Issue 
A&A
Volume 624, April 2019



Article Number  A54  
Number of page(s)  10  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201424881  
Published online  09 April 2019 
Generalized multiplane gravitational lensing: time delays, recursive lens equation, and the masssheet transformation
ArgelanderInstitut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
email: peter@astro.unibonn.de
Received:
29
August
2014
Accepted:
25
February
2019
We consider several aspects of the generalized multiplane gravitational lens theory, in which light rays from a distant source are affected by several main deflectors, and in addition by the tidal gravitational field of the largescale matter distribution in the Universe when propagating between the main deflectors. Specifically, we derive a simple expression for the timedelay function in this case, making use of the general formalism for treating light propagation in inhomogeneous spacetimes which leads to the characterization of distance matrices between main lens planes. Applying Fermat’s principle, an alternative form of the corresponding lens equation is derived, which connects the impact vectors in three consecutive main lens planes, and we show that this form of the lens equation is equivalent to the more standard one. For this, some general relations for cosmological distance matrices are derived. The generalized multiplane lens situation admits a generalized masssheet transformation, which corresponds to uniform isotropic scaling in each lens plane, a corresponding scaling of the deflection angle, and the addition of a tidal matrix (mass sheet plus external shear) to each main lens. The scaling factor in the lens planes exhibits a curious alternating behavior for odd and even numbered planes. We show that the time delay for sources in all lens planes scale with the same factor under this generalized masssheet transformation, thus precluding the use of timedelay ratios to break the masssheet transformation.
Key words: cosmological parameters / gravitational lensing: strong
© ESO 2019
1. Introduction
In strong gravitational lensing systems, in which a galaxy or a galaxy cluster causes multiple images or strong image distortions of background sources, one often neglects the inhomogeneities of the gravitational field between the observer and the lens, and between the lens and the source (see, e.g., Kochanek 2006; Treu 2010; Bartelmann 2010; Kneib & Natarajan 2011, for reviews on strong lensing systems). This usually provides a very good approximation, since the lensing strength of the main lens over the region where strong lensing effects occur is much larger than the typical distortion effects of matter along the lineofsight. The latter is comparable to the typical strength of cosmic shear effects (see, e.g., Bartelmann & Schneider 2001; Schneider 2006; Munshi et al. 2008; Hoekstra 2013), and amounts to about 1 or 2% of the distortion in the stronglens region.
Whereas the propagation effects are thus small, the interest of these weak distortions has been renewed, for at least two different reasons. The first is that strong lens systems may be biased toward showing relatively strong lineofsight structures, thereby increasing their lensing probability. This effect is most likely affecting the lensing cross sections for the formation of giant arcs in clusters (see Bartelmann et al. 1998 for stating the “arc statistic problem”, and Meneghetti et al. 2013 for a recent update of the issue). Cosmological simulations (Puchwein & Hilbert 2009, and references therein) indicate that lineofsight structure can indeed substantially modify the lensing efficiency of clusters. More recently, Bayliss et al. (2014) found a significant overdensity of galaxy groups along the lineofsight toward stronglensing clusters, observationally supporting the presence of this bias.
The second reason for the renewed interest in intervening distortions of stronglens systems is their use for a precise determination of parameters, most noticible the Hubble constant from measured time delays (Refsdal 1964) in multiply imaged quasistellar objects (see, e.g., Kochanek 2003, 2006; Treu 2010; Suyu et al. 2010, 2013). The quality of modern imaging data and the accuracy of time delay estimates allows one to derive estimates of the Hubble constant with a formal error of ∼6% (Suyu et al. 2013). At this level of precision, lineofsight effects may become highly relevant in these stronglensing systems (e.g., Wong et al. 2011; Collett et al. 2013; Greene et al. 2013, and references therein).
With additional deflectors along the lineofsight, the lens equation, which relates the true source position to the observed positions of images, needs to be generalized to include deflection at more than one distance from the observer. Blandford & Narayan (1986) established the theory of multiplane gravitational lensing (see also Chap. 9 of Schneider et al. 1992), where the mapping between images and their source is affected by the action of several deflectors at different redshifts. This multiplane lensing is employed in raytracing simulations – see Refsdal (1970) for the earliest raytracing simulations, Jain et al. (2000), Hilbert et al. (2009), and references therein – where the threedimensional mass distribution between observer and source is partitioned into separate slices, and the mass distribution in each slice is treated as a gravitational lens plane.
The full theory of multiple deflection lensing is required if there are two or more strong deflectors along the same lineofsight toward a background source. For galaxyscale lensing, such systems are rare, since such an alignment is not very probable. However, in the sample of several hundred galaxyscale strong lens systems currently known, there are examples of multiple lenses at two different redshifts (Chae et al. 2001; Gavazzi et al. 2008). These systems may be of particular interest, since they may be used in principle to determine cosmological distance ratios (Collett & Auger 2014), and therefore to constrain the cosmic expansion history, although in practice the masssheet transformation renders this a difficult task (Schneider 2014).
