The effects of structure anisotropy on lensing observables in an exact general relativistic setting for precision cosmology
aa r X i v : . [ a s t r o - ph . C O ] A p r Prepared for submission to JCAP
The effects of structure anisotropy onlensing observables in an exact generalrelativistic setting for precisioncosmology
M. A. Troxel a Mustapha Ishak a Austin Peel a a Department of Physics, The University of Texas at Dallas,Richardson, TX 75080, USA
Abstract.
The study of relativistic, higher order, and nonlinear effects has become necessaryin recent years in the pursuit of precision cosmology. We develop and apply here a frameworkto study gravitational lensing in exact models in general relativity that are not restrictedto homogeneity and isotropy, and where full nonlinearity and relativistic effects are thusnaturally included. We apply the framework to a specific, anisotropic galaxy cluster modelwhich is based on a modified NFW halo density profile and described by the Szekeres metric.We examine the effects of increasing levels of anisotropy in the galaxy cluster on lensingobservables like the convergence and shear for various lensing geometries, finding a strongnonlinear response in both the convergence and shear for rays passing through anisotropicregions of the cluster. Deviation from the expected values in a spherically symmetric structureare asymmetric with respect to path direction and thus will persist as a statistical effect whenaveraged over some ensemble of such clusters. The resulting relative difference in variousgeometries can be as large as approximately 2%, 8%, and 24% in the measure of convergence(1 − κ ) for levels of anisotropy of 5%, 10%, and 15%, respectively, as a fraction of total clustermass. For the total magnitude of shear, the relative difference can grow near the center ofthe structure to be as large as 15%, 32%, and 44% for the same levels of anisotropy, averagedover the two extreme geometries. The convergence is impacted most strongly for rays whichpass in directions along the axis of maximum dipole anisotropy in the structure, while theshear is most strongly impacted for rays which pass in directions orthogonal to this axis, asexpected. The rich features found in the lensing signal due to anisotropic substructure arenearly entirely lost when one treats the cluster in the traditional FLRW lensing framework.These effects due to anisotropic structures are thus likely to impact lensing measurementsand must be fully examined in an era of precision cosmology. ontents As the quality of astrophysical data rapidly improves over the next decades, we are presentedwith rich opportunities to consider and constrain models of both large- and small-scale struc-ture in the universe which take into account the true, inhomogeneous and anisotropic natureof structure. With goals to constrain cosmology at the percent level utilizing upcoming sur-vey results, the consideration of second-order or nonlinear effects is becoming a priority [1–8].One such avenue of exploration is in employing structure models, which are exact solutionsto Einstein’s field equations in general relativity but are not restricted to homogeneity andisotropy, and developing the methods necessary to compare observational predictions of thesemodels to new and improving data [9, 10]. This includes the identification of what impactinhomogeneities and anisotropies will have on astrophysical observables such as lensing anddynamical mass estimates. While this is worthwhile in its own right as an examination ofwhat more general exact models of structure in general relativity predict, it also informs us ofbiases in the interpretation of observational data which symmetry assumptions might cause.These models also take into account the full nonlinearity of general relativity without as-sumptions or simplifications, so that higher order and relativistic effects which have recentlyattracted attention in the literature are automatically included.– 1 –ne of the most promising probes of the universe is gravitational lensing, either inthe form of strong and weak lensing by galaxies and clusters of galaxies or of weak lensingby large-scale structure in the universe (cosmic shear). The combination of new lensingmeasurements from large ongoing and planned surveys with other probes like the cosmicmicrowave background and the distance-redshift relation from type Ia supernovae promisesto significantly constrain cosmological information. Including cosmic shear, for example, canimprove constraints of cosmological parameters by factors of 2-4 (e.g. [11]). Developing aframework in which we can identify and investigate observables like the lensing convergenceand shear within these general inhomogeneous and anisotropic exact solutions is oftentimesnot straightforward, and has been the topic of much recent work. We continue this hereby describing a framework in which to study gravitational lensing in such general metrics,using the Szekeres metric as an example, and discuss how it can be associated with thetraditional lensing convergence and shear used in studies of the universe in the concordanceLambda cold dark matter (ΛCDM) model, which is based upon the Freidmann-Lemaˆıtre-Robertson-Walker (FLRW) metric. The lensing effects of anisotropy on cluster scales haveprimarily been studied recently in terms of the potential triaxiality of a dark matter halo,where determinations of mass through gravitational lensing have been shown to be affectedby up to 50% for cases where the halo is significantly elongated along the line of sight [12].In weak lensing mock catalogues, determinations of mass and concentration can be impactedat the 5% level [13]. Inhomogeneous structure has also been used to demonstrate potentialchallenges to precision cosmological constraints, for example, due to lensing of very smallbeam sources like supernovae [14].The Szekeres metric [15, 16] is an exact solution which has been used in cosmology, but isalso able to model realistic, small-scale structures with a natural FLRW limit as background.The Szekeres models have an irrotational dust source with no symmetries (i.e., no Killingvectors [17]), and thus can represent structure with no assumptions of spherical or axialsymmetry. Previous attempts to compare Szekeres models to cosmological observations haveincluded the growth of large scale structure [18, 19], the expansion history of the universe[20–22], as well as some cosmic microwave background constraints [23, 24]. On smaller scale,they have been studied as models of exact structures like clusters of galaxies, primarily as ameans to test whether realistic cluster-sized densities can evolve in a Szekeres universe fromreasonable initial conditions and without singularities [25–28]. In the past, it has been shownthat structure growth on both small and large scales is enhanced in the nonlinear Szekeresmodels relative to ΛCDM [18, 19, 29, 30]. A systematic consideration of gravitational lensingin the Szekeres metric has not yet been attempted, though some initial work has been donefor the LT metric [31–34]. [35], for example, have considered the impacts of voids (usingSzekeres models) on the measured magnitude of astronomical objects such as supernovaetype Ia, while [36] have commented on the cosmological constant and lensing in class IISzekeres models.This paper follows as part of a series which seeks to explore the effects of inhomogeneityand anisotropy on astrophysical observables in exact, relativistic models, using the Szekeresmodels as a test case. In an era of precision cosmology, such effects have been demonstratedto be significant (e.g. [19, 29] and references). The Szekeres models possess both the homo-geneous FLRW models and the spherically symmetric Lemaˆıtre-Tolman (LT) models [37, 38]as natural limits, which aides in the systematic exploration of the effects due to inclusionof anisotropy in structures. In [29], we developed a realistic Szekeres cluster model, whichreproduces the density profile of a cluster of galaxies at t (the current age of the universe)– 2 –ased upon the Navarro-Frenk-White dark matter profile [39]. We showed that the inclusionof anisotropy relative to a reference spherically symmetric LT model produces a strong, non-linear response in the rate of gravitational clustering in anisotropic regions of the structure,which contributes to a greater total infall velocity. Here we develop a framework in whichto calculate the lensing properties of such a Szekeres cluster model with varying levels ofanisotropy by examining the deviation of neighboring geodesics as they propagate throughand past the