MULTIPLE WEAK DEFLECTIONS IN GALAXY–GALAXY LENSING

Galaxy–galaxy lensing is the systematic weak gravitational lensing of background galaxies by foreground galaxies. Unlike weak lensing by massive galaxy clusters, where the only important lens in the problem is the cluster itself, galaxy–galaxy lensing involves multiple weak deflections. For example, it is common for a distant source galaxy at redshift z_s to be weakly lensed by a more nearby galaxy at redshift z_l1, and for both of these galaxies to then be lensed by another (even more nearby) galaxy at redshift z_l2. Thus, the galaxy with redshift z_l1 serves simultaneously as a lens for the galaxy at z_s and a source for the galaxy at z_l2. In addition, the galaxy at z_s is lensed by two independent foreground galaxies. The importance of such multiple deflections in galaxy–galaxy lensing was first noted by Brainerd et al. (1996; hereafter BBS). Since the work of BBS, galaxy–galaxy lensing has been detected with impressively high statistical significance by a number of different groups. This has enabled constraints to be placed on the nature of the dark matter halos that surround the lens galaxies as well as the bias between mass and light in the universe (see, e.g., Fischer et al. 2000; Guzik & Seljak 2002; Hoekstra et al. 2004, 2005; Sheldon et al. 2004; Heymans et al. 2006; Kleinheinrich et al. 2006; Mandelbaum et al. 2006a, 2006b; Limousin et al. 2007; Parker et al. 2007; Natarajan et al. 2009; Tian et al. 2009).

The purpose of the present investigation is to (1) quantify the frequency of multiple weak lensing deflections in a relatively deep galaxy–galaxy lensing data set, (2) determine the effect of multiple deflections on the net shear for distant source galaxies that have been weakly lensed by foreground galaxies, and (3) demonstrate that galaxy–galaxy lensing alone may contribute a substantial amount to the "cosmic shear" signal on small angular scales. To do this, theoretical shear fields are constructed using a set of observed galaxies with known redshifts and known rest-frame blue luminosities. A simple halo model is used to assign masses to the observed galaxies and Monte Carlo simulations are then used to lens various theoretical source galaxy distributions by the observed galaxies. Theoretical shear fields for full, multiple-deflection calculations are computed; i.e., each source galaxy in the simulation is lensed by all foreground galaxies. In addition, theoretical shear fields for naive, single-deflection calculations (where the closest lens on the sky is assumed to be the only lens) are also computed. The results of the single-deflection calculations are compared to those of the full, multiple-deflection calculations in order to assess the effects of multiple deflections in galaxy–galaxy lensing. Throughout, the weak lensing of an entire source galaxy by a single foreground lens galaxy will be referred to as a "deflection."

The paper is organized as follows. The Monte Carlo simulations of galaxy–galaxy lensing are described in Section 2, the frequency of multiple weak deflections in galaxy–galaxy lensing is computed in Section 3, the effects of multiple weak deflections on the galaxy–galaxy lensing shear are computed in Section 4, the contribution of galaxy–galaxy lensing to cosmic shear is computed in Section 5, and a discussion of cosmic variance in relation to the field size in presented in Section 6. The conclusions are summarized in Section 7.

To investigate the frequency and effects of multiple weak deflections in galaxy–galaxy lensing, Monte Carlo simulations are constructed. The lens galaxies in the Monte Carlo simulations are relatively bright galaxies that are contained within a circle of radius 4', centered on the Hubble Deep Field-North (HDF-N; Williams et al. 1996). This region of sky was the subject of a deep redshift survey (Cohen et al. 1996; Steidel et al. 1996; Lowenthal et al. 1997; Phillips et al. 1997; Cohen et al. 2000) as well as an extensive multicolor photometric investigation (Hogg et al. 2000). As a result, both the redshifts (Cohen et al. 2000, Tables 2A and B) and rest-frame blue luminosities, L_B (Cohen 2001, Table 1), of 590 galaxies in this region of the sky are known. For the simulations, then, the locations of the lenses in redshift space are known very accurately, and the relative strengths of the different lenses can be inferred quite well from their relative luminosities. Therefore, it is possible to make detailed theoretical predictions for the weak galaxy–galaxy lensing shear field that should be expected in this region of sky.

For simplicity, the dark matter halos of the lens galaxies are taken to have a mass density given by

$$\begin{equation} \rho (r) = \frac{\sigma _v^2 s^2}{2\pi G r^2 (r^2 + s^2)}, \end{equation} \tag{ 1 }$$

where σ_v is the velocity dispersion of the halo, G is Newton's constant, and s is a characteristic halo radius (see e.g., BBS; Hudson et al. 1998; Fischer et al. 2000; Hoekstra et al. 2004). It is then convenient to scale the depths of the potential wells of lens galaxies with differing luminosities, L_B, according to a Tully–Fisher or Faber-Jackson type of relation

$$\begin{equation} \frac{\sigma _v}{\sigma _v^\ast } = \left(\frac{L_B}{L_B^\ast } \right)^{1/4}, \end{equation} \tag{ 2 }$$

where σ*_v is the velocity dispersion of the halo of a lens galaxy with rest-frame blue luminosity L*_B. Again for simplicity, it is assumed that the mass-to-light ratio of a galaxy is constant independent of its luminosity. Therefore, the characteristic radii of the halos of galaxies with L_B ≠ L*_B scale with the radii of the halos of L*_B galaxies as

$$\begin{equation} \frac{s}{s^\ast } = \left(\frac{L_B}{L_B^\ast } \right)^{1/2}. \end{equation} \tag{ 3 }$$

Under these assumptions, then, the mass of the halo of an L*_B galaxy is given by

$$\begin{equation} M^\ast = \frac{\pi s^\ast (\sigma _v^\ast)^2}{G} \end{equation} \tag{ 4 }$$

and the deflection of a light ray emanating from a source galaxy is given by

$$\begin{equation} \alpha (X) = \frac{4\pi \sigma _v^2 D_{\rm{ls}}}{D_s X c^2} [ 1 + X - (1 + X^2)^{1/2} ]. \end{equation} \tag{ 5 }$$

Here D_s is the angular diameter distance between the observer and the source, D_ls is the angular diameter distance between the lens and the source, and X is the ratio of the impact parameter of the light ray and the characteristic radius, s, of the lens (i.e., X ≡ R/s; see BBS).

