Gravitational microlensing as a probe for dark matter clumps

0.97 MB·

by  E. Fedorova

Mon.Not.R.Astron.Soc.000,1–14(2014) Printed28January2016 (MNLATEXstylefilev2.2)
Gravitational microlensing as a probe for dark matter
clumps
6 E. Fedorova1,2⋆, V.M. Sliusar2, V.I. Zhdanov2†,
1
0 A.N. Alexandrov2, A. Del Popolo3,4,5, J. Surdej1.
2

n 1 Institute of Astrophysics and Geophysics of Li`ege University,QuartierAgora All´ee du 6 aouˆt 19C, B-4000 Li`ege, Belgium
a 2 Astronomical Observatory of Taras Shevchenko National University of Kyiv,Observatorna 3, Kiev,04053, Ukraine
J 3 Dipartimento di Fisica e Astronomia, Universityof Catania, Viale A.Doria 6, 95125, Catania, Italy
7 4 INFN sezione di Catania, via S.Sofia, 95123 Catania, Italy
2 5 International instituteof physics, Universidade Federal do Rio Grande do Note, 59012-970, Natal, Brazil

]
A
G
.
h
ABSTRACT
p
Extended dark matter (DM) substructures may play the role of microlenses in the
-
o Milky Way and in extragalactic gravitational lens systems (GLSs). We compare mi-
r crolensingeffects causedby pointmasses (Schwarzschildlenses)andextended clumps
t
s ofmatterusingasimplemodelforthelensmapping.Asuperpositionofthepointmass
a and the extended clump is alsoconsidered.For special choicesof the parameters,this
[
model may represent a cusped clump of cold DM, a cored clump of self-interacting

1 darkmatter(SIDM)oranultracompactminihaloofDMsurroundingamassivepoint-
v like object. We built the resulting micro-amplification curves for various parameters
8 of one clump moving with respect to the source in order to estimate differences be-
2 tween the light curves caused by clumps and by point lenses. The results show that
4 it may be difficult to distinguish between these models. However, some region of the
7
clump parameters can be restricted by considering the high amplification events at
0
the present level of photometric accuracy.Then we estimate the statistical properties
.
1 of the amplification curves in extragalactic GLSs. For this purpose, an ensemble of
0 amplificationcurvesis generatedyieldingthe autocorrelationfunctions (ACFs)ofthe
6 curves for different choices of the system parameters. We find that there can be a
1 significantdifference betweenthese ACFs ifthe clumpsize is comparablewith typical
:
v Einstein radii; as a rule, the contribution of clumps makes the ACFs less steep.
i
X Key words: astrophysics,cosmology – Gravitational microlensing, dark matter.
r
a
1 INTRODUCTION of 26.8% non-baryonic dark matter (DM) and 68.3% dark
energy(representedbytheΛterm),andjust4.9% ordinary
Since the beginning of the last century, when some as-
matter (Adeet al. 2013).In this context, therole of Gravi-
tronomersstartedtostudythemattercontentofourneigh-
tationalLensing(GL)canhardlybeoverestimated.GLpro-
borhood(O¨pik1932;Kapteyn1922;Jeans1923;Oort1932;
videsimportantevidencesinfavorofDMexistenceingalac-
Zwicky 1933), many evidences have been collected leading
ticclusters(Markevitch et al.2003;Clowe et al.2004,2006;
one to believe that the Universe in which we live is mainly
Bradaˇc et al. 2006). The other application of GL, following
constituted by non-luminous matter, whose existence is in-
the idea by Paczyn´ski (1986b), deals with the searches for
ferred through its gravitational effects on the remaining
compact objects in the Galactic halo and inside the Milky
constituents of the Universe. This “missing matter”, was
Way(Alcock et al.1993;Udalski et al.1993;Aubourg et al.
dubbed“dunklematerie” (dark matter), by Zwicky (1933).
1993). GL effects over a wide range of lens masses can give
Nowadays, we know, according to Plank’s data mission fit-
usthepossibility of analyzingtheDMsubstructurecharac-
ted with the ΛCDM model, that the Universe is composed
teristics; especially it can give us a clue to solve the “miss-
ingsatellite”problem(Klypin et al.1999;Moore et al.1999;
DelPopolo et al.2014).Hereweanalyzethepossibilities to
⋆ [email protected]
† [email protected]
(cid:13)c 2014RAS
2 E. Fedorova, V.M. Sliusar, V.I. Zhdanov, A.N. Alexandrov, A. Del Popolo, J. Surdej
study some properties of DM using mainly photometric in- massobjectscanmimicthelight curvescausedbyordinary
formation induced by gravitational microlensing effects. compact objects. Therefore, more detailed investigation is
Following the terminology adopted by the GL commu- needed in order to rule out (or to confirm) the existence of
nity, at least three kinds of GL phenomena, characterized theseDMclumpsthatmayeithernotbeverynumerous(so
by different lens masses and typical timescales, exist: that we do not have these clumps within our Solar system)
or not so denseto be observable.
(i) macrolensing and the weak lensing by galaxies or
Photometric signatures of DM substructure via grav-
groups of galaxies;
itational microlensing have been widely discussed; see
(ii) mesolensing:lensesareglobularclusters,dwarfgalax-
Mao et al. (2004); McKean et al. (2007); Oguri (2005) and
ies or DM clusters with mass in the range 103 to 109 M⊙ references therein. There are also some investigations of
(Baryshev & Bukhmastova1997);
variability of spectral line profiles (Metcalf et al. 2003).
(iii) microlensing: lenses are stellar-mass objects.
Paczyn´ski & Wambsganss (1989) derived amplification dis-
As distinct from (i) and (ii) dealing with almost tributionsforstaticgravitationalmacro-lensingwithanon-
static situations or very slow processes, the characteristic constant surface mass density, which includes the cases
timescales of microlensing events in the Milky Way are of of a stochastic system of Gaussian clumps and clumps
the order of weeks and in case of extragalactic microlens- in the form of truncated singular isothermal spheres.
ing events, they are of the order of months or years. This Zakharov & Sazhin (1999), and Zakharov (2010) have con-
makes image brightness variation effects due to (iii) quite sidered a model of microlensing due to a non-compact neu-
observable. tralino star.
Basedonthelensmassrange/timescales,thisterminol- InthispaperweuseseveralothermodelsofDMclumps
ogy determines at the same time the possible observational toanalyzetheobservationalappearanceofDMmicrolensing
appearance of these effects, i.e. multiple correlated static within different cosmological models. We focus our atten-
images with signal arrival time delays in the macrolensing tion on item (iii) tocomparetheobservational appearances
case, slowly-varying distorted images due to mesolensing, of microlensed sources due to point-like and finite-size DM
andhighamplificationeventsonlightcurveswhenthesource clump deflectors and determinehow one can distinguish an
crossesthecausticofamicrolens.Darkmattercanmanifest extendedDMclumpmicrolensingeventfroma“regular”one
itself at thethree mentioned levels, namely: in case of Galactic and extragalactic structures. The main
questionwewanttoanswerconcernsthepossibilities tode-
(i) largehaloesofDMwithmassgreaterthan109M⊙can tect signals from these putative extended mass structures
playtheroleofmacrolenses;themostprominentexampleis
and to estimate the accuracy needed to characterize such
the”Bullet Cluster”(Markevitch et al. 2003);
effects mainly from observed light curves.