Far more common are situations where there is a single strong lens in the lineofsight to a distant source, and several other deflectors at different redshifts situated sufficiently far away from the stronglensing region of the main deflector such that their deflection angle can be linearized across this stronglensing region. Kovner (1987) considered this case of one main lens, combined with linearized deflections at different redshifts. The effects of the linear deflectors can be summarized into a set of matrices which describe the mapping between angles at the vertices of light cones to the separation vectors along the light cone at the redshifts of lens and source. Schneider (1997) reconsidered this situation, using the general formalism of light propagation in an inhomogeneous universe (Seitz et al. 1994; hereafter SSE), and related this generalized gravitational lens equation to the one where the deflection occurs in only a single plane, with a tidal deflection matrix added (see also Keeton et al. 1997). McCully et al. (2014) further generalized this theory to consider several main lenses, together with linearized deflections between the main lens planes. In particular, McCully et al. (2014) emphasized the advantages of this hybrid framework for the modeling of strong lens systems with multiple main deflectors along the lineofsight (see also McCully et al. 2017).
One of the prime motivations for the work of McCully et al. (2014) was the derivation of the timedelay function for such generalized lensing systems, as this is required for relating the time delay to the scalelength of the Universe, i.e., the Hubble radius. Using the Millennium Simulation, Jaroszynski & KostrzewaRutkowska (2014) investigated the impact of the lineofsight matter distribution on strong lensing properties, including the time delay; they concluded that the intermediate mass distribution leads to a spread of ∼6% in the product of Hubble constant and time delay, for strong lensing systems with source redshifts z_{s} ∼ 2. Arguably, the largest obstacle for the timedelay method to obtain accurate estimates for the Hubble constant is the degeneracy of the mass model due to the masssheet transformation (MST; Falco et al. 1985; see Schneider & Sluse 2013 for a recent discussion) or the more general source position transformation (Schneider & Sluse 2014). Recently, Schneider (2014) has shown that an MST also exists for the case of lenses at two different distances from us. In particular, this MST leads to a scaling of all time delays (from sources on both source planes) by the same factor, thus precluding that the degeneracy due to the MST can be broken by timedelay ratios.
In this paper, these results are further generalized to the case of arbitrarily many main lens planes, together with linear deflections between the main lens (and source) planes. After a short summary of the propagation equations in an inhomogeneous universe, as they apply to the case under consideration here, we derive the timedelay function for the case of several main lens planes. Whereas this has been done before by McCully et al. (2014), our expression for the light travel time function is expressed in a form which allows us to apply Fermat’s theorem in gravitational lensing. Thus, the lens equation can be obtained from requiring that the light travel time function is stationary with respect to all impact vectors in the main lens planes. With this procedure, we derive an alternative form of the lens equation, which involves the impact vectors of three consecutive lens planes and the deflection angle in the middle one of them. We show that this iterative form of the lens equation is equivalent to the more standard form; in order to do so, we first obtain a very general relation between distance matrices.
We then turn to the MST in this case, and find a curious property of its behavior: The uniform, isotropic scaling factor which characterizes the MST alternates between a free parameter λ and unity from one lens/source plane to the next. The corresponding modification of the deflection angle in the main lens planes corresponds to a scaling of the deflection, plus the addition of a tidal deflection matrix (which in the absence of linear deflections between lens planes reduces to the addition of a uniform mass sheet). Finally we show that all time delays, for sources located on any plane, scale by the same factor under an MST, precluding the possibility of breaking the degeneracy from the MST by measured timedelay ratios.
2. Generalized multiplane lens equation
2.1. Optical tidal equation
The propagation of light rays follows from the geodesic equation, specialized to a perturbed Robertson–Walker metric (see Schneider et al. 1992; Seitz et al. 1994; Bartelmann 2010, and references therein). In particular, for infinitesimally small light bundles, the separation vector ξ of a ray from the reference (or “central”) ray of the bundle is given by the optical tidal equation
where 𝒯 is the optical tidal matrix, evaluated at the affine parameter λ along the central ray. Here and throughout this paper, a prime denotes differentiation with respect to the affine parameter λ, which is related to the redshift z, the cosmic scale factor a = 1/(1 + z), or cosmic time t through
where we have assumed that λ increases with redshift, i.e., decreases with cosmic time,
is the Hubble function, H_{0} is the Hubble constant, and Ω_{m} and Ω_{Λ} are the cosmic density parameters in matter and vacuum energy.
We consider 𝒯 to consist of three separate components, 𝒯 = 𝒯_{bg} + 𝒯_{sm} + 𝒯_{cl}. The first is the optical tidal matrix of the homogeneous background universe, and is given by (see SSE)
where ℐ is the twodimensional unit matrix. If we consider a light bundle with vertex at redshift z = 0 and affine parameter λ = 0, which is only subject to 𝒯_{bg}, then the solution of Eq. (1) is ξ(λ)=D(λ)θ, where θ is the angle that the light ray under consideration encloses with the fiducial ray at the vertex, and D(λ) is the solution of the differential equation
with initial condition D(0)=0 and D′=1. Here, D is the angulardiameter distance of the homogeneous universe, as a function of affine parameter (or redshift).
