Probing planetary mass dark matter in galaxies: gravitational nanolensing of multiply imaged quasars
aa r X i v : . [ a s t r o - ph . GA ] N ov Mon. Not. R. Astron. Soc. , 1–14 (2010) Printed 4 June 2017 (MN L A TEX style file v2.2)
Probing planetary mass dark matter in galaxies:gravitational nanolensing of multiply imaged quasars
H. Garsden, N. F. Bate and G. F. Lewis ⋆ Sydney Institute for Astronomy, School of Physics, A28, The University of Sydney, NSW, 2006, Australia
Draft: 9 Sep 2011
ABSTRACT
Gravitational microlensing of planetary-mass objects (or “nanolensing”, as it has beentermed) can be used to probe the distribution of mass in a galaxy that is acting as agravitational lens. Microlensing and nanolensing light curve fluctuations are indicativeof the mass of the compact objects within the lens, but the size of the source isimportant, as large sources will smooth out a light curve. Numerical studies have beenmade in the past that investigate a range of sources sizes and masses in the lens. Weextend that work in two ways – by generating high quality maps with over a billionsmall objects down to a mass of 2 . × − M ⊙ , and by investigating the temporalproperties and observability of the nanolensing events. The system studied is a mockquasar system similar to MG 0414+0534. We find that if variability of 0.1 mag inamplitude can be observed, a source size of ∼ . . × − M ⊙ masses, and larger, in the microlensing lightcurve. Our investigation into the temporal properties of nanolensing events finds thatthere are two scales of nanolensing that can be observed – one due to the crossingof nanolensing caustic bands, the other due to the crossing of nanolensing causticsthemselves. The latter are very small, having crossing times of a few days, and requiringsources of size ∼ Key words: galaxies: structure – gravitational lensing: micro – dark matter – quasars:individual: MG0414+0534 – methods: numerical
Gravitational lensing occurs when light is deflectedby gravity. The first instance of gravitational lens-ing in a cosmological context was observed in 1979,where the quasar Q0957+561 is lensed by by a fore-ground galaxy, producing two magnified, distorted, im-ages (Walsh, Carswell, & Weymann 1979). Since then manyother multiply-imaged lensed quasars have been found (e.g.Vanderriest, Wlerick, & Felenbok 1983; Huchra et al. 1985;Lawrence et al. 1995; Myers et al. 1999; Eigenbrod et al.2006). The properties of the lensed images, for exampletheir number, separation on the sky, relative brightness,and others, are determined mostly by treating the galaxy ⋆ as a smooth gravitational lens. Galaxies are not smooth,however, being comprised of many compact objects in ad-dition to smoothly distributed matter. The objects insidethe galaxy can exert an influence on the quasar images be-cause the quasar and galaxy are in transverse motion rel-ative to our line of sight. As the quasar moves over time,this motion shifts the light paths relative to the galaxycomponents, making the quasar images vary in magnitudeover time scales of weeks to years (Shalyapin 2001). Thisphenomenon, called gravitational microlensing, was first ob-served in 1985 (Huchra et al. 1985) and is believed to oc-cur in most lensed quasars (Witt, Mao, & Schechter 1995).It can be used as a probe of the source quasar becausethe quasar’s size and shape affects the microlensing vari-ability (e.g. Mortonson, Schechter, & Wambsganss 2005;Bate et al. 2008; Morgan et al. 2010; Garsden & Lewis2010). For example, when the source is large, the changes inthe light curve are not as prominent, since the larger sourcesmooths out the variability.Microlensing can also be used to probe the mass dis- c (cid:13) H. Garsden, N. F. Bate & G. F. Lewis tribution in the lensing galaxy. Initial studies of quasar mi-crolensing indicated that microlensing was not significantlydependent, within certain constraints, on the mass spec-trum within the lens (e.g. Wambsganss 1992; Lewis & Irwin1995), in particular how much mass is in compact objectsand how much is in smooth matter. However, during in-vestigations of anomalous flux ratios in the lensed quasarMG 0414+0534 (Schechter & Wambsganss 2002), evidenceagainst this view was found, and numerical modelling pre-sented by Schechter, Wambsganss, & Lewis (2004) showedthat it must be false. Beginning with a large number of solar-mass (1M ⊙ ) stars for the lens galaxy, they replaced a sub-stantial fraction of these with smooth matter, keeping thetotal mass the same, and found that the microlensing vari-ability was enhanced. This also introduced the “bi-modal”mass distribution into computational microlensing, wheretwo very distinct mass components in the lens are modelled(e.g. Pooley et al. 2010).Lewis & Gil-Merino (2006) replaced the smooth matterwith many compact objects, all of the same mass, and sig-nificantly less than 1M ⊙ . The objects produced small-scalemicrolensing variability that was not present when smoothmatter was used, as expected. However, it was found that thevariability produced by the small objects could be smoothedout enough so that they could not be distinguished fromsmooth matter, if the source size was over a threshold. It wasalso found that, if source size was taken into account, theamplitude of these variations was an indicator of the massof the objects in the lens. Recently, Chen & Koushiappas(2010) conducted similar modelling, expanding the bi-modaldistribution by using a Salpeter (1955) mass distribution forthe stars and a Navarro, Frenk, & White (1997) power-lawdistribution for the compact objects, considering these aspotential dark matter candidates. They conclude that theirsmall compact objects will add detectable small microlensingevents (“nanolensing” events) to a microlensing light curveof no more than about 0.1 mag over a time scale smallerthan a year. Chen & Koushiappas (2010) were less inter-ested in source size and how this diminishes the detectabilityof nanolensing events, but did confirm that larger sourceswould smooth out the light curves and make