Gravitational lenses in arbitrary space-times
IIFT-UAM/CSIC-20-155
Gravitational lenses in arbitrary space-times
Pierre Fleury ∗1 , Julien Larena †2 , and Jean-Philippe Uzan ‡3,4 Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid,Cantoblanco, 28049 Madrid, Spain Department of Mathematics and Applied Mathematics, University of Cape Town,Rondebosch 7701, South Africa Institut d’Astrophysique de Paris, CNRS UMR 7095, Sorbonne Universités,98 bis Boulevard Arago, 75014 Paris, France Institut Lagrange de Paris, Sorbonne Universités,98 bis, Boulevard Arago, 75014 Paris, France
November 20, 2020
Abstract
The precision reached by current and forthcoming strong-lensing observa-tions requires to accurately model various perturbations to the main deflector.Hitherto, theoretical models have been developed to account for either cos-mological line-of-sight perturbations, or isolated secondary lenses via themulti-plane lensing framework. This article proposes a general formalismto describe multiple lenses within an arbitrary space-time background. Thelens equation, and the expressions of the amplification and time delays, arerigorously derived in that framework. Our results may be applied to a widerange of set-ups, from strong lensing in anisotropic cosmologies, to line-of-sightperturbations beyond the tidal regime. ∗ pierre.fl[email protected] † [email protected] ‡ [email protected] a r X i v : . [ g r- q c ] N ov ravitational lenses in arbitrary space-times Contents1 Introduction 22 Preliminary discussion: reference space-time and lenses 33 One lens in an arbitrary reference space-time 5 N lenses in an arbitrary reference space-time 13 N lenses . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Amplification matrix for N lenses . . . . . . . . . . . . . . . . . . . . 154.3 Time delay for N lenses . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Example: lenses in an under-dense Universe . . . . . . . . . . . . . . 18 (45) for the time-delay matrix 211 Introduction Gravitational lensing (Schneider et al., 1992) has become a key probe of the content,structure, and physical laws of our Universe. While weak lensing teaches us aboutthe distribution of matter on cosmic scales (e.g. Asgari et al., 2020; Gatti et al.,2020), strong lensing lies amongst the best tools to measure today’s cosmic expansionrate H (Refsdal, 1964; Wong et al., 2019), and encapsulates valuable informationon the small-scale distribution and nature of the dark matter (Diego et al., 2018;Harvey et al., 2020; Blum et al., 2020).The quality of current imaging and data analysis requires an equally elaboratetheoretical modelling of the strong lenses. That means complex models for the massdistribution of the main deflector, responsible for, e.g., the formation of multipleimages of the same source, but also of the secondary deflectors which may perturbthe effect of the main lens (Keeton et al., 1997; Tihhonova et al., 2018; Çağan Şengület al., 2020; Li et al., 2020). Such perturbations are referred to as line-of-sight effects.Two distinct frameworks have been developed to model line-of-sight perturbationsin strong gravitational lensing. The first approach (Kovner, 1987; Seitz and Schneider,1994; Bar-Kana, 1996; Schneider, 1997; Birrer et al., 2017) consists in treatingsecondary deflectors in the tidal regime, i.e., adding external convergence and shear ravitational lenses in arbitrary space-times a priori unable to deal with several lenses in amore general cosmological model. The present article proposes to fill this gap with a generalised multi-plane formal-ism, where one or several lenses may be placed in any space-time background. Thisformalism encapsulates all the key lensing observables into a single versatile language.Our results may be applied to a wide range of set-up; three specific examples will beprovided in this article: (1) one lens with cosmological perturbations; (2) one lensin anisotropic cosmology, which has benefited of renewed interest recently (Migkaset al., 2020; Secrest et al., 2020; Fosalba and Gaztañaga, 2020); and (3) multiplelenses in an otherwise under-dense Universe. Furthermore, this work establishesthe necessary tools to accurately describe line-of-sight corrections in strong lensingbeyond the standard convergence and shear (Fleury et al., 2019b, 2020).The article is organised as follows. We start in section 2 with a discussion on thenature of the gravitational fields encountered by light beams, thereby defining thedichotomy between reference space-time and lenses. In section 3 we consider thecase where a single lens is placed within an arbitrary reference space-time, beforemoving to the general N -lens case in section 4.We adopt units for which the speed of light is unity. Bold symbols ( α , β , . . . ) referto two-dimensional vectors, and bold calligraphic symbols ( A , D , . . . ) to matrices. Let a source of light and an observer be placed in an arbitrary space-time. We call light beam the set of light rays that connect the source to the observer; a beam mayhave several components if the source is multiply imaged. Hereafter, we shall restrictour analysis to light propagation in locally weak gravitational fields. This meansthat we do not consider the vicinity of compact objects such as neutron stars orblack holes. It does not mean, however, that space-time is nearly Minkowskian; acounter-example is FLRW.As a beam of light propagates from the source to the observer, it may bedeflected, distorted, and somehow split, by the gravitational field that it experiences.The formalism proposed in this article relies on the broad distinction between twocategories of gravitational fields: smooth fields on the one hand, and rough fields on the other hand. In smooth-field regions, the light beam is continuously (andpossibly strongly) distorted, but its integrity is preserved. In rough-field regions, the Note however that McCully et al. (2014, 2017) proposed a hybrid framework where some ofthe lens planes are turned into “tidal planes”, in order to model, e.g., cosmological perturbationstogether with isolated lenses. ravitational lenses in arbitrary space-times r (cid:29) r E smooth fieldsrough field r E r E Figure 1.