We consider two different kinds of inhomogeneities here. The first of them is related to smallscale density inhomogeneities, such as galaxies and their dark matter halos. For those, the optical tidal matrix 𝒯_{cl} is a strong function of position, and there is no longer a linear relation between enclosed angle θ and separation ξ of a light ray, for finite θ. In stronglensing applications, we are typically interested only in that region of these smallscale inhomogeneities where multiple images can be formed, which corresponds to a few tens of kiloparsecs for galaxy lenses. These smallscale inhomogeneities will be considered explicitly as “main lenses” in the following. The second kind of inhomogeneities is due to the largescale mass distribution of the universe. We assume that the corresponding gravitational field is sufficiently smooth, so that the tidal effects can be considered approximately constant across the region where stronglensing effects occur. Hence we assume that over such a region, 𝒯_{sm} can be considered to depend only on λ, not on the actual position of the light ray. According to SSE, this contribution of the optical tidal matrix is
where ϕ is the Newtonian potential sourced by the density inhomogeneity, i.e., satisfying the (threedimensional) Poisson equation , where is the mean matter density in the Universe, and we assumed that the light ray propagates in the ξ_{3}direction.
2.2. Generalized multiplane lens equation
We consider a direction in the sky with has a set of main lenses along the lineofsight to a distant source, located at redshifts z_{i} or affine parameters λ_{i}. As shown in SSE (see also Schneider 1997), the separation vector ξ(λ) then becomes
where θ is the angle the light encloses with the fiducial ray at the observer. The distance matrix 𝒟(λ) solves the optical tidal equation
with initial conditions 𝒟(0) = 0 and 𝒟′(0) = ℐ, and the distance matrices 𝒟_{i}(λ) solve the same differential equation, but with initial conditions 𝒟_{i}(λ_{i}) = 0 and (d𝒟_{i}/dλ)(λ_{i}) = (1+z_{i})ℐ. They are the distance matrices which apply for light rays having their vertex at λ_{i}. In Eq. (7), H(λ − λ_{i}) is the Heaviside step function. The deflection angle is given in terms of the surface mass density Σ_{i}(ξ_{i}) as
and denotes the deflection angle of the fiducial ray in the ith lens plane. If we now perform the translation , with
then satisfies Eq. (7) with the set to zero. In the following, we always assume this (unobservable) translation, and drop the tilde on henceforth. For the impact vectors ξ_{j} in the jth plane, we then obtain
2.3. Calculation of the distance matrices
The distance matrices 𝒟i(λ) depend on the largescale matter distribution around the lineofsight, which is dominated by dark matter and thus difficult to determine observationally. One possibility to estimate 𝒯_{sm} from observations is to assume that galaxies provide a good tracer of the total matter distribution on large scales. From an observed distribution of galaxies (or a particular kind of galaxies, like luminous red galaxies) around the lineofsight, an estimate of the tidal field can be obtained; this is the strategy proposed in Collett et al. (2013) and Smith et al. (2014).
The propagation of light through the largescale structure is the subject of cosmological weak lensing, or cosmic shear (see, e.g., Schneider 2006). In cosmic shear, one usually describes the propagation matrices in comoving coordinates. In order to connect the formulation given here, where the 𝒟_{i} relate angles to proper transverse separation vectors (which is appropriate, as the matter distribution of the main lenses – galaxies or clusters – are most conveniently described in physical scales), to that used in weak lensing, we show in the appendix that
where χ is the comoving distance, f_{k}(χ) is the comoving angular diameter distance, D_{i}(χ)=a(χ)f_{k}(χ − χ_{i}), which satisfies the differential equation
with initial conditions f_{k}(0) = 0 and df_{k}(0)/dχ = 1. Furthermore, is the spatial curvature of the universe, and H(ϕ) is the twodimensional Hessian of the gravitational potential ϕ, evaluated in comoving transverse coordinates. Provided the perturbations are small, so that 𝒟_{i} deviates only slightly from D_{i}ℐ, we can replace 𝒟_{i}(χ′) by D_{i}(χ′) in the integrand,
In weak lensing, this approximation is called “the neglect of lenslens coupling”, often also termed as “Bornapproximation”. This approximation is very accurate and certainly sufficient for the purposes discussed in the current context^{1}. Thus, the deviation of 𝒟_{i} from D_{i}ℐ is given as a lineofsight integral over the tidal force field, with a distancedependent weighting. We also note that in the approximation (13), the distance matrices 𝒟_{i} are symmetric, which is generally not the case if the exact expression (11) is used.
For statistical studies, instead of trying to obtain the tidal field along the lineofsight toward the sources from observations of the galaxy distribution in those directions, one can also derive the probability distribution for the distance matrices from cosmological simulations, as has been done in Suyu et al. (2013).
3. Timedelay function and the iterative lens equation
In this section, we first derive the light travel time function (LTTF) corresponding to the generalized multiplane lens equation considered in the previous section. An alternative form of the lens equation is then derived from Fermat’s principle, which in the current context states that the lens equation is equivalent to setting the gradient of the LTTF with respect to all impact vectors equal to zero (i.e., real light rays correspond to stationary points of the LTTF). Then we show that this iterative form of the lens equation is equivalent to Eq. (10).