nanolensingharder to detect.The term “nanolensing”, used pre-viously by Walker & Lewis (2003) andSchechter, Wambsganss, & Lewis (2004), is gaining inusage and refers to light deflections on much smallerscales than microlensing, due to planetary-size objectsor possible dark matter objects (Zakharov 2009). Thework of Walker & Lewis (2003) involved the detection ofcosmological planetary masses by the nanolensing of gammaray bursts (Walker & Lewis 2003), and nanolensing is alsoreferred to in exoplanet searches (Zakharov et al. 2010).We use it here to refer to lensing variability produced byobjects in a lens with masses far below that of 1M ⊙ stars,e.g. the low-mass objects in a bi-modal mass distribution.This paper expands on the work of Lewis & Gil-Merino(2006), Lewis (2008) and Chen & Koushiappas (2010) ininvestigating nanolensing events due to small masses, andtheir interaction with source size. Using bi-modal mass dis-tributions, we use several mass values down to 2 . × − M ⊙ for the size of the small objects, while increasing their num-ber to over a billion – far more than has been modelled in the past. We use a mock lensed quasar system thathas been modelled by Schechter & Wambsganss (2002) andChen & Koushiappas (2010), similar to the lensed quasarMG 0414+0534. MG 0414+0534 has not been used in thisand past works due to the high magnification of the sourcequasar, produced by a very large number of objects in thelens galaxy, which are difficult to deal with in numericalmodels. Simple statistics allow us to use the amplitude ofnanolensing events to infer small objects in a lens, based ona range of source sizes. We then follow this with an investi-gation into the duration of the nanolensing events and thesource sizes needed to resolve them – something has not notbeen done in past studies. We show that there are two timeand source scales involved in these events and indicate howthey may be observed. We discuss how our investigationsprovide direction to further work that can be conducted inthis area. The structure of the paper is as follows: In Section2 we introduce numerical modelling of microlensing, and thelensing model and parameters used for this study, in Section3 we present the results of nanolensing of bi-modal mass dis-tributions with different source sizes, including event ampli-tudes and durations. In Section we 4 discusses the resultsand Section 5 contains our conclusions.Throughout this paper, a cosmology with H =70kms − Mpc − , Ω m = 0 . Λ = 0 . The properties of an image in a multiply-imaged quasarare determined mostly by the mass distribution of the lens-ing galaxy, and the relative locations and distances of thegalaxy and quasar (Schneider, Ehlers, & Falco 1992). Forimage modelling, two parameters are used to specify themass in the lensing galaxy at the image positions: the con-vergence ( κ ), and shear ( γ ). The convergence specifies theeffect of mass close to the light path, and the shear is theeffect of the overall mass of the galaxy. Within the conver-gence, a mass spectrum can be chosen for models, we willbe using bi-modal distributions as described above, where asmall amount of mass is in 1M ⊙ stars, and the rest in eithersmall objects, or smooth matter. The distances to the lensand source are subsumed into a distance unit used withinlensing models: the Einstein Radius ( η ). If a point sourceis perfectly in line with a point lens object, usually chosento be 1M ⊙ , the source will appear as a ring around the lens.Projected onto the source plane, the ring radius is given by η = r GMc D os D ls D ol , (1)where M is the mass of the lens and D xy refers to the angulardiameter distance between x and y ; the subscripts s, l, and o representing source, lens, and observer respectively.Microlensing and nanolensing requires the sourceto change location behind the lens (Wyithe 2001;Poindexter & Kochanek 2010) to produce variation in thesource magnification, so a region of the source plane, wherethe source may lie, is defined. Each point in the region hasa magnification value, indicating how a point source willbe (de)magnified at that location. This is a “magnification c (cid:13) , 1–14 ravitational nanolensing of quasar dark matter Table 1.
Parameters for mock lens imagesImage convergence ( κ ) shear ( γ )M 0.475 0.425S 0.525 0.575Parameters for the mock lensed quasar images used in this study.The convergence ( κ ) specifies the mass close to a light ray, theshear ( γ ) specifies the effect of all the mass in the lensing galaxy.They are the same as the M10 and S10 parameters used inSchechter & Wambsganss (2002). map” (Schmidt & Wambsganss 2010), examples of whichare seen in Figure 1. Bright areas in a map indicate locationswhere the source will be magnified and darker regions arewhere it will be demagnified, relative to the average for theimage being modelled. The regions of light and dark are de-lineated by lines called “caustics”, where the magnificationis formally infinite (Blandford & Narayan 1986). There willbe different maps for each image in a lensed quasar, sincethe mass distribution producing each image is different.A magnification map is divided into pixels, which en-forces a lower bound on the size of sources that can be stud-ied. Different source sizes and shapes are studied by creatinga pixelized source profile and convolving this with the map,producing a map for the microlensing of that source. Lightcurves can be obtained by taking the magnification along aline across the map, the line corresponding to the path alongwhich a source may travel. For the lensing galaxy, compactmasses and smooth matter are laid down on a lens plane andrays are fired through them to hit the map. The lens massesare projected onto on a plane because, in a cosmological situ-ation where the distances are great, the lensing galaxy can beapproximated as flat (Schneider, Kochanek, & Wambsganss2006). Rays are fired in the inverse direction, i.e. fromobserver through the lens to source plane, for reasonsof computational ease and efficiency. Because microlens-ing