Illustrating the dichotomy between smooth and rough gravity fields. light beam experiences sudden deflections, which may give rise to multiple images.Thus, rough-field regions correspond to what is usually referred to as lenses , whilesmooth-field regions constitute a reference space-time , where light propagates fromone lens to the next one. In the standard multi-plane lensing framework, the referencespace-time is either Minkowski or FLRW. We shall not make this assumption here. The following considerations are illustrated in fig. 1.Specifically, we shall say that a gravitational field is smooth if the light beam canbe considered infinitesimal within that field. In other words, a field is smooth if thecorresponding space-time curvature is mostly homogeneous within the light beam’scross section (Fleury et al., 2017, 2019a,b). Otherwise, we shall say that the fieldis rough. Let us illustrate this terminology with the example of a point mass M .The curvature that it produces at a distance r reads R µνρσ R µνρσ = 12(2 GM/r ) ;hence the typical scale over which it changes appreciably is r . As a light beamwith cross-sectional area A passes next to the mass, it experiences a smooth fieldif A (cid:28) r , and a rough field otherwise. The picture gets slightly more complicatedif we recall that the beam’s cross-sectional area A ( r ) to depend on r , due to lightfocussing. Denoting A the unlensed area of the beam, its lensed counterpart reads A ( r ) = µ ( r ) A , where µ ( r ) ≡ (1 − r /r ) − is the point-lens magnification, and r E its Einstein radius. The smooth-field condition then becomes A (cid:28) r (1 − r /r ).Another example is light propagating through a diffuse distribution of matter,such as a gas cloud or a dark-matter halo. In that situation, space-time curvatureis effectively dominated by its Ricci component (Fleury et al., 2017), which ismostly controlled by the matter density field ρ . Consequently, the correspondinggravitational field is smooth if ρ is almost homogeneous on the scale of the beam’scross-section, and rough otherwise.Finally, we note that gravitational fields are not always directly generated bymatter distributions in a Newtonian-like manner. The most immediate example isgravitational waves, which are nothing but propagating curvature. In our terminology,a gravitational wave is a smooth field if its wave-length is much larger than thebeam’s diameter, and rough otherwise. Perhaps even more relevant to lensing arethe infinite-wavelength gravitational waves that characterise anisotropic cosmologies. Rigorously speaking, the set of all smooth-field regions traversed by a light beam does notconstitute a physically well-defined space-time. In particular, it does not necessarily satisfy Einstein’sequation, whose right-hand side should also include the matter clumps of the rough-field regions. ravitational lenses in arbitrary space-times α s o u r ce p l a n e θ observer l e n s p l a n e β x si fiducial ray unlensed rayphysical ray α r σ undeflected ray Figure 2.
Lensing by a single thin lens in an arbitrary reference space-time. All the linesare geodesics of the reference space-time. The unlensed ray (dashed line) connects theobserver to the source (star). Its angular separation with the arbitrary fiducial ray (dottedline) is denoted β . The physical ray (thick solid line) is separated from the fiducial ray by θ = β + α ; it is deflected by ˆ α in the lens plane. The undeflected ray (light solid line) isthe continuation of the physical ray if it were not deflected. The lens plane and sourceplane (gray) are orthogonal to the fiducial ray. The lens is not represented in the figure. For instance, the anisotropic expansion of Bianchi I models produces a homogeneousWeyl curvature, i.e. a smooth tidal field, which continuously shears and rotates lightbeams as they propagate (Fleury et al., 2015).In the following, we shall concretely incorporate the distinction between smoothand rough fields, i.e., reference space-time and lenses, in the multi-plane lensingframework, first with a single lens (section 3), and then with N lenses (section 4). Consider a light beam only propagating in smooth-field regions, except one localisedrough-field region modelled as a thin gravitational lens. This situation, depictedin fig. 2, has been studied by several authors in order to evaluate the impact ofcosmological perturbations on the properties of a strong lens. Kovner (1987) refersto it as a thick lens in the telescope approximation ; Schneider et al. (1992); Seitz andSchneider (1994) calls it generalised quadrupole lens ; Bar-Kana (1996) writes about lensing with large-scale structure ; and Schneider (1997), whose presentation is theclosest to ours, simply calls it the cosmological lens . The corresponding formalismhas been recently applied to the weak lensing of Einstein rings by Birrer et al.(2017), with the perspective of novel synergies between weak- and strong- lensingobservations (Kuhn et al., 2020).The results derived in this section include, or are formally equivalent to, theaforementioned works’. However, we insert them in a broader context and extendtheir range of application — a novel example is proposed in section 3.5. Furthermore,to our knowledge we propose in section 3.3 the first rigorous proof of the expressionof the time delay for lensing within an arbitrary reference space-time. ravitational lenses in arbitrary space-times First consider an arbitrary null geodesic of the referencespace-time, called the fiducial ray . We take it to be close enough to the actual ray sothat their angular separation θ can be treated as a small angle. At each point of thefiducial ray, we define a portion of plane orthogonal to the ray. In each plane, weassume that all the distances that will be considered are small enough for positionsto be defined unambiguously. We call s the physical transverse position of the sourcein the relevant plane (source plane).The unlensed ray is the geodesic of the reference space-time connecting the sourceto the observer; we call β its angular separation with respect to the fiducial ray.Hence, β represents the direction in which the source would be observed in thereference space-time, without the lens.The physical ray is made of two portions of geodesics of the reference space-time,which form an angle ˆ α in the lens plane, called deflection angle . Importantly, ˆ α isdefined in the rest frame of the lens; it is thus subject to aberration effects whenevaluated in another frame. We denote with θ the separation between the physicalray and the fiducial ray, i.e. the actual observed direction of the image, in thepresence of the lens. The difference α ≡ θ − β , not to be confused with ˆ α , is the displacement angle . The analogue of α in the source plane, i.e. the difference betweenthe emission directions of the physical and unlensed rays, is denoted σ .We finally introduce a couple of relevant transverse vectors: x (resp. r ) representsthe intersection between the lens plane and the physical ray (resp. unlensed ray); and i denotes the intersection between the source plane and the undeflected continuationof the physical ray. In other words, i is the position of a source that would beobserved in the direction θ in the absence of the lens. In the reference space-time, by definition, the aforementionedrays are considered infinitesimally close to each other. Thus, the relative behaviourof any two of these rays can be described by the Sachs framework (see, e.g., Fleury,2015, Chapt. 2), which relies on the geodesic deviation equation. A key featureof that formalism is the existence of a matrix, called the