It is worth noting that galaxy–galaxy lensing has, of course, been detected in the HDF-N (e.g., dell'Antonio & Tyson 1996; Hudson et al. 1998); however due to the very small number of galaxies in the HDF-N, the galaxy–galaxy lensing signal can only be detected with relatively low significance. In particular, there are simply too few actual source galaxies to carry out a detailed investigation of the effects of multiple deflections using only the observed sources. It is for this reason that Monte Carlo simulations are adopted here.

The completeness limits of the redshift survey are, unfortunately, different for the HDF-N itself and the surrounding area of the sky, the survey being deeper in the region of the HDF-N. This gives rise to a somewhat different redshift distribution for galaxies with measured redshifts in the center of the field versus galaxies with measured redshifts in the outer region of the field. In order to make an accurate prediction for the theoretical shear field, it is important that the redshift completeness limit for the lenses in the Monte Carlo simulations be uniform across the field. Therefore, a conservative completeness limit of R = 23 is imposed here, and the lenses in the Monte Carlo simulations consist of the 427 galaxies with R ⩽ 23 in Cohen et al. (2000) and Cohen (2001) for which spectroscopic redshifts and rest-frame blue luminosities are known. The median redshift of the lens galaxies is therefore z_med = 0.55.

Two approaches are taken to model the redshifts of the source galaxy population: (1) source galaxies are simply placed in a single plane of redshift z_s and (2) source galaxies are distributed in redshift space according to the observed redshift distribution of faint galaxies. The first approach allows an investigation of the frequency of multiple weak deflections as a function of discrete source redshift. The second approach demonstrates the overall effect that would be expected to occur in a deep galaxy–galaxy lensing data set.

Each Monte Carlo simulation includes 10 million source galaxies that are assigned random positions (R.A. and decl.) within a circle of radius 25, centered on the HDF-N. The sources are contained within a smaller area than the lenses because, as will be shown below, about 10% of the time the most important lens for a given source may be more than an arcminute away. By restricting the sources to a smaller area than the lenses, edge effects (in which sources are not properly lensed by all foreground galaxies) are avoided. In simulations where the sources are restricted to a single plane in redshift space, each source is assigned the identical redshift, z_s. In simulations where the sources are broadly distributed in redshift space, the apparent magnitudes of the sources are taken to be in the range 19 < I < 25, and the number of sources per unit magnitude is chosen to match the observed number counts in the I band (e.g., Smail et al. 1995). These sources are assumed to follow a redshift distribution of the form

$$\begin{equation} P(z|I) = \frac{\beta z^2 \exp [ - (z/z_0)^\beta ]}{\Gamma (3/\beta) z_0^3}, \end{equation} \tag{ 6 }$$

which is in good agreement with the redshift surveys of LeFévre et al. (1996, 2004). Assuming β = 1.5 and extrapolating the results of LeFévre et al. (2004) to a sample of galaxies with 19 < I < 25 yields

$$\begin{equation} z_0 = 0.8 [0.86 + 0.15(I-23.35)] \end{equation} \tag{ 7 }$$

(see, e.g., BBS). The median redshift of the sources in this case is z_med = 0.96.

Throughout, we will consider only the weak lensing regime. That is, we will restrict our analysis to the case that the surface mass density of the lenses is very much less than the critical surface mass density for strong lensing ($\Sigma (\theta) \ll \Sigma _c \equiv \frac{c^2}{4\pi G} \frac{D_s}{D_d D_{\rm{ls}}}$), the deflection angle, α, the modulus of the shear, γ, and the convergence, κ, are all small, and γ ≃ κ. Given that the physical size of each lens is very much smaller than the distances between the observer, lens, and source, we will adopt the standard thin lens approximation (e.g., Blandford & Narayan 1986; Schneider et al. 1992). Further, we will perform all calculations in the framework of the Born approximation, in which integrations are performed along an undeflected light ray. This standard weak lensing formalism is valid even in the limit of multiple weak deflections (e.g., Bartelmann & Schneider 2001). Indeed, investigations into the degree to which the Born approximation may affect predictions of cosmic shear (where the weak lenses consist of all the mass along the line of sight), have shown that corrections due to the Born approximation are 2–3 orders of magnitude smaller than the cosmic shear signal itself (e.g., Cooray & Hu 2001; Hilbert et al. 2009).

For each Monte Carlo simulation, specific values of the velocity dispersion, σ*_v, and characteristic radius, s*, for L*_B galaxies are chosen. Velocity dispersions, σ_v, and characteristic radii, s, are then assigned to each lens galaxy based upon the above scaling relations. The redshifts of the lenses, z_l, are taken to be the observed spectroscopic redshifts, and the positions of the lenses in the field (R.A. and decl.) are taken to be the observed positions on the sky. The Monte Carlo simulation then proceeds by computing the weak lensing shear, $\vec{\gamma }$, that is induced as the light rays emanating from the background sources encounter the foreground lenses. In the case of single-deflection calculations, the lensing of each source is computed solely for the lens which is nearest to the source in projection on the sky. That is, the "closest" lenses are the only lenses that are used in the single deflection calculations, and the resulting shear for each source is simply the shear induced by the closest lens. In the case of full, multiple-deflection calculations, the lensing of each source by all foreground lenses is computed. The resulting shear for each source is then the net shear due to all foreground lenses. This is straightforward to compute in the weak lensing regime since all weak deflections may be considered to be independent (e.g., Bartelmann & Schneider 2001).

