Gravitational lensing is most often used as a tool to investigate the
distribution of (dark) matter in the universe, but, if the mass distribution is
known a priori, it becomes, at least in principle, a powerful probe of gravity
itself. Lensing observations are a more powerful tool than dynamical
measurements because they allow measurements of the gravitational field far
away from visible matter. For example, modified Newtonian dynamics (MOND) has
no relativistic extension, and so makes no firm lensing predictions, but
galaxy-galaxy lensing data can be used to empirically the deflection law of a
point-mass. MONDian lensing is consistent with general relativity, in so far as
the deflection experienced by a photon is twice that experienced by a massive
particle moving at the speed of light. With the deflection law in place and no
invisible matter, MOND can be tested wherever lensing is observed. The
implications are that either MONDian lensing is completely non-linear or that
MOND is not an accurate description of the universe.Comment: PASA (OzLens edition), in press; 5 pages, 1 figur

Daniel J. Mortlock

Edwin L. Turner

Tadros

English

arXiv

Crossref

Gravitational Lensing and Modified Newtonian Dynamics

Gravitational lensing is most often used as a tool to investigate the distribution of (dark) matter in the universe, but, if the mass distribution is known a priori, it becomes, at least in principle, a powerful probe of gravity itself. Lensing observations are a more powerful tool than dynamical measurements because they allow measurements of the gravitational field far away from visible matter. For example, modified Newtonian dynamics (MOND) has no relativistic extension, and so makes no firm lensing predictions, but galaxy–galaxy lensing data can be used to empirically constrain the deflection law of a MONDian point-mass. The implied MONDian lensing formalism is consistent with general relativity, in so far as the deflection experienced by a photon is twice that experienced by a massive particle moving at the speed of light. With the deflection law in place and no invisible matter, MOND can be tested wherever lensing is observed

