Generalized Multi-Plane Gravitational Lensing: Time Delays, Recursive Lens Equation, and the Mass-Sheet Transformation

A&A 624, A54 (2019) https://doi.org/10.1051/0004-6361/201424881 Astronomy c ESO 2019 & Astrophysics

Generalized multi-plane gravitational lensing: time delays, recursive lens equation, and the mass-sheet transformation

Peter Schneider

Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany e-mail: [email protected] Received 29 August 2014 / Accepted 25 February 2019

ABSTRACT

We consider several aspects of the generalized multi-plane 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 large-scale matter distribution in the Universe when propagating between the main deflectors. Specifically, we derive a simple expression for the time-delay 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 multi-plane lens situation admits a generalized mass-sheet 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 mass-sheet transformation, thus precluding the use of time-delay ratios to break the mass-sheet transformation. Key words. cosmological parameters – gravitational lensing: strong

1. Introduction mination of parameters, most noticible the Hubble constant from measured time delays (Refsdal 1964) in multiply imaged quasi- In strong gravitational lensing systems, in which a galaxy or a stellar objects (see, e.g., Kochanek 2003, 2006; Treu 2010; galaxy cluster causes multiple images or strong image distortions Suyu et al. 2010, 2013). The quality of modern imaging data of background sources, one often neglects the inhomogeneities and the accuracy of time delay estimates allows one to derive of the gravitational field between the observer and the lens, and estimates of the Hubble constant with a formal error of 6% between the lens and the source (see, e.g., Kochanek 2006; Treu (Suyu et al. 2013). At this level of precision, line-of-sight eff∼ects 2010; Bartelmann 2010; Kneib & Natarajan 2011, for reviews may become highly relevant in these strong-lensing systems on strong lensing systems). This usually provides a very good (e.g., Wong et al. 2011; Collett et al. 2013; Greene et al. 2013, approximation, since the lensing strength of the main lens over and references therein). the region where strong lensing effects occur is much larger than With additional deflectors along the line-of-sight, the the typical distortion effects of matter along the line-of-sight. lens equation, which relates the true source position to the The latter is comparable to the typical strength of cosmic shear observed positions of images, needs to be generalized to effects (see, e.g., Bartelmann & Schneider 2001; Schneider 2006; include deflection at more than one distance from the observer. Munshi et al. 2008; Hoekstra 2013), and amounts to about 1 or 2% Blandford & Narayan(1986) established the theory of multi- of the distortion in the strong-lens region. plane gravitational lensing (see also Chap. 9 of Schneider et al. Whereas the propagation effects are thus small, the interest of 1992), where the mapping between images and their source is these weak distortions has been renewed, for at least two differ- affected by the action of several deflectors at different redshifts. ent reasons. The first is that strong lens systems may be biased This multi-plane lensing is employed in ray-tracing simula- toward showing relatively strong line-of-sight structures, thereby tions – see Refsdal(1970) for the earliest ray-tracing simulations, increasing their lensing probability. This effect is most likely Jain et al.(2000), Hilbert et al.(2009), and references therein – affecting the lensing cross sections for the formation of giant arcs where the three-dimensional mass distribution between observer in clusters (see Bartelmann et al. 1998 for stating the “arc statis- and source is partitioned into separate slices, and the mass tic problem”, and Meneghetti et al. 2013 for a recent update of distribution in each slice is treated as a gravitational lens plane. the issue). Cosmological simulations (Puchwein & Hilbert 2009, The full theory of multiple deflection lensing is required if and references therein) indicate that line-of-sight structure can there are two or more strong deflectors along the same line-of- indeed substantially modify the lensing efficiency of clusters. sight toward a background source. For galaxy-scale lensing, such More recently, Bayliss et al.(2014) found a significant overden- systems are rare, since such an alignment is not very probable. sity of galaxy groups along the line-of-sight toward strong-lensing However, in the sample of several hundred galaxy-scale strong clusters, observationally supporting the presence of this bias. lens systems currently known, there are examples of multiple The second reason for the renewed interest in intervening lenses at two different redshifts (Chae et al. 2001; Gavazzi et al. distortions of strong-lens systems is their use for a precise deter- 2008). These systems may be of particular interest, since they

Article published by EDP Sciences A54, page 1 of 10 A&A 624, A54 (2019) may be used in principle to determine cosmological distance equivalent to the more standard form; in order to do so, we first ratios (Collett & Auger 2014), and therefore to constrain the cos- obtain a very general relation between distance matrices. mic expansion history, although in practice the mass-sheet trans- We then turn to the MST in this case, and find a curious prop- formation renders this a difficult task (Schneider 2014). erty of its behavior: The uniform, isotropic scaling factor which Far more common are situations where there is a single characterizes the MST alternates between a free parameter λ and strong lens in the line-of-sight to a distant source, and sev- unity from one lens/source plane to the next. The correspond- eral other deflectors at different redshifts situated sufficiently far ing modification of the deflection angle in the main lens planes away from the strong-lensing region of the main deflector such corresponds to a scaling of the deflection, plus the addition of that their deflection angle can be linearized across this strong- a tidal deflection matrix (which in the absence of linear deflec- lensing region. Kovner(1987) considered this case of one main tions between lens planes reduces to the addition of a uniform lens, combined with linearized deflections at different redshifts. mass sheet). Finally we show that all time delays, for sources The effects of the linear deflectors can be summarized into a located on any plane, scale by the same factor under an MST, set of matrices which describe the mapping between angles at precluding the possibility of breaking the degeneracy from the the vertices of light cones to the separation vectors along the MST by measured time-delay ratios. 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; 2. Generalized multi-plane lens equation hereafter SSE), and related this generalized gravitational lens 2.1. Optical tidal equation equation to the one where the deflection occurs in only a single plane, with a tidal deflection matrix added (see also Keeton et al. The propagation of light rays follows from the geodesic equa- 1997). McCully et al.(2014) further generalized this theory to tion, specialized to a perturbed Robertson–Walker metric (see consider several main lenses, together with linearized deflec- Schneider et al. 1992; Seitz et al. 1994; Bartelmann 2010, and tions between the main lens planes. In particular, McCully et al. references therein). In particular, for infinitesimally small light (2014) emphasized the advantages of this hybrid framework for bundles, the separation vector ξ of a ray from the reference (or the modeling of strong lens systems with multiple main deflec- “central”) ray of the bundle is given by the optical tidal equation tors along the line-of-sight (see also McCully et al. 2017). One of the prime motivations for the work of McCully et al. ξ00(λ) = (λ) ξ(λ), (1) T (2014) was the derivation of the time-delay function for where is the optical tidal matrix, evaluated at the affine param- such generalized lensing systems, as this is required for T relating the time delay to the scale-length of the Universe, eter λ along the central ray. Here and throughout this paper, a i.e., the Hubble radius. Using the Millennium Simulation, prime denotes differentiation with respect to the affine parame- Jaroszynski & Kostrzewa-Rutkowska(2014) investigated the ter λ, which is related to the redshift z, the cosmic scale factor impact of the line-of-sight matter distribution on strong lens- a = 1/(1 + z), or cosmic time t through ing properties, including the time delay; they concluded that the c dt c da c dz intermediate mass distribution leads to a spread of 6% in the dλ = = = , (2) product of Hubble constant and time delay, for strong∼ lensing −(1 + z) − H H (1 + z)2 systems with source redshifts z 2. Arguably, the largest obsta- s where we have assumed that λ increases with redshift, i.e., cle for the time-delay method to∼ obtain accurate estimates for decreases with cosmic time, the Hubble constant is the degeneracy of the mass model due to the mass-sheet transformation (MST; Falco et al. 1985; see p H = a˙/a = H Ω (1 + z)3 + (1 Ω Ω )(1 + z)2 + Ω (3) Schneider & Sluse 2013 for a recent discussion) or the more gen- 0 m − m − Λ Λ eral source position transformation (Schneider & Sluse 2014). is the Hubble function, H0 is the Hubble constant, and Ωm and Recently, Schneider(2014) has shown that an MST also exists Ω are the cosmic density parameters in matter and vacuum ff Λ for the case of lenses at two di erent distances from us. In partic- energy. ular, this MST leads to a scaling of all time delays (from sources We consider to consist of three separate components, on both source planes) by the same factor, thus precluding that T = bg + sm + cl. The first is the optical tidal matrix of the the degeneracy due to the MST can be broken by time-delay homogeneousT T T backgroundT universe, and is given by (see SSE) ratios. In this paper, these results are further generalized to the 2 3 H0 5 case of arbitrarily many main lens planes, together with linear bg = Ωm(1 + z) , (4) deflections between the main lens (and source) planes. After a T −2 c I short summary of the propagation equations in an inhomoge- where is the two-dimensional unit matrix. If we consider a neous universe, as they apply to the case under consideration light bundleI with vertex at redshift z = 0 and affine parameter here, we derive the time-delay function for the case of sev- λ = 0, which is only subject to bg, then the solution of Eq. (1) eral main lens planes. Whereas this has been done before by is ξ(λ) = D(λ)θ, where θ is theT angle that the light ray under McCully et al.(2014), our expression for the light travel time consideration encloses with the fiducial ray at the vertex, and function is expressed in a form which allows us to apply Fer- D(λ) is the solution of the differential equation mat’s theorem in gravitational lensing. Thus, the lens equation can be obtained from requiring that the light travel time function d2D(λ) 3 H 2 = 0 Ω (1 + z)5 D(λ), (5) is stationary with respect to all impact vectors in the main lens dλ2 −2 c m planes. With this procedure, we derive an alternative form of the lens equation, which involves the impact vectors of three con- with initial condition D(0) = 0 and D0 = 1. Here, D is the secutive lens planes and the deflection angle in the middle one angular-diameter distance of the homogeneous universe, as a of them. We show that this iterative form of the lens equation is function of affine parameter (or redshift).

