# Is it possible to detect gravitational lensing behind the Moon?

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Eddington waited for an eclipse to happen to be able to observe gravitational lensing of the stars behind the Sun. And nowadays, amateurs can do the same thing.

Of course, the Moon is much lighter than the Sun, so it doesn't bend light as much. But since Eddington's time, our observation capabilities have greatly increased. Is it possible to detect the gravitational lensing of stars that are behind the Moon?

The angular deflection caused by the lensing of a distant background object by the Moon is given by $$heta simeq 4 frac{GM}{Rc^2},$$ where $$M$$ is the mass of the lensing object and $$R$$ is the closest projected distance of the ray from the centre of the mass.

For the Moon, the maximum deflection would occur when the ray just grazes the limb of the Moon and would equal 26 micro-arcseconds.

At present the most accurate instruments for measuring precision positions are VLBI radio observations of point sources, where 10 micro-arcsecond relative positional precision can be "routinely achieved" (Reid & Honma 2014). To use this technique you need bright, radio point sources to be close in position to the Moon. This is certainly possible and has been done many times using the Sun as the lensing object, although the claimed accuracies of a short time-series of measurements on single sources is an order of magnitude lower and so probably wouldn't work for the Moon (e.g. Titov et al. 2018 ).

The Gaia astrometry satellite has likely end-of-mission positional precisions of about 5 micros-arcseconds for bright stars, but it is unclear what the precision is for a single scan (which would be needed because the Moon moves!). However, to counterbalance that, one could also average the lensing effect over many stars surrounding the Moon and of course average this for different stars with the Moon observed at different times over the entire Gaia mission.

Unfortutunately, although in principle this could be done, because of the way that Gaia scans the sky, it is always pointing away from the Moon and so there will be no observations towards the Moon that would be capable of revealing these small deflections (e.g. see here).

Conclusion: Measuring the gravitational deflection of light by the Moon is just out of reach of current observational techniques.

Using our Moon as a gravitational lens might prove extremely challenging if not less than the angular resolution of positional measurements allow. It's a small effect with the Sun, and the Moon is 100 million times less massive and 100 times smaller in radius. The angular deviation from the Sun is less than one arcsecond and scales as M/R. So the deviation due to the Moon would be around $$10^{-6}$$ that of the Sun. Current technology allows angular resolution in astronomy of a few micro-arcseconds. In essence the deviation the Moon makes on light's path is about 10 - 100 times less than currently detectible with the best telescopes.

However, gravitational lensing can be used to actually detect exoplanets and exo-moons (i.e. moons around exoplanets).

The basics of that have been worked out around theoretically around 10 years ago (e.g. see this article). Meanwhile there is a few planets confirmed found by this method via the OGLE collaboration - I'm not aware of any exomoon found so far.

Using this formula from Wikipedia, $$heta = frac{4GM}{rc^2}$$ and plugging in the mass and radius of the Moon gives a deflection angle of $$1.2567 imes 10^{-10}$$ radians = $$2.592 imes 10^{-5}$$ arc-seconds. In comparison, the deflection for the Sun is just under 1.75 arc-seconds, using the Sun's equatorial radius.

I expect that it would be very difficult to observe such a tiny deflection from the Earth's surface, even using adaptive optics.

## Gravitational Lensing Caught By Amateur Telescope

Just a few short years ago, even the thought of capturing an astronomy anomaly with what’s considered an “amateur telescope” was absolutely unthinkable. Who were we to even try to do what great minds postulated and even greater equipment resolved? I’ll tell you who… Bernhard Hubl. Come on inside to meet him and see what he can do!

One of the first great minds to consider the effects of gravitational lensing was Orest Chwolson in 1924. By 1936, Einstein had upped the ante on its existence with his theories. A year later in 1937, the brilliant Fritz Zwicky set the idea in motion that galaxy clusters could act as gravitational lenses. It was not until 1979 that this effect was confirmed by observation of the so-called “Twin QSO” SBS 0957+561… and now today we can prove that it can be observed with a 12″ telescope under the right conditions and a lot of determination.

Bernhard Hubl of Nussbach, Austria is just the kind of astrophotographer to try to capture what might be deemed impossible. “Abell 2218 is a galaxy cluster about 2.1 billion light-years away in the constellation Draco. Acting as a powerful gravitational lens, it magnifies and distorts galaxies lying behind the cluster core into long arcs, as predicted by the General Theory of Relativity.”

Say’s Berhard, “I wanted to know, if I could detect signs of these arcs with a 12″ Newtonian at f=1120mm. After over 12 hours of exposure time under excellent conditions, I know that this is a hard job, but I am glad that I could identify the three brighter arcs.”

Many thanks to Bernhard Hubl for his outstanding sense of curiosity and excellent astrophotography… and to the NorthernGalactic community for the heads up!

## Galaxy Shapes in the Frontier Fields Observations

We can learn a lot about galaxies by analyzing their light, through computer modeling, and using other complex techniques. But at the most basic level, we can learn about galaxies by studying their shapes.