3.1. Timedelay function
We first consider a single lens plane at z_{1} with a source at z_{2}, in which case the lens equation reads
where we set . We define the LTTF τ(ξ_{1}, ξ_{2}) to be the excess light travel time from a source at ξ_{2} to the observer caused by the deflection in the lens plane at ξ_{1}. This excess travel time has two components, a geometrical one (a bent ray is longer than an unbent one), and a potential one caused by the retardation of photons in the gravitational potential of the deflector.
In standard lens theory, with unperturbed angulardiameter distances D_{ij}, the potential part of the time delay function takes the form cτ_{pot} = −(1 + z_{1})(D_{1}D_{2}/D_{12})ψ(θ), where ψ(θ) is the deflection potential, which satisfies . Now we define the potential such that it satisfies . This potential is a multiple of ψ, i.e., , up to an irrelevant additive constant. To find k, we consider
yielding
Hence, the potential part of the LTTF takes the form . As expected, it does not depend on the cosmological distances, since it is caused solely by the local effect of propagating through a gravitational field, and the corresponding time interval is then redshifted by a factor (1 + z_{1}) to the observer. Accordingly, also in the case of perturbed light propagation between the lens planes, the potential part of the time delay must have the same form, since it is unaffected by propagation effects.
We find an explicit expression for τ(ξ_{1}, ξ_{2}) by requiring that ∇_{ξ1}τ(ξ_{1}, ξ_{2})=0 is equivalent to the lens Eq. (14). This fixes τ up to a multiplicative constant and terms which depend solely on ξ_{2}. The multiplicative constant is fixed by the explicit expression given above for the potential part of τ, and the additive constant (which is irrelevant for time delay measurements) is fixed by requiring that the geometrical part of τ should vanish if the light ray is undeflected by the main lens. This then yields
where we defined
As shown in Schneider (1997) – see also Kovner (1987) – the matrix C_{ij} is symmetric. Indeed, we see that
is equivalent to the lens Eq. (14), as is easily verified by multiplying the foregoing expression by 𝒟_{12} from the left. We also note that τ = 0 if and if the ray is unbent, i.e., if , as we required for τ. McCully et al. (2014) obtained a somewhat different form for τ, which they showed to be equivalent to the expression given here. However, it must be pointed out that this equivalence applies only to physical light rays, i.e., those which satisfy the gravitational lensing Eq. (14). For those rays, we could write the light travel time as
where the lens Eq. (14) was used to eliminate ξ_{2}. However, Eq. (16) is more general, as it yields the light travel time for all kinematically possible rays, not only for those for which the bend by the main lenses equals the actual deflection angle as calculated as the gradient of . This more general form of the τ is needed if the lens equation is to be derived from Fermat’s principle.
In case of several main lens planes, the LTTF is obtained by considering the replacement of the actual light ray by successively straighter rays, i.e., by removing the bends of the ray and the gravitational potentials they traverse (see Sect. 9.2 of Schneider et al. 1992). Removing the bend and deflection potential in the first plane leads to the contribution (16) of the LTTF. Subsequent removal of the bend and potential in the second lens plane yields a similar contribution, with the indices (1,2) replaced by (2,3). Iterating this consideration, we obtain for the general case
as the sum over terms of the form (16) for the individual planes.
Any ray connecting the source at ξ_{N + 1} and the observer is fully characterized by the impact vectors ξ_{i}, 1 ≤ i ≤ N in the lens planes, since between the planes, it follows the propagation Eq. (1) whose solution is uniquely determined by the two impact vectors at consecutive planes. Therefore, the actual light rays are singled out as those for which the LTTF is stationary, with respect to variations of the impact vectors in the N lens planes. Thus, the lens equation is obtained by setting the derivative of τ with respect to the ξ_{j} equal to zero. For each j ≥ 2, two terms of the above sum contribute, namely the terms i = j and i = j − 1. We obtain
or
This equation relates the position vectors in three consecutive planes to the deflection angle in the middle plane, quite in contrast to the lens Eq. (10) which contains all impact vectors ξ_{i} for a given ξ_{j}, 1 ≤ i ≤ j − 1. Hence, this new lens equation is more “local” than the original one.
In the following we explicitly show that these two forms of the lens equation are equivalent. For this, we first need to derive a general relation between distance matrices.
3.2. A relation between distance matrices
Consider the pairs of light rays sketched in Fig. 1, where the first has a vertex at λ_{q} and encloses an angle θ with the fiducial ray. At the affine parameter λ_{s}, its separation vector from the fiducial ray is ξ_{s}. The second light ray has its vertex at λ_{r} and intersects the first ray at λ_{s}; this then specifies its direction ϑ relative to the fiducial ray. At the intersection point, the two rays enclose an angle φ. From the geometry of the figure, we find
Fig. 1. Sketch of two light rays through four consecutive planes, with 0 ≤ λ_{q} < λ_{r} < λ_{s} < λ_{t}. The rays are not deflected in the lens planes. The first ray has its vertex at λ_{q} and encloses an angle θ with the fiducial ray; the second ray with vertex at λ_{r} encloses an angle ϑ with the fiducial ray. Both rays intersect at λ_{s}. The geometry of this figure yields the relation (25) between distance matrices. 