structure. This is a first step to a more general examination of gravitationallensing in the Szekeres metric.The paper is structured as follows. In section 2, we introduce the lensing framework forinhomogeneous cosmologies, which has been derived from a general treatment of geometricoptics in general relativity. Section 2.2 then describes how this general treatment can berelated to the classic lensing formalism in the case that there exists a background FLRWcosmology, and we derive the convergence and shear for our particular cluster density modelin section 2.3. In section 3.1.2 we discuss the calculation of general lensing properties inan exact, anisotropic model. The Szekeres metric is introduced in section 3.1.3, where webriefly describe how anisotropies can be systematically incorporated into the model, and wespecialize the general lensing formalism to calculations in the Szekeres metric in section 3.2.As an example of the process, we examine what effects the introduction of anisotropies ina structure have on its lensing properties in section 3.3. Finally, we conclude in section 4.Units are chosen throughout the paper such that c = G = 1. The framework for the exact study of lensing in general inhomogeneous cosmologies dif-fers substantially from the traditional lensing framework in the Lambda Cold Dark Matter(ΛCDM) paradigm, where several simplifying assumptions can safely be made (see for exam-ple [40, 41] and references therein). In a general curved spacetime, even the electromagneticwave equation obtains a first-order Ricci tensor component, (cid:3) A µ = − µ J µ + R µν A ν , (2.1)which causes the self-interaction of the wave propagation with the curvature it produces inthe spacetime. Thus the treatment of a light ray’s propagation as geodesic and affinely pa-rameterized is only possible as a zeroth order approximation. This approximation is satisfiedin the short-wave (WKB) limit, which we employ here, such that to an observer the elec-tromagnetic waves appear sufficiently planar and monochromatic over large scales comparedto a typical wavelength, while remaining small compared to the radius of curvature of thespace. In this limit, we can ignore the contribution of the curvature to the wave equation.However, it is important to keep this initial assumption in mind, as it impacts in an exacttreatment even the well-known approximation for the redshift of light, where in the FLRWmetric, for example, the redshift can be related to the scale factor a by1 + z = k µ u µ | e k µ u µ | o = a a , (2.2)where k µ is the null tangent vector of the light and u µ is the 4-velocity of the observer.In order to study gravitational lensing by mass in a general cosmology, we will considerthe geodesic deviation of adjacent light rays γ in a bundle, which will be parameterized– 3 –elative to some fiducial ray γ . This treatment follows the work of [41], and we will usetheir notation where possible. It is described in terms of our specific use, but the reader isdirected to [41] for a thorough treatment of the formalism. An alternative approach usingthe geodesic light-cone gauge has also recently been discussed in [42], where some of thefollowing equations can be given explicit solutions. The fiducial ray will be described bysome null tangent vector k µ = ∂x µ /∂λ , which is affinely parameterized by a parameter λ .For a past-directed light cone, λ = 0 at the observer and increases going backward in time.At the observer, the following is then true for a metric signature ( − + ++):( u µ u µ ) = − u µ k µ = 1 . (2.4)The bundle of light (taken to represent, for example, the image of a distant galaxy) isthen propagated along the path of the fiducial ray, and the deviation of nearby rays relativeto the fiducial ray describes the lensing properties of any intervening mass. To do this, wedefine a ‘deviation vector field’ or Jacobi field Y µ ( ~θ, λ ) = γ µ ( θ, λ ) − γ µ ( θ = 0 , λ ) , (2.5)such that Y represents the (changing) separation between some ray γ and the fiducial ray,from which the angular position θ is identified. Y is parameterized by a space-like basis( E , E ) which span the plane orthogonal to both u µ , k µ to complete the tetrad along γ ,where u µ is the vector resulting from parallely propagating u µ along γ . We can then rewrite Y as Y µ = ξ E µ + ξ E µ + ξ k µ . (2.6)The components of Y along this orthogonal screen ( ξ , ξ ) can then be described by thedeformation equation, which in matrix form is˙ ξ = S ξ , (2.7)where at each point λ along the fiducial ray S = (cid:18) θ − ℜ σ ℑ σ ℑ σ θ + ℜ σ (cid:19) (2.8)is the optical deformation matrix, composed of the Sachs optical scalars [43]: the rate ofexpansion, θ = 12 k µ ; µ , (2.9)and the complex rate of shear, σ = 12 k µ ; ν ǫ ∗ µ ǫ ∗ ν , (2.10)where ǫ µ = E µ + iE µ .Taking the derivative of equation (2.7) w.r.t. λ gives a more useful description of theevolution of ( ξ , ξ ) along the fiducial ray, however, which can be written in terms of theRicci and Weyl curvatures of the spacetime. We can then write¨ ξ = T ξ , (2.11)– 4 –here the optical tidal matrix T is given by T = (cid:18) R − ℜF ℑFℑF R + ℜF (cid:19) . (2.12)The source of convergence is written in terms of the Ricci curvature, R = − R µν k µ k ν , (2.13)while the source of shear is written in terms of the Weyl curvature, F = − C αβµν ǫ ∗ α k β ǫ ∗ µ k ν . (2.14)Once one has defined a spacetime, the Ricci and Weyl curvatures can then be calculated, andequation (2.11) describes the deformation (or lensing) of such an infinitesimal light bundleas it propagates as ξ i on the screen at each event parameterized by λ along the fiducial ray. Λ CDM
One can use a Jacobi mapping between ξ i and the initial angle θ i between the ray of interestand fiducial ray, ξ ( λ ) = D ( λ ) θ , due to the linearity of equation (2.11). The evolution of D as a function of λ is then given by ¨ D = T D , (2.15)where at λ = 0, D = and ˙ D = . This Jacobi matrix D is related to the traditionalmagnification matrix A of lens theory when properly scaled.In an FLRW model, there exists no Weyl curvature and so T = R , which implies that D = ˜ D and ˜ D becomes the angular diameter distance D A . In a general space, one findsinstead that D A = √ det D . From the field equations, it can be shown that in FLRW thesource of convergence simplifies to R = − πρ bg (1 + z ) = − H Ω m (1 + z ) . (2.16)In the weak field limit, we can also include an explicit, isolated mass density (e.g. a clusterof galaxies) and express the optical tidal matrix directly as a function of the density (oralternately the Newtonian or lensing potential). In this case, the source of convergencebecomes simply the sum of these two masses R = − π ( ρ bg + ρ cl )(1 + z ) . (2.17)Over large distances, the cluster density can be neglected, as it acts only over a very smallpart of the total null path of the rays. The source of shear can be written in terms of theNewtonian potential as F = − (2Φ , ij + δ ij Φ , )(1 + z ) . (2.18)Writing the optical tidal matrix in the FLRW case as above is one method of comparingresults between the standard lens theory and the resulting information on geodesic deviationwhich comes from the solution of equation (2.15). Another method is to instead relate thelensing convergence ( κ ) and shear ( γ ) to the Jacobi matrix D as mentioned above. It is– 5 –traightforward to show that when we consider an inhomogenous model with a matching orlimiting background homogeneous cosmology, this relationship is given by D = ˜ D A A , (2.19)where ˜ D A is the angular diameter distance in the background model and A is A = (cid:18) − κ − γ γ γ − κ + γ (cid:19) . (2.20)Thus given a solution to equation (2.15) D = (cid:18) D D D D (cid:19) (2.21)and assuming a general model which can be decomposed into a discrete mass structure (orstructures) with an associated homogeneous background cosmology, we can then write theconvergence and shear due to a structure as (e.g. [14]) κ = 1 − D + D D A (2.22) γ = D − D D A (2.23) γ = D ˜ D A = D ˜ D A (2.24) γ = q γ + γ . (2.25)In a purely FLRW model where A = , this result is consistent with the above assertion that D = ˜ D A , with ˜ D A just being the angular diameter distance in the model. Λ CDM background
In [29], we explored the kinematic impact of introducing anisotropies into the density profileof a cluster of galaxies. We chose the truncated NFW density model of [46], but for reasonsof regularity at the origin in the relativistic model, we modified this to include a maximumcentral density. We can also evaluate analytically the lensing properties of such a cluster inthe ΛCDM model, where Ω m = 0 .