Jacobi matrix D , whichgeneralises the notion of angular-diameter distance. If two rays of the referencespace-time emerge from, or converge to, a point (a) with angular separation θ a , thentheir transverse separation x b at another point (b) reads x b = D a h b ˙ x a = D a h b ω a θ a . (1)In eq. (1), ˙ x ≡ d x / d λ is the derivative of x ( λ ) with respect to an affine parameter λ along the fiducial ray. The cyclic frequency ω a of light measured by (a) appears toconvert affine-parameter derivatives into observed angles, ˙ x = ω θ .The Jacobi matrix D a h b and the product ω a θ a are frame-independent. Thepresence of ω a is thus essential to account for aberration effects in θ a . We stress thata,b are not indices; they represent the positions where x , θ are evaluated. Technically speaking, this plane corresponds to the Sachs screen space, spanned by the Sachsbasis associated with the fiducial ray. See, e.g., Fleury (2015, Chapt. 2) for further details. ravitational lenses in arbitrary space-times h b” in D a h b is designed to clearly indicate that thetwo rays merge at (a). If the roles of (a) and (b) were swapped, i.e. if we consideredtwo rays merging at (b) instead of (a), then their separation at (a) would read x a = D a i b ω b θ b , with D a i b = − D Ta h b , (2)where the T superscript indicates the matrix-transpose operation. Equation (2) isknown as Etherington’s reciprocity law (Etherington, 1933); see Fleury (2015, § 2.2.3)for a derivation using the same conventions as this article. From the definition of the Jacobi matrix, we can express theposition of the image i with respect to the source s in two different ways, i − s = ω o D o h s α = ω d D d h s ˆ α , (3)where o, d, s respectively refer to the observer, deflector (or lens), and source positions.The deflection angle ˆ α ( x ) depends on the position x where the physical ray piercesthe lens plane. For a thin lens, ˆ α is directly related to the surface mass density Σ( x )in the lens plane (Schneider et al., 1992),ˆ α ( x ) = Z d y G Σ( y ) x − y | x − y | , (4)where | . . . | denotes the Euclidean norm. The deflection angle can also be expressedas the gradient of ˆ ψ , which is twice the projected gravitational potential of the lens,ˆ α ( x ) = d ˆ ψ d x , ˆ ψ ( x ) ≡ Z d y G Σ( y ) ln | x − y | . (5)The latter indeed satisfies the projected Poisson equation ∆ ˆ ψ = 8 πG Σ( x ), where ∆denotes the two-dimensional Laplacian.Since x = D o h d ω o θ , we conclude from eq. (3) that the lens equation reads β ( θ ) = θ − (1 + z d ) D − h s D d h s ˆ α ( D o h d ω o θ ) , (6)with z d the observed redshift of the lens, 1 + z d = ω d /ω o . Equation (6) is fully general, provided that light indeedencounters only one rough-field region, which can be modelled as a thin lens. Noassumption is made on the reference space-time apart from the smoothness of itscurvature. Hence, the deflector’s redshift z d must not be understood as a cosmologicalredshift in general.Since ˆ α represents the deflection angle in the rest frame of the lens, it is bydefinition independent of the lens’ motion. In that context, the redshift term 1 + z d encodes aberration effects. For instance, if the lens recedes from the observer, thenits redshift increases, and the net observed deflection (1 + z d ) ˆ α increases as well.Let us finally stress that β is fundamentally linked to the reference space-time.Indeed, β represents the direction in which one would observe the source without the ravitational lenses in arbitrary space-times β and the new notion of unlensed direction.This will be illustrated with two concrete examples in sections 3.4 and 3.5 From the lens equation (6), we immediately derive the amplification matrix of thesystem, i.e. the Jacobian matrix of the lens mapping θ β ( θ ), A ( θ ) ≡ d β d θ = − ω d D − h s D d h s ˆ H ( D o h d ω o θ ) D o h d , (7)where ˆ H ij ≡ ∂ ˆ ψ/∂x i ∂x j is the Hessian matrix of the deflection potential. Note that,contrary to the latter, A is generally not symmetric, due to the coupling betweenthe lens and the reference space-time. The time delay between the images of strong-lensing systems is a key observablein astronomy and cosmology (Wong et al., 2019). Its expression in the presenceof cosmological perturbations can be found in Schneider et al. (1992); Bar-Kana(1996); Schneider (1997), but without a direct proof. Here we propose a rigorousderivation of the time-delay formula, inspired from the wave-front method introducedby Schneider et al. (1992, § 5.3).Let ∆ t denote the time delay between the physical signal and the unlensed signal.In other terms, if a signal emitted by the source reached the observer at t in theabsence of the lens, then it would reach it at t = t + ∆ t in the presence of the lens.Note that ∆ t is not directly observable, because it involves two signals that propagatein different space-times. It is, however, a convenient theoretical intermediate.The time delay is conveniently parameterised as ∆ t ( θ , β ), because it generallydepends on the source position s = ω o D o h s β , and on the point x = ω o D o h d θ where the physical ray pierces the lens plane. In terms of that parameterisation,the observable time delay between two images A and B of the same source reads∆ t AB ( β ) = t A ( β ) − t B ( β ) = ∆ t ( θ A , β ) − ∆ t ( θ B , β ). The time delay between two signals emitted simulta-neously in different directions depend on the source’s position s . Indeed, as depictedin fig. 3, if the source lies at s + d s , then everything happens as if the two signalswere emitted from s but with a slight relative delay d t s = σ · d s . Thus, if two signalsemitted from s are observed with a time delay ∆ t , then shifting the source by d s results in an additional delayd∆ t = (1 + z s )d t s = (1 + z s ) σ ( s ) · d s , (8)where the redshift factor 1 + z s = d t s / d t o allows for the fact that d t s and d∆ t weredefined in distinct frames. ravitational lenses in arbitrary space-times σ d t s d s wave-fronts Figure 3.
