Gravitational Lensing by Intercluster Filaments in MOND/TeVeS Martin Feix SUPA, School of Physics and Astronomy, University of St Andrews ATM workshop Toulouse November 8th 2007
Outline 1 Introduction 2 Basics of TeVeS 3 Lensing in TeVeS 4 Filamentary Lenses 5 Summary
Introduction MOdied Newtonian Dynamics Standard paradigm of GR fails to describe observations in the extragalactic regime (Newtonian gravitational theory) For example, predicted velocity curves due to the Newtonian eld generated by visible matter are much smaller than the observed ones "missing mass", "acceleration discrepancy" Two ways of solving the problem: Consider GR to be true and introduce CDM to explain observational data Change the the law of gravity itself
Introduction MOdied Newtonian Dynamics "Simple" MOND formula for unifying extragalactic dynamics phenomenology WITHOUT invoking CDM (Milgrom 1983) ( ) g µ g = Φ N with and From empirical data: a 0 g = v 2 r = GM r 2 µ µ(x) x for x 1 µ(x) 1 for x 1 a 0 1 10 10 m/s 2
Introduction MOdied Newtonian Dynamics
Introduction MOdied Newtonian Dynamics Using the MOND paradigm, one is able to explain much of extragalactic dynamics (e.g. Tully-Fisher, rotation curves, tidal dwarf galaxies) Believing in CDM or not, it is obviously worth to have a closer look at such modications of gravity However, MOND is not a theory (Violation of conservation laws, no cosmological model, no specication of gravitational light deection) To develop a serious MONDian framework, we need a fully relativistic theory Aether-type theories (Zlosnik et al. 2007) TeVeS (Bekenstein 2004; Zlosnik et al. 2006)
Outline 1 Introduction 2 Basics of TeVeS Physical Metric, Fields and Actions The Free Function Quasi-static Systems Cosmology 3 Lensing in TeVeS 4 Filamentary Lenses 5 Summary
Basics of TeVeS Physical Metric, Fields and Actions TeVeS is based on three dynamical gravitational elds: an Einstein metric g µν a 4-vector eld U µ and a scalar eld Φ Introduce a physical metric g µν : g µν = exp( 2Φ)g µν 2U µ U ν sinh(2φ) Coupling to matter through g µν, thus spacetime delineated by matter dynamics has the metric g µν
Basics of TeVeS Physical Metric, Fields and Actions Total action given by S S g + S s + S v + S m S g = 1 g µν R µν g d 4 x 16πG S s = 1 [ˆσ 2 h µν φ,µφ,ν + V (k ˆσ 2 ) ] g d 4 x, 16πG S v = K [F µν F µν λ(g µν U µ U ν + 1)] g d 4 x, 32πG S m = L m g d 4 x
Basics of TeVeS The Free Function It is useful to introduce a new function y(µ) related to the potential V such that µ(y) = µ(kl 2 h µν φ,µ φ,ν ) = k ˆσ 2 /8π There is no theory for y(µ) great freedom of choice In order to obtain both a MONDian and a Newtonian limit for quasi-static situations (y > 0), however, we must have: y(µ) bµ 2, µ 1 y(µ), µ 1 k, K 1 where l and G N are given by ( ) bk l, G N = 1 + k G G 4πa 0 4π
Basics of TeVeS Quasi-static Systems For quasi-static systems, we can apply the non-relativistic approximation In this case, the total gravitational potential is given by Φ G Φ N + Φ Neglecting temporal derivatives and the pressure term, the equation of the scalar eld reduces to [ µ (kl 2 ( Φ) ) ] 2 Φ = kgρ, and Φ N is calculated from standard Poisson's equation
Basics of TeVeS Cosmology Modied Friedmann equation: ( ) 2 ȧ = 8πG a 3 (ρe 2φ + ρ φ ) K a + Λ 2 3, ( ) 1 dã = e φ ȧ ã d t a φ, d t = e φ dt, ã = e φ a Scalar eld φ evolves slowly throughout cosmological history It is consistent to assume φ 1 (Bekenstein 2004) Contribution of ρ φ small compared to matter ρ φ 0: H 2 H 2 H 2 0 ( Ωm (1 + z) 3 + Ω Λ + Ω K (1 + z) 2)
Basics of TeVeS Cosmology Flat µhdm cosmology (Skordis et al. 2006): Ω Λ 0.78, Ω m 0.22 Open minimal-matter cosmology (Zhao et al. 2006): Ω Λ 0.46, Ω m 0.04 Flat minimal-matter cosmology: Ω Λ 0.95, Ω m 0.05 Minimal-matter models roughly valid up to redshifts of z 3 in the context of gravitational lensing
Outline 1 Introduction 2 Basics of TeVeS 3 Lensing in TeVeS Preliminaries Lensing Formalism 4 Filamentary Lenses 5 Summary
