Gen Relativ Gravit (2010) 42:2301–2322DOI 10.1007/s10714-010-1021-5

REVIEW ARTICLE

Lensing by exotic objects

Alexander Zakharov

Received: 11 February 2010 / Accepted: 17 May 2010 / Published online: 11 June 2010© Springer Science+Business Media, LLC 2010

Abstract Many exotic astronomical objects have been introduced. Usually theobjects have masses, therefore they may act as gravitational lenses. We briefly discussgravitational lensing with cosmic strings. As is well-known, dark matter is one ofthe most important components of the Universe. Recent computer simulations indi-cate that dark matter may form clumps. We review gravitational lensing (includingmicrolensing) for the clumps.

Keywords Gravitational lensing · Microlensing techniques (astronomy) ·Gravitational lenses and luminous arcs · Dark matter

1 Exotic objects

At the beginning of this article it is necessary to clarify our understanding of exoticobjects. This is important because we want to distinguish between objects that are“not exotic”, “exotic” and “too exotic”. We think that objects are not exotic if there aretheoretical plus solid observational arguments for their existence. Objects are exotic ifwe have indirect evidence and solid theoretical arguments for their possible existence.

A. ZakharovNational Astronomical Observatories of Chinese Academy of Sciences,Beijing 100012, China

A. Zakharov (B)Institute of Theoretical and Experimental Physics,B. Cheremushkinskaya, 25, 117259 Moscow, Russiae-mail: zakharov@itep.ru

A. ZakharovBogoliubov Laboratory for Theoretical Physics,JINR, 141980 Dubna, Russia

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Too exotic objects can exist only if we introduce marginal assumptions. Accordingto our understanding dark matter (including the formation of dark matter clumps) isexotic, but not too exotic. Objects such as naked singularities and wormholes may betreated as too exotic because for their existence one has to introduce additional theo-retical assumptions, besides generally accepted ones. We do not consider objects suchas monopoles [1–3], boson stars [4–13], fermion stars [14–30] etc as gravitationallenses, in spite of the fact that these objects are very interesting from a theoreticalpoint of view and their existence does not contradict basic principles; at the moment,however, there are no observational signatures for their existence.

It is our personal point of view that black holes are not exotic at all. Scalar fields asgravitational lenses in astrophysics are too exotic, whereas cosmic strings and clouds(clumps) of dark matter are exotic, but not too exotic. In this review we will concentrateon gravitational lensing (including microlensing) of dark matter clumps.

Gravitational lensing in the strong gravitational field approximation was analyzedin [31–39]. These models may be used for studies of gravitational lens phenomenanear black holes in cases where the impact parameter for a background source is rathersmall (comparable with the Schwarzschild radius). Note that the probability for almostperfect alignment of source, lens and observer is small, or a source position should beconnected with the black hole. The formation of shadows is discussed in a number ofpapers [40–53].

2 Lensing with cosmic strings

Cosmic strings are very interesting objects and they are on the borderline betweenexotic and too exotic objects. They were introduced in modern cosmology byKibble [54,55] (see also [56] for a more recent review), and after that they have beena subject of intensive studies, beginning with papers by Zeldovich [57] and Vilenkin[58]. Moreover, Vilenkin suggested to search for cosmic strings in present studies ofgravitationally lensed systems [58] (see also [59,60]). In particular, people discussedthe possibility of explaining the pair of 19th magnitude quasars 1146+111B,C (dis-covered earlier [61]) by a gravitational lens with a cosmic string [62–66]. One of theimportant lensing features of cosmic strings is the occurrence of background pairs(galaxies or quasars) with the same spectra and redshifts, as it was reported in [67].However, further analysis showed that a gravitational lens model with a cosmic stringhardly ever may be accepted [68,69].