Two signals, emitted simultaneously with anangle σ from s + d s , are equivalent to two signals emittedwith a relative delay d t s = σ · d s from s . Here, every-thing happens as if the physical signal (solid) was emittedslightly before the unlensed signal (dashed), so that d t s < σ and d s . From eq. (8), we see that the expression of the timedelay ∆ t may be obtained by a line integral of the vector field σ ( s ), between anarbitrary reference point and s . In general, the line integral of a vector field dependson the path along which the integral is performed; except if the vector field is agradient, which turns out to be the case. Namely, the emission angle σ between thelensed and unlensed rays depends on the source position s as(1 + z s ) σ ( s ) = d T d s , (9)where the scalar function T reads T ( θ , β ) ≡
12 ( θ − β ) · T ( θ − β ) − (1 + z d ) ˆ ψ [ ω o D o h d θ )] , with T ≡ ω o D To h d D − h s D o h s . (10)(11)In eq. (9), the derivative d T / d s must be understood as a total derivative, in thesense that it accounts for the variation of both natural variables θ , β of T under avariation of s . A detailed proof of eq. (9) is provided in appendix A. Combining itwith eq. (8), we immediately conclude that d∆ t = d T , that is ∆ t ( θ , β ) = T ( θ , β ) + cst . (12)Therefore, the time delay between two different images A, B of the same source reads∆ t AB ( β ) = T ( θ A , β ) − T ( θ B , β ). Note that any function of β could be added to theexpression of T ( θ , β ) without changing the observable ∆ t AB .The time-delay matrix T generalises the more common notion of time-delaydistance to an arbitrary reference space-time. Indeed, if the latter is chosen asthe FLRW space-time, then T = ¯ τ , where ¯ τ ≡ (1 + z d ) ¯ D o h d ¯ D o h s / ¯ D d h s is usuallycalled the time-delay distance. Contrary to the Jacobi matrices that compose it, thetime-delay matrix is symmetric, T T = T . (13)The time-delay matrix is related to, but different from, the telescope matrix in-troduced by Kovner (1987), and then used by Schneider et al. (1992); Seitz and Having noticed that σ ( s ) is a gradient makes our derivation of the time-delay formula simplerand more general than the one originally proposed in Schneider et al. (1992). In particular, we donot need to introduce a reference source whose contribution would be later set to zero on a caustic. ravitational lenses in arbitrary space-times Just like time-like and space-like geodesics can be definedfrom a stationary-time and stationary-length principle, null geodesics may be definedfrom Fermat’s principle; see e.g. Schneider et al. (1992). In the present context,Fermat’s principle states that the arrival time of a physical ray is stationary withrespect to small variations of the position x where it pierces the lens plane. In otherwords, all things being fixed (notably s , r ), physical rays must satisfy ∂T /∂ x = .This property can be checked explicitly as follows. We first rewrite the first termof T as α · T α = (1 + z d ) ˆ α · ˆ T ˆ α , where ˆ T = ω d D o h d D − h s D d h s is also a symmetricmatrix. Then, using the identity ˆ T ˆ α = x − r we immediately find ∂ T∂ x = (1 + z d ) ˆ α ( x ) − d ˆ ψ d x , (14)so that physical light rays are indeed those whose deflection angle ˆ α is dictated bythe physics of the lens plane. Suppose that the reference space-time can be treated as a weakly perturbed homogeneous-isotropic FLRW model. The associated Jacobi matrix takes the form ω a D a h b = ¯ D a h b A a h b . (15)Recompile 28In eq. (15), ¯ D a h b = (1 + z b ) − f K ( χ b − χ a ) denotes the angular diameter distanceof (b) measured from (a) in the FLRW space-time, χ being the radial comovingdistance, and f K ( χ ) ≡ sin( √ Kχ ) / √ K , with K is the spatial-curvature parameter.The second quantity, A a h b = " − κ a h b − Re( γ a h b ) − Im( γ a h b ) − Im( γ a h b ) 1 − κ a h b + Re( γ a h b ) , (16)is the amplification matrix due to cosmological perturbations about FLRW, i.e.caused by the large-scale matter inhomogeneities in the Universe. Its key componentsare the convergence κ a h b and complex shear γ a h b ; we have neglected the anti-symmetricpart of A a h b , which represents the rotation of light beams with respect to paralleltransport, because it is of the order of γ if γ (cid:28) δ ( η, χ, x ) ≡ ( ρ − ¯ ρ ) / ¯ ρ , the convergence and ravitational lenses in arbitrary space-times κ a h b ( θ ) = 32 Ω m0 H Z χ b χ a d χ (1 + z ) f K ( χ b − χ ) f K ( χ − χ a ) f K ( χ b − χ a ) δ [ η − χ, χ, f K ( χ ) θ ] , (17) γ a h b ( θ ) = −
32 Ω m0 H Z χ b χ a d χ (1 + z ) f K ( χ b − χ ) f K ( χ − χ a ) f K ( χ b − χ a ) × Z R d x πx e ϕ δ [ η − χ, χ, f K ( χ ) θ + x ] , (18)where η denotes today’s conformal time, and ϕ is the polar angle of x = x (cos ϕ, sin ϕ ). In the lens equation (6), β stands for the direction in whicha source would be observed without the lens, but still in the presence of the weakcosmological perturbations. In the lensing literature, however, it is understandablymore common to write the lens equation in terms of the completely unlensed sourceposition. That direction, which we shall denote ¯ β = A o h s β , would be the observedposition of the source without both the strong lens and the cosmological perturbations,i.e. in an ideal FLRW Universe. In terms of ¯ β , eq. (6) reads¯ β ( θ ) = A o h s θ − f K ( χ s − χ d ) f K ( χ s ) A d h s ˆ α ( ¯ D o h d A o h d θ ) . (19)Equation (19) had already been obtained in the literature under various forms; forinstance, it is equivalent to eq. (6.7) of Kovner (1987), eq. (14) of Bar-Kana (1996),eq. (35) of McCully et al. (2014); it also coincides with eq. (3.3) of Birrer et al. (2017)in the critical-mass-sheet approximation defined therein. In the absence of cosmological perturbations, i.e. for A a h b = ,the lens equation would take the form¯ β = θ − ¯ α ( θ ) , (20)with ¯ α ( θ ) = [ f K ( χ s − χ d ) /f K ( χ s )] ˆ α ( ¯ D o h d θ ). In the presence of perturbations, thelens equation (19) is formally equivalent to eq. (20), if one replaces ¯ α ( θ ) by α