Galaxy appearance immediately reveals certain characteristics. Elliptical galaxies contain a wealth of old stars. Spiral galaxies are full of gas and dust. Distorted galaxies have likely experienced a gravitational interaction with another galaxy that warped their structure.

The Mice, as these distorted colliding galaxies are called, are a pair of spiral galaxies seen about 160 million years after their closest encounter. Gravity has drawn stars and gas out of the galaxies into long tails. Credit: NASA, H. Ford (JHU), G. Illingworth (UCSC/LO), M.Clampin (STScI), G. Hartig (STScI), the ACS Science Team, and ESA

The Frontier Fields project adds another dimension to this simple analysis. When we look at extremely distant galaxies with the magnification of gravitational lensing, we see new detail that was previously obscured by distance. Their shapes are clues to what occurred within those galaxies when they were very young.

Galaxies viewed through the gravitational lenses of the Frontier Fields clusters can be seen at a resolution 10 times greater than non-lensed galaxies. That means those tiny red dots that so thrill astronomers in normal Hubble images actually have some structure in Frontier Fields imagery.

Previous studies, such as the Hubble Ultra Deep Field, The Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey, or even adaptive optics-enhanced studies by ground telescopes have shown that young, star-forming galaxies at about a redshift of 2 (existing when the universe was about 3.3 billion years old) appear to have a certain lumpiness. But without gravitational lensing, we lack the resolution to say for sure whether those lumps were massive clusters of newly forming stars, or whether some other factor was causing those galaxies to have a clumpy appearance.

Frontier Fields has revealed that yes, many of those galaxies have star-forming knots that really are quite large, implying that star formation occurred in a very different way in the early universe, perhaps involving greater quantities of gas in those young galaxies than previously expected.

Frontier Fields has also given us a better grasp of the physical size of gravitationally lensed young galaxies even farther away, at a redshift of 9 (when the universe was around 500 million years old). Observations show that these galaxies are actually quite small – perhaps 200 parsecs across, while a typical galaxy you see today is closer to 10,000 parsecs across. These observations help plan future observations with the Webb Space Telescope, picking out what will hopefully be the best targets for study.

This composite image shows examples of galaxies similar to our Milky Way at various stages of construction over a time span of 11 billion years. The galaxies are arranged according to time. Those on the left reside nearby those at far right existed when the cosmos was about 2 billion years old. The Frontier Fields project is collecting galaxies from the earliest epochs of the universe to add to such comparisons. Credit: NASA, ESA, P. van Dokkum (Yale University), S. Patel (Leiden University), and the 3D-HST Team

Galaxy shape also plays a role in discoveries in the Frontier Fields’ six parallel fields, which are unaffected by gravitational lensing but provide a view into space almost as deep as Hubble’s famous Ultra Deep Field, with three times the area.

It’s well known that galaxies collide and interact, drawn to one another by gravity. Most galaxies in the universe are thought to have gone through the merger process in the early universe, but the importance of this process is an open question. The transitional period during which galaxies are interacting and merging is relatively short, making it difficult to capture. A distant galaxy may appear clumpy and distorted, but is its appearance due to a merger – or is it just a clumpy galaxy?

Collision-related features — such as tails of stars and gas drawn out into space by gravity, or shells around elliptical galaxies that occur when stars get locked into certain orbits – are excellent indicators of merging galaxies but are hard to detect in distant galaxies with ordinary observations. Frontier Fields’ parallel fields are providing astronomers with a collection of faraway galaxies with these collision-related features, allowing astronomers to learn more about how these mergers affected the galaxies we see today.

As time goes on and the cluster and parallel Frontier Fields are explored in depth by astronomers, we expect to to learn much more about how galaxy evolution and galaxy shapes intertwine. New results are on the way.

## Gravitational Lensing Provides Rare Glimpse into Interiors of Black Holes

The observable Universe is an extremely big place, measuring an estimated 91 billion light-years in diameter. As a result, astronomers are forced to rely on powerful instruments to see faraway objects. But even these are sometimes limited, and must be paired with a technique known as gravitational lensing. This involves relying on a large distribution of matter (a galaxy or star) to magnify the light coming from a distant object.

Using this technique, an international team led by researchers from the California Institute of Technology’s (Caltech) Owens Valley Radio Observatory (OVRO) were able to observe jets of hot gas spewing from a supermassive black hole in a distant galaxy (known as PKS 1413 + 135). The discovery provided the best view to date of the types of hot gas that are often detected coming from the centers of supermassive black holes (SMBH).

The research findings were described in two studies that were published in the August 15th issue of The Astrophysical Journal. Both were led by Harish Vedantham, a Caltech Millikan Postdoctoral Scholar, and were part of an international project led by Anthony Readhead — the Robinson Professor of Astronomy, Emeritus, and director of the OVRO.

This OVRO project has been active since 2008, conducting twice-weekly observations of some 1,800 active SMBHs and their respective galaxies using its 40-meter telescope. These observations have been conducted in support of NASA’s Fermi Gamma-ray Space Telescope, which has been conducting similar studies of these galaxies and their SMBHs during the same period.

