New Astronomy Reviews 42 (1998) 153-1.56 .I 1

XSEWIER

New Astronomy Reviews

Gravitational Lens Magnification:

An Analysis of Abel1 1689

Andy Taylor I and Simon Dye

IfA, University of Edinburgh, Royal Observatory, Edinburgh, EH9 3HJ, Uh

Abstract

The absolute mass of a cluster can be measured by the gravitational lens magnifica-

tion of a background galaxy population. We have made the first application of this

method to measure mass in the cluster Abel1 1689, taking into account the uncer-

tainties introduced by intrinsic clustering of the background galaxies, shot-noise,

and the effects of strong magnification near the cluster center. We present mass maps

and the mass profile. In a cylinder of radius 0.24/~-~Mpc (2’.2) around the peak, we

find the projected mass is nJzd(< 0.24h-‘Mpc) = (0.50 f 0.09) x 10’5h-1h/Io. Our

results are in good agreement with those from the shear analysis.

h’ey words: Cosmology, gravitational lensing, clusters, dark matter

1 Introduction

As the largest condensed gravitationally bound structures, clusters of galaxies

make particularly interesting objects for study. Estimates of cluster masses

and abundances place strong constraints on the amplitude of mass fluctu-

ations, while the ratio of mass to light yields some information about the

degree of galaxy bias on small to intermediate scales. Gravitational lensing is

a particularly clean way to probe the total mass distribution in clusters since

it does not rely on any assumptions about dynamical states.

Gravitational lensing manifests itself in two distinct ways; shearing a distant

source image, and magnifying it. Inversion of the shear distortion pattern to

estimate mass distribution is now a well established process (Tyson & Fischer

1995, Kaiser & Squires 1993). H owever, since shear is not locally related to

the surface density, it can only be calculated up to a constant. An exception

’ E-mail: A. TaylorBroe . ac .uk

1387-6473/98/$ - see front matter 0 1998 Elsevier Science B.V. All rights reserved PII: Sl387-6473(98)00039-6

IV A. Taylor. S. Dye I New Astronomy Reviews 42 (1998) 153-156

to this is when caustic features appear, and one can pin the shear mass to

these (see, for instance, the paper by AbdelSalam in this volume).

Inversion of the magnification effect was first suggested by Broadhurst, Taylor

,G. Peacock (1995; hereafter BTP). Th ere it was shown that lens magnifica-

tion is directly related to the surface mass density and so breaks the mass- normalisation degeneracy. Taylor et al. (1998) extended these results to the strong magnification regime so that surface mass could be normalised every- where, and applied them to Abel1 1689. We refer the reader to these papers

for a more detailed account.

2 The Magnification Effect

The observed number of galaxies seen in projection on the sky is (BTP) n’(r) = nOAp-‘(~)(l+O(~)) h > w ere no is the expected mean number of galax-

ies at a given magnitude. Variations about the mean arise from angular per- turbations in galaxy density, O(z), d ue to galaxy clustering and gravitational

lens magnification. The lensing amplification factor is A = I(1 - K)” - y21-‘, where K is the desired surface mass density and y is the shear. If the galaxy

luminosity function is approximated by n(L) N L-p, then the index of ampli- fication, p - 1, accounts for an increase in both source separation and number as fainter sources come into view.

To estimate the surface mass density, K, from the amplification, we need to

say something about the shear. van Kampen (see this volume) has shown that

realistic clusters lie somewhere between the extremes of a sheet, where y = 0

and an isothermal approximation, where y = K. This allows one to bound the true solution. In addition one can use a local parabolic solution, y - &, which lies somewhere between the two. Taylor et al. (1998) have also shown that for axisymmetric clusters there is an exact non-local solution for y and R profiles, given the amplification.

One could directly calculate the surface mass from the amplification, were it

not for the complicating factors of galaxy clustering in the background source

distribution and shot-noise. The effects of clustering can be eliminated with

redshift information (BTP), but here we account for it by modeling the angular counts by a Lognormal-Poisson model (BTP). We find that this reproduces the distribution of counts in deep fields very well. The only parameter is the variance of the lognormal field, the amplitude and evolution of which

we estimate from the Canada-France Redshift Survey (CFRS; Le Fevre et al. 1996). We use this distribution for a maximum likelihood analysis of the

surface mass density.

A. Taylor, S. Dye I New Astronomy Reviews 42 (1998) 153-156 155

3 Application to Abel1 1689

The data we used comprises 10,000 seconds integration in the V & I bands to I< 24, taken with ESO’s NTT and covering about 30 square arcmin. The offset field, which we used to normalise counts and measure /3, were from the

Keck (Smail et al 1995). W e used colour cuts in the colour-magnitude plane

to select galaxies redder, and so at higher redshift, than the cluster E/SO sequence. The cluster galaxies were then masked over to avoid contamination,

and the background galaxies binned in annuli around the peak of the light

distribution (see left hand side of Figure I). The number counts in each bin are shown in the right hand side of Figure 1, and show a clear caustic feature (zero count) at 0.75’.

X (arm-h) 0 (arcmin)

Fig. 1. Left: Annular bins centered around the peak of the light distribution. Open dots are red background galaxies. Grey patches are cluster members, which are masked over when calculating the mass. Right: Radial fractional number counts of red background galaxies, allowing for obscuration by cluster members. Error bars are from shot-noise. The dotted line is for an isothermal sphere normalised at the caustic point.

Figure 2 (left) h s ows the surface mass density profile, K(O), derived from max-

imum likelihood analysis. As with the number counts, the surface density is well fitted by an isothermal shape. Beyond 3’ the profile is dominated by noise.

The right hand side of Figure 2 shows a 2-D mass map, which appears to agree with general features in maps produced by the shear method.

To convert K to mass we take the redshift of Al689 to be z = 0.18 and the mean

redshift of the background population to be z = 0.8. The projected surface

mass density interior to a radius of 0.24h-lMpc (2’.2) is AJI~~(< 0.24h-r Mpc) =

(0.50f0.09) x lO”h-‘Ma. Th’ IS a g ain is in good agreement with Kaiser (1996) and Tyson & Fischer (1995) h w o use the shear method normalised to the caustic line.

1.56 A. Taylor, S. Dye I New Astronomy Reviews 42 (1998) 153-156

Y

In

d

e (arcmin) x (arcmin)

Fig. 2. Left: The surface mass density profile of A1689. The dark region is bounded

by the sheet and isothermal estimators. The lighter shaded region is the uncertainty

derived from the maximum likelihood analysis, due to clustering and shot-noise. The

solid line is an isothermal fit to the caustic feature. Right: The 2-D surface mass

distribution reconstructed from maximum likelihood analysis

4 Conclusions

We have shown that the surface mass profile of a cluster can be measured via

the magnification effect on background sources, and applied it to Abel1 1689.

We have taken into account both the uncertainty due to intrinsic clustering

and shot-noise in the background galaxy distribution and the effect of strong

magnification near the cluster center.

References

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Kaiser N., Squires G., 1993, ApJ, 404, 441

Kaiser N., 1996, in “Gravitational Dynamics”, Proc. of the 36thh Herstmonceux

Conf., eds 0. Lahav, E. Terlevich, R.J. Terlevich (Cambridge University Press)

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