Let us recall some basics facts of gravitational lensing for cosmic strings. Straightstrings are defined by one dimensionless parameter Gμ � 1, where μ is the linearmass density for strings [58,59]. Therefore, for a cosmic string along the z directionVilenkin [58,59] derived the metric

ds2 = dt2 − dz2 − dr2 − (1 − 8Gμ)r2dφ2. (1)

If we introduce a modified azimuthal angle [58,59]

φ′ = (1 − 4Gμ)φ, (2)

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Lensing by exotic objects 2303

such that φ′ ∈ [0, 2π(1 − 4Gμ)], we see that the metric (1) is locally flat, because itis locally the Minkowskian metric in modified cylindrical coordinates (r, z, φ′), but itis not globally Minkowskian, because the range for φ′ is determined by the relation(2). Light in the (x − y)-plane is deflected by the angle

δ = 4Gμ, (3)

as it was shown by Vilenkin [58,59]. Therefore, the separation angle between twoimages is θ < 2δ = 8Gμ [58,59].

A very interesting pair of galaxies was found, namely the so-called CSL-1 (Capodi-monte–Sternberg Lens candidate) object. These are two distant galaxies with the sameredshifts (z = 0.46) and with very similar flux ratios in different spectral bands [70].The observations were performed with the WideField Imager at the ESO-MPI 2.2-mtelescope, and additional observations in the H-band filter were taken at the TelescopioNazionale Galileo (TNG) with the Near Infrared Camera and Spectrograph (NICS).Initially, the object was considered to be a double image lensed by a cosmic string.In 2005 new spectra of CSL-1 were taken with the Very Large Telescope operatedby the European Southern Observatory and these observations were consistent withgravitational lensing by a cosmic string [71]. However, further observations with theHubble space telescope showed that CSL-1 is a pair of very peculiar interacting ellipti-cal galaxies [72,73]. The knowledge about the existence of galaxies that are so similaris very important to improve algorithms to search for gravitationally lensed systems(in particular by cosmic strings) with present and future surveys.

Now people analyze high resolution surveys to find features of cosmic strings inthem [74] based on theoretical predictions of event rates [75].

3 Regimes of gravitational lensing

There is a number of reviews and books on gravitational lensing [76–82].Let us recall a couple of basic facts about the simplest point-like lens model (the so-

called Schwarzschild lens model). Assume that Ds is the distance between source andobserver, Dd is the distance between lens and observer, Dds is the distance betweenlens and source. The plane passing through the gravitational lens, perpendicular to thevector from the observer to the lens, is called the gravitational lens plane. Similarly aparallel plane passing through the source is called the source plane. We then have thegravitational lens equation [76]

η = Dsξ/Dd − Dds�(ξ), (4)

where the vectors η and ξ determine positions in the source plane and in the lens plane,respectively. The angle � is determined by the following relation for the point-likelens model (the Schwarzschild lens case),

�(ξ) = 4G Mξ/c2ξ2. (5)

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2304 A. Zakharov

Table 1 Different regimesof gravitational lensing [77]

Prefix/name Deflection Mass Lens Timeangle (arcsecond) m/M� delay

Kilo-lensing 103 1018 Supercluster

Macro-lensing 100 1012 Galaxy Months

Milli-lensing 10−3 106 MBH Min.

Micro-lensing 10−6 100 Star 10−4 s

Nano-lensing 10−9 10−6 Planet 10−10 s

Pico-lensing 10−12 10−12 ??? 10−16 s

Femto-lensing 10−15 10−18 Comet 10−20 s

The right-hand side of (4) vanishes if source, lens and observer are on a straight line(η = 0). The corresponding length in the lens plane ξ0 = √

4G M Dd Dds/(c2 Ds) iscalled the Einstein–Chwolson radius. One can introduce the Einstein–Chwolson anglewhich is θ0 = ξ0/Dd . In the framework of the simple lens model, angular distancesbetween images are roughly about 2θ0 and the angle is proportional to the square rootof the lens mass if all other parameters (including distances) are kept fixed. Differentregimes of gravitational lensing are shown in Table 1. If a gravitational lens has atypical galactic mass of about 1012 M�, a typical angular distance between imagesis about a few arcseconds (corresponding to the standard gravitational macrolensingregime); if a gravitational lens has a typical stellar mass of about M�, a typical angulardistance between images is about 10−6 arcseconds (corresponding to the gravitationalmicrolensing regime); if a gravitational lens has a typical Earth-like planet mass ofabout 10−6 M�, a typical angular distance between images is about 10−9 arcseconds(corresponding to the gravitational nanolensing regime). More generally speaking,searches for planets through their impacts on gravitational lensing may be named“gravitational nanolensing”. Really, 10−9 arcseconds is a very small angle; to imagineit one can think of looking at a one inch coin from a distance of about 4.5 × 109 km(or about 30 AU), which is roughly equal to the distance between Sun and Neptune.