Lensing in TeVeS Preliminaries Similar to GR, most of the TeVeS light bending still occurs within a small range around the lens Substitute Φ N by Φ = Φ N + φ Integrating the total potential along the line of sight, we obtain the deection potential Apply common lensing formalism Major task: Determine the scalar potential φ Certain symmetries: simple eld equation, sometimes analytic solutions General: Solve non-linear Poisson's equation in three dimensions
Lensing in TeVeS Lensing Formalism - Deection angle, deection potential The deection angle ˆα can be expressed as ˆα = 2 Φdl, Introduce the deection potential Ψ: Ψ( θ) = 2 D ls Φ(D lθ, z)dz D s D l which is related to the deection angle through α = θ Ψ, α = D ls D s ˆα
Lensing in TeVeS Lensing Formalism - Lens mapping, amplication Linearized lens mapping: A( θ) = ( ) β 1 κ θ = γ1 γ 2, γ 2 1 κ + γ 1, ( 2 Ψ γ 1 = 1 2 γ = θ 2 1 ) 2 Ψ, γ θ 2 2 = 2 Ψ, θ 2 1 θ 2 γ 2 1 + γ2 2, κ = 1 2 θ Ψ. Lensing preserves surface brightness (Liouville's theorem) Flux ratio between source and image given by amplication A: A 1 = (1 κ) 2 γ 2
Lensing in TeVeS Weak Lensing - Reduced shear Introduce the complex shear G, G = γ 1 + iγ 2, and dene the reduced shear g as follows: g = G 1 κ This quantity is the expectation value of the (complex) ellipticity χ of galaxies weakly distorted by the lensing eect Therefore, g corresponds to the signal which can actually be observed If κ 1, we have g γ
Lensing in TeVeS Special Lensing Geometries Consider a deection potential which depends on one coordinate only, e.g. the y-coordinate Ψ = Ψ(y), ˆα = ˆα(y) Using the previous equations for ˆα and κ, we have κ(y) = 1 D l D ls ˆα(y) 2 y Furthermore, we nd the relation D s κ = γ = 1 A 1 2 If the κ = 0, the lens mapping turns to identity and A = 1
Outline 1 Introduction 2 Basics of TeVeS 3 Lensing in TeVeS 4 Filamentary Lenses The ΛCDM simulation by Colberg et al. Filaments in MOND/TeVeS Uniform Model Oscillation Model 5 Summary
Filamentary Lenses - arxiv:0710.4935 The ΛCDM simulation by Colberg et al. (2005) Figure: Two orthogonal projections of the dark matter between two of the clusters in the GIF simulation. The plots show the projected overdensities, smoothed with a Gaussian of radius 0.5h 1 Mpc. The contour levels show overdensities ranging from 0.0 (mean density,black) to 19.0 (white). The y-axis cuts through both cluster centers. The region shown here excludes the clusters themselves, matter follows a lamentary pattern (Colberg et al. 2005).
Filamentary Lenses - arxiv:0710.4935 The ΛCDM simulation by Colberg et al. (2005) Figure: Left: Enclosed overdensity prole of a straight lament. The bold vertical line shows the radius at which the prole starts to follow an r 2 power law. Right: Longitudinal density prole of straight laments, averaged over straight laments that are longer than 5h 1 Mpc. Shown is the enclosed overdensity as a function of the positions along the cluster-cluster axis for all material that is contained within 2h 1 Mpc from the axis.(colberg et al. 2005).
Filamentary Lenses - arxiv:0710.4935 The ΛCDM simulation by Colberg et al. (2005) Straight laments connect almost every two neighboring clusters Generally correspond to overdensities of about 10 30 Cigar-like structures with a length of 20 30Mpc Typically much longer than their diameter (can be approximately treated as innitely elongated objects) Low density gradient along their axis Nearly uniform core and well-dened core radius R f beyond which the density tapers to zero R f 2 2.5h 1 Mpc for the majority of laments
Filamentary Lenses - arxiv:0710.4935 Filaments in MOND/TeVeS But: Filaments are among the lowest-density structures Typical (Newtonian) accelerations: 0.01 0.1 a 0 Expect substantial MONDian eect However, we have no MOND/TeVeS structure formation simulation so far Filaments are generic, HDM and CDM simulations show similar behavior (Knebe et al. 2002; Bode et al. 2001) Baryons in MOND simulations create a similar pattern at low redshifts (Knebe & Gibson 2004) Assume that laments have roughly the same properties (e.g. overdensity, shape) in MOND/TeVeS cosmologies
Filamentary Lenses - arxiv:0710.4935 Uniform Model