As the team indicated in their two studies, these observations have provided new insight into the clumps of matter that are periodically ejected from supermassive black holes, as well as opening up new possibilities for gravitational lensing research. As Dr. Vedantham indicated in a recent Caltech press statement:

We have known about the existence of these clumps of material streaming along black hole jets, and that they move close to the speed of light, but not much is known about their internal structure or how they are launched. With lensing systems like this one, we can see the clumps closer to the central engine of the black hole and in much more detail than before.

While all large galaxies are believed to have an SMBH at the center of their galaxy, not all have jets of hot gas accompanying them. The presence of such jets are associated with what is known as an Active Galactic Nucleus (AGN), a compact region at the center of a galaxy that is especially bright in many wavelengths — including radio, microwave, infrared, optical, ultra-violet, X-ray, and gamma ray radiation.

These jets are the result of material that is being pulled towards an SMBH, some of which ends up being ejected in the form of hot gas. Material in these streams travels at close to the speed of light, and the streams are active for periods ranging from 1 to 10 million years. Whereas most of the time, the jets are relatively consistent, every few years, they spit out additional clumps of hot matter.

Back in 2010, the OVRO researchers noticed that PKS 1413 + 135’s radio emissions had brightened, faded and then brightened again over the course of a year. In 2015, they noticed the same behavior and conducted a detailed analysis. After ruling out other possible explanations, they concluded that the overall brightening was likely caused by two high-speed clumps of material being ejected from the black hole.

These clumps traveled along the jet and became magnified when they passed behind the gravitational lens they were using for their observations. This discovery was quite fortuitous, and was the result of many years of astronomical study. As Timothy Pearson, a senior research scientist at Caltech and a co-author on the paper, explained:

It has taken observations of a huge number of galaxies to find this one object with the symmetrical dips in brightness that point to the presence of a gravitational lens. We are now looking hard at all our other data to try to find similar objects that can give a magnified view of galactic nuclei.

What was also exciting about the international team’s observations was the nature of the “lens” they used. In the past, scientists have relied on massive lenses (i.e. entire galaxies) or micro lenses that consisted of single stars. However, the team led by Dr. Vedantham and Dr. Readhead relied on what they describe as a “milli-lens” of about 10,000 solar masses.

This could be the first study in history that relied on an intermediate-sized lens, which they believe is most likely a star cluster. One of the advantages of a milli-sized lens is that it is not large enough to block out the entire source of light, making it easier to spot smaller objects. With this new gravitational lensing system, it is estimated that astronomers will be able to observe clumps at scales about 100 times smaller than before. As Readhead explained:

The clumps we’re seeing are very close to the central black hole and are tiny — only a few light-days across. We think these tiny components moving at close to the speed of light are being magnified by a gravitational lens in the foreground spiral galaxy. This provides exquisite resolution of a millionth of a second of arc, which is equivalent to viewing a grain of salt on the moon from Earth.

What’s more, the researchers indicate that the lens itself is of scientific interest, for the simple reason that not much is known about objects in this mass range. This potential star cluster could therefore act as a sort of laboratory, giving researchers a chance to study gravitational milli-lensing while also providing a clear view of the nuclear jets streaming from active galactic nuclei.

## Is it possible to detect gravitational lensing behind the Moon? - Astronomy

We cannot see through the disk of the Milky Way, so how can we tell that there are not any close galaxies just on the other side of it?

You ask a good question! In fact, the most recent galaxy in the Local Group that was discovered is the Sagittarius Dwarf ( Note: scroll to the end for an update!) in the late 1990s, a tiny galaxy that is very close to the Milky Way but hidden behind it from our vantage point on Earth.

There are a few ways that astronomers deduce the presence of a small galaxy right behind the Milky Way, and they all involve looking at other wavelengths than in the optical for the reason you state: visible light doesn't get through to us from the other side of the Milky Way because it gets absorbed by the dust in the disk. However, this isn't as big a problem if you look at light at longer wavelengths which our eyes can't see but which we can detect with specially designed telescopes. In fact, the longer the wavelength of the light the less impeded you are by the dust in the Milky Way, and at radio wavelengths there is no obscuration from the Milky Way at all! So to detect galaxies on the other side of the Milky Way's disk, the first step is to use a radio telescope to detect gas whose motion is different from the Milky Way's. This doesn't work if the galaxy is too close to the Milky Way, though, because then its motion gets mixed up with the motion of the gas in our own galaxy, and it becomes difficult to tell them apart (that's why nobody saw the Sagittarius Dwarf with radio telescopes at first). Another approach is to try and detect the small galaxy's stars in the infrared (longer wavelengths than what our eyes can see but shorter than radio waves). There is some obscuration of dust at these wavelengths to worry about, but with a sensitive telescope it still may be possible to find stars in another nearby galaxy that lies behind the disk of our own.

There still may be galaxies smaller and closer than the Sagittarius Dwarf lurking just behind the Milky Way, that we haven't yet seen. Detecting them is a challenge for future, super-sensitive radio and infrared telescopes!

Update: In a press release on November 4th, 2003, astronomers announced that they have found a new galaxy lurking behind the Milky Way's disk, even closer than the Saggitarius Dwarf: Canis Major. We truly are still finding our closest extragalactic neighbors!