(ii) DM subhaloes of intermediate masses can play the
In Section 2, we briefly describe theexisting DMmod-
role of mesolenses, causing the anomalous flux ratios, im-
els.InSection3,weproposea”toymodel”,whichdescribes,
age distortions (Chen et al. 2007) and additional time de-
forspecialchoicesoftheparameters,microlensingbyvarious
lays (Keeton & Moustakas 2009) in extragalactic gravita-
typesofextendedstructures.Themodelisbasedonthede-
tional lens systems. Several candidates to show anomalous flectionangleα r(r2+r2)−a/2,wherer characterizesthe
fluxratiosareknowntoday,e.g.,B2045+265(McKean et al. effectivesizeofa∼clumpcorse1.Fordifferentschoicesofparam-
2007), RX J1131–1231, B1608+656, WFI 2033 – 4723
eters, this model reproduces the effects either due to point
(Congdon et al. 2010), B1938+666 (MacLeod et al. 2012),
microlenses,orCDMminihaloes withacuspeddensitypro-
and we would especially like to stress here the famous GLS
file,orSIDMclumpswithacoreddensityprofile.InSection
Q2237+0305 for its sharp high-amplification microlensing
4wediscuss thesuperpositions ofextendedand point-mass
events(Metcalf et al. 2003).
deflecting objects. In the case of isolated microlenses these
(iii) Clumps of DM with stellar masses can play the role
models can be used to describe microlensing events within
ofmicrolenses,leadingtobothphotometric(highamplifica-
theGalaxy.
tion events) and astrometric (jump-like shifts of brightness
Weperformanumericalcomparisonoftheseeventscon-
centroid of the images of the microlensed source) appear-
sideringdifferentmicrolensmodels.Namely,wecalculatethe
ances. The main difference between DM microlensing and
”amplification curves”, that is the dependence of the total
the ”usual” one due to stars or black holes lies in the non-
amplification of a microlensing system versus time, as the
negligible size of the DM clumps (non-DM microlenses are
lens moves with respect to the line-of-sight to the source.
always considered as point masses).
Wealsopaysomeattentiontothetrajectoriesofthebright-
In this paper, we pay attention to the last item, i.e. ness centroid of the microlensed images, because the fast
microlensingbyextendedclumpsofstellarmasses.Wenote progress in the positional accuracy measurements2 gives us
that continuous observations by EROS, OGLE and other a hope to detect theastrometric signatures of gravitational
groups(Tisserand et al.2007;Wyrzykowskiet al.2009)nei- microlensing in thenear future.
therrevealedanysignoftheseextendedclumpsintheGalac-
tichalo,norprovidedanyproofofsuchstructuresinsidethe
1 Thecorresponding surface mass density is ρ∼(1−a/2)(r2+
Galaxy. The overwhelming majority of the light curves ob-
r2)−a/2+(a/2)r2(r2+r2)−(1+a/2)
servedforGalacticmicrolensingeventsarewelldescribedby s s s
2 e.g.,projectsofradiointerferometryinspace.Notethatastro-
thegravitationalfieldofstars(and,sometimes,ofplanetary
metricaccuracy istypicallyseveral times largerthan the resolu-
massobjects). Therefore, it seemsthatthereisnoroom for tion; i.e. one can achieve a microarcsecond level in positioning,
thestellarmassextendedclumps.However,inthispaperwe though different images of the microlensed source cannot be re-
showthatthelightcurvesduetoextendedclumpsofstellar solvedatthislevel.
(cid:13)c 2014RAS,MNRAS000,1–14
Gravitational microlensing as a probe for dark matter clumps 3
In Section 5 we consider the observational behav-
Table1.DarkmatterhierarchywithintheCDM/SIDMcosmolo-
ior of clumps in extragalactic GLSs. Namely, we study gies.
the statistical effects of gravitational microlensing due to
a stochastic system composed of extended DM clumps Term Objects/scales Massrange
and point masses. There is a number of papers dealing
with statistical subjects of microlensing systems; see, e.g., Superhalo Galaxy clusters, 100 1013-1016M⊙
Mpc
Schneideret al. (1992);Seitzet al.(1994);Neindorf(2003);
Dobler et al. (2007); Schmidt & Wambsganss (2010). Our Halo Galaxies,Mpc 109-1011M⊙
technique is essentially the same as that used by
Paczyn´ski & Wambsganss (1989) and Wambsganss et al. Subhalo Dwarf galaxies or 104-109M⊙
satellites,kpc
(1990), followed by a number of authors; cf. especially
considerations by Metcalf & Madau (2001); Chiba (2002); Minihalo or Stars 10−3-10M⊙
Dalal & Kochanek(2002);Schechter& Wambsganss(2002) clump
dealingwithbrightnessvariationsinconnectiontotheprob-
Microhalo Planets 10−8-
lem of anomalous brightness ratios. Unlike in the previous 10−4M⊙
works we deal with the autocorrelation functions (ACFs)
of the amplification curves for our concrete model of a mi- Nanohalo or 10−18-
crolensingsystem.Wegenerateanensembleofamplification primordial 10−9M⊙
halo
curvesforafixedinputofrandomlydistributedclumpsand
point mass microlenses in the total optical depth; this en-
ables us to derive ACFs for these amplification curves as a
function of different clump contribution, size and in pres- WDM (Berezinsky et al. 2008, 2013; Diemand et al. 2005;
ence of an external shear. Finally, in Section 6 we discuss Springel et al. 2008). To clear out the situation and the
theresults. terminology used to characterize masses and scales of DM
substructures, we have summarized here in Table 1 the
used nomenclature.
DMsubstructuredependsverysignificantlyonthecos-
2 PROPERTIES OF VARIOUS DM MODELS
mologicalmodel.WithinCDMmodels,thesubstructurefor-
DM microstructure still remains the subject of de- mationprocesscanbedescribedas“bottom-up”:thesmall-
bates; there are many hypotheses about DM particles est DM clumps were formed first (therefore the nanohaloes
(Schneider& Weiss 1987; Zackrisson & Riehm 2010). Cold with masses < 10−11M⊙ are often referred as primordial)
dark matter is supposed to contain heavy particles, weakly and do not contain any finer substructure (Contini et al.
interacting with each other and with baryonic matter 2011). Then, in the merging process more and more mas-
(axions or WIMPs), and warm dark matter (WDM) sive structures up to galactic and cluster scales (1010-1015
consists of light, fast-moving particles (sterile neutrinos, M⊙) were gradually formed. Duetovariousdamping scales
CHAMPs, neutralinos or gravitinos), which tend less to the lower limits on substructure mass can vary over a wide
form small-scale compact structures (Schneideret al. 2012; range:from10−4-10−12 M⊙ forvariouskindsofWIMPs(in-
Del Popolo 2014). Self-interacting dark matter can be con- cluding neutralinos), down to 10−18-10−20 M⊙ for axions.
sidered as a particular kind of cold one with a nonzero The mass density distribution in DM structures within the
impact distance for the interaction between DM particles CDM model is often assumed to follow the cusped NFW
(Kormendy & Freeman 2003; Rocha et al. 2012). profile of Navarro et al. (1996).