Each source galaxy is assigned a random intrinsic position angle and an intrinsic ellipticity that is drawn at random from the observed ellipticity distribution of the HDF-N galaxies. The intrinsic shape parameters of the source galaxies are then given by

$$\begin{equation} \vec{\chi }_{\rm in} = \epsilon _{\rm in} e^{2i \phi _{\rm in}} \end{equation} \tag{ 8 }$$

where _in = (a − b)/(a + b) is the intrinsic ellipticity of the source and ϕ_in is its intrinsic position angle. Since we are dealing only with the weak lensing regime, the final image shape of each source galaxy in the multiple-deflection calculations is given by

$$\begin{equation} \vec{\chi }_f = \vec{\chi }_{\rm in} + \Sigma _{j=1}^{N_{\rm lens}} \vec{\gamma }_j, \end{equation} \tag{ 9 }$$

where $\vec{\gamma }_j$ is the shear induced by foreground lens galaxy, j. In the case of the full, multiple-deflection calculations, the net shear due to all lenses with z_l < z_s is used to obtain $\vec{\gamma }_f$ for each source galaxy. In the case of the single deflection calculations, the sum over all foreground lenses is simply replaced by $\vec{\gamma }_{\rm close}$, the shear induced by the lens that is closest to the source in projection on the sky.

Shown in Figure 1 is a zoomed-in image of one of the simulations. The image is centered on the HDF-N, and the locations of chips 2, 3, and 4 on WFPC-2 are shown by the black lines. Here a fiducial lens halo model with σ*_v = 150 km s⁻¹ and s = 100h⁻¹ kpc has been adopted, and the source galaxies have been distributed in redshift space according to Equation (6) above. A flat Λ-dominated cosmology with H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_m0 = 0.3, and Ω_Λ0 = 0.7 has been also been adopted. The left panel of Figure 1 shows the magnitude of the net shear, and for clarity the orientation of the net shear is not shown. Red peaks in the shear field (i.e., locations of the largest net shear) correspond to the locations of the most important weak galaxy lenses in the field. The right panel of Figure 1 shows the surface mass density of the lens galaxies. Note that some very luminous (and, therefore very massive) galaxies do not show up in the shear field due to the fact that their redshifts place them well beyond the median redshift of the sources. A good example of this is the galaxy located at (−23.20, + 56.70) in Figure 1. This galaxy has coordinates on the sky of R.A. = 12^h36^m5272, decl. = +62°13'5470 (J2000). Its rest-frame blue luminosity is 2.95L*_B and, hence, its halo mass is 6.9 × 10¹²M_☉ (for the fiducial model). The center of this galaxy has a high surface mass density (indicated by red in the right panel of Figure 1). However, since the redshift of this galaxy is z = 1.355, it cannot act as a lens for the majority of the sources. Therefore, it does not contribute substantially to the net shear field. By contrast, the two smaller galaxies that are immediately to the east and west of this intrinsically very bright and massive galaxy do show up quite prominently in the shear field. These galaxies have coordinates on the sky of R.A. = 12^h36^m5407, decl. = +62°13'5420 and R.A. = 12^h36^m5177, decl. = +62°13'5370, corresponding to locations of (−32.65, + 56.20) and (−16.56, + 55.70) in Figure 1. These two galaxies have luminosities of L_east = 0.70L*_B and L_west = 0.87L*_B, and redshifts of z_east = 0.851 and z_west = 0.557. Both of these galaxies are assigned very similar halo masses in the simulation (since their luminosities are very similar), and both are clearly visible in the shear field as red peaks. However, the easternmost of these two galaxies corresponds to a smaller peak in the shear field than the westernmost because the redshift of the easternmost galaxy is only slightly less than the median redshift of the sources, while the redshift of the westernmost galaxy is of order half the median redshift of the sources. For a color image of the HDF-N in which the redshifts of the galaxies are indicated, the reader is encouraged to see Figure 2 of Cohen et al. (2000).

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The probability that a given source galaxy will have been weakly lensed by one or more foreground galaxies is, of course, a strong function of the actual value of the shear, γ, induced by a given weak lensing deflection. That is, it is much more likely for a distant galaxy to be lensed by a foreground galaxy which produces an insignificant weak shear of γ ∼ 10⁻⁶ than, say, a relatively large weak shear of γ ∼ 0.01. Therefore, in order to discuss the total number of weak deflections that a given source galaxy is likely to encounter, a decision has to be made as to what value of γ qualifies as a "significant" value of the shear. A typical value of the shear induced by a single weak galaxy lens is γ ∼ 0.005 (see, e.g., BBS) and this value of γ will be used as a baseline for computing the number of weak lensing deflections that source galaxies have undergone in the Monte Carlo simulations.