#### Kristine Spekkens

Kristine studies the dynamics of galaxies and what they can teach us about dark matter in the universe. She got her Ph.D from Cornell in August 2005, was a Jansky post-doctoral fellow at Rutgers University from 2005-2008, and is now a faculty member at the Royal Military College of Canada and at Queen's University.

## Gravitational lensing brings extrasolar planets into focus

A major breakthrough in the search for new worlds beyond our solar system happened recently with the discovery of a planet using a technique known as gravitational lensing. The new planet is about 1.5 times the mass of Jupiter, and is about half way between the Sun and the centre of the Milky Way. This makes it the most distant extrasolar planet detected by astronomers to date.

The discovery, made by Ian Bond and co-workers in the MOA and OGLE collaborations, represents the debut of a new and faster technique for discovering cool planets that orbit stars at large distances from the Sun (Astrophys. J. at press). Most significant of all, however, is the unique capability of gravitational lensing to discover Earth-like planets from the ground.

### Cool planet detector

Today the encyclopedia of extrasolar planets contains about 120 gas giants with masses between about 0.5 and 10 Jupiter masses. Almost all of these were found using the “Doppler wobble” method in which the gravitational pull of a large planet swings the star around the centre of mass of the star–planet system (see “Extrasolar planets”). This wobble introduces periodic Doppler shifts of a few tens of metres per second in the light from the star, and such shifts can now be routinely detected by large telescopes equipped with high-precision spectrographs. A handful of extrasolar planets have also been found using the “transit method”: if the orbit of the planet is such that it passes directly between the host star and the Earth, the planet can be detected by measuring the tiny amount of starlight it blocks as it passes in front of the host star.

However, both the Doppler and transit methods are only sensitive to large planets that orbit relatively close to their parent stars. Smaller and cooler planets that are similar to the Earth escape discovery because their Doppler signatures are small and because they take several years to orbit their host star. On the other hand, gravitational lensing – in which light from a distant star is bent by a massive intervening object – can reveal small, cool planets without having to wait for them to complete an orbit.

The bending of starlight by a massive object is one of the most important predictions of general relativity. In particular, Einstein predicted that the Sun would cause a grazing light ray from a distant star to be deflected by 1.7 arcseconds – a prediction that was famously confirmed by Eddington’s measurements during the total solar eclipse of 1919. Einstein later showed that more distant stars can act as gravitational lenses, and today astronomers routinely use this technique to “weigh” a distant object by simply looking at the effect it has on light.

However, gravitational lenses are imperfect because the rays that pass closest to the lensing mass are deflected more than rays passing further away. This spherical aberration means that an observer looking at a background star through a gravitational lens sees two magnified and distorted images on opposite sides of the lensing star. When the background star, the lens star and the Earth are all perfectly aligned, the two images expand to form an “Einstein ring”.

Stars in our galaxy typically bend light from more distant stars by only a few milliarcseconds, which means that it is not possible to resolve the two images with conventional telescopes. However, the observed brightness of the distant star displays a symmetric rise and fall over a few weeks as the lens star slides past the line of sight. It is this magnification that allows extrasolar planets to be detected (figure 1).

### Gravitational microlensing

Planets close to the lens star act like smaller gravitational lenses that can briefly increase or decrease the magnification of the lens. A cool planet in the “lensing zone” – which is typically between 1.5 and 6 times the Earth-Sun distance – can therefore be detected without having to wait for it to complete its orbit. Furthermore, both the duration and probability of planetary-lensing events scale as the square root of the mass of the planet, which means that the technique is also sensitive to low-mass planets.

Large planets like Jupiter, for example, have a 10% probability of being in the right place to act as lenses for a few days, while Earth-mass planets have a 1% probability of alignment and only act as lenses for a few hours. Unlike other methods, the magnification signal in a microlensing event can be large even though the planet is small. However, the finite angular sizes of the source stars mean that gravitational microlensing is not sensitive to planets smaller than Earth.

The MOA (Microlensing Observations in Astrophysics) and OGLE (Optical Gravitational Lensing Experiment) teams used small dedicated telescopes in Chile and New Zealand to scan rich star fields at the centre of our galaxy, and identified over 500 gravitational-lensing events during 2003. Ian Bond of Edinburgh University identified the new planet by noticing a sudden and unexpected increase in the brightness of one of the lensed stars. The signature of the planet was a pair of spikes in the brightness curve: the first occurs when the lensing effect of the planet causes two new images of the source star to appear, and the second spike – which appeared seven days later – happens when the extra images merge and disappear.

The main challenge in planet searches based on microlensing is to detect the brief and rare planet-lensing anomalies while scanning vast areas of the sky. Moreover, to be able to detect a Jupiter-sized planet the light curves must be measured at least twice per day – and at least twice per hour for Earth-sized planets. Bond’s vigilance enabled him to spot the planet anomaly in its early stages, and therefore to step-up the observational campaign before it was too late.