Recent cosmological N-body simulations within ThebestacceptedalternativetotheCDMcosmologyis
CDM (Schneideret al. 2010; Diemand et al. 2004, 2005; thevelocity-dependentSIDM(self-interacting-dark-matter).
Springel et al. 2008; Stadel et al. 2009; Vogelsberger et al. Thismodelpredictsidentical behaviorof theDMsubstruc-
2012)andWDMmodels(Knebeet al.2008;Schneideret al. ture over various scales and identical mass fractions, thus,
2012) have shown that dark matter is not distributed in theonlydifferenceliesinthedensityprofileofaDMclump.
spacehomogeneously,insteaditformsmoreorlesscompact TheSIDMclumphasacoreanditsdensityfollows aBurk-
structuressurroundedbylessdensecontinuouslydistributed ert profile (Rocha et al. 2012), or pseudo-isothermal sphere
DM. The most massive structures havebeen recognized for (Kormendy& Freeman 2003), contrarily to thecuspy core-
quite a long time already: these are galactic and galaxy less NFW profiletypical for CDM clumps.
cluster DM haloes. But numerical simulations demonstrate Thesetofwarm darkmattermodelsdifferscompletely
that these massive structures are also inhomogeneous: they fromthetwomentionedabovewhenwekeepinmindgravi-
contain smaller compact substructures over a wide range tationalmicrolensingprocesses.Thesubstructureformation
of masses. Such massive haloes containing a hierarchical in WDM models is a complex hybrid; its leading role is
substructure of dark matter are called host haloes. These played by the “top-down” formation process (Knebeet al.
haloesalsocontainsmallersubstructures,etc.However,one 2008),i.e.biggerhaloesandfilamentswereformedfirst,and
should note that only the existence of the more massive smaller structures formed later, as a result of fragmenta-
membersofthishierarchyofDMstructureshasbeenproven tion processes in WDM filaments (Knebeet al. 2003). The
from the results of observations. The lower estimated limit lowerlimitonthesubhalomassissignificantlyhigherthanin
on the substructure masses depends on the particular kind CDM/SIDMones:e.g.,106 108M⊙forgravitinosandeven
−
of considered hypothetical DM particles, varying over a 1011M⊙ for axinos (Hisano et al. 2006). Thus in the WDM
very wide range from 10−12M⊙ for CDM up to 108M⊙ for cosmology,wemaypredictthenonexistenceofDMgravita-
(cid:13)c 2014RAS,MNRAS000,1–14
4 E. Fedorova, V.M. Sliusar, V.I. Zhdanov, A.N. Alexandrov, A. Del Popolo, J. Surdej
tionalmicrolensing.Evendespitethatitwasrecentlyshown like an eclipse of the source image. The role of optical
(Paduroiu et al. 2015) that the WDM structure formation depth/convergence in microlensing processes was analyzed
ratherfollowsthehybridscenariothanthe“top-down”one, by Paczyn´ski (1986a). In the simplest case such an object
the effects of fragmentation and collapse play a significant canbemodeledupbyaSchwarzschildorChang-Refsdallens
role only in massive haloes formation, and thus the situa- with overcritical convergence.
tionwithdarkmatterinducedgravitationalmicrolensingin
WDMcosmology is not significantly altered.
TheDMsubhalomassfunction(SHMF)n(m,M0),such 3 EXTENDED CLUMP MICROLENSING IN
that n(m,M0)dm represents thespace numberof substruc- THE GALAXY
tures with a mass in the range m,m+dm in the host
{ } 3.1 The lens equation
DM halo with a mass M is strongly sensitive to the DM
0
context (i.e. warm, cold, collisionless, repulsive and so on). Speaking about the Milky Way systems, we confine our-
The SHMF appears to provide the essential source of cru- selves to the simplest cases of circular symmetric mass dis-
cial information both in cosmology and elementary par- tributionsthoughinvestigationsofdoublestarsorplanetary
ticle physics. The numerical simulations like “Millenium” systems are also important. One can get some insight into
(Springel et al. 2005) or “Via Lactea” (Contini et al. 2011) thelattercasefromconsiderationsofclumpmodelswithan
allow us to determine it only at higher subhalo masses external shear (Sliusar et al. 2015). In this Section, we use
(mainly > 106M⊙) than typical microlens masses (i.e. not the lens equation in its normalized form; the distances will
greater than100M⊙).Howeverfor lower masses of thesub- be expressed in units of a typical length scale R∗, where
structure we can use here the extrapolated mass function for the case of the point mass M or cored clump microlens
obtained byLee et al. (2009): R2 = 4GMD /(c2D D ) (in case of a point mass this is
∗ ds d s
the radius of the Einstein ring) and for the cusped clump
log(M /m)
f(m)=0.1 max Ra = 8πGρ D /[c2D D (2 a)]; here D is the distance
log(Mmax/Mch) ∗ 0 ds d s − s
between the source and the observer, D the distance be-
d
where Mmax =0.01M⊙ and Mch =107M⊙. Thus, the total tween the deflector and the observer, and Dds the distance
massinsubstructureswithintheCDMmodel(withalower between the source and the deflector. Furthermore y is the
limit of 10−6 M⊙) is 52%, following Lee et al. (2009). This normalized angular sourceposition, r= x1,x2 is thenor-
{ }
value is in good agreement with the 50% of dark matter malized angular image position, r = r. We remind here
| |
in theGalactic halo in substructuresobtained from thehy- that (after normalization) both these vectors are defined in
drodynamical simulations by Diemand et al. (2004). Using thelens plane.
this formula, one can easily find that for the lower limit of The normalized lens equation is:
10−3M⊙ (which can beconsidered as a reasonable limit for
y=(1 σ)r α(r), (1)
microlensmass)itis43%,andfor10M⊙(letusconsiderthis − −
value as an upper limit on microlens mass) it is 30%. Thus where thedeflection angle
wtuirtehsinwtehheadvees1ir3a%bloefrtahnegteo(t1a0l−D3M−m10a)ssM(⊙i.eo.f∼th1e.3s·u1b0s1t1rMuc⊙- α(r)= rr2 rdr′r′ρ(r′) (2)
is in compact stellar-mass subhaloes). If we take into ac- Z0
count thefact that the larger subhaloes (with mass greater issupposedtobenormalizedtoacorrespondinglengthscale,
than 10M⊙)alsocontainfinersubstructures,thisvaluecan σ=conststandsforaconvergence(opticaldepth)thatcan
beeven larger. bedueto, e.g., a background DM; y= y1,y2 .