To begin this section, the Monte Carlo simulations will be restricted to the fiducial halo lens model from the previous section in which σ*_v = 150 km s⁻¹ and s* = 100h⁻¹ kpc, and sources will be placed in single planes in redshift (i.e., all sources will be assigned identical redshifts). Figures 2–4, then, show the probability, P(N_L), that a given source with redshift z_s will be lensed by N_L foreground galaxies, where each individual deflection gives rise to a shear of γ>0.0025, γ>0.005, and γ>0.01, respectively (i.e., the minimum shear in these figures corresponds to half the baseline value, the baseline value, and twice the baseline value, respectively). Here P(N_D = 2) is the probability that a given source will be lensed by two individual foreground galaxies, each of which lensed the source galaxy at a level that is comparable to or greater than the minimum shear value. Since the minimum values adopted in Figures 2–4 are "substantial" values of the galaxy–galaxy lensing shear, the results shown in these figures are conservative estimates of the frequency of multiple deflections. The line types in Figures 2–4 correspond to different values of the cosmological parameters. In all cases H₀ = 70 km s⁻¹ Mpc⁻¹ is adopted. Solid lines show results for a flat Λ-dominated universe with Ω_m0 = 0.3 and Ω_Λ0 = 0.7, dashed lines show results for an open universe with Ω_m0 = 0.3 and Ω_Λ0 = 0.0, and dotted lines show results for an Einstein–de Sitter universe.

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Figures 2–4 demonstrate two fully expected results. First, the frequency of multiple deflections in galaxy–galaxy lensing is a function of the source redshift: the higher the redshift, the more likely multiple deflections are to occur. Second, the frequency of multiple deflections in galaxy–galaxy lensing depends upon the minimum shear value that is adopted: the lower the value of the minimum shear, the more likely that multiple deflections of at least the minimum value will occur. Figure 2 shows that multiple deflections in which each individual deflection results in a shear of γ>0.0025 are highly probable. The probability ranges from 67% for sources with z_s = 0.75% to 100% for sources with z_s = 2.0. Similarly, Figure 3 shows that multiple deflections in which each individual deflection results in a shear of γ>0.005 are highly probable. In this case, the probability ranges from 23% for sources with z_s = 0.75% to 92% for sources with z_s = 2.0. Multiple deflections in which each individual deflection results in a shear of γ>0.01 are relatively rare for sources with z_s ⩽ 1.0, but the probability of such very large multiple deflections increases to 45% for sources with z_s = 2.0 (Figure 4).

In addition to the frequency of multiple deflections, Figures 2–4 make an important point about the role of the cosmological parameters in galaxy–galaxy lensing. By and large, the number and magnitude of individual weak lensing deflections is unaffected by the choice of the cosmological parameters. That is, galaxy–galaxy lensing primarily provides information about the potentials of the lens galaxies, not the cosmology per se (see also BBS). Therefore, for the remainder of the paper a flat Λ-dominated universe with H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_m0 = 0.3, and Ω_Λ0 = 0.7 will be adopted.

While galaxy–galaxy lensing is largely insensitive to the values of the cosmological parameters, it is quite sensitive to masses of the halos of the lens galaxies. The dependence of galaxy–galaxy lensing on the physical radii of the halos of the lens galaxies is rather weak (see, e.g., BBS; Hoekstra et al. 2004; Kleinheinrich et al. 2006); however, the dependence of galaxy–galaxy lensing on the velocity dispersions of the halos of the lens galaxies is quite strong. The effect of varying the characteristic lens parameters on the frequency of multiple weak deflections is shown in Figures 5–7. In contrast with Figures 2–4, here the source galaxies have been distributed broadly in redshift space (as in Figure 1), with a median source redshift of z_s ∼ 0.96. The characteristic halo parameters for L*_B galaxies are varied as follows: σ*_v = 135 km s⁻¹, σ*_v = 150 km s⁻¹, σ*_v = 165 km s⁻¹; s* = 50h⁻¹ kpc, s* = 100h⁻¹ kpc, s* = 200h⁻¹ kpc. As in Figures 2–4, the shear produced by each individual deflection is restricted to γ>0.0025 (Figure 5), γ>0.005 (Figure 6), and γ>0.01 (Figure 7).

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For the adopted source redshift distribution, then, the probability of multiple weak deflections increases as the characteristic mass of the halos of L*_B lens galaxies increases. That is, the larger is the mass of the lens, the wider is its aperture of influence on the sky. For the adopted source redshift distribution, there is a high probability of multiple deflections in which each individual deflection results in a shear of γ>0.0025. The probability ranges from 54% for the lowest characteristic halo mass (Figure 5, top left) to 82% for the highest characteristic halo mass (Figure 5, bottom right). Similarly, there is a high probability of multiple deflections in which each individual deflection results in a shear of γ>0.005. The probability ranges from 26% for the lowest characteristic halo mass (Figure 6, top left) to 59% for the highest characteristic halo mass (Figure 6, bottom right). From Figure 7, instances of multiple deflections in which each individual deflection results in a very substantial shear of γ>0.01 are relatively rare for low values of σ*_v and s*. However, for large values of σ*_v and s*, the probability can exceed 20%. Note that, at fixed impact parameter, the deflection angle, α, caused by L*_B lenses scales as essentially s*(σ*_v)². So, for a lens with a given velocity dispersion, the deflection angle scales approximately linearly with s*. This naturally leads to larger induced shear for larger values of s*, and a correspondingly larger number of individual deflections that exceed the minimum shear thresholds used in Figures 5–7.

The previous section explored the frequency with which source galaxies undergo multiple weak deflections in a deep galaxy–galaxy lensing data set. This section will explore how the net shear, γ_net, obtained from a full, multiple-deflection calculation compares to the shear obtained solely from the closest lens in projection on the sky (γ_close), as well as how the net shear compares to the shear resulting from the largest individual deflection in the multiple-deflection calculation (γ_max). In particular, the following questions will be addressed:

• 1.  
  Is the closest weak lens (in projection on the sky) necessarily the most important weak lens?
• 2.  
  Is the net shear for a given source galaxy in the multiple-deflection calculation larger or smaller than the shear induced by the closest lens?
• 3.  
  Is the net shear for a given source galaxy in the multiple-deflection calculation larger or smaller than the shear resulting from the largest individual weak deflection?
• 4.  
  What effect does the inclusion of multiple deflections have on the mean tangential shear measured about the lens centers?