is produced by a large number of objects in the lensgalaxy, it is difficult to study analytically, and numeri-cal techniques are used; we use the inverse ray-shootingmethod of Wambsganss (1990, 1999) and Garsden & Lewis(2010). Numerical approaches have allowed the study ofmany aspects of quasar microlensing, such as: the sizeof accretion disks (Chartas et al. 2002; Bate et al. 2008;Blackburne et al. 2011); the structure of quasar broadline emission regions (Lewis & Ibata 2004; Keeton et al.2006; O’Dowd et al. 2011); the structure of microlensed wa-ter masers (Garsden, Lewis, & Harvey-Smith 2011); chro-matic effects in microlensing (Wambsganss & Paczynski1991); the nature of dark matter in lensing galaxies(Pooley et al. 2009, 2010; Bate et al. 2011); and the effect ofsource size on microlensing (Bate, Webster, & Wyithe 2007;Mortonson, Schechter, & Wambsganss 2005), among others. We use a mock lensed quasar system that has beenused in other work (Schechter & Wambsganss 2002;Chen & Koushiappas 2010). The convergence and shear pa-rameters for each image are listed in Table 1; these valuesare based on models of the quadruply-imaged quasar MG 0414+0534. The images in the mock lens system are des-ignated M and S; image M is positive-parity, S is negativeparity, the latter meaning the image is mirror-symmetric tothe source. However, for the purposes of this study, the in-ternal structure of the images is not relevant, only the mag-nification of the image relative to the source. Images M andS produce different-looking magnification maps and slightlydifferent behaviour in our models, as will be seen later.Recent studies suggest the dark matter fraction in mi-crolensed quasars at the image positions is high. Bate et al.(2011) preferred 50 +30 − % in MG 0414+0534. Pooley et al.(2009) determined 90% for PG 1115+080, and now(Pooley et al. 2010) suggests >
95% for MG 0414+0534 andH1413+117. For our bi-modal mass distribution, we set twoper cent of the mass in the lens in 1M ⊙ objects, the other98% being in either smooth matter or small compact ob-jects. The small objects are all of the same mass for eachmap. The mass of the small objects can be reduced, whichmeans their number increases to maintain the same totalmass in the lens. The 1M ⊙ objects stay at the same locationsas the other masses change. Following Chen & Koushiappas(2010), we use z = 0.3 for the lens and z = 2 for the source.The Einstein Radius (ER) for a 1M ⊙ object using these dis-tances is 0.0236 pc. The magnification maps cover a region inthe source plane of 20 ×
20 ER (0.47 × ), at a reso-lution of 10000 × , or width 0.002 ER (4.7 × − pc) per pixel. This also means the smallest source size thatcan feasibly be studied with such maps is 0.002 ER.The maps used in this study are shown in Figure 1.The first column has maps for image M, the third columnfor image S. Note how the large scale caustic structure isconsistent within each column, but looks different betweencolumns. This is because of the opposite parity of the quasarimages (Schechter & Wambsganss 2002). The first map forimage M was generated from a lens of 609 1M ⊙ objectsand 14,758,999 objects of 2 . × − M ⊙ , and the secondlast map for image M represents 1,195,478,868 objects of2 . × − M ⊙ . The number of compact objects used to gen-erate each map is listed next to the map in column two.The bottom map was generated from a lens with 609 1M ⊙ objects and the rest of the mass in smooth matter, beingthe same mass as all the small objects combined. The thirdcolumn is similar to column one, but for image S, reachinga limit of 1,318,703,785 small objects, with the number ofobjects listed in column four. To ensure the maps are of highquality with good magnification resolution, we shoot severaltrillion light rays through the lens, over a million per pixel. The computational work required to generate and analysethe maps presents considerable challenges, and it is only inrecent years that they have been overcome. With the adventof supercomputers it is now possible to use large numbers ofobjects in the lensing galaxy, and in this work over 1 billionare used. We also fire rays in parallel, using multiple parallelprocesses. Using the method of Garsden & Lewis (2010), themap containing the largest number of masses can be gener-ated in ∼
14 days on a supercomputer using 16 parallel pro-cesses. For the maps with the least number of lens masses,the total time is ∼
24 hours. About 18% of the compute timeis used to generate the lens objects, and the rest is for firing c (cid:13) , 1–14 H. Garsden, N. F. Bate & G. F. Lewis
Figure 1.
The microlensing maps used in this study, all of size 20 ×
20 ER , with a resolution of 10000 × . Column 1 containsthe maps for image M, generated for a lens that contains 2% of the mass in 609 1M ⊙ objects, and the rest in either compact objects,or smooth matter. The mass of the compact objects is indicated on the left side of each map. Column 3 contains the maps for image S,generated for a lens that contains 2% of the mass in 672 1M ⊙ objects, and the rest in either compact objects, or smooth matter, thecompact object mass being the same as used for the image M map. The mass of the compact matter objects decreases, and the numberof compact objects increases, going down the rows. The second column indicates the number of compact objects for the image M maps,and the 4th column is for the image S maps. c (cid:13) , 1–14 ravitational nanolensing of quasar dark matter the rays, with many rays fired in parallel. The need to placethe lens information in large data files, of about 150Gb for 1billion objects, means ideal speedups are not achievable foreither phase of the program, but the situation is improvedwith the use of 16 Gb of computer memory for caching thefiles; such memory is only now usable because 64-bit RAMaddressing has become available. Sources in our simulations have a simple 2-D Gaussian pro-file, with a radius of 3 σ . Their radii are based on estimatedsizes of quasar structure. For example, accretion disks areof order 0 .