{ }
However,estimatingtheprobabilityoftheDMsubhalo Let us write α(r) = r/ra, where for the cusped clump
microlensing we should take into account that up to 90% we assume a<2 to provide the convergence in the integral
of theprimordial subhaloes had to befully or partially dis- of Eq.(2), and for the point microlens a= 2. For the cored
rupted by tidal forces of stars, thus the percentage of the clumpweuseageneralizedexpressionα(r)=r/(r2+r2)a/2
s
clumped matter in the areas containing stars cannot ex- that formally covers the previous cases if we put rs = 0.
ceed 10% (Schneideret al 2011; Schaw et al. 2007). Thus Therefore, we consider thelens equation
we can expect to trace the DM-induced high amplification
y=r 1 σ R−a , R= r2+r2, a>0. (3)
microlensing events in the Galactic DM halo (as well as in − − s
extragalacticsystems)ratherthaninitsluminousparts.The T(cid:0)he determina(cid:1)nt of theplens mapping (3) is D =
probability of the Galaxy stars being microlensed by a DM det ∂y /∂x . It factorizes as follows:
i j
{ }
microhalo is much lower then. That is why we can hardly
D(r)=ϕ (r)ϕ (r), (4)
expect to find compact DM clumps in the vicinity of the 1 2
Solar system. where
However, at the same time, in the luminous parts of a 1 a 1 ar2
galaxywecanfacethesituationwhenastar(orastar-mass ϕ1(r)=1−σ− Ra, ϕ2(r)=1−σ+ R−a − Ra+s2. (5)
black hole) is surrounded bya densecloud, formed by dark
The image parity is defined bythe sign of D.
matterofaformerclump,disruptedbystar/blackholetidal
The relative amplification3 of the image at r is µ(r) =
force and trapped by its gravity. If such a clump is dense
(1 σ)2/D(r).
enough and thus has an optical depth large enough (over- − | |
critical convergence), we can observe the UCMH (Zhang
2011; Berezinsky et al. 2013), and one of the possible re- 3 thisisnormalizedtotheamplification(1−σ)2 whenthesource
sults of gravitational microlensing by such an object looks isfarfromthelens.
(cid:13)c 2014RAS,MNRAS000,1–14
Gravitational microlensing as a probe for dark matter clumps 5
Forthesecond factor ϕ (r)in Eq. (4) wealso havethe
2
|f(r)| condition4 1 σ <r−a (i.e. r <r ) for the existence of
− s s s,cr
r <r such that ϕ (r )=0. Indeed, under this condition,
c z 2 c
ϕ (0)=1 σ r−a <0andϕ (r )>0havedifferentsigns.
2 − − s 2 z
f Thisisanecessaryandsufficientconditionforauniquevalue
ofr toexiston(0,r ).Theproofoftheuniquenessissome-
c z
what different for 0 < a < 1 and a > 1. For 0 < a < 1 the
function ϕ (r) is monotonically increasing. For a > 1 one
2
e should use Eq. (7) and take into account the existence of

the unique inflection point for f: we observe that ϕ (r) is
2
monotonically increasing for r < r (therefore there can
infl
d c be only one zero in this region provided that rs < rs,cr);
b after passing through a maximum at r = r it decreases
a infl
and it is positive: ϕ (r)>ϕ ( )=1 σ>0.
2 2
∞ −
Itiseasytoseethatr <r ,whenitexists,istheradius
c z
of a circular critical curve that is mapped onto a circular
r
caustic with radius y = f(r ). There exist two additional
c | c |
images of a point source for the case y < y . The problem
c
can be easily treated using the graph of f(r) (Fig. 1, b,c)
Figure 1. Qualitative behavior of the r.h.s. of Eq. (6) for the | |
subcriticalcase(06σ<1:solid;a,b,c)andtheovercriticalcase andtakingintoaccountthefactthatd|f|/dr iseitherequal
(σ > 1: dashed; d,e,f). Here we show the subcritical cases (a) to ϕ2ortoϕ2.Forrs>rs,cr thefunction f(r) startingat
− | |
a > 1,rs = 0, (b) 0 < rs < rs,cr, (c) rs > rs,cr; in case of r=0ismonotonically increasing,therefore,foranyythere
(b)and(c)theremaybeaninflectionpointthatdoes notaffect is a unique solution r(y) of Eq. (6). The (unique) solution
the number of images. The overcritical cases: (d) 0<a<1; (e) of Eq. (3) is r=nyr(y) havingpositive parity.
a>1,rs<rs,cr1;(f)rs=0,a>1. For rs <rs,cr we have f(r)<0, r (0,rz). The func-
∈
tion f(r) has only one maximum y = f(r ). For r > r
c c z
| | | |
the r.h.s. of (Eq. 6) is a monotonically increasing function.
Taking theabsolute valueof both sides of Eq.(3) yields Therefore, for 0 < y < y there are three solutions for
c
Eq. (6): two solutions r = r (0,r ),i = 1,2; r < r ,
y= f(r) , (6) i ∈ z 1 2
| | which yield two solutions of the vectorial lens equation (3)
wheref(r)≡r[1−σ−(r2+rs2)−a/2], y=|y|.Evidently,if r=−nyri(r1 withpositiveparity,r2 withnegativeparity);
risasolutionofEq.(6),thenthesolutionofEq.(3)iseither andonesolutionr3 >rz yieldingthesolutionr=nyr3with
r=nyr or r= nyr, where ny =y/y. positive parity for Eq.(3).
−
Our first goal will be to estimate the number of solu- To sum up, for 0 6 σ < 1 we have a unique image for
tions of Eq.(3), i.e. the number of microlensed images for thecaser >r ;inaddition,fory=0thereisanimage5
s s,cr
different source positions, and to find the caustics of the atr=0.Allimageshaveafinitebrightness.Intheopposite
lens mapping (3) where the lensed image of a point-source case,for0<r <r thereisacircularcausticwithradius
s s,cr
gets infinitely amplified. The problem is reduced to inves- y=y andtherearethreeimages,ifthesourceisinsidethis
c
tigate the monotonicity of the r.h.s. of Eq. (3) and the ze- caustic 0 < y < y , and one image, if y > y . Two images
c c
ros of the determinant D(r) in Eq. (4). Note that any root acquire an infinite amplification when y y 0 and then
c
→ −
of ϕ2(r), if it exists, is smaller than the root of ϕ1(r) (for disappear after crossing yc. In addition, there is a caustic
a > 0) because of the monotonicity of ϕ1(r) and due to point y=0 corresponding to a ring image like the Einstein
ϕ2(r)=ϕ1(r)+ar2/Ra+2. ring emerging in case of a point mass lens.