Throughout this section, source galaxies in the Monte Carlo simulations will be taken to have the broad redshift distribution used in Figure 1 (e.g., Equation (6) above) and a flat Λ-dominated universe will be used.

The first question of this section is addressed in Figure 8. Here the probability distribution for the distance between the strongest individual weak lens, θ_max, and the closest weak lens, θ_min, is shown. The different panels correspond to different characteristic halo parameters that have been adopted for the halos of L*_B galaxies (and appropriately scaled for all lenses according to Equations (2) and (3) above). Figure 8 shows that, in general, the closest lens in projection on the sky is not the strongest individual weak lens. That is, of order 50% of the time the closest lens is not the "most important" weak lens. Figure 8 also shows the importance of performing the multiple-deflection calculation using sources that are contained within an area that is smaller than the area covered by the lenses, since the angular separation between the closest lens to a given source and the most important lens for that same source can reach scales of more than 2'. In particular, ∼35% of the strongest lenses have angular separations ≳20'' from the sources and ∼10% of the strongest lenses have angular separations ≳60'' from the sources.

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Figures 9 and 10 address the second and third questions of this section. That is, how does the net shear experienced by source galaxies in a full, multiple-deflection calculation compare to the shear due to only the closest lens (Figure 9) and to the shear due to the strongest individual weak lens (Figure 10)? Figure 9 shows that the net shear due to all foreground lenses is generally larger than the shear induced by the closest lens on the sky. The ratio of the shears, γ_net/γ_close, is weakly dependent upon the specifics of the lens halo parameters. The probability that the net shear in the full multiple-deflection calculation exceeds the shear due to the single closest lens ranges from 78% (lens halos with small physical extents, s* = 50h⁻¹ kpc) to 82% (lens halos with large physical extents, s* = 200h⁻¹ kpc). Figure 10 shows that the net shear due to all foreground lenses is also generally larger than the shear induced by the strongest individual weak lens in the full, multiple-deflection calculation. As in Figure 9, the ratio of the shears, γ_net/γ_max, is weakly dependent upon the specifics of the lens halo parameters. The probability that the net shear in the full multiple-deflection calculation exceeds the shear due to the single strongest weak lens ranges from 69% (lens halos with small physical extents, s* = 50h⁻¹ kpc) to 76% (lens halos with large physical extents, s* = 200⁻¹ kpc). Figures 9 and 10, then, show that for any given distant source galaxy, the net shear that its image experiences due to all foreground lenses exceeds the shear due solely to the closest lens, as well as the shear due solely to the strongest individual weak lens.

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It may seem somewhat counter-intuitive that the net shear experienced by the images of distant source galaxies in the multiple-deflection calculations generally exceeds the shear due to a naive single-deflection calculation. That is, at first glance one might expect that multiple weak galaxy–galaxy lensing deflections should, on average, cancel each other. For a given source this would, indeed, be the case if all the foreground lenses were located at precisely the same angular separation from the source, had identical gravitational potentials, and had identical redshifts, z_l. Such an idealized situation is, of course, not the case in the real universe. That is, we cannot think in terms of a single lens plane for the galaxy–galaxy lensing problem and to a certain extent the solution has to be understood numerically. This is due to the fact that there are a wide range of lens-source separations, the lenses have a wide range of gravitational potentials, and the lenses are distributed broadly in redshift. These, in combination, result in increased shear in the multiple-deflection calculation for galaxy–galaxy lensing, much as the non-uniformities in the mass distribution along the line of sight give rise to a net "cosmic shear" (see, e.g., reviews by Bartelmann & Schneider 2001; van Waerbeke & Mellier 2003; Refregier 2003; Munshi et al. 2008). That is, like galaxy–galaxy lensing, cosmic shear is inherently a multiple-deflection problem in which the deflections do not simply cancel. A detailed investigation of how the shear experienced by a given source galaxy is affected as one successively adds in more and more weak galaxy lenses will be presented in P. Howell & T. G. Brainerd (2010, in preparation).

The last question of this section, the effect of multiple deflections on the mean tangential shear about the lens centers, is addressed in Figure 11. In Figures 9 and 10, we have computed quantities (net shear, shear due to the closest weak lens, and shear due to the strongest individual weak lens) that cannot, in practice, be measured in an observational data set. That is, without precise knowledge of the intrinsic shape of a source galaxy, the angular diameter distances of the source and all possible foreground lens galaxies, as well as the details of the gravitational potentials of all foreground lens galaxies, it is not possible to deduce γ_net, γ_close, and γ_max for any one source galaxy. Indeed, galaxy–galaxy lensing yields such a small value of γ_net that it can only be detected via an ensemble average over the images of many source galaxies. Therefore, Figure 11 demonstrates the effect of multiple deflections on the observable galaxy–galaxy lensing signal: the mean tangential shear about the lens centers.

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Shown in Figure 11 is the mean tangential shear, γ_T(θ), measured as a function of lens-source angular separation. Since the sources are restricted to a circle of radius 25, while the lenses are restricted to a circle of radius 4'0, it is not possible to compute γ_T(θ) around all of the lenses. Instead, γ_T(θ) is computed using only those lenses that are within a distance r = (150 − θ)'' of the center of the field. This allows the average to be computed in complete circular annuli, centered on each lens galaxy, and avoids edge effects. Solid squares in Figure 11 show γ_T(θ) for the full multiple-deflection calculations in which each source has been lensed by all foreground lenses. Open circles in Figure 11 show γ_T(θ) for single deflection calculations in which each source is lensed by only the closest lens on the sky. Shown in Figure 12 is the ratio of the mean tangential shears that are plotted in Figure 11. That is, Figure 12 shows the ratio of the mean tangential shear obtained from the full multiple-deflection calculations to that obtained from the single-deflection calculations.