At the moment there is no way to resolve micro- or nano-images but there is a wayto discover photometrical features of these phenomena by monitoring light curvesof background sources [83,84]. Moreover, there are projects planning to reach angu-lar resolutions at a microarcsecond level (in different spectral bands), such as theNASA Space Interferometry Mission (SIM), the ESA Global Astrometric Interferom-eter for Astrophysics (Gaia) [85], the NASA MicroArcsecond X-Ray Imaging Mission(MAXIM) [86,87], and the Russian RadioAstron. It is planned to reach even the nano-arcsecond level in the mm band with space–ground interferometry techniques withthe future Millimetron mission.1

4 Gravitational microlensing

There is a number of reviews on gravitational microlensing [80,84,88–97]. We willgive numerical estimates for parameters of the microlensing effect. If the distance

1 See, http://www.asc.rssi.ru.

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Lensing by exotic objects 2305

between a dark body and the Sun is equal to ∼ 10 kpc, then the angular size of theEinstein cone of the dark body with a solar mass is equal to ∼ 0.′′001 and the linearsize of the Einstein cone is equal to about 10 AU. It is clear that, since angular distancesbetween two images are very small, it is very difficult to resolve images by groundbased telescopes, at least in an optical band. This was a reason why Einstein notedthat, if gravitational lenses and sources are stars, the separation angle between imagesis very small and gravitational lens phenomena could hardly ever be detectable [98].2

However, recently a direct method to measure the Einstein–Chwolson angle θ0 wasproposed to resolve double images generated by microlensing with an optical interfer-ometer (such as the Very Large Telescope Interferometer (VLTI)) [99]. Moreover, asit was mentioned earlier it was planned to launch astrometrical space missions, suchas US SIM3 and European Gaia.4 These instruments will have accuracies of about10 microarcseconds and could resolve image splitting in the case of microlensingevents. Applications of future space missions for astrometrical microlensing searchesare discussed in [101,102].

Microlensing for distant quasars was considered in [103] (soon after the first gravita-tional lens discovery [104]) and discovered by [105] in gravitationally lensed systemssince optical depths5 for such systems are highest. Later on, features of microlensing indifferent bands were found in gravitationally lensed systems [106,107], in particular,microlensing event signatures were found with the 1.5-m RTT-150 telescope for thegravitationally lensed system SBS 1520+530 [108]. The optical depth of microlensingfor distant quasars was discussed for different locations of microlenses [109–112]. Aninfluence of microlensing on spectral lines and spectra in different bands was ana-lyzed in [113,114]. These investigations were inspired by discoveries of microlensingfeatures in the X-ray band for gravitationally lensed systems [115–118]. These resultswere obtained owing to an excellent angular resolution in the X-ray band of the Chan-dra satellite, enabling us to resolve different images of gravitationally lensed systemsand to study their luminosities separately.

Basic criteria for microlensing event identification are that a light curve shouldbe symmetrical and achromatic. If we consider a spherically symmetric gravitationalfield of a lens, a point source and a short duration of a microlensing event, then thestatement about the symmetrical and achromatic light curves will be a correct claim,but if we consider a more complicated distribution of a gravitational lens field or anextended source, then some deviations from a symmetric light curve may be observedand (or) the microlensing effect may be chromatic [79].

Many years ago it was found that the density of visible matter is about a few percent-age of the total matter density in galactic halos [119,120] and the invisible componentis called dark matter (DM). It is now known that the matter density (in critical density

2 However, the microlensing effect is observed analyzing luminosity variations of a background source asit was originally proposed by [83] using gravitational focusing of the light.3 http://sim.jpl.nasa.gov/whatis/.4 http://astro.estec.esa.nl/gaia, see also [85,100].5 This quantity characterizes the probability for an analyzing phenomenon; for instance, for the simplestpoint-like lens model the fraction of the solid angle covered with their Einstein rings is called the opticaldepth for gravitational microlensing [89].