We use the latter equation to derive a relation between the 𝒟’s, by expressing all vectors in terms of ξ_{s}. Using the first of (22), we get . Furthermore, the figure shows that Δξ_{t} = 𝒟_{st}φ. On the other hand, ξ_{r} = −𝒟_{s}(λ_{r})φ.
We now have to relate 𝒟_{s}(λ_{r}), the backward extension of the solution 𝒟_{s}(λ) of Eq. (8), to 𝒟_{rs}. For that, we consider two solutions of Eq. (8), 𝒟_{r}(λ) and 𝒟_{s}(λ), with their appropriate initial conditions at λ_{r} and λ_{s}, respectively, and define the matrix
which is the Wronskian of the differential Eq. (8); here, the superscript “t” denotes the transpose of a matrix. The derivative of W vanishes, due to Eq. (8); hence, W is a constant. Evaluating Eq. (23) at λ = λ_{r} and making use of the initial conditions of 𝒟_{r}(λ) yields 𝖶 = (1 + z_{r}) 𝒟_{s}(λr). Similarly, at λ = λ_{s} we find . Thus, we obtain
which is Etherington’s theorem in matrix form (Etherington 1933). With this relation, we then find that
Using Eq. (22) and collecting terms,
follows. Since this relation is valid for all ξ_{s}, a general relation between distance matrices is obtained:
where we multiplied the resulting equation by 𝒟_{qs} from the right^{2}. This result generalizes the corresponding relation in Schneider (2016) derived for the case that the distance matrices 𝒟 reduce to scalars.
Indeed, a relation of this kind is expected to hold: Consider λ_{t} ≡ λ as a variable. The two matrixvalued functions 𝒟_{q}(λ) and 𝒟_{r}(λ) are linearly independent solutions of the transport Eq. (8), provided λ_{r} ≠ λ_{q}. Therefore, the solution 𝒟_{s}(λ) can be written as a linear combination of the other two. This combination should be of the form
which satisfies one of the initial conditions, 𝒟_{s}(λ_{s}) = 0. The matrix X is determined from the second initial condition; our result (25) shows that
3.3. Equivalence of Eqs. (10) and (21)
We shall now show that the two forms (10) and (21) of the lens equation are equivalent. As a first step, we rewrite Eq. (21) in a form that admits a simple geometrical interpretation. Specializing Eq. (25) to q = 0, r = j − 1, s = j, t = j + 1 yields
We next consider the prefactor of ξ_{j} in Eq. (21). Using , which is obtained from the definition (17) of C, we find that this prefactor becomes
where in the last step we made use of Eq. (28) and the fact that the inversion and transposition operations on a matrix commute. Thus, we can rewrite Eq. (21) in the form
Fig. 2. Propagation of a light ray (thick bent line) between three consecutive planes. The vertical line is the optical axis, with respect to which the separation vectors ξ are measured. The geometry of this figure yields the lens Eq. (30) – see text. 
We note that this generalizes equation (4.47) of SSE to the case of general distance matrices between main lens planes. This form of the lens equation can be immediately interpreted geometrically. For this, we consider Fig. 2, from which we read off
Furthermore, η = 𝒟_{j,j+1}φ; on the other hand, ξ_{j−1} = −𝒟_{j,j−1}φ. Eliminating φ from these two relations and making use of Eq. (24), we find
Together, these two equations reproduce Eq. (30). In this form, the equation not only is confined to three consecutive lens planes, but all distance matrices occurring here are those between these three planes.
In order to show the equivalence of Eqs. (10) and (21), it is useful to rewrite the prefactor of ξ_{j − 1} in Eq. (30) in a different form. Making use again of Eq. (28), we obtain
We prove the equivalence by induction; for j = 1, this equivalence is seen from Eq. (18). Hence we assume that it is true for all planes up to j. Then, taking the difference between Eq. (10) for ξ_{j + 1} and Eq. (31),
we need to show that Δ = 0. We first replace ξ_{j − 1} and ξ_{j} by their expressions from Eq. (10), which holds because of the induction assumption,
From this equation, one sees immediately that the terms ∝θ cancel each other. Second, the two terms add up to zero. Third, also the sum of the two terms is zero. Thus, what remains to be shown is that the prefactor of the terms ,
for i ≤ j − 2 vanish. For this, we consider again Eq. (25), setting r = j − 1, s = j, t = j + 1, once with q = 0, and once with q = i. This then yields
After multiplying by 𝒟_{i,j−1}, we see that the final equality shows that K_{i} = 0, which proves that Δ = 0, and thus the equivalence of the two forms of the lens equation. The equation K_{i} = 0 itself provides an interesting relation between distance matrices.