002 pc (Mosquera & Kochanek 2011, estimated inthe I-band from H β ), and broad line regions of order 0 . . × − to 0.033 pc. Beginning at a size of 0.002ER, the next size is 0.01 ER, then 0.05 ER, then increasingby 0.05 ER to 0.4 ER, then in steps of 0.1 ER to 1.0 ER,then in steps of 0.2 ER to 1.4 ER, thus providing more datapoints at the smaller sizes. This is necessary as will be seenfrom the figures presented later on. We always compare a map generated using stars+small ob-jects in the lens, to a map using stars+smooth matter inthe lens, and only maps generated for the same image (Mor S). This means that each map in the first column of Fig-ure 1 will be compared with the bottom one in the column,and the same for the third column. It can be seen that themaps using small objects become more like the correspond-ing smooth matter map as the size of the small objects getssmaller and their number increases; this is expected. How-ever, it is also necessary to consider source size; if these mapsare convolved with larger sources then they may become sim-ilar to the smooth matter case when the small object massis not so small.A simple measure is used to indicate that one map issimilar to another. The maps are converted from magnifica-tions to magnitudes. To avoid edge effects, a margin aroundthe map of width equal to the source radius of the largestsource (1.4 ER) is ignored, and within the margin the mapsare normalized to have an average magnitude of 0. If sucha map generated from smooth matter is subtracted from amap generated from compact matter the result is an approx-imately Gaussian distribution of residual magnitude. Theroot-mean-squared (RMS) of the residual values is used asthe measure of similarity – termed the “difference measure”.It is the average difference between the two maps in unitsof magnitude.
Firstly we demonstrate how two maps, one generated fromcompact objects and the other from smooth matter, become similar as the source size increases. Figure 2 shows a com-parison of such maps; all were generated from a star fieldof 609 1M ⊙ masses at the same fixed locations, plus eithersmooth matter or small objects of 2 . × − M ⊙ . Thefirst column shows maps for a lens containing these objects,the second column shows maps for a lens with smooth mat-ter. The 1M ⊙ masses are responsible for the identical largescale caustics in the maps in each column. The map in thefirst column shows that between the large caustics there issmall-scale structure, produced by the small objects. Thesmooth matter map (second column) does not exhibit this.The third and fourth columns display a light curve extractedfrom a horizontal cut across the middle of the maps in thefirst and second columns respectively. The light curves makeplain the small scale structure in the compact matter map,producing a high degree of variability in the top row that al-most masks the peaks in the light curve from the large scalecaustics, clearly visible in the smooth matter light curve. Go-ing down the row the source size increases, indicated on theleft side of the first map. The fifth column is the differencemeasure between the maps, described previously.As the source size increases, the compact matter andsmooth matter maps and their light curves become similar,and the difference measure decreases. Both of these thingsindicate that the increased source size is reducing the small-scale variation in the compact matter map such that it be-comes like the smooth matter map, and perhaps indistin-guishable from it. Our next step is to execute this sameprocedure for all the compact matter maps in Figure 1. Figure 3 shows how the difference measure decreases assource size increases, for all the compact matter maps. Fig-ure 3 (a) is for the maps for image M, and (b) is for imageS. Both plots are similar and show a smooth decrease in dif-ference (vertical axis) as the source size (horizontal axis) isincreased. The difference measure falls faster early on andthen flattens out at a value below ∼ .
05 for all the maps.The difference starts out smaller for image M, indicatingthat the compact object maps are slightly more similar tothe smooth matter maps in this image. Maps made withlarger masses begin at a higher difference measure, indicat-ing larger masses produce higher amplitude variability be-tween the maps. The discriminating power of the differencemeasure appears to be highest in the middle of the massrange used. Overall, the behaviour is smooth and consistentwithin each image, and the trend is the same for each image.
As Figure 2 has shown, the difference between the maps isdue to nanolensing variability that appears in a light curveas “wiggles” that are smaller than microlensing events. Byobserving this variability, it is possible to infer that small ob-jects are in the lens, and so distinguish these objects fromsmooth matter. Observing nanolensing requires that the am-plitude of nanolensing variability be above the threshold ofobserving sensitivity, which we define here as above a certainmagnitude threshold. Once nanolensing is found it then de-pends on its amplitude in relation to the observing threshold c (cid:13) , 1–14 H. Garsden, N. F. Bate & G. F. Lewis
Figure 2.
Demonstrates how a map generated from compact objects becomes similar to a map generated from smooth matter, as thesize of the microlensed source increases. Column 1 uses the map generated from 2 . × − M ⊙ small compact masses for image M inFigure 1. Column 2 uses the smooth matter map for image M. Column 3 contains a light curve cut horizontally across the middle of thecompact mass map; column 4 contains a light curve from the smooth matter map. Going down the rows from top to bottom, both mapsare convolved with sources of increasing size, and the size is indicated on the left side of the compact mass map. Column 5 contains thedifference measure quantifying the difference between the two maps in the row. as to whether it discriminates between masses for the smallobjects. If the threshold is high, then only high amplitudevariations can be seen, and from Figure 3 this means thatonly large masses can be inferred in the lens. If the thresholdis low, but still only high amplitude variation is seen, it ispossible to conclude that smaller masses are not in the lensbecause smaller variations are not seen. Added to this is thesize of the source, which reduces the amplitude of nanolens-ing (and microlensing) variability, so a knowledge of sourcesize is important. We first look at some properties of the map analysis and then proceed to look at the duration of small-scale changes in light curves, whether these are nanolensingevents, and what source sizes are needed to resolve them. It is evident that the simple measure used for the differencebetween the compact mass and smooth matter maps ex-hibits smooth regular behaviour, which we attribute to the c (cid:13)000
Demonstrates how a map generated from compact objects becomes similar to a map generated from smooth matter, as thesize of the microlensed source increases. Column 1 uses the map generated from 2 . × − M ⊙ small compact masses for image M inFigure 1. Column 2 uses the smooth matter map for image M. Column 3 contains a light curve cut horizontally across the middle of thecompact mass map; column 4 contains a light curve from the smooth matter map. Going down the rows from top to bottom, both mapsare convolved with sources of increasing size, and the size is indicated on the left side of the compact mass map. Column 5 contains thedifference measure quantifying the difference between the two maps in the row. as to whether it discriminates between masses for the smallobjects. If the threshold is high, then only high amplitudevariations can be seen, and from Figure 3 this means thatonly large masses can be inferred in the lens. If the thresholdis low, but still only high amplitude variation is seen, it ispossible to conclude that smaller masses are not in the lensbecause smaller variations are not seen. Added to this is thesize of the source, which reduces the amplitude of nanolens-ing (and microlensing) variability, so a knowledge of sourcesize is important. We first look at some properties of the map analysis and then proceed to look at the duration of small-scale changes in light curves, whether these are nanolensingevents, and what source sizes are needed to resolve them. It is evident that the simple measure used for the differencebetween the compact mass and smooth matter maps ex-hibits smooth regular behaviour, which we attribute to the c (cid:13)000 , 1–14 ravitational nanolensing of quasar dark matter (a) image M (b) image S Figure 3.