Filamentary Lenses - arxiv:0710.4935 Uniform Model Model laments as innite uniform cylinders with radius R f The integral for the deection angle can be rewritten in terms of cylindrical coordinates ˆα(y) = 4y y Φ r 2 y 2 dr The Newtonian acceleration g is given by (λ = M/L = ρπr 2 f ) g N (r) = Φ N (r) = Gλ 2π r R 2 f Gλ 1 2π r,, r < R f r > R f
Filamentary Lenses - arxiv:0710.4935 Uniform Model - Newtonian case As for Newtonian gravity, we can easily calculate deection angle and convergence of our model For y > R f, integration yields For y < R f, we nd ˆα N (y) = 2Gλ π κ N (y) = 2 D l D ls D s ˆα N (y) = Gλ = const, κ N = 0 y R 2 f y 2 Gλ πr 2 f R 2 f R 2 f y 2 ( ) y + arcsin, R f
Filamentary Lenses - arxiv:0710.4935 Uniform Model - MONDian case Choosing a certain form of the free function µ, the total gravitational acceleration in TeVeS can be expressed as (Zhao et al. 2006) g M (r) = Φ M (r) = g N (r) + g N (r)a 0 For y > R f, we arrive at the following: ˆα M (y) = Gλ + Γ(1/4) 2Gλa0 y, Γ(3/4) Gλa 0 κ M (y) = D l D ls D s Γ(1/4) Γ(3/4) Fundamental dierence between MOND and Newtonian dynamics 8y
Filamentary Lenses - arxiv:0710.4935 Uniform Model - MONDian case For y < R f, we eventually obtain ˆα M (y) = ˆα N (y) + + 2Gλa0 y 3/2 π 2Gλa0 y π R f 4 R 2 f y 2 R f y B (0,y 2 /R 2 ) (1/4, 1/2), f B (y 2 /R 2,1) (3/4, 1/2) f where B (p,q) (a, b) is the generalized incomplete Beta function, B (p,q) (a, b) = q p t a 1 (1 t) b 1 dt, Re(a), Re(b) > 0
Filamentary Lenses - arxiv:0710.4935 Uniform Model - Flat µhdm cosmology 50 45 40 0.03 0.025 κ M 10κ N 35 0.02 30 ˆα [arcsec] 25 20 κ = γ 0.015 15 0.01 10 5 ˆα M 10 ˆα N 0.005 0 0 1 2 3 4 5 6 7 8 y [Mpc] 0 0 1 2 3 4 5 6 7 8 y [Mpc]
Filamentary Lenses - arxiv:0710.4935 Uniform Model - Flat minimal-matter cosmology 30 0.02 κ M 25 10κ N 0.015 20 ˆα [arcsec] 15 κ = γ 0.01 10 0.005 5 ˆα M 10 ˆα N 0 0 1 2 3 4 5 6 7 8 y [Mpc] 0 0 1 2 3 4 5 6 7 8 y [Mpc]
Filamentary Lenses - arxiv:0710.4935 Oscillation Model More realistic approach: Describe lament and surrounding area by uctuating density prole including voids Choose a simple toy model for the density contrast: ( ) 1 ( ) δ πr 0 R sin πr f R f, r < 2R f δ(r) = 0, r > 2R f In this case, the Newtonian acceleration is given by ( ( )) Gρ 0 δ 0 R 2 f πr 1 cos, r < 2R π g N (r) = 2 f r R f 0, r > 2R f
Filamentary Lenses - arxiv:0710.4935 Oscillation Model - Flat µhdm cosmology 4 0.007 3.5 ˆα M 10 ˆα N 0.006 0.005 κ M 10κ N 3 0.004 2.5 0.003 ˆα [arcsec] 2 1.5 κ = γ 0.002 0.001 0 1-0.001 0.5-0.002 0 0 1 2 3 4 5 6 7 8-0.003 0 1 2 3 4 5 6 7 8 y [Mpc] y [Mpc]
Filamentary Lenses - arxiv:0710.4935 Oscillation Model - Flat minimal-matter cosmology 1.8 0.005 ˆα M κ M 1.6 10 ˆα N 0.004 10κ N 1.4 0.003 1.2 ˆα [arcsec] 1 0.8 κ = γ 0.002 0.001 0.6 0 0.4 0.2-0.001 0 0 1 2 3 4 5 6 7 8-0.002 0 1 2 3 4 5 6 7 8 y [Mpc] y [Mpc]
Outline 1 Introduction 2 Basics of TeVeS 3 Lensing in TeVeS 4 Filamentary Lenses 5 Summary
Summary We have investigated gravitational lensing by inter-cluster laments in MOND/TeVeS Analyzing toy models of such laments, we were able to provide order-of-magnitude estimates for the lensing signal In Newtonian gravity, the lensing signal generated by laments is negligibly small In MOND/TeVeS, however, lamentary structures can create a lensing signal on the order of κ γ 0.01 as well as an amplication bias at a 2% level, A 1 1.02 Assuming this alternative frame of gravity, laments might cause/contribute to anomalous lensing signals in recent observations, e.g. the "cosmic train wreck" in Abell 520 which has a weak shear signal at a level of 0.02 (Mahdavi et al. 2007)
Summary In principle, the predicted dierence in the weak lensing signal could also be used to test MOND/TeVeS and its variations Several failed attempts to detect laments by means of weak lensing methods, e.g. the analysis of Abell 220 and 223 by Dietrich et al. (2005), might already be a rst hint to problems within such modications of gravity However, shear signals around γ 0.01 are still rather small to be certainly detected Without a MOND/TeVeS N-body structure formation simulation, we cannot be sure about how laments form and how they look like in a MONDian universe compared to the ΛCDM case