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2306 A. Zakharov

units) is �m = 0.3 (including baryonic matter �b ≈ 0.05−0.04, but luminous matter�lum ≤ 0.005), and the -term density is � = 0.7 [121–123]. Thus, the baryonicmatter density is a small fraction of the total density in the Universe. Probably galactichalos are “natural” places to store not only baryonic DM, but non-baryonic DM aswell. If DM forms objects with masses in the range 10−5 −10 M�, microlensing couldhelp to detect such objects.

As it was mentioned before, the possibility to discover microlensing by monitoringbackground stars was proposed for the first time by Byalko [83]. (However, to increasethe probability, in the original paper it was proposed to detect very faint flashes forthe background star light curves and in this form the idea is hardly ever realizable.)Systematic searches for dark matter using typical variations of light curves of indi-vidual stars from millions observable stars started after Paczynski’s discussion of thehalo dark matter discovery using monitoring stars from the Large Magellanic Cloud(LMC) [84]. At the beginning of the 1990s new computer and technical facilities pro-viding the storage and processing capabilities for the huge volume of observationaldata enabled the rapid realization of Paczynski’s proposal (the situation was differ-ent in times of Byalko’s paper). Griest suggested to call these microlenses Machos(Massive Astrophysical Compact Halo Objects) [124]. Besides, MACHO is the nameof the US–English–Australian collaboration project which observed the LMC andthe Galactic bulge using the 1.3-m telescope of the Mount Stromlo observatory inAustralia.6 Since one can monitor several million stars for several years by microlenssearches, the ongoing searches have focused on two targets: a) stars in the Large andSmall Magellanic Clouds (LMC and SMC) which are the nearest galaxies having linesof sight which go out of the Galactic plane and well across the halo; b) stars in theGalactic bulge which allow us to test the distribution of lenses near the Galactic plane.The first papers about the microlensing discovery were published by the MACHOcollaboration [125] and the French collaboration EROS (Expérience de Recherched’Objets Sombres) [126] (however, the first EROS events were not so reliable asMACHO’s).7

The first papers about the discovery of microlensing towards the Galactic bulge werepublished by the US–Polish Optical Gravitational Lens Experiment (OGLE) collabo-ration, which used the 1.3-m telescope at the Las Campanas Observatory. Since June2001, after a second major hardware upgrade, OGLE entered into its third phase,OGLE III, and as a result the collaboration observed more than 200 million stars reg-ularly once every 1–3 nights. During last years OGLE III detected several hundredmicrolensing event candidates each year [128,129]. The OGLE-III phase has endedon May 3rd, 2009.8 During the previous observing seasons the Early Warning System(EWS) of OGLE-III discovered thousands of microlensing event candidates.

To investigate the Macho distribution in another direction one could use searchestoward M31 (Andromeda Galaxy) lying at 725 kpc, which is the closest galaxy for anobserver in the Northern hemisphere [130–133]. On the other hand, there are severalsuitable telescopes concentrated in this hemisphere. In the 1990s two collaborations,

6 MACHO terminated at the end of 1999.7 The EROS experiment terminated in 2002 [127].8 http://www.astrouw.edu.pl/ogle/ogle3/ews/ews/html. OGLE collaboration started the phase OGLE IV.

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namely the AGAPE (Andromeda Gravitational Amplification Pixel Experiment, Picdu Midi, France)9 and the Vatican Advanced Technology Telescope (VATT) started tomonitor pixels instead of individual stars [127,137]. These teams reported discoveriesof several microlensing event candidates [138,139]. Results of Monte Carlo simula-tions for these observations and differences between pixel and standard microlensingare discussed [140–144]. The possibility to find exoplanets with pixel lensing towardsM31 was discussed in [145–147]. It was found that there is a possibility to find evenlow mass exoplanets with this technique. Moreover, the PA-99-N2 event (the so-calledPA-99-N2 anomaly) can be interpreted as a system with an exoplanet mass of about6–7 MJ (MJ is the Jupiter mass) [145–147].10

The matter of gravitational microlenses is unknown till now, although the mostwidespread hypothesis assumes that they are compact dark objects such as browndwarfs. Nevertheless, they could be other objects. In particular, the existence of darkobjects consisting of supersymmetrical weakly interacting particles (neutralinos) hasbeen discussed in [153–156]. The authors have shown that such stars could be formedin the early stages of the Universe and be stable on a cosmological timescale.