Now that gravitational lensing has secured its first new planet, we can expect the pace of discovery of extrasolar planets to increase as teams of astronomers join forces to build a network of dedicated observing facilities. The 1.3 m OGLE telescope at the Las Campanas Observatory in Chile and the MOA 0.6 m telescope at the Mount John observatory in New Zealand find 500-700 microlensing events each year. In addition, the PLANET and microFUN collaborations operate a series of smaller telescopes in Israel, South Africa, Chile and Australia to provide 24 hour coverage of the most promising lensing events. The RoboNet experiment in the UK has joined the quest this month, linking three 2 m robotic telescopes – the Liverpool Telescope on La Palma in the Canary Islands and the two Faulkes telescopes in Hawaii and Australia.

Beyond the excitement of discovering new worlds, the scientific goal of this work is to measure the abundance and mass distribution of cool planets with masses similar to the Earth and above. If cool Earth-like planets turn out to be relatively abundant, gravitational microlensing could uncover them within 3-5 years.

Keith Horne is in the School of Physics and Astronomy, University of St Andrews, UK

## Modeling Gravitational Lenses

Gravitational lenses are easiest to understand if we employ a simple model, similar to the one used to understand conventional lenses in optical systems. We assume that the light travels in a straight line until it reaches the lens, and then its direction is altered by some angle. After that the light travels in a straight line to the observer. The lens itself is assumed to be infinitely thin and to act on the light only as it crosses the plane of the lens. This model captures the essential behavior of the lens system: the lens causes the source to appear to be displaced from the point of view of the observer. At the same time, it avoids the complicated mathematical treatment needed to find the actual path of the light through the region of strong gravity that creates the lens in the first place.

The assumptions used for all lenses, including gravitational ones, are the following:

1. Light travels in a straight line from the source to the infinitely thin lens.
2. The light is bent at the lens plane (and only at the lens plane).
3. The light then travels in a straight line to the observer.

Students who are familiar with the geometrical approach to optics or who have read Going Further 12.1: Optical Lenses will see some similarities between these assumptions and those used in the treatment of glass lenses. The geometry of a lens system is shown in Figure 12.2. A source located at point S is emitting light that is deflected by a lens located between the source and some observer, located at point O. The lens could be anything with mass: a star or planet, or a galaxy or cluster of galaxies. If the lens did not deflect the light, then the observer would see the source at its true position. However, due to the deflection of the light caused by the lens&rsquos gravity, the image of the source is observed to be at position I.

Figure 12.2 The figure shows a side view of the basic elements of a gravitational lens system. The observer is at point O, the source is at S. These two are separated by a distance DSO . The angle &alpha is equal to the deflection angle. The deflection of the light makes the image appear to be at position I as seen by the observer. The image has thus been offset by a small amount compared to the source, i.e. the case if no lens were present. The angle &theta is the angular separation of the image from the observer. The lens is taken to be somewhere along the path of the light a distance DLO from the observer and DLS from the source. These two distances do not have to be the same. Credit: NASA/SSU /Aurore Simonnet

In Figure 12.2, the line connecting the observer to the lens is called the optical axis. We can imagine the optical axis extending behind the lens indefinitely. The source can be located on the optical axis, but in general it does not have to be. That is why we have located the point S slightly off the optical axis. Since gravity always bends a light ray toward the optical axis, the position of the image, point I, will always be located farther from the optical axis than the actual source of the light for the configuration shown. There is an additional image formed below the optical axis for which this is not true, but we will consider that image later. The distance between the source and the observer is DSO, the distance between the lens and the observer is DLO, and the distance between the lens and the source is DLS. The angle &theta in Figure 12.2 is the angular separation between the observer, the image, and the optical axis.

In Going Further 12.2: The Lens Equation, we examine the geometry of gravitational lenses in more detail, and derive a general equation relating the angles and distances for several cases.

It turns out that if the source, lens, and observer are all lined up along the optical axis, the image of the lensed object makes a ring around the lens, as shown in Figure 12.3. The ring-like image is called an Einstein ring.

Figure 12.3 This object (known as SDSS J073728.45+321618.5) is one of several Einstein rings discovered as part of the Sloan Lens ACS Survey (SLACS). The width of this image is about 8 arcsec. For more information, see the press release. Credit: NASA, ESA, A. Bolton (Harvard-Smithsonian CfA) and the SLACS Team

The angular radius, &thetaE, of an Einstein ring is given below.

This is called the Einstein radius of the lens system. We distinguish it by adding the subscript E to the angle &theta from Figure 12.2. As usual, G and c are the gravitational constant and the speed of light. The size of the Einstein ring varies only with the geometry of the lens, as expressed inside the parentheses on the right-hand side, and on the total projected mass within the impact parameter, M(b). For example, if the mass within the impact parameter is larger, then &thetaE will be larger. If the distance between the source and observer (DSO) is bigger, then &thetaE will be smaller.

The following activities will give you an idea of the size of the Einstein radius for objects with masses relevant to astrophysics.

In this activity, you will be able to adjust the parameters in the lens equation in order to determine their effect on the size of the Einstein ring.