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From Figures 11 and 12, then, it is clear that on very small scales, galaxy–galaxy lensing reduces to a single-deflection problem. That is, on scales θ ∼ 1'', there is relatively little difference between the mean tangential shear obtained from the full, multiple-deflection calculations and the single deflection calculations. On scales of θ ≳ 2'', however, the multiple deflection calculations yield a higher value of the mean tangential shear. The difference between γ_T(θ) from the multiple deflection calculations and γ_T(θ) from the single deflection calculations depends somewhat on the characteristic parameters adopted for the halos of L*_B galaxies. On scales θ ∼ 20'', the multiple deflection calculation for the lowest mass halos (top left panel of Figure 12) yields a mean tangential shear that is a factor of ∼1.4 larger than the mean tangential shear from the single deflection calculation. The multiple deflection calculation for the highest mass halos (bottom right panel of Figure 12) yields a mean tangential shear that is a factor of ∼1.7 larger than the mean tangential shear from the single deflection calculation for θ ∼ 20''. It is also interesting to note that γ_T(θ) becomes roughly constant on scales θ ≳ 10''. This is due to the fact that the mean angular separation between the lens galaxies is 107, which corresponds to a comoving distance of 48h⁻¹ kpc at the median redshift of the lenses. That is, on angular scales comparable to and larger than the mean angular separation of the lenses, the halos of nearby lens galaxies are overlapping one another in projection on the sky. This is the primary reason that galaxy–galaxy lensing is not terribly sensitive to the radii of the halos of the lens galaxies.

It should be kept in mind that here we have only modeled the masses of the halos of individual galaxies. In particular, we have not included the fact that many of the larger, brighter galaxies are probably contained within group environments that have a substantial dark matter component over and above the dark matter halos of the individual galaxies. The circularly averaged tangential shear (as we have computed here) is related to the surface mass density of a circular lens through

$$\begin{equation} \gamma _T \Sigma _c = \overline{\Sigma }(< \theta) - \overline{\Sigma }(\theta) \equiv \Delta \Sigma \end{equation} \tag{ 10 }$$

(e.g., Miralda-Escudé 1991). In the case of completely isolated lens galaxies, ΔΣ above is the surface mass density of the dark matter halo of the lens galaxy. In the case of lenses that reside within groups and clusters, ΔΣ includes the mass due to the dark matter halos of the lens galaxies, as well as the mass of the larger dark matter halo that surrounds the group or cluster. In the case of bright, massive galaxies that reside in groups and clusters, the tangential shear shown in Figure 11 does not properly correlate with all of the mass that one would actually expect to contribute to the net shear in a observational data set. This is simply because we have neglected the additional mass associated with the environments in which those galaxies tend to reside.

Observations of galaxy–galaxy lensing have shown that the dependence of the tangential shear on projected distance from the lens is a function of the stellar mass and luminosity of the lens. In particular, the tangential shear measured around lens galaxies with low stellar masses and low luminosities is approximately constant at very large projected distances (e.g., Mandelbaum et al. 2006b, Figures 1 and 2), consistent with the results in Figure 11 above. However, the tangential shear measured around lens galaxies with high stellar masses and high luminosities declines monotonically at large projected distances (e.g., Mandelbaum et al. 2006b, Figures 1 and 2). This is due to the contribution of the overall mass within the relatively higher density environments in which the most massive, most luminous galaxies tend to reside. That is, in practice observed galaxy–galaxy lensing includes the effects of all individual galaxy lenses, as well as the effects of the mass within the local environment that surrounds the lenses. Here we have simply considered the effects of the individual halos of bright galaxies and have not included environmental (e.g., group/cluster) contributions to the net shear.

The implications of Figures 11 and 12 are straightforward. If one wishes to use observations of γ_T(θ) to constrain the fundamental parameters associated with the halos of the lens galaxies (i.e., σ*_v and s* for the model adopted here), it is vital to use full, multiple-deflection Monte Carlo simulations for the parameter fitting. If simple, single-deflection calculations are used, the inferred halo masses will be systematically too large. That is, in order to reproduce an observed galaxy–galaxy lensing signal on angular scales greater than a few arcseconds using a single-deflection calculation, one would need systematically larger halo masses than are required in the full multiple-deflection calculation.

The galaxy–galaxy lensing contribution to cosmic shear is investigated in this section. Cosmic shear is often equated to weak lensing by the large-scale structure of the universe, but in practice cosmic shear is the result of photons from distant source galaxies being deflected by all mass along the line of sight. The mass along the line of sight includes large galaxy clusters, galaxy groups, and filaments, as well as objects with smaller masses such as individual galaxies. In the case of the galaxy–galaxy lensing, we are considering the specific contribution of the highly nonlinear, large k contribution of the power spectrum of density fluctuations, P(k), to the cosmic shear signal.

The source galaxies in the Monte Carlo simulations are assumed to have orientations that are intrinsically uncorrelated (i.e., each source is assigned an initially random position angle). Galaxy–galaxy lensing will, of course, slightly change both the ellipticity and the orientation of each source. In the presence of a number of high-mass lens galaxies that cause multiple weak deflections over large angular scales, the images of the source galaxies may acquire a net preferred orientation due to galaxy–galaxy lensing. This is the signature of cosmic shear, albeit in this case the shear is caused solely by the lens galaxies, not the entire large-scale structure of the universe.