4. Masssheet transformation
In standard gravitational lensing, with a single deflector between the source and observer, there is a transformation of the mass distribution of the lens which keeps most observables invariant, the masssheet transformation (MST, see Falco et al. 1985). Since this transformation is accompanied by a uniform isotropic scaling in the source plane, all magnifications are scaled by the same factor, so that magnification (and thus observable flux) ratios are unchanged. The MST changes the product of time delay and Hubble constant, though, and the corresponding degeneracy can thus be broken by measuring the time delay in lens systems, assuming the Hubble constant to be known from other cosmological observations (see Schneider & Sluse 2013, and references therein). S14 has recently shown that a MST also exists in the case of two lens planes and two source planes. In this section, we show that also for perturbed gravitational lens systems, as considered in this paper, such a MST does exist.
4.1. Single main lens plane
We start with the case of a single lens plane, using the lens Eq. (14), and modify the deflection angle to the new form
where λ is a real number, and G_{1} is a matrix^{3}. Throughout this section, a prime denotes a masssheet transformed quantity. Thus, the modified deflection angle is a scaled version of the original one, plus a term linear in the impact vector. If G_{1} is symmetric, this linear term corresponds to a tidal matrix, i.e., adding a uniform mass sheet to the scaled lens mass distribution, plus an external shear. The modified lens equation then becomes
As for the orignal MST, we require that the modified impact vector is related to the original one by a uniform, isotropic scaling, , where ν_{2} is the scaling factor. Thus we require
In order to have the terms equal on both sides of Eq. (38), we need to set ν_{2} = λ, as is also the case for the MST in standard lensing – the scaling of the source plane (here plane number 2) is the same as that of the deflection angle. The remaining terms are all ∝θ, and setting them equal on both side leads to 𝒟_{2} − 𝒟_{12}𝖦_{1}𝒟_{1} = λ𝒟_{2}, or
Since C_{12} is symmetric (see Schneider 1997), G_{1} is indeed a tidal matrix. This singlemain plane MST was also derived by McCully et al. (2014). Thus, in a generalized gravitational lens situation, the MST requires a shear in addition to a uniform mass sheet^{4}.
4.2. Two main lens planes
We now consider a second lens plane at z_{2}, with the source plane being located at z_{3}. The modified lens equation then reads
and we require the modified deflection angle to be chosen such that the 3plane is just uniformly scaled relative to the original one, i.e., . This condition then yields
In order to account for the term on the r.h.s. of Eq. (41), the modified deflection has to be of the form^{5}
This choice then yields equal terms on both sides. Equating the terms leads to the condition −λ𝒟_{13} + λ𝒟_{23}𝖦_{2}𝒟_{12} = −ν_{3}𝒟_{13}, or
Using the same arguments as in the Appendix of Schneider (1997), it is straightforward to show that any combination of distance matrices of the form
for 0 ≤ z_{r} < z_{s} < z_{t}. Hence, G_{2} is symmetric, and thus corresponds to a tidal matrix. Equating the terms ∝θ in Eq. (41) then leads to
which has the unique solution ν_{3} = 1. Thus, as is the case for the standard multiplane lens discussed in Schneider (2014), the MST does lead to no scaling in the plane j = 3. Therefore,
The implied scaling of the mass distribution in the plane i = 2 that follows from Eq. (42) is discussed in Schneider (2014); in short, the surface mass density distribution giving rise to needs to be scaled in amplitude and scalelength to yield a deflection .
4.3. Arbitrary number of planes
Here we generalize the MST to an arbitrary number of source/lens planes. It turns out that the lens equation in the form (31) is better suited for that purpose. We write it in the form
where B_{j} is the term in parenthesis in Eq. (31). We now assume a scaling in every plane, and set the scaled deflection angles to be
Note that Eqs. (36) and (42) are special cases of the relation (48) for j = 1, 2, respectively, and ν_{1} = 1, ν_{2} = λ, ν_{3} = 1. Then, from , we obtain from Eq. (47)
The terms cancel each other. The terms ∝ξ_{j − 1} yield the condition ν_{j + 1} = ν_{j − 1}, and equating the terms ∝ξ_{j} yields
which is symmetric according to Eq. (44) and thus represents a tidal matrix. Thus, we obtain ν_{j} = λ for j even, and ν_{j} = 1 for j odd. Correspondingly,
We note that Eqs. (39) and (43) are special cases of Eq. (51).
Hence we find that the MST in multiple (lens and source) plane gravitational lensing exhibits a curious behavior: The scaling factor in every second plane is just unity, whereas it is λ in the other half of the planes. In particular that means that a “standard candle” or “standard rod” in one of the planes with j odd cannot be used to break the degeneracy related to the MST, as the images of these sources are unaffected by the MST. The prefactor in the tidal matrices G_{j} are positive on every other plane, and negative on the remaining ones. If one disregards the perturbations between lens planes, so that the distance matrices 𝒟_{ij} reduce to angular diameter distances D_{ij}, then the G_{j} become scalars proportional to the density of uniform mass sheets; in this case, positive and negative densities of these sheets alternate.
4.4. Transformation of the time delay
We next consider how the MST affects the time delays. For that, we assume to have a source on plane number N + 1, with its light being deflected in N main lens planes. The corresponding LTTF is given in Eq. (19), where ξ_{N + 1} is the position of the source in its source plane.