A54, page 2 of 10 P. Schneider: Multi-plane gravitational lensing

˜ We consider two different kinds of inhomogeneities here. tilde on ξ henceforth. For the impact vectors ξ j in the jth plane, The first of them is related to small-scale density inhomo- we then obtain geneities, such as galaxies and their dark matter halos. For those, X j 1 X j 1 − − the optical tidal matrix is a strong function of position, and ξ = (λ j) θ i(λ j) αˆ i(ξ ) j θ i j αˆ i(ξ ). cl j i= i i= i T θ D − 1 D ≡ D − 1 D there is no longer a linear relation between enclosed angle (10) and separation ξ of a light ray, for finite θ. In strong-lensing applications, we are typically interested only in that region of 2.3. Calculation of the distance matrices these small-scale inhomogeneities where multiple images can be formed, which corresponds to a few tens of kiloparsecs for The distance matrices i(λ) depend on the large-scale matter D galaxy lenses. These small-scale inhomogeneities will be con- distribution around the line-of-sight, which is dominated by dark sidered explicitly as “main lenses” in the following. The second matter and thus difficult to determine observationally. One possi- kind of inhomogeneities is due to the large-scale mass distribu- bility to estimate sm from observations is to assume that galax- T tion of the universe. We assume that the corresponding gravi- ies provide a good tracer of the total matter distribution on large tational field is sufficiently smooth, so that the tidal effects can scales. From an observed distribution of galaxies (or a particular be considered approximately constant across the region where kind of galaxies, like luminous red galaxies) around the line- strong-lensing effects occur. Hence we assume that over such a of-sight, an estimate of the tidal field can be obtained; this is the region, sm can be considered to depend only on λ, not on the strategy proposed in Collett et al.(2013) and Smith et al.(2014). actual positionT of the light ray. According to SSE, this contribu- The propagation of light through the large-scale structure is tion of the optical tidal matrix is the subject of cosmological weak lensing, or cosmic shear (see,   e.g., Schneider 2006). In cosmic shear, one usually describes the (1 + z)2  ∂2φ ∂2φ propagation matrices in comoving coordinates. In order to con- =  + δ  , ( sm)i j 2 2 i j 2  (6) T − c ∂ξi ∂ξ j ∂ξ nect the formulation given here, where the i relate angles to 3 proper transverse separation vectors (which isD appropriate, as the where φ is the Newtonian potential sourced by the density inho- matter distribution of the main lenses – galaxies or clusters – are mogeneity, i.e., satisfying the (three-dimensional) Poisson equa- most conveniently described in physical scales), to that used in 2 tion ξ φ = 4πG(ρ ρ¯), whereρ ¯(z) is the mean matter density weak lensing, we show in the appendix that ∇ − Z χ in the Universe, and we assumed that the light ray propagates in 2 a(χ) χ = χ χ0 χ χ0 H φ χ0 χ0 , the ξ3-direction. i( ) Di( ) 2 d fk( ) ( ( )) i( ) (11) D I− c χi a(χ0) − D

where χ is the comoving distance, fk(χ) is the comoving angular 2.2. Generalized multi-plane lens equation diameter distance, D (χ) = a(χ) f (χ χ ), which satisfies the i k − i We consider a direction in the sky with has a set of main lenses differential equation z 2 along the line-of-sight to a distant source, located at redshifts i d fk(χ) or affine parameters λ . As shown in SSE (see also Schneider = K fk(χ) (12) i dχ2 − 1997), the separation vector ξ(λ) then becomes with initial conditions fk(0) = 0 and d fk(0)/dχ = 1. Furthermore, X h i 2 2 0 K = (Ω + ΩΛ 1)H /c is the spatial curvature of the universe, ξ(λ) = (λ) θ i(λ) αî(ξi) αˆ i H(λ λi), (7) m 0 D − iD − − and H(φ) is the− two-dimensional Hessian of the gravitational where θ is the angle the light encloses with the fiducial ray at potential φ, evaluated in comoving transverse coordinates. Pro- the observer. The distance matrix (λ) solves the optical tidal vided the perturbations are small, so that i deviates only slightly D from D , we can replace (χ ) by D (χD) in the integrand, equation iI Di 0 i 0 Z χ 2 λ h i 2 a(χ) d ( ) i(χ) = Di(χ) dχ0 fk(χ χ0)H(φ(χ0))Di(χ0). D = bg(λ) + sm(λ) (λ), (8) D I − c2 a(χ ) − dλ2 T T D χi 0 (13) with initial conditions (0) = 0 and 0(0) = , and the distance matrices (λ) solve theD same differentialD equation,I but with ini- In weak lensing, this approximation is called “the neglect of Di tial conditions i(λi) = 0 and (d i/dλ)(λi) = (1 + zi) . They lens-lens coupling”, often also termed as “Born-approximation”. are the distanceD matrices which applyD for light rays havingI their This approximation is very accurate and certainly sufficient for 1 vertex at λi. In Eq. (7), H(λ λi) is the Heaviside step function. the purposes discussed in the current context . Thus, the devia- − tion of from D is given as a line-of-sight integral over the The deflection angle αˆ i(ξi) is given in terms of the surface mass i i tidal forceD field, withI a distance-dependent weighting. We also density Σi(ξi) as note that in the approximation (13), the distance matrices are Z Di 4G 2 ξi ξ0 symmetric, which is generally not the case if the exact expres- α ξ = ξ0 Σ ξ0 − , ˆ i( i) 2 d i( ) 2 (9) c ξ ξ0 sion (11) is used. | i − | For statistical studies, instead of trying to obtain the tidal field 0 along the line-of-sight toward the sources from observations of and αˆ i denotes the deflection angle of the fiducial ray in the ith lens plane. If we now perform the translation ξ(λ) = ξ˜(λ) + η(λ), the galaxy distribution in those directions, one can also derive the with probability distribution for the distance matrices from cosmolog- X ical simulations, as has been done in Suyu et al.(2013). 0 η(λ) = i(λ) αˆ H(λ λi), iD i − 1 ItmustbestressedherethatthisBorn-approximationonlyappliestothe distance matrices which are governed by the smooth part of accord- ˜ 0 Di T then ξ satisfies Eq. (7) with the αˆ i set to zero. In the following, ing to Eq. (8); only this smooth contribution is contained in Eq. (11). we always assume this (unobservable) translation, and drop the No such approximation is made with regards to the main deflectors.