27, Ω = 0 .
73, and H = 69 km s − Mpc − . The densityof the cluster is isotropic and given by ρ BMO ( x ) = δ c ρ c ( ǫ c + x ) (1 + x ) (cid:18) τ x + τ (cid:19) , (2.26)where ρ c is the critical density of the universe, x = r/r s and τ = r t /r s are dimensionlesslengths, r s = r /c is a scale radius, c = 3 is the concentration of the cluster, r = 1 .
75 Mpcis the radius at which the cluster density is 200 times the critical density, r t = 3 Mpc is thetruncation radius, ǫ c = 0 . δ c ρ c , and the characteristicoverdensity is δ c = 2003 c log (1 + c ) − c c . (2.27)– 6 –rom this density profile, we can evaluate the convergence κ ( x ) = Σ( x ) / Σ crit and shear γ ( x ) = ¯ κ ( x ) − κ ( x ), where Σ( x ) is the surface mass density, ¯ κ ( x ) = M proj ( x ) /πr Σ crit , M proj ( x ) is the projected mass enclosed within r , andΣ crit = D s πD ℓs D ℓ . (2.28)The surface mass density isΣ( x ) = r s Z ∞−∞ dℓρ BMO ( p ℓ + x ) (2.29)= δ c ρ c r s τ ( τ + 1) ( − x −
1) ( ǫ c − h − ǫ c (cid:0)(cid:0) τ − (cid:1) x + 4 (cid:1) + τ + x + 5 (cid:0) − x (cid:1) i F ( x ) − τ ( τ + x ) ( ǫ c + τ ) h − ǫ c (cid:0) τ + (cid:0) τ − (cid:1) x (cid:1) − τ ǫ c (cid:0) τ + 2 τ (cid:0) x + 2 (cid:1) + 2 x (cid:1) + ǫ c (cid:0) τ + τ (cid:0) x − (cid:1) − τ x + x (cid:1) + τ (cid:0) τ + 3 τ (cid:0) x − (cid:1) − τ (cid:0) x + 1 (cid:1) − x (cid:1) i F (cid:16) xτ (cid:17) − π τ + x ) / ( ǫ c + τ ) h ǫ c (cid:0) τ + 2 τ (cid:0) x − (cid:1) − x − (cid:1) + ǫ c (cid:0) τ + 2 τ (cid:0) x − (cid:1) − x (cid:1) + ǫ c (cid:0) τ + 2 τ (cid:0) x − (cid:1) − τ (cid:0) x + 1 (cid:1) − x (cid:1) + 2 τ (cid:0) τ + 4 x + 1 (cid:1) i + (cid:0) − τ − (cid:1) h − ǫ c + ( ǫ c + 1) − τ ( τ + x ) ( ǫ c + τ ) + 2( x −
1) ( ǫ c − i) − δ c ρ c r s τ ( ǫ c − ( ǫ c + τ ) F (cid:18) xǫ c (cid:19) , (2.30)and the projected mass density is then given by M proj ( x ) = r s Z x dx ′ πx ′ Σ( x ′ ) (2.31)= 2 δ c ρ c r s τ ( τ + 1) x ( ǫ c − h ǫ c (cid:0) τ (cid:0) x − (cid:1) − x + 2 (cid:1) + τ + 4 x − i F ( x ) − τ ( ǫ c + τ ) h − τ ǫ c (cid:0) τ + (cid:0) τ + 1 (cid:1) x (cid:1) + 2 ǫ c (cid:0) − τ + τ (cid:0) − x (cid:1) + x (cid:1) + ǫ c (cid:0) τ (cid:0) x − (cid:1) + 2 τ (cid:0) − x (cid:1) + x (cid:1) + 2 τ + 3 τ (cid:0) x − (cid:1) − τ x − τ x i F (cid:16) xτ (cid:17) − π √ τ + x ( ǫ c + τ ) h ǫ c (cid:0) τ + 2 τ (cid:0) x − (cid:1) − x + 1 (cid:1) + 2 ǫ c (cid:0) τ + τ (cid:0) x − (cid:1) − x (cid:1) + ǫ c (cid:0) τ + 2 τ (cid:0) x − (cid:1) − τ (cid:0) x + 1 (cid:1) − x (cid:1) + 2 τ (cid:0) τ + 4 x − (cid:1) i + π τ ( ǫ c + τ ) h (cid:0) τ − (cid:1) τ ǫ c + (cid:0) τ − τ − (cid:1) τ ǫ c + (cid:0) τ − τ + 1 (cid:1) ǫ c + 6 τ − τ i + 2( ǫ c + τ ) h − τ (4 ǫ c + 3) + 2 (cid:0) − τ (cid:1) ǫ c + (cid:0) − τ (cid:1) ǫ c + τ i log( τ ) ) + 4 δ c ρ c r s τ ( ǫ c − ( ǫ c + τ ) x ( (cid:0) ǫ c − x (cid:1) F (cid:18) xǫ c (cid:19) − ǫ c log ( ǫ c ) ) , (2.32)– 7 – .1 1 0.0010.010.1 Sh e a r ( γ ) r (Mpc)0.0010.010.1 0.1 1 C o n v e r g e n ce ( κ ) r (Mpc) With peak central densityWithout peak central density Figure 1 . The convergence ( κ ) and shear ( γ ) resulting from Eqs. (2.30-2.32) for the cluster modeldescribed by the truncated NFW density profile in equation (2.26) is shown compared to that for thesame profile with no central peak density as given in [46]. The convergence and shear are evaluatedfor a source at redshift z = 1 . z = 0 .
5. As expected, the limiting central peak density inour profile causes a reduction in the magnitude of both κ and γ near the center of the structure, aswell as a small decrease in the total shear which extends out to large radii. where F ( x ) = ( arctan √ x − √ x − x > arctanh √ − x √ − x x ≤ F ( x ) = arccoth √ x √ x . (2.34)In the limit that ǫ c →
0, this results in the lensing properties found for the profile of [46].Also letting τ → ∞ , these properties agree with the classic NFW results. The resultingconvergence and shear is shown in Fig. 1 for the cluster profile both with [29] and withoutthe central peak density for a source at z = 1 . z = 0 .
5. The two agree wellexcept for near the center of the cluster, where the convergence and shear are lower in thecluster with a limiting peak density, and in the shear, where the peak density causes a smalldecrease in the total shear over a large range of radii.