Simple calculations yield Thecausticswith sourcetracksandcorrespondingam-
plification curvesare shown in Fig.2.
df(r) dϕ (r) ar
=ϕ (r), 2 = (a 1)r2 3r2 , (7)
dr 2 dr −Ra+4 − − s
whence for a >1, rs >0 we infer t(cid:2)he existence of t(cid:3)he only 3.3 Cored clump, overcritical σ
inflection point r =r 3/(a 1) of the curvey =f(r)
infl s
and there is no inflection for a<−1. For σ >1 (rs =0), let us first consider the case 0<a<1.
p Inthiscasein6equation(6) f(r) r[σ 1+(r2+r2)−a/2]
| |≡ − s
isamonotonically increasingfunction(seeFig.1,d,dashed
3.2 Cored clump, subcritical σ curve). Its derivative is df(r)/dr = ϕ2(r) > σ 1 > 0.
| | − −
The lens mapping has no caustics and critical curves; for
Let us first consider the case r = 0 and the subcritical
valuesofσ:06σ<1.Furtherwesd6enoter =(1 σ)−1/a. any y there is always a unique solution r for Eq.(6) and a
s,cr − uniquesolution r= nyr (positive parity)for thevectorial
Forthecaser <r ,thefirstfactorϕ (r)ofther.h.s. −
s s,cr 1 lens equation (3).
in Eq. (4) equals zero for r = r [(1 σ)−2/a r2]1/2.
z ≡ − − s
Thiscorrespondstoacircularcriticalcurvewitharadiusr
z
aroundtheorigin oftheimageplane,whichismappedonto 4 Thisconditionisformallythesameastheonefortheexistence
an isolated caustic point y = 0 in the source plane. Thus,
ofarootofϕ1(r)
thereisasolution forEq.(3)with y=0.Forrs>rs,cr the 5 for completeness, we note that the trivial solution r = 0 for
function ϕ1(r) is always positive. y=0exists,ifrs6=0,∀a>0,andifrs=0,a<1.
(cid:13)c 2014RAS,MNRAS000,1–14
6 E. Fedorova, V.M. Sliusar, V.I. Zhdanov, A.N. Alexandrov, A. Del Popolo, J. Surdej
B increases for r < r = [(1 a)/(1 σ)]1/a and then de-
max
− −
creases to zero for r r . The value y = f(r ) =
z1 c max
a[(1 a)/(1 σ)]1/a−→1 is the radius of a circu|lar caus|tic.
− −
The behavior of the graph of this function is roughly the
sameasthatillustratedbythesolidcurveinFig.1,b.There
n arethreesolutions forEq.(6)with 0<y<y :namely,two
o c
ti solutions ri, i = 1,2, r1 < r2 for Eq. (6), ri < rz1 corre-
a
c sponding to images at r = nyri (with positive (r1) and
fi negative (r ) parity, and one−solution r > r correspond-
mpli ing to imag2e at r = nyr3 with positive3 parizt1y. The latter
A solution remains valid for y>yc.
For 0<σ <1, a>1, the function f(r) is negative for
r (0,r ),themodulus f(r) decreasesalongthisinterval
z1
∈ | |
and it increases for r > r . The behavior of f(r) is rep-
z1
| |
resented by the curve (a) in Fig. 1. In this case there are
alwaystwosolutionsofEq.(6)forr <r correspondingto
1 2
Time images at points r= nyr1, r=nyr2.
−
Figure 2. Qualitative behavior of the amplification curves, the For σ > 1 and 0 < a < 1 the function f(r) increases
| |
subcritical case, 0 < rs < rcr. The upper curve (dashed): the for all values of r > 0 starting from zero (like the dashed
point source moves along the straight line with the impact pa- curve(d)inFig.1).Thereisonlyonesolutionandoneimage
rameter y < yc yielding two caustic crossings. The lower curve for all y values.
(solid): larger y > yc, no caustic crossings. In the right upper For σ > 1 and a > 1 the function f(r) shows a
corner:thetrajectorieswithrespecttothecircularcausticofthe minimum at r = [(a 1)/(σ 1)]1/a; th|e m|inimum is
lens. y = f(r ) m=ina[(σ 1−)(a 1)]−1−1/a (theradius of a cir-
c min
| | − −
cular caustic). There are two solutions for Eq. (6) if y >y
c
For σ > 1, a > 1 there is an inflection point of the and no solutions if y < y . This case is represented by the
c
function f(r) f(r) according to Eq. (7). Taking into dashed curve(f) in Fig. 1.
| | ≡ −
accountthesignofthisinflection,weseethatthereexistsa
minimum for df(r)/dr at r=r , which is equalto
infl
| |
2 a 1 a/2+1 3.5 Analytical solutions
f′ =σ 1 − .
min − − rsa a+2 Inthissubsection,welistseveralsituationswheneverasim-
(cid:16) (cid:17)
Then new roots of df(r)/dr appear when pleanalytical treatment is possible.
| |
The point mass lens model (Schwarzschild lens) is well
2 1/a a 1 (a+2)/(2a)
rs<rs,cr1= σ 1 a−+2 . kfonrotwhne,iimtcaogrerepsopsoitnidonsstoarae=r=2,yrs(y=0,σy=2+0.4T)/h2eysoalnudtiothnes
Ifr >r ,(cid:16)the−func(cid:17)tion(cid:16)f(r) i(cid:17)smonotonicallyincreasing total amplification of the two lense±d images is µ = (y2 +
s s,cr1 | | p
(not shown in Fig. 1). Let these new roots be rc,1 and rc,2 2)/(y y2+4).
with rc,1 < rc,2, rc,1 being a point of a local maximum For a single SIDM cored clump, let us assume a = 2,
p
of f(r) and rc,2 being a point of a local minimum (see σ = 0: critical curves for such a system exist only when
| |
Fig. 1, e, dashed). The roots correspond to the radii of the r < 1. Under this condition, two circular critical curves
s
critical curves, and |f(rc,1)|,|f(rc,2)| are the radii of the exist. The first one has a radius rz = √1−rs2 correspond-
circular caustics. For f(rc,1) < y < f(rc,2)) Eq.(6) has ing to the caustic point y = 0 (the same as for the point-
| | | |
threesolutionsr1<r2<r3correspondinglywithapositive, mass lens), and the second one has a radius rc such that
negativeandpositiveparityyieldingthreelensedimagesofa r2 = 1 √1+8r2 1 r2 < r2 (this expression is posi-
c 2 s− − s z
pointsource,andthereisonlyoneimagewithpositiveparity tive for r < 1), corresponding to a caustic with a radius
s
otherwise. The solution r1(y) is extended for y < |f(rc,1)| yc =rc (cid:0)1−rc2−rs2 /(cid:1)rc2+rs2 ). Incase of a cored clump
and the corresponding image has an infinite amplification microlens,thelensequationyields:r3 yr2+r(r2 1) yr2 =
when y f(rc,1) 0; in this case r1(y)and r2(y) tendto 0. The r(cid:0)oots can be(cid:1)fo(cid:0)und usin(cid:1)g the∓Cardanosm−eth∓od.sFor
→| |−
rc,1 and then disappear after y crosses the value f(rc,1). the lower plus, two roots must be taken in the interval
| |
Thesolution r3(y)isextendedtoall valuesof y>|f(rc,2)|; 0 < r < rz for y < yc, and for the upper minus, one must
it has analogous properties near the other caustic for y take the root for r > r . If r > 1 (low-density clump), no
→ z s
f(rc,2) +0. In this case r2(y) and r3(y) tend to rc,2 and caustic exists and only one lensed image can be seen.
| |
then disappear as y decreases. We have for the solutions of FortheUCMHmodel,itisinterestingtoconsideranan-
Eq.(3), r=−nyri, i=1,2,3. alyticsolutionfortheovercriticalcaseσ>1(rs=0,a=2).