Computer simulations showed that dark matter may form bound objects (clumps orclouds) with masses of more than 10−6 M� [165] (see also the semi-analytical studiesin [166]). It was noted that the dark matter particles (neutralinos) generate a γ -flux bytheir annihilations. In this case the clumpiness of the dark matter distribution is veryimportant and can significantly enhance the flux since neutralino annihilation cross-sections strongly depend on the neutralino density [166]. Remarkably, mass densitydistributions for these objects obtained by [165] are in agreement with analytical stud-ies [153]. If the size of such an object is more than the Einstein–Chwolson radius,then the properties of lensing by these objects are different from the correspondingproperties induced by compact lenses (Machos). Therefore, it is reasonable to discussthe lensing properties for objects in the mass range from planetary up to stellar masses.Below we summarize these results of studies for microlensing by noncompact objects.

5 Microlensing by noncompact bodies

5.1 The singular microlens model

We consider microlensing by a star in the framework of a rough model which is ratherclear, and we obtain analytical expressions. Of course, a more exact model of thegravitational field of a neutralino star may be considered [156]; nevertheless, we thinkthat our qualitative estimates of the effect are correct.

We approximate the mass density of a neutralino star in the form

ρNeS(r) = ρ0a0

2

r2 for r ≤ a0, (6)

9 The POINT-AGAPE collaboration started in 1999 with the 2.5-m Isaac Newton Telescope (INT) [134,135], the new robotic project Angstrom was proposed as well [136].10 The effectiveness of gravitational lensing for exoplanet searches was shown in [148–152].

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2308 A. Zakharov

where r is the distance from the stellar center, ρ0 is the mass density at distance a0,and a0 is the “radius” of the neutralino star. The r -dependence is an approximation ofthe r -dependence considered in [153], namely

ρNeS(r) = Kr−1.8. (7)

It is not difficult to obtain the surface mass density from the expression (6),

�(ξ) = 2ρ0

√a0

2−ξ2∫

0

a02

ξ2 + h2 dh = 2ρ0a0

2

ξarctan

√a0

2 − ξ2

ξ. (8)

If a0 ξ (we assume that the mass density distribution equation (6) is correct only

for small impact parameters), then �(ξ) −→ πρ0a0

2

ξ.

To derive the gravitational lens equation for the mass density distribution (6) weuse an approach developed for the singular isothermal sphere (SIS) model [76].

The lens equation reads

η = Ds

Ddξ − Dds αNeS(ξ), (9)

where

αNeS(ξ) =∫

R2

d2ξ ′ 4G�(ξ ′)c2

ξ − ξ ′

|ξ − ξ ′|2 . (10)

We use the radius a0, which determines the microlens “mass” M = 4πρ0a03, to

introduce the dimensionless variables

x = ξ

a0, y = η

η0, η0 = a0

Ds

Dd.

With the abbreviation

R0 = M

a02

2πG Dav

c2 , Dav = Dd Dds

Ds(11)

the lens equation reads

y = x − R0x

|x | . (12)

If we normalize distances in the lens plane and in the source plane using R0, namely ifwe introduce the variables y = y

R0, x = x

R0, then the lens equation has the following

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Lensing by exotic objects 2309

simple form:

y = x − x

|x | . (13)

The symbol ∧ will not be written below. Naturally, (13) is the same as the gravitationallens equation for the SIS model in dimensionless form.

We can see the difference between the magnifications for compact and noncompactmicrolenses. At first we write the magnification μ for the neutralino star, namely

μ(y) = 2

yfor y < 1 and μ(y) = 1 + 1

yfor y > 1, (14)

which are simultaneously the precise equations being suitable for the approximation.The total magnification produced by a compact body (Schwarzschild lens) is

μSch(y) = y2 + 4

y√

y2 + 4. (15)

So the difference in the magnifications is an essential feature and, in principle, it ispossible to distinguish compact from noncompact microlenses.

The general discussion of noncompact microlens properties in the framework ofthe singular model is given in [157–159].