Mortlock, Daniel J

Turner, Edwin L

Spiral - Imperial College Digital Repository

arXiv:astro-ph/0103208v1  14 Mar 2001Gravitational lensing and modified Newtonian dynamicsDaniel J. Mortlock1,2 Edwin L. Turner31 Astrophysics Group, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UnitedKingdommortlock@ast.cam.ac.uk2 Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, United Kingdom3 Princeton University Observatory, Peyton Hall, Princeton, NJ 08544, U.S.A.elt@astro.princeton.eduAbstractGravitational lensing is most often used as a tool to investigate the distribution of (dark)matter in the universe, but, if the mass distribution is known a priori, it becomes, at least inprinciple, a powerful probe of gravity itself. Lensing observations are a more powerful toolthan dynamical measurements because they allow measurements of the gravitational fieldfar away from visible matter. For example, modified Newtonian dynamics (MOND) has norelativistic extension, and so makes no firm lensing predictions, but galaxy-galaxy lensingdata can be used to empirically constrain the deflection law of a MONDian point-mass. Theimplied MONDian lensing formalism is consistent with general relativity, in so far as thedeflection experienced by a photon is twice that experienced by a massive particle movingat the speed of light. With the deflection law in place and no invisible matter, MOND canbe tested wherever lensing is observed.Keywords: gravitational lensing – gravitation – dark matter1 IntroductionIn the time since the discovery of the first multiply-imaged quasar (Walsh, Carswell & Weymann1979), gravitational lensing has become one of the more powerful tools available to astronomers.The main advantage of lensing as an astronomical probe lies in its simplicity – the light deflectionproperties of a lens depend on just two things: its mass distribution and the nature of gravitationalphysics.The success of general relativity (GR) – and, in the appropriate limits, Newtonian gravity – issuch that lensing is almost always used to investigate the distribution of matter in the universe.The fact that, for example, cluster lensing cannot be explained by the visible galaxies and gasis then taken to be strong evidence for the preponderance of dark matter. There is, of course, agreat deal of evidence that the universe is dark matter-dominated (e.g., Trimble 1987), but thelack of any non-gravitational detection of dark mass makes it an assumption that should continueto be tested by all available means.The alternative possibility is that GR is only valid in the small-scale, large-acceleration regimesin which it has been experimentally (as opposed to observationally) tested. The majority of direct1tests have been within the Solar system, but measurements of time delays in binary pulsar systems(Taylor et al. 1992) have also verified GR. The degeneracy between the form of the gravitationalacceleration and the distribution of mass is such that GR remains an assumption on galactic (andgreater) scales. If it is hypothesised that there is no dark matter, then it is possible to determinethe nature of gravity on these large scales from lensing or dynamical observations. Aside fromtheir tendency to rely on assumptions of equilibrium, dynamical measurements are subject to themore fundamental limitation that the gravitational field can only be probed in regions where thereis visible matter. By contrast lensing can be used to measure gravitational effects well beyondthe luminous extent of the deflector(s). Further, if such measurements can be made sufficientlyfar from the lens1, its internal structure becomes unimportant, and the lensing data uniquelyconstrains the deflection law of a point-mass. This is a potentially powerful technique that onlybreaks down at angular scales so large that the effects of other deflectors along the line-of-sightbecome important.These methods can be used to determine the deflection law of a point-mass empirically (Mort-lock & Turner 2001a) or to test a particular theory. For example Mortlock & Turner (2001b) in-vestigated gravitational lensing within the framework of modified Newtonian dynamics (MOND).A combined approach is taken here, using MOND as an illustrative example. After explaining theprinciples of the theory (Section 2), a robust deflection law is derived from galaxy-galaxy lensingdata (Section 3). The future prospects for more rigorous tests are then discussed in Section 4.2 MONDMOND (Milgrom 1983) hypothesises that the inertial mass of a particle is decreased when it issubject to an acceleration much weaker than a critical value, a0 ≃ 1.2± 0.1× 10−10 m s−2. In itspurest form a MONDian universe contains no dark matter, and, remarkably, this simple model canexplain the dynamics of galaxies and clusters (McGaugh & de Blok 1998), as well as the observedpower spectrum of cosmic microwave background anisotropies (McGaugh 2000).MOND is, of course, highly unconventional and the variation of inertia would be very difficultto integrate into the current broader understanding of the physical world. This fact alone issufficient to convince many that it is extremely unlikely to be correct. However, even if one takesthis (anti-empirical) point of view, MOND provides a potentially revealing opportunity to testthe depth and robustness of modern science’s observational knowledge of the universe. If it is notpossible to contradict such a simple but seemingly outlandish and improbable hypothesis, howmuch confidence should be placed in more conventional explanations of the observed universe?Another difficulty with MOND is that it is not a complete physical theory, lacking a relativisticextension, and thus making no definite predictions for light deflection (Milgrom 1983; Bekenstein &Milgrom 1984). The natural Ansatz for MONDian lensing is, as in GR, that a photon experiencestwice the deflection of a massive particle moving at the speed of light (Qin, Wu & Zou 1995). Thishypothesis gives qualitatively reasonable predictions (Mortlock & Turner 2001), but the effects ofextended and multiple deflectors are somewhat ambiguous. However, the gravitational propertiesof an isolated point-mass, M , are well defined: the effective force law matches the Newtonian formfor r ≪ rM = (GM/a0)1/2, but falls off as r−1 for r ≫ rM. The details of the physics for a ≃ a0are unspecified, but unimportant in the absence of high precision measurements. Integratingthis acceleration along the line-of-sight gives the (reduced) bending angle, α(θ), which matchesthe Schwarzschild form for small impact parameters [i.e., α(θ) ∝ θ−1], but is constant beyondθ = rM/dod. (Here dod is the angular diameter distance from observer to deflector, which is not1Note that, in the absence of dark matter, it is only the visible extent of the deflector that is relevant.2formally defined in MOND.) The deflection angle is not directly measurable, but the distortionof images and the (total) magnification of sources can be calculated directly from α(θ), and theseare observable.3 Observational constraintsGravitational lensing observations range from light deflection by the Sun, through microlensingin the local group, to the