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Anti-lensing: the bright side of voids Krzysztof Bolejko1, Chris Clarkson2, Roy Maartens3,4, David Bacon3, Nikolai Meures4 and Emma Beynon4 1Sydney Institute for Astronomy, The University of Sydney, School of Physics A28, NSW 2006, Australia 2Centre for Astrophysics, Cosmology & Gravitation, and, Department of Mathematics & Applied Mathematics, University of Cape Town, Cape Town 7701, South Africa 3Physics Department, University of the Western Cape, Cape Town 7535, South Africa 4Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK MorethanhalfofthevolumeofourUniverseisoccupiedbycosmicvoids. Thelensingmagnifica- tioneffectfromthoseunder-denseregionsisgenerallythoughttogiveasmalldimmingcontribution: objectsonthefarsideofavoidaresupposedtobeobservedasslightlysmallerthanifthevoidwere 2 not there, which together with conservation of surface brightness implies net reduction in photons 1 received. This is predicted by the usual weak lensing integral of the density contrast along the 0 line of sight. We show that this standard effect is swamped by a relativistic Doppler term that is 2 typicallyneglected. Contrarytotheusualexpectation,objectsonthefarsideofavoidarebrighter p than they would be otherwise. Thus the local dynamics of matter in and near the void is crucial e andisonlycapturedbythefullrelativisticlensingconvergence. Therearealsosignificantnonlinear S corrections to the relativistic linear theory, which we show actually under-predicts the effect. We use exact solutions to estimate that these can be more than 20% for deep voids. This remains an 4 importantsourceofsystematicerrorsforweaklensingdensityreconstructioningalaxysurveysand 1 forsupernovaeobservations,andmaybethecauseofthereportedextrascatteroffieldsupernovae located on the edge of voids compared to those in clusters. ] O C Introduction Lensingphenomenaaremeasurednot The usual form (2) is an approximation to the first . h only around virialized clusters of galaxies but also term on the right of (3), p through and around unvirialized cosmic voids, which oc- o- cupywellabovehalfthevolumeoftheUniverse. Herewe κ =(cid:90) χSdχ(χS −χ)χ∇2Φ. (4) r show how the standard lensing magnification effect can ∇2Φ 0 χS ⊥ t s beoverwhelmedbyrelativisticcorrectionstothesizeand a brightness of sources in and near voids. The screen-space Laplacian is ∇2 = ∇2 − (n · ∇)2 − [ ⊥ Magnification in the linear approximation The 2χ−1n·∇,wherenistheunitdirectionfromthesource. 1 lensing magnification effect can be expressed in terms of The radial derivatives lead to terms proportional to Φ, v the convergence κ, which corrects the background angu- Φ(cid:48) and Φ(cid:48)(cid:48) [2], which are much smaller than the term 2 lar diameter distance (d¯ ) ∇2Φ on the sub-Hubble scales of interest. Thus in (4), 4 A we may replace ∇2Φ by ∇2Φ, which is given in terms 1 d (z)=d¯ (z)[1−κ(z)]. (1) ⊥ 3 A A of the density contrast δ by the Poisson equation. The . generalrelativisticPoissonequationinvolvesalsothepe- 9 The convergence in a perturbed ΛCDM universe is usu- culiar velocity (v =∂ v): 0 ally given as a line of sight integral over the density con- i i 2 trast δ, v:1 κ=κ = 3H2Ω (cid:90) χSdχ(χS −χ)χ(1+z)δ(χ), (2) ∇2Φ = 3H202aΩm(cid:0)δ−3aHv(cid:1), (5) Xi δ 2 0 m 0 χS v = − 2a (cid:0)Φ(cid:48)+aHΦ(cid:1). (6) 3H2Ω r where dχ=dz/H =−dη, χ is the comoving distance, η 0 m a conformaltimeandS denotesthesource. Infactthefull By(6),aHv isoforderΦandmaybeneglectedin(5)on relativistic expression is [1, 2] therelevantscales. Then(4)reducestotheusuallensing term (2). For an under-density, δ < 0, so that κ < 0 κ=κ +κ +κ +κ , (3) δ ∇2Φ v SW I if the underdensity is the dominant structure along the where the Sachs-Wolfe term κ is given by the dif- line of sight. Then (1) implies that the angular distance SW ference in gravitational potential Φ between source and should be larger than the background value for a fixed observer, and κ is a line of sight integral over Φ and z, and objects should consequently be observed to be I its conformal time derivatives Φ(cid:48),Φ(cid:48)(cid:48). These two terms smaller. Since surface brightness is conserved in lensing, are sub-dominant [1], and we will not discuss their de- thetotalnumberofphotonsarrivingfromtheobjectper tailed form, although we do include them in our nu- unittimeshouldbefewer. Wecanquantifythisusingthe merical calculations below. (The perturbed metric is changetothedistancemodulus,∆m=5log d /d¯ , so 10 A A ds2 =a2(cid:2)−(1+2Φ)dη2+(1−2Φ)dx2(cid:3).) that a positive ∆m corresponds to a fainter source. 