A54, page 3 of 10 A&A 624, A54 (2019)

3. Time-delay function and the iterative lens where we defined 1 1 equation C = − − . (17) i j Di j D jDi In this section, we first derive the light travel time function As shown in Schneider(1997) – see also Kovner(1987) – the (LTTF) corresponding to the generalized multi-plane lens equa- matrix Ci j is symmetric. Indeed, we see that tion considered in the previous section. An alternative form of 1 + z the lens equation is then derived from Fermat’s principle, which τ , = 1 C 1 = ξ1 (ξ1 ξ2) 12ξ1 12− ξ2 αˆ 0 (18) in the current context states that the lens equation is equivalent to ∇ c − D − setting the gradient of the LTTF with respect to all impact vec- is equivalent to the lens Eq. (14), as is easily verified by multiply- tors equal to zero (i.e., real light rays correspond to stationary ing the foregoing expression by 12 from the left. We also note ˆ D 1 points of the LTTF). Then we show that this iterative form of the that τ = 0 if ψ = 0 and if the ray is unbent, i.e., if ξ2 = 2 1− ξ1, lens equation is equivalent to Eq. (10). as we required for τ. McCully et al.(2014) obtainedD aD some- what different form for τ, which they showed to be equivalent to the expression given here. However, it must be pointed out that 3.1. Time-delay function this equivalence applies only to physical light rays, i.e., those We first consider a single lens plane at z1 with a source at z2, in which satisfy the gravitational lensing Eq. (14). For those rays, which case the lens equation reads we could write the light travel time as 1 + z 1 1 t 1 ˆ ξ = θ αˆ (ξ ) = − ξ αˆ (ξ ), (14) cτ = αˆ C12− αˆ 1 (1 + z1)ψ(ξ), 2 D2 − D12 1 D2 D1 1 − D12 1 2 1 − where the lens Eq. (14) was used to eliminate ξ2. However, where we set αˆ (ξ1) αˆ 1(ξ1). We define the LTTF τ(ξ1, ξ2) to be the excess light travel≡ time from a source at ξ to the observer Eq. (16) is more general, as it yields the light travel time for all 2 kinematically possible rays, not only for those for which the bend caused by the deflection in the lens plane at ξ1. This excess travel time has two components, a geometrical one (a bent ray is longer by the main lenses equals the actual deflection angle as calculated ˆ than an unbent one), and a potential one caused by the retarda- as the gradient of ψ. This more general form of the τ is needed if tion of photons in the gravitational potential of the deflector. the lens equation is to be derived from Fermat’s principle. In standard lens theory, with unperturbed angular-diameter In case of several main lens planes, the LTTF is obtained distances Di j, the potential part of the time delay function by considering the replacement of the actual light ray by suc- takes the form cτpot = (1 + z1)(D1D2/D12)ψ(θ), where ψ(θ) cessively straighter rays, i.e., by removing the bends of the ray is the deflection potential,− which satisfies ψ = α(θ) and the gravitational potentials they traverse (see Sect. 9.2 of ∇θ ≡ (D12/D2)αˆ (D1θ). Now we define the potential ψˆ(ξ1) such that Schneider et al. 1992). Removing the bend and deflection poten- it satisfies ξ ψˆ = αˆ . This potential is a multiple of ψ, i.e., tial in the first plane leads to the contribution (16) of the LTTF. ∇ 1 ψˆ(ξ1) = k ψ(ξ1/D1), up to an irrelevant additive constant. To Subsequent removal of the bend and potential in the second lens find k, we consider plane yields a similar contribution, with the indices (1,2) replaced by(2,3).Iteratingthisconsideration,weobtainforthegeneralcase k k k D12 " ψˆ ψ , XN t αˆ = ξ1 = θ = α = αˆ 1 1 ∇ D1 ∇ D1 D1D2 cτ(ξ , ξ ,..., ξ ) = (1 + zi) Ci,i+1ξ − ξ 1 2 N+1 i=1 2 i − Di,i+1 i+1 yielding # 1 1 C− C ξ − ξ ψˆ (ξ ) (19) × i,i+1 i,i+1 i − Di,i+1 i+1 − i i D D ˆ 1 2 ψ(ξ1) = ψ(ξ1/D1). (15) as the sum over terms of the form (16) for the individual planes. D12 Any ray connecting the source at ξN+1 and the observer is Hence, the potential part of the LTTF takes the form cτpot = fully characterized by the impact vectors ξi, 1 i N in the lens planes, since between the planes, it follows≤ the≤ propaga- (1 + z1)ψˆ(ξ1). As expected, it does not depend on the cosmological− distances, since it is caused solely by the local effect of tion Eq. (1) whose solution is uniquely determined by the two propagating through a gravitational field, and the correspond- impact vectors at consecutive planes. Therefore, the actual light ing time interval is then redshifted by a factor (1 + z1) to the rays are singled out as those for which the LTTF is stationary, observer. Accordingly, also in the case of perturbed light prop- with respect to variations of the impact vectors in the N lens agation between the lens planes, the potential part of the time planes. Thus, the lens equation is obtained by setting the deriva- ff tive of τ with respect to the ξ equal to zero. For each j 2, two delay must have the same form, since it is una ected by propa- j ≥ gation effects. terms of the above sum contribute, namely the terms i = j and τ ξ , ξ i = j 1. We obtain We find an explicit expression for ( 1 2) by requiring that − ξ τ(ξ , ξ ) = 0 is equivalent to the lens Eq. (14). This fixes τ h 1 i 1 1 2 (cτ) = (1 + z ) C , + ξ − ξ αˆ (ξ ) up∇ to a multiplicative constant and terms which depend solely on ∇ξ j j j j 1 j − D j, j+1 j+1 − j j t ξ . The multiplicative constant is fixed by the explicit expression 1 1 1 2 + (1 + z j 1) −j 1, j C−j 1, j −j 1, jξ j ξ j 1 given above for the potential part of τ, and the additive constant − D − − D − − − (which is irrelevant for time delay measurements) is fixed by = 0, (20) requiring that the geometrical part of τ should vanish if the light or ray is undeflected by the main lens. This then yields " # 1 + z j 1 t − 1 1 1 ξ j+1 = j, j+1 C j, j+1 + −j 1, j C−j 1, j −j 1, j ξ j D + D − − D − 1 + z1 1 t 1 1 1 z j cτ(ξ1, ξ2) = C12ξ1 12− ξ2 C12− C12ξ1 12− ξ2 2 − D − D 1 + z j 1 t − 1 j, j+1 −j 1, j ξ j 1 j, j+1αˆ j(ξ j). (21) (1 + z ) ψˆ(ξ ), (16) − 1 + z D D − − − D − 1 1 j A54, page 4 of 10 P. Schneider: Multi-plane gravitational lensing