To investigate the degree to which galaxy-galaxy lensing may contribute to the cosmic shear signal, the image correlation function

$$\begin{equation} C_{\chi \chi } (\theta) = \big\langle \vec{\chi }_{f,i} \cdot \vec{\chi }_{f, j}^\ast \big\rangle_\theta \end{equation} \tag{ 11 }$$

is computed. Here the mean is computed over all galaxy pairs i, j separated by an angle θ ± dθ/2, $\vec{\chi }_{f,i}$ is the final shape parameter of source galaxy i, and $\vec{\chi }_{f,j}^\ast$ is the complex conjugate of the final shape parameter of galaxy j (see, e.g., Blandford et al. 1991). The correlation function measures the extent to which galaxy images "point" in the same direction on the sky. If C_χχ(θ) is positive, the images of the galaxies are aligned with each other. If C_χχ(θ) is zero, the images of the galaxies are randomly oriented. If C_χχ(θ) is negative, the images of the galaxies tend to be oriented perpendicular to each other (i.e., they are anti-aligned).

Shown in Figure 13 is the image correlation function for the source galaxies, where the galaxies have again been broadly distributed in redshift space and a flat Λ-dominated universe has been adopted. Solid squares show the results for the full, multiple-deflection calculations and open circles show the results for the single-deflection calculations in which the sources have been lensed by only the closest lenses. From Figure 13, then, it is clear that for angular separations θ ≳ 5'', the single-deflection calculations yield essentially no contribution of galaxy–galaxy lensing to the cosmic shear. That is, if multiple deflections were not important in galaxy–galaxy lensing, one would expect that on scales greater than 5'', galaxies alone would not contribute to cosmic shear. Hence, cosmic shear on scales greater than 5''would be expected to be largely independent of the gravitational potentials of the halos of field galaxies. However, the full, multiple-deflection calculations in Figure 13 show that galaxy–galaxy lensing can, indeed, induce substantial correlations in the source images on scales greater than 5''. Furthermore, the degree of lensing-induced image alignment is strongly affected by the characteristic parameters that are adopted for the halos of L*_B lenses.

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In addition to the image correlation function, the top hat shear variance

$$\begin{equation} \langle \gamma ^2 \rangle = \frac{2}{\pi \theta ^2}\int _0^\infty \frac{dk}{k} P_\kappa (k) [ J_1 (k\theta) ]^2 \end{equation} \tag{ 12 }$$

is common measure of cosmic shear. Here P_κ is the power spectrum of the projected mass density of the universe, J₁ is a Bessel function of the first kind, and θ is the radius of the circular aperture over which the mean is computed. In an observational data set, the function is computed as

$$\begin{equation} \langle \gamma ^2 \rangle = \frac{1}{N (N-1)} \sum _{i\ne j} \vec{\gamma }_i \cdot \vec{\gamma _j}^\ast \end{equation} \tag{ 13 }$$

for all galaxies within a circular aperture of radius θ on the sky. Solid squares in Figure 14 show the shear top hat variance due to galaxy–galaxy lensing alone, obtained from full, multiple-deflection calculations. Again, sources have been broadly distributed in redshift and a flat Λ-dominated universe is adopted. Also shown for comparison (crosses connected by dotted line) are the measured values of <γ² > obtained by Fu et al. (2008) for galaxies in the CFHT Legacy Survey with a median redshift z_m = 0.83. Although this is somewhat lower than the median redshift of the source galaxies in the Monte Carlo simulations, it is sufficiently similar that it is reasonable to compare the observational and theoretical results directly.

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From Figure 14, the small-scale contribution of galaxy–galaxy lensing to <γ² > depends quite strongly on the parameters adopted for the halos of L*_B galaxies, and scales roughly with the relative masses of the halos. For example, the lowest mass L*_B lenses (top left panel) have masses that are a factor of 2.5 smaller than those of the fiducial halo with σ*_v = 150 km s⁻¹ and s* = 100h⁻¹ kpc (middle panel). Similarly, the highest mass L*_B lenses (bottom right panel) have masses that are a factor of 2.4 larger than those of the fiducial halo. At θ = 1', <γ² > for the lowest mass lenses is a factor of 3 smaller than it is for the fiducial halo, and <γ² > for the highest mass lenses is a factor of 3.5 larger than it is for the fiducial halo.

Comparing the squares in Figure 14 (simulation results) to the crosses (observational results), it is clear that depending upon how deep the potential wells of L*_B galaxies are, galaxy–galaxy lensing alone may contribute a substantial amount to cosmic shear. In the case of the lowest mass L*_B halos, galaxy–galaxy lensing alone would be expected to contribute only ∼5.5% of the value of <γ² > measured by Fu et al. (2008) for an aperture of radius θ = 1'. In the case of the fiducial L*_B halos, the contribution of galaxy–galaxy lensing alone increases to ∼16% for θ = 1', while for the highest mass L*_B halos ∼58% of the signal seen by Fu et al. (2008) at θ = 1' would be due to galaxy–galaxy lensing alone. If one were to extrapolate the Fu et al. (2008) results to scales θ < 1', the results shown in the bottom right panel of Figure 14 suggest that the halos of L*_B galaxies are probably not as large adopted in this particular panel. That is, a simple extrapolation of the Fu et al. (2008) result to θ < 1' leads to an expectation of much less observed cosmic shear than is predicted by galaxy–galaxy lensing by our highest mass lenses.