The example cluster profile given in section 2.3 was spherically symmetric and assumed toreside within a background FLRW spacetime. Because the derivation assumes a thin lensand utilizes the resulting surface mass densities, the background cosmology only impacts thecalculation as part of deriving the angular diameter distances used in the critical surfacedensity in equation (2.28), which are calculated ignoring the presence of the structure. Re-laxing these assumptions is necessary in order to explore the degree to which introducing– 8 –nisotropic structure into lensing calculations will impact the resulting convergence or shear,and whether there exist potential biases that this might introduce into our conclusions aboutcosmological or astrophysical quantities in our models.Here we explore this by defining both a spherically symmetric and inhomogeneous ref-erence model for the cluster of galaxies, which is identical in its resulting density to that usedin the ΛCDM calculation in section 2.3, as well as an anisotropic model which allows us tocontrol the degree of anisotropy. The impact of including anisotropies can then be quantifiedby utilizing the process described in section 2 for both models and comparing any discrep-ancies between the resulting lensing properties. To do this, we will use the Lemaˆıtre-Tolman(LT) [37, 38] and Szekeres [15, 16] models, as discussed below.
The reference LT model and Szekeres models through which we vary the levels of anisotropyare described in detail in [29], but we review them here for completeness. Both models arebased on an exact metric representation of the density in the universe that includes a localoverdensity with radial profile consistent with a truncated NFW galaxy cluster model. TheLT model is inhomogeneous, but isotropic about a single point, while the Szekeres modelsrelax that isotropy and are fully general (i.e., they possess no Killing vectors [17]). The LTmodels are a natural limit to the Szekeres models, and the FLRW models can be naturallimits to both.The Szekeres metric is an exact solution to Einstein’s field equations with an irrotationaldust source, and its lack of symmetries makes it ideal to model general asymmetric structurein the universe. We will utilize the LT form of the Szekeres metric, which can be written insynchronous and comoving coordinates as [44] ds = − dt + (Φ , r − Φ E , r / E ) ǫ − k ( r ) dr + Φ E ( dp + dq ) , (3.1)where , α represents partial differentiation with respect to the coordinate α . Depending onthe dependencies of the metric functions on the coordinate r , the Szekeres models fall intotwo classes. Here we use the more general Class I metric, with all metric functions havingan r -dependence.The geometry of the spatial sections in the metric is governed by k ( r ), which generallydepends on r , with open, closed and flat sections potentially existing. The parameter ǫ =0 , ± p, q ) 2-surfaces. In the LT form of the metric, theentire space is foliated by 2-surfaces with a single geometry, but in more general forms of themetric, this geometry can also be a function of r . We will limit our discussion to the quasi-spherical case ( ǫ = 1), since they are better studied and understood. In the quasi-sphericalcase, the ( p, q ) 2-surfaces are spheres with Φ = Φ( t, r ) as their areal radius.Φ( t, r ) is defined by (Φ , t ) = − k + 2 M Φ + Λ3 Φ , (3.2)where M = M ( r ) represents the total active gravitational mass within a sphere of constant r . We will choose Λ = 0 for simplicity, which lets us write the solution of equation (3.2)in a simple parametric form. Equation (3.2) has the same form as the Friedmann equation,but where each surface of constant r evolves independently of the others. The function– 9 – = E ( r, p, q ) in equation (3.1) is E ( r, p, q ) = S ( r )2 "(cid:18) p − P ( r ) S ( r ) (cid:19) + (cid:18) q − Q ( r ) S ( r ) (cid:19) + ǫ , (3.3)and S , P , and Q describe the stereographic projection from the p and q plane onto the unitsphere such that ( p − P ( r ) , q − Q ( r )) S ( r ) = (cos( φ ) , sin( φ ))tan( θ/ . (3.4)When E = E ( p, q ) ( S , P , and Q are constants), equation (3.1) is just the LT metric, andthe spheres of constant t and r are concentric about the origin. The contribution of E , r / E to the Szekeres metric acts to offset these spheres relative to the LT model through the r dependence of S , P , and Q .The density in the Szekeres metric is given by κρ ( t, r, p, q ) = 2( M, r − M E , r / E )Φ (Φ , r − Φ E , r / E ) , (3.5)where κ = 8 π in units where c = G = 1. In an isotropic structure E , r = 0, and thedensity depends only on the total mass function M ( r ) and areal radius Φ( t, r ) (and thusthe coordinates t & r ). In this case, the Szekeres and LT densities are identical. Indeed atany radius where E , r = 0, this is true and the density along that 2-sphere is homogeneous.Substructure in the form of anisotropic perturbations on this corresponding LT structure arethen due to the influence of E , r / E 6 = 0, which introduces a dipole contribution to the densityat a given r [16]. This means that along any surface of constant r where E , r / E 6 = 0, thereexists a single density peak which decreases monotonically to a density minimum located onthe opposite side of the 2-sphere.
Our cluster model is fully defined by the Szekeres metric functions, which have been chosen toreproduce at t the density profile of a truncated NFW cluster that is set within a backgroundFLRW model with Ω m = 1. The resulting density profile, which extends beyond the cluster,then has the form ρ LT ( r ) = ρ BMO ( r ) + ρ c . (3.6)In addition to the functions M ( r ) and k ( r ), there exists a final free function, the Bang time t B ( r ), which results from integrating equation (3.2) t − t B ( r ) = Z Φ0 d Φ ′ q − k ( r ) + M ( r )Φ ′ . (3.7)The function t B ( r ) gives the local time of the Big Bang at some r . Though the function t B ( r ) is arbitrary in general, a homogeneous model requires that t B ( r ) = const.The process to construct the necessary metric functions is discussed in [29], which wewill not reproduce here, since we are not modifying the models used. The resulting functions M ( r ), k ( r ), and t B ( r ) are shown in [29]. The curvature k is everywhere positive, and at r >
50 Mpc k ∝ r and M ∝ r as required in a homogeneous (FLRW) space. The Bangtime function t B is chosen such that the structure is near the middle of its collapsing phase,– 10 – ρ / ρ c r (Mpc)Szekeres - 10%LT Figure 2 . The density profile of the δρ |D /ρ LT |D = 10% Szekeres anisotropic model is compared to theisotropic LT spherically symmetric model. The total mass at each r is identical in both models. TheSzekeres curves are measured through the directions of maximum | ρ − ρ LT | . Solid lines look throughthe φ = π/ φ = − π/ θ = π/ where it has not yet collapsed to a point where pressure or rotation should be a dominantcomponent of the kinematics of the cluster. At large r , t B becomes constant as part ofensuring that the metric becomes homogeneous outside the cluster.The isotropic density profile in equation (2.26) we use for the cluster is based on atruncated version of the NFW profile. The classic NFW profile has two undesirable features:1) its density diverges at small r , which will violate the conditions of regularity at the originnecessary for the Szekeres metric; 2) its enclosed mass diverges at large r , which makesmatching to an FLRW space with homogeneous density difficult. To resolve these problems,we introduced a maximum density at very small r , and a truncation radius r t following [46].The constant density component of equation (3.6) is included so that the model can take onthe appropriate density for an Ω m = 1 FLRW model at large r . To introduce substructure into the galaxy cluster, and thus cause the density profile tobecome anisotropic, we choose a specific form for the functions S ( r ), P ( r ), and Q ( r ) whichcompose the metric function E ( r, p, q ). This function controls the anisotropy in the structure,but must also satisfy several physical requirements as discussed in [29] in order to avoid theformation of singularities during the evolution of the structure. We will also enforce anFLRW background by requiring that S, r = P, r = Q, r = 0 at large r . Once these requirementsare addressed, we define the functions as S ( r ) = const., P ( r ) = 0, and Q ( r ) = 27 e − r r . (3.8)Varying the value of S allows us to control the strength of the anisotropy. This choiceof functions produces a pair of overdense/underdense dipoles in the structure’s density atdifferent values of r , which is shown relative to the LT density in Fig. 2. The boundarybetween these two anisotropic regions occurs at r = 0 . Q, r = 0. Since S and– 11 – are constant, this particular 2-sphere is entirely LT-like (spherically symmetric) and has aconstant density.The strength of the anisotropy is defined as in [29]. This is expressed as the totalfractional displaced mass present in the anisotropies at t , δρ |D /ρ LT |D , where δρ |D is the totaldisplaced mass of the Szekeres structure relative to the LT reference density and ρ LT |D is thetotal LT density bounded by a 2-sphere at r = 2 r . A discussion of the volume integral, X |D , of a quantity X over some domain D in the Szekeres metric is given in appendix A of[29]. We choose anisotropies that represent realistic estimates of the amount of the halomass likely to be present in substructure [45], with δρ |D /ρ LT |D of 5%, 10%, and 15%. Theseanisotropies correspond to S = 4 .