This equation relates the position vectors in three consecutive planes to the deflection angle in the middle plane, quite in con- ∆ξ ✲ ✲t trast to the lens Eq. (10) which contains all impact vectors ξ for t ☞ ✡ i ☞ a given ξ j, 1 i j 1. Hence, this new lens equation is more ξt ✡ ≤ ≤ − ☞ ✡ “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 ϕ✡ ☞ ✡ ☞✡ Consider the pairs of light rays sketched in Fig.1, where the first ☞ has a vertex at λq and encloses an angle θ with the fiducial ray. ✡ ✡☞ At the affine parameter λs, its separation vector from the fiducial s ✲ ✡☞ ray is ξs. The second light ray has its vertex at λr and intersects ξ ☞ λ ϑ s ✡ the first ray at s; this then specifies its direction relative to ✡ ☞ the fiducial ray. At the intersection point, the two rays enclose ✡ ☞ an angle ϕ. From the geometry of the figure, we find ✡ϕ☞ ✡ ☞ 1 1 1 ☞ θ = qr− ξr = qs− ξs = qt− ξt; ✡ D D D ☞ 1 1 ✡ ϑ = − ξ + ∆ξ = − ξ . (22) ☞ Drt t t Drs s ✡ ✡ ☞ We use the latter equation to derive a relation between the ’s, ϑ ☞ by expressing all vectors in terms of ξ . Using the first of (D22), ✡ s ✡ ☞ ξ = 1ξ ∆ξ = we get t qt qs− s. Furthermore, the figure shows that t ✡ ξr✲ ☞ ϕ. On theD otherD hand, ξ = (λ )ϕ. r ☞ Dst r −Ds r We now have to relate s(λr), the backward extension of the r ☞ D ☞ solution s(λ) of Eq. (8), to rs. For that, we consider two solutions of Eq.D (8), (λ) and D (λ), with their appropriate initial ☞ Dr Ds ☞ conditions at λr and λs, respectively, and define the matrix θ ☞ t ☞ d r t d s W(λ) = D (λ) s(λ) (λ) D (λ), (23) ☞ dλ D − Dr dλ q ☞ which is the Wronskian of the differential Eq. (8); here, the Fig. 1. r superscript “t” denotes the transpose of a matrix. The derivative Sketch of two light rays through four consecutive planes, with 0 λq < λr < λs < λt. The rays are not deflected in the lens planes. The of W vanishes, due to Eq. (8); hence, W is a constant. Evaluating first≤ ray has its vertex at λ and encloses an angle θ with the fiducial ray; Eq. (23) at λ = λ and making use of the initial conditions of q r the second ray with vertex at λr encloses an angle ϑ with the fiducial ray. r(λ) yields W = (1 + zr) s(λr). Similarly, at λ = λs we find Both rays intersect at λs. The geometry of this figure yields the relation D t D t W = (1 + zs) (λs) = (1 + zs) . Thus, we obtain (25) between distance matrices. − Dr − Drs 1 + z (λ ) = s t , (24) Ds r −1 + z Drs Schneider(2016) derived for the case that the distance matrices r reduce to scalars. which is Etherington’s theorem in matrix form (Etherington D Indeed, a relation of this kind is expected to hold: Consider 1933). With this relation, we then find that λ λ as a variable. The two matrix-valued functions (λ) t ≡ Dq and r(λ) are linearly independent solutions of the transport 1 + zr 1 D 1 t − Eq. (8), provided λr , λq. Therefore, the solution s(λ) can ∆ξt = stϕ = st −s (λr) ξr = st rs ξr. D D −D D 1 + zs D D be written as a linear combination of the other two. This combination should be of the form Using Eq. (22) and collecting terms, h 1 1 i s(λ) = r(λ) − (λs) q(λ) − (λs) X, (26) + z 1 r q 1 1 1 r t − 1 D D D − D D rt rs− ξs = ξt+∆ξt = qt qs− ξs+ st rs qr qs− ξs D D D D 1 + zs D D D D which satisfies one of the initial conditions, s(λs) = 0. The matrix X is determined from the second initialD condition; our follows. Since this relation is valid for all ξs, a general relation result (25) shows that between distance matrices is obtained: h 1 1 i 1 + zs 1 t + z 1 (λ) = (λ) − (λ ) (λ) − (λ ) − . (27) 1 r t − 1 s r r s q q s qs qr rs st rs qr = rt rs− qs qt, (25) D D D − D D 1 + zr D D D 1 + zs D D D D D D − D where we multiplied the resulting equation by from the 3.3. Equivalence of Eqs. (10) and (21) Dqs right2. This result generalizes the corresponding relation in We shall now show that the two forms (10) and (21) of 2 We explicitly point out that the only geometrical relation used in this the lens equation are equivalent. As a first step, we rewrite derivation is the one between angles and transverse separations, i.e., the Eq. (21) in a form that admits a simple geometrical interpreta- definition of the distance matrices. tion. Specializing Eq. (25) to q = 0, r = j 1, s = j, t = j + 1 − A54, page 5 of 10 A&A 624, A54 (2019)