In this case there exists a critical curve r = 1/√σ 1
c
−
and the radius of the corresponding caustic is described
3.4 Cusped clump: r =0
s by y = 2√σ 1. Two images for y > y are produced
c c
Here we have f(r) = r(1 σ r−a). For 0 < σ < 1, the at the position−s: r = y[y y2 4(σ 1)]/[2y(σ 1)].
functionf(r)<0forr<r− =−(1 σ)−1/a;r istheradius The total amplification−of t±hese tw−o lense−d images is−µ =
z1 z1
of a critical curvecorresponding t−o thecaustic point y=0. [y2 2(σ 1)]/[y y2 4(σ p1)]. For y <y there are no
c
− − − −
For 0 < σ < 1 and 0 < a < 1, the modulus f(r) images and we have an ”occultation”. The lightcurves cor-
| | p
(cid:13)c 2014RAS,MNRAS000,1–14
Gravitational microlensing as a probe for dark matter clumps 7
respondingtothiseffect,bothforpoint-likeandcontinuous thesituation(discussedbelow)whenanextendedclumpsur-
sources, were shown in our previous work (Fedorova et al. rounds the star. Correspondingly, at this stage we concen-
2014). trate on the most simple models of extended microlenses;
For the case of the cusped lens model, the solutions andwechoosetheparametersoftheirmotionwithoutcaus-
can be written analytically for a = 3/2. The lens equation tic crossings. Namely, we consider the clump models with
y=r 1 r−3/2 hastwosolutions.Thefirstsolutionmust twolensedimagesthatwillbecomparedwiththe”fiducial”
−
bewritten separately for ξ =3√3/(2y3/2)<1 and ξ>1: modeloftheusualpointmass microlens(theSchwarzschild
(cid:0) (cid:1)
lens).For Galactic systems weassume thebackgroundcon-
2/3 2 tinuous matter density to be σ = 0. The critical curves of
ξ+ ξ2 1 +1
y − the two-image clump models appear to be very similar in
r= 3(cid:20)(cid:16) p (cid:17) 2/3 (cid:21) , ξ>1; case of the Schwarzschild microlens and the cusped clump
ξ+ ξ2 1 with 0<σ<1 and a>1. In these cases there is onecircu-
−
lar critical curve of unit radius; and also, one caustic point
and (cid:16) p (cid:17) at y = 0 exists. The case of a cusped clump with a < 1 is
4 1 topologicallydifferent,howeveritwillbealsodifficulttodis-
r= y cos2 arccosξ , ξ<1.
3 3 tinguish its light curve from that due to the Schwarzschild
h i lens.
The second solution is
Belowweshowamplificationcurvesgeneratedwithdif-
2/3 2 ferent models corresponding to a straight line motion of
ξ+ ξ2+1 1
y − themicrolensingsystemwithrespecttotheline-of-sightto-
r=−3(cid:20)(cid:16) p (cid:17) 2/3 (cid:21) wards the remote point source. It turns out that the shape
ξ+ ξ2+1 oftheamplificationcurvesinducedbytwo-imageclumpsare
qualitativelysimilartotheonesinducedbyaSchwarzschild
for any valu(cid:16)e ofpy. The(cid:17)magnification curves for these microlens, and thus they can hardly be distinguished from
images can be found using the formula (4): µ = one another. Indeed, by an appropriate choice of the im-
1 1/r3/2 1+1/(2r3/2) −1. pact parameterand thevelocity with respect totheline-of-
−
sight, the Schwarzschild lens light curve can be well fitted
(cid:12)(cid:0) (cid:1)(cid:0) (cid:1)(cid:12)
(cid:12) (cid:12) by microlensing due to a cusped clump (see Fig. 3). For a
wide range of parameters that may be considered as typi-
4 COMPARISON BETWEEN THE CLUMP calforthisproblem,thedifferencebetweentheamplification
MODELS AND THE POINT MASS LENS curvesonaplotissometimesnotvisibletotheeye.Onecan
extract an additional information from the image centroid
4.1 Observational mimicry of the extended mass (IC)6 tracks in the reference frame of the source (see Fig.
microlenses and the source image tracks
3, lower panel). This, however,requires an accuracy for the
How can one distinguish between the different lens mod- positioning measurements at the microarcsecond level; this
els discussed above on thebasis of observations? Ofcourse, is typical for microlensing by Galactic objects and is not
the most correct comparison must come after the fitting of accessible at present. E.g., the modern accuracy achieved
concrete observational data. However, let us first note the with HST (WFPC3 camera) is around 20-40 microarcsec-
qualitative differences mainly arising from the topological onds(Riess et al. 2014).
properties of the corresponding lens mappings, i.e. the ex-
istence of the critical curves, caustics and number of im-
ages of a point source. From the above discussion it is easy 4.2 Numerical estimates
to see that, for different parameters of the source/lens mo-
Thus,weproceedinmoredetailwiththedifferencesbetween
tion, the caustic intersections can occur leading to appear-
theamplificationcurvescorrespondingtodifferentmicrolens
ances/disappearances of the lensed images. These events
models.Notethatweperformedafittingoftheamplification
could lead to such observable effects as high amplifications
curves in Figs. 3,4,5 so as to provide the best overlapping
of the lensed image flux, ”occultations” and sudden jumps
near the maximum of the curves (where they can be well
of the average image positions (image centroids).
approximated by a parabola, and where one can expect a
There exist many observational data compiled by
good measurement accuracy). Careful inspection of these
EROS, OGLE and other groups (Tisserand et al. 2007;
figures shows that there is a slight deviation between the
Wyrzykowskiet al. 2009) hunting for microlensing events
curvesinthewingsoftheseparateHAEs.Inthisconnection,
caused byGalactic objects.
to compare the different microlens models, we proceed as
Some of these events show a complicated behavior like
follows.
thosecharacteristicofcausticcrossings.Thisistypicallyre-
For a moving extended microlens characterized by an
lated to the existence of double star systems or planetary
impact parameter p with respect to the line-of-sight, and
systems(Kainset al.2013;Shinet al.2007).However,most
a transverse velocity (assumed to be V = 1), we generate
typical HAEs (Tisserand et al. 2007; Wyrzykowskiet al.
2009) can be interpreted using the isolated point mass lens
model. It is important to note that there exist some high 6 The image centroid here is a weighted average of positions of
amplification eventswith nodetector identified. alltheimages(withtheexceptionofthelensesthatweassumeto
Ontheotherhand,formostoftheeventsthedeflector- benotvisible);theweightsareproportionaltotheamplifications
starhasbeenidentified;howeverthiscasedoesnotruleout ofthecorrespondinglensedimages.