eq ( θ ) = ( A o h s − ) θ + A d h s ¯ α ( A o h d θ ) (21) ≈ " ( A o h s − ) + d ¯ α d θ ( A o h d − ) θ + A d h s ¯ α ( θ ) , (22)which may be called the equivalent lens. This shows that, in full generality, a lensmodel ¯ α must be supplemented with 9 real parameters (3 κ a h b and 3 complex γ a h b )to properly account for cosmological perturbations. Degeneracies between theseparameters might occur if ¯ α enjoys symmetries.In general, α eq ( θ ) cannot be written as a gradient, which means that it does notderive from a potential. An alternative approach (Schneider, 1997) which circumventsthis issue consists in first applying a transformation β ˜ β ≡ A o h d A − h s β to thesource plane. The resulting equivalent lens then derives from a potential. ravitational lenses in arbitrary space-times Following the discussion of section 3.4.1, we shall con-sider amplifications with respect to the homogeneous Universe, rather than theamplification due to the sole lens. The corresponding “total” amplification matrix isdefined as A tot = d ¯ β / d θ = A o h s A , and reads A tot ( θ ) = A o h s − f K ( χ s − χ d ) f K ( χ d )(1 + z d ) f K ( χ s ) A d h s ˆ H ( ¯ D o h d A o h d θ ) A o h d (23)= A o h s − A d h s h − A ( A o h d θ ) i A o h d , (24)where A ≡ d ¯ α / d θ is the amplification matrix that would characterise the lens in theabsence of cosmological perturbations, i.e. in an ideal FLRW reference space-time.Equation (24) confirms that line-of-sight perturbations do not only add to the effectof a lens, but they also modify the effect of the lens itself. For a perturbed FLRW reference space-time, the general expres-sion (10) of the time delay becomes T ( θ , ¯ β ) = ¯ τ θ − A − h s ¯ β ) · A o h d A − h s A o h s ( θ − A − h s ¯ β ) − (1 + z d ) ˆ ψ ( ¯ D o h d A o h d θ ) (25)= ¯ T ( θ , ¯ β ) + δT ( θ , ¯ β ) (26)at first order in the cosmological perturbations, where, on the one hand¯ T ( θ , ¯ β ) ≡ ¯ τ (cid:20) | θ − ¯ β | − ¯ ψ ( θ ) (cid:21) (27)would be the time-delay function if the reference space-time were strictly homogeneousand isotropic, with ¯ τ = (1 + z d ) ¯ D o h d ¯ D o h s / ¯ D d h s the FLRW time-delay distance and¯ ψ ( θ ) = ( ¯ D d h s / ¯ D o h s ) ˆ ψ ( ¯ D o h d θ ) the background lensing potential; on the other hand, δT ( θ , ¯ β ) ≡
12 ¯ τ ( θ − ¯ β ) · h ( δ A o h s − δ A o h d )( θ + ¯ β ) − δ A d h s ( θ − ¯ β ) i , (28)where we denoted δ A a h b ≡ A a h b − for short, gathers all the corrections due tocosmological perturbations.Taken at face value, the correction δT thereby induced is quite complex. However,for practical analyses of time-delay observations, these may be reduced to a singleexternal convergence and shear . First, since the source position ¯ β is unknown andhence a free parameter in such analyses, it does not make any difference whetherone considers β = A − h s ¯ β instead. Second, if the lens model ¯ ψ ( θ ) is general enough, then it may effectively account for the corrections due to A o h d in the argument of ˆ ψ .In that context, the time-delay model that must be used reads T mod ( θ , β ) = ¯ τ (cid:20)
12 ( θ − β ) · A ext ( θ − β ) − ψ mod ( θ ) (cid:21) , (29) Note that we substituted the lens equation to obtain this expression of δT . A perhaps surprisingconsequence is that ∂T /∂ θ = using that expression. In particular, an elliptic model may not suffice, since γ o h d is generally not aligned with theintrinsic ellipticity of the lens. ravitational lenses in arbitrary space-times A ext ≡ A o h d A − h s A o h s ≈ − " κ ext + Re( γ ext ) Im( γ ext )Im( γ ext κ ext − Re( γ ext ) , (30)featuring an external convergence κ ext = κ o h d + κ o h s − κ d h s and shear γ ext = γ o h d + γ o h s − γ d h s . While the external convergence is routinely implemented in current time-delayanalyses (Gilman et al., 2020), we are not aware of any practical implementation ofthe external shear, although its relevance was suggested by McCully et al. (2017). As a second illustration, consider the case of a lens placed in a homogeneous butanisotropic Universe. If its homogeneity hyper-surfaces have no intrinsic curvature,then it may be described by the Bianchi I geometry (Ellis and MacCallum, 1969). Inthe Bianchi I space-time, cosmic expansion is the same everywhere, but it is fasterin some directions than in others. In comoving coordinates, its metric readsd s = a ( η ) h − d η + e β x ( η ) d x + e β y ( η ) d y + e β z ( η ) d z i , (31)where a ( η ) is the volume scale factor, and the three β i ( η ), which sum to zero, encodethe anisotropy of expansion.The propagation of light in Bianchi I cosmologies has been thoroughly investi-gated in Fleury et al. (2015, 2016), thereby extending previous works on the sametopic (Saunders, 1969). Let us restrict for simplicity to the weak-anisotropy limit( β i (cid:28) A BIo h s = + B ( η s ) − B ( η o ) − η o − η s Z η o η s d η (cid:20) B ( η ) + 12 tr B ( η ) (cid:21) . (32)We used the matrix B AB ( η ) ≡ s o A · diag[ β x ( η ) , β y ( η ) , β z ( η )] s o B where s o1 , s o2 denotethe Sachs basis at the observer. Note that in A BIo h s the source (s) and observer (o) aredefined from their conformal time; this does not account for the change in redshiftdue to the anisotropic expansion, which reads 1 + z = (1 + ¯ z )[1 + tr B ( η s ) − tr B ( η o )].Therefore, all the results of section 3.4 can be used to describe lensing in anweakly anisotropic Universe, if one uses the A BIo h s as amplification matrix all along. N lenses in an arbitrary reference space-time Let us now turn to the more involved case where light travels through an arbitrarynumber N of rough-field regions. The corresponding set-up is depicted in fig. 4. Toour knowledge, such a situation had never been considered in full generality, althoughthe hybrid framework proposed by McCully et al. (2014) may allow one to treat themost relevant cases in practice. ravitational lenses in arbitrary space-times α W ( l + 1 ← l ) x l ˆ α l x l +1 x ... θ β ... x N +1 = s p l a n e l = p l a n e l p l a n e l + s o u r ce p l a n e Figure 4.
