Cosmology. The Origin and Evolution of Cosmic Structure - Coles P., Lucchin F

19

Gravitational

Lensing

In this chapter we shall discuss the cosmological applications of one of the predictions of general relativity. Although it is only recently that the idea of gravitational lensing has found applications in cosmology, the idea that massive bodies could deflect light rays actually furnished the first experimental test of Einstein’s theory in 1919. The story of this test has some interesting lessons for modern cosmology so, before going onto the technical applications of gravitational lensing, we begin with a small amount of history.

19.1 Historical Prelude

The idea that gravity might bend light did not originate with Einstein. It had been suggested before, by Isaac Newton for example. In a rhetorical question posed in his Opticks, Newton wrote:

Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is not this action. . . strongest at the least distance?

In other words, he was arguing that light rays themselves should feel the force of gravity according to the inverse-square law. As far as we know, however, he never attempted to apply this idea to anything that might be observed. Newton’s query was addressed in 1801 by Johann Georg von Soldner. His work was motivated by the desire to know whether the bending of light rays might require certain astronomical observations to be adjusted. He tackled the problem using Newton’s corpuscular theory of light, in which light rays consist of a stream of tiny particles. It is clear that if light does behave in this way, then the mass of each particle

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must be very small. Soldner was able to use Newton’s theory of gravity to solve an example of a ballistic scattering problem.

A small particle moving past a large gravitating object feels a force from the object that is directed towards the centre of the large object. If the particle is moving fast, so that the encounter does not last very long, and the mass of the particle is much less than the mass of the scattering body, what happens is that the particle merely receives a sideways kick which slightly alters the direction of its motion. The size of the kick, and the consequent scattering angle, is quite easy to calculate because the situation allows one to ignore the motion of the scatterer. Although the two bodies exert equal and opposite forces on each other, according to Newton’s third law, the fact that the scatterer has a much larger mass than the ‘scatteree’ means that the former’s acceleration is very much lower. This kind of scattering e ect is exploited by interplanetary probes, which can change course without firing booster rockets by using the gravitational ‘slingshot’ supplied by the Sun or larger planets. When the deflection is small, the angle of deflection

where r is the distance of closest approach between scattering object and scattered body.

Unfortunately, this calculation has a number of problems associated with it. Chief amongst them is the small matter that light does not actually possess mass at all. Although Newton had hit the target with the idea that light consists of a stream of particles, these photons, as they are now called, are known to be massless. Newton’s theory simply cannot be applied to massless particles: they feel no gravitational force (because the force depends on their mass) and they have no inertia. What photons do in a Newtonian world is really anyone’s guess. Nevertheless, the Soldner result is usually called the Newtonian prediction, for want of a better name.

Unaware of Soldner’s calculation, in 1907 Einstein began to think about the possible bending of light. By this stage, he had already formulated the equivalence principle, but it was to be another eight years before the general theory of relativity was completed. He realised that the equivalence principle in itself required light to be bent by gravitating bodies. But he assumed that the e ect was too small ever to be observed in practice, so he shelved the calculation. In 1911, still before the general theory was ready, he returned to the problem. What he did in this calculation was essentially to repeat the argument based on Newtonian theory, but incorporating the equation E = mc2. Although photons do not have mass, they certainly have energy, and Einstein’s theory says that even pure energy has to behave in some ways like mass. Using this argument, and spurred on by the realisation that the light deflection he was thinking about might after all be measurable, he calculated the bending of light from background stars by the Sun.

For light just grazing the Sun’s surface—i.e. with r equal to the radius of the Sun, R , and where M is the mass of the Sun M —Equation (19.1.1) yields a deflection of 0.87 seconds of arc; for reference, the angle in the sky occupied by the Sun

is around half a degree. This answer is precisely the same as the Newtonian value obtained more than a century earlier by Soldner. The predicted deflection is tiny, but according to the astronomers Einstein consulted, it could just about be measured. Stars appearing close to the Sun would appear to be in slightly di erent positions in the sky than they would be when the Sun was in another part of the sky. It was hoped that this kind of observation could be used to test Einstein’s theory. The only problem was that the Sun would have to be edited out of the picture, otherwise stars would not be visible close to it at all. In order to get around this problem, the measurement would have to be made at a very special time and place: during a total eclipse of the Sun.

In 1915, with the full general theory of relativity in hand, Einstein returned to the light-bending problem. And he soon realised that in 1911 he had made a mistake. The correct answer was not the same as the Newtonian result, but twice as large. Einstein had neglected to include all e ects of curved space in the earlier calculation. The origin of the factor two is quite straightforward when one looks at how a Newtonian gravitational potential distorts the metric of space–time. In flat space (which holds for special relativity), the infinitesimal four-dimensional space–time interval ds is related to time intervals dt and distance intervals dl via

light rays follow paths in space–time defined by ds2 = 0, which are straight lines in this case. Of course, the point about the general theory is that light rays are no longer straight. In fact, around a spherical distribution of mass M the metric changes so that, in the weak field limit, it becomes

Since the corrections of order GM/rc2 are small, one can solve the equation ds2 = 0 by expanding each bracket in a power series.

Einstein’s original calculation had included only the first term, which corresponds to the R00 part of the field equations. The second doubles the net deflection. Not only does energy gravitate, so does momentum and this appears in the second term in the metric. The angular deflection predicted by Einstein’s equations in the Newtonian limit is therefore

which yields 1.74 arcsec for M = M and r = R , compared with the 0.87 arcsec obtained using Newtonian theory. Not only is this easier to measure, being larger, but it also o ers the possibility of a definitive test of the theory, since it di ers from the Newtonian value.