For both images, and for all maps, this is a plot of the difference between the compact matter map and smooth matter mapas source size increases. Data for image M is in (a), image S in (b). In both graphs there is one curve for each of the five compactmatter maps, for each image, in Figure 1. The top curve corresponds to compact objects of size 2 . × − M ⊙ , the one below for6 . × − M ⊙ , and so on in decreasing order of mass. The horizontal axis indicates the size of the source convolved with compact andsmooth map for an image, and the vertical axis is an average of the difference between the compact matter and smooth map.(a) image M (b) image S Figure 4.
Indicates the maximum size a source can be so that the RMS difference between compact and smooth matter maps is greaterthen 0.12 mag (bottom line) to 0.04 mag (top line), as a function of the mass of the compact objects in the lens. Data for image M isin (a), and image S in (b). The horizontal axis is the mass of the compact objects, and the vertical axis is the size of the source beinglensed.c (cid:13) , 1–14
H. Garsden, N. F. Bate & G. F. Lewis physics of the situation, the high quality of the maps, andthe very large number of small objects. As the behaviouris similar in both images, our difference measure is usefulfor both positive and negative parity images. The measureis an average of the amplitude of nanolensing variation overthe whole map, and hence also an average of light curve de-viations, e.g. those seen in Figure 2. Chen & Koushiappas(2010) measured nanolensing from light curve deviationsdirectly, by finding peaks indicating small-scale changes,Lewis & Gil-Merino (2006) used a magnification map anal-ysis similar to ours, based on fractional differences betweenmagnification maps. Using Figure 3 we can determine themass of the small objects in the lens based on the ampli-tude of nanolensing variation and an assumption of sourcesize. For example, because all the curves flatten out to below0.047 mag at a source size of 1.4 ER (0.033 pc in our mocksystem), small compact objects will not be distinguishablefrom each other, or from smooth matter, for a source largerthan 1.4 ER, unless the observation threshold is smaller than0.047 mag.Figure 3 can also be used to correlate nanolensing masswith source size if the amplitude of nanolensing is above acertain threshold. Figure 4 shows the largest source size (ver-tical axis) that will allow events with amplitudes of 0.12, 0.1.0.8. 0.6, or 0.04 mag (separate lines) to be observed, for thedifferent masses (horizontal axis). The plots are obtained bydrawing horizontal lines through the graphs in Figure 3. Theplot is log-log, which produces a linear trend, as found byLewis & Gil-Merino (2006). These show that small sourcesare needed to observe high amplitude variations, while alarge source can only produce low magnitude variation. Thesource sizes in Figure 4 range from about 0.07 ER to 1.34ER (0.0017 to 0.032 pc). From Chen & Koushiappas (2010)we adopt a value of 0.1 mag as a threshold difference mea-sure for observing nanolensing. Then, to infer small objectsof mass 2 . × − M ⊙ in the lensing galaxy, the source mustbe of size 0.09 ER (0.0021 pc) or smaller; for objects of2 . × − M ⊙ , a source size of 0.94 ER (0.022 pc) orsmaller will do. The two plots in Figure 4 also show thatcompact masses are slightly harder to find using image M,because the source sizes needed for this are a bit lower over-all. This may also be seen in light curves shown later on (Sec-tion 4.2), where the nanolensing variability appears smallerin image S than that in image M, for the same source.We may use Figures 3 and 4 to determine the lower limitof the masses that could be discriminated, if the source wasquasar sub-structure such as an accretion disk. For example,Blackburne et al. (2011) estimate the size of the λ = 590 nmemission region in MG 0414+0534 to be 10 . cm, based onmicrolensing analysis of image flux ratios. This size is 0.23ER in the mock lens system; using 0.1 mag as the thresh-old for observing nanolensing in Figure 4, variability couldbe seen for small masses down to about 1 . × − M ⊙ . Tosee variability caused by 2 . × − M ⊙ masses, the sourceneeds to be 0.09 ER. The λ = 814 nm emission region inMG 0414+0534 is estimated, from H β flux, to be 1 . × cm = 6 . × − pc in MG 0414+0534, or 0.0026 ER in themock system (Mosquera & Kochanek 2011). As this is be-low 0.09 ER a source of this size can produce nanolensingthat indicates and discriminates between all the small massvalues we have used. In fact from Figure 3, with a source this small we would observe nanolensing variations of 0.4mag or better for all the masses.Lewis & Gil-Merino (2006) and Lewis (2008) used dif-ferent parameters for the lens (e.g. κ = 0.2, γ = 0.5) and adifferent criterion for differentiating compact