Same as fig. 2, but with N lenses labelled by l . The transverse vector x l represents the position where the physical ray intersects the l th lens plane. The observerwould correspond to l = 0 ( x = 0) and the source to l = N + 1 ( x N +1 = s ). N lenses For N lenses, the most direct derivation of the lens equation differs from the single-lens case in its structure. Here, we shall explicitly introduce a past-oriented affineparameter λ , such that λ = 0 at the observer and increases towards the source.Between two successive lens planes, x ( λ ) evolves smoothly with λ according to thegeodesic deviation equation of the reference space-time. However, when light reachesa lens plane, its sudden deflection implies a discontinuity of the derivative ˙ x ≡ d x / d λ . Let us denote ∆ ˙ x l ≡ ˙ x ( λ + l ) − ˙ x ( λ − l ) the discontinuity of˙ x in the l th plane. This quantity is not exactly the deflection angle ˆ α l , because thelatter represents the discontinuity in the way x changes with proper distance ‘ inthe lens’ rest frame. Proper distance and affine parameter are related by d ‘ = ω d λ [see, e.g., Fleury (2015, § 1.3.1)], and hence∆ ˙ x l = d ‘ d λ ! l ∆ d x d ‘ ! l = − ω l ˆ α l , (33)where the minus sign comes from the conventional orientation of ˆ α . Because x ( λ ) satisfies the geodesic deviation equation,which is a linear second-order differential equation, it is linearly related to its initialconditions. Specifically, introducing the “phase space” vector X ( λ ) ≡ " x ( λ )˙ x ( λ ) , (34)there exists a 4 × Wronski matrix W such that for any λ , λ , X ( λ ) = W ( λ ← λ ) X ( λ ) . (35)Although the Wronski matrix will not appear in the final lens equation, it is a veryconvenient tool for its derivation. The full expression of W ( λ ← λ ) is not needed ravitational lenses in arbitrary space-times × D h . Indeed, if two rays cross at λ , then X = ( , ˙ x ), and X = ( D h ˙ x , ˙ x ). Another important property of W , which trivially follows fromits definition (35), is the product law W ( λ ← λ ) = W ( λ ← λ ) W ( λ ← λ ) . (36)Here we denote by W ( l + 1 ← l ) the Wronski matrix that relates X − l +1 ≡ X ( λ − l +1 )before its deflection in the ( l + 1)th plane, to X + l ≡ X ( λ + l ) after its deflection inthe l th plane, X − l +1 = W ( l + 1 ← l ) X + l . (37) Since x ( λ ) is continuous at λ l , thediscontinuity of the phase-space vector X reads∆ X l ≡ X + l − X − l = " ∆ x l ∆ ˙ x l = " − ω l ˆ α l . (38)Denoting X l ≡ X − l (just before deflection at λ l ) for short, eqs. (37) and (38) yieldthe recursion relation X l +1 = W ( l + 1 ← l )( X l + ∆ X l ), which is solved as X l = W ( l ← o) X o + l − X m =1 W ( l ← m )∆ X m . (39)Finally, isolating the first two components, x l , of X l in the above, noting that X = ( , ω o θ ), and expressing the result in terms of β l ≡ ( ω o D o h l ) − x l , we find β l = θ − l − X m =1 (1 + z m ) D − h l D m h l ˆ α m ( x m ) (40)for any l , which only involves Jacobi matrices. The angle β l represents the directionin which a source at x l in the l th plane would be observed in the absence of foregroundlenses m < l . The case l = N + 1, with β N +1 = β yields the explicit lens equation β = θ − N X l =1 (1 + z l ) D − h s D l h s ˆ α l ( x l ) . (41)Note that the single-lens equation (6) is recovered for N = 1. N lenses Just like the lens equation is a recursion relation, the amplification matrix for N lenses takes a recursive form. From β l we shall define the intermediate amplificationmatrix A l ≡ d β l / d θ , which characterises the distortions of an infinitesimal sourcein the l th plane due to the foreground lenses m < l . By differentiating eq. (40), wefind the recursion relation A l = − l − X m =1 ω m D − h l D m h l ˆ H m ( x m ) D o h m A m , (42)where ˆ H mij ≡ ∂ ˆ ψ m /∂x i ∂x j , and with initial condition A = since β = θ . Thecomplete amplification matrix, accounting for all the lenses, is A ≡ ∂ β /∂ θ = A N +1 . ravitational lenses in arbitrary space-times β = θ l = l = l = s o u r ce p l a n e σ σ σ = σ sβ β β ˆ α ˜ α x x x O Figure 5.