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 ❇ ❉ ✁ − ❇ η ξ j+1 ❉ ✁ form. Making use again of Eq. (28), we obtain ❉ ❇ ✁ 1 ❇ ❉ ✁ ξ j+1 = j 1, j+1 −j 1, jξ j D − D − ❇ ❉ ✁ 1 1 1 ϕ ❉ + j+1D−j 1 j 1, j+1 −j 1, j j −j 1 ξ j 1 ❇ ✁ D − − D − D − D D − − ❉αˆ j ❇ ✁ j, j+1αˆ j(ξ ). (31) ❇ ❉ ✁ − D j ❇ ✲❉✁ We prove the equivalence by induction; for j = 1, this equiva- ✁ ❇ ξ j ✁ lence 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) ❇ ✁ ϕ ✁ for ξ and Eq. (31), ❇ ✁ j+1 ❇ ✁ 1 ∆ = j 1, j+1 −j 1, jξ j ❇ ✁ D − D − ❇ ✁ 1 1 1 + j+1D−j 1 j 1, j+1 −j 1, j j −j 1 ξ j 1 j, j+1αˆ j(ξ j) ❇ ✁ D − − D − D − D D − − − D ❇ ✁ X j j+1 θ + i, j+1 αˆ i(ξ ), (32) ✲❇✁ − D i=1D i ξ j 1 we need to show that ∆ = 0. We first replace ξ and ξ by their − j 1 j expressions from Eq. (10), which holds because− of the induction Fig. 2. Propagation of a light ray (thick bent line) between three consec- assumption, utive planes. The vertical line is the optical axis, with respect to which the separation vectors ξ are measured. The geometry of this figure yields X j 1 1 − the lens Eq. (30) – see text. ∆ = j 1, j+1 −j 1, j jθ i, j αˆ i(ξi) D − D − D − i=1 D X j 2 1 − 1 + j+1 j 1, j+1 −j 1, j j θ −j 1 i, j 1 αˆ i(ξi) yields D − D − D − D − i=1 D − D − X j j, j+1αˆ j(ξ ) j+1 θ + i, j+1 αˆ i(ξ ). (33) 1 + z j 1 1 j i=1 i − t − 1 − D − D D j, j+1 j 1, j j 1 = j 1, j+1 −j 1, j j j+1. (28) 1 + z D D − D − D − D − D −D j From this equation, one sees immediately that the terms θ ∝ cancel each other. Second, the two terms αˆ j add up to zero. We next consider the prefactor of ξ j in Eq. (21). Using ∝ 1 1 1 Third, also the sum of the two terms αˆ j 1 is zero. Thus, what C−j , j −j , j = j 1 −j , which is obtained from the definition ∝ − 1 D 1 D − D remains to be shown is that the prefactor of the terms αˆ i, (17−) of C−, we find that this prefactor becomes ∝ 1 Ki = i, j+1 j 1, j+1 −j 1, j i, j 1 + z j 1 t D − D − D − D 1 − 1 1 1 j+1 −j + j, j+1 −j 1, j j 1 −j = j 1, j+1 −j 1, j, (29) 1 1 D D 1 + z j D D − D − D D − D − j+1 j 1, j+1 −j 1, j j −j 1 i, j 1, (34) − D − D − D − D D − D − where in the last step we made use of Eq. (28) and the fact that 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. the inversion and transposition operations on a matrix commute. − Thus, we can rewrite Eq. (21) in the form This then yields

1 1 + z j 1 ξ = ξ 1 t − 1 1 j+1 j 1, j+1 −j 1, j j − , + = , + − + − D − D − j j 1 j 1, j j 1 j 1 j 1, j j j 1 j 1 1 + z j D D − D − D − D − D D − 1 + z j 1 t − 1 j, j+1 −j 1, j ξ j 1 j, j+1αˆ j(ξ j). (30) 1 1 − 1 + z D D − − − D = j 1, j+1 −j 1, j i, j i, j+1 i−, j 1. j D − D − D − D D − (35) We note that this generalizes equation (4.47) of SSE to the case of general distance matrices between main lens planes. This After multiplying by i, j 1, we see that the ﬁnal equality shows D − form of the lens equation can be immediately interpreted geo- that Ki = 0, which proves that ∆ = 0, and thus the equiv- metrically. For this, we consider Fig.2, from which we read o ﬀ alence of the two forms of the lens equation. The equation K = 0 itself provides an interesting relation between distance 1 i η + ξ j+1 = j 1, j+1 −j 1, jξ j j, j+1αˆ j. matrices. D − D − − D

Furthermore, η = j, j+1ϕ; on the other hand, ξ j 1 = j, j 1ϕ. D − −D − Eliminating ϕ from these two relations and making use of 4. Mass-sheet transformation Eq. (24), we find In standard gravitational lensing, with a single deflector between 1 + z j 1 t the source and observer, there is a transformation of the mass − 1 η = j, j+1 −j 1, j ξ j 1. 1 + z j D D − − distribution of the lens which keeps most observables invariant, the mass-sheet transformation (MST, see Falco et al. 1985). Together, these two equations reproduce Eq. (30). In this form, Since this transformation is accompanied by a uniform isotropic the equation not only is confined to three consecutive lens planes, scaling in the source plane, all magnifications are scaled by but all distance matrices occurring here are those between these the same factor, so that magnification (and thus observable three planes. flux) ratios are unchanged. The MST changes the product of