(cid:13)c 2014RAS,MNRAS000,1–14
8 E. Fedorova, V.M. Sliusar, V.I. Zhdanov, A.N. Alexandrov, A. Del Popolo, J. Surdej
clump (q=1, r=1)
1.6 a=1.5, p=1 s Schwarzschild fit 1,8 cusp+point mass
Schwarzschild fit
a=1.0
ation 1.4 cation 1,6 a=1.3a=1.6 q=0.5
plific 1.2 mplifi 1,4 a=1.9
m A
A
1,2
1.0
1,0
-10 -5 0 5 10
Time -3 0 3 6 9
Time
2,2 cusp+point mass
Image centroid trajectories Schwarzschild fit
0.8 2,0 q=0.8
on1,8 q=0.6 a=1
Y ati q=0.4
c1,6
0.4 plifi q=0.2
m1,4
A
1,2
0.0
1,0
-0.8 -0.4 0.0 0.4 0.8
X -3 0 3 6 9
Time
Figure3.Amplificationcurves(upperpanel)andimagecentroid Figure4.Examplesofamplificationcurves(solid)formicrolens-
trajectories(lowerpanel)forthecasesofacoredclump(solid)and ing systems that consist of point masses surrounded by cusped
apointmassmicrolens(markedbythesmallcircles).Theclump clumps with q = 0.5, a = 1.0,1.3,1.6,1.9 (upper panel) a = 1,
parameters shown in the figure correspond to Eq. (9) (q = 1 and q = 0.2,0.4,0.6,0.8 (lower panel). The impact parameter
corresponds to a pure clump, p = 1 is the impact parameter of of the microlens with respect to the line-of-sight is p = 1. The
the clump center with respect to the line-of-sight, its transverse curvesarefittedbymeansofpointmassmicrolensmodels(small
velocityV =1).Theimpactparameterandtransversevelocityof circles).
thepointmassmicrolensarederivedfromfittingtheamplification
curve for the clump model. The image centroid trajectory (in
the rest frame of the source) in case of the Schwarzshild lens is 4.3 Microlensing by a point mass and a clump
rescaled in order to clearly indicate the difference between the
It is natural to consider a situation where the cusped or
models.
cored DMclump is formed in the same region as where the
point mass is situated (and vise versa). The case of a pure
clump microlens and the Schwarzschild lens represent two
extreme cases of this situation. In this connection we con-
its amplification curve µ(t). Then we look for the best fit
sideramicrolens modelthatisrepresentedbythefollowing
near the maximum of this curve using the Schwarzschild
lens equation:
lens model light curve (µ (t)); see Appendix A for de-
Schw
q 1 q
tsaerilvsa.tNioontael tdhaattaawsedihstaivnectafrsoimmpdleeralipnrgobwleitmh.tIhnetrheaelroeba-l y=r 1− Ra − r−2 , R= r2+rs2, (9)
case, the fitting parameters, besides the transverse velocity where(cid:16)thecoefficientq d(cid:17)escribestpherelativecontributionof
and theimpact parameter of the Schwarzschild lens (which theclump,1 q representsthatofthepointmasslens,and
−
are different from the assumed parameters p,V = 1 of the we omitted the contribution due to the background optical
clump+pointmasscomplex),are:thetimeofmaximumam- depth σ; 0 6 q < 1, rs > 0. The amplification is µ(r) =
plification and the maximum intensity of the image (or the 1/D(r), where the determinant D(r) is also given by the
fluxwhenthelensisfarfromtheline-of-sight).Inourcaseof product D(r)=ϕ1(r)ϕ2(r) with
the artificial amplification curve, the latter two parameters
q 1 q
arefixed.Thusthedifferencebetweenthetwoamplification ϕ1(r)=1− Ra − r−2 , (10)
curvesis:
q(a 1) aqr2 1 q
δ=max [µ(t)]−1 µ(t) µ (t) . (8) ϕ2(r)=1+ R−a − Ra+s2 + r−2 . (11)
Schw
t | − |
Furthermore, we assume such numerical values for the
(cid:8) (cid:9)
This is used to compare the amplification curves due to a parameters of the microlens as to provide the qualita-
somewhat more complicated model which corresponds to tive behavior of the lens mapping (9) to be like that of
a combination of a point mass microlens and an extended the Schwarzschild lens with two lensed images, and so
clump. as to provide a considerable amplification (up to 10 and
(cid:13)c 2014RAS,MNRAS000,1–14
Gravitational microlensing as a probe for dark matter clumps 9
cored clump+point mass
Schwarzschild fit 0.16 rs = 1
1,6
e
plification1,4 q=0.2q=0.4q=0.6 q=0.r8s=1, a=1.5 e differenc 0.12 a = 1, p = 0.1 a = 1, p =1
m v 0.08
A1,2 ati
el a = 1.5, p = 0.1 a = 1.5, p = 1
R
1,0 0.04
-5 0 5 10 a = 2, p = 0.1a = 2, p = 1
Time
0.00
0.0 0.2 0.4 0.6 0.8 1.0
q
1,5
cored clump+point mass
Schwarzschild fit
1,4
r =5, a=1.5
s
n 0.32
o
Amplificati111,,,123 q=0.2 q=0q.4=0.6q=0.8 e difference 000...222048 p = 0.1 a = 1.0, rs a= =5 1.0, rs = 10
v
1,0 ati 0.16
-5 0 5 10 Rel 0.12 a = 1.5, rs = 5
Time
0.08
a = 1.5, rs = 10
Figure5.ThesameasinthelowerpanelofFig.4forcoredclump 0.04 a = 2.0, rs = 10
models with a=1.5, rs =1 and rs =5. The impact parameter 0.00
ofthemicrolenswithrespecttotheline-of-sightisp=1. 0.0 0.2 0.4 0.6 0.8 1.0
q
higher), which can be the best to see the signals of the ex-
Figure6.Relativedifferencesδoftheamplificationcurvesfora
tended microlens structure. We shall then estimate the ac-
differentcoredclumpcontributionsqinthemicrolensingcomplex
curacyofmeasurementsneededtodifferentiatebetweenthe
”point mass + clump”. Upper panel: a = 1.0,1.5,2.0, impact
lightcurves of these models. The dependence of this differ-
parameter p = 0.1 (solid curves), p = 1.0 (dashed curves), the
ence upon the contribution q of the clump in the lensing clumpsizers=1;theamplificationiswithinthelimits∼10÷16.
complex ”cored clump +point mass” is shown in Fig. 6 for Forp=0.03wehave almostthe samecurves as forp=0.1,but
some values of a and rs. theamplificationinthiscasevariesfrom∼30to50.Lowerpanel:
Incaseoftheconfigurationofapointmassandacusped the same values of a, p= 0.1, rs = 5 (solid), rs = 10 (dashed).
clump we assume q (0,1), rs =0. The relative differences Thecurvewitha=2,rs=5issuperimposedonthecurvewith
betweentheamplific∈ation curveofthiscomplexandthatof a=1.5,rs=10anditisomitted.Amplificationiswithin3÷10.
thefitted Schwarzschild lens are shown in Fig. 7.