There are three sliders, one for each of the following:

• The mass of the lens (M)
• The distance to the source from the observer (DSO)
• The relative positioning of lens, source, and observer (the ratio DLS/DLO)
###### DETERMINING THE MASS OF A LENS

In this activity, you will analyze gravitationally lensed images to determine the mass of the lens.

1. The image in Figure A.12.1 is an Einstein ring from the SLACS survey.

Figure A.12.1 This object, SDSS J0029-0055, is one of the Einstein rings discovered as part of the Sloan Lens ACS Survey (SLACS). Credit: Adapted from Bolton et al. (2008), Astrophysical Journal, 682, 964

In their paper, the SLACS team has determined that the radius of the Einstein ring is 1.11 arcsec, the distance

to the source is 9.67 × 10 25 m, and the distance to the lens is 2.84 × 10 25 m. Based on this information, what is the mass of the lens?

• Given: &thetaE = 1.11 arcsec, DSO = 9.67E25 m, DLO = 2.84E25 m
• Find: M, the mass of the lens
• Concept(s):

First, get the distance between the lens and the source: DLS = DSO - DLO = 9.67E25 m - 2.84E25 m = 6.83E25 m

Now rearrange the equation to solve for M.

Plugging in numbers, we get the solution.

We can express this in solar masses if we divide by 2E30 kg.

(M = 1.97 imes 10^<11>) solar masses

In the previous activities you saw how changing the geometry or mass of the lens changed the size of the Einstein ring. However, in all cases you might have noticed that the ring is extremely small. For typical Galactic lenses the Einstein radius is only a few milli-arcseconds, too small to be seen. For extragalactic lenses the size is larger, though still usually less than an arcsecond

GOING FURTHER 12.2: THE LENS EQUATION

## Particle Physics Beyond the Standard Model

Pierre Binétruy , in Les Houches , 2006

### 4.3 Cosmic Microwave Background [CMB]

We will recall briefly in Section 5 the history of the Universe (see Table 1 below). We will see that, soon after matter-radiation equality, electrons recombine with the protons to form neutral atoms of hydrogen, which induces the decoupling of matter and photon. From this epoch on, the universe becomes transparent. The primordial gas of photons produced at this epoch cools down as the universe expands and forms nowadays the cosmic microwave background. It is primarily homogeneous and isotropic but includes fluctuations at a level of 10 −5 , which are of much interest since they are imprints of the recombination and earlier epochs.

Table 1 . The different stages of the cosmological evolution in the standard scenario, given in terms of time t since the big bang singularity, the energy kT of the background photons and the redshift z. The double line following nucleosynthesis indicates the part of the evolution which has been tested through observation. The values (h0 = 0.7, ΩM = 0.3, ΩΛ = 0.7) are adopted to compute explicit values.

Before discussing the spectrum of CMB fluctuations, we introduce the important notion of a particle horizon in cosmology.

Because of the speed of light, a photon which is emitted at the big bang (t = 0) will have travelled a finite distance at time t. The proper distance ( 4.13 ) measured at time t is simply given by the integral:

where, in the second line, we have used ( 4.14 ). This is the maximal distance that a photon (or any particle) could have travelled at time t since the big bang. In other words, it is possible to receive signals at a time t only from comoving particles within a sphere of radius dh(t). This distance is known as the particle horizon at time t.

A quantity of relevance for our discussion of CMB fluctuations is the horizon at the time of the recombination i.e. zrec

1100 . We note that the integral on the second line of ( 4.20 ) is dominated by the lowest values of z: z

zrec where the universe is still matter dominated. Hence

We note that this is simply twice the Hubble radius at recombination H −1 (zrec), as can be checked from ( 4.8 ):

This radius is seen from an observer at present time under an angle

where the angular distance has been defined in ( 4.19 ). We can compute analytically this angular distance under the assumption that the universe is matter dominated (see Exercise 4-1). Using ( 4.34 ), we have

Thus, since, in our approximation, the total energy density ΩT is given by ΩM,

We have written in the latter equation ΩT instead of ΩM because numerical computations show that, in case where ΩΛ is non-negligible, the angle depends on ΩM + ΩΛ = ΩT.

We can now discuss the evolution of photon temperature fluctuations. For simplicity, we will assume a flat primordial spectrum of fluctuations: this leads to predictions in good agreement with experiment moreover, as we will see in the next Section, it is naturally explained in the context of inflation scenarios.

Before decoupling, the photons are tightly coupled with the baryons through Thomson scattering. In a gravitational potential well, gravity tends to pull this baryon-photon fluid down the well whereas radiation pressure tends to push it out. Thus, the fluid undergoes a series of acoustic oscillations. These oscillations can obviously only proceed if they are compatible with causality i.e. if the corresponding wavelength is smaller than the horizon scale or the Hubble radius:

Starting with a flat primordial spectrum, we see that the first oscillation peak corresponds to λ

RH (trec), followed by other compression peaks at RH (trec)/n (see Figure 8 ). They correspond to an angular scale on the sky:

Fig. 8 . Evolution of the photon temperature fluctuations before the recombination. This diagram illustrates that oscillations start once the corresponding Fourier mode enters the Hubble radius (these oscillations are fluctuations in temperature, along a vertical axis orthogonal to the two axes that are drawn on the figure).