5.2 Polarization for a singular model for noncompact microlens

Gravitational lensing is an achromatic phenomenon. However, if a source is a giantstar, there is a significant color distribution on the stellar disk. Since we have dif-ferent amplifications for different elements of the stellar disk we may observe colorsignatures of gravitational microlensing. We have the same effect for the polarization.Typically, the total polarization of a star vanishes but different segments of the starhave different local polarizations. The segments have different amplifications in micr-olensing and, as a result, the total polarization is non-vanishing for microlensing. Onceagain, if a source has a finite size, there are variations of gravitational lens amplifica-tion and a polarization in the source plane; as a result one has to expect polarizationvariations in microlensing events [167–170]. The effect is most significant for giantstars as background sources and when the source is crossing caustics (folds [76] andcusps [171,172]) as a consequence of its proper motion.

The polarization was considered in the framework of a singular model for a noncom-pact lens in [175]. There we see two peaks of the polarization curve for a noncompactobject. As it was shown in [163] the occurrence of the peaks is connected with a deriv-ative discontinuity of the amplification factor for a noncompact object. Therefore thephenomenon is more complicated than it was claimed earlier [175]. However, thediscontinuity follows from a singularity of the gravitational lens equation at x = 0.This singularity is a result of the singularity of the density distribution (6) at r = 0.

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2310 A. Zakharov

Gurevich et al. [156] noted that the density distribution can be described by a nonsin-gular distribution,

ρNeS(r) =

⎧⎪⎨

⎪⎩

ρ0, 0 < r < rc

ρ0

(rrc

)−α

rc < r < Rx

0, r > Rx

(16)

where α = 1.8, rc ∼ (0.05 − 0.1)Rx . We will consider below a nonsingular densitydistribution which approximates the expression (16). It is obvious that nonsingulardensity distributions are more realistic under astrophysical conditions than singulardensity distributions.

5.3 Non-singular model for non-compact microlens

We approximate the mass density of a neutralino star in the form

ρNeS(r) = 2ρ0rc

2

r2 + r2c

for r ≤ Rx , (17)

where r is the distance from the stellar center, ρ0 is the mass density at the distancerc from the center, rc is the radius of the core, and Rx is the radius of the neutralinostar. So, we practically use the non-singular isothermal sphere model (or the modelof an isothermal sphere with a core). The r -dependence is an approximation of ther -dependence which was considered in [156], where the authors used the model of anoncompact object with a core. It is clear that the singular (degenerate) dependence(6) is the limiting dependence of (17) for rc → 0.

The surface mass density corresponding to (17) is

�(ξ) = 4ρ0r2c

√Rx

2−ξ2∫

0

dh

ξ2 + h2 + r2c

= 4ρ0rc

2√

ξ2 + r2c

arctan

√Rx

2 − ξ2√

ξ2 + r2c

. (18)

This relation is correct for the entire range of impact parameters. For small impact

parameters, in the case Rx ξ, we have �(ξ) −→ 2πρ0rc

2√ξ2+r2

c. Below we consider

only small impact parameters and, therefore, we use the following approximation forthe surface mass density,

�(ξ) = 2πρ0rc

2√

ξ2 + r2c

. (19)

We calculate the microlens mass as

Mx = 8πρ0r2c

Rx∫

0

r2dr

r2 + r2c

= 8πρ0r2c

(Rx − rc arctan

Rx

rc

)≈ 8πρ0r2

c Rx . (20)

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Lensing by exotic objects 2311

We use the radius rc to introduce the dimensionless variables

x = ξ

rc, y = η

η0, η0 = rc

Ds

Dd, (21)

such that

�cr = c2 Ds

4πG Dd Dds, k(x) = �(a0x)

�cr, α(x) = 1

π

∫

R2

d2x ′k(x′)x − x′

|x − x′|2 .

In these dimensionless variables the lens equation (9) may be written in scalar form[76] as

y = x − α(x) = x − m(x)

x, m(x) = 2

x∫

0

x ′k(x ′)dx ′.

We recall that

k(x) = k0√1 + x2

, (22)

where

k0 = 2πρ0r0

�cr= 2π Mx

rc Rx

G

c2

Dd Dds

Ds= π

4rc Rx

4G Mx

c2

Dd Dds

Ds= π

4

ξ20

rc Rx. (23)

Hence, the lens equation has the following form,

y = x − D

√x2 + 1 − 1

x, (24)

where D = 2k0.