A54, page 6 of 10 P. Schneider: Multi-plane gravitational lensing time delay and Hubble constant, though, and the correspond- and we require the modified deflection angle αˆ 20 to be chosen ing degeneracy can thus be broken by measuring the time delay such that the 3-plane is just uniformly scaled relative to the orig- in lens systems, assuming the Hubble constant to be known inal one, i.e., ξ30 = ν3ξ3. This condition then yields from other cosmological observations (see Schneider & Sluse 3θ 13 λαˆ 1(ξ1) + G1ξ1 23αˆ 20 (ξ20 ) 2013, and references therein). S14 has recently shown that a D − D − D = ν θ αˆ (ξ ) αˆ (ξ ) . (41) MST also exists in the case of two lens planes and two source 3 D3 − D13 1 1 − D23 2 2 planes. In this section, we show that also for perturbed gravi- In order to account for the term αˆ 2 on the r.h.s. of Eq. (41), the tational lens systems, as considered in this paper, such a MST modified deflection has to be of∝ the form5 does exist. αˆ 20 (ξ20 ) = ν3αˆ 2(ξ20 /λ) + G2ξ20 = ν αˆ (ξ0 /λ) + λG θ αˆ (ξ ) . (42) 4.1. Single main lens plane 3 2 2 2 D2 − D12 1 1 This choice then yields equal terms αˆ 2 on both sides. Equating We start with the case of a single lens plane, using the lens Eq. the terms αˆ leads to the condition∝ λ + λ G = α ξ 1 13 23 2 12 (14), and modify the deflection angle ˆ 1( 1) to the new form ν , or∝ − D D D − 3D13 1 1 αˆ 10 (ξ1) = λαˆ 1(ξ1) + G1 ξ1, (36) G = (1 ν /λ) − − . (43) 2 − 3 D23 D13D12 3 where λ is a real number, and G1 is a matrix . Throughout Using the same arguments as in the Appendix of Schneider this section, a prime denotes a mass-sheet transformed quan- (1997), it is straightforward to show that any combination of dis- tity. Thus, the modified deflection angle is a scaled version of tance matrices of the form the original one, plus a term linear in the impact vector. If G1 1 1 − − is symmetric, (44) is symmetric, this linear term corresponds to a tidal matrix, i.e., Dst DrtDrs adding a uniform mass sheet to the scaled lens mass distribu- for 0 zr < zs < zt. Hence, G2 is symmetric, and thus corre- tion, plus an external shear. The modified lens equation then sponds≤ to a tidal matrix. Equating the terms θ in Eq. (41) then becomes leads to ∝ 1 1 ξ0 = θ αˆ 0 (ξ ) = θ λαˆ (ξ ) + G θ . (37) (1 ν3) 3 = (1 λ) 13 − 2 + λ(1 ν3/λ) 13 − 2, (45) 2 D2 − D12 1 1 D2 − D12 1 1 1 D1 − D − D D12 D − D D12 D As for the orignal MST, we require that the modified impact vec- which has the unique solution ν3 = 1. Thus, as is the case for tor ξ0 is related to the original one by a uniform, isotropic scal- the standard multi-plane lens discussed in Schneider(2014), the 2 MST does lead to no scaling in the plane j = 3. Therefore, ing, ξ20 = ν2ξ2, where ν2 is the scaling factor. Thus we require G = /λ 1 1. θ λαˆ (ξ ) + G θ = ν θ αˆ (ξ ) . (38) 2 (1 1 ) 23− 13 12− (46) D2 − D12 1 1 1 D1 2 D2 − D12 1 1 − D D D The implied scaling of the mass distribution in the plane i = 2 In order to have the terms αˆ equal on both sides of Eq. (38), ∝ 1 that follows from Eq. (42) is discussed in Schneider(2014); in we need to set ν2 = λ, as is also the case for the MST in standard short, the surface mass density distribution giving rise to αˆ 2(θ2) lensing – the scaling of the source plane (here plane number 2) needs to be scaled in amplitude and scale-length to yield a deflec- is the same as that of the deflection angle. The remaining terms tion αˆ 2(λθ2). are all θ, and setting them equal on both side leads to 2 G ∝ = λ , or D − D12 1D1 D2 1 1 4.3. Arbitrary number of planes G1 = (1 λ) 12− 2 1− = (1 λ)C12. (39) − D D D − Here we generalize the MST to an arbitrary number of Since C12 is symmetric (see Schneider 1997), G1 is indeed a source/lens planes. It turns out that the lens equation in the form tidal matrix. This single-main plane MST was also derived by (31) is better suited for that purpose. We write it in the form McCully et al.(2014). Thus, in a generalized gravitational lens 1 situation, the MST requires a shear in addition to a uniform mass ξ j+1 = j 1, j+1 −j 1, jξ j + B jξ j 1 j, j+1αˆ j(ξ j), (47) D − D − − − D sheet4. where B j is the term in parenthesis in Eq. (31). We now assume a scaling ξ0j = ν jξ j in every plane, and set the scaled deflection 4.2. Two main lens planes angles to be α ξ = ν α ξ /ν + G ξ = ν α ξ + ν G ξ . We now consider a second lens plane at z2, with the source plane ˆ 0j( 0j) j+1 ˆ j( 0j j) j 0j j+1 ˆ j( j) j j j (48) being located at z3. The modified lens equation then reads Note that Eqs. (36) and (42) are special cases of the relation (48) for j = 1, 2, respectively, and ν1 = 1, ν2 = λ, ν3 = 1. Then, from ξ30 = 3θ 13αˆ 10 (ξ10 ) 23αˆ 20 (ξ20 ), (40) D − D − D ξ0j = ν jξ j, we obtain from Eq. (47) 3 The use of the same symbol λ for the affine parameter and the MST 1 parameter is due to the conventions in the literature, but should not lead ξ0j+1 = ν j j 1, j+1 −j 1, jξ j + ν j 1B jξ j 1 D − D − − − to any confusion; in particular, in this section λ is exclusively used as ν j+1 j, j+1αˆ j(ξ j) ν j j, j+1G jξ j (49) MST parameter, and we use redshift z to label lens planes. − h D − D i 4 1 In case the distance matrices are proportional to the unit matrix, the = ν j+1 j 1, j+1 −j 1, jξ j + B jξ j 1 j, j+1αˆ j(ξ j) . transformation reduces to the known one in standard lensing. Note that D − D − − − D ˆ ˆ 5 the transformation (36) implies the transformation ψ10 (ξ1) = λψ1(ξ1) + This relation is unique for a general deflection law αˆ 2. However, if t ξ1G1ξ1/2 for the deflection potential. For isotropic distance matri- the “strong lens plane” at z2 only yields a deflection linear in the impact ˆ ˆ ces, G1 = (1 λ)D2/(D1D12) , so that ψ10 (ξ1) = λψ1(ξ1) + (1 vector (i.e., if this lens plane is in fact just a “weak” deflector), then the −2 I − λ)(D1D2/D12) θ /2. According to (15), this then implies for the scaled form of αˆ 20 no longer is uniquely determined. This comment also applies | | 2 deflection potential ψ0(θ) = λψ(θ) + (1 λ) θ /2, as in standard lens to the more general case discussed below. Hence, we implicitly assume theory. − | | that all “strong lens planes” yield indeed a non-linear deflection law.

A54, page 7 of 10 A&A 624, A54 (2019)

The terms αˆ j cancel each other. The terms ξ j 1 yield the Taking the difference of τ0 λτ, we first note that the terms ∝ ∝ − − ∝ condition ν j+1 = ν j 1, and equating the terms ξ j yields ψˆ i drop out. Second, we note that the terms containing products − ∝ of ξi and ξi+1 also cancel, since νiνi+1 = λ. Thus we find that 1 1 G j = (1 ν j+1/ν j) −j, j+1 j 1, j+1 −j 1, j, (50) − D D − D − X ( N (1 + zi) t h 2 2 i c(τ0 λτ) = ξ ν λ Ci,i+1 ν Gi ξ which is symmetric according to Eq. (44) and thus represents a − i=1 2 i i − − i i tidal matrix. Thus, we obtain ν j = λ for j even, and ν j = 1 for j ) 2 t 1 t 1 1 odd. Correspondingly, + ν λ ξ − C− − ξ . (55) i+1 − i+1 Di,i+1 i,i+1Di,i+1 i+1 1 1 G j = (1 1/λ) −j, j+1 j 1, j+1 −j 1, j for j even; − D D − D − 1 1 The term i = 1 of the first sum vanishes, since ν1 = 1, and G j = (1 λ) −j, j+1 j 1, j+1 −j 1, j for j odd. (51) − D D − D − Eq. (39) holds. The final term (i = N) of the second sum depends only on the source position, and thus corresponds to the function We note that Eqs. (39) and (43) are special cases of Eq. (51). F(ξ ) previously mentioned. Since this term is of no interest, Hence we find that the MST in multiple (lens and source) N+1 we simply drop it from now on. Relabeling the index of the sec- plane gravitational lensing exhibits a curious behavior: The scal- ond sum as i i 1, we then get ing 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 "X # t N (1 + zi) 2 candle” or “standard rod” in one of the planes with j odd can- c(τ0 λτ) = ξi (νi λ)Ki ξi, (56) not be used to break the degeneracy related to the MST, as the − i=2 2 − images of these sources are unaffected by the MST. The pref- where the matrices K are given as actor in the tidal matrices G j are positive on every other plane, i and negative on the remaining ones. If one disregards the per-  ν2  turbations between lens planes, so that the distance matrices i j 1 + zi 1 1 t 1 1  i  D Ki = − i− 1,i Ci− 1,i i− 1,i + Ci,i+1   Gi. (57) reduce to angular diameter distances Di j, then the G j become +  2  1 zi D − − D − − νi λ scalars proportional to the density of uniform mass sheets; in this − case, positive and negative densities of these sheets alternate. 1 1 1 Since, according to the definition (17), Ci− 1,i i− 1,i = i 1 i− , − D − D − D we can rewrite Ki as 4.4. Transformation of the time delay  ν2  1 + zi 1 1 t 1  i  ff K = − − − + C   G . We next consider how the MST a ects the time delays. For that, i i 1,i i 1 i i,i+1  2  i (58) 1 + zi D − D − D − ν λ we assume to have a source on plane number N +1, with its light i − being deflected in N main lens planes. The corresponding LTTF From Eq. (29), is given in Eq. (19), where ξN+1 is the position of the source in its source plane. To obtain the corresponding function cτ0 after the MST, we 1 + zi 1 1 t 1 1 1 1 1 − i− 1,i i 1 i− = i−,i+1 i 1,i+1 i− 1,i −i,i+1 i+1 i− first need to consider the transformation of the potential time 1 + zi D − D − D D D − D − −D D D delay. That is, we need to find the transformed deflection poten- (59) ψˆ ξ ψˆ ξ = α ξ tial i0( i0), which needs to satisfy ξi0 i0( i0) ˆ i0( i0). For this, ˆ ˆ ∇ t we make the ansatz ψi0(ξi0) = aψi(ξi0/νi) + ξi0 Giξi0/2. Taking the is obtained. Noting that the final term is Ci,i+1, we obtain with gradient yields ξ ψˆ 0(ξ0) = (a/νi)αˆ i(ξ0/νi) + Giξ0. This is seen to Eq. (50) that ∇ i0 i i i i agree with αˆ i0(ξi0) in Eq. (48), provided a = νiνi+1 = λ. Thus,   ν2  !   i  νi+1  1 1 Ki = 1   1  −, + i 1,i+1 − , . (60) 1 t   2  ν  i i 1 − i 1 i ψˆ 0(ξ0) = λψˆ (ξ0/ν ) + ξ0 G ξ0. (52) − νi λ − i D D D − i i i i i 2 i i i − ν = ν = λ We then obtain for the transformed LTTF However, the prefactor vanishes: if i is odd, i 1, i+1 , and thus " XN t 1 1 1  2  ! ! cτ0 = (1 + zi) νiCi,i+1ξi νi+1 −, + ξi+1 C−, +  ν  νi+ 1 i=1 2 − Di i 1 i i 1  i  1 = λ = . 1  2  1 1 (1 ) 0 − ν λ − νi − 1 λ − 1 i ν C ξ ν − ξ − − × i i,i+1 i − i+1Di,i+1 i+1 ν2 # If i is even, νi = λ, νi+1 = 1, and ˆ i t λψi(ξi) ξiGiξi . (53) − − 2  2  ! 2 ! !  ν  νi+ λ 1  i  1 = = . 1  2  1 1 2 1 0 − ν λ − νi − λ λ − λ We now show that i − −