Same as fig. 2, but with N = 3 lenses. Dashed lines represent rays of thereference space-time that connect the positions x l of the physical ray to the observer. N lenses4.3.1 Expression of the time delay. The N -lens case can be intuitively deduced fromthe single-lens case as follows. First connect each point x l to the observer with afictitious ray of the reference space-time, as depicted in fig. 5 for N = 3 lenses. Wemay then identify N triangles formed by the points O , x l , x l +1 . Let us apply thesingle-lens time-delay formula (10) in each of these triangles. Precisely, in the l thtriangle, two signals emitted simultaneously at x l +1 , the first one being undeflected(observed in the direction β l +1 ) and the other one deflected in the l th plane (observedin the direction β l ), would be received with a delay T l ( β l , β l +1 ) = 12 ( β l − β l +1 ) · T l ( l +1) ( β l − β l +1 ) − (1 + z l ) ˆ ψ l ( x l ) , (43)up to a constant, with for any l, m such that 0 < l < m , T lm ≡ ω o D To h l D − l h m D o h m . (44)We note that the time-delay matrix satisfies the following unfolding relation; ; forany three planes l, m, n such that 0 < l < m < n , T − ln = T − lm + T − mn . (45)See appendix C for a proof. Although we shall not use it here, eq. (45) is very usefulto derive the expression of time delays in the context of multi-plane lensing with adominant lens (Fleury et al., 2020).The delay between the actual ray (observed in the direction θ ) and the undeflectedray (observed in the direction β ) is then the sum of all these partial delays:∆ t = T ( β , . . . , β N ) + cst ,T ( β , . . . , β N ) ≡ N X l =1 T l ( β l , β l +1 ) . (46)(47) Equation (45) generalises eq. (6.21) of Petters et al. (2001), which was also reported andexploited by McCully et al. (2014). ravitational lenses in arbitrary space-times Although it yields the correct result, the above intuitivederivation of eq. (46) is actually incomplete. Indeed, we applied the single-lens timedelay formula (10) to two non-physical rays. In particular, the “deflection angle”of the l th triangle, which is formed by the rays observed in the directions β l , β l +1 ,and which we may denote ˜ α l , is not the physical deflection angle ˆ α l = ∂ ˆ ψ l /∂ x l (see fig. 5). This is not a mere detail, because the equality between the deflectionangle and the gradient of the lensing potential was a key of the derivation of eq. (10)proposed in appendix A. Therefore, nothing guarantees in principle that eq. (10) canbe applied to rays that do not exhibit the correct deflection angle.Fortunately, despite this weakness in the way that eq. (46) was obtained, theformula itself turns out to be correct. Let us prove this point. First of all, we notethat the differential expression (8), i.e. d∆ t = (1 + z s ) σ · d s , still applies here becauseit relies on strictly local arguments in the source plane. Therefore, if we can showthat (1 + z s ) σ = d T / d s , then it would imply d∆ t = d T just like in the single-lenscase, and hence eq. (46) would follow.Under a small variation of the source position s , all the intermediate positions x l (and thus β l ) change accordingly, so that each contribution T l of T varies asd T l d s = d β l d s ! T ∂ T l ∂ β l + d β l +1 d s ! T ∂ T l ∂ β l +1 . (48)A similar calculation has already been performed in appendix A, except that (i) whatwas denoted s there is now x l +1 ; and (ii) the deflection angle must be replaced bythe geometrical angle ˜ α l . In other words, eq. (72) applied to the configuration of the l th triangle readsd T l d x l +1 = (1 + z l ) d x l d x l +1 ! T ( ˜ α l − ˆ α l ) + (1 + z l +1 ) σ l , (49)from which it is easy to get the derivative with respect to s by multiplying bothsides with (d x l +1 / d s ) T ; the variable with respect to which one differentiates is justa matter of parameterisation here. The last step consists in noticing the identity˜ α l + σ l − = ˆ α l , (50)which clearly appears in fig. 5, and from which we finally getd T l d s = (1 + z l +1 ) d x l +1 d s ! T σ l − (1 + z l ) d x l d s ! T σ l − . (51)When summing the d T l / d s , all the terms cancel two by two, except the first oneproportional to σ ≡ , and the last one proportional to σ N ≡ σ . Therefore,d T d s = N X l =1 d T l d s = (1 + z s ) σ , (52)which concludes our proof. ravitational lenses in arbitrary space-times Like in the single-lens case, Fermat’s principle states thata light ray is physical if and only if the time-delay function is stationary for this ray.Considering T as a function of x , . . . , x N instead of β , . . . , β N , one gets ∂ T∂ x l = (1 + z l ) ˆ α l ( x l ) − d ˆ ψ l d x l (53)with similar calculations as in appendix A. Therefore, the function T is stationarywith respect to changes of x , . . . , x N for and only for the physical ray. As an illustration of the framework developed in this section, consider the situationwhere a fraction f ∈ [0 ,
1] of the matter in the Universe is homogeneously distributed,while the rest is under the form of lenses. Such a scenario is comparable to theEinstein-Straus ‘Swiss-cheese’ model (Einstein and Straus, 1945; Kantowski, 1969;Fleury et al., 2013), although in the latter the point-masses are surrounded byempty regions, while here we rather consider lenses placed within a homogeneous butunder-dense cosmos. This under-dense Universe stands for our reference space-time,for which the Jacobi matrix is scalar, ω a D a h b = D a h b , (54)where D a h b is given by the Kantowski-Dyer-Roeder distance (Kantowski, 1969; Dyerand Roeder, 1973; Dyer, 1973; Dyer and Roeder, 1974; Fleury, 2014) with smoothnessparameter f . For z <
2, it may be approximated up to a few percent precision bythe standard FLRW distance corrected by a negative convergence (Kainulainen andMarra, 2009), D a h b ≈ ¯ D a h b (1 + κ a h b ), with κ a h b = 32 Ω m0 H ( f − Z χ b χ a d χ (1 + z ) f K ( χ b − χ ) f K ( χ − χ a ) f K ( χ b − χ a ) , (55)which was obtained from eq. (17) with δ = f − β l = θ − l − X m =1 D m h l D o h l ˆ α m ( D o h m β m ) , (56)which is identical to the original multi-plane recursion (Blandford and Narayan,1986), up to the expression of the distances. Such an approach efficiently meets thesomehow heavier formalism developed by McCully et al. (2017, § 2.2.2) to describelenses separated by cosmic voids.Time delays are also affected by the under-density of the reference space-time;the associated function is identical to the one of the standard multi-plane framework, T ( β , . . . β N ) = N X l =1 T l ( β l , β l +1 ) , (57)with T l ( β l , β l +1 ) = τ l ( l +1) | β l +1 − β l | − (1 + z l ) ˆ ψ l ( D o h l β l ) , (58)except that the involved distances are changed, in particular τ l ( l +1) = (1 + z l ) D o h l D o h l +1 D l h l +1 ≈ ¯ τ l ( l +1) (1 + κ o h l + κ o h l +1 − κ l h l +1 ) . (59) ravitational lenses in arbitrary space-times α s o u r ce p l a n e θ o b s e r v e r p l a n e l e n s p l a n e β x siα r σy ˆ α Figure 6.
