Determination of the redshift of an

invisible lens

Christophe Jean and Jean Surdej

Institut d'Astrophysique, Université de Liège, Belgium

1. Introduction

2. Fitting

3. Monte Carlo Simulations

4. Conclusions

Observer

Introduction

Deflectorz r, Ra,

Quasar

Flux received bv the observer from component i at the wavelength À:

F^,i : F| An"-Ext(À" Rv)Lt

with { the flux emitted by the quasar;

Ai the magnification of component i;\

trxt(À/ , Rv) the extinction (À' : , , , Rv is a parameter);' I * zt,and Li the path length of component i through the deflector.

We assume no contamination, no intrinsic colour variations between

the components, no micro-iensing effects and that the extinction is

taking place in a single lensing plane.

Colour difference between component i and a reference component is

then:

(lVlsr,t - fuIb,n) - (Mn,ref - M^r,,.f) :)5

ft LilBxr(Àî, Ârr) - Ext (À'r, Rv))

with Mx, - M^, : -2.5f "* (+)\- ^2,/

and Lt : Li - Lr.r.

Fitting

(Mxr,,i - Msr,n) - (M>,r,ref - M^r,r.f) :

Experimental colour

frfr Lillxt(À'r, Rr) - Ext (^!r, Rr)l

For the fitted colour:

o Ext(Ài , Rv) - Ext (À'r, Rv) is computed for different contiguous

values of Rv and z1;

. L,; is fitted by u routine for each selected (Rv,rt) point in order

to minimize:

(trxperimental colour - Fitted colour)2x'-D ,()r- E-x perrmenl al colour

Ax'

ZS

map is thus obtained:

21

0.01

06

Fitted colour

r.05 RV

Monte Carlo Simulations

Study of the errors affecting the determination of the lens redshift

from a set of 100 simulations. Gaussian noise was added to the

simulated magnitudes of the 4 selected components.

5 filters: UBVRI

0.1 5:t:i--tr

norse.noise:noise:noise:noise:

0.010.020.030.040.05

----x---magmagmagmagmag

I/ll

,r-ry-,.x,

. -l'i!rrr!Y:Ufl-N!, rç*'^liiâé?:r". *]!.t:,' //-' . t':!!!1,'

, t EY.,-** ç1;,;--çi"-o 7-?*^=I: :Ï:::

t'l'1lotrul.-t

la-xxx-ri

0.1b

0.05

00 0.5 1 1.5 2 2.5 3

z1

Gaussian noise. 0.01 mao

0.1

b

0.01

0.0010.5 1.5

Z1

2.5

Gaussian noise: 0.01 mag

0.05

0.04

0.03

0.02

0.01

01 1.5 2 2.5

z1

XJ

^xxxx)<xxv,.x

X XX--

UBVR| _IJHKL

k\ t- ' "--

z-r.X\,

X.,^X-X X-X X-X

x*-**"**l

*:-!--++

5 filters: UBVRI7 filters: UBVRIJH

9 filters: UBVRIJHKL

Ii

T

\/,*\',,,x i*r\

,ï\

0.5

Conclusions

Monte Carlo simulations indicate that the redshift of an invisible

lens can be retrieved with this method provided that we have

accurate photometric observations of the macro-lensed images.

This method is particularly sensitive to the redshift of the deflector

and thus offers an original way of determining the redshift of an

invisible lens.

High quality photometric data (o = 0.01 mag) are badly needed.

Photometry for at least 5 broad band filters is necessary.