Because of the relatively small area of the sky that is covered by the sources in the Monte Carlo simulations, it is not possible to compute <γ² > on large angular scales. However, the rms value, <γ² > ^1/2, decreases linearly with θ and it is, therefore, straightforward to extrapolate the results from Figure 14 to an angular scale at which the contribution of galaxy–galaxy lensing to cosmic shear vanishes. From this extrapolation, then, the contribution of galaxy–galaxy lensing to cosmic shear vanishes at θ = 50 for the lowest mass halos, θ = 52 for the fiducial model, and θ = 54 for the highest mass halos. Therefore, although the contribution of galaxy–galaxy lensing to cosmic shear on small angular scales is very sensitive to the details of the gravitational potentials of galaxies, cosmic shear on scales θ ≳ 5' should not be affected by galaxy–galaxy lensing to any significant degree.

The shear field in the Monte Carlo simulations comes from a set of lenses that are contained within an area of 50 arcmin² on the sky and, therefore, one might be concerned that the results shown above could be compromised by cosmic variance. Here some of the results above are recomputed using subdivisions of the data in order to explore potential small field effects. To do this, we use the fiducial model in which the halos of L*_B galaxies have velocity dispersions of σ*_v = 150 km s⁻¹ and characteristic radii of s* = 100h⁻¹ kpc, we take the sources to be broadly distributed in redshift as above, and we use a flat Λ-dominated cosmology.

Shown in Figure 15 is a comparison of the frequency of multiple deflections that give rise to individual shear values of γ>0.005 (top panels), the mean tangential shear about the lens centers, γ_T(θ) (middle panels), and the image correlation function, C_χχ(θ) (bottom panels) using different subdivisions of the data. The left-hand panels show a comparison of results obtained from data in the northern half of the field and results obtained from data in the southern half of the filed. The right-hand panels show results obtained from data in the eastern half of the field and results obtained from data in the western half of the field. While there are some differences that result when the size of the field is reduced by a factor of 2, the differences are small and suggest that the results obtained from the full field should not suffer dramatically from effects of cosmic variance.

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The frequency and effects of multiple weak lensing deflections in galaxy–galaxy lensing have been investigated using Monte Carlo simulations. The lenses in the simulations are modeled using observed galaxies with magnitudes R ⩽ 23, contained within a circle of radius of 4', centered on the HDF-N. The lenses have known redshifts and known rest-frame B-band luminosities. By adopting a simple halo mass model it is possible to determine the relative strengths of each of the lenses using scaling relations.

The Monte Carlo simulations reveal a number of expected results: (1) the frequency of multiple deflections depends upon the minimum value of the shear (i.e., the lower is the minimum value, the more likely it is that multiple deflections will be experienced by a given source), (2) the frequency of multiple deflections depends upon the source redshift (i.e., the higher is the source redshift, the more likely it is that it will experience multiple deflections) and (3) the higher are the masses of the lenses, the more likely it is that multiple deflections will occur. For a deep galaxy–galaxy lensing data set in which the sources have a median redshift z_s ∼ 1 and the lenses have a median redshift z_l ∼ 0.6, the probability that a given source galaxy will have experienced more than one weak lens that induces a "typical" shear of γ = 0.005 ranges from 26% to 59%, depending upon the masses adopted for the lenses.

The Monte Carlo simulations also reveal a number of results that may seem counter-intuitive at first glance: (1) of order 50% of the time, the closest lens in projection on the sky is not the most important weak lens for a given source, (2) for a given source, the net shear due to all foreground lenses generally exceeds the shear due to the strongest individual weak lens, and (3) multiple deflections give rise to a larger tangential shear around the lens galaxies than a simple, single-deflection calculation in which the closest lens is assumed to be the only lens. This emphasizes the importance of using full, multiple-deflection calculations when using observations of galaxy–galaxy lensing to constrain the parameters of the dark matter halos of the lens galaxies. If multiple deflections are not incorporated into the calculation, this will result in halo masses that are systematically too large.

Lastly, the Monte Carlo simulations reveal that galaxy–galaxy lensing alone can give rise to a cosmic shear signal on small angular scales. This is unsurprising because cosmic shear occurs when photons from distant galaxies are deflected by all mass along the line of sight. In the case of galaxy–galaxy lensing, it is the very large k end of the power spectrum of density fluctuations that contributes to the cosmic shear by inducing correlated image shapes for the distant galaxies. On scales θ ∼ 1', the degree to which galaxy–galaxy lensing contributes to cosmic shear is quite sensitive to the masses of the lens galaxies. Changing the mass of the halo of a fiducial L*_B galaxy by a factor of ∼2.5 changes the contribution to the top hat shear variance, <γ² >, by a factor of ∼3. Comparing the theoretical values of <γ² > at θ = 1' to the value observed by Fu et al. (2008) for sources with a similar redshift distribution, galaxy–galaxy lensing alone could account for as little as ∼5% or as much as ∼58% of the observed value, depending upon the halo mass for L*_B galaxies.

While the small-scale contribution of galaxy–galaxy lensing to cosmic shear is quite sensitive to the masses of the lenses, the scale at which galaxy–galaxy lensing becomes unimportant to cosmic shear is relatively independent of the lens masses. If the results for the galaxy–galaxy lensing contribution to cosmic shear are extrapolated to large scales, the contribution of galaxy–galaxy lensing to cosmic shear should vanish for scales θ ≳ 5', largely independent of the lens masses.

I am pleased to thank Judy Cohen and her collaborators, especially Mike Pahre and David Hogg, for all of their work on the Caltech Faint Galaxy Redshift Survey. Without their efforts, the work presented in this paper would not have been possible. I am also deeply grateful to Judy Cohen for providing the B-band luminosities of the lens galaxies to me in electronic form. In addition, I am very pleased to thank the referee for constructive comments and helpful suggestions that improved the manuscript. This work was supported by the National Science Foundation under NSF contracts AST-0406844 and AST-0708468.