τ0 = λτ + F(ξN+1), (54) Therefore, Ki = 0 for all i, which completes our proof of the validity of Eq. (54). Since the result is independent of the plane which means that the transformed LTTF just scales by a factor on which the source is located – the time delay scales for all λ, independent of the plane on which the source is located, plus source planes with λ – we see that all time delays are scaled by a function which only depends on the location of the source, the factor λ under the MST. In particular this implies that the and thus cancels when considering time delays, i.e., diﬀerences degeneracy due to the MST cannot be broken from measuring between τ for pairs of multiple images of the source. time delay ratios.

A54, page 8 of 10 P. Schneider: Multi-plane gravitational lensing

5. Discussion plane to plane. Since such a mass sheet changes the slope of the total mass distribution, it means that this slope change also In this paper we have considered several aspects of the general- alternates. If one now makes the perhaps plausible assumption ized multi-plane gravitational lensing equation. In contrast to the that the shape of the mean mass profiles of lenses is the same, treatment in Schneider et al.(1992) and more recent papers (e.g., this alternating slope change would violate the universality of McCully et al. 2014), we treat the light propagation between the mean mass profile. Thus, in multi-plane lensing, the mass- ff main lens planes with a continuous formalism, o ered by the sheet generacy may be more easily lifted than in the case of a optical tidal equation, instead of slicing up the matter into sev- 6 single lens plane only. eral “weak-lensing” lens planes . For this, we made use of the We hope that the results obtained here will be useful for formalism of light propagation in arbitrary spacetimes, as given further theoretical studies of generalized multi-plane lensing, as in SSE. As a result, the distance matrices between lens planes well as for modeling lens systems in which more than one main are not written in terms of recursion relations, but as solutions deflector affects the imaging properties between observer’s sky of the optical tidal equation; the explicit solution in terms of an and the source plane. integral over the tidal field caused by large-scale density inhomogeneities along the line-of-sight is provided in Eq. (11). The time-delay function for generalized multi-plane lens- Acknowledgements. The author thanks Thomas Collett, Dominique Sluse, and ing was derived, using the same arguments as employed in Sherry Sutu for helpful comments and discussions, and the anonymous referee for a constructive report. This work was supported in part by the Deutsche Schneider et al.(1992) for the derivation of the time delay in Forschungsgemeinschaft under the TR33 “The Dark Universe”. ordinary multi-plane 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 References the deflection angles. In other word, our expression for τ yields Bartelmann, M. 2010, Classical Quantum Gravity, 27, 233001 the light travel time of a kinematically possible ray with specified Bartelmann, M., & Schneider, P. 2001, Phys. Rep., 340, 291 impact vectors ξ in the main lens planes, up to an additive con- Bartelmann, M., Huss, A., Colberg, J. M., Jenkins, A., & Pearce, F. R. 1998, i A&A, 330, 1 stant. Physical light rays are those for which the light-travel time Bayliss, M. B., Johnson, T., Gladders, M. D., Sharon, K., & Oguri, M. 2014, is stationary; this allows the derivation of an iterative lens equa- ApJ, 783, 41 tion which relates the impact vectors of three consecutive main Blandford, R., & Narayan, R. 1986, ApJ, 310, 568 lens planes to the deflection angle in the middle one of those. Chae, K.-H., Mao, S., & Augusto, P. 2001, MNRAS, 326, 1015 We have shown that this form of the lens equation is equivalent Collett, T. E., & Auger, M. W. 2014, MNRAS, 443, 969 Collett, T. E., Marshall, P. J., Auger, M. W., et al. 2013, MNRAS, 432, 679 to the more standard one which contains the impact vectors and Etherington, I. M. H. 1933, Philos. Mag., 15, 761 deflection angles of all earlier lens planes. This consecutive lens Falco, E. E., Gorenstein, M. V., & Shapiro, I. I. 1985, ApJ, 289, L1 equation is probably preferable for the use in ray-tracing sim- Gavazzi, R., Treu, T., Koopmans, L. V. E., et al. 2008, ApJ, 677, 1046 ulations (see, e.g. Petkova et al. 2014; Jaroszynski´ & Skowron Greene, Z. S., Suyu, S. H., Treu, T., et al. 2013, ApJ, 768, 39 7 Hilbert, S., Hartlap, J., White, S. D. M., & Schneider, P. 2009, A&A, 499, 31 2016) . Hoekstra, H. 2013, ArXiv e-prints [arXiv:1312.5981] Finally, we showed that the generalized multi-plane lens- Jain, B., Seljak, U., & White, S. 2000, ApJ, 530, 547 ing admits a mass-sheet transformation (MST) which leaves all Jaroszynski, M., & Kostrzewa-Rutkowska, Z. 2014, MNRAS, 439, 2432 observables but the time delay invariant. In contrast to ordinary Jaroszynski,´ M., & Skowron, J. 2016, MNRAS, 462, 1405 lensing, the MST corresponds to adding a tidal matrix in each Keeton, C. R., Kochanek, C. S., & Seljak, U. 1997, ApJ, 482, 604 Kneib, J.-P., & Natarajan, P. 2011, A&ARv, 19, 47 main lens plane. We obtained the curious behavior that the uni- Kochanek, C. S. 2003, ApJ, 583, 49 form isotropic scaling of the source/lens planes, which is the key Kochanek, C. S. 2006, in Saas-Fee Advanced Course 33: Gravitational Lensing: aspect of the MST, alternates between planes; in every second Strong, Weak and Micro, eds. G. Meylan, P. Jetzer, P. North, et al., 91 plane, the scaling corresponds to the MST parameter λ, in the Kovner, I. 1987, ApJ, 316, 52 McCully, C., Keeton, C. R., Wong, K. C., & Zabludoff, A. I. 2014, MNRAS, other half of the planes, the scaling is unity. In particular, this 443, 3631 implies that the magnification of sources living on the odd planes McCully, C., Keeton, C. R., Wong, K. C., & Zabludoff, A. I. 2017, ApJ, 836, 141 with scaling factor unity, is unaffected by the MST. All time Meneghetti, M., Bartelmann, M., Dahle, H., & Limousin, M. 2013, Space Sci. delays – i.e., for sources in all main planes – scale as λ under Rev., 177, 31 the MST. Munshi, D., Valageas, P., van Waerbeke, L., & Heavens, A. 2008, Phys. Rep., 462, 67 This curious behavior of the MST in multi-plane lensing Petkova, M., Metcalf, R. B., & Giocoli, C. 2014, MNRAS, 445, 1954 may indeed offer a way to break the corresponding degener- Puchwein, E., & Hilbert, S. 2009, MNRAS, 398, 1298 acy, at least in a statistical way. As we discussed before, in the Refsdal, S. 1964, MNRAS, 128, 307 case of vanishing perturbations between the main lens planes, Refsdal, S. 1970, ApJ, 159, 357 Schneider, P. 1997, MNRAS, 292, 673 the MST corresponds to mass sheets of alternating sign from Schneider, P. 2006, in Saas-Fee Advanced Course 33: Gravitational Lensing: 6 Strong, Weak and Micro, eds. G. Meylan, P. Jetzer, P. North, et al., 269 Whereas these two treatments are equivalent (indeed, as shown in Schneider, P. 2014, A&A, 568, L2 SSE, the slicing into weak-lensing planes corresponds to a discretized Schneider, P. 2016, A&A, 592, L6 version of the optical tidal equation), the continuous formalism is more Schneider, P., & Sluse, D. 2013, A&A, 559, A37 convenient for analytical calculations – for example, obtaining a result Schneider, P., & Sluse, D. 2014, A&A, 564, A103 such as (25) using the discretized version is probably extremely tedious. Schneider, P., Ehlers, J., & Falco, E. E. 1992, Gravitational Lenses (Berlin: 7 The advantage of Eqs. (21) and (10) is twofold: First, in order to cal- Springer-Verlag) culate the impact vectors in all N planes requires of order N2/2 multipli- Seitz, S., & Schneider, P. 1994, A&A, 287, 349 cations for each light ray when Eq. (10) is used, compared to about 3N Seitz, S., Schneider, P., & Ehlers, J. 1994, Classical Quantum Gravity, 11, 2345 Smith, M., Bacon, D. J., Nichol, R. C., et al. 2014, ApJ, 780, 24 multiplications for Eq. (21). More significant, however, is the fact that Suyu, S. H., Marshall, P. J., Auger, M. W., et al. 2010, ApJ, 711, 201 Eq. (21) allows one to save on memory: whereas Eq. (10) requires the Suyu, S. H., Auger, M. W., Hilbert, S., et al. 2013, ApJ, 766, 70 information of all lens planes for each ray, one can treat with Eq. (21) Treu, T. 2010, ARA&A, 48, 87 a large set of rays, tracing them from plane to plane, and in each step Wong, K. C., Keeton, C. R., Williams, K. A., Momcheva, I. G., & Zabludoff, A. require only the information on a single lens plane. I. 2011, ApJ, 726, 84