To obtain the corresponding function cτ′ after the MST, we first need to consider the transformation of the potential time delay. That is, we need to find the transformed deflection potential , which needs to satisfy . For this, we make the ansatz . Taking the gradient yields . This is seen to agree with in Eq. (48), provided a = ν_{i}ν_{i + 1} = λ. Thus,
We then obtain for the transformed LTTF
We now show that
which means that the transformed LTTF just scales by a factor λ, independent of the plane on which the source is located, plus a function which only depends on the location of the source, and thus cancels when considering time delays, i.e., differences between τ for pairs of multiple images of the source.
Taking the difference of τ′−λτ, we first note that the terms drop out. Second, we note that the terms containing products of ξ_{i} and ξ_{i + 1} also cancel, since ν_{i}ν_{i + 1} = λ. Thus we find that
The term i = 1 of the first sum vanishes, since ν_{1} = 1, and Eq. (39) holds. The final term (i = N) of the second sum depends only on the source position, and thus corresponds to the function F(ξ_{N + 1}) previously mentioned. Since this term is of no interest, we simply drop it from now on. Relabeling the index of the second sum as i → i − 1, we then get
where the matrices K_{i} are given as
Since, according to the definition (17), , we can rewrite K_{i} as
From Eq. (29),
is obtained. Noting that the final term is C_{i, i + 1}, we obtain with Eq. (50) that
However, the prefactor vanishes: if i is odd, ν_{i} = 1, ν_{i + 1} = λ, and thus
If i is even, ν_{i} = λ, ν_{i + 1} = 1, and
Therefore, K_{i} = 0 for all i, which completes our proof of the validity of Eq. (54). Since the result is independent of the plane on which the source is located – the time delay scales for all source planes with λ – we see that all time delays are scaled by the factor λ under the MST. In particular this implies that the degeneracy due to the MST cannot be broken from measuring time delay ratios.
5. Discussion
In this paper we have considered several aspects of the generalized multiplane gravitational lensing equation. In contrast to the treatment in Schneider et al. (1992) and more recent papers (e.g., McCully et al. 2014), we treat the light propagation between main lens planes with a continuous formalism, offered by the optical tidal equation, instead of slicing up the matter into several “weaklensing” lens planes^{6}. For this, we made use of the formalism of light propagation in arbitrary spacetimes, as given in SSE. As a result, the distance matrices between lens planes are not written in terms of recursion relations, but as solutions of the optical tidal equation; the explicit solution in terms of an integral over the tidal field caused by largescale density inhomogeneities along the lineofsight is provided in Eq. (11).
The timedelay function for generalized multiplane lensing was derived, using the same arguments as employed in Schneider et al. (1992) for the derivation of the time delay in ordinary multiplane lensing. The explicit form deviates from that obtained in McCully et al. (2014), in that our result depends only on the impact vectors in the various main planes, but not on the deflection angles. In other word, our expression for τ yields the light travel time of a kinematically possible ray with specified impact vectors ξ_{i} in the main lens planes, up to an additive constant. Physical light rays are those for which the lighttravel time is stationary; this allows the derivation of an iterative lens equation which relates the impact vectors of three consecutive main lens planes to the deflection angle in the middle one of those. We have shown that this form of the lens equation is equivalent to the more standard one which contains the impact vectors and deflection angles of all earlier lens planes. This consecutive lens equation is probably preferable for the use in raytracing simulations (see, e.g. Petkova et al. 2014; Jaroszyński & Skowron 2016)^{7}.
Finally, we showed that the generalized multiplane lensing admits a masssheet transformation (MST) which leaves all observables but the time delay invariant. In contrast to ordinary lensing, the MST corresponds to adding a tidal matrix in each main lens plane. We obtained the curious behavior that the uniform isotropic scaling of the source/lens planes, which is the key aspect of the MST, alternates between planes; in every second plane, the scaling corresponds to the MST parameter λ, in the other half of the planes, the scaling is unity. In particular, this implies that the magnification of sources living on the odd planes with scaling factor unity, is unaffected by the MST. All time delays – i.e., for sources in all main planes – scale as λ under the MST.
This curious behavior of the MST in multiplane lensing may indeed offer a way to break the corresponding degeneracy, at least in a statistical way. As we discussed before, in the case of vanishing perturbations between the main lens planes, the MST corresponds to mass sheets of alternating sign from plane to plane. Since such a mass sheet changes the slope of the total mass distribution, it means that this slope change also alternates. If one now makes the perhaps plausible assumption that the shape of the mean mass profiles of lenses is the same, this alternating slope change would violate the universality of the mean mass profile. Thus, in multiplane lensing, the masssheet generacy may be more easily lifted than in the case of a single lens plane only.
We hope that the results obtained here will be useful for further theoretical studies of generalized multiplane lensing, as well as for modeling lens systems in which more than one main deflector affects the imaging properties between observer’s sky and the source plane.