Since photons decouple at trec, we observe the same spectrum presently (up to a redshift in the photon temperature) 18 .

Experiments usually measure the temperature difference of photons received by two antennas separated by an angle θ, averaged over a large fraction of the sky. Defining the correlation function

averaged over all n1 and n2 satisfying the condition n1 · n2 = cos θ, we have indeed

We may decompose C(θ) over Legendre polynomials:

The monopole (l = 0) related to the overall temperature T0, and the dipole (l = 1) due to the Solar system peculiar velocity, bring no information on the primordial fluctuations. A given coefficient Cl characterizes the contribution of the multipole component l to the correlation function. If θ ≪ 1, the main contribution to Cl corresponds to an angular scale 19 θ

200 ◦ /l. The previous discussion (see ( 4.25 ) and ( 4.27 )) implies that we expect the first acoustic peak at a value l ∼ 200 Ω T − 1/2 .

The power spectrum obtained by the WMAP experiment is shown in Figure 9 . One finds the first acoustic peak at l

200, which constrains the ΛCDM model used to perform the fit to ΩT = ΩM + ΩΛ

1. Many other constraints may be inferred from a detailed study of the power spectrum [ 32 ].

Fig. 9 . This figure compares the best fit power law Λ CDM model to the temperature angular power spectrum observed by WMAP. The gray dots are the unbinned data [ 32 ].

## Gravitational Lensing

Gravitational lensing is the bending of light by the gravitational field of matter. It’s predicted by Einstein’s general theory of relativity, and in relativity space and time are combined into an entity called space-time which can be curved. Part of Einstein’s theory tells us how space-time is curved and it tells us that matter curves space-time. To get an idea, if you imagine a rubber sheet and you place a large heavy ball on it, then the rubber sheet will distort. The other half of general relativity is that space-time tells matter how to move, so if you imagine firing a small ball bearing onto this rubber sheet, then its orbit gets changed, it doesn’t go in a straight line.

The second effect is that sources which are gravitationally lensed will get brighter typically. We see that when stars go behind other stars, then the background star gets brighter and then fades back to its normal brightness. We can use that to do various things. One is that if the star in the foreground has a planet, then there’s an additional piece of lensing by the planet and you can detect planets this way. The other thing that we can do with that is to try and work out what the dark matter in the Milky Way is. We conclude that it’s not predominantly in the form of dark objects such as black holes or brown dwarfs because there isn’t enough of this so-called microlensing that takes place in the galaxy.

The third effect that lensing has is to change the shape of the images. We see that in quite extreme circumstances when, for example, if you have a very rich cluster of galaxies that may have a thousand galaxies in an agglomeration, the mass is so grave that the light gets distorted enormously and it can get distorted into very long thin arcs. Sometimes you can even see the same galaxy twice, that light can get bent through two or more directions and still arrive at the telescope. That can tell us a lot about the intervening lensing object.

One of the uses that’s been put to is to determine something about the nature of dark matter.

There’s a cluster called the Bullet Cluster which consists of two clusters which collided with each other about 150 million years ago. The stars in the galaxies just passed through each other. The chances of an interaction are very very small, so we see these two clusters, the separated galaxies having gone through each other. But clusters of galaxies also contain a lot of gas and when the gas interacts, then it collides, gets shock heated and stays in the middle. So we see hot gas in the middle and we see the galaxies on either side and we can use gravitational lensing to work out where the dominant mass contribution is. It turns out to be associated with the galaxies. So the dark matter has passed through in a collisionless way and has ended up where the galaxies are, it hasn’t stayed in the centre. There’s a lot of it: most of the mass is in that dark form. That’s a challenge for other theories of gravity which try to do away with dark matter because if there was no dark matter, we would expect the lensing effect to be associated with the gas because it has much more mass in it than the galaxies. But we don’t see the mass there, we see it associated with the galaxies. So those are very strong lensing effects.

What we also see is very weak lensing effects that occur all over the sky. In fact, if you look at a distant galaxy, then the light that you see has been distorted a little bit, the shapes have been changed by one or two per cent typically and that’s happening all over the sky. It’s a bit of a nuisance but on the other hand, it’s a source of signal because the distortion pattern tells us a lot about the Universe. It’s quite challenging because we don’t know what shapes the galaxies should be before being lensed but we can look for a statistical signal that is all the way across the sky. The light from these distant objects has been travelling for many billions of years during which time the Universe has evolved, so the Universe has got more clumpy as gravity has pulled objects in, so it tells us about how quickly the growth of structure has taken place. That depends on the theory of gravity and other things such as the amount of dark energy in the Universe which is the dominant contributor to the Universe’s energy budget. So it’s a very good way of telling us about the constituents of the Universe and also about the gravity law.

This is a relatively young subject. The first time this was discovered was in about the year 2000 but it’s now a standard tool for probing cosmology. We can also test the gravity law because light and slow-speed particles respond to gravity in different ways and in Einstein’s theory there’s a very simple relationship between those two. So if we compare the distribution of mass from lensing and we look at the distribution of matter from slow-speed objects like galaxies, we can test this theory.