2 FIG. 1: Change in distance modulus due to a 50 Mpc radius void, vs observed redshift, based on: relativistic convergence (3) (solid), usual weak lensing formula (2) (dotted), Doppler term (7) (dot-dashed), exact model (dashed) (top panels). There is a brightening for objects on the far side of the void, whereas the usual weak lensing predicts a dimming of much smaller magnitude (see insets). The left panel shows the effect for a small void well in the linear regime, and the right panel shows a verydeepvoidwherenon-linearcontributionsincreasetheeffect. Thebottompanelsshowthevoiddensitycontrasts. (Asterisk marks the far edge of the void.) The usual formula is a good approximation to three of zero shear. The effect is stronger for a deep void with the terms in (3): κ ≈κ (cid:29)κ ,κ . The remaining δ = −0.95 (Fig. 1, right panel), where the linear ap- δ ∇2Φ SW I min Doppler term, which arises from a shift in the redshift proximation underestimates the anti-lensing effect – we from its background value, consider this in more detail below. Modelling voids via nonlinear solutions Real (cid:20) (cid:21) κ = 1− 1+zS v ·n, (7) voids typically have δ (cid:46) −0.8 [3], and we need to ex- v χSHS S tend the relativistic perturbative analysis to deal with such voids. We can gain some insight via exact solutions is typically ignored – but it cannot be neglected as em- oftheEinsteinfieldequations,whereavoidregionisem- phasised by [1], and as seen in Fig. 1. bedded in a homogeneous ΛCDM solution. We consider Figure 1 (left) shows the correction to the distance 3 models: modulus for a spherical void of radius 50 Mpc in the Spherical void, using an LTB model, with the observer linear regime, δ = −0.05, located at z = 0.1. It is min looking through the centre. The void can be compen- clear that κ predicts a completely wrong magnitude for δ sated by a spherical shell of matter, or uncompensated. sources in or near the void – and furthermore it predicts In the compensated case we choose the wrong sign. By contrast, κ gives a very good ap- v proximationtothefullrelativisticκ. Nearthefaredgeof 1 r (cid:54) 1R, the void there is a significant positive magnification sig- δ(r) −1r−2R(r−R)e3/2−6[(r−R)/R]2 1R(cid:54)2 r (cid:54) 3R, = 2 2 2 nal: objects are brighter than they would be otherwise, δ −r−2(r−2R)2 3R(cid:54)r (cid:54)2R, an effect which extends far beyond the edge of the void. min 0 r2(cid:62)2R, This is the opposite effect one expects based on a naive (8) predictionusingtheusuallensingformula. Wealsoshow and in the uncompensated case, the prediction of an exact model of the void, using the Lemaitre-Tolman-Bondi (LTB) solution (see below). It 1 r (cid:54) 1R, δ(r)  2 demonstrates that the linear relativistic κ is accurate for = −1r−2R(r−R)e3/2−6[(r−R)/R]2 1R(cid:54)r (cid:54)R, δ 2 2 this amplitude of void. Since there is no background for min 0 r (cid:62)R. the LTB case, the effect is not due to peculiar velocity, (9) but rather to the extra redshifting of photons as they (Weonlyconsidergrowingmodes,i.e. auniformbigbang pass through a region of higher expansion rate and non- time.) 3 [4], with the same mass distribution M(r) as the compensated LTB model. In addition there is a dipole- like contribution of the form −(S(cid:48)/S)cosθ where S = r−0.99e−0.99r/(2R),forr <2RandS =(2R)−0.99e−0.99 = const, for r ≥ 2R. This generates a compensated inhomogeneity extending to r =2R. Cylindrical voidwiththeobserverlookingalongthesym- metry axis, using a Szekeres type II model [5]. The density profile along the symmetry axis is compensated and extends to about r =2R. Westandardizethevoidsineachmodelradius50Mpc, with centre at z = 0.1. This corresponds to a slightly different comoving distance in each model, and the redshift of the near and far sides are slightly different. The depths we choose are δ =−0.95 for the spherical and min quasi-sphericalcases, andδ =−0.8forthecylindrical min case. Our model void size is typical in the SDSS survey, where void radii are in the range 5 to 135h−1Mpc [3]. Figure 2 shows the change in distance modulus for each type of void. In all cases we see the same qualitative behaviour as in the perturbative case: a relative dimming ofobjectsonandneartheclosersideofthevoid,through to a relative brightening of objects located on and near the far side of the void. Details of the signal depend on the void shape and the nature of compensating regions, but the usual weak lensing prediction (2) is always completely wrong, unless the source is far from the void (cf the insets in Fig. 1). Thus the predictions of the relativistic perturbative magnification persist in the nonlinear regime and are generic for different void configurations. But nonlinear effects can be large, and the linear relativistic analysis canbewrongbymorethan∼20%comparedtoanexact sphericalvoidmodel. ThisisshowninFig. 3,whichalso includes the error for unvirialized over-densities. Discussion Our results illustrate a general princi- ple: themeasuredmagnitudeofastronomicalobjectsde- pends not only on internal properties of the source and FIG.2: Differenceinmagnitude(toppanels)anddensitycon- statistics of large-scale structure, but also on environ- trast(bottompanels)betweenthebackgroundΛCDMmodel ment and the nature of inhomogeneity along the line of and the exact void models. We show results for an observer sight, which causes (de-)magnification. (See also [6] and lookingthroughsphericalcompensated(top),quasi-spherical [7].) We have used the full relativistic perturbative anal- (middle, shown for both horizontal – blue, dashed – and ver- ysis to show that magnification of objects located in and tical – red, dotted – lines of sight) and cylindrical (bottom) voids. The insets show the equatorial density contrast, with near cosmic voids is dominated by the Doppler term (7), eachsquare160Mpcacross. Inthesphericalcasewealsoshow which overwhelms the lensing effect (2) – and is of op- theuncompensatedcase(reddottedline)discussedinFig.1. posite sign near the far edge of the void (see Fig. 1). In The standard weak lensing prediction is shown in green in other words, we have uncovered an anti-lensing effect for each case, which predicts the wrong sign and amplitude of sources near the far side of voids. The usual weak lens- the effect. ing analysis fails completely to predict the foreground dimming and the background brightening from a void. Using exact solutions to model different voids, we have shown that this qualitative behaviour persists for non- Quasi-spherical void, with a mass concentration off to linear voids and for different shapes and ray directions one side, with the observer looking either through the (Fig. 2). Nonlinear corrections to the relativistic linear concentration, or along a line of sight not containing predictions can be large (Fig. 3). the concentration. We use the Szekeres type I model of Thefailureofstandardweaklensingforobjectsinand 4 FIG. 3: (Left) Accuracy of the lensing approximation compared to the full relativistic approximation for the magnification near the far edge of a spherical compensated void (δ <0) and a unvirialized over-dense lump (δ >0). (Right) Comparison of thefulllinearapproximationtotheexactresult: maximumdifferencesof>20%areseenfordeep–butneverthelessrealistic– voids. Voids/lumps of radius 100, 50, 25, and 10 Mpc are used. near under-dense void regions extends also to over-dense will move the redshift of a sheared source, which is a lump regions, provided that they are unvirialized. Com- correction that should be taken into account in making mon to both cases is the coherent flow into or out of the density maps from shear data. However, the sign of the region, which sources a large velocity contribution. The shearisunchanged,astheDopplertermdoesnotdistort key difference is that the lump occupies a much smaller an object’s shape, but only changes its inferred size. As (and shrinking) volume. As the lump continues to con- pointed out in [1], measurement of the convergence and dense it tends to virialize and the correction then dies lensing shear will provide a powerful probe of peculiar away, whereas the correction for the void increases with velocities. time. For a source near a cluster of galaxies, there is no large net inflow or outflow since the structure is close Could this effect have already been detected? It is to virialized, and so the usual weak lensing analysis is reported in [8] that the scatter in the Hubble diagram accurate. of SNIa magnitudes depends on the environment of the host galaxy. They find that galaxies which have a lower The relativistic linear analysis is accurate for comput- star formation rate are associated with SNIa which have ingthevelocitycontributionduetolarge-scalestructure, a smaller scatter in the Hubble diagram (at 2−3σ). As provided the voids have δ (cid:38) −0.2 (see Fig. 3). The min lower star formation rates are associated with galaxies velocity contribution from large-scale structure was esti- in clusters compared to those in the field – the latter mated for the angular power spectrum at fixed redshift being more likely to be on the edge of a void – it may in [1] – but without taking account of the nonlinear ef- be that this extra scatter is due to the large relativistic fects from voids that we have identified, which we have anti-lensing effect we have described here. This deserves shown amplify the effect. The velocity contribution Cv (cid:96) further investigation. was predicted to exceed the usual Cδ for z (cid:46) 0.2, to be (cid:96) about50%forz ≈0.5andtobenegligibleforz >1. The nonlinearcorrectionsthatwehaveidentifiedduetovoids Acknowledgments: with δ (cid:46) −0.2 will introduce a systematic error into min We thank Mat Smith for discussions. DB and RM are the perturbative calculation. A key question is: how to supported by the UK Science & Technology Facilities estimate the nonlinear void correction and thus correct Council (grant no. ST/H002774/1). DB, CC, RM, NM forthissystematicingalaxysurveys? Asimilarquestion are supported by a Royal Society (UK)/ National Re- applies also to supernovae magnitudes [6]. search Foundation (SA) exchange grant. RM is sup- While the void effect on convergence is large and of ported by the South African Square Kilometre Array the opposite sign to what is expected from lensing, the Project. CC and RM are supported by the National Re- same is not true for lensing shear. The Doppler effect search Foundation. 5 N.MeuresandM.Bruni,Mon.Not.R.Astron.Soc.419, 1937 (2012). [6] C. Clarkson, G. Ellis, A. Faltenbacher, R. Maartens, [1] C. Bonvin, Phys. Rev. D78, 123530 (2008). O. Umeh and J. -P. Uzan, Mon. Not. R. Astron. Soc., [2] C. Bonvin, R. Durrer and M. A. Gasparini, Phys. Rev. in press (2012) [arXiv:1109.2484]. D73, 023523 (2006). 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