5.4 Qualitative analysis

We will show that the gravitational lens equation has only one solution if D < 2 andhas three solutions if D > 2 and y > ycr (we consider the gravitational lens equationfor y > 0), where ycr is a local maximum of the right-hand side in Eq. (24). It ispossible to show that the value xcr , which corresponds to ycr according to

ycr = xcr − D

√1 + x2

cr − 1

xcr, (25)

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2312 A. Zakharov

is given by the following expression,

x2cr = 2D − 1 − √

4D + 1

2. (26)

It is easy to see that, according to (26), x2cr > 0 if and only if D > 2. If we choose

xcr < 0, then ycr > 0. So, if D ≤ 2 the gravitational lens equation has only onesolution (for all y > 0). If D > 2 the gravitational lens equation has one solution (ify > ycr ), three distinct solutions (if y < ycr ), or one single solution and one doublesolution (if y = ycr ).

It is possible to show that the gravitational lens equation is equivalent to the fol-lowing equation

x3 − 2yx2 −(

D2 − y2 − 2D)

x − 2y D = 0, (27)

jointly with the inequality

x2 − yx + D > 0. (28)

Thus, it is possible to obtain the analytical solutions of the gravitational lens equationin a well-known way. We transform to the new variable z = x − 2y

3 and obtain theincomplete equation of third degree

z3 + pz + q = 0, (29)

where p = 2D − D2 − y2

3 and q = 2y3 (

y2

9 − D(D + 1)), so we have the followingexpression for the discriminant,

Q =( p

3

)3 +(q

2

)2 = D2

27

[−y4 + y2(2D2 + 10D − 1) + D(2 − D)3

]. (30)

If Q ≥ 0, then Eq. (29) has a unique real solution (therefore, the gravitational lensequation (24) has a unique real solution). We use the Cardano expression for thesolution,

x = 3√

−q/2 + √Q + 3

√−q/2 − √

Q + 2y

3. (31)

The right-hand side of the gravitational lens equation is shown, for different valuesof the parameter D, in Fig. 1.

We assume that D > 2. If y > ycr the gravitational lens equation has a uniquesolution. If Q ≥ 0 we use the expression (31) for the solution. If Q < 0 we have thefollowing expression,

x = 2

√

− p

3cos

α + 2kπ

3+ 2y

3, (k = 0, 1, 2), (32)

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Lensing by exotic objects 2313

Fig. 1 The right-hand side ofthe gravitational lens equationfor D = 1.8, 2, 2.2.

where

cos α = − q

2√−(p/3)3

. (33)

We select only one solution which corresponds to the inequality (28) and to k = 0 inEq. (32) because, if the gravitational lens equation has only one real solution, then wehave a positive solution x for a positive value of the impact parameter y; therefore wehave the inequality x > y which is easy to see from Eq. (25). It is possible to checkthat the maximal solution of (27) corresponds to k = 0; therefore, it is the solution ofEq. (25).

If y < ycr the gravitational lens equation has three distinct real solutions and weuse the Eqs. (32–33) to get the solutions.

We consider now the case D < 2. We know that the gravitational lens equationhas a unique solution in this case. If Q ≥ 0 we use the expression (31) for the solu-tion. If Q < 0 we have the expressions (32–33); we select only one solution whichcorresponds to the inequality (28) and to k = 0 as in the previous case.

It is known that the magnification for the solution xk to the gravitational lens equa-tion is given by the expression [174]

μk =∣∣∣∣∣

(

1 − D(√

1 + x2 − 1)

x

) (

1 + D

√1 + x2 − 1

x2 − D1√

1 + x2

)∣∣∣∣∣

−1

, (34)

for x = xk , so the total magnification is

μtot(y) =∑

μk, (35)

where the summation is taken over all solutions of the gravitational lens equation fora fixed value y.