vs smooth, butthey suggest 0.2 ER as the source size below which nanolens-ing could be distinguished. If we use an observing thresholdof 0.1 mag then 0.2 ER sits nicely within our range, of 0.09to 0.95 ER, indicating this source size is characteristic of theproblem.The mock lens used here is similar to that ofMG0414+0534, but using MG 0414+0534 for our analysiswould require 10 billion objects in the lens, if other parame-ters are held the same. Nevertheless we may make compar-isons with MG 0414+0534 by setting the Einstein Radiusin the mock lens to the Einstein Radius of MG 0414+0534:0.0136 pc (z L = 0.9584, z S = 2.639; Lawrence et al. 1995;Tonry & Kochanek 1999). In that case, the parsec sizesquoted in the previous discussion scale down by a factorof 0.58, so that the range in Figure 4 becomes 0.0009 to0.018 pc. Bate, Webster, & Wyithe (2007) find a value of0.007 pc for the I-band emission region in MG 0414+0534,which is within this range, and the Einstein Radius wouldbe 0.52 ER based on 1 ER = 0.0136. For a source of size0.52 ER, and consulting Figure 4, light curve variability dueto masses down to ∼ × − M ⊙ in image S could be ob-served, at a threshold of 0.1 mag. For an Einstein Radius of0.0236 pc in the mock lens, no variability over this thresholdwould be seen for any mass. We therefore expect that MG0414+0534 will provide more opportunity for determiningsmall mass size, based on nanolensing amplitude, than themock lens system.Chen & Koushiappas (2010) used a uniform disk for thesource profile. A uniform disk with radius R has a half lightradius of 0.71 R, a disk with a Gaussian profile and radiusof R = 3 σ has a half-light radius of 0.28 R. Therefore oursources are equivalent to a uniform disk that is 0.4 the sizeof the size stated here. We have not yet discussed the temporal characteristics ofthe nanolensing variation – whether it is observable, andhow to observe it. In this section we examine these ques-tions, without embarking on a full statistical analysis, whichis reserved for future work. We assume a velocity of thequasar source over the magnification maps of 600 km s − ,which has been used in the past for the mock lens system(Chen & Koushiappas 2010). This gives a time of 38 yearsto cross the Einstein Radius of a solar mass star, and 71days = 0.00012 ER for the Einstein Radius of our smallestmass object (2 . × − M ⊙ ). In a lens where there are manyobjects and a complex magnification map, such as we havehere, we expect events to be seen on timescales smaller thanthis.In what follows we have used horizontal light curvesrelative to all the magnifications maps. The variability isat its greatest, i.e. highest frequency and shortest period,in that direction, because the shear operates in the verticaldirection on the magnification maps, and the caustics areextended in that direction. Sources moving diagonally across c (cid:13) , 1–14 ravitational nanolensing of quasar dark matter Figure 5.
Light curves representing a time period of 100 years (2.6 ER, 0.061 pc at 600 km s − ), cut horizontally through the middleof the maps in Figure 1, with a source of size 0.02 ER (0.0005 pc). The mass of the small objects used to generate the light curve isindicated in the top right of each panel, and each panel corresponds to the map at the same position in Figure 1. the caustics will show a longer time period for nanolensingvariations, compared to a horizontally moving source.We now turn to an analysis of the nanolensing varia-tions. Figures 5 and 6 show light curves taken from the imageM and image S maps in Figure 1 in the horizontal direction,over a time period of 100 years. Figure 5 uses a fixed sourcesize of 0.02 ER and varies the mass of the small objects, sothat each panel represents a different mass, indicated in thetop of each panel. The top row represents the largest mass,2 . × − M ⊙ , the next row is 6 . × − M ⊙ , and so onin the same order reading down the page as in Figure 1. Thebottom row is the smooth matter map with a source size of0.02 ER. Figure 6 uses the maps that have small objects of 2 . × − M ⊙ , but varies the source size, indicated in thetop of each panel. The top row represents a source of sizeof 0.05 ER, and the next row is 0.01 ER, with the bottomrow being the smooth matter map at the pixel resolutionof the maps – 0.002 ER. In both Figures the left column isimage M, the right column is image S. The light curves con-firm the results of the map analysis, that the amplitude ofthe nanolensing variability decreases as the small objects orsource size decreases. However we have the addition of newtemporal information: as the mass of the small objects or thesource size decrease, the frequency of small-scale variationincreases.It appears, by looking, that the nanolensing variability c (cid:13) , 1–14 H. Garsden, N. F. Bate & G. F. Lewis
Figure 6.