ofeveryimage(i.e.,separatedfromproperbrightnessvaria-
tions that are intrinsic to the source), which arises because
5 STATISTICAL EFFECTS OF DM CLUMPS
ofthesourcemotion7.Asbefore,foratheoreticaltreatment
MICROLENSING IN EXTRAGALACTIC GLS
wedealwiththeamplificationcurves,thatisthedependence
InatypicalextragalacticGLSwehaveseveralmacro-images of theamplification coefficients upon thetime.
of one quasar. Comparison of the amplification curves of Our aim is to estimate the statistical effect of the ex-
theseimagesmakesitpossibletoseparatetheproperbright- tended masses (clumps) on the autocorrelation functions
ness variations of the quasar and to derive variations due of the amplification curves. To compare the amplification
to the microlensing processes. Contrary to the case of the curves in microlensing systems with a different content of
Galaxy, because of the considerable microlensing optical theseclumps,weconsiderasimplemodelofstochasticpoint
depth in extragalactic systems, instead of an isolated point andextendedmasses.Weconfineourselvestoaspecialcase
mass or putative DM clump, we must deal with an ag- of the cored clumps according to Eq. (3) with a = 2 (cf.
gregate of masses in the lensing galaxy. In this case we Zhdanovet al. (2012)). An external shear owing to the av-
have a complicated caustic network generated by unknown erage field of the lensing galaxy will also be taken into ac-
masses with unknown positions. Therefore, the problem count. Thus, we use the lens equation for N masses in one
takes a statistical shade (Paczyn´ski & Wambsganss 1989; lens plane
Wambsganss et al. 1990; Schechter& Wambsganss 2002).
Observationally, the main source of information in a con-
crete GLS is at present an effective microlensed light curve 7 whichisagainassumedtobeauniformstraightlinemotion.
(cid:13)c 2014RAS,MNRAS000,1–14
10 E. Fedorova, V.M. Sliusar, V.I. Zhdanov, A.N. Alexandrov, A. Del Popolo, J. Surdej
0.16
e Point mass + cusped clump
nc 0.14
e
er 0.12 a = 1.0 p = 0.1
diff 0.10 a = 1.0 p = 1
e
v
ati 0.08 a = 1.4 p = 0.1
Rel 0.06
0.04 a = 1.4 p = 1
a = 1.8 p = 0.1
0.02
a = 1.8 p = 1
0.00
0.0 0.2 0.4 0.6 0.8 1.0
q
Figure 7. Relative differences δ of the amplification curves for
different cusped clump contributions q in the microlensingcom- Figure 8.Amplificationmapforσcl=0,γ=0,σtot=0.3.
plex”pointmass+clump”.Herea=1.0,1.5,1.8.Fortheimpact
parameterp=0.1(solid)theamplificationiswithin∼10÷20;for
p=1(dashed)theamplificationis∼1.3÷2.Forp=0.03wehave
µ(y)= I[Y(x) y]dx dx ,
1 2
practically the same curves as for p=0.1, but the amplification −
Z Z
variesfrom∼30to60.
where y is the position of the source center in the source
plane projected in the lens plane and y = Y(r) represents
the lens mapping r y. We used a Gaussian model for
N R2 (r r ) →
y=Aˆr E,i − i (12) I(y); the source half-brightness radius was R1/2 = 0.2, its
− i=1 |r−ri|2+rs2,i motionhasbeenassumedtobeuniform.Duetothemotion
X of thesource we havean amplification curveµ(t).
where Aˆ = diag 1 γ, 1+γ is the two-dimensional ex- Havingalarge numberof realizations of themicrolens-
{ − }
ternal shear matrix (the optical depth of the background ingfieldandthecorrespondingamplificationcurvesµ(t),we
continuous matter is taken to be zero), rs,i is a character- havecalculated the ACFs
istic size of the i-th cored clump with mass M , r is the
i i A(τ)=(∆µ)−2 <[µ(t) µ ][µ(t+τ) µ ]>, (13)
position of its center on the lens plane, and RE,i its Ein- − 0 − 0
stein ring radius: RE2,i = 4GMiDds/(c2DdDs). Obviously, where
for r = 0 we have ordinary point microlenses with mass
s,i µ =<µ(t)>, ∆µ= <[µ(t) µ ]2 >, (14)
Mi. 0 − 0
In our simulations, the microlens masses and posi- the brackets < ... > denpote the averaging over all the real-
tions were chosen in a random way. The positions r were izationsoftheamplificationcurvesforthefixedvalueofthe
i
distributed homogeneously inside a circle, which was big opticaldepthofcontinuousmassesσ andpointmassesσ .
cl p
enoughtominimize theboundaryeffects;also, tocheckthe To consider the possible observational manifestations
resultconvergencyweconsidereddifferentsizesofthecircle. of the extended microlenses, their size R has been cho-
c
The mass distribution followed theSalpeter’s law (Salpeter sen to be comparable with a typical Einstein ring radius
1955) with a power-law index 2.35 within the mass range R of the microlenses; otherwise in the limiting situations
E
−
Mi [0.4;10]M⊙. In every set of numerical experiments, we approach to either the well-known case of a continuous
∈
the input parameters (that were controlled to be the same background matter for R >> R , or we have a standard
c E
in all the realizations of the set) were as follows: the to- microlensing of point masses. Here we present the results
tal optical depth of the microlensing field σ , the opti- for σ = 0.3, and κ = 5. The parameter σ has been
tot tot cl
cal depths σ , σ of the point masses and the extended varied from zero to 0.3. The length unit corresponds to
p cl
clumps(σcl+σp =σtot),andtherelativesizeoftheclumps RE,⊙ =[4GM⊙Dds/(c2DdDs)]1/2 =1. Furthermore, t and
κ= rs,i/RE,i. To introduce the clumps into the microlens- τ are in units of RE,⊙ (i.e. assuming a velocity V =1); the
ing field, we replaced some randomly chosen part of point normalization to M⊙ is just conventional.
massesbyextendedclumpswithoutchangingtheirpositions The examples of the amplification maps in the source
r ,massesM ;thesizeparameterwasr =κR .Then,for plane are shown in Figs. 8 - 10. The amplification dis-
i i s,i E,i
each realization of the microlensing field we obtain an am- tributions for various fractions of clumps (and the same
plification map by means of the ray shooting method com- σ = 0.3, γ = 0) noticeably differ one from another. As
tot
binedwithourGPU-enabledmicrolensingC++codebased canbeexpected,thepresenceoftheclumpsmakesthisdis-
on the hierarchical tree algorythm (Schneideret al. 1992; tribution smoother. The presence of non-zero γ (Fig. 10)
Wambsganss et al. 1992). Convolution of the amplification stretches the maps in the direction of the shear. For every
map with the normalized brightness distribution over the set of parameters σ ,σ ,γ we generated typically several
tot cl
initialsourceI(y)yieldstheamplificationcoefficientµ.The hundredmaps and thecorresponding amplification curves.
mathematical representation of this procedure can be writ- Using the ensemble of these curves, the averaging pro-
ten as (Alexandrov& Zhdanov 2011; Zhdanovet al. 2012) cedureyieldsACFA(τ).ThesefunctionsareshowninFigs.
(cid:13)c 2014RAS,MNRAS000,1–14