehavi et al. (2011) measured the same function in the SDSS and found similar correlation lengths between 4.2ℎ⁻¹Mpc and 10.5ℎ⁻¹Mpc. These correlation lengths correspond to redshift differences between 0.001 and 0.005 at the median redshift of GAMA of𝑧=0.21. We assume that galaxies separated by more than twice the correlation length are only weakly correlated.

Therefore our choice of𝜎_𝑧 =0.01 seems appropriate.

The second argument relates to the distribution of lens galaxy pairs with their redshift difference.

The blue histogram in Fig. 3.1 shows the number of galaxy pairs per redshift difference𝛿 𝑧with fixed angular separation between 4.⁰5 and 5.⁰5 in our lens sample from the MR (see Sect. 3.3). This distribution has a prominent peak for small𝛿 𝑧and a broad background distribution. Thus, most galaxy pairs that appear close on the sky are also close in redshift space. These physical pairs make up the peak. However, the background distribution shows that there are also many galaxy pairs with small angular separation whose redshift difference is large. The optimal redshift weighting function should preserve pairs inside the peak but suppress the background.

The other histograms in Fig. 3.1 show different weighted distributions, where the number of galaxy pairs is multiplied by the redshift-weighting function from Eq. (3.1). These distributions correspond to the effective number of galaxy pairs per redshift difference bin considered for the improved ˜Gestimator. Here, the effect of different𝜎_𝑧is visible. The weighting preserves the peak when we use𝜎_𝑧 = 0.1 and 0.05. However, a high percentage of the background is still present in the weighted distribution. Weighting with𝜎_𝑧 =0.005 and𝜎_𝑧 =0.001 removes the background but also suppresses parts of the peak. A middle ground is found for𝜎_𝑧 =0.01.

Here, the tails of the peak still contribute, whereas most of the background galaxy pairs are suppressed. Consequently, we adopt this value for the measurement of ˜Gand subsequently hN N Mi.

The third argument for our choice of𝜎_𝑧 considers the peculiar velocities of galaxies in clusters, which can cause redshift differences of correlated galaxy pairs inside the same halo. The weighting function𝑍 needs to be broad enough to avoid discarding galaxy pairs whose redshift differences are induced simply by their peculiar motion. Velocities of galaxies inside halos can reach up to 1000 km s⁻¹, leading to redshift differences of up to 0.006. This value is a lower bound for𝜎_𝑧, therefore choosing𝜎_𝑧 =0.01 appears valid.

3.2 Methods

10 ³ 10 ² 10 ¹

Redshift difference z between galaxies 0.0

0.2 0.4 0.6 0.8

Npairs(z<|z1z2|<z+z)/z

×10¹⁰

Not Weighted

z= 0.1

z= 0.05

z= 0.01

z= 0.005

z= 0.001

Figure 3.1:Weighted number of lens galaxy pairs in our sample from the MR with fixed angular separation between4.⁰5and5.⁰5per redshift difference between the pairs. Different colours indicate different widths of the Gaussian weighting function. The blue histogram shows the unweighed distribution, and the green histogram shows the distribution with the weighting chosen for the G3L measurements.

0 0

1 1

1 2

1 2 3

Triplets Average of filled bins

1 1

1 2

1 2 3

Tesselated Bins

ϑ1

ϑ₂

ϑ1

ϑ2

Correlation Function in Bins

ϑ₂

ϑ₁ ϑ2

Correlation Function in Tesselated Bins

ϑ1

ϑ₂

ϑ1

(2)

(1)

(2)

(2)

Figure 3.2:Illustration of the old (1) and new (2) binning scheme for the calculation ofG. In the old˜ binning scheme,G˜was calculated directly from the lens-lens-source triplets inside a given bin. In the new binning scheme, at first, we calculate the average of the lens-lens-source triplets in a bin. We use these averages as seeds for a Voronoi tessellation of the parameter space. Then, we consider each Voronoi cell as a new bin for which we estimateG. We obtain the aperture statistics by integrating over˜ the new bins. We show only two dimensions here, but for the measurement, we also tesselated the third parameterφ.