multiple imaging of high-redshift sources. If it is assumed that thereis no dark matter then all of these observations place constraints on the nature of gravitationallight deflection, but only some represent clean and powerful probes of MOND, whereas others areclearly within the Newtonian regime, or require the untangling of the combined effects of multipledeflectors. These issues are explored more fully by Mortlock & Turner (2001a), but it is clear thatgalaxy-galaxy lensing observations offer by far the best opportunity to make interesting inferencesfrom available data.Galaxy-galaxy lensing, the weak tangential alignment of distant galaxies caused by their morenearby counterparts, has been used to weigh the dark matter distributions of the foregrounddeflectors (e.g., Brainerd, Blandford & Smail 1996; Fischer et al. 2000). The results are allconsistent with galaxies having large isothermal haloes extending to at least several hundred kpc.Interestingly, no upper limit has been placed on the halo size, despite the fact that a systematicdistortion has been measured out to ∼ 1000 arcsec, far beyond the visible extent of the foregroundgalaxies (only a few arcsec). Thus, under the no-dark matter hypothesis, these data represent anideal means of measuring the deflection law of what is effectively an isolated point-mass. Evenwithout performing any further analysis, the fact that the lensing data are consistent with rotationcurve measurements in GR implies that the relationship between the deflection of massive andmassless particles must be the same in MOND (or any other theory).More quantitatively, Fischer et al. (2000) fit the shear signal around ∼ 3 × 104 foregroundgalaxies byγtan(θ) = γtan,60(60 arcsecθ)η, (1)with γtan,60 = 0.0027 ± 0.0005 and η = 0.9 ± 0.1. Figure 1 shows this fit to the data, alongwith the predictions of GR (assuming no dark matter) and the MONDian lensing formalismdescribed in Section 2. The MONDian results (which imply η = 1) are clearly consistent withthe data,2 although there is some ambiguity in the normalisation, as the mass-to-light ratio of thedeflectors are not known with great certainty (Mortlock & Turner 2001b). Thus the properties ofMONDian point-mass lenses must be considered completely (if approximately) defined. It is notclear, however, that the entire derivation of the deflection law given in Mortlock & Turner (2001b)is verified, and so the lensing properties of more complex deflectors remain unknown, althoughfuture investigations should shed some light on this matter as well.4 ConclusionsMOND, an alternative theory of gravity (or inertia), has been able to explain the dynamics ofmassive bodies in terms of visible matter where Newtonian physics requires large amounts ofdark matter. However MOND has no relativistic extension and so makes no predictions aboutgravitational lensing. Here the opposite approach has been taken, using observations of lensing2This agreement also provides quantitative support for the hypothesis that, in MOND (or indeed any alternativegravity theory), photons experience twice the deflection of massive particles moving at the speed of light.3Figure 1: The mean shear, γtan(θ), around foreground galaxies in the g′, r′ and i′ bands, asmeasured by Fischer et al. (2000) is compared to various theoretical predictions. In each bin thedata in the three bands (which are offset for clarity) are strongly correlated as the errors aredominated by the sample noise in the orientations of the background galaxies. The three modelsshown are: the best-fit power law (solid line); the MONDian prediction (dashed line); and theNewtonian result if there is no dark matter (dotted line).to constrain the form of MONDian light deflection. The cleanest way of doing this is to usegalaxy-galaxy lensing data to show that the deflection of photons is simply twice the deflectionthat would be experienced by a massive particle moving at the speed of light. Thus, a relativisticMONDian theory must have a great deal in common with GR.With one more part of a complete theory in place a number of new observational tests becomeavailable. The clearest relate to “simple” lenses: situations in which there is a single, isolateddeflector that has no internal structure on the scales of interest. Microlensing within the local groupfulfills these criteria, although the typical impact parameters are so small that it is only the low-magnification tails of the light-curves that are of interest. However microlensing by cosmologicallydistant deflectors offers a good opportunity to confirm or reject MOND. Several programmesto measure low optical depth microlensing of high-redshift quasars are underway (Walker 1999;Tadros, Warren & Hewett 2001), but thus far no events have been observed. The only (probable)example of cosmological microlensing by an isolated deflector observed to date is the serendipitousdetection of a peak in the light curve of gamma ray burst 000301C (Garnavich, Loeb & Stanek2000), but the photometry is not of sufficient quality to differentiate between MOND and GR.Further observations of galaxy-galaxy lensing offer the most likely tests of MONDian lensing inthe near future. Any asymmetry in the distortions (c.f. Natarajan & Refregier 2000) would be atodds with a no-dark matter theory, and there is also the possibility of measuring an outer cut-offin the shear signal. This must eventually come from the influence of secondary deflectors alongthe line-of-sight, but could also signify a putative return to a Newtonian regime in the ultra-weakacceleration limit. If MOND is not contradicted by any of the above observations, more complexsituations should be investigated. Multiple or extended deflectors must also be able to explain4observed shear fields and cases of multiple imaging without recourse to invisible mass.AcknowledgementsDJM is funded by PPARC, and this work was supported in part by NSF grant AST98-02802.ReferencesBeckenstein, J., Sanders, R.H., 1994, ApJ, 429, 480Brainerd, T.G., Blandford, R.D., Smail, I.S., 1996, ApJ, 466, 623Fischer, P., et al., 2000, AJ, 120, 1198Garnavich, P.M., Loeb, A., Stanek, K.Z., 2000, ApJ, 544, L11McGaugh, S.S., 2000, ApJ, 541, L33McGaugh, S.S., de Blok, W.J.G., 1998, ApJ, 499, 66Milgrom, M., 1983, ApJ, 270, 365Mortlock, D.J., Turner, E.L., 2001a, MNRAS, submittedMortlock, D.J., Turner, E.L., 2001b, MNRAS, submittedNatarajan, P., Refregier, A., 2000, ApJ, 538, L113Trimble, V., 1987, ARA&A, 25, 425Qin, B., Wu, X.P., Zou, Z.L., 1995, A&A, 296, 264Sanders, R.H., 1994, A&A, 284, L31Walker, M.A., 1999, MNRAS, 306, 504Tadros, H., Warren, S.J., Hewett, P.C., 2001, in Cosmological Physics with Gravitational Lensing,eds. Kneib, J.-P., Mellier, Y., Moniez, M., Tran Thanh Van, J., Edition Frontiers, in pressTaylor, J.H., Wolszczan, A., Damour, T., Weisberg, J.M., 1992, Nature, 355, 132Walsh, D., Carswell, R.F., Weymann, R.J., 1979, Nature, 279, 3815

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Gravitational lensing and modified Newtonian dynamics

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Gravitational lensing and modified Newtonian dynamics

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