In case the distance matrices are proportional to the unit matrix, the transformation reduces to the known one in standard lensing. Note that the transformation (36) implies the transformation for the deflection potential. For isotropic distance matrices, 𝖦_{1} = (1−λ)D_{2}/(D_{1}D_{12})ℐ, so that . According to (15), this then implies for the scaled deflection potential ψ′(θ)=λψ(θ)+(1 − λ)θ^{2}/2, as in standard lens theory.
This relation is unique for a general deflection law . However, if the “strong lens plane” at z_{2} only yields a deflection linear in the impact vector (i.e., if this lens plane is in fact just a “weak” deflector), then the form of no longer is uniquely determined. This comment also applies to the more general case discussed below. Hence, we implicitly assume that all “strong lens planes” yield indeed a nonlinear deflection law.
Whereas these two treatments are equivalent (indeed, as shown in SSE, the slicing into weaklensing planes corresponds to a discretized version of the optical tidal equation), the continuous formalism is more convenient for analytical calculations – for example, obtaining a result such as (25) using the discretized version is probably extremely tedious.
The advantage of Eqs. (21) and (10) is twofold: First, in order to calculate the impact vectors in all N planes requires of order N^{2}/2 multiplications for each light ray when Eq. (10) is used, compared to about 3N multiplications for Eq. (21). More significant, however, is the fact that Eq. (21) allows one to save on memory: whereas Eq. (10) requires the information of all lens planes for each ray, one can treat with Eq. (21) a large set of rays, tracing them from plane to plane, and in each step require only the information on a single lens plane.
We note that B ≡ B_{0} corresponds to the matrix B in McCully et al. (2014), and the B_{i} corresponds, up to a prefactor, to their matrices C.
Acknowledgments
The author thanks Thomas Collett, Dominique Sluse, and Sherry Sutu for helpful comments and discussions, and the anonymous referee for a constructive report. This work was supported in part by the Deutsche Forschungsgemeinschaft under the TR33 “The Dark Universe”.
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Appendix A: Distance matrices in terms of peculiar gravitational potential
In this appendix we derive the expression (11) for the distance matrices in an inhomogeneous Universe. As usual in cosmological weak lensing, we work in comoving coordinates, and therefore replace the affine parameter λ by the comoving distance χ, where dχ/da = c/(a^{2}H), and a is the cosmic scale factor normalized to unity today. From the Robertson–Walker metric and the condition for radial null geodesics, we have a dχ = −c dt, and with Eq. (2) follows dχ = dλ/a^{2}. These relations then imply that
The angulardiameter distance D_{i} is related to the comoving angular diameter distance by D_{i} = a f_{k}(χ − χ_{i}), where χ_{i} is the comoving distance corresponding to the affine parameter λ_{i}. Applying Eq. (5), we get
With da/dλ = (dχ/dλ)(da/dχ)=H/c, and
and making use of the relations (A.1), Eq. (A.2) reduces to Eq. (12).
Next we write the distance matrix as 𝒟_{i} = D_{i} 𝖡_{i}, so that B_{i} describes the deviation of 𝒟_{i} from the unperturbed distance matrix D_{i}ℐ^{8}. This factorization transforms Eq. (8) into
Subtracting from this the transport Eq. (5) for D_{i}, we are left with
Inserting D_{i}(λ)=a(λ) f_{k}(χ − χ_{i}), and using the differentiation rules (A.1), this turns into
We this relation, we find for X_{i}(χ):=f_{k}(χ − χ_{i})B_{i}(χ):
where we made use of Eq. (12). This differential equation can be transformed into an integral equation, using the method of Green’s functions, to read
By differtiating twice, one can easily show that Eq. (A.7) indeed is a formal solution of Eq. (A.6), with the correct initial condition X_{i}(χ_{i})=0, d𝖷_{i}(χ_{i})/dχ = ℐ. If we now use the expression (6) for 𝒯_{sm}, neglecting the final term which is a derivative along the lineofsight and thus cancels in the integration, and replacing the derivatives w.r.t. ξ by those w.r.t. comoving transverse coordinates, we get
Using 𝒟_{i} = a 𝖷_{i}, we arrive at Eq. (11). We also note that the largescale structure component of the optical tidal matrix can alternatively written in the form (see Seitz & Schneider 1994)
with Γ_{1} = (ϕ_{,11} − ϕ_{,22})/2 and Γ_{2} = ϕ_{,12}, where the partial derivatives are with respect to transverse comoving coordinates. Thus, one can replace H(ϕ) in Eq. (A.8) by the bracket in Eq. (A.9).
All Figures
Fig. 1. Sketch of two light rays through four consecutive planes, with 0 ≤ λ_{q} < λ_{r} < λ_{s} < λ_{t}. The rays are not deflected in the lens planes. The first ray has its vertex at λ_{q} and encloses an angle θ with the fiducial ray; the second ray with vertex at λ_{r} encloses an angle ϑ with the fiducial ray. Both rays intersect at λ_{s}. The geometry of this figure yields the relation (25) between distance matrices. 

In the text 
Fig. 2. Propagation of a light ray (thick bent line) between three consecutive planes. The vertical line is the optical axis, with respect to which the separation vectors ξ are measured. The geometry of this figure yields the lens Eq. (30) – see text. 

In the text 
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