87, 2 .
43, and 1 .
65 Mpc, respectively.Figure 2 shows the resulting density anisotropies produced in the Szekeres model throughthe angles of largest deviation in the dipoles from the spherically symmetric LT model. Weconsider these anisotropies to be conservative, given that we are simply shifting mass in theunderlying isotropic NFW halo with the Szekeres dipole and not adding the substructuremass to the original density.
The geodesic deviation of a null ray was presented and discussed in section 2 for a generalcosmology. However, we must now specialize this framework to the Szekeres metric. In orderto calculate the geodesic deviation, we must first define the components of the optical tidalmatrix in equation (2.15). The most important components of the optical tidal matrix arethe Ricci and Weyl curvatures of the Szekeres spacetime. The Ricci curvature tensor can beexpressed in terms of the Reimann curvatur tensor or Christoffel symbols as R µν = R αµαν = 2Γ αµ [ ν,α ] + 2Γ αβ [ α Γ βν ] µ , (3.9)and the Weyl curvature tensor as C αβµν = R αβµν − (cid:0) g α [ µ R ν ] β − g β [ µ R ν ] α (cid:1) + 13 Rg α [ µ g ν ] β , (3.10)where R = R αα is the Ricci scalar, g αβ is the metric tensor, and square brackets denote theantisymmetric part. The Christoffel symbols and the resulting Ricci and Weyl curvatures forthe Szekeres metric are given in appendix A.In addition to the curvatures listed above, the optical tidal matrix also depends on boththe null tangent vector k µ and the screen basis vectors E µ and E µ . The null tangent vectorcan be computed from the null geodesic equation ( k µ )˙ + Γ µαβ k α k β = 0, which produces theset of equations [21]0 = ˙ k t + HH, t ( k r ) + F F, t [( k p ) + ( k q ) ] (3.11)0 = H ˙ k r + H ˙ Hk r − HH, r ( k r ) − F F, r [( k p ) + ( k q ) ] (3.12)0 = F ˙ k p + F ˙ F k r − HH, p ( k r ) − F F, p [( k p ) + ( k q ) ] (3.13)0 = F ˙ k q + F ˙ F k r − HH, q ( k r ) − F F, q [( k p ) + ( k q ) ] , (3.14)where we have defined H = (Φ , r − Φ E , r / E ) / p ǫ − k ( r ) and F = Φ / E , and ˙ represents differ-entiation w.r.t. the affine parameter λ . In the LT limit, H = Φ , r / p ǫ − k ( r ).– 12 –inally, the screen basis vectors E µ and E µ are defined at the observer in terms of thenull vector k µ = ( − , k r , k p , k q ), the comoving velocity of the observer u µ = (1 , , , E µ and E µ obey the following orthogonality conditions0 = k µ E µa (3.15)0 = u µ E µa (3.16) δ ab = E aµ E µb , (3.17)where a ∈ { , } . These conditions result in the initial conditions E µ = , FH q ( k p ) + ( k q ) , − HF k r k p p ( k p ) + ( k q ) , − HF k r k q p ( k p ) + ( k q ) ! (3.18) E µ = , , F k p p ( k p ) + ( k q ) , − F k q p ( k p ) + ( k q ) ! , (3.19)for nonzero k p and k q (null vectors that are not initially radial). In the appropriate LT limit,this agrees with the previous results of [31]. Once E µ and E µ are known at the observer,they are parallely propagated along with u µ and k µ along the path of the fiducial ray beingconsidered. Thus, at other positions along the path parameterized by λ , we must similarlysolve the differential equation ( E µa )˙+ Γ µαβ E αa k β = 0 for each of E µ and E µ . This results in aset of equations which are similar in form to equations (3.14).In general, equation (2.15) has no analytic solution. In order to numerically evaluateit, we must instead simultaneously solve the second order differential equations for D and k µ , along with the first order differential equations for E µ and E µ . In sum, this requiresthe simultaneous numerical evaluation of 24 first order differential equations. To integratethis system, we utilize a modified fifth-order Runge-Kutta algorithm with adaptive step-size[47]. The solution follows the propagation of the null ray past the center of the structurefrom a position within the exterior FLRW region ( r >
50 Mpc). Beyond this particularexample of its use, it is also important to note that the evaluation of the Jacobi matrix D is an alternative method to calculate the angular diameter distance in such general models,without the need to evaluate partial derivatives of the null vector.As a verification that this process successfully reproduces the expected lensing profilefor the truncated NFW profile, we evaluate the geodesic deviation for the LT structure withdensity profile described in equation (3.6) and scale it by the equivalent FLRW angulardiameter distance to compare to the classic analytic expressions for the convergence andshear given by equations (2.30) & (2.32). The results of both methods are shown in figure3, where we have a lens at D ℓ ≈
50 Mpc and source at D s ≈
100 Mpc. At small r , boththe convergence and shear are less in the analytical model, which is consistent with thereduction in shear and convergence in the profile with peak central density. This is due todensity evolving as the ray passes through the cluster in the case of the numerical calculation,combined with the ray passing the center of the structure only approximately at t . In theevolving model, the density continues to grow in the center of the structure, and surpassesthis peak density, which causes an amplification of the convergence and shear relative to theanalytic model. In the analytical case, of course, the shear goes to zero near the center ofthe structure where the average enclosed mass is equal to the mass at some r .However, it is clear that the numerical calculation is successful due to the close agree-ment between the two shear values at larger r , and the agreement in the values of the– 13 – .1 1 0.00010.0010.01 Sh e a r ( γ ) r (Mpc)0.00010.0010.01 0.1 1 C o n v e r g e n ce ( κ ) r (Mpc) Geo. deviationAnalytical result Figure 3 . The convergence ( κ ) and shear ( γ ) are compared as a function of impact parameter r for the classic analytical result of a truncated NFW profile with peak central density as described insection 2.3 and that found through the geodesic deviation of a ray passing through an identical andevolving structure as described in section 3.2. In both cases, the source is located at D s ≈
100 Mpcand the lens is at D ( ℓ ) ≈
50 Mpc. The two methods diverge at small r , which is due in the secondcase to the structure evolving as the ray passes through it. For example, the shear should analyticallygo to zero at the center of a structure with a fixed, peak central density within some r , but will notdo so in a structure that evolves, even though the density of the structure is nearly identical whenthe ray passes the center at approximately t . At larger r , the shear and convergence are consistent,though inaccuracy in the determination of the equivalent FLRW angular diameter distance used tocalculate κ and γ in equations (2.25) causes the convergence to again deviate when it becomes verysmall. The shear does not suffer from this, as it is less sensitive to errors in ˜ D A . convergence. The deviation at large r in the convergence is due to a small systematic error inchoosing the appropriate equivalent FLRW distance, which the convergence is more sensitiveto at very small values (large r ). These results lend confidence to the method, however, inagreement with previous work with the LT metric which indicate good agreement betweenclassical lensing results and those due to exact numerical calculations of the lensing propertiesof spherically symmetric structures [33, 34]. Any systematic in the rescaling of the valuesdue to the choice of FLRW distance will also not impact the results of this investigation,where we seek to examine the relative differences due to including anisotropy and are lessinterested in the absolute values of the convergence. To quantify impacts in the lensing properties due to the inclusion of anisotropies, we repeatthe calculations described in the previous section for varying values of S . This producesanisotropies with δρ |D /ρ LT |D of 5%, 10%, and 15% for S = 4 .