Light curves representing a time period of 100 years (2.6 ER, 0.061 pc at 600 km s − ), cut horizontally through the middleof the 2 . × − M ⊙ maps in Figure 1. The left column uses the 2 . × − M ⊙ map for image M, the right column the same mass mapfor image S. Each row represents a different source size, with the size indicated in the top right of each panel. The bottom row is fromthe map generated from smooth matter in place of the 2 . × − M ⊙ masses, and the source size is the pixel resolution of the maps –0.002 ER. is easier to distinguish from the large scale (microlensing)variability in image S compared to image M, because themicrolensing variability is greater in S. To confirm this wefind and compare the nanolensing and microlensing variabil-ity for image M and S. The nanolensing variability is thedifference measure we have been using. The microlensingvariability is the range of magnifications within the smoothmatter map. We present the results for the smallest massobjects, 2 . × − M ⊙ , and a source size of 0.002 ER, themap pixel resolution. The nanolensing variability for these is31% of the microlensing variability, in image M, in terms ofamplitude. For image S, the value is 16%; hence it appearseasier to separate the nanolensing from the microlensing inimage S, at this scale. Naturally this is partly because wehave used a bi-modal mass distribution, and not a continu-ous mass spectrum. We will mention this again, below.The source size used in Figure 5, 0.02 ER, crosses itsown radius in 140 days, which is above the crossing timefor the Einstein Radius of a 2 . × − M ⊙ object (71 days)and therefore probably above the time-scale of nanolensingcaustics. So what is producing the events shown in Figures 5,and in 6? To see this we show in Figure 7(a) the square regionof 100 ×
100 years around the light curve for the map with2 . × − M ⊙ objects, image M, and a source size of 0.05ER (0.0012 pc). These are mass and source sizes that sit inthe middle of our ranges. The “wiggles” in the light curve arecaused not by individual nanolensing caustics but by bandsof densely packed caustics. The nanolensing caustics are notresolvable at the sources sizes and time periods of Figures 6and 5. To highlight this point, we generate a map of 0.2 ERin width, at the same pixel resolution as the maps in Figure 1, and use a source of size 0.0001 ER (2 . × − pc). Asquare segment of this map, of width 1 year (6 . × − pc),is shown in Figure 7(b). At this resolution we can see thenanolensing caustics. Figure 7 (c) shows a light curve cutacross the map in (b). The highest peak in the light curvehas a width of about 8 days (5.7 × − ER), however thereis a small peak just to the left of it which has a width ofonly 2 days. A small source crossing these will produce truenanolensing events . The events are resolvable for this scaleof mass with this scale of source size, due to the fact that thesize of 0.0001 ER is comparable to the Einstein Radius ofthe small objects (0.00012 ER) and smaller than the widthof the caustics: 8 days = 0.00057 ER.The fact that we have so far been looking at causticbands may explain why the nanolensing variability, com-pared to the microlensing variability, appears to be smallerin image S compared to M. The formation of the causticsbands may be on a similar scale in image M and S, whereasthe microlensing caustics are not – they appear to have alarger magnitude range in image S. However, if only truenanolensing and microlensing caustics were compared, wemay find that the relationship of nanolensing to microlens-ing comes out the same for both images. We leave this forfuture investigation.We noted in Section 2.4 that when a map generatedfrom stars+smooth matter is subtracted from the corre-sponding map generated from stars+small masses, the resid-ual magnitudes have a roughly Gaussian distribution. Theresidual magnitudes are the nanolensing variability, sug-gesting that Gaussian noise in microlensing observationswill be indistinguishable from nanolensing. However, sub- c (cid:13) , 1–14 ravitational nanolensing of quasar dark matter Figure 7. (a) shows a square region of width 2.6 ER (100 years) cut from the image M map in Figure 1 that was generated with smallobjects of 2 . × − M ⊙ . The map has been convolved with a source of size 0.05 ER. The region shows caustic bands which producevariability in the light curve (e.g. Figures 5 and 6) due to the presence of the small objects, but do not represent nanolensing caustics,which cannot be resolved at these scales. (b) shows a square region of width 0.026 ER (1 year) cut from a map of width 0.2 ER thatwas generated with small objects of 2 . × − M ⊙ . The map has been convolved with a source of size 0.0001 ER. The region showsindividual nanolensing caustics. (c) contains the light curve cut horizontally through the region in (b). traction of the maps in Figure 1 resolves only the caus-tic bands, which produce the variability seen in Figures 5and 6. If variability due to nanolensing caustics could beresolved, we expect the magnitude distribution to assumea shape more typical of microlensing (Lewis & Irwin 1995;Schechter, Wambsganss, & Lewis 2004). This would make itpossible to distinguish nanolensing from instrument noise.Another method of distinguishing noise be may be to exam-ine the power spectrum of the lightcurves in Figures 5 and6. By inspection, the caustics bands produce a periodicityin the lightcurve that should appear as a peak in a powerspectrum; such a spectrum will distinguish nanolensing fromwhite and other “colours” of noise. A detailed investigationof noise issues will be set aside for future work. As seen in Figures 5 and 6, we find that events have a timescale on the order of a year or less. Figure 8 shows a 10- year long segment of the light curve for the case of the2 . × − M ⊙ masses, source size 0.01 ER, for image M(top row) and image S (bottom row); that is, correspondingto the middle row in Figure 6. The light curves have beensampled at regular intervals of 90 days, and the resultantpoints plotted in crosses on the light curve. This indicatesthat an observing cadence of 90 days is enough to see thenanolensing variability produced by the caustic bands.To find out how often observations need to be madeto observe true nanolensing caustic crossings, we use lightcurves from the map of width 0.2 ER, created with small ob-jects of 2 . × − M ⊙ . We use two source sizes, an accretiondisk of 0.002 ER, and the smallest source size used previ-ously: 0.0001 ER. Light curves are cut horizontally acrossthe map and a representative segment extracted, showingnanolensing events due to nanolensing caustics. These are inFigure 9, where the top row is for image M, the bottom forimage S; the left column is the source of 0.002 ER, the rightcolumn is 0.0001 ER. The time span is 1 year in the left col- c (cid:13) , 1–14 H. Garsden, N. F. Bate & G. F. Lewis
Figure 8.
Segment of a horizontal light curve taken from the maps in Figure 1 for image M and S, that were generated from smallobjects of mass 2 . × − M ⊙ (Figure 1). The top row is from the map for image M, the bottom row, image S. The source size is 0.01ER. The segment shown covers a time period of 10 years. The light curve has been sampled at intervals of 90 days, indicated by thecrosses, showing that such an interval is sufficient to reconstruct the light curve. The line labelled 0.01 ER indicates the time periodrequired to cross that distance, i.e. for the source to cross its own radius. Figure 9.