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Appendix A: Distance matrices in terms of peculiar Subtracting from this the transport Eq. (5) for Di, we are left gravitational potential with 2 In this appendix we derive the expression (11) for the distance d Bi dBi dDi D + 2 = B D . (A.4) matrices in an inhomogeneous Universe. As usual in cosmolog- dλ2 i dλ dλ Tsm i i ical weak lensing, we work in comoving coordinates, and there- Inserting D (λ) = a(λ) f (χ χ ), and using the differentiation fore replace the affine parameter λ by the comoving distance χ, i k − i where dχ/da = c/(a2H), and a is the cosmic scale factor nor- rules (A.1), this turns into malized to unity today. From the Robertson–Walker metric and d2B dB d f χ = i χ χ + i k = B 4 χ χ . the condition for radial null geodesics, we have a d c dt, 2 fk( i) 2 sm ia fk( i) (A.5) and with Eq. (2) follows dχ = dλ/a2. These relations then imply− dχ − dχ dχ T − that We this relation, we find for X (χ):= f (χ χ )B (χ): i k − i i d dχ d 1 d = = , d2X d2B dB d f dλ dλ dχ a2 dχ i = i f (χ χ )+2 i k KX = KX + a4X , (A.6) dχ2 dχ2 k − i dχ dχ − − i Tsm i d2 1 d2 2 H d = , (A.1) dλ2 a4 dχ2 − a3 c dχ where we made use of Eq. (12). This differential equation can be transformed into an integral equation, using the method of d2 2H d 1 d2 a + = Green’s functions, to read dλ2 c dλ a3 dχ2 · Z χ 4 X (χ) = f (χ χ ) + dχ0 f (χ χ0) (χ0)a (χ0)X (χ0). (A.7) The angular-diameter distance Di is related to the comoving i k − i I k − Tsm i angular diameter distance by D = a f (χ χ ), where χ is χi i k − i i the comoving distance corresponding to the affine parameter λi. By differtiating twice, one can easily show that Eq. (A.7) indeed Applying Eq. (5), we get is a formal solution of Eq. (A.6), with the correct initial condi- X χ = X χ / χ = 2 2 2 2 tion i( i) 0, d i( i) d . If we now use the expression d Di d fk da d fk d a 3 H0 Ωm I = a + 2 + fk = fk. (A.2) (6) for sm, neglecting the final term which is a derivative along dλ2 dλ2 dλ dλ dλ2 −2 c a4 the line-of-sightT and thus cancels in the integration, and replac- With da/dλ = (dχ/dλ)(da/dχ) = H/c, and ing the derivatives w.r.t. ξ by those w.r.t. comoving transverse coordinates, we get 2 2 d a dH da dH 1 dH Z χ = = = 2 2 2 X χ = χ χ χ0 χ χ0 H φ χ0 X χ0 . dλ c dλ dλ c da 2c da i( ) fk( i) 2 d fk( ) ( ( )) i( ) (A.8) 2 ! − I − c χi − H Ω + ΩΛ 1 3Ω = 0 2 m − m , 2c2 a3 − a4 Using i = a Xi, we arrive at Eq. (11). We also note that the large-scaleD structure component of the optical tidal matrix can and making use of the relations (A.1), Eq. (A.2) reduces to alternatively written in the form (see Seitz & Schneider 1994) Eq. (12). " !# Next we write the distance matrix as i = Di Bi, so that 2 2 Γ1 Γ2 D sm = 2πGa (ρ ρ¯) + , (A.9) Bi describes the deviation of i from the unperturbed distance T −c2 a4 − Γ2 Γ1 matrix D 8. This factorizationD transforms Eq. (8) into − iI with Γ1 = (φ,11 φ,22)/2 and Γ2 = φ,12, where the partial deriva- 2 2 − d Bi dBi dDi d Di tives are with respect to transverse comoving coordinates. Thus, D + 2 + B = + B D . (A.3) dλ2 i dλ dλ i dλ2 Tbg Tsm i i one can replace H(φ) in Eq. (A.8) by the bracket in Eq. (A.9).

8 We note that B B corresponds to the matrix B in McCully et al. ≡ 0 (2014), and the Bi corresponds, up to a prefactor, to their matrices C.

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