87, 2 .
43, and 1 .
65. To simplifydiscussion of the geodesic deviation, we will define several quantities in analogy to the lensing– 14 –onvergence and shear, D κ = D − D D (3.20) D γ = D + D D (3.21) D γ = D D = D D (3.22) D γ = q D γ + D γ . (3.23)Since we are interested primarily in relative differences, we simplify the discussion by choosinga single D = 100 Mpc to be approximately the magnitude of the FLRW ˜ D A , which is usedto make the quantities dimensionless. The numerical Szekeres results for these quantities arein fact noisy due to limitations in the numerical accuracy of the integration along rays whichpass through the structure at different impact parameters, and thus are smoothed using aBezier algorithm to improve legibility of the figures that follow. We first consider a geometry that should cause maximal effect on the deviation of neighboringgeodesics, where the null ray passes through the cluster parallel to the axis of maximumSzekeres dipole. This corresponds to rays initially at angular coordinates θ = π/ φ = ± π/
2, which matches the directions for which the density profile is shown in figure 2.We first propagate the null rays through the structure at varying impact parameters r andplot the resulting ‘convergence’ D κ ≈ − κ and total magnitude of ‘shear’ D γ ≈ γ . This isshown in figure 4 for both absolute and relative comparisons with the LT structure.It is clear that the presence of anisotropy causes a strong effect on this measure ofconvergence (left panels) in both directions parallel to the dipole, and that the effects arenot symmetric. This second effect can be easily understood, since one dipole substructureis nearer to the center of the structure and denser. This will either increase or decrease themagnitude of convergence based on whether its positive mass contribution is on the sameside of the structure as the source. The increase in D κ corresponds to when the positive halfof this dipole is nearest the source. While the difference appears dramatic, the net effect ofthe anisotropy on a randomly aligned sample of such clusters would be a much more modestincrease in D κ , but still non-zero. The change from a larger increase in D κ at low impactparameter r to a larger decrease at high r between the two paths coincides with the locationsof the density dipole peaks in the structure as a function of r . In regions beyond the Szekeresdipoles, the convergence again agrees well with the LT result, indicating that the effects arelocal and will not impact measurements of convergence outside regions of anisotropy.The total shear (right panels), represented by D γ , is also affected by the anisotropy. Inthis case, though, the shear is primarily diminished in both directions, though less stronglyin the direction were the larger positive mass dipole contribution is on the side of the source.There may be a similar ‘turnover’ in the net effect on D γ between low and high r , but it isdifficult to say based on these results if the shear does have a net increase near r = 1 Mpc.Like the convergence, the total shear also matches well the LT value beyond r = 2 Mpc.While the effects of anisotropy on the total shear are interesting, we can also consider thetwo components of shear D γ ≈ γ (left panel) and D γ ≈ γ (right panel) separately. These– 15 – .1 1 -0.4-0.200.20.40.60.8 ∆ D γ / D L T γ r (Mpc)-0.4-0.200.20.40.6 0.1 1 ∆ D κ / D L T κ r (Mpc) 0.00010.001 D D γ D D κ Szekeres - 15%Szekeres - 10%Szekeres - 5%LT
Figure 4 . For the parallel geometry, where the null ray follows a path parallel to the axis of maximaldipole anisotropy in the structure, the convergence D κ ≈ − κ (left panels) and shear D γ ≈ γ (rightpanels) components of the geodesic deviation are shown. Three levels of anisotropy in the Szekeresmodel, defined as δρ |D /ρ LT |D of 5%, 10%, and 15%, are compared to the spherically symmetric LTreference model. In both cases there are significant deviation from the spherically symmetric (LT)case, which persists even assuming observations would be averaged over a large sample of similar,randomly aligned clusters. Top panels show deviations in magnitude while bottom panels show therelative deviations compared to the LT reference model. Solid lines represent null rays with a sourceat θ = π/ φ = π/
2, while dotted lines have source at φ = − π/ are shown in figure 5, which compares again the three Szekeres models to the sphericallysymmetric LT model. In the LT model, D γ = γ = 0, and the only component of the shearis γ . D γ follows a very similar trend to D γ , and tends to become less than in the LT case as r decreases. The γ component begins as zero in the Szekeres models for large r , consistentwith the LT model, but takes on a large negative value as the ray passes closer to the centerof the structure. In fact, the magnitude of γ becomes larger than that of γ for the 15%anisotropic model and dominates the contribution to total shear at small r . In all cases, γ isnegative for a source at φ = π/ φ = − π/
2. This indicates thatthe anisotropies are in fact causing more complex changes to the shear than simply scalingits magnitude. – 16 – .1 1 1e-050.00010.001 D | D γ | r (Mpc)1e-050.00010.001 0.1 1 D D γ r (Mpc) Szekeres - 15%Szekeres - 10%Szekeres - 5%LT Figure 5 . The magnitude of the components of shear D γ (left panel) and D γ (right panel) for thethree anisotropic models in the parallel geometry. D γ is zero for the LT model, but D γ of the LTmodel is shown for comparison. In each case, D γ is negative for a source at φ = π/ φ = − π/
2, while D γ is always positive. The D γ component is dominant for all modelsbut the 15% anisotropic model with source at φ = − π/
2, where D γ is largest at small r . Solid linesrepresent null rays with a source at θ = π/ φ = π/
2, while dotted lines have source at φ = − π/ To compare to the results in section 3.3.1, we also choose an orthogonal geometry, which oneexpects to cause minimal effects on the deviation of neighboring geodesics. In this geometry,the null ray passes through the cluster perpendicular to the axis of maximum Szekeres dipoleand thus is exposed to the least amount of deviation in the density compared to the LTcase. This corresponds to rays initially at angular coordinates θ = π/ φ = 0 , π . Thepath of the ray passing directly through the center of the structure in this geometry willthus experience the least possible change in density relative to the spherically symmetric LTreference structure. As in the case of paths parallel to the dipole axis, we first propagatethe null rays through the structure at varying impact parameters r and plot the resulting D κ ≈ − κ and D γ ≈ γ . This is shown in figure 6 for both absolute and relative comparisonswith the LT structure.The left panels showing D κ are consistent with our picture of minimal interference inthe lensing properties of the null ray along this direction. In fact, there appears to be noeffect on the convergence of the ray within the accuracy of the numerical calculations, and wecan rule out any effects greater than a few percent relative to the expected LT convergence.The total shear (right panels), however, indicate a much stronger impact on the shear inthis geometry than in the geometry parallel to the dipole axis. It is also consistent witha reduction of the total magnitude of shear, but at much stronger levels and with betteragreement between the two directions, since they are approximately symmetric in this case.However, the shear again agrees with the LT result beyond r = 0 . .1 1 -0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.1 ∆ D γ / D L T γ r (Mpc)-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.1 0.1 1 ∆ D κ / D L T κ r (Mpc) 0.00010.001 D D γ D D κ Szekeres - 15%Szekeres - 10%Szekeres - 5%LT