Same as fig. 2, except that we added the vector y , which is obtained bycontinuing the physical ray without deflection from the source plane to the observer plane. In this article, we have proposed a comprehensive and efficient framework to modelgravitational lensing by one or several deflectors placed in an arbitrary referencespace-time. Our formalism relies on the dichotomy between smooth-field regions ,which form our reference space-time where light beams can be considered infinitesimal,and rough-field regions which can be described as thin lenses. In that context, wehave derived the lens equations, and the expressions of the amplification matrix andtime delays. We illustrated our general results to: a single lens with cosmologicalperturbations; a single lens in an anisotropic Universe; and to N lenses in anunder-dense Universe.Lensing by multiple deflectors has already been actively studied in the literature.The specific additions of the present work can be summarised as follows. In section 3,we extended the description of a single lens with cosmological perturbations (e.g.Schneider, 1997) to a lens within any reference space-time. In particular, we proposedin section 3.3 the first rigorous derivation of the expression of time delays in thatgeneral context. Section 4 further extended the results of section 3 to an arbitrarynumber of lenses, thereby generalising the standard multi-plane lensing formalism,which was hitherto limited to the Minkowski or FLRW reference space-times.Our work establishes a firm basis for the description of gravitational lensing byseveral deflectors; in particular, it shall be applied to the accurate treatment ofline-of-sight effects in strong gravitational lensing beyond the tidal approximation(Fleury et al., 2020, in prep.). Acknowledgements
PF thanks Sherry Suyu for kindly drawing his attention to the high-quality works ofMcCully et al., during a workshop in Benasque in 2019. PF received the supportof a fellowship from “la Caixa” Foundation (ID 100010434). The fellowship code isLCF/BQ/PI19/11690018. ravitational lenses in arbitrary space-times A Derivation of eq. (9)
The goal of this appendix is to explicitly prove that the angle σ defined in section 3.3can be written as a gradient,(1 + z s ) σ = d T d s , (60)where T = 12 α · T α − (1 + z d ) ˆ ψ ( x ) , (61)and T ≡ ω o D To h d D − h s D o h s = T T . (62)The set-up is reminded in fig. 6, where we added the useful quantity y = − ω d D d h o ˆ α = ω s D s h o σ in the observer plane.The two natural variables of T being θ , β , we decompose the total derivative asd T d s = d θ d s ! T ∂ T∂ θ + d β d s ! T ∂ T∂ β . (63)Let us first compute ∂T /∂ θ and ∂T /∂ β . Thanks to the symmetry of T , derivativesof α · T α can be considered as hitting only the first α , while cancelling the pre-factor1 /
2, so that ∂ T∂ θ = T α − (1 + z d ) d ˆ ψ d θ , (64) ∂ T∂ β = − T α . (65)To express T α , we note from fig. 6 that the image displacement i − s can bewritten as a function of either α or ˆ α as i − s = ω o D o h s α = ω d D d h s ˆ α . This yields D − h s D o h s α = (1 + z d ) ˆ α , and thus T α = ω o D To h d D − h s D o h s α = ω d D To h d ˆ α . (66)As x = ω o D o h d θ , we get ω o D o h d = d x / d θ , and hence the first term of eq. (63) reads d θ d s ! T ∂ T∂ θ = (1 + z d ) d x d s ! T ˆ α − d ˆ ψ d x ! = (67)for a physical ray. Therefore, the only non-zero contribution to d T / d s in eq. (63)consists of the second term. From eqs. (65) and (66) and s = ω o D o h s β , we have d β d s ! T ∂ T∂ β = − (cid:16) ω o D To h s (cid:17) − (cid:16) ω d D To h d (cid:17) ˆ α (68)= − (1 + z s ) (cid:16) ω s D o i s (cid:17) − (cid:16) ω d D o i d ˆ α (cid:17) using eq. (2) (69)= − (1 + z s ) (cid:16) ω s D o i s (cid:17) − ( − y ) (70)= (1 + z s ) σ , (71) ravitational lenses in arbitrary space-times T d s = (1 + z d ) d x d s ! T ˆ α − d ˆ ψ d x ! + (1 + z s ) σ = (1 + z s ) σ , (72)which concludes the proof. We chose to explicitly keep the term proportional toˆ α − d ˆ ψ/ d x in the result in order to apply it more easily to the N -lens case. B Symmetry of the time-delay matrix
Let us prove that the time-delay matrix T ≡ ω o D To h d D − h s D o h s = − ω o D o i d D − h s D o h s (73)is symmetric, i.e., T = T T . For that purpose, we shall express the vector y inthe observer’s plane (see fig. 6) in two different ways. The proof will only rely onEtherington’s reciprocity law, D a h b = − D Ta i b .A first option consists in following the chain y → ˆ α → i − s → α , invoking thecorresponding Jacobi matrices. The computation goes as follows, y = − ω d D o i d ˆ α (74)= − D o i d D − h s ( i − s ) (75)= − D o i d D − h s ω o D o h s α (76) y = T α . (77)The second option follows y → σ → x − r → α and reads y = ω s D o i s σ (78)= D o i s D − i s ( x − r ) (79)= D o i s D − i s ω o D o h d α (80)= − ω o D To h s (cid:16) D − h s (cid:17) T D To i d α (81) y = T T α . (82)Comparing eqs. (77) and (82), which are valid for any α , concludes the proof. C Unfolding relation (45) for the time-delay matrix
Let us prove that, for any three lens planes l, m, n such that 0 < l < m < n , thetime-delay matrix satisfies the unfolding relation T − ln = T − lm + T − mn . (83)This relation is illustrated in fig. 7, which is constructed as follows: (i) consider anarbitrary position y in the observer plane; (ii) connect that position to the origin N of the n th lens plane with a ray of the reference space-time, and call L , M the ravitational lenses in arbitrary space-times p l a n e n ζ m o b s e r v e r p l a n e p l a n e l y O p l a n e m NML ζ l Figure 7.
Geometrical meaning of the unfolding relation (45) for the time-delay matrix.Every line is a geodesic of the reference space-time. intersection between that ray with the two intermediate planes l, m ; (iii) connect
L, M to the observer with rays of the reference space-time, which make angles ζ l , ζ m with respect to the fiducial ray.Proceeding just like in appendix B, we may then relate y to the angles ζ l , ζ m as y = T ln ζ l in triangle ( OLN ), (84) y = T mn ζ m in triangle ( OM N ), (85) y = T lm ( ζ l − ζ m ) in triangle ( OLM ). (86)It follows that ( T − ln − T − lm − T − mn ) y = , and therefore eq. (83) because the aboveholds for any y in the observer plane. ReferencesKiDS
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