Segments of light curves taken from maps generated using 2 . × − M ⊙ small objects, with a width of 0.2 ER and a resolutionof 10000 × . The top row represents image M, the bottom row image S. The left column is for a source of size 0.002 ER, theright column for 0.0001 ER. The time period is 1 year for the left column, and 4 weeks for the right column. The crosses indicate samplestaken with an interval of 14 days (left column), or 2 days (right column), showing that observations at these intervals are sufficient toreconstruct the light curve. The lines labelled 0.002 ER and 0.0001 ER indicate the time period required to cross that distance, i.e. forthe source to cross its radius. The right column shows true (albeit smoothed) nanolensing caustic peaks. The left column does not, thepeaks have been smoothed and combined. umn, and 4 weeks in the right column. The light curves aresampled at 14 days (left column) and 2 days (right column)which we suggest is the longest sampling time that will accu-rately reconstruct these light curves. Only the right columnshows true (albeit smoothed) nanolensing caustic crossings– the left column does not. In the left column, nanolens-ing caustics have been smoothed and combined. Thereforethe source of 0.002 ER is still too large to truly resolve thenanolensing caustics – a size of order 0.0001 ER is needed.At 814 nm (271 nm in the rest frame), the pre-dicted flux size of the accretion disk in MG 0414+0534is 0.002 ER = 2 × cm (Mosquera & Kochanek 2011). We assume that the disk is a Shakura-Sunyaev thin disk(Shakura & Sunyaev 1973) where the size of the disk ( R ) isgiven by R = R (cid:16) λλ (cid:17) / (2)where R is the radius at λ .Using this equation and the flux size at 814 nm as areference, we determine that an accretion disk of size 0.0001ER in the mock lens system would be emitting at a wave-length of 25 nm, i.e. the extreme UV. At shorter wave-lengths such as these the emitting region is likely to be c (cid:13) , 1–14 ravitational nanolensing of quasar dark matter highly variable, and such intrinsic variability could inter-fere with observations of nanolensing events. On the otherhand, the time delay between images in MG 0414+0534 is < × M ⊙ for the super-massive black hole(SMBH), which means the smallest innermost stable circu-lar orbit (ISCO), using an extremal co-rotating Kerr blackhole ( R ISCO = GM/c ), is 2.7 × cm = 8.7 × − pc =0.003 ER for the mock lens. This is above the source sizeswe have used in Figure 9.If the Einstein Radius were altered to be the same asthat in MG 0414+0534, the parsec source sizes given previ-ously will scale down by a factor of 0.58. The source velocityin MG 0414+0534 is 270 km s − , and with the different Ein-stein Radius means the light curve durations scale up by afactor of 1.27. In Figure 9 the durations in the left and rightcolumns become 463 days and 35 days respectively, withthe sampling points occurring every 17 days and 2.5 days.The source size for the left column becomes 0.0011 ER andthe right column 0.00006 ER. The observing cadence hasnot altered a great deal, but the source needed to resolvenanolensing caustics is now becoming improbably small. We have shown that high-quality maps generated from verylarge numbers of compact masses can be analysed with sim-ple statistical measures. Using a root-mean-square measure,we show that if the source size is increased the microlens-ing by compact objects and microlensing by smooth matterbecome indistinguishable, but below a threshold it is pos-sible to discriminate between small masses in a lens usingthe amplitude of nanolensing variability. It will require smallsources, of order 0.1 ER, to distinguish very small massesfrom smooth matter at some reasonable level of detectionmagnitude, e.g. 0.1 mag. In future work we will increase thenumber of data points in Figure 4 by generating maps withother small mass values; this is computationally expensivecompared to fleshing out the range of sources, because mapgeneration is more expensive compared to source convolu-tion. However, with new tools (Garsden & Lewis 2010), thismay be feasible. The method also needs to be applied notonly to MG0414+0534, which will require billions of lensmasses, but to other lensed quasars. Rather than using a bi-modal mass function, other distributions such as Salpetercan be tested, and the percentage of small to large objects(Pooley et al. 2009) can be varied.It will also be necessary to obtain a statistical varianceof the results by generating maps that are identical to thosein Figure 1 but with different randomly-generated locationsfor the objects. These maps can be passed through the anal-ysis and the results compared to those presented here. Aslight difference is expected as the large-scale caustics in thecompact matter maps may shift slightly due to a differentfield of small objects surrounding them. However, these dif-ferences should not be significant. The analysis of the light curves show that there arethree scales or levels of lensing in this study – microlens-ing due to the 1M ⊙ stars, variability due to the bands ofnanolensing caustics, and peaks due to the nanolensing caus-tics themselves. Further work needs to be done to determinethe scale at which the nanolensing caustics resolve out of thecaustic bands, and if, at the different scales, and in differentimages, the amplitude and duration of nanolensing variabil-ity remains the same. It is possible that such analyses willprovide tighter constraints on the masses within the lens,rather than the general analysis here.Nanolensing events are very small. Even sources assmall as the optical accretion disk of MG 0414+0534 willnot be enough to resolve individual nanolensing caustics ac-curately, and smaller regions will be necessary to do this.Such regions are less than the expected ISCO of the SMBH,but if they exist, and are not too intrinsically variable, anobserving cadence of 2 days will resolve nanolensing caustics.An observing proposal could be made on this basis, eitherusing the HST or Magellan telescopes, for observations ofnanolensing events in a quasar such as MG 0414+0534. ACKNOWLEDGMENTS
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