Figure 6 . For the orthogonal geometry, where the null ray follows a path orthogonal to the axisof maximal dipole anisotropy in the structure, the convergence D κ ≈ − κ (left panels) and shear D γ ≈ γ (right panels) components of the geodesic deviation are shown. Three levels of anisotropyin the Szekeres model, defined as δρ |D /ρ LT |D of 5%, 10%, and 15%, are compared to the sphericallysymmetric LT reference model. The convergence shows no detectable deviation from the sphericallysymmetric (LT) case, while the shear is significantly reduced for sources in both directions. Top panelsshow deviations in magnitude while bottom panels show the relative deviations compared to the LTreference model. Solid and dotted lines represent null rays with a source at θ = π/ φ = 0 , π . considers the two components independently, which is shown in figure 7 . While null raysfrom sources at both φ = 0 , π have positive γ , the magnitude grows much more strongly atlow r in this case and provides the dominant contribution to γ in the 5 and 10% anisotropiccases. The magnitude of γ still dominates at larger r , but as r decreases, γ is positive anddecreases before becoming negative and increasing in absolute magnitude once again. Thisoccurs at higher r the larger the anisotropies in the structure, with | γ | growing large enoughto provide the dominant contribution to γ again at the center of the structure. In sections 3.3.1 & 3.3.2, we explored the effects on convergence and shear of null rayswhich pass through specific orientations of dipole substructure anisotropy in a galaxy cluster.However, the effects depended strongly on which geometry was used, and the effects onconvergence, for example, depended on which direction the dipole anisotropy was orientedrelative to the source. To consider a more realistic measure of how the convergence and– 18 – .1 1 1e-050.00010.001 D | D γ | r (Mpc)1e-050.00010.001 0.1 1 D | D γ | r (Mpc) Szekeres - 15%Szekeres - 10%Szekeres - 5%LT Figure 7 . The magnitude of the components of shear D γ (left panel) and D γ (right panel) for thethree anisotropic models in the orthogonal geometry. D γ is zero for the LT model, but D γ of theLT model is shown for comparison. In each case, D γ is positive, while D γ is decreases, becomingnegative before decreasing in absolute magnitude again at small r . The D γ component is dominantat small r for all models but the 15% anisotropic model, where D γ grows negative enough to againdominate. Solid and dotted lines represent null rays with a source at θ = π/ φ = 0 , π . shear would be impacted when studying a statistical ensemble of randomly oriented galaxyclusters, we average these effects for each geometry among the two directions which the dipoleanisotropy might be oriented relative to the source. This provides a more realistic and cleanerpicture of the impacts on the convergence and shear measures that each geometry would haveand allows us to clearly quantify the effects of varying the strength of the anisotropy.We show these averages in figure 8 for the two geometries, where the dipole is parallel(top panels) or orthogonal (bottom panels) to the propagation of the light from the source.For the parallel geometry, we see that the average impact on D κ around r = 0 . r = 0 . D γ in the parallel geometry, there appears to be little net effect in the 5% model,though for anisotropies of 10% and 15%, there is an average decrease in the shear of 8% and17% for r < . r = 0 . r = 0 . .1 1 -0.7-0.6-0.5-0.4-0.3-0.2-0.10 ¯ ∆ D γ / D L T γ r (Mpc)-0.7-0.6-0.5-0.4-0.3-0.2-0.10 0.1 1 ¯ ∆ D κ / D L T κ r (Mpc) -0.15-0.1-0.0500.050.10.150.20.250.3 ¯ ∆ D γ / D L T γ -0.15-0.1-0.0500.050.10.150.20.250.3 ¯ ∆ D κ / D L T κ Szekeres - 15%Szekeres - 10%Szekeres - 5%LT
Figure 8 . The average relative difference from the spherically symmetric LT reference model ofthe two directions of sources, φ = ± π/ φ = 0 , π , for D κ and D γ of the parallel (top panels)and orthogonal (bottom panels) geometries, respectively. This simplifies the information found infigures 3.3.1 & 3.3.2. The strong nonlinear dependence on the strength of anisotropy found in [29]is evident in the effects on the convergence for the parallel geometry, while there is no discernableimpact on convergence in the orthogonal geometry. The shear tends to be less than the sphericallysymmetric case in both geometries, but doesn’t share the particular nonlinear dependence found forthe convergence or infall velocities in the models. these results that a net effect will persist and will not be averaged out, due to the nonlinearand asymmetric nature of the deviations due to anisotropies. For example, when taking anaverage over such clusters which are aligned in various directions with their dipole axis eitherparallel and orthogonal to the source and observer, we find relative differences of up to 1%,4%, and 12% in the convergence, and of 15%, 32%, and 44% in the magnitude of shear.In a traditional lensing framework, where the mass is considered as a projection ontosome surface orthogonal to the line of sight, this information on anisotropy is naturally lost.This occurs in two ways. First, when one projects a structure with the Szekeres dipole axisalong the line of sight, there is no difference in the projected mass Σ from a sphericallysymmetric structure. That is, the overdensity on one side of the structure is cancelled out bythe underdensity on the opposite side. This is at odds with the real lensing results shown inFig. 