In 1912, an Argentinian expedition had been sent to Brazil to observe a total eclipse. Light-bending measurements were on the agenda, but bad weather prevented them making any observations. In 1914, a German expedition, organised

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by Erwin Freundlich and funded by Krupp, the arms manufacturer, was sent to the Crimea to observe the eclipse due on 21 August. But when World War I broke out, the party was warned o . Most returned home, but others were detained in Russia. No results were obtained. The war made further European expeditions impossible. One wonders how Einstein would have been treated by history if either of the 1912 or 1914 expeditions had been successful. Until 1915, his reputation was riding on the incorrect value of 0.87 arcsec. As it turned out, the 1919 British expeditions to Sobral and Principe were to prove his later calculation to be right. And the rest, as they say, is history (Dyson et al. 1920).

19.2 Basic Gravitational Optics

In general it is a di cult problem to determine the trajectories of light rays in curved space–times. However, in the cosmological setting, we can simplify the task by applying some assumptions. For a start we assume that the global background geometry is well described by the Robertson–Walker metric we introduced in Chapter 1. Next we make use of a Newtonian approximation for the light trajectories, similar to the discussion of the previous section. We assume that a light ray travels unperturbed from a background source until it is very close to the lens, whereupon it is deflected by some angle we shall assume to be small. It then follows an unperturbed trajectory from the lens to the observer. In doing this we are obliged to require that the e ective gravitational potential of the lens Φ is such that |Φ2| c2 and that the lens is moving with respect to a cosmological frame with a velocity v which is much less than that of light. If these conditions apply, then the deflection produced by the lens is going to be small.

The deflection of a light ray, αˆ, will in general be given by

2

αˆ = c2 Φ dl, (19.2.1)

where the gradient of the Newtonian potential is taken perpendicular to the light path and the integral is taken along photon trajectory. With the simplification mentioned above, the gradient can be taken to be perpendicular to the original (unperturbed) light ray rather than the actual (perturbed) one. In this case we only need to consider the impact parameter b of the light ray as it crosses the lens plane. The relevant potential for a point lens can be written

where z is the distance along the ray. For the case (19.2.2) we therefore find

L

η

β θ

O

Ds

Figure 19.1 Gravitational lensing. A light ray travels from the source S to the observer O passing the lens at an impact parameter ξ. The transverse distance from the optic axis is η. The light ray is deflected through an angle αˆ; the angular separations of source and image from the optic axis are denoted β and θ, respectively. The angular-diameter distances between observer and source, observer and lens and lens and source are Ds, Dd and Dds, respectively. Picture courtesy of Mathias Bartelmann.

in a direction at right angles to the unperturbed ray. The deflection angle is then

This is exactly the result we described in Section 19.1.

If we now assume that the deflection occurs as a kind of ‘impulse’ delivered by the lens within a distance ±b along the original light ray, then we can simplify matters even further. This approximation corresponds to the assumption that the lens is infinitely thin compared with the distances from source to lens and from observer to lens. One then considers the lens to be a mass sheet lying in a plane usually called the lens plane. The relevant property of the sheet is its surface mass density, Σ, where

Σ(ξ) = ρ(ξ, z) dz, (19.2.5)

in which the integral is taken over the photon path as before. It is then straightforward to show that the net deflection (now written as a vector to show its direction in the lens plane) is given by

Now let us generalise to the case of a circular lens with an arbitrary mass profile. The lens Equation (19.2.10) then becomes

If the mass density is su cient, then a source with β = 0, i.e. one that lies on the optic axis, is lensed into a ring with radius θE, where

The two solutions correspond to two images, one lying on either side of the source. One image is always inside the Einstein ring and the other outside it. If the source is moved further from the optic axis (i.e. if β increases), then one image gets closer to the lens and the other gets nearer the source. Gravitational lensing changes the apparent solid angle of the source and therefore results in a magnification by a factor equal to the ratio of the image area to the source area. For a circular lens the magnification factor µ is easily seen to be

19.3 More Complicated Systems

The preceding section dealt with simple lens systems. In the following we shall look at some examples of how to deal with the more general case without any special symmetry. To simplify the notation let us start by defining a scaled potential

ψ(θ) by

We can also use the elements of ψij to construct components of a shear tensor. First define

This notation is useful because it allows a simple visual interpretation of the e ects of lensing. A pure convergence κ corresponds to an isotropic magnification of the source in such a way that a circular source becomes a larger but still circular image. The components γ1 and γ2 represent shear in such a way that

γ = γ12 + γ22 (19.3.15)

represents the magnitude of the shear and φ its orientation. A non-zero shear transforms a circular source into an elliptical image.

In some places the mapping between source and image plane becomes singular. These singularities are normally called caustics and they lead to interesting optical e ects owing to the non-uniqueness of the mapping between image and source planes to which they correspond. Basically a given (extended) lens will generate a set of caustics in the source plane. When a source crosses such a caustic a new pair of images is produced in the image. An extended lens can produce many images, depending on the mass distribution in the lens plane, while a point-mass lens only produces two. Near the caustics the shape of the images can be complicated, producing near-circular giant arcs. These can be very bright, owing to the magnification e ect which is formally infinite at a caustic.

The consequences of these can be spectacular but complicated and, generally, considerable modelling is needed to understand the complex images obtained.