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5.5 Types of polarization and light curves for the nonsingular model

Polarization curves were reproduced for microlensing with noncompact objects usinga singular lens model [175]. It was concluded that polarization curves for compact andnoncompact objects demonstrated very distinguishable shapes, but light curves hadsimilar shapes for compact and noncompact objects in these cases. In the framework ofa more realistic model this conclusion is no longer true. So, the problem is much moredifficult. Without any doubt there are some differences between polarization curvesfor compact and noncompact objects, but the differences are not so great as it wasearlier concluded [175]. We also conclude that it is possible to distinguish compactfrom noncompact objects in principle only if light paths pass through the noncom-pact object. Roughly speaking, we must have the inequality Rx >∼ ξ0 for the radiusof the noncompact body and the Einstein–Chwolson radius; otherwise, if Rx � ξ0,the lens may be treated as a compact one. The polarization curves were simulated inthe framework of a more reliable non-singular model [163]. The light curves werereproduced for these cases as well. Therefore, some properties of the singular modelof a noncompact lens are degenerate and we do not observe these properties for thenonsingular model. For example, it is impossible to get the polarization curve with twopeaks when the respective light curve has only one maximum. When light curves forcompact and noncompact objects have different shapes, the polarization curves havedifferent shapes and it is possible to distinguish the objects in principle. However, theproblem is not as simple as it was noted in [175].

We will show two light curves simulated with a non-singular model of a noncom-pact lens. The light curve corresponding to a non-compact microlens is presented inFig. 2 (for D = 1.9). It resembles the usual light curves derived from the simpleSchwarzschild microlens model. If we consider the finite size of a source (the non-point source) for D = 4, the light curves are shown in Fig. 3; the finiteness of a

Fig. 2 The light curve formicrolensing by a non-compactobject for D = 1.9. The lightcurve resembles the standardlight curve for microlensing by acompact object. A time scale forthe duration of the event ischosen such that thecorresponding dimensionlesstransverse velocity is equal to 1

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Fig. 3 The light curves formicrolensing by a non-compactobject for D = 4 and differentsizes of a non-point source. Themaximal values of the lightcurves decrease with increasingsource radius, which is chosenas Rs = 0.01, 0.03, 0.1,

respectively

source has a smoothening effect on the variations of amplifications near caustics,similarly to microlensing in gravitationally lensed systems and exoplanet searches[113,114,145,146,146]. One can see that the light curves in Fig. 3 resemble the lightcurve of the OGLE # 7 event candidate that is usually interpreted in terms of a binarylens model [176]. It would be reasonable to discuss similarities and differences in thelight curves of a binary lens and a transparent lens model. For the transparent lensmodel one has to expect a symmetrical shape of the light curve, but for a binary lensmodel a light curve may be asymmetrical for a choice of parameters choices. However,if we consider a more general model for a transparent lens (one can choose non-spher-ical distributions of clumps) then one could expect violations of the symmetry in thelight curve. Generally speaking, the two-peak structure indicates two crossings ofcaustics, but the caustics may be produced in different ways, either by a binary lens orby a transparent lens. For calculating the light curve presented in Fig. 3 we assumedthat the source disk has a uniform brightness (this means that the limb darkeningwas ignored), taking into account (34), and we numerically calculated the total fluxfor three images (this region corresponds to the light curve between two peaks) andfluxes for wings of the light curve (we have only one solution of the gravitational lensequation in this region).

We recall that the appearance of two types of light curves for a toy density distri-bution model for a noncompact object was discussed by Ossipov and Kurian [177]. Amore detailed analysis of the nonsingular model and its consequences is presented in[162].

6 Conclusions

Reliable (and verified) signatures of cosmic strings and neutralino clouds have notyet been found. However, theoretical studies of possible observational aspects (and

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corresponding observational activities) may help to discover such exotic objects in thefuture.

Acknowledgments It is a pleasure to acknowledge S. Calchi Novati, A.M. Cherepaschchuk, F. De Paolis,G. Ingrosso, P. Jovanovic, Ph. Jetzer, A.A. Nucita, L.C. Popovic and M. V. Sazhin for useful discussionsand clarifications. The author is grateful to V. Perlick for his kind attention to this contribution. The authoris grateful to the National Natural Science Foundation of China (NNSFC) (Grants # 10703007, 10873020,G10573025, 40674081, 10603008, 40890161, and 10733020) and National Basic Research Program ofChina (G2006CB806303) for a partial financial support of the work.

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