4, where there is clearly a non-negligible impact on convergence and shear measures dueto the anisotropy. Second, one can consider the case of a projection of the structure wherethe Szekeres dipole axis is orthogonal to the line of sight. There is a residual anisotropy in– 20 –he surface density Σ, but any of the nonlinearity exhibited by the true lensing results isdestroyed in the traditional lensing framework. In the latter, one expects that the impacton the surface density due to the projected anisotropy is equal and opposite on either sideof the structure’s center in the plane of projection. Thus, any average over a large ensemblewill result in a statistical lensing signal which is consistent with a spherically symmetric setof cluster lenses while it should not. Gravitational lensing has been demonstrated to be a powerful probe for studying the universe,either through strong and weak lensing by galaxies and clusters of galaxies or by the weaklensing or cosmic shear due to large-scale structure. The swiftly growing and improving setsof data we will collect over the coming decades beg to be met with ever more sophisticatedmodels, where inhomogeneities, anisotropies, and nonlinearities are properly accounted foras present in the true lumpy universe. One way of doing this is by developing exact models ingeneral relativity that are both inhomogeneous and anisotropic to represent realistic structurein the universe, and which are naturally nonlinear. The development of these exact modelsfor use in comparing to cosmological and astrophysical observation is an ongoing process. Theidentification and development of observables like infall velocities and gravitational lensingin such general exact models is not always straightforward, but it is a necessary step in orderto utilize and constrain these models.We have presented here a general framework for studying gravitational lensing basedon the inhomogeneous and anisotropic Szekeres models, one of the best candidates of exactsolutions discovered thus far for studying general structure in the universe. As a test of theframework, we used a realistic galaxy cluster model developed in [29], which has a modifiedNFW density profile, to examine the impact of including anisotropies in the exact structureon lensing observables related to the convergence and shear. The results from this processare consistent in the spherically symmetric limit with previous results for the LT models andthe convergence and shear for the same density profile in the classic lensing formalism. Wefound that the introduction of anisotropies in the structure modifies the usual lensing resultsin a significant quantitative way, as summarized below.Specifically, we compared geometries with the paths of null rays intersecting the clusterat various impact parameters r both parallel and orthogonal to the axis of maximal Szekeresdipole anisotropy. We find that for anisotropies of 5%, 10%, and 15% of the total mass, thereis a net reduction in the magnitude of convergence and shear through anisotropic regions inthe model compared to the usual spherically symmetric case. The convergence is impacted toa large degree when the light passes parallel to the dipole, and the effect is strongly nonlinearwith respect to the amount of anisotropy. This is consistent with findings in [29] for the infallvelocity. The effect is also asymmetric with respect to the direction of the light propagation.This causes the impact to persist even when averaged over path direction at the levels ofabout a 2%, 8%, and 24% increase near r = 0 . r = 0 . r in the orthogonal geometry, whilethe zero component in the spherically symmetric case can become the dominant contributorto the total magnitude in either geometry at low r , and can also change sign depending onthe direction of the light propagation.This work represents a necessary first step to exploring gravitational lensing in moregeneral, anisotropic exact models. Further refinement of the process is necessary, and includesimprovements of numerical accuracy and stability in such sophisticated models. The workserved to test the framework and indicates interesting implications for lensing by exact struc-ture models when anisotropy is included with effects both on convergence and shear. Thework constitutes the basis for future analysis of potential biases to statistical measurementsof lensing by clusters and galaxies, with an initial indication that the effects on convergenceand shear in anisotropic regions will persist at some level even when averaged statisticallyover all geometries. Similarly, additional applications of the framework to other lensing ob-servables like the bending angle, time delays, and mass estimates are also left for a follow-uppaper. Acknowledgments
We thank L. King for useful comments during the preparation of this work. MT acknowledgesthat this work was supported in part by the NASA/TSGC Graduate Fellowship program. MIacknowledges that this material is based upon work supported in part by by NSF under grantAST-1109667 and by NASA under grant NNX09AJ55G, and that part of the calculations forthis work have been performed on the Cosmology Computer Cluster funded by the HoblitzelleFoundation.
A The Christoffel symbols and Ricci and Weyl curvatures in the Szekeresmetric
In order to calculate the geodesic deviation in the Szekeres spacetime, one must parallelypropagate both a fiducial null ray k µ and the screen basis vectors E µ and E µ , which, alongwith the comoving velocity of the observer u µ , form a tetrad at each point along the pathof the fiducal ray affinely parameterized by λ . One must also know the Ricci and Weylcurvature of the spacetime at each point along the path. This first requires the calculation– 22 –f the nonzero Christoffel symbols. For the Szekeres metric they areΓ trr = HH, t Γ rpp = Γ rqq = − F F, r H Γ ppr = Γ prp = H, r F (A.1)Γ tpp = F F, t Γ rrt = Γ rtr = H, t H Γ ppq = Γ pqp = H, q F (A.2)Γ tqq = F F, t Γ rrp = Γ rpr = H, p H Γ qpp = − Γ qqq = − F, q F (A.3)Γ rrr = H, r H Γ rrq = Γ rqr = H, q H Γ qqt = Γ qtq = F, t F (A.4)Γ prr = − HH, p F Γ ppp = − Γ pqq = F, p F Γ qqr = Γ qrq = H, r F (A.5)Γ qrr = − HH, q F Γ ppt = Γ ptp = F, t F Γ qqp = Γ qpq = H, p F . (A.6)Once these are known, the Ricci and Weyl curvatures can be calculated. The nonzerocomponents of the Ricci tensor are R tt = − (cid:18) H, tt H + 2 F, tt F (cid:19) (A.7) R rr = − (cid:18) F, rr F − H, r F, r HF + H (cid:20) − H, tt H + H, pp HF + H, qq HF − F, t H, t HF (cid:21)(cid:19) (A.8) R pp = F (cid:18) F, tt F + F, t F + H, t F, t HF − H (cid:20) F, rr F + F, r F − H, r F, r HF (cid:21)(cid:19) (A.9) − F, pp F − F, qq F + F, p F + F, q F + H, p F, p HF − H, q F, q HF − H, pp HR q = F (cid:18) F, tt F + F, t F + H, t F, t HF − H (cid:20) F, rr F + F, r F − H, r F, r HF (cid:21)(cid:19) (A.10) − F, pp F − F, qq F + F, p F + F, q F − H, p F, p HF + H, q F, q HF − H, qq H , and the nonzero components of the Weyl tensor are C trtr = − H F, p F − H F, q F + H F, pp F + H F, qq F − H F, t F − HH, pp F (A.11) − HH, qq F + F, r F + H F, tt F + H, r F, r HF + HH, t F, t F − F, rr F − HH, tt C tptp = 14 HF [ F ( H, qq − H, pp ) + 2 ( H, p F, p − H, q F, q )] − F H C trtr (A.12) C tqtq = − HF [ F ( H, qq − H, pp ) + 2 ( H, p F, p − H, q F, q )] − F H C trtr (A.13) C rprp = H F [ F ( H, qq − H, pp ) + 2 ( H, p F, p − H, q F, q )] + F C trtr (A.14) C rqrq = − H F [ F ( H, qq − H, pp ) + 2 ( H, p F, p − H, q F, q )] + F C trtr (A.15) C pqpq = − F H C trtr , (A.16)with the additional components due to symmetries in the Weyl tensor being suppressed.– 23 – eferences [1] F. Bernardeau, C. Bonvin, F. Vernizzi, Full-sky lensing shear at second order , Phys. Rev.,D81:083002, (2010).[2] F. Bernardeau, C. Bonvin, N. Van de Rijt, F. Vernizzi,
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