So I look at the statistical analysis of lensing and we devise ways of trying to do this in a principled statistical way. It’s a complicated statistical problem because the signal is very small, we need large volumes and large numbers of objects to do the scientific inference. But from the physics point of view it’s a very beautiful signal which depends essentially only on gravity. In detail it’s much more complicated but the basic physics is really really nice and simple. So I think we tend to fall into two camps: half of us would like to find something new and something wrong with Einstein’s gravity and the other half would like to say, well, it’s really fun to be working at a time when you really establish that the theory is right and our understanding of the Universe is right. I think I probably lean slightly towards the latter.

## Gravitational Lensing Measures the Universe

Data from the Keck telescope (Mauna Kea), the Hubble Space Telescope and the Very Large Array have been used in conjunction with the findings of the Wilkinson Microwave Anisotropy Probe to offer up a new way to measure the size of the universe, as well as how rapidly it is expanding and how old it is now. By determining a value for the Hubble constant, the work confirms the age of the universe within a span of 170 million years as 13.75 billion years old.

I’m always fascinated with work involving gravitational lensing — just yesterday we looked at using the Sun’s lensing effects for potential SETI investigations — and here we have a classic case of measuring how light traveled from a bright, active galaxy along different paths to reach the Earth. A strong gravitational lens like the one used in this study, called B1608+656, creates multiple images of the same galaxy lying behind the lensing object. Studying the time the light took along each path, it was possible to gather information about the distance of the galaxy as well as the age of the universe and details about its expansion.

Image: When a large nearby object, such as a galaxy, blocks a distant object, such as another galaxy, the light can detour around the blockage. But instead of taking a single path, light can bend around the object in one of two, or four different routes, thus doubling or quadrupling the amount of information scientists receive. As the brightness of the background galaxy nucleus fluctuates, physicists can measure the ebb and flow of light from the four distinct paths, such as in the B1608+656 system imaged above. Credit: Sherry Suyu/Argelander Institut für Astronomie, Bonn.

The lensing effect produced four images of the background galaxy. What’s fascinating about lensing is that the time it takes a light ray to travel a short path can be longer than the time it takes to travel a longer path due to the gravitational time delay caused by the lensing object. This short video with physicist Sherry Suyu (University of Bonn) discusses the effect by analogy with travel times on Earth, and explains how the scientists were able to use the multiple images of the background galaxy to compute the tightened value for the Hubble constant — 21 kilometers per second per million light years. In other words, a galaxy that is a million light years away is moving away from us at about 21 kilometers per second.

An international team is behind this work, which is just out in the Astrophysical Journal. Having made a physical measurement of Hubble’s constant, says Phil Marshall (Stanford University), gravitational lensing “has come of age as a competitive tool in the astrophysicist’s toolkit.” The new value for Hubble’s constant is considered the best estimate of the uncertainty in the constant. Beyond that, however, is the fact that lensing is producing an estimate for the universe’s age that gibes well with other methods of analysis, meaning that we’re learning how to harness this remarkable natural tool for future investigations.

The paper is Suyu et al., “Dissecting the Gravitational Lens B1608+656. II. Precision Measurements of the Hubble Constant, Spatial Curvature, and the Dark Energy Equation of State. Astrophysical Journal 711 (1 March 2010), pp. 201-221 (abstract).

Comments on this entry are closed.

The Planetary Society has a podcast on a study that uses gravitational lensing (stars occulting other stars) to detect planets. If I understand right, in this technique the planet(s) are orbiting the foreground star, which causes a measurable distortion in the lensed image:

‘Size of the universe’, I presume, means the size of the *observable* universe, not the entire universe (also including everything beyond our event horizon).

Could such a method, or another, also give us an idea about the size of the *entire* universe, for instance by estimating the curvature? Or is this inherently impossible?

The only size estimates for the entire universe, that I know of, are purely theoretical and lower bounds (such as 10^23 to 10^26 times the observable universe, based on Alan Guth’s inflation theory).

Perhaps the most significant aspect of this finding is that the value this group measured for the Hubble constant agrees so well with other groups who have used methods of measurement based on completely different physics concepts (Cepheid variables, supernovae, etc.). This essentially means that we are zeroing in on the actual value of the Hubble constant after decades of hard work and lingering uncertainty surrounding this fundamental parameter. I agree with those who maintain that we have learned much about the large scale structure and evolution of the Universe, but we still have many mysteries left to solve. Great work my friends!

Concerning the ‘total’ or ‘entire’ universe, and the multiverse, this is just in at Next Big Future (my other favorite website):
http://nextbigfuture.com/2010/03/research-that-considers-multiverse.html

I think, however, that there is some confusion concerning the different concepts: observable universe (unto *our* event horizon), total/entire universe (everything, also beyond our event horizon, stemming from the same original Big Bang, and hence possessing the same physical laws), and the multiverse (different universes originating from as many BB events).

But the confusion may be entirely mine. For a simple provincial boy like me, being mainly interested in the Local Group or maybe the Local Supercluster, all this is quite mind-boggling -)

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