3.2 Methods

triplets, the measured ˜Gis associated with the average𝜗₁,𝜗₂and 𝜙of the corresponding bin.

We use the averages of the triplets in filled bins as seeds to divide the parameter space by a Voronoi tessellation, using the libraryvoro++by Rycroft (2009). We consider each Voronoi cell as a new bin for ˜G. These bins, by definition, contain at least one triplet. We obtain the aperture statistics by integrating over the𝑁_binnew bins, using the numerical approximation of Eq. (2.143),

hN N Mi (𝜃) +i hN N M_⊥i (𝜃) =

𝑁_bin

∑︁

𝑖=1

𝑉(𝐵_𝑖)G˜_est(𝐵_𝑖)𝐴_{N N M}(𝐵_𝑖|𝜃) , (3.11) where𝐵_𝑖is the𝑖th bin,𝑉(𝐵_𝑖) is the volume of this bin, and 𝐴_{N N M}(𝐵_𝑖|𝜃) is the kernel function of Eq. (2.143) evaluated at the seed of𝐵_𝑖. We estimate ˜Gon a grid with 128×128×128 bins with𝜗₁and𝜗₂between 0.⁰15 and 320⁰for the data based on the MR (see Sect. 3.3.1) and between 0.⁰15 and 200⁰for the simple mock data (see Sect. 3.3.2). The tessellation reduces the number of bins by approximately 3 % in both cases.

3.2.3 Conversion into physical units

With the lens redshifts𝑧₁ and𝑧₂, we can transform the projected angular separation vectors𝝑₁ and𝝑₂into physical separations𝒓₁ and𝒓₂ on a plane midway between the two lenses, using

𝒓_1,2= 𝐷_A(0,𝑧₁₂) 𝝑_1,2 =: 𝐷₁₂𝝑_1,2 , (3.12) with the angular diameter distance𝐷_A(𝑧_𝑎,𝑧_𝑏)between redshifts𝑧_𝑎and𝑧_𝑏and the average lens redshift𝑧₁₂ = (𝑧₁+𝑧₂)/2.

The correlation function ˜G𝑍 can therefore be estimated in physical scales in the bin𝐵of𝑟₁,𝑟₂ and𝜙as

G˜𝑍,est(𝐵) =− Í

𝑖 𝑗 𝑘𝑤_𝑘𝜖_𝑘e^{−i(𝜑𝑖 𝑘+𝜑𝑗 𝑘)}

1+𝜔 |𝜽𝑖−𝜽𝑗| 𝑍(Δ𝑧_{𝑖 𝑗})Δ^ph

𝑖 𝑗 𝑘(𝐵) Í

𝑖 𝑗 𝑘𝑤_𝑘𝑍(Δ𝑧_{𝑖 𝑗})Δ^ph

𝑖 𝑗 𝑘(𝐵) , (3.13)

with

Δ^ph

𝑖 𝑗 𝑘(𝐵) =

(1 for 𝐷_𝐴(0,𝑧_{𝑖 𝑗}) |𝜽𝑘 −𝜽𝑖|,𝐷_𝐴(0,𝑧_{𝑖 𝑗}) |𝜽𝑘 −𝜽𝑗|,𝜙_{𝑖 𝑗 𝑘}

∈ 𝐵

0 otherwise , (3.14)

This ˜G𝑍 still depends on the redshift distribution of sources because the gravitational shear𝛾_t depends on the lensing efficiency, which in turn depends on the distances between observer and source and lens and source. To compare the measurements of different surveys with varying source redshift distributions, it is therefore useful to correlate the galaxy number density not with the tangential shear𝛾_t, but instead with the projected excess mass densityΔΣ, given by

ΔΣ(𝜽,𝑧_d,𝑧_s) =











𝛾_t(𝜽)

Σ⁻_crit¹ (𝑧_d,𝑧_s) for 𝑧_d < 𝑧_s

0 else , (3.15)

Im Dokument Testingmodelsofgalaxyformationand evolutionwithgalaxy-galaxy-galaxy lensing (Seite 41-154)